§8.3

From Elasticity to Pricing Decisions

Estimating elasticity is not the goal — setting a price is. The bridge from a regression coefficient to a price recommendation is one of the most elegant results in microeconomics: the inverse-elasticity (or Lerner) rule, which says that the profit-maximizing markup over marginal cost is determined entirely by the absolute value of the own-price elasticity. The same rule also makes vivid the danger of acting on a confounded elasticity, because the markup formula amplifies bias in the elasticity estimate directly into the recommended price.

This article derives the optimal markup rule, shows where the formula breaks down (and why that breakdown is informative), illustrates the markup curve graphically, and works through a data case that quantifies the cost of pricing on a naive elasticity estimate.

The Executive Question: What Price, with What Guardrails?

A pricing analyst announces that the formula recommends cutting price from $3.50 to $2.00 because elasticity is $-2.00$ . A senior manager should never accept this on its face. The right executive question is two questions at once:

What is the formal economic argument behind the optimal price, and what operational guardrails should bracket how we test it?

The first half is the math we are about to derive. The second half is what separates a candidate test price from a national price change.

Deriving the Optimal Markup Rule

Start with the standard profit function for a firm choosing a single price $P$ , with constant marginal cost $MC$ and demand $Q(P)$ :

Profit

\Pi(P) \;=\; (P - MC) \cdot Q(P)

Take the derivative with respect to price and set to zero (using the product rule):

\frac{d\Pi}{dP} \;=\; Q(P) + (P - MC) \cdot \frac{dQ}{dP} \;=\; 0

Divide both sides by $Q(P)$ :

1 + (P - MC) \cdot \frac{1}{Q} \cdot \frac{dQ}{dP} \;=\; 0

Multiply the right-hand fraction by $P/P$ so the elasticity $\epsilon = (dQ/dP)(P/Q)$ appears explicitly:

1 + \frac{P - MC}{P} \cdot \epsilon \;=\; 0

Rearrange to obtain the Lerner index: the profit-maximizing markup as a fraction of price equals the negative inverse of the elasticity:

Lerner index

\frac{P^* - MC}{P^*} \;=\; -\frac{1}{\epsilon}

Solving for the optimal price:

Constant-elasticity optimal pricing rule

P^* \;=\; \frac{MC}{1 + \dfrac{1}{\epsilon}}, \qquad \text{valid for } \epsilon < -1

Two readings of this formula are immediately useful:

As elasticity grows in magnitude ( $\epsilon \to -\infty$ ), the term $1/\epsilon \to 0$ and $P^* \to MC$ . In a hyper-competitive market the firm has no pricing power and must price at cost.
When $-1 < \epsilon < 0$ (inelastic zone), the denominator becomes negative or zero and the formula collapses — mathematically recommending infinite price. The right interpretation is that a profit-maximizing firm should never be in the inelastic zone: raising price would simultaneously raise revenue and reduce volume-driven costs. The formula breaking is the formula telling you the regression must be wrong.

The Markup Curve, Visually

The relationship between $|\epsilon|$ and the optimal markup is non-linear and steep at low values.

The inverse-elasticity rule: optimal markup falls with |ε|

Figure 1. The optimal markup over marginal cost as a function of the absolute value of own-price elasticity. Highly elastic categories (|ε| > 4) earn thin markups. Moderately elastic categories (|ε| ≈ 2) carry meaningful markups. As |ε| approaches 1, the markup explodes — which is also where the formula stops being trustworthy.

The curve does two pedagogically useful things. First, it makes vivid how much a small change in elasticity matters for the recommended price. Moving from $|\epsilon| = 2.0$ to $|\epsilon| = 2.5$ shifts the optimal markup from 100% over MC down to roughly 67%. Second, it shows how dangerous the formula becomes near $|\epsilon| = 1$ : the markup goes to infinity in a way that is mathematically defined but operationally meaningless.

Interactive: The Profit Curve

The profit function itself is the right object for executives to see, because it shows not just the optimum but the penalty of being off-target. A nearly flat peak means small mis-pricing is forgiven; a sharp peak means precision matters.

