Random-Coefficient Models for Customer Heterogeneity

Why Model a Distribution of Price Responses?

Two customers can face the same premium change and respond differently. A single price coefficient describes one average response, but it cannot show how widely customers’ price sensitivities vary. A random-coefficient model represents the price coefficient as a customer-specific quantity drawn from a population distribution. This helps distinguish the response of a typical customer from the average demand change across all customers.

The model’s interpretation depends on the demand outcome and link function. With log demand, a customer’s price coefficient is that customer’s price elasticity. With logistic purchase demand, the coefficient is a change in log odds, and the probability elasticity also depends on the customer’s purchase probability.

Definition: A random price coefficient is a model coefficient that varies across customers according to a specified probability distribution. Write \(B_i\) for customer \(i\)’s price coefficient, \(\mu_B=E[B_i]\) for its population mean, and \(\sigma_B^2=\operatorname{Var}(B_i)\) for its population variance. The distribution describes customer-level heterogeneity; it is not, by itself, the sampling uncertainty in an estimate of the coefficient.

Specifying a Random Price Coefficient

For a continuous demand outcome, consider a log-log model in which \(Q_i(p)\) is customer \(i\)’s demand at price \(p\), and \(p_0\) is a reference price:

\log Q_i(p)=A_i+B_i\log(p/p_0)+\varepsilon_i.

Here \(A_i\) is a customer-specific intercept and \(B_i\) is the random price coefficient. Assume the model has a well-defined conditional mean, with \(E[\varepsilon_i\mid A_i,B_i,p]=0\). The price elasticity of the modeled log demand for customer \(i\) is \(B_i\): a 1% price increase corresponds approximately to a \(B_i\)% change in demand for a small price change. A negative coefficient represents lower demand at higher prices.

One common specification is \(B_i\sim N(\mu_B,\sigma_B^2)\). This is a modeling assumption, not a requirement: a discrete distribution or another continuous distribution may be more appropriate. A normal distribution allows convenient calculations, but it also assigns some probability to positive price coefficients, even when positive individual elasticities are implausible. That implication should be checked rather than ignored.

The random coefficient may be independent of customer characteristics, or its mean may depend on observed characteristics. For example, one can write \(B_i=\mu_B+\gamma z_i+U_i\), where \(z_i\) is an observed characteristic and \(U_i\) is residual customer-level variation. In that specification, \(\gamma z_i\) represents systematic differences associated with \(z_i\), while \(U_i\) represents remaining heterogeneity. A random intercept and random slope can also be dependent; if so, their joint distribution must be specified.

For a binary purchase outcome \(Y_i\), a logistic specification is:

P(Y_i=1\mid B_i,p)= \frac{\exp\{A_i+B_i\log(p/p_0)\}} {1+\exp\{A_i+B_i\log(p/p_0)\}}.

In this model, \(B_i\) is the change in log odds per unit increase in log price, not the purchase-probability elasticity itself. If \(P_i\) denotes the modeled purchase probability, differentiating the logistic function gives the customer’s probability elasticity:

\frac{\partial \log P_i}{\partial \log p} =B_i(1-P_i).

Thus, even with a fixed \(B_i\), the probability elasticity changes with the customer’s purchase probability. This distinction matters when interpreting a distribution of logistic price coefficients.

Individual Distributions and Population-Average Demand

A distribution for \(B_i\) answers questions about customers: what proportion have an elasticity below a threshold, and how much variation is there among them? A population-average demand prediction answers a different question: what is the average outcome after each customer responds according to their own coefficient?

For a random outcome \(Y_i\) and price \(p\), the population-average prediction is the expectation over the coefficient distribution (and any other modeled customer-level variation). In general, for a nonlinear response function \(g\),

E[g(B_i)]\ne g(E[B_i]).

Equality does hold when \(g\) is linear in \(B_i\). It generally fails for exponential and logistic response functions. Therefore, substituting the mean coefficient into a nonlinear demand curve usually does not give the average of customer-level predictions.

Worked Example: Interpreting an Elasticity Distribution

A hypothetical insurer uses a log-log demand model and assumes customer elasticities \(B_i\sim N(-1.00,0.50^2)\). Calculate the proportion of customers with elasticity below \(-1.50\), the proportion with a positive elasticity, and the central 90% interval of the customer-level distribution. Interpret the results.

Method and assumptions. The stated normal distribution applies to individual price elasticities, and the second parameter is the variance. Thus the standard deviation is \(\sigma_B=0.50\). Standardize each threshold using \(Z=(B_i-\mu_B)/\sigma_B\), where \(Z\sim N(0,1)\).

For the first proportion,

P(B_i<-1.50)=P\left(Z<\frac{-1.50-(-1.00)}{0.50}\right) =P(Z<-1.00)=0.1587.

