Joint profit maximizing quantity and price in duopoly.
A duopoly faces an industry demand curve of P=115-2Q. If the industry charge the same price, they share the demand so that each face a demand curve of P=115-4qi for i=1,2. Assume firms are profit maximizers.
a) Suppose the firms share the same marginal cost function of MC=25+qi.
(i) Solve for firm 1's preferred quantity and price.
(ii) Solve for firm 2's preferred quantity and price. Are they the same as or different from those of firm 1? Why?
b) Now suppose that the 2 firms have different marginal costs as follows: mc1=15+2q1, mc2=7+q2. (They still face a common demand curve)
(i) Solve for firm 1's preferred quantity and price.
(ii) Solve for firm 2's preferred quantity and price. Are they the same as or different from those of firm 1? Why?
c) Using the case of different marginal costs:
(i) Solve for the joint-profit maximizing quantity and price.
(ii)Using the information given and your answer above, solve for (the value of) MC at the joint-profit maximizing quantity.
(iii) How much would each firm produce if it charged the joint-profit maximizing price? Compute. Explain your answer.