Cross-Sectional Dynamics in a Two-Sided

Matching Model∗

Lones Smith†

Department of Economics

University of Michigan

this version: May 27, 2009original version: April, 1992

Abstract

This paper studies a dynamic search-theoretic model of two-sidedmatching with ex ante heterogeneous agents and nontransferable utility.There is a continuum of agents’ types in (0, 1), with both parties to thematch (x, y) receiving flow output xy. Foregone wages are the only costof search, and all productive relationships are mutual ‘tenant-at-will’.

Despite a continuum of distinct intertwined dynamic optimizations,I characterize the search equilibrium starting with everyone unmatched:There is a growing pool of permanently employed on [θt, 1), and anelaborate web of temporary matches amongst lower types on (0, θt),where θt ↓ 0. Thus, anyone is eventually is unwilling to accept a tem-porary match. I develop an equilibrium concept addressing credibilityconstraints, but argue that no matches are lost due to the absence ofbinding contracts. I relate this to dynamic properties of flow valuesthat imply that no one ever quits a match more than once in his life.

∗This was first presented in 1992 at Boston University, MIT/Harvard, Michigan, Montreal,Princeton, Windsor, as well as Summer in Tel Aviv 1992 and the 1993 North American SummerEconometric Society Meetings at Boston University.

†e-mail address: lones.smith@gmail.com

1 INTRODUCTION

The equilibrium search literature incorporates natural transactions frictions intothe idyllic formally static models that occupy centerstage in economics. The searchand matching literature initially proceeded by assuming that the heterogeneitywhich motivates the search process is entirely or partly realized ex post. That is,after two individuals have met, a stochastic element is realized, like the price/wageoffer, or some type; only then do they decide upon the wisdom of their potentialmatch or trade. This ‘representative agent’ paradigm has yielded the majority ofinsights we have about the stylized effect of search frictions on trade and matching.

This paper by contrast is part of an overall effort to develop a general theory ofsearch-theoretic matching with ex ante heterogenous agents. The basic allocationquestion is: Who matches with whom? Shimer and Smith (2000) (henceforthSS) have investigated models of employment matching (namely, with transferableutility) and attempted to characterize and compare the search equilibria and socialoptima. Smith (2006) and Burdett and Coles (1998) has focused exclusively on thesearch equilibria for the nontransferable utility (NTU) paradigm in steady-state.

This paper works within the latter framework without transferable currency.This aptly captures the world of social matches, and might roughly proxy for othersettings where wages do not fully reflect values. For instance, academic wages aregenerally forced into a far narrower range than scholarly impact. In this context, Iexplore nonstationary time dynamics. Not only do I ask who matches with whom,but also “when” and “for how long”? This question is surely too bold for any onepaper, and I simply scratch the surface here, asking what happens when a newmatching market opens up. This alone turns out to be a rich enough questionfor study. With heterogenous agents, average and flow present unmatched valuesdiverge, and this leads to explicitly temporary matching. I hope that this analysissheds equilibrium light on behavior in many contexts in the “real world”.

The singular focus of this paper on time dynamics is made possible by focusingon a model whose steady-state equilibrium is straightforward, by now. Supposethere is a continuum of agents indexed by ability levels in (0,1), where the flowoutput xy of the match (x, y) is evenly split. This is equivalent to a world wherethe flow payoff of anyone is the type of his partner if matched, and zero otherwise.So there is really only ‘one side’ to the market. The steady-state matching in thisworld is then a coarse approximation of Becker’s perfect sorting in a frictionlessworld: The interval (0, 1) partitions into subintervals in which all agents matchwhen they meet, as seen in Smith (2006) and Burdett and Coles (1998). Thereason for this “block segregation” is that every agent employs a standard thresholdstrategy — a reservation partner — and different agents employ the same thresholdif they face the same matching opportunity set. Absent a transferable currency asin SS, the equilibrium exhibits the discontinuous step structure.

There is a general appreciation by economists that dynamic opportunities candramatically enrich the range of possible equilibrium outcomes in many contexts.

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Since the steady-state sorting outcome is so simple, this particular model offersan ideal framework to isolate the role of time dynamics, and in particular, theevolution of long-term relationships. For I modify the above setting in three ways:First, all agents are initially unmatched. Second, there is no countervailing forcelike exogenous match break-ups or deaths that can sustain a steady-state. Third,I allow match quits at any moment, thereby admitting temporary matches. Thisassumption is clearly moot in a steady-state model, since no one desires to quit.My principal contributions here are both substantive and methodological. This isto my knowledge the first search paper that systematically investigates the cross-sectional dynamics that arises when a market with heterogeneous agents opens up.This work differs essentially from much of the search literature, for I explain thenature of short-term relationships that precede long-term relationships.

As in any freshly opening matching market, such as a ‘fraternity rush’, a matchscramble arises. Potential partners for new matches arrive in continuous timewith a known Poisson arrival hazard rate. The resulting nonstationary dynam-ics partially admits analytic resolution because everyone still employs thresholdstrategies. I describe the search equilibrium with discounting, and deduce that agrowing cohort of permanently matched, and an elaborate web of deliberately tem-porary matches beneath them endogenously arises. In the model, high types ini-tially only match among themselves in one equivalence class. As the best membersof this class match, the unmatched remainder continuously drops their standard.The unworthy lesser agents bide their time, entering into explicitly temporarymatches. Interestingly, higher agents will often entertain lower standards on tem-porary matches. Both parties to these matches are fully aware that the strongeragent will eventually jilt the weaker at an opportune moment, and move on.

Absent binding contracts, it might come as a surprise that individuals can agreeupon a mutually beneficial match in which one party quits on the other. For whatif this occurs at an unfavorable moment for the other party? With the perfectforesight of the deterministic world, might the match then be declined? This issuedoes not arise here since individuals ineligible for permanent employment havedecreasing flow unmatched values and are always willing to match with themselves.As the paper stands, I simply illustrate these matches by simulation, since thericher characterization of these partial differential equations is exceedingly hard.

My major methodological innovation is the general program that I follow, tryingto transfer the more easily secured insights about average (Bellman) values todeduce what is happening to the flow values. There is a rich dynamic and cross-sectional interplay amongst these quantities and the all-important notion of areservation wage. Smith (1999) shows how values and flow values weakly exceed thelatter, which happen to be the reservation partner in a world of purely permanentmatches, but not in a world of temporary matching. While higher type agentssurely have higher match values, their flow values can be inversely related and thisoccasions the temporary matches and the aforeknown quits.

The proof of existence of the nonstationary equilibrium dynamics for permanent

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employment is also somewhat enlightening, for the dynamical system here is three-dimensional. This precludes the standard application of phase plane diagrams. Myproof instead succeeds by an essentially geometric argument that only considersthe level surfaces of the system, and effectively reduces the dimensionality of theproblem by one. This technique, although not universally applicable, should workin many other unrelated models as well.

The temporary matching problem by contrast is almost too daunting: Searchequilibrium must reconcile a continuum of inherently distinct and yet intertwinedindividual dynamic optimizations. For this reason, I simplify the model as muchas possible — including choosing the most tractable search technology. The easeof matching is linearly proportional to the number of searchers with the quadratictechnology, while finding partners is no easier with more searchers than with less forthe linear search technology. I work exclusively with the quadratic technology, dueto a decision-theoretic advantage: The matching decisions of agents are separablefrom those of agents with whom they do not plan to match. This knife-edgeanalytic property renders potentially horrendous equilibrium calculations ratherpedestrian — for this allows agents planning permanent matches to derive theircommon equilibrium strategy independent of those engaged in temporary matches.

Link to the Search Literature. I believe that Sattinger (1985) and (1992)are the first search-theoretic matching papers with purely ex ante heterogene-ity. Standard search theory is also largely stationary, whereas I explore the non-steady-state analysis. Agents do not have recourse to ex post damages if partnersquit (‘breach’) their relationships, as in Diamond and Maskin (1979) and (1982).Rather, in the NTU spirit of social matching, I assume that all relationships are‘tenant-at-will’. In my paradigm, the only decision left to the agent is when toagree to a match, and when to quit. Thus, one might imagine there is no easilyverifiable way of ascertaining who broke up a relationship. More plausibly — andthe implicit background story in this paper — one could simply suppose that thereis no enforcement mechanism for ensuring that any contracted damages are actu-ally paid. Quitting individuals depart ‘like a thief in the night.’ As is often thecase in social and other matching settings, quits are not mutual decisions.

Well after the first version of this paper, Li and Damiano (2005) wrote a nicesequel. The payoffs and timing are similar, but their objective is instead to explorehow small matching costs prevent sorting in equilibrium.

Outline. The next section briefly analyzes the competitive equilibrium andfirst-best matching without search frictions. Section 2 is entirely self-contained,and provides a flavour of the equilibrium and stylized predictions in analogous twoand three period versions of the model. Section 3 carefully presents the continuous-time model, and some basic tools are then developed in section 4. What followsis an analysis of the permanent matching that arises in section 5, and the tempo-rary matching in section in section 6. Some concluding remarks and an appendixcontaining the less economic proofs follow.

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Figure 1: Discrete-Time Truncated Model.

period t = 1, 2, . . . , T − 1 Tactivity matched =⇒ produce xt (then quit?) produce xT

unmatched =⇒ search again (then accept?)payoff δtxt if matched; else 0 δTxT /(1− δ)

2 DISCRETE-TIME SEARCH: EXAMPLES

Suppose there is a continuum of individuals with types distributed uniformly in(0,1). If individuals x and y are matched, they produce a flow product of xy. In thethe Parteto optimum (or the core), each type x ∈ (0, 1) is paired with another x.For if we instead match some y 6= x, then output would fall, as

xy − x2/2− y2/2 = −(x− y)2/2 < 0.

