This problem involves a production function $T = 30K^{0.4}L^{0.7}$, and we are tasked with minimizing the cost of producing 15,000 thneeds. To achieve this, we need to use cost minimization under a production constraint, a typical optimization problem in economics.

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Step 1: Define the problem

1. Production function:
   $T = 30K^{0.4}L^{0.7}$
2. Cost function:
   $C = p_K K + p_L L = 2000K + 4000L$
3. Output constraint:
   $T = 150 \quad (\text{since output is in 100s, 15,000 thneeds = 150 in this context}).$

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Step 2: Use the Lagrangian method

The Lagrangian is defined as: $\mathcal{L} = 2000K + 4000L + \lambda\left(150 - 30K^{0.4}L^{0.7}\right)$ where $\lambda$ is the Lagrange multiplier.

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Step 3: Solve the system of equations

Take partial derivatives of $\mathcal{L}$ with respect to $K, L,$ and $\lambda$, and set them to zero:

1. Partial with respect to $K$: $\frac{\partial \mathcal{L}}{\partial K} = 2000 - \lambda \cdot 30 \cdot 0.4 K^{-0.6}L^{0.7} = 0$ Simplify: $\lambda = \frac{2000}{12 K^{-0.6} L^{0.7}} \quad \text{(Eq. 1)}$

2. Partial with respect to $L$: $\frac{\partial \mathcal{L}}{\partial L} = 4000 - \lambda \cdot 30 \cdot 0.7 K^{0.4}L^{-0.3} = 0$ Simplify: $\lambda = \frac{4000}{21 K^{0.4} L^{-0.3}} \quad \text{(Eq. 2)}$

3. Partial with respect to $\lambda$: $\frac{\partial \mathcal{L}}{\partial \lambda} = 150 - 30K^{0.4}L^{0.7} = 0$ Simplify: $150 = 30K^{0.4}L^{0.7} \quad \text{(Eq. 3)}$

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Step 4: Equate $\lambda$ from Eq. 1 and Eq. 2

$\frac{2000}{12 K^{-0.6} L^{0.7}} = \frac{4000}{21 K^{0.4} L^{-0.3}}$ Cross-multiply: $2000 \cdot 21 K^{0.4} L^{-0.3} = 4000 \cdot 12 K^{-0.6} L^{0.7}$ Simplify the coefficients: $42 K^{0.4} L^{-0.3} = 48 K^{-0.6} L^{0.7}$ Divide through by $K^{-0.6}L^{-0.3}$: $42 K^{1} = 48 L$ Simplify: $K = \frac{48}{42} L = \frac{8}{7}L \quad \text{(Eq. 4)}$

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Step 5: Substitute $K = \frac{8}{7}L$ into Eq. 3

From Eq. 3: $150 = 30 \left(\frac{8}{7}L\right)^{0.4}L^{0.7}$ Simplify: $150 = 30 \cdot \left(\frac{8}{7}\right)^{0.4} \cdot L^{1.1}$ Divide through by 30: $5 = \left(\frac{8}{7}\right)^{0.4} \cdot L^{1.1}$ Approximate $\left(\frac{8}{7}\right)^{0.4}$ (using a calculator or numerical methods): $\left(\frac{8}{7}\right)^{0.4} \approx 1.049$ Thus: $5 = 1.049 \cdot L^{1.1}$ Solve for $L^{1.1}$: $L^{1.1} = \frac{5}{1.049} \approx 4.766$ Take both sides to the power of $\frac{1}{1.1}$: $L \approx 4.766^{\frac{1}{1.1}} \approx 4.4$

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Step 6: Solve for $K$

Using $K = \frac{8}{7}L$ from Eq. 4: $K = \frac{8}{7} \cdot 4.4 \approx 5.03$

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Final Results:

• $K^* \approx 5.03$ (units of capital)
• $L^* \approx 4.4$ (units of labor)

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Let me know if you’d like more details or clarifications!

Follow-up Questions:

1. How is the Lagrangian method used in constrained optimization?
2. What is the economic intuition behind $K = \frac{8}{7}L$?
3. How do changes in $p_K$ and $p_L$ affect the optimal values of $K$ and $L$?
4. Why does the Cobb-Douglas function ensure diminishing marginal returns?
5. Can you explain how the elasticity of output relates to capital and labor shares?

Tip:

When solving cost-minimization problems with Cobb-Douglas functions, equate the marginal product per dollar spent across all inputs to find optimal input proportions!