Chapter 4: Production Theory

Transcription

1 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 mathematical representaiton of produciton units, which we refer simply as firms. Second, we characterize the 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 commodities. 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 of aranges, in pounds, by z 1 and the generic amount of arange juice, in liters, by y 1. A given firm is able to produce from each pound of oranges, half a liter of juice, and cannot produce any oranges from orange juice. Moreover, the firm can dispose of any excess of orange juice. The mathematical representation of this firm is the set Y = ( z 1,y 1 ) 2 R 2 : z 1 0,y 1 apple 1 2 z 1 Definition. (97). A production set is a subset of R L. The generic production set is Y. We refer to the elements of Y as feasible vectors (for Y ). For each y 2 Y, and each l 2{1,...,L}, commodity l is an input at y whenever y l < 0; symmetrically, it is an output whenever y l > 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 function that tells us which vectors are feasible. No restriction is imposed on whether a commodity is an input or an output. Let F : R L! R be a function. The production set associated with F is Y F y 2 R L : F(y) apple 0 1

2 We refer to F as a transformation function and to the set F y 2 R L : 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: MRT lk (y) F y (y) l F y (y) = k Dy k Dy l With l on the x-axis and k on the y-axis, the MRT is a measure of how much the net output of good k can decrease if the firm increases the net output of good l by one marginal unit. The MRT of commodity l for commodity k at y is the absolute value of the slope of the transformation frontier on 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 be produced with a given input vector. For simplicity, we restrict to the single-output case. Let f : R L + 1! R + be a function. The production set associated with f is n o Y f ( z,q) 2 R L : z (z 1,...,z L 1 ) 2 R L + 1 and q f (z 1,...,z L 1 )=0 We refer to f as a production function and to the set n o f ( z,q) 2 R L : z 2 R L + 1 and q f (z 1,...,z L 1 )=0 as the production frontier. We define the Marginal Rate of Technical Substitution (MRTS) of input l for input k at z 2 R L 1 + as the ratio: MRT S lk (z) f z (z) l f z (z) = k Dz k Dz l The single-output model admits an alternative representation of the production set by means of isoquants. With l on the x-axis and k on the y-axis the MRTS measures the amount of input k that must 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. Page 2 of 12

3 4.2 Properties of production sets Let Y R L be a production set. Then Y is (or satisfies): a. Non-empty iff Y 6= /0. b. Closed iff Y is closed in R L. c. No-free-lunch iff 8 y 2 Y : y 0 =) y = 0. Alternatively, Y \ R L + {0}. d. Possibility of inaction iff 0 2 Y. e. Free-disposal iff 8 y 2 Y, 8 x apple 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,y 0 } Y : y + y 0 2 Y. k. Convexity iff Y is convex. Example. Let Y R 2 R (single-output model). Let the production be defined as f (z 1,z 2 )=z a 1 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 Y satisfies NI returns to scale. Proposition. (48). Let Y R L. a. Suppose that L = 2 and Y R R (single-output model). If Y satisfies free-disposal and CR of scale, then Y is convex. b. Suppose that L = 3 and Y R 2 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 R L. If Y is additive and satisfies NI returns to scale, then Y is a convex cone, i.e., for each pair {y,y 0 } Y and each pair {a,b}2r +, ay + by 0 2 Y. Proof. Let {y,y 0 } Y and {a,b}2r +. 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, a n y 2 Y. By additivity, ay = a n y + + a n y 2 Y. The symmetric argument shows that by0 2 Y. By additivy, ay + by 0 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. Page 3 of 12

4 4.3 Profit maximization In this seciton we characterize the behavior of a price-taker profit maximizing firm. Consider a firm represented by production set Y. Assume that prices are p 2 R L +. The profit of firm Y when producing at a feasible vector y 2 Y is p y  p l y l l2l A price-taker profit maximizing firm that observed prices p solves the Profit Maximization Problem max y2y p y The solution of this problem and the maximum profit attained by the firm define the firm s supply correspondence and profit function. Supply correspondence Definition. (98). Let Y be a production set. The supply correspondence associated with Y is defined by Profit function y Y : R L +! R L p 7! y Y (p) argmax p y y2y Definition. (99). Let Y be a production set. The profit function associated with Y is defined by: P Y : R L +! R p 7! P Y (p) max y2y p y Alternatively, P Y (p) p y for some y 2 y Y (p) Profit maximizaiton: Transformation Function model If production set Y is described by means of a transformation function F, the firm s profit maximizaiton problem becomes max p y {y2r L :F(y)apple0} We simplify notation and denote y F and P F the supply correspondence and profit function associated with Y F. Proposition. (50). Let F be a differentiable transformation function. For each p 2 R L ++ and each pair {l,k} {1,...,L} p l p k = MRT lk (y F (p)) Page 4 of 12

