Question

You are a manager for Herman Miller, a major manufacturer of office furniture. You recently hired an economist to work with engineering and operations experts to estimate the production function for a particular line of office chairs. The report from these experts indicates that the relevant production function is

$\mathbf{Q}\mathbf{=}\mathbf{2}{\mathbf{\left(}\mathbf{K}\mathbf{\right)}}^{\mathbf{1}\mathbf{/}\mathbf{2}}{\mathbf{\left(}\mathbf{L}\mathbf{\right)}}^{\mathbf{1}\mathbf{/}\mathbf{2}}$

where K represents capital equipment and L is labor. Your company has already spent a total of \(8,000 on the 9 units of capital equipment it owns. Due to current economic conditions, the company does not have the flexibility needed to acquire additional equipment. If workers at the firm are paid a competitive wage of \)120 per day and chairs can be sold for $400 each, what is your profit maximizing level of output and labor usage? What is your maximum profit?

Short answer

The profit maximizing input mix, in case the capital is fixed at 9 units, is capital $\mathrm{K}=\mathbf{9}$ and labor$\mathrm{L}=\mathbf{100}.$

The profit-maximizing output level is $\mathbf{60}$ units

The maximum profit earned is $\text{\$4,000}$.

Step-by-step solution

Find the production function with respect to labor

The derivative of the production function with regard to labor is the marginal product of labor:

$\begin{array}{c}{\mathrm{MP}}_{\mathrm{L}}=\frac{\partial \mathrm{Q}}{\partial \mathrm{L}}\\ =\frac{\partial \left(2{\mathrm{K}}^{\frac{1}{2}}{\mathrm{L}}^{\frac{1}{2}}\right)}{\partial \mathrm{L}}\\ =2\times \frac{1}{2}\times {\mathrm{K}}^{\frac{1}{2}}\times \frac{1}{{\mathrm{L}}^{\frac{1}{2}}}\\ =\frac{\sqrt{\mathrm{K}}}{\sqrt{\mathrm{L}}}\end{array}$

Find the marginal product of labour

The capital equipment of the company is set at 9 units. As a result, the marginal product for labor i.e., ${\mathrm{MP}}_{\mathrm{L}}$ is

$\begin{array}{c}{\mathrm{MP}}_{\mathrm{L}}=\frac{\sqrt{\mathrm{K}}}{\sqrt{\mathrm{L}}}\\ =\frac{\sqrt{9}}{\sqrt{\mathrm{L}}}\\ =\frac{3}{\sqrt{\mathrm{L}}}\end{array}$

Find the profit maximizing input

Calculating the optimal number of workers by multiplying the wage rate by the value of the marginal product of labor:

$\begin{array}{c}{\text{VMP}}_{\mathrm{L}}\text{=w}\\ 400\times \frac{3}{\sqrt{\mathrm{L}}}=120\\ \sqrt{\mathrm{L}}=10\\ \mathrm{L}=100\end{array}$

Therefore, the profit maximizing input mix, in case the capital is fixed at 9 units, is capital $\mathbf{K}\mathbf{=}\mathbf{9}$ and labor $\mathbf{L}\mathbf{=}\mathbf{100}\mathbf{.}$

At this input mix, the output produced is:

To figure out profit, total income and expense accountability is needed.

$\begin{array}{c}\mathrm{Q}=2{\mathrm{K}}^{\frac{1}{2}}{\mathrm{L}}^{\frac{1}{2}}\\ =2\times \sqrt{9}\times \sqrt{100}\\ =2\times 3\times 10\\ =60\end{array}$

Thus, the profit-maximizing output level is $\mathbf{60}$ units

Profit is the difference between revenue and expense. Thus, the furniture company made $24000 in total income by selling 60 chairs for $400 each. Again, the total cost of paying$\mathbf{\$}\mathbf{120}$ each to $\mathbf{100}$ units of labor and$\mathbf{\$}\mathbf{8}\mathbf{,}\mathbf{000}$each for 9 units of capital is$\mathbf{\$}\mathbf{20}\mathbf{,}\mathbf{000}\mathbf{.}$

Therefore, the maximum profit earned is $\mathbf{\$}\mathbf{4}\mathbf{,}\mathbf{000}\mathbf{=}\mathbf{(}\mathbf{\$}\mathbf{24}\mathbf{,}\mathbf{000}\mathbf{-}\mathbf{\$}\mathbf{20}\mathbf{,}\mathbf{000}\mathbf{)}$.