Demand Outcomes and Elasticity Estimands

Demand Outcomes and the Question an Estimate Answers

A price-response estimate is meaningful only after two things are clear: what outcome is being measured and for which population. A purchase probability among newly quoted households is not the same outcome as a renewal probability among current policyholders. Nor does an average response for a portfolio necessarily describe each segment within it.

These distinctions matter when interpreting a demand estimate or using it in a pricing decision. The estimate is not simply “the price effect.” It is a quantity defined for a particular outcome, population, price comparison, and set of conditions. The earlier material on portfolio objectives makes clear why aggregate results matter; here the emphasis is on what the aggregate response represents and how it relates to segment-level responses.

Definition: A demand outcome is the response being measured, such as whether an eligible customer purchases a quoted policy, whether a current policyholder renews, or the quantity of coverage purchased. An estimand is the precise population quantity of interest, such as a purchase probability at a stated price or the average change in that probability for a defined target population. An estimator is a method or calculation used to estimate that quantity.

Define the Outcome and Its Eligible Population

For a purchase outcome, define the eligible population as the customers or quote opportunities that could have purchased at the offered terms. For each eligible opportunity, a binary purchase outcome can be recorded as 1 if a purchase occurs within a specified period and 0 otherwise. The purchase probability is the proportion expected to have outcome 1 in the stated population. The observation period and the definition of a completed purchase must be consistent.

For renewal, the eligible population is generally current policyholders who reach a specified renewal decision point. The outcome is 1 if the policy renews under the stated definition and 0 otherwise. A renewal probability therefore has a different denominator from a quote purchase probability: a person who never held the policy is not ordinarily an eligible renewal opportunity. Reporting “retention” without specifying the eligible population and period can conceal this distinction.

An outcome can also be a quantity rather than a binary indicator. Examples include expected units of coverage purchased per eligible quote, or expected insured exposure per customer. The quantity must be defined in consistent units. A probability is dimensionless and lies between 0 and 1; a quantity may be measured in vehicles, coverage units, or another stated unit. A change of 0.01 in probability means a one-percentage-point change, not necessarily a one-percent relative change.

Definition: For a positive price $p$ and a positive demand measure $q(p)$, the own-price elasticity at that price is the proportional rate of change in demand with respect to a proportional rate of change in that same product’s price: $$ \varepsilon(p)=\frac{d\ln q(p)}{d\ln p} =\frac{p}{q(p)}\frac{dq(p)}{dp}. $$ For a purchase probability, $q(p)$ is that probability. For a quantity outcome, $q(p)$ is the expected quantity in its stated units. The derivative is interpreted with the relevant customer characteristics and target-population definition held fixed. This local elasticity is undefined if $p$ or $q(p)$ is zero.

When a price increase reduces demand, the derivative and elasticity are usually negative. The derivative’s units depend on the outcome: for a purchase probability and a price in dollars, it is probability points per dollar. Elasticity has no units. Locally, an elasticity of $-0.25$ means that a 1% price increase corresponds to approximately a 0.25% decrease in demand, under the conditions defining that estimate. It does not mean a decrease of 0.25 percentage points.

Conditional and Population-Average Responses

A conditional response describes demand for a specified type of customer or for customers with specified characteristics. An average response describes the average over a defined target population. If segments have different demand levels or price sensitivities, the average depends on how much each segment contributes to that population.

Suppose a target population has segments $s=1,\ldots,k$, with fixed proportions $w_s$, where the weights are nonnegative and sum to 1. Let $q_s(p)$ be the segment-specific expected outcome at price $p$. For an outcome measured on the same scale in every segment, the population-average outcome is:

\bar q(p)=\sum_{s=1}^{k}w_s q_s(p).

