# Marginal revenue

In microeconomics, marginal revenue (R') is the additional revenue that will be generated by increasing product sales by one unit. It can also be described as the unit revenue the last item sold has generated for the firm. In a perfectly competitive market, the additional revenue generated by selling an additional unit of a good is equal to the price the firm is able to charge the buyer of the good. This is because a firm in a competitive market will always get the same price for every unit it sells regardless of the number of units the firm sells since the firm's sales can never impact the industry's price. However, a monopoly determines the entire industry's sales. As a result, it will have to lower the price of all units sold to increase sales by 1 unit. Therefore the Marginal Revenue generated is always less (lower) than the price the firm is able to charge for the unit sold since each reduction in price causes unit revenue to decline on every good the firm sells. The marginal revenue (the increase in total revenue) is the price the firm gets on the additional unit sold, less the revenue lost by reducing the price on all other units that were sold prior to the decrease in price!

## Definition

Marginal revenue is equal to the ratio of the change in revenue for some change in quantity sold to that change in quantity sold. This can also be represented as a derivative when the change in quantity sold becomes arbitrarily small. More formally, define the revenue function to be the following

$R(q)=P(q)\cdot q$ .

By the product rule, marginal revenue is then given by

$R'(q)=P(q)+P'(q)\cdot q$ .

For a firm facing perfect competition, price does not change with quantity sold ($P'(q)=0$ ), so marginal revenue is equal to price. For a monopoly, the price decreases with quantity sold ($P'(q)<0$ ), so marginal revenue is less than price (for positive $q$ ).

## Marginal revenue curve

The marginal revenue curve is affected by the same factors as the demand curve - changes in income, change in the prices of complements and substitutes, change in populations. These factors can cause the R curve to shift and rotate.

## Relationship between marginal revenue and elasticity

The relationship between marginal revenue and the elasticity of demand by the firm's customers can be derived as follows:

$R'={\frac {dR}{dQ}}$ $R'=\langle R\rangle +\left({\frac {d\langle R\rangle }{dQ}}\right)Q$ as $\langle R\rangle =P$ $R'=\langle R\rangle \left[1+{\frac {d\langle R\rangle }{dQ}}{\frac {Q}{\langle R\rangle }}\right]$ $R'=\langle R\rangle (1+1/e_{\langle R\rangle })$ where e is the price elasticity of demand. If demand is inelastic (e<R> < 1) then R' will be negative, because to sell a marginal (infinitesimal) unit the firm would have to lower the selling price so much that it would lose more revenue on the pre-existing units than it would gain on the incremental unit. If demand is elastic (e<R> > 1) R' will be positive, because the additional unit would not drive down the price by so much. If the firm is a perfect competitor, so that it is so small in the market that its quantity produced and sold has no effect on the price, then the price elasticity of demand is negative infinity, and marginal revenue simply equals the (market-determined) price.

## Marginal revenue and rule of thumb pricing

Profit maximization requires that a firm produces where marginal revenue equals marginal costs. Firm managers are unlikely to have complete information concerning their marginal revenue function or their marginal costs. Fortunately the profit maximization conditions can be expressed in a “more easily applicable form” or rule of thumb.

R' = C'
R' = <R>(1 + 1/e<R>)
MC = <R>(1 + 1/e<R>)
MC = <R> + <R>/e<R>
(<R> - C')/ <R> = - 1/e<R>

Markup is the difference between price and marginal cost. The formula states that markup as a percentage of price equals the negative of the inverse of elasticity of demand. Alternatively, the relationship can be expressed as:

<R> = C'/(1 + 1/e<R>)

Thus if e<R> is - 2 and mc is $5.00 then price is$10.00.

(<R> - C')/ <R> = - 1/e<R> is called the Lerner index after economist Abba Lerner. The Lerner index is a measure of market power - the ability of a firm to charge a price that exceeds marginal cost. The index varies from zero to 1. The greater the difference between price and marginal cost the closer the index value is to 1. The Lerner index increases as demand becomes less elastic.

Real life example: if you can sell 10 units at $20 each or 11 units at$19 each, then your marginal revenue from the eleventh unit is (11 × 19) - (10 × 20) = \$9.