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Problem Set #2（我就会做a，后面的不知如何下手啊） due on December 9, 2005, in class 1 Signalling in a Cournot Setting Consider a market in which a homogeneous good is sold, and in which the demand function is p = 18-Q; where Q is the quantity offered by the firm(s). The market exists during two periods. In the first period, there is only one established firm, firm 1, so it is in a monopoly position. Firm 1's production costs are either CG(q) = 0, or CB(q) = 6q. In the second period there is a second competitor, firm 2, that must decide whether or not to enter the market. Firm 2's production costs are Ce(q) = 3q, but this firm must also pay a fixed cost of 20 if it decides to enter the market. If the decision is to enter, then the two firms compete in the second period in Cournot fashion. The discount rate is 1, or in other words, firm 1 maximizes the sum of profts from the two periods. (a) Assume that everyone knows the costs of the established firm. Calculate the quantity decisions of firm 1, q1Gm , q1Bm(in function of marginal costs) when this firm is in a monopoly position, and the decisions q1Gd ,q1Bd , q2Gdand q2Bd of firms 1 and 2 respectively when they are in a duopoly situation. Also, calculate the profits $1Gd, $1BD and $2Gd and $2Bd. Will firm 2 enter in the second period?

(b) If the costs of firm 1 are unknown to firm 2, show that the above decisions are no longer reasonable, since firm 1 has incentives to make firm 2 believe that it is type G independently if this is true or not (that is, no separating equilibrium exist in which each type behaves as if the information were symmetric). (c) Show that a situation in which the production decisions in the first period for each type of firm are q1G = 12 and q1B= 6 is a separating equilibrium. Determine the beliefs of the entrant as to firm 1's type when firm 1's decision is q = 7, q = 8, q = 9, q = 10, or q = 11. (d) Assume that the probability that firm 1 is type G is 1/2. Would a situation in which both types of firm decide q1 = 9 be a pooling equilibrium? (e) Would the situation of (d) be a pooling equilibrium if the probability that firm 1 is type G where 9=10?

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关键词：Signalling Signal sign ING Lin 题目 Signalling

I would like give you some hints rather than doing them for you.

In (a), in accordance with the conditions provided, firm 2 will enter the market to compete with firm 1. Firm 2's cost function is solely determined, while firm 1 has two choices. Just calculate the payoffs in two equilibra and this can determine firm 1's best respond.

In (b), now the cost funcion of firm 1 is unknown to firm 2, and in another words, firm 2 does not know the payoff if it enters the market. It is easily to get that firm 1 can earn more if firm 2 give up entering, and it is natural for firm 1 to convince firm 2 not to enter. To calculate the payoff of firm 2 when it enters the market and firm 1's type is G, and the result shows that firm 2 should not enter the market. Thus, firm 1 has incentives to make firm 2 believe that it is type G for the sake of its own profit. To proove there is no separating equilibrium, consider that firm 2 believes that firm 1 is type G but actually not, then firm 2 will, plausibly, make an adverse selection.

In (c), similarly, you can do it according to the definition of separating equilibrium.

In (d), just calculate payoffs when q1=9. By comparison, check whether it is an equilibrium.

In (e), it seems that the question give you a probability of type G. calculate the payoffs.