EC487 Advanced Microeconomics, Part I: Lecture 2

Transcription

1 EC487 Advanced Microeconomics, Part I: Lecture 2 Leonardo Felli 32L.LG.04 6 October, 2017

2 Properties of the Profit Function Recall the following property of the profit function π(p, w) = max x p f (x) w x = p f (x(p, w)) w x(p, w) Hotelling s Lemma: π p = y(p, w) 0 and π w i = x i (p, w) 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

3 Hessian of the Profit Function Consider now the Hessian matrix of the profit function π(p, w): 2 π 2 π p 2 p w 1 2 π p w h H = 2 π w 1 p. 2 π w h p 2 π w π w 1 w h π w h w 1 2 π wh 2 By Hotelling s Lemma the matrix H is: y y p w 1 H = y w h x 1 p x 1 w 1 x 1 w h x h p x h w 1 x h w h Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

4 Properties of the Hessian of the Profit Function Result Let F : A R where A R n is a convex open set. Let F ( ) be twice differentiable. Then the function F ( ) is convex if and only if the Hessian matrix of F ( ) is positive semi-definite on A. Recall that the profit function π(p, w) is convex in (p, w). This together with Young theorem implies that H is symmetric and has a non negative main diagonal. Assume now that π(p, w) is defined on an open and convex set S of prices (p, w) then convexity of π(p, w) is equivalent to the Hessian Matrix H being positive semi-definite. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

5 Profit Maximization Profit maximization can be achieved in two sequential steps: 1. Given y, find the choice of inputs that allows the producer to obtain y at the minimum cost; this generates conditional factor demands and the cost function; 2. Given the cost function, find the profit maximizing output level. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

6 Comments Notice: step 1 is common to firms that behave competitively in the input market but not necessarily in the output market. Only in step 2 we impose the competitive assumption on the output market. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

7 Cost Minimization We shall start from step 1: min x s.t. w x f (x) y The necessary first order conditions are: y = f (x ), w λ f (x ) and [w λ f (x )] x = 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

8 Cost Minimization (cont d) or for every input l = 1,..., h: with equality if x l > 0. w l λ f (x ) x l The first order conditions are also sufficient if f (x) is quasi-concave (the input requirement set is convex). Alternatively, a set of sufficient conditions for a local minimum are that f (x) is quasi-concave in a neighborhood of x. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

9 Cost Minimization (cont d) This can be checked (sufficient condition) by means of the bordered hessian matrix and its minors. In the case of only two inputs f (x 1, x 2 ) we have: with equality if x l > 0 w l λ f (x ) x l, l = 1, 2 and SOC: 0 f 1 (x ) f 2 (x ) f 1 (x ) f 11 (x ) f 12 (x ) f 2 (x ) f 21 (x ) f 22 (x ) > 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

10 Cost Minimization (cont d) In the case the two first order conditions are satisfied with equality (no corner solutions) we can rewrite the necessary conditions as: and MRTS = dx 2 dx 1 = f (x )/ x 1 f (x = w 1 )/ x 2 w 2 y = f (x ) Notice a close formal analogy with consumption theory (expenditure minimization). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

11 Conditional Factor Demands and Cost Function This leads to define: the solution to the cost minimization problem: z 1 (w, y) x = z(w, y) =. z h (w, y) the conditional (to y) factor demands (correspondences). the minimand function of the cost minimization problem: the cost function. c(w, y) = w z(w, y) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

12 Properties of the Cost Function 1. c(w, y) is non-decreasing in y. Proof: Suppose not. Then there exist y < y such that (denote x and x the corresponding solution to the cost minimization problem) w x w x > 0 If the latter inequality is strict we have an immediate contradiction of x solving the cost minimization problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

13 Properties of the Cost Function (cont d) If on the other hand w x = w x > 0 then by continuity and monotonicity of f ( ) there exists α (0, 1) close enough to 1 such that f (α x ) > y and w x > w αx which contradicts x solving the cost minimization problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

