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Article Excerpt
Abstract This work introduces a set-theoretic foundation of deterministic bilateral matching processes and studies their properties. In particular, it formalizes a link between matching and informational constraints by developing a notion of anonymity that is based on the agents' matching histories. It also explains why and how various matching processes generate different degrees of "informational isolation" in the economy. We illustrate the usefulness of our approach to modeling matching frameworks by discussing the classical turnpike model of Townsend.
Keywords Bilateral matching * Frictions * Anonymous trading * Spatial interactions
JEL Classification Numbers C78 * E00
1 Introduction
A large segment of economics is concerned with the study of allocations when markets are not functioning well. Market frictions are often seen as involving scarcity of information or geographical separation or inadequate institutions, and have been modeled in a variety of ways. A well-established research program has made these frictions explicit, motivating their presence by modeling trade as occurring in small groups, often pairwise matches. The central assumption is that some technology exists that exogenously selects agents from the population and matches them together. (1)
Matching frameworks have been used to answer basic questions in a variety of settings. For instance, how market frictions affect equilibrium output and unemployment, as in Diamond (1982), the cyclical behavior of job creation and destruction, as in Mortensen and Pissarides (1994), or the relative value of currencies in an international setting, as in Camera and Winkler (2003). A limitation of these frameworks is that the matching technology is insufficiently formalized and mostly descriptive. One is often confronted with various explanations as to how (and to what extent) the assumed frictions are an implication of the mechanism by which agents interact with each other. In short, a unifying theoretical structure is missing. This prevents a clear understanding of the exact connection between the constraints imposed by the meeting technology, the types of obstacles faced by market participants, the trades they can execute or the information they can access, and the possible allocations. Amore structured formalization of these links can improve the formulation of models whose central trait is markets with impaired functioning. Indeed, a comprehensive theory of exchange should clarify how the constraints assumed to be in place originate from the underlying physical environment.
In this paper, we take a step toward advancing the theoretical foundations of matching frameworks by considering physical environments with technologies that exogenously pair agents deterministically. The major contributions are the development of an explicit set-theoretic representation of bilateral matching technologies and a formalization of their method of operation. We describe different matching processes and explain how they can facilitate (or obstruct) the interactions among agents. Especially, we focus on the informational aspects since matching frameworks are used to motivate the existence of spatial separation as well as more general obstacles to economic interactions. (2) Indeed, we introduce a map between properties of the matching process and the degrees of informational openness (that is, the degrees of anonymity) that are consistent with the physical description of the environment.
Our work complements two strands of literature that concern matching environments. One strand includes a growing research on network games and network formation (for a survey see Jackson 2005), a line of work dealing with endogenous matching and its allocative consequences, as in Gale and Shapley (1962) and Roth and Sotomayor (1999) to cite a few, and recent efforts on endogenizing matching frictions and "matching functions" in models with spatially separated agents, as in Lagos (2000). A second strand of literature comprises research directed at building solid mathematical foundations for random matching models; examples include studies on random meetings between agents drawn from countable or uncountable populations, as in Alos-Ferrer (1999), Gilboa and Matsui (1992), or Aliprantis et al. (2004), and work on the exact law of large numbers for random pairwise matching, as in Duffie and Sun (2004a,b).
This paper--that focuses on exogenous matchings and abstracts from their allocative implications--is more closely related to this second strand of literature. However, our approach differs from the foundations of matching studies cited above in that we remove all stochastic elements. We can thus offer a basic conceptual framework that complements both strands of literature not only by contributing to developing a common language and basic notions for the mechanics of matching, but also by allowing us to explore some links between matching dynamics and possible information flows.
The layout of the paper is as follows. In Section 2 we familiarize the reader with our notation. Then, in Section 3 we describe the technical procedure that we use to pair agents in any population, during a single period. To do so we define a notion of matching technology--which we call a bilateral matching rule--and then present a theorem that establishes the structure of matching on any population. Subsequently, in Section 4, we discuss matching over time--as a sequence of matching rules--and then characterize matching processes according to the levels of informational isolation they impose on the economy. To do so, we develop a taxonomy of anonymity that is based on the agents' matching histories. Then, in Section 5, we demonstrate how to construct economies where the matching process provides each agent with an infinite sequence of deterministic pairings, while imposing an extreme degree of informational isolation. Finally, in Section 6 we offer some concluding remarks.
2 Preliminaries
If A is any set, then the symbol |A| denotes the cardinality of A. As usual, |A| = [[ALEPH].sub.0] means that A is a countable set and |A| = c indicates that the cardinality of A is the continuum. If a set A is a union of a pairwise disjoint family of sets {[A.sub.i]}[.sub.i[member of]I], i.e., A = [[union].sub.i[member of]I] [A.sub.i] and [A.sub.i] [intersection] [A.sub.j] = [empty set] if i [not equal to] j, then we denote this by the symbol A = [[??].sub.i[member of]I] [A.sub.i]. Throughout the paper we denote by X a non-empty set representing the population in the economy. We assume these agents are infinitely lived and time is discrete.
3 Bilateral matching rules
In this section, we study how to pair agents in a period. Naturally, the first step we must take is to formalize a general notion of a pairwise matching technology. To do so we introduce the concept of bilateral matching rule.
Definition 1 A bilateral matching rule for the population X is a function [phi] : X [right arrow] X satisfying [[phi].sup.2](x) = x for all x [member of] X, i.e., [[phi].sup.2] = I, the identity mapping on X.
According to the above definition, if [phi] : X [right arrow] X is a bilateral matching rule, then the function [phi] is invertible--and so [phi] is a permutation of X since [phi] is a surjective function that is also one-to-one. However, [phi] belongs to the special class of permutations whose inverses coincide with themselves, i.e., [[phi].sup.-1] = [phi]; these functions are known in mathematics as "involutions". This simply says that any way of pairing agents in the population must be such that the partner of an agent's partner is the agent himself.
Thus, if [phi] is a matching rule and agent x is matched to agent [phi](x), then we call [phi](x) the partner of x. Symmetrically, x = [phi]([phi](x)) is the partner of [phi](x) so that we can call the set {x, [phi](x)} a match. For concreteness, we think of matches as distinct pairs of agents that are spatially separated.
Here are two simple examples of matching rules.
a. Let X = N = {1, 2, ...}, the set of natural numbers, and define [phi] : X [right arrow] X by
[phi](2a) = 2a - 1 and [phi](2a - 1) = 2a.
b. Let X = (0, [infinity]) and...
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