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P erfect M atching and Search in

E conom ic M od els

Ja n E e c k h o u t

Thesis submitted for the degree of Doctor of Philosophy

The London School of Economics and Political Science

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UMI Number: U 615810

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A b stract

T his thesis uses general m atching techniques - b o th perfect m atching and search - to stu d y some problem s in economies th a t are characterised by h et­ erogeneity of th eir agents. Here, m atching in its broadest sense is interpreted as a form of tra d e th a t is strictly lim ited betw een two partners: tran saction s are one-to-one, betw een one buyer and one seller exactly.

The first p a rt proposes a framework th a t integrates two well docum ented strand s of th e existing economic literature. I t is a search m odel th a t gener­ alises th e frictionless perfect m atching m odel to a context w here tra d e does not occur instantaneously. A general m ethodology w ith proof is given w hich allows us to derive th e unique equilibrium allocation of agents. T hough th e lim it case w ith o u t friction reproduces th e perfect m atching result, w ith friction results deviate substantially from conclusions in b o th th e perfect m atching literatu re and th e search literature.

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C on ten ts

A b s t r a c t 2

L is t o f F ig u r e s 5

A c k n o w le d g e m e n ts 6

I n t r o d u c t i o n 7

I

S ea rch

15

1 B il a te r a l S e a rc h a n d V e r tic a l H e te r o g e n e i ty 16

1.1 The Basic M o d e l ...20

1.2 The R esults: Existence and U n iq u e n e ss ...25

1.3 D iscussion of th e R e s u l t s ... 32

1.4 Perfect M atching E q u iv a le n c e ...37

1.5 Concluding R e m a r k s ... 39

1.6 A p p e n d i x ... 41

2 S o m e A p p lic a tio n s 51 2.1 B a c h e l o r s ... 52

2.2 One R eason W hy You D o n ’t W ant to Be a M ember of a Club th a t W ants You as a M e m b e r ... 57

II

P e r fe c t M a tc h in g

62

3 U n iq u e n e s s a n d N e g a tiv e A s s o r t a tiv e M a ti n g in T w o -S id e d M a tc h in g 63 3.1 Uniqueness in a two-sided perfect m atching m o d e l... 65
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3.3 The Assignment g a m e ... 73

3.4 Concluding R e m a r k s ...75

4 W o rk in g fo r a B e t t e r J o b 77 4.1 The Basic Model and R elated L i t e r a t u r e ... 81

4.2 The M ain R e s u lts ... 90

4.3 Efficiency and D is trib u tio n ...97

4.4 Com paring Turnover R e g i m e s ... 102

4.5 Extension: Life. C y c le ...107

4.6 Concluding R e m a r k s ... 109

4.7 A p p e n d i x ...112

5 E d u c a tio n a l M o b ility : T h e E ffe c t o n E ffic ie n c y a n d D i s t r i ­ b u t i o n 114 5.1 The Basic M o d e l ... •...117

5.2 The R e s u l t s ... 122

5.3 The repeated g a m e ... 128

5.4 C o nclusio n... 135

5.5 A p p e n d i x ...137

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List o f F igures

1.1 R eservation S t r a t e g i e s ...29

1.2 P a r t i t i o n i n g ...33

1.3 D ow nw ard Sloping R eservation S tra te g ie s ... 36

1.4 No P a rtitio n in g out off Steady S t a t e ...48

2.1 B a c h e l o r s ... 55

2.2 R ejection of M atches w ith E q u a l s ... 60

5.1 Cases w ith Different E q u ilib ria ... 124

5.2 M ultiple N a sh E q u ilib r ia ... 126

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A cknow ledgem ent s

I would like to th a n k my supervisor Professor K evin R oberts. He h as gener­ ously given me exp ert advice and consistently guided me in th e m ore fruitful direction. I have been greatly influenced by his thinking. B ut above all, his concern and em pathy have given me th e feeling of security and confidence from th e s ta rt u n til th is stage of th e project.

I have also benefitted from discussions and com m ents on different p a rts of this thesis from a num ber of people. I would like to th a n k in p articu lar Steve A lpern, T im Besley, Francis Bloch, Michele Boldrin, K en n eth B u rd ett, D avid Cass, M arco Celentani, M elvyn Coles, Leonardo Felli, R aq u el F ernan­ dez, Francisco Ferreira, R ichard Freem an, M aitreesh G hatak, U rs Haegler, Nobuliiro K iyotaki, George M ailath, M arco M anacorda, A lan M anning, Jo h n H ardm an Moore, G ustavo N om bela, Canice P ren d erg ast, R afael R ob, Sher- w in Rosen, Lones Sm ith, R an d all W right.

I have received financial su p p o rt in th e form of a Junior Fellowship from th e Royal Econom ic Society an d a grant from th e B ritisch Council.

My thesis w as m ainly w ritte n while I was a t th e C entre for Econom ic Perform ance. I would like to th a n k R ichard Jack m an for encouraging me to enrol in th e program m e. T he C entre provided a w arm atm osphere an d I re­ m ain w ith num erous friendships and good memories. I have also enjoyed th e generous hospitality of U niversidad Carlos III de M adrid and th e E u ro p ean University In s titu te in Florence in th e Sum m ers of 95 and 96 respectively.

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In trod u ction

M any behavioural and m arket interactions studied by econom ists are char­ acterised by a one-to-one transaction. In a m onogamic m arriage m arket for example, exactly one p a rtn e r of one sex ” trad es” w ith one p a rtn e r of th e other sex. A n individual’s decision in th e m arket is th e choioe of a p a rt­ ner, ra th e r th a n th e mere choice of th e quantity consum ed from whoever obtained. A ssociated w ith a p artn er is a consum ption level. Similarly, th e one-to-one tran sac tio n accurately describes job m arkets. Firm s an d workers b o th choose exactly one p artn er to engage in production. T his also applies to a large num ber of more trad itio n al economic choioe situations. T he choice of a college for education, for exam ple, or th e purchase of housing: a n addi­ tion al un it (an ex tra course or an additional square m eter) cannot be bought from a separate seller. All these tran sactio n s are characterised by th e same feature: exchange is betw een two p artn ers only.

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parties have to be willing to trade. B oth features, heterogeneity an d th e bi­ lateral agreem ent of a transaction, are present in th e W alrasian m arket, b u t because of triv ial allocation decisions, agents are indifferent. T he indifference is entirely generated by th e price mechanism th a t h a s equal un it prices across agents. In th e presence of B ertran d com petition, th e outside optio n of a n epsilon un it disciplines prices across agents.