5 Proof. Let F be a differentaible transformation function. Take p 2 R L ++ 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 = 0:p l y l L = 0:p k y k l F(y) y l = 0 l F(y) y k = 0 Thus, we have p l p k = F(y) y l F(y) y k = MRT lk (y F (p)) Profit maximization: Production Function model In the production function single-output model we denote input prices by w (w 1,...,w L 1 ) 2 R L 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 maximization problem becomes max z2r L 1 + pf(z) w z We simplify notation and denote y f and P f the supply correspondence and profit functions associated with Y f. Proposition. (51). Let f be a differentiable production function and Y its associated production set. For each w 2 R L 1 ++, each p 2 R ++, and each pair {l,k} {1,...,L}, w l w k = MRT S lk (y f (p)) Proof. Let f be a differentiable production function and Y its associated production set. Take w 2 R L ++ 1, p 2 R ++, and {l,k} {1,...,L}. The Lagrangian of the profit maximization problem is L = pf(z) w z + lz Page 5 of 12

6 The first-order conditions are L f = p l (z) z l z l w l apple 0 with equality if z l > 0 L f = p k (z) z k z k w k apple 0 with equality if z k > 0 When z is an interior solution, we have w l w k = f z l (z) f z k (z) = MRT S lk(y f (p)) 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. P Y is homogeneous of degree one. b. P Y is convex. c. If Y is convex, then it can be recovered from P Y as follows: Y y 2 R L : 8 p 2 R L +, p y apple P Y (p) d. y Y is homogeneous of degree zero. e. If Y is convex, then y Y is convex-valued. If Y is strictly convex, i.e., for each pair {x,y} Y and each a 2 (0,1), ax +(1 a)y belongs to the interior of Y, then y Y is single-valued. f. (Hotteling s Lemma) Let p 2 R L +. If y Y (p) = 1, then P is differentiable for each l 2 {1,...,L}, P Y p l (p)=y Y l (p) g. If y Y is a differentiable function in a neighborhood of p 2 R L +, then 1. Dy Y (p)p = 0 2. Dy Y (p) is symmetric and positive semi-definite. h. (Law of Supply) For each pair {p, p 0 } R L +, (p 0 p) y Y (p 0 ) y Y (p) 0 Proof. Will prove. Check Mas-Colell Proposition 5.C.1 Page 6 of 12

7 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 and only if f is homogeneous of degree one. Conclude that in the single-input, single-output case CR implies that f is linear. Proof. Let f : R L 1! R + be the production function associated with a single-output technology, and let Y be the associated We prove that Y ( z,q) 2 R L : z 2 R L 1 and q f (z) apple 0 8 y 2 Y, 8 z 2 R L 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 R L 1. We want to show that a f (z)= f (az). For this z, let f (z)=q2r +. By definition of the production frontier, ( z,q) 2 f ( z,q) 2 R L : z 2 R L 1 and q f (z)=0 Y. Since Y satisfies CRS, then ( az,aq) 2 Y. So, aq apple f (az), and thus, q apple q a. Let f (az) = q 2 R +. By definition, ( az, q) 2 Y. Since Y satisfies CRS, then z, q q a 2 Y. So, a apple f (z)=q. Now, q apple q a and q q a imply that q = q a. Thus, a f (z)= f (az). Therefore, 8 a 2 R +, 8 z 2 R L 1 : a f (z)= f (az). (() Assume 8 a 2 R +, 8 z 2 R L 1 : a f (z) = f (az). Take y 2 Y and a 2 R +. We want to show that ay 2 Y. Now, y 2 Y implies that y =( z,q) 2 R L such that z 2 R L 1 and q f (z) apple 0. So, q apple f (z), and multiplying both sides by a yields aq apple a f (z). Since f (z) is homogeneous of degree one, then aq apple 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) is concave. Problem 2. Consider the single-input case in which f is defined by: for each z 2 R +, f (z) 1 pz 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 combinations under technology Y (there is free entry to this market). Denote this new production set by AY. Formally, AY {y y n : n 2 N, 8 i = 1,...,n, y i 2 Y } Page 7 of 12

