Discussion Papers in Economics

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1 scusson Ppers n Economcs No. 10/0 Alloctve Effcency nd n Incentve Scheme for Reserch By Anndy Bhttchry, Unversty of York; Herbert Newhouse, Unversty of Clforn eprtment of Economcs nd Relted Studes Unversty of York Heslngton York, YO10 5

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3 Alloctve Effcency nd n Incentve Scheme for Reserch 1 Anndy Bhttchry eprtment of Economcs nd Relted Studes, The Unversty of York, UK Herbert Newhouse eprtment of Economcs Unversty of Clforn, Sn ego Frst Verson: October 005 Ths Verson: Jnury 010 Abstrct In ths pper we exmne whether n ncentve scheme for mprovng reserch cn hve dverse effect on reserch tself. Ths work s mnly motvted by the Reserch Assessment Exercse (RAE) nd the Reserch Excellence Frmework (REF) n UK. In gme theoretc frmework we show tht scheme lke RAE/REF cn ctully result n deterorton of the over-ll reserch n country though t my crete few solted centres of excellence. The centrl ssumpton behnd ths result s tht hgh blty reserchers produce postve externltes to ther collegues. We ssume these externltes hve declnng mrgnl beneft s the number of hgh blty reserchers n deprtment ncreses. Becuse of ths declnng mrgnl beneft n ncentve scheme lke the RAE or REF my led to over concentrton of the hgh blty reserchers n few deprtments. JEL Clssfcton No: I, I8, C71, C7. Keywords: reserch, RAE, REF, coltons, strong nsh equlbrum. 1 We re ndebted to Alessndr Cnep, Vncent Crwford, Kren Mumford, Arunv Sen nd semnr udences t Cen, Lncster nd York for useful comments nd dscussons. Of course, the errors nd shortcomngs re ours.

4 1 Introducton In ths pper we exmne whether ncentve schemes for reserch cn hve n dverse effect on reserch tself. Ths work s mnly motvted by the Reserch Assessment Exercse (RAE) nd the Reserch Excellence Frmework (REF) n UK. In gme theoretc frmework we show tht scheme lke RAE/REF cn ctully result n deterorton of the over-ll reserch n country though t my crete few solted centres of excellence. To cpture the pyment from such ncentve schemes, better deprtment my hve tendency to replce persons wth reltvely lower blty by persons wth reltvely hgher blty. A worse deprtment my be unble to keep ts persons wth reltvely hgher blty nd forced to rely on persons wth reltvely lower blty. The ntuton behnd ths s s follows. We ssume tht resercher gns from the externl effects of her cdemc surroundngs. Furthermore, we ssume tht for n cdemc professonl ths effect declnes s her reserch blty ncreses. Therefore, overll reserch output my ncrese when hgher blty reserchers re dstrbuted more eqully mong reserch nsttutons. However, n ncentve scheme lke RAE/REF wrds n entre deprtment some lump-sum pyment on the bss of some over-ll reserch output of the deprtment nd every member n the deprtment hs ccess to ths rewrd. So, to cpture ths pyment, deprtment wll hve tendency to replce persons wth reltvely lower blty by persons wth reltvely hgher blty. Thus, strct herrchy of deprtments would emerge few very good deprtments followed by strng of bd deprtments. For RAE 001 such phenomenon hs been observed n the deprtments of UK (see Hre (003)). ue to the declnng returns of externltes n blty, overll reserch my ncrese f we move hgh blty resercher from the top deprtment n ths herrchy to lower-rnked deprtment. Ths pper s smply forml modellng of ths de to hghlght ths possble phenomenon n precse mnner. Our nterest s on the peer effects between the reserchers rther thn the qulty of the mtch between prtculr resercher nd her nsttuton. For ths reson we dopt the model of strtegc colton formton of Hrt nd Kurz (1983) rther thn the more stndrd mtchng model (for exmple Bulow nd

