Consumer Demand with Price Aggregators and Low-Rank Cross-Price Effects
Estimating demand for many goods usually means estimating many cross-price effects. We show that low-rank cross-price structure is equivalent to a small number of "price aggregators," and characterize the preferences behind them. Download Paper Abstract Estimating consumer demands is a bread-and-butter undertaking in applied economics. In general, demand for each good depends on the prices of all goods and services, but for most applications it is impractical to estimate models of such high dimension. In this paper, we consider consumer demand with a low rank of the matrix of cross-price effects, a property implicitly assumed in most empirical settings. First, we show that imposing a low rank is equivalent to introducing functions that we call “aggregators”, where each aggregator maps information from an arbitrarily large vector of prices (and perhaps income) into a scalar. We then provide a complete characterization of the preferences that rationalize demand systems with such aggregators. These results can be used to derive new and flexible forms of demand that can be tailored to applications in various fields of economics. Most commonly-used demand systems (including directly-additive, indirectly-additive, non-homothetic CES and Kimball preferences) can be described with one or two aggregators where the price index may coincide with one of the aggregators. Nested and mixed logit require as many aggregators as nests or consumer types. Aggregators can also be naturally expressed as a function of observed product attributes. Using barcode data on purchases of ready-to-eat cereals, we illustrate how to estimate a simple yet flexible specification of such a demand system with K aggregators, with or without using information on product attributes. ...