How closely can optimal selection by individuals in a two-sided matching modelapproximate this socially optimal frictionless outcome? That is, suppose that thismarket is newly opened, and that all individuals are initially unmatched. Let theper period discount rate be δ < 1. Consider the following two distinct paradigms:

1. Matches are proposed to every unmatched agent in each period;

2. In all but the last period, each unmatched agent has a match proposed toher with probability equal to the unemployment rate. In the final period, allremaining unpaired are randomly matched.

In each paradigm, either party to a proposed match can veto it. Output is equallysplit. Further assume that search is a time-consuming process, so that if one ismatched in a given period, one cannot simply quit that match for another thevery next period; rather, a one period sabbatical is required to search. One wayto resolve this model is to consider the following finite-horizon truncation: Letthe matching process extend T periods, with the period-T payoff equal to theflow reward divided by (1 − δ). Equivalently, all unmatched agents after periodT are randomly paired forever, with no further quits or vetoes permitted; payoffs,however, accrue over the infinite horizon. Figure 1 outlines the timing and payoffs.

There is no closed form solution of the discrete time infinite horizon problem.But much of the economic insight is found in the two- and three-period truncations.

2.1 A Two Period Example

Search occurs in two periods. Here there is no distinction between the two matchingparadigms. In the first period, each individual randomly meets another potential

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partner. If either vetoes the match, each receives zero payoff that period. Other-wise, they split the output produced equally. In the second period, all individualsremaining unmatched are randomly paired, once for all. No vetoes are permitted.1

Clearly, no one will ever opt to terminate a match. Thus, individuals effectivelyhave two strategies: match or don’t match in the first period.

Note that the first period decision criterion of all individuals is identical: Giventhe multiplicatively separable production function, one’s own parameter has noeffect on preferences. Each individual simply seeks to maximize the discountedexpected parameter of her partner. Thus, in period one, everyone will accept anyparameter above some threshold θ equal to their discounted expected period twopartner’s parameter. Given that everyone uses the same threshold,

θ = δθ(θ/2) + (1− θ)θ(1 + θ)/2

θ + (1− θ)θ

so that (2−δ)θ2−(4−δ)θ+δ = 0 or θ = 0. There is also an equilibrium with θ = 0that is possible: Everyone who meets in a given period agree to match, knowingthat no one is around next period. I ignore this trivial outcome.2 Thus,

θ =4− δ −

√5δ2 − 16δ + 16

2(2− δ) ,

and there is an equivalence class [θ, 1) of individuals willing to pair in the firstperiod, and no one with an efficiency parameter below θ will pair until period two.

2.2 A Three Period Example

Now suppose that there are three periods, with those matches proposed in the finalperiod necessarily consummated. Individuals now enjoy four strategies: (i) matchin period one (if given the opportunity) and stay matched; (ii) match, and thenquit; (iii) refuse the first period match and then match; and finally (iv) refuse thematches offered in the first two periods. I shall soon rule out the last strategy.

I proceed by backward induction. Independent of non-trivial first period be-haviour, the optimal second period strategy is to accept a proposed partner exactlywhen her index is at least θ2 > 0, where θ2 equals the discounted expected periodthree partner’s parameter.3 Thus, there is some equivalence class (θ2, 1) in periodtwo of mutually desirable individuals. Moreover, by the rationale of the previousexample, no individual will ever opt to terminate a match after this period.

1Or, vetoes are allowed, but would never be used since the alternative payoff equals zero.2One can also argue that it is not stable either, in the sense that it collapses if some individuals

“tremble,” and reject their first period partner.3This expectation depends in an obvious fashion upon the common period two behaviour.

Thus, whether such a θ2 exists or is unique is at issue. I shall, however, shortly exhibit a solutionwhich happens to be unique.

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Next consider the first period behaviour. Since θ2 is the lower threshold foraccepting any new matches in period two, anyone initially paired in period onewill quit her match in period two (and thus forego period two payoffs) if and onlyif her partner’s parameter lies below θ2. Who in the first period then will agreeto match? Clearly, an individual declines a match exactly when her discountedexpected payoff from being eligible for next period’s matching exceeds that of theproposed match. Since those with indices below θ2 will always be turned down (ordumped, if already matched) in the second period, the option of being eligible forthe next period is worthless. They will therefore accept any proposed match inthe first period. (I have now ruled out strategy (iv) above.)

Consider now those with indices at least θ2. Since they are all in the sameequivalence class in the second period, they have the same initial threshold θ1. Notethat θ1 equals the discounted expected eventual partner’s parameter, conditionedon θ2 — since these matches are not quit. Then:

Lemma 1 θ1 > θ2

Proof Denote by Pt(a, b) the fraction of eligible individuals at the outset ofperiod t with types Y in (a, b), and by Et the expectation operator with respectto the measure Pt. Since θ1 = P2(0, θ2)θ2 + P2(θ2, 1)E2(Y | Y ≥ θ2),

(θ1 − θ2)/δ = [P2(0, θ2)θ2 + P2(θ2, 1)E2(Y | Y ≥ θ2)]− E3(Y )

= [P2(0, θ2)θ2 + P2(θ2, 1)E2(Y | Y ≥ θ2)]−[P3(0, θ2)E3(Y | Y < θ2) + P3(θ2, 1)E3(Y | Y ≥ θ2)]

> P2(0, θ2)θ2 + [P2(θ2, 1)− P3(θ2, 1)]θ2 − P3(0, θ2)θ2 = 0

as the second period matching equivalence class implies that in the first paradigm

P3(θ2, 1) =P2(θ2, 1)− P2(θ2, 1)2

1− P2(θ2, 1)2=

P2(θ2, 1)

1 + P2(θ2, 1)< P2(θ2, 1)

while in the second paradigm, P3(θ2, 1) = P2(θ2, 1)− P2(θ2, 1)2 < P2(θ2, 1).

Given θ1 > θ2, the histogram diagram of figure 2 — with agents’ types on thehorizontal axis and fractions matched equal to the shaded height — depicts thematches that occur. A few stylized facts emerge from the above.

• Non-Assortative but Temporary Matches: In period one, the very best and thevery worst agree to match: The two (forward shaded) equivalence classes are(0, θ2) and [θ1, 1); the former consists of purely temporary matches.

• Quits Occur : It is common knowledge that matches consummated within (0, θ2)are only temporary, with those individuals ineligible for second period matches;

• A Groucho Marx Matching Result : Among those not seeking permanent matches,in the first period, no one in (θ2, θ1) is willing to match with anyone who’s willingto match with her — namely those in (0, θ2).

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Matched in Period 1

1

θ2

0

θ1

0 θ2 θ1 1

Matched in Period 2

1

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0

θ1

φθ1

0 θ2 θ1 1

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Figure 2: Three Period Model Histogram of Matched Types. The measuresof agents of each type who are matched in the first two periods are shaded.

• Falling standards. As time progresses, there is less self-segregation: More of thevery best and fewer of the very worst agree to match: In period two, there isone such (backward shaded) equivalence class [θ2, 1), and in the final period theequivalence class constitutes all of (0, 1).4

Having developed some simple intuitions in discrete time, to push the modelmuch further, I shall switch to continuous time; this will permit a more craftedanalytic discussion of the nature of equilibrium. Observations 1, 2, and 4 willprove robust in the continuous time infinite horizon context, while investigatingthe generality of observation 3 will prove pivotal for the analysis of temporarymatching in the continuous time model.

3 THE CONTINUOUS TIME MODEL

Agents. Agents are indexed by their type x ∈ R. Let λ be the time-invariant Borelmeasure on types, with support Σ = (x, 1), for some x ∈ [0, 1). I shall normalizethe measure of everyone to unity, so that λ(R) = λ(Σ) = 1.5 So as to avoid thecomplications occasioned by an atomic type distribution, I simply assume that λ isdifferentiable with respect to Lebesgue measure, with Radon-Nikodym derivativeℓ (a.e.). I shall let L be the associated c.d.f.

It may help to imagine that individuals actually belong to R2, with a continuumof agents of each type x having (one-dimensional) measure ℓ(x). In that case, Imay still without loss of generality simply refer to an agent by her type x as I

4See the Appendix ?? for a derivation of these thresholds.5So long as

∫ ∞

0xλ(dx) < ∞, all results of this paper obtain for supp(λ) = [0,∞). But the

proofs are more tedious, and so this minor piece of generality is avoided.

6

implicitly show that one’s optimal strategy is solely a function of one’s type. Forinstance, in the earlier histograms, λ was Lebesgue measure on (0, 1), and ℓ(x) ≡ 1.

Action Sets. At any instant in continuous time, an individual is eithermatched (employed) or unmatched (unemployed). Only the unmatched engagein (costless) search for a new partner.6 When two agents meet, their efficiencyparameters are perfectly observable to each other. Matches can be thought of asmutual tenant-at-will : Either party may veto the proposed match; if both approve,it is consummated and remains so either until one partner quits. The equilibriumconcept shall rule out ‘imperfect’ behaviour, in which one player vetoes becauseshe anticipates that the other will; such strategic considerations are simply notthe focus of this paper. At the moment the match is severed, both individualsinstantaneously re-enter the pool of searchers.

Match Payoffs. For now, assume the production technology is bilinear: If xis paired with y then the flow output of that relationship is simply the product xy— which is equally divided. Together, the above assumptions on joint productionand output division allow me to normalize players’ payoffs by their own indices,and redefine the flow payoff at time t of each individual x to be her partner’s indexpt(x). If x is unmatched at that moment, let pt(x) = 0.

Turning to the agents’ infinite-horizon objective function, everyone discountsfuture payoffs at the common interest rate β > 0. Thus, each x maximizes herexpected infinite-horizon discounted average payoff E[β

∫ ∞

0e−βtpt(x)dt].