8 1. Characterize and graph AY. Is there a function f A such that AY {( z,q) : z 0, q apple f A (z)}? 2. Is AY closed? Is it additive? Does it satisfy Non-Increasing Returns to scale? Does it satisfy Non-Decreasing returns to scale. Proof. Graphs will not be provided in the solution to this problem, only descriptions about the procedures. Also, Part A will be skipped. Part B. The production set associated with the given production function is Y f ( z,q) 2 R 2 : z 2 R + and q f (z) apple 0 Since Y f contains its boundary points, then it is a closed set. Y f satisfies non-increasing returns to 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 = and ȳ = Then, y + ȳ = /2 Y. Thus, Y is not additive. 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, f A (z)= n r p z ( 2 n z/n)+1 1 = 2/n As n!, then by L Hoptial s rule ( 1/2)(1/ p (z/n)+1)(z/n lim 2 ) z = lim n! ( 2/n 2 ) n! 4 p (z/n)+1 = 1 4 Since f A (z)= 1 4 z is only the upper limit and it is not well defined, then there is no function f A such that AY {( z,q) : z 0, q apple f A (z)}. Important to note, the industry as a whole, i.e., AY, faces CR to scale due to free entry, but each firm i still faces the production technology f (z). Part D, (ii) Since f A (z)= 1 4z is not part of the set, then AY is not closed. Let {y,ỹ}2ay such that y =( z,q) and ỹ =( z, q). Now, q + q < (1/4)(z + z). Thus, y + ỹ 2 AY, so AY is additive. Problem 3. Consider a production set Y R L. Let p Y be the profit function associated with set Y, i.e., for each p 2 R L +, p Y (p) max y2y p y a. Define free disposal and no free lunch. Draw a production set that satisfies these two properties. Illustrate the profit maximization for some price vector p. Page 8 of 12

9 b. Is p Y concanve, convex, or neither of them? c. Suppose now that a firm with produciton set Y may face an uncertain market. The first is an expected profit maximizer and is risk neutral. There are two markets in which the firm may participate. 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 Market 2. Proof. Consider a production set Y R L. Let p Y be the profit function associated with set Y. Part A. Y satisfies no-free-lunch iff 8 y 2 Y : y free disposal iff 8 y 2 Y, 8 x apple y : x 2 Y. Part B. 0 =) y = 0. Alternatively, Y \R L + {0}. Y satisfies We claim that p Y (p) is convex. Take { ˆp, p} 2R L + and a 2 [0,1]. Let ŷ argmax y2y ˆp y and ỹ argmax y2y a) p. py (a ˆp +(1 a) p) =(a ˆp +(1 a) p) ȳ where ȳ arg max y2y a) p) y. Then, a ˆp ȳ+(1 a) p ȳ apple apy ( ˆp)+(1 a)p Y ( p). Thus, 8{p, p 0 } R L +, 8 a 2 [0,1] : p Y (ap +(1 a)p 0 ) apple ap Y (p)+(1 a)p Y (p 0 ). Therefore, p Y (p) is a convex function. Part C. Market 1: ˆ p Y (p)df(p) for some distribution of random price. Market 2: p Y ˆ Since p Y (p) is a convex function, then by Jensen s inequality, p Y ˆ ˆ pdf(p) apple pdf(p). p Y (p)df(p). Problem 4. Let C be a convex and closed set of R L and z 2 R L a vector that is not in C, i.e., z 2 C c. Prove that ther is a closet vector in C to z, that is, there is x 2 C such that for each x 2 C, kz x k apple kz xk. Hint: the function f : R L! R defined by x 2 R L 7! f (x) kz xk 2 is continuous. Proof. Will prove. Problem 5. Let C be a convex and closed set of R L and z 2 R L a vector that is not in C. Let x 2 C. Prove that x is such that for each x 2 C, kz x k apple kz xk if and only if for each x 2 C, (z x ) (x x ) apple 0 Moreover, there is a unique closet vector in C to z. Do it in two steps. Page 9 of 12

