A more general form of the Cobb Douglas production function is q = f(L, K) = AL^aK^b where A, a,...
Question:
A more general form of the Cobb Douglas production function is {eq}q = f(L, K) = AL^aK^b {/eq} where A, a, b > 0 are constants.
Use calculus to solve for the marginal product of capital (MPK).
Production Function :
Production Function depicts the relationship between the factors of production with the production output. The production function can help in determining the returns to scale of production. It also helps in ascertaining the level of output to be produced.
Answer and Explanation:
Become a Study.com member to unlock this answer! Create your account
View this answerSee full answer below.
Ask a question
Our experts can answer your tough homework and study questions.
Ask a question Ask a questionSearch Answers
Learn more about this topic:

from
Chapter 4 / Lesson 2In Economics, there are three factors involved in the theory of production: total product, average product, and marginal product. Explore this theory and learn how to maximize the efficiency of these production tools.
Related to this Question
- A more general form of the Cobb Douglas production function is q = f(L, K) = AL^aK^b where A, a, b > 0 are constants. Use calculus to solve for the marginal product of labor (MPL).
- A more general form of the Cobb Douglas production function is q = f(L, K) = AL^aK^b where A, a, b > 0 are constants. Suppose that A = 20, a = b = 0.5. Derive an equation for the isoquant q = 100 and graph it with labor on the x-axis and capital on the y-
- A more general form of the Cobb Douglas production function is q = f(L, K) = AL^aK^b where A, a, b > 0 are constants. What is the MRTSL,K?
- Suppose the production function is Cobb-Douglas f(x_1, x_2) = x_1^\frac{1}{2}x_2^\frac{3}{2}. a) Write the expression for the marginal product of x_1 at the point (x_1,x_2). b) Does the marginal produ
- Assume you have the following Cobb-Douglas production function: F(K,L) = AK^a' L^(1-a') A) Using this production function, write out the equation that represents the marginal product of capital. B) Us
- Using Calculus, derive the formula for the Marginal Product of Labor(MPL) for one day's work. Cobb-Douglas production function: Y=8K^(1/4)*L^(3/4).
- For the following Cobb-Douglas production function, q=f(L,K)=L^(0.45) K^(0.7), derive expressions for marginal product of labor MP_L and marginal product of capital, MP_K.
- A more general form of the Cobb Douglas production function is q = f(L, K) = AL^aK^b where A, a, b > 0 are constants. In one sentence, interpret what is MRTSL,K tells you if A = 20, a = b = 0.5; and L = 10; K = 20.
- Derive the marginal rate of technical substitution for the Cobb-Douglass production function Q = cL^{alpha}K^{beta}.
- For the following Cobb-Douglass production function, q = f ( L , K ) = L 0.45 K 0.7 a. Derive expressions for marginal product of labor and the marginal product of capital, M P L and M P K . b. D
- Recall that a Cobb-Douglas production function is expressed as a. What is the marginal product of labor? b. What is the marginal product of capital? c. What is the Technical Rate of Substitution (tre
- Assume a firm has a Cobb-Douglas production function Y = L^{0.5} K^{0.5} . Assume (w) wage = $1, (r) rental = $2 and price of output (p) = $5 and firm has linear cost function. What is the marginal
- Given the Cobb-Douglas production function: (a) Graph the function. (b) Show that the function displays diminishing marginal product. Explain what you are doing. (c) Suppose the capital stock incre
- Use the Cobb-Douglas production function to show that a one-unit increase in the labor input will reduce the marginal product of labor and increase the marginal product of capital. Explain each of the
- Show that in the case of Cobb-Douglas production function with constant returns to scale, if each factor of production is paid its marginal product, it exhausts the output (the Euler Theorem).
- Consider a Cobb-Douglas production function: f(L,K)=0.5K^0.5L^0.5. Using this production function, solve a short-run profit maximization problem for a fixed capital stock K=4, output price p=8, wage rate w=2, and capital rental rate r=4.
- Write the equations for the marginal product of capital, marginal product of labor, and marginal rate of technical substitution for the long-run production function q = 5L^0.5K.
- Write the equations for the marginal product of capital, marginal product of labor, and marginal rate of technical substitution for the long-run production function q = 10L + K.
- Write the equations for the marginal product of capital, marginal product of labor, and marginal rate of technical substitution for the long-run production function q = K^2 L.
