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ECONOMICS 8451–MICROECONOMIC THEORY 3

Lecture 1: Introduction

Recall the circular flow of income:

Figure 1.1: The Circular Flow of Income

Basic microeconomics develops theories of the product and factor markets. A good theory has three

properties:

1. As simple as possible

2. Generates refutable hypotheses–i.e., has predictive power

3. NOT about “truth” in some metaphysical sense

Our basic behavioral postulate is:

“Rational actors make choices that maximize their objective.”

• These actors are usually either consumers or producers

• They may or may not be perfectly informed

• They may or may not be perfect competitors (price-takers)

4 LECTURE 1: INTRODUCTION

This leads immediately into the mathematical theory of maximization:

• objective is f(x, p) (e.g., utility, profit)

• x is a vector of choice variables (e.g., quantities to consume, factor inputs to production)

• p is a vector of parameters (e.g., prices, income)

• g(x, p) = 0 is a constraint (this is the general form, examples of which are budget constraints or

production feasibility)

There are two results from this maximization:

• The optimal choices x∗(p) (e.g., demands, supplies)

• The optimal value f∗(p) = f(x∗(p), p) (e.g., indirect utility, cost, or profit)

We ask:

“What does the fact that x∗(p) and f∗(p) are consequences of optimization imply about these functions?”

The answers are the refutable hypotheses from a theory of optimizing behavior.

We find x∗(p) by setting up a Lagrangian function

L(x, λ; p) = f(x, p)− λg(x, p)

and solving the first order conditions (assuming an interior optimum):

Lxi(x, λ; p) = 0, i = 1, . . . , n

Lλ(x, λ; p) = 0

• With explicit functions: solve these equations for x∗ and λ∗.

• More generally: these equations implicitly define x∗ and λ∗.

In either case, they are identities that hold for all p (again, assuming an interior optimum):

Lxi(x∗(p), λ∗(p); p) ≡ 0, i = 1, . . . , n

Lλ(x∗(p), λ∗(p); p) ≡ 0

At the maximum, g(x∗(p), p) ≡ 0. So

L(x∗(p), λ∗(p); p) ≡ f(x∗(p), p) ≡ f∗(p)

and we can differentiate this identity to find how the optimal value f∗ varies with pj :


f∗pj(p) =

n∑i=1

Lxi(x∗(p), λ∗(p); p)

∂x∗i

∂pj+ Lλ(x∗(p), λ∗(p); p)

∂λ∗

∂pj+ Lpj (x

∗(p), λ∗(p); p)

= Lpj (x∗(p), λ∗(p); p)

This result is known as the envelope theorem. It potentially yields refutable hypotheses about f∗.

We can differentiate again to obtain refutable hypotheses about x∗:

f∗pjpk

(p) =n∑i=1

Lpjxi

∂x∗i

∂pk+ Lpjλ

∂λ∗

∂pk+ Lpjpk

There is potentially interesting information here about how x∗i changes with pk–a refutable hypothesis. Three

facts usually help us extract this information from this equation:

1. Concavity of the function f∗ − L in p: this gives the sign of f∗pjpj

− Lpjpj , so the remaining part of the

equation has a known sign when j = k, and hence maybe ∂x∗i

∂pkdoes too. Such results become the “laws”

of supply and demand.

2. Equality of mixed second derivatives of f∗ − L: this gives equality of the two remaining parts of the

two equations when the order of differentiation is reversed, and hence maybe some ∂x∗i

∂pk’s are equal. Such

results become the “reciprocity” or “symmetry” results.

3. Presence of many zero second derivatives of L, which simplifies the above expression.

This is the general view. We will repeatedly encounter special cases of it.

6 LECTURE 1: INTRODUCTION

Exercises for Lecture 1 (Introduction)

1. Give a verbal statement of the envelope theorem.


Lecture 2: Technologies

Readings: Chapter 1 of Varian; Sections 5.A and 5.B of MWG.

We begin the study of product markets with the supply side. The actors that supply final products are

called firms. Before considering their objectives, we first consider their constraints. These take the form of

the physical limitations on their ability to transform inputs into outputs. That is, their constraint is the

technical capabilities, or technology, currently at their disposal to use in producing output. To make any

progress, we must place some minimal structure on these “capabilities.” This is an engineering black box for

us–we seek only to describe its essential properties for economic purposes.

Often, we simply write down a production function. But this approach does not make it clear exactly

what we are assuming about technical capabilities, and also does not accommodate multiple outputs. We

can begin with a more primitive concept, called the production set, and then build up to the more convenient

representation in the form of a production function if that is convenient for some purpose.

Suppose there are n goods in the world. Let z ∈ Rn specify an amount of each of the goods. A production

set Z ⊂ Rn specifies the combinations of the n goods that can be produced. That is, Z is the set of all

“feasible” combinations of the goods. The convention is to measure outputs positively and inputs negatively.

So, for example, the vector (2,−1,−3) indicates that one unit of good 2 and three units of good 3 are used

to produce two units of good 1. This notation does not capture intermediate goods that the firm produces

and then uses to produce final outputs. It is only the net use of goods or production of goods that is

represented by a vector z. Hence these vectors are called netput vectors, rather than input-output vectors.

Some examples of production sets are illustrated in Figure 2.1.

The minimal properties we frequently place on a production set Z are:

1. Z is nonempty–something is feasible.

2. Z is closed–this assures existence of optima in some cases.

• Together, items 1 and 2 are what Varian calls “regularity.”

3. There is no free lunch: Z ∩ Rn+ ⊂ {0}.

4. Inaction is always possible: 0 ∈ Z. Note that this makes item 1 redundant and, in conjunction with item

3, implies Z ∩ Rn+ = {0}. When this is violated, we say there is a “sunk” input.

8 LECTURE 2: TECHNOLOGIES

Figure 2.1: Production Sets. In (a), both z1 and z2 can be both inputs and outputs. In(b), z1 can only be an input, while in (c) z2 can only be an input.

5. There is free disposal: z ∈ Z ⇒ z′ ∈ Z ∀z′ ≤ z. This implies Varian’s “monotonicity,” and is an

assumption that has more content than 1-3 (think of radioactive material as an input to electricity

production–can it be disposed of freely?). Free disposal may seem to imply infinite resources, but it

doesn’t. The assumptions here are only about technical ability, not about availability of resources. It is

up to the market to ration potentially scarce inputs by placing positive prices on them.

The examples in Figure 2.1 possess all five of these properties. If the boundary of Z passes to the southwest

of the origin then the possibility of inaction is violated and we have a sunk input. Such an input can be

“partially” sunk as well. A variant of this is a “fixed” input.

Two other properties we sometimes place on a production set Z are:

6. Global returns to scale:

a. nonincreasing: z ∈ Z ⇒ αz ∈ Z ∀α ∈ [0, 1].

b. nondecreasing: z ∈ Z ⇒ αz ∈ Z ∀α ≥ 1.

c. constant: z ∈ Z ⇒ αz ∈ Z ∀α ≥ 0.

7. Convexity: Z is a convex set.

Often we are concerned with technologies that have only one output. This simplest case is our focus in

this class. We can think of Z for this case as being a subset of Rn+1 and for convenience label the first


element of a netput z as the output. Then, for any z we can define y and x by (y,−x) = z, thereby defining a

separate notation for the output y and the n-dimensional input vector −x, and also conveniently measuring

inputs x as positive rather than negative quantities. Z is a single-output technology if (y,−x) ∈ Z ⇒ x ≥ 0.

In the single-output case we can describe a production set Z in two other ways that are sometimes more

convenient:

• Input requirement sets: Let V (y) = {x ∈ Rn+ : (y,−x) ∈ Z} for y ≥ 0. This is the input requirement set

for output level y.

• Production function: Let f(x) = max{y ∈ R1 : (y,−x) ∈ Z} for x ≥ 0. This is the production function

for the technology Z. Its existence relies on this set being nonempty (assured by possibility of inaction

and free disposal since then (0,−x) ∈ Z, but just Z ⊃ Rn+1− is enough), closed (assured by Z closed), and

bounded (assured by convexity of Z–see the exercises following this lecture).

Note that V (y) ⊂ {x ∈ Rn+ : f(x) ≥ y}, by definition. The converse can fail without free disposal.

Free disposal, and the presence or absence of fixed or sunk inputs, can be conveniently illustrated in terms

of both input requirement sets and production functions.

Occasionally we make use of an additional assumption for the single-output case:

8. Lower hemicontinuity:

a. Lower hemicontinuity in inputs: If (y0 ,−x0) ∈ Z and xi → x0 then there exists yi → y0 such that

(yi,−xi) ∈ Z. This rules out production sets like this:


Figure 2.2: A production set that violates lower hemicontinuity in inputs

This property implies that the production function is lower semicontinuous (see the exercises following

this lecture).

b. Lower hemicontinuity in output: If (y0,−x0) ∈ Z and yi → y0 then there exists xi → x0 such that

(yi,−xi) ∈ Z. This rules out production sets like this:

Figure 2.3: A production set that violates lower hemicontinuity in output


Let f be the production function for a single-output technology Z. The isoquant for output level y is

then defined as {x ∈ Rn+ : f(x) = y}. It is all input vectors that are capable of producing the output level y,

but no more. None of our assumptions rule out the possibility that an isoquant is “thick,” but free disposal

implies that the boundaries of an isoquant cannot slope upward.

Isoquants can be conveniently illustrated on a graph of input requirement sets.

If f is strictly increasing, the isoquants have no thickness and the equation f(x) = y implicitly defines

any one component of x, say xi, as a function of the other components, so we can think of the isoquant for

output level y as this implicit function. In this case, if f is also differentiable then the slope of the implicit

function with respect to some other component is

∂xi(x−i)∂xj

= −fj(x)fi(x)

< 0.

This slope is called the marginal rate of technical substitution of input i for input j (MRTSij), since it gives

the rate at which it is technically possible to substitute input i for input j while maintaining constant output.

Convexity implies that the MRTS is nondecreasing in x−i.


Exercises for Lecture 2 (Technologies)

1. Determine whether each of these technologies satisfies: i) no free lunch, ii) possibility of inaction, iii) free

disposal, iv) convexity (Assume V (0) = R2+ in (e)):


In (e) and (f) the figures show the typical shape of input requirement sets.

2. Give a geometric example of a single-output production set Z that is closed and satisfies no free lunch,

free disposal, and possibility of inaction; yet the corresponding production function can take on infinite

values. Prove that convexity solves this problem.

3. Assume a single-output production set Z is closed and satisfies no free lunch, free disposal, and possibility

of inaction. Prove the following properties of the corresponding production function f :

i. f(0) = 0.

ii. f is nondecreasing.

iii. For any x, f(αx) ≥ αf(x) ∀α ∈ [0, 1] when Z has nonincreasing returns to scale.

iv. For any x, f(αx) ≥ αf(x) ∀α ≥ 1 when Z has nondecreasing returns to scale.

v. For any x, f(αx) = αf(x) ∀α ≥ 0 when Z has constant returns to scale.

vi. A function f is called quasiconcave if, for any two points x and x′ in its domain, the following property

holds:

f(λx + (1− λ)x′) ≥ min{f(x), f(x′)} ∀λ ∈ (0, 1).

Show that the production function f is quasiconcave if and only if V (y) is convex for every y.

vii. A function f is called upper semicontinuous at a point x0 if the function is not much larger than f(x0)

for x near x0. Formally, for every δ > 0 we require an ε > 0 such that ‖x−x0‖ < ε ⇒ f(x) < f(x0)+δ.


Similarly, a function f is lower semicontinuous at x0 if the function is not much smaller than f(x0) for

x near x0. That is, for every δ > 0 we require an ε > 0 such that ‖x− x0‖ < ε ⇒ f(x) > f(x0) − δ.

Together, upper and lower semicontinuity are equivalent to continuity. Also, upper semicontinuity is

equivalent to {x : f(x) ≥ y} being a closed set for every y, and lower semicontinuity is equivalent to

{x : f(x) ≤ y} being a closed set for every y.

a. Show that f is upper semicontinuous.

b. Assume that Z satisfies lower hemicontinuity in output. Show that f is lower semicontinuous (This is

difficult–the important thing is to know that production functions are lower semicontinuous when Z has

this property).

4. Four common production functions for single-output technologies with two inputs are:

a. Cobb-Douglas: f(x1 , x2) = Axα11 xα2

2 , where A, α1, α2 > 0.

b. Leontief (fixed input proportions): f(x1, x2) = min{a1x1, a2x2}, where a1, a2 > 0.

c. Perfect substitutes: f(x1, x2) = a1x1 + a2x2, where a1, a2 > 0.

d. Constant Elasticity of Substitution (CES): f(x1 , x2) = [a1xρ1 + a2x

ρ2]ε/ρ, where ρ ≤ 1 (ρ �= 0) and

a1, a2, ε > 0.

For each of these, give the corresponding production set Z, input requirement sets V (y), isoquants, and

MRTS. Sketch a typical isoquant for each production function. For the CES production function, what

happens to the isoquants as ρ approaches 1? What happens as ρ approaches 0? What happens as ρ

approaches −∞?

5. Let f(x1 , x2) = (a1xρ1+a2x

ρ2)

ερ . For what values of ε does this have constant returns to scale, nondecreasing

returns to scale, nonincreasing returns to scale? Why?


Lecture 3: Profit Maximization: Setup, Calculus, and Traditional Comparative Statics

Readings: Chapter 2 of Varian; pp. 135-139 of MWG.

Now we continue with the supply side of product markets by adding an objective to the firm’s technological

constraint. We begin with the behavioral postulate that firms choose a netput vector to maximize profit,

subject to technical feasibility of the netput vector. We will also assume for now that the firm is a perfect

competitor.

Hence, let Z be the firm’s technology. Assume it satisfies properties 1-5. Since there are n goods, a

nonnegative n-dimensional vector p can represent the prices for the n goods. Then the profit generated by

a netput z is just π(z; p) = p · z, and hence the firm’s problem is

max{z}

p · z subject to z ∈ Z.

Note the role of the sign convention on inputs and outputs here. Inputs are measured negatively, and so

contribute negatively to the profit expression, while outputs are measured positively and thus contribute

positively to profit. The choice variables are z and the parameters are p. This leads to an optimal choice

z∗(p) and an optimal value π∗(p), which in this context are the net supply/demand function and profit

function, respectively, of the firm.

In the two dimensional case, the maximization can be illustrated as in Figure 3.1. The firm’s objective is

to choose that value of z that places it on the highest isoprofit line while still staying in the production set

Z.

With one output, the firm’s problem is equivalently written

max{(y,x)}

(p, w) · (y,−x) subject to x ∈ V (y),

where we have partitioned the price and netput vectors into their corresponding output and input compo-

nents. We will henceforth assume for notational convenience that x and w are n-dimensional (so there are

really n+ 1 goods in the world). If the output price is positive, this is equivalent to

max{x}

π(x; p, w) = pf(x) −w · x subject to x ≥ 0,

16 LECTURE 3: PROFIT MAXIMIZATION (BASICS)

Figure 3.1: Profit Maximization. The intercept of the isoprofit line is optimal profit inunits of good 2.

since for any given x, no choice of y that is less than f(x) can be optimal. Note that this way of incorporating

the constraint into the profit maximization problem only depends on free disposal to ensure that f is defined

on Rn+. It suffices to assume Z ⊃ R

n+1− instead.

The optimal choices for the single-output case are z∗(p, w) = (y∗(p, w),−x∗(p, w)). We call y∗(p, w) the

output supply and x∗(p, w) the input or factor demand.

If we further assume that f is differentiable, then calculus can be used to describe the maximum. The

first order conditions are:

pfi(x∗) ≡ wi for i = 1, . . . , n,

which just says that the MRP must equal the wage for every input. Dividing any two of these yields

fi(x∗)fj(x∗)

=wiwj

,

which is the familiar condition that the MRTS between any two inputs must equal their price ratio (slope

of the isocost line) at the firm’s optimal choice. That is, the firm must choose its inputs so that its isoquant

is tangent to its isocost.

These first order conditions are the basis for our investigation of how the optimal choices x∗(p, w) and

y∗(p, w) = f(x∗(p, w)) change when a price changes. These responses are of primary interest to economists


because they tell the shape of the input demand and output supply functions of the firm. The jargon for

this type of exercise is comparative statics. Since the first order conditions are identities when evaluated at

x∗, we can differentiate them with respect to a price to obtain:

p

n∑i=1

f1i(x∗)∂x∗

i

∂wj= 0 or 1

...

p

n∑i=1

fni(x∗)∂x∗

i

∂wj= 0 or 1.

Organizing these equations into matrices yields:

pf11(x∗) . . . pf1n(x∗)...

...pfn1(x∗) . . . pfnn(x∗)

∂x∗1

∂wj

...∂x∗

n

∂wj

=

0...1...0

← jth line.

By Cramer’s Rule:

∂x∗i

∂wj=

∣∣∣∣∣∣∣∣∣∣∣∣

pf11(x∗) . . . 0 . . . pf1n(x∗)...

......

... 1...

......

...pfn1(x∗) . . . 0 . . . pfnn(x∗)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣pf11(x∗) . . . pf1n(x∗)

......

pfn1(x∗) . . . pfnn(x∗)

∣∣∣∣∣∣∣,

where the 1 appears in the jth row and ith column of the numerator.

The matrices involved in these derivatives are composed of second derivatives of the objective function π

with respect to the choice variables x. Hence, if the maximum is a proper interior maximum then second order

conditions convey some information about the signs of various pieces. Briefly, the second order conditions

for a proper interior unconstrained optimization of a function π(x) are as follows. The Hessian Matrix of π

is the matrix of second derivatives:

Hπ(x) =

∂2π

∂x1∂x1. . . ∂2π

∂x1∂xn

......

∂2π∂xn∂x1

. . . ∂2π∂xn∂xn

.

A principal submatrix of Hπ(x) is any matrix obtained from Hπ(x) by deleting any number of rows and the

same columns. The order of a principal submatrix is just the size of the submatrix. For example, if 2 rows


and columns are deleted then the order is n−2. The determinant of a principal submatrix is called a principal

minor of H , with order the same as the principal submatrix. The principal minors obtained by deleting the

last row and column, and then the last two rows and columns, and then the last 3 rows and columns, until

only the (1, 1) element is left, are called the naturally ordered principal minors. The sufficient second order

conditions for an unconstrained maximum are that the sign of all principal minors of Hπ(x) of order k be

(−1)k, for k = 1, . . . , n. So, for example, the diagonal elements of Hπ(x) must be negative and |Hπ(x)| must

have sign (−1)n. It can be shown that an equivalent condition is that only the naturally ordered principal

minors of order k have sign (−1)k. Since many fewer determinants are involved, this latter condition is

convenient for checking second order conditions in any particular problem. However, the knowledge that all

principal minors have a known sign at a proper interior maximum is useful for performing comparative statics

and so we state the second order conditions in terms of all principal minors. These conditions are really just

an algebraic way of checking that the objective function π is strictly concave in the choice variables x, for

which the matrix Hπ(x) must be negative definite, meaning that a′Hπ(x)a < 0 for every conformable vector

a �= 0. Similarly, the sufficient second order conditions for an unconstrained minimum are that all principal

minors of Hπ(x) be positive regardless of order. An equivalent condition is that all naturally ordered principal

minors be positive. This is just an algebraic way of assuring that the objective function π is strictly convex

in the choice variables x, for which the matrix Hπ(x) must be positive definite, meaning that a′Hπ(x)a > 0

for every conformable vector a �= 0.

Returning to the profit maximization problem, the objective function is π(x; p, w) = pf(x)−w · x, so the

Hessian is

Hπ(x) =

pf11(x∗) . . . pf1n(x∗)...

...pfn1(x∗) . . . pfnn(x∗)

,

which is exactly the matrix in the denominator of our expression for ∂x∗i

∂wj(NOTE: the Hessian we are

examining here is only with respect to the choice variables). Hence we know the sign of our denominator is

(−1)n, assuming we have a proper interior maximum. For the numerator, let Hji denote the submatrix of

Hπ(x) of order n − 1 that results when row j and column i are deleted. Then, evaluating the numerator by

cofactor expansion yields

∂x∗i

∂wj=

(−1)i+j |Hji||Hπ(x)| .


What is the sign of |Hji|? In general, we don’t know because Hji is not a principal submatrix. But,

if i = j then it is a principal submatrix and so the minor has sign (−1)n−1. Hence the sign of ∂x∗i

∂wiis

(−1)i+i(−1)n−1

(−1)n < 0. This is natural: own price effects are negative, while cross-price effects are indeterminate

in general. Hence the law of demand for the input demand of profit-maximizing firms is a direct consequence

of the behavioral postulate, and nothing more. The ambiguous cross-price effects are usually categorized by

comparison with the negative own-price effect. If x∗j moves in the same direction as x∗

i when wi changes

(i.e., if ∂x∗j

∂wi< 0) then we say inputs i and j are gross complements in production. If the two inputs move in

opposite directions (i.e., if ∂x∗j

∂wi> 0) then we say inputs i and j are gross substitutes in production.

These methods can be used to study the effect of a change in the output price p on x∗, and also to study

the effects of changes in p and w on y∗, by noting that

dy∗ =n∑i=1

fi(x∗(p, w))dx∗i .

This is pursued further in the exercises and, like the results derived here, only the own-price effect of p on y

is known in general (it is positive). If ∂x∗i

∂p ≥ 0 we say input i is normal in production, while if ∂x∗i

∂p < 0 we

say input i is inferior in production.

We have proceeded here as if we have a proper interior maximum. There are several cautions that must

be adhered to regarding this:

1. Since x ≥ 0 is required, we can have corner solutions of the form x∗i = 0. Then the first derivative can be

negative: pfi(x∗) < wi.

2. Existence is a real issue for the profit maximization problem. With nondecreasing RTS it is possible for

there to be no maximum, depending on the prices. For convenience, we will henceforth assume π∗ is

defined on p ≥ 0 and w >> 0 when discussing profit functions, but the domain of π∗ can actually be

something smaller than this.

3. Uniqueness is also a real issue unless the technology is strictly convex.


Exercises for Lecture 3 (Profit Maximization: Basics)

1. Let π be the profit-maximizing function for a firm, p be the output price, (w1, w2) be the input price

vector, and f(x1, x2) = x1x2. Describe the input choices this firm makes.

Try the following problems from the books:

2. MWG #5.C.1.

3. MWG #5.C.6(a) and (b).

4. MWG #5.C.9.

5. Varian #2.3.

6. Varian #2.7.


Lecture 4: Profit Maximization: Envelope Properties of π∗ and y∗ and x∗


In this lecture we continue our study of the properties of a profit maximum, but make use of the envelope

theorem rather than using the somewhat cumbersome traditional comparative statics. The optimal choices

are y∗(p, w) and x∗(p, w), and the optimal value of the profit function is

π∗(p, w) = pf(x∗(p, w))− w · x∗(p, w), for p ≥ 0 and w >> 0.

These functions have some remarkable properties due solely to the fact that they are the maximizing choices

and maximal value, respectively, of pf(x) −w · x.

Theorem (Hotelling’s Lemma–Relationship between the Profit Function and the Supply/Factor

Demand). If π∗ is differentiable at (p, w) (almost assured by convexity) then ∂π∗∂p

= y∗(p, w) and ∂π∗∂wi

=

−x∗i (p, w).

Proof. Differentiate π∗(p, w) with respect to p, using the envelope theorem so that this is obtained by

differentiating π and then evaluating at the optimum, to get y∗(p, w). Likewise, differentiate with respect to

wi to get −x∗i (p, w). �

Theorem (Properties of the Profit Function).

1. (Homogeneity) π∗ is homogeneous of degree 1.

2. (Convexity) π∗ is convex (and hence continuous at (p, w) >> 0).

3. (Monotonicity) π∗ is nondecreasing in p and nonincreasing in w.

4. (Nonnegativity) π∗ ≥ 0 ∀(p, w) and π∗(0, w) = 0 ∀w >> 0.

Proof.

1. We will show below that y∗ and x∗ are homogeneous of degree zero. Substitution then yields

π∗(λp, λw) = (λp)y∗(λp, λw)− (λw) · x∗(λp, λw)

= (λp)y∗(p, w)− (λw) · x∗(p, w)

= λπ∗(p, w).

22 LECTURE 4: PROFIT MAXIMIZATION (ADVANCED)

2. Fix (p, w) and (p′, w′). By definition,

π∗(p, w) ≥ pf(x) − w · x ∀x ≥ 0,

π∗(p′, w′) ≥ p′f(x) −w′ · x ∀x ≥ 0.

Multiply the first of these by λ and the second by (1− λ) and add to get

λπ∗(p, w) + (1 − λ)π∗(p′, w′) ≥ (λp+ (1− λ)p′)f(x) − (λw + (1− λ)w′) · x ∀x ≥ 0

= π(x; λp+ (1− λ)p′, λw+ (1− λ)w′) ∀x ≥ 0,

so the left side is an upper bound of the right side profit objective across all feasible input vectors x.

Hence λπ∗(p, w) + (1− λ)π∗(p′, w′) ≥ π∗(λp + (1− λ)p′, λw + (1− λ)w′).

3. Select p′ ≥ p and w′ ≤ w. By definition,

π∗(p′, w′) ≥ p′f(x) −w′ · x ≥ pf(x) −w · x ∀x ≥ 0.

Hence, using the upper bound logic again, π∗(p′, w′) ≥ π∗(p, w).

4. By the possibility of inaction, f(0) ≥ 0. Hence π∗(p, w) ≥ pf(0)−w ·0 ≥ 0. Also, 0f(x)−w ·x ≤ 0 ∀x ≥ 0,

so π∗(0, w) ≤ 0. Combining this with π∗(p, w) ≥ 0 yields π∗(0, w) = 0. �

• Note that the convexity of π∗ implies continuity of π∗, at least on (p, w) >> 0.

Theorem (Properties of the Supply and Factor Demand).

1. (Homogeneity) y∗ and x∗ are homogenous of degree 0.

2. (Symmetry and Semidefiniteness) The matrix of slopes∂y∗∂p

∂y∗∂w1

. . . ∂y∗∂wn

−∂x∗1

∂p− ∂x∗

1∂w1

. . . − ∂x∗1

∂wn

......

. . ....

−∂x∗n

∂p −∂x∗n

∂w1. . . − ∂x∗

n

∂wn

is symmetric and positive semidefinite for (p, w) >> 0.

3. (Nonnegativity) py∗(p, w) − w · x∗(p, w) ≥ 0 ∀(p, w), (y∗(0, w), x∗(0, w)) = ((−∞, 0], 0) ∀w >> 0, and

(y∗, x∗) ≥ 0 ∀(p, w) >> 0.

Proof.


1. λπ(x; p, w) is a monotonic transformation of the objective function π(x; p, w) for λ > 0, and due to

linearity is equal to π(x; λp, λw). Hence y∗(λp, λw) = y∗(p, w) and x∗(λp, λw) = x∗(p, w).

2. Differentiating Hotelling’s Lemma, this matrix is just the matrix of second derivatives of the profit func-

tion: ∂y∗∂p

∂y∗∂w1

. . . ∂y∗∂wn

−∂x∗1

∂p − ∂x∗1

∂w1. . . − ∂x∗

1∂wn

......

. . ....

−∂x∗n

∂p −∂x∗n

∂w1. . . − ∂x∗

n

∂wn

=

∂2π∗∂p∂p

∂2π∗∂p∂w1

. . . ∂2π∗∂p∂wn

∂2π∗∂w1∂p

∂2π∗∂w1∂w1

. . . ∂2π∗∂w1∂wn

......

. . ....

∂2π∗∂wn∂p

∂2π∗∂wn∂w1

. . . ∂2π∗∂wn∂wn

.

Order invariance of second partial derivatives implies symmetry of this matrix. Convexity of the profit

function implies positive semidefiniteness.

3. py∗ − w · x∗ ≥ 0 and x∗(0, w) = 0 for w >> 0 are just nonnegativity of the profit function again. Then,

by possibility of inaction, 0 is a feasible value for y∗(0, w), and by no free lunch no positive value of y is

feasible. Free disposal then yields y∗(0, w) = (−∞, 0] ∀w >> 0. x∗ ≥ 0 is by assumption that x ≥ 0 for

any (y,−x) ∈ Z (the single-output case). y∗ ≥ 0 is due to π∗ ≥ 0 since, with strictly positive prices and

nonnegative inputs, profit is strictly negative if y∗ < 0. �

• Note, in particular, that cross-price effects on supply/demand are symmetric and that own-price effects

are nonnegative for supply and nonpositive for demand. Hence the comparative statics results we derived in

the previous lecture arise as a consequence of the general properties of the maximum here.

• Second, homogeneity is the mathematical statement of the claim that “only relative prices matter.”

• Third, Euler’s Theorem for the homogeneous of degree zero supply/factor demand yields

p∂y∗(p, w)

∂p+

n∑j=1

wj∂y∗(p, w)

∂wj= 0

−p∂x∗

i (p, w)∂p

−n∑j=1

wj∂x∗

i (p, w)∂wj

= 0 for i = 1, . . . , n.

Hence

Hπ∗(p,w)

[pw

]=

∂y∗

∂p∂y∗

∂w1. . . ∂y∗

∂wn

−∂x∗1

∂p − ∂x∗1

∂w1. . . − ∂x∗

1∂wn

......

. . ....

−∂x∗n

∂p−∂x∗

n

∂w1. . . − ∂x∗

n

∂wn

pw1...

wn

=

00...0

,

and we see that the Hessian of the profit function with respect to (p, w) is singular, possessing a zero

eigenvalue.


• Note finally that the only property of the technology we really rely on here is that a maximum exists

(related to regularity, but regularity isn’t sufficient), except that nonnegativity relies in addition on the

possibility of inaction and no free lunch. We used free disposal as well for the definition of f on all of Rn+

but this is not essential – we could work directly with the netput vectors instead. Hence, as long as the

maximum exists, the profit function and the supply/demand function have these basic properties, essentially

just because they maximize.


Exercises for Lecture 4 (Profit Maximization: Advanced)

1. Let z∗i (p, w) be one component of the supply/factor demand vector. Use Euler’s Theorem to obtain

n∑j=1

εij + εip = 0,

where εij =∂z∗i∂wj

wj

z∗iis the elasticity of z∗i with respect to wj and εip = ∂z∗i

∂ppz∗i

is the elasticity of z∗i with

respect to p. This shows that the price elasticities of any profit-maximizing supply/demand function sum

to zero.


2. Varian #3.1.

3. Varian #3.2.

4. Varian #3.5.


Lecture 5: Cost Minimization: Setup, Calculus, and Traditional Comparative Statics


Now we begin consideration of an alternative behavioral postulate for firms. Sometimes the profit maxi-

mization behavioral postulate is not useful, for example if there is:

• Nondecreasing RTS with π > 0 possible, or

• Imperfect competition in the output market.

In these cases we can still study the implications of cost minimization:

min{x}

c(x;w) = w · x subject to (y,−x) ∈ Z,

for y that is feasible (i.e., V (y) �= ∅). If y is feasible then y ∈ [0, y] is feasible, by free disposal. Thus the set

of feasible y’s is an interval of the form [0, y] or [0, y), where y = ∞ is possible. We henceforth denote the

upper endpoint of the interval of feasible y’s by y, and denote the interval by Y .

The constraint can be written in several equivalent forms. First is x ∈ V (y), by definition of V (y).

Or, again using free disposal, the constraint can be written f(x) ≥ y and x ≥ 0 (assuming possibility of

inaction, so that f is defined on Rn+). To see that this version is equivalent to the first two, note that

V (y) ⊂ {x ∈ Rn+ : f(x) ≥ y} by definition of f . Conversely, if f(x) ≥ y for some x ≥ 0 then x ∈ V (f(x))

(by regularity) and V (f(x)) ⊂ V (y) by free disposal. Hence {x ∈ Rn+ : f(x) ≥ y} = V (y). Finally, if f is

also continuous and w >> 0 then the constraint can be written without loss of generality as f(x) = y and

x ≥ 0. This last version follows from continuity of f since we already know f(x) ≥ y for any x ∈ V (y), but

if f(x) > y then by continuity there exists x′ < x (note that x > 0 by no free lunch since f(x) > y ≥ 0) such

that f(x′) > y and w · x′ < w · x (since w >> 0). Hence if f(x) > y then x cannot minimize w · x subject to

f(x) ≥ y, and therefore nothing is lost by just assuming the constraint is f(x) = y.

The choice variables in the cost minimization problem are x and the parameters are (w, y). The fact that

y is a parameter is what distinguishes this behavioral postulate from the profit maximization behavioral

postulate. Note, however, the following:

Theorem. Every profit maximizing firm also minimizes cost.

So, the cost minimization behavioral postulate is worth studying just as a way of better understanding profit

maximization. The proof is as follows:

28 LECTURE 5: COST MINIMIZATION (BASICS)

Proof. Let (y∗, x∗) be profit-maximizing choices, and suppose cost is not minimized at these choices. Then

there exists x ∈ V (y∗) such that w · x < w · x∗. But then py∗ − w · x > py∗ − w · x∗ = π∗ while x ∈ V (y∗),

which contradicts that (y∗, x∗) maximizes profit. �

Cost minimization leads to an optimal choice x∗(w, y) and an optimal value c∗(w, y), which are the

conditional factor demand function and cost function, respectively, of the firm. Note that this x∗ is different

from the profit-maximizing x∗, in that the former is conditional on y while the latter is not (the latter is

sometimes called an unconditional factor demand).

With two inputs, the minimization can be illustrated as in Figure 5.1. The firm’s objective is to choose

that value of x that places it on the lowest isocost line while still staying in the input requirement set V (y).

Figure 5.1: Cost Minimization

Assuming that f is strictly increasing, and further assuming that f is differentiable, calculus can be used

to describe the minimum. The Lagrangian function is then

L(x, λ;w, y) = w · x− λ(f(x) − y),

so the first order conditions are

wi − λ∗fi(x∗) ≡ 0 for i = 1, . . . , n

−(f(x∗)− y) ≡ 0.


Dividing any two of these yields

fi(x∗)fj(x∗)

=wiwj

,

which again is the familiar tangency condition that the MRTS between any two inputs must equal their price

ratio (slope of the isocost line) at the firm’s optimal choice. That is, the firm must choose its inputs so that

its isoquant is tangent to its isocost.

These first order conditions are the basis for our investigation of how the optimal choices x∗(w, y) change

when either an input price or the output level changes. These responses tell the shape of the conditional

factor demand functions of the firm. Once again noting that the first order conditions evaluated at the

maximum are identities, we can differentiate with respect to an input price wj to obtain:

− ∂λ∗

∂wjf1(x∗)− λ∗

n∑i=1

f1i(x∗)∂x∗

i

∂wj= 0 or − 1

...

− ∂λ∗

∂wjfn(x∗)− λ∗

n∑i=1

fni(x∗)∂x∗

i

∂wj= 0 or − 1

−n∑i=1

fi(x∗)∂x∗

i

∂wj= 0.

Organizing these equations into matrices yields:

−λ∗f11 . . . −λ∗f1n −f1

......

...−λ∗fn1 . . . −λ∗fnn −fn−f1 . . . −fn 0

∂x∗1

∂wj

...∂x∗

n

∂wj

∂λ∗∂wj

=

0...0−10...0

← jth line.

Once again, this is just

HL(x,λ)

∂x∗

1∂wj

...∂x∗

n

∂wj

∂λ∗∂wj

=

0...0−10...0

,


so by Cramer’s Rule

∂x∗i

∂wj=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

−λ∗f11 . . . 0 . . . −λ∗f1n −f1...

......

...... −1

......

......

......

−λ∗fn1 . . . 0 . . . −λ∗fnn −fn−f1 . . . 0 . . . −fn 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣|HL(x,λ)| .

Once again, if the minimum is a proper interior minimum then second order conditions will help us sign

this, but we must recall what these conditions are for constrained optimization. The Hessian matrix of

the Lagrangian function L is known as a bordered Hessian. The “border” contains second derivatives of

L with respect to the Lagrangian multiplier λ. If there is more than one constraint then there is more

than one Lagrangian multiplier and hence the “border” has more than one row and column. We can define

principal minors of this matrix in the same way that we do for unconstrained optimization, but the sufficient

second order conditions apply only to border-preserving principal minors, and are affected by the number of

constraints. The sufficient second order conditions for a proper interior constrained maximum are that all

border-preserving principal minors of order k > 2r have sign (−1)k−r, where r is the number of constraints

and k counts the rows and columns in the border. These conditions are really just an algebraic way of checking

that the objective function c is strictly concave in the choice variables subject to constraint (i.e., along the

path of choices that the constraint allows), or negative definiteness of the Hessian of the objective subject

to constraint. When the objective is linear, this is equivalent to strict quasiconcavity of the constraint in

the choice variables: full concavity isn’t needed because y isn’t being chosen. Similarly, the sufficient second

order conditions for a proper interior constrained minimum are that all border-preserving principal minors of

order k > 2r have sign (−1)r. This is just an algebraic way of checking that the objective c is strictly convex

in the choice variables subject to constraint, or positive definiteness of the Hessian of the objective subject

to constraint, which is strict quasiconvexity of the constraint in the choice variables when the objective is

linear. NOTE: These conditions agree with our statement of second order conditions for unconstrained

optimization if we just set r = 0. Also, as in the unconstrained case an equivalent condition is that only the

naturally-ordered border preserving principal minors of order k > 2r have sign (−1)k−r (maximum) or (−1)r

(minimum). Since many fewer determinants are involved, these latter conditions are convenient for checking


second order conditions in any particular problem. However, the knowledge that all principal minors have

a known sign at a proper interior optimum is useful for performing comparative statics and so we state the

second order conditions in terms of all principal minors.

Applying these conditions to the cost minimization problem, if we have a proper interior minimum then

|HL(x,λ)| must have sign (−1)1 since there is one constraint. Using cofactor expansion on the numerator of

our expression for ∂x∗i

∂wjyields (−1)(−1)i+j |Hji|, where Hji is the submatrix of HL(x,λ) of order n that results

when row j and column i are deleted. This has unknown sign when i �= j because |Hji| is not a principal

minor (or because Hji is not border-preserving if either i or j is n + 1). However, if i = j < n + 1 then

|Hji| has sign (−1)1, so ∂x∗i

∂wi< 0. Again, this is natural: own price effects are negative, while cross-price

effects are indeterminate in general. Hence the law of demand holds for the conditional factor demand of a

cost-minimizing firm and is a direct consequence of the behavioral postulate, and nothing more. As with the

unconditional demands, the ambiguous cross-price effects are usually categorized by comparison with the

negative own-price effect. If x∗j moves in the same direction as x∗

i when wi changes (i.e., if∂x∗

j

∂wi< 0) then we

say inputs i and j are (net) complements in production. If the two inputs move in opposite directions (i.e.,

if ∂x∗j

∂wi> 0) then we say inputs i and j are (net) substitutes in production. These labels are “net” because

the changes under study here are net of any changes in output, but usually the complement/substitute

nature of conditional factor demands is described without using the word “net” (as compared to the “gross”

complements or substitutes characterization of profit maximizing demands, which are gross of changes in

the optimal output level, and which are usually described by explicit use of the word “gross”).

These methods can be used to study the effect of a change in the output level y on x∗, and for finding

the effects of changes in w and y on λ∗. This is pursued in the exercises, but none of these effects are known

in general. If ∂x∗i

∂y ≥ 0 we say input i is normal in production, while if ∂x∗i

∂y < 0 we say input i is inferior in

production. Applying the chain rule to differentiate x∗∗j (p, w) ≡ x∗

j (w, y∗∗(p, w)) with respect to p (where

the double asterisk indicates profit-maximizing choices), and recalling that y∗ is increasing in its own-price

p, we see that these labels are consistent with the normal/inferior labels given for the unconditional factor

demands.

We have proceeded here as if we have a proper interior minimum. There are several cautions that must


be adhered to regarding this:

1. Since x ≥ 0 is required, we can have corner solutions of the form x∗i = 0. Then the first derivative can be

positive: wi − λ∗fi(x∗) > 0.

2. Existence is not much of a problem here, because we have a continuous objective on a nonempty, closed

and bounded constraint set (when w >> 0 and y ∈ Y , if we throw out the upper part of V (y)). Henceforth

we will assume w >> 0 and y ∈ Y .

3. Uniqueness is a real issue unless the technology is strictly convex.


Exercises for Lecture 5 (Cost Minimization: Basics)

1. Use the comparative statics methodology to derive expressions for the slope of the conditional factor

demand with respect to output, and the slope of the optimal Lagrange multiplier with respect to input

price i and also with respect to output. Can the sign of any of these be determined? Why?

2. Assume that the conditional factor demand and unconditional demand/supply choices are unique.

a. Prove the following identity:

x∗∗(p, w) ≡ x∗(w, y∗∗(p, w)).

b. Use this identity to show that complements that are either both normal or both inferior in production

are gross complements, but the converse need not hold (Hint: Use symmetry of the demand/supply

derivatives). Similarly, show that substitutes, when one is normal and the other is inferior, are gross

substitutes, but the converse need not hold. Can you begin with a gross characterization and infer a

net characterization if certain normal/inferior combinations hold?

3. Suppose f(x1, x2) = x1x2 is a production function. Find x∗ and c∗.


4. MWG #5.C.10.

5. Varian #5.4.

6. Varian #5.17.


Lecture 6: Cost Minimization: Envelope Properties of c∗ and x∗

Readings: Sections 5.4-5.6 of Varian; pp. 139-143 of MWG.

In this lecture we continue our study of the properties of a cost minimum, but make use of the envelope

theorem rather than using the somewhat cumbersome traditional comparative statics. The optimal choice

is x∗(w, y) and the optimal value of the cost function is

c∗(w, y) = w · x∗(w, y), for w >> 0 and y ∈ Y.

Like the profit and supply/demand functions, these functions have some remarkable properties due solely to

the fact that they are the minimizing choices and minimal value, respectively, of w · x.

Theorem (Shephard’s Lemma–Relationship between the Cost Function and the Conditional

Factor Demand). If c∗ is differentiable at (w, y) (almost assured by concavity) then ∂c∗∂wi

= x∗i (w, y) and

∂c∗∂y = λ∗(w, y).

Proof. Differentiate c∗(w, y) with respect to wi, using the envelope theorem so that this is obtained by

differentiating the Lagrangian function and then evaluating at the optimum, to get x∗i (w, y). Likewise,

differentiate with respect to y to get λ∗(w, y). �

Theorem (Properties of the Cost Function).

1. (Homogeneity) c∗ is homogeneous of degree 1 in w.

2. (Concavity) c∗ is concave in w.

3. (Monotonicity) c∗ is nondecreasing in (w, y). If f is lower semicontinuous, the monotonicity in y is strict.

4. (Continuity) c∗ is lower semicontinuous in (w, y) and is fully continuous in w for fixed y at w >> 0. If Z

is lower hemicontinuous in output, the continuity is full in (w, y) at w >> 0.

5. (Nonnegativity) c∗ > 0 ∀w >> 0 and ∀y ∈ Y − {0}, and c∗(w, 0) = 0 ∀w >> 0.

Proof.

1. We will show below that x∗ is homogeneous of degree 0 in w. Substitution then yields

c∗(λw, y) = (λw) · x∗(λw, y) = (λw) · x∗(w, y) = λc∗(w, y).

36 LECTURE 6: COST MINIMIZATION (ADVANCED)

2. Fix (w, y) and (w′, y). By definition,

c∗(w, y) ≤ w · x ∀x ∈ V (y),

c∗(w′, y) ≤ w′ · x ∀x ∈ V (y).

Multiply the first of these by λ and the second by (1− λ) and add to get

λc∗(w, y) + (1− λ)c∗(w′, y) ≤ (λw + (1− λ)w′) · x ∀x ∈ V (y)

= c(x; λw+ (1− λ)w′) ∀x ∈ V (y),

so the left side is a lower bound of the right side cost objective across all feasible input vectors x. Hence

λc∗(w, y) + (1− λ)c∗(w′, y) ≤ c∗(λw + (1− λ)w′, y).

3. Select w′ ≤ w and y′ ≤ y. By definition,

c∗(w′, y′) ≤ w′ · x ≤ w · x ∀x ∈ V (y′).

By free disposal, V (y) ⊂ V (y′), so

c∗(w′, y′) ≤ w · x ∀x ∈ V (y).

Hence, using the lower bound logic again, c∗(w′, y′) ≤ c∗(w, y). To establish that the monotonicity in y

is strict when f is lower semicontinuous, let y′ < y and suppose w · x∗(w, y) = c∗(w, y) ≤ c∗(w, y′). Since

V (y) ⊂ {x ∈ Rn+ : f(x) ≥ y}, we know f(x∗(w, y)) ≥ y > y′. But, since f is lower semicontinuous and

w >> 0, we know the constraint must be satisfied with equality at the minimum. Therefore, x∗(w, y)

does not minimize w · x subject to x ∈ V (y′): it is possible to achieve lower cost. This contradicts that

c∗(w, y′) is a minimum.

4. Suppose c∗ is not lower semicontinuous at (w0, y0), where w0 >> 0. Then there exists ε > 0 and a sequence

(wi, yi) → (w0, y0) such that c∗(wi, yi) ≤ c∗(w0, y0) − ε for every i = 1, 2, . . . . A cost minimum always

exists when the input price vector is strictly positive and the output level is feasible, so there exists a

sequence xi such that (yi,−xi) ∈ Z and wi ·xi = c∗(wi, yi). The sequence xi is bounded because (1) wi ·xi

is bounded by c∗(w0, y0)− ε, and (2) wi is bounded away from zero since it converges to w0 >> 0. Hence

xi has a convergent subsequence xki → x0, implying c∗(wki , yki) = wki · xki → w0 · x0 ≤ c∗(w0, y0) − ε.


But (y0 ,−x0) ∈ Z because Z is closed, so w0 ·x0 ≤ c∗(w0, y0)−ε contradicts that c∗(w0, y0) is a minimum.

This establishes lower semicontinuity of c∗. Full continuity of c∗ in w for fixed y is implied by concavity.

Full continuity of c∗ in (w, y) when Z is lower hemicontinuous in output is a consequence of the Theorem

of the Maximum (see MWG Theorem M.K.6).

5. By no free lunch, x∗(w, y) > 0 when y > 0. Hence c∗(w, y) = w · x∗(w, y) > 0 when (w, y) >> 0. By the

possibility of inaction, 0 ∈ V (0). Hence c∗(w, 0) ≤ w · 0. But x, w ≥ 0 for all feasible x, so c∗(w, y) ≥ 0

for any (w, y) ≥ 0. Combine to get c∗(w, 0) = 0. �

Theorem (Properties of the Conditional Factor Demand).

1. (Homogeneity) x∗ is homogenous of degree 0 in w.

2. (Symmetry and Semidefiniteness) The matrix of slopes∂λ∗∂y

∂λ∗∂w1

. . . ∂λ∗∂wn

∂x∗1

∂y∂x∗

1∂w1

. . .∂x∗

1∂wn

......

. . ....

∂x∗n

∂y∂x∗

n

∂w1. . .

∂x∗n

∂wn

is symmetric for y ∈ (0, y), and the lower right (n× n) submatrix is negative semidefinite.

3. (Monotonicity) w · x∗ is nondecreasing in (w, y). If f is lower semicontinuous, the monotonicity in y is

strict.

4. (Continuity) w · x∗ is lower semicontinuous in (w, y) and is fully continuous in w for fixed y at w >> 0.

If Z is lower hemicontinuous in output, the continuity is full in (w, y) at w >> 0.

5. (Nonnegativity) x∗ ≥ 0 ∀w >> 0 and ∀y ∈ Y , w · x∗(w, 0) = 0, and x∗(w, y) > 0 ∀w >> 0 and

∀y ∈ Y − {0}.

Proof.

1. λc(x;w) is a monotonic transformation of the objective function c(x;w) for λ > 0, and due to linearity is

equal to c(x; λw). Since w is not part of the constraint, x∗(λw, y) = x∗(w, y).

2. Differentiating Shephard’s Lemma, this matrix is just the matrix of second derivatives of the cost function:∂λ∗∂y

∂λ∗∂w1

. . . ∂λ∗∂wn

∂x∗1

∂y∂x∗

1∂w1

. . .∂x∗

1∂wn

......

. . ....

∂x∗n

∂y∂x∗

n

∂w1. . .

∂x∗n

∂wn

=

∂2c∗∂y∂y

∂2c∗∂y∂w1

. . . ∂2c∗∂y∂wn

∂2c∗∂w1∂y

∂2c∗∂w1∂w1

. . . ∂2c∗∂w1∂wn

......

. . ....

∂2c∗∂wn∂y

∂2c∗∂wn∂w1

. . . ∂2c∗∂wn∂wn

.


Order invariance of second partial derivatives implies symmetry of this matrix. Concavity of the cost

function in w implies negative semidefiniteness of the lower (n × n) submatrix.

3. This is just monotonicity of the cost function.

4. This is just continuity of the cost function.

5. x∗(w, y) ≥ 0 because x∗ is chosen from Rn+. w · x∗(w, 0) = 0 is just c∗(w, 0) = 0 again. x∗(w, y) >

0 ∀w >> 0 and ∀y ∈ Y − {0} is just c∗(w, y) > 0 again for these (w, y) points. �

• Note, in particular, that cross-price effects on the conditional factor demand are symmetric and that

own-price effects are nonpositive. Hence the comparative statics results we derived in the previous lecture

arise as a consequence of the general properties of the minimum here.

• Second, marginal cost is equal to the optimal Lagrange multiplier, which is nonnegative due to mono-

tonicity.

• Third, homogeneity is again the mathematical statement of the claim that “only relative prices matter.”

• Fourth, Euler’s Theorem for the homogeneous of degree zero conditional factor demands yields

n∑j=1

wj∂x∗

i (w, y)∂wj

= 0 for i = 1, . . . , n.

Hence

Hc∗(w)w =

∂x∗

1∂w1

. . .∂x∗

1∂wn

.... . .

...∂x∗

n

∂w1. . .

∂x∗n

∂wn

w1

...wn

=

0...0

,

and we see that the Hessian of the cost function with respect to w is singular, possessing a zero eigenvalue.

• Note also that the main property of the technology we really rely on here is that a minimum exists, which

is guaranteed by regularity, w >> 0, and y ∈ Y . Thus, the cost function and the conditional factor demand

function have these basic properties, essentially just because they minimize, except that monotonicity in y

relies on free disposal and no free lunch (strict monotonicity relies in addition on the possibility of inaction

and lower hemicontinuity in inputs), while nonnegativity relies on no free lunch and the possibility of inaction.


Exercises for Lecture 6 (Cost Minimization: Advanced)

1. Use Euler’s Theorem to show that the price elasticities of a conditional factor demand sum to zero.

2. Suppose c∗(w1, w2, y) = 2√yw1w2 and x∗(w1, w2, y) =

(√yw2w1

,√

yw1w2

). Verify that c∗ has the properties

of a cost function and that Shephard’s Lemma holds.


3. Varian #4.8.

4. Varian #5.6.

5. Varian #5.12.


Lecture 7: Duality of the Cost Function

Readings: Chapter 6 of Varian; Section 3.F of MWG.

We have now studied two optimization problems for firms: profit maximization and cost minimization.

In each case we found that the optimal value function necessarily has some properties (homogeneity, concav-

ity/convexity, monotonicity, continuity, and nonnegativity), mostly for no reason other than its definition

as an optimum of a particular objective. Similarly, in each case we found that the optimal choice func-

tion necessarily has some properties (homogeneity, symmetry/semidefiniteness, monotonicity and continuity

(for conditional demands), and nonnegativity), again basically because of its definition as an optimizer of a

particular objective. Hence, these properties are necessary conditions for cost, profit, supply, and demand

functions.

A natural question is to ask now whether these properties are sufficient for their respective functions.

That is, for example, given a function possessing the 5 properties of a cost function, is that function a cost

function for some firm? This question is really of more importance for economic research than the derivation

of the necessary conditions, because its affirmative answer enables us to use cost functions without regard

for the underlying production function/technology. In each of the cases we have studied, the properties we

have derived are both necessary and sufficient for the function. To get the idea of the sufficiency argument,

we will focus on the cost function. The sufficiency is referred to as duality, since it says that the cost function

is dual to the technology, in the sense that for every well-behaved technology there is a cost function with

the 5 properties, and vice-versa.

The algebra of the argument can get involved, but the geometry is straightforward. Given a cost function

c∗, we wish to find the underlying input requirement sets V (y). To do so, fix output at y and, for some input

price vector w, draw the isocost line for cost level c∗(w, y). The input requirement set must lie everywhere

above this isocost, for otherwise a cost level smaller than c∗(w, y) would be possible. The input requirement

set must also touch the isocost somewhere, for otherwise a cost level as small as c∗(w, y) would not be

attainable. Repeat this argument for all different input price vectors and then intersect the areas to obtain

V (y). This is illustrated in Figure 7.1.

42 LECTURE 7: DUALITY OF THE COST FUNCTION

Figure 7.1: Recovering the Input Requirement Set from the Cost Function

The algebraic statement of this is to form a “hypothetical” input requirement set

H(y) ≡ {x ∈ Rn+ : w · x ≥ c∗(w, y) ∀w >> 0}.

We then must ask whether we can formally show that H(y) is equal to V (y). To do so, we must show that

each set contains the other. The containment H(y) ⊃ V (y) is easy, for if x ∈ V (y) then w · x can be no

smaller than c∗(w, y), by definition of c∗. Since this statement holds for every w, x ∈ H(y). But the opposite

containment is not true in general. The problem that can arise here is due to potential nonconvexity of V (y),

as illustrated in Figure 7.2.

If V (y) is convex then it can be shown that H(y) ⊂ V (y), so the two sets are actually equal. In this case,

H recovers the true underlying technology from c∗, even if all one knows is c∗. The tool for showing this is

an important theorem from convex analysis, known as the separating hyperplane theorem:

Separating Hyperplane Theorem. Let V be a closed, convex, nonempty set in Rn and x be a point not

in V . Then there exists a hyperplane that separates V and x.

Algebraically, a hyperplane is determined by a slope vector w �= 0 and an intercept scalar c. It is the set of

points that satisfy the linear equation defined by w and c:

H(w, c) = {x ∈ Rn : w · x = c}.


Figure 7.2: The Input Requirement Set can be a Proper Subset of the “Hypothet-

ical” Version Recovered from the Cost Function

Hence, a hyperplane is a point in R1, a line in R

2, a plane in R3, and the n-dimensional generalization of a

plane in Rn. Note that many different (w, c) pairs define the same hyperplane, since if x satisfies w · x = c

then x also satisfies λw · x = λc for any scalar λ �= 0. Thus (λw, λc) defines the same hyperplane as (w, c).

Hyperplanes are illustrated in Figure 7.3.

Figure 7.3: Hyperplanes in R1

and R2

A hyperplane separates two sets if all points in one set lie on one side of the hyperplane while all points in


the other set lie on the other side of the hyperplane. Algebraically, a point x lies on one side of the hyperplane

if w · x > c and lies on the other side if w · x < c. So, the theorem claims that there exists an n-dimensional

vector w �= 0 and a scalar c such that w · x > c > w · x ∀x ∈ V . Note that it makes no difference which

direction of inequality we use here, since we can always multiply by the scalar λ = −1 and still be describing

the same hyperplane. Note also that the conditions of the separating hyperplane theorem are sufficient

for the existence of a separating hyperplane, but are not necessary. It is easy to draw counterexamples to

necessity. Also, the theorem is only about existence. There is no claim that the separating hyperplane is

unique (indeed, it is not unique). Figure 7.4 illustrates the separating hyperplane theorem.

Figure 7.4: Separating Hyperplane Theorem in R2. In (a), the conditions of the theorem

hold and a hyperplane H that separates x and V is illustrated. In (b), no hyperplane can bedrawn between x and V because V is not convex. In (c), no hyperplane can be drawn between xand V because V is not closed (here, x is on the boundary of V ). By requiring that V be closed,we obtain strict inequalities in the statement of the theorem. Clearly a hyperplane can be drawnin (c) that passes through x and is just tangent to V . Such a hyperplane would “separate” Vand x with a weak inequality.

How does this theorem prove H(y) ⊂ V (y) when V (y) is convex? Suppose otherwise. Then there exists

x ∈ H(y) that is not in V (y). So, by the separating hyperplane theorem there exists w �= 0 and c such that

w · x > c > w · x ∀x ∈ V (y).

Note that we must have w > 0, for otherwise we can use free disposal to set x so large that these inequalities

must be violated. Moreover, we may assume w >> 0, since doing so can only increase the left side while


increasing the right side an arbitrarily small amount. Since x ∈ H(y), w · x ≥ c∗(w, y). Hence

w · x > c > c∗(w, y) ∀x ∈ V (y).

This inequality says that the objective w ·x cannot be smaller than c > c∗(w, y) on x ∈ V (y). Thus c∗(w, y) is

a cost level that cannot be attained (or even approached) among the feasible input vectors. This contradicts

that c∗(w, y) is the minimum of w · x on V (y), whence the supposition that H(y) is not contained in V (y)

must be incorrect.

What we have just shown is:

If c∗ is a cost function, in that it is the minimum of w · x subject to x ∈ V (y), then we can recover

V (y) even if all we know is c∗ (assuming V (y) is convex).

This presumes that c∗ is indeed a cost function. What if all we have is a function c∗(w, y) that is known to

have some properties? We can try to derive H(y) from c∗, but do we then know that H(y) is a legitimate

input requirement set? And, if so, do we know that c∗ is the cost function one gets by minimizing w ·x subject

to x ∈ H(y)? The answers to these questions are “yes,” but to show them we must avoid any reference to

V (y) or to the interpretation of c∗ as a cost function. Rather, we must rely solely on the properties of c∗.

So, assume only that we have a real-valued function c∗(w, y) of a strictly positive n-dimensional vector w

and a scalar y in an interval Y whose minimum is zero; and that this function satisfies the homogeneity,

concavity, monotonicity, continuity, and nonnegativity properties of a cost function (but that it may not

be a cost function–at least, we do not know this in advance). For convenience we will also assume that c∗

is continuously differentiable, although this assumption is not really needed. Then, the properties of our

“hypothetical” technology Z = {(y,−x) : x ∈ H(y) and y ∈ Y } are:

1. Z is nonempty. This is automatic from the possibility of inaction (item 4 below).

2. Z is closed. Consider any sequence (yi,−xi) → (y0 ,−x0) such that (yi,−xi) ∈ Z for i = 1, 2, . . .. We

must show w · x0 ≥ c∗(w, y0) for every w >> 0. Fix ε > 0. By definition of Z and lower semicontinuity

of c∗ in y, there exists Iε such that w · xi ≥ c∗(w, yi) > c∗(w, y0) − ε for i ≥ Iε. Taking the limit as

i → ∞ yields w · x0 ≥ c∗(w, y0) − ε. This holds for arbitrary ε > 0, so taking the limit as ε ↑ 0 yields

w · x0 ≥ c∗(w, y0).

3. H(y) satisfies no free lunch. Select any w >> 0. When y > 0 we have c∗(w, y) > 0 by nonnegativity.


Hence w · x > 0 ∀x ∈ H(y). This rules out 0 as an element of H(y).

4. H(y) satisfies the possibility of inaction. By nonnegativity, c∗(w, 0) = 0 ∀w >> 0. Hence w · 0 ≥

c∗(w, 0) ∀w >> 0, so 0 ∈ H(0).

5. H(y) satisfies free disposal. Let x ∈ H(y), and suppose x′ ≥ x and y′ ≤ y. Then, using monotonicity of

c∗ in y, w · x′ ≥ w · x ≥ c∗(w, y) ≥ c∗(w, y′) ∀w >> 0. Hence, x′ ∈ H(y′).

7. H(y) is always convex. Let x, x′ ∈ H(y). Then w · x ≥ c∗(w, y) and w · x′ ≥ c∗(w, y), both for all w >> 0.

Multiply the first by λ and the second by (1−λ) and add to obtain w · [λx+(1−λ)x′] ≥ c∗(w, y) ∀w >> 0.

Thus λx+ (1− λ)x′ ∈ H(y).

8. If c∗ is strictly increasing in y, then Z is lower hemicontinuous in inputs. That is, Z has the property:

x0 ∈ H(y0) and xi → x0 implies existence of yi → y0 such that xi ∈ H(yi). The proof is tedious, and is

relegated to the Appendix.

So, H(y) has all of the properties of an input requirement set, and thus we can define a cost function as

the minimum of w · x subject to x ∈ H(y). Is the cost function we define in this way the same function we

started with? That is, do we have

c∗(w, y) = min{x}

w · x subject to x ∈ H(y)?

To show this, fix w0 arbitrarily. By definition, w0 · x ≥ c∗(w0, y) ∀x ∈ H(y). That is, c∗(w0, y) is a lower

bound for w0 · x among those x vectors in H(y), so certainly

c∗(w0, y) ≤ min{x}

w0 · x subject to x ∈ H(y).

It remains to show the opposite inequality. Since c∗ is homogeneous of degree one in w, by Euler’s Theorem

we haven∑i=1

w0i∂c∗(w0, y)

∂wi= c∗(w0, y),

so by setting x0i = ∂c∗(w0,y)∂wi

we have a vector x0 (nonnegative by monotonicity of c∗ in w) such that

w0 · x0 = c∗(w0, y). By concavity of c∗ in w,

c∗(w, y) ≤ c∗(w0, y) +n∑i=1

∂c∗(w0, y)∂wi

(wi − w0i)

= c∗(w0, y) + (w −w0) · x0

= w · x0.


Since this holds for every w >> 0, we have x0 ∈ H(y). That is, we have derived an x0 such that w0 · x0 =

c∗(w0, y) and x0 ∈ H(y). Hence

min{x}

w0 · x subject to x ∈ H(y) ≤ w0 · x0 = c∗(w0, y).

Hence c∗ is indeed the cost function for the well-behaved input requirement sets H(y). That is, the 4

properties of a cost function are sufficient as well as necessary. Hence we can use any function satisfying

these properties as a cost function without actually deriving it from some underlying technology. This also

shows that H(y) is the economically relevant technology for a firm with cost function c∗, even if H(y) differs

from V (y). Nonconvexities in production are irrelevant for the cost function.

Appendix to Lecture 7: Lower Hemicontinuity of Z

Assume c∗ strictly increasing in y on Y . To show: If x0 ∈ H(y0) and xi → x0 then there exists yi → y0

such that xi ∈ H(yi), where

H(y) = {x ∈ Rn+ : w · x ≥ c∗(w, y) ∀w >> 0}.

Fix xi and w >> 0. By monotonicity of c∗ in y, {y ∈ Y : w · xi ≥ c∗(w, y)} is an interval from 0 to some

upper bound (perhaps ∞), with 0 included (by nonnegativity of c∗) and the inclusion of the upper bound

unimportant. As we vary w >> 0, the collection of these upper bounds is nonempty and bounded below

by zero, so it has an infimum (perhaps ∞). Let yi denote the infimum, so that xi ∈ H(y) ∀y < yi and

xi /∈ H(y) ∀y > yi at which c∗ is defined.

Now consider the claim: For every ε > 0 there exists Iε such that

i ≥ Iε ⇒ yi > y0 − ε.

To show this, fix ε > 0 and suppose otherwise. Then there exists a subsequence yki ≤ y0 − ε ∀i. So, by

definition of yki

, there exists wki

>> 0 such that

c∗(wki

, y) > wki · xki ∀y > y0 − ε.

By homogeneity of c∗, this statement holds for λki

wki

as well, where λki

=(∑n

j=1 wki

j

)−1

> 0. Thus,

we may assume that wki

is on the unit simplex for every i. Then wki

has a convergent subsequence


w�ki → w0. Since w�ki · x�ki → w0 · x0 ≥ c∗(w0, y0), we can use continuity1 of c∗ in w to conclude

c∗(w0, y) ≥ c∗(w0, y0) ∀y > y0 − ε. However, c∗(w0, y) < c∗(w0, y0) for y < y0 by strict monotonicity of c∗,

a contradiction. This proves existence of Iε.

Now define yi = min{yi, y0}− 1i . Then yi < yi, so xi ∈ H(yi). And for any ε > 0, y0 −ε− 1

i < yi ≤ y0− 1i

for i ≥ Iε. Hence yi → y0. �

1If w0j = 0 for some j then we don’t have continuity of c∗. In fact, the present treatment doesn’t even establish existence of

c∗ at such a w0. To deal with this requires defining c∗ as an infimum on the boundaries of Rn+ and then utilizing the limiting

continuity of c∗ at these edges.


Exercises for Lecture 7 (Duality of the Cost Function)


1. Varian #5.16.

2. Varian #6.2.

3. Varian #6.3.


Lecture 8: Duality of Supply, Demands, and Profit

Readings: Chapter 6 of Varian; Section 3.F of MWG.

Having now studied the sufficiency of the properties of a cost function, we can proceed to show that the

properties of the conditional factor demand are sufficient as well. Assume we have an n-dimensional vector-

valued function x∗(w, y) of a strictly positive n-dimensional vector w and a scalar y in an interval Y whose

minimum is zero; and that this function satisfies the homogeneity, symmetry/semidefiniteness, monotonicity,

continuity, and nonnegativity properties of a conditional factor demand function (but that it may not be a

conditional factor demand function–at least, we do not know this in advance). Once again we will assume

differentiability for convenience only. Can we show that x∗ is indeed the conditional factor demand for some

firm?

Begin by setting c∗(w, y) = w · x∗(w, y). If this defined function c∗ has the 5 properties of a cost function

then we know from our previous sufficiency proof that c∗ is actually a cost function for the input requirement

sets H(y). If so, then by Shephard’s Lemma the conditional factor demands associated with this cost function

are

∂c∗

∂wi= x∗

i (w, y) +n∑j=1

wj∂x∗

j

∂wi

= x∗i (w, y) +

n∑j=1

wj∂x∗

i

∂wjfor i = 1, . . . , n, by symmetry.

Note that we are not using the envelope theorem here, since we don’t yet know that x∗ is the optimal choice

for minimizing w · x subject to x ∈ H(y). Hence we can only differentiate by brute-force at this point. But,

by homogeneity of x∗ and Euler’s Theorem (this time for a HOD 0 function) we have

n∑j=1

wj∂x∗

i (w, y)∂wj

= 0.

Hence ∂c∗∂wi

= x∗i . That is, x

∗ is the conditional factor demand associated with the cost function c∗, provided

c∗ is indeed a cost function.

All that remains, then, is to use the properties of x∗ to verify that c∗ satisfies the 5 properties of a cost

function.

1. c∗ is homogeneous. By homogeneity of x∗, c∗(λw, y) = (λw) · x∗(λw, y) = λw · x∗(w, y) = λc∗(w, y).

52 LECTURE 8: DUALITY OF SUPPLY, DEMANDS, AND PROFIT

2. c∗ is concave. Since ∂c∗∂wi

= x∗i , the Hessian of c∗ in w is the matrix of first derivatives of x∗ with respect

to w, which is negative semidefinite by the semidefiniteness property of x∗.

3. c∗ is monotonic. Again, since ∂c∗∂wi

= x∗i and x∗ is a nonnegative function, we have the monotonicity in w.

Moreover, monotonicity in y follows directly from the monotonicity property of x∗. If the monotonicity

property of x∗ is strict, then c∗ is strictly increasing in y and so H(y) is lower hemicontinuous in inputs.

4. c∗ is lower semicontinuous. This follows directly from the continuity property of x∗.

5. c∗ is nonnegative. When w >> 0 and y ∈ Y − {0} we have c∗ = w · x∗ > 0 because x∗ > 0 due to

nonnegativity of x∗. Similarly, when y = 0 we have c∗ = 0 because w · x∗ = 0 by nonnegativity of x∗.

That is, the 5 properties of a conditional factor demand function are sufficient as well as necessary. Hence

we can use any function satisfying these properties as a conditional factor demand function without actually

deriving it from some underlying technology.

These same techniques can be used to show that the 4 properties of a profit function and the 3 properties of

the demand/supply function are sufficient. Assume we have a real-valued function π∗(p, w) of a nonnegative

scalar p and a strictly positive n-dimensional vector w, and that this function satisfies the homogeneity,

convexity, monotonicity, and nonnegativity properties of a profit function (but that it may not be a profit

function–at least, we do not know this in advance). From π∗, define a hypothetical production set

H = {(y, x) ∈ R1 ×Rn

+ : (p, w) · (y,−x) ≤ π∗(p, w) ∀p ≥ 0 and w >> 0}.

The geometry of this is identical to the cost function case. Draw the isoprofit line for some p. The production

set must lie below this. Repeat this argument for all different price vectors p and then intersect the areas to

obtain H . This is illustrated in Figure 8.1. The H so obtained has the properties of a production set:

1. H is nonempty. By nonnegativity of π∗, π∗(p, w) ≥ 0 ∀(p, w). Hence (0, 0) ∈ H .

2. H is closed. This is again trivial, since any sequence of (y, x)’s that satisfies py − w · x ≤ π∗(p, w) must

also satisfy this inequality in the limit.

3. H need not satisfy no free lunch. The set H can have a positive y-intercept without violating the properties

of π∗.

