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Consider a differentiated duopoly market in which firms compete by selecting prices and produce to fill orders. Let p1 be the price chosen by firm 1 and let p2 be the price of firm 2. Let q1 and q2 denote the quantities demanded (and produced) by the two firms. Suppose that the demand for firm 1 is given by q1 = 22 - 2p1 + p2 , and the demand for firm 2 is given by q2 = 22 - 2p2 + p1 . Firm 1 produces at a constant marginal cost of 10 and no fixed cost. Firm 2 produces at a constant marginal cost of c and no fixed cost. The payoffs are the firms' individual profits.
Now suppose that firm 1 does not know firm 2's marginal cost c. With probability 1/2 nature picks c = 14, and with probability 1/2 nature picks c = 6. Firm 2 knows its own cost (that is, it observes nature's move), but firm 1 only knows that firm 2's marginal cost is either 6 or 14 (with equal probabilities). Calculate the best-response functions of player 1 and the two types (c = 6 and c = 14) of player 2 and calculate the Bayesian Nash equilibrium quantities.
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