Let the inverse demand curve be given by P = 60 - 4Q, Q = q1 + q2. Costs for each firm are a constant variable cost of 6, a unit capacity charge of 6, and setup costs of f . The incumbent and the entrant play the game of Dixit.
(a) What is the incumbent's marginal cost function for a given capacity? Derive the incumbent's marginal revenue function.
(b) For k1 = 5, what is the incumbent's best-response function? Why? Derive the entrant's best-response function.
(c) What are the Nash equilibrium quantities in the quantity subgame when k1 = 5? Characterize the equilibrium to the quantity subgame for any k1.
(d) For fixed costs of 25 and 64 find the subgame perfect equilibrium to the game of Dixit.
(e) Why is it that the equilibrium is not the same as if the two firms had just played the simple Cournot game. Explain by showing that one firm would want to deviate from the simple Cournot equilibrium in the quantity subgame. What is the strategic move made by firm 1? Is it important that the costs of capacity be sunk?
(f) Suppose that firm 1 built capacity equal to 7. Its motivation for building this is that 7 is the limit output. Would the entrant be deterred from entry by this limit output? Why?