M atching is of course not new. There is a long tra d itio n in economic re­ search dealing w ith m atching. Two m ain stran d s of th e economic literatu re have independently focussed on m any related issues: th e work on perfect m atching and th e search literature. Perfect M atching was introduced by Gale and Shapley [27] in th e early sixties. T hey observed some stro n g be­ havioural regularities in th e m arket for hospital physicians. T heir pioneering work involved m odelling th is m atching m arket. In subsequent research1, it was concluded th a t th e preference stru ctu re is of crucial im portance for th e equilibrium allocation, more so th a n th e presence of transferability of u tility 2. The stan d a rd model, w hether it be w ith or w ith o u t transferable utility, features b o th th e ingredients m entioned above: b ilateral agreem ent and heterogeneity. The second stra n d of th e literatu re, search or im perfect m atching (i.e. m atching w ith frictions), acknowledges th a t tra d in g opportu­ nities arrive a t a certain cost. T he im plication is th a t th e intrinsic utility from consum ption is no longer a perfect indicator of value. Value in addition is determ ined by th e cost of creating th e tra d in g opportunity. Such a cost can be in terpreted as th e cost of gathering inform ation, or as th e opportunity cost of w aiting for th e right tra d in g opportunity. Search does n o t intrinsi­ cally and necessarily involve heterogeneity and bilateral agreem ent3. This

tyor an overview, see Rotli and Sotomayor [62]. 2As in tlie model proposed by Becker [7].

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however implies there is no decision of choice: u p o n m eeting, a tra d e always takes place. T he subsequent search literatu re has incorporated b o th these features separately4.

The aim of th is thesis is to make contributions in two areas. In P a r t I, a unifying approach is proposed to b o th stran d s of th e literatu re, perfect m atching and search. T he objective is to establish how th e characteristics of a generalised m atching m odel w ith search frictions differ from th e existing results in th e literature. In P a rt II, perfect m atching m odels are studied. T his thesis tries to make a contribution in th e area of endogenous choice of characteristics of heterogeneous types. T his is new in th e lite ra tu re and th e presence of m atching proves to yield results which differ su b stan tially from th e W alrasian benchm ark. T hroughout, th e theory is employed to explain economically relevant phenom ena. It is constantly argued th a t m atching pro­ vides a good description of m any phenomena, and th a t it highlights issues of heterogeneity w hich are no t present in th e stan d ard neoclassical approach. It provides a theoretically rigorous framework in which thin k in g a b o u t h et­ erogeneity is natu ral. T he applications th e n are always to be in terp reted in term s of d istrib utio n of heterogeneous types. T he m ain underlying social agenda is to stud y th e effect of distributional considerations on efficiency.

P a rt I consists of two C hapters. The first C h ap ter describes th e m odel and derives th e m ain results. The second C hapter considers some applica­ tions of th e m odel w ith surprising results. A search m odel is proposed of th e m arriage m arket betw een two sets, m ales and females, and it incorporates th e

4Heterogeneity has been used, amongst others, in Jovanovic [32], Diamond [18] and

Pissarides [56]. Bilateral agreement is prominent in the literature on money and search

(Iviyotaki and Wright [33]) where a double coincidence of wants is necessary for trade. In

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two m ain features present in perfect m atching models (i.e. w ith o u t search frictions). B ilateral search (i.e. search by both th e m ales and th e females) and ex-ante vertical heterogeneity (some types are preferred to others and all agents rank types identically) ensure th a t th is m odel is th e generalised version of th e perfect m atching model. I t differs from tra d itio n a l search mod­ els because it jointly incorporates b o th these features. A s m entioned above, one approach (Kiyotaki and W right [33]) does have b o th features, b u t th eir specific notion of horizontal heterogeneity (i.e. each ty p e prefers th e types nearest to her own type) implies th a t all agents have an identical strategy. The novelty of th e results is threefold. F irst, a general m ethodology follows from th e proof of existence and uniqueness of th e equilibrium . T he m ethod allows for th e derivation of th e equilibrium allocation w hatever th e u tility fu n ctio n This includes th e emergence of disconnected sets of m atching for certain preferences. Second, a set of preferences is derived (i.e. satisfying m ultiplicative separability) th a t results in th e p artitioning of b o th distribu­ tions. T hird, it is shown th a t th e lim it case of th e search m odel w ith th e friction disappearing is th e perfect m atching model. T his is new in th e sense th a t it deviates entirely from th e existing models of search. N ot only are ex­ isting models no t th e generalised version of th e perfect m atching model, they could never exhibit phenom ena such as disconnected sets since heterogeneity only exists ex-post5. In addition, our m odel differs in more th a n one respect from research th a t was conducted sim ultaneously and th a t emerged after our results were found6. F irst, th e proof is more general and in addition, it provides an intuitive m ethod for solving for th e equilibrium allocation. Sec­ ond, th e m odel allows us to show th e equivalence w ith th e perfect m atching

5See for example Diamond [18] and Pissarides [56].

6See for example McNamara and Collins [42], Burdett and Coles [13], Blocli and Ryder

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model. Third, it has a more general result on partitio n in g th a n B u rd e tt and Coles [13]7. One result th a t appears in th e other work and th a t is absent in ours is the derivation of multiple endogenous distributions of singles.

In th e second C hapter, th e model of C hapter 1 is used to stu d y two applications th a t have surprising results. F irst, it is shown th a t bachelors may exist. Bachelors are th e lowest types of one of th e tw o distributions th a t rem ain eternally unm atched. This is surprising since in th e perfect m atching literatu re8, no one remains unm atched if b o th populations are of equal size (and provided a m atch yields more utility th a n being single). T he result here is due to th e difference in average length of search of b o th sexes. If one sex, say th e females, on average searches longer th a n th e males, th e n th e lowest male types will never be matched. T hey involuntarily rem ain bachelors. T his follows from th e fact th a t m atches are pairwise and th a t as a result, th e num ber of agents m atched per un it of tim e is equal in b o th sets. T he second application abandons th e assum ption th a t m atched p artn ers are draw n from two disjoint sets and considers pairwise m atching from one set. T he example used to illustrate th e argum ent is the m atching of tennis players as sparring partners. The result derived is th a t for certain preferences, it m ay be th e case th a t some types refuse to play w ith types identical to themselves. The reason is th a t w hen higher types are more im patient, th ey will be willing to accept m atches w ith low types. Such a low type, being more p atient, can afford to w ait u ntil th e more preferred higher types arrive an d can refuse to m atch w ith equals. This provides one reason why you m ay n o t w ant to be a m em ber of th e club th a t w ants you as a member.

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perfect m atching models. In C hapter 3, an extension is m ade to th e exist­ ing literatu re on perfect m atching w ith ordinal preferences9. T his literatu re has derived a large num ber of results o n the existence of stable m atchings for general preferences. The contribution of th is C hapter is to identify a class of preferences for which the stable m atching is unique. T his is useful from a purely theoretical point of view, since quite a large share of th e non- cooperative game theory is concerned w ith uniqueness. More im p o rtan tly however is th a t th e class of preferences identified is wide and includes th e ones w ith th e m ost economic relevance. Moreover, th e result bears some resemblance to single-peakedness, even th o u g h th is is n o t entirely equivalent in a model w ith agents from disjoint sets who have preferences over different objects (i.e. th e types of th e other set). This C h ap ter also m akes a more philosophical point. It is shown th a t if assortative m atin g is defined on th e preferences, th e n an allocation can never be negatively assorted. I t is argued th a t assortative m ating, i.e. th e m ating of likes, necessarily has to be defined on th e individuals’ preferences.

