Why Consumer Heterogeneity Matters
When products differ in price and attributes, consumers need not value those differences in the same way. A feature that strongly attracts one consumer may matter little to another; a price increase may also affect consumers differently. A model that assigns one set of tastes to everyone can miss these patterns, even if it describes the average consumer well.
Random-coefficient utility represents this variation by separating a product’s common utility component from the part that depends on an individual consumer’s tastes. This distinction matters for substitution: when one product becomes less attractive, which alternatives gain depends on the tastes of the consumers who are most likely to switch.
A Utility Specification
Consider market \(t\), products \(j=1,\ldots,J_t\), and an outside option \(j=0\). Let \(q_{jt}\) be a vector of observed product attributes, \(p_{jt}\) the product’s price, and \(\xi_{jt}\) a product-market component not observed by the researcher. Consumer \(i\)’s utility from product \(j\) is
Here \(\beta_i\) is the consumer’s vector of tastes for observed attributes, and \(\alpha_i\) is the consumer’s marginal disutility of price. We take \(\alpha_i>0\), so a higher price reduces utility, holding other terms fixed. The term \(\varepsilon_{ijt}\) is an idiosyncratic utility shock. A common location normalization sets the outside option’s non-idiosyncratic utility to zero; this fixes the utility reference, not the utility scale, and is not a claim that choosing outside has no value.
One useful parameterization makes demographics and unobserved tastes explicit. Let \(d_i\) be a vector of consumer demographics, centered so that its population mean is zero, and let \(\nu_i\) be a vector of unobserved taste draws with mean zero. Then, for example,
The matrices and vectors \(\Pi,\Sigma,\pi\) describe how tastes vary with demographics and unobserved draws. The scalar \(\nu_{i,\alpha}\) is the component of the taste draw used for price sensitivity. The specification must keep \(\alpha_i>0\) on the support of the consumer distribution. In applications, this can be imposed through parameter restrictions or a transformation such as a model for \(\log(\alpha_i)\). A normal draw for an unrestricted \(\alpha_i\) does not guarantee a negative price coefficient for every consumer.
With centered demographics and mean-zero taste draws, the population means of \(\beta_i\) and \(\alpha_i\) are \(\beta\) and \(\alpha\). Define mean utility and the consumer-specific deviation by
Thus \(u_{ijt}=\delta_{jt}+\mu_{ijt}+\varepsilon_{ijt}\). Mean utility is the component built from population-mean tastes and the common product-market term. It is not the utility of every consumer, nor is it the expected utility including the idiosyncratic shock. Conditional on a demographic group, consumers can have a nonzero average \(\mu_{ijt}\), even though the deviations average to zero over the centered population.
Worked Example: Constructing Consumer-Specific Utility
Suppose, hypothetically, there are two observed product attributes: quality \(q_{1j}\), and an indicator \(q_{2j}\) that equals one for a product with a particular feature. There are two products, with \((q_{1A},q_{2A},p_A)=(2,1,3)\) and \((q_{1B},q_{2B},p_B)=(1,0,1)\). For one consumer, suppose the taste coefficients are \(\beta_i=(1.2,0.4)\), \(\alpha_i=0.5\), and \(\xi_A=\xi_B=0\). The outside option has non-idiosyncratic utility zero.
For this consumer, the non-idiosyncratic utility of product A is
For product B it is
The consumer-specific values \(1.3\) and \(0.7\) are not mean utilities unless these happen to equal the population-mean components. If the population-mean coefficients were \(\beta=(1,0.2)\) and \(\alpha=0.4\), then \(\delta_A=2(1)+1(0.2)-0.4(3)=1.0\), and \(\delta_B=1(1)-0.4(1)=0.6\). The differences, \(0.3\) and \(0.1\), are this consumer’s taste deviations \(\mu_{iA}\) and \(\mu_{iB}\).
From Individual Choices to Market Probabilities
To obtain a familiar conditional choice formula, assume that, conditional on the consumer’s tastes and observed product characteristics, the shocks \(\varepsilon_{ijt}\) are independent across products and consumers and follow the type I extreme-value distribution with a common scale normalized to one. Under this assumption, the probability that consumer \(i\) chooses product \(j\), conditional on their coefficients, is
The \(1\) in the denominator is the exponential of the outside option’s normalized non-idiosyncratic utility. This formula is conditional on a particular consumer’s coefficients: it is not yet the market share. The market-level choice probability for product \(j\) is the population average,
The expectation averages over the joint distribution of demographics and unobserved tastes in the market. If demographics are observed for consumers, the average can instead be understood as averaging conditional probabilities over those observed demographics and the remaining taste distribution. In either case, the population’s composition is part of the demand model.
