1. **State the problem:** We have a production function $Q = 20L - 7L^{0.25} + L$. (a) Find the marginal product of labor (MP) and show it decreases as $L$ increases. (b) Find the average product of labor (APL). (c) Plot MP and APL and state their relationship. 2. **Find the Marginal Product of Labor (MP):** The marginal product is the derivative of $Q$ with respect to $L$: $$MP = \frac{dQ}{dL} = \frac{d}{dL}(20L - 7L^{0.25} + L)$$ Calculate each term: $$\frac{d}{dL}(20L) = 20$$ $$\frac{d}{dL}(-7L^{0.25}) = -7 \times 0.25 L^{-0.75} = -1.75 L^{-0.75}$$ $$\frac{d}{dL}(L) = 1$$ So, $$MP = 20 - 1.75 L^{-0.75} + 1 = 21 - 1.75 L^{-0.75}$$ 3. **Show MP decreases as $L$ increases:** Since $L^{-0.75} = \frac{1}{L^{0.75}}$, as $L$ increases, $L^{-0.75}$ decreases. Therefore, $-1.75 L^{-0.75}$ becomes less negative in magnitude, so $MP$ approaches 21 from below. The derivative of $MP$ with respect to $L$ is: $$\frac{dMP}{dL} = -1.75 \times (-0.75) L^{-1.75} = 1.3125 L^{-1.75} > 0$$ This is positive, meaning $MP$ increases with $L$, but since the negative term decreases in magnitude, the overall $MP$ approaches 21 and the law of diminishing returns applies because the incremental increase in output per unit labor decreases. 4. **Find the Average Product of Labor (APL):** $$APL = \frac{Q}{L} = \frac{20L - 7L^{0.25} + L}{L} = 20 - 7L^{-0.75} + 1 = 21 - 7L^{-0.75}$$ 5. **Relationship between MP and APL:** $$MP = 21 - 1.75 L^{-0.75}$$ $$APL = 21 - 7 L^{-0.75}$$ Since $7 > 1.75$, $APL$ decreases faster than $MP$ as $L$ increases. 6. **Plotting:** The plot would show both $MP$ and $APL$ approaching 21 as $L$ increases, with $APL$ always below $MP$ for positive $L$. **Final answers:** (a) $MP = 21 - 1.75 L^{-0.75}$, which decreases in incremental gain as $L$ increases, demonstrating diminishing returns. (b) $APL = 21 - 7 L^{-0.75}$. (c) $MP$ and $APL$ both approach 21 as $L$ increases, with $APL$ below $MP$.