4. H satisfies the possibility of inaction. This is already done in item 1.


5. H satisfies free disposal. Let (y, x) ∈ H and y′ ≤ y, x′ ≥ x. Then py′ −w · x′ ≤ py −w · x ≤ π∗(p, w), so

(y′, x′) ∈ H .

7. H is always convex. Let (y, x), (y′, x′) ∈ H . Then (p, w) · (y,−x) ≤ π∗(p, w) and (p, w) · (y′,−x′) ≤

π∗(p, w), both for all (p, w). Multiply the first by λ and the second by (1 − λ) and add to obtain

(p, w) · [λ(y,−x) + (1− λ)(y′,−x′)] ≤ π∗(p, w) ∀(p, w).

Figure 8.1: Recovering the Production Set from the Profit Function

Just as in the duality arguments used for cost functions, if we began by assuming that π∗ is a profit

function then the production set H just obtained would be identical to the original production set Z from

which π∗ was derived, provided Z is convex. The proof is the same as the cost function case. For (y,−x) ∈ Z

we have π∗(p, w) ≥ py−w ·x by definition of π∗, so (y,−x) ∈ H . On the other hand, if there is a (y,−x) ∈ H

that is not in Z, then by the separating hyperplane theorem there exists a slope vector (p, w) �= 0 and an

intercept π such that (p, w) · (y,−x) < π < (p, w) · (y,−x) ∀(y,−x) ∈ Z. By free disposal (p, w) > 0 since

otherwise these inequalities would be violated for some (y,−x) ∈ Z, and we may assume w >> 0 since

this only decreases the left side while decreasing the right side an arbitrarily small amount. Then, since

(y,−x) ∈ H ,

(p, w) · (y,−x) < π < π∗(p, w), ∀(y,−x) ∈ Z,

which contradicts that π∗(p, w) is the maximum of (p, w) · (y,−x) over Z.


Irrespective of whether we begin by assuming π∗ is a profit function, we can show by relying only on the

properties of π∗ that π∗ is the optimal value function for maximizing profit subject to the production set H .

That is,

π∗(p, w) = max{(y,−x)}

py −w · x subject to (y,−x) ∈ H.

For convenience assume π∗ is continuously differentiable. Fix (p0, w0) >> 0. As with cost functions, the

fact that π∗ is at least as large as this maximum is trivial: By definition of H , π∗(p0, w0) ≥ p0y − w0 · x

for any (y,−x) ∈ H , so π∗(p0, w0) is an upper bound for the objective and hence is at least as large as the

maximum. The opposite inequality requires more effort. Let

y0 =∂π∗(p0, w0)

∂p

x0i = −∂π∗(p0, w0)∂wi

for i = 1, · · · , n,

and note that (y0, x0) ≥ 0 by monotonicity of π∗. By homogeneity of degree one of π∗, Euler’s Theorem

yields

p0∂π∗(p0, w0)

∂p+

n∑i=1

w0i∂π∗(p0, w0)

∂wi= π∗(p0, w0).

That is, (p0, w0) · (y0,−x0) = π∗(p0, w0). Since π∗ is a convex function of (p, w),

π∗(p, w) ≥ π∗(p0, w0) +∂π∗(p0, w0)

∂p(p − p0) +

n∑i=1

∂π∗(p0, w0)∂wi

(wi − w0i) ∀(p, w)

= (p0, w0) · (y0,−x0) + (p− p0, w− w0) · (y0,−x0) ∀(p, w)

= (p, w) · (y0,−x0) ∀(p, w).

Hence (y0,−x0) ∈ H , and this means that the maximum of p0y − w0 · x over H is at least as large as

p0y0 − w0 · x0 = π∗(p0, w0). Combining inequalities shows that π∗(p0, w0) is the maximum of p0y − w0 · x

over H .

That is, the 4 properties of a profit function are sufficient as well as necessary. Hence we can use any

function satisfying these properties as a profit function without actually deriving it from some underlying

technology. This also shows that H is the economically relevant technology for a firm with profit function π∗,

even if H differs from Z. Nonconvexities in production are irrelevant for the profit function.

NOTE: This proof does not work if p0 = 0, since then we cannot simply differentiate π∗. However, we

know from nonnegativity of π∗ that π∗(0, w0) = 0 ∀w0 >> 0. On the other hand, since (0, 0) ∈ H the


maximum of (p0, w0) · (y,−x) over H is at least (p0, w0) · (0, 0) = 0, so the maximum of (0, w0) · (y,−x) over

H is at least as large as π∗(0, w0). That is, π∗(p0, w0) is the maximal profit for the production set H even

when p0 = 0.

Finally we consider the sufficiency of the 3 properties of the supply and (unconditional) factor demand.

Assume we have an n + 1-dimensional vector-valued function (y∗(p, w),−x∗(p, w)) of a nonnegative scalar

p and a strictly positive n-dimensional vector w, and that this function satisfies the homogeneity, symme-

try/semidefiniteness, and nonnegativity properties of a combined supply/factor demand function (but that it

may not be a combined supply/factor demand function–at least, we do not know this in advance). Once again

we will assume differentiability for convenience only. Can we show that (y∗,−x∗) is indeed the combined

supply/factor demand for some firm?

Begin by setting π∗(p, w) = (p, w) · (y∗(p, w),−x∗(p, w)). If this defined function π∗ has the 4 properties

of a profit function then we know from our previous sufficiency proof that π∗ is actually a profit function

for the production set H . If so, then by Hotelling’s Lemma and symmetry the supply and factor demands

associated with this profit function are

∂π∗

∂p= y∗ + p

∂y∗

∂p−

n∑j=1

wj∂x∗

j

∂p= y∗ + p

∂y∗

∂p+

n∑j=1

wj∂y∗

∂wj

∂π∗

∂wi= −x∗

i + p∂y∗

∂wi−

n∑j=1

wj∂x∗

j

∂wi= −x∗

i − p∂x∗

i

∂p−

n∑j=1

wj∂x∗

i

∂wjfor i = 1, . . . , n.

Again, we have not used the envelope theorem here because we do not yet know that (y∗,−x∗) is the

optimal choice for maximizing py −w · x subject to (y,−x) ∈ H . However, we do know that y∗ and x∗i are

homogeneous of degree zero functions, so Euler’s Theorem yields

p∂y∗

∂p+

n∑j=1

wj∂y∗

∂wj= 0

p∂x∗

i

∂p+

n∑j=1

wj∂x∗

i

∂wj= 0 for i = 1, . . . , n.

Hence ∂π∗∂p = y∗ and ∂π∗

∂wi= −x∗

i for i = 1, . . . , n. That is, (y∗,−x∗) is the combined supply/factor demand

associated with the profit function π∗, provided π∗ is indeed a profit function.

All that remains, then, is to use the properties of (y∗,−x∗) to verify that π∗ satisfies the 4 properties of

a profit function.


1. π∗ is homogeneous. By homogeneity of (y∗,−x∗), π∗(λp, λw) = λpy∗(λp, λw) − (λw) · x∗(λp, λw) =

λpy∗(p, w)− λw · x∗(p, w) = λπ∗(p, w).

2. π∗ is convex. Since ∂π∗∂p

= y∗ and ∂π∗∂wi

= −x∗i , the Hessian of π∗ is the matrix of first derivatives of (y∗,−x∗)

with respect to (p, w), which is positive semidefinite by the semidefiniteness property of (y∗,−x∗).

3. π∗ is monotonic. Again, since ∂π∗∂p = y∗ and ∂π∗

∂wi= −x∗

i , and y∗ and x∗ are nonnegative functions, we

have the monotonicity.

4. π∗ is nonnegative. π∗ ≥ 0 is immediate from nonnegativity of (y∗,−x∗). Also, since x∗(w, 0) = 0 we have

π∗(0, w) = 0y∗(0, w)−w · 0 = 0.

That is, the 3 properties of a combined supply/factor demand function are sufficient as well as necessary.

Hence we can use any function satisfying these properties as a combined supply/factor demand function

without actually deriving it from some underlying technology.

NOTE: This proof does not work if p = 0, since then we cannot simply differentiate (y∗,−x∗) and

π∗. Differentiability is used in two places in this proof. First, it is used to show that (y∗,−x∗) is the

combined supply/factor demand associated with the profit function π∗. If p = 0 then by nonnegativity

(y∗,−x∗) = ((−∞, 0], 0), which is the value that the combined supply/factor demand associated with any

profit function must take on at p = 0. Hence (y∗,−x∗) is indeed the combined supply/factor demand

associated with the profit function π∗ even at p = 0. Second, differentiability is used to show that π∗ is

convex, because convexity is inferred from the Hessian of π∗. To show convexity when p = 0, first observe

that the proofs of nonnegativity, monotonicity, and homogeneity of π∗ are all valid for p = 0. Given these,

fix (p, w) and (p′, w′) with p = 0. Then:

λπ∗(p, w) + (1− λ)π∗(p′, w′) = (1− λ)π∗(p′, w′) by nonnegativity of π∗

≥ (1− λ)π∗(p′, w′ +

λ

1− λw

)by monotonicity of π∗

= π∗((1− λ)p′, (1− λ)w′ + λw) by homogeneity of π∗

= π∗(λp+ (1− λ)p′, λw + (1− λ)w′).

That is, π∗ is convex even at p = 0.

The applications of Euler’s Theorem used here reveal another useful property of the optimal value and


optimal choice functions. From the homogeneity, by Euler’s Theorem we have

Hπ∗(p,w)

[pw

]=

∂y∗∂p

∂y∗∂w1

. . . ∂y∗∂wn

−∂x∗1

∂p− ∂x∗

1∂w1

. . . − ∂x∗1

∂wn

......

. . ....

−∂x∗n

∂p −∂x∗n

∂w1. . . − ∂x∗

n

∂wn

pw1...

wn

=

p∂y

∗∂p +

∑nj=1 wj

∂y∗∂wj

−p∂x∗

1∂p −∑n

j=1 wj∂x∗

1∂wj

...−p

∂x∗n

∂p −∑nj=1 wj

∂x∗n

∂wj

=

00...0

.

Thus [ p w′ ]Hπ∗(p,w)

[pw

]= 0 ∀(p, w), and we see that the matrix H is not positive definite. In particular,

Hπ∗(p,w)

[pw

]= 0

[pw

],

so[pw

]is an eigenvector of Hπ∗(p,w) and 0 is an eigenvalue. Since Hπ∗(p,w) has a zero eigenvalue, its

determinant is zero and the Hessian matrix is therefore singular. Similar arguments show that the Hessian

matrix of the cost function with respect to w, multiplied by the input price vector w, is the zero vector; and

therefore 0 is an eigenvalue and the matrix is singular.


Exercises for Lecture 8 (Duality of Supply, Demands, and Profit)

1. Suppose

x∗(w1, w2, y) = [ey − 1]

((w2

w1

)α,

(w1

w2

)β).

a. For what values of α and β is x∗ a conditional factor demand function?

b. For those values of α and β, find the

i. cost function.

ii. supply/demand function.

iii. profit function.

iv. production function.

2. Suppose

y∗(p, w) =p

2min{w1, w2} , x∗1(p, w) =

0, w1 > w2

[0, p2/4w21], w1 = w2

p2

4w21, w1 < w2

, x∗2(p, w) =

p2

4w22, w1 > w2

p2

4w21− x∗

1, w1 = w2.

0, w1 < w2

a. Verify that (y∗,−x∗) is a supply/demand function for some technology.

b. What is the profit function?

c. What is the production function?

d. What is the conditional demand function?

e. What is the cost function?

3. Suppose

π∗(p, w1, w2) =p2

4A,

where A = w1−α1 wα

2 + wβ1w

1−β2 .

a. For what values of α and β is π∗ a profit function?

b. For those values of α and β, find the

i. supply/demand function.

ii. conditional demand function.

iii. cost function.

iv. production function.


Lecture 9: Cost Analysis

Readings: Sections 5.1-5.3 of Varian; Section 5.D of MWG.

In the previous two lectures we studied the cost function c∗(w, y) and discovered that it has the properties

of homogeneity, concavity, monotonicity, continuity, and nonnegativity. The cost function is very useful for

studying imperfectly competitive output markets. If the firm is not a price-taker in its output market then

the firm’s profit objective is

max{(x,y)}

R(y) −w · x subject to x ∈ V (y)

for some revenue function R(y), where R(y) may not take the simple form R(y) = py as in the perfectly

competitive case. Although this is a new objective for us, it is not necessary to start all over and analyze

this problem from the beginning because we know that every profit maximizing firm also minimizes cost.

Thus, we can summarize the firm’s technology by simply using its cost function, rather than by going all the

way back to the production function or the production set. Then the objective becomes

max{y}

R(y) − c∗(w, y).

Really, for economic purposes the only sense in which properties of the technology are interesting is when they

lead to some property of the cost function. In imperfectly competitive output markets the most important

aspect of cost is its shape as a function of output. Hence, we usually just suppress the input price vector

w (i.e., assume these prices aren’t changing) and write c∗(y) or just c(y). For purposes of this lecture we

will assume that we have nice, unique, proper interior cost minima and that all needed derivatives exist.

Nonnegativity and monotonicity tell us that c∗(y) is zero when y = 0 and is increasing, so c∗(y) appears as

in Figure 9.1.

There are several useful cost concepts that are defined from c∗(y):

1. Average cost: ac(y) = c∗(y)/y for y > 0.

2. Marginal cost: mc(y) = ∂c∗∂y for y > 0.

3. Fixed cost: fc is that part of c∗(y) that doesn’t vary with y (if any). A fixed cost can be sunk, partially

sunk, or not sunk.

60 LECTURE 9: COST ANALYSIS

Figure 9.1: Cost as a Function of Output

4. Variable cost: vc(y) is that part of c∗(y) that varies with y.

So, c∗(y) = fc + vc(y)

mc(y) = vc′(y).

5. Average fixed cost: afc(y) = fc/y for y > 0.

6. Average variable cost: avc(y) = vc(y)/y for y > 0.

So, ac(y) = afc(y) + avc(y)

The relationship between average and marginal cost is seen from ac′(y) = mc−acy for y > 0, as in Figure

9.2. The same relationship holds between avc and mc, since mc is the derivative of vc.

Fixed costs appear as in Figure 9.3. Here, c∗(y) = w1x1 + w1(x∗1(w, y) − x1), so fc = w1x1 and vc =

w1(x∗1(w, y)− x1). If the fixed cost were sunk, then the production set would not have the horizontal segment

and the cost function would take on the value w1x1 at y = 0. Similarly, if the fixed cost were partially sunk

then the production set would have only part of the horizontal segment and the cost function would take on

a value between zero and w1x1 at y = 0.

If fc > 0 then afc is everywhere strictly decreasing as a function of y. Thus, if in addition avc is eventually

rising then the relationship ac = afc + avc shows that ac curves are roughly U-shaped. This is illustrated


Figure 9.2: Average and Marginal Cost

Figure 9.3: Fixed Cost

in Figure 9.4. The smallest output at which ac reaches its minimum is called the minimum efficient scale

(mes). There can be more than one output at which ac takes on its minimum value, and they are all efficient

scales, but only the smallest such output is the mes.

Usually we think of sunk costs as arising in the short- or intermediate-run, when some input(s) cannot

be freely varied. In this scenario, we can study the relationship between short-run and long-run costs.

Presumably the fixed input level(s) were chosen with some feasible output level in mind. Call this output


Figure 9.4: U-Shaped Average Cost and the MES

level y and suppose the fixed input is input 1, fixed at x∗1(w, y). Then the short-run problem is

min{x−1}

w1x∗1(w, y) + w−1 · x−1 subject to (x∗

1(w, y), x−1) ∈ V (y).

Thus, the sunk cost is w1x∗1(w, y), and since x1 is fixed the solution will generally be different from x∗−1(w, y).

For example, in two dimensions the choice of x2 might appear as in Figure 9.5.

Figure 9.5: Short-Run Cost Minimization


So, let x∗−1(w, y|x∗

1(w, y)) be the solution. Then the short-run cost function is

c∗(w, y|x∗1(w, y)) = w1x

∗1(w, y) + w−1 · x∗

−1(w, y|x∗1(w, y)).

There are four relationships between the short-run and long-run cost functions. These hold for all feasible

y:

1. c∗(w, y|x∗1(w, y)) ≥ c∗(w, y) ∀y (feasible).

2. c∗(w, y|x∗1(w, y)) = c∗(w, y).

3. mc(y|x∗1(w, y)) = mc(y).

4. mc′(y|x∗1(w, y)) ≥ mc′(y).

Item 1 is by definition: The short-run problem has an additional constraint, so the minimum must be no

smaller. Items 2 and 3 are also immediate: If y = y then x∗−1(w, y|x∗1(w, y)) and λ∗(w, y|x∗

1(w, y)) satisfy the

same set of first order conditions as x∗−1(w, y) and λ∗(w, y) (with x∗

1(w, y|x∗1(w, y)) = x∗

1(w, y)), so they are

equal. This makes the two costs equal at y. And, since λ∗ is marginal cost in both cases, the two marginal

costs are equal as well. Item 4 requires some work with second-order Taylor Series representations of the

cost functions:

c∗(y) = c∗(y) +mc(y)(y − y) +12mc′(y)(y − y)2

c∗(y|x∗1(w, y)) = c∗(y|x∗

1(w, y)) +mc(y|x∗1(w, y))(y − y) +

12mc′(y|x∗

1(w, y))(y − y)2

for some y and y between y and y. Subtract the second from the first and note that the first two terms of

each right side are equal by items 2 and 3, while the difference on the left side is nonpositive by item 1. Thus

mc′(y) −mc′(y|x∗1(w, y)) ≤ 0 ∀y.

Let y → y and note that y → y and y → y as well. Thus, if the cost functions have continuous second

derivatives then item 4 follows. With these four items established, we know a lot about the relative positions

of the two cost functions, as illustrated in Figure 9.6.


Figure 9.6: The Short- and Long-Run Cost Functions

From the definition of average cost, these same relationships must hold between the short- and long-run

average cost curves, as in Figure 9.7.

Figure 9.7: The Short- and Long-Run Average and Marginal Cost Functions

Now that we know the shape of the cost curves, we can return to the firm’s optimization problem stated

in terms of the cost function:

max{y}

R(y) − c∗(y).


The first order condition is R′(y) = mc(y). This condition defines the optimal output choice y∗ provided

that the solution is an interior maximum. Since R′(y) is the change in revenue resulting from a change in

output, we call it the firm’s marginal revenue. The first order condition thus states the familiar condition

that a profit-maximizing firm chooses its output level to equate marginal revenue with marginal cost. The

second order condition is R′′(y) < mc′(y) at y∗. This says that the marginal cost curve must cross the

marginal revenue curve from below as y increases, at the optimal y.

If the firm is a perfect competitor in its output market then R(y) = py for some fixed price p. In this case

marginal revenue is R′(y) = p, a constant for all y, and we have R′′(y) = 0. Thus the first order condition

is p = mc(y) and the second order condition for a proper maximum is mc′(y) > 0. Again, the first order

condition defines the optimal choice y∗ if we have a nice interior maximum. Since the output price is a

parameter for the perfectly competitive firm the y∗ choice is a function of p, the supply function, implicitly

defined by

p ≡ mc(y∗(p)).

This shows that a perfectly competitive firm chooses the output level that equates price with marginal cost.

The comparative statics of price changes require that the firm adjust its supply choice to maintain the p = mc

condition as price changes. Thus, geometrically, the perfectly competitive firm’s supply curve is its marginal

cost curve, as illustrated in Figure 9.8. Algebraically, the supply function is the inverse of the marginal cost

function evaluated at the price:

y∗(p) = mc−1(p).

This statement presumes that the inverse exists (i.e., mc is strictly monotonic) and that the second order

condition holds. Since marginal revenue is constant, the second order condition requires that mc be strictly

increasing.

A major exception to this depiction of a perfect competitor’s supply curve occurs if there is a non-sunk or

partially sunk fixed cost. In that case, as illustrated in Figure 9.3, the cost function has an upward disconti-

nuity and thus it cannot be convex. This means that the profit objective cannot be concave, and therefore

there is a chance that the local first and second order conditions do not describe the global maximum. Figure

9.9 illustrates this case. Here, the ac and avc curves are U-shaped because there is a fixed cost and increasing


Figure 9.8: Supply for a Perfectly Competitive Firm. At price p0, the firm’s optimaloutput choice is y0 because mc equals p0 at this output level. At a different price, say p1, thefirm’s optimal output choice adjusts to y1 so that the p = mc condition is maintained. Thus, asp varies the mc curve traces out the optimal output choices.

marginal cost. At prices below p2, the local conditions indicate that the optimal supply choice is along the

mc curve, for example at y1 when price is p1. But at this output choice revenue per unit is p1, which is

below cost per unit of ac(y1), and so the firm is earning negative profit. This negative profit is due to the

fixed cost. Since the fixed cost is not fully sunk, the firm might be better off choosing an output of zero and

just incurring whatever part of the fixed cost is sunk.

Figure 9.9: Shutdown for a Perfectly Competitive Firm with Fixed Costs


The simplest case of this occurs when all of the fixed cost is sunk. In that case, the firm avoids only

its variable costs when it chooses zero output. If a positive output is chosen, the first and second order

conditions indicate that it must be along the mc curve. So the decision about whether to produce positive

or zero output is based on whether the price line intersects the mc curve above the avc curve. If so, then

revenue is more than variable cost and so the firm can recoup at least some of its fixed cost by producing

something. This is the situation at price p1 in Figure 7.9. However, if price is below p0 then revenue is less

than variable cost and so the firm is better off (incurs a smaller loss) choosing zero output. In this situation,

price p0 is called the shutdown point for the firm. Note that the avc curve may lie below the mc curve at all

output levels, since fixed cost is not included in avc. If this is the case and all of the fixed cost is sunk, then

the firm always chooses output to satisfy p = mc (assuming this is possible and the second order condition

holds).

The other polar extreme occurs when none of the fixed cost is sunk. Then the firm avoids both its variable

and fixed costs when it chooses zero output. Hence, the firm chooses zero output whenever price is below

p2, and p2 is the shutdown point.

If the fixed cost is partially sunk, then the firm avoids some but not all of it by choosing zero output. In

this case there is a shutdown price between p0 and p2, depending on how much of the fixed cost is sunk.

In all cases, the optimal supply behavior of the firm is to choose y along the mc curve when price is

above the shutdown price and y = 0 when price is below the shutdown price. When price is exactly equal

to the shutdown price the firm is indifferent between y = 0 and the output level where p = mc, but the

intermediate output levels are not optimal because they generate less profit than the two extremes (unless mc

has a horizontal segment at the shutdown price). This is illustrated in Figure 9.10.


Figure 9.10: Supply for a Perfectly Competitive Firm with Fixed Costs. ps is theshutdown price for this firm. The supply curve is the dark line. It is discontinuous, jumping fromzero to ys at ps. Output levels between 0 and ys are not optimal at price ps.


Exercises for Lecture 9 (Cost Analysis)

1. Consider a firm that produces one output.

a. Prove that average cost is nonincreasing as a function of output when there is global nondecreasing

returns to scale. Draw a production set, average cost/marginal cost graph, and total cost graph that all

reflect global nondecreasing returns to scale. Now draw a production set and total cost graph indicating

why marginal cost can increase even when there is global nondecreasing returns to scale.

b. Redo question a for the case of global nonincreasing returns to scale (nondecreasing average cost).

c. What do average and marginal cost look like when there is global constant returns to scale?

d. It is possible to define local returns to scale through a measurement called the elasticity of scale. Read

the discussion on elasticity of scale on pages 16-17 and 88-89 of Varian. Using the relationship between

average and marginal cost, compare the characterization of local returns to scale provided by the cost

function with the global characterizations given in parts a-c above.

2. Suppose a single-output perfectly competitive firm has cost function

c∗(y) ={

F + y2, y > 0αF, y = 0.

Here, F > 0 is fixed cost and α ∈ [0, 1] is the proportion of fixed cost that is sunk.

a. Graph the average, average variable, and marginal cost curves for this firm.

Now find the supply curve for this firm when

b. α = 1.

c. 0 < α < 1.

d. α = 0.

In each case, graph the supply curve.

3. Given a production function f(x1, x2) = x1x2, cost function c∗(w, y) = 2√yw1w2, and conditional factor

demand x∗1(w, y) =

√yw2/w1:

a. Find short-run cost c∗(w, y|x1) for some fixed x1.

b. Verify c∗(w, y|x1) = c∗(w, y) (hint: find y first!).


4. Varian #5.2.


5. Varian #5.8.

6. MWG #5.D.2.

7. MWG #5.D.4(a).


Lecture 10: Applied Production Analysis

Readings: None.

Suppose we are faced with the empirical problem of estimating the technology of a firm or group of firms

in an industry. For example, we might want estimates of how input usage varies when input prices change,

the returns to scale of the technology, or the efficiency of the firm(s). One approach is to collect data on

input quantities (x1, . . . , xn) and output levels (y) and then use the data to directly estimate the production

function y = f(x1, . . . , xn). This would involve assuming a functional form for f , defining an “error”

e = y − f(x1, . . . , xn) in the production relationship based on that functional form, making assumptions

about the statistical properties of e, and then using some estimation procedure (such as Ordinary Least

Squares or Maximum Likelihood) to estimate the parameters of the assumed functional form f .

Before proceeding, it should be noted that a crucial statistical assumption typically made is that

E(e|x1, . . . , xn) = 0,

which says that the error in the assumed relationship between y and (x1, . . . , xn) is not related to the

value of the input vector x = (x1, . . . , xn). This is usually implausible: The observed values of inputs and

outputs all result from optimizing behavior by the same economic actor(s), hence any errors of observation

or specification by the analyst, or mistakes by the economic actor(s), are very likely related to each other

in a given observation on (y, x1, . . . , xn). If so, technology parameter estimates obtained by using simple

estimation procedures will be biased. Moreover, the statistical “fixes” for this problem are unlikely to be

helpful in this setting. One would need instruments for ALL inputs to effectively perform instrumental

variables estimation, or a complicated fully-specified joint distribution for the n + 1 observed variables

(y, x1, . . . , xn) to effectively perform maximum likelihood.

Nonetheless, production functions have been (and still are) estimated by many researchers, and much

of the jargon surrounding empirical discussions of production stems from this practice, hence it is worth

reviewing.

Simple Parametric Forms for Production Functions;

Hicks Elasticity of Substitution; Elasticity of Scale

Four common functional forms for the production function f are:

72 LECTURE 10: APPLIED PRODUCTION ANALYSIS

1. Cobb-Douglas: f(x1 , . . . , xn) = Axα11 · · ·xαn

n , which is equivalent to ln f = lnA +∑n

i=1 αi lnxi.

2. Leontief (fixed proportions production): f(x1 , . . . , xn) = [min{α1x1, . . . , αnxn}]ε.

3. Perfect Substitutes: f(x1, . . . , xn) = [α1x1 + · · ·+ αnxn]ε.

4. Constant Elasticity of Substitution (CES): f(x1, . . . , xn) = [α1xρ1 + · · ·+ αnx

ρn]ε/ρ.

Here, αi > 0 and ε > 0 are unknown parameters of each assumed functional form. In the Cobb-Douglas

form, A > 0 is an efficiency parameter (a larger value of A means that any given input vector x produces

a larger amount of output). A is actually a redundant parameter since it can be normalized to unity by

rescaling the units in which output is measured, but the Cobb-Douglas form has the empirically convenient

property that it is linear in logarithms and in that form lnA is convenient because it gives the function a

non-zero intercept. αi in the Cobb-Douglas form is the (constant) elasticity of output with respect to input

i.2 In the Leontief form, αi

αjis the (constant) ratio of xj over xi in any efficient production plan. In the

perfect substitutes form, αi

αjis the (constant) slope of the isoquant between inputs j and i. In the CES

form, αi

αjenters the slope of the isoquant in the same was as the Cobb-Douglas form and ρ ≤ 1 (ρ �= 0) is

an unknown parameter that measures the (constant) elasticity of substitution (discussed below). ε > 0 is

an unknown parameter that measures economies of scale in all forms (an exponent ε could be placed on the

Cobb-Douglas function but it would be empirically indistinguishable from the αi exponents).

All four of these functional forms severely limit the substitution and returns to scale possibilities. To see

the limitations, define the following two measures that are often of interest in empirical work.

Definition. Let f(x1, . . . , xn) be a production function. Then (i) the (Hicks) elasticity of substitution

of input i for input j at input vector x is

σij(x) ≡d(xi

xj

)(xi

xj

) /d(fj(x)fi(x)

)(fj(x)fi(x)

) = − fij(x)[fi(x)xi + fj(x)xj ]xixj[fjj(x)f2

i (x) − 2fij(x)fi(x)fj(x) + fii(x)f2j (x)]

,

and (ii) the elasticity of scale at x is(df(λx)f(λx)

)(dλλ

)∣∣∣∣∣∣λ=1

=df(λx)dλ

λ

f(λx)

∣∣∣∣λ=1

=∑n

i=1 fi(x)xif(x)

.

2Recall that every elasticity can be expressed as a logarithmic derivative so that ∂ ln y∂ ln xi

= αi is the elasticity of y with

respect to xi. To see that an elasticity is a logarithmic derivative, suppose z is a function of w: z = g(w). Then the point

elasticity of z with respect w at the point (w0, z0) is defined to be g′(w0) w0

z0 . Now suppose we express the relationship between

w and z in terms of logarithms by writing w = eln w and z = eln z , so that eln z = g(eln w), or ln z = ln g(eln w), and then

regard the logarithms as the variables. Then ∂ ln z∂ ln w

= 1g(eln w)

g′(eln w)eln w = 1g(w)

g′(w)w. Evaluating at w0 yields g′(w0) w0

z0 ,

the elasticity.


The (Hicks) elasticity of substitution is the percent change in an input mix that results from a percent

change in the (absolute value of the) corresponding marginal rate of technical substitution, evaluated at

input vector x and holding output constant. It measures the curvature of the isoquant at x. To see this,

consider two possible isoquants at a point x, having the same MRTS at x, but one of which is more curved

than the other:

Figure 10.1: The Hicks Elasticity of Substitution

The slope of the line from the origin to x is the input ratio x2x1

at x. If this input ratio changes to the

steeper slope illustrated above, and the isoquant is the curve through x0, then the change in the MRTS

(slope of the isoquant) for the given change in the input ratio is relatively small. On the other hand, if the

isoquant is the curve through x1, then the change in the MRTS for the given change in the input ratio is

relatively large. Hence the elasticity of substitution at x for the isoquant through x1 is smaller than the

elasticity of substitution at x for the isoquant through x0 (recall that the percent change in the MRTS is in

the denominator of the elasticity of substitution). That is, the more curved isoquant is less elastic – a given

percent change in the MRTS along the more curved isoquant yields a smaller percent change in the input

ratio. Intuitively, it is “harder” to change the input ratio along the more curved isoquant (i.e., it would take

a larger change in relative prices).

In the limit, if the elasticity of substitution approaches infinity at all x on an isoquant then the isoquant


approaches a straight line. Conversely, if the elasticity of substitution approaches zero at some x while

approaching infinity at all other x on the isoquant then the isoquant approaches an L-shaped Leontief form.

Now consider the elasticity of scale. The elasticity of scale is the percent change in output that results

from a percent change in the scale of all inputs, evaluated at input vector x. It is a local measure of returns

to scale that can be used to describe a technology at a particular input vector even though the technology

may not have any specific global returns to scale. It is clear from the definition that the elasticity of scale

may change as x changes, so it can only be a measure of returns to scale for infinitesimal changes in scale at

a particular input vector x. If the technology possesses global constant returns to scale then the production

function is linear homogenous, so Euler’s Theorem yields∑n

i=1 fxi(x)xi = f(x). Hence, global constant

returns to scale implies a unit elasticity of scale everywhere, and so a technology is said to possess constant

returns to scale at input vector x if the elasticity of scale is 1 at x. Similarly, a technology is said to possess

decreasing returns to scale at input vector x if the elasticity of scale is less than 1 at x, and increasing returns

to scale at input vector x if the elasticity of scale is greater than 1 at x.

Now return to the four functional forms presented above. The Leontief form has L-shaped isoquants and

the perfect substitutes form has linear isoquants, hence the restrictions on substitution possibilities imposed

by these two functional forms are obvious: The Leontief form imposes a zero elasticity of substitution and

the perfect substitutes form imposes an infinite elasticity of substitution. For the Cobb-Douglas form, it

is straightforward to derive MRTS21 = −α1x2α2x1

when there are two inputs (n = 2). Hence d(x2/x1)d|MRTS| = α2

α1.

Multiplying by |MRTS|x2/x1

yields 1. That is, the Cobb-Douglas form imposes a unit elasticity of substitution.

Similar manipulations reveal that the elasticity of substitution for the CES form is 1 − ρ. This is more

flexible than Cobb-Douglas, in that the CES elasticity of substitution can range from zero to −∞ depending

on the value of the parameter ρ, which presumably would be estimated based on data (so the data can

determine the elasticity of substitution in the CES form). However, the elasticity of substitution in the CES

form is still a constant across all output levels, and across all input vectors along a given isoquant (hence the

name “Constant Elasticity of Substitution”). In short, all four of these functional forms are quite restrictive

regarding the ability to estimate the elasticity of substitution from the data. The first three assume a

particular value that is unrelated to the data, while the CES form assumes the elasticity is a constant but


permits that constant to be estimated.

The CES form is sometimes described as encompassing the other three functional forms under discussion

here. This is because the CES is exactly the perfect substitutes form if ρ = 1, and the CES elasticity of

substitution approaches the Leontief elasticity of substitution (∞) as ρ → −∞ and approaches the Cobb-

Douglas elasticity of substitution (1) as ρ → 0 (even though the CES function is not defined if ρ is exactly

zero). The CES form essentially lets the data determine which of these forms, or some intermediate case of

constant elasticity, applies to the technology under study.

These functional forms impose similar restrictions on the elasticity of scale. It is straightforward to

differentiate each of the four functional forms to show that the elasticity of scale for the Cobb-Douglas is∑ni=1 αi, while the elasticity of scale for the other three forms is ε. In short, all four functional forms impose

a constant elasticity of scale across all output levels and across all input vectors along a given isoquant;

and the elasticity of scale for the Cobb-Douglas form is not a separate parameter from the various output

elasticities. The elasticity of scale for these functional forms can be estimated from the data but the estimate

is constrained to be the same everywhere on the production surface. We cannot, for example, estimate a

U-shaped AC curve from a data set using any of these functional forms.

“Flexible” Functional Forms

The limitations of the simple functional forms considered above led to development of more flexible

functional forms, particularly functional forms that permit the elasticities of substitution and scale to be

different at different input vectors. The two most common functional forms with additional flexibility for

production functions are:

1. Translog: lnf(x1 , . . . , xn) = α0 +∑n

i=1 αi lnxi +∑n

i=1

∑nj=1 αij lnxi lnxj.

2. Generalized Linear: f(x1, . . . , xn) =[∑n

i=1

∑nj=1 αijx

1/2i x

1/2j

]ε.

The translog (short for “transcendental logarithmic”) form is quadratic in the logs of output and the inputs.

This form naturally permits elasticities that vary with the input vector, since elasticities are logarithmic

derivatives and the first logarithmic derivatives depend on the (log) input levels when the function is quadratic

in logs. Indeed, a quadratic in logs is the simplest functional form that allows for nonlinearity in logs. The

generalized linear form is the same idea except it is quadratic in levels rather than logs.


For reasons that will be explained below, we will not derive the Hicks elasticities of substitution for these

two functional forms, but the elasticities of scale are straightforward. Letting z denote the right side of the

translog, we have f(x1, . . . , xn) = ez so

fk = ez∂z

∂xk=

f(x1 , . . . , xn)xk

[αk +

n∑i=1

(αik + αki) lnxi

].

Using this in the definition of the elasticity of scale yields

n∑k=1

[αk +

n∑i=1

(αik + αki) lnxi

],

which clearly depends upon the entire (log) input vector. Similarly, letting z denote the bracketed term on

the right side of the generalized linear form we have

fk =ε

2zε−1x

−1/2k

n∑i=1

(αki + αik)x1/2i .

Therefore

n∑k=1

fkxk =ε

2zε−1

n∑k=1

x1/2k

[n∑i=1

(αki + αik)x1/2i

]= εzε−1

n∑k=1

n∑i=1

αkix1/2k x

1/2i = εzε.

Hence the elasticity of substitution for the generalized linear form is ε, which is still invariant across input

vectors despite the added complexity of the quadratic functional form.

Using Duality Theory in Applied Production Analysis

Recall that duality theory guarantees there is a well-behaved technology underlying every cost function

provided the cost function satisfies homogeneity, concavity, monotonicity, continuity, and nonnegativity; and

vice-versa. Therefore any technology we might want to consider can be estimated by estimating a cost

function that has these five properties rather than by estimating a production function. To do so, we collect

data on input prices (w1, . . . , wn), output levels y and total costs c; and use the data to estimate the cost

function c = g(w1, . . . , wn, y) by assuming a functional form for g, defining the error u = c−g(w1, . . . , wn, y)

in the cost relationship based on that functional form, making assumptions about the statistical properties

of u, and then using some estimation procedure (such as Ordinary Least Squares or Maximum Likelihood)

to estimate the parameters of the assumed functional form g taking care to ensure that the estimated g


function has the five properties of a cost function. From a procedural perspective, this is not much different

from estimating a production function except that we are working with input price and cost observations

rather than input quantity observations; and we have some known properties to impose on the estimated g.

Data availability may create an advantage to estimating a cost function rather than a production function.

Input prices may be easier to observe than input quantities. However, this is not always true.

Even when there is no data advantage, cost estimation is superior because of its statistical properties. As

in the estimation of production functions, the key statistical assumption for unbiased parameter estimators

is that the expected value of the error in the cost relationship be unrelated to the values of the observable

variables on the right side of the relationship. That is, E(u|w1, . . . , wn, y) = 0. If the firm(s) under study can

be reasonably regarded as price-takers in the markets for their inputs (if not, a cost function is not a correct

representation of the technology even from a purely theoretical perspective) then the assumption that input

prices are exogenous to errors of observation or specification by the analyst, and mistakes by the economic

actor(s), is much more plausible than the assumption that input quantities are exogenous to unobservable

errors. In short, it is much more likely the analyst will obtain unbiased estimates of technology parameters

using cost function estimation than production function estimation.

Cost estimation has the further advantage that it leads very directly to estimates of factor demands. Once

a cost function is estimated, the implied estimates of factor demands are immediately obtained through

simple differentiation via Shephard’s Lemma. Estimates of factor demands can, of course, be derived from

an estimated production function as well but doing so requires that the cost minimization problem be solved

explicitly, and the difficulty of deriving a closed-form solution increases with the flexibility of the assumed

functional form.

Note there may still be an endogeneity problem in cost estimation because output is on the right side of

the cost relationship and output is often chosen by the same economic actor(s) that make cost decisions.

However, the endogeneity problem is greatly reduced since inputs are no longer an issue. It may be possible

to find an instrument for output, or it may be plausible to argue that output is not under the firm’s control in

the short-run. In some cases, such as a regulated public utility with carrier-of-last-resort obligations, output

is clearly not chosen by the firm. If endogeneity of output cannot be addressed in these ways then the


same arguments in favor of cost estimation over production function estimation suggest the analyst should

collect data on output prices and estimate a profit function rather than a cost function, assuming the firm

is a price-taker in its output market. This removes output as a variable in the analysis and relies, through

duality, on the profit representation of technology.

Another way to estimate a production relationship using duality theory is to estimate the conditional

factor demands (or, if endogeneity of output is an issue, the unconditional factor demands and output

supply). Duality theory guarantees there is a well-behaved technology underlying every set of conditional

factor demands that satisfies homogeneity, symmetry and semidefiniteness, monotonicity, continuity, and

nonnegativity; and vice-versa. Estimating demands requires data on input quantities as well as input prices

and output, assumptions about the functional forms of the demands, and assumptions about the statistical

properties of the errors; after which some appropriate estimation procedure is used and restrictions are

imposed to ensure that the estimated demands satisfy the five properties of conditional factor demand

functions.

A once-common functional form assumption is that the conditional factor demands are linear in logs:

lnxi = αi +n∑j=1

αij lnwj + αiy lny + ui for i = 1, . . . , n,

where the α’s are the demand parameters to be estimated and ui is the error in the assumed relationship

for input i. This functional form has the appealing feature that the demand elasticities are constant and

directly estimated: αij is the elasticity of demand for input i with respect to price j and αiy is the elasticity

of demand for input i with respect to output. This appeal is misleading, however. It is instructive to apply

duality theory to this functional form. Doing so reveals that this functional form imposes severe unappealing

restrictions on demand.

Consider first the implications of homogeneity. If each input price shifts from wj to λwj then demand at

the new input price vector is

αi +n∑j=1

αij ln(λwj) + αiy ln y + ui = αi +n∑j=1

αij lnwj + αiy lny + ui + λ

n∑j=1

αij.

Homogeneity requires that this shift in input prices leave demand unchanged for every λ > 0; hence the

parameters of the ith equation must satisfy∑n

j=1 αij = 0 in order for this function to be a conditional


factor demand. Some computer programs have options to impose this type of parameter constraint on the

estimation. Alternatively, the constraint can be substituted for one of the price elasticities, say the nth, to

obtain

lnxi = αi +n−1∑j=1

αij(lnwj − lnwn) + αiy ln y + ui = αi +n−1∑j=1

αij(ln(wj/wn) + αiy ln y + ui.

This way of writing the equation essentially declares input n as the numeraire. Estimation proceeds by

defining new, normalized, price observations wj = wj/wn and using the logs of these along with logged

output as independent variables.

Note that xi > 0 with this functional form since xi = exp[αi +∑n

j=1 αij lnwj + αiy lny + ui], and the

exponential function is always positive. Hence the nonnegativity property is automatically satisfied by this

functional form, except the part requiring that cost be zero when output is zero. The latter cannot be

addressed when log output appears in the specification because the log function is not defined at zero.

Note also that this functional form is continuous in all strictly positive input prices and output except,

again, it is not defined at the boundary where output is zero. Hence the continuity property is automatically

satisfied if all n demands have this functional form, except at zero output where cost is not defined.

Some empirical projects focus on the demand for only one input. The only other requirement imposed by

duality theory on the parameters of a single conditional factor demand is the part of semidefiniteness that

ensures the demand is downward-sloping in its own price. That is, αii ≤ 0 is required. All other duality

requirements involve the parameters of multiple demands and therefore cannot be imposed when only one

demand is under study. As αii ≤ 0 is an inequality constraint, it cannot be imposed as part of a linear

estimation procedure. The usual practice is to estimate the demand without imposing this inequality and

then check that the estimated value of αii is non-positive, with much concern generated about the assumed

specification and the data if the estimated value of αii is positive.

Hence, if only one demand is under study then the dictates of duality theory can be incorporated with

only minor inconsistencies. If, however, the demand for more than one factor is under study then symmetry,

the remainder of semidefiniteness, and monotonicity must be considered; and these properties reveal serious

weaknesses of demands that are assumed to be linear in logs.

Consider conditional factor demands for inputs i and j that are linear in logs. Symmetry requires ∂xi

∂wj=


∂xj

∂wi, while the functional form yields ∂xi

∂wj= αij

xi

wj, so symmetry with this functional form requires αij

αji= wixi

wjxj

for all input price vectors and output levels. That is, wixi

wjxj, which is the amount spent on input i relative

to the amount spent on input j, must be invariant to input prices and output levels. There is no reason to

expect this in general and, indeed, it would be a peculiar feature of a demand system. It is straightforward

to take the log of this ratio and substitute the demands to conclude that the following expression would have

to be constant across all input prices and output levels:

(1 + αjj − αji) lnwj − (1 + αii − αij) lnwi +n∑

k=1 (k �=i,j)(αjk − αik) lnwk + (αjy − αiy) ln y.

Since the input prices and output levels can vary independently, this expression is constant if and only if all

of the following hold:

1. 1 + αjj − αji = 0,

2. 1 + αii − αij = 0,

3. αjk = αik for all k �= i, j, and

4. αjy = αiy.

That is, the cross-price elasticities of inputs i and j with respect to any other price must be equal, the output

elasticities of inputs i and j must be equal, and the own- versus cross-price elasticities of each of these inputs

must satisfy a special relationship. If all factor demands were linear in logs then these observations must hold

for every i, j pair. Adding item 1 across i and using homogeneity would then yield αjj = −1. Substituting

back into item 1 would then give αji = 0 for every i. In other words, the functional form would impose

that all demands are unit elastic with respect to the own-price and all cross-price elasticities are zero. The

only parameter left to estimate would then be the (common) output elasticity. In short, assuming the linear

in logs functional form for the entire demand system amounts to assuming particular values for all price

elasticities and that all inputs have the same output elasticity, irrespective of the data.

This is hardly a useful demand system but it would be very difficult to perceive the weaknesses without

duality theory. This is one example of the power of duality theory. The theory can guide our choice of

functional form to avoid absurd assumptions, even when a functional form has superficial appeal. There is

some now-dated empirical work that purports to “reject” neoclassical demand theory by estimating demand

systems with particular functional forms, some of which were linear in logs, and then observing that the


results easily reject the null hypothesis that the estimated parameters satisfy the duality restrictions. We

now see that the problem with some (but not all) of this literature is that the assumed functional forms did

not permit sensible demand estimation, so the apparent “rejection” of neoclassical demand theory was, in

some cases, nothing more than unreasonable assumptions in the empirical specification.

These problems led to development of functional forms for cost functions (and implicitly, conditional

factor demands, by Shephard’s Lemma) that are sufficiently flexible to avoid de facto absurdities. Four such

functional forms are:

1. Linear Expenditure System: g(w1, . . . , wn, y) =∑n

i=1 αiwi + y[wβ1

1 wβ22 · · ·wβn

n

].

2. Generalized Leontief: g(w1, . . . , wn, y) = h(y)∑n

i=1

∑nj=1 αijw

1/2i w

1/2j .

3. Translog: ln g(w1, . . . , wn, y) = α0 +∑n

i=1 αi lnwi + 12

∑ni=1

∑nj=1 αij lnwi lnwj + αy ln y + αyy(ln y)2 +∑n

i=1 αiy lnwi lny.

4. Almost Ideal Demand System: lng(w1, . . . , wn, y) = α0 +∑n

i=1 αi lnwi + 12

∑ni=1

∑nj=1 αij lnwi lnwj +

yβ0

[wβ1

1 wβ22 · · ·wβn

n

].

Here, the α’s and β’s are unknown parameters of each assumed functional form. These parameters have

different interpretations in each functional form and must satisfy various restrictions in each form to ensure

that each cost function has the five properties required by duality theory (homogeneity of degree 1 and

concavity in input prices, monotonicity, continuity, and nonnegativity). h in the generalized Leontief form

is a continuous and strictly increasing function satisfying h(0) = 0. These four functional forms for cost are

all obviously continuous in all strictly positive input prices and output except, as above, the translog is not

defined at the boundary where output is zero. They are all also strictly increasing in y under obvious minor

assumptions, except the translog which cannot be globallymonotonic in y (unless αyy = α1y = · · · = αny = 0)

because the derivative with respect to y involves logarithms, which can be arbitrarily large in both the

positive and negative directions at positive input prices and output levels. This is a shortcoming of the

translog functional form.

Consider first the linear expenditure system. Applying Shephard’s Lemma, the implied conditional factor

demands are

∂g

∂wj= xj = αj +

yβjwj

[wβ1

1 wβ22 · · ·wβn

n

]for j = 1, . . . , n.


Multiplying both sides by wj and substituting g yields

wjxj = αjwj + βj

[g(w1, . . . , wn, y)−

n∑i=1

αiwi

],

which shows why this is called the “linear expenditure system”: Total spending on each input is a linear

function of all input prices and total expenditure. Substituting λwi for wi in g shows that g is homogeneous

of degree 1 if and only if∑n

i=1 βi = 1. Concavity of g is checked by examining the Hessian matrix with

respect to the input prices. The ith diagonal element is βi(βi−1)w2

iy[wβ1

1 wβ22 · · ·wβn

n

]and the (i, j) off-diagonal

element is βiβj

wiwjy[wβ1

1 wβ22 · · ·wβn

n

]. Concavity requires that the naturally-ordered principal minor of order k

have sign (−1)k. Every element has y[wβ1

1 wβ22 · · ·wβn

n

]so this part factors out and can be ignored. The k = 1

minor is β1(β1−1)w2

1which is negative if and only if β1 ≥ 0 (using the homogeneity requirement

∑ni=1 βi = 1).

The k = 2 minor is β1β2(1−β1−β2)w2

1w22

which, given homogeneity and β1 ≥ 0, is positive if and only if β2 ≥ 0.

This pattern continues for k = 3, . . . , n; with the end result being that g is concave if and only if βi ≥ 0 for

all i (given homogeneity). Note that βi ≥ 0 implies all inputs are substitutes (∂xj/∂wi ≥ 0 for i �= j); the

functional form does not permit complementary inputs. Looking at ∂g∂wj

= xj, we see that αj ≥ 0 is required

to ensure that g is nondecreasing in wj at all values of input prices and output (the second term is positive

when y > 0 but tends to zero as y → 0, so there exist values of y and the input prices that make the entire

expression negative if αj < 0). Hence, given homogeneity and concavity, g is monotonic if and only if αj ≥ 0

for all j. Nonnegativity of g is obvious except for the fact that g is not zero at y = 0 unless αi = 0 for

all i. Essentially, αi measures the amount of input i that is sunk; if it is positive then there is a sunk cost

and total cost is therefore not zero even when output is zero. Given this interpretation of αi, looking at the

expression for wjxj reveals that βj measures how total spending on input j responds to changes in variable

cost. To summarize, g is a cost function provided βi ≥ 0 for all i,∑n

i=1 βi = 1, and αi ≥ 0 for all i; with

the further restriction that αi = 0 for all i if we want to insist that the technology satisfies the possibility of

inaction.

Now consider the generalized Leontief. This function is automatically homogeneous of degree 1 in input

prices. The implied conditional factor demands are

∂g

∂wj= xj =

h(y)

2w1/2j

n∑i=1

(αij + αji)w1/2i for j = 1, . . . , n.


If αij = 0 when i �= j then the conditional factor demand is xj = h(y)αjj , which is the functional form of

the factor demands for a Leontief production function, hence the name “generalized Leontief.” Note that

the coefficient on w1/2i w

1/2j is always αij + αji, so we lose nothing but gain parsimony by calling this one

parameter or, equivalently, by assuming αij = αji. The demands can then be written

xj =h(y)

w1/2j

n∑i=1

αijw1/2i .

Since this is ∂g∂wj

, it is clear that g is monotonic in input prices if and only if αij ≥ 0. For concavity, note

that g is a sum of terms that take the form h(y)αijw1/2i w

1/2j and recall that a sum of concave functions is

concave; hence it suffices to show that an arbitrary term of this form is a concave function. The Hessian of

this one term is

h(y)αij4

[−w

−3/2i w

1/2j w

−1/2i w

−1/2j

w−1/2i w

−1/2j −w

1/2i w

−3/2j

].

The (1, 1) element is nonpositive at all positive input prices because αij ≥ 0, and the determinant is

identically zero, hence the monotonicity restriction implies that h(y)αijw1/2i w

1/2j , and hence g, are concave

functions. Finally, note that the restrictions on h(y) assure that g is nonnegative. In short, g is a cost

function provided αij ≥ 0, where we have eliminated redundant parameters by assuming αij = αji (we

could, equivalently, define a new set of parameters by βij = αij + αji and require βij ≥ 0). Similar to

the linear expenditure system, the requirement αij ≥ 0 implies all inputs are substitutes; once again the

functional form does not permit complementary inputs. Estimation requires that some functional form be

assumed for h. A simple functional form that satisfies the assumptions and that has an easy interpretation

(discussed below) is h(y) = yε.

The translog functional form is more complicated. Differentiating yields the factor demands:

xj =∂g

∂wj=

g

wj

∂ ln g

∂ lnwj=

g

wj

[αj +

12

n∑i=1

(αij + αji) lnwi + αjy ln y

]for j = 1, . . . , n.

These factor demands are often written as “cost share” equations:

sj ≡ wjxjg

= αj +12

n∑i=1

(αij + αji) lnwi + αjy lny,

since the fraction on the left side of this equation is the share of total cost spent on input j. The cost share

form is convenient because the right side is linear in parameters and involves no endogenous variables (unless


endogeneity of output is an issue); hence it can be easily estimated with a linear estimation procedure. As

with the generalized Leontief, αij and αji are redundant parameters; we could just as well assume αij = αji

and write the demands in the form:

xj =g

wj

[αj +

n∑i=1

αij lnwi + αjy lny

].

Substituting λwi for wi in the demand equations and factoring out the terms involving λ reveals that,

assuming momentarily g is homogeneous of degree 1 so that g(λw1, . . . , λwn, y) = λg(w1, . . . , wn, y), the

demand for input j is homogeneous of degree 0 if and only if∑n

i=1 αij = 0. Looking back at g and imposing

the condition∑n

i=1 αij = 0 for j = 1, . . . , n, we see that g is homogeneous of degree 1 if and only if∑ni=1 αi = 1 and

∑ni=1 αiy = 0 (in addition to

∑ni=1 αij = 0 for each j).

For concavity of the translog, differentiate xj and collect terms to obtain:

∂g2

∂wj∂wi=

{ gwjwi

[sisj + αij] i �= j

gw2

j[sj(sj − 1) + αjj] i = j

.

The coefficients gwiwj

are not relevant to negative semidefiniteness of the Hessian (g factors out of the

entire matrix and every term of each principal minor has the same product of w’s in the denominator so

the denominator factors out of each principal minor). Hence negative semidefiniteness is determined by

examining two matrices: One with elements sj(sj − 1) on the diagonal and elements sisj off of the diagonal;

and one with elements αij. If each of these matrices is negative semidefinite then the Hessian is negative

semidefinite because the sum of negative semidefinite matrices is negative semidefinite. The naturally-ordered

principal minor of order k = 1 of the first matrix is s1(s1 − 1), which is nonpositive because, by definition,

all cost shares are nonnegative and they sum to one. The naturally-ordered principal minor of order k = 2

is s1s2(1−s1 −s2) ≥ 0. The naturally-ordered principal minor of order k = 3 is s1s2s3(s1 +s2 +s3 −1) ≤ 0.

This pattern continues for k = 4, . . . , n; hence the first matrix is negative semidefinite. This argument is

essentially the same argument used to establish negative semidefiniteness of the Hessian for the generalized

linear form. Therefore a necessary and sufficient condition for global concavity of g is that the matrix of αij

parameters be negative semidefinite (this is clearly sufficient; it is also necessary because the cost shares vary

between zero and one on the input price space, so there will be values of the input price vector for which the

Hessian is not negative semidefinite if the matrix of αij parameters is not negative semidefinite). We have


assumed the matrix of αij parameters is symmetric. Still, the constraint that it be negative semidefinite

is nonlinear and therefore cannot be imposed within a linear estimation procedure. The usual practice is

to estimate the translog cost function without imposing concavity and then check whether the estimated

matrix of αij parameters is negative semidefinite. Note that there is no requirement imposed on the sign

of αij for i �= j (negative semidefiniteness of course imposes that αii be nonpositive so that the demand for

input i slopes downward in its own price). Hence the translog has an advantage over the other functional

forms considered so far: The translog does not impose any restrictions on whether inputs are substitutes

or complements. Indeed, whether two inputs are substitutes or complements can vary across different parts

of the input price/output space due to the presence of the cost shares sisj in the cross-price derivatives of

demands.

Turning now to monotonicity, we noted above that the translog cannot be globally monotonic in y. The

same problem arises regarding monotonicity in each wj, for the same reason (the derivatives xj involve

logs, which can vary over the entire real line). This guaranteed lack of global monotonicity is the most

serious weakness of the translog functional form; all that can be done is to check whether the estimated first

derivatives are nonnegative at each point in the data set, in which case we at least know that the estimated

cost function satisfies the monotonicity requirement locally at relevant data points.

Finally, as with the log-linear demand, the translog cost function is an exponential and is therefore

automatically nonnegative except that it is not defined at y = 0 and therefore cannot satisfy the possibility

of inaction.

To summarize, the translog form is a cost function provided∑n

i=1 αi = 1,∑n

i=1 αiy = 0,∑n

i=1 αij = 0

for each j, and the matrix of αij parameters is negative semidefinite; where we have assumed without loss of

generality that αij = αji; and with the understanding that it is not globally monotonic but the monotonicity

can be checked over the range of observed data.

The almost ideal demand system (AIDS) is a hybrid of the linear expenditure system and the translog.

The factor demands are:

xj =∂g

∂wj=

g

wj

∂ lng

∂ lnwj=

g

wj

[αj +

12

n∑i=1

(αij + αji) lnwi + βjyβ0

[wβ1

1 · · ·wβnn

]]for j = 1, . . . , n.

As with the translog, it may be more convenient to write these demands as cost share equations by multiplying


through by wj

g . αij and αji are once again redundant parameters, so we assume αij = αji without loss of

generality, and add and subtract the first part of the cost function to obtain

sj =wjxjg

= αj +n∑i=1

αij lnwi + βj ln(

g

α0 +∑n

i=1 αi lnwi + 12

∑ni=1

∑nk=1 αik lnwi lnwk

).

The AIDS is sometimes estimated in this form by replacing α0+∑n

i=1 αi lnwi+ 12

∑ni=1

∑nk=1 αik lnwi lnwk,

which is a weighted quadratic of logged prices, with a pre-specified price index and then estimating sj as

a linear function of the lnwi’s and the logged ratio of total cost to the pre-specified price index. This is

only an approximation to the AIDS cost function since the parameters to be estimated are actually in the

true price index. Some type of complication is unavoidable, however, because the true AIDS is not linear

in the parameters even in cost share form. Hence the alternative to the price index approximation is a

nonlinear estimation procedure. A perhaps more serious problem with the rewritten cost shares is that

the algebraic manipulation replaces the potentially endogenous variable y with the certainly endogenous

variable g on the right side of the estimating equation, which is undesirable from an econometric viewpoint;

an instrumental variable is needed (note, however, that this would not be a problem in a consumption

application involving consumers whose budgets are exogenous). This system of demands is called “Almost

Ideal” primarily because the functional form is consistent with quantities and expenditures being aggregates

over groups of maximizing economic actors. In other words, there is a theoretical foundation for using this

functional form with data that is reported for groups rather than individuals.

Substituting λwi for wi in the demand equations and factoring out the terms involving λ reveals that,

assuming momentarily g is homogeneous of degree 1, the demand for input j is homogeneous of degree 0 if

and only if∑n

i=1 αij = 0 and∑n

i=1 βi = 0. Looking back at g and imposing these two conditions for all j,

we see that g is homogeneous of degree 1 if and only if∑n

i=1 αi = 1 (in addition to∑n

i=1 αij = 0 for each j

and∑n

i=1 βi = 0).

For concavity of the AIDS cost function, differentiate xj and collect terms in a manner similar to the

translog to obtain:

∂g2

∂wj∂wi=

g

wjwi

[sisj + αij + βiβjyβ0

[wβ1

1 · · ·wβnn

]]i �= j

gw2

j

[sj(sj − 1) + αjj + β2

j yβ0

[wβ1

1 · · ·wβnn

]]i = j

.

This is very similar to the translog and negative semidefiniteness can again be studied by examining the

Hessian in multiple parts. We have already discussed that the matrix with diagonal elements sj(sj − 1) and


off-diagonal elements sisj is negative semidefinite; and that negative semidefiniteness of the matrix with

entries αij can be checked with the estimated values or imposed on the estimation in a more ambitious

nonlinear estimation procedure. The new consideration here is a third matrix with entries βiβj (the product

yβ0

[wβ1

1 · · ·wβnn

]factors out of the entire matrix). The diagonal elements of this third matrix are squares,

and therefore the matrix cannot be negative semidefinite except in the trivial case of all βj ’s equal to zero.

Hence there will necessarily exist points in the price/output space at which the Hessian of the AIDS cost

function is not negative semidefinite. In short, the AIDS cost function cannot be globally concave and

therefore cannot be consistent with theory (unless the last term of the function is reduced to β0y). This

is a shortcoming of the AIDS functional form, but it is still possible for an estimated AIDS cost function

to be negative semidefinite at all values of prices and output observed in the data and to therefore regard

the estimated AIDS function as consistent with duality theory over a relevant range of prices and output.

As with the translog, the AIDS is flexible enough to permit both substitutes and complements among the

inputs and for this relationship to vary over the price/output space.

Turning now to monotonicity, the AIDS cost function is strictly increasing in output provided only that

β0 > 0. Like the translog, however, the AIDS cannot be globally monotonic in input prices because of the

presence of logged prices in the first derivatives xj . This is another weakness of the AIDS and, again, all

that can be done is to check whether the estimated first derivatives are nonnegative at each point in the

data set.

Nonnegativity is automatic with the AIDS cost function since it is an exponential, except that cost cannot

be zero at zero output (even though the AIDS function is defined at y = 0) since the exponential function

is never zero.

To summarize, the AIDS form is a cost function provided∑n

i=1 αi = 1,∑n

i=1 αij = 0 for each j, and∑ni=1 βi = 0; where we have assumed without loss of generality that αij = αji; and with the understanding

that it is neither globally concave nor globally monotonic in input prices but these can be checked over the

range of observed data (negative semidefiniteness of the estimated αij matrix limits the concavity issue to

the estimate βiβj matrix).

Most statistical software packages include options for imposing linear constraints on the parameters to


be estimated. Hence the parameter restrictions that involve sums of parameters (mostly the homogeneity

restrictions) can usually be imposed on the estimation of each of these four functional forms with little

difficulty. Some of the homogeneity restrictions are sometimes referred to as “adding up” or “cost exhaustion”

restrictions. This terminology arises when the presentation is exclusively in terms of the demand or share

equations, in which case it is necessary to ensure that the shares sum to unity (or the demands, weighted

by their own-prices, sum to total cost) in addition to homogeneity of degree zero of the demands. For

example, looking exclusively at the translog demands without thinking explicitly about the cost function,

the demands are homogeneous of degree zero if∑n

i=1 αij = 0 for every j provided we presume the cost

function g is homogeneous of degree 1. But the associated cost function is not homogeneous of degree 1

unless, in addition,∑n

i=1 αi = 1 and∑n

i=1 αiy = 0. These last two constraints could alternatively be derived

by examining∑n

j=1 sj and noting that these shares do not sum to unity for every value of output unless

the two additional conditions we identified hold. Inequality constraints, including those that would ensure

negative semidefiniteness of a matrix, cannot be imposed within a linear estimation algorithm. The options

are to pursue a nonlinear estimation, such as maximum likelihood, or estimate without imposing inequality

constraints and hope that the estimates satisfy the relevant inequalities.

Let us return to a discussion of elasticities of substitution and scale in the context of a technology

estimation based on a cost function. The Hicks elasticity of substitution measures the percent change in

an input mix resulting from a percent change in a technical rate of substitution along a two-dimensional

isoquant. Although this describes the curvature of the isoquant in the particular two-dimensional plane, it

does not describe optimal input mix changes for a cost-minimizing firm because the Hicks elasticity does

not permit all inputs to change at once, which is generally what happens to the cost-minimizing input

bundle in response to a relative price change. Relating an input mix change to a change in a technical

rate of substitution restricts the firm to a two-dimensional isoquant rather than the n-dimensional isoquant

on which a cost-minimizing firm actually substitutes inputs. Thus, while the technical rate of substitution

equals the price ratio in the (xi, xj) plane at an optimal input mix, this slope may not be the relevant

direction for calculating a change in an optimal input mix. R. G. D. Allen proposed partial (also called

Allen) elasticities of substitution for the case of n inputs and it has been common practice in some empirical


studies to report estimates of these partial elasticities. Partial elasticities of substitution acknowledge that

all inputs may change in response to a relative price change but do not measure the input mix change as

a response to a change in a particular price. For the Hicks elasticity, it is enough to simply examine the

consequences of a change in an input price ratio (optimally, a technical rate of substitution) because the

direction of change is presumed to be along the isoquant in a two-dimensional plane. This is not sufficient for

a partial elasticity; all inputs are allowed to change hence there is no presumption of a direction of change.

The change in an optimal input mix will generally depend on which price is changing. This led Morishima

to propose the following measure of substitution.

Definition. The Morishima elasticity of substitution of input i for input j (in the direction wi) at any

(w, y) � 0 is

σ∗ij(w, y) ≡

d(x∗

i (w,y)x∗

j (w,y)

)(x∗

i (w,y)

x∗j (w,y)

) /d(wj

wi

)(wj

wi

) =∂x∗

j (w, y)∂wi

wix∗j (w, y)

− ∂x∗i (w, y)∂wi

wix∗i (w, y)

,

where it is understood that the differentiation is with respect to wi.

Note that in general σ∗ij(w, y) �= σ∗

ji(w, y) since the latter is in a different direction, wj , than the former.

It is straightforward to apply this definition to any of the conditional factor demands derived above. For

example, after simplification we obtain for the translog σ∗ij(w, y) = 1+ αij

sj+ αii

si. Note that the translog cost

functional form is sufficiently flexible to impose no prior restrictions on the elasticity of substitution and to

permit the elasticity of substitution to vary over the input price/output space.

The elasticity of scale of the underlying technology, at an input x∗(w, y) that minimizes cost for some

(w, y) � 0, can be calculated directly from the cost function as the ratio of average to marginal cost. Hence

there is no need to derive the production function in order to investigate local returns to scale, at least at any

optimal (i.e., economically relevant) input vector. To see this, recall that the elasticity of scale at input x can

be expressed as∑n

i=1 fxi(x)xi

f(x) , where f is the production function. So, at an optimal input vector x∗(w, y),

fxi(x∗) = wi/λ∗ can be substituted from the first order conditions to obtain w·x∗(w,y)

λ∗(w,y)f(x∗(w,y)) . Since x∗(w, y)

is cost minimizing, c∗(w, y) = w · x∗(w, y) and by Shephard’s Lemma (applied to output), λ∗(w, y) = mc(y).

Finally, we have y = f(x∗(w, y)) due to continuity of f , so substituting yields the elasticity of scale at

x∗(w, y) as c∗(w,y)mc(y)y = ac(y)

mc(y) . This ratio is easily calculated for any of the functional forms considered above.


For example, the elasticity of scale for the translog is [αy + 2αyy lny +∑n

i=1 αiy lnwi]−1. Note that the

translog cost functional form is sufficiently flexible to impose no prior restrictions on the elasticity of scale

and to permit the elasticity of scale to vary over the input price/output space.


Exercises for Lecture 10 (Applied Production Analysis)


Lecture 11: Preferences

Readings: Section 7.1 of Varian; Sections 1.A-1.B, 2.A-2.C, 3.A-3.C of MWG.

Until now we have been studying the supply side of product markets and the demand side of factor

markets. The economic agent, or actor, has been the firm. Now we switch to the demand side of the product

markets and the supply side of the factor markets, and begin studying consumers, which is the name we use

for these final demanders and factor suppliers. The study of consumer supply behavior in factor markets

is specialized and is studied in courses on general equilibrium and labor economics. Thus, we focus here

only on consumer demand behavior in product markets. As with all economic agents, there are two basic

pieces to consumer behavior: The objective and the constraints. When studying firms, we began with the

technological constraint since it was the primitive that required explanation. For consumers the constraint

is relatively straightforward, merely requiring that the consumer stay within a budget. But the objective

requires some explanation, so we begin with it.

There are 2 basic components of a consumer’s objective:

• The consumption set X. This is the set of all possible bundles the consumer might consume. This

set includes everything that exists or might conceivably exist, not merely those things that satisfy some

individual or aggregate resource constraint. The description here is only about preferences, not about

availability of resources. It is up to the market to ration potentially scarce commodities by placing positive

prices on them.

• The preference ordering �∼. This is known as a binary relation. It expresses the preference of the consumer

between two bundles in X. x �∼

y means “x is at least as desirable as y” for the consumer under study.

For convenience, we sometimes write y ≺∼

x to denote x �∼

y. Note that the preference ordering is weak. We

can define strict preference and indifference from �∼, rather than introducing them as new primitive concepts,

as follows:

x � y means not(y �∼

x) (“x is (strictly) prefered to y”)

x ∼ y means both x �∼

y and y �∼

x (“indifference” between x and y).

For convenience, we sometimes write y ≺ x to denote x � y.

94 LECTURE 11: PREFERENCES

We must make some assumptions about X and �∼

in order to get anywhere. The two main assumptions

are

1. (Completeness) For every x, y ∈ X, either x �∼

y or y �∼

x, or both. Note that this includes reflexivity as

a special case.

2. (Transitivity) For x, y, z ∈ X, if x �∼

y and y �∼

z, then x �∼

z.

These seem obvious, but are in fact assumptions of substance. For example, set containment is a binary

relation that is not complete, and “defeated” among teams in a sports league is a binary relation that need

not be transitive.