The Measures of Matched and Unmatched Agents. At any time t, Imust be able to precisely describe the distribution of existing matches, which willturn out to be the state variable for the model. Closely following SS, let µt(X, Y )be the mass of agents with types in X ⊂ Σ who are matched with agents withtypes in Y ⊂ Σ. Since obviously µt(X, Y ) ≡ µt(Y,X), each matching measure µt

lies in Ms, the space of nonnegative measures on Σ2 that are uniformly boundedabove by 1 and are symmetric on Σ × Σ. The resulting measure-valued function〈µt〉 : [0,∞) → Ms is absolutely continuous with respect to λ2 ≡ λ × λ, themeasure of agents with types in the pair of sets (so that λ2(X, Y ) = λt(X)λt(Y ))— so let mt ≡ dµt/dλ

2 be the Radon-Nikodym derivative. Hence,

µt(X, Y ) =

∫

X

∫

Y

mt(x, y)λ(dy)λ(dx)

Here, mt : R2 7→ R+ is the match density function, and mt(x, y) may be inter-

preted as the normalized density of existing matches (x, y), where the normaliza-tion accounts for the frequency of the types x and y in the population. Note thatmt(x, y) ≡ mt(y, x), for λ2-a.e.-(x, y), since µt is symmetric.

I shall let νt(X) denote the mass of agents with types in X ⊂ Σ that are

6Neither search costs nor on-the-job search enrich the analysis so much as complicate it, atleast for the few points that I wish to make.

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unmatched at time t. This unemployment measure satisfies

νt(X) = λ(X)− µt(X,Σ) ≡ λ(X)−∫

Σ

∫

X

mt(x, y) λ(dx)λ(dy). (1)

Once again, if ut ≡ dνt/dλ denotes the Radon-Nikodym derivative, then νt(X) ≡∫

Xut(x)λ(dx). Then (1) implies that ut(x) = 1 −

∫

Σmt(x, y) λ(dy) is the point

unemployment rate of type x. The overall unemployment rate (and mass, sinceλ(Σ) = 1) in the economy is denoted ut = νt(Σ) ≡ 1−

∫

Σ

∫

Σmt(x, y)λ(dy)λ(dx).

The Search Technology. With search frictions, match creation takes time.I wish to use the simplest possible search technology involving Poisson arrivaltimes. In the terminology of SS, I shall only consider anonymous search technolo-gies, whereby individuals meet one another in direct proportion to their mass inthe unmatched pool. But what then is the constant of proportionality? In a linearsearch technology, potential partners for unmatched individuals arrive with con-stant flow probability (or ‘hazard rate’) ρ > 0, independent of the unemploymentrate. Thus, an unmatched individual can expect to have a match proposed to herwithin time t > 0 with probability 1− e−ρt. Call ρ the rendezvous rate.

By contrast, in the quadratic search technology, the arrival rate of potentialpartners for unmatched individuals equals ρ times the unemployment rate. Putdifferently, were it possible, an unemployed individual would meet another unem-ployed or employed partner according to a Poisson process with arrival rate ρ.But as it is presumed technically infeasible to match with someone who is alreadyemployed, the overall arrival rate of potential partners equals ρut. Simply put, therate at which a searcher meets any subset of individuals is directly proportional tothe measure of searchers in that subset. Paradigms 1 and 2 in the discrete-timeexamples correspond to the linear and quadratic technologies, respectively.

The analysis of this paper shall be confined to the quadratic technology inlight of a crucial analytic knife-edge advantage. Consider the following slightlyaltered story: Invitations to possible meetings arrive at fixed rate ρ, but now with achance 1−u the other individual is already paired, and thus misses the meeting. Socall the quadratic matching technology strategically separable, since an individualonly cares about the arrival rate of acceptable agents, and with the quadratictechnology (only), this is purely a function of their measure. That is, her currentmatch payoffs are unaffected by any matching decisions made by those with whomshe is unwilling to match. (Whether she is permanently unaffected is anothermatter entirely that this paper must grapple with.) While such an absence of‘crowding out’, or congestion effects, has been conveniently exploited in the past,it is absolutely essential here in this nonstationary environment.

Strategies. I make the major simplifying assumption that strategies are de-pendent only on payoff-relevant state variables, and are not otherwise conditionedon history. This will be my justification for analyzing the model purely in ag-gregate distributional terms. In this model, the sequence of measures 〈µt〉 is theonly payoff-relevant state variable. Because 〈µt〉 evolves deterministically without

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aggregate uncertainty, it therefore suffices to simply condition strategies on timealone.

Pure Strategies.. At any moment in time, an unmatched individual mustchoose which matches to agree to should a meeting occur, whereas someone alreadymatched must decide which matches are worth continuing. I shall suppose thatthese two sets coincide — a restriction which is innocuous in the absence of fixedcosts of forming or dissolving matches, and not tenable otherwise.

Hence, pure strategies can be represented simply by the graph of the acceptancecorrespondence A·(x) : R+ Σ, with time on the horizontal axis, and types x onthe vertical axis. So the acceptance set At ⊆ Σ specifies whom x is willing to bematched with at time t, and is assumed right (upper and lower hemi-) continuous intime; this rules out a large but uninteresting class of weakly dominated strategies.

A possibly more appealing representation of strategies takes horizontal ratherthan vertical slices of A. Define the quitting time τt(x, y) = sups ≥ t | y ∈ As′(x)for all s′ ∈ [t, s), i.e. the first time after t that x plans to quit her match withy, given that she is, or has the opportunity to be, matched with y at time t. Inconcordance with my assumption about acceptance sets, I assume that τt(·, ·) isright continuous in time t.

Mixed Strategies.. The space of time-0 mixtures over acceptance corre-spondences or stopping times T is simply too intractable to represent dynamics.Fortunately there is a simpler restricted mixed strategy space with many niceproperties. The resulting mixtures are analogous to behaviour strategies in gametheory, insofar as they instruct players how to behave at each moment in time.A Markovian quitting rule specifies a chance Qs(t; x, y) that x does not quit thematch (x, y) by time t given that at time s ≤ t, either (i) the match already ob-tains or (ii) a meeting occurs, and given that the match is not otherwise dissolvedin (s, t). This dual usage reflects some of the cutting power of the Markovianspirit. By analogy to the properties of pure strategies, assume that Qs(t; x, y) isright continuous in t ≥ s. SS show that Qs(t; x, y) is weakly monotonically de-creasing in t ≥ s, for fixed s, and that it satisfies the Bayes-Markov property :Qt0(t2; x, y) = Qt0(t1; x, y)Qt1(t2; x, y), for t0 < t1 < t2.

Joint Quitting Chances. Markovian quitting rules by both parties deter-mine the rate at which any given match is being destroyed. Intuitively, sinceplayers’ randomizations are generally assumed to be independent in equilibrium,Ps(t; x, y) ≡ Qs(t; x, y)Qs(t; y, x) is the joint quitting probability. That is, this isthe symmetric chance that neither party quits the match (x, y) by time t giventhat at time s ≤ t, either the match already obtains or a meeting occurs.

Much of the analysis demands that I essentially gloss over the behaviour ofindividual types in the model, and instead focus on positive masses of agents. Solet Πs(t;X, Y ) be the fraction of matches (x, y) ∈ (X, Y ) that are not quit byeither party by time t given that the match already obtains at time s ≤ t. ThenΠs(t) is absolutely continuous with respect to µs, with Radon-Nikodym derivative

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Ps(t; x, y) at (x, y). Consequently,

Πs(t;X, Y ) =

∫

X

∫

Y

Ps(t; x, y)µs(dx, dy) =

∫

X

∫

Y

Ps(t; x, y)ms(x, y)λ(dx)λ(dy)

Stocks and Flows. Match creation and quits determine the time path of〈µt〉, and thus of 〈mt(x, y)〉. Since all strategies are right continuous in time, soalso are 〈µt〉. To begin with, I assume that all individuals are unmatched, so thatµ0 ≡ 0. Then because current matches are just those that have been formed atsome time in the past and have not been destroyed in the interim, I have

µt(X, Y ) =

∫ t

0

ρνs(X, Y )Πs(t;X, Y )ds (2)

Note that Πs(t) determines the ‘flows’ into and out of the current ‘stock’ µt

of matches. To prove the existence of equilibrium later on, I must eliminate anyreference to the flows. This will be possible because SS shows that for any t > s,

Πs(t;X, Y ) = e∫ ts

δ−ρνs′ (X,Y )/µs′ (X,Y )ds′µt(X, Y )/µs(X, Y ) (3)

SS also proves that the matching measure 〈µt〉 belongs to the space DfMs [0,∞)

of right-continuous functions [0,∞) → Ms with left hand limits, meeting thefeasibility conditions

0 ≤ µt(X, Y ) ≤ µs(X, Y )e∫ t

sρνs′ (X,Y )/µs′ (X,Y )ds′ (4a)

µt(X, Y ) ≤ ρνt(X, Y ) (4b)

4 EQUILIBRIUM ANALYSIS

4.1 Value Equations

Let Wt(x) be the expected average present value for x of remaining unmatchedat time t and thereafter behaving optimally. Let Wt(x| y) be the correspondingaverage value to x of either matching with y, or having the opportunity to do so,at time t. So Wt(x| y) ≥ Wt(x). There is a natural relationship between thesetwo quantities: An unmatched individual x receives a zero payoff until her firstpotential match y arrives at time s, whereupon she gets an average utilityWs(x|y),discounted back to the present. Hence,

Wt(x) =

∫ ∞

t

e−∫ s

t (β+∫

Σ ρus′ (y) λ(dy))ds′ ·∫

Σ

ρus(y)Ws(x|y)λ(dy) ds (5)

