s. farshad fatemi microeconomics i

Microeconomics IChapter Three

S. Farshad Fatemi

Introduction

Preference andUtility

The UtilityMaximizationProblem

The ExpenditureMinimizationProblem

Relationshipsbetween Demand,Indirect Utility, andExpenditureFunctions

Integrability

Welfare Evaluationof EconomicChanges

The Strong Axiomof RevealedPreferences

Microeconomics I44715 (1396-97 1st Term) - Group 1

Chapter ThreeClassical Demand Theory

Dr. S. Farshad Fatemi

Graduate School of Management and EconomicsSharif University of Technology

Fall 2017

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Introduction

Definitions (Rationality, desirability and convexity): The preferencerelation % on X is

Rational: if it is complete and transitive.

Monotone: if x , y ∈ X and y � x then y � x .

Strongly Monotone: if x , y ∈ X and y ≥ x and y 6= x theny � x .

Locally non-satiated:∀x ∈ X and ∀ε > 0, ∃y ∈ X such that ||y − x || ≤ ε and y � x .

It can be shown that strong monotonicity is a strongerassumption than monotonicity; and monotonicity is also astronger assumption than local non-satiation.

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Introduction

Convex: if ∀x ∈ X , the upper contour set {y ∈ X : y % x} isconvex.or ∀x ∈ X , if y % x and z % x , then αy + (1− α)z % x for∀α ∈ [0, 1].

Strictly Convex:if ∀x ∈ X , y % x , z % x , and y 6= z then αy + (1− α)z � x for∀α ∈ [0, 1].

Convexity can be interpreted in terms of diminishing marginalrates of substitution.

Can we deduce the consumer entire preference relation from a singleindifference set?

Sometimes.

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Introduction

Definition (homotheticity): A monotone preference relation % onX = RL

+ is homothetic if all indifference sets are related by propor-tional expansion along rays: this is if x ∼ y then αx ∼ αy for ∀α ≥ 0.

Definition (quasilinearity): A preference relation %on X = (−∞,∞)×RL−1

+ is quasilinear with respect to commodity 1 (numeraire commod-ity) if:

i) All the indifference sets are parallel displacements of each otheralong the axis of commodity 1; this is if x ∼ y thenx + αe1 ∼ y + αe1 for e1 = (1, 0, ..., 0) and ∀α ∈ R .

ii) Commodity 1 is desirable; x + αe1 � x for ∀x and ∀α > 0.

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Preference and Utility

For analytical purposes it is useful to interpret consumers’preferences by a utility function.

However, our assumptions so far do not guarantee a rationalpreference relation is not necessarily representable with a utilityfunction.

The assumption that is needed to ensure the existence of autility function for a rational preference relation is continuity.

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Definition (continuity): The preference relation % on X is continu-ous if it is preserved under limits; this is

for any sequence of pairs {(xn, yn)}∞n=1

with xn % yn for all n, x = limn→∞

xn

and y = limn→∞

yn,

we have x % y .

OR

A preference relation % on X is continuous if whenever a � b, thereare neighborhoods Ba and Bb around a and b, respectively, such thatfor all x ∈ Ba and y ∈ Bb, x � y .

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Example for non-continuity: lexicographic preferences.

Proposition (MWG 3.C.1): Suppose that the rational preferencesrelation % on X is continuous, then there is a continuous utility func-tion u(x) that represents % .

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Note:

The utility function u(.) that represents a preference relation isnot unique.

Any strictly increasing transformation of u(.) is also a utilityfunction.

Not all utility representation of a preference relation iscontinuous (one can consider a strictly increasingtransformation which is not continuous).

For analytical purposes, usually we consider the utility functionsto be differentiable (not always e.g. Leontief preferences).

We usually assume that the utility functions to be twicecontinuously differentiable, but we do not formally discuss thenecessary assumptions on preferences that implies this property.

How do restrictions on preferences imply restrictions on Utilityfunctions?

