You manage a plant that mass-produces engines by teams of workers using assembly machines. The technology is summarized by the production function q = 5KL where q is the number of engines per week, K is the number of assembly machines, and L is the number of labor teams. Each assembly machine rents for r = \(10,000 per week, and each team costs w = \)5000 per week. Engine costs are given by the cost of labor teams and machines, plus $2000 per engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its design.

1. What is the cost function for your plant—namely, how much would it cost to produce q engines? What are average and marginal costs for producing q engines? How do average costs vary with output?

2. How many teams are required to produce 250 engines? What is the average cost per engine?

3. You are asked to make recommendations for the design of a new production facility. What capital/ labor (K/L) ratio should the new plant accommodate if it wants to minimize the total cost of producing at any level of output q?

Since the plant has fixed installation of 5 assembly machines, the capital units are 5, and the production function is:

$$\begin{gathered} q=25Lor \\ L=\frac{q}{25} \end{gathered}$$

The total cost function will aggregate fixed cost from the capital and variable cost from labor and raw materials per engine for producing q engines.

$$\begin{gathered} TC=rK+wL+2000q \\ =\$50000+\$5000L+2000q \\ =\$50000+\$5000\frac{q}{25}+2000q \\ =\$50000+\$2200q \end{gathered}$$

The following formula gives the average cost for q units of output:

$$\begin{gathered} \mathit{A}\mathit{C}\mathbf{=}\frac{\mathbf{T}\mathbf{C}}{\mathbf{q}} \\ =\frac{\$50,000+2,200q}{q} \\ =\frac{\$50,000}{q}+2,200 \end{gathered}$$

The marginal cost for each of the q engines is:

$$\begin{gathered} \mathit{M}\mathit{C}\mathbf{=}\frac{\mathbf{d}\mathbf{T}\mathbf{C}}{\mathbf{d}\mathbf{q}} \\ =\frac{d\left(\$50,000+2,200q\right)}{dq} \\ =\$2,200 \end{gathered}$$

The average total cost will decrease with increasing quantity because the average fixed cost will decline with increasing output. However, the average variable cost and marginal cost will remain constant at $2,200.

The plant has to make 250 engines. The number of labor required to produce 250 engines is:

$$\begin{gathered} L=\frac{q}{25} \\ =\frac{250}{25}=10 \end{gathered}$$

So, 10 labor teams are required to produce 250 engines.

The average cost for 250 engines is:

$$\begin{gathered} AC=\frac{\$50,000}{q}+2,200 \\ =\frac{\$50,000}{250}+2,200 \\ =\$200+2,200 \\ =\$2,400 \end{gathered}$$

Thus, the average cost per engine is $2,400.

To minimize the cost, the ratio of marginal productivity of input to its price for labor and capital should be equal.

$\frac{\mathbf{M}{\mathbf{P}}_{\mathbf{L}}}{\mathbf{w}}\mathbf{=}\frac{\mathbf{M}{\mathbf{P}}_{\mathbf{K}}}{\mathbf{r}}$

The marginal productivity of labor is:

$M{P}_L=\frac{d\left(5LK\right)}{dL}=5K$

The marginal productivity of capital is:

$M{P}_K=\frac{d\left(5LK\right)}{dK}=5L$

The following results are obtained by placing the values of w, r, and marginal productivity of labor and capital in the cost minimization rule.

$$\begin{gathered} \frac{5K}{w}=\frac{5L}{r} \\ \frac{K}{L}=\frac{w}{r} \\ \frac{K}{L}=\frac{5000}{10000} \\ \frac{K}{L}=\frac{1}{2} \end{gathered}$$

Thus, the cost-minimizing capital-labor ratio for the new plant to produce q engines at the prevailing prices of inputs and production function is 1:2.