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 for the firm is a decreasing function of the rental rate r.
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Consider a profit maximizing firm that uses a Cobb-Douglas
production function Y = AKαL 1−α and...
Question 2: Production Function and Profit Maxi- mization Consider a production function of Cobb-Douglas form: for...
Question 2: Production Function and Profit Maxi- mization Consider a production function of Cobb-Douglas form: for some α, β E (0, 1) (a) Plot the isoquant of F (b) Derive that technical rate of substitution of F. Does F exhibit diminishing technical rate of substitution? (c) Does F exhibit diminishing marginal productivity of labor? What about marginal (d) Find out the conditions for α and β such that F is increasing return to scale, (e) Suppose that F does not...
A firm has a Cobb-Douglas production function of Q = K^(0.25) L^(0.75) (a) Does this production...
A firm has a Cobb-Douglas production function of Q = K^(0.25) L^(0.75) (a) Does this production technology exhibit increasing, constant, or decreasing returns to scale? (b) Suppose that the rental rate of capital is r = 1, the wage rate is w = 1, and the ?rm wants to produce Q = 3. In the long-run, what combination of L and K should they use? (It would be good to practice doing this with the Lagrangian, even if you can...
Consider a representative firm that produces using capital (K) and labor (N). The firm hires workers...
Consider a representative firm that produces using capital (K) and labor (N). The firm hires workers at wage w and rent capital at rental rate r. Suppose the firm has the production function: F(K, N) = K™NB with a + ß < 1 and a, ß E (0,1) ii) Set up the firm's profit maximization problems and solve for the optimal K*, L*, and 7* (profits) in terms of parameters(12 points) iii) What's the ** when a + b =...
1. Consider the following production functions. In each case determine if: • the function is Cobb...
1. Consider the following production functions. In each case determine if: • the function is Cobb Douglas (Y = AK 11-a). If the function is Cobb Douglas, what is the value of the parameter a? • Do capital and labor exhibit diminishing returns. Explain your thinking using algebra / calculus /a graph etc. (a) F(K, L) = 27K+15VL (b) F(KL) = 5K + 3L (c) F(KL) = K0.5 0.5 (a) F(KL) - VK2 + L2 2. Suppose that the production...
All firms produce according to a Cobb-Douglas production function. This production function should look familiar to...
All firms produce according to a Cobb-Douglas production function. This production function should look familiar to you. It says that output Q is related to inputs K and L as: This production function implies that the cost-minimizing demand for capital will be Ou where w is the wage rate, r is the cost of capital, Q is output level, and α and β are parameters We will assume that α + β 1; this is the constant-returns-to-scale assumption we saw...
3. Consider a firm with the production function F(KL)=1/31/3 You will be solving the profit maximization...
3. Consider a firm with the production function F(KL)=1/31/3 You will be solving the profit maximization for this firm with both the two step and I step methods and proving that the final answers are identical. This big problem is broken up into the following smaller parts: (a) Setup and solve the long run cost minimization problem for the long run optimal amount of capital K*(w,,9) and labor L*(w,r.9), and the long run minimized cost C*(w, 5,9). (Hint: reduce the...
2. Consider a Solow growth model with Cobb-Douglas production function Y Ko (AN)-a with constant savings...
2. Consider a Solow growth model with Cobb-Douglas production function Y Ko (AN)-a with constant savings rate s, depreciation rate d and no growth in productivity or labor (gA = gN = 0) (a) Suppose A = 1, a = 1/3, s = 0.2 and 5 = 0.1 (annual). Calculate the steady state capital per worker and steady state output per worker (b) Suppose that the real wage w and real return to capital r are equal to the marginal...
2. For the following Cobb-Douglas production function, q = f(L,K) = _0.45 0.7 a. Derive expressions...
2. For the following Cobb-Douglas production function, q = f(L,K) = _0.45 0.7 a. Derive expressions for marginal product of labor and marginal product of capital, MP, and MPK. b. Derive the expression for marginal rate of technical substitution, MRTS. C. Does this production function display constant, increasing, or decreasing returns to scale? Why? d. By how much would output increase if the firm increased each input by 50%?