5 Levn (006)). We nlyze the equlbr of deprtment-formton gme nd show tht t s possble to obtn the somewht perverse result tht overll reserch my decrese for socety f deprtments re rewrded for ther ndvdul reserch outputs. Our purpose s smply to show tht n ncentve scheme such s the RAE/REF my hve undesrble consequences on the llocton of reserchers cross deprtments. We mke wht we feel re resonble ssumptons nd show tht such n ncentve scheme for reserch lowers the overll reserch output for one prtculr exmple. The vldty of our ssumptons nd whether or not our exmple s relevnt re both emprcl questons nd re beyond the scope of ths pper. In Secton we provde smple numercl exmple tht llustrtes the pont mde n ths pper. Our model s explned n detl n Secton 3. The results re collected n Secton 4. Secton 5 concludes. An Introductory Exmple To llustrte the mn pont of our pper we strt wth smplfed exmple. The full model s formulted n Secton 3. Suppose there re twelve reserchers n n economy, sx wth hgh blty nd sx wth low blty. Reserch must be performed n deprtment nd ech deprtment requres exctly four reserchers. An ndvdul s reserch output s bsed on her ntrnsc blty nd on the bltes of her collegues. Specfclly n ndvdul s reserch output ncreses ddtvely by fxed mount f exctly one of her collegues s of hgh blty. Her output ncreses by lrger mount f two of her collegues re of hgh blty but there s no ddtonl ncrese f ll three of her collegues re of hgh blty. Arrnge the twelve reserchers nto three deprtments. efne perfectly stble deprtment s one where ech resercher s producng her mxmum possble reserch output. An unstble deprtment s one where t lest one resercher s producng below Our frmework could be dpted to nclude number of reserchers nd deprtment tht my form colton. However, for smplcty, here we model deprtment solely s collecton of ts cdemc members. 3

6 her mxmum possble reserch output. Assumng ech resercher s pd strctly ncresng functon of her own reserch output, no resercher t perfectly stble deprtment cn strctly gn by formng new deprtment wth other reserchers. The reserchers re dvded nto the followng deprtments: Confgurton A (HHHH), (HLLL), (HLLL). The frst deprtment gven here s perfectly stble; ech resercher n ths deprtment hs t lest two collegues of hgh blty. The other two deprtments re not; ech resercher t these deprtments hs zero or one collegues of hgh blty. From these two deprtments, two hgh blty reserchers nd two low blty reserchers would ll strctly gn f they formed new deprtment. Recll tht strong Nsh equlbrum s coltonl equlbrum concept where no colton of gents cn ontly devte so tht ech member of ths colton becomes strctly better off. For our exmple strong Nsh equlbrum wll consst of perfectly stble deprtment, followed by deprtment wth the remnng hgh blty reserchers, followed by deprtment consstng entrely of low blty reserchers. (The generl cse s proved n Proposton below.) Two strong Nsh equlbrum confgurtons re Confgurton B (HHHH), (HHLL), (LLLL) nd Confgurton C (HHHL), (HHHL), (LLLL). Now suppose n ncentve scheme rewrds members of deprtment bsed on the verge level of reserch tht occurs t tht deprtment. The ncentve scheme pyment s strctly ncresng wth regrds to the verge reserch level of deprtment. Confgurton C s no longer n equlbrum. The three hgh blty reserchers from the frst deprtment nd one of the hgh blty reserchers from the second deprtment could form new deprtment nd ll of them would strctly gn. The only strong Nsh equlbrum gven ths ncentve scheme s Confgurton B. Totl reserch output s lower n Confgurton B thn t s n Confgurton C. The fourth hgh blty resercher n the frst deprtment does not dd to the externlty wheres thrd hgh blty resercher n the second deprtment would dd to the externlty. In the remnder of ths pper we nlyze more generl model. We show, frst, tht under some ssumptons, such n ncentve scheme wll result n strct herrchy s the unque equlbrum outcome. Next we show tht wth such n ncentve scheme, the totl equlbrum reserch output my fll. 4

