A production function is a mathematical relation between a firm or country's inputs (capital and...
Question:
A production function is a mathematical relation between a firm or country's inputs (capital and labor) and outputs. Use the Cobb-Douglas Production Function
{eq}Y = K^ \alpha L ^{1 - \alpha}{/eq}
where Y is output (GDP or national income), K is capital, and L is labor (number of workers).
a) Show the production function can be written as {eq}ln(Y) = \alpha ln(K) + (1 - \alpha)ln(L){/eq}.
b) Find the marginal product of capital and labor by taking the derivative of Y with respect to K and L.
Cobb-Douglas Function:
It is a type of production function where input factors labor and capital are implicit variables of the function. The elasticities of labor and capital are also part of the functional parameters. It represents the technological relationship between two-factor inputs.
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answerThe production function is given as:
{eq}{\rm{Y = }}{{\rm{K}}^{\rm{\alpha }}}{{\rm{L}}^{{\rm{1 - \alpha }}}} {/eq}
a) Assuming logarithm on both...
See 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:
Get access to this video and our entire Q&A library
The Cobb Douglas Production Function: Definition, Formula & Example
from
Chapter 1 / Lesson 7
270K
Learn the definition of a production function in economics, understand the definition of a Cobb-Douglas production function and its formula, and explore some examples of Cobb-Douglas production function.
Related to this Question
- 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
- An economy's total labor income is $2 trillion, and total capital income is $1 trillion. In the Cobb-Douglas production function, the exponent on capital is ________. (Show work.)
- An economy has a Cobb-Douglas production function: Y=K alpha (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 popula
- 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
- 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
- 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
- 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
- The following is a Cobb-Douglas production function. TP = 85K^{0.75}L^{0.25} If the firm employs 35 units of capital and 125 units of labor, find: A. the average product of capital (AP_K), and the average product of labor (AP_L). B. The marginal product o
- Suppose that an economy's production function is cobb-douglas with parameter α = 0.3 a. What fractions of income do capital and labor receive? b. Suppose that immigration increase the labor force by 10 percent. What happens to total output (in pre
- In a Cobb-Douglas production function with constant returns to scale for a firm that produces using only labor and capital, if the share of income that goes to labor is 42%, what does alpha equal? (En
- 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
- Suppose that an economy's production function is Cobb-Douglas with parameter alpha = 0.3. What fractions of income do capital and labor receive?
- 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
- 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 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 total output (in percent)?
- Suppose output is produced according to the following Cobb-Douglas production function q = K0.8L0.5. a. (4 pts) If K = 1 and L = 4, what level of output is produced? If K = 3 and L = 12
- 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
- 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?
- If a business has 'L' units of labor (e.g. workers) and K units of capital (e.g. production machines), then its production can be modeled by a Cobb-Douglass Production Function of: P(L,L) = beta L^alp
- 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
- In a Cobb-Douglas production function with constant returns to scale, if the share of income that goes to labor is 42%, what does alpha equal? Enter your number in decimal form as opposed to percentag
- 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?
- Consider a Cobb-Douglas production function with three inputs. a. K is capital (the number of machines), b. L is labor (the number of workers), and c. H is human capital (the number of college degr
- 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
- If total production in the economy is described by the Cobb-Douglas production function Y= AKaL1-a and a = 0.25, then the share of output going to capital is: a. 0.75 b. 0.25Y c. 0.25 d. 0.75Y
- In the steady state of an economy, described by the Cobb-Douglas production function, the: a) Amount of the capital/labor ratio is constant, that is, unchanging b) The output/labor ratio is positive but decreasing c) Population growth and depreciation rat
- In a Cobb-Douglas production function, can capital, K, and labour, L, be negative?
- 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
- 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
- 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
- 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
- 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
- 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.
