Point and Arc Elasticity for Insurance Demand

Why Measure Demand Elasticity?

A premium change can affect demand in two different ways: it may change the chance that a quoted customer buys, or it may change the number of policies sold across a portfolio of quotes. Elasticity puts a price change and a demand change on a common relative scale. For example, a signed elasticity of $-1.2$ means that, near the price being studied, a 1% increase in price is associated with approximately a 1.2% decrease in demand.

There are two useful versions. Point elasticity describes the local response at one price using the slope of a differentiable demand curve. Arc elasticity summarizes the response between two observed prices. The distinction matters: a finite price change need not be small enough for a local slope to represent the whole interval.

Definition: Let $P>0$ be price and let $D(P)>0$ be a demand outcome at price $P$. If $D$ is differentiable at $P$, its signed point price elasticity is $$ \varepsilon(P)=\frac{dD(P)}{dP}\frac{P}{D(P)}. $$ For two observations $(P_1,D_1)$ and $(P_2,D_2)$, the signed midpoint arc elasticity is $$ \varepsilon_{\mathrm{arc}}= \frac{(D_2-D_1)/[(D_1+D_2)/2]} {(P_2-P_1)/[(P_1+P_2)/2]}. $$ The elasticity is signed by convention. For ordinary downward-sloping demand it is negative; its absolute value is sometimes reported as the elasticity magnitude.

The demand outcome must be defined before calculating either measure. It may be a purchase probability, a purchase rate, or a number of policies, provided its meaning and denominator are consistent across prices. A change in purchase probability from 0.40 to 0.36 is a decrease of 4 percentage points, but a relative decrease of 10%. Elasticity uses the relative change, not the percentage-point change.

Why the Point Formula Uses Relative Changes

The point formula is the limiting ratio of a relative demand change to a relative price change. For a small price increment $\Delta P$, differentiability gives

D(P+\Delta P)-D(P)\approx D'(P)\Delta P.

Divide the approximate demand change by the current demand to obtain its relative change. Divide that relative change by the relative price change, $\Delta P/P$. The common increment $\Delta P$ cancels:

\frac{[D(P+\Delta P)-D(P)]/D(P)}{\Delta P/P} \approx \frac{D'(P)\Delta P/D(P)}{\Delta P/P} = D'(P)\frac{P}{D(P)}.

Taking the limit as $\Delta P$ approaches zero yields $\varepsilon(P)=D'(P)P/D(P)$. The slope $D'(P)$ alone is not an elasticity: it has units of demand per dollar. Multiplying by $P/D(P)$ converts the slope to a dimensionless relative response.

Point Elasticity for a Purchase Probability

When demand is the probability of purchase per quote, $D(P)$ lies between 0 and 1. The point formula still applies as long as that probability is positive and the curve is differentiable at the price considered. A probability curve should also be meaningful over the price range in which it is used; a simple linear curve may not remain between 0 and 1 if extrapolated too far.

Worked Example: Local Price Sensitivity of a Purchase Probability

In a hypothetical insurance quote market, the purchase probability is modeled over prices from $50 to $150 by $q(P)=0.72-0.003P$, where $P$ is the quoted premium in dollars. Calculate the signed point elasticity at $P=\$80$, then interpret it. Treat the stated curve as the demand function over this price range.

Step 1: Find the probability at the specified price. At $P=80$,

$$ q(80)=0.72-0.003(80)=0.72-0.24=0.48. $$

The probability is 0.48, or 48%. It is positive and within the valid probability range.

Step 2: Differentiate with respect to price.

$$ q'(P)=-0.003 $$

This slope means the modeled purchase probability changes by $-0.003$ per dollar: a one-dollar price increase corresponds locally to a decrease of 0.3 percentage points.

Step 3: Apply the point formula.

\varepsilon(80)=q'(80)\frac{80}{q(80)} =(-0.003)\frac{80}{0.48} =\frac{-0.24}{0.48} =-0.500.

The signed point elasticity is $-0.500$. Near $80, a 1% price increase corresponds to approximately a 0.5% relative decrease in purchase probability. This is not a decrease of 0.5 percentage points: the elasticity describes a relative change in the 48% probability. The calculation uses the probability as the outcome, not the slope by itself.

