پیامدهای آزمونپذیر از نظریه تعادل عمومی: یک روش مشتق پذیر

Is general equilibrium theory empirically testable? Our perspective on this question differs from the standard, Sonnenschein–Debreu–Mantel (SDM) viewpoint. While the SDM tradition considers aggregate (excess) demand as a function of prices, we suppose that what is observable is the equilibrium price vector as a function of the fundamentals of the economy. We apply this perspective to an exchange economy where equilibrium prices and individual endowments are observable. We derive necessary and sufficient conditions that characterize the equilibrium prices, as functions of initial endowments. Furthermore, we show that, if these conditions are satisfied, then the economy can generically be identified. Finally, we show that when only aggregate data are available, observable restrictions vanish. We conclude that the availability of individual data is essential for the derivation of testable consequences of the general equilibrium construct.

Is general equilibrium theory empirically testable? This question has attracted considerable attention for at least 30 years; that is, at least since the statement of the “Sonnenschein problems”. In two seminal papers, Sonnenschein Sonnenschein, 1973 and Sonnenschein, 1974 posed the question whether the individualistic foundations of general equilibrium theory could generate non-trivial testable restrictions on the aggregate excess demand or market demand functions of an exchange economy. The case of excess demand was solved by Mantel (1974) and Debreu (1974); the market demand problem was solved by Andreu (1983) for finite sets of data, and, recently, by Chiappori and Ekeland (1999a) for analytic demand functions. In all cases, the answer is negative, provided there are enough individuals in the economy—a conclusion that confirmed Sonnenschein’s intuition and initial arguments. These (by now classical) results have been widely interpreted as pointing out a severe weakness of general equilibrium theory, namely its inability to generate empirically falsifiable predictions. A prominent illustration of this stand is provided for instance by Kenneth Arrow, who, in a recent survey, listed among the main developments of utility theory the result that in the aggregate, the hypothesis of rational behavior has in general no implications”, and drew the conclusion that “if agents are different in unspecifiable ways, then […] very few, if any, inferences can be made” ( Arrow, 1991, p. 201). The main claim of the present paper is that this view is overly pessimistic, and that general equilibrium theory can actually generate strong testable predictions, even for large economies. The main idea is in the line of recent contributions by Brown and Matzkin (1996) and Brown and Shannon (2000), and can be summarized as follows. The approach by Sonnenschein, Debreu and Mantel concentrates on the properties of excess (or market) demand as a function of prices only. There are, of course, deep theoretical reasons for the investigation of the structure of aggregate demand as a function of prices; for instance, the Sonnenschein–Debreu–Mantel (SDM) result has strong implications for the convergence of tâtonnement processes. However, this viewpoint is not the only possible one, and actually not the most adequate for assessing the testability of general equilibrium theory. As far as testable predictions are concerned, the structure of aggregate excess demand is not the relevant issue, if only because excess demand is, in principle, not observable, except at equilibrium prices—where, by definition, it vanishes. However, prices are not the only variables that can be observed to vary. Price movements reflect fluctuations of fundamentals, and the relationship between these fundamentals and the resulting equilibrium prices is the natural object for empirical observation. One of the goals of general equilibrium theory is precisely to characterize the properties of this relationship. As it turns out, this characterization generates strong testable restrictions. We develop our claim in the simple but natural context of an exchange economy, where excess demand depends on both prices and initial endowments. The equilibrium equations then relate prices to endowments; the equilibrium manifold is defined as the set of prices and endowments for which excess demand is zero. We are interested in the local structure of that manifold; that is, we study equilibrium prices, locally, as a smooth function of initial endowments. We derive two main results. First, there exist strong restrictions on the local structure of the equilibrium manifold. Some of these restrictions come from the individualism assumption (the aggregate demand arises as the sum of individual demands each of which is a function solely of prices and individual income), and others stem from the rationality assumption (each individual is a utility maximizer). In other words, although none of these assumptions constrains the shape of excess demand as a function of prices (the SDM conclusion), they do restrict the form of the equilibrium manifold, which is of empirical relevance. Second, and perhaps more surprisingly, we prove that, if income effects do not vanish, observing equilibrium prices as a function of initial endowment generically identifies the underlying economy, in the sense that individual preferences can be recovered without ambiguity. In a way, this result is the exact opposite of the SDM conclusion. In the SDM perspective, all the structure due to individual utility maximization is lost by aggregation. Adopting the equilibrium manifold perspective, we reach the opposite conclusion that all the relevant structure is generically preserved, in the sense that the initial economy can be recovered from the local structure of the equilibrium manifold. These results indicate that the two lines contrasted above—the ‘manifold’ point of view versus the SDM excess demand approach—generate different (and in a sense opposite) conclusions. How can this striking discrepancy be explained? Our interpretation emphasizes a crucial difference: in the manifold approach, individual data (initial endowments) are available, whereas only aggregate variables can be observed in the SDM setting. In other words, we understand our results as suggesting the important conclusion that it “whenever data are available at the individual level, then utility maximization generates very stringent restrictions upon observed behavior, even if the observed variables (equilibrium prices in our case) are aggregate”. From this perspective, whether individual transactions can be observed is irrelevant. Individual determinants of individual choices (such as initial endowments or individual incomes) may do just as well. A natural question, then, is whether the converse claim also holds: Is it the case that, when aggregate variables only are observed, no testable restriction can be generated, at least if the number of individuals is “large enough”? Specifically, assume that only aggregate endowments Ω can be recorded. These aggregate endowments are redistributed among individuals in the economy according to some rule that is not observed. In particular, fluctuations in Ω generate changes in individual endowments that are not recorded. What is observed, however, are the corresponding movements of equilibrium prices. In this new context, the equilibrium manifold is observed as a function of aggregate endowments only. Is there any restriction on the form of this relationship? We show that, under an analyticity condition, when the number of individuals is at least equal to the number of commodities, any (sufficiently smooth) manifold can be (locally) rationalized as the equilibrium manifold of an exchange economy with utility maximizing individuals, for some ‘well chosen’ redistribution rule. This result closes the argument by confirming that the Walrasian framework cannot generate restrictions on the local structure of the equilibrium manifold when only aggregate data are observable. In this sense, although our results emphasize a new aspect of aggregation theory, they remain fully consistent with the conventional wisdom of the field. Our work is in the line of a former contribution by Brown and Matzkin (1996), who study the restrictions on the structure of the equilibrium manifold from a “non-parametric”, revealed preferences perspective. In their paper, Brown and Matzkin derive a set of necessary and sufficient conditions under the form of linear equalities and inequalities that have to be satisfied by any finite data set, and they show that these relationships are indeed restrictive. This approach has been recently extended by Kübler, (2003), Snyder (1999) and Brown and Shannon (2000). Our work complements these results in three ways. First, we adopt a differentiable viewpoint, so that our necessary and sufficient conditions take the somewhat more familiar form of a system of partial differential equations, reminiscent of Slutsky conditions. In particular, our conditions can readily be imposed on a parametric estimation of the equilibrium manifold; hence they can be tested using the standard econometric tools of consumer analysis. We provide an example of such a parametric analysis in Section 3. Secondly, the result that these restrictions, if fulfilled, are sufficient to generically recover the underlying economy is original. Thirdly, we extend the analysis to the case where only aggregate endowments are observable, and provide a formal non testability result.

