Stronger core results with multidimensional prices

TL;DR

This paper introduces a generalized equilibrium concept with multidimensional prices for one-sided matchings without money, ensuring existence and convergence properties that improve upon traditional core concepts.

Contribution

It proposes a new solution concept with multidimensional prices that always exists and converges to competitive equilibria, strengthening core stability results.

Findings

The rejective core always exists and is within the weak core.

Rejective core allocations are in the weak core, but not vice versa.

Rejective core converges to competitive equilibria as the economy grows.

Abstract

We study one-sided matchings with endowments in the absence of money. It is well-known that a competitive equilibrium may not always exist and that the strong core may be empty in this setting [Hylland and Zeckhauser, 1979]. We propose a generalization of competitive equilibria that associates each item with a multi-dimensional price. We show that this solution concept always exists and resides within the rejective core [Konovalov, 2005]. Rejective core stability is strictly stronger than weak core stability: allocations in the rejective core are elements of the weak core, but the opposite is not true. Moreover, we show that the rejective core always converges to the set of competitive equilibria with multi-dimensional prices as the economy grows, demonstrating core convergence in a setting without non-satiation.