QUESTION 3 Suppose the current level of output is $10,000. If the elasticities of output with respect to capital and labor are 0.4 and 0.6, respectively, a 13.8% increase in capital combined with a 14.9% increase in labor and a 5% increase in productivity would increase the current level of output to (approximately) $

QUESTION 3 Suppose the current level of output is $10,000. If the elasticities of output with respect to capital and labor are 0.4 and 0.6, respectively, a 13.8% increase in capital combined with a 14.9% increase in labor and a 5% increase in productivity would increase the current level of output to (approximately) $

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter10: Cost Functions

Section: Chapter Questions

Problem 10.11P

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Over the past year, output grew 9%, capital grew 3%, and labor grew 7%. If the elasticities of output with respect to capital and labor are 0.4 and 0.6, respectively, how much did productivity change in percent terms (approximately)? (Submit your answer with up to two decimals, i.e., -10.22 for -10.22% and 11.44 for 11.442%.)?
Suppose that the following production finetion is given: Q= =D4KL For the above production function, find the elasticity of substitution? Find the Retams to Scale. Using the above production funetion: find the labor demand and capital demand as functions of output (Q). price of labor (w) and price of capital (r). Does the Law of Demand hold for each inpur? Are these inputs normal or infenior inputs in the production process? Are inputs complements or substitutes? Why? Find the cost function for the above production finction. Verify the properties of the cost function. Suppose that a fim wants to produce 144 units of output and w-1, r-l. Find long run total cost. Suppose now that wage goes up to 4. Find the new long run total cost. Does fim substitute capital for labor? What is the percentage of cost saving relative to the case where firm is not able to substitute? Suppose that w=1, r-1 and a fimm has fixed amount of capital K 16 in short nun (SR). Find the short run total cost, average…
1. If the estimated Cobb-Douglas production function is Q = 10K0408 a. What are the output elasticities of capital and labor? If the firm increases only the quantity of capital or only the quantity of labor used by 10%, by how much would output increase? ( b. What type of returns to scale does this production function indicate? If the firm increases at the same time both the quantity of labor and the quantity of capital used by 10%, by how much would output increase? (
Suppose the current level of output is $10,000 and use the approximate methods learned in class. If the elasticities of output with respect to capital and labor are 0.3 and 0.7, respectively, a 7% increase in capital combined with a 12% increase in labor and a 5% increase in productivity would increase the current level of output to (approximately) $______.
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Find the elasticity of scale and the elasticity of substitution for the CES production function: 1 1 f(x₁, x₂) = (x³ + x2)³. Solution: We first calculate the marginal products: 2 fx₁ = 3 + = x1fx1 -+ 2 2 10 - ² ( x² + x ) ( + x^²) = ( + + + + ² ) 15 ²0 x² -2/3 x₂ + x2fxz Elasticity of scale = _1₁_+_*__*(+)´<°¸«d«)*;»_ f(x1, f(x1, ‹2)² = -2/3 r2/3 = TRS = t (where TRS = =t). ⇒ r = t³/² ⇒ ln(r) = ln (t) and o = -2/3 dln (r) dln (t) To get elasticity of substitution, we first need TRS and denote r = 1 x3 X1 TMS - F (+4+2)0 = fx₁ fx₂ 2 1 + x²) x₂² MIN x2 3 -2/3 (x² -2/3 2 2/3 2/3 x1 -2/3 = r²/3 x¹/3 + 1/3 = 1.
Show that the following production function has the constant elasticity of substitution: * = f(1, k) = akº (1 – a)!-o, -1/0 where x is total output obtained by using / and k units of labour and capital respectively and a and 0 are constants.
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Imagine you own a business firm with a friend of yours. During the first quarter you come up with the following information that: • Output elasticity of Capital is 0.2 • Output elasticity of Labour is 0.8 • Rate of Technological advancement is 12 a) What kind of business firm is the one that you own? Mention 2 of its characteristics. b) Derive the marginal productivity of capital and labour, by means of Cobb- Douglas production function. c) If Y = F (K, L) where K= 1000 and L= 400, determine the values of MPx and %3D MPL.
Answer 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?
Suppose a firm has a production function with two inputs, capital (K) and labor (L). The production function takes the form: Q = L2K2. Further, let the wage be given by w, the rental rate of capital be given by r, and suppose that the firm wishes to produce Q0 units of output. Determine the elasticity of substitution for this production function. Explain your answer. Determine the returns to scale for the production function. Solve for the long-run optimal input demand functions for capital and labor as a function of exogenous variables only. Derive an expression for long-run total cost as a function of the exogenous variables. Let w=16, r=25, and Q0 = 10,000. Solve for the long-run cost-minimizing input combination.