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Monotonicity of preference relation ⇒ Utility function isincreasing.

y � x ⇒ u(y) > u(x)

Convexity of preference relation ⇒ Utility function isquasiconcave.

u(αx + (1− α)y) ≥ Min(u(x), u(y)) ∀x , y & α ∈ [0, 1]

Strict Convexity of preference relation ⇒ Utility function isstrictly quasiconcave.

u(αx+(1−α)y) > Min(u(x), u(y)) ∀x , y & x 6= y & α ∈ (0, 1)

Note: Quasiconcavity is a weaker property than concavity

u(αx + (1− α)y) ≥ αu(x) + (1− α)u(y)

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Homotheticity of a continuous preference relation ⇔ Utilityfunction is homogenous of degree one

y(αx) = αu(x) ∀α > 0

Quasilinearty of a preference relation with respect to x1 ⇔Utility function is quasilinear function

u(x) = x1 + f (x2, ...xL)

3D demonstrations of utility function:

http://www2.hawaii.edu/∼fuleky/anatomy/anatomy.html

http://www2.hawaii.edu/∼fuleky/anatomy/anatomy2.html

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The Utility Maximization Problem

The consumer problem can be written as:

maxx≥0

u(x)

s.t p.x ≤ w

where p � 0 and w > 0

Proposition (MWG 3.D.1): If p � 0 and u(.) is continuous, thenthe utility maximization problem has a solution.

Proof: A continuous function always has a maximum value on anycompact set.

The Walrasian Demand Correspondence/FunctionThe rule that assigns the set of optimal consumption vectors in theutility maximization problem to the price-wealth pairs.

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Proposition (MWG 3.D.2): Suppose that u(.) is a continuous utilityfunction representing a locally nonsatiated preference relation % de-fined on the consumption set X = RL

+. Then the Walrasian demandcorrespondence x(p,w) possesses the following properties

i) Homogeneity of degree zero in (p,w)

x(αp, αw) = x(p,w) for ∀p,w & α > 0

ii) Walras’ lawp.x = w ∀x ∈ x(p,w)

iii) Convexity/UniquenessIf % is convex (strictly convex) then x(p,w) is a convex set(singleton)

Proof:i) The feasible set remains the same.ii) Simple; using the fact that % is nonsatiated.iii) Can be proved using the fact that u(.) is quasiconcave (strictly

quasiconcave) and any linear combination of two optimalbundles is feasible. 12 / 43


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If u(.) is continuously differentiable and x∗ is a solution to the utilitymaximization problem, then the Kuhn-Tucker conditions say:

∂u(x∗)

∂xl≤ λpl if x∗l = 0 and

∂u(x∗)

∂xl= λpl if x∗l > 0

or

x∗l (∂u(x∗)

∂xl− λpl) = 0

in matrix notation

x∗.(5u(x∗)− λp) = 0

where λ ≥ 0 is the Lagrange multiplier.

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Then for an interior solution x∗ � 0 we must have 5u(x∗) = λpAnd for any two goods l and k :

∂u(x∗)

∂xl− λpl =

∂u(x∗)

∂xk− λpk = 0

So

∂u(x∗)∂xl

∂u(x∗)∂xk

=plpk

MRSlk(x∗) =

∂u(x∗)∂xl

∂u(x∗)∂xk

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In contrast for a boundary solution x∗ ≥ 0 and x∗l = 0 for some l wemight have:

MRSlk(x∗) 6= plpk

λ is the marginal (shadow) value of relaxing the budget constraint.

Example: The demand function for a Cobb-Douglas utility function.

It can be shown that if preferences are continuous, strictly convex,and locally nonsatiated then the Walrasian demand function is alwayscontinuous at all (p,w) � 0. Furthermore, if the determinant ofthe bordered Hessian of u(.) is nonzero at x∗, the Walrasian demandfunction is differentiable.

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The Utility Maximization ProblemThe Indirect Utility Function

The Indirect Utility FunctionThe utility value of the utility maximization problem.

