Chapter 4: Production Theory

Need to complete: Proposition 48, Proposition 52, Problem 4, Problem 5, Problem 6, Problem 7,Problem 10.

In this chapter we study production theory in a commodity space. First, we develop the mathe-matical representaiton of produciton units, which we refer simply as firms. Second, we characterizethe “behavior” of firms whose objective is the maximizaiton of profits.

4.1 Representation of firms

We study an L-commodity world. Here, a firm is an economic entity that transforms commodi-ties. We present a firm by means of the collection of bundles in the commodity space that are“technically feasible” for it.

Example. (13). Assume that L = {oranges, orange juice}. We denote the generic amount ofaranges, in pounds, by z1 and the generic amount of arange juice, in liters, by y1. A given firm isable to produce from each pound of oranges, half a liter of juice, and cannot produce any orangesfrom orange juice. Moreover, the firm can dispose of any excess of orange juice. The mathematicalrepresentation of this firm is the set

Y =

⇢

(�z1,y1) 2 R2 : z1 � 0,y1 12

z1

�

Definition. (97). A production set is a subset of RL. The generic production set is Y .

We refer to the elements of Y as feasible vectors (for Y ). For each y2Y , and each l 2 {1, . . . ,L},commodity l is an input at y whenever yl < 0; symmetrically, it is an output whenever yl > 0.

The following two examples will guid our study of production sets.

Example. (14). (Transformation Function). We define a production set by means of a functionthat tells us which vectors are feasible. No restriction is imposed on whether a commodity is aninput or an output.

Let F : RL! R be a function. The production set associated with F is

Y F ⌘�

y 2 RL : F(y) 0

1

We refer to F as a transformation function and to the set

∂F ⌘�

y 2 RL : F(y) = 0

as the transformation frontier for F .

We define the Marginal Rate of Transformation (MRT) of commodity l for commodity k at

y 2 ∂F as the ratio:

MRTlk(y)⌘∂F∂yl

(y)∂F∂yk

(y)=�Dyk

Dyl

With l on the x-axis and k on the y-axis, the MRT is a measure of how much the net output of goodk can decrease if the firm increases the net output of good l by one marginal unit. The MRT ofcommodity l for commodity k at y is the absolute value of the slope of the transformation frontieron the l� k “slide through y” of the consumption space.

The next example focuses on the single-output model.

Example. (15). (Single-output model). We divide the commodity space into inputs and outputs.We define a production set by means of a funciton that tells us the amount of output that can beproduced with a given input vector. For simplicity, we restrict to the single-output case.

Let f : RL�1+ ! R+ be a function. The production set associated with f is

Y f ⌘n

(�z,q) 2 RL : z⌘ (z1, . . . ,zL�1) 2 RL�1+ and q� f (z1, . . . ,zL�1) = 0

o

We refer to f as a production function and to the set

∂ f ⌘n

(�z,q) 2 RL : z 2 RL�1+ and q� f (z1, . . . ,zL�1) = 0

o

as the production frontier.

We define the Marginal Rate of Technical Substitution (MRTS) of input l for input k at

z 2 RL�1+ as the ratio:

MRT Slk(z)⌘∂ f∂zl(z)

∂ f∂zk

(z)=�Dzk

Dzl

The single-output model admits an alternative representation of the production set by means of iso-quants. With l on the x-axis and k on the y-axis„ the MRTS measures the amount of input k thatmust be reduced to keep output level q = f (z) when the amount of input l in increased marginally.The MRTS of input l for input k at z is the absolute value of the slope of the iso-quant at z.

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4.2 Properties of production sets

Let Y ✓ RL be a production set. Then Y is (or satisfies):

a. Non-empty iff Y 6= /0.

b. Closed iff Y is closed in RL.

c. No-free-lunch iff 8 y 2 Y : y� 0 =) y = 0. Alternatively, Y \RL+ ✓ {0}.

d. Possibility of inaction iff 0 2 Y .

e. Free-disposal iff 8 y 2 Y, 8 x y : x 2 Y .

f. Irreversible iff [y 2 Y ^ y 6= 0] =) �y /2 Y .

g. Non-increasing returns to scale (NI) iff 8 y 2 Y, 8 a 2 [0,1] : ay 2 Y .

h. Non-decreasing returns to scale (ND) iff 8 y 2 Y, 8 a 2 [1,+•) : ay 2 Y .

i. Constant returns to scale (CR) iff Y satisfies both NI and ND returns to scale.

j. Additivity iff 8 {y,y0}✓ Y : y+ y0 2 Y .

k. Convexity iff Y is convex.