Just as in SS, corresponding to the average value Wt(x), I shall define theflow value ψt(x) = Wt(x) − Wt(x)/β. This turns out to be the instantaneousreservation partner of x at time t: the type with whom x is indifferent between

10

remaining unemployed and matching for an arbitrarily short period of time, duringwhich she cannot search. Solving the differential equation for Wt(x) produces theimplicit equation Wt(x) ≡

∫ ∞

tβe−β(s−t)ψs(x) ds. This theoretical construct plays

a key role in much of my analysis. Indeed, differentiation of (5) yields

ψt(x) = (ρ/β)

∫

Σ

ut(y) (Wt(x|y)−Wt(x)) λ(dy) (6)

Results in SS imply that the value to x of the match with y is

Wt(x|y) =

∫ ∞

t

βe−β(s−t) (Pt(s; x, y)y + (1− Pt(s; x, y))ψs(x)) ds (7)

The average value to x of a match with y at time t is the discounted averageof flow playoff y, weighted by the probability the match has not been dissolvedby time s, plus the discounted average of the future (hypothetical) instantaneousreservation partner ψs(x), weighted by the probability the match has dissolved bytime s. Next substitute ψs(x) = βWs(x) − Ws(x) into the above to find that theown-surplus St(x|y) ≡ Wt(x|y)−Wt(x) satisfies:

St(x|y) =

∫ ∞

t

e−β(s−t) Pt(s; x, y)(

β (y −Ws(x)) + Ws(x))

ds (8a)

=

∫ ∞

t

βe−β(s−t)Pt(s; x, y)(y − ψs(x)) ds (8b)

Now, while unmatched for an instant, x foregoes the ephemeral flow chance ofmeeting an individual such as y who would provide her with own-surplus St(x|y).Substituting (8b) into (6) thus yields the following accounting equation:

ψt(x) = (ρ/β)

∫

Σ

ut(y)

(∫ ∞

t

βe−β(s−t)Pt(s; x, y)(y − ψs(x)) ds

)

λ(dy) (9)

So the flow value of x is increasing in both her expected rendezvous probabilityand the expected surplus she will derive from future matches over her unmatchedvalue. The derivation of this implicit flow value equation is much more direct herethan in the TU model of SS, who must first consider the Nash bargaining solution.

I also must consider the twin notion of a permanent reservation or thresholdpartner θt(x) for type x. By this, I mean the type with whom x is indifferentabout matching — assuming that that individual will never quit the match. Thus,if x is willing to match with y, then y ≥ θt(x), while conversely, if y ≥ ψt(x)then x is willing to match with y. The theory describing the relationship betweenWt(x), ψt(x), and θt(x) is developed in Smith (1999). To summarize, all threecoincide in steady-state. But in general, θt(x) ≤ min〈Wt(x), ψt(x)〉, because thereis an ‘option to renew’ bundled with θt(x). Moreover, θt(x) = Wt(x) < ψt(x)when this option is forever exercised, as when θt(x) or Wt(x) is at a strict globalmaximum on [t,∞); conversely, θt(x) = ψt(x) <Wt(x) exactly when this option isnot worthless, because θt(x) is locally increasing. In the intermediate case, I haveθt(x) < min〈Wt(x), ψt(x)〉, with no general ordering among Wt(x) and ψt(x).

11

4.2 Which Matches are Mutually Agreeable?

I now identify the sets of mutually agreeable matches, which shall be the basis forthe equilibrium notion. Intuitively, a match (x, y) is mutually agreeable at time t if(⋆) holds: Wt(x| y) >Wt(x) andWt(y| x) >Wt(y), or equivalently St(x|y) > 0 andSt(y|x) > 0. Later on, I may wish to further nuance and call (x, y) weakly mutuallyagreeable if (x, y) = limn→∞(xn, yn) for mutually agreeable matches (xn, yn). ButI really need a reformulation in terms of the more primitive unmatched values, andthis leads to an interesting discovery.

Observe that for (x, y) to be mutually agreeable, it is necessary that y >θt(x) and x > θt(y). But this is by no means sufficient because it is crucial thatthere exist some common time interval [t, s) that both individuals wish to remainmatched for, and this restriction is not reflected in that assertion. Fortunately,Smith (1999) proves that y > θt(x) iff Wt(x) < y + e−β(s−t)(Ws(x) − y) for somes > 0. So, reformulating (⋆), the more primitive assertion about average valuesrequires that there exist some s > t, such that

Wt(x) < y + e−β(s−t)(Ws(x)− y) (10a)

Wt(y) < x+ e−β(s−t)(Ws(y)− x) (10b)

It so happens that even this dual condition is not sufficient. Indeed, one player,say x, could conceivably wish to quit at a previous time, rendering the match onbalance ex ante strictly unprofitable for y; or if not, y may then wish to quit atan even earlier time, etc. (See figure 3.) Say that a match (x, y) has contractablesurplus if it satisfies (10) for some time s > t. One could simply posit that bindingcontracts exist as a remedy for any such self-imploding yet matches with con-tractable surplus; however, it is not at all obvious which outcome would be agreedupon, as the different parties have different preferences. Resolving this conflictwould likely entail side payments, and this would allow one to circumvent theNTU model entirely — for individuals would then have an incentive to maximizethe joint surplus, as in the TU model.

I shall therefore respect the mutual tenant-at-will paradigm, and carefully for-mulate the correct notion of mutual agreeability. Denote by St(s; x, y) the runningown-surplus to type x if her match with y lasts from time t until time s. Then

St(s; x, y) ≡ (1− e−β(s−t))y + e−β(s−t)Ws(x)−Wt(x)

=

∫ s

t

βe−β(r−t)(y − ψr(x)) dr

I now formulate mutually agreeability indirectly (almost) in terms of primitives.It requires that both iterated incentive compatibility constraints and individualrationality be met.

Lemma 2 (Mutually Agreeable Matches) At time t, the match (x, y) is mu-tually agreeable until time s > t exactly when

12

Figure 3: How Tenant-at-Will Incentive Compatibility Constraints CanUnravel a Potential Match with Contractable Surplus. From the twographs of running own-surplus, it is apparent that x ideally would wish the matchto last until time t1, but y prefers to quit at time t2 < t1. Rationally foreseeingthis, x would then prefer to quit at time t3 < t2. But then y would wish to quit attime t4 < t3. The mutually agreeable outcome subject to incentive compatibility isfor the match (x, y) to last until time t1. Had it been the case that St(s; x, y) < 0until time (t4 + t5)/2, the match would not be mutually agreeable.

St(s; x, y)

St(s; y, x)

t1

t2

t3

t4

t5t

t

$/t

$/t

s

s

• [IC] s = sn = sn−1 where the sequence 〈sk〉 satisfies s0 = ∞ and sk+1 =min〈arg supt≤s<sk

St(s; x, y), arg supt≤s<skSt(s; y, x)〉 for all k;

• [IR] St(s; x, y) ≥ 0 and St(s; y, x) ≥ 0 with strict inequality for at least one.

If the IR constraint binds for both parties (and yet s > 0), simply call the matchweakly mutually agreeable. All other matches are mutually disagreeable. LetMAt

be the set of all mutually agreeable matches (x, y) at time t, and let MAt(x) =y | (x, y) ∈ MAt. Similarly define MDt andMDt(x) for mutually disagreeablematches (x, y) at time t.

4.3 Search Equilibrium

I now adapt the notion of search equilibrium developed for the purely TU modelof SS. In a word, I demand that all mutually agreeable matches (x, y) ∈ (X, Y ) ⊆MAt be maintained and all mutually disagreeable matches (x, y) ∈ (X, Y ) ⊆MDt

be quit at once. In order to establish existence of equilibrium, I shall have to expressthis notion wholly in terms of primitives. To this end, notice that St(s; x, y) is

13

decreasing in s exactly when her then flow surplus y − ψs(x) < 0. Hence, searchequilibrium will ask that a match continue so long as both individuals have positive‘continuation surplus’:

min⟨∫ ∞

te−β(s−t)Pt(s; x, y)(y − ψs(x)) ds,

∫ ∞

te−β(s−t)Pt(s; x, y)(x− ψs(y)) ds

⟩

≷ 0=⇒ 〈Pt(t; x, y) = 1, Pt(t; x, y) = 0〉

for all t, and λ2-a.e. (x, y). No conditions on joint match quitting probabilities areimposed for the nongeneric set of matches which are neither mutually agreeablenor disagreeable. As in SS, I may further reduce this to

min

⟨∫ ∞

t

e−

∫ s

tβ+

ρus′

(x)us′

(y)

ms′

(x,y)ds′

ms(x, y)/mt(x, y)(y − ψs(x)) ds,∫ ∞

t

e−

∫ st

β+ρu

s′(x)u

s′(y)

ms′

(x,y)ds′

ms(x, y)/mt(x, y)(x− ψs(y)) ds

⟩

≷ 0

=⇒ 〈mt(x, y) = ρut(x)ut(y), mt(x, y) = 0〉

(11)

Theorem 1 (Characterization) A search equilibrium is a triple: a matchingmeasure 〈µt〉 whose density satisfies the mutual agreeability condition (11), an un-employment measure 〈νt〉 obeying (1), and a flow value system 〈ψt〉 satisfying (9).

Conjecture 1 (Existence) A search equilibrium exists. One also exists in purestrategies, i.e. for all t, mt(x, y) = ρut(x)ut(y) if y ∈ At(x) and mt(x, y) = 0 if not.

The proof is not yet done, but I am very close. Essentially, the problem is veryclose to the point where the Fan-Glicksberg Fixed Point Theorem will apply. Asfor the purification, intuitively, the only matches that can be randomized over arethe weakly mutually matches that are not mutually agreeable.

