The Slutsky matrix and homogeneity in intertemporal consumer theory

A dual vista of an intertemporal model of the consumer yields insights into its fundamental qualitative structure that previous research was heretofore unable to uncover. For example, the intertemporal Slutsky matrix and its properties of symmetry and negative semidefiniteness apply to the integral of the discounted open-loop demand and supply functions, not to the instantaneous open-loop demand and supply functions. The results are established in a general dynamic model of the consumer which accounts for durable and nondurable goods, installation costs for the durable goods, and a leisure/labor choice.     

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.     

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.     

A discrete characterization of Slutsky symmetry

Summary A smooth demand function is generated by utility maximization if and only if its Slutsky matrix is symmetric and negative semidefinite. Slutsky symmetry is equivalent to absence of smooth revealed preference cycles, cf. Hurwicz and Richter (Econometrica 1979). To observe such a cycle would require a continuum of data. We characterize Slutsky symmetry by means of discrete antisymmetric revealed preference cycles consisting of either three or four observations.The first author's research was supported in part by NSF grant DMS-9401355, and the second author's by the Deutsche Forschungsgemeinschaft, SFB 303, and by a SUNY Faculty Research award.     

Comparative Statics of Money-goods Specifications of the Utility Function

The introduction of real-cash balances into the neoclassical model of the consumer wrecks havoc, in general, on the empirically observable refutable comparative statics properties of the model. We provide the most general solution of this problem to date by deriving a symmetric and negative semidefinite generalized Slutsky matrix that is empirically observable and which contains all other such comparative statics results as a special case. In addition, we clarify and correct two aspects of Samuelson and Sato's (1984) treatment of this problem. Received March 9, 2001; revised version received September 3, 2001     

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 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.     

When is a tax neutral?

A proportional income tax is said to be neutral if variations in the tax rate do not affect the optimal composition of the consumption commodities of a utility maximizing consumer. It is shown that neutrality obtains if and only if the indirect utility function of the consumer is groupwise homothetic. If the income tax rates on labor and property incomes are allowed to differ, then neutrality obtains if and only if the indirect utility function of the consumer is homothetically separable. If, in addition, it is required that the proportional income tax be neutral with respect to not only consumption commodities but also to leisure, then 'the indirect utility function must have the form: $\text{V = V(f(w)+H}{}^1\text{(p))}$, where w and p are the normalized prices of leisure and consumption respectively, and H¹(p) is homogeneous of degree one in p.     

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.     

Surplus measures and luenberger Hicks–Moorsteen productivity indicator

This paper extends to production theory some notions of compensated and equivalent variation analyzed in Luenberger (Econ Theory 7:445–462, 1996) in a consumer context. Along this line the Luenberger–Hicks–Moorsteen productivity indicator introduced in Briec and Kerstens (Econ Theory 23:925–939, 2004) is derived from these concepts for multi-output production technologies. The dual properties of this productivity indicator are analyzed and an aggregate indicator is introduced inspired from the resource function proposed in Luenberger (1996). A connection to a suitable Slutsky matrix is established.     

The implications of a third-order translog demand system and some empirical results

In this paper a homogeneity and symmetry unconstrained variant of the “third-order translog”, representing a third-order approximation to an indirect utility function, is derived. Using Christensen, Jorgenson and Lau's (1975) data, and subsequent observations to 1986, we estimate this third-order translog, and show that it dominates the homogeneity and symmetry unconstrained translog. The restrictions implied by homogeneity and symmetry, conditional on our third-order translog are also rejected. These results are of interest since they demonstrate that standard restrictions implied by the theory of consumer behaviour can be rejected even in the context of a fairly general model which has been estimated using aggregate, time-series data. However, it has been found that the inclusion of dynamic effects sometimes leads to non-rejection of theoretical restrictions (Anderson and Blundell 1983). Unfortunately, the use of aggregate, time-series data to estimate a dynamic, third-order translog would be problematic, since exact aggregation restrictions are imposeda priori and observations are limited. This suggests that it would be more appropriate to estimate the third-order translog using cross-sectional micro-data sets with many observations. This type of data is now available, and can be used to estimate systems of consumer demands. Furthermore, these data sets contain extensive information on demographic characteristics which, having recently been found to be important determinants of demand (Barnes and Gillingham 1984) can be included in general demand systems of the third-order translog type.     

A Monte Carlo test of preference independence

Theil (1987) recently developed distribution-free Monte Carlo testing procedures for demand homogeneity and Slutsky symmetry which do not use asymptotic theory. In this paper we introduce a similar procedure to test preference independence. The application of this procedure is illustrated with Dutch data.     