- Suppose the production function is given by Y=AK^{1/3}L^{2/3} (a) What is the marginal product of capital given the production function? (b) Given your answer to part (a), why might an investor exp
- 1) The general form of the Cobb-Douglas production function is q = L^aK^b. For each of the specific Cobb-Douglas production functions listed below, determine whether it can be characterized byincreasi
- Given a Cobb-Douglas production function Q = 100K^(0.4) L^(0.6), the price of labor per unit is $60, and the price per unit is $40. Use the Lagrangian method to answer this question. You need to show
- The Cobb-Douglas production function and the steady state. Suppose that the economy's production function is given by Y = K^alpha} N^1 - alpha. a. Is this production function characterized by constant
- Write the equations for the marginal product of capital, marginal product of labor, and marginal rate of technical substitution for the long run production function q =12K^0.5 L^0.5.
- The following is another Cobb-Douglas production function: Q = 60 K^{1/4} L^{3/4}. Derive the reduced form of: a. MP_K b. MP_L c. AP_K d. AP_L
- Suppose that the production function is Q = L^{2 / 3} K^{1 / 2}. a. What is the average product of labour, holding capital fixed? b. What is the marginal product of labour? c. Determine whether the production function exhibits diminishing marginal product
- Let y = output, K = capital, L = labor, and W = wood. The Cobb-Douglass production function is y = AL^a K^b W^c, where A, a, b, and c are constants. Using statistical techniques, we can estimate the e
- 2. Prove that if the more general form of the Cobb-Douglas production function Y = AK^{\alpha}L^{\beta} exhibits constant returns to scale^2 than \beta = 1 - \alpha. 3. Given the equation for the Cobb-Douglas production function Y = AK^{\alpha}L^{1-\alpha
- Suppose a firm's production function is given by Q = F(L, K) = 5LK where L is the amount of Labor and K is the amount of capital. For this particular Cobb-Douglas production function, MRTS(L,K) = K/L. The wage rate is $100 per unit of labor and the rental
- Consider the following Cobb-Douglas production function Y = 30K1/2L1/2. Calculate de marginal product of labor. Find the numerical value of MPL when K = 32 and L = 4. In the equilibrium, if we consider that the economy employs 8 workers, what would be the
- This question will walk you through finding the profit-maximizing level of output for a firm with Cobb-Douglas production. Suppose the firm's production function for output y is given by The firm is i
- Consider a Cobb-Douglas production function of: q(L,K) = 30K^0.3*L^0.7 where q is the production level, K is the quantity of capital, and L is the amount of labor. Suppose that a firm is interested in
- Consider the Cobb Douglas production function f (L,K) = L^alpha K^1/3. Suppose the input prices are w_L = 2 and w_K = 3. a) Formally write the long run cost minimization problem. For each value of the
- A firm has the following weekly production function: Q = 20KL - 0.025KL^2. Suppose the firm is in the short-run with K fixed at 20. a. What is the equation for the marginal product? Explain whether the production function is consistent with the Law of Di
- Consider the case when output in the Solow model is produced according to Cobb Douglas production function with share of capital . Derive the formulas for steady state values of capital, output, investment and consumption in this case.
- Suppose a firm's production function is given by Q = L1/2*K1/2. The Marginal Product of Labor and the Marginal Product of Capital are given by: MPL = (K^1/2)/(2L^1/2), and MPK =(L^1/2)/(2K^1/2). a) S
- Given cobb-douglas production function Q = 50L^0.3 k^0.5 Q = 50L^0.4 k^0.6 Q = 50L^0.5 k^0.6 Calculate the output when L=10 and K=15 then double the input when L= 20 and K=30
- Consider a production function given by: Q=KL^2-1/3L a. Derive the equation for the average product of labor and the marginal product of labor. b. Let K=5. Find the level of L at which the marginal
- How do you find the marginal product given a production function?
- A firm produces output that can be sold at a price of $10. The Cobb-Douglas production function is given by Q = F(K,L) = K^1/2 L^1/2. If capital is fixed at 1 unit in the short run, how much labor should the firm employ to maximize profits if the wage rat
- Different cost functions derived from a constant returns to scale Cobb-Douglas production function
- Assume that a manufacturer faces a Cobb-Douglas production function, q= 40K^0.5L^0.5 where q is output per period, L is labor, K is capital A.Specify and illustrate the MP l and AP l for L = 5 to 30 u
- Consider the case when output in the Solow model is produced according to Cobb Douglas production function with share of capital alpha: Show that marginal product of capital at the steady state when savings rate s = alpha will be equal to depreciation ra
- The Cobb-Douglas production function has the following general form: F(K,L)=ZK^{ \alpha}L^{1- \alpha} where Z > 0 is a parameter that represents overall productivity and \alpha is any constant between
- Suppose that a firm has the following Cobb-Douglas production function of Q = K^.25 L^.25. A) What type of returns to scale does it exhibit? B) What must its long-run average total cost curve look lik
- Suppose a firm's production function is given by Q = L^{1/2}*K^{1/2}. The Marginal Product of Labor and the Marginal Product of Capital are given by: MP_L = 1/2L^{-1/2}K^{1/2} and MP_K = 1/2L^{1/2}K
- A firm has a production function of y = f(L, k) = ( sqrtL + sqrtk)^2 a) Find expressions for the marginal product of labor and capital (b) Find the cost function
- Cobb-Douglas production function is: Q = 1.4*L6{0.6}*K^{0.5}. What would be the percentage change in output (%ChangeQ) if labor grows by 3.0% and capital is cut by 5.0%? (Hint: %ChangeQ = (E_L * %
- Consider the Cobb-Douglas production function. K is the amount of capital, and L is the amount of labor. The isoquant associated with this function reflects the levels of capital and labor that yield
- Suppose that an economy's production function is Cobb-Douglas with parameter alpha = 0.3. Suppose that a gift of capital from abroad raises the capital stock by 10 percent. What happens to the rental price of capital?
