Auctions, Matching and the Law of Aggregate Demand

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1 Auctions, Matcing an te Law of Aggregate eman Jon William atfiel an Paul Milgrom 1 February 12, 2004 Abstract. We evelop a moel of matcing wit contracts wic incorporates, as special cases, te college amissions problem, te Kelso-Crawfor labor market matcing moel, an ascening package auctions. We introuce a new law of aggregate eman for te case of iscrete eterogeneous workers an sow tat, wen workers are substitutes, te law is satisfie by te emans of profit maximizing firms. Wen workers are substitutes an te law is satisfie, trutful reporting is a ominant strategy for workers in a worker-offering auction/matcing algoritm. We also parameterize a large class of preferences satisfying te two conitions. I. Introuction Since te pioneering US spectrum auctions of 1994 an 1995, relate ascening multi-item auctions ave been use wit muc fanfare on six continents for sales of raio spectrum an electricity supply contracts (Milgrom (2004)). 2 Package biing, in wic biers can place bis not just for iniviual lots but also for bunles of lots ( packages ), as foun increasing use in procurement applications. Recent proposals in te US to allow package biing for spectrum licenses incorporate ieas suggeste by Ausubel an Milgrom (2002) an by Porter, Rassenti, Roopnarine an Smit (2003). Matcing algoritms base on economic teory are also influencing practice. Rot an Peranson (1999) explain ow a certain two-sie matcing proceure, wic is similar to te college amissions algoritm introuce by Gale an Sapley (1962), as been aapte to matc 20,000 octors per year to meical resiency programs. Abulkairoğlu an Sönmez (2003) avocate a variation of te same algoritm for use by scool coice programs. Tis paper ientifies an explores certain similarities among all of tese auction an matcing mecanisms. To illustrate one similarity, consier te labor market auction moel of Kelso an Crawfor (1982), in wic firms bi for workers in simultaneous ascening auctions. Te Kelso-Crawfor moel assumes tat workers ave preferences over firm-wage pairs an tat all wage offers are rawn from pre-specifie finite set. If tat set inclues only one wage, ten all tat is left for te auction to etermine is te matc of workers to firms, so te auction is effectively transforme into a matcing algoritm. Te auction algoritm begins wit eac firm proposing employment to its 1 We tank Atila Abulkairoğlu, Feerico Ecenique, aniel Lemann, Jon Levin, an Alvin Rot for elpful iscussions. Tis researc is supporte by a National Science Founation researc grant. 2 For example, a New York Times article about te a spectrum auction in te Unite States was ealine Te Greatest Auction Ever. (NYT, Marc 16, 1995, page A17). Te scientific community as also been entusiastic. In its 50 t anniversary self-review, te US National Science Founation reporte tat [f]rom a financial stanpoint, te big payoff for NSF s longstaning support [of auction teory researc] came in 1995 [wen t]e Feeral Communications Commission establise a system for using auctions. 1

2 most preferre set of workers at te one possible wage. Wen some workers turn it own, te firm makes offers to oter workers to fill its remaining openings. Tis proceure is precisely te ospital-offering version of te Gale-Sapley matcing algoritm. ence, te Gale-Sapley matcing algoritm is a special case of te Kelso-Crawfor proceure. Te possibility of extening te National Resient Matcing Program (te Matc ) to permit wage competition is an important consieration in assessing public policy towar te Matc, particularly because tere is some teoretical support for te position tat te Matc may compress an reuce octors wages relative to a perfectly competitive stanar (Bulow an Levin (2003)). Te practical possibility of suc an extension epens on many etails, incluing importantly te form in wic octors an ospitals woul ave to report teir preferences for use in te Matc. In its current incarnation, te matc can accommoate preferences tat encompass affirmative action constraints an a subtle relationsip between internal meicine an its subspecialties, so it will be important for any replacement algoritm to encompass tose as well. We aress some of tese preference encoing issues later in tis paper. A secon important similarity is between te Gale-Sapley octor-offering algoritm an te Ausubel-Milgrom proxy auction. Explaining tis relationsip requires restating te algoritm in a ifferent form from te one use for te preceing comparison. We sow tat if te ospitals in te Matc consier octors to be substitutes, ten te octoroffering algoritm is equivalent to a certain cumulative offer process in wic te ospitals at eac roun can coose from all te offers tey ave receive at any roun, current or past. In a ifferent environment, were tere is but a single ospital or auctioneer but te octors contracts nee not be substitutes an can contain general terms, a formally ientical cumulative offer process coincies exactly wit te Ausubel- Milgrom proxy auction. espite te close connections among tese mecanisms, previous analyses ave mostly treate tem separately. In particular, analyses of auctions typically assume tat biers payoffs are quasi-linear. No corresponing assumption is mae in analyzing te meical matc or te college amissions problem; inee, te very possibility of monetary transfers is exclue from tose formulations. As iscusse below, te quasilinearity assumption combines wit te substitutes assumption of matcing teory in a subtle an restrictive way. Tis paper presents a new moel tat subsumes, unifies an extens te moels cite above. Te basic unit of analysis in our formulation is te contract. To reprouce te Gale-Sapley college amissions problem, we specify tat a contract ientifies only te stuent an college; all oter terms of te relationsip are exogenous. To reprouce te Kelso-Crawfor moel of firms biing for workers, we specify tat a contract ientifies te firm, te worker, an te wage. Finally, to reprouce te Ausubel-Milgrom moel of package biing, we specify tat a contract ientifies te bier, te package of items tat te bier will acquire, an te price to be pai for tat package. Many aitional variations can be encompasse by te moel. For example, a contract migt specify te particular responsibilities tat a worker will ave witin te firm. Our analysis of te Gale-Sapley an Kelso-Crawfor moels an teir extensions empasizes two conitions tat restrict te preferences of te firms/ospitals/colleges: a 2