For a positive elasticity,

P(B_i>0)=P\left(Z>\frac{0-(-1.00)}{0.50}\right) =P(Z>2.00)=1-0.9772=0.0228.

The central 90% interval uses the standard-normal 5th and 95th percentiles, approximately \(-1.645\) and \(1.645\):

-1.00\pm1.645(0.50)=[-1.8225,-0.1775].

Rounded to two decimals, the interval is \([-1.82,-0.18]\). The model assigns about 15.87% of customers elasticity below \(-1.50\), and about 2.28% positive elasticity. Those positive values may warrant review if the application assumes all customers reduce demand as price rises.

Check. The central interval is centered at the mean \(-1.00\), and its endpoints are 1.645 standard deviations from that mean. The positive-elasticity probability is the upper tail beyond two standard deviations, not the lower tail.

Deriving the Average Response for Log Demand

Suppose customer-level demand changes by a price ratio \(r=p/p_0\), with no price-varying residual factor, so that the demand ratio for customer \(i\) is \(r^{B_i}=\exp(B_i\log r)\). If \(B_i\sim N(\mu_B,\sigma_B^2)\), the population-average demand ratio can be derived directly from the normal density:

\begin{aligned} E[r^{B_i}] &=\int_{-\infty}^{\infty} \exp(b\log r)\, \frac{1}{\sigma_B\sqrt{2\pi}} \exp\left(-\frac{(b-\mu_B)^2}{2\sigma_B^2}\right)\,db\\ &=\exp\left(\mu_B\log r+\frac{1}{2}\sigma_B^2(\log r)^2\right). \end{aligned}

To see the second equality, combine the exponents and complete the square in \(b\). The terms involving \(b\) become a normal density centered at \(\mu_B+\sigma_B^2\log r\); integrating that density gives 1. The remaining constant is \(\mu_B\log r+\tfrac12\sigma_B^2(\log r)^2\). This is the normal moment-generating-function result applied at \(t=\log r\).

If instead one evaluates the demand ratio at the mean coefficient, the result is \(r^{\mu_B}\). For a nonzero price change and positive variance, the population-average ratio differs by the factor \(\exp\{\tfrac12\sigma_B^2(\log r)^2\}\). This factor is at least 1. It follows from the curvature of the exponential response, not from a change in the mean elasticity.

Worked Example: Average Demand After a Price Doubling

For a hypothetical continuous-demand portfolio, suppose \(B_i\sim N(-1.00,0.50^2)\), and each customer’s demand ratio when price doubles is \(2^{B_i}\). Calculate the population-average demand ratio and compare it with the ratio obtained by using the mean coefficient.

Method and assumptions. Apply the normal random-coefficient formula derived above, with \(r=2\), \(\mu_B=-1.00\), and \(\sigma_B^2=0.25\). Demand is measured relative to each customer’s baseline, so the calculated ratio is the average of the individual ratios.

First, \(\log 2=0.693147\) and \((\log 2)^2=0.480453\). The exponent for the population-average ratio is

(-1.00)(0.693147)+\frac12(0.25)(0.480453) =-0.693147+0.060057=-0.633090.

Therefore the average ratio is \(\exp(-0.633090)=0.5309\), rounded to four decimal places. Using only the mean coefficient gives

2^{E[B_i]}=2^{-1}=0.5000.

The average of the customer-level demand ratios is 0.5309, compared with 0.5000 at the mean coefficient. The former is about 3.09 percentage points higher. It is still a demand decrease relative to baseline; the higher ratio reflects averaging the nonlinear individual responses.

Check. Since the dispersion adjustment is \(\exp\{\tfrac12(0.25)(\log 2)^2\}>1\), the average ratio must exceed \(2^{-1}\). The computed value satisfies that check.

Logistic Demand: Average Probabilities Are Not Plug-In Probabilities

For binary purchase demand, suppose the price coefficient has a distribution \(F_B\). At price \(p\), the population-average purchase probability is the average of the individual probabilities:

E[P_i(Y_i=1\mid B_i,p)] =\int P(Y_i=1\mid B_i=b,p)\,dF_B(b).

A plug-in calculation instead evaluates the logistic function at \(E[B_i]\). Because the logistic function is nonlinear, the two results need not agree. Moreover, the distribution of \(B_i\) describes log-odds slopes; converting it into a distribution of probability elasticities requires evaluating \(B_i(1-P_i)\) at the price and customer characteristics of interest.

