Alvin E. Roth** iVIarilda Sotomayor

24 Pages·2010·0.46 MB·English

STABLE OUTCOMES IN DISCRETE AND CON­
TINUOUS MODELS OF TWO-SIDED MATCHING:
A UNIFIED TREATMENT*
AlvEi.nR oth**
iVIaSroitlodmaa yor***
Resumo
Aprescl1l1t1a1I1l ltolrSla ltlaidlfelHi lIcclcanWldt aoods esl el ioddeem lam,;t ching
ded ailsa dqonshe l cmloldleidl iossc r(ettaCoiO:;sU llOl loddoe icaa sadmeGe anlteo
0
eS haplee ym)o decloanst i(ntlaCliOosI}; l mloOd edleda e sigdneaS Ghaaop el ey
0::; 0
Slllel mboidke dleJa e siggncauGearoa dleDi CL;laldHoeGl allleEg)uCc.! o ntrcaosmt c
0
ast raltlacluI:tUolst earsic oOrlelscp,ia nrsaolpcea!lriaaa sds o iC:-;O lljdleml oldletloasH
sadoe rivaaqdldalosmi e slmloldcoo paa rdtaiisrU CShlilplo(;1t!Ie)vs Ieoss.tg rlatlel lOS
asr esuletmaq dlolsc ssetganopc rlaltli,c C1dmalc 1uitped6,etq elslnee( lccloeiou cide
0
COllllili cdleefoi puoiJrdo om iufar<;:aaNoco al .l lOddoce alsoa lIhlicpuotdtoee fic
0 a
qlalsep referseaneocs itar[sia ct�oa mqs H ees tdeasiC sO lljClolilllltecolisld qallllaln to
nomso decloonst luJlolcioOsSI ljcnolilutcpoiosdr eqmna eg eltletpmer se fel'encias
01) 01)
cOllteio uspl rleavpsoo sd eSmCa Ij'l lsctOaldlotsi ullalllellte.
*
Partiaslulpyp ortLeyd. 1.CSl.I ggeuhe:'.iJleUl lloFroiuanid atiCo?\nP,q -COllsel',i'haoc iollal
deD esenvolvimento Cei eTuetclnfoilc6og iBcroa,z ialwj, by grantfsr omt heA lfrePd. S loan
Fouuuatiaonnd t he?\ atiolSlcaile nFcOel illualt."iSoAl.l ,
** Departllo1feE ucto nomicUsn,i versoiftP yi ttsburPgiht,t .suuPrA g1h5,2 00U,S A
***
Il1stidteuE tcoo llomIinad ustrUilalli,v ersiFueaddeerd aolR iod eJ aneirAov,.P astel2l5l0' ,
-UrcaR,i od e. JaneiRr.Jo,B, r asil.
Q
R.d eE conometridaeJ anReiiovr .1o 6 n, 2, 1-24N ovem1b9r9o6
pp.
Two-smiadtecdh ing
Abstract
Vvep reseal lttm ifiterde atmeolfla t c lasosft wo-sidleldl atclmioidlelltgsh ati u­
clllcleh' umLo<;ucerl(e:.;st uec ash them arrialglel odoefG la lea ndS hapleayn)d c Olltillll­
OilmSo del(ss llacsht hem isigullllueoluleoLlfS hapleayu dS hllbaimkI t heg enerali�eu
assignmelnltl odoefDl e mangea udG ale)I.I cIo ntrawsitt hp reviotwr,e atllltehuet s,
parallel cOllfcolrtn hsei otlwlos o sfle ltlso daerleds e rivheedr iel Llh es amew ay from
thes amea ssllluptVvieo lslhSo.wt hatt he resilllqlll ttse stailofllo ll low clforsoelllyl
thea ssllluptthiaottl hle c orec oillciwdietsht hec ored efinbeyd weakd ominatioll.
Int he marrimaogdee lt,h ea sslllllopfts itorliplcr t;e ferenccaenss etsh estew os ettso
coinciwdhei,l ien t hec OlltiulmloOdlelltSsh et wos etcso iucibdeec allasgee nthsa ve
cOlltillplrteoflelrSe nacued:,; pricesb ea cda.ni nsctoeudL ill1lOnsly.