SHOW ALL WORK!!! 2. For the following Cobb-Douglas production function, q=f(L,K) = _0.45 0.7 a. Derive expressions for m...
SHOW ALL WORK!!!
2. For the following Cobb-Douglas production function, q=f(L,K) = _0.45 0.7 a. Derive expressions for marginal product of labor and marginal product of capital, MP, and MPK. b. Derive the expression for marginal rate of technical substitution, MRTS. C. Does this production function display constant, increasing, or decreasing returns to scale? Why? d. By how much would output increase if the firm increased each input by 50%?
18. A firm operating on the production function of ? = ? ?? ? pays w...
18. A firm operating on the production function of ? = ? ?? ? pays w and r for the labor and capital respectively in a competitive market. Write the profit maximizing first order condition for optimal level of K and L (you don’t need to solve it !). Check whether the sufficient condition for a maximization hold. Assume that; ? = ? ??? 0 < ? < 0.5 .
Question 2: Production Function and Profit Maxi- mization Consider a production function of Cobb-Douglas form: for some α, β E (0, 1) (a) Plot the isoquant of F (b) Derive that technical rate of substitution of F. Does F exhibit diminishing technical rate of substitution? (c) Does F exhibit diminishing marginal productivity of labor? What about marginal (d) Find out the conditions for α and β such that F is increasing return to scale, (e) Suppose that F does not...
Consider a representative firm that produces using capital (K) and labor (N). The firm hires workers at wage w and rent capital at rental rate r. Suppose the firm has the production function: F(K, N) = K™NB with a + ß < 1 and a, ß E (0,1) ii) Set up the firm's profit maximization problems and solve for the optimal K*, L*, and 7* (profits) in terms of parameters(12 points) iii) What's the ** when a + b =...
1. Consider the following production functions. In each case determine if: • the function is Cobb Douglas (Y = AK 11-a). If the function is Cobb Douglas, what is the value of the parameter a? • Do capital and labor exhibit diminishing returns. Explain your thinking using algebra / calculus /a graph etc. (a) F(K, L) = 27K+15VL (b) F(KL) = 5K + 3L (c) F(KL) = K0.5 0.5 (a) F(KL) - VK2 + L2 2. Suppose that the production...
All firms produce according to a Cobb-Douglas production function. This production function should look familiar to you. It says that output Q is related to inputs K and L as: This production function implies that the cost-minimizing demand for capital will be Ou where w is the wage rate, r is the cost of capital, Q is output level, and α and β are parameters We will assume that α + β 1; this is the constant-returns-to-scale assumption we saw...
3. Consider a firm with the production function F(KL)=1/31/3 You will be solving the profit maximization for this firm with both the two step and I step methods and proving that the final answers are identical. This big problem is broken up into the following smaller parts: (a) Setup and solve the long run cost minimization problem for the long run optimal amount of capital K*(w,,9) and labor L*(w,r.9), and the long run minimized cost C*(w, 5,9). (Hint: reduce the...
2. Consider a Solow growth model with Cobb-Douglas production function Y Ko (AN)-a with constant savings rate s, depreciation rate d and no growth in productivity or labor (gA = gN = 0) (a) Suppose A = 1, a = 1/3, s = 0.2 and 5 = 0.1 (annual). Calculate the steady state capital per worker and steady state output per worker (b) Suppose that the real wage w and real return to capital r are equal to the marginal...
2. For the following Cobb-Douglas production function, q = f(L,K) = _0.45 0.7 a. Derive expressions for marginal product of labor and marginal product of capital, MP, and MPK. b. Derive the expression for marginal rate of technical substitution, MRTS. C. Does this production function display constant, increasing, or decreasing returns to scale? Why? d. By how much would output increase if the firm increased each input by 50%?
SHOW ALL WORK!!!
2. For the following Cobb-Douglas production function, q=f(L,K) = _0.45 0.7 a. Derive expressions for marginal product of labor and marginal product of capital, MP, and MPK. b. Derive the expression for marginal rate of technical substitution, MRTS. C. Does this production function display constant, increasing, or decreasing returns to scale? Why? d. By how much would output increase if the firm increased each input by 50%?
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