Large Indivisibles: An Analysis with Respect to Price Equilibrium and Fairness

Abstract

An exchange economy is considered in which there are a finite number of individuals, the same number of indivisibles, and a fixed amount of a divisible good. Each individual consumes exactly one of the indivisibles and a certain quantity of the divisible good. The existence of prices corresponding to Pareto-efficient allocations is proved. It is also shown that this economy possesses fair allocations, income-fair allocations, coalition-fair allocations, and Pareto-efficient egalitarian-equivalent allocations. IN THIS STUDY the effect of including large indivisibles in an exchange economy will be analyzed. The analysis will be carried out with respect to the existence of price equilibrium and with respect to the existence of different forms of fairness. The economy consists of a finite number of individuals, the same number of indivisibles, and a fixed amount of a divisible good. It is assumed that each individual consumes exactly one of the indivisibles and some quantity of the divisible good. We may interpret the indivisibles as being professions or houses. The relationship between price equilibrium and a Pareto-efficient allocation is well-known and proved in a number of cases under different assumptions about preferences and commodities. For a standard economy see, e.g., Debreu [5]. In case of nonconvexities, which may also describe a certain type of indivisibility, we may have an exact relationship in economies with a continuum of individuals (see, e.g., Hildenbrand [11]), but otherwise an approximate relationship obtains. The degree of approximation depends on the degree of nonconvexities in preferences, and not on the number of individuals (see, e.g., Arrow and Hahn [1]). We will prove the existence of prices characterizing a Pareto-efficient allocation in a case where the indivisibles play an essential role and where approximate results would be too approximate. The indivisibles may be considered large for the individual compared to his total consumption. The theory of fairness has developed considerably during the seventies. Different kinds of concepts of fairness have been introduced and analyzed; see Daniel [41, Foley [81, Pazner and Schmeidler [121, and Schmeidler and Vind [141, among others. Normally, these authors have considered models with divisible goods or an infinite number of individuals. For exchange economies the existence of fair allocations (an envy-free and Pareto-efficient allocation) has been proved in