Liquidity Constrained Markets versus Debt Constrained Markets *

Transcription

1 Lqudy Conraned Marke veru De Conraned Marke * Tmohy J Kehoe Deparmen of Economc, Unvery of Mnneoa and Reearch Deparmen, Federal Reerve Bank of Mnneapol Davd K Levne Deparmen of Economc, UCLA Fr Veron: May 2,997 Th Veron: Ocoer 5, 2 Th documen copyrhed y he auhor You may freely reproduce and drue elecroncally or n prn, provded drued n enrey, ncludn h copyrh noce Arac: Th paper compare wo dfferen model n a common envronmen The fr model ha lqudy conran n ha conumer ave a nle ae ha hey canno ell hor The econd model ha de conran n ha conumer canno orrow o much ha hey would wan o defaul, u oherwe a andard complee marke model Boh model hare he feaure ha ndvdual are unale o compleely nure aan doyncrac hock and ha nere rae are lower han ujecve dcoun rae In a ochac envronmen, he wo model have que dfferen dynamc propere, wh he de conraned model exhn mple ochac eady ae, whle he lqudy conraned model ha reaer perence of hock * The auhor are raeful for fnancal uppor from NSF ran SES and SBR and he UCLA Academc Senae We would alo lke o hank Tom Saren and Bll Zame, a well a workhop parcpan a he IV Coloquo Naconal de Economía Maemáca y Economería n Guadalajara, Mexco, he IXX Smpoo de Anál Económco n Barcelona, Span, he Wner Economerc Socey Meen n Anahem, he Lan Amercan Economerc Socey Meen n Mexco Cy, he Unverdad de Alcane, he Unvera Pompeu Fara, he Unvery of Wconn a Mlwaukee, he Unvery of Houon, Camrde Unvery, and Caholc Unvery of Louvan Jahyun Nahm poned ou an error n an earler veron of Propoon 6 The vew expreed heren are hoe of he auhor and no necearly hoe of he Federal Reerve Bank of Mnneapol or he Federal Reerve Syem

2 Inroducon There conderale emprcal evdence ha oh ndvdual conumer and larer ene uch a counre ear more doyncrac rk han conen wh complee and frconle Arrow-Dereu marke Evdence a he level of he ndvdual conumer dcued, for example, n Hayah [985] and Zelde [989], who how ha ndvdual conumpon poorly correlaed wh areae conumpon Evdence a he nernaonal level dcued, for example, n Backu, Kehoe, and Kydland [992], who pon ou he low correlaon eween conumpon level acro counre Tha ndvdual ear doyncrac rk can e capured y many deparure from he Arrow-Dereu framework Three mporan example of uch model are ncomplee marke model, where here are no enouh ecure o nure aan all even; model of lqudy conran n whch ndvdual conumer are aumed unale o orrow a much a hey would lke n loan marke; and model of advere elecon and moral hazard Incomplee marke model are dcued y Radner [972], Har [975], and Duffe and Shafer [985], for example Example of model of lqudy conran can e found n Bewley [98], Duma [98], Townend [98], Schenkman and We [986], Ael [99], Kehoe, Levne, and Woodford [992] and Heaon and Luca [997] Model of lqudy conran ypcally nvolve ncomplee marke, a no only are here hor ale conran on ecure, u ecure are lmed n numer a well Thee paper have larely focued on he compuaon of pecal ype of equlra n econome where he oude ae a fa money of no nrnc value In hee equlra hock have lon erm conequence We how ha h alo he cae n he ncomplee marke model condered n h paper Model of advere elecon and moral hazard, wh he noale excepon of Preco and Townend [984], are no ordnarly eneral equlrum model, o fall oude he cope of h paper, u he nereed reader hould conul Green [987] who how ome of he lnk eween ae marke model and model of advere elecon Model wh ncomplee marke and/or lqudy conran ypcally have he propere ha n equlrum ndvdual ear doyncrac rk, and nere rae are lower han ujecve dcoun rae There alo a fourh model ha hare hee propere: a model wh ndvdually raonal de conran Here he eup dffer

3 2 from ha of Arrow-Dereu only n he aumpon ha a poron of he endowmen nalenale and canno e ezed f a conumer oe ankrup Th model ha een uded y Schechman and Ecurdero [977], Manuell [986], Marce and Marmon [992], and Kehoe and Levne [993] Kocherlakoa [996], and Aluquerque and Hopenhyn [999] I ha een appled o he udy of exn ae marke y Kehoe and Perr [998], Krueer and Perr [998], and Alvarez and Jermann [2] I worh non ha here are wo dnc model of de conran: hoe n whch rader can e excluded from po marke, or hoe, a n Kehoe and Levne [993] where hey canno The laer poly lead o a falure of he welfare heorem, and concepually more lke he ncomplee marke model In he nle ood model uded here, and wdely ued n applcaon, ncludn he paper ced aove, however, here no po marke, and a a reul he welfare heorem hold In conra o he eneral equlrum approach employed n h paper, model wh de conran can alo e analyzed un he ool of opmal conrac See, for example, Kocherlakoa [996] and Aluquerque and Hopenhyn [999] Th paper drecly compare he de conraned model o he ncomplee marke/lqudy conraned model n he ame phycal envronmen n whch conumer alernae eher deermncally or randomly eween havn hh and low endowmen The oom lne ha he de conraned model, larely ecaue nvolve a much maller deparure from he Arrow-Dereu framework, lead o a valy mpler and more racale model of equlrum n he ochac cae, u neverhele ncorporae he man feaure of equlrum doyncrac rk earn and nere rae lower han ujec dcoun rae 2 The Envronmen There are an nfne numer of dcree me perod =,, In each perod here are wo ype of conumer = 2,, and a connuum of each ype of conumer There a nle conumpon ood x; he repreenave conumer of ype conume x n perod The nfne vecor of conumpon ( x, x ) ++, where ++ he e of equence ha are ounded and pove Boh conumer have he common aonary addvely eparale uly U( x, x ) = ( δ) δ u( x ) The perod uly funcon = wce connuouly dfferenale wh Du( x)>, afe he oundary condon

