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 profitmaximizing levels of \(L, q,\) and profit.

Short answer

Using calculus and the given production function, profit function, wage rate, and unit price, we can find the values for labor input and output quantity that maximize profit. Changes like taxes and subsidies are reflected in the profit function, which we then optimize again to find new values for labor input, output quantity, and profit that take these changes into account. When the firm has to pay a tax on profits, the profit function is adjusted so that only 80% of the profits are kept by the firm, and the problem is optimized again. This gives us the profit-maximizing inputs and output under all the given scenarios.

Step-by-step solution

Find the profit-maximizing quantity of L

The objective of the firm is to maximize profit, which is revenue (R) minus costs (C). In this case, revenue is price(P) times output(q) and costs is wage(w) times labor(L). So the profit \(\pi = PQ - WL\). Substitute \(q=8 \sqrt{L}\), P=150 and w=75, profit can be expressed as a function of labor, \(\pi = (150 \times 8 \sqrt{L}) - (75 \times L)\). To find the maximizing quantity of L, we take the derivative of the profit function with respect to L, set the derivative equal to zero, and solve for L. This gives us the value of L that maximizes profit for a given price and wage rate.

Find the profit-maximizing quantity of q

From the previous step, we have the optimal labor input. We substitute this value into the production function \(q=8 \sqrt{L}\) to obtain the output quantity that maximizes profits.

Find the maximum profit

Substitute the optimal L from step 1 and optimal q from step 2 into the profit function to get the maximum profit.

Evaluate the impact of taxes and subsidies

When the firm is taxed $30 per unit of output and the wage rate is subsidized at a rate of $15 per hour, you need to adjust your profit function to reflect these costs and benefits. The new profit function will include a tax on each unit of output, reducing the price per unit, and a subsidy on the wage rate, reducing the wage cost. Repeat steps 1 to 3 with the new profit equation.

Evaluate the impact of a profit tax

Next, we are told that the firm has to pay a 20 percent tax on its profits. This means that the firm keeps only 80% of the profits it generates. You have to adjust your profit function to reflect these costs. Repeat steps 1 to 3 with this new profit equation.

Key concepts

Production Function

The production function is a mathematical representation that describes the relation between inputs used in production and the resulting output. In our exercise, the production function is described as \(q=8 \sqrt{L}\), where \(q\) is the quantity of output, and \(L\) is the amount of labor used. The square root in this function suggests that diminishing returns to labor set in — a common concept in microeconomics, indicating that as you keep adding labor, the output increases but at a decreasing rate.

Understanding the production function is crucial since it allows a firm to know how much output can be produced with given amounts of inputs, which in turn helps in making decisions about resource allocation for maximizing profit. Normally, more complex production functions may consider multiple inputs like labor, capital, and technology, but in our example, we have a single-input scenario for simplicity.

Revenue and Cost Analysis

Revenue is the income that a firm generates from selling its products or services, calculated as the price per unit multiplied by the number of units sold. In the exercise, it is calculated as \(PQ\), where \(P\) is the price per unit, and \(Q\) is the quantity sold. Cost analysis involves all costs incurred in the production process. In our case, the cost is summed up as the wage rate, \(w\), times the quantity of labor, \(L\), so the cost function is \(wL\).

To analyze profit, which is the ultimate goal for most firms, we need to use the formula: Profit (\(\pi\)) = Revenue (\(R\)) - Cost (\(C\)). Firms aim to achieve the highest possible profit by increasing revenue, reducing costs, or employing a combination of both strategies. The given exercise helps illustrate this through the profit-maximizing quantities of labor and output.

Impact of Taxes and Subsidies on Profit

Taxes and subsidies are external factors that directly impact a firm's profit. A tax per unit of output reduces the revenue per unit, while a subsidy, such as a reduced wage rate, decreases production costs. In our exercise, a \(\(30\) tax on output lowers the revenue from each unit sold, and a \(\)15\) subsidy on wages reduces labor costs.

These changes require a recalculation of the profit-maximizing levels of labor and output, as the firm aims to adjust to these financial incentives or burdens. The analysis of taxes and subsidies is an essential part of microeconomics since such policies can influence a firm's production and pricing strategies significantly.

Derivative Application in Economics

In economics, derivatives are used to find optimization points, such as maximizing profit or minimizing costs. They provide the rate at which one variable changes with respect to another, helping to make decisions about resource allocation.

In the context of the exercise, taking the derivative of the profit function with respect to labor (\(L\)) and setting it to zero gives us the level of labor that maximizes profit. This process, known as marginal analysis, is a cornerstone of optimization in microeconomics. It helps in determining the best possible outcomes given the constraints and is used to analyze how changes in input levels affect overall profitability.

Profit Maximization Strategies

Profit maximization strategies involve choosing the levels of inputs and outputs that lead to the highest profit, taking into consideration the cost structure and the market conditions. In the textbook exercise, we calculate the optimal amount of labor and output using the data provided for prices, wages, and the production function.

The firm selects the level of labor that maximizes profit by setting the derivative of the profit function to zero, as explained earlier. This calculation is critical for understanding how much of an input is required to achieve maximum profit and is applicable to both short-term and long-term planning. Additionally, performing an analysis of different market conditions, such as changes in taxes and subsidies, allows the firm to adjust its strategies to maintain profitability in a dynamic economic environment.