8. In the constant elasticity of substitution (CES) production function q = A[&K° + (1- 6)L°]P, find the degree of homogeneity of the marginal product of capital MPK and the marginal product of labour MP. Show also that Euler's Theorem applies. 9. Find the global maximum and minimum values of
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![8. In the constant elasticity of substitution (CES) production function
q = A[SKP +(1-8)LP]/P, find the degree of homogeneity of the marginal
product of capital MPk and the marginal product of labour MP,. Show also that
Euler's Theorem applies.
9. Find the global maximum and minimum values of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14099dcb-4ed0-4caa-b5e5-2c6c5647f45d%2F5643f957-ec0f-42ae-a369-384c8d45e444%2Figj0s1_processed.jpeg&w=3840&q=75)
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- Consider the following production functions, to be used in this week’s assignment:(A) F(L, K) = 20L^2 + 20K^2(B) F(L, K) = [L^1/2 + K^1/2]^2For each of production functions (A) and (B) given above, do the following steps.(i) Calculate the marginal product of labor MPL(L, K) = ∂F(L, K) / ∂L.(ii) Calculate the marginal product of capital MPK(L, K) = ∂F(L, K) / ∂K.(iii) Calculate the absolute value of the technical rate of substitution as the ratio of marginal products andsimplify as far as possible: |TRS(L, K)| = MPL(L, K) / MPK(L, K). PLEASE SHOW ALL WORKConsider a Cobb-Douglas production function:f(l, k) = Alα k1−α,where A is the total factor of productivity (a constant greater than 1), 0 < α < 1, lrepresentslabor, and k represents capital. The following sub-questions will guide you through showing thatthe elasticity of substitution is constant.a) Find the marginal product of labor. Verify that this production function exhibits diminishingmarginal productivity of labor. b) Find the marginal product of capital. Verify that this production function exhibits diminishingmarginal productivity of capital. c) Find the marginal rate of technical substitution. Write your answer as MRT S = . . . d) In part (C), you should’ve found the MRTS as a function of the input ratio, kl. Take theabsolute value of both sides and solve for the input ratio, so that the expression gives theinput ratio as a function of MRTS (i.e. kl = . . .). Take the log of both sides, then take thederivative with respect to the log of MRTS. Is the elasticity of…The marginal rate of technical substitution of labor for capital (MRTSLK) is defined as the rate at which the quantity of ______.A. capital can be increased for every one unit increase in the quantity of laborB. capital can be increased if the quantity of labor remains the sameC. labor can be reduced as capital costs increaseD. capital can be reduced for every one unit increase in the quantity of labor
- Assume the following production function (photo aswell) q = [0.3k^(-1)+ 0.7l ^(-1)] ^(-1) a) Determine the rate of technical substitution, RTS. b) Determine the elasticity of substitution. c) What would happen to the capital intensity if the relative factor price w/v were to increase by 10%? d) If the elasticity of substitution were higher than what you calculated in (b), would the effect of a 10% increase in the relative factor price w/v on the capital intensity be higher or lower than what you calculated in (c)? Explain your reasoning.With its current levels of input use, a firmʹs Marginal Rate of Technical Substitution is 5 (when capital is on the vertical axis and labour is on the horizontal axis). This impliesA. the firm could produce 5 more units of output if it increased its use of capital by one unit (holding labour constant).B. the marginal product of labour is 5 times the marginal product of capital.C. the marginal product of labour is 1/5 times the marginal product of capital.D. if it used one more unit of both capital and labour, the firm could produce 5 more units of output.From the following production functions 1. Q= a1H + a2L + a3H2 + a4 L2 + a5HL, where ai> 0 2. Q = aH@ Ly, where a, @, y > 0 a. Derive the equation of the relevant isoquant. b. Find out whether the production function is well behaved. c. Examine whether the isoquant is well behaved and therefore represents the behaviour of a rational production. d. Derive equation which describes MRTS of 9ne factor. e. For each equation examine whether the production is homogeneous and if so, what is the degree of homogeneity. Is the equation characterized by IRTS, DRTS or CRTS.
- 10. If over the range of positive marginal product, the short-run total product of factor A in the production of X is given by the equation: X=aA +b A2 – A3 Where a, and b are positive constants, and if the average product of A is maximized when X=100, then the marginal product of A must be greater than the average product when X=50. True or False ?. Prove your answer.Answer the Constrained Optimization: Cobb-Douglas Production Function:3. Solve for the formulas of the Marginal Product of Labor (MPL), and Marginal product of Capital (MPK)4. Using your knowledge of the tangency condition in Producer’s theory, find the combination of K and L that the firm should use to produce the maximum possible output. Do not solve the problem using the Lagrangian method.Note: The tangency conditions just states that the slope of the production function must beequal to the slope of the isocost function.5. What is the maximum possible output that the firm could earn given the constraint it facesAnswer the Constrained Optimization: Cobb-Douglas Production Function:1. Based from the factor shares of the two inputs, what will happen to the number of output ifit the firm decides to triple both the amount of labor and capital?2. State the optimization problem of the firm.3. Solve for the formulas of the Marginal Product of Labor (MPL), and Marginal product ofCapital (MPK)4. Using your knowledge of the tangency condition in Producer’s theory, find the combinationof K and L that the firm should use to produce the maximum possible output. Do not solvethe problem using the Lagrangian method.Note: The tangency conditions just states that the slope of the production function must beequal to the slope of the isocost function.5. What is the maximum possible output that the firm could earn given the constraint it faces?
- Given a constant elasticity of substitution (CES) production function as follows:y = [a1x1ρ + a2x2ρ]1/ρ 1. Find the marginal rate of technical substitution (MRTS)2. Derive the expression for Elasticity of SubstitutionF (L, K) = L0.2K0.7， The wage rate (price per unit of labour) is w = 2 and the capital rental rate (price per unit of capital) is r = 7. Derive the equation of the isoquant for y = 2 (with K in the vertical axis and L in the horizontal axis). Use the first and second derivative to show that this curve is decreasing and convex. Provide a graphical representation of the isoquant indicating at least one combination of labour and capital in this curve.Show diagrammatically how a labour supply curve can be backward bending, fully explaining using income and substitution effects. Explain why a researcher may choose to use Roy’s Identity to derive estimable labour supply functions.
