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Supply functions and short-run equilibrium An industry consists of two firms. Firm 1 has a total cost function given by TC 1 (91 ) = q1 + 91 , while firm 2 has a total cost function given by TC2 (92) = 392 +592 1 for q > 0. (a) Let P denote the (exogenous) price at which each firm can sell its output. Write down each firm's profit-maximization problem and the associated first-order conditions (FOCs). (b) Derive the firms' supply functions q1(P) and q1(P) and verify that these functions are linearly increasing in P. (c Derive the industry supply curve S(P). [Hint: Draw a picture and remember the notion of horizontal summation. You should demonstrate that the industry supply curve is a piecewise function in P] (d) Again assuming that the firms act as price takers, find the industry equilibrium when the industry demand curve is given by QD (P) = = --P [Hint: It may be useful to add the demand curve to the graph considered in part (c)]. (e) Calculate the output and profit of each firm under the equilibrium characterized in part (d)