Expressing Profit as a Function of Employment Only via Substitution

A company produces widgets. The number of widgets produced (q) is determined by the number of labor hours (L) hired, according to the function q = 10√L. The company sells each widget for $20 and the wage rate per labor hour is $50. To find the profit-maximizing number of labor hours, the firm's profit must first be expressed as a function of the single variable L. Which of the following equations correctly represents the company's profit (π) as a function of L?

The First Step in Profit Optimization

A firm wants to determine the number of labor hours (L) that will maximize its profit. It knows its revenue depends on the quantity of goods produced (q), and the quantity produced depends on the labor hours used. To prepare the problem for optimization, several algebraic steps are needed. Arrange these steps in the correct logical order.

Identifying an Error in a Profit-Maximization Strategy

Evaluating Optimization Strategies

Deriving a Profit Function via Substitution

A firm's profit is determined by the revenue from the quantity of output it sells (q) and the cost of the labor it hires (L). The quantity of output produced is itself dependent on the amount of labor hired. To find the specific amount of labor that will maximize profit, it is a necessary first step to express the profit equation as a function of only the quantity of output (q).

Justifying the Substitution Method for Profit Optimization

A firm sells its product for a price (P) of $50 per unit. It hires labor (L) at a wage rate (w) of $200 per hour. The firm's production of output (q) is determined by the amount of labor it hires, according to the function q = 20L. Match each economic concept below with its correct mathematical expression based on the information provided, as a step toward finding the optimal amount of labor to hire.

To solve a firm's profit maximization problem algebraically, we often start by expressing profit as a function of a single choice variable, like the amount of labor to hire. The substitution method is used to replace one variable (e.g., quantity of output) with an expression involving another variable (e.g., labor). This key step simplifies the problem by reducing the number of ______ in the profit equation to one, making it possible to find the maximum point.