I now rule out a class of particularly nasty search equilibria. Intuitively, thesimultaneous quitting en masse of a positive measure of matches at any time (aftertime-0) represents an incredible coordination problem, arguably accomplished onlyby means of a public coordinating device. Yet I am explicitly ruling out ‘sunspots’:The model must evolve deterministically. Still, the formulation of strategies admitsa possible dependence on time, and so conceivably this could occur. With thismotivation, I shall define a bootstrap search equilibrium to be one in which 〈µt〉is discontinuous at some positive time i.e. 〈µt〉 /∈ CMs[0,∞), the space of weak-continuous functions R

+ 7→ Ms. This notion clearly only bites out of steady-state.

Conjecture 2 (Continuity) Bootstrap search equilibria do not exist.

The name ‘bootstrap’ ought to be suggestive of the circular logic underlying thediscontinuity. This logic goes as follows. Suppose by way of contradiction thatat time t∗ a positive mass of agents X did quit their matches. Then it must be

14

true that the flow values ψt(x) of agents x ∈ X have discontinuously jumped up.Indeed, one is always willing to stay matched if one is earning a strictly positiveflow surplus; conversely as the horizon T ∗ approached, the option value of stayingmatched must have been quickly vanishing, so that x would not stay matched unlessher flow surplus was not too negative. The idea is then to deduce a contradiction,by arguing that one could have done better earlier by adopting the matching rulethat is optimal only starting at t∗.

5 PERMANENT EMPLOYMENT

5.1 Preference Monotonicity

I shall hereafter focus on a particular pure strategy search equilibrium with impliedacceptance sets At(x) for all types x.

This section shall essentially assume away the technical difficulties posed bypartners quitting on an individual by looking for the matches that are never ter-minated. Without this threat, the problem everyone confronts here is tantamountto the model of Smith (1999): An individual faces a constant flow arrival rate ofprizes from a deterministically evolving absolutely continuous ‘prize distribution’.In that set up, individuals accept any match yielding a payoff at least their flowvalues. More generally, a similar argument reveals that any two individuals havingthe same strategy are comparable, or

Lemma 3 (Monotonicity) At any time s ≥ t, if x is willing to match with y,then x is strictly willing to match with z > y, so long as x ∈ As(y)⇐⇒ x ∈ As(z)for all s ≥ t.

Under the above conditions, x is not indifferent between (and thus will not mixover) any z > y at any time s ≥ t if x is willing to match with either.

5.2 Cross-Sectional Monotonicity: A Poverty of Riches?

The problem of this paper belongs to a general equilibrium environment. Assuch, individual preferences feed back rather naturally into all unmatched values.To compare values, I shall now define the inverse of At as in Smith (2006). Theopportunity set Ωt(x) = y | x ∈ At(y) consists of all individuals y who are willingto match with x at time t. Whereas Ωt(x) ≡ At(x) in a TU model, this need notobtain in the NTU context here. Characterizing the equilibrium entails knowingwhether reciprocal monotonicity obtains: Is Ωt(y) ⊆ Ωt(x) for all x > y.

So consider two agents y < x. The following seductive argument is false:“Monotonicity implies that anyone who is willing to match with y is strictly willingto match with x, provided x emulates the acceptance strategy of y. So x couldassure herself of Ωt(x) if she so desires, and achieve an average value Wt(x) =Wt(y). Allowing x to behave optimally, we have Wt(x) ≥ Wt(y) for all t.”

15

Of course the flaw in this logic is that x might not be able to credibly committo employing the acceptance strategy of y. In fact, x could conceivably have suchbright prospects in the near future, that she would plan to quit many matchesthat y would not. This fact being foreseen, it could be the case that no onewishes to match with x. Truly, future riches may beget current poverty. As itturns out, while the search equilibrium of this paper will contain such individualswith great expectations, I shall argue that the sorry conclusion will never arisesin equilibrium.

5.3 Cross-Sectional Aggregation

I now give a slightly modified spin on the basic logic underlying the perfect seg-regation result of Smith (2006): If starting at some time, types x and y have thesame preferences and always enjoy identical future opportunities (those types will-ing to match with them, Ω·(x) and Ω·(y)), then they must henceforth make thesame current and future choices. This simple observation establishes

Lemma 4 (Coincident Choices) Fix y 6= x. Suppose that for all s ≥ t, anyonewilling to match with x is also willing to match with y, and vice versa. Then xand y have the same strategy. That is, if Ωs(x) = Ωs(y) for all s ≥ t, thenAs(y) ≡ As(x), Ws(y) ≡ Ws(x), and hence θs(y) = θs(x) for all s ≥ t.

As transparent as it is, Lemma 4 encapsulates the core insight of this section,and in fact the paper, as it allows me to amalgamate groups of individuals whencalculating equilibrium strategies. For this is a context with both single person andcross-sectional dynamics, and this insight allows me to refrain (only temporarily,as it happens) from seriously tackling the two simultaneously.

Call a set Ct of individuals with the same acceptance strategy a coincident choiceset. Note that Lemma 4 shows that Ct cannot contract over time, or Cs ⊆ Ct for allt > s. Let At(Ct) denotes the well-defined common acceptance set of the coincidentchoice set Ct. An especially salient coincident choice set is an equivalence class Et,whose members will only agree to match with other members of Et at time t, i.e. forwhich At(Et) = Et for all t. Clearly, an equivalence class Et ⊆ (0, 1) cannot contractover time, or Es ⊆ Et for all t > s, because this property was already demonstratedfor coincident choice sets. Observe also that any match by individuals within anequivalence class Et ⊆ (0, 1) is never quit. I now show that such a notion is notwithout example.

Theorem 2 (Permanent Employment) In any search equilibrium, there ex-ists a coincident choice set Ct that includes [ρ/(ρ+β), 1). Moreover, Ct is a convexinterval [θt, 1), and in fact Ct is an equivalence class, i.e. θt is weakly decreasing.

Proof Because

θt(x) ≤ Wt(x) <

∫ ∞

t

ρe−(β+ρ)(s−t)ds = ρ/(ρ+ β),

16

if individuals in [ρ/(ρ+ β), 1) plan to stay matched to one another, such matchesare forever mutually agreeable. Conversely, any two types x, y ≥ ρ/(ρ + β) havea strict disincentive to quit a match with one another because their respectiveunmatched values are both strictly less than ρ/(ρ+ β) in any search equilibrium.Hence, they are not permitted to plan to quit. Search equilibrium neatly precludesany imperfect behaviour in which one party quits knowing the other plans to doso, when it is really in the interest of neither to quit. Simply put, a match mustcontinue if it still provides both parties with positive own-surplus.

Now let’s push this argument a little further by looking to dynamic program-ming for intuition. Suggestively, denote by θ ∈ (0, 1) the unique solution of7

θ =

∫ ∞

0

e−∫ s

0 (β+∫ 1θ

ρ λ(dy))ds′∫ 1

θ

ρy λ(dy) ds =

∫ 1

θy λ(dy)

(β/ρ) +∫ 1

θλ(dy)

By searching further, individuals cannot hope to do better than achieve an expectedpayoff of θ. So the previous logic applies, and any match by individuals x, y ≥ θis mutually agreeable and will never be quit.

The above argument can now be reapplied at every moment in continuoustime; however, as time progresses, the threshold θ must fall because the measuresof remaining unmatched high types is obviously falling. This is one place wherethe choice of quadratic search technology bites. So write θ = θt, and it is possibleto solve for the resulting law of motion of 〈θt〉

θt =

∫ ∞

t

e−

∫ s

t

(

β+∫ 1θs′

ρus′ (y) λ(dy))

ds′∫ 1

θs

ρyus(y)λ(dy) ds

as well as the evolution equation for 〈ut(y)|y ≥ θt〉 under the presumption thatlower types will quit any extant match as soon as y = θt. I defer this exercise,as it is the substance of Lemma 5. This exercise establishes that θt is a smoothdecreasing function of time. Observe that we can rule that the flow value

I must justify two key assumptions. First, must any type x quit any extantmatch (with y < θt, perforce) as soon as x = θt, and second, must any x ≥ θt

always refuse to match with any type y < θt? Note that these are one and thesame question, by the presumed nature of the strategy set: The one acceptance setdetermines which ongoing matches to quit and which potential ones to accept. Alsoobserve that such a common strategy certainly constitutes equilibrium behaviourbecause Wt(x) = θt(x) = θt for any x ≥ θt by construction, in light of the Bellmanoptimality principle. Namely, whenever a type finds herself above the threshold,

7Note that there is a solution to this equation because both sides are continuous in θ ∈ (0, 1),and as θ increases from 0, the left hand side (LHS) ranges from 0 to 1 while the RHS rangesfrom a positive number down to 0. Moreover, the LHS has slope 1, while the derivative of theRHS continuously tends to E[y − θ|y ≥ θ] < 1 as ρ/β → ∞. Hence, the solution is unique forlarge enough ρ or small enough β — that is, when the search frictions are not too bad. I shallmaintain such an assumption.

17

she knows that she may match with any type in [θt, 1) without fear of her partnersubsequently quitting; she therefore behaves optimally by accepting precisely thosetypes in [θt, 1).

I now turn to the necessity of the above two assumptions. To do so, I firstremark that θt — having been calculated assuming the smallest possible band ofmutually agreeable matching [θt, 1) — is certainly an upper bound for the averagevalue Wt(y) for any y ∈ (0, 1) in any search equilibrium. For any other searchequilibrium can only result in less choosy behaviour by types in [θt, 1); this will inturn yield a lower mass of unemployment types in [θt, 1), and thus a lower impliedvalue for them too.

