Common Errors in Interpreting Price Elasticity

Why Similar-Sounding Effects Are Not the Same

A price-response model may report a coefficient, an odds ratio, a probability change, or an elasticity. These quantities describe different scales. Treating them as interchangeable can turn a correctly estimated response into a seriously misinterpreted business result. In particular, a coefficient from a logistic model is a change in log odds, not a direct change in probability and not, by itself, a price elasticity.

The distinctions matter when communicating results and when comparing customers or price changes. The examples below use hypothetical demand probabilities. They illustrate how to interpret model output; whether a fitted price relationship supports a causal conclusion depends on the study design and assumptions, including the issues addressed in the earlier material on confounding and instrumental variables.

Definition: For an event with probability $p$, the odds are $p/(1-p)$, and the log odds are $\log\{p/(1-p)\}$. The odds ratio comparing probabilities $p_1$ and $p_0$ is $[p_1/(1-p_1)]/[p_0/(1-p_0)]$. A probability change is $p_1-p_0$; expressed in percentage points, it is $100(p_1-p_0)$. The relative percentage change in probability is $100(p_1-p_0)/p_0$, provided $p_0>0$.

From a Coefficient to an Odds Ratio

For a binary outcome, such as whether a hypothetical customer chooses a quoted product, a logistic model writes the probability $p$ as

\log\left(\frac{p}{1-p}\right)=\eta, \qquad p=\frac{e^\eta}{1+e^\eta}.

If a predictor $x$ has coefficient $\beta$, changing $x$ by $\Delta x$ changes the fitted log odds by $\beta\Delta x$. Exponentiating both sides gives the odds ratio:

\frac{\text{odds at }x+\Delta x}{\text{odds at }x} =e^{\beta\Delta x}.

The odds ratio depends on the predictor’s units and the size of the change. If $x=\log P$, where $P$ is price, multiplying price by $r$ changes the odds by a factor $r^\beta$. If $x$ is price measured in dollars, a change of $\Delta P$ dollars changes the odds by $e^{\beta\Delta P}$. State the predictor scale and the change before interpreting the coefficient.

Worked Example: A Percentage Price Increase in a Log-Price Model

A hypothetical model has $\log\{p/(1-p)\}=\alpha-1.2\log(P/100)$. At $P=\$100$, the fitted demand probability is $0.4000$. Calculate the effect of a 10% price increase on the odds and probability. Also compare the exact probability change with the local elasticity approximation.

Method. The log-price coefficient is $\beta=-1.2$, and a 10% increase multiplies price by $r=1.10$. The change in log odds is $-1.2\log(1.10)=-0.11437$. Thus the odds ratio is $e^{-0.11437}=1.10^{-1.2}=0.8919$: odds decrease by $1-0.8919=0.1081$, or 10.81%.

At the starting probability, the odds are $0.4000/0.6000=0.6667$. The new odds are $0.6667(0.8919)=0.5946$, so the new probability is $0.5946/(1+0.5946)=0.3729$.

The probability change is $0.3729-0.4000=-0.0271$, a decrease of 2.71 percentage points. The relative probability change is $-0.0271/0.4000=-0.0678$, or a 6.78% decrease. At the starting probability, the local elasticity is $-1.2(1-0.4000)=-0.7200$; a small 10% price change would therefore suggest about a 7.20% probability decrease. The exact finite-change decrease is 6.78%, not 7.20%.

Check. The price increase reduces both odds and probability, as the negative coefficient implies. The new probability $0.3729$ is between zero and one. Common mistake avoided: interpreting $-1.2$ as a 120% probability reduction, or applying the local elasticity as an exact finite-change formula.

Probability Changes Depend on the Starting Probability

An odds ratio alone does not determine a probability-point change. The starting probability is also required. To see why, let the starting odds be $o_0=p_0/(1-p_0)$ and let the odds ratio be $R$. Then the new odds are $Ro_0$, so converting those odds back to a probability gives

p_1=\frac{R\,p_0}{1-p_0+R\,p_0}.

Even when two customers have the same odds ratio, different starting probabilities generally produce different probability changes. Also keep two kinds of percentages separate: a 20% decrease in odds is a statement about odds, while a decrease of 20% in probability is a relative change from the starting probability. Neither means a decrease of 20 percentage points.