Proposition (MWG 3.D.3): Suppose that u(.) is a continuous util-ity function representing a locally non-satiated preference relation %defined on the consumption set X = RL

+. Then the indirect utilityfunction v(p,w) possesses the following properties:

i. Homogeneity of degree zero in (p,w)

v(αp, αw) = v(p,w) ∀p,w & α > 0

ii. Strictly increasing in w and non-increasing in pl

p.x = w ∀x ∈ x(p,w)

iii. Quasiconvex

{(p,w) : v(p,w) ≤ v} is a convex set ∀v

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The Expenditure Minimization Problem

The consumer’s problem can also be written as (which is the dualproblem to the utility maximization problem):

minx≥0

p.x

s.t u(x) ≥ u

where p � 0 and u > u(0).

The EMP reverses the roles of objective function and constraint.

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Proposition (MWG 3.E.1): Suppose that u(.) is a continuous util-ity function representing a locally non-satiated preference relation %defined on X = RL

+ and that the price vector is p � 0. We have:

i) If x∗ is optimal in the UMP when wealth is w > 0, then x∗ isoptimal in the EMP when the required utility level is u(x∗).Moreover, the minimized expenditure level in this EMP isexactly w .

ii) If x∗ is optimal in the EMP when the required utility level isu > u(0), then x∗ is optimal in the UMP when wealth is p.x∗.Moreover, the maximized utility level in this UMP is exactly u.Students need to go through the proof in MWG

The Expenditure FunctionGiven the prices and the required utility, the value of the EMP isdenoted e(p, u) which is called the expenditure function. Its value isp.x∗ where x∗ is any solution to the EMP.

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+. Then the expenditure function e(p, u) is

i. Homogeneous of degree one in p

e(αp, u) = αe(p, u) ∀p, u & α > 0

ii. Strictly increasing in u and nondecreasing in pl ; ∀liii. Concave in p

iv. Continuous in p and u

Proof:

i. The constraint set remains unchanged and the minimizationproblem is equivalent to the original one.

ii. Use the continuity of u(.) for the first part and the optimizationcondition for the second part.

iii. Can be proved using the definition of expenditure function.

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Note: For any p � 0, w > 0 and u > u(0)

e(p, v(p,w)) = w and v(p, e(p, u)) = u

are inverse to one another.

The Hicksian (Compensated) Demand FunctionThe set of optimal commodity vectors in the EMP is denoted byh(p, u) ⊂ RL

+.

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+. Then for any p � 0, the Hicksian demandh(p, u) is

i. Homogeneous of degree zero in p

h(αp, u) = h(p, u) ∀p, u and α > 0

ii. No excess Utility

∀x ∈ h(p, u) : u(x) = u

iii. Convexity/UniquenessIf % is convex (strictly convex) then h(p, u) is a convex set(singleton)

Note: the Walrasian and Hicksian demand functions are equivalentas:

h(p, u) = x(p, e(p, u)) and x(p,w) = h(p, v(p,w))

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The Hicksian demand function is also called compensated demandfunction since it can show the level of wealth compensation requiredto keep the consumer at the same level of utility after a change inprices.

The Hicksian Demand and the Compensated Law of Demand

Proposition (MWG 3.E.4): Suppose that u(.) is a continuous utilityfunction representing a locally nonsatiated preference relation % andthat h(p, u) consists of a single element for all p � 0. Then h(p, u)satisfies the compensated law of demand:

∀p′, p′′; (p′′ − p′).[h(p′′, u)− h(p′, u)] ≤ 0

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Relationships between Demand, Indirect Utility,and Expenditure Functions

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Proposition (MWG 3.G.1): (Shephard’s lemma) Suppose that u(.)is a continuous utility function representing a locally non-satiated andstrictly convex preference relation % defined on X = RL

+. Then

∀p, u ∴ h(p, u) = 5pe(p, u)

Or

∀p, u ∴ hl(p, u) =∂e(p, u)

∂pl∀l = 1, .., L

The Economic interpretation of this proposition can be seen if we lookmore carefully at the following:

∂e(p, u)

∂pl=

∂

∂pl

∑k

pkhk(p, u) = hl(p, u) +∑k

pk∂hk(p, u)

