# 12.2 Total, Average, and Marginal Revenue

## Total and average revenue

Because we’re assuming that the firm sells every unit at the same price $p$, the total revenue for a firm facing inverse demand curve $p(q)$ is just \(\text{Total revenue from selling }q\text{ units at price }p(q) = r(q) = p(q) \times q\) Just as the average cost is the total cost divided by the quantity, average revenue is the total revenue divided by the quantity: \(\text{Average revenue }AR(q) = {r(q) \over q} = p(q)\) Note that since every unit is being sold at price $p$, the average revenue is just the price.

For example, if a firm faces the inverse demand curve \(p(q) = 20 - q\) then its revenue function is \(r(q) = p(q)q = 20q - q^2\) and its average revenue is \(AR(q) = {r(q) \over q} = 20 - q\) which is the same as the inverse demand function.

Visually, this means the average revenue curve is the same as the inverse demand curve $p(q)$, and the total revenue is the area represented by price times quantity:

## Marginal revenue

We can define marginal revenue as the increase in revenue from increasing output by a bit. Using calculus and the product rule, we have that \(\text{Marginal revenue =}MR(q) = {dr \over dq} = {dp \over dq} \times q + p\) Let’s see what this means. The ratio $dp/dq$ is the slope of the inverse demand curve. If the inverse demand curve is downward sloping, this is negative: that is, if the firm wants to sell $dq$ more units, the price it could charge would drop by $dp$. Note that if we multiply both sides by $dq$, we get \(dr = dp \times q + dq \times p\) This says that if the firm wants to sell $dq$ more units, the resulting change in revenue $dr$ will be composed of two changes:

**Price effect:**Revenue will*decrease*by $dp \times q$, because the firm is dropping its price by $dp$ on the $q$ units it was previously selling; and**Output effect:**Revenue will*increase*by $dq \times p$, because the firm will sell $dq$ additional units for price $p$.

Visually, we can see these two effects as the red area (lost revenue due to the price effect) and green area (new revenue due to the output effect) in the diagram below:

## Marginal revenue and elasticity

Try dragging the $(q,p)$ point to the right along the demand curve. Note that even though this demand curve is linear, so $dp/dq$ is constant, the relative sizes of the red and green areas change as quantity increases. Indeed, beyond a certain quantity, the red area is greater than the green area, meaning that revenue *decreases* if you increase the quantity. This happens when

\(\begin{aligned}
\text{Price effect} &> \text{Output effect}\\
|dp| \times q &> dq \times p\\
{|dp| \over p} &> {dq \over q}\\
\text{\% change in price} &> \text{\% change in quantity}\\
\end{aligned}\)
Recall that our definition of the price elasticity of demand was
\(\epsilon_{q,p} = {\text{\% change in quantity} \over \text{\% change in price}} = {dq \over dp} \times {p \over q}\)
This says that *marginal revenue will be positive when demand is elastic, and negative when demand is inelastic.*

In fact, we can express marginal revenue *in terms of elasticity*. Let’s start with the formula for marginal revenue,
\(MR(q) = {dp \over dq}\times q + p\)
If we multiply the first term by $p/p$, we get
\(MR(q) = \left[{dp \over dq} \times {q \over p}\right] \times p + p = p \left(1 + {1 \over \underbrace{\frac{dq}{dp} \times {p \over q}}_\epsilon}\right)\)
Notice that the denominator of the last term is just our expression for the price elasticity of demand, which we know is negative because the demand curve is downward sloping ($dq/dp < 0$). Therefore we can write this as
\(MR = p \left(1 - {1 \over |\epsilon|}\right)\)
Note that the more elastic demand is (i.e., the higher $|\epsilon|$ is), the closer marginal revenue is to the price; and that marginal revenue will be negative if $|\epsilon| < 1$ (that is, if demand is inelastic.) In the extreme, if demand is *perfectly elastic*, as in the case of a competitive firm, the price effect is zero, so the marginal revenue is exactly equal to the price:

## Relationship between total and marginal revenue

Mathematically, marginal revenue is just the derivative of total revenue; so if, for example, we have the total revenue function \(r(q) = 20q - q^2\) then the marginal revenue will be \(MR(q) = r'(q) = 20 - 2q\)

Visually, we can see the relationship between total and marginal revenue by plotting them together. Since MR is the derivative of TR, the height of the MR curve is the *slope* of the TR curve. Where TR is increasing, MR is positive; and where TR is decreasing, MR is negative:

Again, for a competitive firm, the diagram is much simpler: since $r(q) = pq$, the total revenue function is just a line with slope $p$; and $AR = MR = p$, so both of those curves are horizontal lines with a height of $p$:

Having analyzed the firm’s revenue as a function of $q$, let’s now combine that with our cost function to analyze its profit.