Log-Log Demand Models and Constant Elasticity

Why Use a Log-Log Demand Model?

Demand for an insurance product may be recorded as a positive continuous quantity, such as a coverage amount selected or an expected number of units purchased. When the response is positive and its relationship with price is naturally proportional, a log-log regression offers a useful interpretation: the price coefficient measures the percentage response in demand to a percentage change in price. This is a constant-elasticity specification, not a claim that demand changes by a constant number of units for each dollar of price change.

The model is appropriate only for observations with strictly positive demand. It also needs careful prediction: a fitted value for log demand is not automatically the conditional arithmetic mean of demand. These distinctions are important when using demand estimates in pricing analysis. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: For positive continuous demand $D>0$ and positive price $P>0$, a log-log regression specifies $$ \ln D=\alpha+\beta\ln P+\boldsymbol{\gamma}^{\mathsf T}\mathbf{x}+\varepsilon, $$ where $\mathbf{x}$ is a vector of other predictors, $\alpha$ is an intercept, $\beta$ is the price coefficient, and $\varepsilon$ is an error with conditional mean zero. If the predictors other than price are held fixed, the model-implied demand has constant price elasticity $\beta$.

It is often useful to express price relative to a reference value $P_0$, for example, by replacing $\ln P$ with $\ln(P/P_0)$. This changes the interpretation of the intercept but not the price coefficient: $\beta$ remains the elasticity. For a demand variable measured in units, taking $\ln D$ means its numerical value must be positive; the unit of measurement affects the intercept, but not the elasticity.

Elasticity and Finite Price Changes

To interpret the coefficient, first hold the other predictors fixed and consider the model-implied demand $D(P)=C P^\beta$, where $C$ collects the fixed terms. The point elasticity is the derivative of demand with respect to price, scaled by the ratio of price to demand.

\frac{\partial D}{\partial P} =\beta C P^{\beta-1} =\frac{\beta D}{P}, \qquad \frac{\partial D}{\partial P}\frac{P}{D}=\beta.

This derivation shows why the coefficient is an elasticity: the proportional response is $\beta$ at every price, provided the other predictors are held fixed. A negative coefficient indicates that higher price corresponds to lower demand. For a finite price change from $P_1$ to $P_2$, the exact model-implied demand ratio is $(P_2/P_1)^\beta$; the coefficient times the percentage price change is only a small-change approximation.

\frac{D(P_2)}{D(P_1)} =\left(\frac{P_2}{P_1}\right)^\beta, \qquad \%\Delta D =100\left[\left(\frac{P_2}{P_1}\right)^\beta-1\right]\%.

Worked Example: Translating Elasticity Into a Demand Change

In a hypothetical portfolio, a fitted log-log model has price coefficient $\beta=-1.2$. At a price of $500, its fitted geometric-mean demand is 40 units. Estimate the demand at $550, holding all other predictors fixed, and calculate the model-implied percentage change.

Method. Use the finite-change ratio from the log-log model. The price ratio is $550/500=1.10$, so the demand multiplier is $$ 1.10^{-1.2} =\exp[-1.2\ln(1.10)] =\exp[-1.2(0.0953102)] =\exp(-0.1143722) \approx 0.89193. $$

The new fitted geometric-mean demand is $40(0.89193)=35.6772$, or 35.68 units to two decimal places. The percentage change is $100(0.89193-1)=-10.807\%$, or a 10.81% decrease to two decimal places. The multiplier is below one, consistent with the negative price coefficient.

Common mistake avoided: multiplying the elasticity by the 10% price increase and reporting an exact 12% demand decrease. That is a small-change approximation, not the finite-change result. The exact model-implied decrease is 10.81%.

Retransformation: From Log Demand to Demand

A fitted log-scale value $\widehat{\eta}$, where $\eta=\alpha+\beta\ln P+\boldsymbol{\gamma}^{\mathsf T}\mathbf{x}$, is a prediction of the conditional mean of $\ln D$ when the model has $E[\varepsilon\mid P,\mathbf{x}]=0$. Exponentiating it gives $\exp(\widehat{\eta})$. Under a symmetric normal error model, this is the conditional median of $D$, not its conditional arithmetic mean. More generally, it is the geometric-scale prediction implied by the fitted log response.