∂pl

∂e(p, u) = hl(p, u)∂pl +∑k

pk∂hk(p, u)

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Walrasian Demand Function Proposition (MWG 3.G.2): Supposethat u(.) is a continuous utility function representing a locally non-satiated and strictly convex preference relation % defined on X = RL

+.Suppose also that h(., u) is continuously differentiable at (p, u). Then

i. Dph(p, u) = D2pe(p, u)

ii. Dph(p, u) is a negative semidefinite matrix.

iii. Dph(p, u) is a symmetric matrix.

iv. Dph(p, u)p = 0

Note:(ii) implies that compensated own price effects are nonpositive.(iii) implies that for compensated price cross derivatives we must have:

∂hk(p, u)

∂pl=∂hl(p, u)

∂pk

We also define two goods as substitutes and complements based on

the sign of ∂hk (p,u)∂pl

. And (iv) implies that every good has at least onesubstitute.

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Proposition (MWG 3.G.3): Suppose that u(.) is a continuous util-ity function representing a locally non-satiated and strictly convexpreference relation % defined on X = RL

+. Then

∀p,w , u = v(p,w) ∴ Dph(p, u) = Dpx(p,w)+Dwx(p,w)x(p,w)T

Or

∀p,w , u = v(p,w) ∴∂hl(p, u)

∂pk=∂xl(p,w)

∂pk+∂xl(p,w)

∂wxk(p,w) ∀l , k

Note:The Slutsky equation describes the relationship between the slope ofthe ordinary and compensated demand functions.

If the good is normal: ∂hl (p,u)∂pl

< ∂xl (p,w)∂pl

and

If the good is inferior: ∂hl (p,u)∂pl

> ∂xl (p,w)∂pl

.

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Proposition (MWG 3.G.4): (Roy’s Identity) Suppose that u(.) isa continuous utility function representing a locally non-satiated andstrictly convex preference relation % defined on X = RL

+. Supposealso that v(.) is differentiable at (p, w)� 0. Then

x(p, w) = − 1

5wv(p, w)5p v(p, w)

Or

xl(p, w) = −∂v(p,w)

∂pl∂v(p,w)

∂w

∀l = 1, ..., L

Note:Roy’s Identity and Shepard’s lemma are parallel results for UMP andEMP.

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If we observe a demand function which

is homogeneous of degree zero,

satisfies the Walras’ law, and

has a substitution matrix that is symmetric and negativesemidefinite

can we find preferences that rationalize that?

The answer to this question is important because:

1) Whether these conditions are sufficient conditions as well?

2) Completes the comparison of preference-based and choice-basedtheories of demand.

3) When can we recover the customer preferences from consumerbehavior observations for welfare analysis?

4) Simple form demand functions may not correspondent to simpleform preferences.

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Integrability

The problem of integrability can be divided to two questions:

1. Recovering an expenditure function from the demand function.

2. Recovering the preferences from the expenditure function.

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Welfare Evaluation of Economic Changes

Definition: Money metric indirect utility function:

e(p, v(p1,w))− e(p, v(p0,w)) ∀pIn particular if p = p0:

Equivalent variation:

EV = e(p0, v(p1,w))− e(p0, v(p0,w))

= e(p0, u1)− e(p0, u0)

And if p = p1:

Compensating variation:

CV = e(p1, v(p1,w))− e(p1, v(p0,w))

= e(p1, u1)− e(p1, u0)

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The representation of equivalent variation in terms of Hicksian de-mand (when only p1 changes):

EV = e(p0, u1)− e(p0, u0)

= e(p0, u1)− w

= e(p0, u1)− e(p1, u1)

EV (p0, p1,w) =

∫ p01

p11

h1(p1, p−1, u1)dp1

The representation of compensating variation in terms of Hicksiandemand (when only p1 changes):

CV (p0, p1,w) =

∫ p01

p11

h1(p1, p−1, u0)dp1

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Example:u(x1, x2) = 0.6ln(x1) + 0.4ln(x2)

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x1(p1, p2,w) = 0.6w

p1

h1(p1, p2, u) = u − 0.4ln(2p1

3p2)