Pala1l'l"G.s-Chiav:lvaet:c isItianbgll,le l atchcionrgel,,a ttiocpet,i lllllaliatchillg
C6digJoE L: C78
1.I ntroduction.
Oneo ft hpee rsipsutzezinlnte stt hheeoo rfty w o-smiadtecdh ing
itsh ger esaitm iloaftr hirete ys uclotnsci enltg'1h c1eo roeft hgea me
thahta vbee eonb taifnoterhd de i scmroedtee(l es.t gh.me a rriage
modeolfG alaen Sdh ap(l1e9y6 a2n)fd)o c ro ntinmuoodue(sle s. g.
thaes signmmoedneotlfS hapalneSdyh ub(i1k9 7a2nt)dh gee ner­
aliazsesdi gnmmoednoetflD emanagneGd a le (1985)), even though
thtew koi nodfms o dealpsp etaohr a ve importanFtu rd­ifferences.
thermtohrpeer ,o boyfw sh ich rtehseuhslaetvs seof abre eonb tained
remaqiuni dtieff eraennsdte ,et mor eloynd ifferpernitn ciApsl es.
BalliannsGdka il( e1 9p9.02 ,7 4n)o te:
"Theirseb, y n owa, s ubstalnittiearloa ntt uhrees e
problaenmdos n,ie ss trubcytk h fea ctth aatl moaslrtle ­
sulptrso vfeotdrh oer dicnaashlea vaen aloigntu hecesa r­
dincaalsa el,t hotuhtgeeh c hnoifqp ureoisont f h tew coa ses
arieng eneqruaidlti eff eraenntdth ,ei rases y ento " unified
theowrhyi"cc ohv ebrost h."
2 RevidseEt cao nomet1r6i( a2N )o vem1b9r9o6
AlvEi.nR ot&h M ariSlodtao mayor
Them ospte rpleaxsipneogcft t h peu zzilste h atth sei milar
conclufsotirho tenw sco l asosfge asm reesq uire atshsuaamtrp et ions
nomte rely dbiuffater rieent n htfe,o llowing siemnvpseoerr yt ant
nearolpyp osTiodt eer.im voesot ft hree suwlewt isbl elc oncerned
withhe rient, h mea rrimaogdewe eln eetdoa ssutmheap tl ayhearvse
strpircetf erie.netc.he asnt,op layiesir n diffebreetnwteb eeni ng
matchteoad n pya iorfp ossmiabtleeIs nt. h ceo ntinmuooduesl s,
howevtehares, s umpittsih oaentv eprlya yebrem acdaienn different
betweepna iorfp ossmiabtlebesy a ppropardijautset ments of
any
prices.
Perhtahpmeso sdtr amactoincc lcuosmimootnno t htew moo d­
elisst haitnt, h mea rrimaogdeew lh enno a genatrsie n different
betwaeletne rnmaatticvhaeen sid,n t hceo ntinmuoodue(sli psna r­
ticutlhaaers signmmoednetl ) alwalhg eennm tasyb em adien dif­
ferbeenttw aenetynw moa tchtehsee,rx ei sftoesra, c shi doeft he
markaep to,i ntth ceio nrw eh iicsoh p tifmoaarll a lg enotnts h at
sidoeft he mairntk hesete ,n tshean toa geonntt hasti doeft he
l
markdeoteb se tattea rn oyt hceorro eu tco.Im tie as lrekandoyw n
thatth emraeyn oetx iosntoe f t hoep tipmoailni tnts h ceo re if
indiffeirsae lnlcoeiwsnte hdme a rrimaagrek (esteR,eo tahn d So­
tomay(o1r9 90A)f)tp.er re setnhtfeio nrgmm aold ewlew, i slhlo w
bye xamptlheat th iasl shoa ppeinfis n diffeirspe rneccel iunad ed
modeilnw himcohn eiysc ontinduiovuisslTiyhb iilsset .y piocfa l
howt hvee rsyi micloanrc luasbiootunhtste w moo dedlois n f act
depend odni svsieamrsiysl uamrp tions.
