Monopoly and Duopoly with a Manager: Suppose there are two rms competing in the Cournot oligopoly model with price in the market 10(91: 92) Z 4 (I1 (12- Suppose that each rm 3' = 1, 2 has cost function 001,-) = qi so the rm's prots are 773-01142) : @(qhqg) c)q,. Suppose that Firm 1's quantity is set by a manager whose goal is not simple prot maximization but to maximize 7r1(q1, Q2) l dql. That is, the manager may care about market share for ego reasons or make money o the production process, etc. (a) For any a E [0, 3) solve for the quantity rm 1 would produce if rm 2 produces 92- (b) Suppose rst that rm 2 is not in the market, and so rm 1 is a monopoly. How do the quantity, price, and prots of the monopolist depend on a? (c) Suppose now that rm 2 is also in the market so we have a duopoly. Determine the equilibrium quantities for any a E [0, 3) when Firm 2 seeks to maximize 7r2(q1, Q2). (d) What are Firm 1's prots as a function of o: in the duopoly? (e) Suppose Firm 1 could choose as for its manager. What a would it choose to maximize its own prots? (f) Provide an economic intuition for your answer in part (e)