Suppose the errors are conditionally normal with mean zero and constant variance $\sigma^2$: $\varepsilon\mid P,\mathbf{x}\sim N(0,\sigma^2)$. Then $D=\exp(\eta)\exp(\varepsilon)$. The normal moment-generating function gives $E[\exp(\varepsilon)\mid P,\mathbf{x}]=\exp(\sigma^2/2)$. Therefore the conditional arithmetic mean includes a multiplicative correction.

E[D\mid P,\mathbf{x}] =\exp(\eta)E[\exp(\varepsilon)\mid P,\mathbf{x}] =\exp\left(\eta+\frac{\sigma^2}{2}\right).

The correction does not change the price elasticity if $\sigma^2$ is constant across prices and predictors: it multiplies predicted demand by the same factor. If the error distribution is not normal, the factor $\exp(\sigma^2/2)$ need not be correct. A common alternative is Duan’s smearing estimate: calculate the average of $\exp(\widehat{\varepsilon}_j)$ across fitting observations, then multiply $\exp(\widehat{\eta})$ by that average. If the error distribution changes with predictors, a single overall smearing factor may also be inadequate; conditional retransformation should reflect that variation.

Worked Example: Adjusting a Prediction for Log-Scale Error

For a hypothetical segment at a reference price of $800, a fitted log-log model gives $\widehat{\eta}=\ln(30)$, and the estimated conditional error variance is $\sigma^2=0.16$. The price coefficient is $-0.8$. Estimate the conditional median and mean demand at $1,000, assuming normally distributed, homoskedastic errors.

Method. First apply the exact price multiplier. The price ratio is $1{,}000/800=1.25$, giving $$ 1.25^{-0.8} =\exp[-0.8\ln(1.25)] =\exp[-0.8(0.2231436)] \approx 0.8365116. $$

The new fitted log-scale value is $\ln(30)+\ln(0.8365116)$. Its exponential is the conditional median: $30(0.8365116)=25.09535$, or 25.10 units to two decimal places. Relative to 30 units, the decrease is $100(1-0.8365116)=16.34884\%$, or 16.35% to two decimal places.

For the conditional mean, the retransformation factor is $\exp(\sigma^2/2)=\exp(0.16/2)=\exp(0.08)=1.0832871$. Thus the mean prediction is $25.09535(1.0832871)=27.18547$, or 27.19 units to two decimal places. The mean exceeds the median because the lognormal demand distribution is right-skewed.

Common mistake avoided: treating $\exp(\widehat{\eta})$ as the arithmetic mean. Under the stated normal-error assumption it is the median; omitting the factor $\exp(\sigma^2/2)$ understates the mean.

Zero-Valued Outcomes Need Separate Treatment

A log-log model cannot be fitted directly to observations with $D=0$, because $\ln(0)$ is undefined. Simply discarding zero observations changes the target: the fitted relationship then describes positive demand among observations with $D>0$, not average demand across all quotes or customers. Adding an arbitrary small constant before taking logs is not a neutral fix. It changes the response scale and generally removes the simple constant-elasticity interpretation for the original demand.

One practical structure separates whether demand is positive from its amount when positive. Let $q(P,\mathbf{x})=P(D>0\mid P,\mathbf{x})$. A log-log model can describe the conditional positive amount, while a separate model or estimate describes $q$. The unconditional expected demand is the product of these quantities.

E[D\mid P,\mathbf{x}] =P(D>0\mid P,\mathbf{x})E[D\mid D>0,P,\mathbf{x}].

This identity follows by splitting the expectation over the events $D=0$ and $D>0$: the zero event contributes zero, and the positive event contributes its probability times the mean conditional on that event. The price elasticity of unconditional expected demand can therefore differ from the coefficient in the positive-demand log-log model if the probability of positive demand also changes with price.

Worked Example: Combining Positive Demand With Participation

A hypothetical quote study estimates that, at a $600 price, the probability of positive demand is 0.80 and mean demand conditional on being positive is 15 units. At $660, the corresponding probability is estimated to be 0.72. A log-log model for positive demand has price coefficient $-0.5$. Calculate the unconditional expected demand at both prices and its percentage change.

