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Consider the market with inverse demand p(y) = 10-y. There are infinitely many firms that could enter this market, numbered i = 1, 2, 3, . . . , ∞. The game proceeds in two stages: 1. The firms simultaneously decide whether to enter the market or not. At the end of this stage, all firms observe which firms entered. 2. The firms that entered simultaneously choose their quantities. The cost of entry is F = 2. Each firm that enters then may produce any quantity at zero cost. The firms have identical products. We will use backward induction to find the Subgame Perfect Nash Equilibrium of this game.
(a) Consider only stage 2. Suppose N firms entered this market. Find a symmetric Nash equilibrium among the N firms that entered.
(b) In the Nash equilibrium you found above, what are the post-entry profits of each individual firm that entered, as a function of N?
(c) Now consider stage 1. Suppose every firm knows that if N firms enter in stage 1, then in stage 2 those firms will play the Nash equilibrium you characterized above. Also suppose that each firms knows how many firms will enter the market in equilibrium. If every firm is making an optimal entry decision, what is the equilibrium N∗ ?
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