Thipsa pperre saeu nntisfi terde atomfae c nlta osfts w o-sided
matchmiondget lhsai tn cltuhdmeea sr riaangde continuous models,
sucthh atthp ea raclolnecll ufsotirho tenw sso e otfsm odealrsde e -
1
Fore xamplien:a markewti tehq naulllllloufie uresl lticaia nsuie Jlclnetrixuc nayie n;:o fs ome
homogeneous inJg.oiovdio,.u sticuoimeei slt lh ec oraer ec ltaracteLryia z seidn gplrei cbee,t ween
ther eservatpiroinco eft heb uyerasn ut hes elieartws b,i cahl tlr adaerse J IIauTeh.eu uyeorp timal
COroeu tcomeisst heo nea tw hicht hep riciesa sb igha sp ossibil.ee,a. t t heN eller reservation
pricaen,d t hes elloeprt imaclor oel ltcoimste h eo nea tw hicthh ep riciesa sl owa sp ossible.
RevidsetE ac onomet1r6(i 2aN) o vem1b9r9o6 3
Two-sdiedm atching
rivientd h sea mwea yf rotmh sea maes sumpTthieao pnp.r opriate
assumptioount b tetus oru nbbstu lstei mptlhere:e suilnqt use stion
alflo lflroowtm h aes sumptthiatothnc e o rceo incwiidttehhsc e o re
definbeywd e adko minaItnti homena .r rimaogdeet lha,es sumption
ofs trpircetf erceanucsteehtssew soe ttosc oincwihdiielnt,e h ceo n­
tinumooudset lhste w soe ctosi nbceicdaeau gseenh tasvc eo ntinuous
preferaennpdcr eiscc aebnse a djusctoendtu isnluyo2.
Ino ufro rmtarle atmweedn etv,e al colpa osfms o dewlhsi ch
generbaoltithzh meea rriaangcdeo ntinmuoodueIsln st .hm iosd el,
the cormemsoufnlo ttrhs e tsweco l asosfme osd ewlisfl oll flroow m
thaes sumptthiaoutnt si lairtiein ecsr eianms oinnegty h,at th see t
ofi ndivirdautailtolrnyaa nls ifsbe orusn daenddt, h atth ceo ries
non-emapntdcy l osteodg,e twhiettrhh aes sumptthiatothn ce o re
coincwiidttehhs ce o re dbeyfiw neeaddko mina(tAiloonlft. h ese
assumpetxicoefnpostt rh lea asrtse a tibsyfit ehdme a rrimaogdee l
evewnh epnr eferaernneoc stet sr ict.)
Two-sided mmoadteaclrhsnei o nmtge reolfty h eoretical inter­
estth;e ayra el spoo werdfeuslc rmiopdteiDlvises. c mroedteehl asv e
provteobd e approprwiiatwthhe i tcthooa onlaslm yasrek ient s
which cwaabnge et sr eaatspe adro tft hjeo dbe scrisputcaihso n,
thmea rkfeotmrse diicnatlei rntn hsUe . S-R.o t(h1 9a8,14 9 8a6n)d
thUe. K-R.o t(h1 919909,o1 rn) o,n -wmaagtec hpirnogc essesee s,
MongaenldRl o t(h1 9a9n1dS) o toma(y1o9cr9) I6.ts eelmisk ely
2 (1989)
Anan alogouunsi ficawast ioanc ocmplishienR do tahn dS otorrnyor concertnhien g
rrnrriagrmde el (whiicnhV OlovnESe -torrnt-cohnien ga)n dt hec olladmiesgsei onrsmde l (which
involmmvye;- Io-onerrnt ching)T.h at psapehro wtehda tth ceo roeft hceo lladmisesige onsgarre
shartESh per opertiESi ntf horrnreur nida gtroed ele vewnh ecno llemagyES b ei ndiffbeer""te"nnt
diffeernetnetr cliasnsesg ,so lonasg thehya vset ripcrte fereonvceiesrn divisdtuuadle nTthse.
