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Sagot :
Certainly! Let's tackle this problem using optimization techniques, specifically the method of Lagrange multipliers. Here's a step-by-step solution:
### Problem Breakdown
- We are given a production function: [tex]\( P(L, K) = 1500 L^{2/3} K^{1/3} \)[/tex].
- A unit of labor costs [tex]$100. - A unit of capital costs $[/tex]400.
- Total budget for labor and capital is [tex]$360,000. We need to find: (a) The optimal amounts spent on labor and capital to maximize production. (b) The value of the Lagrange multiplier. ### Steps to Solve 1. Define the Objective Function and Constraint: - Objective Function: \( P(L, K) = 1500 L^{2/3} K^{1/3} \) - Cost Constraint: \( 100L + 400K = 360,000 \) 2. Set Up the Lagrangian: - The Lagrangian \( \mathcal{L}(L, K, \lambda) = 1500 L^{2/3} K^{1/3} + \lambda (360,000 - 100L - 400K) \) 3. Take Partial Derivatives and Set Them to Zero: - \(\frac{\partial \mathcal{L}}{\partial L} = 1000 L^{-1/3}K^{1/3} - 100 \lambda = 0 \) - \(\frac{\partial \mathcal{L}}{\partial K} = 500 L^{2/3} K^{-2/3} - 400 \lambda = 0 \) - \(\frac{\partial \mathcal{L}}{\partial \lambda} = 360,000 - 100L - 400K = 0 \) 4. Solve the System of Equations: From \(\frac{\partial \mathcal{L}}{\partial L} = 0 \): \[ 1000 \frac{K^{1/3}}{L^{1/3}} = 100 \lambda \rightarrow \lambda = 10 \frac{K^{1/3}}{L^{1/3}} \] From \(\frac{\partial \mathcal{L}}{\partial K} = 0 \): \[ 500 \frac{L^{2/3}}{K^{2/3}} = 400 \lambda \rightarrow \lambda = \frac{5}{4} \frac{L^{2/3}}{K^{2/3}} \] Set the expressions for \(\lambda\) equal to each other: \[ 10 \frac{K^{1/3}}{L^{1/3}} = \frac{5}{4} \frac{L^{2/3}}{K^{2/3}} \rightarrow 40 K^{1/3 + 2/3} = 5 L^{2/3 + 1/3} \rightarrow 40K = 5L \rightarrow 8K = L \] 5. Substitute into the Constraint: \[ 100L + 400K = 360,000 \] Substituting \( L = 8K \): \[ 100(8K) + 400K = 360,000 \rightarrow 800K + 400K = 360,000 \rightarrow 1200K = 360,000 \rightarrow K = 300 \] Therefore, \( L = 8 \times 300 = 2400 \). 6. Calculate the Spending: - Amount spent on labor: \( 2400 \times 100 = \$[/tex]240,000 \)
- Amount spent on capital: [tex]\( 300 \times 400 = \$120,000 \)[/tex]
### Results
(a) The optimal amount spent on labor is [tex]\(\$240,000\)[/tex].
The optimal amount spent on capital is [tex]\(\$120,000\)[/tex].
(b) The value of the Lagrange multiplier is [tex]\(\lambda = 10\)[/tex].
Thus, here are the answers:
- How much should be spent on labor? [tex]\(\$240,000\)[/tex]
- How much on capital? [tex]\(\$120,000\)[/tex]
- The value of the Lagrange multiplier? [tex]\(10\)[/tex].
### Problem Breakdown
- We are given a production function: [tex]\( P(L, K) = 1500 L^{2/3} K^{1/3} \)[/tex].
- A unit of labor costs [tex]$100. - A unit of capital costs $[/tex]400.
- Total budget for labor and capital is [tex]$360,000. We need to find: (a) The optimal amounts spent on labor and capital to maximize production. (b) The value of the Lagrange multiplier. ### Steps to Solve 1. Define the Objective Function and Constraint: - Objective Function: \( P(L, K) = 1500 L^{2/3} K^{1/3} \) - Cost Constraint: \( 100L + 400K = 360,000 \) 2. Set Up the Lagrangian: - The Lagrangian \( \mathcal{L}(L, K, \lambda) = 1500 L^{2/3} K^{1/3} + \lambda (360,000 - 100L - 400K) \) 3. Take Partial Derivatives and Set Them to Zero: - \(\frac{\partial \mathcal{L}}{\partial L} = 1000 L^{-1/3}K^{1/3} - 100 \lambda = 0 \) - \(\frac{\partial \mathcal{L}}{\partial K} = 500 L^{2/3} K^{-2/3} - 400 \lambda = 0 \) - \(\frac{\partial \mathcal{L}}{\partial \lambda} = 360,000 - 100L - 400K = 0 \) 4. Solve the System of Equations: From \(\frac{\partial \mathcal{L}}{\partial L} = 0 \): \[ 1000 \frac{K^{1/3}}{L^{1/3}} = 100 \lambda \rightarrow \lambda = 10 \frac{K^{1/3}}{L^{1/3}} \] From \(\frac{\partial \mathcal{L}}{\partial K} = 0 \): \[ 500 \frac{L^{2/3}}{K^{2/3}} = 400 \lambda \rightarrow \lambda = \frac{5}{4} \frac{L^{2/3}}{K^{2/3}} \] Set the expressions for \(\lambda\) equal to each other: \[ 10 \frac{K^{1/3}}{L^{1/3}} = \frac{5}{4} \frac{L^{2/3}}{K^{2/3}} \rightarrow 40 K^{1/3 + 2/3} = 5 L^{2/3 + 1/3} \rightarrow 40K = 5L \rightarrow 8K = L \] 5. Substitute into the Constraint: \[ 100L + 400K = 360,000 \] Substituting \( L = 8K \): \[ 100(8K) + 400K = 360,000 \rightarrow 800K + 400K = 360,000 \rightarrow 1200K = 360,000 \rightarrow K = 300 \] Therefore, \( L = 8 \times 300 = 2400 \). 6. Calculate the Spending: - Amount spent on labor: \( 2400 \times 100 = \$[/tex]240,000 \)
- Amount spent on capital: [tex]\( 300 \times 400 = \$120,000 \)[/tex]
### Results
(a) The optimal amount spent on labor is [tex]\(\$240,000\)[/tex].
The optimal amount spent on capital is [tex]\(\$120,000\)[/tex].
(b) The value of the Lagrange multiplier is [tex]\(\lambda = 10\)[/tex].
Thus, here are the answers:
- How much should be spent on labor? [tex]\(\$240,000\)[/tex]
- How much on capital? [tex]\(\$120,000\)[/tex]
- The value of the Lagrange multiplier? [tex]\(10\)[/tex].
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