14 Properties of the Cost Function (cont d) 2. c(w, y) is non-decreasing with respect to w l for every l = 1,..., h. Proof: Consider w and w such that w l w l for every k l. but w k = w k Let x and x be the solutions to the cost minimization problem with w and w respectively. Then by definition of c(w, y) c(w, y) = w x w x w x = c(w, y). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

15 Properties of the Cost Function (cont d) 3. c(w, y) is homogeneous of degree 1 in w. Proof: The feasible set of the cost minimization problem f (x) y does not change when w is multiplied by the factor t > 0. Hence t > 0, minimizing (t w) x on this set leads to the same answer as minimizing w x. Let x be the solution, then: c(t w, y) = (t w) x = t c(w, y). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

16 Properties of the Cost Function (cont d) 4. c(w, y) is a concave function in w. Proof: Let ŵ = t w + (1 t) w for t [0, 1]. Let ˆx be the solution to the cost minimization problem for ŵ. Then c(ŵ, y) = ŵ ˆx = t w ˆx + (1 t) w ˆx t c(w, y) + (1 t) c(w, y) by definition of c(w, y) and c(w, y) and f (ˆx) y. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

17 Properties of the Conditional Factor Demands 5. Shephard s Lemma: If z(w, y) is single valued with respect to w then c(w, y) is differentiable with respect to w and c(w, y) w l = z l (w, y) Proof: By constrained Envelope Theorem and the definition of cost function c(w, y) = w z(w, y) = w z(w, y) λ(w, y) [f (z(w, y) y] Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

18 Properties of the Conditional Factor Demands (cont d) 6. z(w, y) is homogeneous of degree 0 in w. Proof: By Shepard s Lemma and the following result. Result If a function G(x) is homogeneous of degree r in x then ( G/ x l ) is homogeneous of degree (r 1) in x for every l = 1,..., L. Proof: Differentiate with respect to x l the identity that defines homogeneity of degree r: G(k x) k r G(x) k > 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

19 Properties of the Conditional Factor Demands (cont d) We obtain: k G(k x) x l = k r G(x) x l k > 0 or G(k x) x l (r 1) G(x) = k k > 0 x l This is the definition of homogeneity of degree (r 1) of the G(k x) function. x l Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

20 Properties of the Conditional Factor Demands (cont d) Notice that 6. The Lagrange multiplier of the cost minimization problem is the marginal cost of output: c(w, y) y = λ(w, y) Proof: By constrained Envelope Theorem applied to the cost function: c(w, y) = w z(w, y) λ(w, y) [f (z(w, y) y] Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

21 Law of Supply 7. Let the set W of input prices w be open and convex, if z(w, y) is differentiable in w then: H = 2 c w c w 1 w h c w h w 1 2 c wh 2 = z 1 z 1 w h w z h w 1 z h w h is a symmetric and negative semi-definite matrix. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

22 Law of Supply Proof: Symmetry follows from Shephard s lemma and Young Theorem: z l = ( ) c(w, y) = ( ) c(w, y) = z i w i w i w l w l w i w l While negative semi-definiteness follows from the concavity of c(w, y), the fact that W is open and convex and the following result. Result Let F : A R where A R n is a convex open set. Let F ( ) be twice differentiable. Then the function F ( ) is concave if and only if the Hessian matrix of F ( ) is negative semi-definite on A. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

23 Euler Theorem Result (Euler Theorem) If a function G(x) is homogeneous of degree r in x then: r G(x) = G(x) x Proof: Differentiating with respect to k the identity: G(k x) k r G(x) k > 0 we obtain: G(kx) x = rk (r 1) G(x) k > 0 for k = 1 we obtain: G(x) x = r G(x). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