These assumptions are primitive, but cumbersome. It would be much more convenient if we could replace

�∼

with a system of index numbers that expresses the same information. Such a system of index numbers is

called a utility function. It is a real-valued function on X whose values give the same ordering as �∼. If there

is a utility function for any given preference relation then we can use all of our standard optimization tools

on the utility function in order to describe consumer behavior, thereby avoiding direct consideration of the

preference relation. Hence our first task is to determine what properties a preference relation must possess

in order to assure that a utility function exists.

Definition. Let X be a consumption set and�∼be a preference ordering onX. We say a function U : X → R

1

represents �∼

if and only if, for every x, y ∈ X,

U(x) ≥ U(y) ⇔ x �∼

y.

Theorem. If U represents �∼, then �

∼is complete and transitive.

Proof. Fix x, y, z ∈ X. Since ≥ is complete on R1, either U(x) ≥ U(y) or U(y) ≥ U(x). Hence, either

x �∼

y or y �∼

x (completeness). Now assume x �∼

y and y �∼

z. Then U(x) ≥ U(y) and U(y) ≥ U(z), so

U(x) ≥ U(z) since ≥ is transitive on R1. Hence x �

∼z (transitivity).

A more difficult question is whether there is a converse to this theorem. That is, if we have a preference

relation that is complete and transitive, is there necessarily a utility function that represents it? The answer

is NO, as established by the following famous counter-example.


Example: Lexicographic Preference Relation. Let X = R2+ and define �

∼as follows:

(x1, x2) �∼ (y1, y2) if and only if either x1 > y1

or x1 = y1 and x2 ≥ y2.

The basic idea of the lexicographic relation is that it assigns a unique position to every point in R2+, so

that there is no indifference. The lexicographic ordering does this by proceeding hierarchically: If the

first component is decisive, then the ordering is established. If not, proceed to the second component.

Geometrically, it appears as follows:

Figure 11.1: Lexicographic Preference Relation

Any attempt to represent this with a real-valued utility function is trying to map R2+ into unique values of

R1, which cannot be done because R

2 is an order of magnitude “larger” than R1. However, this preference

relation is complete and transitive. Thus, this example shows that some additional property must be imposed

on a preference relation if we want to ensure existence of a utility representation.

For most applications it suffices to assume X = Rn+. Then an element x ∈ X is an n-dimensional vector,

and we can think of each component as being the quantity of a commodity that is included in the bundle.

Thus, there are n commodities. This notion of commodities is quite general. Each component can have a

place and time stamp, so that spatial and dynamic problems can be encompassed. In some contexts it is


necessary to consider continuous space and/or time, in which case an infinite-dimensional commodity space

must be considered. This makes things more complicated. For our purposes, we will henceforth assume

X = Rn+ for n finite. With this assumption on X, the representation problem can be solved with a third

assumption on �∼:

3. (Continuity) Let xi be a sequence in Rn+ and y be any element of R

n+. If xi → x0 and xi �∼ y then x0 �

∼y.

In other words, {x ∈ Rn+ : x �

∼y} is a closed set. Similarly, {x ∈ R

n+ : x ≺

∼y} is a closed set.

Note that the lexicographic preference relation violates this assumption.

Theorem (Debreu Representation Theorem). If �∼is complete, transitive, and continuous on X = R

n+

then there exists a continuous utility function U that represents �∼.

The proof is difficult, and actually does not require that X be a Euclidean space. Any topological space will

do. For those interested, the proof is on pp. 56-59 of Theory of Value by G. Debreu, Cowles Foundation for

Research in Economics Monograph 17, Wiley: New York (1959).

The Debreu Representation Theorem shows that we need only assume X and �∼

satisfy completeness,

transitivity, and continuity to know that we can discuss consumer choice in terms of a utility function. Note,

however, that the representation is not unique: If U represents �∼and h is a strictly increasing function from

R1 to R

1, then h(U) also represents �∼. This is to be expected. �

∼merely orders the points in X. It does

not express a strength of preference. Hence the distance between U(x) and U(y) cannot be informative, for

this would mean that the utility representation of �∼

conveys more information than the original preference

ordering. Only the ordering that U(x) ≥ U(y) or U(y) ≥ U(x) is informative, whence we say that utility

is an ordinal, not a cardinal, concept. Note also that, while the conditions of the Debreu Representation

Theorem are sufficient for existence of a utility representation, the continuity condition is not necessary

(our first theorem showed that completeness and transitivity are necessary, however). A counter-example to

necessity of continuity is included in the exercises.

We sometimes need to speak of the commodity bundles that are all at least as desirable as a given bundle,

or of the bundles that are the same as a given bundle. Hence we define the “preferred-to” or “upper contour”

set, and also the “indifference curve:”

{x ∈ X : x �∼

y} (the set of bundles that are (weakly) preferred to a given y)


{x ∈ X : x ∼ y} (the indifference curve for a given y)

An indifference curve need not be a “curve” at all. By definition, it consists of all points x in Rn+ such that

U(x) = U(y), so it is a level curve of U just as an isoquant is a level curve of a production function. But a

level curve of U is only a curve in some conventional sense if U is a “nicely-behaved” function.

Even though completeness, transitivity, and continuity are sufficient for a utility representation of a

preference ordering, we sometimes need other assumptions. Let x, x′, y ∈ X. The most important additional

assumptions are:

4. (Free Disposal, or Weak Monotonicity) x ≥ y ⇒ x �∼

y.

5. (Strong Monotonicity) x > y ⇒ x � y.

6. (Local Nonsatiation) For every x ∈ X and ε > 0, there exists y ∈ X such that ‖y − x‖ ≤ ε and y � x.

7. (Convexity) x �∼

y and x′ �∼

y together imply λx+ (1− λ)x′ �∼

y for every λ ∈ (0, 1).

8. (Strict Convexity) x �∼

y and x′ �∼

y (x �= x′) together imply λx+ (1− λ)x′ � y for every λ ∈ (0, 1).

These properties can be conveniently illustrated on a graph of preferred-to sets and indifference curves. Note

that free disposal implies that the boundary of a preferred-to set cannot slope upward.

If U is differentiable and its partials are nonzero (strong monotonicity), then the indifference curve for y

is indeed a curve that defines any one component of x, say xi, as a function of the other components, so we

can think of the indifference curve for y as this implicit function. Since this implicit function is defined by

U(xi(x−i), x−i) ≡ U(y), its slope in the (xi, xj) space at some point x is found by implicit differentiation to

be

∂xi(x−i)∂xj

= −Uj(x)Ui(x)

< 0.

This slope is called the marginal rate of substitution of good i for good j (MRSij), since it gives the rate at

which the consumer can substitute xi for xj while maintaining a constant level of satisfaction. Hence strong

monotonicity is associated with a negative MRS, while convexity is associated with a nondecreasing MRS

in x−i. Note that the MRS is a basic property of the preference relation, in that the same MRS arises no

matter which utility representation is utilized.


Exercises for Lecture 11 (Preferences)

1. Let X be a consumption set and �∼

be a complete and transitive preference relation on X. Define x � y

to mean “not(y �∼

x)” and x ∼ y to mean “both x �∼

y and y �∼

x.” Using only the completeness and

transitivity assumptions, show for x, y, z ∈ X:

a. x � y and y � x cannot both hold.

b. x � x cannot hold (i.e., � is irreflexive).

c. � is transitive.

d. x �∼

y and y � z together imply x � z.

e. x � y and y �∼

z together imply x � z.

f. x ∼ x (i.e., ∼ is reflexive).

g. ∼ is transitive.

h. If x ∼ y then y ∼ x (i.e., ∼ is symmetric).

2. Let X = Rn+, �∼ be a complete, transitive, and continuous preference relation on X, and U represent �

∼.

Prove the following:

a. x � y if and only if U(x) > U(y).

b. x ∼ y if and only if U(x) = U(y).

c. If �∼

satisfies free disposal, then U is nondecreasing.

d. If �∼

satisfies strong monotonicity, then U is strictly increasing.

e. If �∼satisfies local nonsatiation, then for every x ∈ X and ε > 0 there exists y ∈ X such that ‖y−x‖ ≤ ε

and U(y) > U(x) (i.e., U is locally non-maximal).

f. If �∼

satisfies convexity, then U is a quasiconcave function.

g. If �∼

satisfies strict convexity, then U is a strictly quasiconcave function.

3. Let �∼

be the lexicographic preference relation on R2+:

(x1, x2) �∼ (y1, y2) if and only if either x1 > y1

or x1 = y1 and x2 ≥ y2.

Show that �∼

is strongly monotonic and strictly convex.



4. MWG #3.B.3.

5. MWG #3.C.4.


Lecture 12: Utility Maximization: Setup, Calculus, and Traditional Comparative Statics

Readings: Sections 7.2, 8.4 of Varian; Sections 2.D, 3.D of MWG.

Last lecture, we described basic properties of a preference ordering on a consumption set X. Now it is

time to add the constraint to the problem and then study the maximum. We continue to assume X = Rn+, so

that a vector x ∈ X is a commodity bundle. We also assume the preference ordering is complete, transitive,

and continuous on X; so that we can proceed by using a utility representation U of the preference ordering.

Let p ∈ Rn+ be a vector of prices, one for each of the commodities, and m ≥ 0 be (fixed) money income.

We begin with the behavioral postulate that a perfectly competitive (price-taking) consumer chooses the

commodity bundle x ∈ X that maximizes U , subject to x being affordable. Here, “affordable” means that

p · x ≤ m, or in set notation that x ∈ B(p,m) where B(p,m) ≡ {x′ ∈ Rn+ : p · x′ ≤ m}. B is called the budget

set of the consumer for price p and income m, and x ∈ B(p,m) is the constraint for the consumer. So, we

must study the problem

max{x}

U(x) subject to x ∈ B(p,m).

As in all optimal choice problems, there are two results from this that we may be able to derive refutable

hypotheses about: the optimal choice x∗(p,m) and the optimal value U∗(p,m). The former is known as the

Marshallian demand and the latter is called the indirect utility function.

Geometrically, the objective is to get on the highest indifference curve possible while staying in the budget

set. When there are two commodities the maximization can be illustrated as in Figure 12.1.

Without more assumptions, not much can be said about this. Usually we assume, in addition to com-

pleteness, transitivity, and continuity, at least local nonsatiation. This implies that the optimal choice is on

the budget line (see the exercises following this lecture). Hence, with local nonsatiation we lose no generality

by just considering the equality-constrained problem

max{x∈R

n+}

U(x) subject to p · x = m.

If this is a nice well-behaved calculus problem the Lagrangian function is

L(x, λ; p,m) = U(x)− λ(p · x−m),

102 LECTURE 12: UTILITY MAXIMIZATION (BASICS)

Figure 12.1: Utility Maximization. The budget set must appear as illustrated, but withoutadditional assumptions the indifference curves and preferred-to sets could take almost any shape.

and so the first order conditions are

Ui(x∗(p,m))− λ∗(p,m)pi ≡ 0 for i = 1, . . . , n

−p · x∗(p,m) +m ≡ 0.

Forming a ratio yields the condition that the negative of each MRS must equal the corresponding price ratio:

Ui(x∗)Uj(x∗)

=pipj

,

which is just the algebraic statement that the indifference curve must be tangent to the budget line at

the maximum (assuming a nice interior calculus maximum). Note that only the MRS is involved in this

condition, not the individual utility derivatives. Hence, this description of the maximal choice is independent

of the utility representation used. Another way of saying this is to note that different representations are

just monotonic transformations of each other, so the maximal choice is independent of which representation

is used.

As always, the first order conditions can be thought of as defining x∗ and λ∗, as denoted above. Then

there are n + 1 equations in the n + 1 unknowns x∗ and λ∗, so we can at least hope to solve for these

unknowns. But it is really the slopes of x∗ and λ∗ in which we are most interested. We can study these

slopes by doing the usual comparative statics exercise of differentiating the entire system with respect to pj


(for example):

HL(x,λ)

∂x∗

1∂pj

...∂x∗

n

∂pj

∂λ∗∂pj

=

0...0

λ∗(p,m)0...0

x∗j (p,m)

← jth row,

where

HL(x,λ) =

U11 . . . U1n −p1...

......

Un1 . . . Unn −pn−p1 . . . −pn 0

is the (n + 1) × (n + 1) bordered Hessian of the maximization problem. Since this is a maximization

problem with r = 1 constraints, at a proper interior maximum the border-preserving principal minors of

order k > 2r = 2 have sign (−1)k−r. Applying Cramer’s Rule to solve for ∂x∗i

∂pjand then performing cofactor

expansion on the ith column of the numerator yields:

∂x∗i

∂pj=

(−1)j+iλ∗|Hji|+ (−1)n+1+ix∗j |Hn+1,i|

|HL(x,λ)| .

From the second order conditions |HL(x,λ)| has sign (−1)n+1−1, but neither of the minors in the numerator are

principal unless i = j. When i = j, |Hji| has sign (−1)(n+1−1)−1, which is opposite that of the denominator.

But even then Hn+1,i is neither border-preserving nor a principal submatrix, so its sign is unknown. Hence,

in general we cannot sign the slopes of Marshallian demand curves, even with respect to the own-price.

That is, the law of demand need not hold here. The reason is that the price change brings about a change

in spending of x∗j∂pj , and this income effect introduces another term, whose sign is unknown, into the

expression for ∂x∗i

∂pj.

To see that this term really is due to an implicit change in income, investigate the effects of a change in

income m via comparative statics:

HL(x,λ)

∂x∗

1∂m...

∂x∗n

∂m∂λ∗∂m

=

0...0−1

.

Hence

∂x∗i

∂m=

(−1)n+1+i(−1)|Hn+1,i||HL(x,λ)| .


In general, the sign of this is unknown since Hn+1,i is neither border-preserving nor a principal submatrix.

If ∂pj > 0, for example, then spending increases by x∗j∂pj, so it’s as if the consumer received a decrease in

income of ∂m = −x∗j∂pj . Substituting this in place of ∂m above shows that the extra term in the original

expression for ∂x∗i

∂pjis precisely the effect of a change in income of −x∗

j∂pj , whose sign is not known in general.

Similarly, we can find (but cannot sign) expressions for ∂λ∗∂pj

and ∂λ∗∂m . See the exercises following this

lecture.

As always with these optimization problems, there are three cautions to worry about:

1. (Existence) If p >> 0 then we are maximizing a continuous function U on the closed and bounded set

B(p,m), so a maximum exists (note the role of having a continuous, or at least upper continuous (so

U is upper semicontinuous, as shown by T. Rader in “The Existence of a Utility Function to Represent

Preferences,” Review of Economic Studies 30 (1963), pp. 229-232), representation U here). If pi = 0 for

some i then existence can fail, as for example with a Cobb-Douglas utility function.

2. (Uniqueness) This is determined by the convexity of �∼. If �

∼is not convex, or is convex but not strictly

convex, then there can be multiple optima. A complete failure of convexity is somewhat problematic

because this leads to discontinuous demand curves, which can cause an existence failure for equilibrium

prices (this comment applies to firms’ demands and supplies as well).

3. (Corners) Since we did not explicitly incorporate the n constraints x ≥ 0, the above calculus conditions

can fail if the maximum occurs where some xi = 0. This can lead to Ui(x∗) − λ∗pi < 0. The calculus

condition is suggesting that a decrease in xi would improve things, but the consumer cannot pursue this

because of the constraint.


Exercises for Lecture 12 (Utility Maximization: Basics)

1. Use comparative statics methodology to find expressions for ∂λ∗∂pj

and ∂λ∗∂m . Can either of these be signed?

Why/why not? Give an economic interpretation of ∂λ∗∂m

.

2. (From W. Nicholson, Microeconomic Theory, 6th edition (1994), p. 758) This exercise introduces how

standard utility theory can be used to study the choice of labor and leisure for a family. A welfare

program for low-income people offers a family a basic grant of $6,000 per year. This grant is reduced by

$0.75 for each $1 of other income the family has.

a. How much in welfare benefits does the family receive if it has no other income? If the head of the

family earns $2,000 per year? How about $4,000 per year?

b. At what level of earnings does the welfare grant become zero?

c. Assume that the head of this family can earn $4 per hour and that the family has no other income.

What is the annual budget constraint for this family if it does not participate in the welfare program?

That is, how are dollars of consumption (C) and hours of leisure (H) related?

d. What is the budget constraint if the family opts to participate in the welfare program (the grant cannot

be negative)?

e. Graph your results from parts (c) and (d).

f. Suppose the government changes the rules of the welfare program to permit families to keep 50 percent

of what they earn. How would this change your answers to parts (d) and (e)?

g. Using your results from part (f), can you predict whether the head of this family will work more or

less under the new rules described in part (f)?

3. (From E. Silberberg, The Structure of Economics, 2nd edition (1990), p. 358) This exercise introduces

how standard utility theory can be used to study the allocation of consumption over time. Suppose a

consumer will have income $m1 this year and $m2 next year. He or she consumes $x1 this year and $x2

next year, and is able to borrow or lend at interest rate r provided the two-period budget constraint is

met. Assume the consumer maximizes the utility U(x1, x2) of consumption over these two years.

a. Write the consumer’s budget constraint.

b. Derive the comparative statics for this problem. Will an increase in this year’s income necessarily lead


to an increase in consumption this year?

c. Prove that the consumer is better off (worse off) when the interest rate rises if he or she was a net

saver (dissaver) this year before the interest rate increase.

4. Suppose a consumer of two goods, x1 and x2, has utility function U(x1, x2) =√x1 + 4 + x2.

a. Carefully draw a typical indifference curve for utility level U (note: U ≥ 2 since U(0) = 2).

b. Find the Marshallian demands.


5. Varian #7.2 (ignore the expenditure function part).

6. Varian #7.5.

7. Varian #8.5.

8. Varian #8.13.

9. MWG #2.E.1.

10. MWG #2.E.2.

11. MWG #2.E.8.


Lecture 13: Utility Maximization: Envelope Properties of U∗ and x∗

Readings: Section 7.3 of Varian; Sections 2.E, 3.D of MWG.

In this lecture we continue our study of the properties of a utility maximum, but make use of the envelope

theorem rather than using the cumbersome traditional comparative statics. The optimal choice is x∗(p,m)

and the optimal value of the utility function is U∗(p,m) = U(x∗(p,m)), both for p >> 0 and m ≥ 0.

There is an interval of feasible utility levels that plays a role analogous to the interval of feasible output

levels from producer theory. So, we use the producer theory notation and denote this interval by Y , and

let y = sup{x} U(x) (possibly ∞) denote the upper end of Y . By continuity, Y is an interval from U(0)

(inclusive) to y (perhaps inclusive, perhaps not, depending on whether U achieves its supremum). Actually,

since we have not assumed monotonicity of �∼, there may be feasible utility levels smaller than U(0), but

they are economically irrelevant since no utility-maximizing consumer would ever choose them, so we omit

them from Y .

The optimal choice x∗ and optimal value U∗ functions have some remarkable properties due solely to the

fact that they are the maximizing choice and maximal value, respectively, of U(x) subject to p · x ≤ m.

Theorem (Roy’s Identity–Relationship between the Indirect Utility Function and the Mar-

shallian Demand). If U∗ is differentiable at (p,m) then ∂U∗∂m

= λ∗(p,m) and, if ∂U∗∂m

�= 0 at (p,m), then:

−∂U∗∂pi

∂U∗∂m

= x∗i (p,m).

Proof. Differentiate U∗(p,m) with respect to m, using the envelope theorem so that this is obtained by dif-

ferentiating the Lagrangian function and then evaluating at the optimum, to get λ∗(p,m). Then differentiate

U∗(p,m) with respect to pi, using the envelope again, to get −λ∗(p,m)x∗i (p,m). Solve this for x∗

i and then

substitute ∂U∗/∂m for λ∗. �

• Note that λ∗ is the marginal utility of income.

Theorem (Properties of the Marshallian Demand).

1. (Homogeneity) x∗ is homogeneous of degree 0.

2. (Symmetry and Semidefiniteness) To be announced.

3. (Nonnegativity) x∗(p,m) ≥ 0 ∀(p,m).

108 LECTURE 13: UTILITY MAXIMIZATION (ADVANCED)

4. (Walras’ Law) If the underlying preference relation is locally nonsatiated, then p · x∗(p,m) = m ∀(p,m).

Proof.

1. U is unaffected by p and m. Moreover, B(p,m) = B(λp, λm). So, the maximization problem is the same

when the parameters are (λp, λm) as when the parameters are (p,m), whence x∗(p,m) = x∗(λp, λm).

2. Note that x∗i is not a derivative of U∗, so we do not automatically have symmetry of slopes of x∗. There

is, however, another symmetry property that will arise later in the study of expenditure minimization.

3. x∗(p,m) ≥ 0 since x∗ is chosen from Rn+.

4. By definition, x∗(p,m) ∈ B(p,m), so p · x∗(p,m) ≤ m. If p · x∗(p,m) < m, then by local nonsatiation

there exists x ∈ Rn+ such that U(x) > U(x∗(p,m)) and ‖x − x∗(p,m)‖ < ε, where ε = m−p·x∗(p,m)

‖p‖ > 0.

Hence, using the Cauchy-Schwarz inequality,

p · (x− x∗(p,m)) ≤ ‖p‖ ‖x− x∗(p,m)‖ < ‖p‖ε = m− p · x∗(p,m),

or p · x < m. That is, x ∈ B(p,m), which contradicts that x∗(p,m) maximizes U . �

Theorem (Properties of the Indirect Utility Function).

1. (Homogeneity) U∗ is homogeneous of degree 0.

2. (Quasiconvexity) U∗ is quasiconvex.

3. (Monotonicity) U∗ is nondecreasing in m and nonincreasing in p. If the underlying preference relation is

locally nonsatiated, then U∗ is strictly increasing in m.

4. (Continuity) U∗(p,m) is continuous, and U∗(p,R1+) = Y ∀p >> 0.

Proof.

1. By homogeneity of x∗, U∗(λp, λm) = U(x∗(λp, λm)) = U(x∗(p,m)) = U∗(p,m).

2. Fix (p,m), (p′, m′) ≥ 0. By definition,

[λp+ (1− λ)p′] · x∗(λp+ (1− λ)p′, λm+ (1− λ)m′) ≤ λm+ (1− λ)m′.

Hence, either p · x∗(λp+ (1− λ)p′, λm+ (1− λ)m′) ≤ m or p′ · x∗(λp+ (1− λ)p′, λm+ (1− λ)m′) ≤ m′,

or both. That is, x∗(λp + (1 − λ)p′, λm + (1 − λ)m′) is feasible when prices and income are either


(p,m) or (p′, m′). Hence, either U∗(p,m) ≥ U(x∗(λp + (1 − λ)p′, λm + (1 − λ)m′)) or U∗(p′, m′) ≥

U(x∗(λp+ (1− λ)p′, λm+ (1− λ)m′)). Either way,

U(x∗(λp+ (1− λ)p′, λm+ (1− λ)m′)) ≤ max{U∗(p,m), U∗(p′, m′)}.

3. Let m′ ≥ m and p′ ≤ p. Then p′ · x∗(p,m) ≤ p · x∗(p,m) ≤ m ≤ m′. So, x∗(p,m) is feasible when

prices and income are (p′, m′), whence U∗(p′, m′) ≥ U(x∗(p,m)) = U∗(p,m). If the underlying preference

relation is locally nonsatiated and m′ > m, then x∗(p,m) cannot maximize U when prices are (p′, m′),

since the inequality p′ · x∗(p,m) < m′ becomes strict while Walras’ Law requires p′ · x∗(p′, m′) = m′.

Hence U∗(p′, m′) > U(x∗(p,m)) = U∗(p,m).

4. Continuity follows from the Theorem of the Maximum (see MWG Theorem M.K.6). Clearly U∗(p,R1+) ⊂

Y . Moreover, for any p >> 0 and U ∈ Y , we have U∗(p, 0) = U(0) ≤ U since B(p, 0) = {0}; and since

U ∈ Y there exists x ∈ Rn+ such that U(x) ≥ U . So for m = p · x we have x ∈ B(p,m), and therefore

U∗(p,m) ≥ U(x) ≥ U . Thus, by continuity of U∗ we have U ∈ U∗(p,R1+), establishing U∗(p,R1

+) ⊃ Y . �

• Note, in particular, that there is no semidefiniteness property for x∗ here, because we only know

quasiconvexity of U∗, not full convexity, so we do not know the slopes of x∗ and λ∗ in general (also, x∗ is

not a derivative of U∗, as noted above). This is consistent with the lack of comparative statics results we

noted in the previous lecture.

• Homogeneity means that consumers do not have “money illusion.” If all prices and income are inflated

by the same amount, neither behavior nor satisfaction change. Again we see that “only relative prices and

income matter.”

• Note also that the only property of preferences that we really rely on here is that a maximum exists

(guaranteed by completeness, transitivity, and continuity for p >> 0 and m ≥ 0), except for Walras’ Law

and strict monotonicity of U∗ in m, which rely on local nonsatiation.

• Note finally that Euler’s Theorem for the homogeneous of degree zero Marshallian demand yields

n∑j=1

pj∂x∗

i (p,m)∂pj

+m∂x∗

i (p,m)∂m

= 0 for i = 1, . . . , n.


Hence ∂x∗

1∂p1

. . .∂x∗

1∂pn

∂x∗1

∂m

.... . .

......

∂x∗n

∂p1. . .

∂x∗n

∂pn

∂x∗n

∂m

p1...pnm

=

0...00

.


Exercises for Lecture 13 (Utility Maximization: Advanced)

1. Suppose a consumer’s preferences are homothetic, in addition to being complete, transitive, continuous,

and locally nonsatiable on Rn+. This means that the utility function is a monotonic transformation of

a function that is homogeneous of degree r > 0 (i.e., U(x) = f(g(x)), where f : R1 → R

1 is strictly

increasing and g : Rn+ → R

1 is homogeneous of degree r).

a. Show that there is no loss of generality in just assuming that the consumer’s utility function is homo-

geneous of degree 1. Assume this henceforth.

b. Use Euler’s Theorem to show that U∗(p,m) = λ∗(p,m)m.

c. Differentiate to show that ∂λ∗∂pi

= −λ∗x∗i

m .

d. Now use part c and the usual approach to deriving symmetry results to show that the matrix of price

derivatives of the Marshallian demand is symmetric when preferences are homothetic (even though it

is not symmetric in general).


2. Varian #7.4(a).

3. MWG #3.D.1.

4. MWG #3.D.2 (show the quasiconvexity only as a function of p – as a function of m it is more difficult).

5. MWG #3.D.5 (again, show the quasiconvexity only as a function of p).


Lecture 14: Expenditure Minimization:

Setup, Calculus, Traditional Comparative Statics, and Envelope Properties of e∗ and h∗

Readings: Sections 7.3, 8.3 of Varian; Section 3.E of MWG.

In this lecture we consider an alternative behavioral postulate for consumers. As we saw, a Marshallian

demand has few definite comparative statics properties that arise naturally from the utility maximization

behavioral postulate. Marshallian demand does in fact possess a symmetry/semidefiniteness property, that

we can derive by studying the expenditure minimization behavioral postulate:

min{x}

e(x; p) = p · x subject to U(x) ≥ U and x ≥ 0.

That is, the behavioral postulate is that a consumer chooses the commodity bundle that minimizes the

expenditure required to achieve a given level of utility. The consumer is still perfectly competitive since this

minimization assumes price-taking behavior, and we continue to assume the consumer’s preference relation

is complete, transitive, and continuous so that we can represent it with a utility function.

The parameters of this optimization are p ∈ Rn++ and a feasible utility level U ∈ Y . Thus the resulting

optimal choice of x and optimal value of e depend on the parameters (p, U) ∈ Rn++ × Y . We depart

from convention here and denote the optimal choice of x by h∗(p, U), as a reminder that the expenditure-

minimizing demands are called Hicksian, or compensated, demands. The optimal value of e is denoted

e∗(p, U) = p · h∗(p, U), and is called the expenditure function.

Geometrically, the objective is to get on the lowest isoexpenditure line while staying on or above the

indifference curve for utility level U . With two commodities, the minimization can be illustrated as in

Figure 13.1.

As usual, we can simplify somewhat by writing the constraint as an equality. In this problem, we can

do this for essentially the same reasons we could write the constraint as f(x) = y in the cost minimization

problem. U is continuous (in particular, lower semicontinuous), so for any x such that U(x) > U we can

reduce x slightly to x′ (note that x > 0 since U ≥ U(0)) and still obtain utility U(x′) ≥ U . Since x′ < x

we have p · x′ < p · x as long as p >> 0. Hence, the minimum never occurs at an x that yields more utility

than U , and so nothing is lost by just assuming the constraint is U(x) = U . Using this way of writing the

constraint, if U is differentiable we can write the Lagrangian function L(x, λ; p, U) = p ·x−λ(U(x)− U ) and

114 LECTURE 14: EXPENDITURE MINIMIZATION

Figure 14.1: Expenditure Minimization. The indifference curve and the preferred-to setneed not have the conventional shape.

proceed to describe the minimum with calculus. Alternatively, we can use the envelope theorem to derive

properties of e∗ and h∗.

It is unnecessary to grind through all of this, however, because in fact the problem is identical to cost

minimization in all important respects, and we already know the properties of a cost minimum. Recall that

the cost minimization problem is

min{x}

w · x subject to f(x) = y and x ≥ 0

for some lower semicontinuous production function f(x) that is defined on Rn+. Hence, all that is changed is

the symbols: we are now using p in place of w, U in place of f , and U in place of y. Therefore, any properties

of a cost minimum that derive solely from that fact that we are minimizing are immediately present here.

In particular, with regard to the standard calculus approach, a nice proper interior calculus minimum is

characterized by tangency between the indifference curve for utility level U and an isoexpenditure line, which

is algebraically described by equality between each MRS and the corresponding price ratio. Also, the law of

demand holds here: ∂h∗i

∂pi≤ 0, since this comparative static result depends only on the first- and second-order

conditions. Finally, the same cautions are relevant here about existence (assured since the objective e is

continuous on a constraint set that is nonempty (as long as U ∈ Y ), closed (by continuity of U), and bounded

(provided p >> 0 so that we can throw out the upper part of the preferred-to set)), uniqueness (assured


when the preferred-to set is strictly convex, i.e., �∼strictly convex), and corners (pi−λ∗Ui(h∗) > 0 is possible

if h∗i = 0).

Similarly, with regard to the envelope approach, we immediately get Shephard’s Lemma, since it is

obtained by just differentiating the Lagrangian function.

Theorem (Shephard’s Lemma–Relationship Between the Expenditure Function and the Hick-

sian Demand). If e∗ is differentiable at (p, U) (almost assured by concavity) then ∂e∗∂pi

= h∗i (p, U) for

i = 1, . . . , n and ∂e∗∂U

= λ∗(p, U).

And we immediately get the properties of e∗ and h∗ as well.

Theorem (Properties of the Expenditure Function).

1. (Homogeneity) e∗ is homogeneous of degree 1 in p.

2. (Concavity) e∗ is concave in p (and hence continuous at p >> 0).

3. (Monotonicity) e∗ is nondecreasing in p, and is strictly increasing in U .

4. (Continuity) e∗ is fully continuous in (p, U) at p >> 0.

5. (Nonnegativity) e∗ > 0 ∀p >> 0 and ∀U ∈ Y − {U(0)}, and e∗(p, U(0)) = 0 ∀p >> 0.

Proof. Properties 1 – 5 of a cost function depend mainly on existence of a minimum, which is guaranteed

here by continuity of U , p >> 0, and U ∈ Y . Thus properties 1 and 2 of e∗ hold simply because the cost

function has these properties whenever the cost minima exist.

A few additional observations are needed to show properties 3 – 5 of e∗, because strict monotonicity in

output, continuity, and nonnegativity of cost rely on some additional properties of the technology. Specifi-

cally; free disposal, the possibility of inaction, no free lunch, and lower hemicontinuity in inputs and output

are used to show these properties of cost. Free disposal need not hold here (unless we assume monotonicity

of �∼), but the only role played by free disposal in showing strict monotonicity in output, continuity, and

nonnegativity of cost is to assure existence of f on all of Rn+. In the present context, existence of U on all

of Rn+ is assured by the Debreu Representation Theorem and the assumption that X = R

n+. The possibility

of inaction and no free lunch hold in the present context provided U(0) is regarded as output level 0. To see

this, simply note that U(0) is by assumption a feasible utility level since x = 0 is in the consumption set.


This is the possibility of inaction. Since U is a function, to achieve a utility level U > U(0) some bundle

other than x = 0 must be consumed. Since the consumption set is Rn+, this means x > 0 is required in order

for U(x) ≥ U > U(0). This is no free lunch.

Continuity of the cost function is an application of the Theorem of the Maximum (see MWG Theorem

M.K.6), which requires that the constraint correspondence be both lower and upper hemicontinuous. Lower

hemicontinuity in output (i.e., in utility levels U) is assured in the present context by the continuity of

�∼, which implies continuity of U as shown by the Debreu Representation Theorem. Lower hemicontinuity

in inputs (i.e., in commodity bundles x) is assured in the present context by local nonsatiation of �∼, as

follows. Given a sequence U i → U0 and a point x0 satisfying U(x0) ≥ U0, we must show existence of a

sequence xi → x0 such that U(xi) ≥ U i ∀i. By local nonsatiation, there exists a sequence yj → x0 such that

U(yj) > U(x0) ∀j, and it can be assumed without loss of generality that U(yj ) ≥ U(yj+1) for j = 1, 2, . . ..

Consider any yj . As U i → U0, there exists kj such that U(yj) ≥ U i ∀i ≥ kj , and it can be assumed without

loss of generality that k1 < k2 < · · · . Now construct the xi sequence as follows:

xi ={

any x satisfying U(x) ≥ U i for i = 1, . . . , k1 − 1 (if any – such x exist provided U i ∈ Y ∀i)yj for i = kj, . . . , kj+1 − 1; j = 1, 2, . . .

This xi has the required properties. By construction, U(xi) = U(yj) > U i when kj ≤ i < kj+1. Moreover, for

any ε > 0 there exists J such that∣∣yj − x0

∣∣ < ε for j ≥ J . Letting I = kJ , we have i ≥ I ⇒ i ≥ kJ ⇒ xi = yj

for some j ≥ J . So i ≥ I ⇒ ∣∣xi − x0∣∣ < ε.

Hence, by regarding U(0) as output level 0 we obtain strict monotonicity of e∗ in U , continuity of e∗, and

nonnegativity of e∗ directly from the analogous properties of the cost function. �

Theorem (Properties of the Hicksian Demand).

1. (Homogeneity) h∗ is homogeneous of degree 0 in p.

2. (Symmetry and Semidefiniteness) The matrix of slopes∂λ∗∂U

∂λ∗∂p1

. . . ∂λ∗∂pn

∂h∗1

∂U

∂h∗1

∂p1. . .

∂h∗1

∂pn

......

. . ....

∂h∗n

∂U

∂h∗n

∂p1. . .

∂h∗n

∂pn

is symmetric for U ∈ (U(0), y), and the lower right (n× n) submatrix is negative semidefinite.

3. (Monotonicity) p · h∗ is nondecreasing in p, and is strictly increasing in U .


4. (Continuity) p · h∗ is fully continuous in (p, U) at p >> 0.

5. (Nonnegativity) h∗ ≥ 0 ∀p >> 0 and ∀U ∈ Y , p · h∗(p, U(0)) = 0, and h∗(p, U) > 0 ∀p >> 0 and

∀U ∈ Y − {U(0)}.

Proof. Properties 1 – 5 of a conditional factor demand depend only on existence of a minimum over some

subset of Rn+, Shephard’s Lemma, and the properties of a cost function. Since all of these conditions hold in

the present context once U(0) is regarded as output level 0, properties 1 – 5 of h∗ hold simply because the

conditional factor demand has these properties. �

• So, as with cost minimization, cross-price effects on Hicksian demands are symmetric and own-price

effects are nonpositive, and the optimal Lagrange multiplier is the marginal cost of obtaining higher utility.

• Also as with cost minimization, Euler’s Theorem for the homogeneous of degree zero Hicksian demand

yieldsn∑j=1

pj∂h∗

i (p, U)∂pj

= 0 for i = 1, . . . , n.

Hence

He∗(p)p =

∂h∗

1∂p1

. . .∂h∗

1∂pn

.... . .

...∂h∗

n

∂p1. . .

∂h∗n

∂pn

p1

...pn

=

0...0

,

and we see that the Hessian of the expenditure function with respect to p is singular, possessing a zero

eigenvalue.

• Finally, homogeneity again shows that “only relative prices matter.”


Exercises for Lecture 14 (Expenditure Minimization)

1. Let e∗(p, U) = pU be an expenditure function, where p is a scalar.

a. Verify that e∗ has the properties of an expenditure function.

b. Find the Hicksian demand, h∗.

c. Verify that h∗ has the properties of a Hicksian demand.


2. Finish Varian #7.2.

3. Varian #8.8.

4. MWG #3.E.2.

5. MWG #3.E.6.

6. MWG #3.G.3 except parts (c) and (e).

7. MWG #3.G.14 (note that when they say “Walrasian” substitution matrix they mean the substitution

matrix of the Hicksian demands).

8. MWG #3.G.15.


Lecture 15: Relationships between Utility Maximization and Expenditure Minimization

Readings: Sections 7.4, 8.2, 8.5 of Varian; Section 3.G of MWG.

We now know that Hicksian demands slope downward, while Marshallian demands may not. Unfortu-

nately, Hicksian demands are not observable since they depend on unobservable utility. However, we can

use the information about Hicksian demands to further explore Marshallian demands, by establishing the

relationship between utility maximization and expenditure minimization. We continue to assume the con-

sumer has a complete, transitive, and continuous preference relation, so that it is represented by a continuous

utility function U ; and that p >> 0, m ≥ 0, and U ∈ Y .

First we show that a utility-maximizing consumer also minimizes the expenditure of achieving the optimal

level of utility, provided preferences are locally nonsatiated. This means

1. x∗(p,m) = h∗(p, U∗(p,m)), and

2. m = e∗(p, U∗(p,m)).

This is illustrated in Figure 15.1.

Figure 15.1: Utility maximization implies expenditure minimization. x∗ is the optimalchoice for income m. If the light shading is the preferred-to set for x∗ then we obtain the lowestpossible isoexpenditure line subject to this preferred-to set by choosing x∗ as our Hicksian demandpoint, in which case expenditure minimization coincides with utility maximization. Note, however,that if the indifference curve for utility level U∗(p,m) were “thick,” for example the dark shadedarea, then x∗ still maximizes utility but expenditure minimization occurs at the illustrated h∗.Thick indifference curves are inconsistent with local nonsatiation. Thus the graph shows that localnonsatiation is central to the coincidence of utility maximization and expenditure minimization.

120 LECTURE 15: UTILITY AND EXPENDITURE RELATIONSHIPS

Then we show the converse, that an expenditure-minimizing consumer also maximizes the utility from

spending the expenditure-minimizing level of income. This means

3. h∗(p, U) = x∗(p, e∗(p, U)), and

4. U = U∗(p, e∗(p, U)).

This is illustrated in Figure 15.2.

Figure 15.2: Expenditure minimization implies utility maximization. If the preferred-to set for utility level U is both the light and dark shaded areas, then the lowest possible iso-expenditure line subject to this preferred-to set occurs at h∗. Now given income level e∗, thechoice h∗ maximizes utility since it is not possible to attain any other point in the preferred setto this point (i.e., in the shaded areas). Hence utility maximization coincides with expenditureminimization. If the indifference curve is “thick” like the dark shaded area then there may beother points above the isoexpenditure line for e∗ that deliver the same utility at these prices, forexample the illustrated x∗, but such points cost more than e∗ (m > e∗) and thus are not in thebudget set for income e∗. Thus local nonsatiation is not needed to establish that expenditureminimization implies utility maximization. However, as the proof in the text shows, lower semi-continuity of utility is needed instead of local nonsatiation (otherwise, there could be points alongthe isoexpenditure line e∗ that deliver utility U but that deliver less utility than some other pointalong the isoexpenditure line).

Equations 1–4 hold for every p >> 0, m ≥ 0, and U ∈ Y , and so are identities that describe the equivalence

of utility maximization and expenditure minimization (at the corresponding points).

To show the first two identities, suppose that x∗(p,m) does not minimize p ·x subject to U(x) ≥ U∗(p,m).

Then there exists x ∈ Rn+ such that p · x < p · x∗(p,m) and U(x) ≥ U∗(p,m). Let ε = p·(x∗(p,m)−x)

‖p‖ > 0.

By local nonsatiation, there exists x′ ∈ Rn+ such that U(x′) > U(x) and ‖x′ − x‖ < ε. By feasibility,


p · x∗(p,m) ≤ m. So

p · x′ = p · (x′ − x) + p · x ≤ ‖p‖‖x′ − x‖+ p · x by the Cauchy-Schwarz Inequality

< ‖p‖ε+ p · x

= p · (x∗(p,m)− x) + p · x

= p · x∗(p,m) ≤ m.

That is, x′ can be purchased with income m at price p, and delivers utility U(x′) strictly greater than

U(x∗(p,m)). This contradicts that U∗(p,m) is the maximal utility. Hence x∗(p,m) must minimize p·x subject

to U(x) ≥ U∗(p,m). That is, x∗(p,m) = h∗(p, U∗(p,m)), establishing identity 1 (if the optimal choices

are non-unique, this argument shows that any utility-maximizing choice must also minimize expenditure).

Moreover, by definition e∗(p, U∗(p,m)) = p · h∗(p, U∗(p,m)) = p · x∗(p,m) = m, where the last equality is

Walras’ Law (local nonsatiation), establishing identity 2.

To show the second two identities, consider first the U = U(0) case. Then, since p >> 0 we have

h∗(p, U) = 0 and e∗(p, U) = 0. Hence, the budget set B(p, e∗(p, U)) is the singleton {0} (again using p >> 0).

Therefore x∗(p, e∗(p, U)) = 0 = h∗(p, U) and U∗(p, e∗(p, U)) = U(x∗(p, e∗(p, U))) = U(0) = U . Now consider

the U ∈ Y − {U(0)} case. Suppose that h∗(p, U) does not maximize U(x) subject to p · x ≤ e∗(p, U). Then

there exists x ∈ Rn+ such that U(x) > U(h∗(p, U)) ≥ U and p · x ≤ e∗(p, U). Note that, since U > U(0),

we know x > 0. Therefore, by lower semicontinuity of U , there exists x′ < x such that U(x′) > U , and so

p·x′ < e∗(p, U) (using p >> 0). But this contradicts that e∗(p, U) is the minimal expenditure. Hence h∗(p, U)

must maximize U(x) subject to p · x ≤ e∗(p, U). That is, h∗(p, U) = x∗(p, e∗(p, U)), establishing identity 3

(again, if the optimal choices are non-unique, this argument shows that any expenditure-minimizing choice

must also maximize utility). Moreover, by definition U∗(p, e∗(p, U)) = U(x∗(p, e∗(p, U))) = U(h∗(p, U)) = U ,

where the last equality follows because the Hicksian demand must satisfy the constraint with equality (by

lower semicontinuity), establishing identity 4.

We can use these identities to further investigate the slopes of Marshallian demands. Differentiating

identity 3 for demand x∗i with respect to pj yields:

∂h∗i (p, U)∂pj

=∂x∗

i (p,m)∂pj

∣∣∣∣m=e∗(p,U)

+∂x∗

i (p, e∗(p, U))

∂m

∂e∗(p, U)∂pj

.


By Shephard’s Lemma ∂e∗∂pj

= h∗j (p, U), so

∂x∗i (p,m)∂pj

∣∣∣∣m=e∗(p,U)

=∂h∗

i (p, U)∂pj

− h∗j (p, U)

∂x∗i (p, e

∗(p, U))∂m

.

Now evaluate this at U = U∗(p,m) for some arbitrary income level m ≥ 0, and substitute identities 1 and 2

to obtain:

∂x∗i (p,m)∂pj

=∂h∗

i (p, U)∂pj

∣∣∣∣U=U∗(p,m)

− x∗j(p,m)

∂x∗i (p,m)∂m

. (SD)

When we consider the case i = j this equation shows explicitly why we cannot sign ∂x∗i

∂pi. The own-price

slope of the Marshallian demand is composed of two terms, the first of which is known to be nonpositive

but the second of which has unknown sign. The second term is the income effect of the price change, since

it tells how much the Marshallian demand for good i changes when income changes by ∂m = −x∗i ∂pi. We

already knew from the comparative statics analysis that this term represents the effect on demand of the

implicit income change that occurs whenever a price changes. However, the current notation illuminates the

role of the first term as well. It is the change in demand that occurs when we do not hold income fixed as

the price changes, but instead allow income to change in whatever way is necessary (i.e., according to e∗) to

keep utility constant. This substitution effect is known to be nonpositive, simply because the law of demand

holds for expenditure-minimizing (i.e., cost-minimizing) demands. The very important two-term expression

(SD) for the slope of a Marshallian demand is known as the Slutsky Decomposition of the price change. It is

illustrated in Figure 15.3.


Figure 15.3: The Slutsky Decomposition. When p1 increases from p′1 to p′′1 the budgetline swings down, as indicated by the arrow. The new price ratio determines the slope of the newbudget line. At this new price ratio, the consumer cannot afford any bundle that yields the originalutility level U . Rather, at the new price, income would have to be increased enough to make pointh∗(p′′1 , p2, U) affordable in order for the consumer to maintain the original level of utility. Hencethe price increase imposes an implicit income decrease corresponding to the distance between thetwo parallel budget lines, and the change in demand from point h∗(p′′1 , p2, U) to point x∗(p′′1 , p2, m)that would occur from such an income change is the income effect of the price change. The changein demand from x∗(p′1, p2, m) to h∗(p′′1 , p2, U) that would occur if there were no implicit incomechange is the pure substitution effect of the price change, and it is always nonpositive on the goodwhose price is changing (note that x1 decreases as we move from x∗(p′1, p2, m) to h∗(p′′1 , p2, U)).Point h∗(p′′1 , p2, U) is the Hicksian demand point for the original utility level U and the new higherprice. Since the consumer would have to be compensated for the implicit income loss in orderto buy point h∗(p′′1 , p2, U), the Hicksian demand is sometimes called the (income) compensateddemand (and, likewise, the Marshallian demand is sometimes called the uncompensated demand).

The Slutsky Decomposition allows us to characterize the situations in which Marshallian demands will

indeed obey the law of demand:

• If ∂x∗i

∂m ≥ 0 we say xi is a normal good, and we then know that its Marshallian demand slopes downward.

The Hicksian demand curve is at least as steep as the Marshallian demand curve in this case (the Hicksian

demand changes no more than the Marshallian demand when price changes, but since price is on the

vertical axis in a conventional demand graph this means the Hicksian curve is at least as steep as the

Marshallian curve). In Figure 14.3, x1 decreases as we move from h∗(p′′1 , p2, U) to x∗(p′′1 , p2, m), so x1 is

normal in that graph. The corresponding demand curves are illustrated in Figure 15.4.


Figure 15.4: Hicksian and Marshallian Demands for a Normal Good

• If ∂x∗i

∂m < 0 we say xi is an inferior good. If this income derivative, weighted by x∗i , is smaller in absolute

value than ∂h∗i

∂pithen the Marshallian demand still slopes downward but is now steeper than the Hicksian

demand. This is illustrated in Figure 15.5. If not, then the Marshallian demand slopes upward, and we

say xi is a Giffen good. A Giffen good is illustrated in Figure 15.6.

• Note that it is not possible in general to classify goods as substitutes or complements based on the

Marshallian demands. Due to income effects, which need not be symmetric between two goods, the price

derivatives of Marshallian demands are not symmetric. Hence it is possible that ∂x∗i

∂pj> 0 while ∂x∗

j

∂pi< 0, in

which case any classification is arbitrary. Since the Hicksian demand slopes are symmetric, it is possible

to classify substitutes and complements based on Hicksian demands (or equivalently based on the Slutsky

matrix, which is observable). This approach essentially removes the ambiguous income effects, but is not

pursued very frequently.


Figure 15.5: Hicksian and Marshallian Demands for an Inferior (non-Giffen) Good


Figure 15.6: Hicksian and Marshallian Demands for a Giffen Good

Another use of the Slutsky decomposition is that it allows us to express the slope of the Hicksian demand

in terms of entirely observable components:

∂h∗i (p, U)∂pj

∣∣∣∣U=U∗(p,m)

=∂x∗

i (p,m)∂pj

+ x∗j(p,m)

∂x∗i (p,m)∂m

.

Hence, from the properties of the Hicksian demand we now know a symmetry/semidefiniteness property of


the Marshallian demand:

Theorem (Addendum to Properties of the Marshallian Demand).

2. (Symmetry and Semidefiniteness) The matrix of slopes(∂x∗

1∂p1

+ x∗1∂x∗

1∂m

). . .

(∂x∗

1∂pn

+ x∗n∂x∗

1∂m

)...

......(

∂x∗n

∂p1+ x∗

1∂x∗

n

∂m

). . .

(∂x∗

n

∂pn+ x∗

n∂x∗

n

∂m

)

is symmetric and negative semidefinite for m > 0.

We can summarize the relationships between utility maximization and expenditure minimization that we

have derived so far with the following diagram:

Figure 15.7: Relationships between Utility Maximization and Expenditure Mini-

mization. The dark arrows at the bottom are identities 1 and 3, respectively. The other darkarrows are other relationships already established. The gray arrows are relationships not yetestablished.

There are two types of relationships illustrated in this diagram that we have not yet addressed. First, how

do we derive U∗ from e∗, and vice-versa? Identities 2 and 4 establish the relationship. Given U∗, by strict

monotonicity inm (from local nonsatiation) we know there is an inverse function U∗−1 on Y (U∗−1 is defined

on Y for any p >> 0, by continuity of U∗) such that U∗−1(U∗(p,m)) = m (the dependence of U∗−1 on p is

suppressed for convenience). So, from identity 4 we have U∗−1(U) = U∗−1(U∗(p, e∗(p, U))) = e∗(p, U) for

U ∈ Y . That is, if we set the given U∗ equal to U and invert the resulting expression to solve for m, the


end result is e∗(p, U). Similarly, given e∗, by strict monotonicity in U (from lower semicontinuity of U) we

know there is an inverse function e∗−1 on R1+ (assuming local nonsatiation, so that e∗(p, U) is onto R

1+ as

a function of U ∈ Y , for every p >> 0) such that e∗−1(e∗(p, U)) = U (dependence of e∗−1 on p is again

suppressed). So, from identity 2 we have e∗−1(m) = e∗−1(e∗(p, U∗(p,m))) = U∗(p,m) for m ≥ 0. That is, if

we set the given e∗ equal to m and invert the resulting expression to solve for U , the end result is U∗(p,m).

The second type of relationship we have not yet addressed is more substantial, involving movement from

x∗ to U∗. This is known as the integrability problem and is quite important, since the only truly observable

function here is the Marshallian demand x∗. The easiest approach is to convert x∗ to e∗ and then just invert

e∗, as above, to find U∗. It is tempting to imagine that we could convert x∗ to h∗ and then use e∗ = p · h∗,

but this logic is circular because we have to know e∗ in order to convert x∗ into h∗, and if we knew e∗ we

would essentially already know U∗ by inversion. We are interested in finding a way to derive U∗ (and e∗

and h∗) from x∗, when all that we know to start with is x∗. To do this, use identity 3 along with Shephard’s

Lemma to get

x∗i (p, e

∗(p, U)) = h∗i (p, U) =

∂e∗(p, U)∂pi

for i = 1, . . . , n.

That is, we know that the given x∗ vector forms a system of partial differential equations of the e∗ function

(involving both e∗ and its partial derivatives). To find e∗ we must solve this system, whence the term

“integrability.” There are known necessary and sufficient conditions for there to be a solution for such

systems, which, as we will see in Lecture 16, can be expected to hold here. Thus, we can generally expect

that the expenditure function can be derived from the Marshallian demands. Actually performing the

integration can be quite tedious, however.


Exercises for Lecture 15 (Relationships between Utility Maximization and Expenditure Minimization)

1. Suppose a consumer has expenditure function e∗(p1, p2, U) = U√p1p2. Find the Marshallian demands.

2. Suppose preferences are homothetic. Can we classify complements and substitutes based on Marshallian

demands? (Refer to exercise 1 of Lecture 12)


3. Varian #7.3.

4. Varian #7.4(b) and (c).

5. Varian #8.1.

6. Varian #8.15. Note: The question should read “What is the slope of the supply function of labor?”

7. MWG #2.F.10(a).

8. MWG #3.G.2.

9. MWG #3.G.3(c) and (e).

10. MWG #3.H.5.

11. MWG #3.H.6.


Lecture 16: Consumer Welfare

Readings: Chapter 10 of Varian; Section 3.I of MWG.

The welfare a consumer receives from a particular price vector and income endowment is measured by

the indirect utility function. This welfare measurement is merely a utility index, stated in arbitrary units

depending on the particular utility representation we happen to use for the consumer’s preferences. For

many purposes it is convenient to obtain a monetary measure of welfare. Such a measure allows us to

calculate the fixed-dollar (lump-sum) tax or subsidy we would have to impose on a consumer in order to

achieve a particular welfare goal. The most pressing need for such calculations is in evaluating the welfare

consequences of a change in the price of one good, so we focus here on calculating monetary measures of

such price changes. For notational ease we suppress the subscripts and other prices, and just denote the

price of the good in question by p, its Marshallian demand by x∗(p,m), its Hicksian demand by h∗(p, U),

the indirect utility function by U∗(p,m), and the expenditure function by e∗(p, U).

Consider Figure 16.1, which illustrates an increase in the price of a normal good from p0 to p1. Quantity

demanded drops from x∗(p0, m) to x∗(p1, m) and indirect utility drops from U∗(p0, m) to U∗(p1, m). Thus,

in utility terms, the damage to welfare from the price increase is U∗(p1, m) − U∗(p0, m) < 0. One way to

place this change in utility into monetary units is to ask

“By how much would we have to reduce the consumer’s income, rather than increasing the price,

in order to impose the same welfare loss as the price increase?”

On the graph, the lower utilityU∗(p1, m) places the consumer on the lower Hicksian demand h∗(p, U∗(p1, m)).

Utility is constant at the new lower level all along this Hicksian demand, so our new income level must shift

the Marshallian demand at the old price p0 left to this lower Hicksian demand. The income level that

accomplishes this is e∗(p0, U∗(p1, m)), so the reduction in consumer’s income that is equivalent in welfare

terms to the price increase is e∗(p0, U∗(p1, m)) −m. Since m = e∗(p1, U∗(p1, m)), this change is

EV ≡ e∗(p0, U∗(p1, m)) − e∗(p1, U∗(p1, m))

=∫ p0

p1

∂e∗(p, U∗(p1, m))∂p

dp

= −∫ p1

p0h∗(p, U∗(p1, m)) dp.

132 LECTURE 16: CONSUMER WELFARE

This shows that the change in income that is equivalent in welfare terms to the price increase is the (negative)

area to the left of the Hicksian demand for the new utility level U∗(p1, m). Since this area gives a monetary

equivalent to the price increase, it is called the equivalent variation (EV) of the price increase.

Figure 16.1: Equivalent and Compensating Variations from a Price Increase. EVis the (negative) lightly shaded area. CV is both the lightly and darkly shaded areas.

Another way to place the change in utility from the price increase into monetary units is to ask

“By how much would we have to increase the consumer’s income, after the price increase, in order

to neutralize the welfare effect of the price increase?”

On the graph, this means we must give the consumer income that returns utility to U∗(p0, m) despite

the fact that the price has risen to p1, so we must place the consumer on the original Hicksian demand

h∗(p, U∗(p0, m)) at price p1. Thus our new income level must shift the Marshallian demand at p1 right to

the original Hicksian demand. The income level that accomplishes this is e∗(p1, U∗(p0, m)), so the increase

in consumer’s income that compensates in welfare terms for the price increase is e∗(p1, U∗(p0, m))−m. Since

m = e∗(p0, U∗(p0, m)), this change is

CV ≡ e∗(p1, U∗(p0, m)) − e∗(p0, U∗(p0, m))

=∫ p1

p0

∂e∗(p, U∗(p0, m))∂p

dp

=∫ p1

p0h∗(p, U∗(p0, m)) dp.


This shows that the change in income that compensates in welfare terms for the price increase is the area

to the left of the Hicksian demand for the original utility level U∗(p0, m). Since this area gives a monetary

compensation for the price increase, it is called the compensating variation (CV) of the price increase.

For a price increase, the CV is positive (we must give income to the consumer in order to “compensate”

for the price increase) while the EV is negative (we must take income from the consumer in order to generate

the same utility decrease as the price increase would generate). Also, in this case the CV is larger in absolute

value than the EV.

Both the equivalent and compensating variations give theoretically correct monetary comparisons of the

welfare levels at the two prices. They differ in the utility level they use as the basis for comparison. The

equivalent variation uses as a basis the utility level that would occur if the price change were imposed, while

the compensating variation uses as a basis the utility level that occurs with no price change. The former

corresponds to using the new price as the base and the latter corresponds to using the original price as the

base. There are, in principle, many other monetary measures that could be proposed, each using a different

price for the base and therefore a different base utility level. EV and CV simply use the most natural bases,

since they are the bases actually under consideration.

Notice, however, that the area to the left of the Marshallian demand does not generally give a theoretically

correct monetary comparison of the two welfare levels. This is because welfare varies along the Marshallian

demand as the consumer continuously shifts across Hicksian demands, each one corresponding to a different

utility level. In other words, the welfare basis is constantly changing as we move along a Marshallian

demand. This is unfortunate, since the Marshallian demand is the observable component here. It would

be very convenient if we could simply examine the area to the left of a Marshallian demand to obtain a

monetary measure of a welfare change. Indeed, there is a long tradition in economics of studying consumer’s

surplus, which is usually defined to be exactly the area to the left of a Marshallian demand.

The fact that consumer’s surplus measures the wrong area is not an insurmountable problem, as there are

several approaches to overcoming this. First, we can simply note that consumer’s surplus is bounded above

and below by the EV and CV. Thus, consumer’s surplus is between two theoretically correct measures. It

is possible to state bounds on the size of the error committed when consumer’s surplus is used in lieu of


either the EV or CV, and thus to state upper and lower bounds on a theoretically correct monetary measure.

Second, we can use the relationships between utility maximization and expenditure minimization discussed

in the last lecture to derive the EV and CV from a known Marshallian demand. To do this, we must integrate

the Marshallian demand to obtain the expenditure function, and then find the difference between the two

relevant expenditure levels. Third, we can identify a condition under which consumer’s surplus coincides

with the EV or CV, or is at least very close to these correct measures. Then, assuming this condition holds,

consumer’s surplus is indeed a theoretically correct measure. Since the entire problem here arises because

the Marshallian and Hicksian demands do not coincide, the needed condition is that the source of difference

between Marshallian and Hicksian demands be zero. From the Slutsky Decomposition, we know that the

difference between these two demands is the income effect. The two demands coincide at the original utility

level and price, and the Slutsky Decomposition tells us that they have the same slope as we move away

from this point whenever the income effect is zero. So the two demands must coincide over the entire range

of prices at which the income effect is zero. Thus, if there is no income effect on the good under study

over the range of prices and income under study, then consumer’s surplus is an exact monetary measure of

the welfare consequences of a price change. Moreover, if the income effect is “small” over this range, then

consumer’s surplus is an approximate monetary measure of the welfare change. The preference structure

that results in a zero income effect is the quasilinear structure U(x) = xi + g(x−i) for some commodity xi

and some function g. That is, if the utility function can be factored into two terms, one of which is linear in

one commodity (the ith), then the income effect on all other commodities is zero (assuming that the optimal

choice of xi is positive).

These area-based measures of welfare lead to area-based measures of the so-called “deadweight loss”

associated with a price change. Consider Figure 16.2. If we forego the price increase and instead impose an

equivalent variation on the consumer, we take from the consumer area EV on the graph. On the other hand,

if we impose the price increase then consumption drops from the original level to x∗(p1, m), and so we take

from the consumer only the extra revenue from these sales at the higher price p1. That is, our take from the

consumer is just (p1 − p0)x∗(p1, m). Because the Hicksian demand is downward-sloping, our revenue from

the consumer is necessarily larger when we impose an equivalent variation than when we simply raise the


price, even though the consumer is left with the same utility in either case. In other words, the EV is a more

efficient way for us to impose the utility decrease on the consumer. The difference between the two areas is

the dark shaded triangle on the graph, and this triangle is called the deadweight loss of the price increase.

Figure 16.2: Deadweight Loss from a Price Increase based on the EV. EV is boththe (negative) light and dark shaded areas. Revenue from the price increase is only the lightshaded area. Hence the deadweight loss of the price increase is the dark shaded area.

The deadweight loss can be illustrated in terms of the CV as well, as in Figure 16.3. If we impose

the price increase and then compensate the consumer for the price increase, consumption still drops to

x∗(p1, e∗(p1, U∗(p0, m))) due to the negative slope of the Hicksian demand. We collect extra revenue from

the consumer of (p1 − p0)x∗((p1, e∗(p1, U∗(p0, m))), but the compensation to the consumer costs the area

CV. So we have to pay a net amount equal to the difference between the two areas, illustrated as the dark

shaded triangle on the graph, even though the consumer’s utility is unchanged. This dark shaded area is

once again the deadweight loss of the price increase.

These deadweight losses may or may not be offset by other savings, for example cost savings from lower

production. If such savings exceed the deadweight loss then the price increase may be beneficial overall.

The welfare analysis of a price decrease is not substantially different. But, since utility increases with a

price decrease, and since EV is based on the new utility level while CV is based on the original utility level,

the two measures switch positions on the graph when the price change is a decrease rather than an increase.


Figure 16.3: Deadweight Loss from a Price Increase Based on the CV. CV is boththe light and dark shaded areas. Revenue from the price increase is only the light shaded area.Hence the deadweight loss of the price increase is the dark shaded area.

In this case EV is larger in absolute value than CV, and CV is negative (we must take income from the

consumer in order to “uncompensate” for the price decrease) while EV is positive (we must give income to

the consumer in order to generate the same utility increase as the price decrease would generate).

The welfare analysis of an inferior good is also not substantially different. For a price increase, the EV is

still negative and the CV is still positive, but when the good is inferior the EV is larger in absolute value

than the CV.


Exercises for Lecture 16 (Consumer Welfare)

1. Carefully graph the EV and CV for a price decrease of a normal good. Place the Marshallian demands

after both variations on your graph. Illustrate the deadweight losses based on both the EV and CV.

2. Carefully graph the EV and CV for a price increase of an inferior good. Place the Marshallian demands

after both variations on your graph. Illustrate the deadweight losses based on both the EV and CV.


3. MWG #3.D.4.

4. Varian #10.2.

5. MWG #3.I.2.


Lecture 17: Duality of Expenditure, Indirect Utility, and Marshallian and Hicksian Demands

Readings: Sections 8.5 and 8.6 of Varian; Section 3.H of MWG.

Our preceeding discussions of utility maximization and expenditure minimization presume that the func-

tions U∗, x∗, e∗, and h∗ arise from some optimization problem with a well-behaved underlying preference

relation. That is, all of the properties we have derived are necessary for their respective functions. We want

to know whether these properties are sufficient as well, so that we can regard any function with these prop-

erties as an indirect utility, Marshallian demand, expenditure, or Hicksian demand function, respectively,

even if we never derived it from some underlying optimization.

Suppose we begin with a real-valued function e∗(p, U) of a strictly positive n-dimensional vector p and a

scalar U in some interval Y from some U(0) (inclusive) to some y (perhaps inclusive, perhaps not); and that

this function satisfies the homogeneity, concavity, monotonicity, continuity, and nonnegativity properties of

an expenditure function (but that it may not be an expenditure function–at least, we do now know this in

advance). For convenience we will also assume that e∗ is continuously differentiable, although this assumption

is not really needed. This is the same list of properties that a cost function possesses (provided we regard

U(0) as output level 0), and we know that these properties are sufficient for a cost function. So for U ∈ Y ,

as we have already shown,

H(U) ≡ {x ∈ Rn+ : p · x ≥ e∗(p, U) ∀p >> 0}

is a well-behaved “input requirement set,” possessing the following properties: {(x, U) : x ∈ H(U)} is

nonempty and closed; there is no free lunch (an inability to “produce” “output” levels greater than U(0) with

x = 0, i.e., 0 /∈ H(U) for U > U(0)), possibility of inaction (that U(0) can be “produced” from x = 0, i.e.,

0 ∈ H(U(0))), free disposal, and convexity of H(U); and lower hemicontinuity in “inputs” x and “output”

U .

If e∗ indeed came from minimization of p · x subject to U(x) ≥ U for some “production function” U , then

H(U) ⊃ {x ∈ R+n : U(x) ≥ U} just as in the production context. Unlike the production context, however,

convexity of the preference relation does not imply H(U) ⊂ {x ∈ R+n : U(x) ≥ U} because we have not

assumed monotonicity of �∼, which is the preference counterpart to free disposal. Figure 17.1 shows why a

lack of monotonicity allows {x ∈ R+n : U(x) ≥ U} to be a proper subset of H(U) even when �

∼is convex.

140 LECTURE 17: CONSUMER DUALITY

Formally, the application of the separating hyperplane theorem that we used in the production context can

fail here because we used free disposal to assure that the hyperplane had a strictly positive slope vector.

Without this, we do not know that the slope vector is one of the price vectors used in the definition of

H and so there can be points in H(U) that are not in {x ∈ Rn+ : U(x) ≥ U}. However, as illustrated in

Figure 16.1, if there are extra bundles included in H such bundles are economically irrelevant in that no

maximizing consumer would ever choose them at positive prices. Most importantly, the proof from the

production context that

e∗(p, U) = min{x}

p · x subject to x ∈ H(U)

in no way relies upon H recovering the true underlying input requirement sets. Rather, all that matters is

that H recovers economically relevant input requirement sets. Hence e∗ is indeed the minimum cost for the

technology H , irrespective of whether we know in advance that e∗ is a minimum over {x ∈ Rn+ : U(x) ≥ U}

for some function U . That is, H(U) are the “input requirement sets” underlying the “cost function” e∗, and

therefore a “production function” U(x) underlying e∗ is defined by U(x) ≡ max{U : x ∈ H(U)}. Since H(U)

has the properties of an input requirement set, this derived U has the properties of a production function

(including continuity, since H(U) satisfies lower hemicontinuity in inputs), except that U(0) is the value of

U where the original e∗ is zero, rather than being zero.

Now suppose we just change our interpretation of the symbols and regard the sets H(U) as preferred-to

sets and U(x) as a utility function for some preference relation �∼. That is, define a preference relation �

∼on

Rn+ as follows: For any x, y ∈ R

n+, x �

∼y if and only if U(x) ≥ U(y). This preference relation is complete,

transitive, and continuous on Rn+ since it is represented by a continuous function U on R

n+. And since U is

the production function for H , for U ∈ Y we have e∗(p, U) = min{x} p·x subject to U(x) ≥ U , so e∗ is indeed

the expenditure function for the utility function U (i.e., the preference relation �∼). That is, the properties of

an expenditure function are sufficient for e∗ to be an expenditure function for some well-behaved underlying

preference relation on Rn+.

This discussion indicates exactly how to proceed when we begin with an n-dimensional vector-valued

function h∗(p, U) of a strictly positive n-dimensional vector p and a scalar U in some interval Y ; that

satisfies the homogeneity, symmetry/semidefiniteness, monotonicity, continuity, and nonnegativity properties


Figure 17.1: Failure of H(U) to Recover the True Preferred-to Set when the

Preference Relation is Convex but not Monotonic. The true preferred-to set forutility level U is the lightly shaded area. H(U) includes the dark shading as well, because H isonly defined in terms of strictly positive prices. The true preferred-to set is convex, but omits thedark shaded areas because �

∼is not monotonic.

of a Hicksian demand (but may not be a Hicksian demand–at least, we do not know this in advance). Again

we assume differentiability for convenience only. Since this is the same list of properties that are sufficient for

a conditional factor demand (provided we again regard U(0) as output level 0), they are sufficient to ensure

that the derived function e∗(p, U) ≡ p · h∗(p, U) is the expenditure function for the underlying preferred-to

sets H(U) and that h∗ is the Hicksian demand corresponding to this expenditure function. We simply note

that U(0) is output level 0 and proceed as above. So h∗ is indeed a Hicksian demand for the utility function

U above, and we see that the properties of a Hicksian demand are sufficient for h∗ to be a Hicksian demand

for some well-behaved underlying preference relation on Rn+.

Now we can proceed to establish sufficiency of the properties of the indirect utility function. Start with

a real-valued function U∗(p,m) of a strictly positive n-dimensional vector p and a nonnegative scalar m;

that satisfies the homogeneity, quasiconvexity, monotonicity, and continuity properties of an indirect utility

function (but that is not known to be an indirect utility function). Since U∗(p,R1+) = Y ∀p >> 0, we can

implicitly define a proposed expenditure function e∗ according to identity 4 from Lecture 14. That is, for


U ∈ Y , define e∗ implicitly by

U∗(p, e∗(p, U)) ≡ U .

