# Marginal Revenue and the Demand Curve

How to Calculate Them and to Represent Them Graphically

Marginal revenue is the additional revenue that a producer receives from selling one more unit of the good that he produces. Because profit maximization happens at the quantity where marginal revenue equals marginal cost, it's important not only to understand how to calculate marginal revenue but also how to represent it graphically:

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## Demand Curve

The demand curve shows the quantity of an item that consumers in a market are willing and able to buy at each price point.

The demand curve is important in understanding marginal revenue because it shows how much a producer has to lower his price to sell one more of an item. Specifically, the steeper the demand curve is, the more a producer must lower his price to increase the amount that consumers are willing and able to buy, and vice versa.

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## Marginal Revenue Curve versus Demand Curve

Graphically, the marginal revenue curve is always below the demand curve when the demand curve is downward sloping because, when a producer has to lower his price to sell more of an item, marginal revenue is less than price.

In the case of straight-line demand curves, the marginal revenue curve has the same intercept on the P axis as the demand curve but is twice as steep, as illustrated in this diagram.

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## Algebra of Marginal Revenue

Because marginal revenue is the derivative of total revenue, we can construct the marginal revenue curve by calculating total revenue as a function of quantity and then taking the derivative. To calculate total revenue, we start by solving the demand curve for price rather than quantity (this formulation is referred to as the inverse demand curve) and then plugging that into the total revenue formula, as done in this example.

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## Marginal Revenue Is the Derivative of Total Revenue

As stated before, marginal revenue is then calculated by taking the derivative of total revenue with respect to quantity, as shown here.

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## Marginal Revenue Curve versus Demand Curve

When we compare this example inverse demand curve (top) and the resulting marginal revenue curve (bottom), we notice that the constant is the same in both equations, but the coefficient on Q is twice as large in the marginal revenue equation as it is in the demand equation.

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## Marginal Revenue Curve versus Demand Curve Graphically

When we look at the marginal revenue curve versus the demand curve graphically, we notice that both curves have the same intercept on the P axis, because they have the same constant, and the marginal revenue curve is twice as steep as the demand curve, because the coefficient on Q is twice as large in the marginal revenue curve. Notice also that, because the marginal revenue curve is twice as steep, it intersects the Q axis at a quantity that is half as large as the Q-axis intercept on the demand curve (20 versus 40 in this example).

Understanding marginal revenue both algebraically and graphically is important, because marginal revenue is one side of the profit-maximization calculation.

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## Special Case of Demand and Marginal Revenue Curves

In the special case of a perfectly competitive market, a producer faces a perfectly elastic demand curve and therefore doesn't have to lower its price to sell more output. In this case, marginal revenue is equal to price as opposed to being strictly less than price and, as a result, the marginal revenue curve is the same as the demand curve.

This situation still follows the rule that the marginal revenue curve is twice as steep as the demand curve since twice a slope of zero is still a slope of zero.

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