3 substitutes conition an an law of aggregate eman conition. We fin tat tese two conitions are implie by te assumptions of earlier analyses, so our unifie treatment implies te central results of tose teories as special cases. In te traition of eman teory, we efine substitutes by a comparative statics conition. In eman teory, te exogenous parameter cange is a price ecrease, so te callenge is to exten te efinition to moels in wic tere may be no price tat is allowe to cange. In our contracts moel, a price reuction correspons formally to expaning te firm s opportunity set, tat is, to making te set of feasible contracts larger. Our substitutes conition asserts tat wen te firm cooses from an expane set of contracts, te set of contracts it rejects also expans (weakly). As we will sow, tis abstract substitutes conition coincies exactly wit te eman teory conition for stanar moels wit prices. It also coincies exactly wit te Rot an Sotomayor (1990) substitutable preferences conition for te college amissions problem, in wic tere are no prices. Te law of aggregate eman is similarly efine by a comparative static. It is te conition tat wen a college or firm cooses from an expane set, it amits at least as many stuents or ires at least as many workers. 3 Te term law of aggregate eman is motivate by te relation of tis conition to te law of eman in proucer teory. Accoring to proucer teory, a profit-maximizing firm emans (weakly) more of any input as its price falls. For te matcing moel wit prices, te law of aggregate eman requires tat wen any input price falls, te aggregate quantity emane, wic inclues te quantities emane of tat input an all of its substitutes, rises (weakly). Notice tat it is tricky even to state suc a law in proucer teory wit ivisible inputs, because tere is no general aggregate quantity measure wen ivisible inputs are iverse. In te present moel wit inivisible workers, we measure te aggregate quantity of workers emane or ire by te total number of suc workers. A key step in our analysis is to prove a new result in eman teory: if workers are substitutes, ten a profit maximizing firm s employment coices satisfy te law of aggregate eman. Since firms are profit maximizers an regar workers as substitutes in te Kelso-Crawfor moel, it follows tat te law of aggregate eman ols for tat moel. Tus, one implication of te stanar quasi-linearity assumption of auction teory is tat te biers preferences satisfy te law of aggregate eman. We fin tat responsive preferences, wic are commonly assume in matcing teory analyses, also satisfy te law of aggregate eman. We ten prove some new results for te class of auction an matcing moels tat satisfy tis law. Te paper is organize as follows. Section II introuces te matcing-wit-contracts notation an caracterizes te stable sets of contracts or core allocations in terms of te solution of a certain system of two equations. Section III introuces te substitutes conition an uses it to prove tat te set of core allocations is a non-empty lattice, an tat a certain generalization of te Gale-Sapley 3 In teir stuy of a moel of sceule matcing, Alkan an Gale (2003) inepenently introuce a similar notion, wic tey call size monotonicity. 3

4 algoritm ientifies its maximum an minimum elements. Tese two extreme points are caracterize as a octor-optimal/ospital-pessimal point, wic is a point tat is te best in te core for every octor an tat is worst in te core for every ospital, an a ospitaloptimal/octor-pessimal point wit te reverse attributes. Section III also proves several relate results. First, if tere are at least two ospitals an if some ospital as preferences tat are not substitutes, ten even if all oter ospitals ave just a single opening, tere exists a profile of preferences for te stuents an colleges suc tat no core allocation exists. Tis result is important for te construction of matcing algoritms. It means tat no matcing proceure wic permits stuents an colleges to report preferences tat are not substitutes can be guarantee always to select a core allocation wit respect to te reporte preferences. Anoter result concerns vacancy cain ynamics, wic traces te ynamic ajustment of te labor market wen a worker retires or a new worker enters te market an te ynamics are represente by te operator we ave escribe. Te analysis extens tat of Blum, Rot an Rotblum (1997) but wit a larger class of preferences in wic firms o not ave an exogenously fixe number of vacancies an te number of positions tat are fille can cange uring te ajustment process. We fin tat, starting from a core allocation, te vacancy ajustment process converges to a new core allocation. Section IV introuces te law of aggregate eman, verifies tat it ols for a profitmaximizing firm wen inputs are substitutes, an explores its consequences. Wen bot te substitutes an te law of aggregate eman conition are satisfie, ten (1) te set of workers employe an te set of jobs fille is te same at every stable collection of contracts an (2) it is a ominant strategy for octors (or workers or stuents) to report teir preferences trutfully in te octor-offering version of te extene Gale-Sapley algoritm. We also emonstrate te necessity of a weaker version of te law of aggregate eman for tese conclusions. Our conclusion about tis ominant strategy property substantially extens earlier finings about incentives in matcing. Te first suc results, ue to ubins an Freeman (1981) an Rot (1982), establise te ominant strategy property for te marriage problem, wic is a one-to-one matcing problem tat is a special case of te college amissions problem. Similarly, emange an Gale (1985) establis te ominant strategy property for te worker-firm matcing problem in wic eac firm as singleton preferences. Tese results generalize to te case of responsive preferences, tat is, to te case were eac ospital (or college or firm) beaves just te same as a collection of smaller ospitals wit one opening eac. For te college amissions problem, Abulkairoğlu (2003) as sown tat te ominant strategy property also ols wen colleges ave responsive preferences wit capacity constraints, were te constraints limit te number of workers of a particular type tat can be ire. All of tese moels wit a ominant strategy property satisfy our substitutes an law of aggregate eman conitions, so te earlier ominant strategy results are all subsume by our new result. In section V, we aress te practicality of te generalize algoritm by asking ow te ospitals migt express teir complex preferences in a neatly parameterize way. We introuce te extene assignment preferences an sow tat tey subsume certain 4