Example. Let Y ✓ R2�⇥R (single-output model). Let the production be defined as

f (z1,z2) = za1 zb

2

If a+b < 1, then Y satisfies NI returns to scale. If a+b > 1, then Y satisfies ND returns to scale.If a+ b = 1, then Y satisfies CR to scale. Important to note, if Y is convex and 0 2 Y , then Ysatisfies NI returns to scale.

Proposition. (48). Let Y ✓ RL.

a. Suppose that L = 2 and Y ✓ R�⇥R (single-output model). If Y satisfies free-disposal andCR of scale, then Y is convex.

b. Suppose that L = 3 and Y ✓ R2�⇥R (single-output model). Then, free-disposal and CR of

scale may not imply that Y is convex.

Proof. Will prove.

Proposition. (49). Let Y ✓ RL. If Y is additive and satisfies NI returns to scale, then Y is a convexcone, i.e., for each pair {y,y0}✓ Y and each pair {a,b} 2 R+, ay+by0 2 Y .

Proof. Let {y,y0} ✓ Y and {a,b} 2 R+. If a 2 [0,1], then NI returns to scale imply that ay 2 Y .If a � 1, let n 2 N be such that a

n < 1. Then, by NI returns to scale, an y 2 Y . By additivity, ay =

an y+ · · ·+ a

n y 2 Y . The symmetric argument shows that by0 2 Y . By additivy, ay+by0 2 Y .

Corollary. (11). Additivity and NI returns to scale imply CR returns to scale and convexity.

Proof. This is a direct consequence of the proof and result of Proposition 49.

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4.3 Profit maximization

In this seciton we characterize the behavior of a price-taker profit maximizing firm. Consider afirm represented by production set Y . Assume that prices are p 2 RL

+. The profit of firm Y whenproducing at a feasible vector y 2 Y is

p · y⌘Âl2L

plyl

A price-taker profit maximizing firm that observed prices p solves the Profit Maximization Prob-

lem

maxy2Y

p · y

The solution of this problem and the maximum profit attained by the firm define the firm’s supplycorrespondence and profit function.

Supply correspondence

Definition. (98). Let Y be a production set. The supply correspondence associated with Y isdefined by

yY : RL+ ! RL

p 7! yY (p)⌘ argmaxy2Y

p · y

Profit function

Definition. (99). Let Y be a production set. The profit function associated with Y is defined by:

PY : RL+ ! R

p 7! PY (p)⌘maxy2Y

p · y

Alternatively, PY (p)⌘ p · y⇤ for some y⇤ 2 yY (p)

4.3.1 Profit maximizaiton: Transformation Function model

If production set Y is described by means of a transformation function F , the firm’s profit maxi-mizaiton problem becomes

max{y2RL:F(y)0}

p · y

We simplify notation and denote yF and PF the supply correspondence and profit function associ-ated with Y F .

Proposition. (50). Let F be a differentiable transformation function. For each p 2 RL++ and each

pair {l,k}✓ {1, . . . ,L}pl

pk= MRTlk(yF(p))

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Proof. Let F be a differentaible transformation function. Take p 2 RL++ and {l,k} ✓ {1, . . . ,L}.

The Lagrangian to the profit maximization problem is

L = p · y�l(F(y))

The first-order conditions are given by

∂L∂yl

= 0 : pl�l∂F(y)∂yl

= 0

∂L∂yk

= 0 : pk�l∂F(y)∂yk

= 0

Thus, we have

pl

pk=

∂F(y)∂yl

∂F(y)∂yk

= MRTlk(yF(p))

4.3.2 Profit maximization: Production Function model

In the production function single-output model we denote input prices by w⌘ (w1, . . . ,wL�1) 2RL�1+ and output price by p 2 R+.

If the production set Y is described by means of a production funciton f , the firm’s profit maxi-mization problem becomes

maxz2RL�1

+

p f (z)�w · z

We simplify notation and denote y f and P f the supply correspondence and profit functions asso-ciated with Y f .

Proposition. (51). Let f be a differentiable production function and Y its associated productionset. For each w 2 RL�1

++ , each p 2 R++, and each pair {l,k}✓ {1, . . . ,L},

wl

wk= MRT Slk(y f (p))

Proof. Let f be a differentiable production function and Y its associated production set. Takew 2 RL�1

++ , p 2 R++, and {l,k}✓ {1, . . . ,L}. The Lagrangian of the profit maximization problemis

L = p f (z)�w · z+lz

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The first-order conditions are

∂L∂zl

= pl∂ f∂zl

(z)�wl 0 with equality if zl > 0

∂L∂zk

= pk∂ f∂zk

(z)�wk 0 with equality if zk > 0

When z⇤ is an interior solution, we have

wl

wk=

∂ f∂zl(z)

∂ f∂zk

(z)= MRT Slk(y f (p))

4.3.3. Profit maximization: general model

The following are properties of a supply correspondence and a profit function.

