Consider a market with an incumbent (I) and a potential entrant (E). The incumbent has innite capacity. Market demand is D(p) = 100 - p. Firms set prices and if the two firms have equal prices then I gets the entire demand. If E has the lower price, it will serve the entire market if it has enough capacity. If E has a lower price and it does not have enough capacity, the consumers with the highest valuations buy from E and I has demand D(pI ) - kE where kE is the capacity of the incumbent. If I has the lower price, I gets the entire demand and E gets nothing. The firms play the following game. Both firms have zero marginal costs of both production and building capacity.
Stage 1: E can choose to enter by paying a sunk entry cost of 100. If it enters, E chooses its capacity and price. It can choose between (capacity,price) combinations of (45,10), (25,15) or (10,40). [i.e., these are its only possible choices]
Stage 2: I sets its price.
(a) Find the subgame perfect Nash equilibrium of this game. Explain why E makes the choice it does.
(b) Would this still be an equilibrium if E set its capacity in stage 1, but sets its price simultaneously with I in stage 2? Explain.