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 toestimate the production function for a particular line of office chairs. The report from these experts includes that the relevant production function is

Q = 2(K)^1/2 (L)^1/2

where K represents capital equipment and L is labor. Your company has already spent a total of $10,000 on the 4 units of capital equipment it owns. Due to currenteconomic conditions, the company does not have the flexibility needed to acquire additional equipment. If workers at the firm are paid a competitive wage of $100 andchairs can be sold for $200 each, what is your profit-maximizing level of output and labor usage? What is your maximum profit?

Answer 1

Given information:

- Production function: \(Q=2 K^{\frac{1}{2}} L^{\frac{1}{2}}\)

\(\cdot K=4\)

- Price of labor \((w)=\$ 100\)

- Price of capital \((r)=\$ 10,000 / 4=\$ 2500\)

- Price of chairs \(=\$ 200\)

At the Pareto optimal point, MRS=price ratio

That is,

$$ \begin{array}{l} \frac{M P_{L}}{M P_{K}}=\frac{w}{r} \\ \frac{\left(\frac{\delta Q}{\delta L}\right)}{\left(\frac{\delta Q}{\delta K}\right)}=\frac{w}{r} \\ \frac{2 K^{\frac{1}{2}} L^{-\frac{1}{2}}}{2 K^{-\frac{1}{2}} L^{\frac{1}{2}}}=\frac{100}{2500} \\ \frac{K}{L}=\frac{1}{25} \\ \frac{4}{L}=\frac{1}{25} \\ L^{*}=100 \\ \end{array} $$

Thus profit maximizing labor usage \(=100\) units.

Profit maximizing level of output \(=2 \mathrm{~K}^{1 / 2} \mathrm{~L}^{1 / 2}=2(4)^{1 / 2}(100)^{1 / 2}=40\) units

Calculate profit as follows:

$$ \begin{array}{l} \text { Profit }=\text { Total revenue }-\text { Total cost } \\ \text { Profit }=(\text { Price } \times \text { Quantity })-(w L+r K) \\ \text { Profit }=(\$ 200 \times 40)-(\$ 100 \times 100+\$ 2500 \times 4) \\ \text { Profit }=\$ 8,000-\$ 20,000 \\ \text { Profit }=-\$ 12000 \end{array} $$

The firm is incurring loss of \(\$ 12000\).