Consider a Cournot duopoly with two firms (firm 1 and firm 2) operating in a market with linear inverse Demand P(Q) = x Q where Q is the sum of the quantities produced by both rms and x is a parameter that captures the level of demand in this industry. Firms have 0 cost of production. The demand for the good is uncertain: it is high, x = x_H, with probability 1/3 and it is low, x = xL (< x_H), with probability 2/3. Furthermore, information is asymmetric: rm 1 knows whether demand is high or low but rm 2 does not. All this is common knowledge. The two rms simultaneously choose quantities.

(a) What are the actions, types, beliefs and payos of both rms?

(b) Suppose that 7x_L > x_H. Find the Bayesian Nash equilibrium of the game.

(c) Compare the quantities chosen by rms 1 and 2. Interpret the result.

Now, call q₁^-H and q₂^-H the quantities chosen by rms 1 and 2 if the demand is high under complete information (that is, when both know it). Also, call q₁^-L and q₂^-L  the quantities chosen by rms 1 and 2 if the demand is low under complete information (that is, when both know it).

(d) Find (q₁^-H ,  q₂^-H, q₁^-L , q₂^-L) Compare these quantities with the quantities determined in the case of incomplete information. Interpret the result