Presenter: Alex Lamon
In this lesson students solve an optimization problem based on the real-world example of profit maximization. Students analyze a revenue and profit report for Apple, Inc. to explore profits and profit maximization. They view a video to strengthen their understanding of profits and profit-maximizing behavior. Students use an online graphing tool to plot a quadratic total revenue function. They also analyze the graph of a cubic total cost function. Students then construct a profit function based on a total revenue function and total cost function. Finally, students solve for the optimal quantity of output that would maximize profits based on a given profit function. They determine the appropriate critical value by taking the first derivative of the profit function and then verify that this value yields a maximum by applying a second derivative test to the profit function.
The primary goal of a business is arguably to maximize its profits. Profit is the net financial gain a business receives from producing and selling a good or service. Profits are calculated as the difference between total revenue (TR) and total cost (TC). Businesses earn a profit (π) when total revenue is greater than total cost. They earn a loss when total revenue is less than total cost, and they break-even when total revenue equals total cost. Profit maximization or loss minimization, if the firm cannot earn a profit, occurs at the level of profit where marginal revenue (MR) equals marginal cost (MC). (Note: an exception occurs if variable costs are greater than marginal revenue, in which case the firm would shut down immediately). Marginal revenue is additional revenue received from selling an additional unit of output and would likely shrink as more output is produced. Marginal cost is the additional cost incurred from producing an additional unit of output and would likely rise as more output is produced. A business can add to its profits by increasing production as long as marginal revenue is greater than marginal cost. Therefore, the quantity of output that maximizes profits is the quantity at which marginal revenue equals marginal cost. If a business would produce output greater than this quantity, marginal revenue would be less than marginal cost, which would diminish its profits.
Mathematically, the profit function is constructed as the difference between the total revenue function and the total cost function. Total revenue and total cost can be considered functions of one variable, the quantity of output (Q). Marginal revenue is represented by the slope of the total revenue function; using calculus, the slope – and therefore marginal revenue – is estimated by the first derivative of the total revenue function. Similarly, marginal cost is equal to the slope of the total cost function, which can be estimated as the first derivative of the total cost function. For many businesses, total revenues are a quadratic function based on the selling price (P), which can be expressed as a function of the quantity demanded, and the quantity of output sold. Total costs are often represented as a cubic function because costs typically exhibit the behavior of initially increasing at a decreasing rate and then increasing at an increasing rate. This behavior of total costs is explained by the concept of diminishing marginal returns. As a business increases its production of output by using more of a variable input in combination with a fixed input, initially the marginal cost of production decreases. This happens because the addition of resources used in production initially increases, which effectively causes the production of additional units of output to become less costly per unit of output. At some point, however, the addition of more variable resources adds challenges that make production more difficult, resulting in higher costs per unit of output. In other words, at some point of production, marginal cost increases.
The quantity of output that maximizes profits can be determined mathematically by solving an optimization problem using calculus. This is done by taking the first derivative of the profit function, setting it equal to zero, and solving for the quantity of output. To verify that this point occurs at the maximum of the profit function, a second derivative test is done on the profit function. To be a maximum, the second derivative must be less than zero, which indicates that the profit function is concave downward and the point is a relative maximum.
This lesson assumes students are familiar with polynomial equations, inflection points, critical values, the quadratic formula, and differentiation. The lesson provides a practical application of mathematics and economics, and it should help students to understand a connection between economics and mathematics.
Presenter: Alex Lamon
Presenter: Tawni Hunt-Ferrarini
Grades 3-5, 6-8
Presenter: Minnesota Council on Education