7 3 The Model The Agents: The fnte set of n plyers (ech n cdemc) s denoted by N. Ech n N hs n ntrnsc blty for reserch lyng n the ntervl [0, 1]. Wthout loss of generlty, we order the plyers ccordng to ther blty for reserch,.e., > f nd only f <. We model the producton of reserch s two-stge gme. In the frst stge cdemcs form deprtments s descrbed below. In the second stge ech chooses level of effort, 0, e e to produce reserch. The cost of effort for ech s gven by c, wth c c 0, 0 c c. If > then, for ech e,,,. The Envronment for Reserch: Frst, we represent the dfferent spects of reserch (volume, qulty etc.) s composte sclr vrble. Next we ssume tht reserch cn be conducted only n n nsttutonl settng sy, n n cdemc deprtment. A deprtment s non-empty subset of N. We ssume tht ech fesble deprtment must hve exctly k ( n) members 3. As we hve mentoned bove, plyer gets some postve externlty n reserch from the presence of other members n deprtment 4. We ssume tht ths externlty for plyer n deprtment s gven by: q () : 1. 3 In the next subsecton we expln the precse menng of fesble deprtment. We cn generlze ths ssumpton bt n the followng wy. Suppose the mnml sze of deprtment must be k (ths s ntutve s deprtment consstng of only one member (sy) s rdculous!) nd there s congeston cost f the deprtment sze exceeds k. Then we cn show tht n equlbrum every deprtment wll hve exctly k members. However, lttle of mportnce s gned by ths ddtonl complcton. 4 Ths externlty s emprclly observed n U.S. unverstes n Km, Morse nd Zngles (006) lthough they fnd the effect hs dmnshed over tme. 5

8 The totl reserch output for person n deprtment s gven by: x e,, q where e s person s level of effort, s her ntrnsc blty nd q () s the level of the x externlty obtned by n deprtment. For ll, 0 wth x 0 x e nd 0. For ll, x 0. We sometmes express the output of by x wth no possblty of confuson. A person s reserch output s contnuous nd lso strctly ncresng n q, the level of externlty, untl fxed q <1. In [ q,1], m m x s x constnt wth respect to q. Addtonlly we ssume 0 s ncresng n q untl In [ q,1], m x s constnt wth respect to q. We ssume tht f > then m q. q m q m, tht s, the less the ntrnsc blty of resercher, the further she s helped by externlty from her collegues. We denote the mxmum vlue of m q (cross ll s) by q. Ths ssumpton expresses the de tht the effect of externlty ceses t some pont. Ths s crtcl ssumpton for our result. Next we ssume tht f plyer s more ble thn plyer then the mrgnl reserch output of s more thn tht of t ny gven effort level gven the sme coworkers. We stte ths ssumpton s Condton A. Condton A: If > x x e q e q. then for ll q, e,,,,, Next we ssume tht f plyer s more ble thn plyer then the mrgnl reserch output of s more thn tht of t ny gven effort level f they re n the sme deprtment. 6

