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Two firms are competing in the market for widgets. The output of the two firms are perfect substitutes, so that the demand curve is given by Q = max{ a− p; 0}, where p is low price.

Firm i has constant marginal cost of production 0 < ci <a , and no capacity constraints. Firms

simultaneously announce prices, and the lowest pricing firm has sales equal to total market demand.

The division of the market in the event of a tie (i.e., both rms announcing the same price) depends

upon their costs: if firms have equal costs, then the market demand is evenly split between the two

firms; if firms have dierent costs, the lowest cost firm has sales equal to total market demand, and

the high cost firm has no sales.

1. Suppose c1 = c2 = c (i.e., the two firms have identical costs). Restricting attention to pure

strategies, prove that there is a unique Nash equilibrium. What is it? What are firm profits

in this equilibrium?

2. Suppose c1 < c2 < ( a+ c1)=2. Still restricting attention to pure strategies, describe the set

of Nash equilibria. Are there any in weakly undominated strategies?

3. We now add an investment stage before the pricing game. At the start of the game, both rms

have identical costs of cH, but before the rms announce prices, firm 1 has the opportunity to

invest in a technology that gives a lower unit cost cL of production (where cL < cH <

( a+ cL)=2). This technology requires an investment of k > 0. The acquisition of the technology is public before the pricing game subgame is played. Describe the extensive form

of the game. Describe a subgame perfect equilibrium in which firm 1 acquires the technology

(as usual, make clear any assumptions you need to make on the parameters). Is there a

subgame perfect equilibrium in which firm 1 does not acquire the technology? If not, why

not? If there is, compare to the equilibrium in which firm 1 acquires the technology.

谢谢！。。请写详细些。。

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