Conditions for an optimum require that both partials

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Conditions for an optimum require that both partials be set equal to zero and the resulting equations be solved simultaneously for optimal values of Q C and Q D : Q D * = 150 units and QC* = 175 units. P = 600 – 150 – 175 = $275, so P* = $275.
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Myisha Coleman Week 6 CYU b) Determine the total profits for each firm at the equilibrium output found in Part (a). C * = 25,000 + 500(175) (175) 2 175(150) = $5,625. D * = 20,000 + 475(150) (150) 2 175(150) = $2,500 2. Assume that two companies (A and B) are duopolists who produce identical products. Demand for the products is given by the following linear demand function: P = 200 – QA – QB where QA and QB are the quantities sold by the respective firms and P is the selling price. Total cost functions for the two companies are TCA = 1,500 + 55QA + Q2A TCB = 1,200 + 20QB + 2Q2B Assume that the firms act independently as in the Cournot model (i.e., each firm assumes that the other firm’s output will not change). b) Determine Firm A, Firm B and total industry profits at the optimal solution found in Part (a). a.  A = PQ A TC A = (200 Q A Q B )∙Q A (1500 + 55 Q A + Q A 2 ) A = 1500 + 145Q A 2Q A 2 Q A Q B  A /Q A = 145 4Q A Q B = 0
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Myisha Coleman Week 6 CYU B = PQ B TC B = (200 Q A Q B )∙Q B (1200 + 20 Q B + 2Q B 2 ) B = 1200 + 180Q B 3Q B 2 Q A Q B  B /Q B = 180 6Q B Q A = 0 Solving these two equations simultaneously results in: Q B * = 25 and Q A * = 30, accordingly, P = 200 -30 -25 = $145, or P* = $145 for both firms. b.  A * = 1500 + 145(30) 2(30) 2 30(25) = $300. B * = 1200 +180(25) 3(25) 2 30(25)= $675. * = 300 + 675 = $975. 5. Alchem (L) is the price leader in the polyglue market. All 10 other manufacturers (follower [F] firms) sell polyglue at the same price as Alchem. Alchem allows the other firms to sell as much as they wish at the estimated price and supplies the remainder of the demand itself. Total demand for polyglue is given by the following function (QT = QL + QF): P = 20,000 – 4QT Alchem’s marginal cost function for manufacturing and selling polyglue is MCL = 5,000 +QL The aggregate marginal cost function for the other manufacturers of polyglue is EMCF = 2,000 + 4QF
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