Worked Example: Averaging Across Two Demographic Groups
Suppose, hypothetically, that a market has two equally sized demographic groups and two products, A and B, plus an outside option. Assume the conditional logit shock assumptions above. For group 1, the non-idiosyncratic utilities are \(V_A=-0.8\), \(V_B=-0.4\), and \(V_0=0\). For group 2, they are \(V_A=0.2\), \(V_B=0.1\), and \(V_0=0\). These differences in utilities could arise from demographic interactions with product attributes.
For group 1, the denominator is
Its probabilities are \(P_{1A}=0.449329/2.119649\approx0.2120\), \(P_{1B}=0.670320/2.119649\approx0.3162\), and \(P_{10}=1/2.119649\approx0.4718\). For group 2,
Thus \(P_{2A}\approx0.3672\), \(P_{2B}\approx0.3322\), and \(P_{20}\approx0.3006\). With equal group sizes, the aggregate probabilities are the group averages:
The rounded probabilities sum to one. The aggregate probability is not obtained by assigning each consumer the average of the two groups’ utilities and then applying the logit formula. The group-level probabilities must be averaged because the nonlinear choice formula applies separately to each group.
Why the Taste Distribution Shapes Substitution
Two features of the taste distribution matter beyond its average. First, demographic interactions can make the conditional mean tastes differ systematically across groups. Second, unobserved taste variation can create consumers who place unusually high or low value on particular attributes. Correlation among taste coefficients also matters: consumers who especially value one attribute may tend to value another attribute more, or less.
As a result, when a product changes in price or attributes, the consumers most affected need not be representative of the whole market. If a product is particularly attractive to consumers who also favor a close alternative, that alternative may receive more of the switching demand. A model with a single common coefficient vector cannot represent this sorting of consumers across products. Random coefficients allow substitution patterns to reflect who prefers what, rather than only the market’s average product utilities.
Worked Example: Same Mean Utilities, Different Aggregate Choices
Consider a hypothetical market with two equally sized groups, products A and B, and an outside option. Assume the same conditional logit shock model. Group 1 has non-idiosyncratic utilities \((V_A,V_B,V_0)=(1,0,0)\); group 2 has \((0,1,0)\). The average utility for each inside product across groups is \(0.5\).
For group 1, the denominator is \(\exp(1)+\exp(0)+1=2.71828+1+1=4.71828\). Its probabilities for A, B, and the outside option are approximately \(0.5761\), \(0.2119\), and \(0.2119\). Group 2 has the probabilities reversed for A and B, with the same outside probability. Averaging gives
Now suppose instead that every consumer has utility \(0.5\) for each inside product. The denominator is \(2\exp(0.5)+1=2(1.64872)+1=4.29744\). Each inside product then has probability \(1.64872/4.29744\approx0.3837\), and the outside option has probability \(1/4.29744\approx0.2327\). The mean utilities are the same in these two cases, but the aggregate probabilities differ: heterogeneity changes how often consumers choose an inside product. It also changes which consumers are most likely to choose each product, an important determinant of substitution when product offerings or prices change.
A Short Justification: Why Averaging Utilities Is Not Enough
Let \(V_{ij}\) denote consumer \(i\)’s non-idiosyncratic utility for product \(j\). Under the conditional logit assumptions, the probability conditional on tastes is the softmax function
The law of total probability says that the unconditional probability is \(\mathbb{E}_i[g_j(V_i)]\). It does not say that this is \(g_j(\mathbb{E}_i[V_i])\). The function \(g_j\) is nonlinear because utilities appear in exponentials and in a denominator that also varies across consumers. In general,
The two-group example demonstrates the difference directly. Consequently, a model’s taste distribution is not an optional detail that can always be replaced by its mean. Its spread, dependence structure, and demographic composition affect predicted choices and substitution.
Common Mistake
A frequent error is to treat \(\delta_{jt}\) as the utility experienced by every consumer, or to compute a market share by applying the logit formula to mean utilities alone. That calculation suppresses \(\mu_{ijt}\) and generally loses the variation that produces different consumer-level probabilities. The correction is to specify the consumer-specific coefficients and their distribution, calculate choice probabilities conditional on those tastes, and average those probabilities over the population. Also keep the roles of the terms distinct: \(\xi_{jt}\) is common to consumers facing product \(j\) in market \(t\), whereas \(\nu_i\) represents consumer-level taste variation.
Check Your Understanding
Answer each question using the definitions and assumptions in this tutorial.
- In the utility specification, which terms are common to all consumers choosing a given product in a market, and which terms vary by consumer?
- Why must the price coefficient’s sign or distribution be constrained if the model is intended to represent consumers who dislike higher prices?
- What assumptions on the idiosyncratic utility shocks yield the conditional logit choice formula stated here?
- Why is the market probability generally not the logit probability evaluated at average utility?
- Give one reason why demographic interactions or correlated taste coefficients could change which products gain consumers when another product becomes less attractive.