Worked Example: Averaging Purchase Probabilities Across Two Customer Types

A hypothetical insurer models purchase probability as logistic, with \(A_i=0\). Half the customers have \(B_i=-1\), and half have \(B_i=-3\). At a price \(p=e\,p_0\), calculate each type’s purchase probability, the population-average probability, and the plug-in probability using the mean coefficient. Then calculate the probability elasticity for each type and the aggregate probability elasticity.

Method and assumptions. The coefficient distribution is discrete with equal weights, and the logistic model is correctly specified at the stated price. Since \(\log(p/p_0)=\log e=1\), the two log odds are \(-1\) and \(-3\).

For \(B_i=-1\), the purchase probability is \(1/(1+e^1)=0.2689\). For \(B_i=-3\), it is \(1/(1+e^3)=0.0474\). The average probability is

\bar P=\frac{0.2689+0.0474}{2}=0.1582

to four decimal places using unrounded probabilities. The mean coefficient is \((-1-3)/2=-2\); evaluating the logistic model at that coefficient gives \(1/(1+e^2)=0.1192\). Thus the average of the two type-specific probabilities is 0.1582, not 0.1192.

For each type, the probability elasticity is \(B_i(1-P_i)\). Using unrounded probabilities, these elasticities are \(-1(1-0.268941)=-0.7311\) and \(-3(1-0.047426)=-2.8577\), to four decimals. The aggregate elasticity is the derivative of average purchase probability with respect to log price, divided by average purchase probability:

\frac{d\log\bar P}{d\log p} =\frac{E[B_iP_i(1-P_i)]}{E[P_i]} =\frac{\{(-1)(0.268941)(0.731059)+(-3)(0.047426)(0.952574)\}/2}{0.158184} \approx -1.0499.

The aggregate elasticity is not the unweighted average of the two individual elasticities. That unweighted mean is \((-0.7311-2.8577)/2=-1.7944\), while the aggregate elasticity is approximately \(-1.0499\). The aggregate calculation weights each type’s probability change and scales it by the total purchase probability.

Check. Both types have negative price coefficients and both probability elasticities are negative, so a price increase reduces the modeled purchase probability. The aggregate derivative is also negative, consistent with those individual responses.

Common Mistakes and Practical Interpretation

The most important distinction is between a distribution of individual responses and an average outcome for the population. A coefficient mean summarizes the center of the slope distribution; it does not generally summarize the population-average demand curve.

Plugging the mean coefficient into a nonlinear model. This gives \(g(E[B_i])\), not \(E[g(B_i)]\). Integrate or average the customer-level predictions when the target is population-average demand.
Calling a logistic slope an elasticity. In the logistic model, \(B_i\) is a log-odds coefficient. The probability elasticity is \(B_i(1-P_i)\), which depends on the modeled probability.
Ignoring the support of a chosen distribution. A normal random slope can assign probability to responses contrary to substantive expectations. Assess the implied distribution and consider whether a different specification is needed.
Confusing heterogeneity with estimation uncertainty. The variance \(\sigma_B^2\) describes variation among customers in the model. A standard error for an estimated mean or variance describes uncertainty due to the data; it is a separate quantity.
Reporting only the mean response. Two populations can have the same mean slope and different slope variances, leading to different average outcomes under nonlinear demand models. Report distributional information when customer variation matters to the decision.

In practice, random-coefficient models require enough information to distinguish customer-level variation from unexplained outcome variation. Distributional assumptions and model fit should be examined, and predictions should be averaged over the intended customer population. These distinctions are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, distinguish the individual coefficient distribution from the population-average demand response.

A log-log demand model has \(B_i\sim N(-0.80,0.30^2)\). What proportion of customers have elasticity below \(-1.10\)? (Answer: 0.1587)
Under a log-log model, \(B_i\sim N(-1.20,0.40^2)\). A price doubles. What is the population-average demand ratio, using the normal random-coefficient formula? (Answer: 0.4523, to four decimals)
A logistic purchase model has \(A_i=0\), with equal proportions of customers having slopes \(-0.5\) and \(-1.5\). At \(p=2p_0\), what is the average purchase probability, and what is the probability from plugging in the mean slope? (Answer: 0.3377 average; 0.3333 at the mean slope, to four decimals)
For a log-log model, \(B_i\sim N(-0.90,0.20^2)\), and the price ratio is \(1.50\). Calculate the population-average demand ratio and the ratio at the mean coefficient. (Answer: 0.6966 average; 0.6942 at the mean coefficient, to four decimals)
In a logistic model at a particular price, two equally sized customer types have purchase probabilities 0.40 and 0.10, with slopes \(-1\) and \(-2\), respectively. Using \(B_i(1-P_i)\), find each type’s probability elasticity and the aggregate probability elasticity. (Answer: individual elasticities \(-0.6000\) and \(-1.8000\); aggregate elasticity \(-0.8400\))

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