Check. The price-scaled slope is $-0.003\times80=-0.24$; dividing by the current probability, 0.48, gives $-0.5$, as calculated.

Arc Elasticity for Finite Price Changes

For two distinct price-outcome pairs, the midpoint formula divides the relative change in demand by the relative change in price. Each relative change uses the average of its two endpoint values as its denominator. This midpoint convention treats the two endpoints symmetrically: reversing their order changes the sign of both changes, leaving the ratio unchanged. It does not require a differentiable demand curve.

Arc elasticity is an interval summary, not the exact point elasticity at either endpoint. If demand is nonlinear, the local elasticity can differ across the interval. The midpoint formula is especially useful when the available information consists of two observed prices and outcomes rather than a fitted demand curve.

Worked Example: Arc Elasticity From Hypothetical Quote Counts

A hypothetical insurer offers the same product to two comparable groups, with 2,000 quotes in each group. At a premium of $800, 900 customers buy; at $1,000, 780 buy. Calculate the midpoint arc elasticity of the purchase probability between the two prices. Assume the quoted groups are comparable for this descriptive calculation.

Step 1: Convert counts to purchase probabilities. Using purchases divided by quotes,

q_1=\frac{900}{2{,}000}=0.45, \qquad q_2=\frac{780}{2{,}000}=0.39.

The outcome change is $q_2-q_1=0.39-0.45=-0.06$. This is a decrease of 6 percentage points; the arc calculation will scale it by the average probability.

Step 2: Calculate the midpoint relative change in probability. The average probability is $(0.45+0.39)/2=0.42$, so

\frac{q_2-q_1}{(q_1+q_2)/2} = \frac{-0.06}{0.42} =-0.142857\ldots

Step 3: Calculate the midpoint relative change in price. The average price is $(800+1{,}000)/2=\$900$, giving

\frac{P_2-P_1}{(P_1+P_2)/2} = \frac{200}{900} =0.222222\ldots

Step 4: Divide the relative changes.

\varepsilon_{\mathrm{arc}} = \frac{-0.06/0.42}{200/900} =-0.642857\ldots \approx -0.643.

The midpoint arc elasticity is $-0.643$. Across these endpoints, the purchase probability decreased by about 0.643% for each 1% increase in premium, using midpoint percentage changes. The result describes this finite interval; it is not automatically a local estimate at $800 or $1,000.

Check. The relative probability change is negative and the relative price change is positive, so the elasticity must be negative. Using the two midpoint changes gives $-0.142857/0.222222=-0.643$, to three decimals.

Elasticity for Policy Quantities

Demand can instead be measured as a quantity, such as the expected number of policies sold from a fixed market or quote opportunity. The same formulas apply, but the outcome must use the same quantity definition and exposure basis at both prices. If every quote has the same exposure denominator at each price, converting purchase counts into probabilities simply multiplies both counts by the same constant; that constant cancels in the relative change and elasticity.

Worked Example: Point and Arc Elasticity for Policy Quantity

For a hypothetical fixed market, expected policies sold at premium $P$ are modeled by $Q(P)=1{,}500-4P-0.01P^2$, with $P$ in dollars and $Q$ in policies over the same selling period. Calculate point elasticity at $100, then calculate midpoint arc elasticity from $100 to $120. Use the curve only on this interval, where its predicted quantities are positive.

Step 1: Evaluate quantity at both endpoints.

Q(100)=1{,}500-4(100)-0.01(100^2) =1{,}500-400-100=1{,}000,

and

Q(120)=1{,}500-4(120)-0.01(120^2) =1{,}500-480-144=876.

Both quantities are positive. Demand decreases by $876-1{,}000=-124$ policies over the interval.

Step 2: Differentiate and calculate point elasticity at $100.

Q'(P)=-4-0.02P, \qquad Q'(100)=-4-0.02(100)=-6.

The slope is $-6$ policies per dollar at $100. Scale by price divided by quantity:

\varepsilon(100)=(-6)\frac{100}{1{,}000}=-0.600.

