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2. Consider a market containing four identical firms, each of which makes an identical product. The inverse demand for this product is P = 100 - Q. The production costs for firms 1, 2, and 3 are identical and given by C(q;) = 20qi where qi is the output of firm i, i = 1,2,3. The production cost function of firm 4 is C(q4) = 30q4. Assume that the firms each choose their outputs to maximize profits given they each act as a Cournot competitor. a) Derive the Cournot equilibrium output for each firm, the product price, and the profits of all four firms. Remark: in answering (a), you can assume that firms with identical costs (firms 1, 2, and 3) will produce the same output. Under this assumption, the system of four equations (i.e., best replies) reduces to two equations in which you need to solve for firm 4's output13.8. In a two-tier market, there are two upstream firms, Up and Up, and two downstream firms, D, and D2. Consider the following dynamic game. In stage one, the upstream firms produce y, and y2 quantities of the same good at zero cost and sell it at a wholesale price, w, to the two downstream retailers. In stage two, the two downstream firms sell quantities q, and q2 at a price p to consumers whose inverse market demand is given byp = 180 - O, where ? = 91 + 92. The only costs incurred by the downstream firms is the per-unit cost, w, of acquiring the goods. Find the SPNE quantities9 , and 72, retail price p*, and wholesale price w*. (Hint: Suppose that the upstream firms have set the wholesale price w. Solve 273 for the stage two symmetric downstream Cournot quantity, q*, as a function of w. Then find the total quantity, *, sold downstream; this equals y1 + 12. Invert this function to obtain w as a function ofy, and y2; this is the inverse demand facing the upstream producers in stage one. Finally, solve the stage one upstream symmetric equilibrium)8-2. (Monopoly input supply) An upstream firm is the monopoly supplier of an input which it produces at constant marginal and average cost 50 per unit and sells to two downstream firms. The downstream firms use one unit of the input, in combination with other inputs, to produce a final good that they supply to a market with inverse demand equation p = 1100- - (91 + 92) . The two downstream firms compete as Cournot duopolists. Downstream firm 1 has constant marginal and average cost 100 per unit of output for all inputs except the input produced by the upstream monopolist. Downstream firm 2 has constant marginal and average cost 200 per unit of output for all inputs except the input produced by the upstream monopolist. (a) Find the profit-maximizing input price w if the upstream firm must charge an identical price to both downstream firms. Find the equilibrium outputs of the downstream firms, the final good price, and the profits of all three firms. (b) The profit-maximizing input prices if the upstream firm is allowed to price discriminate are w1 = 525 to the first downstream firm and w2 = 475 to the second downstream firm. Find the equilibrium outputs of the downstream firms, the final good price, and the profits of all three firms