Proposition. (52). Assume that Y is non-empty, closed, and satisfies free disposal. Then:

a. PY is homogeneous of degree one.

b. PY is convex.

c. If Y is convex, then it can be recovered from PY as follows:

Y ⌘�

y 2 RL : 8 p 2 RL+, p · yPY (p)

d. yY is homogeneous of degree zero.

e. If Y is convex, then yY is convex-valued. If Y is strictly convex, i.e., for each pair {x,y}✓ Yand each a 2 (0,1), ax+(1�a)y belongs to the interior of Y , then yY is single-valued.

f. (Hotteling’s Lemma) Let p 2 RL+. If

�

�yY (p)�

� = 1, then P is differentiable for each l 2{1, . . . ,L},

∂PY

∂pl(p) = yY

l (p)

g. If yY is a differentiable function in a neighborhood of p 2 RL+, then

1. DyY (p)p = 0

2. DyY (p) is symmetric and positive semi-definite.

h. (Law of Supply) For each pair {p, p0}✓ RL+,

(p0 � p) ·�

yY (p0)� yY (p)�

� 0

Proof. Will prove. Check Mas-Colell Proposition 5.C.1

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4.4 Problems

Problem 1. Suppose that f is the production functions associated with a single-output technology,and let Y be the associated production set. Prove that Y satisfies Constant Returns to Scale if andonly if f is homogeneous of degree one. Conclude that in the single-input, single-output case CRimplies that f is linear.

Proof. Let f : RL�1! R+ be the production function associated with a single-output technology,and let Y be the associated

Y ⌘�

(�z,q) 2 RL : z 2 RL�1 and q� f (z) 0

We prove that

8 y 2 Y, 8 z 2 RL�1, 8 a 2 R+ : ay 2 Y () a f (z) = f (az)

()) Assume 8 y 2 Y, 8 a 2 R+ : ay 2 Y . Take a 2 R+, z 2 RL�1. We want to show thata f (z) = f (az). For this z, let f (z) = q 2 R+. By definition of the production frontier, (�z,q) 2∂ f ⌘

�

(�z,q) 2 RL : z 2 RL�1 and q� f (z) = 0

✓Y . Since Y satisfies CRS, then (�az,aq)2Y .So, aq f (az), and thus, q q

a . Let f (az) = q 2 R+. By definition, (�az, q) 2 Y . Since Y

satisfies CRS, then⇣

�z, qa

⌘

2Y . So, qa f (z) = q. Now, q q

a and q� qa imply that q = q

a . Thus,

a f (z) = f (az). Therefore, 8 a 2 R+, 8 z 2 RL�1 : a f (z) = f (az).

(() Assume 8 a 2 R+, 8 z 2 RL�1 : a f (z) = f (az). Take y 2 Y and a 2 R+. We want toshow that ay 2 Y . Now, y 2 Y implies that y = (�z,q) 2 RL such that z 2 RL�1 and q� f (z) 0.So, q f (z), and multiplying both sides by a yields aq a f (z). Since f (z) is homogeneousof degree one, then aq f (az). By definition, (�az,aq) 2 Y , which implies that a(�z,q) 2 Y .Thus, ay 2 Y . Therefore, 8 y 2 Y, 8 a 2 R+ : ay 2 Y .

Remark. Also, try to prove that for a single-output technology, Y is convex if and only if f (z) isconcave.

Problem 2. Consider the single-input case in which f is defined by: for each z 2 R+,

f (z)⌘ 12

⇣pz+1�1

⌘

a. Graph Y

b. Graphically show that Y is closed and satisfies Non-increasing returns to scale.

c. Graphically show that Y is not additive and does not satisfy Non-decreasing returns to scale.

d. Imagine now that it is possible to add any number of production input-output combinationsunder technology Y (there is free entry to this market). Denote this new production set byAY . Formally,

AY ⌘ {y1 + · · ·+ yn : n 2 N, 8 i = 1, . . . ,n, yi 2 Y}

Page 7 of 12

1. Characterize and graph AY . Is there a function fA such that AY ⌘ {(�z,q) : z� 0, q fA(z)}?

2. Is AY closed? Is it additive? Does it satisfy Non-Increasing Returns to scale? Does itsatisfy Non-Decreasing returns to scale.