4 3 2 Du( x) a x, and ha Dux ( )< The common dcoun facor δ afe < δ < There are wo ype of capal: human capal (or laor) and phycal capal (ree or land) The ervce of he (one un of) human capal held y ype conumer n perod are denoed w Thee ervce ake on one of wo value, ω and ω, wh ω < ω, correpondn o ad and ood producvy repecvely Moreover, f one conumer ha ood producvy, hen he oher conumer ha ad producvy, o f w = ω hen w = ω (where he ype of conumer who no ype ) We ar y aumn ha producvy alernae eween ood and ad, o f w = ω hen w = + ω Suequenly, we wll allow for a more eneral proce of randomly 2 wchn eween he wo producvy par ( w, w ) = ( ω, ω ) and 2 ( w, w ) = ( ω, ω ) There one un of phycal capal n he economy Th capal durale and reurn r > of he conumpon ood n every perod If r =, phycal capal would e nerpreed a fa money, u we do no allow h cae Snce r > we may nerpre phycal capal a ree, wh r en he amoun of conumpon ood produced every perod y he ree A conumer of ype hold a hare θ of he capal ock a he ennn of me Inal phycal capal holdn are θ The oal upply of he conumpon ood n h economy he um of he ndvdual producvy, plu he reurn on he nle un of phycal capal ω + ω + r We denoe h areae upply a ω The ocal fealy condon for h economy n each perod are 2 x + x ω + ω + r = ω 2 θ + θ 3 Marke Arranemen In h phycal envronmen, we conder wo dfferen model of neremporal rade In he lqudy conraned economy conumer can only carry ou neremporal rade y exchann real capal The conumpon ood aken o e numerare, and he prce of phycal capal n perod denoed y v The ojecve of a conumer of ype o olve he prolem

5 4 max ( δ) δ ux ( ) = ujec o x + vθ w + ( v + r) θ + θ, =,, The crucal feaure of h model ha phycal capal can e held only n nonneave amoun, and ha here are no ecure or oher ae ha can e raded ede phycal capal To underand h eer, ueful o hnk of rade a akn place a dfferen phycal locaon around he crcle, a hown n Fure Only conumer a he ame locaon can rade; he meaure of oh ype of conumer he ame The ype conumer do no move, and ype 2 conumer move counerclockwe The eenal feaure ha ype 2 conumer move n uch a fahon, ay a nle radan each perod, ha hey never reurn o he ame locaon In h model, neremporal rade can e carred on only y exchann phycal capal, and phycal capal can no e held n neave quane, o h explan oh why here only one ecury, and why canno e old hor Laer n he paper, we dcu he conequence of allown phycal capal o e orrowed θ ven locaon arrow denoe movemen of ype 2 Fure

6 5 In he lqudy conraned economy, an equlrum an nfne equence of conumpon level, capal holdn, and capal prce uch ha conumer maxmze uly ven her conran, and uch ha he ocal fealy condon are afed The econd model of neremporal rade ha we examne he de conraned economy Here we allow orrown and lendn and, n he ochac cae ha we dcu laer, he ale and purchae of nurance conrac There are, however, de conran Thee come aou ecaue conumer have he opon of on ankrup, or oherwe opn ou of neremporal rade If hey chooe o do h, hey renee on all exn de They are excluded from all furher parcpaon n neremporal rade, however, and her phycal capal ezed The endowmen of human capal aumed o e nalenale: canno e aken away, nor can conumer e prevened from conumn reurn Noce ha unlke he model of radn phycal capal, whch can e compleely decenralzed, h model requre a cred aency, a overnmen, or ome cenral auhory o keep rack of who ha one ankrup and o aure ha her capal ezed and ha hey do no connue o orrow and lend Formally, h a model n whch conumer face he ndvdual raonaly conran τ τ ( δ) δ ux ( ) ( δ) δ uw ( ) τ = τ Th ay ha n every perod, he value of connun o parcpae n he economy no le han he value of droppn ou In h en, he aence of prvae nformaon mple ha no conumer acually oe ankrup n equlrum: he cred aency wll never lend o much o conumer ha hey wll chooe ankrupcy Th very unlke he ncomplee marke ankrupcy model of Duey, Geanakoplo, and Shuk [988] and Zame [993] In h de conraned economy, nce marke are complee, conumer purchae conumpon n perod for p and hey ell he reurn on her capal w a he ame prce The correpondn opmzaon prolem τ = τ + rθ

7 6 max ( δ) δ ux ( ) = ujec o = = px p( w + θ r) τ ( ) τ δ δ ux ( ) ( δ) δ uw ( ), =,, τ = τ Noce ha we have wren he ude conran n he Arrow-Dereu form A uual n h or of model, and a we how formally n appendx, we can equally well formulae he ude conran a a equence of complee ecure marke, τ = x + vθ w + ( v + r) θ + θ Θ, θ ven =,, The conran θ Θ rule ou Ponz cheme, u unlke he lqudy conraned economy where Θ=, here Θ a pove conan choen lare enouh no o conran o orrown An equlrum of he de conraned economy an nfne equence of conumpon level and conumpon prce uch ha conumer maxmze uly ven her conran and uch ha he ocal fealy condon for conumpon afed We ar y examnn ymmerc eady ae of oh he lqudy and he de conraned economy In a ymmerc eady ae Becaue x x = R S T x x f w f w = ω = ω + x =ω, we can characerze conumpon a a ymmerc eady ae y he nle numer x A uual n eady ae analy, o mplemen he eady ae a an equlrum, we mu creae a ranfer paymen eween he conumer o ha hey afy her ude conran Laer we exend he analy of he de conraned economy o more eneral dynamc equlra 4 Comparon of Lqudy and De Conraned Marke We can now compare he eady ae equlra of he lqudy and de conraned econome Throuhou he analy we ue fr order Euler condon o characerze he opmum of he conumer I well known ha, oeher wh a τ

