# 12.4 Elasticity and Markup Pricing

When we were deriving the formula for marginal revenue, we showed that one form of it was \(MR = p \times \left(1 - {1 \over |\epsilon_{q,p}|}\right)\) where $\epsilon_{q,p}$ is the price elasticity of demand.

In the last section we said that the **profit-maximizing condition** was
\(MR = MC\)

If we plug in the elasticity definition of MR into the profit-maximization condition, we can rewrite that condition as
\(p \times \left(1 - {1 \over |\epsilon_{q,p}|}\right) = MC\)
or
\(p = {MC \over 1 - {1 \over |\epsilon_{q,p}|}}\)
This is known as the **inverse elasticity pricing rule**. We can confirm, for example, that it held in our previous example: for that case, we had the demand function
\(D(p) = 20 - p\)
and found that the optimal quantity to produce was $q^\star = 8$, which it sold for $p^\star = 12$; therefore its price elasticity of demand would be
\(\epsilon_{q,p} = {dq \over dp} \times {p \over q} = -1 \times {12 \over 8} = {3 \over 2}\)
Its marginal cost at $q = 8$ was $q/2 = 4$; so the IEPR would say that its optimal price would be
\(p = {4 \over 1 - {1 \over 3/2}} = {4 \over 1 - {2 \over 3}} = {4 \over {1 \over 3}} = 12\)
which is indeed the optimal price.

## When to use the IEPR

The IEPR is particularly useful in cases where both the marginal cost and elasticity are constant. For example, if a firm has the cost function
\(c(q) = 200 + 4q\)
and faces the demand function
\(D(p) = 6400p^{-2}\)
then its MC is constant at 4, and its price elasticity of demand is constant at $-2$, so the IEPR says it should charge
\(p = {4 \over 1 - {1 \over |-2|}} = 8\)
Plugging this back into the demand function gives us the optimal quantity of
\(q = D(8) = 6400 \times 8^{-2} = 100\)
Of course, it would be possible to go the long way round — solve for the inverse demand, find total and marginal revenue, yada yada yada — but this is *much* faster in cases like this.

## Relationship to market power

If we rearrange the IEPR, a little algebra shows we can write it as
\({p - MC \over p} = {1 \over |\epsilon_{q,p}|}\)
The left-hand side of this equation is the *fraction of the price which represents a markup above and beyond marginal cost*. For example, our firm facing the linear demand curve $D(p) = 20 - p$ charged $p^\star = 12$, but only had a marginal cost of $MC = 4$; hence its markup over marginal cost was $12 - 4 = 8$, which represented $8/12 = 2/3$ of its price. As we saw above, the price elasticity of demand at $p = 12$ was $3/2$.

This ratio of markup to price is known as the *Lerner Index*, and has historically been used in antitrust cases as a measure of market power. Clearly, the less elastic the demand curve faced by a firm, the more it will be able to raise its price above marginal cost. However, the inverse is also true: as the demand curve becomes more elastic, the firm’s ability to raise price over marginal cost gets smaller and smaller; and in the extreme case of a competitive firm, vanishes entirely. We’ll conclude our analysis of profit maximization by looking at this special case of the competitive firm.