Random Coefficients Logit Model

Summary

The random-coefficients (mixed) logit is the demand engine of BLP. Each consumer $i$ chooses the product with the highest indirect utility $U_{ijt}={\delta }_{jt}+{\mu }_{ijt}+{ϵ}_{ijt}$, where ${\delta }_{jt}$ is the mean utility common to all consumers, ${\mu }_{ijt}$ is an individual-specific deviation capturing heterogeneous tastes (random coefficients), and ${ϵ}_{ijt}$ is i.i.d. type-I extreme value. Aggregate market shares are integrals of the logit choice probability over the distribution of consumer types. Because the random coefficients induce correlation in tastes across products, the model relaxes the IIA / independence-from-irrelevant-alternatives restriction of plain logit, yielding realistic, flexible substitution patterns.

Overview

Plain multinomial logit imposes IIA: cross-price elasticities depend only on shares, so a price increase pushes consumers to all other products in proportion to their share — economically implausible. BLP fixes this by letting taste coefficients vary across consumers. Two consumers facing the same products can have very different substitution patterns; when aggregated, this generates rich, data-driven cross-elasticities. McFadden & Train (2000) show that any random-utility model can be approximated by a mixed multinomial logit with a sufficient basis of characteristics.

Main Content

Indirect utility ^utility

Individual $i$ in market $t=1,\dots ,T$ gets utility from product $j$:

$$U_{ijt}={\delta }_{jt}+{\mu }_{ijt}+{ϵ}_{ijt}.$$

• ${\delta }_{jt}$ — mean utility of product $j$ in market $t$ (common across $i$).
• ${\mu }_{ijt}$ — random-coefficient deviation: the consumer-specific taste departure from the mean, parameterized by ${\tilde{\theta }}_2$.
• ${ϵ}_{ijt}$ — idiosyncratic i.i.d. type-I extreme value (Gumbel) error.

Consumers choose among $J_t=\{0,1,\dots ,J_t\}$ alternatives including the outside good $j=0$, normalized to $U_{i0t}={ϵ}_{i0t}$. The choice indicator is

$$d_{ijt}=\{\begin{array}{ll}1& \text{if }{U}_{ijt}>U_{ikt}\text{for all }k\ne j,\\ 0& \text{otherwise.}\end{array}$$

Market shares as integrals ^shares-integral

Aggregate shares integrate the individual choices over heterogeneity:

$$s_{jt}=\int d_{ijt}({\mathbf{\delta }}_t,{\mathbf{\mu }}_{it})\mathrm{d}{\mathbf{\mu }}_{it}\mathrm{d}{\mathbf{ϵ}}_{it}.$$

With i.i.d. type-I extreme value ${ϵ}_{ijt}$, the inner ($ϵ$) integral has the closed-form logit kernel, leaving an integral over consumer types:

$$s_{jt}({\mathbf{\delta }}_t,{\tilde{\theta }}_2)=\int \frac{\exp ({\delta }_{jt}+{\mu }_{ijt})}{{\sum }_{k\in J_t}\exp ({\delta }_{kt}+{\mu }_{ikt})}f({\mathbf{\mu }}_{it}\mid {\tilde{\theta }}_2)\mathrm{d}{\mathbf{\mu }}_{it}.$$

Here $f({\mathbf{\mu }}_{it}\mid {\tilde{\theta }}_2)$ is the mixing distribution over heterogeneous types. This is why the model is called mixed logit or random coefficients logit: each individual’s demand is a multinomial logit kernel, mixed over types. The integral has no closed form and must be approximated numerically — see Numerical Integration and Optimization in PyBLP.

Mean-utility index and the structural error ^delta-index

The key insight of Berry (1994) / BLP (1995) is the nonlinear change of variables ${\mathbf{\delta }}_t\equiv D_t^{-1}({\mathcal{S}}_t,{\tilde{\theta }}_2)$: given observed shares, the share system can be inverted to recover the $J_t$-vector of mean utilities (see The BLP Contraction Mapping). Under an additivity assumption the recovered ${\delta }_{jt}$ is written as a linear index:

$${\delta }_{jt}({\mathcal{S}}_t,{\tilde{\theta }}_2)=[x_{jt},v_{jt}]\beta -\alpha p_{jt}+{\xi }_{jt},$$

with exogenous characteristics $x_{jt}$, exogenous demand-shifters $v_{jt}$, endogenous price $p_{jt}$ (coefficient $\alpha$), and structural unobservable ${\xi }_{jt}$ (the unobserved product quality). Special cases: plain logit has $D_t^{-1}=\log s_{jt}-\log s_{0t}$ (no nonlinear parameters); nested logit has $D_t^{-1}=\log s_{jt}-\log s_{0t}-\rho \log s_{j\mid ht}$.

Random coefficients nested logit (RCNL) ^rcnl

The RCNL of Brenkers & Verboven (2006) relaxes the i.i.d. assumption on ${ϵ}_{ijt}$ to a two-level nested logit, adding a within-nest correlation parameter $\rho$ so that ${\theta }_2\equiv [\alpha ,\rho ,{\tilde{\theta }}_2]$ and

$$U_{ijt}={\delta }_{jt}+{\mu }_{ijt}({\tilde{\theta }}_2)+{ϵ}_{ijt}(\rho ).$$

Shares now involve a consumer-specific inclusive value $I{V}_{iht}({\mathbf{\delta }}_t,{\mathbf{\mu }}_{it})=(1-\rho )\log {\sum }_{j\in J_{ht}}\exp \left(\frac{{\delta }_{jt}+{\mu }_{ijt}}{1-\rho }\right)$. RCNL is popular when the key substitution dimension is categorical (e.g. spirits, beer). It is harder to estimate because the share inversion is no longer a plain contraction and slows as $\rho \to 1$ (see The BLP Contraction Mapping).

Examples

In the cereal application, let the one nonlinear characteristic be sugar content $x_{jt}$ with ${\mu }_{ijt}={\sigma }_x{x}_{jt}{\nu }_{it}$, ${\nu }_{it}\sim N(0,1)$. A consumer with high ${\nu }_{it}$ strongly prefers sugary cereals; when a sugary brand’s price rises she substitutes mainly toward other sugary brands, not toward bran flakes — exactly the non-IIA substitution plain logit cannot produce. The parameter ${\sigma }_x$ (in ${\tilde{\theta }}_2$) governs how dispersed these tastes are; ${\sigma }_x=0$ collapses to plain logit.

Connections

• Discrete Choice Models — random-utility foundation; BLP aggregates individual choices to market shares.
• Market Share Models — RCL is a structural market-share model with micro-founded substitution.
• The BLP Contraction Mapping — how the share integrals are inverted to recover ${\mathbf{\delta }}_t$.
• GMM Estimation and Instruments for Price Endogeneity — uses the linear index in ${\delta }_{jt}$ to form moments.
• Numerical Integration and Optimization in PyBLP — how the share integral is approximated.

See Also

• BLP Demand Estimation - Overview
• Supply Side and Markups
• Functional Forms in Marketing
• _Index