8 7 ranveraly condon, he Euler condon are neceary and uffcen for a pah o e an opmum See Schenkman [976] and Araujo and Schenkman [977] Thee ame paper how ha he ranveraly condon afed f he pah ounded In our analy, he pah we udy all convere o (or even en a) a eady ae, o hey are ounded A a reul we focu our analy on he fr order condon We en y characerzn equlra n he lqudy conraned economy In h economy, x deermned y he fac ha he conumer wh ood producvy free o purchae a much phycal capal a he whe from he conumer wh ad producvy H marnal uly n he curren perod Du( x ), whle nex perod he wll have ad producvy, and marnal uly Du( x ) = Du( ω x ) Conequenly, he fr order condon for he conumer maxmzaon prolem can e wren a Du( x ) v = δ Du( ω x ) + r v + In he appendx we how ha any equlrum prce v, v afyn hee equaon mu e ounded a well Smple alerac manpulaon hen mple ha, f v for all, hen v = v for all The hree condon ha mu e afed are he ude conran n he ood and ad ae, and he fr order condon n he ood ae x + vθ = ω + ( v+ r) θ x + vθ = ω + ( v+ r) θ Du( x ) v = δdu( ω x )( v + r) Mulplyn he fr equaon y Du( x ) and he econd y δdu( ω x ), we ue θ + θ = and x + x =ω o fnd Du( x )( x ω ) + δdu( ω x )( ω x ω ) = Du( x ) v + δdu( ω x )( v + r) + Du( x )(2 v + r) θ δdu( ω x )(2 v + r) θ Suun he fr order for he ood ae no he rh hand de of h equaon, we oan Du( x )( x ω ) + δdu( ω x )( ω x ω ) = Du( x ) δdu( ω x ) rθ ( + )

9 8 I convenen o defne L f ( x ) = Du( x )( x ω ) + δdu( ω x )( ω x ω ) There are wo pole: eher θ > a x = ω / 2 or θ = for x [ ω / 2, ω ] In he laer cae, we have f L ( x ) = In he former cae, f L ( ω / 2) We have demonraed he follown reul Propoon : A ymmerc eady ae x characerzed y of he lqudy conraned economy f L ( ω / 2) and x = ω /2 or ω L > ω /2, f ( x )= and x [ ω / 2, ω ] We urn nex o he de conraned economy We defne he conumpon e for each ndvdual o e he e of nonneave conumpon plan ha are ndvdually raonal Gven h defnon, he model a andard complee marke model wh a fne numer of conumer ype The andard arumen mple ha he equlrum Pareo effcen: Suppoe, o he conrary ha here ex an alernave allocaon ha feale, afe he ndvdual raonaly conran, yeld a lea a much uly o oh conumer, and yeld rcly more uly o a lea one conumer Then h alernave allocaon mu an o he conumer ha rcly eer off a conumpon undle ha co rcly more han h endowmen a he equlrum prce, px ~ > p ( w + θ r ) = = Furhermore, mu an he oher conumer a conumpon undle ha co a lea a much a h endowmen ecaue, f an a conumpon undle ha co rcly le, he conumer could pend he exra ncome, make hmelf eer off, and no volae h ndvdual raonaly conran, px ~ p ( w + θ r ) = = Toeher, hee wo condon mply ha he alernave allocaon co more han he areae endowmen

10 9 2 p ( ~ x + ~ x ) > pω = = A n he model whou de conran, h mple ha he alernave allocaon canno e feale, whch conradc he aumpon ha here a Pareo uperor allocaon Propoon 2: An equlrum allocaon n he de conraned economy Pareo effcen In a ymmerc eady ae, he fr e o equalze conumpon eween he wo conumer, x = ω / 2 I may e mpole o reach h allocaon whou volan he ndvdual raonaly conran, however To acheve a Pareo mprovemen over auarky un a aonary allocaon, conumpon mu e ranferred from he conumer wh ood producvy o he conumer wh ad producvy Evenually, he ndvdual raonaly conran for he conumer wh ood producvy may e volaed: he conumer wh ood producvy would prefer o declare ankrupcy raher han o make he ranfer We conclude ha, f conumpon eween he wo conumer no equalzed, hen he ndvdual raonaly conran for he conumer wh ood producvy mu nd exacly The uly ha he conumer wh ood producvy receve n equlrum proporonal o ux ( ) + δu( ω x ); he uly he would have receved from h endowmen proporonal o u( ω ) + δu( ω ) If we defne D f ( x ) = u( x ) u( ω ) + δ u( ω x ) u( ω ) c h, hen he exac ndn of he ndvdual raonaly conran can e wren f We can ummarze h dcuon D ( x )= Propoon 3: A ymmerc eady ae x characerzed y of he de conraned economy f L f D ( ω / 2) and x = ω /2 or ω D > ω /2, f ( x )= and x [ ω / 2, ω ] We can now compare eady ae of he wo model y udyn he funcon D and f : Concavy of he uly funcon mple ha f D concave Snce f L

11 replace he uly dfference n f D wh he lope of he uly funcon mulpled y he dfference eween he wo conumpon level x and x = ω x, concavy of u D L alo mple ha f ( x ) > f ( x ) Fnally, r > mple ha ω = ω + ω + r > ω + ω, and h mean ha f L ( ω ) > Fure 2 how wha f L D and f look lke n he cae where f D ( ω / 2) < From h fure we can mmedaely ee ha eady ae of oh ype ex: nce each funcon f connuou and pove a ω, eher pove a ω /2, n whch cae ω /2 a eady ae, or mu e zero omewhere on he nerval [ ω / 2, ω ], and ha zero a eady ae Moreover, nce f D concave, can e zero a mo n h nerval, o ha n he de conraned economy he eady ae unque If we calculae Df L L and uue n he neror eady ae condon f ( x )=, we fnd ha a neror eady ae Df L ( x )> A can e een n Fure 2, h oeher wh he oundary condon f L ( ω ) > mple ha n he lqudy conraned economy he eady ae unque We um up our dcuon wh a propoon f D ω /2 ω f L x Fure 2 Propoon 4: A ymmerc eady ae ex oh n he lqudy conraned and n he de conraned economy In each cae here only one ymmerc eady ae In he lmn cae where δ = n he lqudy conraned economy, we can calculae f L ( ω / 2 ) = Du ( ω / 2 ) r > For δ uffcenly cloe o, f L ( ω / 2) >, and o he only lqudy conraned ymmerc eady ae wll e he ymmerc fr e