Method. First update positive demand using the price ratio $660/600=1.10$: $$ 1.10^{-0.5} =\frac{1}{\sqrt{1.10}} \approx 0.9534626. $$

At $660, mean demand conditional on being positive is $15(0.9534626)=14.30194$ units. At $600, unconditional expected demand is $0.80(15)=12$ units. At $660, it is $0.72(14.30194)=10.29740$ units, or 10.30 units to two decimal places.

The percentage change in unconditional expected demand is $100(10.29740/12-1)=-14.1883\%$, or a 14.19% decrease to two decimal places. This differs from the positive-demand decrease alone because the probability of positive demand also falls.

Common mistake avoided: applying the positive-demand multiplier to the baseline unconditional mean while ignoring the change in participation. Both the positive amount and the probability of positive demand enter the unconditional prediction.

Specification, Interpretation, and Practical Checks

Before interpreting a fitted price coefficient, confirm that the response was positive for every observation included in the log-demand regression, that price is positive, and that the units and transformations are consistent. Estimate the model on the log response, then interpret $\beta$ as a conditional association unless the data and study design support a causal price effect. A regression coefficient by itself does not establish that changing price caused the observed demand response.

For a correctly specified log-log relationship, a 1% price increase corresponds approximately to a $\beta$% demand change for a small change. For a large or specified finite price change, use the exact ratio $(P_2/P_1)^\beta$. When reporting a demand level, state whether it is an exponentiated log-scale prediction, a conditional median, or a retransformation-adjusted conditional mean. If zeros occur, state whether the target is demand conditional on being positive or unconditional expected demand.

Define the prediction target.
Determine whether the question concerns positive demand only or demand including zero outcomes.

Specify and interpret the log-log model.
For a positive-demand model, read the price coefficient as its conditional constant elasticity, with other predictors held fixed.

Calculate finite price effects exactly.
Use the price ratio raised to the coefficient, rather than treating the elasticity approximation as exact.

Retransform for the requested summary.
Distinguish $\exp(\widehat{\eta})$ from the conditional mean, and use a suitable correction when estimating the latter.

Key takeaway: In a log-log model for positive demand, the price coefficient is a constant elasticity, and a finite price change has multiplier $(P_2/P_1)^\beta$. Predictions on the original demand scale require attention to retransformation, while zero outcomes require a model or estimand that treats participation separately.

Check Your Understanding

For each problem, state the relevant demand summary and retain enough precision to check the calculation.

A positive-demand log-log model has price coefficient $-0.6$. Calculate the exact demand multiplier and percentage change for a 15% price increase. (Answer: multiplier $1.15^{-0.6}\approx0.9196$; demand decreases about 8.04%.)
A segment has fitted geometric-mean demand of 30 units at a $400 price, with price coefficient $-0.8$. Estimate the fitted geometric-mean demand at $500 and the exact percentage decrease. (Answer: $30(1.25^{-0.8})\approx25.10$ units; decrease about 16.35%.)
At a given price, a log-demand model has $\widehat{\eta}=\ln(16)$ and normal error variance $\sigma^2=0.09$. Find the conditional median and retransformation-adjusted mean. (Answer: median $16.00$ units; mean $16e^{0.045}\approx16.74$ units.)
At the original price, positive demand occurs with probability 0.75 and its conditional mean is 20 units. At a 20% higher price, the probability is 0.60 and the positive-demand log-log coefficient is $-0.5$. Calculate unconditional expected demand at the higher price and its percentage change from the original price. (Answer: $0.60(20(1.20^{-0.5}))\approx10.95$ units, down about 26.97% from 15 units.)
A positive-demand model has price coefficient $-1.3$, baseline geometric-mean demand of 10 units, and normal log-error variance $0.25$. Price doubles. Find the new conditional median and the retransformation-adjusted mean. (Answer: median $10(2^{-1.3})\approx4.06$ units; mean $10(2^{-1.3})e^{0.125}\approx4.60$ units.)

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