To solve the problem of maximizing profit, a firm must first understand the relationship between its profit and the two key variables it controls: the wage (w) it pays and the number of employees (N) it hires. This analysis is the necessary first step in determining the optimal wage and employment combination.

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A firm cannot choose just any combination of wage ($w$) and employment ($N$), as some combinations are not feasible. The firm's primary objective is to solve a constrained choice problem: it must identify the single combination of wage and employment from all its feasible options that yields the maximum possible profit.

The Firm's Constrained Choice Problem of Profit Maximization

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

CORE Econ - The Economy 2.0: Microeconomics

A firm in the wage-setting model confronts a trade-off between its employment level and the wage rate. To hire more workers, it must offer a higher wage. While each new hire may be profitable individually (if their wage is below the revenue they generate, e.g., €800), the necessity of paying this increased wage to all existing employees reduces the profit margin on every worker.

The Firm's Wage-Employment Trade-off

The calculation of profit in the language school model starts from the general principle that profit equals sales revenue minus input costs. This is simplified by two key assumptions: first, that each tutor generates a fixed weekly revenue of `y = €800` from student fees, and second, that the firm's only input cost is the wage paid to tutors.

Assumptions for Calculating Profit in the Language School Model

Figure 6.15 addresses the central question of where the highest profit lies within the feasible set. It does this by overlaying the firm's isoprofit curves on the diagram of the feasible set, which consists of all points on or above the no-shirking wage curve. This combined view allows the firm to visually identify the most profitable combination of wage and employment it can achieve.

Figure 6.15: Finding the Profit-Maximizing Point by Combining Isoprofit Curves and the Feasible Set

An alternative method for solving a firm's profit-maximization problem is to directly calculate the profit for each feasible price-quantity combination along the demand curve. By plotting these profit values against their corresponding quantities, a profit function graph is generated. The peak of this curve reveals the quantity that yields the highest possible profit for the firm.

Deriving the Profit Function from the Demand Curve

Profit as a Function of Wage and Employment

A firm's profit is calculated by subtracting its total input costs from its total sales revenue. This fundamental relationship forms the basis for all profit analysis.

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