BLP Demand Estimation - Overview

Summary

The Berry, Levinsohn, and Pakes (1995) model is the workhorse estimator for differentiated-products demand. It specifies a random-coefficients (mixed) logit demand system that allows flexible substitution patterns, addresses price endogeneity via instruments, and can be paired with a supply side (Bertrand-Nash markups) to recover marginal costs. Conlon & Gortmaker (2020) review the modern literature, derive a slightly different formulation amenable to fixed effects and optimal instruments, and collect concrete best practices implemented as defaults in the PyBLP Python package.

Overview

Empirical supply-and-demand models for differentiated products are central to the New Empirical Industrial Organization (NEIO) literature, used for merger evaluation, valuing new goods, and studying two-sided markets. The BLP approach scales to many products and uses both aggregate and disaggregate data.

The model is “simple to understand, challenging to estimate.” At its core it is a nonlinear change of variables from observed market shares ${\mathcal{S}}_t$ to mean utilities ${\mathbf{\delta }}_t$. After this change of variables, BLP reduces to a linear IV regression (demand alone) or a two-equation linear IV problem (supply and demand). The difficulty is that the parameters ${\theta }_2$ governing the nonlinear change are unknown, producing a non-linear, non-convex GMM optimization with a simulated objective.

The paper organizes its contribution around the major tasks of the BLP estimator: solving the fixed point (share inversion), optimization, numerical integration, instrument construction, and solving counterfactual pricing equilibria. Its headline empirical findings (via Monte Carlo) differ from prior literature: multiple local optima appear rare in well-identified problems, and good finite-sample performance is achievable even in small samples, especially when optimal instruments are used together with supply-side restrictions.

Main Content

The BLP problem at a glance ^blp-pipeline

1. Demand utility (random-coefficients logit): $U_{ijt}={\delta }_{jt}+{\mu }_{ijt}+{ϵ}_{ijt}$ — see Random Coefficients Logit Model.
2. Share inversion: invert observed shares to mean utilities, ${\mathbf{\delta }}_t\equiv D_t^{-1}({\mathcal{S}}_t,{\tilde{\theta }}_2)$, via the BLP contraction mapping.
3. Linear index: ${\delta }_{jt}=[x_{jt},v_{jt}]\beta -\alpha p_{jt}+{\xi }_{jt}$, with structural error ${\xi }_{jt}$.
4. GMM moments: $E[{\xi }_{jt}{Z}_{jt}^D]=0$ using instruments for endogenous price — see GMM Estimation and Instruments for Price Endogeneity.
5. Optional supply side: Bertrand-Nash FOCs give markups, recover marginal cost $c_{jt}=p_{jt}-{\eta }_{jt}$, add supply moments $E[{\omega }_{jt}{Z}_{jt}^S]=0$ — see Supply Side and Markups.
6. Estimation: nested fixed-point (NFXP) GMM with numerical integration and gradient-based optimization — see Numerical Integration and Optimization in PyBLP.

Parameter partition ^theta-partition

The parameter vector $\theta$ is split into three parts:

• ${\theta }_1$ ($K_1\times 1$): linear demand parameters $\beta$.
• ${\theta }_3$ ($K_3\times 1$): linear supply parameters $\gamma$.
• ${\theta }_2$ ($K_2\times 1$): the nonlinear parameters — the price coefficient $\alpha$ and the parameters ${\tilde{\theta }}_2$ governing heterogeneous tastes. These are common to both supply and demand and govern the endogenous objects.

The NFXP algorithm concentrates out the linear $[{\theta }_1,{\theta }_3]$ and searches only over the $K_2$ nonlinear parameters ${\theta }_2$, so the Hessian is only $K_2\times K_2$ and large numbers of (linear) fixed effects are “essentially free.”

What is novel in Conlon & Gortmaker (2020) ^novelty

• A reformulation of the BLP problem (placing $\alpha p_{jt}$ on the LHS of the linear system) that supports simultaneous supply and demand with high-dimensional fixed effects and analytic gradients.
• A characterization of the feasible approximation to optimal instruments (Amemiya 1977 / Chamberlain 1987) that makes exclusion and cross-equation restrictions explicit, paralleling Berry & Haile (2014).
• Concrete, benchmarked best practices: SQUAREM / Levenberg-Marquardt for the inner loop, Gauss-Hermite product rules (sparse grids / scrambled Halton in high dimensions) for integration, gradient-based optimization with box constraints and tight tolerances, and the log-sum-exp trick for numerical stability.

Examples

A canonical setup is the automobile or ready-to-eat cereal application (e.g., BLP 1995; Nevo 2000b/2001):

• Markets $t$ are model-years or city-quarters; products $j$ are car models or cereal brands; the outside good $j=0$ is “buy nothing.”
• Characteristics $x_{jt}$: horsepower, fuel economy, size (cars), or sugar/brand dummies (cereal); price $p_{jt}$ is endogenous.
• A random coefficient on a characteristic (e.g. ${\mu }_{ijt}={\sigma }_x{x}_{jt}{\nu }_{it}$) generates realistic substitution: consumers who like big cars substitute toward other big cars when one’s price rises, breaking IIA.
• The Conlon-Gortmaker Monte Carlo baseline: $T=20$ markets, firms $F_t\in \{2,5,10\}$, products per firm $J_{ft}\in \{3,4,5\}$, structural errors with ${\sigma }_{\xi }^2={\sigma }_{\omega }^2=0.2$, ${\sigma }_{\xi \omega }=0.1$; demand parameters $[{\beta }_0,{\beta }_x,\alpha ]=[-7,6,-1]$, ${\sigma }_x=3$; outside shares $0.8<s_{0t}<0.9$.

Connections

• Discrete Choice Models — BLP is the aggregate-data extension of random-utility discrete choice.
• Market Share Models — BLP is a structural, micro-founded market-share model.
• Method of Simulated Moments — the simulated GMM objective is closely related to MSM (integrals approximated by simulation).
• Instrumental Variables — after share inversion, BLP is a (nonlinear) IV/GMM problem.
• Parameter Estimation in Market Response — situates BLP within structural estimation of demand.

See Also

• Random Coefficients Logit Model
• The BLP Contraction Mapping
• GMM Estimation and Instruments for Price Endogeneity
• Numerical Integration and Optimization in PyBLP
• Supply Side and Markups
• _Index