- Consider the following Cobb-Douglas production function Y = 30K1/2L1/2. Calculate the MPK when and in the equilibrium what would be the real interest rate. Use a graph in your answer.
- The production function is given by Q = K^1/4L^1/4. a. Derive the marginal product of capital. Consider a production manager who must produce 200 units. b. Given this, express labor in terms of the needed output (200) and capital (K). c. From this, derive
- Solve for the marginal product of labor for the following production function. Does the marginal product of labor increase, decrease, or remain constant with increases in Q? Q = (aL^{\rho} + bK^{\rho})^{\delta/\rho}
- Assume a hypothetical economy has a Cobb-Douglas production function given as:Y= and capital-output ratio is 2, saving rate 25 percent, population growth rate 2 percent and depreciation rate is 10 per
- Given the production function q = 1.37LK, what is the marginal product of capital? A) 1.37 B) 0 C) 1.37K D) 1.37L E) Cannot be determined with the information
- What is the equation needed to find the marginal product of capital? a. The change in total output / change in capital b. Change in capital / labor c. Total output / capital d. Total output / labor
- What is the equation for k* based on the values given below? Its per-worker production function is y = f(k) = 5k1/2. The marginal product of capital What is the equation for k* based on the values giv
- Let the production function for the firm be Cobb-Douglass, with fixed capital (K): Y=zF(K,N^d)=z(K)^alpha(N^d)^(1-alpha) where 0 is less than alpha is less than 1. a. Solve for labor demand as a funct
- Suppose the production function for good q is given by q = 3K + 2L where K and L are capital and labor inputs. Consider three statements about this function: I. The function exhibits constant returns to scale. II. The function exhibits constant marginal p
- Consider an economy with the following Cobb-Douglas production function: Y = F(K, L) = K^1/3 L^2/3. A. Derive the equation describing labor demand in this economy as a function of the real wage and the capital stock. B. The economy has 27,000 units of cap
- Given the production function q = 6L + 2K, what is the marginal product of labor when capital is fixed at 15?
- Suppose that the production function of a firm is given by the equation Q = 2K1/2L1/2, where Q represents units of output, K units of capital, and L units of labor. What is the marginal product of labor and the marginal product of capital at K = 40 and L
- For the production function Q = K^{0.5}L ^{0.5}, the slope at any point on any isoquant will be: a. -2. b. -1 c. \frac{-K}{L} d. \frac{-3K}{2L}
- Suppose a firm's production function is given by Q = L^(1/2)K^(1/2). The Marginal Product of Labor and the Marginal Product of Capital are given by: MP_(L) = K^(1/2)/(2L^(1/2)), and MP_(K) = L^(1/2)/(
- The production function for the Gwilmo Firm can be written as Q = 9K^{1/2}L^{1/2}. 1. Graph the isoquant for Q = 1,350. 2. Assume K = 1,600 and L = 225. Calculate the marginal product of L. 3. Assume K = 1,600 and L = 225. Now, decrease L by one unit. By
- Consider the following production function: *Q=100K.4L.6 a) Derive expression for the marginal product of capital, and for the marginal product of capital. b) Compute the marginal products of capit
- Given the production function q = 1.37LK, what is the marginal product of capital? A. 1.37 B. 0 C. 1.37K D. 1.37L E. Cannot be determined with the information given.
- Under what conditions do the following production functions exhibit decreasing, constant, or increasing returns to scale? a. q = L + K, a linear production function, b. q = L^{\alpha}K^{\beta}, a general Cobb-Douglas production function.
- If the U.S. production function is Cobb Douglas with capital share 0.3, output growth is 3 percent per year, depreciation is 4 percent per year, and the capital output ratio is 2.5, what is the saving rate that is consistent with steady-state growth?
- A firm uses two inputs, X and Y and its production function is Q = radical(xy), where here we are using x and y to represent the quantities of the two inputs. (a) Calculate the marginal products of X
- How to calculate the Cobb Douglas production function?
