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Chapter 12 / Profit Maximization

# 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.

Copyright (c) Christopher Makler / econgraphs.org