10 a. Suppose that x 2 C such that for each x 2 C, (z x ) (x x ) apple 0 Prove that x is the unique closet element in C to z. Do it as follows: prove that the function f defined above is strictly convex. Let x 2 R L be such that x 6= x. Prove, using a Taylor expansion of degree two, that f (x) > f (x ). b. Assume that x is a closet element in C to z. Let x 2 C. Let j : R! R be the function defined by t 7! j(t) kz (x +t(x x ))k 2. Prove that j is differentiable and that j 0 (0) 0. Conclude that (z x ) (x x ) apple 0. Proof. Will prove. Problem 6. (Basic Separation Theorem) Suppose that C is a convex and closed set of R L and z 2 R L a vector that is not in C. Prove that there is a non-zero vector p 2 R L and a real number c such that p z > c and for each x 2 C, 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 nonfeasible point at C. Let x 2 C 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 R L and z be a boundary point of C, i.e., z 2 C \C c. (Here if A is a set, A denotes the closure of A, i.e., the set of all limit points of A. In other words, the set of points, a such that there is a sequence in A that converges to a. For instance, the closure of (0,1) is [0,1].) Prove that there is a non-zero vector p 2 R L such that for each x 2 C, p x apple p z. Provide a graphical interpretation when C is a production set and z is a boundary point of C. Proof. Will prove. Problem 8. Let Y R L be a non-empty convex production set that satisfies free disposal. Let y be efficient at Y, i.e., there is no y 0 2 Y such that y 0 y. Prove that there is p 0 such that y is profit maximizing at p. Proof. Let Y R L be a non-empty convex production set that satisfies free disposal. Let y be efficient at 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 profit maximizing at p. First, we show that y is a boundary point of Y, i.e., y 2 Y \ Y c. Suppose for contradiction that y /2 Y \ Y c. Then, y 2 Int(Y ), which by definition is an open set. So, 9 e > 0 such that B e (y) Int(Y ). Let t 2 R L such that 8 l 2{1,...,L} : t l = 1. Define y 0 = y +(e/2)t. Then, Page 10 of 12

11 y 0 2 B e (y), and thus, y 0 2 Y. However, y 0 (! ). Hence, y 2 Y \Y c. y, which contradicts the fact that y is efficient in Y Now, by the Basic Supporting Hyperplane Theorem, 9 p 2 R L, p 6= 0 such that 8 y 0 2 Y, p y 0 apple p y. Define ỹ = y ev l for some l, where v l 2 R L that has 1 in the l-th component and zeros for k 6= l. Since y ỹ and Y satisfies free disposal, then ỹ 2 Y. As a result, p y p ỹ, which implies that p (y ỹ) 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. Problem 9. Let Y R L be a non-empty, convex, closed production set that satisfies free disposal. Prove that Y = y 2 R L : 8 p 0, p y apple p Y (p) Is it always true that Y = y 2 R L : 8 p 0, p y apple p Y (p) Would your answer change if Y also satisfies no-free-lunch? Proof. Let Y R L be a non-empty, convex, closed production set that satisfies free disposal. We prove that Y = y 2 R L : 8 p 0, p y apple p Y (p) A by showing Y A and A Y. (Y A) Take ỹ 2 Y. We want to show that ỹ 2 A. Take ˆp 0. Consider the profit function associated with Y evaluated at ˆp: p Y ( ˆp)=max y2y ˆp y Define y as y argmax y2y ˆp y Important to note, p Y ( ˆp) ˆp y, Since ỹ 2 Y, then ˆp ỹ apple ˆp y. Thus, by definition, ỹ 2 A. (A Y ) Take ỹ 2 A. Then, 8 p 0:p ỹ apple p Y (p), where p Y (p) is the profit function associated with Y p Y (p)=max y2y p y We want to show that ỹ 2 Y. Suppose for contradiction ỹ /2 Y, which means that ỹ 2 Y c. Since Y is a closed and convex set of R L and ỹ 2 Y c R L, then by the Basic Separation Theorem, there exists ˆp 2 R L and c 2 R such that 8 y 2 Y : ˆp y < c < ˆp ỹ. This implies that 8 y 2 Y :ˆp y < ˆp ỹ. We claim that ˆp 0. Suppose for contradiction that ˆp l < 0 for some l 2{1,...,L}. Let ȳ 2 Y. Define ȳ n = ȳ nv l, where n 2 N and v l 2 R L that has 1 in the l-th component and zeros for Page 11 of 12