9 The eprtment Formton Gme: Insted of modellng the recrutng of fculty members s mtchng process (see, e.g., Roth nd Sotomyor (1990)) we model the process of the formton of deprtment b nto. We use the model of strtegc colton formton ntroduced n Hrt nd Kurz (1983). Ths s gme n norml form where the outcome resultng from strtegy profle s confgurton of deprtments formed endogenously s result of the strtegc choce of the members. Gven the well-known flux of the cdemcs pror to the RAEs, such modelng should be cceptble! We ssume perfect nformton. Ths s ustfble s the reserch blty of person s observed qute precsely by publctons, prtcptons n conferences etc. The set of plyers s N. The strtegy of plyer s to nnounce the deprtment she wnts to be n. Therefore, formlly, the strtegy set for plyer, The outcome of strtegy profle { S N S}. ( S ) s prtton of N, C=( 1,..., ). Ech N member of ths prtton s deprtment. If deprtment s of crdnlty k then t s fesble; otherwse t s nfesble. Suppose for n N, C() s the unque element of C tht contns. Then, C( ) { } { N S S }. So, complete greement mong the potentl members concernng who re to be ncluded n the deprtment s necessry for fesble deprtment to be formed 5. If deprtment s nfesble (contnng more members thn k or less) then ech plyer n such deprtment receves py-off of 0. If deprtment s fesble, then we dstngush two regmes. In the orgnl regme, clled O-regme, person n deprtment receves py-off equl to her reserch output mnus her cost of effort: O u e x e,, q c e, s descrbed bove. The totl reserch output of deprtment s the sum of the reserch outputs of ts members. 5 Any prtton tht contns the mxmum number of fesble deprtments s Nsh equlbrum for ths stge of the gme. We chose the strong Nsh equlbrum soluton concept to elmnte the equlbr we feel re vcuous. 7

10 Now suppose there s n ncentve scheme so tht the Government pys lump-sum pyment to the entre deprtment on the bss of ts verge reserch output 6. We model the py-off of the plyers n such regme (we cll t R-regme) s follows. Suppose for deprtment the verge reserch output s x. Then gets n ddtonl pyment nd every n gets n ddtonl lump-sum pyment x x 0 where s strctly ncresng, concve functon. 7 Then the py-off to plyer n deprtment s gven by: R u e x e,, q c e, x Ths completes the descrpton of the gme. The soluton concept we use s hybrd one n the sprt of bckwrd nducton. We ssume tht n the effort subgme ech n ech deprtment plys Nsh equlbrum n pure strteges gven the effort choce of other members. Below we show tht, fortuntely, gven our ssumptons, such Nsh equlbrum s unque. Then, gven the Nsh equlbrum effort choces, we look t the Strong Nsh Equlbr n pure strteges (SNE) (see, e.g., Bernhem et l. (1987)) of the reduced gme of deprtment formton under the two regmes. Recll tht strtegy profle exst ( S ) s n SNE f there does not N such tht the plyers n cn ontly devte (whle those n N\ stck to the equlbrum strteges) nd ech plyer n cn strctly gn by such devton 8. As we hve lredy noted bove, gven the structure of our gme, the power of n ndvdul plyer s mnml. So, SNE, rther thn non-coopertve soluton concept, s n pproprte soluton concept for the deprtment formton subgme. N 6 Although the exct rnkng ccordng to specfc ncentve scheme lke the RAE s much more complcted, the verge reserch performnce of deprtment s good summry ndctor of such rnkngs becuse, for exmple, RAE tkes nto ccount both the totl output of the deprtment s well s the percentge of the cdemc stff ncluded n the RAE submssons. 7 We cn use other mesures of centrl tendency lke the medn or some quntle of the dstrbuton of reserch outputs of the deprtment members s the bss for the Governmentl lump-sum pyment. Then lso, the ntuton of our result would be vld. However, of course, the precse condtons for the results would chnge. 8 The noton of SNE s smlr to the noton of stblty n the mtchng lterture. See, for nstnce, Kelso nd Crwford (198). 8