reasoins t hawth enc ollehgaESv set ripcrte fereonvcesESrt udentthse,wy i lnlo tb ei ndifferent
bewetena nyt wo entercilnagss tESh ecany beas siguedi nt hec oree,v etnh ougthh emayy be
indiffebreetnwteeso nrre paiorfse ntercilnagssES .A ndt hreES ulftostr h rrnre riage trodelf ollow
frotmh ew eakaessru mptiotnh anto a genitsi ndiffebreewetnetn a nyt worrnte; whocan be
assiguetod himi nt hceo re.
4 RevisdteaE coonmetria 16( 2)N ovembro19 96
AlvEi.nR ot&h M ariSlodtao mayor
thacto ntinmuooduemsla syb em orues effouarln alymsairnkge ts
inw hicwha gaersde e termeinndeodg enTohuecs llayos.fms o dels
introdhuecraeeld l owwasgte osb et reaetietdwh aeyri, . eei.t ahse r
parotft hjeo dbe scrioprat sia onen n,d ogendoeutselrymp ianretd
oft hmea rkoeutt come.
2.T he MathematicaMlo del.
Theraert ew fion idties,j soeiotnfspt l ayPe r{=sI ..,.i ., .m}.
andQ = {l,,.,.· n}j·.,T hep layienPr asn dQ wiblelc alPl­ed
playaenrQds- plaryeesrpse cTthiesv eeotlfa y l.pl e rmistsria­bnlse
febrest wteewpnol ayiesars se otfn umbewrhsi wcihbl eld enoted
byS .F omra themactoincvaelnw ieew niacllell aop wl ayteomr a ke
at ranosffs e0r=t oh imsFeulrft.h eirifmt orraen sStf oej tr hse n
wes atyh ajtt rans-fteoir .s
s
Thaitsw ,ew iblela ssumtihnagt
1r0 E S ;
12-E S -8E S .
s =
Thes trucotfpu rreef eriesgn icvebesynt huet ilfiutnyc tions:
Ui:j S anVd.i . ; :T hSuU si .iAtss)h u et itloiii t ny of
-+ R -+ R. P
beimnagt chweidtj ha nrde ceiavm ionnge tpaaryym eonfstf rom
j,i sf 2: 0,or t ransf8et rojr i,isf n :s; g O. A nalogVoiujsi(ls8ty )h e
utiloifjti ynQ fobre imnagt chweidti h.
Ther eservuattiilofinott rih epesl ayiea rnsdj are gbiyv en
Ti= Uii(aOn)Sd j = Vi.ir(eOs)p,e c.Tt hiemv aerlkygeatm iest h en
givbeyn( PQ,,S ,u ,V ). A transffreorim t oj isi ndividually
s
ratiioUfni ail( 2:- T8i)a ndVij( s2:) SiW'e w iblela ssumtihnagt
foarl ili nP anadl jli nQ :
A)-Uia.nidV i ajr set riicntclrye iants hivena gr iSa;bl e
A2-thsee otfa liln divirdautailtolrnyaa nls tfheacrtas bn e m ade
betweia ennjd i sb ound(eImdta. y b ee mp)t.y
RevidsetE ac onomet1r6(i 2aN) o vem1b9r9o6 5
Two-smiadtecdh ing
Lacktihnigs assiutwm oputlbideop no,s sfiobarlp el ayteor
regasrodmm ea tcahs" infingiotoe,dil "nyt hsee ntsheaa tp layer
woubldep repatrope aday n fyi naimtoeu tnota chiietvS.ep ecifi­
calalnyiy,n divirdautailmolanyta cblhe tw·iea enn.di c ourledm ain
possniobm laet theorwl artghere e servuattiilooif.int . y
Definitio1n. A matchiisanb gi j{'e cftrioPom nU Q ontPo U Q
ofo rdtewr(o t hias/Lt,2 (=x x))s ,u ch it"fhC airf:t) Q t heIn" (=i i)
anidIf (.i" ) rf: P theI(.in" ) = . i.
