4 Enter, then Compete, with Many Potential Entrants
Suppose that there are infinitely many potential entrants $i = 1, 2, \dots$ in the "enter, then
compete" game. At the beginning of the game, firms simultaneously decide whether to enter
the market or not, where the entry cost is 10 for each firm. Next, any firm that entered the
market chooses its production level. If multiple firms entered, these firms simultaneously
choose their production levels. A firm that entered the market can observe the other firms
that entered before choosing its production level.
Suppose that firm $i$'s production level $q_i$ is chosen from $[0, \infty)$ and the marginal cost of
production is 2 for each firm. The market price is given by $p(q) = 14 - q$, where $q$ is total
production. If a firm does not enter the market, its payoff is zero, while if a firm does enter
the market, its payoff is revenue minus costs.
a. Find a pure-strategy subgame-perfect equilibrium. How many firms enter the market?
(Hint: use backward induction. That is, solve the subgame where $N$ firms enter the
market for each $N = 1, 2, \dots$, and then consider the entire game.)
b. Now suppose that the entry cost is 3, rather than 10. Find a pure-strategy subgame-
perfect equilibrium. How many firms enter the market?
