Cores, Almost Competitive Prices, and the Approximate Optimality of Walrasian Allocations in Discrete Spaces

Abstract

This paper is devoted to problems arising when all or some goods are indivisible. Part One deals with the properties of the core in such situations, Part Two with the optimality properties of the Walrasian (competitive) allocations. The purpose of Part One is to show how certain relationships between the core and prices can be extended (with modifications) to exchange economies in which all or some commodities are indivisible. The proposition3 stating that a Walrasian allocation is in the core (provided preferences are selfish) remains valid without any additional assumptions when all goods are indivisible, as can be seen by inspecting the standard proofs. (see, e.g., Debreu and Scarf, Th-m 1, p.240, or Arrow and Hahn, Ch. 8, Th-m 1, p.187). [This is largely due to the fact that the blocking definition requires that strict preference hold for each member of a blocking coalition. Hence, in particular, a core allocation is only required to be Weakly Pareto Optimal (i.e., Pareto Efficient in the sense of Arrow and Hahn or Varian) rather than Pareto Optimal as defined, e.g., in Koopmans, p.46.4 If a ‘strict core’ were defined as the set of allocations ‘strongly unblocked’ in the sense of Def. 6, p.196, in Arrow and Hahn, then additional assumptions, such as local non-satiation, would be needed to guarantee that a Walrasian allocation belongs to the strict core.]