1. The production function for a firm is given by the following expression: q = f(L, K) = [1/2 . K1/3 Suppose that each unit of labor (L) can be hired for a wage of w = 3. In addition, assume that each unit of capital (X) can be rented at a rate of r = 2. Solve for the MPL and MPK. b. Solve for the slope of an isoquant at any bundle of capital and labor. c. What is the slope of the isocost curve? d. If q units of output are being produced at the lowest possible cost to the firm, solve for L(q) and K(q), the amounts of labor and capital that would be used to produce q units of output. e. Using your answers to part (d) above, solve for the long-run cost function that specifies the lowest cost the firm would incur to produce q units of output. f. Using your answer to part (e) above, solve for the long-run marginal cost function. g. Suppose the firm had invested in 8 units of capital in the short-run (i.e. K = 8). Solve for the level of labor (L(q)) that would be required to produce q units of output in the short-run. h. Using your answer to (g) above, solve for the short-run cost function. i. Solve for the short-run marginal cost-function. j. For what level of q are the short-run and long-run costs equal? k. At the level of q that equates the short-run and long-run costs, what are the short-run and long- run marginal costs? I. If the price of the good is $20 (p = 20), what is the firm's profit maximizing level of output in the short-run (assuming K = 8)? m. If the price of the good is $20 (p = 20), what is the firm's profit maximizing level of output in the long-run? ARE 100A - Fall 2020 n. In the short-run, what would the price need to be in order for the firm to reduce their output to q = 2? In the long-run, what would the price need to be in order for the firm to reduce their output to q = 2