Proof. Graphs will not be provided in the solution to this problem, only descriptions about theprocedures. Also, Part A will be skipped.

Part B.

The production set associated with the given production function is

Y f ⌘�

(�z,q) 2 R2 : z 2 R+ and q� f (z) 0

Since Y f contains its boundary points, then it is a closed set. Y f satisfies non-increasing returnsto scale iff 8 y 2 Y, 8 a 2 [0,1] : ay 2 Y . Take a 2 [0,1] and y 2 Y . By the construction of f (z),ay 2 Y .

Part C.

Let y =⇥

�3 12

⇤0 and y =⇥

�8 1⇤0. Then, y+ y =

⇥

�11 1.5⇤

/2 Y . Thus, Y is notadditive. Let a = 2. Then, ay /2 Y . Thus, Y does not satisfy ND returns to scale.

Part D, (i)

Suppose n firms enter the market. Then,

fA(z) =n2

✓

r

zn+1�1

◆

=

p

(z/n)+1�12/n

As n! •, then by L’Hoptial’s rule

limn!•

(�1/2)(1/p

(z/n)+1)(z/n2)

(�2/n2)= lim

n!•z

4p

(z/n)+1=

14

Since fA(z) = 14z is only the upper limit and it is not well defined, then there is no function fA such

that AY ⌘ {(�z,q) : z� 0, q fA(z)}. Important to note, the industry as a whole, i.e., AY , facesCR to scale due to free entry, but each firm i still faces the production technology f (z).

Part D, (ii)

Since fA(z) = 14z is not part of the set, then AY is not closed. Let {y, y}2AY such that y= (�z,q)

and y = (�z, q). Now, q+ q < (1/4)(z+ z). Thus, y+ y 2 AY , so AY is additive.

Problem 3. Consider a production set Y ✓RL. Let pY be the profit function associated with set Y ,i.e., for each p 2 RL

+,

pY (p)⌘maxy2Y

p · y

a. Define free disposal and no free lunch. Draw a production set that satisfies these two prop-erties. Illustrate the profit maximization for some price vector p.

Page 8 of 12

b. Is pY concanve, convex, or neither of them?

c. Suppose now that a firm with produciton set Y may face an uncertain market. The first is anexpected profit maximizer and is risk neutral. There are two markets in which the firm mayparticipate.

• Market 1: price is uncertain with some probability distribution.

• Market 2: price is certian; it is exactly the expected price in Market 1.

Prove that the firm always finds particpating in Market 1 as good as participating in Market2.

Proof. Consider a production set Y ✓ RL. Let pY be the profit function associated with set Y .

Part A.

Y satisfies no-free-lunch iff 8 y 2Y : y� 0 =) y = 0. Alternatively, Y \RL+ ✓ {0}. Y satisfies

free disposal iff 8 y 2 Y, 8 x y : x 2 Y .

Part B.

We claim that pY (p) is convex. Take {p, p} 2 RL+ and a 2 [0,1]. Let y ⌘ argmax

y2Yp·y and

y ⌘ argmaxy2Y

p · y. Consider ap+(1�a) p. pY (ap+(1�a)p) = (ap+(1�a)p) · y where y ⌘

argmaxy2Y

(ap+(1�a) p) ·y. Then, ap · y+(1�a)p · y apY (p)+(1�a)pY (p). Thus, 8 {p, p0}✓

RL+, 8 a 2 [0,1] : pY (ap+(1�a)p0) apY (p)+ (1�a)pY (p0). Therefore, pY (p) is a convex

function.

Part C.

Market 1:ˆ

pY (p)dF(p) for some distribution of random price. Market 2: pY✓ˆ

pdF(p)◆

.

Since pY (p) is a convex function, then by Jensen’s inequality, pY✓ˆ

pdF(p)◆

ˆ

pY (p)dF(p).

Problem 4. Let C be a convex and closed set of RL and z 2 RL a vector that is not in C, i.e.,z 2 Cc. Prove that ther is a closet vector in C to z, that is, there is x⇤ 2 C such that for eachx 2C, kz� x⇤k kz� xk. Hint: the function f : RL! R defined by x 2 RL 7! f (x)⌘ kz� xk2 iscontinuous.

Proof. Will prove.

Problem 5. Let C be a convex and closed set of RL and z2RL a vector that is not in C. Let x⇤ 2C.Prove that x⇤ is such that for each x 2C, kz� x⇤k kz� xk if and only if for each x 2C,

(z� x⇤) · (x� x⇤) 0

Moreover, there is a unique closet vector in C to z. Do it in two steps.