12 x D L = ω / 2 Snce f ( x ) > f ( x ) he ame aemen rue n he de conraned economy cae: n oh cae we reach full effcency when conumer are uffcenly paen In a mlar ven, we ee ha n f L ( ω / 2) = n ( ω / 2 ω ) + δ( ω / 2 ω ) Increan r, holdn ω and ω fxed, ha he effec of ncrean ω = ω + ω + r When r uffcenly lare, ω /2 ω and f L ( ω / 2) aan mply ha oh lqudy conraned and de conraned ymmerc eady ae are fr e In oher word, f he ro reurn o he ock of phycal capal uffcenly lare relave o he producvy of human capal, hen marke are fully effcen The nuon for hee reul mple: In he lqudy conraned economy ncrean δ ncreae he eady ae prce of phycal capal v, hu ncrean v + r Increan r doe h drecly The larer v + r, he eaer for conumer o mooh conumpon un rade n phycal capal In he de conraned economy ncrean δ ncreae he penaly for ankrupcy ha a conumer uffer from en excluded from neremporal rade Increan r ncreae he penaly ha he uffer from lon h collaeral, h endowmen of phycal capal The larer are hee penale, he eaer o afy he ndvdual raonaly conran The nere rae a he ymmerc eady ae can e calculaed from a rao of marnal ule Du( x ) = δdu( x ) In he ymmerc fr e h ve he uual complee marke nere rae equal o he ujecve dcoun rae / δ When he ymmerc fr e no reached, x > x, o he nere rae wll e lower han he ujecve dcoun rae The nuon mple: Borrower are conraned, lender are no To keep he level of loan from lender a low a requred n equlrum, he marke mu have a low rae of nere The eneral feaure of oh he lqudy conraned and he de conraned economy can e lluraed y a mple numercal example Suppoe ha uly ven y ux ( ) = lox, and ha he endowmen and dcoun facor are ω = 24, ω = 9, r =, δ = / 2 Here he conumer are que mpaen, and her producvy flucuae

13 2 uanally In addon, human capal much more mporan han phycal capal In he lqudy conraned economy we compue L f ( x ) = ( x 24)/ x + ( 25 x )/( 34 x ) =, 2 from whch follow ha x conraned economy, = 2 63, x = 337 By way of conra, n he de reuln n x c h, D f ( x ) = lo x lo 24 + lo( 34 x ) lo9 = 2 = 8, x = 6 A can e een, he lqudy conraned economy ha le conumpon moohn, and ndeed, he de conraned economy exh a lare deree of conumpon moohn A we hall ee elow, f he hock more peren, he deree of conumpon moohn nfcanly reduced I alo of nere o compue he nere rae The ujecve dcoun rae correpondn o a dcoun facor of /2 percen In he lqudy conraned economy however, he nere rae 296 percen, conderaly lower In he de conraned economy 778 percen Th example alo ueful ecaue llurae how he ymmerc eady ae of he wo model can e mplemened a equlra The prolem ha we mu e around ha dcounn pu he wo conumer n aymmerc poon: he ype of conumer who fr ha ood producvy ha a permanen advanae over he oher ype The eae way o compenae for h advanae and arrve a he eady ae o mpoe a ranfer paymen from one conumer ype o he oher In he lqudy conraned model, we need he ude conran for he conumer ype who fr ha hh producvy, ay ype, o hold n he fr perod, x + v ω + θ ( v+ r) τ Th conran hold wh equaly when v = ω x f τ = θ ( v+ r ) In he de conraned model, we need o ranfer enouh ncome o ha he preen dcouned value of lfeme ncome are equal 2 2 p( w + θ r) p( w r) τ = + + = θ τ =

14 3 Alernavely, n he de conraned model, we could nroduce uncerany efore he fr perod, vn oh conumer ype equal chance of havn he hh producvy fr, and allown hem o wre connen conrac aan h nal uncerany 5 Shor Sale So far we have aumed ha capal mu e held n nonneave amoun Th an mmedae conequence of he locaonal ory ven aove In he deermnc cae, ha here only one ae, phycal capal, mean ha only he naly o orrow preven ae marke from en complee In he nex econ, we conder a ochac economy In he ochac cae, ha here only one ae force ae marke o e ncomplee In he ochac en, he locaonal ory play a more nfcan role, ecaue ve an economcally enle ory of why ae marke are ncomplee I eay o work ou wha happen n he deermnc cae when orrown of phycal capal allowed, even houh uch ale are no compale wh our locaonal ory We aume ha he conran on hor ale of capal ake he form θ d The only chane n he prevou analy ha he conumer wh ad producvy can now pend up o + d un of phycal capal o purchae ω x un of conumpon If we redefne f L ( x ) = Du( x )( x ω + rd) + δdu( ω x )( ω x ω + rd), hen he characerzaon of equlrum n Propoon 3 connue o hold I ovou ha, f d uffcenly lare, f L ( ω / 2) > and he ymmerc fr e he unque ymmerc eady ae Snce a nle ae all ha needed for marke compleene n he deermnc cae, h hould come a no urpre There alo a unque level of de d L o ha f ( x ) =, where x he unque oluon of D f ( x ) = In he numercal example n he prevou econ, en d = 32 reul n a oluon where x = 8 n he modfed lqudy conraned model, ju a n he de conraned model If our welfare creron place equal weh on he wo ype uly and f he de lm d > d (and x < ω / 2 ), hen he lqudy conraned equlrum provde a hher welfare level han he de conraned equlrum (If he dcoun facor cloe enouh o one, hher welfare n h ene wll alo mply Pareo domnance) The

