A Two Stage Model of Assignment and Market

Abstract

This paper studies a two-stage economy where the non-monetary assignments of indivisible objects are followed by market transactions. In this economy, there are finitely many players and finitely many types of indivisible objects and one divisible good called money. Every player demands at most one object besides money. The first stage is governed by a non-monetary assignment mechanism, while the second stage is governed by the market. As a mechanism in the first stage, this paper considers the Boston mechanism and the deferred acceptance algorithm. This paper defines perfect market equilibrium (PME) where the second stage outcome is a market equilibrium both on and off the equilibrium paths, and the first stage strategy profile is a Nash equilibrium of the mechanism, taking the second stage outcomes as given. This paper then analyzes two situations, the economies with and without money. We also applied our analysis to the college admission problem: some players (firms) cannot obtain objects (degrees) in the first stage, waiting for some other players (students) obtaining them, and buy the objects (hire the students with degrees) through the market in the second stage. This paper provides us with some necessary and sufficient conditions under which efficiency and stability are guaranteed.