Why Derivatives Appear in Economics
Calculus becomes especially useful in economics when a quantity changes continuously as the number of units produced changes. A company might want to know how much its cost increases when production rises from 100 units to 101 units, or how much additional revenue it earns by selling one more product.
These are questions about rates of change. And the derivative is precisely the mathematical tool for describing an instantaneous rate of change.
If represents the cost of producing units, then tells us how rapidly the cost is changing as production changes.
In economics, this derivative is called the marginal cost.
A marginal quantity is a derivative. It tells you approximately how much the corresponding total quantity changes when production increases by one additional unit.
Total Cost and Marginal Cost
Suppose a company has a total cost function , where is the number of units produced. The function gives the total cost of producing units.
The marginal cost function is the derivative:
Notice the distinction:
- Total cost tells you the entire cost of producing a given number of units.
- Marginal cost tells you how rapidly total cost is changing at that production level.
Worked Example: Finding Marginal Cost
A company models its total cost, in dollars, for producing units by
Find the marginal cost when the company produces 100 units and interpret your answer.
Marginal Revenue
Cost is not the only economic quantity that can be differentiated. Suppose represents the total revenue earned from selling units.
The derivative of revenue is called marginal revenue:
Marginal revenue measures the approximate additional revenue generated by selling one more unit at a particular sales level.
A company estimates its revenue, in dollars, from selling units as
Find and interpret the marginal revenue at 50 units.
Marginal Profit
Profit is what remains after costs are subtracted from revenue. If is revenue and is cost, then profit is
Taking the derivative gives
Therefore:
This gives an important economic relationship:
- If marginal revenue is greater than marginal cost, producing another unit tends to increase profit.
- If marginal revenue is less than marginal cost, producing another unit tends to decrease profit.
- If marginal revenue equals marginal cost, the profit function has a critical point.
Marginal Revenue vs. Marginal Cost
This comparison is one of the most important applications of derivatives in economics.
At a production level where marginal revenue equals marginal cost, the additional revenue from producing one more unit is approximately equal to the additional cost of producing it.
This does not automatically prove that the company is maximizing profit. As with any calculus optimization problem, you still need to determine whether the critical point corresponds to a maximum or minimum and whether it lies in the meaningful domain.
Worked Example: Maximizing Profit
A company's revenue and cost functions are
where is the number of units produced. Find the production level that maximizes profit.
The Marginal Approximation
The derivative gives an instantaneous rate of change, so marginal cost can also be used to approximate the change in total cost.
If a company is producing units and increases production by a small amount , then
For a one-unit increase, , giving the familiar approximation
So marginal cost is essentially the derivative-based estimate of the cost of producing one additional unit.
A company's total cost is
Estimate the additional cost of increasing production from 200 units to 201 units using marginal cost.
Average Cost vs. Marginal Cost
Students often confuse average cost with marginal cost. They are different quantities.
If total cost is , then average cost is
while marginal cost is
Average cost asks, "How much does each unit cost on average?" Marginal cost asks, "How much is total cost changing if production increases slightly?"
Common Mistakes
If the question asks for marginal cost, you generally need the derivative of the cost function—not the original cost function itself.
Finding is only the first step. If the question asks for marginal cost at 100 units, you still need to calculate .
AP Calculus free-response questions often require interpretation. Don't stop at a number such as 60. State what that number means in terms of dollars, units, revenue, cost, or profit.
Equality identifies a critical point of the profit function. You still need to justify that the critical point gives a maximum, such as by using the first derivative test, second derivative test, or another appropriate argument.
Practice Problems
A company's total cost is . Find the marginal cost at 20 units.
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A company's revenue is . Find and interpret the marginal revenue at 40 units.
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A company has revenue and cost . Find the production level that maximizes profit.
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Suppose represents total cost. Explain what means in context.
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