5 previously ientifie classes an satisfy bot te substitutes conition an te law of aggregate eman. Section VI introuces cumulative offer processes as an alternative auction/matcing algoritm an sows tat wen contracts are substitutes, it coincies wit te octoroffering algoritm of te previous sections. Since te Ausubel-Milgrom proxy auction is also a cumulative offer process, our ominant strategy conclusion of section IV implies an extension of te Ausubel-Milgrom ominant strategy result. For te case wen contracts are not substitutes, te cumulative offer process can converge to an allocation tat is not stable. We sow tat if tere is a single ospital/auctioneer, owever, te cumulative offer process converges to a core allocation, even wen goos are not substitutes. Section VII conclues. II. Stable Collections of Contracts Te matcing moel witout transfers as many applications, of wic te best known among economists is te matc of octors to ospital resiency programs in te Unite States. For te remainer of te paper, we aopt te terminology of octors an ospitals, wic plays te same respective roles as stuents an colleges in te college amissions problem an similar roles to tose of workers an firms in te Kelso- Crawfor labor market moel. Notation Te sets of octors an ospitals are enote by an, respectively, an te set of contracts is enote by X. We assume only tat eac contract x X is bilateral, so tat it is associate wit one octor x an one ospital x. Wen all terms of employment are fixe an exogenous, te set of contracts is just te set of octor-ospital pairs: X. For te Kelso-Crawfor moel, a contract specifies a firm, a worker an a wage, X W. Eac octor can sign only one contract. er preferences over possible contracts, incluing te null contract, are escribe by te total orer. Te null contract represents unemployment, an contracts are acceptable or unacceptable accoring to weter tey are more preferre tan. Wen we write preferences as P : x y z, we mean tat P names te preference orer of an tat te liste contracts (in tis case, x, y an z) are te only acceptable ones. Given a set of contracts X X offere in te market, octor s cosen set C ( X ) is eiter te null set, if no acceptable contracts are offere, or te singleton set consisting of te most preferre contract. We formalize tis as follows: if { x X x =, x } = C ( X ) = (1) { max { x X x }} oterwise = 5

6 Te coices of a ospital are more complicate, because it as preferences over sets of octors. Its cosen set is a subset of te contracts tat name it, tat is, C( X ) { x X x = }. In aition, we o not allow a ospital to coose to sign two contracts wit te same octor. ( )( )(, ( )) X X x x C X x x x x. (2) Let C ( X ) = C ( X ) enote te set of contracts cosen by some octor from set X. Offers in X tat are not cosen are in te rejecte set: R( X ) = X C( X ). Similarly, te ospitals cosen an rejecte sets are enote by C( X ) = C( X ) an R ( X ) = X C ( X ). Core Allocations: Stable Sets of Contracts In our moel, an allocation is a collection of contracts, since tat etermines te payoffs to eac participant. Tere is a subtlety in efining te core for matcing moels tat centers on te efinition of wen a coalition can block a propose allocation. Te resolution most consistent wit te previous literature is to focus on te case were a coalition can block a propose allocation if tere is anoter allocation tat te coalition members can implement by itself tat all coalition members weakly prefer an tat some coalition members strictly prefer. In te usual way for matcing moels, if any coalition of ospitals an octors can block an allocation, ten tere is a subcoalition consisting of a single ospital an its octors (if any) tat can also block, since tey can make te beneficial eviation on teir own. A set of contracts may also be blocke by an iniviual octor, wo fins er assigne contract unacceptable. Wit tese observations in min, we introuce te following efinition. efinition. A set of contracts X X is unblocke if (i) C ( X ) = C ( X ) = X an (ii) tere exists some no ospital an set of contracts X C( X ) suc tat X = C ( X X ) C ( X X ). If conition (i) fails, ten some octor or ospital prefers to reject some contract. If conition (ii) fails, ten tere is an alternative set of contracts tat a ospital strictly prefers an tat its corresponing octors weakly prefer. A core allocation or stable set of contracts is a set of contracts X tat is unblocke. Our first result caracterizes te stable sets of contracts in terms of te solution of a system of two equations. Teorem 1. If ( X, X ) X 2 satisfies te system of equations X = X R ( X ) an X = X R ( X ), (3) 6

7 ten C ( X) = C( X) = X X is a stable set of contracts. Conversely, for any stable collection of contracts X, tere exists some pair ( X, X ) satisfying (3) suc tat X = X X. Proof. Since C( X) X an C( X) X, C( X) C( X) X X. For te reverse inclusion, suppose x X X. Ten, (3) implies x R ( X ) an x R ( X ). ence, x C ( X) C( X). Since x is arbitrary, X X C ( X ) C ( X ). It follows tat X X = C ( X ) C ( X ). Next, we prove tat C ( X) = C( X). Suppose to te contrary tat tere exists some x C ( X) C( X). Since x C( X ), x R ( X ). Ten, x X R ( X) = X. Terefore, since x C ( X ), it follows tat x R ( X ) an ence x X R ( X) = X, so x C ( X ). Tis contraiction implies tat C( X) C( X) =. A symmetric argument implies tat C( X) C( X) =, so C ( X ) = C ( X ). ence, C ( X ) = C ( X ) = X X. To sow tat X X X = C( X) = C( X) is a stable set of contracts, observe first tat by reveale preference, X = C( X ) = C( X ), so conition (i) is satisfie. Next, consier any ospital an set of contracts X C ( X X ). Since X = C ( X ), it follows by reveale preference of te octors tat X R( X) =. Tus, X X R ( X) = X by (3). So if X C( X ), ten by te reveale preferences of ospital, X C( X) = C( X ). ence, X C( X X ), so conition (ii) is satisfie.. So, te set of contracts X is unblocke. For te secon statement of te teorem, suppose tat X is a stable collection of contracts. Since te octors coice sets are singletons, we may efine X to be te set of contracts tat some octor in weakly prefers to er contract in X. By construction, X X. Since X is stable, C( X) = X. Let X = X ( X X ). By construction, C ( X ) = X. ence, ( X, X ) satisfies (3). Teorem 1 is formulate to apply to general sets of contracts. It is te basis of our analysis of stable sets of contracts in te entire set of moels treate in tis paper. III. Substitutes In tis section, we introuce our first restriction on ospital preferences, wic is te restriction tat contracts are substitutes. We use te restriction to prove te existence of a stable set of contracts an to stuy an algoritm tat ientifies tose contracts. Our substitutes conition generalizes te Rot-Sotomayor substitutable preferences conition to preferences over contracts. In wors, te substitutable preferences conition states tat if a octor is not cosen by a ospital from some set of available octors, ten 7

