# Profit Maximization

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## Choosing a Quantity that Maximizes Profit

In most cases, economists model a company maximizing profit by choosing the quantity of output that is the most beneficial for the firm. (This makes more sense than maximizing profit by choosing a price directly, since in some situations- such as competitive markets- firms don't have any influence over the price that they can charge.) One way to find the profit-maximizing quantity would be to take the derivative of the profit formula with respect to quantity and setting the resulting expression equal to zero and then solving for quantity.

Many economics courses, however, don't rely on the use of calculus, so it's helpful to develop the condition for profit maximization in a more intuitive way.

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## Marginal Revenue and Marginal Cost

In order to figure out how to choose the quantity that maximizes profit, it's helpful to think about the incremental effect that producing and selling additional (or marginal) units has on profit. In this context, the relevant quantities to think about are marginal revenue, which represents the incremental up side to increasing quantity, and marginal cost, which represents the incremental down side to increasing quantity.

Typical marginal revenue and marginal cost curves are depicted above. As the graph illustrates, marginal revenue generally decreases as quantity increases, and marginal cost generally increases as quantity increases. (That said, cases where marginal revenue or marginal cost are constant certainly exist as well.)

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## Increasing Profit by Increasing Quantity

Initially, as a company begins increasing output, the marginal revenue gained from selling one more unit is larger than the marginal cost of producing this unit. Therefore, producing and selling this unit of output will add to profit the difference between marginal revenue and marginal cost. Increasing output will continue to increase profit in this way until the quantity where marginal revenue is equal to marginal cost is reached.

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## Decreasing Profit by Increasing Quantity

If the company were to keep increasing output past the quantity where marginal revenue is equal to marginal cost, the marginal cost of doing so would be larger than the marginal revenue. Therefore, increasing quantity into this range would result in incremental losses and would subtract from profit.

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## Profit Is Maximized Where Marginal Revenue Is Equal to Marginal Cost

As the previous discussion shows, profit is maximized at the quantity where marginal revenue at that quantity is equal to marginal cost at that quantity. At this quantity, all of the units that add incremental profit are produced and none of the units that create incremental losses are produced.

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## Multiple Points of Intersection Between Marginal Revenue and Marginal Cost

It is possible that, in some unusual situations, there are multiple quantities at which marginal revenue is equal to marginal cost. When this happens, it's important to think carefully about which of these quantities actually results in the largest profit.

One way to do this would be to calculate profit at each of the potential profit-maximizing quantities and observe which profit is largest. If this isn't feasible, it's also usually possible to tell which quantity is profit maximizing by looking at the marginal revenue and marginal cost curves. In the diagram above, for example, it must be the case that the larger quantity where marginal revenue and marginal cost intersect must result in larger profit simply because marginal revenue is greater than marginal cost in the region between the first point of intersection and the second.

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## Profit Maximization with Discrete Quantities

The same rule- namely, that profit is maximized at the quantity where marginal revenue is equal to marginal cost- can be applied when maximizing profit over discrete quantities of production. In the example above, we can see directly that profit is maximized at a quantity of 3, but we can also see that this is the quantity where marginal revenue and marginal cost are equal at \$2.

You probably noticed that profit reaches its largest value both at a quantity of 2 and a quantity of 3 in the example above. This is because, when marginal revenue and marginal cost are equal, that unit of production doesn't create incremental profit for the firm. That said, it's pretty safe to assume that a firm would produce this last unit of output, even though it's technically indifferent between producing and not producing at this quantity.

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## Profit Maximization When Marginal Revenue and Marginal Cost Don't Intersect

When dealing with discrete quantities of output, sometimes a quantity where marginal revenue is exactly equal to marginal cost won't exist, as shown in the example above. We can, however, see directly that profit is maximized at a quantity of 3. Using the intuition of profit maximization that we developed earlier, we can also infer that a firm will want to produce as long as the marginal revenue from doing so is at least as large as the marginal cost of doing so and won't want to produce units where marginal cost is greater than marginal revenue.

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## Profit Maximization when Positive Profit Is Not Possible

The same profit-maximization rule applies when positive profit is not possible. In the example above, a quantity of 3 is still the profit-maximizing quantity, since this quantity results in the largest amount of profit for the firm. When profit numbers are negative over all quantities of output, the profit-maximizing quantity can be more precisely described as the loss-minimizing quantity.

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## Profit Maximization Using Calculus

As it turns out, finding the profit-maximizing quantity by taking the derivative of profit with respect to quantity and setting it equal to zero results in exactly the same rule for profit maximization as we derived previously! This is because marginal revenue is equal to the derivative of total revenue with respect to quantity and marginal cost is equal to the derivative of total cost with respect to quantity.