? E XERCISE 286.1 (Cournot’s duopoly game with imperfect information) Write down the maximization…

? EXERCISE 286.1 (Cournot’s duopoly game with imperfect information) Write down the maximization problems that determine the best response function each type of each player.      (Denote by q0, q£ , and qh  the outputs of types 0, £, and h of firm 1,

and by qL and qH the outputs of types L and H of firm 2.) Now suppose that the inverse demand function is given by P(Q) = α − Q for Q ≤ α and P(Q) = 0  for

Q > α. For values of cH and cL close enough that there is a Nash equilibrium in which all outputs are positive, find this equilibrium. Check that when π = 0 the equilibrium output of type 0 of firm 1 is equal to the equilibrium output of firm 1

you found in Exercise 285.1, and that the equilibrium outputs of the two types of firm 2 are the same as the ones you found in that exercise. Check also that when π = 1 the equilibrium outputs of type £ of firm 1 and type L of firm 2 are the same

as the equilibrium outputs when there is perfect information and the costs are   c

and cL, and that the equilibrium outputs of type h of firm 1 and type H of firm 2 are the same as the equilibrium outputs when there is perfect information and the

 

 

Figure 287.1 The information structure for the model in Section 9.5.2, in which firm 2 does not know whether firm 1 knows its cost. The frame labeled i: x encloses the states that generates the signal x for firm i.

 

 

costs are c and cH . Show that for 0 1, the equilibrium outputs of types L

and H of firm 2 lie between their values when π = 0 and when π = 1.