Consider a Cournot duopoly with two firms (firm 1 and firm 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 q_{1}^{-H} and q_{2}^{-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 q_{1}^{-L }and q_{2}^{-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 (q_{1}^{-H} , q_{2}^{-H}, q_{1}^{-L} , q_{2}^{-L}) Compare these quantities with the quantities determined in the case of incomplete information. Interpret the result