C hapter 4 considers the perfect m atching m odel w ith transferable utility. This does n o t differ substantially from th e m odel w ith ordinal preferences, b u t it allows for th e derivation of pay-off functions from a jo in t surplus th a t is split betw een th e p artn ers of a m atch. The pay-off functions them selves are entirely derived from th e surplus function and th e allocation in equilibrium for a given surplus. Becker [7] shows th a t for a surplus function w hich has a positive cross p a rtia l derivative w ith respect to th e ty p es of b o th sets (i.e. they are strategic complements), th e equilibrium exhibits positive assortative m ating. T he allocation is negatively assorted w ith a negative cross p a rtia l and agents are indifferent if th e cross p a rtia l is zero. I t is derived in C h ap ter

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4 th a t in th is case, th e distribution of ty p es enters th e pay-off function. T his follows from th e argum ent made above: in a m atching m arket w ith one-to- one transactions, the allocation is non trivial. T he innovative contribution is to consider th e endogenous choice of th e characteristics of a type. The results th a t are shown have a strong bearing on th e dependence of th e pay­ off function o n th e d istribution of types. In th is C hapter th is fram ework is em bedded in th e labour m arket where workers m atch w ith jobs.

The central premise is th a t a d istribution of heterogeneous jobs exists. T he im plication is th a t a worker’s productivity differs in different jobs. This deviates from m ost of th e stan d ard analysis where it is assum ed th a t p ro ­ ductivity is entirely embodied in th e worker. T his implies th a t th e allocation of workers to jo b s is non trivial. Once m atched to a job, a worker chooses th e level of effort to exert. I t is shown th a t in a repeated game, and w ith perfectly observable productivities, th e level of effort is super optim al. This is th e case if current effort affects th e future type: perform ing well now makes a worker more productive in th e future. In effect, current effort en­ dogenously determ ines th e future characteristic w hich in tu r n determ ines th e future equilibrium allocation. The inefficiency result is entirely in contrast w ith th e neoclassical model of effort choice, where th e equilibrium level is optim al. I t does bear some resemblance to th e re p u ta tio n and principle- agent lite ra tu re 10, b u t th e results stan d w ith perfect observability! Here, th e result is derived from th e rank dependence of th e pay-off function in a m atching model. This follows from th e no n triv ial allocation of workers to jobs. W ith endogenous choice of characteristics, workers play a R ank-O rder Tournam ent: current effort increases th e rank in th e future d istrib u tio n of types. In addition, th e policy im plications would be th e opposite: a ta x

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in th e rep u tatio n m odel reduces th e revelation of inform ation and hence is inefficient; an income ta x in th e m atching model, w hich exactly off-sets th e rank effect, improves efficiency.

The m atching framework w ith endogenous choice of characteristics (i.e. the worker’s ability in th e labour m arket) has a nice interpretation. Effort is viewed as a to o l to gain prom otion, i.e. to get a b e tte r job. Two fu rth er results are shown. F irst, inequality has an am biguous effect on th e choice of effort. Second, a higher ra te of turnover increases inefficient effort. All these results are m atched w ith a num ber of stylised facts from th e em pirical effort supply literature.

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P art I

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C hapter 1

B ilateral Search and V ertical

H eterog en eity

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acceptance. Tlie second highest ranked m ale would like to he m atched w ith th e highest type female, b u t she will n o t acoept m arriage. She can do b e tte r being m atched w ith the highest type male. W ith search frictions, th e to p female will accept males over a certain range, since w aiting to o long is costly.

The m ain contribution of th is C h ap ter is to show th a t provided th e dis­ trib u tio n of singles is stationary, a (Nash) equilibrium allocation exists and is unique. T his is tru e for any specification of th e utility function. T his result is surprising in two respects. F irst, uniqueness. T he strateg ies of one sex are m onotonic in th e strategies of th e other sex, so a continuum of s tra te ­ gies would he expected. However, an argum ent of ite ra te d elim ination of dom inated strategies only leaves one strateg y to survive. Second, existence. W hatever preferences are assumed, th e allocation can always be found using th e recursive elim ination m ethod. This can give rise to th e existenoe of quite unexpected m atching sets.

Given existence and uniqueness, three additional results are derived. F irst, th e d istrib u tio n of types is endogenously p artitio n ed for preferences th a t are m ultiplicatively separable. T his is unexpected since preferences are type-dependent, while by definition of endogenous partitio n in g , strategies are not. Second, for some preferences, m atching sets are disconnected. This implies th a t you are rejected w hen you propose a m atch w ith some types, even th o u g h you are accepted by b o th higher and lower types. Finally, th e m odel is show n to be robust. W ith frictions disappearing, th e equilibrium allocation coincides w ith th e equivalent allocation in perfect m atching.

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Mc-N am ara and Collins [42], B u rdett and Coles [13], and Bloch and R yder [12] all have these specific u tiltity functions. All of th e m derive th e p artitio n in g result since th eir preferences are a lim it case of m ultiplicative separability. The result derived here applies to a more general class of u tility functions. The general partitioning result has independently b een discovered by S m ith [69]. Sm ith also looks a t general preferences b u t provides a different proof and solution. The m ain novelty of th e approach here is th e intuitive appeal of th e proof and its wide applicability to any u tility specification. T he fact th a t th e equilibrium is shown to exist in a stro n g concept like iterated dom­ inance provides significant behavioural foundations for b o th th e resulting equilibrium allocation and th e m ethod or algorithm of obtaining it.

T his appealing and intuitive m ethod and solution is derived under th e assum ption of a statio n ary distribution of singles. A sim ilar approach is adopted in M cN am ara and Collins [42], and Bloch and R y d er [12]. Endo- genising th e d istrib u tio n in itself does not pose any problem (this is done in th e A ppendix). The problem is to find a fixed poin t for th e equilibrium distribution and th is goes a t th e expense of th e intuitive derivation of th e equilibrium allocation1. B u rd ett and Coles [13] show th a t due to thick m arket externalities, for some param eter values m ultiple stead y sta te distributions can be supported in equilibrium . O ur contribution is to show th a t given a distribution of singles, th e allocation is unique for any preferences.

The generalisation of th e perfect m atching m odel to th e search m odel is very m uch modelled in th e tra d itio n of th e literatu re. T he m ain aspect how­ ever is th a t b o th sides of th e m arket search (i.e. th ere is b ilateral search), and th a t th ere exists an ex ante heterogeneity of th e types. In th is m arriage model, individuals of one sex will m eet p o ten tial p a rtn e rs a t random and

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there are only a lim ited num ber of m eetings per u n it of tim e. Given perfect inform ation, th e type of th e po ten tial p a rtn e r is observed u p o n m eeting and can be accepted or rejected. If th e ty p e is to o low, it m ay pay to w ait until a higher type is m et. Acoepting, however, only implies th a t a m atch m ate­ rialises provided there is a double coincidence of w ants (i.e. th e other p arty decides to accept as well). In the presence of b ilateral search, th e decision to form a m atch cannot be enforced unilaterally. T he u tility derived from a m atch is represented by any cardinal u tility function th a t satisfies V ertical Heterogeneity. The m odel considered features N on-Transfer able Utility.

