Question

Four firms (A, B, C, and D) play a simultaneous-move pricing game. Each firm (i) may choose any price Pi ∈ [0, ∞) with the goal of maximizing its own profit. (Firms do not care directly about their own quantity or others’ profits.) Firms A and B have MC = 10, while firms C and D have MC = 20. The firms serve a market with the demand curve Q = 100 – P. All firms produce exactly the same product, so consumers purchase only from the firm with the lowest price. If multiple firms have the same low price, consumers divide their prices evenly among the low-priced firms.

a. There are many equilibria in this simultaneous-move pricing game. Provide one equilibrium combination of prices, and argue that no firm has a unilateral incentive to deviate from these prices.

b. Provide a second equilibrium that is distinct from the combination of prices you provided in part (a). As in (a), argue for the absence of a unilateral incentive to deviate from these prices. (For a set of actions to be distinct, one or more prices must be different from the price combination in part a. It is not necessary to have all four prices be different.)

Answer 1

a) Since all firms are setting prices simultaneously, each firm will solve its own profit maximization problem. In the short run for competitive market firms, equilibrium is established when price is equal to the firm's marginal cost. In this case, firms 1 and 2 will have an equilibrium price of 10 and firms 3 and 4 will have an equilibrium price of 20. Since firms 3 and 4 cannot lower their prices as that would lead to negative profits and firms 1 and 2 charge the same lower price of 10, consumers will divide their consumption evenly among firms 1 and 2.

Solving the market demand equation for P = 10, we get Q = 90. Hence, firms 1 and 2 will produce 45 units each and firms 3 and 4 will not produce anything. The equilibrium prices are (P1,P2,P3,P4) = (10,10,20,20)

b) Since firms 3 and 4 cannot decrease their prices, firms 1 and 2 can move together and set a price higher than 10. For instance, if firm 1 and 2 set a price of 11, both will increase profits without reduction in market share. But if only one of them increases the price, the entire market demand would go to the firm with lower price. Hence, (P1,P2,P3,P4) = (11,11,20,20) is a set of equilibrium prices with both firms 1 and 2 producing 44.5 units each.

But this can only be established by a bilateral movement in prices. If the movement were unilateral, the entire market demand would go too the firm with the lower price. Hence, no firm has an incentive to deviate from the equilibrium prices from part a) which are (P1,P2,P3,P4) = (10,10,20,20)