This defines a real-valued function e∗ provided the monotonicity property in m of the original U∗ is strict. If

we can verify that this defined e∗ possesses the properties of an expenditure function, then we know from the

sufficiency discussion above that it is indeed an expenditure function for some underlying preference relation

�∼. And if so, then U∗ must be the corresponding indirect utility function since U∗ is just the inverse of our

defined e∗. However, in verifying that e∗ has the sufficient properties of an expenditure function we have

only the properties of U∗ to work with, since the initial U∗ function is not known in advance to be an indirect

utility function. These properties of U∗ are indeed sufficient for e∗ to possess the following properties:

1. (Homogeneity) U∗(λp, e∗(λp, U)) = U by definition of e∗, while U∗(λp, λe∗(p, U)) = U by homogeneity

of U∗. Hence U∗(λp, e∗(λp, U)) = U∗(λp, λe∗(p, U)). Since U∗ is strictly increasing in m, this implies

e∗(λp, U) = λe∗(p, U).

2. (Concavity) Fix p, p′, and U . Since U∗ is quasiconvex,

U∗(λp+ (1− λ)p′, λe∗(p, U) + (1 − λ)e∗(p′, U)) ≤ max{U∗(p, e∗(p, U)), U∗(p′, e∗(p′, U))} for λ ∈ (0, 1).

By definition of e∗,

max{U∗(p, e∗(p, U)), U∗(p′, e∗(p′, U))} = U = U∗(λp+ (1− λ)p′, e∗(λp + (1− λ)p′, U)).

Hence

U∗(λp+ (1− λ)p′, λe∗(p, U) + (1− λ)e∗(p′, U)) ≤ U∗(λp+ (1− λ)p′, e∗(λp + (1− λ)p′, U)).

By strict monotonicity of U∗ in m,

λe∗(p, U) + (1− λ)e∗(p′, U) ≤ e∗(λp+ (1− λ)p′, U) for λ ∈ (0, 1).

3. (Monotonicity) Let p ≥ p′ and U > U ′. By definition of e∗,

U∗(p, e∗(p, U)) = U > U ′ = U∗(p, e∗(p, U ′)).


Since U∗ is strictly increasing in m, this implies e∗(p, U) > e∗(p, U ′). Similarly, using monotonicity of U∗

in p,

U∗(p, e∗(p, U)) = U = U∗(p′, e∗(p′, U)) ≥ U∗(p, e∗(p′, U)).

So, again using strict monotonicity of U∗ in m, e∗(p, U) ≥ e∗(p′, U).

4. (Continuity) This is immediate from continuity of U∗ and strict monotonicity of U∗ in m.

5. (Nonnegativity) By continuity and strict monotonicity in m of U∗, U∗(p, 0) = U(0) ∀p >> 0. Thus, by

definition U∗(p, e∗(p, U(0))) = U(0) = U∗(p, 0). So strict monotonicity of U∗ in m implies e∗(p, U(0)) = 0.

Moreover, for U ∈ Y − {U(0)} we have U∗(p, e∗(p, U)) = U > U(0) = U∗(p, e∗(p, U(0))) = U∗(p, 0). So

strict monotonicity of U∗ in m again implies e∗(p, U) > 0 for U > U(0).

Since e∗ satisfies the sufficient conditions for an expenditure function, it is indeed an expenditure function.

Inverting this e∗ yields the corresponding indirect utility function, which is precisely the U∗ function we

started with, since e∗ was obtained by just inverting U∗. Thus the properties of an indirect utility function

are indeed sufficient for U∗ to be an indirect utility function for some well-behaved underlying preference

relation on Rn+.

Finally, suppose x∗(p,m) is an n-dimensional vector-valued function of a strictly positive vector p and a

nonnegative scalar m that satisfies the homogeneity, symmetry/semidefiniteness, nonnegativity, and Walras’

Law properties of a Marshallian demand, but is not known to be a Marshallian demand. Can we show

that x∗ is indeed a Marshallian demand for some underlying preference relation? Hurwicz and Uzawa (“On

the Integrability of Demand Functions,” in Preferences, Utility, and Demand, edited by Chipman, Hurwicz,

Richter, and Sonnenschein, New York: Harcourt Brace Jovanovich, 1971) prove the following:

Lemma (Hurwicz and Uzawa, Lemma 1 (p. 124)). Suppose x∗(p,m) is an n-dimensional vector-

valued function defined for p ∈ Rn++ and m ∈ R

1+, satisfying:

1. x∗ is differentiable on Rn++ × R

1+ (one-sided when m = 0).

2.∂x∗

i (p,m)∂m is bounded on A× R

n+, where A is any compact subset of R

n++.

3. The matrix of derivatives∂x∗

i (p,m)∂pj

+ x∗j (p,m)∂x

∗i (p,m)∂m

for i, j = 1, . . . , n is symmetric at every (p,m) ∈

Rn++ × R

1+.


Then, for each (p0, m0) ∈ Rn++ × R

1+ there exists a unique function g(p; p0, m0) such that

x∗i (p, g(p; p

0, m0)) =∂g(p; p0, m0)

∂pifor i = 1, . . . , n; ∀p >> 0, and

g(p0; p0, m0) = m0.

NOTE: This result is a variant of Frobenius’ Theorem for this particular set of partial differential equations.

The variant here gives a unique global solution to the system of partial differential equations. The condition

g(p0; p0, m0) = m0 is the “initial condition” for the system.

Since condition 3 of this lemma is symmetry of our Slutsky matrix, if we just add to the list of properties

of x∗ the assumptions that x∗ is differentiable, and that its income derivatives do not explode, then the

function x∗ satisfies the conditions of Hurwicz and Uzawa’s lemma.

Hurwicz and Uzawa also establish an important property of the solution g:

Lemma (Hurwicz and Uzawa, Lemma 3 (p. 125)). Let (p0, m0), (p1, m1) ∈ Rn++ × R

1+ be two initial

conditions. If g(p2; p0, m0) < g(p2; p1, m1) for some p2 >> 0, then

g(p; p0, m0) < g(p; p1, m1) ∀p >> 0.

Proof. Suppose not, so there exists p3 >> 0 such that g(p3; p0, m0) ≥ g(p3; p1, m1). Then, by continuity of

g (follows from differentiability), there exists p4 between p2 and p3 such that

g(p4; p0, m0) = g(p4; p1, m1).

Since g is a unique solution to the system of partial differential equations (Lemma 1), we then have

g(p; p0, m0) = g(p; p1, m1) for all p >> 0 (i.e., the two solutions g(p; p0, m0) and g(p; p1, m1) to the

system are equal at a point p4, and their values as we depart from that point are determined by the

derivatives, which are the same at all points, so the “two” solutions must be the same). This contradicts

g(p2; p0, m0) < g(p2; p1, m1). �

Now assume x∗ is differentiable and has bounded income derivatives. Suppose we set U = m0 (actually,

we only need a strictly increasing (1-1) mapping U(m0) of income values m0 into utility values U) and define

a hypothetical expenditure function by e∗(p, U ; p0) = g(p; p0, U). Then e∗ is indeed an expenditure function

for any arbitrary p0 >> 0 (suppressed below for convenience) because it satisfies the sufficient conditions:


1. (Homogeneity) Walras’ Law says p · x∗(p, e∗(p, U)) = e∗(p, U). Substituting from the partial differential

equations that define e∗, this isn∑i=1

pi∂e∗(p, U)

∂pi= e∗(p, U).

But this is Euler’s equation for a function e∗ that is homogeneous of degree one in p. Since Euler’s

equation is both necessary and sufficient for homogeneity, e∗ is homogeneous of degree 1.

2. (Concavity) The Hessian matrix of e∗ with respect to p is just the Slutsky matrix, which is negative

semidefinite by the semidefiniteness property of x∗.

3. (Monotonicity) By nonnegativity of x∗, ∂e∗∂pi

= x∗i ≥ 0. For strict monotonicity in U , let U0 < U1 and

note that, from the initial condition defining g,

e∗(p0, U0) = g(p0; p0, U0) = U0 < U1 = g(p0; p0, U1) = e∗(p0, U1).

By Hurwicz and Uzawa’s Lemma 3, this holds at every p >> 0. That is, e∗(p, U0) < e∗(p, U1), or e∗ is

strictly increasing in U .

4. (Continuity) By definition, e∗(p, U) = g(p; p0, U), and g(p; p0, m0) is continuous in (p,m0) by its definition

in Hurwicz and Uzawa’s Lemma 1. Therefore e∗ is continuous.

5. (Nonnegativity) First we show e∗(p, U) ≥ 0 ∀p >> 0 and ∀U ≥ 0 (or, for a different choice of U(m0), for

all U ≥ U(0)). Suppose otherwise. Then there exists p >> 0 and U ≥ 0 such that e∗(p, U) < 0. Select

λ > 0 such that λp > p0. Then, from the definition of e∗ and the initial condition defining g, and using

the monotonicity and homogeneity of e∗, we have

0 ≤ U = e∗(p0, U) ≤ e∗(λp, U) = λe∗(p, U) < 0,

a contradiction. Using this, we now show that e∗(p, 0) = 0 ∀p >> 0, so that U(0) = 0 (or, for a different

choice of U(m0), U(0) = U(0)). Suppose otherwise. Then, since e∗ ≥ 0, there exists p1 >> 0 such that

e∗(p1, 0) = g(p1; p0, 0) > 0. From the initial condition defining g, g(p1; p1, 0) = 0 < g(p1; p0, 0), so by

Hurwicz and Uzawa’s Lemma 3 we have g(p; p1, 0) < g(p; p0, 0) for every p >> 0. But, again using the

initial condition, this implies g(p0; p1, 0) < g(p0; p0, 0) = 0, a contradiction to e∗ ≥ 0. Finally, since e∗ is

strictly increasing in U , e∗(p, 0) = 0 ∀p >> 0 immediately implies e∗(p, U) > 0 for p >> 0 and U > 0

(i.e., Y = R1+ for this choice of U).


Since e∗ is an expenditure function, we know that there is an underlying preference relation such that

e∗(p, U) is the minimum cost of obtaining utility U , that the indirect utility U∗ corresponding to e∗ is

just the inverse of e∗, that the corresponding Hicksian demands are h∗i (p, U) = ∂e∗(p,U)

∂pi, and that the

corresponding Marshallian demands are h∗i (p, U

∗(p,m)) (from identity 1 of the Lecture 14). But, from the

definition of e∗ and identity 2 of Lecture 14, these Marshallian demands are

∂e∗(p, U)∂pi

∣∣∣∣U=U∗(p,m)

= x∗i (p, e

∗(p, U∗(p,m))) = x∗i (p,m),

which are the original x∗i functions that we began with. That is, x∗ is indeed a Marshallian demand for

some underlying preference relation. The properties of a Marshallian demand, along with assumptions 1 and

2 of Hurwicz and Uzawa’s Lemma 1, are sufficient for x∗ to be a Marshallian demand for some well-behaved

underlying preference relation on Rn+.


Exercises for Lecture 17 (Consumer Duality)


1. Varian #9.8.

2. MWG #3.G.10.

3. MWG #3.G.11.


SOLUTIONS TO EXERCISES


Solutions to Exercises for Lecture 1 (Introduction)

1. Envelope Theorem: The derivative of an optimized function when a parameter changes equals the deriva-

tive of the corresponding Lagrangian function with respect to the parameter, evaluated at the maximum.

152 SOLUTIONS TO EXERCISES FOR LECTURE 2 (TECHNOLOGIES)

Solutions to Exercises for Lecture 2 (Technologies)

1.(i) (ii) (iii) (iv)

(a) no yes no yes(b) yes yes yes no(c) yes yes yes yes(d) no yes yes no(e) yes yes yes yes(f) yes no yes yes

2.

A “well-behaved” single-output production set whose production function can

take on infinite values

The boundary of Z has a vertical asymptote at z′1, so f(z1) = max{z2 : (z1 , z2) ∈ Z} = ∞ for any z1 ≤ z′1.

Note that Z is not convex in this figure. If Z is convex, an infinite value for f(z1) cannot occur. To prove

this claim formally, suppose otherwise. That is, suppose Z is convex but there is a value of z1, as in the

figure, for which (z1, z2) ∈ Z for z2 arbitrarily large. Then there is a sequence zi2 such that zi2 > 1 ∀i,

limi→∞ zi2 = ∞, and (z1 , zi2) ∈ Z ∀i. By convexity and the possibility of inaction,

α(z1, zi2) + (1− α)(0, 0) ∈ Z ∀α ∈ [0, 1].

Now set αi = 1/zi2 (αi ∈ (0, 1) since zi2 > 1), so that

αi(z1, zi2) + (1− αi)(0, 0) = (z1/z

i2, 1) ∈ Z.

Letting i → ∞, we get (0, 1) ∈ Z since Z is a closed set. But this contradicts the no free lunch property.


3.

i. f(0) = max{y : (y, 0) ∈ Z} by definition. By possibility of inaction, (0, 0) ∈ Z. Hence f(0) ≥ 0. By no

free lunch, f(0) ≤ 0. Combining, f(0) = 0.

ii. Let x ≥ x′. By free disposal, if (y,−x′) ∈ Z then (y,−x) ∈ Z too. Hence f(x) ≥ f(x′).

iii. Fix x and consider any y such that (y,−x) ∈ Z. Then (αy,−αx) ∈ Z for α ∈ [0, 1]. So, by definition

of f , f(αx) ≥ αy. This holds for any y such that (y,−x) ∈ Z. Hence f(αx) is an upper bound for

{αy : (y,−x) ∈ Z}, implying f(αx) ≥ αf(x).

iv. The proof is identical to part iii., except for a different specification of α.

v. Same as part iii.

vi. Necessity: Fix y and consider any x, x′ ∈ V (y). By quasiconcavity, f(αx+(1−α)x′) ≥ min{f(x), f(x′)}.

Since x, x′ ∈ V (y), f(x) ≥ y and f(x′) ≥ y. Thus f(αx+ (1−α)x′) ≥ y. By free disposal, this implies

αx+ (1− α)x′ ∈ V (y), so V (y) is convex.

Sufficiency: Fix any x and x′. Set y = min{f(x), f(x′)} so that f(x) ≥ y and f(x′) ≥ y. By free

disposal, y′ ≤ y ⇒ V (y) ⊂ V (y′), so y ≤ f(x′) ⇒ V (f(x′)) ⊂ V (y), and thus x′ ∈ V (y). This same

argument yields x ∈ V (y). Then by convexity αx+ (1 − α)x′ ∈ V (y). Finally, by the definition of f ,

f(αx+ (1− α)x′) ≥ y, so f is quasiconcave.

vii. a. We must show that {x : f(x) ≥ y} is a closed set for arbitrary y. Let xi be a sequence that converges

to x0, satisfying f(xi) ≥ y. We must show that f(x0) ≥ y. By free disposal,

{x : f(x) ≥ y} = V (y).

So, xi ∈ V (y) for every i. By closedness of Z, V (y) is closed. Hence x0 ∈ V (y), so f(x0) ≥ y.

b. We must show that {x : f(x) ≤ y} is a closed set for arbitrary y. Let xi be a sequence that

converges to x0, satisfying f(xi) ≤ y. We must show f(x0) ≤ y. Since f(xi) is a bounded sequence,

there exists a convergent subsequence f(xki

) → f0 ≤ y. We know (f(xki

),−xki

) ∈ Z ∀i. Since Z is

closed, this implies (f0 ,−x0) ∈ Z. Thus f(x0) ≥ f0, from the definition of f , and free disposal then

gives (f(x0),−x0) ∈ Z.

Now suppose f(x0) > f0 , and set ε = f(x0) − f0 > 0. Since (f(x0),−x0) ∈ Z and xki → x0,

by lower hemicontinuity of Z in output there exists yki → f(x0) such that (yk

i

,−xki

) ∈ Z. By


convergence, for i large we have yki

> f(x0) − ε2 and, since (yk

i

,−xki

) ∈ Z, we have f(xki

) ≥

yki

> f(x0) − ε2 = 1

2(f(x0) + f0) for i large. But since f(xk

i

) → f0 , for i large we must also have

f(xki

) < f0 = ε2 + 1

2 (f(x0) + f0), a contradiction. Hence f(x0) = f0 ≤ y.

4.

a.

Z = {(y, x1, x2) : y ≤ A(−x1)α1(−x2)α2 and x1, x2 ≤ 0}.

V (y) = {(x1, x2) : y ≤ Axα11 xα2

2 and x1, x2 ≥ 0}.

The isoquant for y is {(x1, x2) : y = Axα11 xα2

2 and x1, x2 ≥ 0}.

The MRTS at (x1, x2) is −f1f2

= −Aα1xα1−11 x

α22

Aα2xα11 x

α2−12

= −α1x2α2x1

.

All isoquants are asymptotic to both axes, except for the y = 0 isoquant (which is both axes). Higher values

of y produce isoquants to the northeast of the one illustrated:

Typical isoquant for Cobb-Douglas technology

b.

Z = {(y, x1, x2) : y ≤ min{a1(−x1), a2(−x2)} and x1, x2 ≤ 0}.

V (y) = {(x1, x2) : y ≤ min{a1x1, a2x2} and x1, x2 ≥ 0}.

The isoquant for y is {(x1, x2) : y = min{a1x1, a2x2} and x1, x2 ≥ 0}.

The MRTS at (x1, x2) is −∞ if x2 > a1a2

x1 and is 0 if x2 < a1a2

x1. The MRTS is undefined if x2 = a1a2

x1.


All isoquants are L-shaped, with the vertex along the line through the origin with slope a1/a2. Higher values

of y produce isoquants to the northeast of the one illustrated:

Typical isoquant for Leontief technology

c.

Z = {(y, x1, x2) : y ≤ a1(−x1) + a2(−x2) and x1, x2 ≤ 0}.

V (y) = {(x1, x2) : y ≤ a1x1 + a2x2 and x1, x2 ≥ 0}.

The isoquant for y is {(x1, x2) : y = a1x1 + a2x2 and x1, x2 ≥ 0}.

The MRTS at (x1, x2) is −a1a2.

All isoquants are straight lines with slope −a1/a2. Higher values of y produce isoquants to the northeast of

the one illustrated:


Typical isoquant for perfect substitutes technology

d.

Z = {(y, x1, x2) : y ≤ [a1(−x1)ρ + a2(−x2)ρ]ε/ρ and x1, x2 ≤ 0}.

V (y) = {(x1, x2) : y ≤ [a1xρ1 + a2x

ρ2]ε/ρ and x1, x2 ≥ 0}.

The isoquant for y is {(x1, x2) : y = [a1xρ1 + a2x

ρ2 ]ε/ρ and x1, x2 ≥ 0}.

The MRTS at (x1, x2) is

−f1

f2= −(ε/ρ)[a1x

ρ1 + a2x

ρ2]ε/ρa1ρx

ρ−11

(ε/ρ)[a1xρ1 + a2x

ρ2]ε/ρa2ρx

ρ−12

= −a1

a2

(x1

x2

)ρ−1

.

The shape of the isoquants depends on ρ. See Varian pp. 19-20 for a discussion.

5. Note that

f(αx1, αx2) = [a1(αx1)ρ + a2(αx2)ρ]ερ ] = αε[a1x

ρ1 + a2x

ρ2]

ερ ] = αεf(x1, x2).

a. We have constant returns to scale when the function is homogeneous of degree one, that is, when

f(αx1, αx2) = αf(x1, x2). Thus αε = α, and ε = 1.

b. We have nondecreasing returns to scale when the function is homogeneous of degree greater than one,

that is, when f(αx1, αx2) ≥ αf(x1, x2). Thus αε ≥ α, and ε ≥ 1.

c. We have nonincreasing returns to scale when the function is homogeneous of degree less than one, that

is, when f(αx1, αx2) ≤ αf(x1, x2). Thus αε ≤ α, and ε ≤ 1.


Solutions to Exercises for Lecture 3 (Profit Maximization: Basics)

1. This is an IRTS Cobb-Douglas function. So, −MRTS = w1w2

⇒ x2x1

= w1w2

, so x2 = w1w2

x1.

π = pw1

w2x2

1 − 2w1x1.

Now let x1 → ∞, which implies that π → ∞. Hence x2 = w1w2

x1 and x2 → ∞.

2. (MWG #5.C.1)

Proof. Suppose not, so that 0 < π∗(p) < ∞. Then there exists a feasible netput z∗(p) such that π∗(p) =

p ·z∗(p). By nondecreasing returns to scale, αz∗(p) ∈ Z ∀α ≥ 1, since z∗(p) ∈ Z. Thus, profit of p ·(αz∗(p)) =

απ∗(p) ≥ π∗(p) can be obtained. Since π∗(p) > 0, limα→∞ απ∗(p) = ∞, contradicting that 0 < π∗(p) < ∞

can be optimal. �

3. (MWG #5.C.6)

a. Since y∗(p, w) = f(x∗(p, w)), we have

∂y∗

∂p=

n∑i=1

fi(x∗)∂x∗

i

∂p.

Differentiating the first order conditions pfi(x∗) −wi = 0 for i = 1, ..., n with respect to p yields

Hπ(x)

∂x∗

1∂p

...∂x∗

n

∂p

= −

f1...fn

.

Multiply both sides of this equation by the row vector[ ∂x∗

1∂p

· · · ∂x∗n

∂p

] ≡ a′ to get

a′Hπ(x)a = −n∑i=1

fi(x∗)∂x∗

i

∂p.

Hence ∂y∗

∂p= −a′Hπ(x)a > 0, since Hπ(x) is negative definite by the sufficient second order conditions.

b. Since 0 < ∂y∗∂p =

∑ni=1 fi(x

∗)∂x∗i

∂p and fi ≥ 0 ∀i, it follows that ∂x∗i

∂p > 0 for at least one i.

4. (MWG #5.C.9)

a. f(z) = (z1 + z2)12 . As seen in the figure below, the isoquants are linear. So, the max occurs at a corner

unless w1 = w2, in which case any point on the isoquant is optimal.

Assume w1w2

> 1. Then z∗1 = 0 and z∗2 = y2, so π = py−w2y2, ∂π

∂y = p− 2w2y, and therefore y∗ = p2w2

.

Similarly, y∗ = p2w1

if w1w2

< 1. If w1w2

= 1, then any point on the isoquant will do, so just use one of the

158 SOLUTIONS TO EXERCISES FOR LECTURE 3 (PROFIT MAXIMIZATION: BASICS)

endpoints to calculate y∗. In general,

y∗(p, w1, w2) =

{p

2w2if w1

w2≥ 1

p2w1

if w1w2

< 1

π∗(p, w1, w2) =

{p2

4w2if w1

w2≥ 1

p2

4w1if w1

w2< 1.

Isoquant for output level y (Perfect substitutes technology)

b. f(z) = [min(z1 , z2)]12 . As seen in the figure above, the isoquants are L-shaped. So, the maximum

occurs at z∗1 = z∗2 = y2 , irrespective of what the input prices are. Thus, π = py − (w1 + w2)y2 , ∂π∂y =

p− 2(w1 +w2)y, and therefore

y∗ =p

2(w1 + w2)

π∗ =p2

2(w1 + w2)− (w1 +w2)

p2

4(w1 + w2)2=

p2

4(w1 +w2).

c. f(z) = (zρ1 + zρ2 )1ρ , which is constant returns CES. Since this is constant returns to scale, either π∗ = 0

or π∗ = ∞. For a given output level y, the firm sets -MRTS = w1w2

:(z1

z2

)ρ−1

=w1

w2.

So, for this given y = (zρ1 + zρ2)1ρ we can substitute to get zρ1 =

(w1w2

) ρρ−1

zρ2 and y =[(

w1w2

) ρρ−1

+ 1] 1

ρ

z2.

Thus, z2 = y

[(w1w2

) ρρ−1

+ 1]− 1

ρ

and similarly z1 = y

[(w2w1

) ρρ−1

+ 1]− 1

ρ

. However, these are only


Isoquant for output level y (Leontief technology)

solutions for z1 and z2 conditional on y. The level of y that is optimal is completely arbitrary if π = 0

and is infinite (giving z∗1 = z∗2 = ∞ too) if π > 0 is possible. Substituting for z1 and z2 in the profit

expression yields

π = py − y

w1

[(w2

w1

) ρρ−1

+ 1

]− 1ρ

+w2

[(w1

w2

) ρρ−1

+ 1

]− 1ρ

.

Setting this to zero yields

p = w1

[(w2

w1

) ρρ−1

+ 1

]− 1ρ

+ w2

[(w1

w2

) ρρ−1

+ 1

]− 1ρ

. (*)

If the prices are related in this way then π∗ = 0 and y∗ is anything. Otherwise, either y∗ = 0 and

π∗ = 0 or y∗ = ∞ and π∗ = ∞. Summary:

y∗(p, w1, w2) =

0 if p < (*)[0,∞) if p = (*)∞ if p > (*)

π∗(p, w1, w2) ={

0 if p ≤ (*)∞ if p > (*).

5. (Varian # 2.3) The FOC is paxa−1 = w, and the SOC is pa(a− 1)xa−2 < 0, which holds for x ≥ 0 since

0 < a < 1. The FOC yields x∗(p, w) =(wap

) 1a−1

. The supply function is given by

y∗(p, w) = f(x∗(p, w)) =(

w

ap

) aa−1

,

160 SOLUTIONS TO EXERCISES FOR LECTURE 3 (PROFIT MAXIMIZATION: BASICS)

and the profit function is given by

π∗(p, w) = py∗(p, w)− wx∗(p, w) = p

(w

ap

) aa−1

−w

(w

ap

) 1a−1

.

To check for homogeneity:

π∗(αp, αw) = αp

(αw

aαp

) aa−1

− αw

(αw

aαp

) 1a−1

= αp

(w

ap

) aa−1

− αw

(w

ap

) 1a−1

= απ∗(p, w).

To check for convexity: Rearrange π∗ so that π∗ = Ap1

1−a wa

a−1 , where A = aa

1−a − a1

1−a . Then

∂π∗

∂p= A

(1

1− a

)p

a1−a w

aa−1

∂π∗

∂w= A

(a

a − 1

)p

11−a w

1a−1 .

The Hessian matrix can now be written as:

Hπ∗(p,w) =

[∂2π∗(p,w)

∂p2∂2π∗(p,w)∂p∂w

∂2π∗(p,w)∂w∂p

∂2π∗(p,w)∂w2

]= A

[a

(1−a)2 p2a−11−a w

aa−1 − a

(1−a)2 pa

1−a w1

a−1

− a(1−a)2p

a1−a w

1a−1 a

(1−a)2 p1

1−a w2−aa−1

].

The naturally-ordered principal minors of this matrix are

Aa

(1− a)2p

2a−11−a w

aa−1 > 0, and 0.

Thus, the Hessian is a positive semidefinite matrix, which implies that π∗(p, w) is convex in (p, w).

6. (Varian # 2.7)

a. We want to maximize 20x− x2 −wx. The first-order condition is 20− 2x− w = 0.

b. For x∗ = 0, the derivative of profit with respect to x must be nonpositive at x = 0: 20 − 2x− w ≤ 0

when x = 0, or w ≥ 20.

c. The optimal x will be 10 when w = 0.

d. The factor demand function is x∗ = 10− w2 or, to be more precise, x∗ = max{10− w

2 , 0}.

e. Profits are 20x− x2 − wx = (20− w − x)x. Substitute x∗ = 10− w2 to get

π∗(w) =

{ (10− w

2

)2, w < 20

0, w ≥ 20.

f. The derivative of profit with respect to w is w2− 10 = −x∗ for w < 20.


Solutions to Exercises for Lecture 4 (Profit Maximization: Advanced)

1. We know z∗i (p, w) is homogeneous of degree 0 in (p, w). So, Euler’s Theorem says

n∑j=1

∂z∗i (p, w)∂wj

wj +∂z∗i (p, w)

∂pp = 0.

Divide this by z∗i to obtainn∑j=1

∂z∗i (p, w)∂wj

wjz∗i

+∂z∗i (p, w)

∂p

p

z∗i= 0.

Therefore,∑n

j=1 εij + εip = 0.

2. (Varian #3.1)

a. ∂2φi

∂w2i≥ 0 due to convexity; ∂φi

∂wi≤ 0 due to monotonicity.

b. −x∗i (w1, w2) = ∂φi

∂wi, so ∂x∗

i

∂wj= 0.

c. FOC’s are:

f1(x∗1, x

∗2) = w1

f2(x∗1, x

∗2) = w2

Differentiate these to get [f11 f12

f21 f22

] [ ∂x∗1

∂w2∂x∗

2∂w2

]=[01

].

So by Cramer’s Rule,

∂x∗1

∂w2=

(−1)f12∣∣∣∣ f11 f12

f21 f22

∣∣∣∣ = 0 from above.

Thus f12 = 0 at all optima. This implies f1 is independent of x2, so f(x1 , x2) must be an additively

separable form: f(x1 , x2) = g(x1) + h(x2).

3. (Varian #3.2)

maxx

pf(x)− wx ={

p lnx− wx if x > 1−wx if x ≤ 1

p 1x− w = 0 ⇒ x∗ = p

w, and −px−2 < 0, so the SOC holds. So, π∗(p, w) = p ln( p

w) − p = p

[ln( p

w)− 1

]when this is > 0:

π∗(p, w) ={ 0 p

w ≤ e

p[ln( p

w)− 1

]pw

> e.

162 SOLUTIONS TO EXERCISES FOR LECTURE 4 (PROFIT MAXIMIZATION: ADVANCED)

4. (Varian #3.5) If w >> 0, the firm will never use more of factor i than it needs to, which implies x1 = x2.

Hence the profit maximization problem can be written as

maxx

pxa1 − (w1 +w2)x1.

The first-order condition is

paxa−11 − (w1 +w2) = 0.

The factor demand function and the profit function are the same as if the production function were

f(x) = xa, but the factor price is w1+w2 rather than w. See Varian #2.3 from Lecture 3 for the solution.

In order for a maximum to exist, a < 1 is needed (or a = 1 with p = w1 + w2). Otherwise this is an IRS

technology and profit is infinite.


Solutions to Exercises for Lecture 5 (Cost Minimization: Basics)

1. Differentiating the FOC’s with respect to y yields:

HL(x,λ)

∂x∗

1∂y

...∂x∗

n

∂y∂λ∗∂y

=

0...0−1

So,

∂x∗i

∂y=

(−1)(−1)i+(n+1)|Hn+1,i||HL(x,λ)|

where Hn+1,i is the submatrix of HL(x,λ) obtained by deleting row n+ 1 and column i. Since i < n+ 1,

Hn+1,i is neither a principal submatrix nor border-preserving. Thus, ∂x∗i

∂ycannot be signed in general.

Similarly,

∂λ∗

∂y=

(−1)(−1)2(n+1)|Hn+1,n+1||HL(x,λ)| ,

which cannot be signed because Hn+1,n+1 is not border-preserving (it is a principal submatrix). Applying

this methodology to the expression derived in class for differentiation of the FOC’s with respect to wi,

∂λ∗

∂wi=

(1)(−1)i+(n+1)|Hi,n+1||HL(x,λ)| .

This suffers from the same problem as ∂x∗i

∂y .

2.

a. By definition,

py∗∗ −w · x∗∗ ≥ py − w · x ∀ (y,−x) ∈ Z.

So, if (y∗∗,−x) ∈ Z, then

py∗∗ − w · x∗∗ ≥ py∗∗ −w · x

⇒ w · x∗∗ ≤ w · x.

This holds ∀ x ∈ V (y∗∗). So w ·x∗∗ is a lower bound for w ·x on V (y∗∗). Hence, w ·x∗∗ ≤ w ·x∗(w, y∗∗).

However, by definition x∗∗ ∈ V (y∗∗), so w · x∗(w, y∗∗) ≤ w · x∗∗. Thus w · x∗ = w · x∗∗. Uniqueness

then implies x∗ = x∗∗.

164 SOLUTIONS TO EXERCISES FOR LECTURE 5 (COST MINIMIZATION: BASICS)

b. Differentiate with respect to wj :

∂x∗∗i

∂wj=

∂x∗i

∂wj+

∂x∗i

∂y

∂y∗∗

∂wj.

By symmetry of the substitution matrix,

∂y∗∗

∂wj= −∂x∗∗

j

∂p.

Now differentiate the identity with respect to p:

∂x∗∗j

∂p=

∂x∗j

∂y

∂y∗∗

∂p.

So,

∂x∗∗i

∂wj=

∂x∗i

∂wj− ∂x∗

i

∂y

∂x∗j

∂y

∂y∗∗

∂p. (*)

We know ∂y∗∗

∂p≥ 0. Complements means ∂x∗

i

∂wj≤ 0. So, if ∂x∗

i

∂yand ∂x∗

j

∂yhave the same sign (either both

normal or both inferior) then

∂x∗∗i

∂wj≤ 0 (Gross complements).

The converse fails: If we have ∂x∗∗i

∂wj≤ 0 and ∂x∗

i

∂y

∂x∗j

∂y ≥ 0 we cannot conclude that

∂x∗i

∂wj≤ 0.

Substitutes case: ∂x∗i

∂wj≥ 0, so from (*) if ∂x

∗i

∂y and ∂x∗j

∂y have opposite signs (one is normal and the other

is inferior) then

∂x∗∗i

∂wj≥ 0 (Gross Substitutes).

The converse fails: If we have ∂x∗∗i

∂wj≥ 0 and ∂x∗

i

∂y

∂x∗j

∂y ≤ 0 we cannot conclude that

∂x∗i

∂wj≥ 0.

Similarly, from (*) if inputs i and j are gross complements and ∂x∗i

∂y

∂x∗j

∂y≤ 0 then inputs i and j are

complements. If inputs i and j are gross substitutes and ∂x∗i

∂y

∂x∗j

∂y ≥ 0 then inputs i and j are substitutes.

3.

MRTS = −x2

x1= −w1

w2or x2 =

w1

w2x1.


x1x2 = y, substitute in x2, to get x1w1

w2x1 = y

⇒ x∗1 =

(yw2

w1

) 12

x∗2 =

(yw1

w2

) 12

c∗ = w1

(yw2

w1

) 12

+w2

(yw1

w2

) 12

= 2 (yw1w2)12

4. (MWG #5.C.10)

a. As shown in the figure below, when the isocost line has steeper than −1 slope we get an optimal choice

at z∗1 = 0 and z∗2 = y; when the isocost slope is flatter than −1 we get z∗1 = y and z∗2 = 0; when the

isocost slope = −1, any point on the isocost is optimal.

Isoquant for output level y is linear with slope −1 (perfect substitutes technol-

ogy)

Summary:

z∗1(w1, w2, y) =

0 if w1

w2> 1

y if w1w2

< 1

[0, y] if w1w2

= 1

z∗2(w1, w2, y) =

0 if w2

w1> 1

y if w2w1

< 1

y − z∗1 if w2w1

= 1

So,

c∗(w1, w2, y) = w1z∗1 +w2z

∗2 =

{w1y if w1

w2≤ 1

w2y if w1w2

> 1.


Isoquant for output level y (Leontief Technology)

b. Isoquant is L-shaped, so minimal (z1, z2) combination is always at the corner z1=z2 .

Summary:

z∗1(w1, w2, y) = y ∀w1, w2 >> 0

z∗2(w1, w2, y) = y ∀w1, w2 >> 0

c∗(w1, w2, y) = w1z∗1 + w2z

∗2 = (w1 + w2)y.

c. If ρ = 1, then it’s (a) and CES isn’t defined for ρ = 0. So, assume ρ < 1 and ρ �= 0. Then the

Lagrangian function is L = w1z1 +w2z2 − λ((zρ1 + zρ2)1ρ − y). FOC’s are:

w1 − λ

ρ(zρ1 + zρ2)

1ρ−1

ρzρ−11 = 0

w2 − λ

ρ(zρ1 + zρ2)

1ρ−1

ρzρ−12 = 0

Forming the ratio yields w1w2

=(z1z2

)ρ−1

or z1 = w1w2

1ρ−1 z2. Substitute into the constraint to get

[(w1

w2

) ρρ−1

zρ2 + zρ2

] 1ρ

= y, or z∗2 = y

[(w1

w2

) ρρ−1

+ 1

]− 1ρ

.

Similar substitution yields z∗1 = y

[(w2w1

) ρρ−1

+ 1]− 1

ρ

. Thus

c∗(w1, w2, y) = y

w1

[(w2

w1

) ρρ−1

+ 1

]− 1ρ

+w2

[(w1

w2

) ρρ−1

+ 1

]− 1ρ

.


5. (Varian #5.4) The lines y = 2x1+x2 and y = x1+2x2 cross at x1 = x2. Above the x2 = x1 line, 2x1+x2

is smaller. Below, x1 + 2x2 is smaller:


Typical isoquant for output level y

Cases:

w1

w2> 2 ⇒ x∗

2 = y and x∗1 = 0

w1

w2= 2 ⇒ x∗

2 ∈[y3, y]and x∗

1 =12(y − x∗

2)

2 >w1

w2>

12⇒ x∗

1 = x∗2 =

y

3

w1

w2=

12⇒ x∗

2 ∈[0,

y

3

]and x∗

1 = y − 2x∗2

12>

w1

w2⇒ x∗

2 = 0 and x∗1 = y.

So factor demands are:

x∗1 =

0 if w1w2

> 2[0, y3

]if w1

w2= 2

y3 if 2 > w1

w2> 1

2[y3 , y]

if w1w2

= 12

y if w1w2

< 12

x∗2 =

y if w1w2

> 2

y − 2x∗1 if w1

w2= 2

y3 if 2 > w1

w2> 1

2

12(y − x∗

1) if w1w2

= 12

0 if w1w2

< 12,

and the cost function is

c∗(w1, w2, y) = w1x∗1 + w2x

∗2.

6. (Varian # 5.17)

a. y = (ax1 + bx2)12


b.

Typical isoquant for output level y

So

x∗1 =

0 if w1

w2> a

b[0, y

2

a

]if w1

w2= a

b

y2

aif w1

w2< a

b

x∗2 =

y2

b if w1w2

> ab

1b

(y2 − ax∗

1

)if w1

w2= a

b

0 if w1w2

< ab

c.

c∗ =

{w2

y2

bif w1

w2≥ a

b

w1y2

a if w1w2

< ab

170 SOLUTIONS TO EXERCISES FOR LECTURE 6 (COST MINIMIZATION: ADVANCED)

Solutions to Exercises for Lecture 6 (Cost Minimization: Advanced)

1. We know x∗i (w, y) is homogeneous of degree zero in w. So, Euler’s Theorem says

n∑j=1

∂x∗i (w, y)∂wj

wj = 0.

Divide this by x∗i to obtain

n∑j=1

∂x∗i (w, y)∂wj

wjx∗i

= 0.

That is,∑n

j=1 εij = 0.

2.

a. HOD 1 in w:

c∗(αw1, αw2, y) = 2(y(αw1)(αw2))12 = α2(yw1w2)

12 = αc∗(w1, w2, y).

b. Shephard’s Lemma:

∂c∗

∂w1= (yw1w2)−

12 yw2 =

(yw2

w1

) 12

= x∗1.

∂c∗

∂w2= (yw1w2)−

12 yw1 =

(yw1

w2

) 12

= x∗2.

c. Concave in w:

Hc∗(w) =

[−1

2 (yw2)12 w

−32

112y

12 (w1w2)−

12

12y

12 (w1w2)−

12 −1

2 (yw1)12w

− 32

2

]

The diagonal elements are less than 0, and |Hc∗(w)| = 14

[y

w1w2− y

w1w2

]= 0. So, Hc∗(w) is negative

semidefinite ⇒ c∗ is concave.

d. Monotonic in y and w: From above, ∂c∗∂w1

≥ 0 and ∂c∗∂w2

≥ 0. Also,

∂c∗

∂y=(w1w2

y

) 12

> 0

e. Continuity: c∗ is the composition of the square root function and a product, both of which are contin-

uous.

f. Nonnegativity:

c∗ = 2(yw1w2)12 > 0 when (y, w1, w2) >> 0.

c∗(w, 0) = 2(0 · w1w2)12 = 0.


3. (Varian #4.8) y = x1x2, so 0 = x2 + x1(∂x2∂x1

) along an isoquant. MRTS= −x2x1

= −w1w2

= −1. So, at

minimum, x∗2 = x∗

1. Thus, c∗ = w1x∗1 +w2x

∗2 = 2x∗

1 = 4, or x∗1 = x∗

2 = 2. Then, y = x∗1x

∗2 = 2 · 2 = 4.

4. (Varian #5.6) a = 12 , c = −1

2 by homogeneity. And b = 3 since ∂x1∂w2

= ∂x2∂w1

.

5. (Varian #5.12)

a.

x1 is an inferior input

b. Constant returns to scale implies HOD 1 of c∗ in y: c∗(w, αy) = αc∗(w, y) (see Varian, p. 67). Since

this holds for all w >> 0, the derivatives with respect to w must also be equal for all w >> 0:

∂c∗(w,αy)∂wi

= α ∂c∗(w,y)∂wi

. By Shepard’s Lemma, x∗i (w, αy) = αx∗

i (w, y). Now differentiate with respect to

α: ∂x∗i (w,αy)∂y y = x∗

i (w, y). Evaluate at α = 1 to get ∂x∗i

∂y = x∗i

y ≥ 0.

c. Use Shepard’s Lemma to show that λ∗(w, y) is marginal cost. Then use symmetry to show ∂λ∗∂wi

= ∂x∗i

∂y .

So, if ∂λ∗∂wi

< 0, then ∂x∗i

∂y < 0.

172 SOLUTIONS TO EXERCISES FOR LECTURE 7 (DUALITY OF THE COST FUNCTION)

Solutions to Exercises for Lecture 7 (Duality of the Cost Function)

1. (Varian #5.16)

a. Homogeneous of Degree 1?

c(αw, y) = y12 ((αw1)(αw2))

34

= α32 y

12 (w1w2)

34 �= αc(w, y), so NO.

Monotonic?

∂c

∂wi= y

1234w

− 14

i w34j > 0, �

∂c

∂y=

12y−

12 (w1w2)

34 > 0 �

Concave?

H =

[−y

12 3

16w−5

41 w

342 y

12 9

16w− 1

41 w

− 14

2

y12 9

16w

−14

1 w−1

42 −y

12 3

16w

341 w

−54

2

]

The (1, 1) “naturally ordered” principal minor is −y12 3

16w−5

41 w

342 < 0. The other one is

|H | = y9162

(w1w2)−12 − y

92

162(w1w2)−

12

=9y162

(w1w2)−12 [1− 9] = −72y

162(w1w2)−

12 < 0.

The principal minor of order 2 is not positive, so c is not concave.

Continuous? c involves only positive exponents and products, so it is continuous.

Nonnegative? c(w, y) > 0 for (w, y) >> 0 and c(w, 0) = 0 are obvious.

b. Homogeneous of Degree 1?

c(αw, y) = y(αw1 + (αw1αw2)12 + αw2)

= αy(w1 + (w1w2)12 +w2) = αc(w, y) �

Monotonic?

∂c

∂wi= y(1 +

12w

−12

i w12j ) > 0, �

∂c

∂y= (w1 + (w1w2)

12 + w2) > 0 �


Concave?

H =

[−y

4w

−32

1 w122

y4w

− 12

1 w− 1

22

y4w

−12

1 w−1

22 −y

4w121 w

− 32

2

]The (1, 1) “naturally ordered” principal is −y

4w

− 32

1 w122 < 0 �. The other one is |H | = (y

4

)2 (w1w2)−1 −(y4

)2 (w1w2)−1 = 0 �.

Continuous? c involves only positive exponents, products, and sums, so it is continuous.


c. Homogeneous of Degree 1?

c(αw, y) = y((αw1)e−αw1 + αw2)

= αy(w1e−αw1 + w2) �= αc(w, y), so NO.

Monotonic?

∂c

∂w1= y[e−w1 −w1e

−w1 ] > 0 for w1 < 1, < 0 for w1 > 1, NO

∂c

∂w2= y > 0 �

∂c

∂y= w1e

−w1 +w2 > 0 �

Concave?

H =[ye−w1 [w1 − 2] 0

0 0

]The (1, 1) “naturally ordered” principal minor is ye−w1 [w1−2], which is positive if w1 > 2 but negative

if w1 < 2, so this function is not concave.

Continuous ? c involves only sums, products, and the exponential function, so it is continuous.


d. Homogeneous of Degree 1?

c(αw, y) = y[αw1 − (αw1αw2)

12 + αw2

]= αy

[w1 − (w1w2)

12 + w2

]= αc(w, y) �

Monotonic?

∂c

∂wi= y

[1− 1

2w

− 12

i w12j

]� 0, NO

∂c

∂y= w1 − (w1w2)

12 +w2 = (w

121 − w

122 )

2 + (w1w2)12 > 0 �

174 SOLUTIONS TO EXERCISES FOR LECTURE 7 (DUALITY OF THE COST FUNCTION)

Concave?

H =

[y4w

− 32

1 w122 −y

4w− 1

21 w

− 12

2

−y4w

−12

1 w−1

22

y4w

121 w

− 32

2

].

The (1, 1) “naturally ordered” principal minor is y4w

− 32

1 w122 > 0, so NO. The other one is |H | =(

y4

)2 (w1w2)−1 − (y4

)2 (w1w2)−1 = 0 �.

Continuous? c involves only positive exponents, products, and sums, so it is continuous.

Nonnegative? From ∂c∂y

, we see that c(w, y) > 0 for (w, y) >> 0. c(w, 0) = 0 is obvious.

e. Homogeneous of degree 1?

c(αw, y) =(y +

1y

)(αw1αw2)

12 = α

(y +

1y

)(w1w2)

12 = αc(w, y) �

Monotonic?

∂c

∂wi=

12

(y +

1y

)w

− 12

i w12j > 0 �

∂c

∂y=(1− 1

y2

)(w1w2)

12

> 0 if y > 1

< 0 if y < 1, so NO.

Concave?

H =

[−1

4(y + 1

y)w−3

21 w

122

14(y + 1

y)w− 1

21 w

−12

2

14(y + 1

y)w− 1

21 w

−12

2 −14(y + 1

y)w

121 w

−32

2

]

The (1, 1) “naturally ordered” principal minor is −14

(y + 1

y

)w

−32

1 w122 < 0 �. The other one is |H | =

0 �.

Continuous? c involves only sums, products, and positive exponents except for the 1/y term. So c is

continuous everywhere 1/y is defined, but we cannot address continuity at y = 0 because c(w, 0) is not

defined.

Nonnegative? c(w, y) > 0 for (w, y) >> 0 is obvious, but c(w, 0) is not defined.

Summary: So, the only function that is a cost function is (b.). By Shephard’s Lemma,

x∗i = y

(1 +

12w

− 12

i w12j

)= y

(1 +

12

(wjwi

) 12)

for i, j = 1, 2.


Use these two equations to solve for y in terms of (x1, x2) by eliminating w2w1

:

4(x∗

1

y− 1)2

=w2

w1and

14

(x∗

2

y− 1)−2

=w2

w1

⇒ (x∗1 − y)(x∗

2 − y) =y2

4.

Use the quadratic formula to get

y =23

[(x1 + x2) +

√(x1 + x2)2 − 3x1x2

].

2. (Varian #6.2) c∗ = y[w1 + w2], x∗1 = ∂c∗

∂w1= y, and x∗

2 = ∂c∗∂w2

= y. So, the production function is

y = x∗1 = x∗

2 at the optimum. This happens when y = min{x1, x2} is the production function.

3. (Varian #6.3) The cost function must be nondecreasing in both prices, so a and b are both nonnegative.

The cost function must be concave in both prices, so a and b are both no greater than 1. Finally, the cost

function must be homogeneous of degree 1, so a = 1− b.

176 SOLUTIONS TO EXERCISES FOR LECTURE 8 (DUALITY OF SUPPLY, DEMANDS, AND PROFIT)

Solutions to Exercises for Lecture 8 (Duality of Supply, Demands, and Profit)

1.

a. ∂x∗1

∂w1= (ey − 1)(−α)w−α−1

1 wα2 ≤ 0 ⇔ α ≥ 0 (semidefiniteness)

∂x∗2

∂w2= (ey − 1)(−β)wβ

1w−β−12 ≤ 0 ⇔ β ≥ 0 (semidefiniteness)

∂x∗1

∂w2= (ey − 1)αw−α

1 wα−12

∂x∗2

∂w1= (ey − 1)βwβ−1

1 w−β2

So, ∂x∗1

∂w2= ∂x∗

2∂w1

(symmetry) ⇔ αw−α1 wα−1

2 = βwβ−11 w−β

2 . This must hold ∀ w >> 0, which can only

work if

α = β, −α = β − 1, α− 1 = −β.

Combine to get α = β = 12 . Also, x

∗ is homogeneous for any α, β, so∣∣∣∣∣∂x∗

1∂w1

∂x∗1

∂w2∂x∗

2∂w1

∂x∗2

∂w2

∣∣∣∣∣ = 0,

verifying the remaining part of semidefiniteness. Monotonicity in y is clear: ∂(w·x∗)∂y = ey2w

121 w

122 > 0.

Continuity is also clear since when α = β = 12, since w · x∗ then involves only positive exponents. For

nonnegativity, x∗ ≥ 0 is obvious for y ≥ 0 and x∗(w, 0) = 0 is also obvious, as is x∗(w, y) > 0 for y > 0.

b. i. c∗(w, y) = w · x∗(w, y) = (ey − 1)2w121 w

122

ii. maxy py − c∗(w, y) = maxy py − (ey − 1)2w121 w

122

∂

∂y= p− ey2w

121 w

122 = 0 ⇒ ey =

p

2w121 w

122

⇒ y = ln

(p

2w121 w

122

)∂2

∂y2= −ey2w

121 w

122 < 0 ⇒ max .

So,

y∗(p, w) =

0, for p < 2w

121 w

122

ln(

p

2w121 w

122

), for p ≥ 2w

121 w

122 .

Then,

x∗(p, w) = x∗(w, y∗(p, w)) =

0, for p < 2w

121 w

122(

p

2w121 w

122

− 1)[(

w2w1

) 12,(w1w2

) 12], for p ≥ 2w

121 w

122


iii.

π∗(p, w) =

0, for p < 2w

121 w

122

p ln(

p

2w121 w

122

)−(

p

2w121 w

122

− 1)2w

121 w

122 , for p ≥ 2w

121 w

122

iv. From x∗, x∗1x

∗2 = (ey − 1)2, thereby eliminating w. So, ey = x

121 x

122 + 1 or y = ln(x

121 x

122 + 1).

2.

a. Homogeneity:

y∗(λp, λw) =λp

2min{λw1, λw2} =p

2min{w1, w2} = y∗(p, w)

x∗1(λp, λw) =

0, for λw1 > λw2[0, (λp)2

4(λw1)2

], for λw1 = λw2

(λp)2

4(λw1)2, for λw1 < λw2

=

0, for w1 > w2[0, p2

4w21

], for w1 = w2

p2

4w21, for w1 < w2

= x∗1(p, w)

x∗2 is the same. So, (y∗,−x∗) is HOD 0.

Nonnegativity:

π∗ = py∗ −w · x∗ =

p2

2min{w1,w2} − p2

4w1, w1 ≤ w2

p2

2min{w1,w2} − p2

4w2, w1 ≥ w2

=p2

2min{w1, w2} − p2

4min{w1, w2}=

p2

4min{w1, w2} ≥ 0.

y∗(0, w) = 0 and x∗(0, w) = 0; and (y∗, x∗) ≥ 0; are obvious.

Symmetry and Semidefiniteness:

∂y∗

∂p=

12min{w1, w2} > 0;

∂y∗

∂w1=

{ − p2w2

1, for w1 < w2

0, for w1 > w2

;∂y∗

∂w2=

{0, for w1 < w2

− p2w2

2, for w1 > w2.

∂x∗1

∂p=

{0, for w1 > w2

p2w2

1, for w1 < w2

;∂x∗

1

∂w1=

{0, for w1 > w2

− p2

2w31, for w1 < w2

;∂x∗

1

∂w2={

0, for w1 > w2

0, for w1 < w2

∂x∗2

∂p=

{p

2w22, for w1 > w2

0, for w1 < w2

;∂x∗

2

∂w1={

0, for w1 > w2

0, for w1 < w2

;∂x∗

2

∂w2=

{− p2

2w32, for w1 > w2

0, for w1 < w2

Except for ∂y∗∂p , none of these are defined when w1 = w2. Clearly ∂y∗

∂p > 0, − ∂x∗1

∂w1≥ 0, − ∂x∗

2∂w2

≥ 0. Also,∣∣∣∣∣∂y∗∂p

∂y∗∂w1

−∂x∗1

∂p− ∂x∗

1∂w1

∣∣∣∣∣ ={

0, w1 > w2

− p2

4w41+ p2

4w41= 0, w1 < w2.

178 SOLUTIONS TO EXERCISES FOR LECTURE 8 (DUALITY OF SUPPLY, DEMANDS, AND PROFIT)

It is not necessary to check the whole 3x3 determinant (it is zero, due to homogeneity). Hence, the

matrix of derivatives is positive semidefinite (when it is defined). For symmetry, ∂y∗∂wi

= −∂x∗i

∂p and

∂x∗1

∂w2= ∂x∗

2∂w1

both hold by inspection.

b. π∗ = p2

4min{w1,w2} , from above.

c. The demands take on only corner values, with the switch occurring at w1 = w2. This means the

isoquant must be linear with slope −1 :

A typical isoquant

The intercepts are obtained by noticing that y∗2 = x∗i . So, y

2 = x1 + x2, or y =√x1 + x2.

d. From c,

x∗1(w, y) =

y2, for w1 < w2

[0, y2], for w1 = w2

0, for w1 > w2

x∗2(w, y) =

0, for w1 < w2

y2 − x∗1, for w1 = w2

y2, for w1 > w2

e. c∗ = w · x∗ = min{w1, w2}y2

3.

a. Homogeneity: π∗(λp, λw) = (λp)2

4[(λw1)1−α(λw2)α+(λw1)β(λw2)1−β ]= λp2

4A= λπ∗(p, w), where

A =[(λw1)1−α(λw2)α + (λw1)β(λw2)1−β

].


So, HOD 1 holds ∀ α, β.

Monotonicity: π∗ is ↑ in p. π∗ is ↓ in w provided α, β ∈ [0, 1]

Nonnegativity: π∗ ≥ 0 and π∗(0, w) = 0 are obvious.

Convexity:

y∗ =∂π∗

∂p=

p

2A; −x∗

1 =∂π∗

∂w1= − p2

4A2

∂A

∂w1; −x∗

2 =∂π∗

∂w2= − p2

4A2

∂A

∂w2.

∂2π∗

∂p2=

12A

;∂2π∗

∂p∂w1= − p

2A2

∂A

∂w1;

∂2π∗

∂p∂w2= − p

2A2

∂A

∂w2

∂2π∗

∂w21

=p2

2A3

[∂A

∂w1

]2− p2

4A2

∂2A

∂w21

;∂2π∗

∂w1∂w2=

p2

2A3

∂A

∂w1

∂A

∂w2− p2

4A2

∂2A

∂w1∂w2.

Here, ∂2A∂w2

1= −α(1−α)w−α−1

1 wα2 +(β−1)βwβ−2

1 w1−β2 ≤ 0 for α, β ∈ [0, 1]. Since A > 0, ∂

2π∗∂p2

> 0. And

∣∣∣∣∣∂2π∗∂p2

∂2π∗∂p∂w1

∂2π∗∂w1∂p

∂2π∗∂w2

1

∣∣∣∣∣ = p2

4A4

(∂A

∂w1

)2

− p2

8A3

∂2A

∂w21

− p2

4A4

(∂A

∂w1

)2

= − p2

8A3

∂2A

∂w21

≥ 0.

It is unnecessary to check the 3x3 determinant due to homogeneity (it is zero), so the Hessian of π∗ is

positive semidefinite and thus π∗ is convex. Therefore, the only restrictions on α, β are α, β ∈ [0, 1].

b. i. See the first derivatives above, by Hotelling’s Lemma. Must substitute

∂A

∂w1= (1 − α)w−α

1 wα2 + βwβ−1

1 w1−β2

∂A

∂w2= αw1−α

1 wα−12 + (1− β)wβ

1 w−β2 .

ii. Note that x∗i = y∗2 ∂A

∂wi. So x∗

i (w, y) = y2 ∂A∂wi

.

iii. c∗ = w · x∗ = y2(w1

∂A∂w1

+ w2∂A∂w2

)= y2

(w1−α

1 wα2 +wβ

1w1−β2

)= y2A.

iv. Use x∗1 = y2

[(1− α)w−α + βwβ−1

], x∗

2 = y2[αw1−α + (1 − β)wβ

], where w = w1

w2is the relative

price. In principle these two equations can be substituted to eliminate w, leaving only the relationship

between y, x1, and x2. But this is messy in practice because of the presence of terms with different

powers of w. This is a good example of being able to verify cost, profit, demand, and supply functions

by duality even though the production function is complicated.

180 SOLUTIONS TO EXERCISES FOR LECTURE 9 (COST ANALYSIS)

Solutions to Exercises for Lecture 9 (Cost Analysis)

1.

a. Assume f(αx) ≥ αf(x) for α ≥ 1, where f is the production function. Fix y′ ≥ y and define α = y′y ≥ 1.

If x∗(y) is an optimal choice when output is y, then by feasibility f(x∗(y)) ≥ y. Hence αf(x∗(y)) ≥

αy = y′, so by nondecreasing returns f(αx∗(y)) ≥ y′. That is, αx∗(y) is feasible for producing y′. Then

by definition of the minimum

c∗(y′) ≤ w · (αx∗(y)) = αc∗(y) =y′

yc∗(y).

Divide by y′ to get ac(y′) ≤ ac(y).

Nondecreasing returns. Note that mc lies below ac, but mc has no particular shape. Notealso that the slope of a ray to c∗ is average cost at that point, so the slope of such rays mustdecrease as y increases.


Nondecreasing returns, nontraditional case. This is nondecreasing returns, since allrays remain in Z once they pass inside Z, as illustrated. Similarly, the slopes of rays from theorigin to c∗ fall as y increases, as illustrated. But the nonconvexity in the boundary of Z creates anonconcavity in c∗, so that mc clearly increases over the range of y values where c∗ is nonconcave.

b. Assume f(αx) ≥ αf(x) for α ∈ (0, 1], where f is the production function. Fix y′ ≤ y and define

α = y′y ≤ 1. By feasibility, f(x∗(y)) ≥ y. Hence f(αx∗(y)) ≥ αf(x∗(y)) ≥ αy = y′. That is, αx∗(y) is

feasible for producing y′. Hence,

c ∗ (y′) ≤ w · (αx∗(y)) = αc∗(y) =y′

yc∗(y).

Divide by y′ to get ac(y′) ≤ ac(y).


Nonincreasing returns. Note that mc lies above ac, but mc has no particular shape. Notealso that the slope of a ray to c∗ is average cost at that point, so the slope of such rays mustincrease as y increases.

Nonincreasing returns, nontraditional case. This is nonincreasing returns, since allrays remain out of Z once they pass outside Z, as illustrated. Similarly, the slopes of rays fromthe origin to c∗ increase as y increases, as illustrated. But the nonconvexity in Z creates anonconvexity in c∗, so that mc clearly decreases over the range of y values where c∗ is nonconvex.

c. Combine a and b to conclude ac is constant. Then use 0 = ac′ = 1y (mc − ac) to conclude ac = mc

everywhere, so mc is constant also.

d. From Varian pp. 88-89, the elasticity of scale at output level y equals ac(y)mc(y) . So, if there is local


increasing returns to scale at output y then ac(y) > mc(y). This means that ac(y) is decreasing locally

at output level y, which is the local counterpart to part a of the question. Similarly, if there is local

decreasing returns to scale at output y then ac(y) < mc(y), so ac(y) is increasing locally, which is the

local counterpart to part b of the question. Finally, if there is local constant returns to scale then

ac(y) = mc(y), so ac(y) is stationary at y, which is the local counterpart to part c of the question.

2.

a.

ac =c∗(y)y

=F

y+ y for y > 0

avc =vc(y)y

=y2

y= y for y > 0

mc =∂c(y)∂y

= 2y for y > 0

Average, Average variable, and Marginal cost curves


b.

Supply curve when α = 1. It consists of the entire marginal cost curve above avc because nofixed cost can be saved by shutting down.

c. When 0 < α < 1, for purposes of determining the shutdown point it is as if fixed cost is (1 − α)F in

the average cost curve, since this is the fixed cost that is avoidable (non-sunk). Thus, the average cost

curve that is relevant for determining the shutdown point is ac = (1−α)Fy + y for y > 0 (Note: this is

NOT the true average cost curve. The true average cost curve is illustrated in part a. This curve is

just a tool for finding the shutdown point when fixed costs are partially sunk). Then the supply curve

is


Supply curve when 0 < α < 1

d. The graph for part c provides a general framework for deriving the supply curve. If we let α = 1 in

that graph we get the supply curve illustrated in part b. If we let α = 0 in that graph we obtain the

usual undergraduate illustration of shutdown, which implicitly assumes that no fixed costs are sunk:

Supply curve when α = 0

3.

a. y = x1x2, so x2 = yx1

. So,

c∗(w, y|x1) = w1x1 + w2

(y

x1

).


b. x1 = x∗1(w, y) =

√yw2w1

, so y = x21w1w2

. Then

c∗(w, y) = 2√

x21

w1

w2w1w2 = 2x1w1.

On the other hand,

c∗(w, y|x1) = w1x1 + w21x1

(x2

1

w1

w2

)= w1x1 + w1x1 = 2w1x1.

4. (Varian #5.2) y = y1 + y2. We want to min{y1,y2} c1(y1) + c2(y2) subject to y = y1 + y2:

miny1

3y21 + (y − y1)2

FOC:

6y1 + 2(y − y1)(−1) = 0

y1 =y

4,⇒ y2 = y − y1 =

3y4

SOC:

8 > 0,⇒ minimum

c∗(y) = c1

(y4

)+ c2

(3y4

)= 3

(y4

)2

+(3y4

)2

=3y2

4.

5. (Varian #5.8) If p = 2, then a nonnegative profit is possible, so set p = c′ to obtain 2 = 2y or y∗ = 1.

Since profit is zero at this price, y∗ = 0 is also optimal (i.e., y∗ = {0, 1}). If p = 1 then positive profit is

impossible, so y∗ = 0. Setting p = c′ to obtain y∗ = 12is incorrect since π∗ = 1

2−((

12

)2 + 1)= −3

4< 0.

6. (MWG #5.D.2) Assume cv(q) is strictly convex, for specificity. The part of cost that can be avoided by

choosing q = 0 is cv(q) + [K − c(0)], so shutdown is determined by looking at the average cost curve from

this. cv(q)q

is upward-sloping whenever it is relevant (i.e., when marginal cost is above average cost) and

1q [K − c(0)] is downward-sloping. Thus a U-shape emerges:

7. (MWG #5.D.4(a)) Fix q and let qj be a partition of q such that∑J

j=1 qj = q (qj > 0; J > 1). Then

J∑j=1

c(qj) =J∑j=1

qjac(qj) >J∑j=1

qjac(q) since qj < q and ac is falling

= ac(q)J∑j=1

qj = ac(q)q = c(q).


Supply with partially sunk costs

Solutions to Exercises for Lecture 10 (Applied Production Analysis)

188 SOLUTIONS TO EXERCISES FOR LECTURE 11 (PREFERENCES)

Solutions to Exercises for Lecture 11 (Preferences)

1.

a. Suppose x � y. By definition of �, we have not(y �∼

x). By completeness of �∼, x �

∼y (NOTE: We

have just shown that x � y implies x �∼

y, a property we will use below). Then, by definition of �, we

cannot have y � x since this means not(x �∼

y).

b. This is a special case of part a, with x replacing y (there is no use of the idea that x �= y in the solution

of part a).

c. Let x � y and y � z. Suppose x � z does not hold. Then not(z �∼

x) does not hold, so z �∼

x. Since

x �∼

y (from above), transitivity of �∼

yields z �∼

y. But we also have not(z �∼

y), a contradiction.

d. Proof is identical to part c.

e. Suppose not. Then z �∼

x. Now apply the result in part d to z �∼

x and x � y to obtain z � y. That

is, not(y �∼

z), a contradiction.

f. For clarity, let y = x. Then, by completeness, either y �∼

x or x �∼

y (or both). If the former fails,

then we have not(y �∼

x), or not(x �∼

x), which violates the latter. Similarly, if the latter fails then the

former is violated. Hence, we have both y �∼

x and x �∼

y, or y ∼ x. Since y = x, x ∼ x.

g. Let x ∼ y and y ∼ z. Then, by definition of ∼, x �∼

y and y �∼

z, so by transitivity of �∼, x �

∼z.

Similarly, z �∼

y and y �∼

x, so z �∼

x. Combining, x ∼ z.

h. If x ∼ y, then both x �∼

y and y �∼

x. Hence y ∼ x.

2. Throughout the solution, assume x, x′, y ∈ X.

a. Let x � y. Then not(y �∼

x). If U(y) ≥ U(x) then, since U represents �∼, y �

∼x, a contradiction. So

U(x) > U(y). Now the converse: assume U(x) > U(y). If y �∼

x then U(y) ≥ U(x), a contradiction, so

not(y �∼

x), or x � y.

b. Let x ∼ y. Then x �∼

y and y �∼

x. Since U represents �∼, U(x) ≥ U(y) and U(y) ≥ U(x), so

U(x) = U(y). Now the converse: assume U(x) = U(y). Then U(x) ≥ U(y) and U(y) ≥ U(x). Since U

represents �∼, x �

∼y and y �

∼x, so x ∼ y.

c. Let x ≥ y (weak vector inequality). Then, by free disposal, x �∼

y. Since U represents �∼, U(x) ≥ U(y).

d. Let x > y Then, by strong monotonicity, x � y, so by part a U(x) > U(y).


e. Fix ε > 0. By local nonsatiation, there exists y ∈ X such that ‖y − x‖ < ε and y � x. Hence

U(y) > U(x) (from above).

f. By completeness, either x �∼

y or y �∼

x. For concreteness, assume x �∼

y. Then, since U represents

�∼, U(x) ≥ U(y), so U(y) = min{U(x), U(y)}. Using completeness, y �

∼y. So, by convexity of

�∼, λx + (1 − λ)y �

∼y (setting x′ = y in the definition of convexity of �

∼). Since U represents �

∼,

U(λx+ (1− λ)y) ≥ U(y) = min{U(x), U(y)}. The argument is the same if y �∼

x–just interchange the

roles of x and y.

g. Assume x �= y and proceed as in part d, obtaining λx + (1 − λ)y � y from strict convexity of �∼, so

that U(λx+ (1− λ)y)) > U(y) (from part a).

3. Strong Monotonicity: Let x > y. If x1 > y1, then x �∼

y, as desired. Moreover, not(y1 ≥ x1), so not(y �∼

x).

Hence x � y. If x1 = y1, then x2 > y2, so again x �∼

y. And again not(y2 ≥ x2), so not (y �∼

x). Hence

x � y.

Strict Convexity: Let x �∼

z and y �∼

z, where x �= y. Hence x1 ≥ z1 and y1 ≥ z1. If either of these is strict,

then λx1 + (1− λ)y1 > z1, so λx + (1− λ)y � z. If x1 = y1 = z1, then x2 ≥ z2 and y2 ≥ z2, and one of

these must be strict since x �= y. So λx2 +(1−λ)y2 > z2 while λx1 +(1−λ)y1 = z1, or λx+(1−λ)y � z.

4. (MWG #3.B.3) Suppose the indifference curves are as follows, with utility increasing in the direction of

the arrows:

This is strictly convex since a convex combination of any two points on or above an indifference curve lies

everywhere (strictly) above that indifference curve. It is also locally nonsatiated since the ε-ball around

any point includes points above the indifference curve through the point. However, it is not monotone

since any point due east of another point is on a lower indifference curve.

5. (MWG #3.C.4) Suppose X = R1+ and x �

∼y if either y = 1 or x �= 1. The utility function U(x) = 1 for

x �= 1 and U(1) = 0 represents �∼. To check this, suppose first that y = 1. Then x �

∼y. Also, U(y) = 0

and U(x) is either 1 or 0, so U(x) ≥ U(y), as desired. Now suppose y �= 1 and x �= 1. Then x �∼

y. And

U(x) = 1 = U(y), so U(x) ≥ U(y), as desired. Finally suppose y �= 1 and x = 1. Then y �∼

x. And

U(y) = 1 while U(x) = 0, so U(y) ≥ U(x), as desired. Hence, by the theorem presented in class we know

�∼

is complete and transitive. However, �∼

is not closed, because the “preferred-to” set {x ∈ X : x �∼

y}

190 SOLUTIONS TO EXERCISES FOR LECTURE 11 (PREFERENCES)

A strictly convex preference relation that is locally nonsatiated but not mono-

tone

for y �= 1 has a “hole” in it at x = 1.


Solutions to Exercises for Lecture 12 (Utility Maximization: Basics)

1. We know, for L = U(x)− λ(p · x−m), that

HL(x,λ)

∂x∗1

∂pj

...

...∂x∗

n

∂pj

∂λ∗∂pj

=

0...0λ∗

0...0x∗j

← jth row

.

