5. Oligopoly Oligopoly: Small number of firms: Firms depend on each other. Identical products: Firms jointly face a downward sloping industry demand No entry: Long-run positive profits are possible

This formula applies to any type of firms in the output market 1. Competitive Output Market Competitive industry: Many firms: Firms are independent of each other in decision making Identical product: Each firm faces a horizontal demand curve at the market price Free entry: Zero profit in the long run A competitive firm takes the market price as given. For a given market price p

Social welfare function W: Rn-R gives social utility W(u1, u2,. un ). W is strictly increasing is socially optimal if it solves max Wu(a1), u2(a2),., un(n) st>Tis>w Proposition 1.29. If is SO, it is PO. I Proposition 1. 30. Suppose that preferences are continuous, strictly monotonic, and strictly convex. Then, for any PO allocation x* with >>0,v i, there exist ai

Equilibrium price. Equilibrium allocation: x=xi(p,p·w2), Note: A p* for any >0 is also an equilibrium price. Offer curve: (p)(p, p. w;). The equilibrium is the intersection point of the offer curves. Excess demand function:

(1)i(p, u) is zero homogeneous in p (2) substitution matrix: Dpi(p, u)<0 (3)symmetric cross-price effects: 23i(p 2=2z(p, u) (4) decreasing:=n≤0

Production Plans with Multiple Outputs Lety≡(m,,…,ym) be a net output vector, YArn be a convex set,G:Y→R be twice differentiable Production possibility set:{y∈Y|G(y)≤0} Assumption 1.1. Gy (y)>0, Vi,yEY. Proposition 1. 12. Production frontier yEY G(y)=0 contains technologically efficient production plans Definition 1.1. Marginal rate of transformation

Thus, to find Nash equilibria in IN=N, A(Si)), uin, we use the conditions: for ach player i (1)he is indifferent among all strategies in St,and (2)any strategy in St is at least as good as any strategy in S Example 3.7.(Meeting in an Airport ). Mr Wang and Ms Yang are to meet in an irport. However, they do not know whether they are to meet at door a or door B. The payoffs are specified in the following

Optimization See Sydsaeter(2005, Chapters 2, 3)and Chiang(1984, Chapters 9, 11, 12 and 21) Positive definite matrix Definite matrices are directly related to optimization. A symmetric matrix A is positive semi- definite(A≥0) if rAr≥0,Vx; positive definite(A>0)

then there exists AE R\ such that (Kuhn-Tucker condition) G(s') =0 and 1. Lagrange Method for Constrained Optimization FOC: D.L(,\)=0. The following classical theorem is from Takayama(1993, p.114). Theorem A-4 (Sufficieney). Let f and, i= ,..m, be quasi-concave, where Theorem A-1. (Lagrange). For f: and G\\, consider the following G=(.8 ) Let r' satisfy the Kuhn-Tucker condition and the FOC for (A.2). Then, x' problem is a global maximum point if max f() (1)Df(x') =0, and f is locally twice continuously differentiable,or

第一讲导论 (一)企业的性质 1、从道德人到经济人 2、经济学的若干基本前提(假设) 3、企业的性质:传统观点 4、企业的性质:基于经济人的讨论