- Given a Cobb-Douglas production function: Y(t)=K(t) alpha L(t)1-alpha where K(t) and L(t) are capital and labor respectively at time t. Assume population growth n and capital depreciation delta. Write
- Consider a Cobb-Douglas production function with three inputs. K is capital (the number of machines), L is labor (the number of workers), and H is human capital (the number of college degrees among th
- Recall the Cobb-Douglas production function for one industry is: Y_t = AL^(beta _1)_t K^(beta^2)_t, where 'A' is total factor productivity, Y_t is output of the industry at date 't,' L_t is labor inpu
- Firm Alpha uses capital K and labor L to produce output q. The firm's production function is F(K,L)= 5K^{0.4}\times L^{0.6}. The prices of capital and labor are r = 4 and w=4, respectively. Moreover,
- The production function for a country is given by Y = F(K, L) = K0.4L0.6. From this production function, we can solve for the marginal products of capital and labor. Total physical capital, K, is 6400, and total labor available, L, is 12,000. a. What is t
- Suppose a firm's technology is represented by the Cobb-Douglas production function F(L, K) = 5 L K. The wage rate is $50 and the rental rate of capital is $10. What is the least-cost combination to produce 100 units of output?
- Suppose that an economy's production function is Cobb-Douglas with parameter alpha = 0.3. Suppose that immigration increases the labor force by 10 percent. What happens to the rental price of capital?
- 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
- Firm Omega uses capital K and labor L to produce output q. The firm's production function is F(K,L) = 12K+3L. The prices of capital and labor are r = 40 and w=4, respectively. In the long-run, when th
- 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 real wage?
- 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
- 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
- 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.
- A firm's production function is given by f(L, K) = LK^1/2. The prices of labor and capital are w = 20 and r = 10, respectively. Suppose the firm's capital is fixed at 25. Find, as functions of output
- The Cobb-Douglas production function for Mabel's factory is Q = (L^0.4)(K^0.7). a) Based on the function above, does Mabel's factory experience economies or diseconomies of scale? Explain. b) If the manager wished to raise productivity by 50% and planned
- 3.Input as a function of wages for a Cobb-Douglas production functions
- 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
- 1. Finding the steady state K/N and Y/N. Suppose an economy's production function is given by: Y = \beta K^{\alpha} N^{1 - \alpha} where Y is output, K is capital, and N is labor. (a) Rewrite this
- Assume that the production function is given by Y A K 0.5, L 0.5 , where Y is GDP, K is capital stock, and L is labor. The parameter A is equal to 10. Assume also that capital is 100, labor is 400, an
- A firm has a production function F(L,K), where K is a fixed amount of capital, and L is the variable amount of labor hired. The equation w= pF_{L}(L,K) determines the amount of labor that the firm hir
- Consider the Solow growth model (without tech.) for the Taiwanese economy: The first equation is a Cobb-Douglas production function with and The second equation states that the labor force grows at 5%
- 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
- Suppose an economy's production function is Cobb-Douglas with parameter = 0.3. A) What fractions of income do capital and labor receive? B) Suppose that immigration increases the labor force by 10 percent. What happens to total output in percent? The rent
- If a Cobb-Douglas production function has alpha = 0.34 and beta = 0.42, what percentage change will occur in outputs if there is a 1% increase in inputs? A) 0.8% B) 8% C) 0.78% D) -0.76%
- Econometric studies of the cotton industry in India indicate that the Cobb-Douglas production function can be applied, and that the exponent of labor 0.92 and the exponent of capital is 0.15. Suppose
- A Cobb-Douglas production function for new company is given by ?? f(x,y) ? 50x 2 5 y 3 5 where x represents the units of labor and y represents the units of capital. Suppose units of Labor and capital
- Suppose that an economy's production function is Cobb-Douglas with parameter alpha = 0.3. Suppose that a technological advance raises the value of the parameter A by 10 percent. What happens to the rental price of capital?