8 tat octor will still not be cosen if te set of available octors is larger. Our substitutes conition is similarly efine as follows: efinition. Elements of X are substitutes for ospital if for all subsets X X X we ave R ( X ) R ( X ). In te language of lattice teory, wic we use below, elements of X are substitutes for ospital exactly wen te function R is isotone. In eman teory, substitutes is efine by a comparative static tat uses prices. It says tat, limiting attention to te omain of wage vectors at wic tere is a unique optimum for te ospital, te ospital s eman for any octor is non-ecreasing in te wage of eac oter octor. Our next result verifies tat for resource allocation problems involving prices, our efinition of substitutes coincies wit te stanar eman teory efinition. For simplicity, we focus on a single ospital an suppress its ientifier from our notation. Tus, imagine tat a ospital cooses octors contracts from a subset of X = W, were is a finite set of octors an W = { w,..., w} is a finite set of possible wages. Assume tat te set of wages W is suc tat te ospitals preferences are always strict. Suppose tat w= maxw is a proibitively ig wage, so tat no ospital ever ires a octor at wage w. In eman teory, it is stanar to represent te ospital s market opportunities by a vector w W tat specifies a wage w at wic eac octor can be ire. We can exten te omain of te coice function C to allow market opportunities to be expresse by wage vectors, as follows: ( ) ( ) {(, ) } w W C w C w (4) Formula (4) associates wit any wage vector w te set of contracts {( w, ) } an efines Cw ( ) to be te coice from tat set. Wit te coice function extene tis way, we can now escribe te traitional eman teory substitutes conition. Te conition asserts tat increasing te wage of octor from w to w cannot reuce eman for any oter octor. efinition. C satisfies te eman-teory substitutes conition if (i), (ii) (, w ) C( w) an (iii) w > w imply tat (, w ) C( w, w ). To compare te two conitions, we nee to be able to assign a vector of wages to eac set of contracts X. It is possible tat, in X, some octor is unavailable at any wage or is available at several ifferent wages. For a profit-maximizing ospital, te octor s relevant wage is te lowest wage, if any, at wic se is available. Moreover, suc a ospital oes not istinguis between a octor wo is unavailable an one wo is available only at a proibitively ig wage. Tus, from te perspective of a profitmaximizing ospital, aving contracts X available is equivalent to facing a wage vector Wˆ ( X ) specifie as follows: 8

9 Wˆ ( X ) = min{ s s = w or (, s ) X }. (5) In view of te preceing iscussion, a profit-maximizing ospital s coices must obey te following ientity: (A1) C( X ) = C( Wˆ ( X )). (6) Teorem 2. Suppose tat X = W is a finite set of octor-wage pairs an tat (A1) ols. Ten C satisfies te eman teory substitutes conition if an only if it contracts are substitutes. {, } j j j Proof. Let i j, ( j, wj ) C( w) an w i > wi. efine Z ( w) ( j w ) w w. Ten, Z ( w i, w i) Z( w). If contracts are substitutes, ten R ( Z( wi, w i) ) R( Z( w) ) By (A1), since ( jw, j ) C( w), it follows tat ( jw, j ) C( Z( w) ), so ( j, wj ) R( Z( w) ). ence, ( jw, j) R( Z( w i, w i) ). So, ( j, wj) Z( w i, w i) R( Z( w i, w i) ) = C( Z( w i, w i) ) an tus ( j, wj) C( w i, w i) by. assumption (A1). Tus, C satisfies eman teory substitutes. Conversely, suppose contracts are not substitutes. Ten, tere exists a set X, an j, w R( X ) j w R X, were element ( iw, i ) X X X {( i, wi )}, an ( j ) suc tat (, j ) ( ) =. Using (A1), w < Wˆ ( X ). Let w = Wˆ ( X ) an w = Wˆ ( X ). Ten, i i w i w i jw, j C w jw, j C w i, w i, so C oes not satisfy te eman teory substitutes conition. In particular, tis sows tat te Kelso-Crawfor gross substitutes conition is subsume by our substitutes conition. > an ( ) ( ), but ( ) ( ) i 1. Substitutes an Stable Matces We now introuce a monotonic algoritm tat will be sown to coincie wit te Gale-Sapley algoritm on its original omain. To escribe te monotonicity tat is foun in te algoritm, let us efine an orer on X X as follows: ( X, X ) ( X, X ) X X an X X. (7) ( ) ( ) Wit tis efinition, ( X X, ) is a finite lattice. Te algoritm is efine as te iterate applications of a certain function F : X X X X, as efine below. F1 ( X ) = X R ( X ) F2 ( X ) = X R ( X ) (, ) = ( ), ( ) ( 1 2( 1 )) F X X F X F F X i (8) 9

