Profit Maximization as Tangency Between an Isoprofit Curve and the No-Shirking Wage Curve

Expressing Profit as a Function of Employment Only via Substitution

A profit-maximizing firm determines that to ensure its employees work effectively, the wage ($w$) it pays must be greater than or equal to a specific value that depends on the number of employees ($N$). This relationship is described by the constraint $w \ge W(N)$, where $W(N)$ is the minimum wage required for any given level of employment. When choosing its wage and employment level to maximize profit, which of the following statements best describes the firm's optimal wage-setting strategy?

A profit-maximizing firm faces a constraint where the wage ($w$) it pays must be greater than or equal to a specific minimum level, $W(N)$, which depends on the number of employees ($N$). The firm will find it optimal to pay a wage strictly greater than $W(N)$ if it believes the extra pay will significantly boost employee morale.

Rationale for Simplifying the Wage Constraint

Evaluating Competing Wage Strategies

Critique of a Suboptimal Wage-Setting Strategy

A firm's wage-setting is constrained by the condition $w \ge W(N)$, where $w$ is the wage and $W(N)$ is the minimum required wage for a given employment level $N$. Match each of the following firm scenarios to the most logical wage-setting outcome that results from its objective.

A firm's objective is to maximize profit, calculated as total revenue minus total wage costs. The firm is subject to a rule that the wage it pays, w, must be greater than or equal to a minimum level that depends on the number of employees, N. This relationship is expressed as w ≥ W(N). Since paying any wage higher than the absolute minimum required for a given N would unnecessarily increase costs and thus reduce profit, the firm's profit-maximization calculation can be simplified by treating this constraint as a strict ________.

A firm seeks to maximize its profits. It understands that the wage ($w$) it pays must be at least as high as a certain minimum level, $W(N)$, which is determined by the number of employees ($N$). This gives the constraint $w \ge W(N)$. Arrange the following steps in the logical order that demonstrates how the firm would reason to simplify this constraint for its profit-maximization calculation.

Analyzing a Firm's Wage Decision

A firm operates under the constraint that its wage, w, must be at least as high as a minimum level, W(N), which is determined by the number of employees, N. The function W(N) represents the lowest possible wage the firm can pay to ensure its employees work effectively at a given employment level. The firm's goal is to maximize profit. For a specific employment level, N*, the firm calculates that W(N*) is $20 per hour. A manager proposes paying $22 per hour for this same employment level, N*. Based on the logic of cost minimization for a given output, what is the direct consequence of this proposal?

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.

Expressing Profit as a Function of Employment Only via Substitution