# Step 1: Check Cost-Minimization Condition To minimize costs, a firm should allocate resources such that the **marginal product per dollar** is equalized across inputs. The condition is: \[ \frac{MP_L}{w} = \frac{MP_K}{r} \] Where: - \( MP_L \) = Marginal Product of Labor = 20 - \( w \) = Wage Rate = $40 - \( MP_K \) = Marginal Product of Capital = 40 - \( r \) = Rental Rate of Capital = $80 Calculate each side: \[ \frac{MP_L}{w} = \frac{20}{40} = .5 \] \[ \frac{MP_K}{r} = \frac{40}{80} = .5 \] **Explanation:** Since \(\frac{MP_L}{w} = \frac{MP_K}{r}\), the firm is currently allocating labor and capital efficiently and is minimizing cost. --- # Step 2: Factor Substitution and Restoring Equilibrium Suppose the ratios above were **not equal**. For example, if \(\frac{MP_L}{w} > \frac{MP_K}{r}\), then each dollar spent on labor yields more output than a dollar spent on capital. **Restoring Equilibrium:** - The firm should **substitute labor for capital** (use more labor, less capital). - This increases the marginal product of capital (since it's used less) and decreases the marginal product of labor (since it's used more), moving the ratios toward equality. **Explanation:** By adjusting the combination of labor and capital, the firm will eventually reach the point where the marginal product per dollar is equal for both inputs, thus minimizing costs. --- # Final Answer - **Yes, the firm is minimizing cost** because the marginal product per dollar is equal for both labor and capital (\(.5\) for each). - If the ratios were not equal, the firm could restore equilibrium by substituting the input with the higher marginal product per dollar for the input with the lower one, until equality is reached.