10 As we ave previously observe, since te octors coices are singletons, a reveale preference argument establises tat te function R : X X is isotone. If te contracts are substitutes for te ospitals, ten te function R : X X is isotone as well. Wen F : X X, X X, is also isotone, tat is, it bot are isotone, te function ( ) ( ) satisfies ( ) ( ) ( X, X X, X ) ( F( X, X) F( X, X )). Tus, F : X X X X is an isotone function from a finite lattice into itself. Using fixe point teory for finite lattices, te set of fixe points is a non-empty lattice an iterate applications of F, starting from te minimum an maximum points of X X, converge monotonically to a fixe point of F. 4 We summarize te particular application ere wit te following teorem. Teorem 3. Suppose contracts are substitutes for te ospitals. Ten, 1. te set of fixe points of F on X X is a non-empty finite lattice, an in particular inclues a smallest element ( X, X ) an a largest element ( X, X ), 2. starting at ( X, X) ( X, ) igest fixe point ( X, X) sup {( X, X ) F( X, X ) ( X, X )} 3. starting at ( X, X) (, X) lowest fixe point ( X, X) = inf {( X, X ) F( X, X ) ( X, X )}. =, te algoritm converges monotonically to te =, an =, te algoritm converges monotonically to te Te facts tat ( X, X ) is te igest fixe point of F an tat ( X, X ) is te lowest in te specifie orer mean tat for any oter fixe point ( X, X ), X X X an X X X. Because octors are better off wen tey can coose from a larger set of contracts, it follows tat te octors unanimously weakly prefer C( X ) to C( X ) to C( X ) an similarly tat te ospitals unanimously prefer C ( X ) to C ( X ) to C ( X ). Notice, by teorem 1, tat C( X) = C( X) = X X an C( X) = C( X) = X X, so we ave te following welfare conclusion. Teorem 4. Suppose contracts are substitutes for te ospitals. Ten, te stable set of contracts X X is te unanimously most preferre stable set for te octors an te unanimously least preferre stable set for te ospitals. Similarly, te stable set 4 Tis special case of Tarski s fixe point teorem can be simply prove as follows: Let Z be a finite lattice wit maximum point z. Let z0 = z, z1 = F( z0), zn = F( zn 1). Plainly, z1 z0 an since F is isotone, z2 = F( z1) F( z0) = z1 an similarly zn+ 1 zn for all n. So, te ecreasing sequence { z n} converges in a finite number of steps to a point ẑ wit F( zˆ) = zˆ. Moreover, for any fixe point z, since z z, n n z = F ( z ) F ( z) = zˆ for n large, so ẑ is te maximum fixe point. A similar argument applies for te minimum fixe point. 10

11 X X is te unanimously most preferre stable set for te ospitals an te unanimously least preferre stable set for te octors. Teorems 3 an 4 uplicate an exten familiar conclusions about stable matces in te Gale-Sapley matcing problem an a similar conclusion about equilibrium prices in te Kelso-Crawfor labor market moel. Tese new teorems encompass bot tese oler moels, an aitional ones wit general contract terms. To see ow te Gale-Sapley algoritm is encompasse, consier te octor-offering algoritm. As in te original formulation, we suppose tat ospitals ave a ranking of octors tat is inepenent of te oter octors tey will ire, so ospital just cooses its n most preferre octors (from among tose wo are acceptable an ave propose to it). Let us interpret X ( t) to be te cumulative set of contracts offere by te octors to te ospitals troug iteration t, an let us interpret X ( t ) to be te set of contracts tat ave not yet been rejecte by te ospitals troug iteration t. Ten, te contracts el at te en of te iteration are precisely tose tat ave been offere but not rejecte, wic are tose in X ( t) X( t). Te process initiates wit no offers aving been mae or rejecte so, X (0) = X an X (0) =. Iterate applications of te operator F escribe above efine a monotonic process, in wic te set of octors make an ever-larger (accumulate) set of offers an te set of unrejecte offers grows smaller roun by roun. Using te specification of F an starting from te extreme point ( X, ), we ave: ( ) ( ) X() t = X R X( t 1) X () t = X R X () t After offers ave been mae in iteration t 1, te ospital s cumulative set of offers is X ( t 1). Eac ospital ol onto te n best offers it as receive at any iteration provie tat many acceptable offers ave been mae; oterwise it ols all acceptable R X t X R X ( t 1) = X ( t). At roun t, if a offers tat ave been mae. Tus, te accumulate set of rejecte offers is ( ( 1) ) an te unrejecte offers are tose in ( ) octor s is being el, ten te last offer te octor mae was its best contract in X ( t ). If a octor s last offer was rejecte, ten its new offer is its best contract in X ( t ). Te contracts tat octors ave not offere at tis roun or any earlier one are terefore tose R X t. So, te accumulate set of offers octors ave mae are tose in in ( () ) ( ) X R X () t = X () t. Accoring to tis analysis, wen X ( t ) an X ( t ) are interprete as suggeste above, te process { X ( t), X( t )} escribe by (9) an te initial conitions X (0) = X an X (0) = coincies wit caracterizes te octor-offering Gale-Sapley algoritm. (9) 11

12 For te ospital offering algoritm, a similar analysis applies but wit a ifferent interpretation of te sets an a ifferent initial conition. We interpret X ( t) to be te cumulative set of contracts offere by te ospitals to te octors before iteration t an X () t to be te set of contracts tat ave not yet been rejecte by te ospitals up to an incluing iteration t. Ten, te contracts el at te en of iteration t are precisely tose tat ave been offere but not rejecte, wic are tose in X ( t+ 1) X( t). Wit tis interpretation, te analysis is ientical to te one above. Te Gale-Sapley ospital offering algoritm is caracterize by (9) an te initial conitions X (0) = an X (0) = X. Te same logic applies to te Kelso-Crawfor moel, provie one extens teir original treatment to inclue a version in wic te workers make offers in aition to te treatment in wic firms make offers. Te wors of te preceing paragraps apply exactly, but a contract offer now inclues a wage so, for example, a ospital wose contract offer is rejecte by a octor may fin tat its next most preferre contract is at a iger wage to te same octor. 2. Wen Contracts are Not Substitutes It is clear from te preceing analysis tat tat efinition of substitutes is just sufficient to allow our matematical tools to be applie. In tis section, we establis more. We sow tat if tere is any ospital for wic contracts are not substitutes, te very existence of a stable set of contracts cannot be guarantee. Teorem 5. Suppose contains at least two ospitals, wic we enote by an. Furter suppose tat R is not isotone, tat is, contracts are not substitutes for. Ten, tere exist preference orerings for te octors in set, a preference orering for a ospital wit a single job opening suc tat, regarless of te preferences of te oter ospitals, no stable set of contracts exists. Proof. We may limit attention to te case wit exactly two ospitals by specifying tat te octors fin te oter ospitals to be unacceptable. Suppose R is not isotone. Ten, tere exists some x, y X an X X suc tat for all x X x x R X R X y. By construction, since, = an suc tat ( ) ( { }) ( { }) x, y C X y, contracts x an y specify ifferent octors, say, 1 x y 2. Let x an y enote te corresponing contracts for octors 1 an 2 in wic ospital is substitute for. We specify preferences as follows: First, for ospital, we take { x} { y} an all oter contracts are unacceptable. Secon, octors in x C X C X y, prefer teir elements of ( ( ) ( { })) { 1 2} C ( X ) C ( X { y} ) to any oter contract. Tir, 1 as { x} 1 { x } oter contracts lower. Fourt, 2 as { y} { y} 2 an ranks all an ranks all oter contracts lower. 12

