Let's solve the problem step by step. ### **Step 1: Define the variables and constraints** Given: - Production function: \( f(x, y) = 50x^{4/5}y^{1/5} \) - \( x \): Number of units of labor (in hours) - \( y \): Number of units of capital (in dollars invested) - Labor cost: $20/hour - Total budget: $4,500,000 The cost constraint is: \[ 20x + y = 4,500,000 \] --- ### **Step 2: Set up the Lagrange function** We want to maximize \( f(x, y) \) subject to the constraint \( 20x + y = 4,500,000 \). Set up the Lagrangian: \[ \mathcal{L}(x, y, \lambda) = 50x^{4/5}y^{1/5} + \lambda(4,500,000 - 20x - y) \] --- ### **Step 3: Find the partial derivatives and set to zero** \[ \frac{\partial \mathcal{L}}{\partial x} = 50 \cdot \frac{4}{5} x^{-1/5} y^{1/5} - 20\lambda = 0 \] \[ \frac{\partial \mathcal{L}}{\partial y} = 50 x^{4/5} \cdot \frac{1}{5} y^{-4/5} - \lambda = 0 \] \[ \frac{\partial \mathcal{L}}{\partial \lambda} = 4,500,000 - 20x - y = 0 \] --- ### **Step 4: Solve the system** From the first two equations: \[ 50 \cdot \frac{4}{5} x^{-1/5} y^{1/5} = 20\lambda \] \[ 40 x^{-1/5} y^{1/5} = 20\lambda \] \[ 2 x^{-1/5} y^{1/5} = \lambda \quad (1) \] \[ 50 x^{4/5} \cdot \frac{1}{5} y^{-4/5} = \lambda \] \[ 10 x^{4/5} y^{-4/5} = \lambda \quad (2) \] Set the two expressions for \( \lambda \) equal: \[ 2 x^{-1/5} y^{1/5} = 10 x^{4/5} y^{-4/5} \] Divide both sides by \( x^{-1/5} \): \[ 2 y^{1/5} = 10 x \] \[ y^{1/5} = 5x \] \[ y = (5x)^5 = 3125x^5 \] --- ### **Step 5: Plug into the constraint** \[ 20x + y = 4,500,000 \] \[ 20x + 3125x^5 = 4,500,000 \] This is a quintic equation. Let's check if \( y \) could be written in terms of \( x \) and solved numerically. However, let's use the ratio approach from the Cobb-Douglas maximization (using the exponents as the optimal ratio): Optimal cost allocation for Cobb-Douglas: Let \( a = 4/5 \), \( b = 1/5 \), and total budget = \( B \). Allocate: - Fraction for labor: \( \frac{a}{a+b} = \frac{4/5}{1} = 0.8 \) - Fraction for capital: \( \frac{b}{a+b} = \frac{1/5}{1} = 0.2 \) So: - Labor: \( 0.8 \times 4,500,000 = 3,600,000 \) - Capital: \( 0.2 \times 4,500,000 = 900,000 \) --- ### **Step 6: Calculate hours of labor (\( x \))** Labor cost per hour = $20 \[ x = \frac{\text{Labor allocation}}{\text{Cost per hour}} = \frac{3,600,000}{20} = 180,000 \text{ hours} \] --- ### **Step 7: Final answers** **a. Hours of labor:** \[ \boxed{180,000} \] **b. Dollars to invest in capital:** \[ \boxed{900,000} \] --- ### **Step 8: Maximum production** Plug values into \( f(x, y) = 50x^{4/5}y^{1/5} \): First, calculate powers: - \( x^{4/5} = (180,000)^{4/5} \) - \( y^{1/5} = (900,000)^{1/5} \) Calculate: - \( \ln(180,000) = 12.1 \implies 4/5 \ln(180,000) = 9.68 \implies e^{9.68} \approx 16,140 \) - \( \ln(900,000) = 13.71 \implies 1/5 \ln(900,000) = 2.74 \implies e^{2.74} \approx 15.5 \) So, \[ f(x, y) = 50 \times 16,140 \times 15.5 \approx 12,523,950 \] --- ### **Step 9: If budget is increased by 1 dollar** Marginal product with respect to budget is the Lagrange multiplier \( \lambda \). For Cobb-Douglas, it's \( f(x, y)/B \): \[ \text{Increase in units per extra \$1:} \quad \frac{f(x, y)}{B} = \frac{12,523,950}{4,500,000} \approx 2.78 \] --- ## **Final Answers** **a. Hours of labor:** \(\boxed{180,000}\) **b. Dollars to invest in capital:** \(\boxed{900,000}\) **c. Maximum production:** \(\boxed{12,523,950}\) units **d. Increase per extra \$1:** \(\boxed{2.78}\) units Let me know if you'd like to see any calculation in more detail!