A formal algebraic proof can be used to verify that lower isoprofit curves correspond to higher profit levels. This verification activity involves applying partial differentiation to the isoprofit equation. Specifically, one analyzes the change in wage ($w$) with respect to a change in the profit level ($\Pi_0$) while keeping the employment level ($N$) constant.

Algebraically Verifying Isoprofit Curve Positions Using Partial Differentiation

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.

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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.

Isoprofit Curve

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

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

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

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

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

In the wage-setting model, the zero-profit line represents combinations of wage and employment where the firm's economic profit is zero, marking the boundary of economic viability. For a firm where revenue per employee is €800, this line is composed of two segments: a vertical line along the wage axis from the origin (0, 0) up to (0, 800), which then becomes a horizontal line extending from that point onwards.

The Zero-Profit Line in the Wage-Setting Model

An isoprofit curve illustrates the various combinations of employment and wages that yield an identical profit for a firm. For instance, a single upward-sloping, concave curve representing a profit of €1,500 connects several specific points. These labeled points include A (10 employees, €650 wage), B (40 employees, €762.50 wage), and C (75 employees, €780 wage), all of which result in the same €1,500 profit.

The €1,500 Isoprofit Curve with Labeled Points (A, B, C)

In the wage-employment model depicted, an isoprofit curve representing a profit of €3,000 is shown. This upward-sloping, concave curve is situated at a lower position on the graph than the curve for a €1,500 profit, illustrating that higher profit levels correspond to lower isoprofit curves.

The €3,000 Isoprofit Curve

The shape of an isoprofit curve can be determined by studying its equation. A key step is to rewrite the equation to express wage ($w$) as a function of employment ($N$), which is useful for graphing since wage is on the vertical axis. This rearranged formula shows that as employment ($N$) increases, the wage ($w$) must also increase to maintain the same profit level. This occurs because the term representing profit per worker, $\Pi_0/N$, decreases as $N$ rises, meaning a smaller amount is subtracted from the fixed revenue per worker.

Algebraic Analysis of the Isoprofit Curve Equation

The geometric shape of isoprofit curves can be understood by analyzing the algebraic properties of their governing equation. For graphical representation in a diagram where price (P) is on the vertical axis and quantity (Q) is on the horizontal axis, it is particularly useful to algebraically manipulate the isoprofit equation to express P as a function of Q.

Determining Isoprofit Curve Shape through Algebraic Rearrangement in a P-Q Model

The isoprofit curve that represents a profit level of zero (Π₀=0) is identical to the firm's average cost curve. This identity occurs because a firm's economic profit is zero precisely when the price per unit is equal to the average cost of producing that unit.

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