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) = COL 1C 2 where co, C₁, and c₂ are constants. The variable L represents the units of input labor and the variable C represents the units of input of capital. (a) In this example, assume co = 5, c₁ = 0.25, and c₂ = 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $65,000 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. L, CZO $ ≤ 65,000 (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 necessary.) -([ at (L, C) =
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) = COL 1C 2 where co, C₁, and c₂ are constants. The variable L represents the units of input labor and the variable C represents the units of input of capital. (a) In this example, assume co = 5, c₁ = 0.25, and c₂ = 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $65,000 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. L, CZO $ ≤ 65,000 (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 necessary.) -([ at (L, C) =
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Transcribed Image Text: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) = CL1C2
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 co = 5, C₁ = 0.25, and C₂ = 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $65,000 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.
L, C ≥ 0
$
≤ 65,000
(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 necessary.)
at (L, C) =
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I am not sure why part b was incorrect
![**Cobb-Douglas Production Function and Optimization Model**
The Cobb-Douglas production function is a fundamental model in economics used to examine output as a function of capital and labor. It is expressed as:
\[ f(L, C) = c_0L^{c_1}C^{c_2} \]
where \(c_0\), \(c_1\), and \(c_2\) are constants. The variable \(L\) represents the units of labor input, and the variable \(C\) represents the units of capital input.
### (a) Problem Scenario
In this specific problem, we are given:
- \(c_0 = 5\)
- \(c_1 = 0.25\)
- \(c_2 = 0.75\)
Assume each unit of labor costs $25, and each unit of capital costs $75. Given a budget of $65,000, our goal is to develop an optimization model that determines how to allocate the budget between capital and labor to maximize output.
**Objective Function:**
Maximize:
\[ 5L^{0.25}C^{0.75} \]
**Subject to:**
\[ 25L + 75C \leq 65,000 \]
\[ L, C \geq 0 \]
### (b) Finding the Optimal Solution
The task is to find the optimal solution value in dollars based on the formulated model.
**Hint:** Introduce bound constraints based on the budget constraint:
- \( L \leq 3,000 \)
- \( C \leq 1,000 \)
Utilize the Multistart option as described in Appendix 8.1.
**Proposed Solution:**
- Maximum Revenue: $3,750
- Optimal Allocation: \( (L, C) = (750, 750) \)
**Note:** The initial values listed here are marked with red crosses, indicating an error or incomplete calculation. Reevaluation may be necessary.
This exercise demonstrates a practical application of the Cobb-Douglas function in optimization and illustrates how constraints and optimization techniques can be used to make economic decisions.](https://content.bartleby.com/qna-images/question/eeaa8045-e0ae-4f8f-bfb9-81e4f69114f4/c3a215b7-eb31-46f8-8235-cab60d399ac3/pj55bb_processed.png)
Transcribed Image Text:**Cobb-Douglas Production Function and Optimization Model**
The Cobb-Douglas production function is a fundamental model in economics used to examine output as a function of capital and labor. It is expressed as:
\[ f(L, C) = c_0L^{c_1}C^{c_2} \]
where \(c_0\), \(c_1\), and \(c_2\) are constants. The variable \(L\) represents the units of labor input, and the variable \(C\) represents the units of capital input.
### (a) Problem Scenario
In this specific problem, we are given:
- \(c_0 = 5\)
- \(c_1 = 0.25\)
- \(c_2 = 0.75\)
Assume each unit of labor costs $25, and each unit of capital costs $75. Given a budget of $65,000, our goal is to develop an optimization model that determines how to allocate the budget between capital and labor to maximize output.
**Objective Function:**
Maximize:
\[ 5L^{0.25}C^{0.75} \]
**Subject to:**
\[ 25L + 75C \leq 65,000 \]
\[ L, C \geq 0 \]
### (b) Finding the Optimal Solution
The task is to find the optimal solution value in dollars based on the formulated model.
**Hint:** Introduce bound constraints based on the budget constraint:
- \( L \leq 3,000 \)
- \( C \leq 1,000 \)
Utilize the Multistart option as described in Appendix 8.1.
**Proposed Solution:**
- Maximum Revenue: $3,750
- Optimal Allocation: \( (L, C) = (750, 750) \)
**Note:** The initial values listed here are marked with red crosses, indicating an error or incomplete calculation. Reevaluation may be necessary.
This exercise demonstrates a practical application of the Cobb-Douglas function in optimization and illustrates how constraints and optimization techniques can be used to make economic decisions.
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