13 Finally, te remaining octors fin all contracts from ospitals an to be unacceptable. Consier a feasible, acceptable allocation X suc tat y X. Since an 2 can ave only one contract in X, x, y X. Ten, s contracts in X form a subset of X, so x is not inclue an 1 as a contract less preferre tan x. Ten, te eviation to x blocks X. by ( ) 1, Consier a feasible, acceptable allocation X suc tat y X. Ten, eiter x, y X or X is blocke by a coalition incluing, 1 an 2 using te contracts x an y. owever, if x, y X to contract y blocks X., ten a eviation by ( ) Since all feasible allocations are blocke, tere exists no stable set of contracts. Togeter, teorems 3 an 5 caracterize te set of preferences tat can be allowe as inputs into a matcing algoritm if we wis to guarantee tat te outcome of te algoritm is a stable set of contracts wit respect to te reporte preferences. Accoring to teorem 3, we can allow all preferences tat satisfy substitutes an still reac an outcome tat is a stable collection of contracts. Accoring to teorem 5, if we allow any preference tat oes not satisfy te substitutes conition, ten tere is some profile of singleton preferences for te oter parties suc tat no stable collection of contracts exists. Tis teory also reaffirms an extens te close connection between te substitutes conition an oter concepts tat as been establise in te recent auctions literature wit quasi-linear preferences. Milgrom (2000) stuies an auction moel wit iscrete goos an transfers an in wic bier values are allowe may be any aitive function an may inclue oter functions as well. 5 e sows tat if goos are substitutes, ten a competitive equilibrium exists. If, owever, tere are at least tree biers an if tere is any allowe value suc tat te goos are not all substitutes, ten tere is some profile of values suc tat no competitive equilibrium exists. Gul an Staccetti (1999) establis te same positive existence result. Tey also sow tat if preferences inclue all values in wic a bier wants only one particular goo as well as any one for wic goos are not all substitutes, an if te number of biers is sufficiently large, ten tere is some profile of preferences for wic no competitive equilibrium exists. Ausubel an Milgrom (2002) establis tat if (i) tere is some bier for wom preferences are not eman teory substitutes, (ii) values may be any aitive function an (iii) tere are at least tree biers in total, ten tere is some profile of preferences suc tat te Vickrey outcome is not stable an te core imputations o not form a lattice. Conversely, if all biers ave preferences tat are eman teory substitutes, ten te Vickrey outcome is in te core an te core imputations o form a lattice. Taken togeter, tese results establis a close connection between te substitutes conition, te cooperative concept of te core, te non-cooperative concepts of Vickrey outcomes, an competitive equilibrium. 2, 5 A valuation function is aitive if te value of any set of items is te sum of te separate values of te elements. Suc a function v is also escribe as moular, aitivity is equivalent to te requirement tat for all sets A an B, va ( B) + va ( B) = va ( ) + vb ( ). 13

14 3. Vacancy Cain ynamics Suppose tat a labor market as reace equilibrium, wit all intereste octors place at ospitals in a stable matc. Suppose some octor ten retires. Imagine a process in wic a ospital seeks to replace its retire octor by raiing oter ospitals to ire aitional octors. If te ospital makes an offer tat woul succee in iring a octor away from anoter ospital, te affecte ospital as tree options: it may make an offer to anoter octor (or several), improve te terms for its current octor, or leave te position vacant. Suppose it makes watever contract offer woul best serve its purposes. In moels witout contracts, tis process in wic octors an vacancies move from one ospital to anoter as been calle vacancy cain ynamics (Blum, Rot an Rotblum (1997)). Tese analyses exploit bot te absence of any ajustments of wages or terms of employment an te fact tat, in te Gale-Sapley moel, te notion of a vacancy is well efine by te formulation. Recall tat, in tat moel, a ospital as n positions. If one of its n octors retires, ten it as one well-efine vacancy. In our more general teory, a ospital migt replace a single octor by multiple oters. espite tis extra complexity, te formal results for te extene moel are quite similar to tose for an environment wit simple responsive preferences. Starting wit a stable collection of contracts X, let X be te set of contracts tat some octor weakly prefers to er current contract in X an let X = X ( X X ). As in te proof of teorem 1, we ave F( X, X ) = ( X, X ) an X = X X. To stuy te ynamics tat results from te retirement of octor, we suppose te process starts from te initial state ( X (0), X(0)) = ( X, X ). Tis means tat te employees start by consiering only offers tat are at least as goo as teir current positions an tat ospitals remember wic employees ave rejecte tem in te past. Te octors rejection function is cange by te retirement of octor to R ˆ, were Rˆ ( X ) = R ( X ) { x X x = }, tat is, in aition to te ol rejections, all contract offers aresse to te retire octor are rejecte. To syncronize te timing wit our earlier notation, let us imagine tat ospitals make offers at roun t 1 an octors accept or reject tem at roun t. ospitals consier as potentially available te octors in X ( 1) ˆ t = X R( X( t 1)) an te octors ten reject all but te best offers, so te cumulative set of offers receive is X () t = X R ( X ( t 1)). efine: Fˆ( X, X ) = ( X R ( X ), X Rˆ ( X )) (10) If contracts are substitutes for te ospitals, ten ˆF is isotone an, since F( X, X ) = ( X, X ), it follows tat F ˆ ( X, X ) ( X, X ). Ten, since ( X (0), X (0)) = ( X, X ), we ave: ( X (1), X (1)) = Fˆ ( X (0), X (0)) ( X (0), X (0)). 14

