Search for Counterparties
This chapter introduces the modeling of search and random matching in large economies. The objective is to build intuition and techniques for later chapters. After some mathematical prerequisites, the notion of random matching is defined. The law of large numbers is then invoked to calculate the crosssectional distribution of types of matches. This is extended to multiperiod search, first in discrete-time settings and then in continuous time. The optimal search intensity of a given agent, given the cross-sectional distribution of types in the population, is characterized with Bellman's principle. We then briefly take up the issue of equilibrium search efforts.
We fix some mathematical preliminaries, beginning with a probability space (Ω,F,P,). The elements of Ω are the possible states of the world. The elements of F are events, sets of states to which we can assign a probability. The probability measure P assigns a probability in [0,1] to each event. We also fix a measure space (G, G, γ) of agents so that γ(B) is the quantity of agents in a measurable subset B of agents. The total quantity γ(G) of agents is positive but need not be 1.
We suppose that the measure γ is atomless, meaning that there is an infinite number of agents none of which has a positive mass. The set of agents is therefore sometimes described as a [continuum.] For example, agents could be uniformly distributed on the unit interval G = [0,1]. Combining the continuum property with a notion of the independence of search across agents will lead in this chapter to an exact law of large numbers, by which the the crosssectional distribution of search outcomes is deterministic (almost surely). For