The Rank of Demand Systems: Theory and Nonparametric Estimation

Abstract

THE RANK O F DEMAND SYSTEMS: THEORY AND NONPARAMETRIC ESTIMATION' Gorman's (1981) concept of Engel curve "rank" is extended to apply to any demand system. Rank is shown to have implications for specification, separability, and aggregation of demands. A simple nonparametric test of rank using Engel curve data is described and applied to U.S. and U.K. consumer survey data. The test employs a new general method for testing the rank of estimated matrices. The results are used to assess theoretical and empirical aggregation error in representative consumer models, and to explain a representative consumer paradox.

Local rank tests in a multivariate nonparametric relationship

“…For example, Gorman (1981) showed that commonly used exactly aggregable demand systems, when derived through a utility maximization principle, have always rank less than or equal to 3. Lewbel (1991) showed that ranks have important implications on functional structure and aggregation of demand systems. Some further theoretical studies related to ranks can be found in Lewbel (1989), Russell and Farris (1993), and Lewbel and Perraudin (1995).…”

“…, z J ) are prices of J goods faced by a consumer. Introduced by Gorman (1981) and later developed by Lewbel (1991), the rank of a demand system can be either local or global. The local rank rk{f (·, z)} at a fixed value of z is defined as the dimension of the function space spanned by the coordinate functions f 1 (x, z), .…”

Factor-GMM estimation with large sets of possibly weak instruments

“…Underlying the factor model is the idea that a large number of economic variables can be adequately modeled by a small number of indicator variables or shocks. Factor analysis has been used fruitfully to model, among other cases, asset returns, macroeconomic aggregates and Engel curves (see, e.g., Stock and Watson (1989), Lewbel (1991) and others).…”

A class of demand systems satisfying global regularity and having complete rank flexibility

“…Gorman (1981) defined the rank of a demand system as the dimension of the space spanned by its Engel curves. Lewbel (1991) extended the definition of rank to non-aggregable systems and also showed that rank is equivalent to the minimum number of price indices in the indirect utility function. Thus, the Cobb-Douglas system is rank one, while, as pointed out above, AIDS is rank two and QUAIDS is rank three.…”

“…This class of demand systems follow the steps of the almost ideal demand system (AIDS) due to Deaton and Muellbauer (1980a), the quadratic almost ideal demand system (QUAIDS) due to Banks et al (1997), the Modified almost ideal demand system (MAIDS) due to Cooper and McLaren (1992) and Cooper and McLaren (1996) , and a more recent rank-four demand system due to Lewbel (2003). Members from this class can be specified to acquire as large a rank as required for empirical work, following the definition of rank, due to Lewbel (1991) which generalizes Gorman's rank to all demand systems. They also exhibit a clear and valid homothetic asymptotic behaviour, as income approaches infinity.…”