Observe that if θt(x) is monotonically decreasing, then Smith (1999) impliesthat Wt(x) = θt(x) for all t. Hence, x never wishes to quit any match, regardlessof how she will be treated by her potential partners. Indeed, since her unmatchedvalue is falling, her incentive to stay in the match only grows with time. But inthat case, x can credibly commit to stay matched with any y ≥ θt(x) and mustturn down all y < θt(x). And since Wt(y) ≤ θt ≤ x, no type y ever wishes to partcompany with x. So all matches between x and y ≥ θt(x) are mutually agreeableforever afterwards. But if θt(x) is monotonically decreasing for all x ≥ θt, thenit must be invariant in x, since any individual can credibly mimic another, andall such types x ≥ θt are desirable matches for anyone else. But in that case, weclearly must have θt(x) = θt for all x, and I am done.

So it remains to rule out θt(x) not being monotonic decreasing, for some orall x ≥ θt. To do so, I shall consider the function θt(x) = sups≥t θs(x). Think ofthis as ‘filling in all the valleys’ of θt(x). By results in Smith (1999), wheneverθt(x) = θt(x), I must additionally have Wt(x) = θt(x) = θt(x). Now assume,by way of contradiction, that θt(x) < θt(x) in some search equilibrium. Smith(1999) shows that θt(x) is always an upper semicontinuous function of time (andso, θt(x) is a continuous function of time). Hence, there are times t1 < t2 such thatθt(x) < θt(x) for all t ∈ (t1, t2), with θt1(x) = θt2(x) = θt(x). Thus Smith (1999)yields Wt1(x) =Wt2(x).

My goal is to argue that irrational behaviour is implied either at t1 or t2. Forinstance, I could suppose by way of contradiction that behaviour is optimal startingat time t2. In that case, there are uniformly fewer unmatched above θt1(x) = θt2(x)available at t2 than at t1, because in the interum, some have matched with oneanother, and those matches are not quit. It seems implauble then that we donot have Wt1(x) > Wt2(x). So let x starting at time t1 adopt the acceptancestrategy which is optimal starting only at time t2. That is, let her substituteAt(x) = At−t1+t2(x) for all t ≥ t1. Being a deviation from optimality, this cannotimprove Wt(x) on [t1,∞), and thus any match that was credible for her before isstill credible.

(to be continued) ♦

18

Remark. The analysis is made especially difficult due to the following fact.Recall that faced with a prospect of one’s partner quitting, we can only say forcertain that y will match with x if x ≥ ψt(x). But I cannot simply conclude thatany type y outside the coincident choice set is always willing to match with x ≥ θt

for that would require showing ψt(y) ≤ θt. But formula (9) will not easily permitsuch a deduction. Conceivably, ψt(x) > θt can occur if future flow values ψt(x)are sufficiently small. In words, an individual doesn’t care about a high immediateflow payoff if her unmatched value is rapidly falling, and she knows she will soonget jilted. In fact, I later wish to rule out ψt(x) > x ever.

5.4 Dynamic Analysis of Permanent Employment

State Variables and Laws of Motion. I now turn to some unresolvedissues from the proof of Theorem 2. With a falling threshold θt, individuals x < θt

will quit any matches precisely at the moment they are absorbed into Ct. In lightof strategic separability, at time t the highest individual only statically cares aboutthe mass ut of unmatched individuals x ≥ θt, and the average type µ(t) amongunmatched individuals above θt. I shall therefore look for an equilibrium thresholdpath parameterized by just these variables. For analytic ease, however, I shallslightly transform them. Replace ut by

ut ≡ L(θt) + ut = L(θt) +

∫ 1

θt

ut(x)λ(dx) (12)

Note that ut is not the global unemployment rate (i.e., the overall fraction of allunmatched individuals at time t), but rather is the fraction of all individuals notin permanent matches at time t. Similarly supplant µ(t) by

πt ≡ µ(t)[ut − L(θt)] +

∫ θt

0

xλ(dx) =

∫ θt

0

xλ(dx) +

∫ 1

θt

xut(x)λ(dx) (13)

This yields the natural definitions ut =∫ 1

0ut(x)λ(dx) and πt =

∫ 1

0xut(x)λ(dx),

mistakenly assuming that ut(x) = 1 for all x < θt. This assumption will not matterif any x quits her match at as soon as x = θt(x), which by assumption holds.

Lemma 5 (Laws of Motion) If θt is monotonically decreasing, then the triple

(θt, ut, πt) is differentiable and satisfies π0 =∫ 1

0xλ(dx), u0 = 1, and

θuπ

=

βθ + ρ(θu− π −∫ θ

0(θ − x)λ(dx))

−ρ[u− L(θ)]2

−ρ[u − L(θ)](π −∫ θ

0xλ(dx))

(14)

Equilibrium Analysis of the Permanent Matching. So long as I canshow that Theorem 2 is true, i.e. equivalence class permanent matching arises in

19

any search equilibrium, I can safely analyze the permanent matching dynamicsgiven by Lemma 5 in isolation of the what is happening outside Ct. For individualsgladly quit their matches the moment they enter Ct.

The fixed points of the dynamical system (14) as parameterized by θ are easilyfound to be

F0 = (θ, L(θ),

∫ θ

0

xλ(dx)),

But observe that

ut − L(θt) ≥ πt −∫ θt

0

xλ(dx) ≥ ut − L(θt)θt ≥ 0. (15)

Thus, the only feasible stationary point of the dynamical system (14) is the ‘all-

matched’ point (0, 0, 0). Does there exist a path starting at (θ, 1,∫ θ

0xλ(dx)) for

some θ ∈ (0, 1) and tending to F0? Note that the nested inequalities in (15) alreadyimply π ≤ 0 by economic feasibility, while the laws of motion (14) guarantee u ≤ 0.

Theorem 3 (Existence and Uniqueness) For any search equilibrium, thereexists a unique pattern of permanent matching, with an equivalence class Ct = [θt, 1)for all t ≥ 0, where 0 < θ0 < ρ/(ρ + β) and θt ↓ 0. The corresponding dynamicalsystem is saddle point stable.

A depiction of a typical threshold path is found in figure 4. The existence argu-ment is of sufficient interest that it soon follows. I omit the uniqueness proof, butit will follow from a careful examination of the system (14) in light of the construc-tive existence argument. The saddle-point character of the equilibrium will alsofollow from that argument, and the Jacobian calculation performed in the proofof Lemma 7.

Remark. In this outcome, types in [θt, 1) only look for permanent matcheswith one another. Thus, if supp(λ) ⊂ (ε, 1), then within finite time, all individualsbelong to a single equivalence class, so that all proposed matches are agreed to.By the same token, for any fixed ε > 0, there exists ρ > 0 small enough that allproposed matches are agreed to from the outset. Thus, the most of the interestingcase is precisely the one I am focusing on, with ε = 0.

5.5 An Existence Proof in R3

A wealth of results in dynamical systems theory is specific to two-dimensionalmodels (eg. the Poincare-Bendixson Theorem). Indeed, the staple tool for studyingdynamics is the phase plane diagram. Clearly, this is not an option with a three-dimensional system. I can, nonetheless, exhibit a geometrical method of proof justfor this case.

Suppose WLOG that λ is uniform Lebesgue measure on (0, 1), so that itsdensity satisfies ℓ(x) = x. Consider the triangular ‘wedge’ in figure 5 defined by

20

Figure 4: Simulated Equilibrium Threshold Path. This is drawn for β = 0.01,ρ = 1, and uniform λ (Lebesgue measure). Notice that θ < 0 and θ > 0.9

θ

time

20 40 60 80 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

temporary matches

permanent matches

0 ≤ θ ≤ 1, 0 ≤ u ≤ θ, 0 ≤ π ≤ 1/2. Let E be the initial ‘edge’ of the wedge definedby π = 1/2 and u = 1, i.e. no where one is yet matched (shaded dark in Figure 5). Ishall show that for the above dynamical system, there exists a unique economicallyfeasible path starting on E that eventually hits the origin (θ, u, π) = (0, 0, 0). Ishould remark that for generic flows through the wedge, such an occurrence wouldbe truly miraculous (i.e. that a path from a given line intersects a given vertex isa zero probability event)! Still, I demonstrate that just such a miracle occurs.

Thinking in R3 is hard. The idea is to reduce the problem to a planar one byfocusing on the faces F of the wedge. That is, I now show that the locus of ‘exits’from the wedge of the dynamical system starting on E passes through the origin.

Lemma 6 Let Π be the set of all paths starting on the upper edge E of the wedgehaving u = 0, π = 1/2. Let Π = Π ∩ F denote the locus of exit points of Π fromthe wedge. Then (0, 0, 0) ∈ Π.

I prove the Lemma in the appendix, but will give a sketch here. Consider apurely geometric exercise that would work if Π were continuous. Denote a paththat starts at (θ, 1, 1/2) ∈ E as x(θ). It’s easy to see that for θ0 near 1, θ(0) isjust below β > 0, and so x(θ0) hits F close to but below (1, 1, 1/2) on the “slopingface” (see figure 5) of the wedge; similarly, for θ0 near 0, θ0 is just above −1/2,and so x(θ0) hits F close to but below (0, 1, 1/2) on the “front face” of the wedge.

Put ρ = 1. All paths of the autonomous system (5) must exit the wedge on thetop face u = 1 where π < (1 + β)θ− θ2/2, or anywhere on the front face θ = 0, orthe lower sloping face u = θ where π < βθ + θ2/2. Hence, Π is confined to thesefaces, and so x(θ0) = (0, 0, 0) for some 0 < θ0 < 1 by continuity of Π.