In th is framework, agents will choose strategies to accept or reject po­ ten tia l p artn ers th a t come along in order to maximise th e value function of searching. E ntirely counterintuitive, th e uniqueness and existence result derives from th e fact th a t th e equilibrium solution can be solved for, using a n iterated strict dominance argum ent. T he in tu itio n is th a t w ith V ertical Heterogeneity, th e to p types of b o th sexes are m ost desired by all, so th ey can be assured to be accepted by all types. Hence, th ey have a n iterated strict dom inant strategy. Given these strategies, th is argum ent equally ap­ plies to th e next b u t to p types, etc. In th e presence of search frictions, th ey have to accept a range of types w ith positive mass, so th a t a finite num ber of iterations will suffice.

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possible equilibrium outcomes in function of th e pay-off specification. The equivalence of th e m odel w ith th e Gale-Shapley-Becker m odel is rigorously proved in Section 1.4. Some concluding rem arks are m ade in Section 1.5. The A ppendix provides a proof for th e m ain Proposition an d derives th e endoge­ nous d istribution of singles. It is also shown th a t even w ith m ultiplicatively separable utility functions, strategies are type dependent o u t of Steady State and th a t th e Steady S tate ’’never arrives” .

1.1

The B asic M odel

Consider two disjoint sets of infinitely lived individuals: females and males. They are intrinsically heterogeneous in type. Only one dim ension of h et­ erogeneity will be considered, so th a t th eir ty p e can be represented by one variable 9. T his type can be interpreted as a m easure of eith er beauty, w ealth, sexual attractio n , etc. or as a composite m easure of all those char­ acteristics. Females and males are distinguished by 9 j and 9m respectively. B oth populations of singles are cumulatively d istrib u ted according to Fi(9) over 0{ = [£i5 #{], i E { /, m ] (w ith fi{9) th e density function) and have equal measure one.

Individuals can be in two possible states. T hey can either be m atched to a p artn er or th ey can be single. W h en single th ey are looking for a p a rtn e r to be m atched to. P a rtn e rs of a different sex m eet randomly, and u p o n m eeting they can perfectly observe th e type of th e o th er sex. A t th a t m om ent, th ey will decide w hether to acoept or reject a m atch w ith th e p a rtn e r m et. A m atch is m aterialised only w hen b o th p artn ers decide to accept each other. The decision is bilateral and cannot be enforced unilaterally.

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state of being m atched on th e other hand brings all p o ten tial pleasure th a t exists in th is world. I t is modelled as a n instantaneous u tility derived a t the mom ent th e m atch is formed, i.e. w hen b o th individuals decide to ac­ cept the m atch th e m arriage is instantaneously ” consum (m at)ed” . T he non- transferable u tility to a n individual of sex i characterised by ty p e 9{ from being m atched to a ty p e 9j is Ui(9j, 9{), w ith u continuous an d in general no sym m etry is required Ui 7^ Uj. Preferences exhibit V ertical H eterogeneity,

ilq. > 0. This implies th a t there is a ranking of th e types of th e o th er sex on which all individuals agree. All m en agree th a t Ju lie tte Binoche is th e m ost beautiful w om an and all w om en have no doubts ab o u t w ho is th e least endowed man. N ote th a t u tility is type dependent w itho u t any restrictions. Showing an equilibrium exists in th e presence of a general u tility specifica­ tio n is exactly th e objective of th is paper. Clearly, utility is cardinal, since a search model intrinsically p u ts a cardinal value on th e tim e of search. The general utility specification allows for any cardinal value of th e vertically h et­ erogeneous preference orderings as long as the values are bounded: u(0i) > 0 , u(9i) < oo2.

Typically, in a search environm ent it is recognised th a t th e instantaneous u tility from ” consum ption” does no t m easure th e exact satisfaction, sinoe it does n o t take into account th e (in) direct cost incurred during search. In a search model w ith prices, price is no longer a perfect indicator of th e value, as is th e case in th e neoclassical model. Search models do however use th e neoclassical tools by collapsing instantaneous u tility and search costs into Value functions, using some m echanistic representation of a environm ent w ith friction. Here, a con stan t re tu rn s to m atching3 search technology is specified

2Iu what follows, tlie notation u(*) may be nsed to signify Ui(-,6i) wlien tliere is no

confusion possible.

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as follows. W h en single, an individual bum ps into someone of th e other sex w ith probability (3. T his arrival rate (3 is distributed according to a Poisson process. Infinitely lived agents are no t m atched for life. W ith probability a , a m atch is dissolved4. For th e purpose of th is chapter, on-the-job search, endogenous separation and polygamy are ruled out.

Crucially, not all poten tial p artn ers m et will yield a match. In th e first place, an individual m ay not be entirely satisfied w ith th e type of th e other sex and will prefer to search further u ntil a more preferred ty p e is met. Second, an individual m ay be very willing to enter a m atch, h u t th e po ten tial p artn er may w ish to postpone th e m atch. A n individual’s stra teg y will be determ ined subject to being accepted, so in th e first instanoe, a strateg y of an individual will be determ ined tak in g th e strategies of all oth er players as given. A n equilibrium will th e n be a rule such th a t a n agent maximises th e value function taking into account th a t all other agents ad o p t such a maximising strategy.

A n individual’s optim ising strateg y will be derived from m axim ising th e value functions Vq and V\ in b o th possible state, th e value for being single and m atched respectively. They will in general be different depending on th e type The value functions of b o th states are w ritte n in th e form of Bellm an equations which give the current option value, given a positive in terest rate r.

rVo(9i) = (3 m ax Ee . [0, u(9j,9i) -f V1(9i) - V0{9{) | given acceptance by 9j\

______________________________ ( i . i )

proportional to tlie number of individuals searching. As a result, tlie number of meetings

per person is constant.

4 Modelling finitely lived agents witli an exogenous inflow of new birtlis yields the same

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rVi {9i) = a[V0(9i) - V i {9i)} (1.2) N ote tlia t being single has a positive option value associated w ith it even th o u gh being single does not yield any intrinsic instantaneous utility. The reason is of course th a t th ere exists th e probability of being m atched a t some future point in tim e. W h en single, a po ten tial p a rtn e r is m et w ith probability /?. The type of th e p a rtn e r is random ly draw n from th e pool of singles. Provided th e ty p e 9j accepts th e m atch, th e instan tan eou s u tility derived is u(9j, 9i). M arriage will be proposed if being m atched to 9j yields a higher utility th a n th e value of looking further until a m ore preferred ty p e is m et. This is th e case w hen u(9j, 9{) -\-V\{9i) is higher th a n th e expected value of rem aining single Vo(9i). Since separation occurs w ith fixed probability a , th e option value of being m atched is given by a tim es th e residual value of switching from being m atched to being single.

