高级微观经济学试卷

武汉大学2011-2012学年第二学期期末考试

经济与管理学院 2010级 数理经济与金融专业 《高级微观经济学（市场与均衡）》试题（A卷）

注意：答题一律写在答题纸上，不要写在试卷或草稿纸上。书写工整，可以用中文或英文答题。第1,2,3题为必做题；第4,5,6题中选作两题。如果6道全做了，只对前5道计分。每道题20分。 必做题:第1,2,3题

1. Consumers are uniformly distributed along a boardwalk that is 1 mile long. Ice-cream prices are

regulated, so consumers go to the nearest vendor because they dislike walking(assume that at the regulated prices all consumers will purchase an ice cream even if they have to walk a full mile).If more than one vendor is at the same location, they split the business evenly.

(a) Consider a game in which two ice-cream vendors pick their locations simultaneously. Show that there exists a unique pure strategy Nash equilibrium and that it involves both vendors locating at the midpoint of the boardwalk.

(b) Show that with three vendors, no pure strategy Nash equilibrium exists.

2. Consider a game in which the following simultaneous-move game is played twice:

Player 2

a1 Player 1 a2

b1

10,10 12,2 13,0 b2

2,12 5,5 0,0 b3

0,13 0,0 1,1 a3

The players observe the actions chosen in the first play of the game prior to the second play. What are the pure strategy subgame perfect Nash equilibria of this game?

3. (The problem of the commons) Lake Ec can be freely accessed by fishermen. The cost of sending a boat out on the lake is r?0.When b boats are sent out onto the lake, f(b)fish are caught in total [so each boat catches f(b)/bfish],where f(0)?0,f'(b)?0and f''(b)?0 at all b?0.The price of fish is p?0,which is unaffected by the level of the catch from lake Ec. (a) Characterize the equilibrium number of boats that are sent out on the lake.

(b) Characterize the optimal number of boats that should be sent out on the lake. Compare this with your answer to (a).

(c) What per-boat fishing tax would restore efficiency?

(d) Suppose that the lake is instead owned by a single individual who can choose how many boats to send out. What level would this owner choose?

选作题:在第4,5,6题中选作两题

4. In a Stackelberg duopoly, one firm is a “leader” and one is a “follower”. Both firms know each other’s costs and market demand. The follower takes the leader’s output as given and picks her own output accordingly (i.e., the follower acts like a Cournot competitor).The leader takes the follower’s reactions as given and picks her own output accordingly. Suppose that firms 1 and 2

2face market demand, p?100?(q1?q2).Firm costs are c1?10q1 and c2?q2.

(a) Calculate market price and each firm’s profit assuming that firm 1 is the leader and firm 2

the follower.

(b) Do the same assuming that firm 2 is the leader and firm 1 is the follower.

(c) Given your answers in parts (a) and (b), who would firm 1 want to be leader in the market?

who would firm 2 want to be the leader?

(d) If each firm assumes what it wants to be the case in part (c), what are the equilibrium

market price and firm profits? How does this compare with the compare with the Cournot-Nash Equilibrium in this market?

5. Consider a market with demand function x(p)?A?Bp in which every potential firm has cost function c(q)?k??q??q2, where ??0and ??0.

(a) Calculate the long-run competitive equilibrium price, output per firm, aggregate output, and number of firms. Ignore the integer constraint on the number of firms. How does each of these vary withA?

(b) Now examine the short-run competitive equilibrium response to a change in A starting from the long-run equilibrium you identified in (a). How does the change in price depend on the level of A in the initial equilibrium? What happens asA??? What accounts for this effect of market size?

6. Consider the following two-period model: A firm is a monopolist in a market with an inverse demand function(in each period) of p(q)?a?bq.The cost per unit in period 1 is c1.In period 2,however,the monopolist has “learned by doing” and so its constant cost per unit of output is

c2?c1?mq1,where q1 is the monopolist’s period 1 output level. Assume a?c and b?m.

Also assume that the monopolist does not discount future earnings. (a) What is the monopolist’s level of output in each of the periods?

(b) What outcome would be implemented by a benevolent social planner who fully controlled

the monopolist? Is there any sense in which the planner’s period 1 output is selected so that “price equals marginal cost”?

(c) Given that the monopolist will be selecting the period 2 output level,would the planner like

the monopolist to slightly increase the level of period 1 output above that identified in (a)?Can you give any intuition for this?

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