1. The problem involves verifying the equality \( \frac{MP_k}{P_k} = \frac{MP_l}{P_l} \) given the production function \( P = 12kl \) and marginal products \( MP_k = \frac{\partial P}{\partial k} = 12l \) and \( MP_l = \frac{\partial P}{\partial l} = 12k \). 2. Given values are \( P_k = 4 \), \( P_l = 3 \), \( k = 12 \), and \( l = 16 \). 3. Calculate \( \frac{MP_k}{P_k} \): $$\frac{MP_k}{P_k} = \frac{12l}{4} = 3l$$ Substitute \( l = 16 \): $$3 \times 16 = 48$$ 4. Calculate \( \frac{MP_l}{P_l} \): $$\frac{MP_l}{P_l} = \frac{12k}{3} = 4k$$ Substitute \( k = 12 \): $$4 \times 12 = 48$$ 5. Since both ratios equal 48, the equality \( \frac{MP_k}{P_k} = \frac{MP_l}{P_l} \) is verified. Final answer: \( \frac{MP_k}{P_k} = \frac{MP_l}{P_l} = 48 \).