Question

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 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. [Note: Be careful, make sure that you characterize the equilibrium for any parameter r ∈ [0;4].]
1
(c) Compare the price level in duopoly to the price level under 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 pricesetting stage, firm B decides whether to enter. To enter the firm has to pay an entry cost K which is sunk. Depending on r calculate the critical sunk cost ˆ K above which firm B would not be willing to enter the market.

Answer 1

(If buys fromA) ax=8-X-PA Ux=8- (1-x) -PB (if buys from ) if A is a monopoly, maximum it will charge will be the price level for which ntility will be geso. of price that'll make 8- X-PA=0. Consumer indifferent between buying or PA = X-8 not buying) to anaxinnise PA consumer will be at x=1 T (profits) = x-8 - 12 Cat x =1) 1-8-/ T = Vas 1 =18 if two firers are there, they both will be charging prices such that utility goes to zero... so for firm a for firm B- 8-(1-x) - Pg 20 PA= x-r Po = (1-x)-7. if consumer is at (x) At they both for nash equilibrium o f should charge equel prices - X-X = (1-x) so equilibrium ax= 12 (PA, PB) = (0.5-88.5-8) X=05
​​​​​​
- as we can clearly see than price price charged under charged lower clearly see price charged under part b is much rice charged under price cher part a PA= 100 P = 0.5-8 (under part a) (under past 6) und d) ere, dropoly prices are recessorily lower than monopoly prices. k= sunkcost. a maximum it can pay = maximuon he'll be able to charge K= 0.5-8. has was the price it can charge. this is the maximum from will be willing to pay we market. be willing to pay to enter the