The α-MEU model: A comment

In Ghirardato et al. (2004) [7], Ghirardato, Macheroni and Marinacci propose a method for distinguishing between perceived ambiguity and the decision-maker?s reaction to it. They study a general class of preferences which they refer to as invariant biseparable. This class includes CEU and MEU. They axiomatize a subclass of α-MEU preferences. If attention is restricted to finite state spaces, we show that any α-MEU preference relation, satisfies GMM?s axioms if and only ifα=0 or 1, that is, the preferences must be either maxmin or maxmax. We show by example that these axioms may be satisfied when the state space is [0,1].     

On the characterization of excess demand functions

Summary. In this paper, we give the necessary and sufficient conditions that characterize the individual excess demand function when it depends smoothly on prices and endowments. A given function is an excess demand function if and only if it satisfies, in addition to Walras law and zero homogeneity in prices, a set of first order partial differential equations, its substitution matrix is symmetric and negative semidefinite. Moreover, we show that these conditions are equivalent to the symmetry and negative semidefiniteness of Slutsky matrix, Walras law and zero homogeneity of Marshallian demand functions.Received: 25 November 2002, Revised: 11 March 2004, JEL Classification Numbers: D11.Marwan Aloqeili: I would like to thank an anonymous referee for helpful comments.     

An almost ideal demand system for alcoholic beverages in British Columbia

An almost ideal demand system for alcoholic beverages in British Columbia is estimated based on five beverage categories. Estimates of the model unrestricted and restricted to satisfy homogeneity and symmetry are presented. The restrictions are tested: as is common in applied demand analysis a number of rejections are encountered, although within-equation tests tend to support homogeneity. The rejections which are encountered are not mitigated by the inclusion of dynamic elements. The Slutsky matrix is used to examine the concavity of the expenditure function, which is found to be mildly violated. Marshallian and Hicksian own-, cross-price, and income elasticities are calculated and are found to be largely consistent with previous findings, although some noteworthy results are obtained.     

A note on the underestimation and overestimation in stochastic input-output models

Here we show that (i) the Leontief inverse (I – A)^-1 is underestimated when the elements of A, the fixed coefficient input–output matrix, are biproportionally stochastic, and (ii) the output vector in a non-linear input–output system is overestimated when the final demand vector is stochastic.     

Optimal Intertemporal Consumption under Uncertainty

We analyze the optimal consumption program of an infinitely lived consumer who maximizes the discounted sum of utilities subject to a sequence of budget constraints where both the interest rate and his income are stochastic. We show that if the income and interest rate processes are sufficiently stochastic and the long run average rate of interest is greater than or equal to the discount rate, then consumption eventually grows without bound with probability one. We also establish conditions under which the borrowing constraints must be binding and examine how the income process affects the optimal consumption program. Journal of Economic Literature Classification Number: D91.     

Conservative belief and rationality

Playersʼ beliefs may be incompatible, in the sense that player i can assign probability 1 to an event E to which player j assigns probability 0. One way to block incompatibility is to assume a common prior. We consider here a different approach: we require playersʼ beliefs to be conservative, in the sense that all players must ascribe the actual world positive probability. We show that common conservative belief of rationality (CCBR) characterizes strategies in the support of a subjective correlated equilibrium where all playersʼ beliefs have common support. We also define a notion of strong rationalizability, and show that it is characterized by CCBR.     

The communication requirements of efficient allocations and supporting prices

We show that any communication finding a value-maximizing allocation in a private-information economy must also discover supporting prices (in general personalized and nonlinear). In particular, to allocate L indivisible items between two agents, a price must be revealed for each of the 2^L-1 bundles. We prove that all monotonic prices for an agent must be used, hence exponential communication in L is needed. Furthermore, exponential communication is needed just to ensure a higher share of surplus than that realized by auctioning all items as a bundle, or even a higher expected surplus (for some probability distribution over valuations). When the utilities are submodular, efficiency still requires exponential communication (and fully polynomial approximation is impossible). When the items are identical, arbitrarily good approximation is obtained with exponentially less communication than exact efficiency.     

Test for the real option in consumer behavior

Consumers become indecisive when facing too many choices. Economic analysis suggests that when a decision involves uncertain outcome, can be delayed and is irreversible, there will be a real option in the cost–benefit analysis. For example, the option to keep alive a consumer's purchasing decision has a significant value. It allows the consumer to take advantage of any future advantageous deals while avoiding the bad choices. This renders the consumer more hesitant. When a consumer decides to exercise his buying decision, he demands a compensation for the loss of this option. Hence, the benefits of a purchase must be over and above its costs by a wide margin (the option value). Data from a survey at a Turkish university on hypothetical purchase decisions confirmed the existence of this real option. We conclude with marketing policy recommendations and future research directions. Connection to the Prospect Theory is briefly explored.Note: Although the 3rd person singular pronoun he/his was used throughout to describe the consumer, he was intended to be gender-neutral.