Worked Example: An Odds Ratio Is Not a Probability Reduction

A hypothetical quoted-product model has a coefficient of $-0.5$ for each \$100 increase in price. A customer’s fitted probability at the original price is $0.6000$. For a \$100 increase, calculate the odds ratio, the new probability, the percentage-point change, and the relative percentage change in probability.

Method. The change in the predictor is one \$100 unit, so the log-odds change is $-0.5(1)=-0.5$. The odds ratio is $e^{-0.5}=0.6065$, meaning the odds fall by $1-0.6065=39.35\%$.

The starting odds are $0.6000/0.4000=1.5000$. The new odds are $1.5000(0.6065)=0.9098$, giving new probability $0.9098/(1+0.9098)=0.4764$. The probability change is $0.4764-0.6000=-0.1236$, or $-12.36$ percentage points. Relative to the starting probability, the change is $-0.1236/0.6000=-0.2060$, or a 20.60% decrease.

Check. The new probability is below $0.6000$, consistent with the negative coefficient. Its decrease is 12.36 percentage points, not 39.35 percentage points; 39.35% describes the decrease in odds. Common mistake avoided: applying the odds percentage change directly to the probability.

Elasticity Is a Local Probability Response

Price elasticity of a probability is the proportional change in probability per proportional change in price. For a differentiable fitted probability $p(P)$, its point elasticity is

\varepsilon(P)=\frac{d p}{dP}\frac{P}{p}.

For a logit model, differentiating $p=e^\eta/(1+e^\eta)$ gives $dp/dP=p(1-p)(d\eta/dP)$. Substitution into the elasticity definition yields

\varepsilon(P)=(1-p)\,P\frac{d\eta}{dP}.

This result explains why the log-odds coefficient is not generally the elasticity. If $\eta=\alpha+\beta\log P$, then $P(d\eta/dP)=\beta$, so $\varepsilon=\beta(1-p)$. The elasticity depends on the current probability. If instead $\eta=\alpha+\beta P$, then $\varepsilon=\beta P(1-p)$; it depends on both price and probability. In either case, the formula is a local, or infinitesimal, response. For a finite price change, calculate the two fitted probabilities and use their actual difference rather than assuming the point elasticity remains constant throughout the change.

Worked Example: Elasticity With Price Measured in Dollars

A hypothetical model is $\log\{p/(1-p)\}=1.5-0.02P$, with $P$ measured in dollars. Calculate the fitted probability and elasticity at $P=\$50$, then calculate the exact probability change when price rises to $\$55$.

Method. At \$50, $\eta=1.5-0.02(50)=0.5$, so $p=e^{0.5}/(1+e^{0.5})=0.6225$. Because $d\eta/dP=-0.02$, the elasticity at \$50 is $(1-0.6225)(50)(-0.02)=-0.3775$. Locally, a 1% price increase corresponds to approximately a 0.3775% decrease in probability.

At \$55, $\eta=1.5-0.02(55)=0.4$, so $p=e^{0.4}/(1+e^{0.4})=0.5987$. The exact probability change is $0.5987-0.6225=-0.0238$, or a decrease of 2.38 percentage points. The relative probability decrease is $0.0238/0.6225=3.82\%$.

As a check on the odds calculation, the price change is five dollars, so the odds ratio is $e^{-0.02(5)}=e^{-0.1}=0.9048$. The starting odds are $e^{0.5}=1.6487$, and multiplying by $0.9048$ gives $e^{0.4}=1.4918$, the odds corresponding to the new fitted probability. Common mistake avoided: calling the coefficient $-0.02$ the elasticity. It is a log-odds change per dollar; the elasticity at \$50 is $-0.3775$.

Customer-Level Effects and Averages

A model’s response may vary by customer because starting probabilities, coefficients, or both differ. For a finite price change, the average customer-level probability effect is the average of the individual probability changes. In general, this is not equal to calculating a probability change for a customer with average characteristics. The logit transformation is nonlinear, so averaging predictors or probabilities at different stages can give different results.

