General Equilibrium with Production

Transcription

1 General Equilibrium with Production Ram Singh Microeconomic Theory Lecture 11 Ram Singh: (DSE) General Equilibrium: Production Lecture 11 1 / 24

2 Producer Firms I There are N individuals; i = 1,..., N There are M goods; j = 1,..., M There are K firms; k = 1,..., K Each firm has a set of production plans, i.e., production set Y k R M The production set Y k is the set of feasible production plans for firm k Examples: Let M = 3 and K = 2. Suppose Firm 1 can produce six units of good 2 by using three units of good 1 and nine units of good 3, i.e., Firm 1: y 1 = ( 3, 6, 9) Firm 2: y 2 = (8, 3 1 2, 14) Economy: Y = (5, 2 1 2, 23) Ram Singh: (DSE) General Equilibrium: Production Lecture 11 2 / 24

3 Producer Firms II A typical production plan for firm k is where y k j y k = (y k 1,..., y k M), > 0 if good j is an output produced by firm k y k j < 0 if good j is an input used by firm k. Let p = (p 1,..., p M ) be the given price vector. For any y k Y k, profit for firm k is π(p, y k ) = p 1 y k p M y k M = p.y k. So, PMP for firm k is: max{p 1 y k y k Y k p M ym}, k i.e., max{p.y k } y k Y k Ram Singh: (DSE) General Equilibrium: Production Lecture 11 3 / 24

4 Producer Firms III Let Π k (p) = max y k Y k π(p, y k ) Π k (p) homogenous function of degree 1 in p. Assume, for each firm k, 0 Y k R M, i.e., firm can always earn Zero profit So, profit is non-negative Y k is closed and bounded and strictly convex For any given price vector p, the profit maximizing choice/production plan is unique For different price vector p, profit maximizing choice/production plan will be different, in general. Ram Singh: (DSE) General Equilibrium: Production Lecture 11 4 / 24

5 Aggregate Production Plans I Let y k = (y1 k,..., y M k ) be a feasible production plan for firm k = 1,..., K. Corresponding to the above production plan, the aggregate production plan for the economy is given by Y = (y 1,..., y K ), where y k = (y k 1,..., y k M). (1) Corresponding to the production plan (1), the total production of good j is given by yj k. So, corresponding to the production plan (1), the total output vector is: ( y1 k,..., ym) k = y k = Y Ram Singh: (DSE) General Equilibrium: Production Lecture 11 5 / 24

6 Aggregate Production Plans II if K y j k > 0, good j is a net output for the economy if K y j k < 0, good j is a net input for the economy. The aggregate production possibility set (for the entire economy) is Y = { Y Y = } y k, where y k Y k. That is, if Ŷ Y, then there exist production plans ŷ1,..., ŷ k,..., ŷ K such that ŷ k Ŷ k, i.e., ŷ k is a feasible plan for firm k = 1,..., K ; and Ŷ = K ŷk Ram Singh: (DSE) General Equilibrium: Production Lecture 11 6 / 24

7 Aggregate Production Plans III Proposition Given the above assumptions on Y k and PMP for firms, 0 Y R M Y is closed and bounded and strictly convex Question Why Y is a bounded set? Proposition Given the above assumptions on Y k, for any given price vector p = (p 1,.., p M ) >> (0,..., 0), the following PMP has a unique solution: max{p.y k } for every,...,k y k Y k Ram Singh: (DSE) General Equilibrium: Production Lecture 11 7 / 24

8 Aggregate Production Plans IV Proposition Given the above assumptions on Y, for any given price vector p = (p 1,.., p M ) >> (0,..., 0), the following PMP has a unique solution: max {p.y} Y Y Ram Singh: (DSE) General Equilibrium: Production Lecture 11 8 / 24

9 Efficient Production: Two definitions I Definition (Definition 1): Production Plan Y = (y 1,..., y K ) Y is efficient if there is no other plan Z = (z 1,..., z K ) such that: Z is feasible, i.e., Z Y, and z k j zj k > y k j, for all goods j y k j, for at least one good j Remark Suppose kth good is a net input. In that case, K zk j > K that the production plan requires smaller quantity of this input. y j k implies Ram Singh: (DSE) General Equilibrium: Production Lecture 11 9 / 24

10 Efficient Production: Two definitions II Suppose Y = (y 1, y 2 ): firm 1: y 1 = ( 2, 4, 6) firm 2: y 2 = (8, 4, 14) So, Y = 2 1 yi = ( 2 1 y 1, 2 1 y 2, 2 1 y 3) = (6, 0, 20) Let Z = (z 1, z 2 ): firm 1: z 1 = ( 2, 4, 6) firm 2: z 2 = (8, 3 1 2, 14) So, Z = 2 1 zi = ( 2 1 z 1, 2 1 z 2, 2 1 z 3) = (6, 1 2, 20) Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

11 Efficient Production: Two definitions III Definition (Definition 2): For given price vector p = (p 1,..., p M ), production Plan Y = (y 1,..., y K ) Y is efficient if it solves max Y Y {p.y }, i.e., p.y p.y for all Y Y Question Does (D1) imply (D2)? Does (D2) imply (D1)? Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

12 Production Equilibrium I Total supply is given by Let N e i + i=1 i=k p = ( p 1,..., p M ) be a price vector. Ȳ = (ȳ 1,..., ȳ K ) be a production plan for the economy. y k Definition (Ȳ, p) is a competitive production equilibrium only if: For all k = 1,..., K, ȳ k solves max y k Y k { p.y k }. Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