21

Figure 5: System Dynamics in (θ, u, π)-space. The diagram depicts thedynamics of (14) for ρ = 1 and uniform λ, i.e. Lebesgue measure.

π

u

θ

(0, 0, 0)

‘all matched point’

(0, 0, 1/2)

(0, 1, 0)

(1, 1, 0) (1, 1, 1/2)

(0, 1, 1/2)

u=θ

π=βθ

+θ2 /2

, u=θ

π = (1 + β)θ − θ2 /2

← fixed point locus F0

5.6 Rate of Convergence to Equilibrium

The search frictions obviously prevent the market from clearing immediately. Itturns out that the convergence to equilibrium with the quadratic search technol-ogy is very slow: Rather than than a (negative) exponential convergence rate,convergence occurs at a hyperbolic rate.

Lemma 7 In the unique equilibrium, the fraction of workers not in permanentmatches satisfies ut = o(1/t).

The proof of the following result is in the appendix. In a word, the slow convergenceis a simple consequence of the fact that no eigenvalue of the associated dynamicalsystem is negative at the origin. The economic intuition is equally simple: Witha quadratic matching technology, even if individuals accept all matches, the flowmatching rate is proportional to the square of the mass of unmatched agents.That is, I can hope for no better (by an order of magnitude) than the above rateof matches. The message of Lemma 7 is that I can in fact asymptotically achievethis best. The appendix also contains an intuitive rationale for this result.

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5.7 Comparative Statics

First note that a rescaling of (β, ρ) only serves to renormalize time; therefore, itsuffices to investigate the comparative statics in β alone, for instance. Of primeinterest is what happens to the unique initial threshold θ0 as β changes, andespecially for β close to 0. Unfortunately, a closed theoretical answer is outsidethe realm of known analytic methods. Still, calculations on Mathematica stronglysuggest (see Figure 6) that the first best is approached as search frictions vanish.

Conjecture 3 The initial threshold θ0 is monotonic decreasing in the β/ρ, withlimβ/ρ→0 θ0(β/ρ) = 1.

I wish to briefly make the case that this yields a rather unintuitive prediction.Consider the world without any explicit source of time discounting, or β = 0. It isnot implausible that the declining quality level in the pool is would provide enoughimplicit discounting to secure an equilibrium. It does not. Using the machinery ofthis section, one can show that there can exist no search equilibrium.

As an application, the undiscounted GNP must approach the frictionless-best∫ 1

0x2λ(dx) as β/ρ vanishes. Indeed, if limβ→0 θ0(β) = 1 near u = 1, π = 1/2, then

(5) implies that limβ→0 θ′β(0) = 0. That is, the threshold descends very slowly, so

that most matches involve individuals who are arbitrarily close to one another. Itis also likely that the present value of GNP tends to

∫ 1

0x2λ(dx), so that the welfare

loss due to search frictions evaporates, but that is harder to show.It is easy to bound θ0(β) away from 1, and thus provide a lower limit on the

welfare loss (ignoring the welfare contribution made by temporary employment).Supposing that ℓ(x) ≡ x, I must have 0 < −2θ0(β) = ρθ0(β)2 − 2(ρ+ β)θ0(β) + ρ,which implies that θ0(β) < 1+β−

√

β2 + 2β. Note that the right hand side tendsto 1 as β ↓ 0. This bound concurs with figure 4, since θ0(.01) < .865.

Figure 6: Simulated Initial Thresholds as a Function of β when ρ = 1.

0.2 0.4 0.6 0.8 1

0.3

0.4

0.5

0.6

0.7

0.8

β

β(0)θ

23

6 TEMPORARY EMPLOYMENT

The analytical power of the identical vNM preference assumption now comes tothe fore as I shall henceforth simply assume the permanent matching of the lastsection is given. This shall serve as an anchor for all temporary matching decisions.Now one could imagine a model whereby temporary matching tautologically couldnot arise. Namely, suppose that xy were a one-shot only utility ‘flash’ instead ofthe flow output of the match (x, y). Then the analysis of the last section would bea complete characterization of the search equilibrium — that is, of who ultimatelymatches with whom. And in fact, the incomplete proof of Theorem 2 could bevery quickly finished.

With the maintained assumption of flow payoffs, individuals awaiting theireventual permanent match will desire to profitably use the interim time period.Given that time is of the essence (as β > 0), those types below Ct match early, andlater quit at least when they are absorbed in Ct. This will play out in a divergencebetween their average and flow unmatched values. The analysis of this section isinspired by the discrete time examples; however, it so happens that the effects Ihave shall describe really only surface in a four or more period model. Below thethreshold, the analysis becomes much richer and more complicated to flesh out.Because θt < 0, different types have distinct absorption times; consequently, thereare no equivalence classes below Ct. But more to the point for this section, everysuch match dissolution (for this reason) of x with y 6= x is non-mutual. Whetherquits arise for other reasons is something I wish to explore, as it is is a crucialdeterminant of the nature of the unemployment dynamics.

Intertemporal Effects. Let θ−1(x) denote the time that θt = x, withθ−1(x) = 0 if x > θ0. Now, an example in Smith (1999) shows that the thresholdpartner θt(x) vanishes as t ↑ θ−1(x). But Smith (1999) also proves that for fixed x,the threshold partner θt(x) is an upper semicontinuous functions of time t. Thus,for any x < θt, θt(x) is decreasing for t close enough to θ−1(x).

With the following result, this can also be extended to flow values.

Lemma 8 (u.s.c. Flow Values) For fixed x, the flow value ψt(x) is an uppersemicontinuous functions of time t.

Proof Look at formula (9). Since Qt(s; x, y) satisfies the Bayes-Markov property,so does Pt(s; x, y). But this implies that Pt(s; x, y) is upper semicontinuous in t:In other words, for fixed s, Pt(s; x, y) can jump up, but not down in t. Intuitively,since the quitting rule is memoryless, matches begun at close by times cannot betreated capriciously differently, unless the first match is simply unprofitable andis thus not okayed (i.e. forgotten about entirely). Moreover, the search dynamicsconstrain the unemployment rate ut(x) to be upper semicontinuous at all x, sinceit is never the case that a positive mass of any type ‘suddenly’ finds a match.After integrating and multiplying, the resulting expression for ψt(x) is also uppersemicontinuous in t.

24

I believe I can use equation (6) to show that the flow value ψt(x) also vanishesas we approach time θ−1(x). Together with upper semicontinuity from Lemma 8,this implies that for any x < θt, the flow value ψt(x) is decreasing for t close enoughto θ−1(x). But since ψt(x) =Wt(x) − Wt(x)/β, I must also have Wt(x) > 0 thentoo. Altogether, I have

Lemma 9 (Intertemporal Monotonicity) For all x < θt, both the flow valueψt(x) and threshold partner θt(x) are continuously decreasing in t and eventuallyvanishing t = θ−1(x), during which time the average value Wt(x) is increasing int, tending towards x.

Recall that any individual x < θt will quit any match at time θ−1(x). But byLemma 9, if a match with y ever makes sense for x, then it will continue to provideflow surplus until time θ−1(x). Hence,

Corollary (Once in a Lifetime Quits) Any x < θt never quits a match beforetime θ−1(x). As a result, she only quits a match at most once in her lifetime.

Observe in passing that with self-preference, if x ever does match with x, then thatmatch is permanent.

As both a result in and of itself, and a key tool in the analysis that follows, Iwould like to be able to make the following conclusion about optimal strategies.

Lemma 10 (Monotonic Preferences) For each x < θt, At(x) = [θt(x), 1), forsome θt(x) ∈ (0, θt).

Let me try to develop the intuition for this one by simply determining the accep-tance set of a given individual. To this end, let θt(x|s) be the minimum type withwhom x will agree to match at time t whom x knows will quit at time s > t. Inother words, it satisfies the dynamic arbitrage equation

Wt(x) ≡ (1− e−β(s−t))θt(x|s) + e−β(s−t)Ws(x)

Think of the threshold partner as coming with an eternal option, whereas θt(x|s)is more akin to the American options which can be exercised at any moment insome finite time period. This quantity is intuitively sandwiched between the flowvalue and threshold partner of x, or θt(x) ≤ θt(x|s) ≤ ψt(x). Lemma 10 is makingthe assertion that θt(x|θ−1(y)) < y for all y ≥ θt(x). In other words, the higherflow reward from x matching with higher types y more than compensates for theshorter period with which they are planning to remain matched with x.

One approach to proving the above result is rather blunt. Any type x is willingto match with any y ≥ ψt(x). So eventually ψt(x) < x, and individuals are willingto match with ‘themselves’. It would clearly suffice to establish that this self-preference holds from the outset for everyone. Since the value of type x starts atWt(x) = x and proceeds to decrease the moment she is absorbed by Ct, this is notat all implausible. On the other hand, low enough types can hope to temporarilymatch with very high types early on, and so ψt(x) could potentially be very high.The following lemma resolves the debate.

25

Conjecture 4 (Self-Preference) For any t and x, the flow value θt(x) ≤ x forall x < θt. Thus, [x, 1) ⊆ At(x), or in particular x is always willing to match withanother x.

My idea for the proof is to proceed by contradiction, and focus on the last timethat ψt(x) continuously slices through x from above. I haven’t yet managed toproperly formulate this.

Note that self-preference in particular rules out the discrete time ‘GrouchoMarx’ effect. It must purely stem from the lumpiness of discrete time.

Observe that no two distinct types have the same future opportunities, as theirdate of absorption into the top equivalence class is different. It is now within mygrasp to establish a result that seemed out of reach in section 5.

Lemma 11 (Value Monotonicity) For any t, if y < x, then Wt(x) ≤ Wt(y)with equality exactly when x > θt.

