Elsevier

Journal of Economic Theory

Volume 165, September 2016, Pages 25-36
Journal of Economic Theory

Random decentralized market processes for stable job matchings with competitive salaries

https://doi.org/10.1016/j.jet.2016.04.003Get rights and content

Abstract

We analyze a decentralized process in a basic labor market where finitely many heterogeneous firms and workers meet directly and randomly in pursuit of higher payoffs over time and agents may behave myopically. We find a general random decentralized market process that almost surely converges in finite time to a competitive equilibrium of the market. A key proposition en route to this result exhibits a finite sequence of successive bilateral trades from an arbitrary initial market state to a stable matching between firms and workers with a scheme of competitive salary offers.

Introduction

Adam Smith's Invisible Hand captures the self-regulating nature of a decentralized market where self-interested market participants, making independent decisions freely, can settle the market on a competitive equilibrium outcome. Traditionally a fictitious Walrasian auctioneer has been used to match the supply and demand of each commodity (service) at its competitive price (wage). However, many competitive markets, labor markets being a leading example, involve mainly uncoordinated bilateral trades and are typically decentralized. The purpose of this paper is to analyze the long-run behavior of a general random market process in a basic labor market where transactions take the form of bilateral trades so as to mimic the decentralized behavior of the labor market.

We consider a labor market where finite heterogeneous firms and workers meet directly and randomly to search for higher payoffs over time. In the market, all agents make their own decisions independently and can behave myopically, perhaps because information is dispersed and agents may not have a complete picture of the entire market. When a worker and a firm match as partners, they generate a joint surplus which is then split within the matched pair. Each agent can dissolve her current partnership unilaterally if standing alone becomes a better option. A worker and a firm, currently not matched, can form a new partnership as long as doing so makes none of the two worse off and at least one strictly better off — in this case the firm fires its previous worker and the worker abandons her previous firm, if any, and the deserted parties can be worse off. We call such transactions bilateral trades or pair improvements. In such a market process, quits and layoffs routinely arise as a result of agents seeking better matches and it is also possible that workers eventually return to their previous employers but with different contracts. The random process proceeds spontaneously and is decentralized, in that every agent acts only according to her own interests without any centralized coordination, and unforeseen and unexpected market outcomes can emerge from the agents' actions under imperfect information about the market.

The basic question we consider is whether the above random, chaotic, and dynamic decentralized process eventually leads the market to efficient assignments of workers to firms and in particular to a competitive equilibrium.1 We establish that this market process converges with probability one to a competitive equilibrium of the market in finite time, so long as each possible bilateral trade conditional on the current market state arises with an arbitrary but positive probability in the process (Theorem 1). An interpretation of this positive probability is that although information is imperfect and dispersed among all market participants, it flows sufficiently freely so that the agents are informed about and can therefore respond to newly arrived opportunities. A crucial step for establishing Theorem 1 is to show that the random process is not trapped in trading cycles indefinitely. To this end, we demonstrate via a novel algorithm the existence of a finite sequence of successive bilateral trades from an arbitrary initial market state to a competitive equilibrium (Proposition 1).

Our study is closely related to the seminal work by Crawford and Knoer (1981), who consider a similar labor market and propose a deterministic salary adjustment process for the market in which firms make offers and workers then accept or reject the offers. The salary adjustment process, a generalization of the deferred acceptance algorithm of Gale and Shapley (1962), converges to a competitive equilibrium of the market.2 Our market process is random and uncoordinated, beginning with any market state and having no central planner to guide transactions. Crawford and Knoer's process is however deterministic and does not involve any uncertainty. In addition, their process starts with a specific market state and requires firms to use retention to maintain payoff monotonicity of every worker.3 Such monotonicity cannot hold in our processes, where a bilateral trade, while improving the welfare of the involved pair, typically makes the abandoned firm and worker worse off. The overall market welfare is hence not necessarily monotone after a sequence of bilateral trades, making a design relying on monotonicity arguments difficult if not impossible.

Our study is also related to the seminal work by Roth and Vande Vate (1990), who reexamine the Gale–Shapley marriage matching model and develop a decentralized process that finds almost surely a stable matching between men and women.4 A key difference here is that our model admits monetary transfers and has a competitive equilibrium supported by competitive prices, while it is known that the marriage matching model generally does not have competitive prices to support any stable matching. Another major difference is that the stability solution in the marriage matching literature is strictly weaker than ours which coincides with competitive equilibrium. Furthermore, a key step of the algorithm in Roth and Vande Vate (1990) maintains strict payoff monotonicity of one side of the market, which fails in our setting because we work with a more general notion of blocking involving both a pair of agents and surplus division.5 Hence, while Roth and Vande Vate (1990) provide a decentralized foundation for stability in the marriage matching, our study offers a decentralized framework for competitive equilibrium in the assignment market.

Finally, it is worth mentioning another strand of literature concerning Feldman (1974) and Green (1974). While their processes are deterministic for certain classes of NTU games, the current process is random and also involves significant indivisibility.

Section snippets

The model

Consider a labor market with heterogeneous firms and workers. Denote the (finite) set of firms as F and the (finite) set of workers as W such that FW=. Each firm hires at most one worker and each worker accepts at most one job.6 A matching or assignment μ is a one-to-one mapping from FW to itself such that for all xFW, x is either self-matched (μ(x)=x

Main results

In this section we present our central result Theorem 1, which demonstrates that starting with an arbitrary initial market state, any random and decentralized process in which every bilateral trade conditional on the current market state occurs with a positive probability will converge with probability one to a competitive equilibrium in finite time. To achieve this goal, we first present a crucial mathematical result Proposition 1 establishing the existence of a finite sequence of successive

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We are grateful to the editor, an associate editor and anonymous referees for constructive comments, criticisms and suggestions, and to Vince Crawford, Chiaki Hara, Jean-Jacques Herings, Ian Jewitt, Atsushi Kajii, Fuhito Kojima, Patrick Legros, Rudolf Müller, Al Roth, Larry Samuelson, Utku Ünver, and many participants at various institutes and workshops for their helpful feedback. The second author's work is supported by JSPS/MEXT KAKENHI 25280004.

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