- Let Y denote output K is capital stock and L is labor input. A denotes total factor productivity. The production function is the following: Y = AF(K, L) = A(K + L). a. Calculate output and the marginal product of capital (MPK) when A=3, K=1 and L = 1. b
- Consider a production function for an economy: Y = 20 (L^{0.5}K^{0.4}N^{0.1}) where L is labor, K is capital, and N is land. In this economy the factors of production are in fixed supply with L = 100, K = 100, and N = 100. a) What is the level of output i
- 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 an economy's production function is Cobb-Douglas with parameter alpha = 0.3. Suppose that a technological advance raises the value of the parameter A by 10 percent. What happens to total output (in percent)?
- 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-
- An assumption of the Cobb-Douglas production function model is that: a. inputs exhibit diminishing returns. b. inputs exhibit increasing returns. c. inputs can be zero and still have a positive value for output. d. There are no assumptions about this mode
- In a generalized Cobb Douglas production function given by Q=CLaKbQ = C L^a K^b, is it possible for L or K to be inferior inputs?
- The Cobb-Douglas Production function Q = AK 1 -a L a where 0 a 1 a = .75 k = 4 is the amount of capital L = 16 is the amount of labor A = 12 is the measure of technology 1. Find the real wage
- Let Y denote output. K is capital stock and L is labor input. The production function is the following: Y = F(K,L) = 4K +2L. a. Calculate output and the marginal product of capital (MPK) when K=1 and L=1. b. Calculate output and the MPK when K=2 and L=1
- Assume that the world works according to the Classical model. In a small open economy, the output is produced according to a Cobb-Douglas production function, consumption is equal to C=40+0.6(Y-T) and the investment function is I=280-10r. You know that th
- An economy with production function Y = F(K) = the square root of K that has 400 units of capital will produce units of output. A. 1,600 B. 400 C. 40 D. 20
- If output is only a function of labor, then: a. the slope of the production function measures the average product of labor. b. a firm cannot be profitable in the long run. c. the slope of a ray from the origin to a point on the production function measur
- Consider the Solow Model. Use the following information to fill in the table and show your work. Cobb-Douglas production function: Y = AK^{1/2}L^{1/2}. C is 1.6.
- A firm uses capital and labor to produce a single output good. The production function is given by F(K, L) = K L^{0.5} where K is the amount of capital and L is the amount of labor employed by the firm. The unit prices of capital and labor are given by, r
- A firm has the production function f(k, l) = 2k sqrt l. Let the price of capital be r = 1, the price labor be w = 2, and the price of output be p. Find the marginal products of capital and labor. Does the firm have constant returns to scale?
- 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
- 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
- Suppose output is a function of skilled and unskilled labor (Cobb-Douglas production function) Y = H^a L^{(1 - a)}. What real-life factors can affect a, i.e. the elasticity of output with respect to
- Assume that the world works according to the Classical model. In a small open economy, output is produced according to a Cobb-Douglas production function, consumption is equal to C = 40 + 0.6(Y - T), and the investment function is I = 280 - 10r. You know
- Two countries, Richland and Poorland, are described by the Solow growth model. They have the same Cobb-Douglas production function, F(K, ^{\alpha} L) = A K^{\alpha} L ^{1-\alpha}, but with different q
- Let the production function of a firm be given by Find the equation of the isoquant associated with a production level of 10 units of output.
- 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
- 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
- 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
- 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).
- Suppose an economy's production possibilities are represented by the function Y = A K L where Y represents total output (i.e GDP), K is capital, L is labor, and A is total factor productivity (TFP
- Some economists believe that the U.S. economy as a whole can be modeled with the following production function, called the Cobb-Douglas production function: where Y is the amount of output, K is the amount of capital, L is the amount of labor, and A is a
- The production function is Y = 3KL. If there are 10 units of capital (K) and 50 units of labor (L), the aggregate output is: A) 1,500 B) 500 C) 3,500 D) 35,010
- Some economists believe that the US. economy as a whole can be modeled with the following production function, called the Cobb-Douglas production function: Y = AK^{1/3}L^{2/3} where Y is the amount of output K is the amount of capital, L is the amount of
Explore our homework questions and answers library
Browse
by subject