Thaits /L,( =i. )im eantsh aattt hmea tchWiiin gsm atchteojd ,
wherIe"a(=si i m) e aniis ss ingle.
Definitio2n. Ano 'Utciosamm ea tchIi"an ngda p aiorfv ectors
m
('Uv,)c altlhepeda yowff'i,t hE IR anvdE IR nA.n o utcowmiel l
'U
bed enobtye( d'Uv ,;I ")'
Definitio3n. Theo utco('mUv,e ;I "i)sf ae siibtflh ee irsaes eotf
numbesriEsjS ,d efionnelidyIf i" )( =s ujc h that
i)'U i= Uii(-.S,)a nVd i= V .ii(SiiIf." 1C=)i j,)
j
and
ii')iU= 7'ii/Lf (=ii)
iii�) r i,v i� S:i( indivriadtuiaoln ality).
·Ui.
Thep ayo(ff' Ui,sfva e)s iib(fl' eU I,"vi);sa f easoiubtlceof moer
some maIt"cI.hnt i hncigas s wees atyh aIti" sc ompawtiittbhhl ee
feaspiabyl(oe'Uffv , ) .
Definitio4n. Thef easoiubtlceCo umv,e;I "i)ss t aibtflh eei rnseo
pai(ri ,EjP ) x Q wit/Lh( irf:).ia nSd E sSu cthh aUti -,Sj)>( 'U i
anVdi :i>(vS i')
6 RevidseEt cao nomet1r6i( a2N )o vem1b9r9o6
AlvEi.nR ot&h M ariSlodtao mayor
Thiisse quivtaolr eenqtu itrhianntogf easoiubtlceoC muve''; , ' )1"
exissutct hh aI"t') ( ija nd
=
,
Viv i'
<
Similwaedr elfiyan s et able payoff.
Thedne fini4ts iaoytnsh a(tv) i ss taibiflit esn odto minated
u,
bya nyf easpiabyloveffi s ao mpea i(rij ,).( Suac pha i(rij ,)i s
callaeb dl ockinWge t paaktiehr re)u .lo efts h gea mteo b et hat
anpya i(rij ,)i n xQ isf rteoem atcthoo naen otahtae nrpy r ice
P
si ft hebyo tahg reaen,da nyi ndiviisd uaftlorr eeem asiinn gle.
Thetnh pea i(rijs,)a rteh oen leys senctoiaallia tnidaop nasy,o ff
iss tabialfne od n liyif ti isn t hceo rweh,i wcihbl eld enobtyeC d.
(Thciosi ncibdeetnwcteehes ene otfs tapbalyeo affnstd h ceo rdeo es
nostu rvtihvgeee neralizationm aftrcohtmio mn oagnn ey--ttoo-­one
manmya tchsienReg. :o atnhSd o tomayor (199(01)9,9 2S otomayor
an1d9 9a6, b)).
Definitio5n. Thes taboluet co(um,V e;/ 1i)ss triscttalibyfl e
theirsne o p ai(rij ,) xQ wit/1h( i§t) a jn ds sucthh at
E P E S
Uij(-'Usia) n Vd ii>(V sio)r U i.i>(U -ias n)Vd ij(Vsi)'
= =
Similwa.edr elfiyan s et riscttaplbayly eo ff.
Defin5is taiytoshn a (tv )i ss triscttalibyifl t ie ss tabalnde
n,
itis n owte adkolmyi nabtyae ndfy e aspiabylvoeffi s ao mceo alition
(ij.,)T hep ai(rij ,)i sc alalw eeda bkl ocpkaiinrg.
Thes eotfs triscttalpbyal yeo cffosi ncwiidttehhsc e o rdee fined
byw eadko minawthiioicnsah,s ubsoefCt a ndw ibleld enobtye d
C.