The decision of an individual of ty p e 9i is either to accept or reject a type 9j th a t is m et. We will represent th is by th e binary variables fti(9j, #{) which is defined as 7Ti(9j,9i) = 1, if a m atch w ith 9j is accepted by 9 and 7Ti(9j,9i) = 0, if it is rejected. Clearly, acceptance does n o t necessarily imply th a t a m atch materialises, given th e bilateral n atu re of th e decision to form a m atch. A type of th e other sex 9j accepts a type 9{ if 7Tj(9^ 9j) = 1. W h eth er a type 9\ is accepted is given by th e inverse function of 7Tj(9i,9j). Hence, once any potential trad in g p a rtn e r is m et, th e m atch is m aterialised w ith probability ijji(9i)

^ ( 0 0 = / 7ri (x,9i)7rj (9i,x)dFj (x) (1.3) JQj

where Fj{9j) is th e cumulative density function of singles in th e m arket.

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measure of people searching is not equal to the measure of the population.

More importantly, since in general not all types have the same strategy (i.e.

the strategy is type dependent), the distribution of singles Fi(9i) is not equal to the distribution of the entire population, say Hi(9i). In the Appendix, the

relation between the H and F is derived. All our results go through with an

endogenously derived distribution of singles, both in the steady state and out

of the steady state.

There are however two reasons why our results are derived under an ex­

ogenously given distribution of singles F . First, we do not have a proof for

a fixed point of this distribution. Second, there is a source of multiplicity of

steady state equilibria which is independent of the potential multiplicity this

paper shows not to exist. Burdett and Coles [13] provide an example where

separate beliefs about the steady state distribution can be supported in equi­

librium. This multiplicity is due to thick market externalities in the search

technology, very much as in Diamond [18]. The main contribution of Propo­

sition 1 below is to show that given a distribution of singles (of which more

than one may exist), there exists a unique equilibrium allocation. Below, it

will become apparent that that in itself is a most nontrivial result.

E qu atio n (1.1) can now be rew ritten in term s of th e binary variables 7t* and 7Tj and th e distribution of single males and females F(9{] and F(9j).

rVo{Oi) = [3 [ 7Ti(xj 9i)nj(9i, x)[u(x, ft) + Vi (ft) - Vo(0i)]dFj(x) (1.1’) J&j

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as a reservation strategy: a type 9j is offered m arriage if u(9j,9i) -fi Vi(9i) — Vo(9i) > 0. The reservation strateg y implies th a t for each 9{ th ere m ust th e n be a critical ty p e 9j = (j)j(9i) w hich solves th e equation

u f a ) > Vo(0i) - V ^ ) (1.5)

R e m a r k 2 The reservation strategy restricts the strategy space since it rules out strategies where lower type males choose to reject a high type female

because they know they will be rejected anyway. That would yield a degenerate

equilibrium where everyone rejects everyone. Because we impose the strategy

”accept all types for which the expected value of a match is higher than the

marginal type”, high types cannot be rejected strategically.

A n optim al strategy 7Ti(9j, 9{) will be determ ined in function of th e critical type (f)j associated w ith th e strateg y (1.5). A n Im perfect M atching Equilib­ rium can now be defined using th e notion of N ash equilibrium . It is a list of optim ising strategies taking into account th a t all other agents use th eir optim ising strategy.

D e fin itio n 1 For given distributions of singles Fi and F j, an Imperfect Matching Equilibrium is a list (^i(9j,9i),iTj(9i,9j)), W9i G 0 i, 6 Oj sat­ isfying:

1. Equations (1.1*) and (1.2);

2. The reservation strategy (1.5).

1.2

The R esults: E xistence and U niqueness

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Lemm a 1 claims th a t a reservation stra teg y implies all types 9j above th e critical type <pj are accepted and all th e types below are rejected. Lem m a 2 prooeeds to prove th a t there is a unique reservation strateg y holding th e strategy of all other players constant. Lem m a 3 shows th a t th e reservation strategy has a n upper and a lower bound. W ith these Lemmas, th e m ain result in Proposition 1 can be shown.

Lemma 1 provides th e relation betw een th e strateg y 7T; and th e reservation type pj.

L e m m a 1 A reservation strategy iTi(9j,9i) satisfies:

*i(9jA) =

L

ifOj >

7n(9j ,9i) = 0, tf 9j <

<Pj(9i)-P r o o f . pj(9{) has to satisfy th e reservation strategy u ( p j,9 j) > Vo{9i). Since ug. > 0 and = 0, the Lemma is always (never) satisfied for 9j > (< ) p j( 9{). I t follows th a t any 9j > (< ) p j( 9 i) will be accepted (rejected), so th a t nTi(9j,9i) = 1(= 0). ■

R e m a r k 3 The use of the decision variables ir* may at this stage appear cumbersome notation, since from Lemma 1, strategies are monotonic in 9j,

so that TTj = 1 always constitutes a connected set in 9j. However, not only do we need 1Ti(x,9i), but also its inverse tti(9 j,x ). In general, 7Tj = 1 is not a connected set in 9{. A s a result, with the decision variables the calcula­

tion of integrals can be made without knowing the internal boundaries of the

disconnected sets.

Lemma 2 shows th a t, given th e strategies of all o th er players, th e reserva­ tio n strategy is unique. Using th e reservation stra teg y (1.5) equations (1.1*) and (1.2 ) can be rew ritten

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w ith 7i(0j) = f&j ~ u {4>j)]dFj(x )- T he first-order condition (1.6) embodies the trade-off m ade by every individual agent. W ith a reservation strategy, all types above th e reservation ty p e (pj(9{) are ac­ cepted and a m atch is m aterialised if you are accepted by th ese types. Given acceptance, V ertical Heterogeneity implies th a t th e higher th e reservation value, th e higher th e expected value of th e m atch. T he cost of increasing th e reservation value though is th a t th e probability of leaving th e pool of singles decreases: being more choosy m eans th a t (utilityless) w aiting tim es increase. In th e lim it, th e prinoe(ss) of your dream s arrives w ith probabil­ ity zero, hence th e expected tim e of being single is infinite and u tility is zero. Moreover, w ithout a direct search cost, th e o p p o rtu n ity cost of w aiting is utility foregone while you could be m atched to a p artn er. Solving (1.6) yields a critical ty p e (pj(9{), V#*, and hence a reservation stra teg y 7Ti(#j,#{), V#{. Lemma 2 shows it is unique.

L em m a 2 Given ttj, and for </>'• = m a x {^ E Q j \ ^ j { 9 j ) = 1}'

(i) (f)j is the unique solution to Ti((pj) =

0

;

(ii) (f)j <

0

'..

P r o o f . F irst, it follows from th e definitions of ipi(9i) and 0 ' th a t for (pj > (p'j, ipi = 0 and as a result 7* = 0. Since u{9f) > 0, (from uq. > 0 and u(9_j) > 0 ), it follows th a t Ti((pj) > 0 for (pj > 0'- an d th a t th e re is no solution to Ti((pj) = 0 in (0j‘,9j). N ext, (T{)(f) = (r 4- a)«^ — j 3 ^ > 0, V0, E Qj since J -f = ~ u9j {(pj'i 9{) f Qj 7ri ( z ,9 i)7ri (9i,x )d F j(x ) < 0 an d ue . > 0. G iven th a t Ti((pj) > 0 for (pj > (pj and th a t (Ti)# > 0, a solution to Ti((pj) = 0 is in

[9j,<P'j\- This establishes (ii) (pj < (p'j.