15 Iterating, ( X ( ), ( )) ˆ n X n = F( X( n 1), X( n 1)) ( X( n 1), X( n 1)) : te octors accumulate offers an te contracts tat are potentially available to te ospitals srinks. A fixe point is reace an, by Teorem 1, it correspons to a stable collection of contracts. Teorem 6. Suppose tat contracts are substitutes an tat ( X, X ) is a stable set of contracts. Suppose tat a octor retires an tat te ensuing ajustment process is escribe by ( X (0), X(0)) = ( X, X ) an ( X ( ), ( )) ˆ t X t = F( X( t 1), X( t 1)). Ten, te sequence {( X ( t), X( t)) } converges to a stable collection of contracts at wic all te unretire octors are weakly better off an all te ospitals are weakly worse off tan at te initial state ( X, X ). 6 Te sequence of contract offers an job moves escribe by iterate applications of ˆF inclues all te complexity escribe above. ospitals tat lose a octor may seek several replacements. ospitals wose octors receive contract offers may retain tose octors by offering better terms or may ire a ifferent octor an later reire te original octor at a new contract. All along te way, te octors fin temselves coosing from more an better options an te ospitals fin temselves marcing own teir preference lists by offering costlier terms, paying iger wages, or making offers to oter octors wom tey a earlier rejecte. IV. Law of Aggregate eman We now introuce a secon restriction on preferences tat allows us to prove te next two results about te structure of te core. We call tis restriction te law of aggregate eman. Rougly, tis law states tat as te price falls, agents soul eman more of a goo. ere, prices falling correspons to more contracts being available, an emaning more correspons to taking on (weakly) more contracts. We formalize tis intuition wit te following efinition. efinition. Te preferences of ospital satisfy te law of aggregate eman if for all X X C X C X., ( ) ( ) Accoring to tis efinition, if te set of possible contracts expans (analogous to a ecrease in some octors wages), ten te total number of contracts cosen by ospital oes not fall. Te corresponing property for octor preferences is implie by reveale preference, because eac octor cooses at most one contract. Just as for te substitutes conition, wen wages are enogenous, we interpret te efinition as applying to te omain of wage vectors for wic te ospital s optimum is unique. Below, te law of aggregate eman allows us to caracterize bot te necessary an sufficient conitions for some of te properties of matcing moels, allowing us to pin own exactly wat preference profiles are allowe for bot te rural ospitals property to ol an to ensure tat trutful revelation is a ominant strategy for te octors. 6 Te teorem oes not claim, an it is not generally true, tat tis new point must be te new octor-best stable set of contracts. 15

16 Previously, in matcing moels witout money, te ominant strategy result was known only for responsive preferences wit capacity constraints (Abulkairoğlu (2003)). We subsume tat result wit our teorem. First, owever, we sow tat te law of aggregate always ols in te Kelso- Crawfor framework of profit-maximizing firms; it is a consequence of te fact tat firm s payoff functions are quasilinear. Teorem 7. If ospital s preferences are quasilinear an satisfy te substitutes conition, ten tey satisfy te law of aggregate eman. Proof. Suppose X W Z w j, w w w. Ten, te law of j j j aggregate eman is te statement tat for any wage vectors wwsatisfying, ˆ w wˆ suc C Z w C Z wˆ. = an let ( ) ( ) tat te coices sets are singletons, ( ) ( ) ( ( )) { } Te proof is by contraiction. Suppose te law of aggregate eman oes not ol. Ten tere exists a wage vector w an a octor suc tat for some (an ence all) ε > 0, C( Z( w + ε, w ) ) > C( Z( w ε, w ) ). Since s preferences are quasi-linear, canging octor s wage can affect te iring of oter octors only if it affects te iring of octor. It follows tat tere are exactly two optimal coices for te ospital at wage vector w; tese are C( Z( w ε, w ) ) at wic octor is ire an C( Z( w + ε, w ) ) at wic is not ire but suc tat two oter octors are ire, tat is, tere exist octors, C( Z( w + ε, w ) ) C( Z( w ε, w ) ). Let te corresponing payoff for te ospital (wen face wit wage vector w) be π. Consier te wage vector w = ( w ε, w 2 ε, w {, }). For ε positive an sufficiently small, te ospital s payoff at wage vector w is π + 2ε if it cooses C( Z( w + ε, w ) ) an it is π + ε if it cooses C( Z( w ε, w ) ), an one of tese coices must be optimal. So, Cw ( ) = C( Z( w + ε, w ) ). But ten, raising te wage of octor from w 2ε to w wile oling te oter wages at w reuces te eman for octor from one to zero, in violation of te eman teory substitutes conition. 1. Rural ospitals Teorem In te matc between octors an ospitals, certain rural ospitals often a trouble filling all teir positions, raising te question of weter tere are oter core matces at wic te rural ospitals migt o better. Rot (1986) analyze tis question for te case of X = an responsive preferences an foun tat te answer is no: every ospital tat as unfille positions at some stable matc is assigne exactly te same octors at every stable matc. In particular, every ospital ires te same number of octors at every stable matc. 16

