Consumption Sets

Abstract

The idea of consumption sets was introduced into general equilibrium theory in July 1954 in Arrow and Debreu (1954, pp. 268–9) and Debreu (1954, p. 588), the name itself appearing only in the latter paper. Later expositions were given by Debreu (1959) and Arrow and Hahn (1971) and a more general discussion by Koopmans (1957, Essay 1). Although there have been several articles concerned with nonconvex consumption sets (e.g. Yamazaki 1978), in more recent years their role in general equilibrium theory has been muted, especially in approaches that use global analysis (see for example, Mas-Colell 1985, p. 69). Such sets play no role in partial equilibrium theories of consumer’s demand, even in such modern treatments as Deaton and Muellbauer (1980). Since general equilibrium theory prides itself on precision and rigour (e.g. Debreu 1959, p. x), it is odd that on close examination the meaning of consumption sets becomes unclear. Indeed, three quite different meanings can be distinguished within the various definitions presented in the literature. These are given below (in each case the containing set is the commodity space, usually Rⁿ): M1 The consumption set C1 is that subset on which the individual’s preferences are defined. M2 The consumption set C2 is that subset delimited by a natural bound on the individual’s supply of labour services, i.e. 24 hours a day. M3 The consumption set C3 is the subset of all those bundles, the consumption of any one of which would permit the individual to survive. Each definition in the literature can (but here will not) be classified according to which of these meanings it includes. In probably the best known of them (Debreu 1959, ch. 4), the consumption set appears to be the intersection of all three subsets C1–C3. M1 is plain. After all, preferences have to be defined on some proper subset of the commodity space, since the whole space includes bundles with some inadmissibly negative coordinates. M2 is also reasonable, although a full treatment of heterogeneous labour services does raise problems for what is meant by an Arrow–Debreu ‘commodity’ (see for example, that of Arrow–Hahn 1971, pp. 75–6). It is M3 that gives real difficulty, both in itself and in relation to the others.