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- Suppose a firm has the following production function: Q(L, K) = 2K¹/2 1/2 Recall that the isocost line is as follows: C=wL+rK 1. What is the (long run) optimal choice of L and K for a given Q, w, and r? In other words, provide a formula for the optimal choice of labor L* (w, r, Q) and capital K* (w, r, Q) as a function of the parameters Q, w, and r. 2. Given Q = 40, w = 16, and r 4, what are the optimal levels of labor and capital, L* and K*? What is the cost of producing Q = 40 at these input prices? = = 3. Suppose now that you are in the short run, Q = 36, w = 1, r 2, and the capital level is fixed at K = 9. What is the optimal level of labor in the short run? What is the cost of producing Q = 36 in the short run at these input prices?Suppose a firm has the following production function: Q(L, K) = min{K, 2L} Recall that the isocost line is as follows: C = WL + rK 1. What is the (long-run) optimal choice of L and K for a given Q, w, and r? In other words, provide a formula for the optimal choice of labor L* (w, r, Q) and capital K* (w, r, Q) as a function of the parameters Q, w, and r. 2. Given Q = 16, w = 6, and r = 2, what are the optimal levels of labor and capital, L* and K*? What is the cost of producing Q = 16 at these input prices? 3. Suppose now that you are in the short run, Q = 16, w = 6, r = 2, and the capital level is fixed at K = 20. What is the optimal level of labor in the short run? What is the cost of producing Q = 16 in the short run at these input prices? Would it be possible to meet Q = 16 if K = 4 in the short run?Assume the Cobb-Douglas production function is Q= L ^0.75 K^0.5 and if price of labor per day is 5 birr and price of capital per day is 10 birr, and if total outlay (cost budget) per day is 400 birr, A.Find L and K that maximize out put B.What is the maximum out put the equilibrium L* and K*
- Suppose a firm has the following production function: Q(L,K) = 2K¹/2 1/2 Recall that the isocost line is as follows: C=wL+rK 3. Suppose now that you are in the short run, = Q = 36, w = 1, r 2, and the capital level is fixed at K = 9. What is the optimal level of labor in the short run? What is the cost of producing Q = 36 in the short run at these input prices?Question 3: Suppose a firm has the following production function: Q(L, K) = 2L + K Recall that the isocost line is as follows: C=wL+rK 1. What is the (long-run ) optimal choice of L and K for a given Q, w, and r? In other words, provide a formula for the optimal choice of labor L* (w, r, Q) and capital K* (w, r, Q) as a function of the parameters Q, w, and r. 2. Given Q = 20, w = 6, and r = = 1, what are L* and K*?The production function for a product is given by q= 10K^(1/2)L^(1/2) where K is capital, and L is labor and q is output d) Now suppose w =30 and r = 120. What is the minimum cost of producing q=1000. (You must show your work by clearly writing the equations that you use to derive the cost minimizing levels of L and K.) e) Now suppose that the firm is in the short run and cannot vary the amount of capital. That is, it must use the same amount of capital as in part d). However, the firm wants to produce 1200 units of output. How much labor should it use to minimize its cost and what is the minimum cost of producing q =1200?
- Question 2: Suppose a firm has the following production function: Q(L,K) = min{K, 2L} Recall that the isocost line is as follows: C=wL+rK 1. What is the (long-run) optimal choice of L and K for a given Q, w, and r? In other words, provide a formula for the optimal choice of labor L* (w, r, Q) and capital K* (w, r, Q) as a function of the parameters Q, w, and r. 2. Given Q = 16, w = 6, and r = 2, what are the optimal levels of labor and capital, L* and K*? What is the cost of producing Q = 16 at these input prices? 3. Suppose now that you are in the short run, Q = 16, w = 6, r = 2, and the capital level is fixed at K = 20. What is the optimal level of labor in the short run? What is the cost of producing Q = 16 in the short run at these input prices? Would it be possible to meet Q = 16 if K = 4 in the short run?The production function of a firm is f : R² → R, (L, K) + 10L + 16K – 2L? – K?, and the per unit costs of the quantities of labor input L and capital input K are 4 and 2 dollars, respectively. The firm wants to maximize its production of output subject to the total cost of production not exceeding 20 dollars. The firm can only employ non-negative quantities of labor and capital. (a) Write the firm's problem as an optimization problem with inequality constraints. (b) Identify any points in the constraint set that violate the non-degenerate constraint. qualification for the problem in (a). (c) Construct the Kuhn-Tucker Lagrangian by defining L(L, K, X) for the probļem in (a). (d) List the Kunh-Tucker first-order conditions for the problem in (a). (e) Solve the first-order conditions from (d) to find the critical points of the Lagrangian. (f) Justify that the problem in (a) is a concave program and conclude with the values of labor and capital that solve the firm's problem. (g) Based on…A firm's production function is q = 26x^0.33y^0.67, where x and y are the amounts of factors x and y that the firm uses as inputs. If the firm is minimizing unit costs and if the price of factor x is 6 times the price of factor y, the ratio in which the firm will use factors x and y is closest to. A. x/y = 0.08 B. x/y = 0.25 C. x/y = 0.5 D. x/y = 2.4
- Which of these production functions have diminishing marginal returns to labor? a)F(K,L) = 2K+15L b)F(K,L)=√KL c) F(K,L) = 2√K+15√LA firm's production function is given by Q = 8L K + 7L Unit capital and labour costs are $1 and $7 respectively and total input costs are $500. (a) Find the values of L and K which maximise output. (b) Verify that the ratio of marginal product to price is the same for both inputs at the optimum. (a) The value of K = and the value of L = !! (Round your answer to three decimal places as needed.) (b) The ratio of the marginal product of capital to the unit price of capital is MPK PK Work out the ratio of the marginal product of labour to the unit price of labour MP %3! PL (Give your answers to the nearest whole number as needed.) Hence the ratio of marginal product to price is the same for all inputs. Any slight discrepancy in the answers is due to rounding.A firm can manufacture a product according to the production function Q = F(K, L) = K0.5L0.5. (a) What is the average product of labor, APL, when the level of capital is fixed at 36 units and the firm uses 16 units of labor? (b) What is the marginal product of labor, MPL, when capital is fixed at 36 units? (c) Suppose capital is fixed at 36 units. If the firm can sell its output at a price of $100 per unit of output and can hire labor at $40 per unit of labor, how many units of labor should the firm hire in order to maximize profits?