Proof Any y > x can mimic the acceptance strategy of x and assure herself ofx. This is a credible offer because her flow value if falling, by Lemma 9. But ystrictly prefers not to mimic x starting at time θ−1(y). Hence, Wt(y) >Wt(x).

Theorem 4 (Portmanteau Characterization) Any x < θ0 has a thresholdpath 〈θt(x)〉 that is differentiable and strictly monotonic decreasing, and satisfiesθ0(x) ∈ (0, x), θt(x)→ 0 as t ↑ θ−1(x), and θθ−1(x)−(x) = 0.

This theorem is partially summarized in Figure 7.

Figure 7: Conjectured Temporary Employment Dynamics. I claim thatevery type y’s threshold partner lies below that type, and monotonically decreasesto 0 as t ↓ θ−1(y). The schematic diagram is drawn for very low search frictions,so that θ0 is close to 1.

θt

θt(x)

θt(y)

y

x

t0

1−

26

Figure 8: Conjectured Cross-Sectional Temporary Matching. Individualssomewhere between 0 and θt have the highest threshold partner. For individualsx < θ∗t , this renders many individuals opportunity sets the union of two intervals,namely, the very high [bt(x), θt) and very low (0, at(x)) types.

θt(y)

at(x) bt(x) θty

x

θ∗t

0

Cross-Sectional Effects. By Lemma 9, individuals very close to but justbelow θt have very low flow values. Moreover, very low individuals with pooropportunity sets also intuitively have low flow values. On the other hand, thehighest flow value ought to be bounded below θt. This suggests the following basiccross-sectional result:

Conjecture 5 (Opportunity Sets) Let θ∗t = maxx∈(0,θt) θt(x). Then Ωt(x) =(0, θt) for any x ∈ [θ∗t , θt). For any x ∈ (0, θ∗t ), there exists 0 < at(x) < bt(x) < θt

such that Ωt(x) = (0, at(x) ∪ [bt(x), θt).

This result is partially summarized in Figure 8. Note that if self-preference doesnot obtain at time t, then it must be violated by the individual with the highestflow value, i.e. that y < θt for whom θt(y) = θ∗t .

Note that even with all the basic theory I have posited, with a separate thresh-old function θt(x) for all types x < θt, the resulting model only admits resolutionvia partial differential equations. Thus, the identical vNM assumption effectivelyrenders the analysis for much of the model an exercise in differential equationsinstead.

There is one other property worthy of note that follows from the results andconjectures of this section. As apparent in Figure 7,

Conjecture 6 (Single Crossing Property) If x < y < θt and θt(y) = θt(x),then θt(y) < θt(x).

A OMITTED PROOFS AND DISCUSSIONS

A.1 Proof of Lemma 5

I first note the following useful ‘folk result’ on hazard rates.

27

Lemma 12 (Hazard Rates) Consider an event which has a piecewise continu-ous flow probability h(t) of occurring at time t. The chance that it does not occur

by time t equals P (t) = e−ρ∫ t0 h(s)ds.

I have never seen an explicit statement or proof of this simple result. To see whyit is true, observe that P (0) = 1 and, for small ε > 0, P (t+ ε) = P (t)[1− εh(t)].Rearranging terms, and letting ε ↓ 0, I have P (t)/P (t) = −h(t) a.e. This integratesto ln P (t) = −

∫ t

0h(s)ds, yielding the desired result.

Now let’s return to establishing the equations of Lemma 5, which are centralthe paper. I am able to use the simplification Vt = θt since θ is nondecreasing.With no discounting, θt is an optimal threshold for the “eligible” individuals x ≥ θt

if the alternative of waiting provides the same expected eventual partner. But foranyone eligible, any (not necessarily acceptable) match proposed at time t providesan expected eventual partner’s index of

µt(1− L(θt)/ut) + θt(L(θt)/ut) = (πt +

∫ θt

0

(θt − x)ℓ(x)dx)/ut

= θt +

∫ 1

θt

(x− θt)ℓ(x)ut(x)dx

This expression reflects the L(θt)/ut chance that the match will be rejected, plus

the fact that µt ≡ [πt −∫ θt

0xℓ(x)dx]/[ut − L(θt)]. Integrating with respect to the

“next arrival density” implicit in Lemma 12, I discover

θt =

∫ ∞

t

ρuse−

∫ s

tρur+βdr(πs +

∫ θs

0

(θs − x)ℓ(x)dx)/usds

=

∫ ∞

t

ρe−∫ s

tρur+βdr

(

πs +

∫ θs

0

(θs − x)ℓ(x)dx)

ds

By the Fundamental Theorem of Calculus, θt is differentiable, and so the chainrule yields the desired optimality equation:

θt = (β + ρut)θt − ρ(πt +

∫ θt

0

(θt − x)ℓ(x)dx) (16)

Next notice that the flow probability that a given individual x will meet anacceptable partner x ≥ θt at time t equals ρ[ut − L(θt)]. Hence,

ut(x) =

−ρ[ut − L(θt)]ut(x) for x ≥ θt

0 for x < θt(17)

28

if there are no quits. Because individual x quits any other matches by time θ−1(x),ut(θt) ≡ 1. Differentiation of (12) is also permitted, yielding:

ut = θtL(θt) +

∫ 1

θt

ut(x)ℓ(x)dx− θtL(θt)

= −∫ 1

θt

ρ[ut − L(θt)]ut(x)ℓ(x)dx

= −ρ[ut − L(θt)]2

Similarly, application of (17) to (13) yields the law of motion for πt.

πt = θtθtL(θt) +

∫ 1

θt

xut(x)ℓ(x)dx− θtθtL(θt)

= −∫ 1

θt

xρ[ut − L(θt)]ut(x)ℓ(x)dx

= −ρ[π −∫ θt

0

xℓ(x)dx][ut − L(θt)]

A.2 Proof of Lemma 6

As noted, (0, 0, 0) is the only feasible stationary point of the dynamical sys-tem (5.4), and in fact the only point of F0 inside the ‘wedge’. Thus, there areno sinks inside the wedge. Each portion of the dynamical system (5), plus thejoin between the two, is infinitely-differentiable. It is an easy exercise to demon-strate that the resulting system must be Lipschitz. It follows by standard results(eg. Hirsch and Smale (1974)) that the dynamical system is continuous in initialconditions after any fixed finite time interval.

It is then possible to show that if the path x(θ) exits the wedge in some bound-edly finite time τ(θ) < τ for θ < θ, then the locus Π of exit points is continuous onF . But τ(θ) <∞ for some θ ∈ (0, 1), with x(θ) exiting on the front face θ = 0 forsmall θ. By letting τ → ∞, I see that τ must be continuous whenever it is finite.Since the only stationary point of the dynamical system on the front face of thewedge is (0, 0, 0), if τ(θ) −→ ∞ as θ −→ θ′, then necessarily xt(θ) ↑ (0, 0, 0), asrequired.

A.3 Proof of Lemma 7

Intuitive Justification. Consider the following corollary to Lemma 12, whichis an analogue of a familiar probability result.10

10As written, the lemma is not quite in standard form. The garden variety Borel-CantelliLemma, a bulwark of modern probability theory, states that if A1, A2, . . . is an infinite sequence ofindependent events, then Pr[Ai infinitely often] is zero or one as Σ∞

1Pr[Ai] <∞ or Σ∞

1Pr[Ai] =

∞. The result deals with the case where Ak is the event that the individual meets a prize in Sk

at time k.

29

Lemma 13 (Poisson Borel-Cantelli Lemma) Let a sequence of events A orB occur according to some stationary Poisson process. Let the draws be indepen-dent at the realized Poisson times t1, t2, . . .. Assume that conditional on an eventoccurring at any time t ≥ 0, that the chance that it is an A is g(t). Then A isnever observed with positive probability if and only if

∫ ∞

0g(t)dt <∞.

Think of A as the event that an unmatched partner arrives at a potentialmeeting. It is easy to see that the measure g(t) of Ct must satisfy this inequality.Indeed, the fastest that g can vanish is if all individuals in Ct agree to all matcheswith one another, i.e. g′(t) = −ρg(t)2. This yields g(t) = (ρt − g0)

−1, so thatclearly

∫ ∞

0g(s)ds =∞.

Serious Proof. As before, first suppose that L(x) ≡ x for simplicity.Consider the fixed points F0 of the system (5). Next to determine the stabil-ity properties of (0, 0, 0), I shall first linearize everything. That is, write (5) as(θ, u, π) = Dℓ(θ, u, π) · (θ, u, π), where

Dℓ(θ, u, π) =

β + u− θ θ −12(u− θ) −2(u− θ) 0

π + θ(u− 3θ/2) θ2/2− π θ − u

.

This immediately yields

Dℓ(0, 0, 0) =

β 0 −10 0 00 0 0

. (18)

The Jacobian matrix (18) shows that near the origin, the linearized systemexplodes except along the plane π = βθ. Consequently, in the non-linear system(5), in any equilibrium, I must have near the origin π = βθ+ o(θ). Combining thiswith (15) yields u ≥ θ+ π− θ2/2 = (1 + β)θ+ o(θ), so that for t large enough, i.e.for θ small enough,

[ut − L(θt)]/ut ≥ 1− θ/[(1 + β)θ + o(θ)] ≈ β/(1 + β),

for some ε > 0. Given the law of motion (5), (19) yields ut/u2t ⋖−ρ(β/(1+β))2 for

large enough t. Integration then yields ut ⋗ (1 + β)2/(ρβ2t+ c) for some constantc > 1. On the other hand, (14) easily implies that ut/u

2t > −1ρ for all t ≥ 0, i.e.

ut < c1/(ρt) for some constant c1 > 1. As previously alluded to, a higher ρ merelyresults in faster convergence by a factor 1/ρ, yielding a lower unemployment rate.

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