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for some 9^ there is no interior solution Ti((pj) = 0. A n optim ising agent will th e n choose th e unique (pj as th e m inimum Bj E @j, satisfying th e reservation strategy. T his m aximises th e expected value Vq(0») — V\(9{) and th e solution is a corner solution. T his establishes (i) (pj is th e unique solution to Ti((pj) =

0

. ■

The proof of uniqueness of th e reservation value is m ade using th e fact th a t th e value function is monotonic in th e reservation value. P a rt (ii) of th e Lemma shows th a t th e reservation value cannot be above th e highest type th a t is willing to accept you. O n th e other hand, if there is no interior solution below th a t, th e solution is th e corner solution 9j. Together w ith Lem m a 1, it th e n follows th a t any ty p e above th e reservation ty p e is accepted. Like a unique optim al response in a norm al form game does n o t im ply a unique N ash equilibrium , uniqueness of th e reservation strategy, given th e strategies of all other players does no t imply equilibrium is unique. T his is illustrated in Figure 1.1. The lower graph is th e reservation strateg y of all ty p es 0*: above th e reservation type, all 9j are accepted, below th ey are rejected. The upper g rap h is th e reservation value for all types 9j\ to th e right of th e graph, all 9i are accepted, to th e left all are rejected. Clearly, left of 9* n o t all 9j are willing to acoept a m atch. O nly th e types 9j below th e up p er g rap h (i.e. inverse of th e reservation strateg y of th e types 9j) will accept. T he vertical distance betw een th e two graphs is th e n th e range over w hich m atches are materialised. M easured over th e distrib u tio n Fj it determ ines th e probability of acceptance ipj.

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e*

Figure 1.1: Reservation Strategies

the upper one. This is shown rigorously in Lemma 3(ii). As a result, many equilibria may be envisaged: one unique response for each strategy of the other players. However, Lemma 3(i) proves th a t agents accept matches w ith strictly positive mass (if not so, they will become m atched w ith probability zero). This implies th a t, given the strategies of all other players, there is an upper bound to the reservation strategy. P a rt (iii) th e n provides the proof th a t if accepted by some positive mass, there is also a lower bound to the reservation strategy.

R e m a r k 4 For the remainder of the paper, the following notation is used.

• 7tJ (9).) > tT?(9k), k 6 { i , j } means that fo r a given

7 T V 0 / . and with strict inequality fo r some k with positive mass;

*i(Qk) = K?(dk) i f fo r a given 9_k,?r?(03-,0i) = n? V0*;

[image:31.595.148.338.152.345.2]
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. (0},0}) < (91,9}) if at least one of the following two equations holds with strict inequality: 6} < Of or 9} < 9};

. (010}) = (91 9 )) if both 9] = 91 and 9} = 9}.

R e m a r k 5 With every value of (pj(9i), there is associated a value 7Vi(9j,9i). It follows that the whole schedule (pj(9i), V#* is defined by /iri(9j,9i). In terms of notation, 77 = is the reaction function t* yielding the unique solution 7Ti for a given Wj. That is, for a given iTj, TJ is solved V9 j.

L e m m a 3 (i) An optimising agent will always accept matches within a range of agents with strictly positive mass;

(ii) 7r}(0j) > (9j) implies (p} > (p};

(Hi) nj(9j) > 0 implies there is an upper bound on the reservation value (pj ■

P r o o f , (i) A population w itli zero m ass implies t h a t 7* = 0. Since u(9f) > 0, Ti((pj) > 0. Ti((pj) = 0 can only be satisfied for some % = 0. This implies acoepting a population w ith strictly positive mass. This applies to all types of b o th sexes since u(9f) > 0 and u(9{) > 0,

(ii) 7rj(9 j) > ftj(9j) oeteris paribus implies 7 ? > 7 ?, by definition of 7 *. If (p} is th e unique solution to Tj(0] | p j) = 0, i.e. Ti(cp}) = 0 given 7rj, th e n it follows th a t T{(0j | tt?) ^ 0, since '1 (p^ 0 . Tlie unique solution, to Ti((pj | n}) = 0 th e n satisfies (p} > (p}\

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L e m m a 4 For utility functions of both sexes satisfying

(r + a )u di{4>j) ~ (3 [ 'Ki{x,ei)nj (ej , x)[udi{x) - u e X ^ d F ^ x ) = 0 (1.7) J&j

the equilibrium mapping is type-independent, fo r a given ttj and Fj .

P r o o f . From Lem m a 2, th ere exists a unique reservation strategy, given th e strategy of all other players. Type-independence of th e reservation strateg y will occur w hen, tak in g ttj as given, = 0 , i.e. w hen a lower type has th e

same reservation value. W ith > 0, th e im plicit function theorem implies T0i = 0 , or equation (1.7). ■

P r o p o s iti o n 2 For multiplicatively separable utility functions, the distribu­ tions of types are endogenously partitioned.

P r o o f . Consider th e general form ulation of a m ultiplicatively separable u tility function: u(9j,9i) = v(9j)w(9i). T (0 J) = 0 can be re w ritte n as

(r + a)v(9j) — (3 [ a?)[i;(a?) — v((f>j)]dFj(x) = 0 (1.8) J&j

It is easily verifiable th a t T0. = 0, provided ftj{9i, 9j) is independent of 9{, i.e. 7Tj(9*, •) = 'Kj(9?, •), V0? 9f. By requiring th a t Uj is m ultiplicative, th is is autom atically satisfied \/9{ in th e same p artition , provided th a t individuals have tim e invariant strategies. ■

1.3

D iscussion o f th e R esults

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e*

0

Figure 1.2: Partitioning

[image:34.595.150.343.151.346.2]
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revised (type-independent) reservation strategy (the second horizontal line). Again, th e iterated dominance argum ent implies th a t th e second p a rtitio n is formed by all types th a t are accepted w ith certainty. T his goes on u n til all types belong to a p artition.

Before th e algorithm is extended to th e general case, two rem arks. The partitioning result is quite surprising. T hough u tility functions are type- dependent, th e strategies are not. For a special case of m ultiplicatively sep­ arable u tility functions w ith = 0 , th e result is fairly intuitive since th e utility function is type-independent5. I t follows th a t th e first order condition (1.6) is type independent. U tility derived and hence th e o p p o rtu n ity cost are identical ex ante for types w ithin one p artition. Hence, th ey will solve for th e same solution. W ith type dependent u tility functions th is is equally th e case b u t for different reasons. Consider for exam ple th e case w here higher types derive more utility from being m atched w ith a high type of th e other sex (i.e. the u tility exhibits strategic com plem entarities). I t follows th a t th e expected value of being m atched is increasing in type. O n th e one hand, higher types will be more choosy and have higher reservation values. O n th e other hand, w ith o u t direct search costs, th e cost of search is th e o p p o rtu n ity cost of not being m atched. As a result, th e search cost is increasing in type. The higher types are more im patient and choose lower reservation values. For m ultiplicative utility functions, these two effects cancel o u t against each other and th e first-order condition (1.8) is homogeneous of degree zero in th e own type.