If the segment proportions remain fixed as price changes, differentiating gives $\bar q'(p)=\sum_s w_s q_s'(p)$. Thus the aggregate derivative is a population-weighted average of segment derivatives. The aggregate elasticity, however, is not generally the population-weighted average of segment elasticities. Substituting $q_s'(p)=q_s(p)\varepsilon_s(p)/p$ into the aggregate elasticity gives:

\bar\varepsilon(p) =\frac{p}{\bar q(p)}\bar q'(p) =\sum_{s=1}^{k} \frac{w_s q_s(p)}{\bar q(p)}\varepsilon_s(p).

The weights in this identity are each segment’s share of the average outcome, not necessarily its share of customers or quotes. This distinction applies directly to purchase probabilities: segments with more expected purchases contribute more to the aggregate probability’s proportional response.

Worked Example: Aggregating Purchase-Probability Elasticities

A hypothetical insurer evaluates purchase among two equally sized quote segments at an annual premium of $100. In segment A, the purchase probability is 0.80 and its local derivative with respect to premium is $-0.002$ probability units per dollar. In segment B, the probability is 0.40 and the derivative is $-0.0005$ per dollar. Calculate the aggregate probability, the aggregate derivative and elasticity, and the equal-quote-weighted mean of the segment elasticities.

Method and assumptions. Use the fixed target mix $w_A=w_B=0.50$. Treat the given derivatives as local price responses at $100, with the probabilities and derivatives applying to the same purchase outcome and eligible quote population.

The aggregate purchase probability is: $$ \bar q=0.50(0.80)+0.50(0.40)=0.40+0.20=0.60. $$ The aggregate derivative is: $$ \bar q'=0.50(-0.002)+0.50(-0.0005) =-0.001-0.00025=-0.00125 $$ probability units per dollar.

The segment elasticities at $100 are: $$ \varepsilon_A=\frac{100}{0.80}(-0.002)=-0.2500, \qquad \varepsilon_B=\frac{100}{0.40}(-0.0005)=-0.1250. $$ Their equal-quote-weighted mean is: $$ 0.50(-0.2500)+0.50(-0.1250)=-0.1875. $$ The aggregate elasticity, calculated from the aggregate probability and derivative, is: $$ \bar\varepsilon=\frac{100}{0.60}(-0.00125) =-0.208333\ldots\approx -0.2083. $$ The result differs from the equal-quote-weighted mean because the aggregate elasticity weights segment elasticities by expected-purchase shares. Those shares are $0.50(0.80)/0.60=2/3$ for A and $0.50(0.40)/0.60=1/3$ for B. Checking with these weights gives $(2/3)(-0.2500)+(1/3)(-0.1250)=-0.208333\ldots$, as required.

Common mistake avoided: averaging elasticities by quote share and calling the result the aggregate elasticity. Quote-weighted averaging gives $-0.1875$; the aggregate elasticity is $-0.2083$ because it is based on the change in aggregate probability relative to that aggregate probability.

Why the Aggregate Formula Holds

The aggregation identity follows from the definition of an average and the definition of elasticity; it does not require a special demand model. The important assumption is that the segment weights describe the target population and are held fixed for the price comparison.

Average the outcome.
For fixed segment proportions, the average is $\bar q(p)=\sum_s w_s q_s(p)$.

Differentiate with fixed weights.
Since each $w_s$ is constant, $\bar q'(p)=\sum_s w_s q_s'(p)$.

Substitute each segment’s elasticity.
By definition, $q_s'(p)=q_s(p)\varepsilon_s(p)/p$. Therefore $\bar q'(p)=\sum_s w_s q_s(p)\varepsilon_s(p)/p$.

Form the aggregate elasticity.
Multiplying by $p/\bar q(p)$ gives $\bar\varepsilon(p)=\sum_s [w_s q_s(p)/\bar q(p)]\varepsilon_s(p)$. The weights sum to 1 because $\sum_s w_s q_s(p)=\bar q(p)$.

If the mix itself changes with price, the fixed-weight result does not apply. For differentiable weights $w_s(p)$, differentiating $\bar q(p)=\sum_s w_s(p)q_s(p)$ also produces $\sum_s w_s'(p)q_s(p)$. That additional term represents a change in composition, distinct from changes in demand within segments. An analyst must therefore say whether the estimand concerns a fixed target mix or an average over a changing population.