So, ∂λ∗∂pj

= 1|HL(x,λ)|

[(−1)j+(n+1)λ∗|Hj,n+1|+ (−1)2(n+1)x∗

j |Hn+1,n+1|]. This cannot be signed because

neither Hj,n+1 nor Hn+1,n+1 are border-preserving (Hj,n+1 also is not a principal submatrix). Similarly,

HL(x,λ)

∂x∗

1∂m...

∂x∗n

∂m∂λ∗∂m

=

0...0−1

.

So, ∂λ∗∂m = 1

|HL(x,λ)|(−1)2(n+1)(−1)|Hn+1,n+1|, which cannot be signed because Hn+1,n+1 is not border-

preserving. Since λ∗ is the marginal utility of income, the sign of ∂λ∗∂m

determines whether there is

diminishing or increasing marginal utility of income. Either is possible.

2.

a.

g = 6, 000, if no income.

g = 4, 500,when m = 2, 000.

g = 3, 000,when m = 4, 000.

b. 6, 000− 0.75m = 0. Therefore m = 8, 000.

c.

PC + 4× 365H = 24× 4× 365,

where P is a price of consumption and H is hours of leisure per day. This is

PC + 1, 460H = 35, 040.

192 SOLUTIONS TO EXERCISES FOR LECTURE 12 (UTILITY MAXIMIZATION: BASICS)

d.

PC + 1, 460H = 35, 040 + [6, 000− 365× 4× 0.75(24−H)]

subject to 1, 460× 0.75× (24−H) ≤ 6, 000.

This means H ≥ 18.52.e.

Budget constraint under old rule

f. With the grant, the budget constraint is

PC + 1, 460H = 35, 040 + [6, 000− 365× 4× 0.5(24−H)]

subject to 1, 460× 0.5× (24−H) ≤ 6, 000.

This means H ≥ 15.78.


Budget constraint under new rule

g. Whether the head of this family will work more or less under the new rule depends on the utility

function. A complete answer requires that we discuss income and substitution effects, which we do

not cover until Lecture 14, so it may be useful to revisit this solution after reading Lecture 14. In the

graph, we can divide the H-axis into three regions. If the utility function is tangent in region (a), the

new rule does not change anything (assuming the indifference curve is not so flat that it crosses the

kinked part of the new budget constraint). If the tangency is in region (b), the budget line becomes

flatter relative to the part of the budget line the consumer is on under the old rule. Thus, effectively,

the relative price of H (leisure) has decreased. The substitution effect causes the head of the family to

consume more leisure and work less, and the income effect also causes more consumption of leisure if

leisure is normal. Even if leisure is inferior, more leisure is consumed and the head works less unless

leisure is Giffen. If the tangency is in region (c), the budget line becomes steeper relative to the part of

the budget line the consumer is on under the old rule. Thus, effectively, the relative price of H (leisure)

has increased. In this case the substitution effect causes the head of the family to consume less leisure


and work more, and the income effect causes more consumption of leisure if leisure is normal (note

that the implicit change in income is an increase since the budget line moves out). Thus, in region (c)

we cannot predict what happens to the level of work if leisure is normal, but if leisure is inferior then

work increases.

Utility maximization under old and new rule

3.

a.

x1 +1

1 + rx2 = m1 +

m2

1 + r.

b.

L = U(x1, x2)− λ

(x1 +

11 + r

x2 −m1 − m2

1 + r

)FOCs:

U1 = λ

U2 = λ1

1 + r

−x1 − 11 + r

x2 +m1 +m2

1 + r= 0.


Differentiating the FOC’s with respect to m1 yields:

U11∂x1

∂m1+ U12

∂x2

∂m1− ∂λ

∂m1= 0

U21∂x1

∂m1+ U22

∂x2

∂m1− 1

1 + r

∂λ

∂m1= 0

− ∂x1

∂m1− 1

1 + r

∂x2

∂m1+ 1 = 0.

In matrix form, U11 U12 −1U21 U22

−11+r

−1 −11+r 0

∂x1∂m1∂x2∂m1∂λ∂m1

=

00−1

.

So by Cramer’s Rule,

∂x1

∂m1=

∣∣∣∣∣∣0 U12 −10 U22

−11+r

−1 −11+r

0

∣∣∣∣∣∣∣∣HL(x,λ)

∣∣ =U12

11+r

− U22∣∣HL(x,λ)

∣∣ .

This implies that if U12 > 0, ∂x1∂m1

> 0, in which case an increase in this year’s income leads to an

increase in consumption this year. But this need not be true generally, even for a strictly quasiconvex

utility function.

c. By the envelope theorem,

∂U∗(r,m1, m2)∂r

=∂L

∂r

∣∣∣∣x=x∗, λ=λ∗

=λ∗

(1 + r)2(x∗

2 −m2).

Since λ∗ > 0 given the way L is formulated (i.e., the marginal utility of income must be positive), we

have indirect utility increasing in r if x∗2 > m2 and decreasing in r if x∗

2 < m2. These cases correspond

to saving and dissaving, respectively. Geometrically, if r′ > r then the graph is:


Intertemporal choice of consumption. In (a) we have c1 < m1 so the consumer is asaver. When r increases, points above the original indifference curve become feasible, so utilityincreases. Exactly the opposite happens in (b), where the consumer is a dissaver.

4.

a. x2 = U −√x1 + 4

dx2

dx1= −1

2(x1 + 4)−

12 < 0 for x1 ≥ 0

d2x2

dx21

=14(x1 + 4)−

32 > 0 for x1 ≥ 0.

So, the indifference curve is decreasing and convex. At x1 = 0, the value is x2 = U − 2 and the slope

is −14 . At x2 = 0, the value is x1 = U

2 − 4 and the slope is −12U

.

b. If p1p2 ≥ 14 then x∗

1 = 0 and x∗2 = m

p2. When x2 = 0 and x1 = m

p1we have U =

√mp1

+ 4, so if p1p2 ≤ 12√

mp1

+4

then x∗1 = m

p1and x∗

2 = 0. If 14 > p1

p2> 1

2√

mp1

+4then set -MRS=p1

p2:

12√x1 + 4

=p1

p2⇒ x∗

1 =(

p2

2p1

)2

− 4.

Then x∗2 = m

p2− p1

p2x∗

1 = mp2

− p24p1

+ 4p1p2

.


A typical indifference curve

Summary:

x∗1 =

mp1

, if p1p2

≤ 12√

mp1

+4(p22p1

)2

− 4, if 12√

mp1

+4< p1

p2< 1

4

0, if p1p2

≥ 14

; x∗2 =

0, if p1

p2≤ 1

2√

mp1

+4

m+4p1p2

− p24p1

, if 12√

mp1

+4< p1

p2< 1

4

mp2

, if p1p2

≥ 14.

5. (Varian #7.2) A typical indifference curve appears in the graph below. So, the maximum occurs at the

corners. If p1p2

> 1 then only consume x2. If p1p2

< 1 then only consume x1. If p1p2

= 1, then consume either

good, but not both. So,

x∗1(p,m) =

mp1

, if p1p2

< 1{0, mp1

}, if p1

p2= 1 (and x∗

2 ={mp2

, 0}, respectively)

0, if p1p2

> 1

Hence,

U∗(p,m) =

max{mp1

, 0}, when p1

p2≤ 1

max{0, mp2

}, when p1

p2> 1

= max{

m

p1,m

p2

}.


A typical indifference curve

6. (Varian #7.5)

a. Quasilinear preferences.

b. Less than u(1).

c. v(p1, p2, m) = max{u(1)− p1 +m,m}

7. (Varian #8.5) A typical indifference curve is x2 = Ux−3

21 , with slope ∂x2

∂x1= −3

2Ux−5

21 and curvature

∂2x2∂x2

1= 15

4Ux

− 72

1 . So, it has the shape shown in the graph below. This assures us that the FOC’s will

describe the maximum. They amount to setting the MRS equal to the negative price ratio and also using

the budget constraint:

3U

2x521

=p1

p2and p1x1 + p2x2 = m.

So, x521 = 3Up2

2p1= 3x

321 x2p22p1

, or x1 = 3x2p22p1

. Hence, p13x2p22p1

+ p2x2 = m ⇒ x∗2 = 2m

5p2⇒ x∗

1 = 3m5p1

.

When p1 = 3, p2 = 4, and m = 100, we have x∗1 = 20 and x∗

2 = 10.


An indifference curve where axes are asymptotes when U > 0

8. (Varian #8.13)

a. Draw the lines x2 + 2x1 = 20 and x1 + 2x2 = 20. The indifference curve is the northeast boundary of

this X. The area where u ≥ 20 is on and above this X.

b. The slope of a budget line is −p1/p2. If the budget line is steeper than 2, x1 = 0. Hence the condition

is p1/p2 > 2.

c. Similarly, if the budget line is flatter than 1/2, x2 will equal 0, so the condition is p1/p2 < 1/2.

d. If the optimum is unique, it must occur where x2 +2x1 = x1 +2x2. This implies that x1 = x2, so that

x1/x2 = 1.

9. (MWG #r) It follows from the definition that the demand function satisfies homogeneity of degree zero

for β > 0. To check whether the demand function satisfies Walras’ Law, we have to calculate px:

px =p2w + p3w + βp1w

p2 + p3 + p1=

p2 + p3 + βp1

p2 + p3 + p1w.

Hence, px = w iff β = 1. Therefore the demand function satisfies Walras’ Law iff β = 1.

10. (MWG #2.E.2) Multiplying both sides of equation (2.E.4) of Proposition 2.E.2 by pk/w, we obtain

L∑�=1

(p�x�(p, w)/w)∂x�∂pk

(p, w)(pk/x�(p, w)) + pkxk(p, w)/w = 0.


Hence,∑L

�=1 b�(p, w)ε�k(p, w) + bk(p, w) = 0. It follows from (2.E.6) of Proposition 2.E.3 that

L∑�=1

(p�x�(p, w)/w)∂x�∂w

(p, w)(w/x�(p, w)) = 1.

Hence,∑L

�=1 b�(p, w)ε�w(p, w) = 1.

11. (MWG #2.E.8) For the first part, note that

lnx�(p, w) = lnx�(eln p1 , · · · , eln pL , elnw).

Thus, by the chain rule,

d lnx�(p, w)/d lnpk =∂x�

∂pk(p, w)eln pk

x�(p, w)=

∂x�

∂pk(p, w)pk

x�(p, w)= ε�k(p, w).

Similarly,

d lnx�(p, w)/d lnw =∂x�

∂w (p, w)elnw

x�(p, w)=

∂x�

∂w (p, w)wx�(p, w)

= ε�w(p, w).

Since α1 = d lnx�(p, w)/d lnp1, α2 = d lnx�(p, w)/d lnp2, and γ = d lnx�(p, w)/d lnw, the last conclusion

is established.


Solutions to Exercises for Lecture 13 (Utility Maximization: Advanced)

1.

a. A homothetic function is a monotonic transformation of a homogeneous function. So, the preferences

can be represented by a function of the form U(x) = f(g(x)), where f is strictly increasing and g is

homogeneous of degree r. Clearly g(x) ≡ f−1(U(x)) also represents these preferences, since f−1 is

monotonic. Similarly, h(x) ≡ [g(x)]1r also represents these preferences. But h is homogeneous of degree

one:

h(λx) = [g(λx)]1r = [λrg(x)]

1r = λ[g(x)]

1r = λh(x).

b. From part a, assume U is homogeneous of degree one. The first order conditions are Ui(x∗) = λ∗pi for

i = 1, . . . , n. Multiplying by x∗i and summing across i yields

n∑i=1

Ui(x∗)x∗i = λ∗

n∑i=1

pix∗i = λ∗m,

where the last equality is Walras’ Law. By Euler’s Theorem, the left side is U(x∗) = U∗(p,m).

c. Differentiating this with respect to pi yields

∂U∗

∂pi=

∂λ∗

∂pim.

Divide both sides by λ∗ and then note that the resulting left side is −x∗i , by Roy’s Identity. Rearrange

to obtain ∂λ∗∂pi

= −λ∗x∗i

m .

d. Rearranging Roy’s Identity yields ∂U∗∂pi

= −λ∗x∗i . Differentiating this with respect to pj yields

∂2U∗

∂pi∂pj= −∂λ∗

∂pjx∗i − λ∗ ∂x

∗i

∂pj.

Now reverse the order of differentiation and use the invariance of the cross partials of U∗ to the order

of differentiation to obtain

−∂λ∗

∂pjx∗i − λ∗ ∂x

∗i

∂pj= −∂λ∗

∂pix∗j − λ∗ ∂x

∗j

∂pi.

Substitute for the derivatives of λ∗ from part c to obtain

λ∗x∗jx

∗i

m− λ∗ ∂x

∗i

∂pj=

λ∗x∗i x

∗j

m− λ∗ ∂x

∗j

∂pi.

202 SOLUTIONS TO EXERCISES FOR LECTURE 13 (UTILITY MAXIMIZATION: ADVANCED)

Canceling terms yields ∂x∗i

∂pj= ∂x∗

j

∂pi.

2. (Varian #7.4(a)) Use Roy’s Identity:

∂v

∂p1= − m

(p1 + p2)2;

∂v

∂p2= − m

(p1 + p2)2;

∂v

∂m=

1p1 + p2

.

So,

x∗1 = −

− m(p1+p2)2

1p1+p2

=m

p1 + p2and x∗

2 =m

p1 + p2.

3. (MWG #3.D.1) x∗i =

αimpi

; where α1 + α2 = 1, α1, α2 > 0, and U = Kxα11 xα2

2 .

Homogeneous of degree 0?

x∗i (λp, λm) =

αi(λm)(λpi)

=αim

pi= x∗

i (p,m) �

Walras Law?

p · x∗ = p1α1m

p1+ p2

α2m

p2= (α1 + α2)m = m �

Uniqueness? There’s only one (x∗1, x

∗2) pair that maximizes U , because U is strictly quasiconcave. To

verify this, set U = min{Kxα11 xα2

2 , Kyα11 yα2

2 }. We must show

K(λx1 + (1− λ)y1)α1(λx2 + (1 − λ)y2)α2 > U.

The function f(z) = zα is strictly concave because 0 < α < 1. This is verified by noting that f ′′(z) =

α(α− 1)zα−2 < 0 for z > 0. Thus,

(λx1 + (1 − λ)y1)α1 > λxα11 + (1− λ)yα1

1

(λx2 + (1 − λ)y2)α2 > λxα22 + (1− λ)yα2

2 .

Multiplying these together yields

K(λx1 + (1− λ)y1)α1(λx2 + (1− λ)y2)α2

> K[λ2xα11 xα2

2 + (1− λ)2yα11 yα2

2 + λ(1− λ)(xα11 yα2

2 + yα11 xα2

2 )]

≥ [λ2 + (1− λ)2]U +Kλ(1 − λ)(xα11 yα2

2 + yα11 xα2

2 ).


Suppose we can show K(xα11 yα2

2 + yα11 xα2

2 ) ≥ 2U. Then we have

U(λx + (1− λ)y) > [λ2 + (1− λ)2 + 2λ(1− λ)]U

= [λ+ (1− λ)]2U = U,

and we’re done. To show this, use Kxα11 ≥ U

xα22

and Kyα11 ≥ U

yα22

to write

K (xα11 yα2

2 + yα11 xα2

2 ) ≥ U(zα2 +1

zα2),

where z = y2x2

. The expression zα2 + 1zα2 (*) is always ≥ 2 ∀ z > 0. To see this, minimize it:

∂(∗)∂z

= α2zα2−1 − α2z

−α2−1 = 0 ⇒ zα2−1 =1

zα2+1.

This FOC can only hold if z = 1, in which case zα2 + 1zα2 = 1 + 1 = 2. To verify the second order

condition, note that ∂2(∗)∂z2 = α2(α2 − 1)zα2−2 + α2(α2 + 1)z−α2−2, which is 2α2

2 > 0 at z = 1.

Nonnegativity? x∗ ≥ 0 is obvious.

4. (MWG #3.D.2)

i. Homogeneous of Degree 0?

U∗(λp, λm) = [α ln(α) + (1− α) ln(1− α)] + ln(λm) − α ln(λp1)− (1− α) ln(λp2)

= A + ln(λ)− α ln(λ) − (1− α) ln(λ) + ln(m) − α ln(p1)− (1 − α) ln(p2)

= A + ln(m) − α ln(p1)− (1− α) ln(p2)

= U∗(p,m), where A = α ln(α) + (1− α) ln(1− α).

ii. Monotonicity?

∂U∗

∂m=

1m

> 0 for m > 0.

Note that we must define U∗(p, 0) = −∞ to have U∗ defined for m ≥ 0 while preserving monotonicity

(i.e., U(0, 0) = −∞). This is because the representation used (i.e., natural log form) is not bounded

below. It is of no consequence, since another representation like xα1 x1−α2 is bounded below.

∂U∗

∂pi= − α

p1or − 1− α

p2, which are both negative for 0 < α < 1.


iii. Quasiconvex in (p,m)? We will only show this in p. The Hessian is[αp21

0

0 1−αp22

],

which is positive semidefinite since all principal minors are ≥ 0. Thus, U∗ is convex in p, which implies

quasiconvexity automatically.

iv. Continuity? This is obvious, except when m = 0. The definition U∗(p, 0) = −∞ preserves the

continuity at m = 0. Note that we then have U∗(p,R1+) = [−∞,∞) ∀p >> 0, so Y = [−∞,∞).

5. (MWG#3.D.5)

a. An indifference curve is xρ1 + xρ2 = Uρ, so

ρxρ−11 + ρxρ−1

2

dx2

dx1= 0 ⇒ dx2

dx1= −

(x1

x2

)ρ−1

< 0.

d2x2

dx12 = −(ρ − 1)

(x1

x2

)ρ−2[x2 − x1

dx2dx1

x22

]{

> 0, if ρ < 1= 0, if ρ = 1.

So, the slope and curvature of the indifference curves are right for an interior solution, but there are

corners to worry about. If ρ > 0 then the indifference curve looks like the figure above. Since the slope

at the corners is 0 and −∞, the optimal choice will never occur at these corners.

A typical indifference curve for ρ > 0


When ρ < 0 the indifference curves look like the figure below. Once again, corners are not optimal.

So, assuming ρ < 1 we can use the FOCs to describe the maximum. Set -MRS=price ratio and use the

budget line: (x1

x2

)ρ−1

=p1

p2⇒ x1 =

(p1

p2

) 1ρ−1

x2.

A typical indifference curve for ρ < 0

So, p1

(p1p2

) 1ρ−1

x2 + p2x2 = m, or

x∗2 =

m

p1

(p1p2

) 1ρ−1

+ p2

=mp

1ρ−12

pρ

ρ−11 + p

ρρ−12

x∗1 =

(p1

p2

) 1ρ−1 mp

1ρ−12

pρ

ρ−11 + p

ρρ−12

=mp

1ρ−11

pρ

ρ−11 + p

ρρ−12

.

Substitute into U to get

U∗(p,m) =m

pρ

ρ−11 + p

ρρ−12

[p

ρρ−11 + p

ρρ−12

] 1ρ

= m[p

ρρ−11 + p

ρρ−12

] 1−ρρ

.

b. First check the properties of x∗:

Homogeneity?

x∗i (λp, λm) =

λm(λpi)1

ρ−1

(λpi)ρ

ρ−1 + (λpj)ρ

ρ−1=

λρ

ρ−1 mp1

ρ−1i

λρ

ρ−1

[p

ρρ−1i + p

ρρ−1j

] = x∗i (p,m) �


Walras’ Law?

p1x∗1 + p2x

∗2 =

mpρ

ρ−11 +mp

ρρ−12

pρ

ρ−11 + p

ρρ−12

= m �

Uniqueness?

There is only one (x∗1, x

∗2) pair that maximizes U, because U is strictly quasiconcave. We already

verified this geometrically. To see it analytically, consider two cases:

Case 1 ρ > 0. We must show

[(λx1 + (1− λ)y1)ρ + (λx2 + (1− λ)y2)ρ

] 1ρ

> min{(xρ1 + xρ2)1ρ , (yρ1 + yρ2)

1ρ }

for λ ∈ (0, 1) and x �= y. We already argued in question 3 that the function zρ is strictly concave for

0 < ρ < 1. So,

(λx1 + (1− λ)y1)ρ ≥ λxρ1 + (1− λ)yρ1

(λx2 + (1− λ)y2)ρ ≥ λxρ2 + (1− λ)yρ2 ,

and one of these inequalities is strict since x �= y. Hence,

(λx1 + (1− λ)y1)ρ + (λx2 + (1− λ)y2)ρ > λ(xρ1 + xρ2) + (1− λ)(yρ1 + yρ2).

Since ρ > 0, the function z1ρ is strictly increasing in z. So,

[(λx1 + (1− λ)y1)ρ + (λx2 + (1− λ)y2)ρ]1ρ > [λ(xρ1 + xρ2) + (1− λ)(yρ1 + yρ2)]

1ρ

≥ [λmin{xρ1 + xρ2, yρ1 + yρ2}+ (1− λ)min{xρ1 + xρ2, y

ρ1 + yρ2}]

1ρ

= [min{xρ1 + xρ2, yρ1 + yρ2}]

1ρ

= min{(xρ1 + xρ2)1ρ , (yρ1 + yρ2)

1ρ }.

Case 2 ρ < 0. Now zρ is strictly convex, so using the same reasoning as above yields

(λx1 + (1− λ)y1)ρ + (λx2 + (1− λ)y2)ρ < λ(xρ1 + xρ2) + (1− λ)(yρ1 + yρ2)

≤ max{xρ1 + xρ2, yρ1 + yρ2}.

Since ρ < 0, z1ρ is strictly decreasing, so

[(λx1 + (1− λ)y1)ρ + (λx2 + (1− λ)y2)ρ]1ρ > [max{xρ1 + xρ2, y

ρ1 + yρ2}]

1ρ .


Suppose xρ1 + xρ2 ≥ yρ1 + yρ2 . Then

[max{xρ1 + xρ2, yρ1 + yρ2}]

1ρ = (xρ1 + xρ2)

1ρ .

But, since ρ < 0, (xρ1 + xρ2)1ρ ≤ (yρ1 + yρ2)

1ρ , so (xρ1 + xρ2)

1ρ = min

{(xρ1 + xρ2)

1ρ , (yρ1 + yρ2)

1ρ

}. The same

argument applies if xρ1 + xρ2 < yρ1 + yρ2 . Generally,

[max{xρ1 + xρ2, yρ1 + yρ2}]

1ρ = min

{(xρ1 + xρ2)

1ρ , (yρ1 + yρ2 )

1ρ

},

yielding the result.

Nonnegativity? x∗ ≥ 0 is obvious.

Now check the properties of U∗ :

Homogeneity?

U∗(αp, αm) = (αm)[(αp1)

ρρ−1 + (αp2)

ρρ−1

] 1−ρρ

= (αm)α−1[p

ρρ−11 + p

ρρ−12

] 1−ρρ

= U∗(p,m) �

Monotonicity?

∂U∗

∂m=[p

ρρ−11 + p

ρρ−12

] 1−ρρ

> 0

∂U∗

∂pi= m

1− ρ

ρ

[p

ρρ−11 + p

ρρ−12

] 1−2ρρ

(ρ

ρ− 1

)p

1ρ−1i

= −m[p

ρρ−11 + p

ρρ−12

] 1−2ρρ

p1

ρ−1i < 0.

Quasiconvex in (p,m)?

We will only show this in p. Let δ = ρρ−1 so that U∗ = m

[pδ1 + pδ2

]− 1δ . δ ranges between −∞ and 0 as

ρ moves from 1 to 0, and δ ranges from 0 to 1 as ρ moves from 0 to −∞. So, consider two cases:

Case 1 δ ∈ (0, 1). Then, as before, pδ1 + pδ2 is a strictly concave function of (p1, p2), since the Hessian

is [δ(δ − 1)pδ−2

1 00 δ(δ − 1)pδ−2

2

],

which is negative semidefinite. So, for any two price vectors (p1, p2) �= (q1, q2) we have

(λp1 + (1− λ)q1)δ + (λp2 + (1− λ)q2)δ > λ(pδ1 + pδ2

)+ (1− λ)

(qδ1 + qδ2

)for λ ∈ (0, 1).


Thus, since δ > 0,

m[(λp1 + (1− λ)q1)δ + (λp2 + (1− λ)q2)δ

]− 1δ ≤ m

[λ(pδ1 + pδ2) + (1− λ)(qδ1 + qδ2)

]− 1δ .

Also, the function z−1δ is strictly convex since its second derivative is −1

δ (−1δ − 1)z−

1δ −2 > 0. So,

[λ(pδ1 + pδ2) + (1 − λ)(qδ1 + qδ2)

]− 1δ < λ

(pδ1 + pδ2

)− 1δ + (1− λ)

(qδ1 + qδ2

)− 1δ .

Combining inequalities shows that U∗ is convex in p.

Case 2 δ < 0. Then pδ1 + pδ2 is strictly convex in (p1, p2), so

(λp1 + (1− λ)q1)δ + (λp2 + (1− λ)q2)δ < λ(pδ1 + pδ2) + (1− λ)(qδ1 + qδ2).

Since −1δ > 0, this inequality is unchanged when we raise both sides to the −1

δ power:

[(λp1 + (1− λ)q1)δ + (λp2 + (1− λ)q2)

δ]− 1

δ

<[λ(pδ1 + pδ2

)+ (1− λ)

(qδ1 + q2

)δ]− 1δ

≤ [λmax{pδ1 + pδ2, q

δ1 + qδ2

}+ (1− λ)max

{pδ1 + pδ2, q

δ1 + qδ2

}]− 1δ

= max{(

pδ1 + pδ2)− 1

δ ,(qδ1 + qδ2

)− 1δ

},

so U∗ is quasiconvex in p.

Continuity? This is clear by inspection, provided ρ �= 0 and ρ �= 1. Note that U∗(p,R1+) = [0,∞) =

Y ∀p >> 0.

c. For linear utility, the problem is just like problem 5 from Lecture 11 (when α1 = α2–that is, the slope

of the indifference curve is −1), except that all points on the budget line are optimal when p1 = p2.

Hence,

x∗1 =

mp1

, if p1p2

< 1[0, mp1

], if p1

p2= 1

0, if p1p2

> 1

; x∗2 =

0, if p1

p2< 1

mp2

− x∗1, if p1

p2= 1

mp2

, if p1p2

> 1

; U∗ =

{mp1

, if p1p2

≤ 1mp2

, if p1p2

> 1

=m

min{p1, p2} .

For the comparison to CES, write

x∗1 =

m

p1 +(pρ2p1

) 1ρ−1

.


Letting ρ go to 1 yields m

p1+(

p2p1

)−∞ . If p2 > p1 this is mp1

, while p2 < p1 yields 0. If p2 = p1 then

x∗1 = m

2p1, the middle of the budget line. The same arguments work for x∗

2. Similarly, factor out p2 in

U∗ to get

U∗ =m

p2

[(p1

p2

) ρρ−1

+ 1

] 1−ρρ

.

Letting ρ go to 1 sends the exponent ρρ−1 to −∞, so

[(p1p2

) ρρ−1

+ 1]1−ρ

ρ

→ 1 when p1 > p2. Since

1−ρρ → 1, we get m

p2. Just reverse the argument for p1 < p2.

For Leontief utility, U = min{x1, x2} and so the indifference curves are as shown below. So, x∗1 = x∗

2 =

mp1+p2

and U∗ = mp1+p2

.

A typical indifference curve for Leontief utility

For the comparison to CES, note that 1ρ−1

→ 0 as ρ → −∞ and ρρ−1

→ 1. So,

x∗1 =

mp1

ρ−11

pρ

ρ−11 + p

ρρ−12

→ mp01

p1 + p2=

m

p1 + p2,

and the same for x∗2. Also,

1−ρρ → −1 so

U∗ = m[p

ρρ−11 + p

ρρ−12

] 1−ρρ → m[p1 + p2]−1.

d. x∗1x∗2=(p1p2

) 1ρ−1

. So,∂

(x∗1

x∗2

)∂(

p1p2

) = 1ρ−1

(p1p2

) 1ρ−1−1

.

⇒ ξ12 =−1

ρ− 1

(p1

p2

) 1ρ−1−1 (

p1

p2

)(p2

p1

) 1ρ−1

=−1

ρ− 1.


Note that this does not vary with p1, p2.

The linear case is obtained by letting ρ go to 1, so ξ12 = ∞.

The Leontief case lets ρ → −∞, so ξ12 = 0.

The Cobb-Douglas case lets ρ → 0, so ξ12 = 1. Note that x∗1 → m

2p1and x∗

2 → m2p2

as ρ → 0, which

are the C-D demands when the exponents are 12 .


Solutions to Exercises for Lecture 14 (Expenditure Minimization)

1.

a. Homogeneity: e∗(αp, U) = αpU = αe∗(p, U) �

Concavity: ∂2e∗∂p2 = 0 �

Monotonicity: ∂e∗∂p = U ≥ 0 for U ≥ 0, which is where e∗ ≥ 0. ∂e∗

∂U= p > 0 for p > 0 �

Continuity: e∗ only involves products, so it is continuous.

Nonnegativity: e∗ > 0 ∀ p > 0 when U > 0. e∗ = 0 ∀ p when U = 0 �

b. h∗ = ∂e∗∂p

= U

c. Homogeneity: h∗(αp, U) = U = h∗(p, U) �

Symmetry and Semidefiniteness: ∂h∗∂p = 0, which is symmetric and negative semidefinite.

Monotonicity: p · h∗ = pU , which is strictly increasing in U for p > 0.

Continuity: p · h∗ = pU , which only involves products.

Nonnegativity: h∗ = U ≥ 0 ∀ U ≥ 0. ph∗ = 0 when U = 0, and h∗ > 0 when p > 0 and U > 0.

2. (Varian #7.2) For given U, the indifference curve is

So, h∗

1 = 0 and h∗2 = U, when p1

p2> 1

h∗1 = U and h∗

2 = 0, when p1p2

< 1

h∗1 = either 0 or U and h∗

2 = either U or 0, respectively, when p1p2

= 1.

212 SOLUTIONS TO EXERCISES FOR LECTURE 14 (EXPENDITURE MINIMIZATION)

Therefore,

e∗(p, U) =

{p1 · 0 + p2U, when p1

p2≥ 1

p1U + p2 · 0, when p1p2

< 1= U min{p1, p2}.

3. (Varian #8.8) To do this, we must first do problem 8.6:

L = x121 x

132 − λ(p1x1 + p2x2 −m)

The FOC’s are

12x−1

21 x

132 = λp1;

13x

121 x

− 23

2 = λp2.

So

3x2

2x1=

p1

p2

From the budget constraint:

p1x1 + p2

(p1

p2

23x1

)= m ⇒ x∗

1 =3m5p1

; x∗2 =

2m5p2

.

⇒ U∗ =(3m5p1

) 12(2m5p2

) 13

=(m5

) 56(

3p1

) 36(

2p2

) 26

.

Solve for m to get e∗:

(U∗)65 =

m

5

(3p1

) 35(

2p2

) 25

m = 5(U∗)65

(p1

3

)35(p2

2

) 25

e∗(p, U) = 5U65(p1

3

) 35(p2

2

) 25.

Use Shephard’s Lemma to get h∗:

h∗1 =

∂e∗

∂p1= 5U

65(p1

3

)− 25(p2

2

) 25 1335= U

65

(3p2

2p1

) 25

h∗2 =

∂e∗

∂p2= 5U

65(p1

3

) 35(p2

2

)− 35 1225= U

65

(2p1

3p2

) 35

.

Now proceed to problem 8.8:

Since U∗ is a monotonic transform of U, we know x∗1 and x∗

2 are unchanged. We can verify this, if desired:

L =12ln(x1) +

13ln(x2)− λ(p1x1 + p2x2 −m).


The FOC’s are:

12x1

= λp1;1

3x2= λp2.

So

3x2

2x1=

p1

p2.

This is the same condition as before, so x∗1 = 3m

5p1and x∗

2 = 2m5p2

.

⇒ U∗ =12ln(3m5p1

)+

13ln(2m5p2

)= ln

(3m5p1

) 12

+ ln(2m5p2

) 13

.

So eU∗=(

3m5p1

) 12(

2m5p2

) 13, which is U∗ in problem 8.6. To find e∗, we must solve U∗ for m. This involves

taking the exponential above and then solving for m, which is the same derivation we did before. Thus,

it yields

e∗(p, U) = 5(eU )65

(p1

3

) 35(p2

2

) 25.

Obviously, differentiating this yields the same Hicksian demands as before, with U replaced by eU .

4. (MWG #3.E.2) First, check the expenditure function: e∗(p, U) = α−α(1 − α)α−1pα1 p1−α2 U . Note that

U(0) = 0 and Y = [0,∞).

Homogeneous?

e∗(λp, U) = α−α(1− α)α−1(λp1)α(λp2)1−αU

= λα−α(1− α)α−1pα1 p1−α2 U

= λe∗(p, U) �

Monotonicity?

∂e∗

∂p1= α1−α(1− α)α−1pα−1

1 p1−α2 U ≥ 0 for U ≥ 0 �

∂e∗

∂p2= α−α(1 − α)αpα1 p

−α2 U ≥ 0 for U ≥ 0 �

∂e∗

∂U= α−α(1 − α)α−1pα1 p

1−α2 > 0 �

Concavity?

∂2e∗

∂p21

= −α1−α(1− α)αpα−21 p1−α

2 U ≤ 0 for U ≥ 0.


It is unnecessary to check the Hessian determinant (it is zero, by homogeneity). So, e∗ is concave.

Continuity? Obvious by inspection.

Nonnegativity? e∗(p, U) > 0 for (p, U) >> 0 and e∗(p, 0) = 0 are obvious.

Now check the Hicksian demands:

Homogeneity?

h∗1(λp, U) =

(αλp2

(1 − α)λp1

)1−αU =

(αp2

(1− α)p1

)1−αU = h∗

1(p, U) �

h∗2(λp, U) =

((1 − α)λp1

αλp2

)αU =

((1− α)p1

αp2

)αU = h∗

2(p, U) �

Symmetry and Semidefiniteness? We know the Hessian of e∗ is negative semidefinite for U ≥ 0 from

above. Symmetry is verified by checking

∂h∗1

∂p2=(

α

1− α

)1−α(1− α)pα−1

1 p−α2 U =∂h∗

2

∂p1.

Monotonicity?

p · h∗ = p1h∗1 + p2h

∗2 =

(αp2

1− α

)1−αpα1 U +

((1− α)p1

α

)αp1−α2 U

=

[(α

1− α

)1−α+(1− α

α

)α]pα1 p

1−α2 U = e∗,

which we already showed is strictly increasing in U .

Nonnegativity? h∗ ≥ 0 ∀p >> 0 and U ≥ 0; p · h∗(p, 0) = 0; and h∗(p, U) > 0 for U > 0 and p >> 0

are clear by inspection.

Excess Utility?

U(h∗1, h

∗2) =

[αp2

(1− α)p1

]α(1−α)[(1− α)p1

αp2

]α(1−α)

UαU

1−α

=

[αp2

(1− α)p1

]α(1−α)−α(1−α)

Uα+1−α

= U �

Uniqueness? h∗ is unique because U is strictly quasiconcave. We verified this for the Cobb-Douglas

utility function in problem 3 of the exercises for Lecture 12.

5. (MWG #3.E.6) We want to min p1x1 + p2x2 subject to [xρ1 + xρ2]1ρ ≥ U. We verified in problem 5 of the

exercises for Lecture 12 that the solution is interior for ρ �= 1 and ρ �= 0. We also derived the MRS as


−(x1x2)ρ−1 so that the tangency condition gives

x1 =(p1

p2

) 1ρ−1

x2.

Substitute into the indifference curve xρ1 + xρ2 = Uρto get

(p1p2

) ρρ−1

xρ2 + xρ2 = Uρ. So

h∗2 =

U[(p1p2

) ρρ−1

+ 1] 1

ρ

=Up

1ρ−12[

pρ

ρ−11 + p

ρρ−12

] 1ρ

=Upδ−1

2[pδ1 + pδ2

]1−1δ

, where δ =ρ

ρ− 1.

So,

h∗1 =

(p1

p2

) 1ρ−1 U[(

p1p2

) ρρ−1

+ 1] 1

ρ

=U[(

p2p1

) ρρ−1

+ 1] 1

ρ

=Up

1ρ−11[

pρ

ρ−12 + p

ρρ−11

] 1ρ

=Upδ−1

1[pδ1 + pδ2

]1−1δ

.

Therefore,

e∗(p, U) =U[

pδ1 + pδ2]1− 1

δ

[pδ1 + pδ2

]= U

[pδ1 + pδ2

] 1δ .

Properties of e∗: Note first that U(0) = 0 and Y = [0,∞).

Homogeneity?

e∗(αp, U) = U[(αp1)δ + (αp2)δ

] 1δ = Uα

[pδ1 + pδ2

] 1δ = αe∗(p, U) �

Monotonicity?

∂e∗

∂pi=

U

δ

[pδ1 + pδ2

] 1δ −1

δpδ−1i ≥ 0 for U ≥ 0;

∂e∗

∂U=[pδ1 + pδ2

] 1δ > 0.

Concavity?

∂2e∗

∂p2i

= U[pδ1 + pδ2

] 1δ −2

(1δ− 1)δp

2(δ−1)i + U

[pδ1 + pδ2

] 1δ −1

(δ − 1)pδ−2i

= U[pδ1 + pδ2

] 1δ −1

(1− δ)pδ−2i

[(pδ1 + pδ2)

−1pδi − 1] ≤ 0 for U ≥ 0.

It is unnecessary to check the Hessian determinant (it is zero, by homogeneity). So, e∗ is concave.

Continuity? Clear by inspection.

Nonnegativity? e∗(p, U) > 0 for (p, U) >> 0 and e∗(p, 0) = 0 are obvious.

Properties of h∗:

Homogeneity?

h∗i (αp, U) =

U(αpi)δ−1

[(αp1)δ + (αp2)δ]1−1

δ

=αδ−1Upδ−1

i

αδ−1[pδ1 + pδ2

]1−1δ

= h∗i (p, U) �


Symmetry and Semidefiniteness?

∂h∗i

∂pi= U

[pδ1 + pδ2

] 1δ −1

(1− δ)pδ−2i

[(pδ1 + pδ2)

−1pδi − 1] ≤ 0 for U ≥ 0

∂h∗1

∂p2= U

(1δ− 1)[

pδ1 + pδ2] 1

δ −2δ(p1p2)δ−1 =

∂h∗2

∂p1.

So the determinant is

∂h∗1

∂p1

∂h∗2

∂p2− ∂h∗

1

∂p2

∂h∗2

∂p1= U

2[pδ1 + pδ2]

3δ −2(1− δ)2(p1p2)δ−2[(

pδ1pδ1 + pδ2

− 1)(

pδ2pδ1 + pδ2

− 1)− (pδ1 + pδ2)

−2(p1p2)δ]

= U2[pδ1 + pδ2]

3δ −2(1− δ)2(p1p2)δ−2[0] = 0.

Monotonicity? p · h∗ = p1h∗1 + p2h

∗2 = U

[pδ1 + pδ2

] 1δ −1 [

pδ1 + pδ2]= U

[pδ1 + pδ2

] 1δ . This, of course, is e∗,

which we already verified is strictly increasing in U .

Nonnegativity? h∗ ≥ 0 for p >> 0 and U ≥ 0; p · h∗(p, 0) = 0; and h∗(p, U) > 0 for (p, U) >> 0 are all

clear by inspection.

Excess utility?

U(h∗) =[h∗ρ

1 + h∗ρ2

] 1ρ

=U[p

ρρ−11 + p

ρρ−12

] 1ρ

[p

ρρ−11 + p

ρρ−12

] 1ρ

= U �

Uniqueness? Again, h∗ is unique because U is strictly quasiconcave, as verified in problem 5 of the

exercises for Lecture 12.

6. (MWG #3.G.3 except (c) and (e))

a. Since u(x) = (x1 − b1)α(x2 − b2)β(x3− b3)γ , we can use a monotonic transformation of the given utility

function to get lnu(x) = α ln(x1 − b1) + β ln(x2 − b2) + γ ln(x3 − b3). The first order condition of the

EMP yield the Hicksian demand function

h(p, u) = (b1, b2, b3) + u(α/p1)−α(β/p2)−β(γ/p3)−γ(α/p1, β/p2, γ/p3).

Multiplying p · h(p, u), we obtain the expenditure function

e(p, u) = p · b+ u(α/p1)−α(β/p2)−β(γ/p3)−γ ,


where b = ( b1 b2 b3 ) and we have used α+ β + γ = 1. Now we can check Proposition 3.E.2:

i.

e(λp, u) = λpb+ (1/λ)−(α+β+γ)u(α/p1)−α(β/p2)−β(γ/p3)−γ = λe(p, u)

ii.

∂e(p, u)∂u

= (α/p1)−α(β/p2)−β(γ/p3)−γ > 0

∂e(p, u)∂p

= h(p, u) = (b1, b2, b3) + u(α/p1)−α(β/p2)−β(γ/p3)−γ(α/p1, β/p2, γ/p3) ≥ 0.

iii.

D2pe(p, u) = Dph(p, u) = u(p1/α)α(p2/β)β(p3/γ)γ

α(α− 1)/p21 αβ/p1p2 αγ/p1p3

αβ/p1p2 β(β − 1)/p22 βγ/p2p3

αγ/p1p3 βγ/p2p3 γ(γ − 1)/p23

.

We must show that this matrix is negative semidefinite. The diagonal elements are negative because

α, β, γ < 1. The 2× 2 naturally-ordered principal minor is |H2| = αβ(1−α−β)p21p

22

> 0. The 3× 3 naturally-

ordered principal minor is

|H3| = 1p21p

22p

33

[αβγ(α − 1)(β − 1)(γ − 1) + α2β2γ2 + α2β2γ2

− α2βγ2(β − 1)− α2β2γ(γ − 1) − αβ2γ2(α− 1)] = 0.

So H is negative semidefinite.

iv. Continuity of e in p and u is clear by inspection, provided p >> 0.

Finally, check Proposition 3.E.3:

i.

h(λp1, u) = b1 + u(α/λp1)−α(β/λp2)−β(γ/λp3)−γα/λp1 = h(p1, u)

ii. Let A = (α/p1)−α(β/p2)−β(γ/p3)−γ . Then

U(h) =[uA(α/p1)]α[uA(β/p2)]β[uA(γ/p3)]γ

(uA)α+β+γ(α/p1)α(β/p2)β(γ/p3)γ

(uA)1A−1 = u.


iii. h is clearly unique (it only takes on one value for each p and u. To see that this is due to the strict

quasiconcavity of u, we can check the Hessian of lnu:

Hu =

−α

(x1−b1)2−β

(x2−b2)2 −γ(x3−b3)2

.

This matrix is obviously negative definite, so u is strictly concave (and hence strictly quasiconcave).

b.

∂e(p, u)∂p

= h(p, u) = (b1, b2, b3) + u(α/p1)−α(β/p2)−β(γ/p3)−γ(α/p1, β/p2, γ/p3).

d. The Hessian of e derived above is clearly symmetric (the compensated cross-price effects are symmetric),

and all diagonal elements are negative (the own substitution terms are negative).

7. (MWG #3.G.14) Symmetry immediately yields−10 ? 3? −4 ?3 ? ?

.

By homogeneity of degree 0 of h∗ and Euler’s Theorem:

n∑j=1

pj∂h∗

i

∂pj= 0 for i = 1, · · · , n (i.e., (iv) from proposition 3.G.2)

Thus, for i = 1 we have 1(−10) + 2(?) + 6(3) = 0, or ? = −4. Applying symmetry again yields−10 −4 3−4 −4 ?3 ? ?

.

Now, using row 2, 1(−4)+2(−4)+6(?) = 0, so ? = 2. Symmetry and row 3 then gives 1(3)+2(2)+6(?) = 0,

so ? = −76 . Thus we have −10 −4 3

−4 −4 23 2 −7

6

.

This is symmetric by construction and is negative semidefinite because the naturally-ordered principal

minors alternate in sign:

−10 < 0; (−10)(−4) − (−4)(−4) = 24 > 0; (−20)73− 24− 24 + 36 + 40 + 8(7

3) = 0.

8. (MWG #3.G.15) First, check the shape of utility function u(x) = 2x1/21 + 4x1/2

2 :

dx2

dx1= −(1/2)

(x2

x1

)1/2

< 0.


So

d2x2

dx21

= −(1/4)(x2

x1

)−1/2[∂x2∂x1

x1 − x2

x21

]> 0.

Note also that ∂x2∂x1

approaches infinity as x1 → 0 and approaches 0 as x1 → ∞. Therefore, calculus will

always describe the optimum.

a.

max u = 2x1/21 + 4x1/2

2 , s.t. p1x1 + p2x2 = m.

L = 2x1/21 + 4x1/2

2 − λ(p1x1 + p2x2 −m).

FOC’s:

x1 : x−1/21 − λp1 = 0, (1)

x2 : 2x−1/22 − λp2 = 0, (2)

λ : p1x1 + p2x2 = m, (3)

From (1) and (2), x2 = 4x1p21

p22. So

p1x1 + p24x1p

21

p22

= m

x∗1(p,m) =

m

p1 + 4p21/p2

=mp2

p1(p2 + 4p1)

x∗2(p,m) =

m

p2 + p22/4p1

=4mp1

p2(p2 + 4p1).

b.

minp1x1 + p2x2, s.t. 2x1/21 + 4x1/2

2 ≥ u

L = p1x1 + p2x2 − λ(2x1/21 + 4x1/2

2 − u).

FOC’s:

x1 : p1 − λx−1/21 = 0, (1)

x2 : p2 − 2λx−1/22 = 0, (2)

λ : 2x1/21 + 4x1/2

2 = u, (3)


From (1) and (2), x2 = 4x1p21

p22. So

2x1/21 + 4

2p1

p2x

1/21 = u

h∗1(p, u) =

u2

(2 + 8p1/p2)2=

p22u

2

4(p2 + 4p1)2

h∗2(p, u) =

u2

(4 + p2/p1)2=

p21u

2

(p2 + 4p1)2.

c. The expenditure function is

e∗(p, u) = p1h1(p, u) + p2h2(p, u)

= (p1p22 + 4p2p

21)

u2

4(p2 + 4p1)2

=u2p1p2

4(p2 + 4p1).

Differentiating e∗ yields

∂e∗

∂p1=

u2p2(p2 + 4p1) − u2p1p244(p2 + 4p1)2

=u2p2

2

4(p2 + 4p1)2= h∗

1

∂e∗

∂p2=

u2p1(p2 + 4p1) − u2p1p2

4(p2 + 4p1)2=

u2p21

(p2 + 4p1)2= h∗

2

d. Invert e∗ from part c to obtain

u∗(p,m) = 2[m

p1+

4mp2

]1/2.

To verify Roy’s identity, first find the partials of u∗:

∂u∗

∂m= m−1/2

[1p1

+4p2

]1/2∂u∗

∂p1= m1/2

[1p1

+4p2

]−1/2(− 1p21

).

So

−∂u∗∂p1∂u∗∂m

=m

p21

[1p1

+4p2

]−1

=mp2

p1(p2 + 4p1)= x∗

1.

x∗2 is similar.


Solutions to Exercises for Lecture 15 (Utility and Expenditure Relationships)

1. Indirect utility is m = U∗√p1p2, or U∗ = m(p1p2)−12 . Now use Roy’s Identity:

∂U∗

∂p1= −1

2m

p321 p

122

;∂U∗

∂p2= −1

2m

p121 p

322

;∂U∗

∂m=

1

p121 p

122

.

So,

x∗1 = −

−12

m

p321 p

122

1

p121 p

122

=m

2p1and x∗

2 =m

2p2.

2. Yes, because the matrix of price derivatives of the Marshallian demands is symmetric when preferences

are homothetic. So, there is no ambiguity in the definition of a substitute or complement in this case.

3. (Varian #7.3) Invert indirect utility to get the expenditure function: e(p1, p2, u) = umin{p1, p2}. By

Roy’s identity, the Marshallian demands are:

x1 ={ m

p1, p1 < p2

0, p1 > p2

, x2 ={ 0, p1 < p2

mp2

, p1 > p2

Substitute these into indirect utility to get

u(x1, x2) ={

x1, p1 < p2

x2, p1 > p2

The only way utility can take on these values is if the indifference curve is linear with a slope of -1 and an

intercept of u (or strictly concave with intercepts on the two axes both equal to u). So u(x1, x2) = x1+x2

is the simplest utility function that is consistent with the given indirect utility function.

4. (Varian #7.4(b) and (c))

b. U = e∗p1+p2

, so e∗ = (p1 + p2)U .

c. By Shephard’s Lemma, h∗1 = U and h∗

2 = U. So, the system from which we must “eliminate” price is

x1 = U and x2 = U.

Since the prices don’t appear in this system, we know that expenditure minimizing choices must satisfy

x1 = x2 = U no matter what the prices are. This can only happen if U(x1, x2) = min{x1, x2}.

5. (Varian #8.1) We know that

xj(p,m) ≡ hj(p, v(p,m)) ≡ ∂e(p, v(p,m))/∂pj .

222 SOLUTIONS TO EXERCISES FOR LECTURE 15 (UTILITY AND EXPENDITURE RELATIONSHIPS)

Differentiating with respect to m gives us

∂xj∂m

=∂2e(p, v(p,m))

∂pj∂u

∂v(p,m)∂m

.

Since the marginal utility of income, ∂v/∂m, must be positive, the result follows.

6. (Varian #8.15) Let L be the Marshallian demand for leisure, Ls be the Hicksian demand for leisure, w

be the price of leisure (i.e., w is the wage rate), and L be the time endowment. “Income” is m + wL

(see the discussion on pages 145-6 of Varian), which complicates things because a change in the price of

leisure affects money income. So, if we think of L as a function of prices and income in the usual way,

the relevant arguments of L are (w,m+ wL). Thus the total effect of a change in w on L is

∂L

∂w= L1 + L2L.

From Slutsky’s equation,

L1 =∂Ls

∂w− LL2.

Substituting yields

∂L

∂w=

∂Ls

∂w+ L2(L− L).

Now note that the substitution effect is always negative, (L−L) is always nonnegative, and hence if leisure

is inferior, ∂L∂w is necessarily negative. This says that the slope of the demand for leisure is negative when

leisure is inferior. Since the supply of labor is just L − L, the slope of the labor supply curve is positive

in this circumstance.

7. (MWG #2.F.10(a)) Consider x1. Its derivatives are

∂x1

∂p1= −p2

p21

(p1 + p2 + p3)−1 − p2

p1(p1 + p2 + p3)−2 = −1

3− 1

9= −4

9∂x1

∂p2=

1p1

(p1 + p2 + p3)−1 − p2

p1(p1 + p2 + p3)−2 =

29

∂x1

∂p3= −p2

p1(p1 + p2 + p3)−2 = −1

9.

So the first row of the (Marshallian) substitution matrix is:

(∂x1∂p1

∂x1∂p2

∂x1∂p3

)= (−4

929 −1

9 ) .


Similar exercises on x2 and x3 yield the second and third rows, respectively, of the substitution matrix.

The final result is

19

−4 2 −1−1 −4 22 −1 −4

.

Hence the substitution matrix is clearly not symmetric. The diagonal elements are negative, the 2 × 2

naturally-ordered principle minor is |H2| = 2/9 > 0, and the 3 × 3 naturally-ordered principle minor is

|H3| = −1/9 < 0. So, the substitution matrix is negative definite.

8. (MWG #3.G.2) Use example 3.D.1 to get the Marshallian Demands for the general Cobb-Douglas U =

Kxα1x1−α2 :

x∗1 =

αm

p1; x∗

2 =(1− α)m

p2; U∗ = K

(αm

p1

)α ((1 − α)mp2

)1−α.

Use example 3.E.1 to get

h∗1 =

(αp2

(1− α)p1

)1−αU ; h∗

2 =((1− α)p1

αp2

)αU ; e∗ = α−α(1− α)α−1pα1 p

1−α2 U.

Verify Prop. 3.G.1:

∂e∗

∂p1= α−α(1− α)α−1

(p1

p2

)α−1

Uα

=(1− α

α

)α−1(p1

p2

)α−1

U = h∗1 �

∂e∗

∂p2= α−α(1− α)α−1

(p1

p2

)αU(1− α)

=(1− α

α

)α (p1

p2

)αU = h∗

2 �

Verify Prop. 3.G.2:

Hessian of e∗ is (1− α)(

αp2(1−α)p1

)−α ( −αp2(1−α)p21

)U (1− α)

(αp2

(1−α)p1

)−αα

(1−α)p1U

(1− α)(

αp2(1−α)p1

)−αα

(1−α)p1U (−α)

(αp2

(1−α)p1

)−α−1α

(1−α)p1U

.

i. The fact that this is the Jacobian of h∗ is obvious from the definitions, as is (iii) symmetry.

ii. The diagonal elements are clearly negative, and

|H | =(

αp2

(1− α)p1

)−2α−1α3p2U

2

(1 − α)p31

−(

αp2

(1− α)p1

)−2αα2U

2

p21

=(

αp2

(1− α)p1

)−2α[(1− α)p1

αp2

α3p2U2

(1− α)p31

− α2U2

p21

]= 0.


So, H is negative semidefinite.

iv.

Hp =

[−( αp2

(1−α)p1

)−α αp2Up1

+(

αp2(1−α)p1

)−ααp2Up1(

αp2(1−α)p1

)−ααU − ( αp2

(1−α)p1

)−α−1 α2p2U(1−α)p1

]

=[

0(αp2

(1−α)p1

)−α[αU − αU

]=[00

].

Verify Prop. 3.G.3:

From above, ∂x∗i

∂pj= 0 for i �= j and ∂x∗

1∂p1

= −αmp21

; ∂x∗2

∂p2= −(1−α)m

p22. Also, ∂x

∗1

∂m= α

p1and ∂x∗

2∂m

= 1−αp2

.

So, for example, ∂x∗1

∂p2+ x∗

2∂x∗

1∂m = 0 + (1−α)m

p2αp1

. From the Hessian above, ∂h∗1

∂p2=(

αp2(1−α)p1

)−ααUp1

.

Evaluating at U = U∗(p,m) = (αmp1 )α( (1−α)m

p2

)1−α yields

∂h∗1

∂p2=(

αp2

(1 − α)p1

)−αα

p1

(αp2

(1 − α)p1

)αm(1− α)

p2

=αm(1− α)

p1p2=

∂x∗1

∂p2+ x∗

2

∂x∗1

∂m�

The others are verified analogously.

Verify Prop. 3.G.4:

∂U∗

∂p1= α

(αm

p1

)α−1( (1− α)mp2

)1−α(−αm

p21

)= −

(α

p1

)α−1(1− α

p2

)1−αα2m

p1

∂U∗

∂m=(

α

p1

)α(1− α

p2

)1−α.

So,

−∂U∗∂p1∂U∗∂m

=α2m

p21

(α

p1

)−1

=αm

p1= x∗

1 �

Similar for x∗2.

9. (MWG #3.G.3(c) and (e))

c. From exercise 5 of lecture 13,

Dph(p, u) = u(p1/α)α(p2/β)β(p3/γ)γ

α(α− 1)/p21 αβ/p1p2 αγ/p1p3

αβ/p1p2 β(β − 1)/p22 βγ/p2p3

αγ/p1p3 βγ/p2p3 γ(γ − 1)/p23

.

We will verify the Slutsky equation for the slope of the demand for commodity 1 with respect to the

price of commodity 2:

∂x1

∂p2=

∂h1

∂p2− x2

∂x1

∂w.


The first step is to derive the Marshallian demands, which are

x1 = b1 + (w − p · b) αp1

x2 = b2 + (w − p · b) βp2

x3 = b3 + (w − p · b) γ

p3,

where b = ( b1 b2 b3 ). Therefore, for the Slutsky Equation we must check whether

−b2α

p1= u(p1/α)α(p2/β)β(p3/γ)γ

αβ

p1p2−[b2 + (w − p · b) β

p2

]α

p1,

where u = u∗(p, w) = [w− p · b][(α/p1)α(β/p2)β(γ/p3)γ ] (from inversion of the e∗ obtained in exercise

5 of lecture 13). Substituting for u makes the condition to be checked become

−b2α

p1=

[w− p · b]αβp1p2

−[b2 + (w − p · b) β

p2

]α

p1

−b2α

p1= −b2

α

p1,

which is clearly true. The others are verified analogously.

e. Once we have verified the Slutsky equation for all three commodities with respect to all three prices,

the matrix we are dealing with is just the Hessian of e reproduced above. We already showed in exercise

5a of Lecture 13 that this matrix is negative semidefinite. Although we did not explicitly note that

its rank is 2, we observed: 1) the 1 × 1 naturally-ordered principle minor is strictly negative, 2) the

2× 2 naturally-ordered principle minor is strictly positive, and 3) the 3× 3 naturally-ordered principle

minor is zero. This means the rank is 2.

10. (MWG #3.H.5) By equation (3.E.1), we can recover the expenditure function by simply inverting the

indirect utility function. To get the direct utility function, first use Roy’s identity to obtain the Marshallian

demands. Now we have U∗ and x∗. We are then looking for a function U(x) such that U(x∗) = U∗. To

find U , we must eliminate the prices and income from our system of equations. Set x = x∗ and solve this

system of n equations for the n relative prices pi/m, for i = 1, . . . , n. The solutions will depend only on

x, due to the homogeneity of x∗. Now substitute these “solutions” for the relative prices into U∗. Due to

homogeneity, U∗ only depends on relative prices, so after the substitution we have U∗ depending only on

x. This is the direct utility function. Alternatively (actually, equivalently), we can set x = h∗ and solve


this system of n equations for the n prices. The solution will depend on x and U , so call it p(x, U). Then

write the identity p(x, U) · x = e∗(p(x, U), U) and solve this for U . The solution depends only on x, and

is the direct utility function.

11. (MWG #3.H.6) By equation (3.H.2), ∂e(p, u)/∂p� = α�e(p, u)/p�. The derivatives of e bear this relation-

ship with the value of e when e takes the form e(p, u) = β(ΠL�=1p

α�

� ) for some β that does not depend on

prices. A convenient choice of β is β = (ΠL�=1α

−α�

� )u, because then we obtain e(p, u) = (ΠL�=1(p�/α�)

α�)u.

This is the expenditure function of the Cobb-Douglas utility function u(x) = ΠL�=1x

α�

� .


Solutions to Exercises for Lecture 16 (Consumer Welfare)1.

EV for a Price Decrease (Normal Good). The dark shaded area is the Deadweight Loss.EV is both the light and dark shaded areas.

CV for a Price Decrease (Normal Good). The dark shaded area is the Deadweight Loss.CV is both the (negative) light and dark shaded areas.

228 SOLUTIONS TO EXERCISES FOR LECTURE 16 (CONSUMER WELFARE)

2.

EV for a Price Increase (Inferior Good). The dark shaded area is the Deadweight Loss.EV is both the (negative) light and dark shaded areas.

CV for a Price Increase (Inferior Good). The dark shaded area is the Deadweight Loss.CV is both the light and dark shaded areas.

3. (MWG #3.D.4)

a. A utility function that represents the preference relation takes the form (MWG, p. 50)

U(x) = x1 + φ(x2, · · · , xn).


We wish to maximize this subject to p · x = m. Solve the budget constraint for x1 and substitute into

U to obtain the alternative problem

max(x2,··· ,xL)

m

p1−

L∑i=2

pip1

xi + φ(x2, · · · , xL).

Call this objective function f(x2, · · · , xL), and consider the transformation of f given by

h(f) = f − m

p1.

This transformation is strictly increasing in f, so it makes no difference whether we maximize f or

h(f). But,

h(f) = −L∑i=2

pip1

xi + φ(x2, · · · , xL),

which does not depend on m. Hence, the optimal choices of x2, · · · , xL cannot depend on m. This is

because of the linearity of U in x1, which causes m to drop out of the optimization. From the budget

constraint,L∑i=1

pi∂x∗

i

∂m= 1.

Since x∗i is independent of m for i = 2, · · · , L, this means ∂x∗

1∂m = 1

p1.

b. From the alternate formulation of the maximization problem,

U∗(p,m) =m

p1−

L∑i=2

pip1

x∗i + φ(x∗

2, · · · , x∗L).

Since x∗2, · · · , x∗

L do not depend on m, only the first term here depends on m. Hence, just let everything

else be some function of p, say g(p), so that

U∗(p,m) =m

p1+ g(p).

With p1 normalized to one, this is the specified form.

c. In the alternative formulation of the problem, the constraint x1 ≥ 0 becomes mp1

−∑Li=2

pi

p1xi ≥ 0.

Hence, even with the transformation h, m is still present in the problem through this constraint. As

long as the constraint does not bind at the optimum, the demands for x2, · · · , xL will be (locally)

independent of m. In particular, for the L = 2 case we have an FOC of

−p2

p1+ η′(x∗

2) = 0

230 SOLUTIONS TO EXERCISES FOR LECTURE 16 (CONSUMER WELFARE)

(assuming η is increasing and concave). Hence, as long as this yields x∗2 < m

p2the constraint is not

binding and therefore x∗2 is locally independent of m. A sufficient condition for this is η′(mp2 ) < p2

p1,

because then the choice of x2 = mp2

yields a negative derivative, so it is optimal to reduce x2 and

thereby satisfy the constraint automatically. Thus, assuming η is concave and η′(0) > p2p1

(otherwise

the consumer never buys x2), we have

x∗2 =

{mp2

, when mp2

≤ x2

x2, when mp2

> x2,

where x2 is the value of x2 satisfying −p2p1

+ η′(x2) = 0 (unique if η is strictly concave).

4. (Varian #10.2) Ellsworth’s demand functions for the x-good and the y-good take the form

x = y =150

px + py.

Plugging this into the utility function, we find that the indirect utility function takes the form

v(px, py, 150) =150

px + py.

Hence A is the solution to

150−A

1 + 1=

1501 + 2

.

and B is the solution to

1501 + 1

=150 + B

1 + 2.

Solving, we have A = 50 and B = 75. Of course, A is the equivalent variation of the price increase and

B is the compensating variation of the price increase.

5. (MWG #3.I.2) Denote the deadweight loss given in equation (3.I.5) by DW1(t) and the one in equation

(3.I.6) by DW0(t). Then

DW′1(t) = h1(p0

1 + t, p−1, u1)−

(h1(p0

1 + t, p−1, u1) + t

∂h1(p01 + t, p−1, u

1)∂p1

)= −t

∂h1(p01 + t, p−1, u

1)∂p1

.

Hence DW′1(t) ≥ 0 for all t ≥ 0 and DW

′1(0) = 0. It can be similarly shown that DW

′0(t) ≥ 0 for all t ≥ 0

and DW′0(0) = 0. Note that if h1(p, u) is strictly decreasing, then ∂h1(p,u

0)∂p1

< 0 and the derivative above


is then strictly positive for t > 0. If we look at the graph, we see that the deadweight loss comes from the

change in quantity along the Hicksian demand that occurs when the price changes. Hence, there is always

deadweight loss when the Hicksian demand is strictly downward-sloping, but if the Hicksian demand is

vertical then quantity does not change and so there is no deadweight loss.

232 SOLUTIONS TO EXERCISES FOR LECTURE 17 (CONSUMER DUALITY)

Solutions to Exercises for Lecture 17 (Consumer Duality)

1. (Varian #9.8)

a.

∂e∗

∂p= exp(a− bp+ ce∗).

Then invert to get U∗.

b. Using the normalization U0 = m0 in the notes, e∗(p0, U0; p0) = U0 for some given (p0, U0) >> 0.

2. (MWG #3.G.10)

1. Homogeneous of degree 0 in (p,m) :

a(p) must be homogeneous of degree 0 in p

b(p) must be homogeneous of degree −1 in p

2. Monotonicity:

a(p) and b(p) must both be nonincreasing

b(p) must be positive when p >> 0.

3. Quasiconvexity:

Since a(p) is homogeneous, Euler’s equation yields

n∑i=1

∂a(p)∂pi

pi = 0.

Since a(p) is nonincreasing, ∂a(p)∂pi

≤ 0. So, at p >> 0 we have ∂a∂pi

= 0. That is, a(p) is a constant,

independent of p. Thus, for quasiconvexity we must have b(p) quasiconvex.

4. Continuity: Since a(p) is constant, it is continuous. So, we need b(p) continuous. This automatically

yields U∗(p,R1+) an interval from 0 to some y ∀p >> 0.

3. (MWG #3.G.11) We want x∗2 as a function of x∗

1 for a given p, with m varying parametrically in the

background. By Roy’s Identity,

x∗1 = −

∂a∂p1

+m ∂b∂p1

b(p); x∗

2 = −∂a∂p2

+m ∂b∂p2

b(p).

Solving for m:

m =−1∂b∂p1

[x∗

1b(p) +∂a

∂p1

]; m =

−1∂b∂p2

[x∗

2b(p) +∂a

∂p2

].


So, ∂b∂p2

[x∗

1b(p) +∂a∂p1

]= ∂b

∂p1

[x∗

2b(p) +∂a∂p2

].

Solving for x∗2:

x∗2 =

[∂b∂p2∂b∂p1

[x∗1b(p) +

∂a

∂p1]− ∂a

∂p2

]1

b(p)

=1

b(p)

[∂a

∂p1

∂b∂p2∂b∂p1

− ∂a

∂p2

]+

[∂b∂p2∂b∂p1

]x∗

1.

Since b and a depend on neither m nor x∗, this is a linear function of x∗1.


PAST TESTS


Test 3: December 11, 2007

Throughout, assume market inverse (Marshallian) demand is P = 100− Q for 0 ≤ P ≤ 100 and Q = 0 for

P > 100, where P is the price per unit and Q is the number of units sold. Assume also there are two firms

whose only costs are the constant marginal costs c1 = 50 and c2 = 72.

1. (15) Suppose the two firms compete as Cournot duopolists. Carefully draw the reaction curves, labeling

slopes, intercepts, and the equilibrium values. What is the aggregate quantity in equilibrium? What is

the equilibrium price? How much profit does each firm earn in equilibrium?

2. (5) If firm 1 is a monopoly, how much does it produce? What price does it charge? How much profit does

it earn?

3. (5) Carefully draw a graph of market demand. Mark the prices and aggregate quantities from questions

1 and 2 on your graph.

4. (10) Assume consumers’ income effects are zero at all prices under consideration. On your graph from

question 3, shade the equivalent variation when the market moves from the monopoly in question 2 to

the duopoly in question 1. Calculate this equivalent variation. Does the Cournot competition in question

1 improve social welfare relative to the monopoly situation in question 2? Explain why or why not.

5. (5) If the two firms compete as Bertrand duopolists, what price does each firm charge in equilibrium?

How much does each firm produce? What is the aggregate quantity?

6. (15) Put the price and aggregate quantity from question 5 on your graph from question 3. Continuing to

assume consumers’ income effects are zero, is society better off under Bertrand competition than under

the monopoly in question 2? Why? Contrast your answer with your answer about social welfare from

question 4.

7. (10) Suppose the two firms find a way to collude. How much should each firm produce in order to maximize

collusive profit? Suppose each firm believes they will end up in the Cournot equilibrium if they don’t

collude. How should the collusive profit be divided in order to enforce collusion?

8. (15) Suppose the two firms compete as Stackelberg duopolists with firm 1 the leader and firm 2 the follower.

Find the subgame perfect equilibrium. Use isoprofit lines to illustrate this equilibrium on your graph from

question 1. What is the equilibrium price? How much profit does each firm earn in equilibrium? Still

238 TEST 3: DECEMBER 11, 2007

assuming consumers’ income effects are zero, is society better off under this Stackelberg competition than

under the monopoly situation in question 2? Explain why. Contrast your answer with your answer about

social welfare from question 4.

9. (5) Suppose the two firms are each price-takers. What are the equilibrium price and market quantity?

How much does each firm produce in equilibrium?

10. (15) Now change the assumption that income effects are zero. In particular, suppose the good is normal.

Does this change your answer from question 4 about whether society is better off under Cournot competi-

tion than the monopoly in question 2? Does it change your answer from question 6 about social welfare?

Does it change your answer from question 8 about social welfare? Draw graphs to illustrate and explain

your answers.


Test 2: November 6, 2007

1. (20) Consider a price-taking firm with cost function

c(y) = 1 +{

y when y ≤ 12y2 − 1 when y > 1

where y ≥ 0 is the level of output and the input prices are constant and therefore suppressed. Assume

α% of the fixed cost are avoidable.

a. Derive the avoidable, average, marginal, and average avoidable cost expressions. Carefully graph the

latter three curves, labeling important points on the graph.

b. Derive the supply curve for this firm.

2. (25) Consider a price-taking consumer with fixed money income and utility function U(x1, x2) = x1+g(x2)

on the consumption set R2+, where g is a twice differentiable function with g′ > 0 and g′′ < 0.

a. Write the first order conditions for utility maximization.

b. Manipulate the conditions from part a to obtain an equation that implicitly defines the Marshallian

demand for x2 (assuming both x1 and x2 are positive at the optimum).

c. From your equation in part b, how does a change in income affect consumption of x2?

d. What do you conclude from this about the Hicksian demands for x2?

