Theory | blprs

Overview

Berry–Levinsohn–Pakes (BLP) demand estimation models differentiated products by combining aggregate market shares with heterogeneous consumer preferences. The blprs crate keeps the standard BLP structure while exposing ergonomic Rust abstractions for each component: product data validation, integration over random coefficients, the contraction mapping that recovers mean utilities, and a two-step GMM estimator.

This page sketches the key identities that guide the implementation. Wherever possible we point to the corresponding blprs module so theory and code stay in sync.

Consumers and utility

Each consumer draw $i$ who observes product $j$ in market $t$ has indirect utility

[U_{ijt} = \delta_{jt} + \mu_{ijt}(\theta_2, \nu_i) + \varepsilon_{ijt},]

where:

$\delta_{jt}$ is the mean utility of product $j$ in market $t$.
$\mu_{ijt}$ captures deviations from the market mean via random coefficients with parameters $\theta_2$ (called sigma in the library) and consumer draw $\nu_i$.
$\varepsilon_{ijt}$ follows an i.i.d. Type-I Extreme Value distribution.

In blprs, market-level primitives live in data::ProductData, and the heterogeneous draws $\nu_i$ are packaged by integration::SimulationDraws.

Market shares

Conditional on a draw $\nu_i$, the probability that the consumer purchases product $j$ in market $t$ is

[P_{ijt}(\delta, \theta_2, \nu_i) = \frac{\exp\left(\delta_{jt} + \mu_{ijt}\right)}{1 + \sum_{k \in \mathcal{J}t} \exp\left(\delta{kt} + \mu_{ikt}\right)}.]

Aggregate shares integrate these probabilities over the distribution of draws:

[\hat{s}{jt}(\delta, \theta_2) = \int P{ijt}(\delta, \theta_2, \nu) \, dF(\nu) \approx \sum_{i=1}^I w_i P_{ijt}(\delta, \theta_2, \nu_i),]

with weights $w_i$ that sum to one. The approximation above is exactly what demand::predict_shares computes using the draws supplied to the BlpProblem.

Contraction mapping

Observed shares $s_{jt}$ give us equilibrium conditions that pin down mean utilities. BLP proposes iterating on

[\delta^{(m+1)}{jt} = \delta^{(m)}{jt} + \ln s_{jt} - \ln \hat{s}_{jt}(\delta^{(m)}, \theta_2).]

Under standard assumptions this fixed-point iteration converges to the unique vector $\delta(\theta_2)$ that rationalizes the data. In blprs, demand::solve_delta implements this contraction with configurable tolerance, maximum iterations, damping, and minimum share guards exposed through ContractionOptions. Convergence diagnostics are returned as a ContractionSummary.

Linear parameters and unobserved quality

Once we obtain $\delta$, linear taste parameters $\beta$ follow from the moment condition

[E[Z_t^\top \xi_t] = 0, \qquad \xi_t = \delta_t - X_{1t} \beta,]

where $X_{1t}$ are the observed linear characteristics and $Z_t$ are instruments. blprs solves for $\beta$ via two-stage least squares inside estimation::compute_linear_parameters (invoked internally by BlpProblem::estimate). The resulting structural shocks $\xi$ are exposed to users so supply-side models or diagnostics can be layered on later.

GMM objective

The demand-side estimator minimizes the quadratic form

[Q(\theta_2) = g(\theta_2)^\top W g(\theta_2), \qquad g(\theta_2) = \frac{1}{T} \sum_{t} Z_t^\top \xi_t(\theta_2),]

with $W$ a positive-definite weighting matrix. The current implementation offers an inverse $Z^\top Z$ weighting by default and accepts custom positive-definite matrices through WeightingMatrix. Future releases will incorporate heteroskedasticity-robust and optimal weighting strategies.

Connecting theory to the codebase

Theory object	Library representation
Product characteristics $(X_1, X_2)$	`ProductData::x1`, `ProductData::x2`
Market shares $s_{jt}$	`ProductData::shares`
Draws $\nu_i$, weights $w_i$	`SimulationDraws`
Mean utility $\delta$	Output of `solve_delta`, stored in `EstimationResult::delta`
Structural error $\xi$	`EstimationResult::xi`
GMM objective $Q$	`EstimationResult::gmm_value`

The guides page tracks upcoming additions such as supply-side pricing, optimal instruments, and counterfactual engines.