Suppose that a paving company produces paved parking spaces (q) using a fixed quantity of land (T) and variable quantities of cement (C) and labor (L). The firm is currently paving 1000 parking spaces. The firm’s cost of cement is \(4,000 per acre covered, and its cost of labor is \)12/hour. For the quantities of C and L that the firm has chosen, MPC = 50 and MPL = 4.

1. Is this firm minimizing its cost of producing parking spaces? How do you know?

2. If the firm is not cost-minimizing, how must it alter its choices of C and L in order to decrease cost?

The inputs used for parking spaces are labor and cement.The cost minimization rule says that the marginal productivity of the inputs per dollar should be equal. Thus,

$\frac{\mathbf{M}{\mathbf{P}}_{\mathbf{L}}}{\mathbf{w}}\mathbf{=}\frac{\mathbf{M}{\mathbf{P}}_{\mathbf{C}}}{\mathbf{c}}$

For labor, the marginal productivity per dollar is:

$\frac{{\mathrm{MP}}_{\mathrm{L}}}{\mathrm{w}}=\frac{4}{12}=0.3334$

For cement, the marginal productivity per dollar is:

$\frac{{\mathrm{MP}}_c}{c}=\frac{50}{4000}=0.0125$

Since the marginal productivity of the two inputs per dollar does not match, the firm has not recently minimized the cost.

To minimize the cost, the firm can increase the labor, which will lower its marginal productivity per dollar, equal to 0.0125.

$$\begin{gathered} \frac{M{P}_L}{w}=0.0125 \\ \frac{M{P}_L}{12}=0.0125 \\ MP=0.15 \end{gathered}$$

Thus, the firm can increase the labor till its marginal productivity becomes 0.15.

The firm can also choose to decrease the cement employed such that the marginal productivity per dollar equals 0.3334.

$$\begin{gathered} \frac{M{P}_c}{c}=0.3334 \\ \frac{M{P}_c}{4000}=0.3334 \\ M{P}_c=1333.6 \end{gathered}$$

Thus, the firm can also choose to decrease the cement till its marginal productivity increases to 1,333.6.