Exploring the effects of aggregation error in the estimation of consumer demand elasticities

Errors introduced by using aggregate data in estimating a consumer demand model have long been a concern. We study the effects of such errors on elasticity estimates derived from AIDS and QUAIDS models. Based on a survey of published articles, a generic parameterization of the income distribution, and the range of Gini coefficients reported for 28 OECD countries, we generate and analyze a large number of “observations” on the differences between elasticities calculated at the aggregate level and those calculated at the micro level. We suggest a procedure for evaluating the likely range of aggregation error when a model is estimated with aggregate data.     

Approximately rational consumer demand

Summary We define measures of violations of Slutsky symmetry and negative semidefiniteness and relate them to measures of revealed preference inconsistencies exhibited by nonoptimizing demand behavior. The degree of Slutsky asymmetry is shown to restrict the rate at which real income can rise everywhere along smooth loops in income and price space. The largest eigenvalue of the quadratic form of the Slutsky matrix is used to bound violations of the weak axiom. The sizes of the violations of either Slutsky condition are used to bound the distance between the given demand function and approximating functions that satisfy that Slutsky condition exactly.This paper was originally titled Approximate Slutsky Conditions. The authors thank W. Hildenbrand, L. Hurwicz, C. Le Van, W. Neuefeind, R. Pollak, M. K. Richter, W. Shafer and T. M. Stoker for helpful discussions and suggestions. The first author is an Alfred P. Sloan Foundation research fellow. His research was supported in part by NSF grant DMS 8514341 and a Presidential Young Investigator Award. This paper was completed while the first author was visiting the Mathematical Sciences Research Institute, Berkeley, and the second author was visiting the University of Bonn, supported by the Deutsche Forschungsgemeinschaft SFB 303 and by a SUNY Faculty Research Award. Both authors are grateful for their hosts' hospitality.     

Exact aggregation and estimation

In this paper we provide sufficient conditions, which, when combined with recent developments in the theory of exact aggregation, permit estimation of group specific demand systems when micro-level information is unavailable. We illustrate the approach by estimating group specific cost of living indices using only readily available census information and aggregate expenditure data.     

Errors of aggregation and errors of specification in a consumer demand model: a theoretical note

Abstract .  Consumer demand models based on the concept of a representative or average consumer suffer from aggregation error. Misspecification of the underlying micro utility-maximizing model, which is virtually inevitable, also results in error. This note provides a theoretical investigation of the relationship between the two types of error. Misspecified expenditure support functions for demand systems at the micro level induce the same misspecified structure in the corresponding expenditure functions at the macro level, and the errors at the two levels are shown to be of similar order.     

Estimating consumer demand in Korea

In this paper, time series annual data on five consumer goods for Korea are analyzed using the neoclassical model of consumer behavior. The approach taken is the indirect utility function and employs a translog form. Various restrictive specifications of consumer preferences with respect to homotheticity and separability are examined and rejected by the likelihood ratio test. Income and price elasticities are estimated, which appear to be generally high relative to those of developed countries. Growth in real income is primarily responsible for changes in commodity demand. Marginal utilities of income and associated income elasticities are calculated, which are consistent with normal expectations.     

The integrability of consumer demand functions

The purpose of this paper is to characterize the class of systems of consumer demand functions that are representable as ratios of first-order polynomial functions and are integrable. Starting from a general system of consumer demand functions representable as ratios, we impose successively the restrictions corresponding to homogeneity, summability, symmetry, non-negativity, and monotonicity. We find that the only such systems which are capable of modeling arbitrary own- and cross-substitution effects are the systems generated by transcendental logarithmic utility functions.     

A trigonometric flexible consumer demand system

Abstract.  This paper proposes the first ever empirical specification of a trigonometric demand system. The new model is potentially useful because of some attractive features. It is flexible, amenable to exact aggregation over consumers, possessed of trigonometric Engel curves, which can oscillate, and able to have an unusually large regular region. With comparisons between the new model and two other popular models, an illustration is given for Japanese demand for non‐durables and services. The new model shows relatively gentle Engel curves with an inflection point on each of them, which seem reasonable, given that aggregate expenditure is used in parameter estimation. JEL classification: C51, D12     

Intertemporal price discrimination and consumer demand

Under a particular class of utility functions, intertemporal price discrimination (IPD) is not feasible. That is, customers cannot be made to pay different prices for a durable good at different points in time. Other factors such as falling costs, and differing discount rates between buyers and sellers have been found to make intertemporal pricing schemes feasible, or even profitable. None of these factors, however, were fundamental demand differences which give rise to static price discriminations. In this paper we argue that IPD is indeed feasible and sometimes profitable, if only we allow for a nondurable good in the utility function. A simple additively separable utility is examined first, which is then extended to a nonseparable utility function which allows richer substitution/complementary relations between the durable and the nondurable goods. This may help us to better understand the similarities between static and intertemporal price discriminations.     