24 Returns to Scale We now introduce a set of new properties closely related to the ones of the expenditure function in consumer theory. 8. If f (x) is homogeneous of degree one (i.e. exhibits constant returns to scale), then c(w, y) and z(w, y) are homogeneous of degree one in y. Proof: Let k > 0 and consider: c(w, k y) = min x s.t. w x f (x) k y (1) By definition of c(w, y) if x is the solution to the cost minimization problem we have y = f (x ). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

25 Returns to Scale (cont d) Hence by homogeneity of degree 1 of f (x) we obtain: k y = k f (x ) = f (k x ) which implies that k x is feasible in Problem (1). Therefore: k c(w, y) = k [w x ] = w (k x ) c(w, k y). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

26 Returns to Scale (cont d) Let now ˆx be the solution to Problem (1). Necessarily: f (ˆx) = k y or, by homogeneity of degree 1: f [(1/k) ˆx] = (1/k) f (ˆx) = y which implies that [(1/k) ˆx] is feasible in the problem that defines c(w, y). Therefore we get: c(w, k y) = w ˆx = k w [(1/k) ˆx] k c(w, y) which concludes the proof. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

27 Returns to Scale (cont d) 8. In other words, a technology that exhibits CRS has a cost function that is linear in y: c(w, y) = c(w)y. 9. A technology that exhibits CRS has equal and constant marginal ( c(w, y)/ y) and average cost functions ( c(w, y)/ y) = (c(w, y)/y). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

28 Returns to Scale (cont d) Proof: Homogeneity of degree 1 and Euler theorem imply c(w, y) c y (w)y = c(w, y) or c y (w) =. y c y (w) = c(w,y) y y c(w, y) = c y (w) y y Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

29 Returns to Scale (cont d) 10. If f (x) is convex (IRS technology), then the cost function c(w, y) is concave in y. 11. A technology that exhibits IRS has a decreasing marginal cost function ( c(w, y)/ y) and average cost function ( c(w, y)/ y) (c(w, y)/y) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

30 Returns to Scale (cont d) c(w,y) y c(w, y) c y (w, y) y y Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

31 Returns to Scale (cont d) 12. If f (x) is concave (DRS technology), then the cost function c(w, y) is convex in y. 13. A technology that exhibits DRS has an increasing marginal cost function ( c(w, y)/ y) and average cost function ( c(w, y)/ y) (c(w, y)/y) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

32 Returns to Scale (cont d) c y (w, y) c(w,y) y c(w, y) y y Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

33 Profit Maximization with Single Output Assume that the output market is competitive. The profit maximization problem is then: max y p y c(w, y) The necessary FOC are: p c(w, y ) y 0 with equality if y > 0. The sufficient SOC for a local maximum: 2 c(w, y ) y 2 > 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

34 Profit Maximization with Single Output (cont d) Clearly the SOC imply at least local DRS in a neighborhood of y. Notice that if y > 0 the optimal choice of the firm is: p = c(w, y ) y = MC(y ) In words, price equal to marginal cost. This condition defines the solution to the profit maximization problem: the supply function: y (w, p) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

35 Profit Maximization with Single Output (cont d) The two profit maximization problems we have considered so far produce the same outcome for equal (w, p). In fact: max y py c(w, y) where c(w, y) = min x s.t. w x f (x) y yields max x s.t. p y w x f (x) = y which is the very first problem we considered. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

36 Short Run Short run cost minimization arises when one or more inputs may be fixed, x h = x h, while the remaining inputs may be varied at will. The short run variable cost function: Alternatively: h 1 c S (w, y, x h ) = w h x h + min w l x l x 1,...,x h 1 s.t. l=1 f (x 1,..., x h 1, x h ) y c S (w, y, x h ) = min x s.t. w x f (x) y x h = x h Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

37 Short Run (cont d) If z(w, y) denotes the long run conditional factor demands, that solve: c(w, y) = min w x x s.t. f (x) y Let x = ( x 1,..., x h ) be the input vector that achieves the minimum long run cost of producing ȳ with prices w: x = ( x 1,..., x h ) = z( w, ȳ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

