Managerial Economics: Applications, Strategies and Tactics (MindTap Course List)
14th Edition
ISBN: 9781305506381
Author: James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris
Publisher: Cengage Learning
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Chapter 7, Problem 1.1CE
To determine
To evaluate the estimation of the Cobb-Douglas production function.
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Quadratic Production Function
Estimate a quadratic production function where Q = output; L = labour input; K = capital input.
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Estimate the Cobb-Douglas production function Q= ¼ αLβ1Kβ2, where Q = output; L = labour input; K = capital input; and α, β1, and β2 are the parameters to be estimated.
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Consider the following production function:q = (KL)^α, where α > 0.Answer the following questions:(a) Under what conditions (i.e. values of α) will the production function exhibit decreasing returns to scale? Under what conditions will it exhibit constant returns to scale? Under what circumstances will it exhibit increasing returns to scale?
(b) Confirm that the marginal physical product of capital is homogenous of degree zero in the case in which the production function exhibits constant returns to scale.
(c) Derive an expression for the cost function of a firm using the productionfunction to produce output of a good.
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Chapter 7 Solutions
Managerial Economics: Applications, Strategies and Tactics (MindTap Course List)
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- Assuming a Cobb-Douglas production function with constant returns to scale, then, as L rises with K and A constant, it will be the case that: (a) Both the marginal product of labour and the marginal product of capital will fall(b) The marginal product of labour will fall and the marginal product of capitall will rise(c) Both the marginal product of labour and the marginal product of capital will rise(d) The marginal product of labour will rise and the marginal product of capital will fallarrow_forwardThe Cobb - Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C) = COLc1Cc2 where c0, c1, and c2 are constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a)In this example, assume c0 = 5, c1 = 0.25, and c2= 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $70,000 available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. = 0 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in units)? (Hint: When using Excel Solver, use the bounds 0 < = L < = 3,000 and 0 <= C <= 1,000. Round your answers to the nearest integer when necessary.)________ units at (L, C) = Please use Excel to get answers. Show steps on Excel Please!arrow_forwardFor each of the following production functions, determine whether it exhibits increasing, constant or decreasing returns to scale: a) Q = 2K + L b) Q = 3L + L/K c) Q = Min(2K,L) d) Q = L*Karrow_forward
- Consider the production function: Q = 2K + 3L. The MRTSLK is:arrow_forwardThe engineers at Morris Industries obtained the following estimate of the firm's production function: Q = F(K, L) = min {3K, 4L} How much output is produced when 2 units of labor and 5 units of capital are employed?arrow_forwardConsider the following production function: q = (KL)“, where a > 0. Answer the following questions: (a) Under what conditions (i.e. values of a) will the production function exhibit decreasing returns to scale? Under what conditions will it exhibit constant returns to scale? Under what circumstances will it exhibit increasing returns to scale? (b) Confirm that the marginal physical product of capital is homogenous of degree zero in the case in which the production function exhibits constant returns to scale. (c) Derive an expression for the cost function of a firm using the production function to produce output of a good. (d) Find the first and second partial derivatives of the cost function with respect to q. Interpret the second partial derivative and relate the sign of the derivative to the returns to scale.arrow_forward
- How would you determine that a two-input Cobb-Douglas production function has decreasing returns to scale (DRS), increasing returns to scale (IRS) or constant returns to scale (CRS) depending on whether β is larger than, smaller than, or equal to one?arrow_forwardThe engineers at Morris Industries obtained the following estimate of the firm’s production function Q = F ( K, L ) = min { 3K, 4L } How much output is produced when 2 units of labor and 5 units of capital are employed?arrow_forwardA second firm's production function is given by the equation Q = 12L.5K.5. Input prices are $36 per labor unit and $16 per capital unit, and P = $10. a. In the short run, the firm has a fixed amount of capital, K = 9. Create a spreadsheet to model this production setting. Determine the firm's profit - maximizing employment of labor. Use the spreadsheet to probe the solution using your spreadsheet's optimizer.arrow_forward
- The Cobb-Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C) = c₂LC1C²2 C2 are constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a) In this example, assume co 5, C₁ = 0.25, and c2₂ 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $90,000 available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. = = where co, C₁, and Max s.t. L, C ≥ 0 A ≤ 90,000 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in dollars)? Hint: Put bound constraints on the variables based on the budget constraint. Use L ≤ 3,000 and C ≤ 1,000 and use the Multistart option as described in Appendix 8.1. (Round your answers to the nearest integer when…arrow_forwardQ2. Determine the returns to scale of the following production function: Hand written plzz Y = 5K0.5 +3L0.5 (a) constant (b) increasing (c) decreasing (d) indeterminatearrow_forwardThe Cobb - Douglas production function is a classic model from economics used to model output as a function of capital c. and labor. It has the form f(L,C) = c_L" C² where co, c₁, and c₂ are constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a) In this example, assume c₁ = 5, c₁ = 0.25, and c = 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $80,000 available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. 25L +75C 80000 L,C>0 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in units)? (Hint: When using Excel Solver, use the bounds 0 L3,000 and 0 C1,000. Round your answers to the nearest integer when necessary.) units at (L,C) = (x)arrow_forward
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