15 4 mplcaon ha o enforce he repaymen of de n he ncomplee marke model when d > d, wll e neceary o eze human a well a phycal capal 6 A Sochac Envronmen Wh deermnc alernaon eween producve, he lqudy conraned economy and de conraned economy are que mlar: he major dfference ha he de conran allow reaer rade We now how ha when we allow for random producve, equlrum wh de conran connue o e decred y a ochac veron of a eady ae, u he lqudy conraned economy doe no perm h ype of mple equlrum We modfy he phycal envronmen o ha he conumer wh ood producvy choen randomly Le η {,2} denoe he conumer who ha ood producvy a me Th random varale aumed o follow a Markov proce, whch characerzed y a nle numer < π <, he proaly of a reveral, ha, a ranon from he ae where ype ha ood producvy o he ae where ype 2 ha ood producvy, or vce vera When π = we are n he deermnc cae The economy now ake place on a ree raher han over me The roo of he ree denoed y η A ae hory a fne l = ( η,, η ) of even ha have aken place hrouh me ( ), where ( ) he lenh of he vecor, he me a whch occur The hory mmedaely pror o denoed, and f he node follow on he ree, we wre > The counale e of all ae hore denoed S The proaly of a ae hory compued from he Markov ranon proale π pr( η η ) pr( η η ) pr( η η ) = ( ) ( ) ( ) ( ) 2

16 5 η 2 = η = η 2 = 2 η η = 2 η 2 = η 2 = 2 Fure 3 Conumpon and endowmen are now ucrped y ae hory, raher han ( ) y me Uly for conumer he expeced uly ( δ ) δ π ux ( ) Defne θ o e he holdn of capal a he end of ae The opmzaon prolem n he lqudy conraned cae now ecome ( ) max ( δ ) δ π ( ) ux S ujec o x + vθ(, η) w + ( v + r) θ θ, θ fxed In he de conraned economy he opmzaon prolem of he conumer ( ) max ( δ) δ π ( ) S ux S ujec o px p( w + θ r) S ( ) ( ) ( ) ( / ) ux ( ) ( ) ( δ δ π π δ δ ) ( ) ( π / π ) uw ( ) A n he deermnc cae, a propoon appendx how ha h Arrow-Dereu formulaon of he ude conran ha an equvalen equenal marke formulaon S

17 6 x + q(, ) θ(, ) + q(, 2) θ(, 2) w + ( v + r) θ θ Θ, θ fxed, where q (, η ) he prce of he Arrow ecury raded n ae ha prome a un of phycal capal o e delvered a ae (, η ) A andard arrae arumen mple ha q(, ) + q(, 2) = v The equenal marke ude conran for de conraned marke dffer from he lqudy conraned ude conran n wo way Fr, a n he deermnc cae, we have Θ> raher han Θ = Second, and nfcanly, n he lqudy conraned cae conumer are rerced o rade n whch θ (, ) = θ (, 2) In he ochac cae, we defne a ymmerc ochac eady ae y conumpon x when producvy ood and x when producvy ad, and he rule x = R S T x x w = ω w = ω In he de conraned economy, ochac eady ae are much lke deermnc eady ae: we decreae x from ω unl we eher acheve he ymmerc fr e a x = ω /2 or unl he ndvdual raonaly conran en o nd A n he deermnc cae, we defne a funcon proporonal o he dfference eween he uly from he eady ae conumpon plan and conumpon n auarky A recurve calculaon how ha h funcon D f ( x ) = δ( π) u( x ) u( ω ) + δπ u( ω x ) u( ω ) c h c h By exacly he ame arumen a ha leadn o Propoon 3, we oan he follown reul Propoon 5: A ymmerc ochac eady ae x characerzed y of he de conraned economy f D ( ω / 2) and x = ω /2 or ω D > ω /2, f ( x )= and x [ ω / 2, ω ] When π = he funcon f D concave and afe f D ( ω ) >, and we have concluded ha a ymmerc eady ae ex and unque Snce when < π < ll rue ha f D concave and afe f D ( ω ) >, we reach exacly he ame concluon

18 7 Propoon 6: A ymmerc ochac eady ae ex n he de conraned economy There only one ymmerc ochac eady ae An neren queon how he eady ae level of conumpon depend on he parameer π meaurn he perence of he hock From he mplc funcon heorem, n he cae where he de conran nd, we can compue dx f = d( π ) f D D / π / x We already oerved ha a an neror eady ae f D mu nerec he ax from D elow, o f / x pove We can alo rewre f D a When f c h c h D f ( x ) = ( δ) u( x ) u( ω ) + δπ u( ω x ) u( ω ) + u( x ) u( ω ) D ( x )=, nce he fr erm neave, he econd erm pove, and nce f D / π proporonal o he econd erm, alo pove We conclude ha dx >, d( π ) meann ha a more peren hock reul n reaer conumpon y he ood producvy conumer, or equvalenly le rade eween he wo conumer Th reul renforced y reexamnaon of he numercal example Recall ha ux ( ) = lox, whle he endowmen and dcoun facor are ω = 24, ω = 9, r =, δ = / 2 Recall ha n he deermnc cae, π =, we had x = 8, x = 6 By way of conra, f π = / 2, we can compue D 3 f ( x ) = (lo x lo 24) + clo( 34 x ) lo9h =, 4 4 from whch follow ha x = 252, x = 2 48, a conderale reducon n he amoun of conumpon moohn In he de conraned economy, when he economy ecome ochac, conumpon moohn reduced, and conumpon of x and x flucuae randomly Concepually, however, he equlra are very mlar n he deermnc and ochac cae The cae of lqudy conran rknly dfferen A n he deermnc cae, he ood producvy conumer rade ood o he ad producvy conumer n