3. (30) Here is an incompletely defined function that we suspect might be a cost function:

c∗(w1, w2, y) ={ 3w2y when w1

w2≥ 2

3w1y when w2w1

≥ 2.

We want to retrieve an implied input requirement set

H(y) = {(x1, x2) ∈ R2+ : w1x1 +w2x2 ≥ c∗(w1, w2, y) ∀(w1, w2) >> 0}

that would yield c∗ as a cost function.

a. First consider price pairs satisfying w2w1

≥ 2. Which among such pairs make w1x1 + w2x2 smallest for

any given (x1, x2)? Infer from this a condition that every (x1, x2) pair in H(y) must satisfy.

b. Now do the same thing for price pairs satisfying w1w2

≥ 2. Specifically, state a condition that every

(x1, x2) pair in H(y) must satisfy from the w1w2

≥ 2 scenario.

c. Combine your answers to parts a and b to give a proposed H(y) set. Graph your proposed H(y).

240 TEST 2: NOVEMBER 6, 2007

d. Using your proposed H(y), define c∗(w1, w2, y) when 12 < w1

w2< 2 so that c∗ really is the cost function

associated with H(y).

4. (25) Consider a price-taking consumer with utility function on R2+ given by

U(x1, x2) ={

x1 + x2 when x1 �= x2

x1 + x2 + 1 when x1 = x2

.

a. Carefully draw the indifference curve for utility level U > 1. Label the intercepts and any other

important points.

b. Comment on the properties of this consumer’s preferences.

c. Derive the expenditure function for values of U exceeding 1.


Test 1: October 4, 2007

1. (40) Here is a production set: Z = {(0, 0), (−1, 2), (−3, 3), (−4, 5))} (i.e., 4 points in R2).

a. (3) Carefully draw Z.

b. (3) Does Z possess the possibility of inaction property? Why?

c. (3) Does Z possess the no free lunch property? Why?

d. (3) Does Z possess the free disposal property? Why?

e. (18) Let p1 be the price of the first commodity and p2 be the price of the second commodity. Derive

the profit function.

f. (10) Suppose the following points were added to Z:

{(z1, z2) : (z1, z2) < (0, 0) or (z1, z2) < (−1, 2) or (z1, z2) < (−3, 3) or (z1, z2) < (−4, 5)}.

How would your answers to a-e change, if at all?

2. (30) Consider the following optimization problem:

min{K,L}

K subject to g(K,L) ≥ q,

where g is a “well-behaved” real-valued function of the two variables K and L, and q is a parameter.

Suppose we write the Lagrangian function as

L = K − λ[g(K,L) − q].

Let (K∗, L∗, λ∗) be the unique interior solution to this problem. What does λ∗ measure? What property

would the choice of K and L have if λ were above λ∗? What property would the choice of K and L have

if λ were below λ∗? Why?

3. (30) Suppose there are two inputs, x1 and x2, and that w1 and w2 are the prices of one logarithmic unit

of x1 and x2, respectively, so that the cost minimization problem is

min{(x1,x2)≥0}

w1 lnx1 + w2 lnx2 subject to f(x1, x2) ≥ y.

Is the resulting cost function homogeneous and concave in (w1, w2)? Why or why not?



1. (20) Suppose there is a consumer whose preferences do not have the local nonsatiation property. Draw

an Edgeworth box showing why the first welfare theorem might not hold. Fully explain your drawing.

2. (15) A monopolist produces output y with constant marginal cost c ≥ 0. The inverse demand for y is

p = α− βy for y ≥ 0, where α > c and β > 0. Find the monopoly output, price, and profit. Also find the

deadweight loss caused by the monopoly under the assumption that there are no income effects on this

good.

3. (30) Two homogeneous-products Bertrand duopolists each have constant marginal cost c = 4 and compete

for market inverse demand p = 20−y for y ≥ 0. These two sellers choose prices simultaneously (they split

the market if they charge the same price). However, the only possible prices they can choose are either

p = 10 or p = 6 (i.e., each sellers’ strategy space is the doubleton {10, 6}).

a. Describe Nash equilibrium in this model.

b. What, if anything, about equilibrium changes if the two possible prices are p = 10 and p = 7? Why?

c. Now suppose the players move sequentially, with one of them (the leader) credibly committing to a

price that is observed before the other (the follower) makes a price choice. Describe equilibrium if the

only possible prices are p = 10 and p = 6. Describe equilibrium if the only possible prices are p = 10

and p = 7.

4. (35) Consider the following economy. There are two commodities, x1 and x2. Let p denote the price of x1

and assume the price of x2 is normalized to one. There is one firm with production function x1 = 2√x2

for x2 ≥ 0. There is one consumer with utility function U =[x

2/31 + x

2/32

]3/2. This consumer is endowed

with x2 = 3 units of good x2. This consumer is also the sole owner of the firm. Suppose the firm and the

consumer each acts (separately) as a price-taker. Find the equilibrium value of p.



1. (35) Here is a utility function on R2+: U(x1, x2) = x2 − (x1 − 1)2.

a. (10) Carefully graph a few indifference curves, labeling the utility level of each curve, and also labeling

the coordinates of and slopes at points on the axes, if any.

b. (15) State whether the preferences represented by this utility function have each of the following

properties, and explain why:

CompletenessTransitivityContinuityLocal NonsatiationWeak MonotonicityConvexity

c. (10) Find the Marshallian demands.

2. (35) Here is a utility function on R2+: U(x1, x2) = x1x2 + x1. Let pi denote the price of commodity i and

m denote money income. Assume p1 = 1 and m = 50. Find the deadweight loss in consumer welfare due

to an increase in p2 from p2 = 2 to p2 = 4.

3. (30) Consider a consumer whose preferences on R3+ satisfy completeness, transitivity, continuity, and local

nonsatiation. Let x∗i denote the Marshallian demand for good i, h∗

i the Hicksian demand for good i, pi the

price of good i, and m money income. Here is some information about this consumer’s demand behavior

at prices p1 = 1, p2 = 2, p3 = 6 and income m = 10:

∂x∗1

∂p2= −5,

∂x∗1

∂p3= 2, and

∂h∗1

∂p1= −10.

Suppose further that the income effect on the Marshallian demand for commodities 2 and 3 are zero at

these prices and income level. What is the Marshallian quantity demanded of commodity 1 at these prices

and income level?

244 TEST 1: OCTOBER 5, 2006


1. (20) Here is an optimization problem:

f∗(a) = max{x1,x2,x3,x4,x5}

x1x2x3x4x5 + x1/31 x

1/92 x5

4x5 + x1/53 x4x

2/75 + a2x3

subject to5∑i=1

pixi = y and xi ≥ 0,

where a, pi, and y are all positive numbers. Assume there is an interior solution.

a. What happens to the optimal choice of x3 when a changes? Why?

b. What happens to f∗ when a changes? Why?

c. Give an explanation for why f∗′′(a) ≥ 2x∗3(a).

2. (20) Here is a production function for a price-taking seller that can use the inputs x1 and x2:

f(x1 , x2) ={

0 if x1 ≤ 1√min{x1 − 1, x2} if x1 > 1

.

a. Find the cost function c∗(w1, w2, y), where wi is the price of xi and y is the output level.

b. Use the cost function to describe the optimal supply behavior of this seller.

3. (25) Suppose the only thing you know about a production set Z is that it is sufficiently well-behaved to

ensure that a profit maximum exists for every price vector. Which of the standard properties of the profit

function hold, and why? Which properties do not hold, and why?

4. (35) Suppose the real-valued function g(θ, γ) has the following properties: Homogeneous of degree 1,

concave, and nondecreasing in θ; strictly increasing in γ; and differentiable in (θ, γ) (recall that differen-

tiability of a function implies continuity). Here, θ ∈ RN++ and γ ∈ R

1+. Define:

H(γ) = {x ∈ R+n : θ · x ≥ g(θ, γ) ∀θ >> 0}.

Think of H as a hypothetical production technology expressed in terms of input requirement sets.

a. Identify which standard properties of technologies this hypothetical technology necessarily has, and which

properties might not hold. Explain why for each property.

b. Is g(θ, γ) the minimum over x of θ · x subject to x ∈ H(γ)? Why/why not?



1. (45) Consider a market with 10 identical sellers, each of which has cost function c(q) = q2, where q is the

quantity produced by one seller. Market demand is

Q ={

780− p, 0 ≤ p ≤ 7800, 780 < p

,

where Q is aggregate quantity and p is price per unit.

a. (5) Derive the market equilibrium price and quantity when each seller behaves as a price-taker.

b. (10) Derive the market equilibrium price and quantity when each seller behaves as a Cournot competi-

tor.

c. (10) How much deadweight loss is created by Cournot behavior?

d. (5) What will happen in this market if there is free entry and a large number of identical potential

entrants?

e. (15) Suppose additional sellers with identical cost function can enter this market except that each

entrant incurs a $1,000 sunk cost of entry. Each potential entrant anticipates before entry that, after

entry, Cournot equilibrium will occur among the firms that have entered. How many firms will be in

the market in equilibrium?

2. (15) Suppose there is only 1 seller in question 1 who behaves as a monopolist. What price and quantity

will the seller choose? What is the price elasticity of demand at the seller’s choice? Based on the elasticity,

what is the seller’s percent markup (Lerner Index)?

3. (40) Suppose all buyers and sellers in a private-ownership economy have “well-behaved” production sets

and preferences, except that there is a consumer whose preferences do not satisfy local nonsatiation.

a. (20) Is it possible that a competitive equilibrium is Pareto efficient? If so, give an example of a Pareto

efficient competitive equilibrium (a graph will suffice). If not, explain why not.

b. (20) Is it possible that a competitive equilibrium is not Pareto efficient? If so, give an example of a

competitive equilibrium that is not Pareto efficient (a graph will suffice). If not, explain why not.



1. (30) Assume the consumption set is R3+ and the preferences of a consumer are as follows: Utility increases

by x1 units when consumption of good 1 is x1 units; utility increases by√x2 units when consumption of

good 2 is x2 units; utility decreases by x3 units when consumption of good 3 is x3 units.

a. (15) Explain whether the preferences are complete, transitive, continuous, locally nonsatiable, convex,

and monotonic.

b. (15) Find the Marshallian demands and the indirect utility function.

2. (35)

a. (10) Here is a proposed Marshallian demand: x∗(p1, p2, m) =(m3p1

, 2m3p2

). Can this be a Marshallian

demand for a consumer with complete, transitive, and continuous preferences on R2+? Explain why.

b. (10) Here is a proposed Hicksian demand: h∗(p1, p2, U) =(U√

p2p1

, U√

p1p2

). Can this be a Hicksian

demand for a consumer with complete, transitive, and continuous preferences on R2+? Explain why.

c. (15) Can the proposed Marshallian demand given in question a be for the same consumer as the

proposed Hicksian demand given in question b? Explain why.

3. (35) Suppose a consumer has Hicksian demand h∗ as defined in question 2.b. The government is consider-

ing the following two taxes for this consumer. Tax A charges t% on the amount spent on good 1. That is,

if the price received by the seller is p1 then the price paid by the consumer is (1+ t)p1. Tax B reduces the

consumer’s income by an amount T . That is, if income is m before paying the tax then income is m− T

after paying the tax. Suppose prices p1 and p2 are not affected by these taxes, and that the government

calculates T so that the consumer is indifferent between Tax A and Tax B.

a. (10) Derive an expression for the government’s revenue from Tax A in terms of income m, prices p1

and p2, and the tax rate t.

b. (15) Derive an expression for the government’s revenue from Tax B in terms of income m, prices p1

and p2, and the tax rate t.

c. (10) Show which tax collects more revenue for the government.



1. (25) Here is an optimization problem:

g∗(a) = min{x}

g(x; a), where g(x; a) = 2x4 − a3x.

a. (5) Find x∗(a), the optimal choice of x.

b. (5) Graph g as a function of a, with x fixed at an arbitrary positive value. Take care to label the

intercepts and to correctly depict the slope and curvature.

c. (10) Now suppose that the fixed value of x in part b is x∗(a0) for some particular value a0 of a. Add

g∗(a) to your graph in part b without actually deriving g∗(a).

d. (5) Based only on your graph in part c, can you tell whether g∗(a) a concave function? Why/why not?

2. (25) Here is a graph of a production set Z:

a. (20) Which of the properties 1-7 discussed in class does this technology have? Explain why for each

property.

b. (5) Does the “abnormality” in Z affect the profit function? Why/why not?

3. (25) Here is a cost function: c∗(w1, w2, w3, y) =(w1 +

√w2w3

)(ey − 1).

a. (5) Find the conditional factor demands.

b. (20) Verify that the matrix of demand slopes is symmetric and negative semidefinite. Also verify that

the change in marginal cost when a price changes equals the change in the corresponding input when


output changes.

4. (25) Here is a production function:

f(x1, x2) ={

0, 14≤ x1 < 1√

x2, x1 ≥ 1(x1 <

14is impossible).

a. (10) Find the cost function.

b. (15) Suppose w1 = 12 and w2 = 1. Find the supply curve for a firm that is a price-taker in the output

market. Carefully graph the supply curve.



Throughout this test, assume:

• We are studying a market with aggregate inverse demand P = 10− 2Q for 0 ≤ Q ≤ 5, where P is market

price and Q is market quantity.

• All firms produce homogeneous products and are identical with cost function

C ={

6q − q2, 0 ≤ q ≤ 24 + 2q, 2 ≤ q

,

where q is the output of one firm and C is the cost of one firm.

• Consumer welfare can be measured by Marshallian consumer surplus.

1. (5) Carefully draw the demand, marginal cost, and (market) marginal revenue curves on one graph. Label

the intercepts, slopes and points where the curves cross.

2. (5) Is there a pure price-taking equilibrium in this market? If so, what is it?

3. (10) What is the monopoly output and price?

4. (10) What is the socially efficient output level?

5. (5) Compare your answers to questions 2 and 4. Explain why your answers are consistent with the First

Theorem of Welfare Economics.

6. (10) Calculate the deadweight loss in the monopoly outcome.

7. (10) Find the Cournot duopoly symmetric Nash equilibrium output of each firm, and the equilibrium

price.

8. (10) Calculate the deadweight loss in the Cournot duopoly Nash equilibrium (be careful!).

9. (10) Compare your answers to questions 6 and 8, and give an economic explanation.

10. (5) At what price does a single seller make zero profit?

11. (10) Is the price you found in question 10 a Nash equilibrium price in a Bertrand duopoly? Explain why

or why not.

12. (10) Suppose firm 1 is the leader and firm 2 is the follower in a Stackelberg duopoly. Find the subgame

perfect Nash equilibrium strategy profile (state the equilibrium strategies carefully!).



1. (35) Suppose the consumption set is X = R2+, and for (x1, x2) ∈ R

2+ and (y1, y2) ∈ R

2+ we have preference

�∼

defined as follows:

(x1, x2) �∼ (y1 , y2) ⇔ either x1 + x2 ≥ 1 or y1 < 1.

State whether these preferences have each of the following properties, and explain why in each case (5

points each):

a. Complete?

b. Transitive?

c. Continuous?

d. Local Nonsatiation?

e. (Weak) Monotonicity?

f. Convexity?

g. Is there a utility function that represents �∼

on R2+? If so, give one such function. If not, explain why

not.

2. (45) Here is a utility function for a price-taking consumer of two commodities x1 ≥ 0 and x2 ≥ 0:

U(x1, x2) ={

min{x1, x2} when 0 ≤ min{x1, x2} ≤ 1min{2x1 − 1, x2} otherwise

.

Let m be money income, p1 be the price of x1, and p2 be the price of x2.

a. (20) Suppose income is m = 5 and the price of x2 is p2 = 1. Carefully draw a graph illustrating the

substitution and income effects of a decrease in the price of good 1 from p1 = 9 to p1 = 1. Calculate

the substitution and income effects on the quantity demanded of x1.

b. (15) Derive the Hicksian demand functions and the expenditure function for this consumer.

c. (10) Verify that the expenditure function you derived in part b has the 4 properties required of every

expenditure function.

3. (20) Here is a pair of Marshallian demands for a consumer of two goods, in standard notation:

x∗1(p1, p2, m) =

m

4p1, x∗

2(p1, p2, m) =3m4p2

.


Here is an indirect utility function for a consumer of two goods, also in standard notation:

U∗(p1, p2, m) =m3

p1p22

.

Is it possible that the given Marshallian demands are for the same consumer (i.e., same preferences) as

the given indirect utility function? Explain why or why not.



1. (30) Suppose there are two commodities in the world and a production set consists of the following five

pairs of the two commodities:

Z = {(0, 0), (−1, 1), (1,−2), (−3, 2), (−4, 3)}.

a. (15) Determine whether this technology has the following properties, and explain why in each case:

Closed, Nonempty, Possibility of Inaction, Convexity, No Free Lunch, and Free Disposal.

b. (15) Let p1 and p2 denote the prices of commodities 1 and 2, respectively. Find the profit function

π∗(p1, p2) for a profit-maximizing price-taking producer.

2. (20) Here is a Hessian matrix with respect to (w1, w2, w3) of a function that depends on (w1, w2, w3) and

y, evaluated at a particular point where (w1, w2, w3) and y are all positive:−1 1 01 −3 00 0 −1

.

If we interpret (w1, w2, w3) as input prices, can the function be a cost function for a price-taking cost-

minimizing producer? Explain why or why not.

3. (40) Here is a production function for a single-output (y) producer who uses the two inputs x1 ≥ 0 and

x2 ≥ 0:

y ={

min{x1, x2} when 0 ≤ y ≤ 1min{2x1 − 1, x2} when 1 < y

.

a. (10) Draw enough isoquants to make clear the basic shape of the isoquant family.

b. (15) Let w1 denote the price of x1 and w2 denote the price of x2. Find the cost function for a price-taking

cost-minimizing producer.

c. (10) Plot the average and marginal cost functions on one graph. Label important points on the graph.

d. (5) Suppose w1 = 4, w2 = 1, and the price of output is p = 4. How much will a profit-maximizing

price-taking seller produce?

4. (10) Use the envelope theorem to prove that standard profit functions are homogeneous of degree one.



1. (50) Suppose there are a large number N of buyers of a final product. The buyers each buy at most one

unit and are each willing to pay up to $A for that one unit (A is the same for all N buyers).

a. (5) Draw the aggregate demand curve for the final product.

b. (5) Assume sellers are price-takers who all have constant marginal cost C and no other cost. Is there

a competitive equilibrium in this market? Why? If so, what is it?

c. (5) Now assume one of the sellers described in item b is a monopolist. What will the monopolist do?

d. (5) Compare and contrast the efficiency (Marshallian consumer surplus plus profit) of the equilibria in

items b and c.

e. (8) Now assume there are two sellers who compete by choosing quantities simultaneously and nonco-

operatively. Seller 1 has constant marginal cost C1 and seller 2 has constant marginal cost C2. Assume

C1 < C2 < A, there are no other costs, and both sellers know all these parameters. Find all Nash

Equilibria and explain why they are equilibria.

f. (7) Now assume the duopolists described in item e compete by choosing prices simultaneously and

noncooperatively rather than quantities. Find all Nash Equilibria, and explain why they are equilibria.

g. (7) Now assume seller 1 described in item e is a Stackelberg leader and seller 2 is a Stackelberg follower.

Find all Nash Equilibria. What happens if the roles of leader and follower are reversed?

h. (8) Compare and contrast the efficiency (Marshallian consumer surplus plus profit) of the equilibria in

items e, f, and g.

2. (15) Suppose a small city sells licenses to taxi drivers. A taxi cannot operate in the city without a license.

The licensed taxis compete as identical Cournot competitors with aggregate inverse demand P = α− 110y,

where P is the market price and y is aggregate output. Each taxi has total cost c(yi) = y2i , where yi is the

output of taxi i. Suppose the city wants to maximize its revenue from license sales. How many licenses

should the city sell?

3. (35) One way the problems with potential nonexistence of long-run perfectly competitive equilibrium

might be ameliorated is if the sellers’ collective use of scarce inputs drives up the price of those inputs.

This increases all costs as the size of the industry grows, and thereby limits the size of the industry.

To model this, suppose there are many identical potential sellers with cost function c(y) =[F + 1

2y2]n,

where y is the output of one seller and n is the number of sellers who are producing positive quantity.

a. (5) Carefully draw the AC and MC curves when there is one active producer.

b. (5) Carefully draw the AC and MC curves where there are two active producers. What is the long-run

supply of two producers?

c. (10) What is the long-run supply curve of the industry, and how does it differ from the long-run industry

supply that we would have if cost did not depend on n?

d. (15) Suppose market inverse demand is p(Y ) = α−Y , where Y is aggregate industry output. For what

values of α is there a perfectly competitive equilibrium? How does this differ from the values of α for

which there would be a perfectly competitive equilibrium if cost did not depend on n?



1. (35) A consumer has the following utility function on the consumption set R2+:

U(x1, x2) ={ 1

3x1 + 23x2, if x2 ≥ x1

23x1 + 1

3x2, if x2 ≤ x1

.

a. (5) Carefully draw the indifference curve for utility level U = 1.

b. (10) Derive the Marshallian demands and the indirect utility function.

c. (10) Derive the Hicksian demands and the expenditure function.

d. (10) Suppose p2 = 1. Carefully draw a graph that illustrates the substitution and income effects of a

change in p1 from p01 = 1

2 to p11 = 2.

2. (30) A consumer of two goods has Hicksian demands h∗1 =

(p2p1

eU)1/2

and h∗2 =

(p1p2

eU)1/2

. Suppose

p2 = 1 and money income is m = 8. Consider a change in p1 from p01 = 1 to p1

1 = 2. Calculate the

percent error if the change in Marshallian surplus is used to measure the welfare change rather than the

compensating variation.

PLEASE SEE THE BACK FOR QUESTION 3


3. (35) Suppose the consumption set is R2+, which contains the following subsets:

In the graph, C and B ∩ C both include their boundaries, and B is all of R2+ except the part labeled C.

A consumer has the following preferences on R+2 : y �

∼x if and only if either 1) x ∈ B and y ∈ B, or 2)

x ∈ C. State whether these preferences have each of the following properties, and explain why in each

case (5 points each):

a. Complete?

b. Transitive?

c. Continuous?

d. Local Nonsatiation?

e. (Weak) Monotonicity?

f. Convexity?

g. Is there a utility function that represents �∼

on R2+? If so, give one such utility function. If not, explain

why not.



1. (50) A price-taking firm produces output y from inputs x1 and x2 according to the production function

y ={

ln(6x1 + 6x2 + 1), if x2 ≤ 4x1

ln(10x1 + 5x2 + 1), if x2 ≥ 4x1

.

a. (10) Derive the conditional factor demand functions for x1 and x2.

b. (5) Derive the cost function.

c. (10) Derive the supply function.

d. (10) Derive the profit function.

e. (5) Write an expression for the production set Z.

f. (10) Does this firm’s technology possess the 5 basic properties of technologies? Why?

2. (50) Here is a possible cost function for a price-taking firm, in standard notation:

c∗(w1, w2, y) = 8y2(w1w2)1/2.

a. (10) Verify that this function is indeed a cost function.

b. (5) Derive the conditional factor demands.

c. (10) Assume the input prices are w1 = 2 and w2 = 8. Derive and graph the marginal and average cost

curves as functions of output y.

d. (25) Continue to assume the input prices are w1 = 2 and w2 = 8, and take as given that the production

function for this cost function is y = 12 (x1x2)1/4. Derive the short-run cost function when input x1 is

fixed at x1 = x∗1(2, 8, 1). Now derive the corresponding short-run marginal and average cost curves, and

graph them on the same graph with the long-run marginal and average cost curves. Be careful to draw

the curves in the correct relative positions and to label ALL important points of intersection. How

much will this firm supply in the short-run and long-run if the output price is p = 8? Give an economic

explanation for the short-run supply decision, in particular compared to fixed cost the long-run supply

decision.



1. (20) Suppose n identical firms compete by simultaneously choosing quantities of their homogeneous prod-

ucts. Market demand is P = 100−Q for 0 ≤ Q ≤ 100 and the only non-sunk cost is constant marginal

cost equal to 20. How large must n be to assure that the Nash Equilibrium price-cost margin satisfies the

Justice Department’s guideline of being no larger than 5%? Show why.

2. (20) Suppose, in addition to the n firms described in question 1, there is an additional firm (label it

firm 0). This firm has marginal cost of 20 like all of the other firms, but selects its quantity before the

other n firms. In other words, this firm behaves as a Stackelberg leader and the other n firms behave as

simultaneous followers.

a. (15) Let π∗0(n) be the equilibrium profit of firm 0. Use the envelope theorem to show that ∂π∗

0∂n =

−(n + 1)−1π∗0(n) (ignore the integer constraint on n when doing this). Only answers that use the

envelope theorem to show this will receive credit.

b. (5) How much does firm 0 produce in equilibrium?

3. (20) Suppose two firms compete by simultaneously setting prices for their homogeneous products. Market

inverse demand is P = 100− Q for 0 ≤ Q ≤ 100. Firm 1 has constant marginal cost of C1 = 30. Firm

2 has constant marginal cost of C2 = 40. These are the only costs. Find the Nash Equilibrium prices.

Explain why your prices are equilibrium. How much does each firm sell in equilibrium?

4. (20) Suppose we have a perfectly competitive market and that all firms are identical with average cost

ac(y) =

10− y, 0 < y ≤ 55, 5 ≤ y ≤ 6y − 1, 6 ≤ y,

where y is the output of an individual firm. The market inverse demand is p(Y ) = a − Y for 0 ≤ Y ≤ a,

where Y is market quantity.

a. (5) If there is a long-run equilibrium, what must the price be?

b. (15) How large must a be in order to guarantee that there is a long-run equilibrium?

5. (20) Suppose n = 2 in question 1 and there is no possibility of entry by additional firms. How much is

one of the firms willing to pay to acquire the other firm before the quantity-setting occurs? Assume there

are no income effects. How much deadweight loss would be created by an acquisition like this?



1. (20) Suppose utility is U(x1, x2) = x21 + x2. Find the Hicksian demands and the expenditure function.

2. (25) Suppose the consumption set X is the set of nonnegative integers and a consumer has the following

preference relation: For x ∈ X and y ∈ X, x �∼

y if and only if x is an even number.

a. (10) Is this preference relation complete? Why?

b. (10) Is this preference relation transitive? Why?

c. (5) Does this preference relation satisfy free disposal? Why?

3. (25) Suppose U∗ = m(p1+p2)p1p2

is an indirect utility function. Find a utility function U(x1, x2) such that U∗

is the maximum of U(x1, x2) over x1, x2 ≥ 0, subject to p1x1 + p2x2 ≤ m.

4. (15) A consumer has expenditure function e∗ =√p1 for some particular level of utility. The price p1

increases from 1 to 100. What is the compensating variation of this price change? Why?

5. (15) Suppose utility is U(x1, x2) = x2(x1 + 1). Find the Marshallian demands.



NOTATION. y =quantity of output, xi =quantity of input i, p =price per unit of y, wi =price per unit

of xi.

1. (20) Consider a price-taking firm with production function f(x1, x2) = Ax1x2, where A > 0 is a parameter

that captures technical progress (a higher value of A means a more advanced technology). Use the

envelope theorem to show that the effect of a technical advance on the cost of producing output level y is

−mc(y)y/A.

2. (20) A price-taking firm has cost function

c(y) =

{F + y

y+1+ y2

4, y > 0

αF, y = 0

where α ∈ [0, 1] is the proportion of fixed costs that are sunk.

a. (10) Suppose α = 1. Carefully graph the firm’s supply function.

b. (10) Illustrate in qualitative terms what happens to the supply function if α < 1.

3. (30) Suppose a typical input requirement set is:

Comment on any unusual properties of this technology. Then find the profit function for a price-taking

firm with this technology.

260 TEST 1: OCTOBER 10, 2002

4. (30) Suppose c∗(w1, w2, y) = yw1 .

a. (10) Verify that c∗ could be the cost function of a price-taking firm that can use two inputs to produce

output.

b. (10) Draw the economically relevant input requirement set H(y) in R2+ for arbitrary y > 0.

c. (10) Draw another input requirement set in R2+ that is consistent with c∗ but that perhaps violates

some of the standard assumptions on technologies. Which assumption(s) are violated by your input

requirement set?



1. (50) Suppose market demand is

Q ={

100− P 0 ≤ P ≤ 1000 P > 100

where P is the price per unit and Q is the number of units sold. Assume there are two firms whose only

costs are the constant marginal costs c1 and c2.

a. (5) If firm 1 is a monopoly, how much does it produce? What price does it charge? How much profit

does it earn?

b. (10) If the two firms compete as Cournot-Nash duopolists, how much does each firm produce at an

interior equilibrium? What is the aggregate quantity? What is the equilibrium price? How much profit

does each firm earn?

c. (15) Let P1 denote the monopoly price charged by firm 1 in part a. How much does firm 2 produce in

the Cournot-Nash equilibrium if c2 = P1? Assume the area under the Marshallian demand measures

consumer welfare. What happens to aggregate welfare (sum of consumer surplus and profit) in the

Cournot-Nash equilibrium when there is a small decrease in c2 from the level c2 = P1? Does the

introduction of Cournot competition benefit society (compared with monopoly) when the new firm has

marginal cost near the existing monopoly price? Why? What if the new firm has marginal cost near

the marginal cost of the existing monopolist? Why?

d. (10) If the two firms compete as Bertrand-Nash duopolists, what price does each firm charge in equi-

librium? How much does each firm produce? What is the aggregate quantity?

e. (10) Is society better off under Bertrand competition than under monopoly? Why? Is the conclusion

different from when the duopolists compete in quantities? Why?


2. (15) Suppose the most efficient technology looks like this:

a. (5) Is there a long-run perfectly competitive equilibrium? Why?

b. (10) What is the maximum number of firms that can produce positive output in a long-run perfectly

competitive equilibrium? What is the minimum number?

3. (35) Suppose market demand is as given in question 1. Assume there is a dominant firm with cost

function cd(Qd) = Qd and a competitive fringe with n identical firms, each of which has cost function

cf(qf) = 10qf +10q2f , where Qd is the output of the dominant firm and qf is the output of a typical fringe

firm (so Q = Qd + nqf).

a. (15) Assume n < 180. How much does the dominant firm produce, and what price does it charge?

b. (10) What happens to the output of the dominant firm in part a if n increases? Give a graphical and

verbal explanation for this.

c. (10) How much does the dominant firm produce and what price does it charge if n > 180?



1. (20) Consider the following function for U ≥ 0 and pi > 0, in the notation used in class:

e∗(p1, p2, U) =p1p2U

p1 + p2.

Is there a complete, transitive, continuous, and locally nonsatiated preference relation �∼

on R2+ for which

this equation is the expenditure function? Why?

2. (20) Consider the following system of equations for m ≥ 0 and pi > 0, in the notation used in class:

x∗1(p1, p2, m) =

p2m

p1(p1 + p2)x∗

2(p1, p2, m) =p1m

p2(p1 + p2).

Is there a complete, transitive, continuous, and locally nonsatiated preference relation �∼

on R2+ for which

these equations are the Marshallian demands? Why?

3. (15) According to the function given in question 1, what is the Compensating Variation of an increase in

p1 from p01 to p1

1, for given m and p2?

4. (10) Does the consumer in question 1 have the same preferences as the consumer in question 2? Why?

5. (20) Derive a utility function for which the equation given in question 1 is the expenditure function.

6. (15) Suppose X = {Tree,Grass, Iron,Flower} is a consumption set and �∼

is a binary relation defined on

X as follows: For a ∈ X and b ∈ X, a �∼

b if and only if a is a plant.

a. (8) Is �∼

complete on X? Why?

b. (7) Is �∼

transitive on X? Why?

264 TEST 1: OCTOBER 10, 2001


1. (30) Is the following system of equations a set of conditional factor demands for a technology that satisfies

the five basic assumptions? Why? If not, which of the five basic assumption(s) are violated by the

underlying technology? Why?

x∗1(w1, w2, y) =

{ √w2w1

(y − 1) for y ≥ 1

0 for 0 ≤ y < 1; x∗

2(w1, w2, y) =

{ √w1w2

(y − 1) for y ≥ 1

0 for 0 ≤ y < 1.

2. (70) Suppose a price-taking firm has the production set Z = {(y,−x) ∈ R2 : x ≥ 0 & y ≤ −2x & y ≤ 1−x}.

a. (5) Carefully draw Z, labeling slopes and critical points.

b. (5) Does Z satisfy the five basic assumptions of a technology? Why?

c. (5) Characterize the returns to scale of Z.

d. (5) Is Z convex? Why?

e. (5) What are the input requirement sets V (y) for y ≥ 0?

f. (10) Find the supply/demand function and the profit function.

g. (10) Find the conditional factor demand function and the cost function.

h. (10) Verify that your cost function has the properties of a cost function.

i. (10) Use your cost function to find average cost and marginal cost for some fixed value w >> 0 of the

input price. Graph these curves as a function of output y.

j. (5) Where is the supply function on your graph? Reconcile this graphical derivation of the supply

function with the supply function you found in part f.



1. (50) Suppose two firms compete as Cournot duopolists for aggregate inverse demand P = 150−Q. Firm

1 has a non-sunk fixed cost of F1 = $1, 600 and a constant marginal cost of MC1 = $10. Firm 2 has a

non-sunk fixed cost of F2 = $900 and a constant marginal cost of MC2 = $20.

a. (15) Derive the reaction curves of the two firms. Graph them, labeling slopes and all critical points.

b. (10) Find all Cournot-Nash equilibria.

c. Suppose all income effects are zero for this good.

i. (10) If the market is unregulated, what is the best equilibrium aggregate welfare outcome? Why?

ii. (5) Can the outcome you identified in part i. be assured by enacting a law that makes it illegal for

one of the firms to produce, but does nothing else to regulate the firms? Why?

iii. (5) Is the outcome you identified in part i. Pareto Efficient? Why?

d. (5) Discuss equilibrium if the firms were to behave as perfect competitors.

2. (50) Suppose there are k identical price-taking consumers, each with income m and utility function

U(x, y) = lnx+ ln y. Let p denote the price of x and assume the price of y has been normalized to one.

Suppose also there are E identical producers of x, each with constant marginal cost c > 0 and no fixed

costs.

a. (10) Derive the aggregate demand for x.

b. (5) Write the revenue function for the monopolist when E = 1. What is unusual about this revenue

function? What does this imply about the monopolist’s optimal behavior?

c. (5) If the firm(s) behave as perfect competitors, what is the equilibrium price of x?

d. (10) Suppose for the remainder of this problem that E ≥ 2. Derive the Cournot-Nash equilibrium

price of x as a function of the number of firms E. How many firms are required to drive the Cournot

equilibrium price down to within 10% of the perfectly competitive price?

e. (20) Derive an expression for the deadweight loss that is incurred if the government compensates

consumers for the higher price caused by the non-price-taking behavior of the Cournot oligopolists.

(HINT: From your calculations in part a, find the indirect utility function for one consumer and invert

it to obtain the expenditure function. Evaluate the expenditure function at arbitrary p and at the

optimal utility level for the perfectly competitive price, and then multiply by k to obtain the aggregate

expenditure function for the perfectly competitive utility level. Use this to derive deadweight loss.)

Suppose there are k = 1 million consumers who each have income of m = $30 thousand, and there are

enough firms to make the Cournot equilibrium price no more than 10% above the perfectly competitive

price. How large is the deadweight loss?



1. (35) Consider the preference relation defined by the following utility function for (x, y) ∈ R2+:

U(x, y) =√

x2 + (y − 10)2.

a. Carefully draw the indifference curves for utility levels U = 1 and U = 2.

b. Briefly explain why this preference relation is complete, transitive, and continuous on R2+.

c. Is this preference relation (briefly explain why in each case)

i. Locally nonsatiated?

ii. Monotonic (strong or weak)?

iii. Convex?

d. Find the Marshallian demands.

2. (25) Consider the following system of equations defined for (p1, p2) >> 0 and m ≥ 0:

x∗1 =

{m2

p1p2, if m ≤ p2

mp1

, if m > p2

x∗2 =

{mp2

[1− m

p2

], if m ≤ p2

0, if m > p2

Demonstrate whether these equations are or are not Marshallian demands.

3. (20) Given the indirect utility function U∗(p1, p2, m) =(m2

p1p2

)1/4

, find the

a. Expenditure function.

b. Hicksian demands.

c. Marshallian demands.

4. (20) Suppose you have estimated the Hicksian demands h∗1 = U p2

p1and h∗

2 = U p1p2. What is the equivalent

variation of a change in p1 from p1 = 1 to p1 = 31 when income is m = 160 and p2 = 1?



1. (25) Consider the set Z = R1−.

a. (10) Demonstrate whether Z satisfies the five basic properties of a production set.

b. (5) Is Z convex? Why?

c. (5) What returns to scale does Z display? Why?

d. (5) Derive the optimal netput function and the profit function.

2. (15) Consider a firm with production function f(x1, x2) = α ln(x1+1)+(1−α) ln(x2+1), where α ∈ (0, 1).

Take as given that the factor demands for this firm are x∗1 = pα

w1− 1 and x∗

2 = p(1−α)w2

− 1. USE THE

ENVELOPE THEOREM to show that profit increases with a small increase in α if and only if

w2w1

> 1−αα (Note: You will not receive credit for a proof of this unless your proof applies the envelope

theorem).

3. (25) Consider the function π∗(p, w) = p for p ≥ 0 and w >> 0.

a. (10) Verify that π∗ is a profit function.

b. (15) For simplicity, suppose w is a scalar (i.e., w ∈ R1++). Use algebraic and geometric arguments

to draw a production set H such that π∗(p, w) = maxpy − wx subject to (y,−x) ∈ H . Does your

production set satisfy no free lunch?

PLEASE SEE THE BACK FOR THE LAST QUESTION

268 TEST 1: OCTOBER 12, 2000

4. (35) Suppose a price-taking firm has the following production set:

a. (5) What are the input requirement sets V (y) for y ≥ 0?

b. (10) Find the supply/demand function and the profit function.

c. (5) Find the conditional factor demand function and the cost function.

d. (10) Use your cost function to find average cost, average variable cost, and average fixed cost for some

fixed value w >> 0 of the input price. Graph these curves as a function of output y.

e. (5) What is the effective “marginal cost curve” of this firm? Use this to illustrate your supply function

on your graph.



1. (20) Consider a market with inverse demand p = 100− y, where y is market output, and a large number

of potential firms who all behave as perfect competitors and who all have cost function c(yi) = 1 + y2i ,

where yi is the output of firm i.

a. If there is a long-run equilibrium, what is the price?

b. Is there a long-run equilibrium? If so, find it.

2. (30) Consider an economy with one firm and one consumer. The consumer’s welfare is accurately measured

by the Marshallian surplus for good 1. The only endowment is x2 = $m, possessed by the consumer. The

firm produces good 1 from good 2. The consumer’s Marshallian demand and the firm’s costs are:

a. Is the allocation (x1, m− AC(x1)) for the consumer and (x1, AC(x1)) for the firm feasible? Why?

b. Is the allocation in part (a) Pareto Efficient? Why?

c. Is the allocation in part (a) a (competitive) equilibrium allocation? Why?

d. Is the allocation (x1, m) for the consumer and (x1, 0) for the firm feasible? Why?

e. Is the allocation in part (d) Pareto Efficient? Why?

f. Is the allocation in part (d) a (competitive) equilibrium allocation? Why?

g. Relate your answers to parts (b),(c) and parts (e),(f) to the First Fundamental Theorem of Welfare

Economics.


3. (20) Consider a monopolist facing inverse demand p(y) = 100− y with cost

c(y) ={

0, y = 01500 + y2, y > 0

.

How much will this firm produce, and what price will it charge? How much profit will it earn?

4. (30) Suppose we have a symmetric Cournot duopoly with aggregate inverse demand p(y) = 100−y, where

y = y1 + y2, and cost

c(yi) ={

0, yi = 01200 + yi, yi > 0

for i = 1, 2.

a. Derive the reaction curves for these duopolists, and graph them.

b. What do you notice about Nash Equilibrium?



1. Let X = R2+ be a consumption set and consider the following preference relation on X:

(x1, x2) �∼ (y1, y2) if and only if |x1 − 10|+ |x2 − 10| ≥ |y1 − 10|+ |y2 − 10|.

Is this relation (say why in each case)a. Complete?b. Transitive?c. Continuous?d. Monotonic?e. Locally nonsatiated?f. Convex?

2. Consider the following functions

h∗1 =

U2 , if p1

p2< 2

U2 or 0, if p1

p2= 2

0, otherwise

h∗2 =

0, if p1

p2< 2

0 or U , if p1p2

= 2

U , otherwise

a. Verify that these are Hicksian demands.

b. Find the corresponding Marshallian demands.

c. Verify the Slutsky Decomposition for x∗1 in response to a change in p1.

3. Consider a consumer with utility function U(x1, x2, x3) = x1x2 + x3. Assume a fixed amount of good 3

is given to the consumer by the government at no charge, and the consumer cannot buy or sell good 3 at

any price. Also assume that prices p1 and p2 are fixed, and money income is fixed at m.

a. Suppose the government gift is x3 = 0. Find the Marshallian demands for x1 and x2.

b. Use your result from part a to obtain the Marshallian demands when x3 > 0. Be sure your solution

method plainly relies on your solution to part a, and is NOT obtained by simply resolving the problem.

c. Find the indirect utility function for a fixed arbitrary level of x3.

d. Use your result for part c to obtain the expenditure function. Again, DO NOT simply resolve the

problem.

e. Suppose income is m = 10, both prices are p1 = p2 = 1, and the amount of good 3 provided by the

government is x3 = 11. If the government wants to eliminate the gift of x3 and give income instead,

how much income must the government transfer to the consumer in order to leave utility unchanged?

272 TEST 1: OCTOBER 14, 1999


1 (25). Suppose we have a production function for a single-input technology given by f(x) = (50− x)(2 + x).

a (15). Describe the properties of the technology in terms of the standard assumptions on production sets.

b (10). Based on your answer to part a, what standard properties will the profit function possess? Which of

the standard properties will not hold for the profit function? Answer this question by referring to the

reasons that the profit function possesses certain properties, not by deriving the profit function and

checking its properties.

2 (25). Consider a profit-maximizing firm that is not a perfect competitor in its output market. Instead, the firm

faces an inverse demand curve p(y; r), where p is its price, y is its output level, and r is the price charged by

a rival firm whose product is regarded by consumers as a substitute product. Hence the revenue function

is R(y; r) = p(y; r)y.

a (15). Use the envelope theorem to determine the effect on optimal profit of a change in the rival’s price.

b (10). Use the traditional comparative statics methodology to show that the effect on optimal output of a

change in the rival’s price depends on how the change in the rival’s price affects the slope of the inverse

demand (with respect to output).

3 (25). Suppose c∗(w1, w2, y) = (y + y2)(w1 + (w1w2)1/2 +w2). Is c∗ a cost function? Why?

4 (25). Suppose c∗(w1, w2, y|x1 = x1) is a short-run cost function.

a (5). Illustrate the short-run optimal choice of x2 on an isoquant/isocost graph.

b (10). Use your graph to argue geometrically that the value of x1 that minimizes c∗(w1, w2, y|x1 = x1) is the

long-run cost-minimizing choice of x1, for the given w and y.

c (10). Use the result in part b to find the long-run cost function when c∗(w1, w2, y|x1 = x1) = w1x1 + w2yx1

.



1. Suppose aggregate inverse demand is p = 12−y for y ≤ 12 and there are two firms. Firm one has constant

marginal cost of 4 and firm two has constant marginal cost of 10. There are no fixed costs.

a. Find the Cournot-Nash equilibrium quantities and market price.

b. Suppose firm one is a monopolist. Find the quantity and price for this monopolist.

c. Compare your answers to questions a and b, and explain the relationship.

2. Suppose the city of New York requires that each taxicab have a license. Inverse demand for taxi service is

p = 510− y for y ≤ 510, where y is the quantity of taxi services consumed. Each taxi has cost c = 1+ y2,

not including the cost of the license, and each taxi behaves as a perfect competitor. There is a large

number of potential sellers of taxi services.

a. If the city auctions 100 licenses, what will the auction price of the licenses be? How much revenue does

the city receive?

b. If the city issues a license to anybody that wants one, how many licenses will be issued?

c. How much consumer surplus is lost due to the supply restriction of only 100 licenses?

d. Suppose the city gives all of the revenue from selling the 100 licenses to consumers, in a way that does

not depend on the quantity of taxi services consumed. Can the city fully reimburse consumers for their

lost surplus? If not, how much surplus is still lost after the rebate?

e. Now suppose the city wants to maximize revenue from auctioning the licenses. Let n be the number

of licenses offered for sale, so that the city’s problem is to choose the optimal n. For an individual

supplier of taxi services, the price depends on n and so the optimal profit of each supplier depends on

n. Denote this by π∗(n). In this notation, what is the city’s revenue from offering n licenses for sale?

(NOTE: do not attempt to find π∗(n)). Explain how you would use the envelope theorem in finding

the optimal number of licenses for the city (but do not actually try to find the optimal n).

3. Suppose the matrix of derivatives of a consumer’s Hicksian demands is

(a b2 −1

2

),

and the prices are (8, p2). Find a, b, and p2.


4. Suppose income is fixed. Can a Marshallian demand look like this?

Explain why/why not.

5. Consider the following functions:

h1 = U

(b11 + b12

(p2

p1

)1/2)

h2 = U

(b22 + b21

(p1

p2

)1/2)

a. Verify that these form a system of Hicksian demands if the bij parameters satisfy certain restrictions.

What are the restrictions?

b. Find the Marshallian demands.

6. Suppose �∼

is a complete preference relation on the consumption set X = Rn+. Two properties that �

∼

might possess are

NONSATIATION: For every y ∈ X, there exists x ∈ X such that x � y. NOTE: This is NOT local,

since there is no requirement that x be “close” to y.

SEMI-STRICT CONVEXITY: If x � y and x′ �∼

y then λx+ (1− λ)x′ � y for every λ ∈ (0, 1).

Assume �∼has the nonsatiation and semi-strict convexity properties. Show that �

∼then also satisfies local

nonsatiation.



1. Suppose a consumer of two goods, x1 and x2, has utility function U(x1, x2) = x2 − 1x1. Note that U can

be negative at positive values of x1 and x2.

a. Draw a typical indifference curve for a utility level U that is nonnegative.

b. Find the Hicksian demands for values of U that are nonnegative.

c. Use your answer to b. to find the indirect utility function for values of income m and prices p1 and p2

that satisfy a certain condition. What is the condition?

d. Use your answer to c. to find the Marshallian demands for values of income and prices that satisfy

your condition.

2. Suppose �∼

is a complete preference relation on the consumption set X = Rn+. Two properties that �

∼

might possess are

NONSATIATION: For every y ∈ X, there exists x ∈ X such that x � y. NOTE: This is NOT local,

since there is no requirement that x be “close” to y.

SEMI-STRICT CONVEXITY: If x � y and x′ �∼

y then λx+ (1− λ)x′ � y for every λ ∈ (0, 1).

Assume �∼has the nonsatiation and semi-strict convexity properties. Show that �

∼then also satisfies local

nonsatiation.

3. Suppose there are two commodities (x1, x2) ≥ 0, with prices (p1, p2) >> 0. Consider the functions:

x∗1 =

m

ap1and x∗

2 =m

bp2.

For what value(s) of the parameters a and b are these functions Marshallian demands of x1 and x2 for

some consumer with income m and locally nonsatiated preferences?

4. Let x∗i (p,m) be a differentiable Marshallian demand for good i. Fix prices p at p0 >> 0. Suppose Mr.

Giffen makes the following claim:

“There must exist an income level m0 > 0 such that x∗i is not upward-sloping in pi at (p0, m0).”

Is Mr. Giffen right or wrong? Explain why.

276 TEST 1: OCTOBER 15, 1998


1. Suppose the production function for a free disposal technology is f(x1 , x2) = (x1 + x2)2 for (x1, x2) ≥ 0.

a. Draw a typical isoquant.

b. Find the cost function.

c. Suppose input 1 is fixed at x1 = 10. Find the short-run cost function c∗(w1, w2, y|x1).

2. Consider the function c∗(w1, w2, y) = (2wα1 +w2)(ey − 1) for (w1, w2, y) ≥ 0.

a. For what value(s) of α is c∗ a cost function? Why?

b. For these value(s) of α, characterize the returns to scale of the technology underlying c∗. Explain why

you think the returns to scale are as you describe.

c. For these value(s) of α, find the conditional factor demands.

3. Consider the production set Z = {(z1, z2) ∈ R2 : z2 ≤ 1− (1 + z1)2}. Demonstrate geometrically whether

Z has each of the following properties (HINT: Carefully plot the curve z2 = 1− (1 + z1)2):

a. Nonempty.

b. Closed.

c. Possibility of inaction.

d. No free lunch.

e. Free disposal.

f. Convex.

4. Consider the functions (for p ≥ 0 and (w1, w2) >> 0):

y∗(p, w1, w2) =p

2

(1w1

+1w2

)x∗

1(p, w1, w2) =(

p

2w1

)2

x∗2(p, w1, w2) =

(p

2w2

)2

.

a. Demonstrate formally that these are profit-maximizing supply and demand functions.

b. Find the profit function.



1. A consumer has utility function U(x1, x2) = (x1 + x2)2. Find the Hicksian demand functions.

2. Suppose a consumer has Hicksian demands:

h∗1(p1, p2, U) = − 1

U

[(p2

p1

)1/2

+ 1

]

h∗2(p1, p2, U) = − 1

U

[(p1

p2

)1/2

+ 1

].

Find the Marshallian demands (note: U < 0 here).

3. Consider a perfectly competitive market with demand curve D(p) and an arbitrarily large number of

actual and potential firms who all have average cost curve

ac(y) =

11− y, 0 < y ≤ 101, 10 ≤ y ≤ 11y − 10, 11 ≤ y.

Suppose also that c(0) = 0 (i.e., their technologies have the possibility of inaction). Give a lower bound

on demand that ensures existence of a long-run perfectly competitive equilibrium with free entry and exit.

4. Consider a monopolist with inverse demand function p(y) = 7− 12y and cost function

c(y) ={

10y − y2 , 0 ≤ y ≤ 5∞, 5 < y.

Find the optimal output level and price for this monopolist.

5. Suppose there are three commodities in the world: Guns, Butter, and Mayonnaise. Ann, Beth, and Carol

each have complete and transitive preferences over these three goods. Pop U. List is a politician elected

by Ann, Beth, and Carol; who has the following preferences:

”I think commodity 1 is at least as good as commodity 2 if and only if a majority of my

constituents think commodity 1 is at least as good as commodity 2.”

Are Pop’s preferences necessarily complete and transitive? Why or why not?

6. Given x∗1(p1, p2, m) = 2m

3p1and x∗

2(p1, p2, m) = m3p2

, demonstrate that these functions are Marshallian

demands.


7. Suppose prices on three goods are p1 = 1, p2 = 2, and p3 = 4. At these prices, the following observations

are made:

∂h∗1

∂p1= −1

∂h∗1

∂p2= 1

∂h∗2

∂p2= −2.

Find the Hicksian substitution matrix among these three goods at the stated prices.



1. (15) Suppose the production function is CES: f(x1 , x2) = (xρ1 + xρ2)ε/ρ (ρ �= 0). Find the conditional factor

demands and the cost function.

2. (10) Draw a production set that satisfies no free lunch, possibility of inaction, convexity, and regularity; but

violates free disposal.

3. (10) Draw a production set that has decreasing returns to scale but that is not convex.

4. (20) Suppose we have the function c∗(w1, w2, y) = (w1 + w2) ln(y + 2) for w1, w2, y ≥ 0. Check whether c∗

satisfies sufficient conditions for a cost function. If not, identify which of the standard properties the

underlying technology does not possess.

5. (30) Suppose we have the function π∗(p, w) = p2

w for p, w > 0 (both p and w are scalars).

a. (5) Verify that π∗ is a profit function.

b. (10) Find the underlying production function, f(x).

c. (10) Verify that π∗(p, w) = maxpf(x) − wx.

d. (5) Find the corresponding cost function, c∗(w, y).

6. (15) Suppose we have the functions x∗1(w1, w2, y) = x∗

2(w1, w2, y) = ln(y + 1). Check whether x∗1 and x∗

2 are

conditional factor demands.



1. (20) Given

h∗1(p1, p2, U) =

(p2

p1

)1/2

U

h∗2(p1, p2, U) =

(p1

p2

)1/2

U .

a. (5) Verify that h∗1 and h∗

2 are Hicksian demands.

b. (15) Find corresponding Marshallian demands.

2. (5) Suppose the consumption set is the collection of all cars in the world. Alice has no opinion about car 1

and car 2 unless car 1 is a BMW, in which case she thinks car 1 is at least as good as car 2. Is there a

utility function that represents Alice’s preferences? Why/why not?

3. (20) Consider a consumer whose utility function is additively separable:

U(x1, . . . , xn) = f1(x1) + f2(x2) + · · ·+ fn(xn),

where each fi is an increasing, concave function. Show that the Marshallian demands are downward-

sloping. (Hint: determine whether each good is normal).

4. (20) Consider a perfectly competitive industry in which all firms possess the technology summarized by the

average cost function

ac(y) =

y + 1

y , 0 < y < 12, 1 ≤ y ≤ 2(y − 1) + 1

y−1 , 2 < y.

a. (5) Sketch this average cost curve.

b. (5) If there is a long-run equilibrium price pe, what is it?

c. (10) How large must market demand be at price pe in order to guarantee there is a long-run equilibrium?

5. (15) Now suppose there is a monopolist with average cost given in question 4 and market inverse demand

p = 4− y. How much surplus is lost due to the monopoly behavior of this firm?

6. (15) Now suppose there are two firms with average cost ac(y) = y + 1y and market inverse demand given in

question 5. How much surplus is lost if these firms engage in Cournot behavior rather than perfectly

competitive behavior?

7. (5) Give a technology and input price for which the average cost function is that given in question 6.



1. (15) Decide whether each of the following technologies satisfy no free lunch, possibility of inaction, free disposal,

and convexity. Also characterize the returns to scale of each.

a) Z = {(x, y) ∈ R2 : x < 1 and y ≤ ln(−x+ 1)}

b) Z = {(x, y) ∈ R2 : x2 + y2 ≤ 1}

c) V (y) = {x ∈ R2+ : x2

1 + x22 ≥ y}

2. (10) Suppose px = 1 and py = 0. Illustrate the profit maximizing point for a firm with technology b) in

question 1 above. Does this firm earn positive profit?

3. (20) Consider a firm with production function y = ln(x1 + 1) + x2 for x ≥ 0. Find the conditional factor

demands. How much will this firm supply if it maximizes profit with prices satisfying p = w1 = w2?

4. (20) Might c∗(w, y) = ye12 ln(w1)+

12 ln(w2) be a cost function? Why/why not? If so, find the conditional factor

demand functions.

5. (35) Consider a perfectly competitive cost-minimizing firm with a well-behaved technology that has to pay a

tax of t percent on its purchases of input 1. Thus the total cost to the firm of using 10 units of input 1 is

(1 + t)w110. Use the envelope theorem to show that the cost function is concave in the tax rate t.

282 TEST 1: OCTOBER 14, 1995


1. (15) Decide whether each of the following technologies satisfy no free lunch, possibility of inaction, free disposal,

and convexity. Also characterize the returns to scale of each.

a) Z = {(x, y) ∈ R2 : y ≤ (−x)3}

b) V (y) = {x ∈ R2+ : x1 ≥ y

x2}

c)

2. (10) Suppose px = py. Illustrate the profit maximizing point for a firm with technology c) in question 1 above.

Does this firm earn positive profit?

3. (20) Consider a firm with production function y = ex2 −1+x1 for x ≥ 0. Find the conditional factor demands.

How much will this firm supply if it maximizes profit with prices satisfying p = w1 = w2?

4. (20) Might π∗(p, w) = p2

4w be a profit function? Why/why not? If so, find the supply and demand functions.

5. (35) Consider a firm with average cost function ac(y) = (ay−1)2+1 for y > 0, where a > 0 is some parameter.

Suppose that, rather than maximizing profit, this firm chooses its output level y to minimize average cost.

Use the envelope theorem to show that this behavior yields total cost of 1a , no matter what the value of

a is.


SOLUTIONS TO PAST TESTS


Solutions to Test 3: December 11, 2007

1. The revenues are R1 = [100− Q1 − Q2]Q1 and R2 = [100 − Q1 − Q2]Q2, so the marginal revenues are

MR1 = 100− 2Q1 −Q2 and MR2 = 100−Q1 − 2Q2. Setting these equal to the marginal costs gives the

two reaction curves:

Q∗1(Q2) = 25− Q2

2

Q∗2(Q1) = 14− Q1

2.

The simultaneous solution is Q1 = 24 and Q2 = 2. Here is the graph:

Note that the heavy dark lines are the reaction curves for firms 2 and 1 once Q1 exceeds 28 or Q2 exceeds

50, respectively. This will become important in question 8. Aggregate quantity is Q = Q1 +Q2 = 26 and

price is P = 100−Q = 74. Profits are then

π1 = [P − c1]Q1 = [74− 50]24 = 576

π2 = [P − c2]Q2 = [74− 72]2 = 4.

2. Revenue is R = PQ = [100−Q]Q, so marginal revenue is MR = 100− 2Q. Set this equal to marginal

cost c1 = 50 and solve for Q to obtain Q = 25. Price is then P = 100 − Q = 75, and profit is

π1 = [P − c1]Q = [75− 50]25 = 625.

286 SOLUTIONS TO TEST 3: DECEMBER 11, 2007

3.

4. The Hicksian and Marshallian demands coincide due to zero income effects. So the EV is the area left

of the Marshallian demand. The shaded area is 1 × 25 + 12× 1 × 1 = 25.5. Profit in question 2 is 625.

Aggregate profit in question 1 is π1 + π2 = 576 + 4 = 580. So Cournot competition reduces profit by

625 − 580 = 45. Consumer surplus only goes up by 25.5, so society is worse off with the competition.

Even though aggregate quantity goes up and price goes down, thereby eliminating some deadweight loss,

production costs rise because some of the production shifts to a firm that is very inefficient. The new firm

has costs that are much higher than the old monopolist. These higher costs more than offset the gain due

to elimination of deadweight loss.

5. With Bertrand competition the firms undercut each other until price equals the marginal cost of the

high-cost firm. So P1 = P2 = c2 = 72 (note that this is below the monopoly price of firm 1, calculated

in question 2). This assumes that firm 1 sells all Q = 100− 72 = 28 units of output at this price, so the

output of firm 2 is zero. If not, then firm 1 will undercut the price of 72 by an arbitrarily small amount

and firm 1’s output will be above 28 by an arbitrarily small amount.

6. Under Bertrand competition, price is lower and quantity is higher than under Cournot competition. So

there is an additional saving of deadweight loss beyond that saved under Cournot competition (see the

graph). Also, the low-cost firm produces all of the output under Bertrand competition, so there is no rise


in production costs (unlike the Cournot case). Hence society is unambiguously better off.

7. Firm 1 is globally more efficient than firm 2. Hence all production should occur in firm 1. The collusive

optimum is firm 1’s monopoly outcome, derived in question 2. However, firm 2 can obtain profit of $4 by

refusing to collude and thereby forcing the outcome to the Cournot solution. So at least $4 of firm 1’s

profit must be given to firm 2 to buy firm 2’s cooperation. Similarly, firm 1 can obtain profit of $576 by

refusing to collude. So no more than $625 - $576 = $49 of firm 1’s profit can be given to firm 2. Any

division of the $625 profit that gives firm 2 an amount between $4 and $49 is possible, depending on the

bargaining mechanism used to divide the profit.

8. Firm 2 (the follower) behaves as a Cournot competitor. So firm 2’s complete contingent plan is:

Q∗2(Q1) =

{14− Q1

2 when Q1 ≤ 280 when Q1 > 28

.

Firm 1 anticipates this reaction and therefore has objective function

π1 = [100−Q1 −Q∗2(Q1)]Q1 − 50Q1 = [50−Q1 −Q∗

2(Q1)]Q1.

Substituting Q∗2(Q1) = 14 − Q1

2, differentiating with respect to Q1, and setting the derivative to zero

yields Q1 = 36. If we are careless, we might imagine this is the equilibrium strategy for firm 1. But

it exceeds 28, so 14 − Q12

is not the correct expression for Q∗2(Q1) at this quantity. Indeed, Q1 = 36

imagines that firm 1 chooses the light colored tangency between an isoprofit and firm 2’s reaction curve

illustrated above in the graph for question 1. This corresponds to Q2 = −4, which is nonsensical. Instead,

the best isoprofit firm 1 can obtain touches firm 2’s reaction curve at the kink in Q∗2(Q1) that occurs

when it hits the axis at Q1 = 28. This is the darker isoprofit in the graph above, which actually crosses

the line Q2 = 14 − Q12 but does not cross firm 2’s reaction curve because of the kink. So the subgame

perfect equilibrium is (28, Q∗2(Q1)) (note that firm 2’s strategy is the whole function; firm 2’s quantity in

the equilibrium is zero). The price is P = 72 and the profits are π1 = (72 − 50)28 = 616 and π2 = 0.

Note that firm 1 produces more than the monopoly output of 25 and the price is correspondingly lower.

This occurs even though firm 2 produces nothing in equilibrium because it is in firm 1’s interest to push

output up to 28 in order to drive firm 2 from the market. This eliminates more deadweight loss than

the Cournot competition above while still having all production done by the low-cost firm. Hence social

welfare is higher than under either the Cournot or monopoly situations.


9. Firm 1 acting as a price-taker is willing to supply any amount between 0 and ∞ at price P = c1 = 50.

Supply is zero below this price (firm 2 supplies nothing unless price is at least c2 = 72) and is infinite

above this price. So P = 50 and Q = 50, with firm 1 producing the entire amount and firm 2 producing

nothing.

10. Now the Hicksian and Marshallian demands do not coincide. A normal good has a Hicksian demand that

is steeper than the Marshallian demand:

Note that the relevant Hicksian demand for the EV of a price decrease from 75 to 74 is the one through

the point (26, 74) because we are measuring a change in income that is equivalent to the price decrease

so utility is held constant at the level obtained after the price decrease. The EV is now larger by the

amount of the shaded area in the graph. However, this area is less than 1 and is therefore far too small

to make up for the higher production costs that occur under Cournot competition; society is still better

off under the monopoly. Bertrand competition is still better than the monopoly because price is lower in

a Bertrand environment and all production is done by the low-cost firm. Similarly, Stackelberg is better

than monopoly or Cournot (Stackelberg is actually the same as Bertrand in this model).


Solutions to Test 2: November 6, 2007

1.

a.

Avoidable C = α+{

y when y ≤ 12y2 − 1 when y > 1

AC ={ 1

y+ 1 when y ≤ 1

2y when y > 1

MC ={

1 when y < 14y when y > 1

AAC =

{αy+ 1 when y ≤ 1

α−1y + 2y when y > 1

b. Set p = MC when p ≥ AAC. This gives p = 4y, or y = p4 for p > 4. y = 1 is optimal for any p between

1 + α and 4. y = 1 and y = 0 are both optimal when p = 1 + α. y = 0 is optimal when p < 1 + α.

2.

a. The Lagrangian function is L = x1 + g(x2) − λ[p1x1 + p2x2 −m]. So the FOCs are 1 − λp1 = 0 and

g′(x2)− λp2 = 0 (and also −[p1x1 + p2x2 −m] = 0).

b. The first equation gives λ = 1/p1. Substituting this into the second gives g′(x2) = p2p1. This is the

(implicit) solution for x2 since x2 is the only choice variable in this equation.

c. m does not enter the equation for x2, so x2 does not change when m changes (assuming an interior

solution).

290 SOLUTIONS TO TEST 2: NOVEMBER 6, 2007

d. Since the income effect on x2 is zero, the Hicksian and Marshallian demands are the same. The

Hicksian demand coincides with the Marshallian demand no matter what level of U is used in the

Hicksian demand.

3.

a. w2 = 2w1 makes w1x1 + w2x2 smallest. So the defining condition for H(y) requires w1x1 + 2w1x2 ≥

3w1y.

b. w1 = 2w2 makes w1x1 + w2x2 smallest. So the defining condition for H(y) requires 2w2x1 + w2x2 ≥

3w2y.

c. H(y) = {(x1, x2) ∈ R2+ : x1 + 2x2 ≥ 3y and 2x1 + x2 ≥ 3y}. The (x1, x2) pairs that satisfy these two

inequalities are the shaded area:

d. If the isocost slope is steeper than 2 then the cost minimum subject to H(y) occurs at the corner

(0, 3y), which gives c∗ = 3w2y as expected. If the isocost slope is flatter than 1/2 then the cost

minimum subject to H(y) occurs at the corner (3y, 0), which gives c∗ = 3w1y also as expected. All

that remains is to identify the cost minimum when the isocost slope is between 1/2 and 2. Given H(y),

the minimum occurs at the vertex (y, y), which gives c∗ = (w1 + w2)y.

4.

a. The indifference curve is the line with slope −1 from (0, U) to (U , 0), except for the point where this


line crosses the x2 = x1 line, plus the point ((U − 1)/2, (U − 1)/2):

b. The preferences are complete and transitive because they are represented by a utility function. They

are not continuous because of the “hole” in the indifference curve at x2 = x1. They are also not

monotonic or strongly monotonic because all points to the northeast of ((U − 1)/2, (U − 1)/2) and

below the line x2 = U − x1 are on a lower indifference curve (except the points on the x2 = x1 locus).

The preferences are, however, locally nonsatiable since we can increase any bundle by (ε, ε) and thereby

strictly increase utility. The lack of continuity rules out all hope for any type of convexity.

c. The delicate part is to consider when it is optimal to choose the point on the x2 = x1 line. Consider the

line through this point and the lower-right corner (U , 0). This line has slope − U−1U+1

. If the isocost line

is flatter than this line then the minimum occurs at (U , 0), which gives expenditure of p1U . Similarly,

the line through the point on the x2 = x1 line and the upper-left corner (0, U) has slope − U+1U−1

. If the

isocost line is steeper than this line then the minimum occurs at (0, U), which gives expenditure of p2U .

The minimum occurs at ((U − 1)/2, (U − 1)/2) for any other isocost slope, which gives expenditure

(p1 + p2) U−12

. To summarize, the expenditure function is

e∗(p1, p2, U) =

p1U when p1

p2≤ U−1

U+1

(p1 + p2) U−12 when U−1

U+1< p1

p2< U+1

U−1

p2U when p1p2

≥ U+1U−1

.

292 SOLUTIONS TO TEST 1: OCTOBER 4, 2007

Solutions to Test 1: October 4, 2007

1.

a. The production set is only the 4 square points:

b. Yes because the point (0, 0) is in the set.

c. Yes because Z does not contain any points in the first quadrant except (0, 0).

d. No because none of the points below and left of the four illustrated points are elements of Z.

e. The (absolute) slope of the isoprofit lines is p1p2. When this slope is steeper than 2 the highest isoprofit is

attained at (0, 0), so profit is zero. When this slope is between 1 and 2 the highest isoprofit is attained

at (−1, 2), so profit is 2p2 − p1. When this slope is flatter than 1 the highest isoprofit is attained at

(−4, 5), so profit is 5p2 − 4p1. Note that the point (−3, 3) is never optimal because it is “inside” the

line between (−1, 2) and (−4, 5). To summarize:

π∗(p1, p2) =

0, 2 ≤ p1

p2

2p2 − p1, 1 ≤ p1p2

< 2

5p2 − 4p1,p1p2

< 1

.

f. The graph of Z is changed to include all points below and to the left of the 4 points illustrated above.

This changes d to “yes”. Nothing else, including π∗, changes because the interior points are irrelevant

for π∗.

2. λ∗ measures how much a change in q is worth in units of the objective K. If λ is too high then the choice

of K will be such that g(K,L) > q because this decreases L more than the change in K increases the


objective. If λ is too low then the choice of K will be such that g(K,L) < q because this increases L less

than the change in K decreases the objective.

3. Yes. For homogeneity, evaluating the objective at (αw1, αw2) is still a monotonic transformation of the

objective evaluated at (w1, w2), so the optimal choices of x1 and x2 are unaffected. That is, x∗i (αw) =

x∗i (w) (y is suppressed here because it is not changing). Hence

c∗(αw) = αw1 lnx∗1(αw) + αw2 lnx∗

2(αw) = αw1 lnx∗1(w) + αw2 lnx∗

2(w) = αc∗(w).

For concavity, fix w0i and w1

i and note that the definition of a minimum requires, for every (x1, x2) pair

satisfying f(x1, x2) ≥ y,

c∗(w0) ≤ w01 lnx1 +w0

2 lnx2

c∗(w1) ≤ w11 lnx1 +w1

2 lnx2

multiplying the first inequality by α and the second by (1− α) (for α ∈ [0, 1]), and adding, yields

αc∗(w0) + (1− α)c∗(w1) ≤ [αw01 + (1− α)w1

1

]lnx1 +

[αw0

2 + (1− α)w12

]lnx2

for every (x1, x2) pair satisfying f(x1 , x2) ≥ y (this is the same proof used for a “normal” cost function).

