Two firms, Firm 1 and Firm 2, compete by simultaneously choosing prices. Both firms sell an identical product for which each of 100 consumers has a maximum willingness to pay of $10. Each consumer will buy at most 1 unit, and will buy it from whichever firm charges the lowest price. If both firms set the same price, they share the market equally. Costs are given by c1(q1)=4q1. Because of governmental regulation, firms can only choose prices which are multiples of $0.25, and they cannot price above $10.
a) Write down the profit function of Firm i as a function of p1 and p2.
b) If Firm 1 chooses p1 = 6, Firm 2's best response is to set what price?
c) If Firm 1 chooses p1 = 4.5, Firm 2's best response is to choose what price?
d) If Firm 1 chooses p1 = 3, Firm 2's best response is to choose what price?
e) This game has 3 Nash equilibria in pure strategies. Find them all.
f) Now suppose both firms are capacity-constrained and can produce at most 45 units. If firms set different prices, consumers will first buy from the firm charging the lower price. Once that firm's supply is exhausted, consumers will buy from the firm charging the higher price until that firm's supply is exhausted. What are Firm 1's equilibrium profits?