Optimal price under constant elasticity

Move elasticity and marginal cost to see why the same formula is a decision aid, not a policy by itself.

Optimal price

$1.82

Current avg.

$1.65

Elasticity: -2.23Marginal cost: $1.00

Figure 2. Profit as a function of price, with sliders for marginal cost and own-price elasticity. The orange line marks the current price; the green dashed line marks the formula's optimum. Notice how sensitive the peak is to small changes in elasticity, especially when |ε| is close to 1.

The shape of the curve is itself a piece of evidence. A flat-topped profit curve is forgiving — modest deviations from the optimum cost little, so a conservative price is fine. A sharp peak punishes both over- and under-pricing, and the value of more precise elasticity estimates rises accordingly.

Guardrails: The Formula Is a Generator, Not a Command

The optimal-pricing formula has exactly three inputs: own-price elasticity, marginal cost, and the assumption that the world is static. Each of those deserves a check before the recommendation leaves the spreadsheet.

Guardrail	What it protects against	What to check
Causal identification of $\epsilon$	Acting on a confounded elasticity that is wrong by a factor of 1.5×	Was the regression run with appropriate fixed effects and controls? Climb the regression ladder.
Stable marginal cost	Optimizing yesterday's cost while procurement costs are rising	Refresh $MC$ regularly; recompute the price when inputs move materially.
Competitor reaction	Assuming rivals will hold price after a major move	Cross-reference with cross-price matrix; expect retaliation when cross-elasticities are large.
Customer trust	Short-term profit at the cost of long-term LTV	Run a holdout cohort to measure churn at the new price, not just immediate volume.
Empirical validation	Treating a model number as a real-world certainty	Stagger the rollout regionally; require A/B evidence before national deployment.

The formula gives you a candidate price to test, not a price to set globally. That is the single most important framing to carry out of this chapter.

Concept check

Three questions spanning own-price elasticity, the sign of a cross-price coefficient, and the optimal markup rule.

1.
The log-log demand model $\ln Q = \beta_0 + \beta_1 \ln P + \cdots$ is preferred over a linear model for pricing because…
2.
The estimated cross-price elasticity of pastry sales with respect to coffee price is $-0.35$ . The strategic interpretation is…
3.
The Lerner index says the profit-maximizing markup over marginal cost is…

Data Case: The Cost of Acting on the Naive Estimate

To put a dollar figure on the cost of pricing on a confounded elasticity, compare what the formula recommends under the naive and the well-identified estimates from Chapter 8.1.

A standard Progresso can has $MC = \$1.00$ . Two competing elasticity estimates:

Naive bivariate OLS: $\widehat{\epsilon} = -3.21$ .
Store fixed effects: $\widehat{\epsilon} = -2.23$ .

Plug each into the formula:

Optimal price under the naive estimate

P^*_{\text{naive}} \;=\; \frac{\$1.00}{1 + 1/(-3.21)} \;=\; \frac{\$1.00}{0.6885} \;\approx\; \$1.45

Optimal price under the well-identified estimate

P^*_{\text{FE}} \;=\; \frac{\$1.00}{1 + 1/(-2.23)} \;=\; \frac{\$1.00}{0.5516} \;\approx\; \$1.82

The naive recommendation lands exactly at the current price — "we are already optimized, do nothing." The well-identified recommendation says current pricing is roughly 25% below optimum. The mechanical source of the bias is the omitted seasonality: retailers cut prices in low-demand months, so the naive regression sees discounts paired with the noise of low-demand weeks and overstates how much volume the discount actually produces.

The financial consequences are direct. At a marginal cost of $1.00:

The naive model's "optimal" $1.45 gives a per-can margin of $0.45.
The well-identified model's optimal $1.82 gives a per-can margin of $0.82 — an 82% increase in unit margin.

Even after accounting for the volume loss implied by elasticity $-2.23$ , the net change in total profit is strongly positive. The data case is the clearest possible illustration of why identification is the gating step in any pricing analysis: the formula is the same; the answer it produces depends entirely on the elasticity you feed it.