5Tlie partitioning result, obtained in different frameworks, lias always been derived

for a special case of tlie type-independent ntility function: Ui = 6j. See McNamara and

Collins [42], Block and Ryder [12], Burdett and Coles [13]. The exception being Smitk [69]

wko looks at multiplicative pay-offs and derives a similar result to the one in Proposition

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N ote further th a t in case u(9_^) = u(9j) = 0 th e num ber of p artitio n s goes to infinity. A t th e bottom it is always more lucrative to w ait a b it more and not accept th e lower types since th e y yield utility going to zero. The proof is beyond th e purpose of th is C hapter.

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e* 0 j

Figure 1.3: Downward Sloping Reservation Strategies

by all. Hence they will revise their upper bound downwards. However, from Lemma 3(iii), and given acceptance by some, they now also have a lower bound. This holds for b o th sexes. Given the lower bound of th e other sex, they will revise their upper bound and given the upper bound of the other sex, they will revise the lower bound. This goes infinitely until the unique reservation schedule is determined. The panel on the right in Figure 1.3 is merely a variation on the same theme. One schedule is upward sloping, the other downward. Again, by elim inating dominated strategies startin g from th e top (i.e. above (#*, 9*)), th e whole schedule can be constructed uniquely.

R e m a r k 6 Figure 1.3 clearly illustrates that the slope o f the schedule cfj is not only a function of the utility function. It can be shown that fo r util­

ity functions exhibiting log-supermodularity (i.e. U\2u > u^u2) the reserva­

tion value, given acceptance by all, is increasing in type and decreasing if

it is log-submodular (see Sm ith [69]). However, the equilibrium schedule is

not necessarily downward sloping over the whole range even if there is

[image:37.596.58.437.154.334.2]
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submodularity but tt, is type independent. This is fo r example the case at the upper part of the distribution. In Figure 1.3, even though in both cases at least

one of the utility function is log-submodular, at the lower end the schedule is

upward sloping. The reason is that in that range, ttj is type-dependent.

1.4

Perfect M atching Equivalence

In this Section, it is shown th a t th e equilibrium is indeed th e generalisation of th e Perfect M atching model. F irst, th e perfect m atching m odel is defined in more detail. Second, it is shown th a t th e B ilateral Search m odel w ith Vertical Heterogeneity yields th e same outcom e as th e Perfect M atching m odel w hen th e search friction disappears in th e lim it. The search friction disappears when w aiting tim e goes to zero, i.e. w hen th e arrival rate [3 goes to infinity.

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male. Clearly, th is establishes th a t a stable m atching is a core concept and thus a cooperative equilibrium. In C hapter 3 (Corollary 4), it is shown th a t there exists a unique stable m atching, p(9i) = 9j O Fi(9{) = Fj(9j): in equilibrium , only individuals of th e same rank m atch.

The Equivalence betw een Perfect M atching and Search can now be estab­ lished. N ote however th a t there is an entirely different use of equilibrium con­ cept: cooperative versus non-cooperative equilibrium . W h a t will be shown is th a t th e non-cooperative search equilibrium yields th e sam e outcom e as the cooperative stable m atching w ithout friction w hen th e search friction is infinitely sm all (i.e. lim /? —» oo). I t can actually be shown th a t th e sta ­ ble m atching is equivalent to th e trem bling han d equilibrium w hich rules out degenerate non-cooperative equilibria. N ote also th a t our restrictio n to reservation strategies has a similar im pact.

P r o p o s iti o n 3 E q u i v a l e n c e . The Gale-Shapley-Becker Perfect Matching model is the limit case of the search model when trading opportunities arrive

instantaneously (i.e. lim/3 —> oo).

P r o o f . For lim /? —> oo, th e system of equations (1.1) and (1.2) collapses. The sta te of being single now coincides w ith th e sta te of being m atched since a m atch is instantaneously realised. I t follows th a t th e value of being single has to equal th e expected value of being m atched: Vo(9i) — E\V\(9i) | 7Tj = 1]. A n individual 9{ will choose a reservation value (pj such as to maximise th e expected value of being m atched subject to being accepted. T his implies

«rt r t a \ I@j > x )u (x )dFA x ) n .

1 ‘ / a . Wi(x, x)dFj(x)

EV\ is m onotonically increasing uupj, provided acceptance by se x j : > 0, s.t. 7Tj = 1 is derived from

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s.t. 7Tj = 1, w hich is satisfied since Uq. > 0. T he solution to th is problem is a corner solution: E \ \ is maximised w hen (f)j is maximised, s.t. ttj = 1. W ith </>'• = max{^- E Qj \ f^i(9j) = 1}, th e optim al choice of (f)j is (f)j(9i) = ^ -(0»), \/9i. Likewise, 4>i(9j) = <fi'i{9j), V9j-. A pplying th e algorithm in th e proof of Proposition 1 th e n gives the following allocation: a type 9i will m atch w ith 9j if and only if Fi(9i) = Fj(9j). T his is equivalent to th e stable m atching A*(0i) = 9j Fi(9i) = Fj(9j).

1.5

C oncluding Rem arks

In th is paper, th e Perfect M atching m odel is extended to an Im perfect M atch­ ing Model w ith search frictions. A search m odel is proposed featuring vertical heterogeneity and bilateral search. Equilibrium in a concept as stro n g as iter­ ated elim ination of dom inated strategies is shown to exist and is unique, irre­ spective of th e u tility specification of individuals. I t is derived th a t its lim it case w ith o u t friction is th e Gale-Shapley-Becker perfect m atching model. Equilibrium properties can be derived in function of th e pay-off function and as a result, for m ultiplicatively separable u tility functions, equilibrium stra te ­ gies are type-independent even tho u g h th e u tility function does depend on th e type. I t follows th a t b o th distributions of types are endogenously p arti­ tioned, segregating th e m arket into classes.

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w ith a general existence proof of a steady state equilibrium is th a t usual techniques do no t readily generalise w ith non continuous strateg y spaces.

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1.6

A ppendix

P r o p o s i ti o n 1.

P r o o f . Ite ra te d elim ination implies n iterations. Therefore, th e following n o tatio n is introduced. F irst, because of th e argum ent of iterated elimina­ tion, th e variable fj,i(9j,9i) = 7Tj(9i, 9j) is introduced in order to distinguish th e acceptance rule by others from th e strateg y by other players. Clearly, in equilibrium they are th e same. U i(9j,9i’,n) = II;(rz), Vi, j is th e schedule 7Xi(9j,9i) calculated in iteratio n n, provided (ii = 1. 4>j{9i\n) is th e reserva­ tio n value associated w ith !!;(#.;, 0;;7z), provided /i; = 1. Likewise, ^;(rz) is fii(9j,9i), VzJ(7 in iteration n.

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to Lem ma 5. A s a result, after th e 7i- th iteration, all (#*, have a unique iterated strict dom inant strategy.

