Chapter 5, Problem 1CACQ

a)

To determine

To Calculate:The average product of labor when capital is 81units and variable input, labor is 16units and also whenvariable input, labor changes to 256units.

Explanation of Solution

  $\begin{array}{l}A{P}_L=\frac{Q}{L}=\frac{K^{3/4}{L}^{1/4}}{L}={\left(\frac{K}{L}\right)}^{3/4}\\ \text{Average prodcut of labor when capital is 81 and labor is 16}\text{.}\\ A{P}_L={\left(\frac{K}{L}\right)}^{3/4}\\ A{P}_L={\left(\frac{81}{16}\right)}^{3/4}={\left(\frac{3}{2}\right)}^3=3.375\end{array}$

  $\begin{array}{l}A{P}_L\text{ when capital is 81 and labor is 256}\\ A{P}_L={\left(\frac{81}{256}\right)}^{3/4}={\left(\frac{3}{4}\right)}^3=0.422\end{array}$

b)

To determine

To calculate:Marginal product of labor.

Explanation of Solution

  $\begin{array}{l}M{P}_L=\frac{dQ}{dL}=K^{3/4}\left(1/4\right)L^{\frac{1}{4}-1}=\frac{1}{4}{\left(\frac{K}{L}\right)}^{3/4}\\ M{P}_L\text{ When capital is 81}\text{.}\\ M{P}_L=\frac{1}{4}{\left(\frac{81}{L}\right)}^{3/4}=\frac{6.75}{L^{0.75}}\end{array}$

  $\begin{array}{l}M{P}_L\text{ When capital is 81 and labor is 16}\text{.}\\ M{P}_L=\frac{6.75}{{16}^{0.75}}=0.844\\ M{P}_L\text{ When capital is 81 and labor is 81}\text{.}\\ M{P}_L=\frac{6.75}{{81}^{0.75}}=0.250\end{array}$

c)

To determine

To calculate: Number of labor that a firm should hire.

Explanation of Solution

  $\begin{array}{c}VM{P}_L={\text{when capital is 81 = (MP}}_L\text{ when capital is 81)}\times \text{\$200}\\ \text{=}\frac{6.75}{L^{0.75}}\times \$200\\ =\frac{1350}{L^{0.75}}\end{array}$

Number of labors hired to maximize the profit.

  $\begin{array}{c}VM{P}_L\text{ when capital is 81 = \$500}\\ \frac{1350}{L^{0.75}}=50\\ L=81\end{array}$