Worked Example: Changing the Target Population

At an annual price of $100, a hypothetical insurer has purchase probabilities of 0.90 in segment A and 0.50 in segment B. The corresponding local derivatives are $-0.001$ and $-0.003$ probability units per dollar. Calculate the aggregate probability and elasticity for a target population that is 75% A and 25% B. Then repeat for a population that is 25% A and 75% B, and compare the conditional elasticities.

Method and assumptions. Hold each specified population’s segment weights fixed at the stated price. The probabilities and derivatives refer to the same purchase outcome. First, the segment elasticities are: $$ \varepsilon_A=\frac{100}{0.90}(-0.001)=-0.111111\ldots, \qquad \varepsilon_B=\frac{100}{0.50}(-0.003)=-0.6000. $$ These conditional elasticities do not change when the target mix changes.

For the 75% A, 25% B population, the average probability and derivative are: $$ \bar q=0.75(0.90)+0.25(0.50)=0.675+0.125=0.800, $$ $$ \bar q'=0.75(-0.001)+0.25(-0.003) =-0.00075-0.00075=-0.00150. $$ Thus: $$ \bar\varepsilon=\frac{100}{0.800}(-0.00150)=-0.1875. $$

For the 25% A, 75% B population: $$ \bar q=0.25(0.90)+0.75(0.50)=0.225+0.375=0.600, $$ $$ \bar q'=0.25(-0.001)+0.75(-0.003) =-0.00025-0.00225=-0.00250, $$ $$ \bar\varepsilon=\frac{100}{0.600}(-0.00250) =-0.416666\ldots\approx -0.4167. $$ The second target population has a more negative average elasticity because it contains more of segment B, whose conditional elasticity is more negative. The calculation also checks against the outcome-share weights: in the first population, A and B contribute $0.675/0.800$ and $0.125/0.800$; in the second, they contribute $0.225/0.600$ and $0.375/0.600$.

Common mistake avoided: treating an average elasticity as an intrinsic property of the product, independent of the customer mix. It is an estimand for a specified population; changing the population can change the average even when the segment-specific responses stay the same.

Probability Is Not Quantity

A binary purchase probability answers whether an eligible opportunity results in a purchase. It does not, by itself, describe the amount purchased. If some customers purchase multiple coverage units, for example, expected units per quote depend both on the probability of purchase and on the expected number of units among purchasers.

Let $r(p)$ be the purchase probability and let $m(p)$ be expected quantity conditional on purchase. If non-purchasers receive zero units, expected quantity per eligible opportunity is $Q(p)=r(p)m(p)$. For positive differentiable $r$ and $m$, the product rule gives $Q'=r'm+rm'$. Dividing by $Q=rm$ and multiplying by $p$ shows that the quantity elasticity is the sum of the purchase-probability elasticity and the conditional-quantity elasticity. This result applies to this product definition of expected quantity; it does not say that every quantity outcome can be decomposed this way.

Worked Example: Purchase Probability Versus Expected Coverage Units

A hypothetical commercial product is quoted at an annual price of $200. At that price, the purchase probability is 0.40, with local derivative $-0.001$ per dollar. Among purchasers, expected coverage units are 1.50, with local derivative $-0.002$ units per dollar. Calculate the local elasticities for purchase probability, conditional quantity, and expected units per eligible quote. Also calculate the exact expected units after a $5 increase if both stated response functions are linear over that interval.

Method and assumptions. Assume non-purchasers buy zero units, so expected units per eligible quote are the purchase probability times expected units conditional on purchase. The stated slopes are per dollar, and the linearity assumption applies only to the requested $5 interval.

