复旦大学：《高级微观经济学—均衡理论》英文版 Problem Set 3 Prof. Gene Chang

Problem Set 3 Prof gene Chang (For Ph D students, due on Dec. 28th in the class, or TAs office. For M.A. students, due on Dec 26th, in the class or TA's office. Late turn-in may result in a penalty in the grade) 1. Kland is a robinson Crusoe type economy. Robinson, as the consumer and the owner of the firm, has the utility function U(x, D)=ahn x+BInI, where I is leisure. Robinson's endowment of leisure/abor is one unit. The production function is x=cL, where c is a constant and L is the labor input. Find the general (Walrasian) equilibrium for this economy 2. Review the Debreu's cond itions for the existence of general equilibria. Below is a consumer's preference. The shaded area is the indifference set(where the consumer is ind ifferent among the points in the set). Which Debreu s condition(s)is violated? What kind of troubles this preference may cause in the existence of an equilibrium? 3. Consumer A's preference is represented by the utility function u, (x,, x,)=x,+x ,and consumer Bs preference is represented by the utility function u,(x,, x2)=max(x x2j a) Draw a typical indifference curve for each of them in the x-x, space b)Who's preference satisfies with Debreu's cond ition for existence of a general equilibrium? If any preference does not satisfy the cond itions, tell it violates which Debreu's condition c)Show in the edgeworth box the Pareto efficient set (contains Pareto efficient bundles) d) Can the economy still have a competitive equilibrium? If can, illustrate this state in an Edgeworth box. You need to show a possible initial endowment, the budget lines of both consumers, and the equilibrium allocation of the economy 4. Assume that a competitive equilibrium is defined as the excess demand strictly equal to zero in all markets. That is =(p)=0 j=1,,n. Using Brower's fixed point theorem to prove the existence of the competitive equilibrium

Problem Set 3 Prof. Gene Chang (For Ph.D students, due on Dec. 28th in the class, or TA’s office. For M.A. students, due on Dec. 26th, in the class or TA’s office. Late turn-in may result in a penalty in the grade) 1. Kland is a Robinson Crusoe type economy. Robinson, as the consumer and the owner of the firm, has the utility function U(x,l) = a ln x +  ln l , where l is leisure. Robinson’s endowment of leisure/labor is one unit. The production function is x = cL , where c is a constant and L is the labor input. Find the general (Walrasian) equilibrium for this economy. 2. Review the Debreu’s conditions for the existence of general equilibria. Below is a consumer’s preference. The shaded area is the indifference set (where the consumer is indifferent among the points in the set). Which Debreu’s condition(s) is violated? What kind of troubles this preference may cause in the existence of an equilibrium? 3. Consumer A’s preference is represented by the utility function 2 2 1 2 1 2 ( , ) A u x x x x = + and consumer B’s preference is represented by the utility function 1 2 1 2 ( , ) max{ , } A u x x x x = . a) Draw a typical indifference curve for each of them in the 1 x -- 2 x space. b) Who’s preference satisfies with Debreu’s condition for existence of a general equilibrium? If any preference does not satisfy the conditions, tell it violates which Debreu’s condition. c) Show in the Edgeworth box the Pareto efficient set (contains Pareto efficient bundles). d) Can the economy still have a competitive equilibrium? If can, illustrate this state in an Edgeworth box. You need to show a possible initial endowment, the budget lines of both consumers, and the equilibrium allocation of the economy. 4. Assume that a competitive equilibrium is defined as the excess demand strictly equal to zero in all markets. That is ( ) * 0 1,..., j z j n p = = . Using Brower’s fixed point theorem to prove the existence of the competitive equilibrium. X1 X2