Thef ollomwoidneaglr swe e lkln owsnp ecciaasoleft s h meo del
wea rter eahteirneg:
RevidsetE ac onomet1r6(i 2aN) o vem1b9r9o6 7
Two-smiadtecdh ing
The Marrigae Model
HerSe = {O}S.i nSc ei sf initthepe,l aycearnls i tshte ir
acceptpaabrlteni enor rsd,ei rna, f initoefp rleifsetr eLnecte s.
L(id)e notthele i osftp referoefpn lcaeyseT rh·uiLs.( i)( j,0),
=
(Ic0,)" , (·0- ,)i .".,m eantsh aUti ,>;(U Oi)/ ,� ,,(e Ot)cT .h ef act
";
thadto cnso ptl aayn ryo ilnet hgiasm iem pltiheaastn oyu tcome
s
('(u);,/ )v1i cs omplestpeelcyib fyt ihemeda tchfitSn.ug c ahm atch­
inigsc alslteadib tflh ece o rrespoountdcionmsget aibTslh eai.ts ,
fitas s table miatfth ceihrnsieo pn agi Or, i.nPi lQx w itfOht) 01 j
sucthh aUti.i (>O)U ;,'a(niVd)i ('O>;)V ( ,O,)( ;).;(O).
Thu/si",sa s tambaltec hiitfnh ge irsne o p ai(rij ,)i nP Qx,
witchn j01,s ucthh aiat n jdp reefaecroh t hteotr h eriers pective
/"
matuensd /e1r .
Whetnh per eferaernscct ertsih crete sulmtoidneiglst hwee ll
knowmna rrimaogdeei ln,t rodbuyGc aeladen Sdh ap(l1e9y6 2).
The Assginment Game
Int hicsa se andt hearrene u mbeariasjn db if:oiar,l l
S = R
paiCrijs,)i nP Qx,s ucthh at
Ui:ia(i8+i) 1/"(b.8 +)
= s 1.1, = 1.1., s.
Ifw ed efinCtei .im ax{aib:iOij} ,t, h eanf easoiubtlceo me
= +
(u(,);v/ )1i sst aibfl ea nidtf h oenrlneyop aii(sr i i,nPj )Qx ,w ith
ft01( ji,)s ucthh aUti+ viCt ij·
<
Thimso dieeslq uivtaotl heaens ts igngmaemonefSt h apalnedy
Shub(i1k9 7a2nid)ti sa speccaisoaeftl h meo deblel ow.
The GeneralizAesds ginmentG ame
Int hmiosd el antdh uet ifluintcyt airsoetn rsi ctly increas­
S = R
inagn don to( ancdo nsequceonnttlinyIu toiu sts ha)ed. d itional
R
assumptthiatothnu e t ility afroucnn tcott ihoahntas ts h ceo nse­
R
quence thpalta yceeavrbne e mr ayd ien diffbeertewnaetne pyna ir
8 RevidseEt cao nomet1r6i(a 2N )o vem1b9r9o6
AlvEi.nR ot&h M ariSlodtao mayor
ofm atebysa na ppropardijautseto mfpe rnitcT ehsef. a ctth afto r
eveprayi( rji ),t h eirsse o mter anstfheamrta keiis n diffteor ent
8
beiunngm atc(ahnesddo moet hterra ns' tfheamrta ke.jis n differ­
8
en)ti mpltiheatsth see otfi ndivirdautailtolrnyaa nls tfheacrtas n
bem adbee twieae nnjd i sb ounded.
] ]
Ifw ed efifni.ei= Ujiangdi =j V ii,t heanf easoiubtlceo me
(( v)J.l;)i sst aibafln eod n liytf h eirnse op ai(rji ),in P Qx,w ith
u,
J.li(f) j ,s ucthh afti( jU i9)i (iv i0).
+ <
Thimso deilse quivtaolte hnmeto deolfD emanagnedG ale
(198F5o)Dr.e managneGd a ltehc eo ntionftu hiuett yi fluintcyt ions
isar equiroeftm hemeno td el.