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In this case:

p01 = p0

2 = 1 & w = 14.484 ⇒ x01 = 8.690 & x0

2 = 5.793 & u0 = 2

p11 = 2, p1

2 = 1 & w = 14.484 ⇒ x11 = 4.345 & x1

2 = 5.793 & u1 = 1.584

e(p0, u1) = 9.556 & e(p1, u0) = 21.953

EV (p0, p1,w) = 9.556− 14.484 = −4.928

CV (p0, p1,w) = 14.484− 21.953 = −7.469

(recall that x1 is normal): 0 > EV (p0, p1,w) > CV (p0, p1,w)

Example: The deadweight loss from commodity taxation

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Welfare Analysis with Partial Information So far, we learnt howit is possible to calculate the welfare effect of a price change when weknow the consumer’s expenditure function. The latter is not alwaysthe case.Firstly, a test is introduced to evaluate whether a price change im-proves the welfare or not.

Proposition (MWG 3.I.1): Suppose that the consumer has a locallynonsatiated rational preference relation %. If (p1 − p0).x0 < 0 , thenthe consumer is strictly better off under price-wealth situation (p1,w)than under (p0,w).

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Proof: (p1 − p0).x0 < 0 ⇒ p1.x0 < p0.x0 = w which means x0 isaffordable under (p1,w) and is an interior point in the budget con-straint. Then since preferences are nonsatiated then there should bea better option available under (p1,w).

Proposition (MWG 3.I.2): Suppose that the consumer has a dif-ferentiable expenditure function. Then if (p1 − p0).x0 > 0, thereis a sufficiently small α ∈ (0, 1) such that for all α < α we havee((1 − α)p0 + αp1, u0) > w , and so the consumer is strictly betterunder price-wealth situation (p0,w) than under ((1−α)p0 +αp1,w).

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Approximation of the Welfare Effect using the Walrasian De-mand CurveSince Hicksian demand is not directly observable, the Area Variation(AV) has been used extensively in the literature:

AV (p0, p1,w) =

∫ p01

p11

x1(p1, p−1,w)dp1

It can be seen that for a normal good:

EV (p0, p1,w) > AV (p0, p1,w) > CV (p0, p1,w)

And for an inferior good:

EV (p0, p1,w) < AV (p0, p1,w) < CV (p0, p1,w)

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When (p1 − p0) is small, a better approximation is:∫ p01

p11

h1(p1, p−1, u0)dp1

where h1 is a Taylor approximation of h1:

h(p, u0) = h(p, u0) + Dph(p, u0)(p − p0)

This measure is also computable from the knowledge of the observableWalrasian demand function:

h(p, u0) = x(p,w) + S(p,w)(p − p0)

And since only the price of good 1 is changing:

h1(p1, p−1, u0) = x1(p0

1 , p−1,w) + s11(p01 , p−1,w)(p1 − p0

1)

where

s11(p01 , p−1,w) =

∂x1(p0,w)

∂p1+∂x1(p0,w)

∂wx1(p0,w)

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The Strong Axiom of Revealed Preferences

It represents the conditions on consumer choice in the same fash-ion as WARP such that consumer behaviour can be rationalized bypreferences.

Definition (MWG 3.J.1): The market demand function x(p,w)Satisfies SARP if for any list

(p1,w1), ..., (pN ,wN)

with x(pn−1,wn−1) 6= x(pn,wn) for all n ≤ N − 1,

we have:

pN .x(p1,w1) > wN whenever pn.x(pn+1,wn+1) ≤ wn for ∀n ≤ N−1.

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The Strong Axiom of Revealed Preferences

Proposition (MWG 3.J.1): If the market demand function x(p,w)Satisfies SARP then there is a rational preference relation % thatrationalizes x(p,w), that is, such that for all (p,w), x(p,w) � y forevery y 6= x(p,w) with y ∈ Bp,w .

Note: For L = 2 the SARP and the WARP are equivalent. But forL > 2, the SARP is stronger than the WARP.

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