北大光华微观经济学作业 Problem Set-4

Problem Set 4

1．Two firms A and B produce luxury sheepskin auto seat covers. Each firm has a cost function given by 2()20C q q q =+. The market demand for these seat covers is represented by the inverse demand curve 2002P Q =−, where A B Q q q .

=+1) If each firm acts to maximize its profits, taking its rival’s output as given, what will be Cournot equilibrium output level for each firm? What are the profits for each firm?

2) It occurs to the managers of firm A and B that they could do a lot better by

colluding. If the two firms collude, what will be the profit-maximizing choice of output? What is the output and the profit for each firm?

3) The managers of these firms realize that explicit agreements to collude are illegal. Each firm must decide on its own whether to produce the Cournot quantity or the cartel quantity. What output strategy is each firm likely to purse? Why?

2. Two firms compete by choosing price. Their demand functions are

1122220;20Q P P Q P =−+=−+1P L

where P 1 and P 2 are the prices charged by each firm, respectively, and Q 1 and Q 2 are the resulting demands. Marginal costs are zero.

1) Suppose the two firms set their prices at the same time . Find the resulting Nash

equilibrium prices.

2) Suppose Firm 1 sets its price first , and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be? Does the first mover have a first-mover advantage?

3. Suppose a firm’s production function is given by 212Q L =−, for L = 0 to 6, where L is labor input per day, and Q is output per day. Derive and draw the firm’s demand for labor curve if the out sells for $10 in a competitive market. How many workers will the firm hire when the wage rate is $30 per day?

4．Consider the following game, where x > 0:

Firm 2

High Price Low Price

Firm 1

High Price 140, 140 20, 160 Low Price 90+x , 90-x 50, 50

a) For what values of x do both firms have a dominant strategy? What is the Nash equilibrium (or equilibria) in these cases?

b) For what values of x does only one firm have a dominant strategy? What is the Nash equilibrium (or equilibria) in these cases?

c) Are there any values of x such that neither firm has a dominant strategy? Ignoring mixed strategy, is there Nash equilibrium in such cases?

5．考虑下面一个两人静态博弈的报酬矩阵(payoff matrix)：

参与人2

Z

Y

X

A 18, 4 8, 12 15, 10

参与人1 B 4, 6 6, 7 14, 8

C 6, 8 4, 2 6, 11

a)找出上述博弈的纳什均衡，并简要说明理由。

b)假定上述静态博弈采取序贯的方式 (sequential) 进行：参与人1首先选择其策略(选择

之后就不可更改)，然后参与人2在观察到参与人1的策略之后再进行选择。这个博

弈的均衡策略会是怎样的？简要说明理由。

c)现在假定博弈是这样进行的：参与人1首先选择其策略，然后参与人2在观察到参

与人1的策略之后再进行选择。但是，在参与人2做出选择之后，参与人1又可以自

由改变其策略。此时博弈的均衡又是什么？为什么？简要说明理由。