12 k 6= l. Since 8 n 2 N :ȳ n apple ȳ and Y satisfies free disposal, then 8 n 2 N :ȳ n 2 Y. So, we have ˆp ȳ n = ˆp ȳ n ˆp l. Since ˆp l < 0, then as n!, ˆp ȳ n!. Hence, 9 ň 2 N :ˆp ȳň > c. But this contradicts the fact that 8 y 2 Y :ˆp y < c (! ). Thus, ˆp 0. Now, define y as y argmax y2y ˆp y Important to note, p Y ( ˆp) ˆp y. Since y 2 Y, then ˆp y < ˆp ỹ, which is equivalent to p Y ( ˆp) < ˆp ỹ. However, we have just shown 9 p 0:p Y (p) < p ỹ, which means that ỹ /2 A (! ). Therefore, ỹ 2 Y. Is it always true that Y = {y 2 R L : 8 p 0, p y apple p Y (p)} A No, the equality between the two sets does not hold. To explain, here is a counter example. Take ˆp 2 R 2 ++ and z =[0,1] 0. Define Y {(y 1,y 2 ) 2 R 2 : y 2 apple 0}. Note that z /2 Y. Since ˆp 1 > 0, then p Y ( ˆp)=+. So, A {y2r 2 : p 0, p y apple p Y (p)}. Now, ˆp z = ˆp 2 < +, which implies that z 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 {(y 1,y 2 ) 2 R 2 : (y 1,y 2 ) 2 R R}. Since ˆp 2 > 0, then p Y ( ˆp)=+. Also, ˆp z = ˆp 2 < +. Thus, z 2 A, but z /2 Y. Problem 10. (Separate problem attached with Micro Homework Assignment 12). Suppose that there are two inputs with prices (w 1,w 2 ) and one output with price p. Consider a Cobb-Douglas production function f : R 2 +! R, f (z 1,z 2 )=Az a 1 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 demand correspondence, and cost function. d. Verify Hotteling s Lemma for each (w 1,w 2, p) where it holds. Proof. Will prove. Page 12 of 12

13 Chapter 4: Production theory Need to complete: Proposition 53, Proposition 54, Proposition 55, Problem Cost minimization In the context of the Production function model, we can unambiguously refer to inputs and hence keep track of a firm s costs. In this section we characterize a firm s profit maximizaiton problem by means of its cost minimization problem. For simplicity, we will concentrate in the single-ouput case. Consider a firm represented by a production function f. Imagine that the firm decides to produce q units of output and input prices are w 2 R L +. The firm s Cost Minimization Problem for q at w is: min {z2r L + : 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 : R L +! R (w,q) 7! c f (w,q) min {z2r L + : 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 : R L +! R L 1 + (w,q) 7! z f (w,q) arg min {z2r L + : f (z) q} w z Proposition. (53). Let f be a production function, w 2 R L 1 + and q 2 R +. Assume that z f (w,q) = 1 and z f (w,q) 0. Then: 1

14 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 constraint q f (z) apple 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 R L 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 R L + 1 : f (z) q} is convex, then z f is convex-valued. If for each q 2 R +, {z 2 R L + 1 : f (z) q} is strictly convex, then z f is single-valued. f. (Sheppard s Lemma) Let w 2 R L 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 R L 1 +, then 1. D w z f (w,q)w = 0 2. D w z f (w,q)=d 2 wc(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 R L 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

15 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 environment 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 2 R + be the output price and w 2 R + be the input price. In order to simplify notation, we consider w to be fixed and denoted c f (w,q) by c(q). Likewise, we denote y f (w, p) by Q(p). The PMP is then max q 0 pq c(q) Assume that this problem has a unique solution Q(p). If this solution is interior and the cost function 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 Long-run vs. short-run Suppose that a firm is represented by a production set Y. If there is free entry of firms to this market, this firm can be replciated as many times as possible. Thus, in the long-run, the aggregate technical restrictions are given by the set AY {y 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 production functions that is homogeneous of degree one, then the supply correspondence is flat. Page 3 of 5