19 8 exchane for phycal capal Snce he ood producvy conumer hold phycal capal a he end of he perod, h fr order condon vdux ( ) = δ ( v + r)( π) Du( x ) + ( v + r) πdu( ω x ) + + connue o deermne prce, where v + he prce of capal when he ae a + he ame a ha n he prevou perod, and v + he prce when a reveral of he ae ake place In addon, we how n he appendx (ee alo Levne and Zame [996], for example) ha he capal prce v mu e unformly ounded, ay y v, or ele no one would e wlln o hold capal Th oundedne of phycal capal prce poe a dlemma, however The conumer wh ad producvy mu purchae x ω un of conumpon each perod, and o mu expend a lea ( / v)( x ω ) un of phycal capal each perod Snce here only one un of capal n he economy, a conumer can have ad producvy no more han v /( x ω ) perod efore he wll have expended all of h phycal capal If < π <, however, hen here a pove proaly ha a conumer wll have a run of ad luck wh h producvy ha exceed h lenh of me We conclude ha x ω =, ha, he only pole ochac eady ae auarky In auarky, however, he conumer wh ad producvy free o orrow, o prce mu e deermned alo y h fr order condon Du( ω ) v+ r Du( ω ) = δ = ( π) Du( ω ) + πdu( ω ) v ( π) Du( ω ) + πdu( ω ) Th pole only f ω = ω, volan he aumpon ha ω > ω We can ummarze h dcuon wh a propoon Propoon 7: If < π < here no ymmerc ochac eady ae wh lqudy conran There a mple nuon for h reul: Suppoe ha a conumer ha ad producvy for he fr me Then he hould ell ome of h phycal capal o mooh h conumpon If he conumer unlucky enouh o have ad producvy n he uequen perod, he ha le phycal capal and o n a dfferen uaon han when he had ad producvy for he fr me Conequenly, wh lqudy conran, conumpon mu depend no only on he curren ae, u alo on he druon of

20 9 phycal capal eween he wo ype Noce ha h propoon no enve o permn orrown of phycal capal: any fxed de conran wll evenually e exceeded y a very lon run of ad luck 7 Dynamc Analy The de conraned economy uffcenly mple ha we can ve a complee analy even whou he eady ae aumpon Gven an arrary nal condon θ, θ 2 here a unque equlrum Th equlrum can have wo dnc phae: an nal phae and a fnal phae In he deermnc cae he nal phae ju he fr perod; more enerally, he nal phae la unl he wo conumer ype exchane role The equlrum one of wo ype If he parameer are uch ha he ymmerc fr e afe he de conran, hen n he fnal phae, each conumer conumpon over me conan If he ymmerc fr e doe no afy he de conran, hen he fnal phae he ymmerc eady ae The rkn fac n h cae ha, even f he nal condon and nal phae are que aymmerc, once role have revered, from ha pon on conumpon depend only a conumer endowmen, and no on h ype Th characerzaon of dynamc equlrum formally preened n he appendx Recall ha he fr welfare heorem hold for hee econome There are wo eparae cae, he non-ndn cae n whch he de conran doe no nd a he ymmerc fr e and he ndn cae n whch doe In he non-ndn cae, effcency requre ha conumpon reman conan unl he fr pon n me a whch he de conran doe en o nd If he de conran never nd, hen he equlrum a eady ae, alhouh no necearly a ymmerc one If he de conran doe nd a ome pon, hen from ha pon on, he aonary of he model force he economy o a eady ae wh conan conumpon n whch he de conran nd on ju one conumer ype The ndn cae, where he de conran ndn a he ymmerc fr e, more neren Here, eay o how ha n equlrum he de conran evenually nd on oh ype of conumer When nd for he fr me, he equlrum jump mmedaely o he ymmerc eady ae The nuon clear: Effcency requre ha he equlrum allocaon olve he prolem of maxmzn he

21 2 expeced dcouned uly of he unconraned ype from ha dae onward, ujec o he ndvdual raonaly conran for he oher conumer ype If he unconraned ype alway he ype wh he maller endowmen, hen h prolem ymmerc eween he wo conumer and he equlrum afer he de conran nd for he fr me mu e oh ymmerc and aonary The dynamc pah of conumpon and capal can e lluraed y our numercal example For mplcy we dcu he deermnc cae We fr conder an example n he non-ndn cae In any uch example, we hould fr check o ee f he conan allocaon ha afe he ude conran alo afe he ndvdual raonaly conran: f doe hen h he unque equlrum To do h check, we oerve ha n any eady ae wh non-ndn ndvdual raonaly conran he prce of capal v = δr/( δ) Len he fr conumer ype have he hh endowmen fr, he equenal marke ude conran are x + vθ = ω + ( v+ r) θ, x + vθ = ω + ( v+ r) θ We can ealy olve hee wo equaon for θ and x o fnd and mlarly we can fnd x = ω + δ( ω ω )/( δ) + rθ 2 x 2 2 = ω + δ( ω ω )/( δ) + rθ We hen mply check ha ux ( ) + δux ( ) u( ω ) + δu( ω ), for = 2, To make hn neren, le u uppoe ha ux ( ) = lox, ω = 24, ω = 9, r =, and δ = 3/ 4 If θ 2 < 33, hen x 2 < 576, whch can ealy e checked o volae he ndvdual 2 raonaly conran Take hen he cae n whch θ ndvdual raonaly conran ar o nd, we neverhele have x x 2 = Bennn a =, when he = 8 24, = 576 Wha aou =? The fr order condon for ype, who unconraned n orrown and he face ude conran Du( x) v r = + δdu( x ) v x + vθ = ω + ( v+ r) θ

22 2 Thee mu e olved for x and v Snce afer = we wll e a he eady ae, oh conumer mu hold he ame capal hare on no perod ha hey wll hold on no any odd perod Un h fac, we calculae x 2 = 9 27, x = 4 73, v = 37 Nex we examne he ndn cae Suppoe we lower he dcoun facor from δ = 3/ 4 o δ = / 2 Then we can ealy check ha a he ymmerc fr e, he ndvdual raonaly conran nd Compun he ymmerc eady ae, we fnd ha v = 29, θ = 32, θ = 232, x = 8, x = 6 In perod = we ue he ude conran and he fr order condon, o olve for x, v ; n he example we have Du( x) v r = + δdu( x ) v v x 24 + θ = 6 32 θ θ = 6 32 θ 8 Concluon Wh lqudy conran he rucure of equlrum n he ochac cae complcaed: canno e a ochac eady ae In a ene he pcure wore han h Equlra have een compued n a few pecal cae, a n Schenkman and We [986] and Kehoe, Levne, and Woodford [992] There a eneral heorem aou exence of Markov equlrum due o Duffe, Geanakoplo, Ma-Colell, and McLennan [994], and a mehod of compun approxmae equlra due o Levne [993] The equlra are Markov on a very lare ae pace, however, and a far a we know no model ha comne oh doyncrac and areae rk ha een uccefully ued for calraon or emaon The de conraned economy much mpler Sochac eady ae do ex, and are eay o compue Th ecaue n a ochac eady ae hor run hock have no lon run effec