Page 9 of 12

a. Suppose that x⇤ 2C such that for each x 2C,

(z� x⇤) · (x� x⇤) 0

Prove that x⇤ is the unique closet element in C to z. Do it as follows: prove that the functionf defined above is strictly convex. Let x 2 RL be such that x 6= x⇤. Prove, using a Taylorexpansion of degree two, that f (x)> f (x⇤).

b. Assume that x⇤ is a closet element in C to z. Let x2C. Let j : R!R be the function definedby t 7! j(t) ⌘ kz� (x⇤+ t(x� x⇤))k2. Prove that j is differentiable and that j0(0) � 0.Conclude that (z� x⇤) · (x� x⇤) 0.

Proof. Will prove.

Problem 6. (Basic Separation Theorem) Suppose that C is a convex and closed set of RL andz 2 RL a vector that is not in C. Prove that there is a non-zero vector p 2 RL and a real number csuch that p · z > c and for each x 2C, p · x < c. Do it in three steps.

• Prove a graphical interpretation of the theorem when C is a production set and z is a non-feasible point at C.

• Let x⇤ 2C be the closet point in C to z. Let p⌘ z� x⇤. Prove that p · z > p · x⇤.

• Let c 2 R be such that p · x⇤ < c < p · z. Prove that p and c have the desired properties.

Proof. Will prove.

Problem 7. (Basic Supporting Hyperplane Theorem) Let C be a convex set of RL and z be aboundary point of C, i.e., z 2 C\Cc. (Here if A is a set, A denotes the closure of A, i.e., the setof all limit points of A. In other words, the set of points, a such that there is a sequence in Athat converges to a. For instance, the closure of (0,1) is [0,1].) Prove that there is a non-zerovector p 2 RL such that for each x 2C, p · x p · z. Provide a graphical interpretation when C is aproduction set and z is a boundary point of C.

Proof. Will prove.

Problem 8. Let Y ✓RL be a non-empty convex production set that satisfies free disposal. Let y beefficient at Y , i.e., there is no y0 2 Y such that y0 � y. Prove that there is p� 0 such that y is profitmaximizing at p.

Proof. Let Y ✓ RL be a non-empty convex production set that satisfies free disposal. Let y beefficient at Y , i.e., there is no y0 2 Y such that y0 � y We prove that there is p � 0 such that y isprofit maximizing at p.

First, we show that y is a boundary point of Y , i.e., y 2 Y \Y c. Suppose for contradictionthat y /2 Y \Y c. Then, y 2 Int(Y ), which by definition is an open set. So, 9 e > 0 such thatBe(y) ✓ Int(Y ). Let t 2 RL such that 8 l 2 {1, . . . ,L} : tl = 1. Define y0 = y+ (e/2)t. Then,

Page 10 of 12

y0 2 Be(y), and thus, y0 2 Y . However, y0 � y, which contradicts the fact that y is efficient in Y(! ). Hence, y 2 Y \Y c.

Now, by the Basic Supporting Hyperplane Theorem, 9 p 2RL, p 6= 0 such that 8 y0 2Y , p ·y0 p · y. Define y = y� evl for some l, where vl 2 RL that has 1 in the l-th component and zeros fork 6= l. Since y� y and Y satisfies free disposal, then y 2 Y . As a result, p · y� p · y, which impliesthat p · (y� y)� 0. So, we have p · vl � 0, and thus, pl � 0 for all l.

Therefore, there is p� 0 such that y is profit maximizing at p.

Problem 9. Let Y ✓ RL be a non-empty, convex, closed production set that satisfies free disposal.Prove that

Y =�

y 2 RL : 8 p� 0, p · y pY (p)

Is it always true that

Y =�

y 2 RL : 8 p� 0, p · y pY (p)

Would your answer change if Y also satisfies no-free-lunch?

Proof. Let Y ✓ RL be a non-empty, convex, closed production set that satisfies free disposal. Weprove that

Y =�

y 2 RL : 8 p� 0, p · y pY (p)

⌘ A

by showing Y ✓ A and A✓ Y .

(Y ✓ A) Take y 2 Y . We want to show that y 2 A. Take p � 0. Consider the profit functionassociated with Y evaluated at p:

pY (p) = maxy2Y

p · y

Define y⇤ as

y⇤ ⌘ argmaxy2Y

p · y

Important to note, pY (p)⌘ p · y, Since y 2 Y , then p · y p · y⇤. Thus, by definition, y 2 A.