11 4 The Results We proceed s follows. Frst (n Lemm 1) we show tht for both the regmes, there exsts unque Nsh equlbrum level of effort choce for ech plyer n ech deprtment. Gven ths, we cn menngfully nlyze the reduced gme of deprtment formton n the frst stge s ech plyer s the sure bout wht py-off she wll get t the ply n the second stge. Next we show n Proposton 1 tht under Condton A nd ssumng tht s lner, the unque equlbrum confgurton of deprtments tht wll be obtned n stge 1 under the R-regme s strctly herrchcl,.e., the k hghest blty persons would be n one deprtment, the set of next k hghest persons n nother deprtment, etc. Then we show n Proposton tht n generl, more thn one equlbrum confgurtons of deprtments my emerge n the O-regme. Fnlly, n Proposton 3 we provde n exmple where the totl reserch output n one of the equlbrum confgurtons of deprtments for the O-regme s more thn tht n the unque equlbrum confgurton of deprtments n the R-regme. Frst we look t the level of effort chosen n equlbrum by plyer wthn deprtment t the second stge of the gme. Lemm 1 There exsts unque Nsh equlbrum n pure strteges for the effort choce subgme n both regmes. Proof: Note tht for both regmes the py-off functon of ech s contnuous n the profle of effort choces nd strctly concve n e. Snce the strtegy set s compct, Nsh equlbrum n pure strteges exsts. Next, for ether regme, for ny deprtment,, consder the mtrx J such tht the -th entry of J s gven by u. 9

12 Gven tht c 0 e, J cn be esly shown to be negtve defnte. To see ths, for the R-regme, note tht the -th entry of J s gven by: x ( 1 ') c x ' ' x x nd the -th entry (wth dfferent from ) s gven by ''. x Therefore, J cn be wrtten s the sum of two mtrces J nd dgonl mtrx whose -th entry s gven by: x (1 ') e c ' ' J where J s wheres, the -th entry of J s gven by x x. x For ny -dmensonl vector, the product ' ' J s ' '( ) whch, gven the concvty of s non-postve. And gven our ssumptons on the functons x nd c, the mtrx J s negtve defnte. Therefore, the mtrx J s lso negtve defnte. Then by Rosen (1965), there exsts unque pure strtegy Nsh equlbrum n the effort subgme. The effort level e chosen by plyer n equlbrum of the effort-subgme n deprtment n the orgnl (O) regme stsfes the followng frst order condton: x c,, q, [1] Smlrly, the effort level e chosen by plyer n equlbrum of the effort-subgme n deprtment n the ncentve (R) regme stsfes the followng frst order condton: 10

13 x c 1 ' x e,, q e, [] Now we look t the problem of the exstence of n SNE n the reduced gme of deprtment formton. We strt wth nother lemm. We show tht totl reserch output s hgher n better deprtment. Frst we defne better deprtment. efnton 1 eprtment B s better deprtment thn W B W f 1,, k wth B for t lest one 1,, kwhen B nd W re both ordered from the W best member to the worst member (here deprtment ). stnds for the blty of the -th plyer n Lemm Suppose B s better deprtment thn W. Then the totl reserch output n B n equlbrum s more thn tht n W. Moreover, let be lner. Let B be the -th rnked plyer (ccordng to blty) n B nd be the -th rnked plyer n W. Then e * e * where e * W B W s the equlbrum effort choce of the -th rnked plyer n deprtment. Proof: Suppose otherwse. Then W x W x B s.t.. (Here, by x() we denote the totl reserch output n deprtment n equlbrum.) For ths condton to hold, t lest one resercher n W must exert more effort thn the dentclly rnked resercher n B W B. Tht s, nd such tht the equlbrum level of effort by plyer s more thn tht of plyer, whch we wrte smply s e > e. Recll tht resercher s frst order condton for the effort subgme n the R-regme (Equton ): x c 1 ' x e,, q e, 11

