Question

A firm uses a single input, labor, to produce output q according to the production function q = 8√L. The commodity sells for \(150 per unit and the wage rate is \)75 per hour.

1. Find the profit-maximizing quantity of L.
2. Find the profit-maximizing quantity of q.
3. What is the maximum profit?
4. Suppose now that the firm is taxed \(30 per unit of output and that the wage rate is subsidized at a rate of \)15 per hour. Assume that the firm is a price taker, so the price of the product remains at $150. Find the new profit-maximizing levels of L, q, and profit.
5. Now suppose that the firm is required to pay a20-percent tax on its profits. Find the new profit-maximizing levels of L, q, and profit.

Short answer

1. The profit-maximizing quantity of labor will be 64.
2. The profit-maximizing output (q) will be 64 units.
3. The maximum profit will be $4800.
4. The new profit-maximizing quantity of labor will be 64, the output 64 units, and the profit will be $3840.
5. The new profit-maximizing quantity of labor will be 64, the output 64 units, and the profit will be $3840.

Step-by-step solution

Step 1. Explanation for part (a)

The profit-maximizing level of labor is calculated below:

The profit-maximizing quantity of labor will be 64 laborers.

Step 2. Explanation for part (b)

The profit-maximizing output is calculated below:

$$\begin{gathered} q=8\sqrt{L} \\ =8\sqrt{64} \\ =8\times 8 \\ =64 \end{gathered}$$

The profit-maximizing output will be 64 units.

Step 3. Explanation for part (c)

The profit-maximizing quantity of labor is 64 labors, and output is 64 units.

The profit is calculated below:

$$\begin{gathered} \mathrm{\pi }=\mathrm{TR}-\mathrm{TC} \\ =\mathrm{pq}-\mathrm{wL} \\ =150\left(8\sqrt{64}\right)-75\left(64\right) \\ =150\times 64-75\left(64\right) \\ =\left(150-75\right)\times 64 \\ =75\times 64 \\ =\$4800 \end{gathered}$$

The profit will be $4,800.

Step 4. Explanation for part (d)

After-tax, the price received by the producer will reduce; thus, the price received falls by the tax amount, i.e., by $30. Hence, the producer gets $120 (=150 – 30). The wage subsidy reduces labor costs; thus, the per-unit cost for labor will be $60 (=75 – 15).

The profit-maximizing labor is calculated below:

$$\begin{gathered} \mathrm{\pi }=\mathrm{TR}-\mathrm{TC} \\ =\mathrm{pq}-\mathrm{wL} \\ =120\left(8{\mathrm{L}}^{0.5}\right)-60\mathrm{L} \\ \frac{\mathrm{d\pi }}{\mathrm{dL}}=0.5\times 120\times 8{\mathrm{L}}^{-0.5}-60=0 \\ 480{\mathrm{L}}^{-0.5}=60 \\ 60{\mathrm{L}}^{0.5}=480 \end{gathered}$$

The output is calculated below:

$$\begin{gathered} q=8\sqrt{L} \\ =8\sqrt{64} \\ =8\times 8 \\ =64 \end{gathered}$$

The output will be 64 units.

The profit is calculated below:

$$\begin{gathered} \mathrm{\pi }=\mathrm{TR}-\mathrm{TC} \\ =\mathrm{pq}-\mathrm{wL} \\ =120\left(8\sqrt{64}\right)-60\left(64\right) \\ =120\times 64-60\times 64 \\ =\left(120-60\right)\times 64 \\ =60\times 64 \\ =\$3840 \end{gathered}$$

The profit will be $3,840.

Explanation for part (e)

When there is a tax of 20% on profit, there is no change in the product's price and labor cost, i.e., the product's price is $150, and the labor cost is $75.

The profit-maximizing level of labor is calculated below:

$$\begin{gathered} \mathrm{\pi }=\mathrm{TR}-\mathrm{TC} \\ =\mathrm{pq}-\mathrm{wL} \\ =150\left(8{\mathrm{L}}^{0.5}\right)-75\mathrm{L} \\ \frac{\mathrm{d\pi }}{\mathrm{dL}}=0.5\times 150\times 8{\mathrm{L}}^{-0.5}-75=0 \\ 600{\mathrm{L}}^{-0.5}=75 \\ 75{\mathrm{L}}^{0.5}=600 \\ L=\frac{600\times 600}{75\times 75} \\ L=64 \end{gathered}$$

The profit-maximizing quantity of labor will be 64 laborers.

The profit-maximizing output is calculated below:

$$\begin{gathered} q=8\sqrt{L} \\ =8\sqrt{64} \\ =8\times 8 \\ =64 \end{gathered}$$

The profit-maximizing output will be 64 units.

The profit is calculated below:

$$\begin{gathered} \mathrm{\pi }=0.8\left(\mathrm{TR}-\mathrm{TC}\right) \\ =0.8\left(\mathrm{pq}-\mathrm{wL}\right) \\ =0.8\left(150\left(8\sqrt{64}\right)-75\left(64\right)\right) \\ =0.8\left(150\times 64-75\left(64\right)\right) \\ =0.8\left(\left(150-75\right)-64\right) \\ =0.8\times 75\times 64 \\ =\$3840 \end{gathered}$$

The profit will be $3,840.