13 Production Equilibrium II Proposition Take any price vector p = ( p 1,..., p M ). If ȳ k Y k solves max{ p.y k } for k = 1,..., K. y k Y k Let Ȳ, where Ȳ = K ȳk. Then, there is NO other plan Z = (z 1,..., z K ) such that: Z is feasible, i.e., Z Y, and z k j zj k > ȳ k j, for all goods j ȳ k j, for at least one good j Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

14 Production Equilibrium III Proposition Take any price vector p = ( p 1,..., p M ). Suppose production plans ȳ 1,..., ȳ K are such that ȳ k Y k solves max{ p.y k } for all k = 1,..., K y k Y k. Then, Ȳ = K ȳk solves max { p.y} Y Y That is, individual profit maximization leads to total profit maximization. Why? Question What assumption is made about Externality? Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

15 Production Equilibrium IV Proposition Take any price vector p = ( p 1,..., p M ). If Ȳ solves max { p.y} Y Y then there exist production plans ȳ 1,..., ȳ K such that ȳ k Y k, Ȳ = K ȳk, and ȳ k solves max y k Y k { p.y k } for all k = 1,..., K Proof: Let Ȳ solve max Y Y{ p.y}. That is, ( y Y)[ p.ȳ p.y] Let ȳ 1,..., ȳ K are such that ȳ = K ȳk and ȳ k Y k. If possible, suppose for some k, ȳ k does not solve max y k Y k { p.y k } Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

16 Production Equilibrium V So, there exists some ŷ k Y k such that p.ŷ k > p.ȳ k Now, consider the production plan Z such that Z = (z 1,..., z k,..., z K ) = (ȳ 1,..., ŷ k,..., ȳ K ) Clearly, Z Y. It is easy to show that p.z > p.ȳ, a contradiction. Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

17 Privatized Economy I Privatized Economy: (u i (.), e i (.), Y k, θ ik ) i {1,..,N},j {1,...,M},k {1,...,K } All firms are privately owned θ ik is the (ownership) share of kth firm owned by i the individual; 0 θ ik 1 for all i = 1,..., N and k = 1,..., K n i=1 θik = 1 for all k = 1,..., K Now, for any given price vector, p = (p 1,..., p M ), individual i solves max x i R M + u i (x), s.t. p.x i I i (p), where I i (p) = p.e i + θ ik π k (p). Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

18 Privatized Economy II Remark In view of the above assumptions on Production sets, π(p) 0 is bounded above and continuous in p. Moreover, I i (p) is a continuous function of p In view of the assumption on u i (.), the solution of the above UMP is unique. Definition The excess demand function for good j is z j (p) = N i=1 x i j (p, I i (p)) y k j (p) N ej i. i=1 So, excess demand vector is z(p) = (z 1 (p),..., z M (p)). Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

19 Privatized Economy III Definition Feasible Allocation: An allocation (X, Y), where Y = (y 1,..., x K ) is feasible, if: For all k, y k Y k N x i = i=1 N e i + y k. i=1 i=k Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

20 Existence of WE I As before, in view of the assumption on u i (.) and Y k, the excess demand function is homogeneous function of p of degree 0. For equilibrium to exist Y + N i=1 ei >> 0 must hold for some Y Y. As p j 0 for some j, the excess demand becomes unbounded for one of such commodities. Theorem Consider an economy (u i (.), e i, θ ik, Y j ), where i = 1,..., N, j = 1,..., M and k = 1,..., K. If u i (.) and Y j satisfy above assumptions, then there exists a price vector p >> 0 such that z(p ) = 0. Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

21 Efficiency of WE I Definition A feasible allocation ( X, Ȳ) is Pareto optimum if there is no other allocation (X, Y) such that N x i = i=1 N e i + i=1 y k ; i=k u i (x i ) u i ( x) for all i {1,..., N} and u i (x j ) > u j ( x) for some j {1,..., N} Theorem Consider an economy (u i (.), e i, θ ik, y k ), where i = 1,..., N and k = 1,..., K. If u i (.) is strictly increasing, then every WE is Pareto optimum. Proof: Suppose ( X, Ȳ) along with p is WE. Therefore, Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

22 Efficiency of WE II N x i = i=1 N e i + Suppose ( X, Ȳ) is not PO. So, there is some (X, Y), such that i=1 i=k ȳ k N N x i = e i + y k ; i=1 i=1 i=k u i (x i ) u i ( x i ) for all i {1,..., N} and u i (x i ) > u j ( x j ) for some j {1,..., N} Therefore, p.x i p. x i for all i {1,..., N} and p.x j > p. x i for some j {1,..., N} Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

23 Efficiency of WE III N N p. x i > p. x i i=1 M M p. y j + e j i=1 > M M p. ȳ j + j=1 i=1 j=1 j=1 M p. y j > M p. j=1 j=1 a contradiction. Why? ȳ j e j Ram Singh: (DSE) General Equilibrium: Production Lecture / 24

24 Efficiency of WE IV Theorem Consider an economy (u i (.), e i, θ ik, Y j ), where i = 1,..., N and k = 1,..., K. Suppose, u i (.) and Y j satisfy above assumptions, and y + N i=1 ei >> 0 for some y Y. If ( x, ȳ) is any feasible PO allocation, then there exist such that Cash transfers T i, i = 1,.., N such that N i=1 T i = 0 A price vector p = ( p 1,..., p M ) 1 x i maximizes u i (x i ) s.t. p.x i p.e i + K θik π k ( p) + T i for all i = 1,..., N 2 ȳ k maximizes p.y k for all k = 1,..., K 3 z j ( p) = 0 for all j = 1,..., M Ram Singh: (DSE) General Equilibrium: Production Lecture / 24