Consider a market in which consumer-type x is uniformly distributed on the unit interval. Consumers demand 0 or 1 unit (they buy at most one unit overall in the market). Firm A is located at 0 and firm B at 1. Firms incur constant marginal costs of production c = 1/2. There is a mass 1 of consumers. A consumer located at x ∈ [0;1] obtains utility ux = r−x−pA if she buys from firm A; ux = r−(1−x)−pB if she buys from firm B; and 0 if she does not buy. If more than one firm is present, firms simultaneously set prices.

(a) Consider the monopoly problem in which only firm A is present and sets its prices to maximize profits. Calculate the monopoly solution depending on r where r ∈ [0;4]. 

(b) Consider the duopoly problem in which firms compete in prices. Solve for Nash equilibrium depending on r. [Be careful, make sure that you characterize the equilibrium for any parameter r ∈ [0;4].]

(c) Compare the price level in a duopoly to the price level under a monopoly. Are duopoly pricing necessarily lower than monopoly prices? Explain your findings. 

(d) Suppose that firm A is the incumbent and, thus, has already entered the market. Suppose at a stage prior to the price-setting stage, firm B decides whether to enter. To enter the firm has to pay an entry cost K which is sunk. Depending on r calculation the critical sunk cost ˆ K above which firm B would not be willing to enter the market.