A firm uses a single input, labor, to produce output \(q\) according to the production function \(q=8 \sqrt{L}\). The commodity sells for \(\$ 150\) per unit and the wage rate is \(\$ 75\) per hour. a. Find the profit-maximizing quantity of \(L\) b. Find the profit-maximizing quantity of \(q\) c. What is the maximum profit? d. 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. e. Now suppose that the firm is required to pay a 20 percent tax on its profits. Find the new profit-maximizing levels of \(L, q,\) and profit.

Step by step solution

01

- Profit Maximizing Quantity of L

First, we need to calculate the firm's profit function. The profit function is revenue minus cost, where revenue is price times quantity \( R = P*q \), and cost is wage rate times labor \( C = w*L \). Lets substitute the given values into the profit function \( \pi ={P*q - w*L} \). Here, \( P = 150, q = 8\sqrt{L} \) and \( w = 75 \). Substituting these values, we get \( \pi = 150*8\sqrt{L} - 75*L \). Differentiating the profit function with respect to \( L \) and setting equal to zero to find the maximizing value, we have: \[ \frac{d\pi}{dL} = 1200*\frac{1}{\sqrt{L}} - 75 = 0 \]. Solving this if we find \( L=256 \).

02

- Profit Maximizing Quantity of q

Next, we'll find the profit-maximizing quantity of output \( q \) by substituting \( L = 256 \) into the production function \( q = 8\sqrt{L} \). This yields \( q = 8*\sqrt{256} = 128 \) units.

03

- Maximum Profit Value

To find the maximum profit value, we substitute \( L = 256 \) into the profit function \[ \pi = 150*8\sqrt{L} - 75*L \]. This gives us \( \pi = 192000 - 19200 = 172800 \) dollars.

04

- New Profit Maximizing Values with Tax and Subsidy

Now, accounting for the $30 tax per unit of output and $15 per hour wage subsidy, we reformulate the profit function \[ \pi = (P - 30)q - (w-15)L \]. The new profit maximization problem leads to \[ L' = 100, q' = 80, \text{ and } \pi' = 105000 \] dollars.

05

- Tax of 20 percent on Profits

Assuming a 20% tax on profit, the firm's post-tax profit function becomes \[ \pi'' = (1-0.20)*\pi' \]. This yields \( \pi'' = 84000 \) dollars. The optimal levels of \( L \) and \( q \) remain unchanged.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!