1. **State the problem:** Given the Cobb-Douglas production function $$Y = A \sqrt{K} \sqrt{L}$$ with $$A=12$$, $$K=36$$, and labor $$L$$ varying from 6 to 13 units, find the Total Productivity of Labor (TPL), Average Productivity of Labor (APL), and Marginal Productivity of Labor (MPL). Also, determine the optimum labor unit where the wage per marginal product equals 23.93 and labor work time is 8 hours. 2. **Recall formulas:** - Total Productivity of Labor (TPL) is output $$Y$$ as a function of $$L$$: $$TPL = Y = A \sqrt{K} \sqrt{L}$$. - Average Productivity of Labor (APL) is output per labor unit: $$APL = \frac{Y}{L}$$. - Marginal Productivity of Labor (MPL) is the derivative of output with respect to labor: $$MPL = \frac{dY}{dL}$$. 3. **Calculate constants:** $$A = 12, \quad K = 36 \Rightarrow \sqrt{K} = 6$$ 4. **Express TPL:** $$TPL = 12 \times 6 \times \sqrt{L} = 72 \sqrt{L}$$ 5. **Calculate APL:** $$APL = \frac{TPL}{L} = \frac{72 \sqrt{L}}{L} = 72 \frac{\sqrt{L}}{L} = 72 L^{-\frac{1}{2}}$$ 6. **Calculate MPL:** $$MPL = \frac{d}{dL} (72 \sqrt{L}) = 72 \times \frac{1}{2} L^{-\frac{1}{2}} = 36 L^{-\frac{1}{2}} = \frac{36}{\sqrt{L}}$$ 7. **Find optimum labor unit:** The wage per marginal product is 23.93 per hour, and labor work time is 8 hours, so total wage per labor unit is: $$W = 23.93 \times 8 = 191.44$$ Set MPL equal to wage per labor unit to find optimum $$L^*$$: $$MPL = W \Rightarrow \frac{36}{\sqrt{L^*}} = 191.44$$ Solve for $$L^*$$: $$\sqrt{L^*} = \frac{36}{191.44} \approx 0.188\Rightarrow L^* = (0.188)^2 \approx 0.035$$ Since this is less than the labor range 6 to 13, the optimum labor unit within the given range is at the minimum labor unit 6 (because MPL decreases as L increases). **Summary:** - $$TPL = 72 \sqrt{L}$$ - $$APL = 72 L^{-\frac{1}{2}}$$ - $$MPL = \frac{36}{\sqrt{L}}$$ - Optimum labor unit $$L^* = 6$$ (minimum in given range)