When the goal is an average fitted response for a defined group, calculate each customer’s fitted probabilities under both price scenarios, take each customer’s difference, and average those differences using the intended customer weights. An effect evaluated at mean characteristics answers a different question. Be explicit about which quantity is being reported.

Worked Example: Averaging Two Customer-Level Responses

Two hypothetical customer groups are equally large. At the original price, their fitted demand probabilities are $0.8000$ and $0.2000$. A price change multiplies each group’s odds by $0.5$. Calculate the average probability change across customers and compare it with applying the same odds ratio to the average starting probability.

Method. For group 1, starting odds are $0.8000/0.2000=4.0000$. New odds are $4.0000(0.5)=2.0000$, giving new probability $2/(1+2)=0.6667$. Its probability change is $0.6667-0.8000=-0.1333$.

For group 2, starting odds are $0.2000/0.8000=0.2500$. New odds are $0.2500(0.5)=0.1250$, giving new probability $0.1250/(1+0.1250)=0.1111$. Its probability change is $0.1111-0.2000=-0.0889$.

The equal-weight average starting probability is $(0.8000+0.2000)/2=0.5000$, and the average new probability is $(0.6667+0.1111)/2=0.3889$. Thus the average customer-level change is $0.3889-0.5000=-0.1111$, or $-11.11$ percentage points.

If instead the odds ratio is applied to the average starting probability, the starting odds are $0.5000/0.5000=1.0000$, and the new odds are $0.5000$. This gives a new probability of $0.5000/(1+0.5000)=0.3333$, an apparent change of $-16.67$ percentage points. That is not the average of the two customer-level changes. Check. The direct average of the two new probabilities, $0.3889$, agrees with the average of the separately calculated customer-level effects. Common mistake avoided: treating a result at the average probability as the average result for customers.

Interpretation Checks Before Reporting a Result

A careful interpretation names the outcome, predictor scale, comparison, and response scale. For example, “a 10% price increase multiplies fitted odds by 0.8919 at the stated model specification” is precise. “Demand falls 10.81%” is ambiguous unless it explicitly says that the percentage refers to odds, not probability.

Before reporting a result, check these points:

Identify the model scale. A logistic coefficient is on the log-odds scale. Exponentiating the coefficient gives an odds ratio for a one-unit predictor change, not a probability change.
State the predictor units. A coefficient per dollar, per \$100, or per unit of log price implies different calculations.
Give a baseline for probability comparisons. An odds ratio does not specify a unique probability change without a starting probability.
Label the scale of every percentage. Distinguish odds percentage changes, relative probability changes, and percentage-point changes.
Separate local from finite changes. A point elasticity is a derivative at a specified price and customer profile; finite changes should be computed from the fitted probabilities at both prices.
Specify the population summary. State whether a result is for a particular customer, a customer at average characteristics, or an average of customer-level effects.

Making these distinctions is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus: the key skill is not merely obtaining a coefficient, but translating it into the correct quantity for the question being asked.

Check Your Understanding

For each problem, keep odds, probabilities, and elasticity on their own scales.

A log-price model has coefficient $-0.6$. What is the odds ratio for a 5% price increase, and what percentage change in odds does it represent? (Answer: odds ratio $1.05^{-0.6}=0.9712$, a 2.88% decrease in odds.)
A customer has starting probability $0.2500$, and a price comparison has odds ratio $1.8$. Find the new probability and the probability-point change. (Answer: new probability $0.3750$; increase of 12.50 percentage points.)
In a log-price logit model, a customer has starting probability $0.7000$, and the log-price coefficient is $-0.9$. For a 20% price increase, find the odds ratio and the new probability. (Answer: odds ratio $1.20^{-0.9}=0.8487$; new probability approximately $0.6645$.)
Two equally sized groups have starting probabilities $0.6000$ and $0.3000$. Their odds are each multiplied by $0.7$. Find the average new probability and the average probability-point change. (Answer: average new probability $0.3715$; average change $-7.85$ percentage points.)
A logit model has $\eta=\alpha+\beta\log P$. State its probability elasticity at a specified price and explain whether it is generally constant across customers. (Answer: $\varepsilon=\beta(1-p)$; it is generally not constant because the fitted probability $p$ can vary.)

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