This activity guides the analysis of how a minimum wage affects a firm's choices in the no-shirking wage model, as depicted in Figure 6.19. A crucial technique to verify that a new point, such as F, is indeed the profit-maximizing choice is to visually trace the new lower boundary of the firm's feasible set. By observing the isoprofit curves that this boundary crosses, one can determine where the firm's profit is rising or falling, and thereby confirm the location of the new optimum.

Activity: Analyzing the Effect of a Minimum Wage Using the No-Shirking Wage Curve Model

The position of an isoprofit curve on a wage-employment diagram indicates its associated profit level, a relationship derived from the isoprofit curve equation. For any given level of employment ($N$), a higher profit level ($\Pi_0$) can only be achieved with a lower wage ($w$). Consequently, isoprofit curves representing greater profits are located lower on the diagram, meaning they are situated further down in the figure.

Profit Levels and Isoprofit Curve Positions

Isoprofit curves can be viewed as the indifference curves for a firm. An indifference curve typically joins all combinations of goods that provide a constant level of utility. Similarly, a firm is indifferent between the various combinations of price and quantity along a single isoprofit curve, as every point on that curve represents the same level of profit.

Isoprofit Curves as the Firm's Indifference Curves

The wage-setting model demonstrates how firms can set wages to fulfill two key objectives: to recruit and retain the necessary workforce, and to provide employees with a strong incentive to work hard. This model, depicted in diagrams like Figure 6.15, is used to find a firm's profit-maximizing wage and employment level by overlaying its isoprofit curves onto its feasible set of (N, w) combinations. The feasible set consists of all points on or above the no-shirking wage curve, and the optimal choice for the firm is the combination of wage and employment on the highest possible isoprofit curve within this set.

The Wage-Setting Model

This diagram, referenced as both Figure 6.13 and Figure E6.3, visualizes the isoprofit curves for the language school's profit model. The curves connect combinations of wage (w) and employment (N) that yield equal profit, based on the formula `Profit = (800 - w) × N`. The diagram plots employment (0 to 80) against wage (400 to 850), showing upward-sloping, concave isoprofit curves for profits like €1,500, €3,000, and more. Higher profit curves are positioned lower. A zero-profit line is also depicted.

Figure 6.13/E6.3 - Isoprofit Curves for the Language School Model

An isoprofit curve is defined by an equation that holds a firm's profit level constant at a specific value, Π₀. This equation links the variables of price (P), quantity (Q), and total cost (C(Q)). The fundamental relationship is derived from the definition of profit: Profit = Total Revenue - Total Cost. For a constant profit level, this is expressed as Π₀ = P*Q - C(Q). This can be algebraically rearranged to PQ = C(Q) + Π₀, which states that for a given profit level, total revenue must equal the sum of total cost and that profit.

General Equation of an Isoprofit Curve

The slope of an isoprofit curve changes depending on the position in the wage-employment diagram. This can be seen directly from the slope formula, $dw/dN = (y-w)/N$, without calculating specific profit levels. The formula shows that the curve becomes flatter (the slope decreases) at higher levels of employment ($N$) and wages ($w$), corresponding to the top-right of the diagram. The economic intuition is that when N and w are high, the profit gained from an additional worker is smaller. Therefore, only a small wage adjustment is needed to keep total profit constant, resulting in a flatter curve and explaining the concave shape of isoprofit curves.

How Wage and Employment Levels Determine the Isoprofit Curve's Slope

A key distinction between isoprofit curves and indifference curves lies in their geometric properties. Isoprofit curves are characteristically increasing and concave. In contrast, standard indifference curves for desirable goods are typically decreasing and convex.

Shape of Isoprofit Curves vs. Indifference Curves

The specific geometric form of a firm's isoprofit curves is directly determined by the shape of its average cost curve.

Influence of Average Cost Curve Shape on Isoprofit Curve Shape

To determine the slope of an isoprofit curve at any given point within a price-quantity model, one must differentiate the isoprofit curve's equation. This mathematical process typically involves using the quotient rule for differentiation, especially after the equation has been arranged to express price as a function of quantity. [1, 7]

Slope of an Isoprofit Curve in a Price-Quantity Model

The profit margin is the difference between the price of a product (P) and its marginal production cost (c). For instance, at the profit-maximizing point E for Beautiful Cars, the profit margin represents the additional profit the company earns from producing and selling the 32nd car.

Profit Margin

The slope of an isoprofit curve is determined by the profit margin, defined as the difference between the price (P) and the marginal cost (c). A larger profit margin results in a steeper curve. Conversely, as the price approaches the marginal cost, the profit margin shrinks, causing the curve to become flatter.

Profit Margin's Effect on Isoprofit Curve Slope

This example considers a firm where the unit cost, which is the expense for creating one unit of output, remains unchanged at any production level. A key implication of this setup is that the firm's marginal cost is also constant.

A Firm with a Constant Unit Cost

An isoprofit curve, a term derived from the Greek 'iso' meaning 'equal', is a line that connects all the combinations of price (P) and quantity (Q) for a good that yield the same amount of profit for a firm. A common characteristic of these curves in a price-quantity model is that they slope downward. A series of these curves, each corresponding to a different profit level, can be drawn on a graph to visualize the firm's profit landscape.

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A firm's total profit (Π) is a function of both the wage ($w$) and the level of employment ($N$). This is calculated by finding the net profit from each worker, which is the difference between revenue per worker ($y$) and the wage paid ($w$), and then multiplying that amount by the total number of employees ($N$).

Firm's Profit as a Function of Wages and Employment

Isoprofit curves serve as the two-dimensional representation of a company's profit function. They are formed by projecting the contours from a three-dimensional 'profit hill' onto a two-dimensional grid defined by price (P) and quantity (Q). Consequently, each isoprofit curve joins all the price and quantity combinations that result in the same constant level of profit.

Isoprofit Curves as 2D Representations of Profit Hill Contours

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

In the simplified language school model, a firm's total profit is contingent on two main drivers: the wage (w) and the level of employment (N). Generally, profit is high when employment is high and wages are low. More specifically, profitability is achieved as long as the wage is below the revenue generated per tutor (e.g., €800). From this point, total profit increases as the wage paid to tutors decreases and as the number of employed tutors increases.

Drivers of Profit in the Language School Model

Isoprofit Curve

For the language school model, where revenue per employee (y) is established at €800, the firm's profit is determined by the wage (w) and employment level (N) according to the specific formula: `Profit = (800 - w) × N`.

Profit Calculation in the Language School Model

The firm's profit maximization problem can be solved straightforwardly using the substitution method. This algebraic approach is particularly effective because the profit function has a relatively simple mathematical form.

Simple Profit Function Enables Substitution Method for Optimization

A specific example of profit calculation involves a company that sells 50,000 pounds (Q) of cereal at a price (P) of \$4 per pound. With a unit cost of \$2, the company's total profit is \$100,000. This combination of price and quantity is located on the \$100,000 profit contour line in a three-dimensional visualization of the firm's profit.

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