14 1 ' 1 ' x B Now, x W x by concvty of. Also, by Condton A, x,, W B e q,, q B W snce e > e, nd q q. c However,,, c due to our ssumptons bout c,. Then the frst order condton cnnot hold for both nd. Therefore totl output must be hgher n the better deprtment. Ths rgument, clerly, lso works for the O- regme (there, smply, the condton x W 1 ' 1 ' x B s left out). Now, ssume tht s lner, tht s, ts dervtve s constnt. Then, replctng the rgument bove we fnd tht e * e * where e * B W s the equlbrum effort choce of the -th rnked plyer n deprtment s requred n the lemm. 9 Lemm mples tht the equlbrum totl reserch output n better deprtment (s defned bove) s more thn n ny worse deprtment. Next we show tht under Condtons A nd the ssumpton of lnerty of n the R- regme, the unque SNE outcome would be such tht strct herrchy of deprtments would form. The k hghest blty persons would be n one deprtment, the set of next k hghest persons n nother deprtment, etc. Formlly: Proposton 1 Suppose Condton A holds nd let be lner. Then, n the R-regme, the unque SNE outcome tht, k,..., 1, m n k m n s s follows. For every, k N & m, n 1,...,. such 9 The lnerty of s smple suffcent condton, but not necessry for the second prt of Lemm to be vld. 1

15 Proof: Let 1 be the deprtment consstng of the k best reserchers. Let be ny other deprtment wth, '. The deprtment 1 1 s better deprtment thn s specfed n efnton 1. Rnk the reserchers n 1 nd ccordng to blty. Lemm 1 ' shows tht e e for ech person of rnk n ether deprtment. By Lemm we lso get tht x x. 1 ' R ' R 1 Suppose u * ' u * 1, tht s, let plyer s py-off gven her equlbrum choce of effort n deprtment be more thn her py-off gven her equlbrum choce of effort n deprtment 1. R ' R 1 Then u * 1 u * 1. 1 But then e * cnnot be Nsh equlbrum choce for plyer n the effort choce subgme n deprtment 1. Remrk 1: Proposton 1 gves the unque equlbrum for the R-regme. Specfclly ths equlbrum hs the k best reserchers n the frst deprtment, the remnng k best reserchers n the second deprtment nd so on 10. Next we wll demonstrte the possble exstence of other equlbr n the O-regme. Proposton There exsts t lest one SNE for the deprtment formton gme n the O- regme. Proof: Our proof s constructve. Frst we sy tht deprtment s perfectly stble (gven regme) f for every n O O ', u * u * ' for every other deprtment. Gven the SNE soluton concept no plyer would devte out of perfectly stble deprtment. Now we descrbe the constructon. 10 In pper on ths theme L Mnn (008) explns tht ths type of herrchy wll result becuse the better deprtments get more fundng from the RAE whch leds them to hre better reserchers. He then uses relblty theory to determne when such herrchy s desrble. 13

16 Step 1: Form perfectly stble deprtment 1. Then, from N \ 1, form nother perfectly stble deprtment. Contnue ths process s long s possble. Ths process would termnte owng to the fnteness of N. Let ths collecton be ( 1,..., ). (Note tht the set of such deprtments my be empty.) Step : Form 1 by tkng k ``best plyers (n terms of ndvdul ntrnsc blty) from the remnng N {... } nd so on untl no more fesble deprtments cn \ 1 be formed. Note tht plyer from deprtment 1 my gn by formng new deprtment wth plyers from ( 1,..., ). However, by Lemm, none of the plyers n (,..., 1 ) would gn strctly by formng such deprtment. Furthermore, plyer from deprtment 1 cnnot gn by formng new deprtment wth plyers from,..., 1 n k Therefore tkng ( 1,..., ) s gven, no plyer from deprtment 1 wll devte. Smlr logc demonstrtes tht no plyer wll leve the remnng deprtments formed n Step. The resultng set of deprtments s n SNE outcome.. Remrk : Note tht the strct herrchy found s the unque SNE n the R-regme s lso n SNE n the O-regme. Proposton 3 below gves our desred result. It shows tht t s possble to hve strctly hgher totl reserch output n n equlbrum outcome n the O-regme compred to the unque equlbrum outcome n the R-regme. Proposton 3 Gven the bove ssumptons t s possble to hve strctly hgher totl reserch output n n equlbrum outcome n the O-regme compred to the unque equlbrum outcome n the R-regme. Proof: We demonstrte ths wth n exmple. 14

