It seems appropriate to end this volume with a few words about why we have selected and organized the material as we have, and what we think has been learned. To put it another way, now is a good time to explain the title of the book, which includes both "modeling" and "analysis." The fact that the book contains many theorems should suggest what the analysis consists of. And the fact that we began the book with a detailed description of a particular labor market and with a brief description of some auction phenomena, should suggest at least part of what we mean by modeling. But if that were all there was to it, we could have finished our work in many fewer chapters.

Instead, we analyzed a whole family of closely related models, discrete and continuous, with and without complete information, with and without money, with firms employing one worker or many, and with simple or complex preferences. One purpose of these final remarks is to make clear what we think the consideration of all these models together adds to our understanding and interpretation of each of them.

Introduction

Economic data are rarely homogeneous; we seldom have observations that can be regarded as repeated drawings from the same probability distribution. We normally must allow for measured, and possibly for unmeasured, systematic differences between people, firms, or whatever is the unit of observation. In this chapter we shall introduce regressors or covariates explicitly into the hazard function and discuss the various types of covariate that econometricians seem likely to want to consider. We shall see that some of the relations of chapter 1 can fail to hold when covariates are present.

We shall suppose in this section that all covariates are fully observed by the econometrician. In chapter 4 and subsequently in many parts of the book we shall deal with models containing unobserved covariates.

Time-Invariant Covariates

The simplest covariates are those which do not change over time. Two examples might be the sex or the race of a person. More generally, the relevant characteristics of an agent at the time of entry to the state under study, that is, his relevant biography to that point in time, constitute a set of duration-invariant regressor variables. Other variables can and do change as the duration of stay changes, but at a sufficiently slow pace relative to typical durations of stay that, for practical purposes, they can be treated as if they were constants. Two examples might be the age of a person, or the state of the business cycle, in the context of studies of unemployment duration.

Introduction

The models of chapters 1 through 5 emphasised the construction of the hazard function as the basis of model-building and thus implicitly stressed the chance character of movement between states. In economic applications as compared to applications in technology or medical science the element of choice cannot be ignored. It may be luck that an unemployed man is offered a job today, but he must choose whether or not to take it. Both choice and chance enter into the the transition process. In this chapter we shall give an account of an approach to modelling in which the choice element in each transition is emphasised. In this approach people at all times are assumed to occupy the state that they prefer, given the opportunity set that they currently face. The element of chance enters into the transition process because both the desirability of different states and the opportunities open to the economic agent vary in a partly probabilistic way over time.

When econometricians model static discrete choice among K states they find it helpful to associate with each state a utility, uj, j = 1, 2, . . . , K, depending upon the characteristics both of the state and of the choosing individual such that he chooses that state affording the greatest utility. In modelling choice among two-states then, state 2 is chosen if u2 — u1> 0 and state 1 otherwise. In modelling the stochastic process of movement between states it is natural to adapt this approach to a dynamic context in which a sequence of choices is to be made. Thus we associate with each state and each time point an instantaneous utility flow, uj (t) . The objective of the agent is no longer to choose the state with the greatest current utility flow. The problem in a dynamic context is to formulate a rule which tells the agent, given any vector of utility flows at time t, u(t), which state to occupy.

The purpose of this book is two-fold. First, it reviews and integrates the growing literature about a family of models of labor markets, auctions, and other economic environments. Second, we hope it will illustrate the subtle interactions between modeling considerations and mathematical analysis that characterize the use of game theory to explain and predict the behavior of complex “real-world” economic systems.

We will be concentrating on “two-sided matching markets.” The term “two-sided” refers to the fact that agents in such markets belong, from the outset, to one of two disjoint sets - for example, firms or workers. This contrasts with commodity markets, in which the market price may determine whether an agent is a buyer or a seller. Thus whereas the market for gold has both sellers and buyers, any particular agent might be a buyer at one price and a seller at another, so the market is not two-sided in the sense we will speak of. But a labor market often is, since firms and workers are distinct. For example, as the wages of professors fall, some professors may leave the market, but none will become universities. The term “matching” refers to the bilateral nature of exchange in these markets - for example, if I work for some firm, then that firm employs me. This contrasts with markets for goods, in which someone may come to market with a truck full of wheat, and return home with a new tractor, even though the buyer of wheat doesn't sell tractors, and the seller of tractors didn't buy any wheat.

Chapters 7, 8, and 9 present three models of one-to-one matching in which money plays a prominent role. Unlike the model explored in Section 6.2, money will be modeled as a continuous rather than as a discrete variable, and this will let us employ a different set of mathematical tools. An even more important difference between the models of these chapters and those we have dealt with until now is that in these models, we are going to assume that the preferences individuals have for different matchings are basically monetary in nature.

Chapter 7 presents a model of exchange between one seller, with a single object to sell, and many buyers. This model will be simple enough so that the analysis can be conducted without much technical apparatus. We will explore it in enough detail to see that with one or two significant exceptions, the phenomena uncovered in our presentation of the marriage problem will also arise in these models with money. We will also relate this discussion to our discussion in Section 1.2 of strategic behavior in auctions, and particularly to the options available to coalitions of bidders.

Consider a market consisting of a single seller, who owns one unit of an indivisible good, and n buyers, each of whom is interested in purchasing the object if the price is right. Each buyer b places a monetary value $rb on the object, which is the maximum amount he or she is willing to pay, and the seller similarly places a value $rs on the object, which is the price below which he or she will not sell. We call these monetary values the reservation prices of the agents. Each buyer has cash on hand sufficient to pay his or her reservation price.

We may interpret the reservation price as follows, for example. The seller has in hand an offer of $$rs from some outside party (not one of the buyers in the present market), who will buy the object at that price if it is offered by the seller. Each buyer (whom we may think of as a broker, rather than as a final consumer of the object) has in hand an offer of $rb from a client who will purchase the object from the buyer at that price, should he or she obtain it in the market. Since the seller knows that he or she can earn at least $rs, the seller will not sell at a lower price. And since each buyer b knows that he or she can earn $rb (but no more) by reselling the object, the buyer will not buy at a higher price.

We turn now to a different class of questions, which will require different kinds of models and theories. In the previous sections we investigated the marriage market by exploring the kind of matchings we might expect to observe. Now we will investigate how we should expect individual agents to behave. Specifically, we will want to know to what extent it is wise for men and women to be frank about their preferences for possible mates, and how we can expect them to act in the process of courtship and marriage. Lest the analysis seem a little cold-blooded at times, it is good to remember that our primary interest here is in a simple model of labor markets, and that the phenomena we will be studying can perhaps better be thought of as the courtship that takes place between potential employers and employees. But for simplicity we consider these questions first in the context of the marriage problem, and defer to later chapters more realistic models in which firms may employ more than one worker.

To address these questions of individual behavior, we need to model the decisions that individuals may be called upon to make in the course of a marriage market. We have so far considered only the general rules of the game, which state that for a marriage to take place between some man m and woman w, it is both necessary and sufficient that the two of them agree. These rules are reflected in our definition of a stable matching. But these general rules might be implemented by many different particular procedures. These particular procedures, which constitute the detailed rules of the game, will determine what specific decisions each agent faces. They will determine how an agent goes about making his preferences known, and the order in which decisions are taken. They will, in short, be a description of the mechanics of the market.