Now minimize the right side over (x1, x2) subject to the constraint to get

αc∗(w0) + (1− α)c∗(w1) ≤ c∗(αw0 + (1− α)w1),

which is concavity.



1. If consumer 1 does not have locally nonsatiated preferences then indifference curves can be thick. Suppose

point E in the figure below is the endowment and the indifference curves are as shown, with consumer 1’s

indifference curve the entire shaded area, and both consumers having monotonic preferences. Then the

price line shown is an equilibrium, because each consumer is on his/her highest possible indifference curve

given his/her budget set, and both consumers are choosing the same point Q. However, the allocation Q

is not Pareto optimal because allocation R gives consumer 2 higher utility without changing the utility of

consumer 1.

2. The profit objective is

max{y}

π(y) = [p(y) − c]y = [α− c− βy]y.

The FOC is π′(y) = α− c − 2βy = 0 and the SOC is satisfied since π′′(y) = −2β < 0. Solving the FOC

yields monopoly output of ym = α−c2β . Substituting ym into the inverse demand yields the monopoly

price of pm = α+c2 . Substituting ym and pm into π yields the monopoly profit of πm = 1

β

[α−c

2

]2. The

deadweight loss is an area under the Marshallian demand because there are no income effects. It is the

area of the triangle with vertices at (ym , c), (ym, pm), and (p−1(c), c). Since p−1(c) = α−cβ , this area is

DWL =12

[α− c

β− α− c

2β

][α+ c

2− c

]=

12β

[α− c

2

]2.

3.


a. First, both sellers must charge the same price in equilibrium. If they charge different prices, the high-

price seller gets zero profit but can get half of the positive market profit generated at the low price by

switching to the low price (note that the low price exceeds marginal cost). Second, the strategy profile

(6, 6) is a Nash equilibrium since both sellers get positive profit and a unilateral switch to the high

price yields zero profit. So the only question is whether (10, 10) is a Nash equilibrium as well. The

market demand at p = 10 is y(10) = 10 and the market demand at p = 6 is y(6) = 14, so market profit

at each price is π(10) = [10− 4]10 = 60 and π(6) = [6− 4]14 = 28. If both sellers are charging p = 10

then they each get profit of 30, while a unilateral switch to p = 6 yields profit of only 28 for the seller

who switched. Hence (10, 10) is a second Nash equilibrium.

b. The observations that both firms must charge the same price and that a profile with both firms charging

the low price ((7, 7) in this case) is a Nash equilibrium do not depend on the particular prices (assuming

the low price is between the monopoly price and marginal cost). So (7, 7) is a Nash equilibrium. Market

output is y(7) = 13, so market profit is π(7) = [7 − 4]13 = 39. Hence (10, 10) is no longer a Nash

equilibrium, because a unilateral switch to p = 7 increases profit from 30 to 39.

c. If the leader chooses 10 then the follower will choose 10 as well, since doing so yields profit of 30 for

the follower while choosing 6 only yields profit of 28. The leader gets profit of 30 in this case. If the

leader chooses 6 then the follower will choose 6 as well, since doing so yields profit of 28/2 = 14 for

the follower while choosing 10 would yield zero profit since the follower would be the high-price seller.

The leader gets profit of 14 in this case. Hence the Nash equilibrium is for the leader to choose 10 and

the follower to choose the function p(10) = 10 and p(6) = 6. Both sellers will choose 10 in equilibrium.

Things change if the low price is 7. Then a choice of 10 by the leader will cause the follower to choose

7 since doing so give the follower profit of 39 rather than 30. The leader gets profit zero in this case. A

choice of 7 by the leader will cause the follower to choose 7, yielding profit of 39/2 for both the leader

and the follower. Hence the Nash equilibrium is for the leader to choose 7 and the follower to choose

the function p(10) = 7 and p(7) = 7. Both sellers will choose 7 in equilibrium.

4. The profit objective is

max{x2}

π = px1 − x2 = 2p√x2 − x2.


The FOC is π′(x2) = px−1/22 − 1 = 0, yielding x∗

2 = p2 and x∗1 = 2p. The SOC is π′′(x2) = −p

2x−3/22 < 0

so the FOC yields a maximum provided the quantities are positive. Substituting into the objective yields

π∗(p) = p2.

The consumer’s monetary resources are x2 + π∗(p) = 3 + p2. Hence the utility objective is

max{x1,x2}

[x

2/31 + x

2/32

]3/2subject to px1 + x2 = 3 + p2.

The partials of utility are Ui = 32

[x

2/31 + x

2/32

]1/223x

−1/3i . Hence the MRS is −U1

U2= −

[x2x1

]1/3. Setting

this equal to the budget slope of −p yields x2 = x1p3 (it is straightforward but tedious to check that the

indifference curve is strictly convex with infinite slope where it hits the vertical axis and zero slope where

it hits the horizontal axis). Substituting for x2 in the budget line and simplifying yields x∗1 = 3+p2

p+p3 .

The equilibrium price can be found by setting supply equal to demand for either good. For example,

supply of good 1 from the firm’s problem is x∗1 = 2p and demand for good 1 from the consumer’s

problem is x∗1 = 3+p2

p+p3 . Setting these equal and simplifying yields 2p4 + p2 − 3 = 0. This factors into

(2p2 + 3)(p2 − 1) = 0. So the equilibrium price is p = 1.



1.

a. The indifference curves are parabolas with minimum at x1 = 1. The parabola shifts vertically as the

utility level changes. The marginal rate of substitution is

∂x2

∂x1= −U1

U2= 2(x1 − 1).

Note that this slope is −2 at x1 = 0 irrespective of the utility level. So every indifference curve has

slope −2 where it hits the vertical axis. A rough graph is (the parts in the fourth quadrant are for

illustration purposes only, and are not actually part of the indifference curve):

b. Completeness: Yes, because the preferences are represented by a utility function.

Transitivity: Yes, because the preferences are represented by a utility function.

Continuity: Yes, because U is a continuous function, so the sets {(x1, x2) : U(x1, x2) ≥ U} and

{(x1, x2) : U(x1, x2) ≤ U} are closed for every U .

Local Nonsatiation: Yes, because an infinitesimal increase in x2 always increases utility.

Weak Monotonicity: No, because an increase in x1 starting at any x1 ≥ 1 always decreases utility.

Convexity: Yes, because the entire convex combination of any two points that are on or above a

given indifference curve lies entirely above that indifference curve.


c. The slope of 2 is critical to the analysis. Budget lines that have a slope steeper than 2 result in a corner

solution at x1 = 0 and x2 = m/p2, since this point achieves the highest possible indifference curve.

Otherwise, the optimum occurs at a tangency between the indifference curve and the budget line, which

is described by MRS = −p1/p2 or 2(x1 − 1) = −p1/p2. Solving for x1 yields x∗1 = 1 − p1/2p2. The

budget line then yields x∗2 = m/p2 − (p1/p2)x∗

1 = m/p2 − (p1/p2) [1− p1/2p2]. In summary,

x∗1 =

{0 if p1

p2≥ 2

1− p12p2

if p1p2

< 2x∗

2 =

mp2

if p1p2

≥ 2

mp2

− p1p2

[1− p1

2p2

]if p1

p2< 2

2. First we have to find U∗. The MRS is

∂x2

∂x1= −U1

U2= −x2 + 1

x1= − U

x21

.

Differentiating again while holding U constant confirms that the indifference curves are strictly convex, so

any tangency between an indifference curve and a budget line is a maximum. Setting the MRS equal to

−p1p2

yields x1 = p2p1(x2+1). Substituting this into the budget line and using p1 = 1 yields the Marshallian

demands

x∗1 =

m+ p2

2and x∗

2 =m− p2

2p2.

Note that these expressions are an interior solution only if m > p2, which holds for the values given in

the problem (otherwise, the optimum is at the corner x2 = 0 and x1 = m). Substituting x∗1 and x∗

2 into

U and simplifying yields (for m > p2)

U∗ =(m+ p2)2

4p2.

Therefore the initial utility level is U∗(1, 2, 50) = 338. The income level m required to maintain this

utility level when p2 = 4 satisfies

338 =(m+ 4)2

16.

Solving this quadratic yields m = −4+4√338 = 69.54. Quantity demanded at the new price/income pair

is x∗2 = (69.54− 4)/8 = 8.19. So we must spend 69.54− 50 = 19.54 to keep the consumer indifferent and

we receive only (4− 2)(8.19) = 16.38 in new revenue, yielding a deadweight loss of 19.54− 16.38 = 3.16.


3. One implication of homogeneity of x∗1 (of degree zero) is

p1∂x∗

1

∂p1+ p2

∂x∗1

∂p2+ p3

∂x∗1

∂p3+m

∂x∗1

∂m= 0

(this is Euler’s Theorem). Substituting in the given information yields

(1)∂x∗

1

∂p1+ (2)(−5) + (6)(2) + (10)

∂x∗1

∂m= 0.

Since there are no income effects on commodities 2 and 3, but local nonsatiation holds (so we must stay

on the budget line), we know ∂x∗1

∂m = 1p1

= 1. Therefore

∂x∗1

∂p1= −12.

Now use the Slutsky equation:

∂x∗1

∂p1=

∂h∗1

∂p1− x∗

1

∂x∗1

∂m

= −10− (x∗1)(1).

Substituting the two equations together yields x∗1 = 2.



1.

a. Since we are assuming there is an interior solution, that solution must satisfy the first order conditions

of the Lagrangian function L. Differentiating the first order conditions with respect to a yields a system

of equations of the following form:

HL(x1,x2,x3,x4,x5,λ)

∂x∗

1∂a...

∂x∗5

∂a∂λ∗∂a

=

00

−2a000

,

where λ∗ is the Lagrangian multiplier at the optimum and H is the Hessian of the Lagrangian function

with respect to the choice variables (x1, x2, x3, x4, x5, λ). Applying Cramer’s Rule to this system and

then using cofactor expansion on the numerator yields

∂x∗3

∂a=

(−1)3+3(−2a)|H33||H | ,

where H33 is the submatrix of H obtained by deleting the third row and third column. The second

order conditions require that |H | and |H33| have opposite signs, so

∂x∗3

∂a

s= 2a > 0.

b. The envelope theorem gives

f∗′(a) =∂L

∂a

∣∣∣∣xi=x∗

i ,λ=λ∗= 2ax∗

3(a) > 0.

c. Here is an answer based on comparing the optimal value function to the objective function, as we did

several times in class. Fix a at a particular value, say a0, and define the function

g(a) = x∗1(a0)x∗

2(a0)x∗3(a0)x∗

4(a0)x∗5(a0) + x∗

1(a0)1/3x∗2(a0)1/9x∗

4(a0)5x∗5(a0)

+ x∗3(a0)1/5x∗

4(a0)x∗5(a0)2/7 + a2x∗

3(a0).

g is the objective function considered as a function of a with the values of the choice variables fixed at

the values that are optimal when a = a0. By definition, g(a0) = f∗(a0) and the envelope theorem gives


g′(a) = f∗′(a) at a = a0. Moreover, f∗ cannot drop below g as a departs from a0 because f∗ is the

maximum (note that changes in a do not affect the feasibility of the optimal choices x∗i (a0) because a

does not affect the constraint). This means f∗ must be “more convex” than g at a = a0. Note that

g′′(a) = 2x∗3(a0). Hence f∗′′(a) ≥ 2x∗

3(a0) at a = a0. This holds for any a0.

Here is a second, equally acceptable, answer based on the answers to questions b and c above. Differ-

entiating b, we obtain

f∗′′(a) = 2x∗3(a) + 2a

∂x∗3

∂a> 2x∗

3(a).

The inequality follows from question a above.

2.

a. At least 1 unit of x1 must be used in order to produce any output. This usage of x1 is avoidable, so

the fixed cost is not sunk. A typical isoquant for a positive level of output appears as follows:

The isoquant for output level zero is the horizontal axis and the entire area between the vertical axis

and the vertical line x1 = 1. The lowest isocost line for any positive output level is attained at the

vertex of the isoquant where x2 = x1 − 1. So we have y =√x2 =

√x1 − 1 at the minimum, or

x∗1 = y2 + 1 and x∗

2 = y2 . This gives a cost function of

c∗(w1, w2, y) ={

0 if y = 0(w1 +w2)y2 +w1 if y > 0

.


b. The average cost curve is AC = (w1 + w2)y + w1y for y > 0 and the marginal cost curve is MC =

2(w1 + w2)y. It is straightforward to set AC = MC and solve to find that the minimum of the AC

curve is at y =√

w1w1+w2

, and the common value of AC and MC at this intersection is 2√

w1(w1 + w2).

Setting p = MC gives a supply curve of y∗ = p2(w1+w2)

. Since all of the fixed cost is avoidable, this

expression for y∗ is relevant for p ≥ 2√

w1(w1 +w2). y∗ = 0 when p ≤ 2√

w1(w1 +w2).

3. This is a matter of going through the proof of the properties of the profit function and checking on

which properties of π∗ rely on particular properties of the technology. If we know that π∗ exists then

the homogeneity, convexity, and monotonicity all follow merely from the definition of π∗ as a maximum.

Nonnegativity, however, relies on the possibility of inaction (profit can be negative if there is a sunk cost).

4. The function g has all of the standard properties of a cost function except nonnegativity. Therefore we

need only consider what properties of H rely on nonnegativity and whether potential failures of these

properties affect the conclusion that g is the cost function for the technology H .

a. Going through the proof of the properties of H defined in terms of some function g, we see that no free

lunch and possibility of inaction both rely on nonnegativity. So H need not have the no free lunch and

possibility of inaction properties. All of the other properties of H (i.e., that Z = {(y,−x) : x ∈ H(y)}

is nonempty and closed, free disposal, convexity, and continuity) hold. The proofs that these properties

hold are the same as always, except for showing that Z is nonempty (Z nonempty is normally taken as

a consequence of possibility of inaction, so whether Z is nonempty has to be addressed in a different

way here – more on this below).

b. Let c∗ Be the cost function for H . The proof that g(w, y) = c∗(w, y) proceeds as follows: First note

that c∗(w, y) ≥ g(w, y) by definition of g as a lower bound of the cost objective on the feasible set

H(y); then establish existence of an input vector x0 with the properties (1) w · x0 = g(w, y), and (2)

x0 ∈ H(y). Existence of such an x0 establishes c∗(w, y) ≤ g(w, y) by definition of c∗ as a minimum over

H(y). Item (1) is a pure consequence of homogeneity of g. Item (2) is a pure consequence of concavity

of g. Therefore the fact that g may not possess the nonnegativity property has no implications for the

basic result that g is a cost function. Note that existence of x0 ∈ H(y) establishes that Z is nonempty

(assuming there are y values in the domain of g).



1.

a. Marginal cost is c′(q) = 2q, so single-firm supply is q = p2. Therefore market supply is Q = 5p. This

intersects market demand at Q = 650 and p = 130.

b. The profit objective of one firm is π(q) = (780 − q − Q)q − q2, where Q is the aggregate quantity of

all other sellers. The FOC is 780 − 2q − Q − 2q = 0. By symmetry, Q = (n − 1)q in equilibrium.

Substituting this into the FOC with n = 10 and solving yields q = 60. Hence Q = 600. Substituting

this into demand gives p = 180.

c. Here is the graph:

The deadweight loss is the shaded area:(

12

)(50)(60) = 1, 500.

d. This technology has decreasing returns, so every price-taker makes positive profit. Entry will continue

until there are a large number of producers each producing a small amount and price virtually zero.

Technically, there is no free-entry equilibrium because, with any finite number of producers, a new

entrant can always earn a tiny positive profit.

e. Using the FOC from part b with n unspecified yields q = 780n+3

. The equilibrium price with n sellers is

p = 780− nq, so equilibrium profit for one of n sellers is π = pq − q2 = [p − q]q = [780− (n + 1)q]q.

Now substituting in q = 780n+3 and simplifying yields π = 2

(780n+3

)2

. This is strictly decreasing in n.


Each entrant must anticipate that this post-entry profit will exceed the entry cost of $1,000. Setting

π = 1, 000 and solving for n yields n = 780√2/1, 000−3 ≈ 31.9. Thus all sellers make positive variable

profit above $1,000 when there are 31 sellers but the variable profit will not cover the entry cost if there

are 32 sellers. In equilibrium there will be n = 31 sellers.

2. Monopoly marginal revenue is MR = 780 − 2Q. Setting this equal to marginal cost of MC = 2Q and

solving for Q gives Q = 195. Substituting into demand gives P = 585. The elasticity at this point is

ε = |(−1)585195

| = 3. Based on this, the Lerner Index is 33%.

3.

a. Yes, a competitive equilibrium can be Pareto efficient even when a consumer’s preferences do not

have the local nonsatiation property. The proof of the First Theorem of Welfare Economics relies on

local nonsatiation only to ensure that every consumer’s optimum must be on the budget constraint in

equilibrium. This can happen even for a consumer whose preferences do not satisfy local nonsatiation.

For example, suppose the utility function has a local maximum inside the budget set but then increases

to higher utility levels at higher consumption levels, where a competitive equilibrium budget constraint

lies. Then the only maximum is on the budget line and the competitive equilibrium is Pareto efficient

because any reallocation to a point below the budget line of the consumer in question, even to the

interior local maximum, will harm that consumer. In short, the conditions of the First Theorem of

Welfare Economics are sufficient but not necessary for a competitive equilibrium to be Pareto efficient.

b. Yes. An important condition of the First Theorem of Welfare Economics is violated so the conclusion

of the theorem may not hold. In particular, suppose a consumer in a pure exchange economy has

a thick indifference curve and a competitive equilibrium budget constraint passes through that thick

indifference curve, as illustrated in the following Edgeworth box:


This shows an equilibrium because supply equals demand and each consumer is individually maximized

given the budget line. However, the equilibrium allocation is not Pareto efficient because there are other

allocations, like point C, that are strictly better for B and no worse for A.



1.

a. These preferences are represented by the utility function U(x1, x2, x3) = x1 +√x2 − x3. This function

is continuous. Therefore the preferences are complete, transitive, and continuous. The preferences are

also locally nonsatiable since a small increase in x1 or x2 always improves utility. The preferences

are also convex. Convexity can be shown by thinking about the shape of the indifference curves (see

below), or by noting that the utility function is concave since the Hessian is a matrix of zeros except

for the (2,2) element, which is negative. The preferences are not monotonic since increases in x3 cause

utility to decrease.

b. x3 decreases utility, so the demand for x3 is identically zero for all non-negative prices and income. A

typical indifference curve between x1 and x2 for utility level U is:

The slope of the indifference curve is zero at x2 = 0 but is −2U at x1 = 0, so a corner solution is

possible with x1 = 0. Solving the FOCs simultaneously for x1 and x2 yields

x∗1 =

m

p1− p1

4p2and x∗

2 =(

p1

2p2

)2

.

These are the demands provided the solution for x∗1 is positive. Otherwise, x∗

1 = 0 and x∗2 = m

p2. To

summarize:

x∗1 =

{ mp1

− p14p2

, if 4mp2 ≥ p21

0, otherwise, x∗

2 =

(p12p2

)2

, if 4mp2 ≥ p21

mp2

, otherwise, x∗

3 = 0.


Substituting these solutions into U gives indirect utility of

U∗ =

mp1

+ p14p2

, if 4mp2 ≥ p21√

mp2

, otherwise.

2.

a. Check whether the demands have the properties of Marshallian demands. (1) The given demands are

homogeneous of degree zero in prices and income: multiplying both prices and income by some scalar

α does not change anything because the α’s cancel. (2) The Slutsky matrix is

{∂x∗

i

∂pj+ x∗

j

∂x∗i

∂m

}ni,j=1

=

[− 2m9p21

2m9p1p2

2m9p1p2

− 2m9p22

].

This matrix is symmetric, and it is negative semidefinite because the (1, 1) element is negative and

the determinant of the whole matrix is zero. (3) The given x∗ function is nonnegative. (4) p · x∗ =

p1m3p1

+ 2p2m3p2

= m, so Walras’ Law holds. Therefore the given function can be a Marshallian demand

for a consumer with well-behaved preferences.

b. Check whether the demands have the properties of Hicksian demands. (1) The given demands are

homogeneous of degree zero in prices: multiplying both prices by some scalar α does not change

anything because the α’s cancel. (2) The matrix of price slopes is negative semidefinite, because the

(1,1) element is ∂h∗1

∂p1= − U

2 p−3/21 p

1/22 , which is negative, and the determinant of the whole matrix

must be zero due to the homogeneity. The matrix of price slopes is also symmetric because ∂h∗1

∂p2=

∂h∗2

∂p1= U

2√p1p2

. (3) The corresponding expenditure function is e∗ = p · h∗ = 2U√p1p2. This function is

nondecreasing in (p1, p2) provided U ≥ 0 (more on this condition below), and is strictly increasing in U

at positive prices. (4) The e∗ just derived is also continuous. (5) The given demands are nonnegative

provided U ≥ 0, are strictly positive at positive prices provided U > 0, and the expenditure function

is identically zero in prices at U = 0. So the nonnegativity property holds with U(0) = 0. Therefore

the given function can be a Hicksian demand for a consumer with well-behaved preferences, with the

utility representation U of those preferences that appears in h∗ benchmarked so that U ≥ U(0) = 0.

c. Invert e∗ from question b to get indirect utility U∗ = m2√p1p2

. Then use Roy’s Identity to get the

Marshallian demand x∗ =(m2p1

, m2p2

). This is different from the Marshallian demand given in question

a, so the preferences are not the same (cannot be for the same consumer).


3.

a. Using the Marshallian demand from question 3.c, government revenue from Tax A is RA = tp1x∗1((1 +

t)p1, p2, m) = tm2(1+t) .

b. T is calculated so that the consumer is indifferent between Tax A and Tax B. That is, T is the (negative

of the) equivalent variation of the price increase from p1 to (1 + t)p1. Using the expenditure function

from question 2.b and the indirect utility function from question 2.c, government revenue from Tax B

is RB = T = −EV = −[e∗(p1, p2, U∗((1 + t)p1, p2, m)) − m] = −[2U∗((1 + t)p1, p2, m)

√p1p2 −m] =

−m[(1 + t)−1/2 − 1

].

c. RB − RA = m[1− (1 + t)−1/2 − t

2(1+t)

]. Making a common denominator, this has the same sign as

g(t) = 2 + t − 2√1 + t. Note that g(0) = 0 (neither tax produces any revenue if the tax rate is zero)

and g′(t) = 1− (1 + t)−1/2 > 0 for t > 0. Hence RB > RA for every t > 0. Tax B always collects more

revenue for the government when the level of Tax B is set to make the consumer indifferent between

Tax A and Tax B. This is a general property: A lump-sum tax does not distort consumption, and is

therefore more efficient than a tax that changes the relative prices paid by consumers.



1.

a. The FOC is 8x3 − a3 = 0. This yields x∗(a) = a/2. The second derivative is 24x2, which is always

positive. So there is only one extremum, and it is a minimum.

b.

c. See the graph above. Note that a0 must be positive because x = x∗(a0) = a0/2 is positive. Also note

that, for a0 positive, 21/3x < a0 when x = x∗(a0) = a0/2, so a0 is right of the point where g crosses the

a axis. The graph shows a lot of the g∗ function for completeness. Only the local part of g∗ near a0

can actually be graphed without explicitly deriving g∗. The important thing is that optimality requires

that g∗ be tangent to g at a0 and below g everywhere else.

d. g∗ is locally concave because g is locally concave. This argument applies at all a0 (even when a0 < 0,

because then the shape of g is reflected across the vertical axis compared with the graph above), so g∗

is indeed globally concave.

2.

a. The technology is nonempty because there are points in the set. The technology is NOT closed, because

part of the boundary is missing. The technology satisfies no free lunch because the only point in the

first quadrant that is also in Z is the origin. The technology satisfies possibility of inaction because the


origin is included. The technology satisfies free disposal because there is no upward-sloping boundary.

The technology has no particular global returns to scale (the various rays involved in the definitions

will pass outside of the set). The technology is not convex because of the part of the boundary that

curves upward.

b. No. Draw some isoprofit lines that are tangent to the production set. No profit maximizing actor ever

chooses a point on the non-convex part of the boundary, so the fact that part of the boundary is not

included has no implications for maximization. This conclusion is not general. It holds only because

the missing boundary is on a nonconvex part of the boundary.

3.

a. Use Shephard’s Lemma to get

x∗1 =

∂c∗

∂w1= ey − 1

x∗2 =

∂c∗

∂w2= (1/2)w−1/2

2 w1/23 (ey − 1)

x∗3 =

∂c∗

∂w3= (1/2)w1/2

2 w−1/23 (ey − 1).

b. Differentiate the factor demands from part a to get the matrix of demand slopes with respect to prices:

(1/4)(ey − 1)

0 0 00 −w

−3/22 w

1/23 w

−1/22 w

−1/23

0 w−1/22 w

−1/23 −w

1/22 w

−3/23

.

Symmetry is verified by inspection. Negative semidefiniteness is verified by noting that the determinant

of the whole matrix is zero, the determinant of the 2× 2 matrix in the lower-right corner is zero, and

the determinant of the 1× 1 matrix in the lower right corner is negative.

The last part of the question involves derivatives with respect to y. From c∗, marginal cost is (w1 +

√w2w3)ey. Differentiating marginal cost with respect to w1, w2, and w3 yields ey, (1/2)w−1/2

2 w1/23 ey,

and (1/2)w1/22 w

−1/23 ey, respectively. These are exactly the same as the expressions obtained by differ-

entiating x∗1, x∗

2, and x∗3 above with respect to y, respectively.

4.

a. For y = 0, we can choose (x1, x2) = (1/4, 0) to get cost w1/4. For y > 0, we must choose x1 ≥ 1.

However, x1 is not productive beyond 1, so it is cost minimizing to choose x1 = 1. Then we choose


x2 = y2 , and get cost w1 + w2y2. To summarize:

c∗(w1, w2, y) ={

w1/4, y = 0w1 +w2y

2 , y > 0

b. At the stated prices, cost is 12 + y2 for y > 0. At y = 0, cost is 3. So 3 is sunk (out of the total fixed

cost of 12). We ignore the sunk cost when determining shutdown. Hence the non-sunk cost is 9 + y2,

so the average non-sunk cost is 9/y + y. Marginal cost is 2y. Setting marginal cost equal to average

non-sunk cost yields y = 3. Substituting back into either function gives 6. The graph is (the heavily

shaded line is the supply curve):

The equation for the supply curve is

y∗ ={

0, p ≤ 6p/2, p ≥ 6

.



1.

2. No. The technology has increasing returns (average cost is 6 − q for q ≤ 2 and 4/q + 2 for q ≥ 2, which

is strictly decreasing). A price-taking seller will supply an infinite amount at any price above 2, and will

supply zero at any price below 2. So the only hope for an equilibrium price is P = 2. However, profit is

negative at every positive quantity when P = 2 because the higher marginal cost of the first 2 units is

never recovered. So a price-taking seller will supply zero when P = 2. Hence there is no price at which

supply equals demand.

3. From the graph, the monopoly output is 2 and the monopoly price is 6.

4. Marginal cost equals marginal benefit at the social optimum. When Marshallian surplus measures con-

sumer welfare, marginal benefit is the inverse demand curve. From the graph, marginal cost crosses

demand at output 4.

5. The First Theorem of Welfare Economics states (under some conditions) that a price-taking equilibrium is

Pareto Efficient. The theorem does not guarantee existence of a price-taking equilibrium. So, when there

is no price-taking equilibrium, the theorem is vacuous. There is no conflict between non-existence of a

price-taking equilibrium and the theorem because the theorem merely establishes a property of equilibrium

when the equilibrium exists.


6. The monopoly deadweight loss area is shaded in the graph. The area is DWL = 12(4− 2)(6− 2) = 4.

7. For firm 1, the Cournot objective is max{q1≥0} π1(q1|q2) = (10−2q1−2q2)q1−(6q1−q21) for q1 ∈ [0, 2]. It is

clear from the graph that we do not have to think about quantities larger than 2, since the marginal revenue

curve for a Cournot duopolist is left of the monopoly marginal revenue curve, so the duopoly marginal

revenue must cross marginal cost to the left of 2. The first order condition is 10− 2q2− 4q1 − 6+2q1 = 0.

Solving for q1 gives the reaction curve q∗1(q2) = 2 − q2 for q2 ∈ [0, 2]. This reaction curve is unusual in

that its slope is -1. So if we find the reaction curve for the other firm it will be exactly the same line

(the reaction curves coincide). Hence any point on this line is a Nash equilibrium, but only the point

(q1, q2) = (1, 1) is a symmetric Nash equilibrium. The equilibrium market output is 2 (same as monopoly),

so the equilibrium price is 6.

8. It is tempting to conclude that the duopoly deadweight loss is the same as the monopoly deadweight

loss, since aggregate output and price are the same. However, the symmetric Cournot duopoly incurs

higher cost than the monopoly in producing 2 units of output, because of increasing returns. So there is

additional deadweight loss, equal to the extra cost incurred by the duopoly. In the graph, this extra cost

is the difference between the area under MC between 0 and 1 and the area under MC between 1 and 2.

We can depict this area by moving the segment of MC between 1 and 2 to the interval between 0 and 1,

and then looking at the difference between this moved segment and the MC curve. This extra DWL area

is shaded in the graph. The extra area is (6 - 4)(1 - 0) = 2, so the total DWL for the duopoly is 6.

9. Normally we expect the duopoly to be more efficient than the monopoly. But with globally increasing

returns the socially efficient way to produce any given output is to have one firm do all production. So

there is a tradeoff. A duopoly usually produces more than a monopoly, thereby moving the market closer

to the social optimum. But the duopoly incurs inefficiently high costs in doing so, compared to what

could be obtained if the monopoly produced more. So the net effect on welfare of having a duopoly rather

than a monopoly is ambiguous when there is increasing returns. In the particular example under study

here the shape of the MC curve causes the symmetric duopoly output to be the same as the monopoly

output, thereby eliminating the benefit of duopoly. Hence it is unambiguously bad for welfare to have a

duopoly in this particular example.


10. We must set monopoly profit equal to zero and solve for price and quantity. Profit is positive at the

monopoly output of 2 in question 3. So we know the zero profit point will be at a quantity greater

than 2. This means we must use the second part of the cost function. So the zero-profit condition is

8q−2q2−4 = 0. The solutions to this equation from the quadratic formula are q = 2±√2. Since we know

we are looking for a quantity larger than 2, the relevant solution is q = 2 +√2. The price is therefore

P = 10− 2(2 +√2) = 6− 2

√2.

11. Suppose we have two Bertrand competitors, each charging price P1 = P2 = 6−2√2. We must ask whether

either firm has a unilateral incentive to change price. This price gives zero profit for a single-seller. With

increasing returns, this means the two firms make negative profit at this price if they both produce positive

output. Since all costs are avoidable, the firms prefer shut down over negative profit. So the only way

the price pair P1 = P2 = 6 − 2√2 can be an equilibrium is if one firm produces all output (i.e., quantity

2 +√2) and the other firm produces zero output. With this arrangement of the quantities, consider now

the incentives of each firm to change price. The firm that is producing nothing can do no better by raising

price (output and profit will still be zero). This firm will steal the whole market from its rival if it lowers

price, and will make negative profit as the only seller at a lower price because the price we are looking at

is the zero-profit price. Hence this firm has no incentive to change price. The firm that is producing the

market output is making zero profit because we are looking at the zero-profit price. This firm will have

zero output and thus zero profit if it unilaterally raises price because then all consumers will buy from

the other firm. And it will earn negative profit at a lower price. Hence this firm also has no incentive

to change price. Therefore the price pair P1 = P2 = 6 − 2√2 is a Nash equilibrium, provided one firm

produces nothing and the other firm produces the entire market output.

12. From question 7, the reaction curve of the follower is q∗2(q1) = 2 − q1 for q1 ∈ [0, 2], and this is the

only relevant range of q1, for the same reason given in question 7. So the leader’s objective is π1 =

(10− 2q1 − 2(2− q1))q1 − (6q1 − q21) = q2

1 for q1 ∈ [0, 2]. This objective is strictly increasing in q1 on the

relevant interval. So the leader’s maximum is the corner q∗1 = 2. Therefore the Nash equilibrium strategy

profile is (2, 2− q1). Note that the equilibrium strategy for firm 2 is a contingent plan, not just the output

level firm 2 produces in equilibrium (i.e., not 0).



1.

a. Consider any two points (x1, x2) and (y1, y2). If x1 + x2 ≥ 1 then (x1, x2) �∼

(y1, y2). Otherwise

x1 + x2 < 1, which implies x1 < 1, so (y1, y2) �∼ (x1, x2). Hence the preferences are complete.

b. Let x = (0, 0), y = (1/2, 1), and z = (1, 1). Then x �∼

y because y1 < 1, and y �∼

z because y1 + y2 ≥ 1.

However, x �∼

z does not hold because neither x1 + x2 ≥ 1 nor z1 < 1 holds. Hence the preferences are

not transitive.

c. Continuity requires that the sets {x ∈ R2+ : x �

∼y} and {x ∈ R

2+ : y �

∼x} be closed for every y ∈ R

2+.

Suppose y1 + y2 < 1. Then y1 < 1, and the only way y �∼

x can hold is if x1 < 1. So {x ∈ R2+ : y �

∼

x} = {x ∈ R2+ : x1 < 1}:

This set does not include its right-side boundary, so it is not closed. Hence the preferences are not

continuous.

d. If x1 + x2 ≥ 1 and y1 + y2 ≥ 1 then (x1, x2) �∼

(y1, y2) and (y1, y2) �∼

(x1, x2). So all points whose

coordinates sum to at least one are the same according to this preference relation. Therefore local

nonsatiation fails.

e. Suppose (x1, x2) ≥ (y1 , y2). If x1 + x2 ≥ 1 then (x1, x2) �∼ (y1, y2). Otherwise y1 + y2 ≤ x1 + x2 < 1,

which implies y1 < 1, so again (x1, x2) �∼ (y1, y2). Hence the preferences are monotonic.


f. Convexity requires that the preferred-to set {x ∈ R2+ : x �

∼y} be a convex set for every y ∈ R

2+. If

y1 ≥ 1 then the preferred-to set is all points x such that x1 + x2 ≥ 1:

Otherwise y1 < 1, in which case the preferred-to set is all of R2+. Both of these preferred-to sets are

convex, so the preferences are convex.

g. No. Necessary conditions for a utility representation are that the preferences be complete and transitive.

Since transitivity fails, there is no utility function that represents these preferences.

2. This utility function is identical to the production function given in question 3 of the first test of this

year, so the indifference curves have exactly the same shape as the isoquants from that question.

a. The budget line when p1 = 9 is 9x1+x2 = 5. The maximum occurs at x2 = x1, at point A in the graph.

So 9x1 + x1 = 5, giving x1 = x2 = 1/2. Utility is U = x1 = x2 = 1/2. The budget line when p1 = 1 is

x1+x2 = 5. The maximum occurs at x2 = 2x1−1, at point B in the graph. So x1+2x1−1 = 5, giving

x1 = 2 and x2 = 3. Utility is U = 2x1 − 1 = x2 = 3. A budget line with the new slope −1, tangent to

the original indifference curve, passes through the point A because of the L-shaped indifference curves.

So the substitution effect is zero. The income effect is the movement from point A to point B, which

is 1.5 units of x1.

b. This is exactly the same as finding the conditional factor demands and cost function in question 3.b of

the first test of this year. So h∗1 = h∗

2 = U when U ≤ 1. When U > 1, the coordinates of the vertex in


terms of U are h∗1 = U+1

2and h∗

2 = U . To summarize, the Hicksian demands are

h∗1(p1, p2, U) =

{U , when U ≤ 1U+1

2 , when U > 1, h∗

2(p1, p2, U) ={

U, when U ≤ 1U, when U > 1

.

Hence the expenditure function is

e∗(p1, p2, U) =

{[p1 + p2]U when 0 ≤ U ≤ 1[p1

U+12

+ p2U]

when 1 < U.

c. First is homogeneity of degree 1 in prices:

e∗(λp1, λp2, U) =

{[λp1 + λp2]U when 0 ≤ U ≤ 1[λp1

U+12 + λp2U

]when 1 < U

= λ

{[p1 + p2]U when 0 ≤ U ≤ 1[p1

U+12

+ p2U]

when 1 < U

= λe∗(p1, p2, U).

Second is concavity in prices, which can be checked by verifying that the Hessian of e∗ with respect to

prices is negative semidefinite. The Hessian of e∗ with respect to prices is a matrix of zeros because


e∗ is linear in prices. So the Hessian is negative semidefinite trivially. Third is monotonicity. The e∗

function given above is obviously increasing in prices and utility level, since everything in the functions

is positive. And the monotonicity is strict in U when the prices are positive. Fourth is continuity. e∗

involves only sums and products, and both pieces are p1 + p2 when U = 1, so it is continuous. Last is

nonnegativity. U(0) = 0 in this case. Evaluating e∗ at U = 0 clearly gives e∗ = 0, whereas evaluating

at U > 0 clearly gives a positive number when the prices are positive.

3. No, because the Marshallian demands are not related to the indirect utility function in the required way.

Roy’s Identity dictates that the Marshallian demand for good 1, given the indirect utility function, be

x∗1 = −

∂U∗∂p1∂U∗∂m

= −− m3

p21p22

3m2

p1p22

=m

3p1.

This is not the same as the given Marshallian demand x∗1. The same problem arises with the demand for

good 2.



1.

a. Here is a graph of the production set. Only the black dots are included in the set. The gray lines in

the graph are not part of the set.

The technology is nonempty because there are elements in Z. It is closed because it includes all of its

boundaries (each of the 5 points is a boundary). It satisfies possibility of inaction because the point

(0, 0) is in the set. It satisfies no free lunch because Z contains no points in the first quadrant except

the origin. Free disposal is violated because, generally, if we move from any of the 5 points in Z to

the left or below we encounter points that are not in Z. Convexity is violated because the gray lines

between the points are not included in Z (neither are other combinations of the points that have not

been drawn on the graph).

b. For each (p1, p2) pair, the optimal choice has to be one of the 5 points. Which point is determined

by the slope of the isoprofit. Notice that the point (−3, 2) is below the gray line between (−4, 3) and

(−1, 1). So (−3, 2) cannot be optimal (it is always possible to obtain a higher isoprofit line at either

(−4, 3) or (−1, 1)). If the slope is of the isoprofits is flatter than −2/3 then the highest isoprofit is

obtained at (−4, 3). If the slope of the isoprofits is between −2/3 and −1 then the highest isoprofit is

obtained at (−1, 1). If the slope of the isoprofits is between −1 and −2 then the highest isoprofit is


obtained at (0, 0). If the slope of the isoprofits is steeper than −2 then the highest isoprofit is obtained

at (1,−2). So the profit function is:

π∗(p1, p2) =

−4p1 + 3p2 when p1

p2≤ 2

3

−p1 + p2 when 23 ≤ p1

p2≤ 1

0 when 1 ≤ p1p2

≤ 2

p1 − 2p2 when 2 ≤ p1p2

2. Necessary conditions for the function to be a cost function are that the Hessian be negative semidefinite

and symmetric. The given matrix is obviously symmetric. It is also negative semidefinite because the

(1, 1) naturally ordered principal minor is −1 (negative), the (2, 2) naturally ordered principal minor is

(−1)(−3) − (1)(1) = 2 (positive), and the (3, 3) naturally ordered principal minor is (−1)[(−1)(−3) −

(1)(1)] = −2 (negative). However, the function must also be homogeneous of degree 1, which implies that

the Hessian is singular. The given matrix is not singular since its determinant is nonzero. Therefore the

function cannot be a cost function.

3.

a. The gray lines in the graph below are index lines useful for locating the isoquants. The L-shaped dark

line closest to the origin is a typical isoquant for an output level y0 that is less than one. The other

L-shaped line is a typical isoquant for an output level y1 that is greater than one. The isoquant for

output level 1 is L-shaped with its vertex at the point where the two gray lines cross.

b. The cost minimum always occurs at the vertex of the relevant isoquant. So x∗1 = x∗

2 = y when y ≤ 1.


When y > 1, the coordinates of the vertex in terms of y are x∗1 = y+1

2 and x∗2 = y. Hence the cost

function is

c∗(w1, w2, y) ={

[w1 +w2]y when 0 ≤ y ≤ 1[w1

y+12 + w2y

]when 1 < y

c. From the cost function found in part b,

AC(y) ={

w1 + w2 when y ≤ 1w12

+ w2 + w12y

when 1 < y

MC(y) ={

w1 + w2 when y < 1w12

+ w2 when 1 < y

(MC is not defined at y = 1 because the cost function has a kink there). So the graph is:

d. At the stated prices, the upper part of the MC curve is 5 and the lower part is 3. So the output price is

between these two parts. MC “crosses” this output price at y = 1, but the crossing is a local minimum,

not a local maximum. As y → ∞, AC approaches the lower part of MC, so for the stated prices AC

approaches 3. Hence, for y large AC is below the output price of 4, and remains below no matter how

large y is. Thus, a price-taker can make infinite profit by choosing y = ∞.

4. Consider π∗(αp) for some price vector p and scalar α > 0. The underlying objective function is π(z;αp) =

(αp) · z. By the envelope theorem, ∂π∗(αp)∂α = ∂π(z;αp)

∂α

∣∣∣z=z∗

. The right side here is p · z∗(αp), which is

α−1π∗(αp). So we have π∗(αp) = α ∂π∗(αp)∂α

. That is, the function π∗(αp) considered as a function of α is

a linear function of its first derivative. This means π∗(αp) is linear in α. For example, using the product


rule to differentiate again yields ∂π∗(αp)∂α = ∂π∗(αp)

∂α +α ∂2π∗(αp)∂α2 , which implies ∂2π∗(αp)

∂α2 = 0. A zero second

derivative implies the original function is linear.



1.

a.

b. Yes, there is a competitive equilibrium at price equal to C. At this price, supply is horizontal and

demand is N . So supply equals demand with total output given by N .

c. Charge price equal to A and sell N units. Demand is horizontal at price A up to quantity N , so

marginal revenue equals A to the left of N . Marginal revenue is zero to the right of N . So marginal

revenue “crosses” marginal cost at quantity N with price A.

d. They are equally efficient. Both create all possible surplus from this market. The only difference is in

who gets the surplus. Under competition, the consumers get N(A − C) and the sellers get nothing.

Under monopoly, the consumers get nothing and the monopolist gets N(A − C). Monopoly does not

create deadweight loss because demand is perfectly inelastic at N (up to price A).

e. Assume seller 2 produces Q2. Then seller 1’s optimal choice is N − Q2 (or zero, if Q2 exceeds N)

because seller 1 makes the same positive profit margin of A − C1 on every unit produced, provided

aggregate output is not higher than N . When aggregate output exceeds N , price is driven to zero (or,

equivalently from seller 1’s perspective, there are some units with positive production cost that cannot

be sold). So seller 1 does not want to produce more than N−Q2. The same discussion applies to seller


2, taking Q1 as given. So any pair of nonnegative quantities that add to N is a Nash Equilibrium,

because at such a quantity pair neither seller has a unilateral incentive to change quantity (even though

their quantities might not be equal at such a quantity pair).

f. The sellers will undercut each other until the price is driven down to C2. When P1 = C2, seller 2 has

no further incentive to cut price and also has no incentive to raise price since seller 2’s profit is zero

in either case. If seller 2 actually (and arbitrarily, from seller 2’s perspective) chooses P2 = C2, seller

1 also has no incentive to change price provided seller 1 sells all N units of output (which is OK with

seller 2 because seller 2’s profit margin is zero). This is the only equilibrium.

g. The leader chooses Q1 = N and the follower reacts by choosing Q2 = 0. The latter is the optimal

reaction by a Cournot competitor to a choice of N by the rival, from item e above. Given that the

follower will fully accommodate the leader’s quantity choice, the leader wants to sell all the units since

choosing this quantity does not reduce the price compared with choosing some smaller quantity. So

the pair (N,Q2(Q1) = max{N −Q1, 0}) is the only subgame perfect Nash Equilibrium (the follower’s

strategy is a complete contingent plan). If the roles of leader and follower are switched, the only

subgame perfect Nash Equilibrium is (Q1(Q2) = max{N −Q2, 0}, N).

h. In all three cases, all N units are sold so gross consumer surplus of NA is created. In item e when

Q2 = 0, item f, and item g when firm 1 is the leader, all production is done by the low-cost firm so net

surplus plus profit is N(A − C1). This is fully efficient although the allocation of the surplus differs

across these cases. In the other cases at least some production is done by the high-cost firm, so net

surplus plus profit is lower.

2. Assume there are n taxis with licenses. Then the objective function for one of them is

max{yi≥0}

πi =

α− 110

yi +n∑j �=i

yj

yi − y2i .

The FOC is

α− 110

n∑j �=i

yj + 2yi

= 2yi.

By symmetry, in equilibrium all n quantities will be the same, so the common equilibrium quantity yi

satisfies

α− 110

(n+ 1)yi = 2yi,


or yi = 10α21+n . The equilibrium profit of one taxi is therefore

π∗ =(α− 1

10n

10α21 + n

)10α

21 + n−(

10α21 + n

)2

=110α2

(21 + n)2.

The city can charge up to π∗ for each license. So the aggregate revenue for the city is nπ∗. Differentiating

this with respect to n gives

∂nπ∗

∂n=

110α2(21− n)(21 + n)3

.

So n = 21 is the optimal number of licenses for the city.

3.

a. It is straightforward to set AC = MC and find they intersect at y =√2F . This intersection actually

does not depend on n. The value of both AC and MC at this quantity is n√2F .

b. With n sellers the figure is as shown below. The long run supply is the horizontal sum of the MC

curves of n sellers, above the AC curve, and zero below the AC curve.

c. The industry will experience entry and exit according to whether profit can be earned at the existing

price. The figures show supply for n = 1 and for arbitrary n. If we stack the supply for n = 1 on the

supply for n = 2 on the supply for n = 3 ..., we get a line through the origin with slope 1, above√2F ,

which is the lowest level that can occur (i.e., n = 1). This is the long-run industry supply. If cost did


not depend on n, the long-run industry supply would be discrete points at multiples of√2F along the

horizontal line at√2F .

d. We must guarantee that demand intersects long-run industry supply. Long-run industry supply has

only one discontinuity, from y = 0 to y =√2F , at price p =

√2F . So we must guarantee that demand

does not cross in that discontinuity. If α ≤ √2F then supply and demand are equal at a quantity of

zero and a price of α. The more interesting case is when α ≥ 2√2F . Then demand exceeds

√2F at

a price of p =√2F , so industry demand and long-run supply cross somewhere above p =

√2F . If

√2F < α < 2

√2F then there is no equilibrium.



1.

a. Set U = 1 in the following graph of a typical indifference curve:

b. The slope of the line from (0, 3U/2) to (3U/2, 0) is −1. So when the slope of the budget line is steeper

than −1 the highest feasible indifference curve is attained at (0, m/p2) and when the slope of the budget

line is flatter than −1 the highest feasible indifference curve is attained at (m/p1, 0). When the slope

of the budget line is exactly −1 the consumer is indifferent between these two endpoints of the budget

line. So the Marshallian demands are

x∗1 =

0, if p1 > p2

0 or m/p1, if p1 = p2

m/p1, if p1 < p2

and x∗2 =

m/p2, if p1 > p2

m/p2 or 0, if p1 = p2

0, if p1 < p2

.

Substituting these into U gives indirect utility of

U∗ =

{2m3p2

, if p1 ≥ p2

2m3p1

, if p1 < p2

.

c. The same geometric observations yield the Hicksian demands:

h∗1 =

0, if p1 > p2

0 or 3U/2, if p1 = p2

3U/2, if p1 < p2

and h∗2 =

3U/2, if p1 > p2

3U/2 or 0, if p1 = p2

0, if p1 < p2

.

Substituting these into e gives expenditure of

e∗ =

{3Up2

2, if p1 ≥ p2

3Up12 , if p1 < p2

.


d. The original budget line has slope 1/2, like the line from point A to point C in the graph below. The

utility maximizing point is A. The new budget line has slope 2, like the line from point C to point D.

The utility maximizing point is C. When the budget line has slope 2 but utility is held at the original

level, the expenditure minimizing point is B. So the substitution effect is from A to B and the income

effect is from B to C.

2. First we must find the expenditure and indirect utility functions. By definition,

e∗(p1, p2, U) = p1h∗1 + p2h

∗2 = 2

(p1p2e

U)1/2

.

Inverting then gives

U∗(p1, p2, m) = ln(

m2

4p1p2

).

Now, by definition the compensating variation is

e∗(2, 1, U∗(1, 1, 8))− 8 = e∗(2, 1, ln(16))− 8 = 2(32)1/2 − 8 ≈ 3.31.

To find the change in Marshallian surplus, we need the Marshallian demand for good 1. By Roy’s Identity,

x∗1(p1, p2, m) = −

∂U∗∂p1∂U∗∂m

=m

2p1.

So the change in Marshallian surplus is

−∫ 2

1

m

2p1dp1 = − m

2ln(p1)

∣∣∣2p1=1

= −m ln(2)2

= −4 ln(2) ≈ −2.77.


Therefore the percent error is (sign is unimportant):

3.31− 2.773.31

≈ 16%.

3.

a. Consider any 2 points x and y in the consumption set. If x ∈ C then y �∼

x by item 2. If x /∈ C then

we must have x ∈ B, in which case if y ∈ C then x �∼

y by item 2 and if y /∈ C then still x �∼

y by

item 1 (noting that y /∈ C ⇒ y ∈ B). We always get a comparison between x and y. Therefore the

preferences are complete.

b. Consider any 3 points x, y, and z in the consumption set, and assume x �∼

y and y �∼

z. One set of

points that is consistent with this setup is y ∈ B ∩ C, x ∈ C but x /∈ B, and z ∈ B but z /∈ C. Here,

x �∼

y holds by item 2 (using item 2 on y, not on x) and y �∼

z holds by item 1. However, we do not

have x �∼

z because x ∈ C but x /∈ B, while z /∈ C. Therefore the preferences are not transitive.

c. Suppose y /∈ C. Then the set {x ∈ R2+ : x �

∼y} is B. On the graph, this is the entire first quadrant

except the part labeled C, The part labeled C includes its boundaries, so the left, bottom, and right

boundaries of that part are not included in B. Hence the set {x ∈ R2+ : x �

∼y} is not closed. Therefore

the preferences are not continuous.

d. Pick any point on the interior of B (or C). A circle can be draw around such a point small enough

so that the entire circle stays in B (or C). The consumer is indifferent among all points in B (and all

points in C). So local nonsatiation fails.

e. A point in the area labeled B is strictly preferred to a point in the area labeled C, but the latter can

be to the northeast of the former. So monotonicity fails.

f. The points in B are strictly preferred to the points in C but not in B. So, for y ∈ B but y /∈ C the

preferred set {x ∈ R2+ : x �

∼y} is B. B is not a convex set because a segment connecting two points on

opposite sides of the area labeled C passes outside of B. Therefore the preferences are not convex.

g. There is no utility function because the preferences are not transitive (transitivity and completeness

are necessary for there to be a utility representation).



1.

a. Here is a graph of the isoquant for output level y:

From the graph, we conclude by inspection that the conditional factor demands are

(x∗1, x

∗2) =

(0, (ey − 1)/5), if 2 < w1/w2

([0, (ey − 1)/30], (ey − 1)/5− 2x1), if w1/w2 = 2((ey − 1)/30, 4(ey − 1)/30), if 1 < w1/w2 < 2([(ey − 1)/30, (ey − 1)/6], (ey − 1)/6− x1), if w1/w2 = 1((ey − 1)/6, 0), if w1/w2 < 1

.

b.

c∗(w1, w2, y) = w1x∗1 +w2x

∗2 =

w2(ey − 1)/5, if 2 ≤ w1/w2

[(ey − 1)/30][w1 + 4w2], if 1 ≤ w1/w2 ≤ 2w1(ey − 1)/6, if w1/w2 ≤ 1

.

c. Marginal cost is

mc =∂c∗

∂y=

w2e

y/5, if 2 ≤ w1/w2

[ey/30][w1 + 4w2], if 1 ≤ w1/w2 ≤ 2w1e

y/6, if w1/w2 ≤ 1.

Let p be the output price, set p = mc, and solve for y to obtain

y∗(p, w1, w2) =

ln(5p/w2), if 2 ≤ w1/w2

ln(30p/(w1 + 4w2)), if 1 ≤ w1/w2 ≤ 2ln(6p/w1), if w1/w2 ≤ 1

.

This is the supply function provided it is positive. Otherwise, we have a corner solution and therefore

y∗ = 0. The corner solution occurs when p is so low relative to w1 and w2 that positive profit cannot


be obtained. The threshold values of p are p = w2/5 when 2 ≤ w1/w2, p = (w1 + 4w2)/30 when

1 ≤ w1/w2 ≤ 2, and p = w1/6 when w1/w2 ≤ 1.

d.

π∗(p, w1, w2) = py∗ − c∗(w1, w2, y∗)

=

p ln(5p/w2)− p+w2/5), if 2 ≤ w1/w2

p ln(30p/(w1 + 4w2))− p+ (w1 + 4w2)/30, if 1 ≤ w1/w2 ≤ 2p ln(6p/w1)− p+w1/6, if w1/w2 ≤ 1

.

As above, this is the solution when we assume the y∗ values above are positive. Whenever the prices

are such that y∗ = 0 we also have π∗ = 0.

e.

Z ={(y,−x1,−x2) : x1 ≥ 0, x2 ≥ 0, and y ≤

{ln(6x1 + 6x2 + 1), if x2 ≤ 4x1

ln(10x1 + 5x2 + 1), if x2 ≥ 4x1

}.

f. Yes. The properties are 1) Z nonempty, 2) Z closed, 3) (0, 0, 0) ∈ Z (possibility of inaction), 4)

(y, 0, 0) /∈ Z when y > 0 (no free lunch), and 5) (y′, x′1, x

′2) ∈ Z when (y, x1, x2) ∈ Z and (y′, x′

1, x′2) ≤

(y, x1, x2) (free disposal). Z is closed because the natural log is a continuous function and all of the

inequalities used in the definition of Z are weak. The origin is in Z because ln(1) = 0. As usual, this

implies nonemptiness. This also shows that no free lunch holds because the left side of the defining

inequality for Z would be positive while the right side would be zero. Finally, free disposal holds

because the natural log is an increasing function and the arguments inside it are increasing in x1 and

x2, while the left side of the defining inequality decreases with y.

2.

a. The five necessary and sufficient conditions are:

Homogeneous of degree 1 in (w1, w2). c∗(λw1, λw2, y) = 8y2(λw1λw2)1/2 = λ8y2(w1w2)1/2.

Concave in (w1, w2). Differentiate the conditional factor demands from part b below to obtain the

Hessian [−2y2w−3/21 w

1/22 2y2w

−1/21 w

−1/22

2y2w−1/21 w

−1/22 −2y2w

1/21 w

−3/22

].

The diagonal elements of this matrix are negative, and the determinant of the whole matrix is zero, so

the matrix is negative semidefinite.


Monotonic in (w1, w2, y). By inspection, c∗ is nondecreasing in w1, w2, and y (strictly increasing when

all three are positive).

Continuity. c∗ involves only products and positive exponents, so it is continuous.

Nonnegativity. By inspection, c∗ is positive when w1, w2, and y are all positive; and c∗ is zero when y

is zero.

b.

x∗1 =

∂c∗

∂w1= 4y2w

−1/21 w

1/22 and x∗

2 =∂c∗

∂w2= 4y2w

1/21 w

−1/22 .

c. c∗(2, 8, y) = 32y2. So mc = 64y and ac = 32y. The graph is:


d. From the conditional factor demands above, x1 = x∗(2, 8, 1) = 8. So the fixed cost in the short-run is

w1x1 = 16, and the short-run minimization problem is

min{x2≥0}

16 + w2x2 subject to y =12(8x2)1/4.

As there is only one input to choose, we choose the smallest amount of that input consistent with

producing y: x∗2(w1, w2, y|x1 = 8) = 2y4. So the short-run cost function is c∗(w1, w2, y|x1 = 8) =

c∗(2, 8, y|x1 = 8) = 16 + 16y4. Therefore the short-run marginal cost is mc(y|x1 = 8) = 64y3 and the

short-run average cost is ac(y|x1 = 8) = 16y + 16y3. The graph is:


At p = 8, the setting p = mc shows the firm will supply y∗ = 1/8 in the long-run and y∗ = (1/8)1/3 in

the short-run. The latter is larger because p = 8 is below the intersection of the two mc curves, where

the firm has an excess of input 1 in the short-run that it is better to use than let lay unused.



1. Firm i chooses yi to maximize πi(yi|y−i) =(100−∑n

j=1 yj

)yi − 20yi. The FOC with respect to yi is

80−∑nj=1 yj − yi = 0. As all firms are identical, the symmetric Nash Equilibrium involves yj all equal

in equilibrium. Call this common equilibrium output level yi. Then the FOC is 80 − (n + 1)yi = 0, so

yi = 80/(n + 1) is the Nash Equilibrium output for each firm. Therefore aggregate output is 80nn+1 and

price is P = 100− 80nn+1 = 100+20n

n+1 . This gives a price-cost margin of

P −mc

P=

100+20nn+1 − 20100+20nn+1

=80

100 + 20n.

This margin is less than or equal to .05 if and only if n ≥ 75.

2.

a. With the addition of firm 0’s output, the FOC from question 1 becomes 80 − y0 − (n + 1)yi = 0,

so the symmetric Nash Equilibrium quantity of each follower, given that the leader has chosen y0, is

yi = 80−y0n+1 . Therefore the leader’s objective is

π0(y0|y∗i (y0); i = 1, . . . , n) =(100− y0 − (80− y0)n

n+ 1

)y0 − 20y0 =

80− y0

n+ 1y0.

By the envelope theorem,

∂π∗0

∂n=

∂π0

∂n

∣∣∣∣y0=y∗0

= − 80− y0

(n + 1)2y0

∣∣∣∣y0=y∗0

= − (n+ 1)−1π0

∣∣y0=y∗0

= −(n + 1)−1π∗0 .

b. From the expression for π0 above, the FOC for firm 0 is 80−2y0n+1

= 0, so y∗0 = 40.

3. The firms will engage in Bertrand undercutting until it is unprofitable for one firm to undercut the

other. This happens when P1 = P2 = C2 = 40, because at this price firm 2 cannot benefit from further

undercutting. At these prices, firm 2 earns zero profit and firm 1 earns π1 = (40 − 30)Q1 ≥ 0. Neither

firm gains from raising price because that just leads to zero profit. Firm 2 clearly does not gain from

lowering price because then its profit margin is negative. Firm 1 will gain from lowering price unless Q1

is the entire market demand. So for this to be equilibrium we must have Q1 = 100− 40 = 60 and Q2 = 0.

Firm 2 doesn’t care about Q2 because its profit margin is zero. Hence P1 = P2 = 40 with Q1 = 60

is equilibrium provided firm 1 doesn’t want to lower price even more. It is easy to check that firm 1’s

monopoly price is 65, so lowering price would just move firm 1 further from its monopoly price.


4. This question is exactly the same as the 1998 quiz for Lecture 17 (Perfect Competition). See the solution

there.

5. From question 1, with n = 2 we have y1 = y2 = 803 in equilibrium, and the equilibrium price is P = 140

3 .

Therefore each firm earns profit of π1 = π2 =(

1403

− 20)

803=(

803

)2. As a monopolist, one of the firms will

produce the solution from question 1 for n = 1, giving ym = 40, Pm = 60, and πm = (60 − 20)40 = 402.

So the movement from duopoly to monopoly is worth 402 − ( 803

)2 = 5(

403

)2 to the firm that becomes the

monopoly. The graph is:

The deadweight loss from the acquisition is DWL = 12

(60− 140

3

) (1603 − 40

)= 800

9 .



1. A typical indifference curve is:

This indifference curve is strictly concave, so the minimum will be at one of the corners. The slope of the

line from one corner to the other is −√U . Therefore the Hicksian demands are

h∗ =

(0, U

), if p1

p2≥

√U(√

U , 0), if p1

p2≤

√U

This gives the following expenditure function:

e∗ =

{p2U , if p1

p2≥

√U

p1

√U, if p1

p2≤

√U

2.

a. No. The preference relation gives no ranking if both x and y are odd numbers.

b. Yes. Suppose x �∼

y and y �∼

z. The first ranking implies that x is even. Therefore x �∼

z.

c. No. According to the relation, 2 � 3. A correct answer must note that an even number that is less

than an odd number is strictly preferred to the odd number.

3. Invert U∗ to obtain the expenditure function e∗ = Up1p2p1+p2

. Differentiate e∗ to find the Hicksian demands via

Shephard’s Lemma: h∗1 = U

(p2

p1+p2

)2

and h∗2 = U

(p1

p1+p2

)2

. Let p = p1p2

be the relative price. In terms

of the relative price, the Hicksian demands are h∗1 = U

(1

1+p

)2

and h∗2 = U

(p

1+p

)2

. These equations

express the quantities x1 = h∗1 and x2 = h∗

2 in terms of utility U = U and the relative price p. Solving


simultaneously to eliminate p yields U =(√

x1 +√x2

)2. It is straightforward to check that the indirect

utility function obtained by maximizing U is indeed the given U∗.

4. CV is the extra expenditure required to keep utility constant when the price changes. Utility is constant

in e∗. Expenditure increases from 1 to 10. So the CV is 9.

5. The MRS is − x2x1+1

. It is straightforward to check that the indifference curve is strictly convex. So setting

the MRS equal to the negative price ratio and using the budget constraint yields the simultaneous solution

x1 = m−p12p1

and x2 = m+p12p2

. These are the Marshallian demands provided there is not a corner solution.

The MRS at x2 = 0 is zero, so no corner solution is possible with x2 = 0. But the MRS at x1 = 0 is

−x2. So a corner solution with x1 = 0 is possible. The tangency gives x1 > 0 provided m > p1. So the

Marshallian demands are

x∗ =

(0, m

p2

), if m ≤ p1(

m−p12p1

, m+p12p2

), if m > p1



1. The Lagrangian function for cost minimization is L = w1x1 + w2x2 − λ(Ax1x2 − y). By the envelope

theorem,

∂c∗

∂A=

∂L

∂A

∣∣∣∣xi=x∗

i &λ=λ∗= −λx1x2|xi=x∗

i &λ=λ∗ = −λ∗x∗1x

∗2.

Now use the fact that λ∗ is marginal cost and y = Ax∗1x

∗2 to rewrite this derivative as −mc(y)y/A.

2.

a. The average variable cost function is AV C(y) = 1y+1

+ y4. So AV C ′(y) = − 1

(y+1)2+ 1

4and AV C ′′(y) =

2(y+1)3 > 0. Hence setting AV C ′(y) = 0 will yield a minimum. The value of y at which AV C ′(y) = 0

is y = 1 and the value of AV C there is AV C(1) = 34 . Also, AV C(0) = 1.

The marginal cost function is MC(y) = 1(y+1)2

+ y2. So MC(0) = 1. Therefore the graph is

The supply curve is the MC curve above the AV C curve (because all fixed costs are sunk), and the

vertical axis at prices below 3/4.

b. The shutdown decision is made by examining average avoidable costs (AAC). When α < 1 some of the

fixed costs are avoidable (specifically, (1−α)F ). So the average cost curve that is relevant for shutdown

decisions is AAC(y) = (1−α)Fy +AV C(y). This curve must be above AV C(y) and must cross MC(y)


at the minimum of AAC(y), so now the graph is:

The lowest price at which there is positive supply slides up the MC curve and the portion of the supply

curve on the vertical axis likewise slides up.

3. This technology is clearly not convex. More importantly, this technology does not satisfy the possibility of

inaction (because e0 = 1, so V (0) does not include the origin). All other standard properties are holding.

To find the profit function, first find the cost function. The lowest isocost line is always achieved at one

of the corners. So we have either x∗ = (0, ey) or x∗ = (ey, 0). The former occurs when the isocost line

has slope steeper than −1 and the latter occurs when the isocost line has slope flatter than −1. So the

cost function is

c∗(w, y) ={

w1ey, w1

w2≤ 1

w2ey, w1

w2> 1

Therefore the profit objective is

max{y≥0}

π = py − wey,

where w = min{w1, w2}. The FOC for y yields y∗ = ln(pw

). This is indeed a local maximum because the

second derivative is −wey < 0. However, the value of y∗ given by the FOC can be negative. This will

happen when p < w. So we have y∗ = 0 when p ≤ w and y∗ = ln(pw

)when p > w. Substituting into the

objective yields the profit function:

π∗(p, w1, w2) ={ −w, p ≤ w

p[ln(pw

) − 1], p > w


Note that the profit can be negative (even when p > w, because ln(p/w) < 1 when p/w < e). This is

because the possibility of inaction is not holding.

4.

a. The five sufficient properties for a cost function are

Homogeneity of degree 1 in w. Here we have c∗(αw1, αw2, y) = y(αw1) = α(yw1) = αc∗(w1, w2, y).

Concavity in w. c∗ is linear in w, so it is trivially concave (Hessian is a matrix of zeros).

Monotonicity. ∂c∗∂w1

= y ≥ 0; ∂c∗∂w2

= 0; and ∂c∗∂y = w1 > 0.

Continuity. c∗ = yw1 involves only products, so it is continuous.

Nonnegativity. c∗ = yw1 > 0 when y > 0 and w1 > 0. c∗ = 0 when y = 0.

As c∗ satisfies the sufficient conditions for a cost function, we know it is indeed a cost function for some

well-behaved technology.

b. From Shephard’s Lemma, x∗1 = y and x∗

2 = 0. So the optimal choice in R2+ is x∗ = (y, 0) no matter

what the input prices are. Therefore the input requirement set cannot contain any points left of (y, 0)

(otherwise, such points would be chosen as cost minimizing points for some isocost slope). Using free

disposal then yields the input requirement set

c. Intersecting areas above isocost lines over all positive relative prices cannot reveal information about

possible positive slopes to the boundaries of input requirement sets. Yet another input requirement set


that yields the same cost function is

This set does not satisfy free disposal, but this does not affect cost because no points off of the horizontal

axis are ever chosen as cost-minimizing points.



1.

a. MR = 100 − 2Q. Set equal to MC to obtain Q = 100−c12 . Then P = 100 − 100−c1

2 = 100+c12 . This

assumes c1 < 100 (otherwise, Q = 0). Profit is

π1 = [P − c1]Q =[100 + c1

2− c1

]100− c1

2=[100− c1

2

]2.

b. The Cournot objectives are

π1 = [100−Q1 −Q2 − c1]Q1

π2 = [100−Q1 −Q2 − c2]Q2.

So the FOCs are

100− 2Q1 −Q2 = c1

100−Q1 − 2Q2 = c2.

Solving simultaneously for the quantities yields the Nash Equilibrium:

Q1 =13[100 + c2 − 2c1]

Q2 =13[100 + c1 − 2c2].

Adding these gives aggregate quantity of Q = 13 [200−c1−c2], and substituting into the inverse demand

gives the equilibrium price P = 13[100 + c1 + c2]. Then equilibrium profits are

π1 =[100 + c2 − 2c1

3

]2π2 =

[100 + c1 − 2c2

3

]2.

c. If c2 = 100+c12 then Q2 = 0. Aggregate welfare based on Marshallian surplus is

W = CS+π1+π2 =

[∫ Q

0

[100− t]dt− PQ

]+[P − c1]Q1+[P − c2]Q2 =

∫ Q

0

[100− t]dt− c1Q1− c2Q2.

Differentiating with respect to c2 yields

∂W

∂c2= P (Q)

∂Q

∂c2− c1

∂Q1

∂c2− c2

∂Q2

∂c2−Q2

= [P − c1]∂Q1

∂c2+ [P − c2]

∂Q2

∂c2−Q2.


Evaluating at c2 = 100+c12 (so P = c2 and Q2 = 0) yields

∂W

∂c2

∣∣∣∣c2=(100+c1)/2

= [P − c1]∂Q1

∂c2.

From the Nash equilibrium quantities, ∂Q1∂c2

= 13 , so

∂W

∂c2

∣∣∣∣c2=(100+c1)/2

= [P − c1]13=

100− c16

> 0

(again assuming c1 < 100). Therefore a small decrease in c2 from 100+c12

results in a decrease in welfare,

and society therefore does not benefit from Cournot competition when c2 is near 100+c12 . Competition

is harmful in this case because the new competitor is much less efficient than the monopolist (c2 > c1).

