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4. You are operating a firm that uses labor (L) and capital (K) to produce widgets. L is variable in the short run, but capital is fixed at a level K. (a) Write down the firm's short-run profit maximization problem where y denotes output, p denotes the price of the firm's output, w denotes the cost of L, and r denotes the cost of K. (b) Now derive an expression for the firm's short-run isoprofit function in which y is a function of L, K, p, w, and r. (c) Now say that y = f (L, K) = 6L 3 K and p =3, w =6, r = 1, and K = 1. If the firm is maximizing profits, how many units of L will it hire? What is the firm's output? What is the firm's short-run profit? (d) Now say that w falls to 4. How does this reduction in labor costs affect the amount of labor hired by the firm, the firm's output, and the firm's profit level? Assume that p = 3, r = 1, and K = 1.4. You are operating a firm that uses labor (L) and capital (K) to produce widgets. L is variable in the short run, but capital is fixed at a level K. (a) Write down the firm's short-run profit maximization problem where y denotes output, p denotes the price of the firm's output, w denotes the cost of L, and r denotes the cost of K. (b) Now derive an expression for the firm's short-run isoprofit function in which y is a function of L, K, p, w, and r. (c) Now say that y = f (L, K) = 6L 3 K and p =3, w =6, r = 1, and K = 1. If the firm is maximizing profits, how many units of L will it hire? What is the firm's output? What is the firm's short-run profit? (d) Now say that w falls to 4. How does this reduction in labor costs affect the amount of labor hired by the firm, the firm's output, and the firm's profit level? Assume that p = 3, r = 1, and K = 1