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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.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?A Firms production function is given by Q=L?K +1/L when L 2 1 and K 2 1). Find out AP , MPL , MPx and MRTS,x . At what input condition does APL become equal to MP ?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. A firm has the production function X = LK (X = output, L = labour, K = capital); labour and capital prices are Ksh. 8 and Ksh. 10 per unit; and it desires to produce 32 units of output. What is the least cost combination of labour and capital? Consider the production function, Q = 86 L0.61 K0.39Which 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√L
