In a market with annual demand Q = 100 − p, there are two firms, A and B, that make identical products. Because their products are identical, if one charge a lower price than the other, all consumers will want to buy from the lower-priced firm. If they charge the same price, consumers are indifferent and end up splitting their purchases about evenly between the firms. Marginal cost is 10 and there are no capacity constraints. A. What are the single-period Nash equilibrium prices, pA and pB?
B. What prices would maximize the two firms’ joint profits? Assume that one firm cannot observe the other’s price until after it has set its own price for the year. Assume further that both firms know that if one undercuts the other, they will revert forever to the noncooperative behavior you described in (a).
C. If the interest rate is 10%, is one repeated-game Nash equilibrium for both firms to charge the price you found in part (b)? What if the interest rate is 110%? What is the highest interest rate at which the joint profit-maximizing price is sustainable?
D. Describe qualitatively how your answer to (c) would change if neither firm was certain that it would be able to detect changes in its rival’s price. In particular, what if a price change is detected with a probability of 0.7 each period after it occurs?
Return to the situation in part (c), with an interest rate of 10%. But now suppose that the market for this good is declining. The demand is Q = A − p with A = 100 in the current period, but the value of A is expected to decline by 10% each year (i. e., to 90 next year, then 81 the following year, etc).
E. Now, is it a repeated-game Nash equilibrium for both firms to charge the monopoly price from part (b)?