# Marginal Revenue

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Marginal revenue (MR) measures the change in total revenue that occurs when a firm increases output by one unit, i.e., it is the extra income (or revenue) generated by selling one additional unit.

Whenever a firm increases the number of units it sells, this will affect the firm's revenue, because the quantity sold is higher and revenue is equal to the number of units sold times the price they were sold. In a perfectly competitive market, each firm acts as a price taker and can sell as much as it wants to at the market price. This implies that the marginal revenue in a perfectly competitive market is equal to the market price.

For example, if a firm is selling orange juice at â‚¬2 per glass (and it can sell as many glasses as it wants to at this price), then any additional glass of orange juice that the firm sells increases total revenue (TR) by 2 and thus, marginal revenue is equal to 2. That is, if the firm increases quantity from 10 to 11 glasses, total revenue increases from 20 to 22 and the change in total revenue is equal to 2:

MR = TR (11 glasses) â€“ TR (10 glasses) = 11 * 2 â€“ 10 * 2 = 22 â€“ 20 = 2.

If the market is not perfectly competitive, the marginal revenue is not necessarily equal to the market price. Consider a monopolist who faces the entire market demand. In order to sell more units, the monopolist would have to decrease the price for all units (not only for the additional unit that it wants to sell). This means that total revenue could increase by less than the market price if the next unit lowers the sales price by too much.

To be more precise, selling one more unit will increase total revenue by the price at which the good is sold minus the revenue that the monopolist loses from selling all units at a lower price.

Total revenue is a quadratic function which is maximized at the point where marginal revenue is equal to zero (at quantity q *). Mathematically the marginal revenue is the derivative of total revenue with respect to quantity, i.e.

\begin{equation*}
\mathit{MR} = \frac{\mathit{dTR}}{\mathit{dq}} = \mathit{p(q)} + \mathit{q}*\frac{\partial\mathit{p(q)}}{\partial\mathit{q}}
\end{equation*}

where TR is total revenue TR p(q)q. The first part of this derivative p(q) is positive, because when the quantity sold rises total revenue increases by the price at which the extra unit is sold. The second term of the derivative will be negative, because to sell more units, the price per unit must decrease. Multiplied by the total number of units q, this measures the total loss in revenue due to the lower price.

Keep in mind that marginal revenue is not the same as average revenue AR, which is the revenue per unit, i.e.,

\begin{equation*}
\mathit{AR} = \frac{\textit{Total revenue}}{\textit{quantity}} = \frac{\textit{TR}}{\textit{q}}
\end{equation*}

Marginal revenue is an important concept in economics, because together with marginal costs it determines the equilibrium quantity and price in a perfectly competitive market. A firm maximizes its profits when it produces the quantity at which the marginal revenue of the last unit produced is equal to marginal costs of that unit. That is, profit maximization requires MR  =  MC .

If marginal revenue exceeds marginal costs (MR MC), selling one additional unit increases total revenue more than total costs. Therefore, profit will increase if it sells this unit. (Remember that the profit of the firm is total revenue minus total costs). Consequently, a profit- or utility-maximizing firm will produce this additional unit.

If marginal costs exceed marginal revenue (MR MC) total costs increase more than total revenue for the next unit produced, and it is better for the firm not to sell this unit. When MR MC the firm has no incentive to either increase or decrease its quantity as both would lead to a reduction in profits. Therefore, firms stop producing where MR = MC, but before MR < MC.

In economics, we usually assume that the firm's objective is to maximize profits. But, some economists following Baumol (see e.g. â€śBusiness Behavior, Economics and Growthâ€ť, 1967) argue that under certain conditions firms seek to maximize revenue rather than profits. Baumol argues that the managers of a firm, in contrast to the owners and shareholders, are more interested in maximizing sales than profits because there is a direct link between the salary and reputation of a manager and total revenue.

## Good to know

What determines the wage that you can earn? According to economic theory, your salary will depend on how much the total revenue of your employer increases by hiring you.

This theory can be used to explain why some football clubs are willing to pay huge sums for players that they expect to contribute considerably to the success and fame of the club. For example, in Summer 2019 Eden Hazard transferred from Chelsea to Real Madrid for â‚¬150 million. This large sum reflects not only Eden's expected contributions on the field, but also new merchandising opportunities and brand recognition for Real Madrid.

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