Mathematical functions serve as the fundamental building blocks for quantifying and analyzing complex economic relationships. They provide a formal language that allows economists and business analysts to translate conceptual challenges—such as maximizing profit or establishing market price—into precise, solvable models. By representing variables like cost, revenue, and demand as functions of factors like production quantity or price, we can systematically explore outcomes, identify optimal strategies, and understand the underlying dynamics of business operations.

1.1 Modeling Core Business Operations: Cost, Revenue, and Profit

At the heart of any commercial enterprise is the pursuit of profit. A profit function provides a clear, quantitative model of this objective by defining total profit as the difference between total revenue and total cost. These components are themselves functions of key decision variables, such as the price at which a product is sold.

For instance, consider a manufacturer producing radios at a cost of $10 per unit. If the radios are sold for a price of x dollars, the profit per radio is (x – 10). Market analysis suggests that the number of radios consumers will buy each month is a function of the price, approximated by the expression (80 – x). By combining these relationships, we can construct a comprehensive profit function, P(x):

Profit = (Number of Radios Sold) × (Profit per Radio) P(x) = (80 – x)(x – 10)

This quadratic function formally models the manufacturer’s monthly profit based on a single decision variable: the selling price. This allows for a structured analysis to determine the price that will yield the greatest possible profit.

1.2 Determining Viability: Break-Even Analysis

A critical question for any business is determining the point of viability—the level of production and sales at which the enterprise is neither making a profit nor incurring a loss. This is known as the break-even point. Mathematically, it is identified as the intersection point of the Total Cost C(x) and Total Revenue R(x) functions.

A simple model for the Green-Belt Company illustrates this principle. The company has fixed daily costs of $300 and a production cost of $2 per belt. Belts are sold for $3 each. The corresponding functions are:

• Total Revenue: R(x) = 3x
• Total Cost: C(x) = 2x + 300

The break-even point occurs where R(x) = C(x), or 3x = 2x + 300, which solves to x = 300. This means the company must sell exactly 300 belts to cover all its costs. Graphically, the region where the revenue line is below the cost line represents a Loss, while the region where it is above represents a Profit. It is crucial to note that this linear model presumes constant variable costs and selling prices, an assumption that may not hold under significant economies of scale or changing market conditions.

More complex profit models can reveal a profitable range of activity. For a company with the profit function P(x) = -x² + 60x – 500, setting P(x) = 0 reveals two break-even points: x=10 and x=50. This indicates that the company generates a profit only when it operates between these two levels. Producing fewer than 10 or more than 50 units results in a financial loss, defining a clear range of viable production scale.

1.3 Establishing Market Dynamics: Supply, Demand, and Equilibrium

The principles of supply and demand are cornerstones of economic theory, describing the relationship between the price of a commodity and the quantity that producers are willing to sell and consumers are willing to buy. These relationships are modeled with supply and demand functions, where price (p) is typically the independent variable.

• A supply function, S(p), models the quantity of a product that manufacturers will make available at a given price. Generally, as the price increases, producers are willing to supply more.
• A demand function, D(p), models the quantity that consumers will purchase at that price. Typically, as the price increases, consumer demand decreases.

The point where these two forces balance is market equilibrium. It is the price at which the quantity supplied equals the quantity demanded, resulting in a stable market with “neither a surplus nor a shortage of the commodity.” This equilibrium is found by setting the two functions equal to each other.

For a commodity with the following supply and demand functions:

• Supply: S(p) = p² + 3p – 70
• Demand: D(p) = 410 – p

Market equilibrium is found by solving S(p) = D(p):

p² + 3p – 70 = 410 – p p² + 4p – 480 = 0

Solving this equation yields a valid equilibrium price of $20. At this price, the quantity supplied and demanded is D(20) = 410 – 20 = 390 units. This equilibrium point is where the market clears, satisfying both producers and consumers without excess or scarcity.

While the functional models in this section provide a static snapshot of economic conditions, they cannot answer dynamic questions about the rate of change. To analyze the marginal impact of the next dollar spent or the precise price that maximizes profit, we must turn to the more powerful framework of differential calculus.