Problem 4. Consider a Cournot duopoly game. Two firms compete in quantities in a market with demand function p = 1-91-92. Suppose that each firm has a production cost c e {CL, CH), 3 which is firm's private information. Suppose, than joint distribution of costs is defined as follows: Firm 1/Firm 2 CL CH CL CH - 4 where a e [0, !]. That is, both costs are ex ante equally likely and parameter a captures the correlation between the costs. Assume that costs are small enough, so that the outputs are positive. 1. Suppose that firm / 6 {1, 2) has cost realization cy, je {L, H). What is the posterior belief of firm i about firm's -i costs? 2. Define the strategy of each player and set up the payoff function for each player. 3. Compute the best response for each player and find the Bayes-Nash equilibrium of the game. 4. How does the output of each type of the player change with a? Provide an intuitive explanation for this relation