Question

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.

Short answer

Profit-maximizing quantity of \( L = 256 \) hours, profit-maximizing quantity of \( q = 128 \) units, maximum profit = $172800. After a tax of $30 per unit and a wage subsidy of $15 per hour, the new values are \( L' = 100 \) hours, \( q' = 80 \) units, and profit = $105000. When a 20% tax on profit is imposed, the profit becomes $84000 and the quantities \( L \) and \( q \) remain the same.

Step-by-step solution

- 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 \( L=256 \).

- 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.

- 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.

- 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.

- 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.

Key concepts

Production Function

In economics, the production function is a fundamental concept that describes the relationship between inputs and outputs in the production process. It outlines how much output can be generated with a given amount of inputs, typically labor and capital.

For our firm that uses labor as its single input, the production function is given by \( q = 8 \sqrt{L} \), meaning output \( q \) is a function of the square root of labor \( L \).

This function shows how labor is transformed into goods or services. As labor increases, output increases as well, but at a decreasing rate due to the square root. This reflects the principle of diminishing marginal returns, common in production functions.

• More labor means more output, but each additional unit of labor contributes less to total output than the previous one.
• Understanding the production function helps firms determine the optimal amount of inputs needed to maximize production.

Cost Function

The cost function gives you insight into the expenses a firm incurs based on the level of output they produce. For our scenario, the cost is related mainly to wages, since the firm hires labor to produce its goods.

The cost function in our example can be expressed as \( C = w \times L \), where \( w = 75 \) represents the wage rate per hour of labor, and \( L \) is the total labor employed. The firm's costs increase linearly with labor, under the assumption that each unit of labor costs the same.

Understanding cost functions is critical because:

• It helps firms plan and budget their expenditures.
• It allows firms to evaluate how changes in wage rates or labor hours impact overall costs.
• Being aware of costs helps in determining profit by comparing against revenues.

Price Taker

A 'price taker' is a firm or entity in a market structure where it cannot influence the price of the goods it sells. This is usually due to the competitiveness of the market where numerous sellers exist, each selling identical products.

Our firm is considered a price taker, selling each unit of its output for a fixed price of $150. Because it can sell as many units as it desires without affecting the market price, the firm must focus on minimizing costs and maximizing output efficiency to maximize profits.

• Price takers accept the market price as given.
• They lack market power, meaning their actions do not influence the overall market price.
• Their main strategy is to adjust production and costs to achieve optimal profit margins.

Subsidies and Taxes

Subsidies and taxes are governmental financial policies that can significantly influence a firm’s profitability. Subsidies reduce the cost of production by providing financial assistance, while taxes increase costs by taking money from firms.

In this exercise, our firm deals with both:
- **Subsidy:** The firm receives a wage subsidy of \(15 per hour, lowering the effective wage cost to \)60 per hour \( (75 - 15) \). This means employing labor becomes cheaper, encouraging more hiring.
- **Tax:** A \(30 output tax is levied per unit produced, reducing the price received from \)150 to $120 \( (150 - 30) \). This increases the effective cost of production per unit.

These financial adjustments affect the firm’s profit maximization calculations:

• Subsidies can incentivize increased production and hiring by lowering costs.
• Taxes necessitate a recalibration of the profit equation, impacting how the firm optimizes production.
• The firm must assess net gains or losses after subsidies and taxes to determine true profitability.