(A✓Y ) Take y 2 A. Then, 8 p� 0 : p · y pY (p), where pY (p) is the profit function associatedwith Y

pY (p) = maxy2Y

p · y

We want to show that y 2Y . Suppose for contradiction y /2Y , which means that y 2Y c. Since Y isa closed and convex set of RL and y 2Y c ✓RL, then by the Basic Separation Theorem, there existsp 2 RL and c 2 R such that 8 y 2 Y : p · y < c < p · y. This implies that 8 y 2 Y : p · y < p · y.

We claim that p � 0. Suppose for contradiction that pl < 0 for some l 2 {1, . . . ,L}. Let y 2 Y .Define yn = y� nvl , where n 2 N and vl 2 RL that has 1 in the l-th component and zeros for

Page 11 of 12

k 6= l. Since 8 n 2 N : yn y and Y satisfies free disposal, then 8 n 2 N : yn 2 Y . So, we havep · yn = p · y�npl . Since pl < 0, then as n! •, p · yn! •. Hence, 9 n 2 N : p · yn > c. But thiscontradicts the fact that 8 y 2 Y : p · y < c (! ). Thus, p� 0.

Now, define y⇤ as

y⇤ ⌘ argmaxy2Y

p · y

Important to note, pY (p)⌘ p · y⇤. Since y⇤ 2 Y , then p · y⇤ < p · y, which is equivalent to pY (p)<p · y. However, we have just shown 9 p � 0 : pY (p) < p · y, which means that y /2 A (! ).Therefore, y 2 Y .

Is it always true that

Y = {y 2 RL : 8 p� 0, p · y pY (p)}⌘ A

No, the equality between the two sets does not hold. To explain, here is a counter example. Takep 2 R2

++ and z = [0,1]0. Define Y ⌘ {(y1,y2) 2 R2 : y2 0}. Note that z /2 Y . Since p1 > 0, thenpY (p) = +•. So, A⌘ {y 2 R2 : p� 0, p · y pY (p)}. Now, p · z = p2 <+•, which implies thatz 2 A. Thefore, Y 6= A. In general, Y ✓ A, but A * Y .

Would your answer change if Y also satisfies no free-lunch? No, my answer would not change.Consider Y ⌘ {(y1,y2) 2 R2 : (y1,y2) 2 R�⇥R}. Since p2 > 0, then pY (p) = +•. Also, p · z =p2 <+•. Thus, z 2 A, but z /2 Y .

Problem 10. (Separate problem attached with Micro Homework Assignment 12). Suppose thatthere are two inputs with prices (w1,w2) and one output with price p. Consider a Cobb-Douglasproduction function f : R2

+! R, f (z1,z2) = Aza1 zb

2.

a. Define Y f .

b. Characterize the combinations of a and b for which Y f satisfies NI, ND, and CR of scale.

c. Calculate the associated supply correspondence, profit function, conditional factor demandcorrespondence, and cost function.

d. Verify Hotteling’s Lemma for each (w1,w2, p) where it holds.

Proof. Will prove.

Page 12 of 12

Chapter 4: Production theory

Need to complete: Proposition 53, Proposition 54, Proposition 55, Problem 1

4.5 Cost minimization

In the context of the Production function model, we can unambiguously refer to inputs and hencekeep track of a firm’s costs. In this section we characterize a firm’s profit maximizaiton problemby means of its cost minimization problem. For simplicity, we will concentrate in the single-ouputcase.

Consider a firm represented by a production function f . Imagine that the firm decides to produceq units of output and input prices are w 2RL

+. The firm’s Cost Minimization Problem for q at w is:

min{z2RL

+: f (z)�q}w · z

Cost function

Definition. (100). Let f be a production function. The cost function associated with f is defined

by:

c

f : RL

+ ! R(w,q) 7! c

f (w,q)⌘ min{z2RL

+: f (z)�q}w · z

Conditional factor demand correspondence

Definition. (101). Let f be a production function. The conditional factor demand correspondence

associated with f is defined by:

z

f : RL

+ ! RL�1+

(w,q) 7! z

f (w,q)⌘ arg min{z2RL

+: f (z)�q}w · z

Proposition. (53). Let f be a production function, w2RL�1+ and q2R+. Assume that

��z

f (w,q)��=

1 and z

f (w,q)� 0. Then:

1

a. For each pair {l,k}✓ {1, . . . ,L�1}w

l

w

k

= MRT S

lk

(z f (w,q))

b. If l is the Lagrange Multiplier associated with the Cost Minimization Problem for the con-

straint q� f (z) 0, then

∂c

∂q

(w,q) = |l|

Proof. Will prove. Check Mas-Colell, Chapter 5 (Cost Minimization).

The following are properties of a cost function.

