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1. Consider the following production function: Y = A min{2N, K} (1) where A measures productivity,...
9. Suppose the firm's production function is given by f(K,L) min (K",L" (a) For what values...
9. Suppose the firm's production function is given by f(K,L) min (K",L" (a) For what values of a will the firm exhibit decreasing returns to scale? Constant returns to scale? Increasing returns to scale? (b) Derive the long-run cost function and the optimal input choices. (c) Suppose the capital is fixed at R = 10,000 and a =. Assuming that the firm wants to produce less than 100 units, derive 10. Consider the production function: f(K, L) = KLi. Let...
Consider an economy where the aggregate production function is characterized by the following equation: Y=2N^1/2. The...
Consider an economy where the aggregate production function is characterized by the following equation: Y=2N^1/2. The labor supply curve is given by the following equation: Ns=8(W/P). Derive the labor demand curve and calculate the equilibrium real wage and the equilibrium level of labor.
9. Suppose the firm's production function is given by f(K,L) = min (Kº,L"} (a) For what values of a will the firm e...
9. Suppose the firm's production function is given by f(K,L) = min (Kº,L"} (a) For what values of a will the firm exhibit decreasing returns to scale? Constant returns to scale? Increasing returns to scale? (b) Derive the long-run cost function and the optimal input choices. (c) Suppose the capital is fixed at K = 10,000 and a = 1. Assuming that the firm wants to produce less than 100 units, derive 10. Consider the production function: f(K,L)=KLI. Let w...
Consider the Leontief production function F(KL) = min {K,L], where capital K and labor L have...
Consider the Leontief production function F(KL) = min {K,L], where capital K and labor L have respective positive input prices r and w. (a) Why is it that the cost-minimizing firm sets K 5. L? (b) What is the cost function? (c) How would your answer to part (b) change, if at all, if rw 0? Explain.
Consider a profit maximizing firm that uses a Cobb-Douglas production function Y = AKαL 1−α and...
Consider a profit maximizing firm that uses a Cobb-Douglas production function Y = AKαL 1−α and hires labor L at wage rate w and capital K at rental rate r. (1) Set up the profit-maximization problem of the firm and derive the first-order condition for the profit-maximizing choice of capital. (2) Show that the marginal product of capital is a decreasing function of capital. (3) Solve for the optimal choice of capital and show that the optimal choice of capital...
Consider 2 firms with the following 2 different production functions (i.) y(L,K) = aL + bK...
Consider 2 firms with the following 2 different production functions (i.) y(L,K) = aL + bK (ii.) y(L,K) = L^0.5K^0.5 where y denotes the quantity produced and L and K are the amount of labor and capital, respectively. a. Assume K is fixed at 100. Do these production functions exhibit decreasing marginal products of labor? b. Assume K can be freely chosen. Do these production functions exhibit constant returns to scale? c. For each of the production functions, draw the...
Consider the following production functions Y = AK1/2L1/2 Y=AK+3L a. Fixing total factor productivity (A) at...
Consider the following production functions Y = AK1/2L1/2 Y=AK+3L a. Fixing total factor productivity (A) at 2 and labor employment (L) at 16 units, what is the marginal product of capital when capital employment (K) is 25, 35, and 45 for each production function? Do these production functions exhibit diminishing returns to capital employment? Explain. b. Are labor and capital complements under these production functions? Explain. c. Is either production function a “Cobb-Douglas” function? Explain. 3. Describe the difference between...
From now on, let the production function of the firm be Y = 2K*N1-a, where a...
From now on, let the production function of the firm be Y = 2K*N1-a, where a is a parameter between 0 and 1. 4. Verify that this production function has constant return to scale. 5. Derive the marginal product of labor MPx and marginal product of capital MPK. How does MPN change with N and K? 6. Solve the firm's optimization problem analytically. That is, to derive the firm's opti- mal choice as a function of exogenous variables (i.e., in...
1. [30 POINTS] Consider the production function y=f(L,K) = 4/1/2K1/4 where L is labor and K...
1. [30 POINTS] Consider the production function y=f(L,K) = 4/1/2K1/4 where L is labor and K is capital. Price per unit of the labor is w, price per unit of the capital is r, and the price per unit of the output is p. (a) (10 POINTS] In long-run, if the firm's objective is to maximize its profit, what are the factor demand functions of labor and capital? (b) (10 Points) What is the optimal output level y and the...
1. The production function of a firm is f(1,k) = Vlk where l is labor and...
1. The production function of a firm is f(1,k) = Vlk where l is labor and k is capital/machinery. a. In the short run, if the quantity of capital is fixed at 64, derive the short run total cost SC(q), average cost SAC(q), and marginal cost SMC(q) of this firm. Assume each input costs $1 per unit. At what output does the minimum of SAC(q) occur? b. If labor and capital cost r and w respectively, and the quantity of...
9. Suppose the firm's production function is given by f(K,L) min (K",L" (a) For what values of a will the firm exhibit decreasing returns to scale? Constant returns to scale? Increasing returns to scale? (b) Derive the long-run cost function and the optimal input choices. (c) Suppose the capital is fixed at R = 10,000 and a =. Assuming that the firm wants to produce less than 100 units, derive 10. Consider the production function: f(K, L) = KLi. Let...
9. Suppose the firm's production function is given by f(K,L) = min (Kº,L"} (a) For what values of a will the firm exhibit decreasing returns to scale? Constant returns to scale? Increasing returns to scale? (b) Derive the long-run cost function and the optimal input choices. (c) Suppose the capital is fixed at K = 10,000 and a = 1. Assuming that the firm wants to produce less than 100 units, derive 10. Consider the production function: f(K,L)=KLI. Let w...
Consider the Leontief production function F(KL) = min {K,L], where capital K and labor L have respective positive input prices r and w. (a) Why is it that the cost-minimizing firm sets K 5. L? (b) What is the cost function? (c) How would your answer to part (b) change, if at all, if rw 0? Explain.
From now on, let the production function of the firm be Y = 2K*N1-a, where a is a parameter between 0 and 1. 4. Verify that this production function has constant return to scale. 5. Derive the marginal product of labor MPx and marginal product of capital MPK. How does MPN change with N and K? 6. Solve the firm's optimization problem analytically. That is, to derive the firm's opti- mal choice as a function of exogenous variables (i.e., in...
1. [30 POINTS] Consider the production function y=f(L,K) = 4/1/2K1/4 where L is labor and K is capital. Price per unit of the labor is w, price per unit of the capital is r, and the price per unit of the output is p. (a) (10 POINTS] In long-run, if the firm's objective is to maximize its profit, what are the factor demand functions of labor and capital? (b) (10 Points) What is the optimal output level y and the...
1. The production function of a firm is f(1,k) = Vlk where l is labor and k is capital/machinery. a. In the short run, if the quantity of capital is fixed at 64, derive the short run total cost SC(q), average cost SAC(q), and marginal cost SMC(q) of this firm. Assume each input costs $1 per unit. At what output does the minimum of SAC(q) occur? b. If labor and capital cost r and w respectively, and the quantity of...
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