Optimization in Economics: Profit Maximization

What Profit Maximization Is Really Asking

Many economics optimization problems can be translated directly into the language of calculus. A business produces and sells some quantity of a product. Producing more units usually changes both the money earned from sales and the cost of production. The goal is to determine the quantity that produces the greatest possible profit.

The central idea is simple: profit equals revenue minus cost. Once you have a function for profit, maximizing profit becomes an ordinary calculus optimization problem.

The variable usually represents the number of units produced and sold. Your job is to determine the value of that makes as large as possible.

Revenue, Cost, and Profit

Revenue

Revenue is the amount of money a business receives from selling its products. If units are sold at a price of dollars per unit, then revenue is:

If the price itself depends on the number of units sold, then is a function of , and revenue becomes a product of two functions.

Cost

The cost function describes how much it costs to produce units. It can include fixed costs, such as rent and equipment, as well as variable costs that increase with production.

Profit

Profit is what remains after subtracting the cost of production from revenue:

Therefore, if you are given revenue and cost functions, you can immediately construct the profit function and optimize it with derivatives.

The Calculus Behind Profit Maximization

To maximize a differentiable profit function, find its critical points by setting the derivative equal to zero:

This leads to one of the most important relationships in economic optimization: marginal revenue equals marginal cost.

Here, marginal revenue is the derivative of revenue and marginal cost is the derivative of cost:

So the condition for an interior profit maximum can be written as:

Intuitively, if the next unit adds more revenue than cost, producing another unit increases profit. If the next unit costs more than the revenue it generates, producing another unit decreases profit. The maximum occurs where those two marginal effects balance.

A Step-by-Step Method

1. Define the quantity variable. Let represent the number of units produced and sold.
2. Write the revenue function. Determine how much money the business receives from selling units.
3. Write the cost function. Include both fixed and variable costs when appropriate.
4. Construct the profit function. Subtract cost from revenue.
5. Differentiate the profit function. Find .
6. Find critical points. Solve .
7. Verify that the critical point gives a maximum. Use the second derivative, a sign chart, or compare relevant endpoints.
8. Interpret the answer. State the production level and, when requested, the maximum profit.

Worked Examples

Why Marginal Revenue = Marginal Cost Makes Sense

The equation is more than a calculus trick. It describes the economic logic behind the optimum.

Suppose a company is producing at a quantity where marginal revenue is greater than marginal cost. Then producing one additional unit adds more to revenue than it adds to cost. Profit therefore increases.

On the other hand, if marginal cost is greater than marginal revenue, the next unit costs more to produce than it generates in revenue. Producing additional units decreases profit.

Therefore, at an interior profit maximum, the two marginal quantities balance. This is why so many economic optimization problems ultimately reduce to solving an equation involving marginal revenue and marginal cost.

Competitive Firms vs. Firms Facing a Demand Curve

The exact setup depends on the economic model in the problem.

When the Price Is Constant

If every unit is sold for the same fixed price , revenue is simply:

Therefore marginal revenue is constant:

The profit-maximizing condition becomes .

When Price Depends on Quantity

If the price falls as the quantity sold increases, then the price is a function of quantity, such as . Revenue must be calculated as quantity times price:

In this situation, marginal revenue is generally not equal to the price. You must differentiate the revenue function.

Common Mistakes

Practice Problems