Firms 1 and 2 produce differentiated products at no cost. Demands are given by Q d 1 (p1, p2) = 1 − p1 + p2 3 and Q d 2 (p1, p2) = 1 + p1 3 − p2.
(a) Calculate the Bertrand equilibrium prices.
(b) Suppose firm 1 chooses its price first, knowing this will be observed by firm 2 before firm 2 chooses its price. Determine the subgame perfect equilibrium prices and the associated profit per firm (this is the price-setting analogue of the Stackelberg equilibrium discussed in class). Compare each firm’s profit with that earned in the Bertrand equilibrium (part (a)).
(c) In a reaction-function diagram depict the solutions to parts (a) and (b). Label the associated simultaneous-choice and sequential-choice equilibrium points and add representative iso-profit curves through the equilibrium points. (Your diagram need not be exact, but you should correctly depict the directions of the orientations of the reaction functions and iso-profit curves. Label where the reaction functions’s intersections with either axis, and on the axes show the equilibrium outputs.)
(d) Now suppose the firms choose prices in an infinitely-repeated play of the Bertrand stage game. Future profits are discounted at the common per-period discount factor δ.
i. Determine the symmetric prices (p1 = p2 = p ∗ ) that maximize the firms’ combined profit in a single period.
ii. For what values of δ can the joint-profit-maximizing prices p ∗ be sustained in every period as an equilibrium by a Grim Trigger Strategy that threatens reversion to the Bertrand equilibrium.