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Article Excerpt
Summary. We identify conditions under which preferences over subsets of a consumption world can be reduced to preferences over bundles of "commodities". We distinguish ordinal bundles, whose coordinates are defined up to monotone transformations, from cardinal bundles, whose coordinates are defined up to positive linear transformations.
Keywords and Phrases: Commodity, Preference, Ordinal measure, Cardinal measure.
JEL Classification Numbers: D11.
1 Introduction
This note is an attempt to endogenize the notion of commodity. According to the standard view in economic theory, spelled out in Debreu (1959) for instance, a commodity is defined by specifying the objective, intrinsic characteristics it possesses. This includes not only its physical attributes, but also the location, time, and state of the world at which it is available. In Debreu's model, commodities are completely homogenous and finite in number.
Lancaster (1966) emphasized that commodities are not the direct object of utility but that "it is the properties or characteristics of the goods from which utility is derived". In Lancaster's model, commodities are points in a (finite dimensional Euclidean) space of characteristics and thus form a continuum. From Hotelling (1929) to Dixit and Stiglitz (1977), Hart (1979), and many others, the idea of a space of commodity characteristics appears, in one form or another, in all contributions to the theory of monopolistic competition: see Benassy (1991) for a survey. This is no surprise; the problem of defining a commodity is intimately linked to the issue of endogenous product differentiation and the emergence of local monopolies. Differentiated commodities were introduced into purely competitive equilibrium theory by Mas-Colell (1975), Jones (1984), and others: see Aliprantis, Brown, and Burkinshaw (1989) and Mas-Colell and Zame (1991) for a systematic exposition and a detailed survey of the broader literature on infinite dimensional commodity spaces.
In Lancaster's framework, or in its refinements such as Mas-Colell's model, characteristics remain exogenously defined, objective, and measurable in a cardinal sense. Here we take one further step towards endogenizing commodities. Most economic theorists would consider that if two goods are perfect substitutes in the eyes of all economic agents, it is parsimonious and useful modelling practice not to distinguish between them, even if they are intrinsically quite different. Conversely, advertisers often work hard to emphasize product differences that are intrinsically trivial. These casual observations suggest that commodities should perhaps be given a subjective and collective definition: what matters to define a commodity is probably not as much the objective characteristics of it as the ones subjectively perceived and considered relevant by all agents.
We assume that agents' preferences are primitively defined over subsets of an a priori rather unstructured consumption world, and we propose to define or "construct" commodities from these preferences: a commodity is a subset of the world that the agents' preferences reveal to be both sufficiently homogenous and distinct from the rest of the world. Two examples might help understand our approach.
Example 1 Two agents share a cake; they have preferences over the various possible pieces of it. The "world" in this example is the cake, a compact subset W of [R.sup.3], and the preferences are orderings over the [sigma]-algebra [SIGMA] of Borel subsets of the cake. Suppose that agent i's preference admits a numerical representation [u.sub.i] : [SIGMA] [right arrow] R of the type
[u.sub.i](A) = [v.sub.i]([lambda](A [intersection] [W.sup.1]), [lambda](A [intersection] [W.sup.2])),
for i = 1, 2, where [W.sup.1], [W.sup.2] are two compact subsets forming a partition of the cake, [lambda] is Lebesgue measure, and [v.sub.1], [v.sub.2] are arbitrary monotone functions defined on [R.sub.+.sup.2]. Such preferences suggest that the cake is made up of two commodities, [W.sup.1] and [W.sup.2] : agents compare pieces of the cake according to the quantities of these commodities that they contain.
The two endogenous commodities in Example 1 are instances of "cardinal" commodities, whose quantities may be expressed by means of bona fide [sigma]--additive measures over the subsets of the cake. The concept of cardinal commodity is extremely useful: it gives meaning to unit prices, makes it possible to formulate assumptions of convexity on both preferences and technology, and, ultimately, address equilibrium issues. Our second example is meant to introduce the notion...
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