Proposition. (54). Let f be a production function. Then:

a. c

f

is homogeneous of degree one in w.

b. For each q 2 R+, c

f (·,q) is a concave function, i.e., c

f

is a concave function of w

c. If for each q > 0, {z 2 RL�1+ : f (z)� q} are convex sets, then

Y

f ⌘ {(�z,q) : z � 0, 8 w � 0, w · z � c(w,q)}

d. z

f

is homogeneous of degree zero in w.

e. If for each q 2 R+, {z 2 RL�1+ : f (z) � q} is convex, then z

f

is convex-valued. If for each

q 2 R+, {z 2 RL�1+ : f (z)� q} is strictly convex, then z

f

is single-valued.

f. (Sheppard’s Lemma) Let w 2 RL�1+ and q 2 R+. If

��z

f (w,q)�� = 1, then c

f

is differentiable

with respect to w at (w,q) and for each l 2 {1, . . . ,L�1},

∂c

f

∂w

l

(w,q) = z

f

l

(w,q)

or in matrix notation,

—w

c(w,q) = z(w,q)

g. If z

f

is a differentiable function in a neighborhood of w 2 RL�1+ , then

1. D

w

z

f (w,q)w = 02. D

w

z

f (w,q) = D

2w

c(w,q) is symmetric and negative semi-definite.

h. If f is homogeneous of degree one, i.e., Y

f

satisfies CR, then c

f

and z

f

are homogenous of

degree one in q.

i. If f is concave, then for each w 2 RL�1+ , c

f (w, ·) is a convex function, i.e., c

f

is a convex

function of q.

Proof. Will prove. Check Mas-Colell Proposition 5.C.2

Page 2 of 5

4.5.1 Single-input and single-output model

In this section, we study the single-input and single-output model. The simplicity of this envi-ronment allows us to graphically characterize a firm’s cost minimization and profit maximization.

Consider a firm represented by the production function f : R+ !R+. Let p 2R+ be the outputprice and w 2 R+ be the input price. In order to simplify notation, we consider w to be fixed anddenoted c

f (w,q) by c(q). Likewise, we denote y

f (w, p) by Q(p). The PMP is then

maxq�0

pq� c(q)

Assume that this problem has a unique solution Q(p). If this solution is interior and the costfunction is differentiable, then

p = c

0(Q(p))

Definition. (102). Let q 2 R+. Then

• AC(q)⌘ c(q)q

is the average cost at q.

• MC(c)⌘ c

0(q) is the marginal cost at q.

4.5.2 Long-run vs. short-run

Suppose that a firm is represented by a production set Y . If there is free entry of firms to thismarket, this firm can be replciated as many times as possible. Thus, in the “long-run,” the aggregatetechnical restrictions are given by the set

AY ⌘ {y1 + · · ·+ y

n

: n 2 N, 8 i = 1, . . . ,n, y

i

2 Y}

This is the aggregate produciton set, which describes the feasible production plans for an industry,as a whole.

Proposition. (55). If Y satisfies NI returns to scale, then AY satisfies CR to scale.

Proof. Will prove. Check Problem 2, Section 4.4 for a basic outline of this proof.

Under additional technical conditions, one can guarantee that if Y is represented by a productionfunctions that is homogeneous of degree one, then the supply correspondence is flat.

Page 3 of 5

Multiple input case with temporarily fixed inputs

In situations in which there are multiple inputs, some of these may be temporarily fixed. Economistsrefer to these situations as “short-run.” Consistently, they refer to situations in which no input isfixed in the “long-run.”

For instance, suppose that there are two inputs: labor, denoted by L, and capital, denoted by K.The “long-run” production function is simply the production function as we have defined it in ourproduction model

(L,K) 2 R2+ 7! f (L,K) 2 R+

For a fixed amount of capital, say K, the short-run production function is f

K

defined by, for each L,f

K

(L)⌘ f (L, K). The cost function associated with f

K

is the short-run cost function at K. One caneasily graphically characterize the long-run cost and average cost functions, as the lower envelopeof short-run cost and short-run average cost functions. Clearly, under this interpretation, long-runsupply correspondences are not necessarily flat.

