Aggregation and pairwise aggregation of demand when the distribution of income is fixed

This paper derives necessary and sufficient conditions for “pairwise aggregation” of the demand functions of a group of consumers (conditions under which the mean demand for each pair of consumers satisfies the Slutsky restrictions) when the distribution of income is fixed. The sufficient conditions imply existence of a “representative” competitive consumer whose demand is the mean demand of the group. The necessary conditions imply that such a representative consumer exists for every fixed income distribution only if the consumers have homothetic preferences or, alternatively, if for each price vector, all the consumers' income expansion paths lie in the same plane.     

Aggregation and the existence of a social utility function

In a well-known paper Gorman (Econometrica21 (1953)) established that the necessary and sufficient condition for the existence of an aggregate, or social, utility function, independent of the distribution of income, is that all individuals' income consumption paths be parallel straight lines. Recently Chipman (J. Econ. Theory8 (1974)), building on the paper of Hurwicz and Uzawa (in “Preference Utility and Demand”) has shown that if the distribution of income is proportional and individual preferences are homothetic, aggregate consumption behavior obeys the necessary integrability conditions. It is shown here that the consistency of aggregate behavior can be derived from more general conditions than the ones used by Chipman and Gorman. Examples of demand systems from which aggregate behavior implies a social utility function are provided. It is then shown that if individual demand functions are linear in income—a form employed by both Gorman and Chipman—it is not necessary that the distribution of income be fixed.     

Social welfare and the unrepresentative representative consumer

If, for all prices, income distribution is optimal for a planner with a social welfare function, then aggregate demand is the same as that of a single “representative consumer” whose preferences over aggregate consumption are the same as the planner's. This paper shows that the converse is false. Aggregate demand may be the demand function of a representative consumer although the income distribution is not optimal for any social welfare function. The representative consumer may be Pareto inconsistent, preferring situation A to B when all the actual consumers prefer B to A. We give conditions under which existence of a representative consumer implies that the income distribution satisfies first order conditions for optimality. Satisfying the first order optimality conditions for an additively separable social welfare function is essentially equivalent to aggregate demand for every pair of consumers having a symmetric Slutsky matrix.     

Exact aggregation of consumer demand systems

This paper is divided into two parts which deal with closely connected issues. The first section of the paper explores the structure of consumer demand systems necessary and sufficient for exact aggregation. The second section addresses a related empirical question: what, if anything, do the restrictions imposed on exactly aggregable demand systems buy the econometrician engaged in estimating integrable consumer demands? In particular, if the objective of an empirical exercise is to estimate the demand systems of individual utility maximizing consumers and only aggregate expenditure information and information on the income and demographic composition of the population are available, then under what conditions can the parameters of the estimated aggregate expenditure system be used to uniquely identify the parameters of the underlying individual demand systems?     

The econometric modelling of aggregate consumer behaviour

This paper examines several problems involved in modelling the structure of consumer behaviour. A new model of consumer demand is presented which encompasses the indirect translog, the CES and the LES as special cases, enabling nested comparisons to be made. Within this framework we show that acceptance of zero-degree homogeneity and/or symmetry of the Slutsky matrix is heavily dependent upon the structure within which inferential tests are carried out, and that great care must be utilised in formulating nested hypotheses and drawing inferences on the basis of χ² and F-statistics. We then show that conventional econometric work implies that consumer preferences must be homothetic if symmetry of the Slutsky matrix is imposed. An alternative procedure is suggested and utilised. Finally, we suggest that low Durbin–Watson statistics may arise from misspecification of functional form and may not be due solely to omitted dynamics, as many conjecture.     

Aggregation for material balances

In an input-output system, let final demands and gross outputs be iteratively balanced by successive approximations. The speed of convergence will depend, among other things, on the initial choice of gross outputs. Suppose that, using some aggregation weights, aggregate supply is made equal to aggregate demand in the initial plan. The current paper finds the set of aggregation weights that yields speediest convergence. An economic interpretation of the “optimal” aggregation weights is given, and some examples are calculated. J. Comp. Econ., March 1978, 2(1), pp. 1–11. Boston University, Boston, Mass., and Massachusetts Institute of Technology, Cambridge, Mass.     

On some global properties of Gorman class demand systems

This letter explores some global properties of the Gorman class of demand functions. We find that generalizing the homothetic preferences case to non-homotheticity gives rise to an unexpected global problem. In particular, interaction of the bounded budget share condition with either Slutsky symmetry or with homogeneity reduces the set of admissible Gorman class demand functions to but one — the polar form.     

Measures of inequality as an aggregation of individual preferences about income distribution: The arrowian case

An inequality preorder is a preorder on a simplex which satisfies symmetry and strict Schur-convexity (the mathematical equivalent of the principle of transfers of Pigou and Dalton). It is shown that we cannot aggregate individual inequality preorders to a collective one if we are interested in Arrow's aggregation rules. The proof uses an interesting result of Kalai, Muller and Satterthwaite (Public Choice 34 (1979), 87–97). Moreover, we prove further results for the aggregation of individual inequality indices when we allow cardinality and interpersonal comparibility of utility.     

How serious is the aggregation problem? An empirical illustration

Heterogeneity in consumer behaviour may create problems with aggregation across consumers. If so, we may not be able to make correct inferences about behaviour based on aggregated data. However, using micro estimates to predict aggregate demand responses to policy changes may also create a bias if not aggregated properly. This may sound like a Catch 22 situation, but it is not, as it is possible to calculate both micro and aggregate demand responses based on microdata. The size of the aggregation bias is an empirical question. In this article, we show how to calculate theoretically consistent aggregate demand responses. We use both micro and macro data for Norwegian household electricity consumption to illustrate the magnitude and direction of different aggregation biases. We find considerable aggregation biases, in particular, when estimating with macro data.     

