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Exercise 11.3 Consider the following duopoly model. There are two firms sup- plying a market where demand is given by p(Q)- a-bQ. Firm i produces qi units of output and so the total level of production is q1q2. Both firms face the same constant marginal cost, so the cost of producing qi for firm i įs cqỉ. Thus the profit functions of firms 1 and 2 respectively, are given by:(a) Suppose that each firm takes the output of the other firm as fixed, and calcu- lates its own best choice of quantity. In this case, firm i solves 0, taking as given the value of qj. Calculate for the linear demand case the equations: di 兀 0q2 Solve for q1 and q2 and denote the solutions qf and q2, where d denotes duopoly Compare the value of Qd, with the monopoly level of output. (b) Suppose now that we have n firms instead of two. This case is called oligopoly. Let the output of the ith firm be q°, where o denotes oligopoly, and see if you can solve for Q-91 + q + + q%. (c) Suppose now that a government tax per unit of output is imposed on each firm with tax ti per u : nit on firm i. Then the profit functions become R ll epeat the exercise in (a), and solve for q1 and q2. In this case, q1 and q2 wi depend on t1 and t2 (and the other parameters. α,b,c), so we can write q1(k1,t2) and q2(t1,t2). The total tax revenue of the government is T(t1,t2) t1q1(ti,t2)+ t2q2(t1,t2). Maximize T with respect to ti and t2, and check that the second-order conditions for a maximum are satisfied

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