38 Short Run vs. Long Run We can then characterize the relationship between short and long run total costs, or alternatively, short run and long run variable costs (more familiar). Notice that c(w, y) c S (w, y, z h (w, y)) (2) or c(w, y) y cs (w, y, z h (w, y)) y Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

39 Short Run vs. Long Run (cont d) moreover by Envelope Theorem c(w, y) y cs (w, y, z h (w, y)) y (3) We shall now focus on a neighborhood of ( w, ȳ) and set x h = z h ( w, ȳ). From (2) above by Envelope Theorem we get: c(w, y) y = cs (w, y, x h ) y Recall that Envelope Theorem implies that only the first order effect is zero. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

40 Short Run vs. Long Run (cont d) Since (3) is an identity in (w, y) we can differentiate both sides with respect to y: 2 c S (w, y, x h ) y c S (w, y, x h ) z h (w, y) = 2 c(w, y) y x h y y 2 and with respect to w h : 2 c S (w, y, x h ) y w h + 2 c S (w, y, x h ) y x h z h (w, y) w h = 2 c(w, y) y w h Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

41 Short Run vs. Long Run (cont d) Now 2 c S (w, y, x h ) y w h = 0 since is independent of y. c S (w, y, x h ) w h = x h Hence by Shephard s Lemma: 2 c S (w, y, x h ) y x h = z h(w, y)/ y z h (w, y)/ w h Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

42 Short Run vs. Long Run (cont d) which implies by substitution: 2 c S (w, y, x h ) y 2 + ( z h(w, y)/ y) 2 = 2 c(w, y) z h (w, y)/ w h y 2 which delivers: 2 c S (w, y, x h ) y 2 2 c(w, y) y 2 since ( z h (w, y)/ y) 2 z h (w, y)/ w h 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

43 Short Run vs. Long Run (cont d) This allows us to conclude that the loss function: l(w, y) = c(w, y) c S (w, y, x h ) 0 reaches a local maximum at x. Proof: We have shown above that the FOC are satisfied: c S (w, y, x h ) y = c(w, y) y and we just proved that the SOC hold: 2 c(w, y) y 2 2 c S (w, y, x h ) y 2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

44 Le Chatelier Principle In other words: c(w, y) c S (w, y, x h ) A similar approach proves that for every l {1,..., h}: Moving to profit maximization: 0 zs l w l z l w l and 0 x S l w l x l w l 0 y S p y p All these results are known as Le Chatelier Principle. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

45 Aggregation The question we address is when can we speak of an aggregate supply function in similar terms as the aggregate demand you have seen before? The question above is closely related to the question: under which conditions can we speak of a representative producer? Recall that the key problem when constructing an aggregate demand was the presence of income effects in consumer theory. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

46 Aggregate Supply The absence of a budget constraint implies that individual firms supply are not subject to income effects. Hence aggregation of production theory is simpler and requires less restrictive conditions. Consider J production technologies: (Z 1,..., Z J ) Let z j (p, w) = ( x j (p, w) y j (p, w) ) be firm j s production plan. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

47 Law of Supply We have seen that the matrix of cross and own price effects of production plan z j (p.w): Dz j (p, w) is symmetric and positive semi-definite. This is also known as law of supply. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

48 Aggregate Law of Supply Define now the following aggregate optimal production plan: z(p, w) = J j=1 z j (p, w) = ( j x j (p, w) j y j (p, w) ) Does an aggregate law of supply hold? Since both symmetry and positive semi-definiteness are preserved under sum then Dz(p, w) is also symmetric and positive semi-definite. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49

49 Representative Producer Result In a purely competitive environment the maximum profit obtained by every firm maximizing profits separately is the same as the profit obtained if all J firms where they coordinate their choices in a joint profit maximization: J π(p, w) = π j (p, w) j=1 In other words, there exists a representative producer. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, / 49