Wew icloln cenotnmr aartkefe otersi thieasrn sye otfi nteger
S
numbesrast is(fhyo ir1n 2go) r Wew irlelf teoar m arket
S = R.
whicsha tiassfiseusm pAt]ia onndAs2 a sd iscorrec toen tinuous,
accortdowi hnegt hiesars eotfi ntenguemrb eorrs
S S = R.
There suolftt hnsee xste ctmioosntd leyp eonntd h aes sumption
that C.T hicso ndiitssi aotni sifife idsd iscarnedta el l
C = S
preferaernsect ersiI ctit sa. l ssoa tiisnfit ehdce o ntincuaosouefs
DemanagneGd a lfeo,tr h etmh uet ilUiita.inieVd i.s ia rceo ntinuous.
Withotuhtoe p posaistseu mptthiaoptnr so dutchceeo mmon
resuflodtris s carnecdto en tinmuooduetslh so rsees umlatyfs a tiol
holadst, h eex istoefanQ c-eo ptsitmaaoblul tec owmhei,ica sht l east
weakplrye febryar leQld- platyoae nroyst hsetra obultec oSmuec.h
ano utcoemxeii sntt hse mwoidtfiehnl i ntuem boefpr a yments when
preferaernsect reaisnc dit n,t hmeo dewli tcho ntinduiovuissliyb le
paymewnhtestn h uet ilfiutnyc tairooenn st os ot haatn ayg ent
R
cabne m adien diffbeertewnaetne tynw moa tches.
Ift hper eferaernneoc stet sr iintc htme a rrimaogdeet lh,ee x is­
tenocfae m ano ptimmaatlc h(oirna wg o maonp timmaatlc h)ii sn g
nogtu aranatsie ctea dbn,e s eeinnR otahn Sdo toma(y1o9rE9 x0:,
2.1p5.3, 4 )O.n t hoet hhearn idtf,h erneo s iasla trhya mta kes
8
aw orkienrd iffbeertewnbetee inen mgp loaynebdde iunnge mployed
RevidsetE ac onomet1r6(i 2aN) o vem1b9r9o6 9
Two-smiadtecdh ing
themraey b en os taboluet cotmheai tso ptifmoatrlh weo rkers.
Thee xambpelleio lwl ustthrreao tplelesa ybeyid n diffeirnte hnec e
discarnecdto en tinmuooduesl s.
Tos eteh reo pllea ybeyid n diffewrheennc e constihdee r
S = R,
follotwwieonx ga mpilnwe hsi tchhe mraeyb en ow agweh imcahk es
aw orkperre ciselyb eitnwdteiwefdonfi effreeernmetpn tl oyompe­nt
portunIintt hiee sefi.xr asmtpt lhueet ilairtneio ectso ntiniuno us;
thsee coenxda mtphlueet ilairtceio enst inuous.
Counterexampl1e. Consitdhceear s oefo nfier ma ntdw woo rk­
eris.,Pe . U}a ndQ {w]',tV 2L}e.t anlde tth uet ility
= = S = R
funcotfit ohfiner mb e
U(lO(1+S S)
=
Urr(O)0
=
anlde tth uet ilityf oferua ncwchot rikoe(nr )b e
w 'tv = WI'tV,2
foSr� 0
S for 0,a nd
s <
VlO'W(OO.)
=
Theirsne os alatrhya mta keasw orkienrd iffbeertewnbetee inn g
s
emploaynedbd e iunnge mploAynedtd h.e irsne o s taboluet come
thaitso ptifmoatrlh weo rkeTrhse.ar ree ex acttwlsoyt aobulet ­
C0111eS,
",cn= w ];11.r = 1,V I= 1,V 2= 0, and
/.,= '' tV(2uf;f'= ) 1 ,v� = 0,v S= 1,
an'tvd] pre",f etro",s' w hi'tVl2eh atsh oep popsrietfee rence.
Int hiesx amptlhceeo riesn oatl attnincdete hrpe r eferences
oft hweo rkbeercsa (use' ')i sn ostt abCloen.t rtaor y
11 /\ ,11 .V vV
10 RevidseEt ac onomet1r6i( a2N )o vem1b9r9o6