So even though aggregate output increases, price decreases, and consumers are better off; production

costs rise enough to more than offset the gains to consumers. On the other hand, if the new firm has

cost c2 = c1 then the introduction of competition increases aggregate quantity and decreases price with

no offsetting rise in production costs. This is a pure saving of deadweight loss, and therefore improves

welfare.

d. Suppose c1 ≤ c2. Then Bertrand competitors undercut each other until P1 = P2 = c2. At this

price, firm 2 has no further incentive to undercut. Firm 1 will undercut slightly if necessary to obtain

the whole market (unless c1 = c2, in which case they probably split the market), but since firm 2 is

indifferent between selling and not selling it is simplest to just assume firm 1 sells the entire demand

at price P1 = P2 = c2. These prices are a Nash equilibrium provided firm 1 sells the entire demand

(unless c1 = c2, in which case any division of demand is a Nash equilibrium). The aggregate quantity

is Q = 100 − P = 100 − c2. If c1 > c2 then the roles of c1 and c2 reverse, and P1 = P2 = c1 is a

Nash equilibrium (provided firm 2 sells the entire demand) and Q = 100− c1. All of this assumes the

high-cost firm has marginal cost below the low-cost firm’s monopoly price. Otherwise, the low-cost

firm just charges its monopoly price and sells its monopoly quantity, as in part a above.

e. When a high-cost competitor competes in prices, it forces the price down to its own cost provided only

that its cost is below the monopoly price of the low-cost firm. This eliminates deadweight loss. In

equilibrium, the low-cost firm still produces all of the output, so there is no increase in production

costs of the type encountered in the Cournot model. Hence welfare improves provided only that the

high-cost firm is efficient enough to alter the monopoly behavior of the low-cost firm.


2.

a. Yes. We could check the sufficient condition, but since we have actual numbers in this case it is easiest

to just construct an equilibrium. For example, three firms each producing 113 units of output is an

equilibrium.

b. If all firms produce the smallest possible amount (i.e., 2) then six firms produce too much (i.e., 12).

So five firms is the maximum number. Five is an equilibrium if each produces 115 units of output, for

example. If all firms produce the largest possible amount (i.e., 4) then two firms produce too little

(i.e., 8). So three firms is the minimum number.

3.

a. The marginal cost of a typical fringe firm is c′f = 10+ 20qf . So when the dominant firm charges price

P the typical fringe firm chooses qf to satisfy 10 + 20qf = P , or qf = P20 − 1

2 . This is the supply

curve of a typical fringe firm for P > 10. When P ≤ 10 the fringe supplies zero (more on this below).

So aggregate fringe supply is Qf = nqf = n[P20 − 1

2

]for P > 10. Then the residual demand of the

dominant firm is

Qd = Q−Qf = 100− P − n

[P

20− 1

2

]=[100 +

n

2

]−[1 +

n

20

]P.

Invert this to obtain the dominant firm’s inverse residual demand:

P ={[

100 +n

2

]−Qd

} 20n + 20

.

Multiplying this by Qd gives revenue for the dominant firm, and differentiating then gives marginal

revenue MRd = 1n+20

[2000 + 10n− 40Qd]. Setting this equal to marginal cost c′d = 1 and solving for

Qd gives the dominant firm’s quantity Qd = 1980+9n40 . Substitute this into the dominant firm’s inverse

residual demand to get price P = 2020+11n2[n+20]

. This is all contingent on P > 10. The dominant firm’s

price satisfies P > 10 if and only if n < 180.

b. ∂Qd

∂n = 940 > 0, so the dominant firm increases output as the number of fringe firms increases. This

is because the fringe supply becomes more elastic as n increases, which makes the dominant firm’s

residual demand more elastic. The arrows show what happens as n increases:


The marginal revenue curve that accompanies the dominant firm’s residual demand has smaller inter-

cept and flatter slope as n increases. The slope flattens enough to make the point of intersection with

the horizontal line c′d = 1 move to the right as n increases.

c. If n > 180 then the marginal revenue curve that accompanies the dominant firm’s residual demand

crosses the horizontal line c′d = 1 to the right of Q = 90. Of course, the dominant firm’s residual

demand and residual marginal revenue curves are not relevant for quantities above 90, since fringe

supply is zero at such low prices. Hence the dominant firm supplies Qd = 90 and charges P = 10. At

the kink, marginal revenue is below marginal cost for price decreases but above marginal cost for price

increases.



1. Check the five properties of an expenditure function:

Homogeneity of degree one in prices:

e∗(λp1, λp2, U) =(λp1)(λp2)Uλp1 + λp2

= λp1p2U

p1 + p2= λe∗(p1, p2, U).

Increasing in pi; Strictly increasing in U :

∂e∗

∂pi=

p2j U

(p1 + p2)2≥ 0;

∂e∗

∂U=

p1p2

p1 + p2> 0.

Concavity in (p1, p2): The Hessian of e∗ with respect to prices is

2U

[− p22

(p1+p2)3p1p2

(p1+p2)3

p1p2(p1+p2)3

− p21(p1+p2)3

].

The diagonal elements are nonpositive, and the determinant is zero, so this matrix is negative semidef-

inite.

Continuity: e∗ involves only sums, products, and division, so it is continuous except at p1 = p2 = 0,

where it is undefined.

Nonnegativity: By inspection, e∗(p1, p2, 0) = 0 and e∗(p1, p2, U > 0) > 0.

So e∗ is indeed an expenditure function for some complete, transitive, continuous, and locally nonsatiated

preference relation �∼

on R2+.

2. This can be answered by checking whether the given functions possess the properties of Marshallian

demands, but since question 4 asks for a comparison between these Marshallian demands and the expen-

diture function from question 1, we can answer both question 2 and question 4 at once by showing that

these Marshallian demands arise from the expenditure function in question 1. From the derivatives above,

the Hicksian demands via Shephard’s Lemma are

h∗1 =

p22U

(p1 + p2)2; h∗

2 =p21U

(p1 + p2)2.

Inverting the expenditure function yields the indirect utility function U∗ = m(p1+p2)p1p2

. Substitute U∗ into

the Hicksian demands to get the Marshallian demands (or, use Roy’s Identity):

x∗1 =

p22

(p1 + p2)2m(p1 + p2)

p1p2=

p2m

p1(p1 + p2); x∗

2 =p21

(p1 + p2)2m(p1 + p2)

p1p2=

p1m

p2(p1 + p2).


We know these are Marshallian demands for some complete, transitive, continuous, and locally nonsatiated

preference relation �∼

on R2+, because we showed that the function in question 1 from which these are

derived is indeed an expenditure function.

3. CV = e∗(p11, p2, U

∗(p01, p2, m)) −m. Substitute U∗ from above into e∗ to obtain

CV =p11p2

p11 + p2

m(p01 + p2)p01p2

−m = m

[p11(p0

1 + p2)p01(p1

1 + p2)− 1]= m

p2(p11 − p0

1)p01(p1

1 + p2).

4. Yes. They have the same Marshallian demands, so their (economically relevant) preferences are the same.

5. Let p = p1/p2 be the relative price. In terms of p, the Hicksian demands are

h∗1 =

U

(p+ 1)2; h∗

2 =p2U

(p+ 1)2.

These two equations relate quantities (h∗1 and h∗

2) to the utility level those quantities deliver (U), for any

given relative price p. We can substitute them together to eliminate p and thereby obtain the relationship

between quantities and utility. From h∗1, p =

√Uh∗1− 1. From h∗

2, U = h∗2

[p+1p

]2. Substitute the former

into the latter to obtain

U = h∗2

√

Uh∗1√

Uh∗1− 1

2

= h∗2

[ √U√

U −√h∗1

]2

.

This is [√U −√h∗

1

]2= h∗

2 ⇔√

U −√h∗1 =

√h∗

2 ⇔ U =[√

h∗1 +

√h∗

2

]2.

So the utility function is U(x1, x2) =[√

x1 +√x2

]2.6.

a. No. As long as either a or b is a plant, the relation gives a (weak) preference between a and b. But

if a and b are both “Iron,” the relation gives no information about whether a �∼

b or b �∼

a. How

do we know Iron �∼

Iron? The given relation says nothing about this. Completeness requires that all

pairs be comparable, even if the two elements in the pair are the same (Varian lists this separately as

“Reflexivity,” but it’s really just a special case of the Completeness property).

b. Yes. Given that a �∼

b and b �∼

c, the relation tells us that a and b are plants. Thus a �∼

c even if c is

not a plant (i.e., even if c =“Iron”).



1. First check to see if the functions satisfy the five sufficient conditions for factor demands:

a. Homogeneity of degree zero in prices. Changing all input prices by a factor λ obviously does nothing

when y < 1. When y ≥ 1,

x∗i (λw1, λw2, y) =

√λwjλwi

(y − 1) =√

wjwi

(y − 1) = x∗i (w1, w2, y),

so homogeneity of degree zero holds.

b. Symmetry and semidefiniteness of the matrix of price derivatives. For y < 1 the matrix is zero, which

is symmetric and negative semidefinite. For y ≥ 1 the matrix is

[∂x∗

1∂w1

∂x∗1

∂w2∂x∗

2∂w1

∂x∗2

∂w2

]=[−w

−3/21 w

1/22 w

−1/21 w

−1/22

w−1/21 w

−1/22 −w

1/21 w

−3/22

]y − 12

.

This is clearly symmetric and the (1, 1) naturally ordered principal minor is clearly nonpositive. The

second order principal minor is the determinant

| · | = (y − 1)2

4[w−1

1 w−12 −w−1

1 w−12

]= 0.

So the matrix is symmetric and negative semidefinite.

c. w · x∗ is nondecreasing in (w, y). For y < 1, w · x∗ = 0, which is nondecreasing. For y ≥ 1, w · x∗ =

w1x∗1 + w2x

∗2 = 2

√w1w2(y − 1), which is also nondecreasing in all its arguments. Note that this

expression is zero at y = 1, so the nondecreasing property holds across the kink as well.

d. w · x∗ is continuous. The two parts are each individually continuous, and both parts are 0 at y = 1.

e. Nonnegativity. x∗i ≥ 0 for all (w, y) by inspection. As noted above, w · x∗ = 0 when y = 0. These two

parts of the nonnegativity property hold. However, w · x∗ is not strictly positive when 0 < y ≤ 1, so

the last part of the nonnegativity property fails. The given system is not a system of factor demands

for a technology that satisfies the five basic assumptions. Since x∗ = 0 when y > 0, the underlying

technology permits a free lunch. Positive output can be produced with zero inputs.


2.a.

b. Yes. Z is clearly nonempty. Since the defining inequalities are weak, the boundaries are included and

therefore Z is closed. The possibility of inaction is present because (0, 0) is an element of Z. Free

disposal is present because none of the boundaries of Z have a positive slope. No free lunch is satisfied

because no part of Z except the origin is in the first quadrant.

c. Pick any point in Z. A segment from that point to the origin lies entirely in Z. Therefore Z has

nonincreasing returns. There are points in Z (for example, the point (−1, 2)) that cannot be expanded

while staying in Z. Therefore Z does not have nondecreasing returns.

d. Yes, Z is convex. The segment joining any two points in Z remains entirely in Z.

e. V (y) is the (negative of the) set of x values that, together with the given y, form elements in Z. So

V (y) = [0,∞) for y ≤ 0; V (y) = [y/2,∞) for 0 < y ≤ 2; V (y) = [y − 1,∞) for 2 < y.

f. Superimpose the isoprofit line for a given slope −w/p on the graph of Z. If the slope is steeper (in

absolute value) than −2 then the highest isoprofit is achieved at the point (0, 0). If the slope equals

−2 then any point on the segment between (0, 0) and (−1, 2) is on the highest possible isoprofit. If

the slope is between −2 and −1 then the highest isoprofit is achieved at the point (−1, 2). If the slope

equals −1 then any point on the line y = 1−x to the left of (−1, 2) is on the highest possible isoprofit.

If the slope is flatter (in absolute value) than −1 then we can achieve successively higher isoprofit lines


by moving to the northwest along the upper boundary of the set. Hence the highest isoprofit is achieved

at (−∞,∞). These observations lead to the following supply/demand function:

(y∗,−x∗) =

(0, 0) if w/p > 2(−2x∗, [−1, 0]) if w/p = 2(2,−1) if 1 < w/p < 2(1− x∗, (−∞,−1]) if w/p = 1(∞,−∞) if w/p < 1

Multiplying these optimal choices by the prices gives the profit function:

π∗ =

0 if w/p ≥ 22p− w if 1 ≤ w/p < 2∞ if w/p < 1

g. At a positive input price, cost is minimized by using the smallest possible x for any given y. From the

input requirement sets found in part e, the smallest possible x values are

x∗ ={ y

2 if 0 ≤ y ≤ 2y − 1 if y > 2

Multiply this by the input price w to obtain the cost function:

c∗ ={ wy

2 if 0 ≤ y ≤ 2w(y − 1) if y > 2

h. Check the five sufficient conditions for cost functions:

Homogeneity of degree one in prices.

c∗(λw, y) ={ λwy

2if 0 ≤ y ≤ 2

λw(y − 1) if y > 2= λ

{ wy2 if 0 ≤ y ≤ 2w(y − 1) if y > 2

= λc∗(w, y),

so homogeneity of degree one holds.

Concavity in prices. For a given y, c∗ is linear in w with slope either y/2 or y − 1. A straight line is

always concave (second derivative is zero).

Nondecreasing in (w, y). From above, the slope of c∗ with respect to w is positive (y − 1 is the slope

only when y > 2). The slope with respect to y is either w/2 or w, which are both positive. Note that

c∗ is continuous at y = 2 (both parts of c∗ are w), so there is no downward jump between the two

segments of c∗.

Continuous. Each individual part of c∗ is continuous, and both parts are w at y = 2.


Nonnegative; and zero when y = 0. This is clear by inspection (again, the y − 1 segment only applies

when y > 2).

i.

ac(y) = c∗/y =

{ w2 if 0 < y ≤ 2

w(1− 1

y

)if y > 2

mc(y) =∂c∗

∂y={ w

2 if 0 < y < 2w if y > 2

j. Supply is marginal cost above average non-sunk cost. Since there are no fixed costs here, supply is just

marginal cost. If p is below w/2 then supply is the vertical axis (y∗ = 0). If p equals w/2 then supply

is any output level between 0 and 2. If p is between w/2 and w then supply is y∗ = 2. If p equals w

then supply is any output level greater than or equal to 2. If p exceeds w then supply is infinite (price

exceeds marginal cost for all output levels, so profit is always increased by an increase in output). Note

that this is exactly the same as the y∗ function obtained in part f. The break points occur at p = w/2

and p = w, as in part f. At these critical prices, there are ranges of optimal supply levels that coincide

with the ranges identified in part f.



1.

a. Let Qi denote the quantity for firm i. Then revenue for firm i is Ri = [150−Qi −Qj]Qi, so marginal

revenue is MRi = 150− 2Qi − Qj. Setting this equal to MCi and solving for Qi yields Qi = [150−

MCi −Qj]/2. Substituting in the marginal costs yields

Q∗1 = 70− 1

2Q2, Q∗

2 = 65− 12Q1.

These are the reaction curves, ignoring the possibility of shutdown at a positive output level. However,

since there are non-sunk fixed costs, the firms will shut down at positive output levels. To determine

the shutdown points, we must find each firm’s optimal profit as a function of the other firm’s output.

The equilibrium price from firm i’s perspective is

Pi = 150−Qi −Qj = 150− 150−MCi −Qj

2−Qj =

150 +MCi −Qj

2.

So firm i’s profit is

πi = [Pi −MCi]Qi − Fi =[150 +MCi −Qj

2−MCi

][150−MCi −Qj

2

]− Fi

=[150−MCi −Qj

2

]2− Fi. (1)

Setting this equal to zero and solving for Qj gives the level of the rival’s output above which firm i will

shut down:

150−MCi −Qj = 2√

Fi ⇒ Qj = 150−MCi − 2√

Fi

(the other root is not relevant because it makes the intercept of residual demand, 150 − Qj, below

marginal cost). Substituting in marginal and fixed costs yields Q2 = 60 and Q1 = 70. So the reaction

curves are

Q∗1(Q2) =

{70− 1

2Q2, Q2 ≤ 600, Q2 ≥ 60

, Q∗2(Q1) =

{65− 1

2Q1, Q1 ≤ 700, Q1 ≥ 70


Solving the positive parts simultaneously gives an intersection of (Q1, Q2) = (50, 40). Also, on the

positive parts, Q∗1(60) = 40 and Q∗

2(70) = 30. The graph is:

b. It is immediate from the graph that there are three (Q1, Q2) pairs that are Nash Equilibria:

I = (0, 65), II = (50, 40), III = (70, 0).

c.

i. We must calculate the sum of consumer’s surplus and profit in each of the Nash Equilibria (it is

not immediate that the equilibrium with price closest to marginal cost is best, because there are extra

fixed costs incurred when more than one firm operates). The prices in the three equilibria are

P I = 150− 65 = 85, P II = 150− 50− 40 = 60, P III = 150− 70 = 80.

So the consumer’s surpluses in the three equilibria are

CSI =12(65)(150− 85) =

652

2, CSII =

12(90)(150− 60) =

902

2, CSIII =

12(70)(150− 80) =

702

2.

From (1) above, the profits in the three equilibria are

πI1 = 0, πII1 =[150− 10− 40

2

]2−1, 600 = 502−1, 600, πIII1 =

[150− 10− 0

2

]2−1, 600 = 702−1, 600


πI2 =[150− 20− 0

2

]2− 900 = 652 − 900, πII2 =

[150− 20− 50

2

]2− 900 = 402 − 900, πIII2 = 0

So aggregate welfare in each of the three equilibria is:

πI1 + πI2 +CSI =32652 − 900 =

10, 8752

πII1 + πII2 +CSII = 502 + 402 +12902 − 2, 500 =

11, 3002

πIII1 + πIII2 +CSIII =32702 − 1, 600 =

11, 5002

Thus equilibrium III = (70, 0) is best for aggregate welfare. This is better than II, despite the extra

deadweight loss created by the higher price, because the extra deadweight loss is less than the fixed

cost of 900 that is saved when firm 2 does not operate.

ii. There is no way to tell which equilibrium will actually emerge unless we make more assumptions.

However, a law that prohibits firm 2 from operating will produce the best welfare outcome among the

three equilibria because firm 1 will behave as a monopolist under such a law, which is exactly what firm

1 does anyway in equilibrium III. Of course, it would be even better to regulate firm 1 as a monopolist

and thereby force firm 1 to price closer to marginal cost, assuming the regulator has reasonably good

demand and cost information.

iii. No, it is not Pareto Efficient, because there is deadweight loss relative to having price equal MC1.

The best outcome would be to set P = MC1 and then compensate firm 1 for its fixed cost through

some type of lump-sum transfer.

d. Both firms have strictly increasing returns, because a positive fixed cost and a constant marginal cost

result in a strictly downward-sloping ac curve. So, there is no perfectly competitive equilibrium price.

If the firms behave as price-takers, they will supply infinite output at prices above marginal cost and

zero output at prices no greater than marginal cost. But demand is positive and finite at marginal cost

c > 0, so there is no price at which demand equals competitive supply.

2.

a. This is Cobb-Douglas utility. It is straightforward to find the Marshallian demands x∗(p,m) = m2p and

y∗(p,m) = m2 . For future reference, the indirect utility function is therefore

U∗(p,m) = U(x∗, y∗) = ln(

m

2p

)+ ln

(m2

).


Since there are k identical consumers, aggregate demand for x is D(p) = kx∗(p,m) = km2p .

b. Revenue for a monopolist is R = pD(p) = km2 for D(p) > 0. This revenue expression depends on

neither price nor quantity. In other words, the monopolist’s revenue is constant no matter how much

the monopolist chooses to produce. This is because Cobb-Douglas demands have the property that

elasticity is one everywhere on the demand curve. Hence marginal revenue is zero at every positive

output level. However, revenue is positive when D(p) > 0 (there is a discontinuity at D(p) = 0). Since

marginal cost is positive, the monopolist wants to produce the smallest possible positive output.

c. p = c.

d. Let Q denote aggregate quantity and Qi the quantity of firm i. Invert aggregate demand from part

a to obtain inverse demand p(Q) = km2Q . Then the revenue of firm i is Ri = p(Q)Qi = km

2QQi, so

marginal revenue is MRi = km2Q

− km2Q2Qi. Setting this equal to marginal cost yields the optimality

condition km2Q − km

2Q2Qi = c. Noting that the E firms are identical, the symmetric equilibrium outputs

are identical. Call this common output level q, so Qi = q and Q = Eq in equilibrium. Putting these

into the optimality condition yields km2�q − km

2�2q2 q = c, or q = km(�−1)2�2c . Aggregate equilibrium output

is thus Q = Eq = km(�−1)2�c . Putting this into the aggregate inverse demand yields an equilibrium price

of p = km2

2�ckm(�−1)

= �c�−1

. Since the perfectly competitive price is c, the Cournot equilibrium price is

within 10% of the competitive price when ��−1 ≤ 1.1, or E ≥ 11.

e. Inverting indirect utility from part a yields the expenditure function e∗(p, U) = 2√

p · exp(U). We are

seeking a compensating variation measure of deadweight loss of the price increase from p = c to p = �c�−1

(the government is going to “compensate” consumers for the higher price). So the compensation will

restore utility after the price increase to the level consumers would get if p = c. This utility level is

U∗(c,m) = ln(m2c

)+ ln

(m2

).

Putting this level of utility into the expenditure function yields e∗(p, U∗(c,m)) = m√

pc . This is the

expenditure required to maintain utility at the level obtained when price is at the competitive level,

as a function of p. Since there are k consumers, aggregate expenditure is ke∗(p, U∗(c,m)) = km√

pc.

Thus the aggregate expenditure required to keep utility at the perfectly competitive level, when the


firms behave as Cournot oligopolists, is

ke∗(

cE

E− 1, U∗(c,m)

)= km

√E

E− 1.

The compensating variation is the difference between this expenditure and the original aggregate ex-

penditure km:

CV = km

√E

E− 1− km = km

[√E

E− 1− 1

].

To obtain the deadweight loss, we must subtract the extra revenue from CV. The price difference is

c��−1 − c = c

�−1 . The quantity at the higher price, after the compensation, is most easily found from the

Hicksian demand, which is

h∗(p, U∗(c,m)) =∂e∗(p, U∗(c,m))

∂p=

m

2

√1pc

.

Evaluating this demand at the oligopoly price yields quantity h∗(

c��−1

, U∗(c,m))= m

2c

√�−1�. Multi-

plying this by k to get aggregate quantity yields extra revenue of

c

E− 1km

2c

√E− 1E

=km

2

√1

E(E− 1).

Thus the deadweight loss is

CV - extra revenue = km

[√E

E− 1− 1

]− km

2

√1

E(E− 1)= km

[√E

E− 1− 1− 1

2

√1

E(E− 1)

].

With k = 1, 000, 000, m = $30, 000, and E = 11, the deadweight loss is $34,071,561.



1.

a. The indifference curves are semicircles with radii U centered at the point (0, 10). Larger semicircles

correspond to higher utility levels:

b. The utility function is defined on all of R2+. This implies completeness and transitivity of the preference

relation. The utility function is also continuous, which implies continuity of the preference relation.

c. For local nonsatiation, select any point in R2+. Utility is strictly increasing if the x coordinate is

increased while the y coordinate is held fixed, because such a change always moves the consumer to a

larger semicircle (even at (0, 10) and along the x-axis, including the origin). Thus, local nonsatiation

holds. However, no version of monotonicity holds. For example, (0, 10) is worse than (0, 0). Also,

convexity does not hold, as can be seen by joining points A and B in the figure above.

d. From the shape of the indifference curves, the optimal choice must be at one of the corners (m/p1, 0)

or (0, m/p2). The utility at these two corners is U(m/p1, 0) =√(m/p1)2 + 100 and U(0, m/p2) =


|m/p2 − 10|, respectively. So,

x∗1 =

mp1

, if√

(m/p1)2 + 100 > |m/p2 − 10|{mp1

, 0}, if

√(m/p1)2 + 100 = |m/p2 − 10|

0, if√

(m/p1)2 + 100 < |m/p2 − 10|

x∗2 =

0, if

√(m/p1)2 + 100 > |m/p2 − 10|{

mp2

, 0}, if

√(m/p1)2 + 100 = |m/p2 − 10|

mp2

, if√

(m/p1)2 + 100 < |m/p2 − 10|

2. It is straightforward to establish that these equations are homogeneous of degree zero, nonnegative, and

satisfy Walras’ Law. The key is checking symmetry and semidefiniteness of the matrix whose elements

are ∂x∗i

∂pj+ x∗

j∂x∗

i

∂m. The first diagonal element when m < p2 is:

∂x∗1

∂p1+ x∗

1

∂x∗1

∂m= − m2

p21p2

+m2

p1p2

[2mp1p2

]=

m2

p21p2

[2mp2

− 1].

This is positive for 12p2 < m < p2, so negative semidefiniteness fails. The equations are not Marshallian

demands.

3.

a. Invert U∗ to get e∗(p1, p2, U) = U2√p1p2.

b. Differentiate e∗ to get h∗1 = ∂e∗

∂p1= U2

2

√p2p1

and h∗2 = ∂e∗

∂p2= U2

2

√p1p2.

c. Differentiate U∗ to get

x∗1 = −∂U∗/∂p1

∂U∗/∂m= −−m2p−2

1 p−12

2mp−11 p−1

2

=m

2p1

x∗2 = −∂U∗/∂p2

∂U∗/∂m= −−m2p−1

1 p−22

2mp−11 p−1

2

=m

2p2

4. e∗(p1, p2, U) = p1h∗1 + p2h

∗2 = U [p1 + p2]. Invert to get U∗(p1, p2, m) = m

p1+p2. So, utility at the initial

price is U∗(1, 1, 160) = 80 and utility at the new price is U∗(31, 1, 160) = 5. Expenditure at the initial

price and the new utility level is e∗(1, 1, 5) = 10, so the change in income that is equivalent, in utility

terms, to the price change is 10− 160 = −150.



1.

a. This set is just the negative half of the real line, including the origin. Since the origin is included,

the possibility of inaction is satisfied and the set is closed (the only “boundary” is zero). It is also

clearly nonempty. All numbers smaller than any number in the set are also in the set, so free disposal

is satisfied. No positive numbers are included, so no free lunch is satisfied.

b. Z is convex. Take any two nonpositive real numbers z1 and z2, and form the weighted average λz1 +

(1− λ)z2 . The result is a nonpositive real number.

c. Constant returns. Shrink or expand any nonpositive real number z and you still get a nonpositive real

number αz, for α > 0.

d. Profit is pz, where p and z are both real numbers and p ≥ 0 and z ≤ 0. pz is maximized by choosing

z∗ = 0 unless p = 0. When p = 0, z∗ = (−∞, 0]. In both cases, π∗ = 0 ∀p ≥ 0.

2. The profit objective is π = p[α ln(x1 + 1) + (1− α) ln(x2 + 1)]−w1x1 −w2x2. By the envelope theorem,

∂π∗

∂α=

∂π

∂α

∣∣∣∣x1=x∗1

x2=x∗2

= p [ln(x1 + 1)− ln(x2 + 1)]|x1=x∗1

x2=x∗2

= p

[ln(pα

w1

)− ln

(p(1− α)

w2

)].

Since ln is a strictly increasing function, this expression is positive if and only if pαw1> p(1−α)

w2, or w2

w1> 1−α

α .

3.

a. Homogeneity holds since π∗(λp, λw) = λp = λπ∗(p, w). Convexity holds because π∗ is linear. Mono-

tonicity holds since π∗ is strictly increasing in p and does not vary with w. Nonnegativity holds because

π∗ ≥ 0 ∀p ≥ 0, and π∗ = 0 when p = 0.

b. Let H = {(y,−x) ∈ R2 : py − wx ≤ p ∀p ≥ 0 and w >> 0}. By setting p = 0 and considering

any w >> 0 we see that x ≥ 0 for any (y,−x) ∈ H . The defining inequality for H can be written

p(y − 1) ≤ wx. Since the right side of this inequality is nonnegative, we see that the inequality will

hold for any p ≥ 0 provided y ≤ 1. Thus, H is:


With this production set, the highest isoprofit line is achieved at (x, y) = (0, 1) no matter what the

slope of the isoprofit is. If p = 0 then the isoprofits are vertical, in which case any point on the right

boundary of H is optimal. So, (x∗, y∗) = (0, 1) ∀(p, w) >> 0, and (x∗, y∗) = (0, (−∞, 1]) if p = 0. In

either case, π∗ = p.

4.

a. V (y) = [0,∞) for y ≤ 0. V (y) = [1,∞) for y ∈ (0, 1]. V (y) = ∅ for y > 1.

b. If the slope of the isoprofit is less than or equal to −1 then the highest isoprofit is achieved at (x, y) =

(0, 0). If the slope of the isoprofit is greater than or equal to −1 then the highest isoprofit is achieved

at (x, y) = (−1, 1). So the supply/demand function for strictly positive prices is

(x∗, y∗) =

{(0, 0), w

p≥ 1

(−1, 1), wp≤ 1

If p = 0 then x∗ = 0 and y∗ = (−∞, 0]. Thus, profit is

π∗ =

{0, w

p ≥ 1 or p = 0

p− w, wp≤ 1

c. For a given y ≤ 1, the cost minimizing choice of x is to choose the smallest possible value. If y ≤ 0 this

means x∗ = 0, and if y ∈ (0, 1] this means x∗ = 1. So the conditional factor demand is

x∗ ={

0, y ≤ 01, y ∈ (0, 1]


This means the cost function is

c∗ ={

0, y ≤ 0w, y ∈ (0, 1]

d. ac = wy , avc = 0, afc = w

y (all costs are fixed):

Marginal cost is clearly zero for y ∈ (0, 1). It is impossible to produce more than y = 1 with this

technology, so in cost terms it is as if marginal cost has a vertical jump and becomes infinite at y = 1.

This conception of marginal cost is consistent with the supply behavior identified above. If price is at

or above the low point on the ac curve (i.e., w) then the firm chooses the only output where p = mc,

which is y = 1 where the vertical jump occurs. If price is below the minimum of the ac curve then the

firm shuts down since none of the fixed costs are sunk.



1.

a. AC = 1yi+yi and MC = 2yi. Setting these equal gives the minimum average cost at yi = 1. The value

here is MC(1) = 2. So, p = 2 is the only potential long-run competitive equilibrium price.

b. At p = 2, each firm must produce yi = 1. Demand is y = 100− 2 = 98. So, there is an equilibrium

with 98 firms.

2.

a. Yes. We have consumption of good 1 = x1 and production of good 1 = x1. Thus, consumption ≤

production. For good 2, we have m− AC(x1) +AC(x1) ≤ m.

b. No. x1 avoids the deadweight loss. The consumer would gladly pay MC[x1 − x1] for the extra units,

and the firm would be indifferent.

c. No. The price that makes demand = x1 makes supply infinite.

d. Yes. x1 ≤ x1 and m+ 0 ≤ m.

e. Yes. Any deviation involves deadweight loss, from which somebody is made worse off.

f. No. The price that makes demand = x1 makes supply zero.

g. The theorem gives sufficient but not necessary conditions for an equilibrium to be Pareto Efficient. Here,

x1 is Pareto Efficient, but is not an equilibrium. This shows that the conditions are not necessary. x1

is neither an equilibrium nor Pareto Efficient, so it is unrelated to the theorem.

3. MC = 2y and MR = 100− 2y, so the interior solution is y∗ = 25. But π(25) = −250, so the firm chooses

y∗ = 0 and p∗ ≥ 100 to get π∗ = 0 (since the fixed cost is not sunk).

4.

a. The FOC for firm i is 100− 2yi − yj = 1, so y∗i = 99−yj

2. This is the interior solution, which is only

relevant when π∗i (yj) ≡ π(y∗i (yj); yj) ≥ 0 (we know that shutdown is a concern here because of the

non-sunk fixed cost). Substitution and simplification gives the optimal value function for firm i as

π∗(yj) =(

99−yj

2

)2

−1200. Setting this to zero and solving for yj yields yj = 99−40√3 $ 29.72. Thus,

the reaction curve for firm i is

y∗i (yj) ={ 99−yj

2, yj ≤ 29.72

0, yj ≥ 29.72


b. Because of the discontinuity, y∗i does not cross y2 = y1, so there is no (pure strategy) symmetric Nash

Equilibrium. However, there are two non-symmetric equilibria, indicated by the dark squares on the

above graph.



1. The indifference curves are squares:

a. Complete: Yes. Everything is on some indifference curve.

b. Transitive: Yes. Utility increases as we move away from (10, 10).

c. Continuous: Yes. The sets outside each square, including the square, are closed. Same for the sets

inside each square.

d. Monotonic: No.

e. Locally nonsatiated. This is tricky. As the squares intersect the axes we get points that have superior

neighbors if we move out along the axis (when xi > 10) or in along the axis (when xi < 10). But, at

(0, 0) there is no point that is locally superior. So, this preference relation is NOT locally nonsatiated.

f. Convex: No. The segment from A to B in the graph passes inside the square.

2.

a. Homogeneity:

h∗1(λp, U) =

U2 , if λp1

λp2< 2

U2 or 0, if λp1

λp2= 2

0, otherwise

=

U2 , if p1

p2< 2

U2 or 0, if p1

p2= 2

0, otherwise

= h∗1(p, U).

The proof is the same for h∗2.


Symmetry/semidefiniteness: ∂h∗i

∂pj= 0 provided p1 �= 2p2. So the substitution matrix is

[0 00 0

]for p1 �= 2p2,

which is symmetric and negative semidefinite. When p1 = 2p2, the situation is

Monotonicity and Continuity: p ·h∗ = min{p12 , p2

}U , which is strictly increasing in U and continuous.

Nonnegativity: The origin is U = U(0) = 0. Then h∗ > 0 ∀U > U(0) and h∗(p, U(0)) = 0 ∀p (so

p · h∗(p, U(0)) = 0).

b. The expenditure function is

e∗ = p · h∗ =

{p1

U2, if p1

p2≤ 2

p2U , if p1p2

> 2= U min

{p1

2, p2

}.

Invert to obtain indirect utility:

U∗ =m

min{p12, p2

} .Then use the identity

x∗1(p,m) = h∗

1(p, U∗(p,m)) =

m

2min{ p12 ,p2} , if p1

p2< 2

m2min{ p1

2 ,p2} or 0, if p1p2

= 2

0, otherwise

=

mp1

, if p1p2

< 2mp1

or 0, if p1p2

= 2

0, otherwise

.

Likewise,

x∗2(p,m) = h∗

2(p, U∗(p,m)) =

0, if p1

p2< 2

0 or mp2

, if p1p2

= 2mp2

, if p1p2

> 2


c.

∂x∗1

∂p1

?=

∂h∗1

∂p1− x∗

1

∂x∗1

∂m.

For p1p2

< 2:

−m

p21

?=0− m

p1

1p1

. Clearly yes.

For p1p2

> 2:

0?=0− 00. Clearly yes.

3.

a. maxx1x2 subject to p1x1 + p2x2 = m. This is standard Cobb-Douglas. The Marshallian demands are

x∗1 =

m

2p1, x∗

2 =m

2p2.

b. U is a monotonic transformation of x1x2 and adding x3 > 0 does not affect the constraint, so the

Marshallian demands for x1 and x2 are unchanged.

c. U∗ = x∗1x

∗2 + x3 = m2

4p1p2+ x3.

d. By inversion, e∗ =√(U − x3)4p1p2.

e. The target level of utility is U∗(1, 1, 10|x3 = 11) = 1004

+11 = 36. We need e∗ when U = 36 and x3 = 0.

So e∗ =√

(36− 0)4 · 1 · 1 = 12. Existing income is m = 10, so the transfer is 12− 10 = 2.



1. f(x) = 100 + 48x − x2. This function has a positive intercept, is increasing for small x, and then is

decreasing for large x. The production set is Z = {(y,−x) : y ≤ 100 + 48x− x2 and x ≥ 0}:

a. Z is nonempty, closed (since the boundary is included), convex (since f is concave), and satisfies lower

semicontinuity in output (since f is continuous). Also, (0, 0) ∈ Z so Z satisfies the possibility of

inaction. But, f(0) > 0 so no free lunch is violated, and f is non-monotonic so free disposal is violated.

b. π∗ is homogeneous of degree 1, monotonic, and convex because these properties come from the definition

of π∗ as a maximum. The only property we might suspect is violated is nonnegativity, since it relies on

additional properties of the technology. However, in this case we still get π∗(p, w) ≥ 0 and π∗(0, w) = 0

because Z satisfies the possibility of inaction.

2. The firm’s profit objective is π(y; r) = R(y; r) − c∗(y) = p(y; r)y − c∗(y).

a. By the envelope theorem,

∂π∗(r)∂r

=∂π

∂r

∣∣∣∣y=y∗

= p2(y∗; r)y∗ > 0.

The sign follows because substitutes means p2 > 0.

b. The first order condition is π1(y∗(r); r) ≡ 0. Assuming a proper maximum, this equation defines y∗(r).


Differentiate with respect to r to obtain

π11∂y∗

∂r+ π12 = 0,

which implies

∂y∗

∂r= −π12

π11.

By the second order condition, π11 < 0, so ∂y∗

∂rhas the same sign as π12:

π12 = π21 =∂

∂y(π2) =

∂

∂y(p2y) = p2 + p21y = p2 + p12y.

We know p2 > 0, but p12 is the change in the slope of the inverse demand when the rival’s price changes.

We cannot determine the sign of ∂y∗∂r without information on this.

3. This is Varian #5.16b (Lecture 8, Exercise 1) except for y+y2 rather than y. Since we still have c∗ ≥ 0, c∗

strictly increasing in y, and c∗(w, 0) = 0; this function satisfies all sufficient conditions for a cost function.

4.

a.

b. As we vary x1, for short-run cost minimization we must stay on the isoquant labeled y. So the short-

run minimum always occurs where x1 intersects the y isoquant. This occurs on different isocosts as x1


varies. The lowest isocost that this occurs on is when x1 is such that it crosses the y isoquant at the

point where the isocost is tangent to the isoquant. Of course, this point is also the long-run minimum,

so the value of x1 that minimizes short-run cost (i.e., that places us on the lowest isocost while staying

on the y isoquant) is exactly x∗1(w, y).

c.

min{x1}

w1x1 +w2y

x1.

The first order condition is

∂c∗(w, y|x1)∂x1

= w1 − w2y

x21

= 0.

This implies x∗1 =

√w2yw1

. The second order condition is satisfied:

∂2c∗

∂x21

=2w2y

x31

> 0.

Substitute x∗1 into c∗(w, y|x1) to obtain long-run cost:

c∗(w, y) = w1x∗1 +

w2y

x∗1

= w1

√w2y

w1+ w2y

√w1

w2y= 2

√w1w2y.



1.

a.

π1 = (12− y1 − y2)y1 − 4y1 and π2 = (12− y1 − y2)y2 − 10y2.

Differentiate to get

12− 2y1 − y2 = 4 and 12− 2y2 − y1 = 10,

and then solve for y1 and y2, respectively, to get the reaction functions:

y∗1(y2) =8− y2

2and y∗2(y1) =

2− y1

2.

Setting y∗2(y1) to zero yields the y1 intercept of 2. Setting y2 to zero in y∗1(y2) yields the y1 intercept

of 4. Hence these reactions curves do not cross in the positive quadrant. They appear as:

For values of y1 greater than 2, firm 2 reacts by choosing y2 = 0 because firm 2’s costs are too high to

make any profit when y1 ≥ 2. Therefore the Nash equilibrium is (y1, y2) = (4, 0). Substituting 4+0 into the

demand curve yields a price of 8.

b. The monopoly profit function for firm 1 is the same as the duopoly profit function when y2 = 0. So

firm 1 chooses the same quantity of 4 and sets the same price of 8.

c. They are the same, because firm 2’s high cost causes it to drop out, effectively making firm 1 a

monopolist (firm 2’s marginal cost is above firm 1’s monopoly price).


2.

a. Differentiating yields marginal cost of c′ = 2y, and dividing yields average cost of ac = 1y+ y. Setting

these equal shows that they cross at y = 1. The cost curves look like:

So individual firm supply is y = p/2 for p ≥ 2. With 100 firms, market supply is y = 50p. Solving this

simultaneously with market demand yields an equilibrium price of p = 10. Hence individual supply is 5 in

equilibrium and individual profit is

π = (10)(5)− 52 − 1 = 24.

Thus, an arbitrarily large number of bidders will bid the price of the license up to 24, because this is how

much the license is worth in terms of profit potential. The city gets revenue of R = (100)(24) = 2400.

b. Now profit will be bid to zero, so we must have p = 2 in any long-run equilibrium and we must have

each firm producing at the minimum average cost quantity of y = 1. Market demand at this price is

y = 510− 2 = 508, so we will have 508 firms each producing one unit.

c. The lost consumer’s surplus is the shaded area:

So, the area is (8)(500) + (1/2)(8)(8) = 4032.

d. No. The net loss is 4032 - 2400 = 1632. This is the deadweight consumer’s surplus plus profit lost,

since the city can only rebate the profit that is actually transferred from consumers to firms, and then

to the city.


e. R(n) = nπ∗(n). To maximize R, find R′ = π∗(n) + nπ∗′(n). But π∗(n) = π(y∗(n), n) = p(n)y∗(n) −

c(y∗(n)). So, by the envelope theorem, π∗′ = p′(n)y∗(n).

3. Symmetry gives b = 2. Homogeneity of h2 yields

p1∂h2

∂p1+ p2

∂h2

∂p2= 0

(8)(2) + p2

(−12

)= 0,

so p2 = 32. Homogeneity of h1 then yields

p1a + 2p2 = 0

8a+ 64 = 0,

so a = −8.

4. Yes. At high prices, the good is normal (or at least non-Giffen). Demand does not remain bounded away

from 0 for high prices, so there is no implied violation of the budget constraint. At low prices, the good

turns Giffen, so the extra income is simply being devoted to some other good.

5.

a. Homogeneity is obvious. The substitution matrix is −Ub12p1/22

2p3/21

Ub121

2p1/21 p

1/22

Ub211

2p1/21 p

1/22

−Ub21p1/21

2p3/22

.


Hence symmetry requires b12 = b21 and negative semidefiniteness requires Ub12, U b21 ≥ 0. Monotonic-

ity requires that p · h∗ = U[p1b11 + p2b22 + (b12 + b21)(p1p2)1/2

]be strictly increasing in U . Since

p can be arbitrarily large, this requires bij ≥ 0 for i, j = 1, 2, with at least one of these inequalities

strict. Returning to negative semidefiniteness, we then see that U ≥ 0 unless b12 = b21 = 0. With

the bij ’s nonnegative, hi ≥ 0 for U ≥ 0 automatically. Moreover, from the expression just derived,

p · h∗(p, U) = 0 ∀p >> 0 if and only if U = 0 (unless all bij’s are zero, which cannot happen since

at least one of the bij’s must be strictly positive, by monotonicity). Hence U(0) = 0 and so U ≥ 0

is required. With this restriction, we have h∗i (p, U) > 0 ∀p >> 0 and ∀U > 0, provided at least one

bij > 0 for each i. To summarize, h1 and h2 are Hicksian demands provided either b12 = b21 = 0 and

b11, b22 > 0; or b12 = b21 > 0 and b11, b22 ≥ 0.

b. By definition, expenditure is

e∗ = U

[p1

(b11 + b12

(p2

p1

)1/2)

+ p2

(b22 + b21

(p1

p2

)1/2)]

.

Invert to get indirect utility

U∗ = m

[p1

(b11 + b12

(p2

p1

)1/2)

+ p2

(b22 + b21

(p1

p2

)1/2)]−1

.

Differentiate to get

∂U∗

∂m= [·]−1

∂U∗

∂p1= −m[·]−2

(b11 +

12b12

(p2

p1

)1/2

+12b21

p2

(p1p2)1/2

)

= −m[·]−2

(b11 + b12

(p2

p1

)1/2)

∂U∗

∂p2= −m[·]−2

(b22 + b21

(p1

p2

)1/2)

.

So, by Roy’s Identity,

x∗1 = m[·]−1

(b11 + b12

(p2

p1

)1/2)

x∗2 = m[·]−1

(b22 + b12

(p1

p2

)1/2)

.

6. This is question 2 from the November 19, 1998 test. See those solutions for the answer.



1.

a.

b. The indifference curve for U ≥ 0 is strictly convex and never touches the axes, so the optimal choice is

described by the first order conditions

p1 =λ

x21

and p2 = λ.

These yield h∗1 =

√p2p1. Substituting into the constraint then yields h∗

2 =√

p1p2

+ U . These equations

are valid for any U ≥ 0.

c. For U ≥ 0, from above e∗ = p1h∗1 + p2h

∗2 = 2

√p1p2 + p2U . Invert e∗ to get U∗ = m

p2− 2√

p1p2. This

expression is valid for U∗ ≥ 0 (i.e. U ≥ 0). That is, the expression is valid for m ≥ 2√p1p2.

d. Use Roy’s Identity to get x∗1 =

√p2p1

and x∗2 = m−√

p1p2p2

, once again for m ≥ 2√p1p2.

2. Fix y ∈ X and ε > 0. By nonsatiation, there exists x ∈ X such that x � y. By completeness, y �∼

y. So,

by semi-strict convexity, λx + (1 − λ)y � y. Now choose λ close to 0 so that the convex combination is

close to y. By choosing λ “small enough,” we get the convex combination within ε of y, and it is strictly

preferred to y as noted above. In particular, choosing λ = ε2‖y−x‖ yields

‖(λx+ (1− λ)y) − y‖ = λ‖x− y‖ =ε

2< ε


(Note that ‖y − x‖ > 0 because y �= x since x � y).

3. Homogeneity is satisfied for any nonzero a and b (neither a nor b can be zero because this would make

demand infinite, thereby violating Walras’ Law). Nonnegativity requires a > 0 and b > 0. Since we are

looking for a consumer whose preferences are locally nonsatiated, we know Walras’ Law must hold. This

implies 1a + 1

b = 1. Since a and b are positive, we then conclude that a > 1 and b > 1. Taking the price

and income derivatives, we find the Slutsky substitution matrix to be

[ (1−a)ma2p21

mabp1p2

mabp1p2

(1−b)mb2p22

].

This is symmetric for any nonzero a and b. For negative semidefiniteness, the diagonal elements must be

nonpositive, so a ≥ 1 and b ≥ 1. Also, the determinant must be nonnegative, so ab ≥ a + b. Dividing by

ab, we see that we already have this condition holding with equality from Walras’ Law. Hence, symmetry

and semidefiniteness add no new restrictions on a and b. To summarize, any positive values of a and b

satisfying 1 = 1a + 1

b yield valid Marshallian demands.

4. Mr. Giffen is correct. To see why, suppose otherwise. Then x∗i is upward-sloping in pi at (p0, m) for every

m > 0. Since the substitution effect is nonpositive, this means that the income effect is strictly negative

at every m > 0. Since we know that x∗i (p

0, 0) = 0, the strict negative slope of x∗i with respect to m at

every m > 0 implies that x∗i (p

0, m) is strictly negative for m > 0, which violates nonnegativity of the

demand, a contradiction.



1.

a.

b. From the linear isoquant, an optimal choice occurs at the corners, and is (0,√y) if w1

w2≥ 1 and is

(√y, 0) if w1w2

< 1. So,

c∗(w1, w2, y) ={

w2√y, w1

w2≥ 1

w1√y, w1

w2< 1.

c. If√y ≤ 10 (y ≤ 100) then there is more than enough of input 1 at x1 = 10 to produce the required

output, without using any of input 2. If√y > 10 then the amount of input 2 needed is determined

from the isoquant, x2 =√y − x1 =

√y − 10. Hence:

c∗(w1, w2, y|x1 = 10) ={

10w1,√y ≤ 10

10w1 +w2(√y − 10),

√y > 10.

2.

a. The only way homogeneity of degree one can hold is if α = 1.

b. Average cost is ac(y) = (2w1 + w2) ey−1y . So,

ac′(y) = (2w1 + w2)1y2

(1 + ey(y − 1)).

When y = 0 we have 1+ey(y−1) = 0, and this expression is clearly increasing in y. That is, ac′(y) > 0

for all y > 0, so we at least suspect there is decreasing returns to scale.


c. By Shephard’s Lemma,

x∗1 =

∂c∗

∂w1= 2(ey − 1) and x∗

2 =∂c∗

∂w2= (ey − 1).

3. A careful graphing reveals the following production set:

This set is closed since the defining inequality is weak (so the boundary is included in Z), and is obviously

nonempty. Also, the boundary goes through the origin (check that (0, 0) satisfies the equation z2 = 1− (1+

z1)2), so possibility of inaction is satisfied. No points in the strictly positive quadrant are included, so no

free lunch is also satisfied. Free disposal is violated, however, because the boundary on the left side of Z is

downward-sloping. Finally, since the boundary curves downward everywhere, Z is convex.

4.

a. We must check that the three sufficient conditions are holding.

1. (Homogeneity)

y∗(λp, λw1, λw2) =λp

2

(1

λw1+

1λw2

)=

p

2

(1w1

+1w2

)= y∗(p, w1, w2)

x∗1(λp, λw1, λw2) =

(λp

2λw1

)2

=(

p

2w1

)2

= x∗1(p, w1, w2)

x∗2(λp, λw1, λw2) =

(λp

2λw2

)2

=(

p

2w2

)2

= x∗2(p, w1, w2).


2. (Symmetry and Semidefiniteness) The matrix of derivatives of these supply and demand functions is

∂y∗∂p

∂y∗∂w1

∂y∗∂w2

−∂x∗1

∂p − ∂x∗1

∂w1− ∂x∗

1∂w2

−∂x∗2

∂p − ∂x∗2

∂w1− ∂x∗

2∂w2

=

12

(1w1

+ 1w2

)− p

2w21

− p2w2

2

− p2w2

1

p2

2w31

0

− p2w2

20 p2

2w32

This is symmetric by inspection. To check positive semidefiniteness, check that all the naturally ordered

principal minors are nonnegative. The order 1 naturally ordered principal minor is just the (1, 1)

element, which is positive. The order 2 naturally ordered principal minor is p2

4w31

(1w1

+ 1w2

)− p2

4w41> 0.

The order 3 naturally ordered principal minor is p4

8w31w

32

(1w1

+ 1w2

)− p4

8w31w

42− p4

8w32w

41= 0 (alternatively,

we can simply note that the matrix is singular, by homogeneity, so its determinant must be zero).

3. (Nonnegativity) First, the profit function defined from these supply and demand functions must be

nonnegative. It is derived below, and is clearly nonnegative. Second, the demands and supplies

themselves must be nonnegative. This is also clear by inspection (This and the semidefiniteness relies

on p ≥ 0 and w >> 0). Third, the demand and supply must be zero when p = 0, for every w >> 0.

This is also clear by inspection.

b.

π∗(p, w1, w2) = py∗ −w · x∗ =p2

2

(1w1

+1w2

)− w1

(p

2w1

)2

− w2

(p

2w2

)2

=p2

4

(1w1

+1w2

).



1. Indifference curves are linear with a slope of −1. So,

h∗1 =

0, if p1

p2> 1

[0,√U ], if p1

p2= 1

√U, if p1

p2< 1

; h∗2 =

√U, if p1

p2> 1

√U − h∗

1, if p1p2

= 1

0, if p1p2

< 1

.

2.

e∗ = p1h∗1 + p2h

∗2 = − 1

U

[p1

[(p2

p1

) 12

+ 1

]+ p2

[(p1

p2

) 12

+ 1

]]= − 1

U

[2(p1p2)

12 + p1 + p2

].

So, U∗ = − 1m

[2(p1p2)

12 + p1 + p2

].

∂U∗

∂p1= − 1

m

[(p1p2)−

12 p2 + 1

]= − 1

m

[(p2

p1

) 12

+ 1

]∂U∗

∂p2= − 1

m

[(p1

p2

) 12

+ 1

]∂U∗

∂m=

1m2

[2(p1p2)

12 + p1 + p2

].

By Roy’s Identity,

x∗1 = −

∂U∗∂p1∂U∗∂m

= m

(p2p1

) 12+ 1

2(p1p2)12 + p1 + p2

x∗2 = −

∂U∗∂p2∂U∗∂m

= m

(p1p2

) 12+ 1

2(p1p2)12 + p1 + p2

.

3. The condition on the number of firms n is (n−1)y ≥ ny, where y = 10 and y = 11 here. So (n−1)11 ≥ 10n,

or n ≥ 11. Then we must have D(min ac) ≥ (n− 1)y. Since minac = 1, we need D(1) ≥ 10 · 10 = 100.

4. MC = 10− 2y,MR = 7− y. MC is steeper than MR (more negatively sloped). So MC = MR is a local

minimum. Hence we must consider the corners y = 0 and y = 5. Now π = 0 at y = 0 obviously. At

y = 5, p = 7− 52 = 9

2 and c = 50− 25 = 25, so π = 92 · 5− 25 = 45

2 − 502 = −5

2 < 0. So y∗ = 0 is the max

and p∗ ≥ 7.

5. Complete: Yes, because with three voters every pairwise comparison has a majority, provided the voters

can make every pairwise comparison (which they can).

Transitive: Not necessarily. Suppose the voters’ preferences were

Ann: G � B � M ; Beth: M � G � B Carol: B � M � G


Then Pop has G � B and B � M, but M � G.

6.

i. Homogeneous of degree 0:

x∗1(λp, λm) =

2λm3λp1

=2m3p1

= x∗1(p,m) �

x∗2(λp, λm) =

λm

3λp2=

m

3p2= x∗

2(p,m) �

ii. Nonnegativity: Obviously x∗ ≥ 0.

iii. Walras’ Law:

p · x∗ = p12m3p1

+ p2m3p2

= 2m3 + m

3 = m, so Walras’ Law is satisfied.

iv. The substitution matrix must be symmetric and negative semidefinite:

∂x∗1

∂p1= − 2m

3p21

;∂x∗

2

∂p2= − m

3p22

;∂x∗

1

∂m=

23p1

;∂x∗

2

∂m=

13p2

.

So the matrix is[∂x∗

1∂p1

+ x∗1∂x∗

1∂m

∂x∗1

∂p2+ x∗

2∂x∗

1∂m

∂x∗2

∂p1+ x∗

1∂x∗

2∂m

∂x∗2

∂p2+ x∗

2∂x∗

2∂m

]=

− 2m3p21

+(

2m3p1

)(2

3p1

) (m3p2

)(2

3p1

)(

2m3p1

)(1

3p2

)− m

3p22+(m3p2

)(1

3p2

) =

[− 2m9p21

2m9p1p2

2m9p1p2

− 2m9p22

],

which is obviously symmetric. The diagonal elements are less than 0. The determinant is

4m2

81p21p

22

− 4m2

81p21p

22

≥ 0.

7.

−1 1 ba −2 cd e f

a = 1 by symmetry. By homogeneity,

∑3i=1

∂h∗j

∂pipi = 0. So, (−1)(1) + (1)(2) + (b)(4) = 0. So, b = −1

4.

Then d = −14 by symmetry. This leaves −1 1 −1

41 −2 c−1

4 e f

.

So, (1)(1) + (−2)(2) + (c)(4) = 0, so c = 34 . Then e = 3

4 by symmetry. This leaves −1 1 −14

1 −2 34

−14

34

f

.

So,(−1

4

)(1) +

(34

)(2) + (f)(4) = 0, so f = − 5

16 . Finally,−1 1 −14

1 −2 34

−14

34 − 5

16

.



1.

min{(x1,x2)}

w1x1 + w2x2 − λ[(xρ1 + xρ2)

ερ − y

]Take the first derivative to find the FOCs:

w1 − λε

ρ(xρ1 + xρ2)

ερ−1

ρxρ−11 = 0

w2 − λε

ρ(xρ1 + xρ2)

ερ−1

ρxρ−12 = 0.

Divide: w1w2

= xρ−11

xρ−12

⇒ x1 =[w1w2

] 1ρ−1

x2. Now use the constraint:

[[(w1

w2

) 1ρ−1

x2

]ρ+ xρ2

] ερ

= y,

so

xε2

[(w1

w2

) ρρ−1

+ 1

] ερ

= y ⇒ x∗2 = y

1ε

[(w1

w2

) ρρ−1

+ 1

]− 1ρ

and x∗1 = y

1ε

[(w2

w1

) ρρ−1

+ 1

]− 1ρ

.

So c∗ = w1x∗1 + w2x

∗2 = y

1ε

([w−ρ

1

(w2w1

) ρρ−1

+w−ρ1

]− 1ρ

+[w−ρ

2

(w1w2

) ρρ−1

+w−ρ2

]− 1ρ

).

2.


3.

4.

i. HOD 1:

c∗(λw1, λw2, y) = (λw1 + λw2) ln(y + 2) = λ(w1 + w2) ln(y + 2) = λc∗(w1, w2, y) �

ii. Monotonicity in w and y:

∂c∗

∂wi= ln(y + 2) > 0 for y ≥ 0 for i = 1, 2, �

∂c∗

∂y=

w1 + w2

y + 2> 0 for w >> 0 and y ≥ 0 �

iii. Concavity:

∂2c∗∂wi∂wj

= 0 ∀ i, j since c∗ is linear in w1 and w2. �

iv. Continuity: c∗ involves only sums, products, and the log function. So it is continuous provided the

argument of the log function is positive (which it is for y ≥ 0).

v. Nonnegativity:

c∗(w, y) > 0 for w, y >> 0 �.

c∗(w, 0) = 0? NO: ln(2) > 0, so c∗(w, 0) > 0.

Defining H(y) = {x ∈ Rn+ : w · x ≥ c∗(w, y) ∀ w >> 0}, we see that this is regular, has free

disposal, no free lunch, convexity, and lower semicontinuity in output. But possibility of inaction fails

since c∗(w, 0) > 0.


5.

a. HOD 1:

π∗(λp, λw) =(λp)2

λw= λ

p2

w= λπ∗(p, w) �

Monotonicity:

∂π∗

∂p=

2pw

> 0 for (p, w) >> 0 �

∂π∗

∂w= − p2

w2< 0 for (p, w) >> 0 �

Convexity:

∂2π∗

∂p2=

2w;

∂2π∗

∂p∂w= − 2p

w2;

∂2π∗

∂w2=

2p2

w3.

So,

Hπ∗ =[ 2

w − 2pw2

− 2pw2

2p2

w3

].

The diagonal elements are positive and |Hπ∗ | = 4p2

w4 − 4p2

w4 = 0. Hence Hπ∗ is positive semidefinite,

implying that π∗ is convex.

Nonnegativity: π∗ ≥ 0 and π∗(0, w) = 0 are clear by inspection.

b. H = {(y, x) ∈ R2+ : py−wx ≤ π∗(p, w) ∀(p, w) >> 0}. So, max{p,w} py−wx− p2

wto get y− 2p

w= 0 ⇒

y = 2(pw

)and −x+ p2

w2 = 0 ⇒ x =(pw

)2. Substitute to get y = 2x12 .

c. max{x} p2x12 − wx ⇒ px− 1

2 − w = 0, or x =(pw

)2. So maxpy − wx = 2p

(pw

) − w(pw

)2= 2p2

w − p2

w =

p2

w = π∗.

d. Since y = 2x12 (only one input), c∗ = w y2

4 .

6.

HOD 0:

x∗i (λw1, λw2, y) = ln(y + 1) = x∗

i (w1, w2, y) �

Monotonicity: w · x∗ = (w1 + w2) ln(y + 1), so

∂w · x∗

∂y=

w1 + w2

y + 1> 0 �

Symmetry and Semidefiniteness: The substitution matrix is 0, so it is symmetric and negative semi-

definite.


Continuity: The expression for w · x∗ above is continuous at y ≥ 0.

Nonnegativity: x∗i ≥ 0 ∀ w, y ≥ 0 �; w · x∗(w, 0) = 0 �; x∗(w, y) = ln(y + 1) > 0 for y > 0 �



1.

a. Homogeneity is obvious.

Monotonicity: p · h∗ = 2U(p1p2)1/2, so ∂p·h∗

∂U= 2(p1p2)1/2 > 0.

Nonnegativity: h∗ ≥ 0 clearly requires U ≥ 0. Then p · h∗(p, 0) = 0 from above, and h∗(p, U) > 0 for

U > 0 is obvious.

Continuity: The expression for p · h∗ given above involves only products and nonnegative exponents,

so it is continuous.

Symmetry and Semidefiniteness: −12

p122

p321

U 12

p− 1

22

p121

U

12

p− 1

21

p122

U −12

p121

p322

U

.

This is symmetric and negative semidefinite since the diagonal elements are ≤ 0 and the determinant

is U2

4p1p2− U

2

4p1p2= 0.

b. e∗ = p · h∗ = 2U(p1p2)12 . So U∗ = m

2(p1p2)12. So x∗

1 = h∗1(p, U∗) =

(p2p1

) 12 m

2(p1p2)12

= m2p1

and x∗2 =

h∗2(p, U

∗) =(p1p2

)12 m

2(p1p2)12= m

2p2.

2. No, the ranking is not complete since comparisons that don’t involve a BMW are not made.

3. Slutsky: ∂x∗i

∂pi= ∂h∗

i

∂pi− x∗

i∂x∗

i

∂m , which is necessarily negative if ∂x∗i

∂m > 0.

FOC’s are:

∂fi(x∗i )

∂xi− λ∗pi = 0 for i = 1, · · · , n

−p · x+m = 0

So, differentiate with respect to m:

∂2f1∂x2

1−p1

. . ....

∂2fn

∂x2n

−pn−p1 · · · −pn 0

∂x∗1

∂m...

∂x∗n

∂m∂λ∗∂m

=

0...0−1

.


By Cramer’s Rule,

∂x∗i

∂m=

1|H |

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∂2f1∂x2

10 −p1

. . ....

...

0...

.... . .

...0 ∂2fn

∂x2n

−pn−p1 · · · −1 · · · −pn 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣=

1|H |(−1)(−1)n+1+i(−pi)(−1)n+i

[∂2f1∂x2

1· · · ∂2fi−1

∂x2i−1

· ∂2fi+1

∂x2i+1

· · · ∂2fn

∂x2n

]=

1|H |(−pi)[·].

|H | has the sign (−1)n and [·] has sign (−1)n−1, so the whole thing is positive.

4.

a.

Average cost curve

b. pe = 2

c. We need (n − 1)y ≥ ny, or (n − 1)2 ≥ n, yielding n ≥ 2. So all output levels beyond 1 · y can be

produced at minimum ac. So, D(2) ≥ 1 is sufficient.

5. mr = 4 − 2y. So, y∗ = 1 for the monopolist, and p∗ = 3. This is where mr = mc. The area of the

deadweight loss triangle is 12 .


6. πi = (4− y1 − y2)yi − (1 + y2i ) since AC(y) = y + 1

y⇒ c(y) = 1 + y2. ∂πi

∂yi= 4 − 2yi − yj − 2yi = 0. So,

we need to solve two equations:

4− 4y1 − y2 = 0

4− 4y2 − y1 = 0.

Multiplying the second equation by -4, we get

4− 4y1 − y2 = 0

−16 + 4y1 + 16y2 = 0.

Add the two together to get 12− 15y2 = 0, so y2 = 1215 = 4

5 = y1 ⇒ y1 + y2 = 85 . Perfect competition has

individual supplies p = 2yi, so aggregate supply is y1 + y2 = p. Setting this equal to demand, y = 4 − y,

or y = 2. So the comparison is:


Here, consumer’s surplus lost is 25· 2

5· 1

2= 2

25and producer’s surplus lost is 2

25also. Thus, the total lost

surplus is 425 .

7. c(y) = 1 + y2, where 1 represents the fixed (sunk) cost. So, assume px = 1.

The production function for x ≥ 1 is y =√x− 1. This gives c(y) = pxx = 1 · (y2 + 1) = y2 + 1.



1.no free lunch possibility of inaction free disposal convexity

(a) y y y y(b) n y n y(c) y y y n

a.

This has decreasing returns to scale.b.

This has decreasing returns to scale.


c.

This has increasing returns to scale.

2. The isoprofit lines are vertical. So, letting x be the output and y be the input, the firm chooses x∗ = 1

and y∗ = 0, and receives π∗ = 1.

3. y = ln(x1 + 1) + x2. Taking the derivative with respect to x1, we get 0 = 1x1+1

+ dx2dx1

⇒ dx2dx1

= − 1x1+1

=

−w1w2

⇒ x∗1 = w2

w1− 1. Substituting x∗

1 into the production function, we get x∗2 = y − ln(w2

w1). Taking the

second derivative, we get d2x2dx2

1= 1

(x1+1)2 > 0. So the isoquant appears as:


Thus, the conditional factor demands are:

x∗1 =

0, if w1

w2≥ 1

ey − 1, if w1w2

≤ e−yw2w1

− 1, otherwise

; x∗2 =

y, if w1

w2≥ 1

0, if w1w2

≤ e−y

y − ln(w2w1

), otherwise

.

When p = w1 = w2, we have x∗1 = 0 and x∗

2 = y, so π = p[ln 1+ y]−w2y = 0 ∀y. Thus the firm is indifferent

to the choice of y.

4. Note first that c∗(w, y) = yw1/21 w

1/22 . Now check the properties of a cost function:

Homogeneity:

c∗(λw, y) = y(λw1)1/2(λw2)1/2 = yλw1/21 w

1/22 = λc∗(w, y) �.

Monotonicity:

∂c∗

∂wi=

12yw

−1/2i w

1/2j > 0;

∂c∗

∂y= w

1/21 w

1/22 > 0 �.

Concavity:

H =y

4w1/21 w

1/22

[−w2w1

11 −w1

w2

].

To check for negative semidefiniteness, note that the diagonal elements are ≤ 0 and the determinant is

|H | = y2

16w1w2(1− 1) = 0. So, the function is concave.

Nonnegativity: c∗(w, 0) = 0 and c∗(w, y) > 0 for y > 0 are obvious.

Continuity: c∗ involves only products and nonnegative exponents, so it is continuous.

Since c∗ is a cost function, we can find the conditional factor demands from Shephard’s Lemma:

x∗i =

∂c∗

∂wi=

12yw

−1/2i w

1/2j .

5. We are given a tax t on x1. So, our cost function is c∗((1 + t)w1, w2, y). Using the envelope theorem,

∂c∗∂t

= ∂L∂t

|x∗ = w1x∗1. Taking the second derivative, we get ∂2c∗

∂t2= w1

∂x∗1

∂t= w1

∂x∗1((1+t)w1,w2,y)

∂w1· w1 =

w21 · ∂x

∗1

∂w1≤ 0, so the cost function is concave in t.



1.no free lunch possibility of inaction free disposal convexity returns to scale

a) y y y n no particular RTSb) y y∗ y n IRTSc) y y y n non-IRTS

*This answer presumes that 00 = 0. Otherwise, technology b does not satisfy possibility of inaction.

a.

b.


2.

3. The isoquant is y = ex2 − 1 + x1. So MRTS is 0 = ex2 dx2dx1

+ 1 or dx2dx1

= −e−x2 . Thus d2x2dx2

1= e−x2 dx2

dx1=

−e−2x2 < 0. That is, the isoquant is concave. So, cost minimizing choices are at the corners. Graph is:

The slope of the line through the two intercepts is − ln(y+1)y . So,

x∗1 =

{0, if w1

w2>

ln(y+1)y

y, if w1w2

< ln(y+1)y

; x∗2 =

{ln(y + 1), if w1

w2>

ln(y+1)y

0, if w1w2

< ln(y+1)y

If w1w2

= ln(y+1)y , then both corners are cost-minimizing. Note that for any y > 0, ln(y + 1) < y :


So for y > 0 we have x∗1 = 0 and x∗

2 = ln(y+1) when w1w2

= 1. Thus π = py− c∗(y) = py−w2ln(y+1) =

p[y − ln(y + 1)] for y > 0 and p = w1 = w2. Since the difference y − ln(y + 1) → ∞ as y → ∞, we get

y∗ = ∞.

4. First, check the properties of a profit function:

Homogeneity:

π∗(λp, λw) =(λp)2

4λw= λ

p2

4w= λπ∗(p, w) �

Monotonicity:

∂π∗

∂p=

2p4w

≥ 0 �;∂π∗

∂w= − p2

4w2≤ 0 �

Convexity:

H =[ 1

2w− p

2w2

− p2w2

p2

2w3

]The naturally-ordered principal minors are 1

2w > 0 and |H | = p2

4w4 − p2

4w4 = 0 ≥ 0, so H is positive

semidefinite, implying that π∗ is convex.

Nonnegativity: π∗(0, w) = 0 and π∗(p, w) ≥ 0 are obvious.

Since π∗ is a profit function, from Hotelling’s Lemma we get y∗ = ∂π∗∂p = p

2w and x∗ = −∂π∗∂w = p2

4w2 .

5. To find y∗,miny(ay − 1)2 + 1. Taking the derivative with respect to y, we get y∗ = 1a. The Envelope

Theorem says that

∂ac(y∗(a), a)∂a

=∂ac(y, a)

∂a

∣∣∣∣y=y∗

.


So ∂ac(y∗(a),a)∂a

= 2(ay − 1)y|y=y∗ = 0. This shows that ac is constant in a when y is chosen to minimize

ac. So, c(y∗, a) = y∗ac(y∗, a) = 1ak, where k is the constant level of ac(y∗, a). To find k, evaluate

ac(y∗, a) = (a 1a − 1)2 + 1 = 1. So, k = 1 and c(y∗, a) = 1

a .

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