The purchase-probability elasticity is: $$ \varepsilon_r=\frac{200}{0.40}(-0.001)=-0.5000. $$ The conditional-quantity elasticity is: $$ \varepsilon_m=\frac{200}{1.50}(-0.002) =-0.266666\ldots\approx -0.2667. $$ At the current price, expected units per eligible quote are $Q=0.40(1.50)=0.600$. The derivative, using the product rule, is: $$ Q'=(-0.001)(1.50)+(0.40)(-0.002) =-0.0015-0.0008=-0.0023 $$ units per dollar. Therefore: $$ \varepsilon_Q=\frac{200}{0.600}(-0.0023) =-0.766666\ldots\approx -0.7667. $$ This equals the sum $-0.5000-0.266666\ldots=-0.766666\ldots$, checking the product-rule result.

After the $5 increase, the price is $205. Under the stated linearity assumption, purchase probability is $0.40+5(-0.001)=0.395$, and expected units among purchasers are $1.50+5(-0.002)=1.49$. Expected units per eligible quote are therefore $0.395(1.49)=0.58855$, a decrease of $0.60000-0.58855=0.01145$ units. The purchase probability itself fell by 0.005, or 0.5 percentage points; relative to its initial 0.40, that is a 1.25% decrease. These are distinct changes in distinct outcomes.

Common mistake avoided: reporting the 0.005 decline in purchase probability as a 0.5% decline, or treating it as the decline in coverage quantity. The probability change is 0.5 percentage points, while expected units reflect both the purchase probability and quantity among purchasers.

Common Interpretation Traps

Using the wrong denominator. A quote purchase rate uses eligible quote opportunities; a renewal rate uses eligible renewal opportunities. They are not interchangeable rates.
Confusing percentage points with percent. A probability changing from 0.40 to 0.39 falls by 0.01, or one percentage point; relative to 0.40, the decrease is 2.5%.
Calling a slope an elasticity. A derivative such as $-0.002$ probability units per dollar has units. Elasticity also scales by price and divides by the current outcome.
Ignoring the target population. An aggregate estimate depends on the population’s segment mix and the weights used to define the average.
Equating purchase probability with quantity demand. If purchasers can buy different amounts, probability alone does not give expected quantity.
Overstating what an estimate means. An estimate can describe an observed association or a price effect, depending on how it was obtained and the assumptions supporting it. Its outcome and target population should be stated either way.

A useful reporting statement names the outcome, eligible population, price unit and period, and the interpretation of the response. For example: “The local own-price elasticity of purchase probability is estimated for eligible household quote opportunities at an annual premium of $100, standardized to the stated segment mix.” Such wording makes clear what the estimate does—and does not—summarize.

Check Your Understanding

For each problem, identify the outcome and target population before calculating. Round elasticities to four decimal places where needed.

A hypothetical insurer observes 175 purchases among 500 eligible new-business quote opportunities and 208 renewals among 260 eligible renewal opportunities. Calculate each outcome rate and the difference in percentage points. (Answer: purchase rate $0.3500$; renewal rate $0.8000$; renewal rate is 45 percentage points higher.)
At a price of $120, segments A and B have purchase probabilities 0.70 and 0.50, respectively, with local derivatives $-0.002$ and $-0.001$ probability units per dollar. The target mix is 40% A and 60% B. Calculate the aggregate probability, derivative, aggregate elasticity, and quote-weighted mean segment elasticity. (Answer: aggregate probability $0.5800$; derivative $-0.0014$ per dollar; aggregate elasticity $-0.2897$; quote-weighted mean segment elasticity $-0.2811$.)
At a price of $100, segment A has probability 0.85 and derivative $-0.001$; segment B has probability 0.45 and derivative $-0.002$, both per dollar. For a target mix of 20% A and 80% B, calculate the aggregate elasticity. (Answer: aggregate probability $0.5300$, derivative $-0.0018$ per dollar, and elasticity $-0.3396$.)
At a price of $80, purchase probability is 0.50 with derivative $-0.0015$ per dollar. Conditional expected quantity among purchasers is 2.00 units with derivative $-0.004$ units per dollar. Assuming non-purchasers buy zero units, calculate the expected units per eligible opportunity and its local elasticity. (Answer: expected quantity $1.0000$ unit; derivative $-0.0050$ units per dollar; elasticity $-0.4000$.)

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