23 22 Appendx We rea he eneral cae of random producvy The reul alo apply o he pecal cae of he deermnc model of he fr half of he paper when we e he reveral proaly π = Lemma: Equlrum capal prce v are ounded Proof: Le x ++ denoe he equlrum conumpon plan of ype, and le θ e he correpondn plan for holdn capal The raey of proof o conruc, for any ae hory, an alernave conumpon plan x for one of ype ha afe he ude and lqudy conran The fac ha he uly from he equlrum plan a lea a ood a ha from he alernave plan ve re o an nequaly We can hen manpulae h nequaly, un he fac ha equlrum conumpon mu e ocally feale, o derve an upper ound on he capal prce v (Capal prce are ounded elow ecaue hey are nonneave) Fx S One ype, ay, mu hold a lea half he phycal capal ock n equlrum a Conder he alernave plan for ype, x, ha conume x = v /2 a, x = w for ae hore > ha follow, and x = x (he ame a he equlrum plan) for all oher ae hore Snce ype hold a lea half of he phycal capal a h plan afe he ude and lqudy conran f we chooe capal holdn θ = for and θ conumpon plan x λ λ x = ( λ) x + λx = θ oherwe For λ, defne he Snce x λ a convex comnaon of x and x, alo afe he ude and lqudy λ conran f we chooe capal holdn θ = ( λ) θ + λθ Snce a equlrum prce x λ feale for and x opmal, we mu have ( ) ( ) λ ( δ ) δ π ux ( ) ( δ ) δ π ( ) S S ux Snce x and x dffer only alon he ranch of he ree of ae hore ha en a h nequaly hold alo where he um on oh de only over ae he equal or follow Dvdn he reuln nequaly y δ > ( ) π, we can wre ( ) ( ) λ ( ) ( ) ( ) ( δ δ π δ δ ) λ ux ux π ux ( ) ux ( )

24 23 λ λ The concavy of u mple ha Du( x ) x x u( x ) u( x ) Th nequaly mmedae f x x λ λ x, nce hen ux ( ) ux ( ) nonpove If x x λ, hen λ x o for all x, x x x, Du( x ) Du( x ) and he nequaly follow from Taylor heorem Snce n addon, x λ x λω, and for > x ha λ Du( ω ) λω ux ( ) ux ( ) Suun ack no he prevou uly nequaly, h ve ω, we conclude δ ( ) λ π Du( ω ) λω ( δ) δ ( ) π u( x) u( x) Dvdn oh de y δ Snce x ω and x ( ) π λ and akn he lm a λ we fnd δdu( ω ) ω ( δ) Du( x ) x x = v / 2, we conclude ha f v /2 ω δdu( ω ) ω ( δ) Du( ω) v / 2 ω Conequenly v R S T max 2ω,2 L NM δdu( ω ) ω + ω ( δ) Du( ω) OU QP V W Propoon A: If p, x, x 2 an equlrum of he de conraned economy wh Arrow-Dereu ude conran, hen here ex prce q and v and ae holdn θ 2 and θ uch ha q, v, x, x 2, θ 2, θ an equlrum of he economy wh equenal marke conran Converely, f q, v, x, x 2, θ, θ 2 an equlrum of he economy wh equenal marke conran, hen here ex prce p uch ha p, x, x 2 an equlrum of he economy wh Arrow-Dereu ude conran Proof: Conder fr an equlrum of he economy wh Arrow-Dereu ude conran The ude conran px p w r S S + ( θ ) If wealh p w ( r S + θ ) unounded or f p = for ome, he conumer prolem ha no oluon, o n equlrum neher of hee can e he cae From fne

25 24 wealh and he fac ha w +θ r unformly ounded away from zero, we conclude ha he nfne prce vecor p an elemen of ; ha, he equence p ummale Snce p > and p, we can defne q (, η) r = (, η) p η x w (, ) θ (, η ) = pq (, η) p p ( ) If we now plu hee defnon no x + q θ + q θ, we fnd ha (, ) (, ) (, 2) (, 2) x + q(, ) θ (, ) + q(, 2) θ (, 2) = w + ( v + r) θ Snce n equlrum x w ω, we alo fnd from he defnon ha θ ω r (, η) / Conequenly, f Θ ω / r, any ude feale plan wh repec o he Arrow-Dereu ude conran ude feale wh repec o he equenal marke ude conran Moreover, he conumpon alernave defned n he lemma ude feale wh repec o he preen value ude conran, o follow alo ha he ν are unformly ounded Now conder an equlrum of he economy wh equenal marke ude conran We wan how ha, f he prce ν are unformly ounded, a conumpon plan ha feale wh repec o he equenal marke ude conran feale wh repec o he correpondn Arrow-Dereu ude conran Le = ( η,, η ), and le = ( η), 2 = ( η, η2),, = We defne Arrow- Dereu prce y p = q τ = τ ν + r τ Snce q + q = v, and ν unformly ounded, follow ha p (, ) (, 2) We now recurvely work he ude conran forward, olvn he ude conran for ae holdn θ (, η ) x + q θ + q θ w = v + r (, η ) (, η,) (, η,) (, η,2) (, η,2) (, η) (, η )