17 Suppose n = 4 nd k =. 0.7, q 0. m , q 0.3 m 0.4, q 0.6 m , q 0.7 m 4 4, c e e e The followng results hold for ny sensble output functon. Consder two structures of deprtments: Structure A: 1 = (1, ), = (3, 4). Structure B: 1 = (, 3), = (1, 4). Note tht by Proposton 1, Structure A s the unque equlbrum for the R-regme nd by Proposton, s lso n equlbrum for the O-regme. Structure B s n equlbrum for the O-regme; both deprtments re perfectly stble snce ech plyer n ech deprtment receves her mxmum externlty. Consder the two structures under the O-regme. Plyer 1 nd plyer ech receve her mxmum externlty. Therefore ech of them wll produce the sme mount n ether structure. Recll tht by Equton 1, plyer 3 s frst order condton for equlbrum choce of effort for structure A, A e 3, requres tht: x c A A 3 *, 3,0. 3 *, Smlrly, plyer 3 s frst order condton for equlbrum choce of effort for structure B, B e 3, requres tht: x B c B ( e3, 3,0.6) ( e3, 3)

18 Also note tht: B A 3 3 x x B A 3 *, 3,0.6 3 *, 3, e * e * by our ssumpton on the functon c,. B A Smlrly, e4 * e4 *. Therefore the totl output under the O-regme t Structure B s greter thn tht for Structure A. Then, f ' s smll enough, then by the contnuty of the py-off functons of the plyers, the totl output under the O-regme t Structure B s greter thn tht for the unque equlbrum under the R-regme. Remrk 3: Of course, ths result holds for mny other exmples; we ust provde one frly smple cse. 5 Concluson The bove results re not ment s condemnton of ny specfc ncentve scheme such s the RAE or the REF. Rther, we merely demonstrte tht t s possble tht such ncentve schemes my lower totl reserch output gven ndvdully optmzng reserchers. Snce our results re dependent on the vlues of the prmeters, ny specfc ncentve scheme for reserch my or my not lower the overll level of reserch n n economy. One re of future reserch s how probble t s tht such perverse scenro exsts. 16

19 References: 1. Bernhem., B. Peleg nd M. Whnston (1987): Colton-Proof Nsh Equlbr I: Concepts, Journl of Economc Theory, 4, Bulow, J. nd J. Levn (006): Mtchng nd Prce Competton, Amercn Economc Revew, 96, Epple,. nd R. E. Romno (1998): Competton Between Prvte nd Publc Schools, Vouchers, nd Peer-Group Effects, Amercn Economc Revew, 6, Hre, P. (003): The UK's Reserch Assessment Exercse: Impct on Insttutons, eprtments, Indvduls, Hgher Educton Mngement nd Polcy, 15, Hrt, S. nd M. Kurz (1983): Endogenous Formton of Coltons, Econometrc, 51, Kelso, A. nd V. Crwford (198): Job Mtchng, Colton Formton, nd Gross Substtutes. Econometrc, 50, Km, E. H., A. Morse nd L. Zngles (006): Are Elte Unverstes Losng Ther Compettve Edge?, NBER Workng Pper L Mnn, M. (008): Assessng the Assessment. Or the RAE nd the Optml Orgnzton of Unversty Reserch. Scottsh Journl of Poltcl Economy, 55, Rosen, J. B. (1965): Exstence nd Unqueness of Equlbrum Ponts for Concve N-Person Gmes, Econometrc, 33,

20 10. Roth, A. nd M. Sotomyor (1990): Two-sded mtchng: study n gmetheoretc modelng nd nlyss. Econometrc Socety Monogrphs, vol. 18. Cmbrdge Unversty Press, Cmbrdge. 11. Wnston, G. C. nd. J. Zmmermn (003): Peer Effects n Hgher Educton, n College ecsons: How Students Actully Mke Them nd How they Could, C. Hoxby, ed., Unversty of Chcgo Press. 18