4.6 Aggregate production

Let {Y1, . . . ,YJ

} be a family of production sts. The aggregate production set associated with{Y1, . . . ,YJ

} is the set

Y

AG

⌘ Y1 + · · ·+Y

J

⌘ {y1 + · · ·+ y

J

: y1 2 y1, . . . ,yJ

2 Y

J

}

The aggregate supply correspondence associated with {Y1, . . . ,YJ

} is the correspondence that asso-ciated with each p 2 RL, the set

y

AG

(p)⌘ y

Y1(p)+ · · ·+ y

Y

J(p)⌘�

y1 + · · ·+ y

J

: y1 2 y

Y1 , . . . ,yJ

2 y

Y

J

Proposition. (56). Let {Y1, . . . ,YJ

} be a family of production sets. Then, for each p 2 RL

+,

a. y

Y

AG(p) = y

AG

(p)

b. PY

AG(p) = PY1(p)+ · · ·+PY

J(p)

Proof. Let y 2 y

AG

(p). Then, y = y1 + · · ·+ y

J

where y1 2 y

Y1 , . . . ,yJ

2 y

Y

J . Thus, y 2 Y

AG

andconsequently, p · y PY

AG(p). Thus,

p · y = PY1(p)+ · · ·+PY

J(p) PY

AG(p)

We claim that p · y � PY

AG(p). Suppose by contradiction that p · y < PY

AG(p). Then, there isy

0 ⌘ y

01 + · · ·+ y

0J

2 Y

AG

such that p · y < p · y0. Then, there is j 2 {1, . . . ,J} such that p · y0j

> p · yj

for otherwise p · y01 + · · ·+ p · · ·y0J

p · y1 + · · ·+ p · yJ

. Thus, y

j

/2 y

Y

j(p). This is a contradiction.Thus, y 2 y

Y

AG(p) and p · y = PY

AG(p) = PY1(p)+ · · ·+PY

J(p).Now, let y 2 y

Y

AG . Then, p ·y = PY

AG and there are y1 2Y1, . . . ,yJ

2Y

J

such that y = y1+ · · ·+y

J

.Let j 2 {1, . . . ,J}. Then, p · y

j

PY

j(p). We claim that y

j

2 y

Y

j(p). Suppose for contradictionthat for some j

⇤ 2 {1, . . . ,J}, p · yj

⇤ < PY

j

⇤ (p). Then, p · y1 + · · ·+ p · yJ

< PY1(p)+ · · ·+PY

J(p).Thus, PY

AG < PY1(p)+ · · ·+PY

J(p). This is a contradiction.

Page 4 of 5

4.7 Efficient production

Definition. (103). Let Y ✓RL

be a production set. A feasible vector y 2Y is efficient if there is no

y

0 2 Y such that y

0 � y.

Proposition. (57). Let Y ✓ RL

and y 2 Y . Then,

a. If y 2 Y is profit maximizing at some p 2 RL

++, then y is efficient.

b. Suppose that Y is convex. If y 2Y is efficient for Y , then it is profit maxizing for some p� 0.

Proof. Let Y ✓ RL be a production set and y 2 Y .

Part A.

Assume y 2 Y is profit maximizing at some p 2 RL

++. We want to prove that y is efficient.Suppose for contradiction that y is not efficient. Then, there is y

0 2 Y such that y

0 6= y and y

0 �y, Because p� 0, this implies that p · y

00 > p · y, contradicting the assumption that y is profitmaximizing.

Part B.

Let Y ✓ RL be a non-empty convex production set that satisfies free disposal. Let y be efficientat Y , i.e., there is no y

0 2 Y such that y

0 � y We prove that there is p � 0 such that y is profitmaximizing at p.

First, we show that y is a boundary point of Y , i.e., y 2 Y \Y

c. Suppose for contradictionthat y /2 Y \Y

c. Then, y 2 Int(Y ), which by definition is an open set. So, 9 e > 0 such thatBe(y) ✓ Int(Y ). Let t 2 RL such that 8 l 2 {1, . . . ,L} : t

l

= 1. Define y

0 = y+ (e/2)t. Then,y

0 2 Be(y), and thus, y

0 2 Y . However, y

0 � y, which contradicts the fact that y is efficient in Y

(! ). Hence, y 2 Y \Y

c.

Now, by the Basic Supporting Hyperplane Theorem, 9 p 2RL, p 6= 0 such that 8 y

0 2Y , p ·y0 p · y. Define y = y� ev

l

for some l, where v

l

2 RL that has 1 in the l-th component and zeros fork 6= l. Since y� y and Y satisfies free disposal, then y 2 Y . As a result, p · y� p · y, which impliesthat p · (y� y)� 0. So, we have p · v

l

� 0, and thus, p

l

� 0 for all l.

Therefore, there is p� 0 such that y is profit maximizing at p.

4.8 Problems

Problem 1. Prove that the converse of Statement A in Proposition 57 is not necessarily true. Provethat Statement 2 in the same proposition cannot be strengthened to require p� 0.

Proof. Will prove. The second part of this proof is given by the solution to Problem 9, Section4.4.

Page 5 of 5