# Step-by-Step Solution ## 1. **Production Function and Constant Returns to Scale** The production function is: \[ y = L^a K^b \] It is given that the production function exhibits **constant returns to scale**. For any \( t > 0 \): \[ y' = (tL)^a (tK)^b = t^a L^a \cdot t^b K^b = t^{a+b} L^a K^b \] Constant returns to scale means \( y' = t y \): \[ t^{a+b} L^a K^b = t L^a K^b \implies t^{a+b} = t \implies a + b = 1 \] --- ## 2. **Long-run Cost Minimization** Let: - \( w \): wage rate (price of labor) - \( r \): rental rate (price of capital) The **cost minimization problem** for output level \( y \): \[ \min_{L,K} \quad wL + rK \quad \text{subject to} \quad L^a K^b = y \] ### Form the Lagrangian \[ \mathcal{L} = wL + rK + \lambda (y - L^a K^b) \] ### First-order conditions \[ \frac{\partial \mathcal{L}}{\partial L} = w - \lambda a L^{a-1} K^b = 0 \implies w = \lambda a L^{a-1} K^b \] \[ \frac{\partial \mathcal{L}}{\partial K} = r - \lambda b L^a K^{b-1} = 0 \implies r = \lambda b L^a K^{b-1} \] ### Ratio of FOCs \[ \frac{w}{r} = \frac{a L^{a-1} K^b}{b L^a K^{b-1}} \] Simplify numerator and denominator: \[ = \frac{a}{b} \cdot \frac{L^{a-1}}{L^a} \cdot \frac{K^b}{K^{b-1}} = \frac{a}{b} \cdot L^{-1} \cdot K^1 = \frac{a}{b} \cdot \frac{K}{L} \] So, \[ \frac{w}{r} = \frac{a}{b} \cdot \frac{K}{L} \] --- ## 3. **Labor Cost is Twice Capital Cost** Let \(C_L = wL\) and \(C_K = rK\). Given: \(wL = 2rK\) So: \[ wL = 2rK \implies \frac{w}{r} = 2 \cdot \frac{K}{L} \] --- ## 4. **Equate the Two Expressions for \(w/r\)** From above: \[ \frac{w}{r} = \frac{a}{b} \cdot \frac{K}{L} = 2 \cdot \frac{K}{L} \] So, \[ \frac{a}{b} = 2 \implies a = 2b \] But from step 1, \(a + b = 1\): \[ a + b = 1 \\ 2b + b = 1 \\ 3b = 1 \\ b = \frac{1}{3} \] \[ a = 2b = 2 \cdot \frac{1}{3} = \frac{2}{3} \] --- ## 5. **Final Answer** \[ \boxed{a = \frac{2}{3}, \quad b = \frac{1}{3}} \]