HW15_sol

HW15_sol - Intermediate Microeconomics Winter 2008 Problem...

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Intermediate Microeconomics, Winter 2008 Problem Set No 15 Solutions Q1. Suppose a monopolist faces a demand function D ( p ) = 100 - Ap and suppose costs are c . If A = 5 and c = 2, ﬁnd the optimal quantity and the optimal price that the monopolist will set. Find the optimal price and the optimal quantity as a function of general A and c . In order to write proﬁt as a function of quantity only, we need to ﬁnd the inverse demand function faced by the monopolist. The demand function is q = 100 - Ap , so, solving for p , the inverse demand function is p ( q ) = 100 A - q A . The monopolist’s problem is to choose q to maximize π ( q ). max q π ( q ) max q q * p ( q ) - C ( q ) max q q * ( 100 A - q A ) - cq We can conﬁrm that proﬁt is a concave function of q , and therefore the maximum is attained where: π 0 ( q ) = 0 100 A - 2 q A - c = 0 2 q A = 100 A - c 2 q = 100 - Ac q = 50 - Ac 2 Plugging in A = 5 and c = 2, we get q = 45. Q2. Suppose the demand function for AA and UA for ﬂights between Chicago and Detroit is D ( p ) = 100 - Ap . If AA and UA each choose to supply a quantity of 30, what will be the price? If UA chooses a supply of q UA =30, what will be the price as a function of the quantity that AA supplies? Given the choice q UA = 30, and given constant marginal costs c = 1, what is the optimal supply of AA? (You might want to choose

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HW15_sol - Intermediate Microeconomics Winter 2008 Problem...

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