- Suppose the wage is 8, the rental rate of capital is 128, and the firm's CRS Cobb-Douglas production function is q=3L^(1/3)K^(2/3) a. What is the cost-minimizing bundle of labor and capital for produc
- Suppose a production function is given by Q = 4K + 3L. What is the marginal product of capital when 10 units of capital and 10 units of labor are employed?
- Suppose the production function is given by Q = 3K + 4L. What is the marginal product of capital when 5 units of capital and 10 units of labor are employed?
- Suppose the production for good q is given by q=3k+2l, where k and l are capital and labor inputs. Consider three statements function about this function: I. the function exhibits constant returns to scale. II. the function exhibits diminishing marginal p
- A firm has a Cobb Douglas production function q = AL^{\alpha}K^{\beta}, where \alpha + \bet a= 1. On the basis of this information, what properties does its cost function have? The firms long run aver
- Suppose we know that output in the economy is given by the production function: Y_t = A_t K_t^(1/4) L_t^(3/4) a) Use partial derivative techniques to solve for the marginal product of capital (Remembe
- The Cobb-Douglas production function is given by: Y = AK^(a)L^(1-a) where 0 < ? < 1. f.) Does the Cobb-Douglas production function display "constant returns to scale"? (Doubling both inputs doubles ou
- Suppose that a firm has a production function given by q = 10L0.5K0.6. The firm has 10 units of capital in the short run. Which of the following will describe the marginal product of labor (MPL) for this production function? a. increasing marginal returns
- Consider a production function given by: Q = 27K^{2}L^{0.5} - 2K^{4} A. Let L = 16. Find the level of K at which the marginal product of capital reaches a maximum B. Let L = 16. Find the level of K
- Consider the linear production function q=f(K,L)=2L+K . a. What is the short-run production function given that capital is fixed at K=100? b. What is the marginal product of labor?
- Consider the Production Function, Y = 25K1/3L2/3 (a) Calculate the marginal product of labor and capital (b) Does this production function exhibit constant/increasing/decreasing returns to scale? (
- Consider the production function is q = L^0.6 + 4K. A) Starting from the input combination (10,10), calculate the marginal product of adding one worker. B) What is the marginal product of adding anoth
- If the production function is Q = K^(1/2) L^(1/2) and capital is fixed at 100 units, then the marginal product of labor (MPL) will be?
- Given the Production Function Q = 72X + 15X^2 - X^3, where Q = output and X = Input a. What is the Marginal Product (MP) when X = 8? b. What is the Average Product (AP) when X = 6? c. At what value of
- Suppose the production function for a firm is given by q = 5L0.5K0.25. In the short run, the firm has 16 units of capital. Find the marginal product of labor (MPL). Round to 2 decimal places.
- Suppose a firm has the Cobb-Douglas production function Q= f(K, L) = 2K^0.7L^0.8, where K is capital and L is labor. Using this function, show the following: (a) Does this production function exhibit
- Which of the following production functions exhibit constant marginal product of capital, K? In each case y is output and K and L are inputs. a. y=K*L^{2/3} b. y=3K^{1/2}*L^{1/2} c. y=K^{1/2}+L^{1/2} d. y=2K+3L
- In the production function, Q = 10L1/2 K1/2. Calculate the slope of the isoquant when the entrepreneur is producing efficiently with 9 laborers and 16 units of capital. (Hint: The slope of the isoquant = the ratio of the marginal product of labor to the m
- The production function takes the following formY = F(K,N) = zK^0.3N^0.7 (a) Write the expressions for marginal product of labor and marginal product of capital.
- The production function for a product is given by q = K1/2L1/4 where K is capital, L is labor and q is output. a. Find the marginal products of labor and capital. b. Is the marginal product of labor increasing or decreasing with labor? Is the marginal p
- An economy has a Cobb-Douglas production function: Y = Kalpha(LE) 1-alpha. The economy has a capital share of a third, a saving rate of 24 percent, a depreciation rate of 3 percent, a rate of populati
- Prove that the sum of the indices of a Cobb-Douglas production function (Q = AL a K b ) determines the returns to scale. Order Now production function is lnQ = 0.82lnK + 0.47lnL; where K is capital an
- Consider the following production function F(K, L) = { KL } / {K + L } (a) Does it satisfy the Constant Returns to Scale Assumption? Explain. (b) Are marginal products diminishing? Explain. (If you can't find marginal products mathematically, you can com
- Consider a Cobb-Douglas production function with three inputs. K is capital, L is unskilled labor, and H is skilled labor: Y = K^{1/3}L^{1/3}H^{1/3} Find: 1. The marginal product of unskilled labo
- In this problem, you'll compare a short-run and a long-run cost function for a Cobb-Douglas production process. More specifically, assume that a firm uses labor and capital to produce an output accord