At $100, a 1% price increase corresponds locally to an approximately 0.6% decrease in expected policies sold.

Step 3: Calculate midpoint arc elasticity over the full interval. Average quantity is $(1{,}000+876)/2=938$, and average price is $(100+120)/2=\$110$. Therefore,

\varepsilon_{\mathrm{arc}} = \frac{-124/938}{20/110} = \frac{-0.132196\ldots}{0.181818\ldots} =-0.727078\ldots \approx -0.727.

The interval elasticity is more negative than the point elasticity at $100. That is possible because the slope and the demand level both vary with price along this curve. The arc value summarizes the whole interval; the point value applies locally at its stated price.

Check. The endpoint quantity ratio is not used directly as the arc percentage change: the midpoint denominator is 938, while the price midpoint denominator is 110. Substituting those averages gives $-0.727$, with the expected negative sign.

Interpretation, Conditions, and Common Traps

Elasticity has no units: dollars and policies, or dollars and probabilities, cancel through the relative changes. That makes it useful for comparing demand responses across outcomes with different measurement scales. It does not, by itself, establish that a premium change caused a demand change. A causal interpretation requires a price comparison that supports that conclusion; otherwise, the elasticity is descriptive of the observed or modeled relationship.

The standard calculation also has limits. Point elasticity requires a differentiable demand function at the price and a positive demand outcome. Arc elasticity requires positive average price and positive average outcome so the midpoint relative changes are defined. If the measured demand is zero at both endpoints, or the average outcome is zero, the midpoint calculation is not defined. If demand rises with price, the result is positive; do not change the sign merely to conform to the usual downward-sloping pattern.

A frequent exam error is to use the slope as the elasticity, or to divide a probability change by the wrong quantity. For the first worked example, the following is incorrect:

\text{Incorrect: }\varepsilon(80)=q'(80)=-0.003.

That number is a change in probability per dollar, not a dimensionless elasticity. The correction is to multiply the slope by $P/q(P)$:

\text{Correct: }\varepsilon(80)=(-0.003)\frac{80}{0.48}=-0.500.

Another trap is to treat a 6-percentage-point probability drop as a 6% relative drop. In the quote-count example the relative probability change is $-0.06/0.42$, not $-0.06$. A third is mixing price units: if the derivative is per dollar, use price in dollars. If price is converted to hundreds of dollars, the derivative must also be converted to probability per hundred dollars; the final elasticity is unchanged only when the units are consistent.

Define the outcome.
State whether demand is a probability, rate, or quantity, and keep its denominator and population basis consistent.

Choose point or arc.
Use point elasticity for a differentiable curve at a specified price; use midpoint arc elasticity for two finite endpoint observations.

Calculate relative changes.
For point elasticity, use $D'(P)P/D(P)$. For an arc, divide the midpoint relative demand change by the midpoint relative price change.

Interpret and check.
Retain the sign, state whether the result is local or interval-based, and verify the units, denominators, and demand values.

Key takeaway: Elasticity is a ratio of relative demand change to relative price change. Point elasticity uses a local derivative; arc elasticity uses midpoint changes across two prices. Keep the outcome definition and units consistent, and distinguish relative probability changes from percentage-point changes.

Check Your Understanding

For each problem, report signed elasticity and retain enough precision to support your interpretation.

A hypothetical purchase-probability curve is $q(P)=0.70-0.002P$, with $P$ in dollars. Find point elasticity at $100. (Answer: $-0.400$)
At $600, a hypothetical quote group has purchase probability 0.50; at $700, it is 0.42. Find midpoint arc elasticity. (Answer: approximately $-1.130$)
A hypothetical market has expected quantity $Q(P)=2{,}000-5P$ policies, with $P$ in dollars. Find point elasticity at $200. (Answer: $-1.000$)
A fixed hypothetical market sells 500 policies at $80 and 420 policies at $100. Find midpoint arc elasticity of quantity. (Answer: approximately $-0.783$)
Two hypothetical groups each receive 1,000 quotes. At $500, 320 customers buy; at $550, 280 buy. Find midpoint arc elasticity of purchase probability. (Answer: $-1.400$)

What You'll Learn