17 In tis section, we sow by an example tat tis last conclusion oes not generalize to te full set of environments in wic contracts are substitutes. 7 We ten prove tat if preferences satisfy te law of aggregate eman an substitutes, ten te last conclusion of Rot s teorem ols: every ospital signs exactly te same number of contracts at every point in te core, altoug te octors assigne an te terms of employment can vary. Finally, we sow tat any violation of te law of aggregate eman implies preferences exist suc tat te above conclusion oes not ol. Suppose tat = {, } an {,, } maximize, were: 1 2 =. For ospital 1, suppose its coices { } {, } { } { } {, } {, }. (11) Tis preference satisfies substitutes. 8 Suppose 2 as one position, wit its preferences among octors given by Finally, suppose 1 an 2 prefer 1 to 2 X =,,, an wile 3 as te reverse preference. Ten, te matces {( 1 3) ( 2 1) } X {( 1, 1),( 1, 2),( 2, 3) } = are bot stable but ospital 1 employs a ifferent number of octors an te set of octors assigne iffers between te two matces. Tis example involves a failure of te law of aggregate eman, because as te set of,, te number of available contracts expans by te aition of 3 to te set { } octors emane eclines from two to one. Wen te law of aggregate eman ols, owever, we ave te following result. Teorem 8. If ospital preferences satisfy substitutes an te law of aggregate X, X an every an, eman, ten for every stable allocation ( ) ( ) ( ) C X = C X an C( X) = C( X). Tat is, every octor an ospital signs te same number of contracts at every stable collection of contracts. Proof. By efinition, X X 1 2, so by reveale preference, C ( X ) C ( X ). Also, X X, so by te law of aggregate eman, C( X ) C( X) 1, C ( X) = C ( X ) an C ( X) = C ( X), so C ( X ) = C ( X ) an C ( X ) = C ( X ). Combining tese leas to ( ) C X C ( X) = C( X) C( X) = C ( X), wic begins an. By Teorem ens wit te same sum. ence, none of te inequalities can be strict. Te next teorem verifies tat te counterexamples evelope above can always be generalize wenever any ospital s preferences violate te law of aggregate eman. 7 A similar example appears in Martínez, Massó, Neme an Ovieo (2000). 8 Tese preferences, owever, o not isplay te single improvement property tat Gul an Staccetti (1999) introuce an sow is caracteristic of substitutes preferences in moels wit quasi-linear utility. 17

18 Teorem 9. If tere exists a ospital an sets X X X suc tat C( X ) > C( X ) an at least one oter ospital, ten tere exist singleton preferences for te oter ospitals an octors suc tat te number of octors employe by is ifferent for two stable matces. >, tere exists some set Y, X Y X an contract Proof. Since C( X ) C( X ) x suc tat C( Y) > C( Y { x} ). Since x C Y { x} C( Y) C( Y { x} ) contracts yz, R( Y { x} ) R( Y) yz C( Y) y z., ( ) (as oterwise = for te preferences to be rationalizable) tere must exist two,, suc tat y x z y. Moreover, since enoting by te secon ospital wose existence is ypotesize by te teorem, we specify preferences as follows. Let all te octors wit contracts in Y ave tose contracts be teir most favore, an let all oter octors fin any contract wit unacceptable. Let all octors fin any contract not involving ospital or to be unacceptable. In principle, tere are tree cases. If x = y, ten let P : y x an P : z. Ten, tere exist two stable matces, ( { }) an C ( ) C Y x x z Y, wit z employe in te first matc but not in te secon. Te case x = z is symmetric. Finally, if y x z, ten let x, y, z enote contracts wit ospital were te octors (an any oter terms) are te same as in x, yz, respectively. Specify te remaining preferences by Px = x x, Py = y y, Pz = z z, an P' = { y' } { x' } { z }. Ten, tere exist two ifferent stable matces, { x } C ( Y) an { y } C ( Y { x} ), wit z employe in te first matc but not in te secon. Tis sows tat te law of aggregate eman is not only a sufficient conition but, in te sense escribe by te teorem, a necessary one to guarantee tat eac agent as te same number of contracts at every stable matc. 2. Trutful Revelation as a ominant Strategy Te main result of tis section concerns octors incentives to report teir preferences trutfully. For te octor-offering algoritm, if ospital preferences satisfy te law of aggregate eman an te substitutes conition, ten it is ominant strategy for octors to 18

19 trutfully reveal teir preferences over contracts. 9 We ten furter sow tat bot preference conitions play essential roles in te conclusion. We will sow te positive incentive result for te octor offering algoritm in two steps tat igligt te ifferent roles of te two preference assumptions. First, we sow tat te substitutes conition, by itself, guarantees tat octors will not want to exaggerate te ranking of an unattainable contract. More precisely, if tere exists a preferences list for a octor suc tat obtains contract x by submitting tis list, ten can also obtain x by submitting a preference list tat inclues only contract x. Secon, we will sow tat aing te law of aggregate eman guarantees tat a octor oes at least as well as reporting trutfully as by reporting any singleton. Togeter, tese are te ominant strategy result. To unerstan wy submitting unattaine contracts can not elp a octor, consier te following. Let x be te most-preferre contract tat can obtain by submitting any preference list (oling all oter submitte preferences fixe). Note tat all tat accomplises wen reporting tat certain contracts are preferre to x is to make it easier for some coalition to block outcomes involving x. Tus, if x is attainable wit any report, it is attainable wit te report P : x tat ranks x as te only acceptable contract. Tis intuition is capture in te following teorem: Teorem 10. Let ospitals preferences satisfy te substitutes conition an let te matcing algoritm prouce te octor-optimal matc. Fixing te preferences of ospitals an of octors besies, let x be te outcome tat obtains by reporting preferences P : z1 z2... zn x. Ten, te outcome tat obtains by reporting preferences P : x is also x. Proof.. Let X enote te collection of contracts cosen by te algoritm wen octor submits preference P. If tis collection, wic is stable uner te reporte preferences, is not stable uner P, ten tere exists a blocking coalition. Tis blocking coalition must contain, as no oter octor s preferences ave cange, but tat is impossible, since x is s favorite contract accoring to te preferences P. Since X is stable uner P, te octor-optimal stable matc uner P (te existence of wic is guarantee by Teorem 2) must make every octor (weakly) better off tan at X. In particular, octor must obtain x. Some oter octors may be strictly better off wen submits er sorter preference list; tere are fewer collections of contracts tat now objects to, so te core may become larger, an te octor-optimal point of te enlarge core makes all octors weakly better off an may make some strictly better off. Witout te law of aggregate eman, owever, it may still be in a octor s interest to conceal er preferences for unattainable positions. To see tis, consier te case wit two 9 It is, of course, not a ominant strategy for ospitals to trutfully reveal; nor woul it be so even if we consiere te ospital-offering algoritm. For furter iscussion of tis point, see Rot an Sotomayor (1990). 19