16 Multiple input case with temporarily fixed inputs In situations in which there are multiple inputs, some of these may be temporarily fixed. Economists refer to these situations as short-run. Consistently, they refer to situations in which no input is fixed 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 our production model (L,K) 2 R 2 + 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 can easily graphically characterize the long-run cost and average cost functions, as the lower envelope of short-run cost and short-run average cost functions. Clearly, under this interpretation, long-run supply correspondences are not necessarily flat. 4.6 Aggregate production Let {Y 1,...,Y J } be a family of production sts. The aggregate production set associated with {Y 1,...,Y J } is the set Y AG Y 1 + +Y J {y y J : y 1 2 y 1,...,y J 2 Y J } The aggregate supply correspondence associated with {Y 1,...,Y J } is the correspondence that associated with each p 2 R L, the set y AG (p) y Y 1 (p)+ + y Y J (p) y y J : y 1 2 y Y 1,...,y J 2 y Y J Proposition. (56). Let {Y 1,...,Y J } be a family of production sets. Then, for each p 2 R L +, a. y Y AG(p)=y AG (p) b. P Y AG(p)=P Y 1(p)+ + P Y J(p) Proof. Let y 2 y AG (p). Then, y = y y J where y 1 2 y Y 1,...,y J 2 y Y J. Thus, y 2 Y AG and consequently, p y apple P Y AG(p). Thus, p y = P Y 1 (p)+ + P Y J (p) apple P Y AG (p) We claim that p y P Y AG(p). Suppose by contradiction that p y < P Y AG(p). Then, there is y 0 y y0 J 2 Y AG such that p y < p y 0. Then, there is j 2{1,...,J} such that p y 0 j > p y j for otherwise p y p y0 J apple p y p y J. Thus, y j /2 y Y j(p). This is a contradiction. Thus, y 2 y Y AG(p) and p y = P Y AG(p)=P Y 1(p)+ + P Y J(p). Now, let y 2 y Y AG. Then, p y = P Y AG and there are y 1 2 Y 1,...,y J 2 Y J such that y = y 1 + +y J. Let j 2{1,...,J}. Then, p y j apple P Y j(p). We claim that y j 2 y Y j(p). Suppose for contradiction that for some j 2{1,...,J}, p y j < P Y j (p). Then, p y p y J < P Y 1(p)+ + P Y J(p). Thus, P Y AG < P Y 1(p)+ + P Y J(p). This is a contradiction. Page 4 of 5

17 4.7 Efficient production Definition. (103). Let Y R L be a production set. A feasible vector y 2 Y is efficient if there is no y 0 2 Y such that y 0 y. Proposition. (57). Let Y R L and y 2 Y. Then, a. If y 2 Y is profit maximizing at some p 2 R L ++, then y is efficient. b. Suppose that Y is convex. If y 2 Y is efficient for Y, then it is profit maxizing for some p 0. Proof. Let Y R L be a production set and y 2 Y. Part A. Assume y 2 Y is profit maximizing at some p 2 R L ++. 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 profit maximizing. Part B. Let Y R L be a non-empty convex production set that satisfies free disposal. Let y be efficient at 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 profit maximizing at p. First, we show that y is a boundary point of Y, i.e., y 2 Y \ Y c. Suppose for contradiction that y /2 Y \ Y c. Then, y 2 Int(Y ), which by definition is an open set. So, 9 e > 0 such that B e (y) Int(Y ). Let t 2 R L such that 8 l 2{1,...,L} : t l = 1. Define y 0 = y +(e/2)t. Then, y 0 2 B e (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 2 R L, p 6= 0 such that 8 y 0 2 Y, p y 0 apple p y. Define ỹ = y ev l for some l, where v l 2 R L that has 1 in the l-th component and zeros for k 6= l. Since y ỹ and Y satisfies free disposal, then ỹ 2 Y. As a result, p y p ỹ, which implies that p (y ỹ) 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. Prove that 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, Section 4.4. Page 5 of 5