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?