Marginal revenue, simply put, 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 to not only understand how to calculate marginal revenue but also how to represent marginal revenue graphically.

### The Demand Curve

The demand curve, on the other hand, 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 in order to sell one more of an item. Specifically, the steeper the demand curve is, the more a producer must lower his price in order to increase the amount that consumers are willing and able to buy, and vice versa.

### The Marginal Revenue Curve versus the Demand Curve

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

In the case of straight-line demand curves, it turns out that the marginal revenue curve has the same intercept on the P axis as the demand curve but it twice as steep, as illustrated in the diagram above.

### The Algebra of Marginal Revenue

Since 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 the example above.

### 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 in the example above.

(See here for a review of calculus derivatives.)

### The Marginal Revenue Curve versus the 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.

### The Marginal Revenue Curve versus the Demand Curve

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 (since they have the same constant) and the marginal revenue curve is twice as steep as the demand curve (since 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 very important, since marginal revenue is one side of the profit-maximization calculation.

### A Special Case of the 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 at all in order 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.

Interestingly enough, 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.