The negativity condition in consumer demand

The study stresses the importance of one relatively neglected restriction on demand equations: the negativity condition (i.e., the necessity of a negative semidefinite substitution matrix). That condition is implemented by using the Cholesky decomposition. It implies nonlinear combinations of the parameters which are estimated by a maximum likelihood procedure. The method, applied to Dutch and German data, leads to an acceptance of the theoretical restrictions stemming from consumer theory.     

Exact aggregation and Generalized Slutsky Conditions

This article indicates a natural connection between Generalized Slutsky Conditions and demand structures of the exact aggregation form. It is shown that Generalized Slutsky Conditions are assured when the number of consumers is greater than or equal to the number of goods, if and only if demands are of a generalized exact aggregation form. This result provides some understanding of the properties of aggregate demand and allowable differences between consumers, as well as naturally generalizing W. M. Gorman's (Econometrica21 (1953), 63–80) famous conditions for the integrability of aggregate demand.     

Quotas and the structure of consumer demand

In the direct utility function, quantities axe exogenous, while prices and total expenditure are endogenous. Consequently, this utility function is appropriate for analyzing the impact on prices and total expenditure from an exogenous change in the quantity of a commodity. Such an exogenous change in quantity could occur when quotas change. The direct translog utility function is estimated for a four commodity breakdown of U.S. expenditure. Estimates of quantity elasticities of price and expenditure indicate that domestically produced non-durables are necessities. As a result, the imposition of quotas on these goods will be particularly deleterious to lower income consumers.     

An intertemporal system of dynamic consumer demand functions

This paper constructs and estimates a system of dynamic consumer demand equations under the assumption of rational expectations about anticipated human wealth. The traditional one-period budget constraint is replaced by the lifetime anticipated wealth constraint. Lagged dependent variables are rationalized by an adjustment cost argument. In the model presented, both the dynamic adjustment coefficients as well as the parameters characterizing the underlying long-run preferences are identified and can be estimated. We find that a weak version of the REH cannot be rejected on the data. In common with most empirical studies on demand behaviour we have to reject the symmetry restrictions.     

A Bayesian approach to aggregation in demand systems: smoothing with perfect substitution priors

Demand analysis requires aggregation of commodities. Some are imposed at the data collection level, leaving some for the estimation level. When data are collected, the implicit assumption underlying the aggregation is perfect substitutability: one gallon of gasoline is viewed by consumers as equivalent to another gallon; hence, the two are added together. While such aggregation can be carried out further by the data analyst, it is difficult to incorporate perfect substitutability into the estimation of direct demand systems. Perfect substitution in that context implies discontinuous demand functions, which are not nested within standard empirical demand systems. Perfect substitution is much more easily handled in a system of inverse demands, though an empirical method to impose perfect substitutability in an inverse demand system has not previously appeared in the literature. In this article, we develop such a method, which allows perfect substitutability to be imposed as a prior restriction. We use Leamer’s information contract curve as a tool to flexibly impose the substitution restriction and to investigate consistency between the data and prior. We illustrate the method with an application to inverse demands for fish in Korea.     

Exact welfare measurement for double-log demand with partial adjustment

This paper demonstrates that a double-log demand with partial adjustment (DLPA) is consistent with the theory of consumer utility maximization. It offers an approach for calculating the compensating variation (CV), the exact welfare effect of a change in a price series when a DLPA is employed. Significant bias may result if the CV is based on a static double-log demand when a DLPA function is appropriate. We revisit a recent study of demand for gasoline in the U.S., finding that the CV based on the static double-log would overstate the welfare effect of a 6-month temporary gasoline tax by 7.5%.     

Implications of consumer heterogeneity in time-series estimates of US money demand

The money demand function has traditionally been estimated with income and interest rates, typically employing quite lengthy time series. Controversy, however, surrounds the importance of heterogeneous agents in monetary economics and throughout macroeconomics more generally. In particular, if proportions of agents with different traits (and hence, different money demands) are changing over time, ignoring those changes may bias estimated income and interest elasticities. This concern, as well as that of appropriate functional form, is explored here. Controlling for consumer heterogeneity has surprising effects on the estimated elasticities of traditional variables.