A first and obvious conclusion of our work is that the “equilibrium manifold” approach leads to conclusions that differ deeply from the Sonnenschein–Debreu–Mantel excess demand perspective. The main conclusion of the latter literature is that all the structure due to individual utility maximization is lost by aggregation. Adopting the equilibrium manifold perspective, we reach the opposite conclusion that all the relevant structure is generically preserved, in the sense that the initial economy can be recovered from the structure of the equilibrium manifold. In that sense, our results both generalize Brown and Matzkin’s findings and shed a new light on their scope and status. We refer to (Kübler et al., 2002) for an extension to the case of uncertainty and incomplete markets. Also, out interpretation of these results is simple. Rephrasing Arrow’s statement quoted in the introduction, we believe that in the aggregate, the hypothesis of rational behavior and market equilibrium has in general strong implications even if individuals are different in unspecifiable ways; however, the latter can be tested only insofar as data are available at the individual level. In short, rationality may be testable, but not without individual data. Finally, what is the empirical relevance of the restrictions derived in the paper? An obvious qualification is that they rely on the impact of changes in individual endowments on aggregate prices. Obviously, the larger the economy, the smaller such effects, and the more difficult it will be to produce empirical work on them. It should be stressed, however, that general equilibrium does not apply only to ‘large’ economies. On the contrary, the tools of general equilibrium theory have been recently applied, in a very successful way, to the analysis of the behavior of ’small’ groups. For instance, standard demand theory uses data on households or families, most of which gather several individuals. Models aimed at taking into account the ‘non unitary’ nature of the interactions at stake usually rely on a ‘collective’ approach, that postulates only efficiency. With private consumptions—a framework that has been used in most empirical applications—efficient allocations and market equilibria coincide, and general equilibrium theory is a relevant tool (see for instance Chiappori and Ekeland (2002)). The same approach has also been adopted to the analysis of such groups as committees, clubs, villages and other local organizations, which have also attracted much interest. For instance, many micro studies in development, starting with Townsend’s seminal investigation of risk sharing within an Indian village (Townsend, 1994), are based on data collected at the local level; it is not uncommon to observe endowments (say, individual crops) and prices within the village, a context to which our framework directly applies. Even in large economies, our result may still apply directly when individuals belong to a finite (and “small”) number of homogeneous “classes”. Finally, an interesting question is how our results can be extended to production economies. The idea is that, in a production context, changes in factor endowments will have an observable impact on factor prices, and that the corresponding equilibrium manifold can in principle be studied in a similar way. All this shall be the subject of further research.