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1. Consider a firm with the production function: fpl, kq “ l αkβ, for 0 ? α ? 1 and 0 ? β ? 1. Assume that the input prices w and r and the output price p are strictly positive. Let’s first consider the two-step profit maximization: (a) Solve the Cost Minimization Problem for the conditional factor demand functions l c?pw, r, qq and kc?pw, r, qq as well as the Lagrangian multiplier λ?pw, r, qq. (b) Derive the cost function Cpw, r, qq. Derive the marginal cost function and compare it with λ?pw, r, qq. Find the conditions on α and β such that the marginal cost function is increasing, decreasing, or constant in q. (c) Now that we know the cost function Cpw, r, qq, solve the second step of the two-step profit maximization problem for the firm’s supply function q?pw, r, pq. Make sure that for w, r, and p, the firm is indeed maximizing its profit at the q?pw, r, pq you’ve specified. Now let’s move to the one-step profit maximization: (d) Solve the Profit Maximization Problem for the factor demand functions l ?pw, r, pq and k?pw, r, pq. (e) Given the factor demand functions, derive the supply function q?pw, r, pq. Compare your answer in this part to what you’ve got in part (d). If they’re different, then which one is wrong and why? (f) (optional) When α ` β “ 1 and p ? MC, verify if the cost is equal to zero as predicted by Euler’s Theorem. Next, let’s think about change in the environment: (g) (optional) Suppose r “ p “ 1. When the wage rate w increases, we mentioned in class how the factor demand for labor l ?pw, r, pq will respond to this change. Use the l ?pw, r, pq you’ve solved from above to verify if l ?pw, r, pq always change in the direction we predicted. Explain why it could be inconsistent with our prediction. (h) For this question, assume α “ β “ 1 4 . Suppose w “ 1, r “ 1, and the output price 1 increases to 2. What is the factor demand for capital at prices w “ r “ p “ 1? Calculate the increase in the quantity supplied in the short run and in the long run. Finally, let’s draw some cost curves and think about firm and industry supply: (i) Suppose α “ β “ 1 2 , w “ r “ 1. ¯ k “ 1 in the short run and capital is flexible in the long run. Draw the short run average variable cost curve, short run marginal cost curve, long run average total cost curve and long run marginal cost curve. Find the firm’s short run and long run supply function. (j) Suppose we have two identical firm in the industry with all the parameters same as above. Find the industry’s short run and long run supply function.