Marginal Revenue

Marginal revenue (MR) is the additional total revenue generated by selling one more unit of a good or service. In microeconomics, it connects the demand facing a firm with its decisions about output and price. Marginal revenue measures a change in sales receipts, not a change in profit: the latter also depends on marginal cost. Its relationship to the selling price differs between price-taking firms and firms whose output decisions affect that price. (openstax.org)

Definition and mathematical representation

For a discrete change in sales quantity, marginal revenue is calculated as

where $R$ denotes total revenue and $Q$ denotes quantity sold. When sales increase by exactly one unit, it is simply $R(Q+1)-R(Q)$. For a larger increase, the ratio measures the average additional revenue per unit across that interval. (openstax.org)

When revenue is a differentiable function of quantity, marginal revenue is its derivative:

If every unit sells at the same price and the firm faces an inverse demand curve $P(Q)$, then $R(Q)=P(Q)Q$. Differentiation gives

The first term captures revenue from additional sales; the second captures the effect of the accompanying price change on the units sold. Average revenue, by contrast, is $R(Q)/Q$, which equals price under uniform pricing. (live.ocw.mit.edu)

Competition and market power

Under perfect competition, an individual firm takes the market price as given. Its additional output does not change that price, so $dP/dQ=0$ and

The firm’s marginal-revenue curve is therefore horizontal at the prevailing price. This describes the individual firm, not the market-wide demand curve; the market price is determined by supply and demand. (assets.openstax.org)

A uniform-price monopoly instead faces the market demand curve. With downward-sloping demand, selling more requires a lower price. Additional revenue from the extra unit is partly offset by reduced receipts on the other units, making $MR<P$ at positive quantities. This compares alternative price–quantity combinations; it does not imply that completed transactions must be retrospectively repriced. (openstax.org)

The same distinction applies to firms with market power under monopolistic competition. Their individual demand curves slope downward, and their marginal-revenue curves lie below their demand curves under uniform pricing. Entry or exit can shift the demand facing each firm and therefore its marginal revenue. (openstax.org)

Relationship to demand elasticity

Marginal revenue can also be expressed using price elasticity of demand. With signed elasticity

the uniform-price relationship becomes

Equivalently, using the positive absolute elasticity $E=|\varepsilon|$,

Consequently, marginal revenue is positive when demand is elastic ($E>1$), zero when it is unit elastic ($E=1$), and negative when it is inelastic ($E<1$). Negative marginal revenue means that increasing sales reduces total revenue because the price reduction outweighs the gain from additional quantity. These relationships concern movement along a given demand curve, not a shift in demand. (live.ocw.mit.edu)

Linear-demand example

For a linear inverse demand curve

the revenue and marginal-revenue functions are

Thus, marginal revenue has the same vertical intercept as demand but twice its downward slope. Its quantity-axis intercept occurs halfway to demand’s intercept. This geometric property is specific to linear demand. (openstax.org)

As a numerical illustration, suppose $P(Q)=100-2Q$, with price measured in dollars. At $Q=20$, price is $60, total revenue is $1,200, and derivative-based marginal revenue is $20. Increasing sales discretely to 21 units lowers price to $58 and raises revenue to $1,218. The additional unit therefore adds $18, not its $58 selling price. The difference between $18 and $20 reflects the distinction between a finite increment and an instantaneous derivative.

Profit-maximizing output

In mathematical optimization, a firm’s profit can be written as the objective function

Differentiating gives $\pi'(Q)=MR(Q)-MC(Q)$. At a differentiable interior optimum, the first-order condition is $MR=MC$. Equality identifies a candidate optimum; a strict local maximum additionally requires profit to turn downward, for example through $\pi''(Q)<0$. Boundaries and other feasible outputs must also be considered. (openstax.org)

For a uniform-price monopolist, the marginal-revenue–marginal-cost intersection identifies output; the demand curve then identifies the corresponding price. Equating price with marginal cost would generally select a different quantity. Furthermore, $MR=MC$ does not establish that economic profit is positive: profitability depends on total revenue relative to total cost, or price relative to average cost. (openstax.org)

Pricing assumptions

The standard formula $R(Q)=P(Q)Q$ assumes uniform pricing. Under price discrimination, units or customers can face different prices, so revenue must instead be constructed from the actual payment schedule. In the idealized case of perfect first-degree discrimination, each additional unit is sold at its buyer’s willingness to pay without reducing receipts from earlier units. The resulting output can match the competitive quantity, although the seller captures the consumer surplus that buyers would otherwise receive. (assets.openstax.org)