1. A fter n — 1 iterations th e schedules IL (n — 1) and H j(n — 1) are uniquely determ ined for all (9i,9j) > (9*(n—l ) , 9 j ( n — l) ) . I t follows th a t in th e next iteratio n th e schedules fa(n) and fa(n) are uniquely defined in th a t range. For th e other types, m axim al acceptance (i.e. fJ>(n) = 1) allows us to determ ine th e dom inated strategies. Hence, determ ine fa(n) = U j ( n — 1), if 9j > 9 j(n — 1); fa{n) = 1, otherwise. Likewise fa(n) = — 1), if 9i > 9 * ( n — 1); fa(n) = 1, otherwise. A t th e s ta rt of th e procedure (n = 1), fa(n) = 1, and f a {n) = 1, V0*;

2. n i( n ) = Ti(fii(n)) and Uj{n) = Tj{fa{n))

3. Consider all types (9^9j) < { 9 \{n — l ) , 9 j ( n — 1)). Taking into account the unique strategies of all higher types and by determ ining fa(n) and fa(n) in term s of m axim al acceptance (i.e. from step 1, there exists no fa > fa(n), it follows from Lemma 2 th a t all reservation strategies 9j > 4>j{9i\ n) are strictly dom inated for all 9{. From Lemma 1, H{(?i) = 1 for all

Likewise, all reservation strategies 9i > <pi (9j; n) are strictly dom inated for all 9j and n ,(n) = 1 for all 9i > 4>i(9j\ n)\

4. Define

(9t(n),0Uu)) =

(m in^i,m in$j) [ nj(re)II,(re) = 1 and

V0j <

e*i{n -

1): H^n) = 1, Vfy

< 0^(n -

1)

< 0}(n

- 1) : Il^n) = 1, V0{ <

0?(n -

1)

( A l.l)

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th e sets are em pty and th e pair (9*(n),9*(n)) and th e unique ite ra te d stric t dom inant strategies are determ ined according to Lem m a 5 below.

5. From step 3 and from equation ( A l.l) , all types E [ 9 l ( n ) , 9 * ( n —1)]

and 9 j E [ 9 j ( n ) , 9 * ( n —1)] have f i i ( n ) = fJ*j(n) = 1 independently of any other

players strateg y (because th e strategies are dom inated), sinoe fii = Uj and jjij = ]!{. The reservation strateg y of all these ty p es is th u s independent of th e strategy of any other player. By elim inating th e dom inated strategies, all these types have a unique iterated strict dom inant strateg y and U j ( n )

respectively (from Lemm a 2(i));

This iterative procedure is repeated un til U i(N ) = IIi(./V +l) and U j(N ) = H j( N -f 1). Because (9*(n), 9j(n)) < (9*(n — l),0 ? (n — 1)) and from Lemma 3(i), every agent chooses to accept m atches from a pop u latio n w ith strictly positive mass. A s a result, th e populations elim inating strictly dom inated strategies in every iteration have strictly positive mass. I t follows th a t th e equilibrium list (U i( N ) ,U j(N )) is obtained after a finite num ber of N iterations. ■

L e m m a 5 If according to equation ( A l . l ) (9 t(n ),9j(n )) = ( 9 l( n —\),9*j(ri— 1)) a new pair can be defined such that (9^(n),9j(n)) < (9^(n—l ) , 9 j ( n —l) ) and such that there exists a unique iterated strict dominant strategy for all

types in the interval ([9*(n),9*(n — 1)], \9 j(n ),9 j(n — 1)]).

P r o o f . F irst, (9*(ri),9j(n)) = (9*(n—l ) , 9 j ( n —l)) implies th a t b o th /i{(n) > 0 for 9j > (j)j(n— 1) and fJ>j(n) > 0 for 9i > (f>i(n— 1). T his follows from Lemma 3(i) and (ii). If it is not satisfied say for sex z, th ere would exist a range of dom inated strategies w ith strictly positive m ass below 0 j for th e types of sex

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holds. Therefore, th e pair can be redefined such th a t th e set is non-em pty

\

(a>! \ m t \\

I

I

M n)

^ 0.

>

4>An -

!)}>

m in ^ - | fJ>j{n) > 0, V19* > 1)} j

(A1.2)

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There is a unique strateg y Ili(n ) and I Ij(n) for all types G [9*(n),9*(n— 1)] and 9j G [9j(n),9j(n — 1)]. Hence, th e pair (#?(n),#J(n,)) is defined as in (A1.2 ). ■

E n d o g e n o u s d i s t r i b u t i o n o f S in g le s

Let Hi(9i) be th e distrib u tio n of th e entire population of sex i and Hf(9{) th e d istrib utio n of singles. A ll these distributions can be tim e variant. If rii is th e fraction of singles of type 9{ a t a p articu lar m om ent in tim e, th e density function hs{9i) of singles associated w ith th e p o p u latio n density h(9{) is given by

*'<*■> - ( 1 - 4 )

A t any m om ent in tim e, th e law of m otion is given by hi = + a ( l — rii). Clearly, ou t of th e steady state, th e distribu tio n of singles hs changes over tim e. Clearly, in a steady state F = H£. T his is also tru e ou t of steady sta te if players do n o t hold rational expectations and believe th e observed distri­ butio n will not change over tim e6. If agents hold full ratio n al expectations, they will take into account th e change in th e d istrib u tio n over th e expected duratio n (Z^ ) -1 of being single. The belief ab o u t th e d istrib u tio n of singles th e n satisfies F(9{) = H^{9i)dt.

Endogenising th e distrib u tio n of singles leaves th e existence and unique­ ness of th e allocation in ta c t (though th ere is a new souroe of m ultiplicity). In addition, th e characterisation of equilibria for given preferences and th e perfect m atching equivalence still hold. There is however a stro n g implica­ tio n for th e off th e steady sta te characterisation of th e p artitio nin g result. T his is shown below.

6Tliis corresponds to wliat is called a Partial Rational Expectations belief in Burdett

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N o n S te a d y S t a t e T y p e D e p e n d e n c e

In Section Three, it was shown th a t provided th e d istrib u tio n of types is tim e stationary, there is endogenous p artitio n in g of th e d istribu tio n of types. Highly appealing as th is may seem, th e result fails to hold o u t of stead y state: reservation strategies are type dependent. T his implies th a t even a steady state equilibrium can only be shown to exist if th e o u t of steady sta te equilibrium exists. Since our general P rop o sitio n of existenoe of equilibrium out of steady state is shown w ith and w ith ou t type dependence, it follows th a t th e steady state partitioning result can come ab o u t from any in itial condition. All th e other work so far (Sm ith [69], B u rd ett and Coles [13]) could only conjecture th a t th e steady state would come about. However, a proof for th e case of type-dependent strategies as in Proposition 1 is necessary.

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tim e 0 < t < T. By th e same argum ent, all d istributions a t any later d ate are stochastically dom inated by th e earlier ones. The R atio n al E xpectations7 strategy of all types a t t is calculated given Ft = f t +^ 1^ H^dt, w hich is stochastically dom inated by any F a t a n earlier tim e. By counter exam ple, it is now shown th a t th e out-of-Steady-State strategies are ty p e dependent.

P r o p o s iti o n 4 Out of Steady State, equilibrium strategies do not endoge­ nously partition the distributions of types, even with multiplicatively separable

utility functions.

P r o o f . Consider th e highest types. A t t = 0, th eir reservation strateg y is 0o, calculated tak in g

References

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