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     

Empirical regularities in South African consumption patterns

This article investigates a number of empirical regularities in the South African consumption patterns. The data support the following empirical regularities: (1) variability in consumption systematically exceeds the variability in prices; (2) law of demand; (3) income flexibility is about ?0.5; (4) Engel's law; and (5) the demand hypotheses, demand homogeneity and Slutsky symmetry are acceptable. In contrast to the findings for a number of other countries, another important empirical regularity that consumer's utility function is additive is rejected for the South African consumers. Based on the implied demand elasticity estimates from the preferred model, it is found that food, housing and medical care are necessities, and clothing, furniture, transport and recreation are luxuries and demand for all the commodities are price inelastic.     

Demand Aggregation and the Weak Axiom of Stochastic Revealed Preference

We address the problem of aggregating demand across a group of consumers, who are identical in terms of wealth and face identical price vectors, but vary in their chosen consumption bundles. We show that, when a stochastic demand function is constructed to aggregate a number of deterministic demand functions, satisfaction of the weak axiom of stochastic revealed preference by this stochastic demand function is weaker than the restriction that every underlying deterministic demand function satisfy Samuelson's weak axiom of revealed preference. Journal of Economic Literature Classification Number: D11.     

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.     

When do market games have transferable utility?

A pure exchange economy generates a “market game” in which the allocations achievable by any coalition are determined by the initial endowments of its members. Subject to certain regularity conditions, it is shown that for a market game it is possible to find utility representations for each consumer so that the game can be treated as a game with transferable utility if and only if indirect utility of all consumers can be represented in the Gorman polar form. This is the class for which aggregate demand behaves as if it were the demand of a single consumer.     

Location models as models of product choice

Markets in which firm choice of product is an economic decision are commonly analyzed using a variant of the location model pioneered by Hotelling (Econ. J.39 (1929), 41–57). This paper develops a framework for formalizing the idea that a set of consumers with preferences defined over an abstract set of commodities can be represented as if they behaved as is typically assumed in the goods-as-locations models. With this framework in place, we are able to derive the preference assumptions underlying the two most common versions of these models, and also to uncover the foundations of the now well-known principle of minimum differentiation.     

A general theory of demand for information about product quality

The purpose of this paper is to construct a general theory—analogous to the neoclassical theory of consumer demand—of demand for information about product quality. The model proposed here introduces uncertain product quality by assuming that commodities possess attributes which consumers desire and that an uncertain commodity possesses are unknown quantity of some attribute. It is assumed that information about the quality of uncertain products is available. The consumer's utility function of information is derived and his information demand function is obtained.Partial analogues of the neoclassical theorems are proved for information demand. Roughly, we find that the Slutsky matrix is symmetric and that a submatrix is negative definite. The negative definite submatrix contains those terms which measure the effect of (income compensated) changes in information prices on information demand.The analysis employed to obtain these results parallels—with some important modifications—the neoclassical analysis of commodity demand. The method used to obtain expressions for the Slutsky substitution terms is an extension of the approach introduced by McKenzie [10] and later used by Hurwicz and Uzawa [6]. This approach uses an “income compensation” function to arrive at an “income compensated” demand function.     

Hedonic prices and cost/benefit analysis

Hedonic prices have been used to evaluate the willingness to pay for attributes. We reformulate the notion of hedonic price from a composite price on housing to a unit price on traded quantities, in conformity with long run competitive equilibrium theory. This formulation was suggested (but not developed) by Rosen (J. Polit. Econ.82, No. 1 (1974), 34–35). By first characterizing an efficient allocation of consumers to space, we show that hedonic unit prices can be understood as a bid-rent function which supports the efficient allocation. This is despite the fact that the lots over which consumers bid are themselves endogenous. We show that unit hedonic prices reveal preferences in a manner different from composite hedonic expenditures.     

‘Double dipping’: the combined effects of social security and civil service pensions on employee retirement

How do macro variables such as aggregate consumption, aggregate money demand, prices, and interest rates vary in response to government attempts to redistribute income across agents through selective tax-transfer policies? This question is investigated for an overlapping generations model consisting of heterogeneously endowed three-period lived agents. In the presumably most favorable case for invariance (identical log-linear utility functions), it is shown that positive interest rate macro equilibria are invariant with respect to redistributions in social security benefits if and only if all agents initially exhibit qualitatively identical liquidity preference behavior, i.e. positive money holding only in youth, only in middle-age, or never.     

To Commit or Not to Commit: Endogenous Timing in Experimental Duopoly Markets

In this paper we experimentally investigate the extended game with action commitment of Hamilton and Slutsky (1990, Games Econ. Behavior2, 29–46). In their duopoly game firms can choose their quantities in one of two periods before the market clears. If a firm commits to a quantity in period 1, it does not know whether the other firm also commits early. By waiting until period 2, a firm can observe the other firm's period-1 action. Hamilton and Slutsky predicted the emergence of endogenous Stackelberg leadership. Our data, however, do not confirm the theory. While Stackelberg equilibria are extremely rare, we often observe endogenous Cournot outcomes and sometimes collusive play. This is partly driven by the fact that endogenous Stackelberg followers learn to behave in a reciprocal fashion over time, i.e., they learn to reward cooperation and to punish exploitation. Journal of Economic Literature Classification Numbers: C72, C92, D43.     

Continuous utility functions for noninferior demand functions

Summary It is shown that if a demand function with no inferior goods satisfies the Slutsky conditions and has a convex range, then it is generated by a continuous utility function. The same conclusion holds when the Slutsky conditions are replaced by the strong axiom of revealed preference.