26 25 and uun ack no he prevou perod ude conran x + q(, ) θ (, ) + q(, 2) θ (, 2) = w + ( v + r) θ Un he defnon of preen value prce, h yeld a equence of ude conran of he form p( x w θ r) + p S,( ) T S,( ) T < θ = Snce p and n equlrum + Θ θ Θ, he fnal um vanhe a T, and he Arrow-Dereu ude conran afed Propoon B: There a unque equlrum of he de conraned model Durn he nal phae, equlrum conumpon conan In he ndn cae, follown he nal phae, conumpon follow he ymmerc eady ae In he non-ndn cae, follown he nal phae, conumpon conan, alhouh poly dfferen han durn he nal phae Proof: We fr conder he ndn cae Snce he fr welfare heorem hold, uffce o how ha all effcen allocaon have he requred propery Unquene of equlrum follow drecly nce he value of ndvdual conumer allocaon a he upporn prce are monoone n he welfare weh To udy effcen allocaon, we formulae he Pareo prolem recurvely a he prolem of maxmzn he uly of a conumer nally n he ood ae ujec o ocal fealy, ndvdual raonaly and a uly conran for he oher conumer Denoe y U, U he averae preen value ule receved n he ood and ad ae repecvely a he ymmerc eady ae We denoe y U, U he averae preen value ule from he endowmen n he ood and ad ae repecvely Noe ha U = U I convenen alo o defne he funcon u( u) a he uly a conumer receve when he oher conumer receve uly u whn a perod Noce ha h funcon mooh, rcly concave, and ha own nvere Noce fr ha he averae preen value uly a conumer nally n he ad ae receve mu e a lea U, and, nce he conran nd on he oher conumer a he ymmerc eady ae, no more han U Le V ( U ) for U [ U, U ] e he oluon o he prolem of maxmzn he uly of a conumer nally n he ood ae

27 26 ujec o ocal fealy, ndvdual raonaly and a uly conran for he oher conumer The nvere of h funcon denoed y V ( U ) Explon he ymmery eween he wo conumer, le u e he nal uly of he conumer n he ad ae, le U ~ e h econd perod averae preen value f he reman n he ad ae, and le ~ U e h econd perod averae preen value f he wche o he ood ae The Bellman equaon V ( U ) = max ( δ) u( u ) + δ( π) V ( U ~ ) + δπ V ( U ~ ) u, U ~, U ~ ujec o ( δ) u + δ( π) U ~ ~ + δπu U ~ U U ~ U U V ( U ~ ) U ( ~ V U ) U The ojecve funcon rcly concave, o h prolem ha a unque oluon Conequenly, uffce o verfy ha our propoed plan of me conan conumpon n he nal phae, and he ymmerc eady ae hereafer olve h prolem Under h propoal ~ ~ U U, U U = =, and he uly conran hould nd, o ha fr conran hold wh equaly and ued o deermne u δπ u = U + U U d δ Plun hee uee no he Bellman equaon, and oervn ha V ( U ) = U, we can olve o fnd our propoed value funcon V F δπ ( ) u U U U I δ + U HG ( U ) = K J + δπ δ δ( π) and we may alo olve for he nvere funcon V F d δπ ( ) u U U U I δ + U HG ( U ) = K J + δπ δ δ( π) d,

28 27 We need only how ha he fr order condon and conran are afed y our propoed oluon We en y verfyn he conran hold a he propoed oluon The fr conran hold wh equaly y conrucon The econd conran hold ecaue he U mu e n he rane [ U, U ] The hrd conran hold ecaue he ymmerc eady ae afe he ndvdual raonaly rercon; ndeed hold wh equaly Turnn o he fourh conran, nce u rcly decrean, nce U uffce o how ha F I HG d K J + d uu δπ U U δπ + U U U δ δ To how h, oerve ha a he ymmerc eady ae η ( ) ( ) η + + η δ δ π δπ = η u U U U, o ha η η δπ u U ( U η U η = + ) δ Snce U U, = U he nequaly n queon read u( u ) u Snce he ymmerc eady ae ocally feale, n fac u( u ) = u So he fourh nequaly verfed Smlarly for he ffh nequaly, we may wre a F I HG d K J + d uu δπ U U δπ + U U U δ δ Makn ue of he equaon for u η aove and u( u ) = u, h ecome δπ ( U U U δπ ) U U U + + δ d, δ whch follow drecly from he fac ha a he ymmerc eady ae U U To verfy he fr order condon, we ue ha all he Larane mulpler are zero excep for hoe correpondn o he fr and hrd conran So we wre he Laranean ( ) ( ) ( ) ( ~ ) ( ~ δ uu + δ π V U + δπv U ) e ( ) ( ) ~ ~ ~ + λ δ u + δ π U + δπ U + µ U j

29 28 The correpondn fr order condon are λ = Du( u ), λ = DV ( U ~ ), µ = δπ DV ( U ~ ) + λ I uffce o how, herefore, ha Du( u ) = DV ( U ), and Du( u ) DV ( U ) From he defnon of V, V aove, we have F HG F HG F H d I K J d I K J δπ DV ( U ) = Du U + U U δ δπ DV ( U ) = Du U + U U δ I K By conrucon, δπ u = U + U U d δ Th ve Du( u ) = DV ( U ) mmedaely Suun no he fnal nequaly, we mu how d F I HG d K J δπ Du( U U U ) Du U δπ + + U U δ δ Snce U U U, h follow from he fac ha Du rcly decrean Turnn o he non-ndn cae, oerve ha when he uly conran do no nd, he unque effcen allocaon for each conumer o have a conan conumpon ream n all perod If we ncreae he uly of he conumer nally n he ad ae, evenually he conran nd on he conumer n he ood ae Snce h an effcen allocaon, he conumer nally n he ad ae can receve no hher uly n any feale allocaon afyn he uly conran In he oppoe cae, where we reduce he uly of he conumer nally n he ad ae, evenually he conran nd on ha conumer n he fr perod follown he nal phae To reduce h uly furher, we mply reduce h conumpon n he nal phae only, leavn he conran afer he nal phae ju ndn I eay o how ha effcency demand a conan conumpon ream durn he nal phae, and h allocaon can ealy e verfed o e effcen un exacly he ame ype of dynamc prorammn arumen ued aove

30 29 Reference Ael, A B [99]: Ae Prce Under Heeroeneou Belef, Amercan Economc Revew, 8, Aluquerque, R and H A Hopenhyn [999]: Opmal Dynamc Lendn Conrac wh Imperfec Enforcealy, Unvery of Rocheer Alvarez, F and U Jermann [2]: Effcency, Equlrum, and Ae Prcn wh he Rk of Defaul, Economerca, 68, Araujo, A and J A Schenkman [977]: Smoohne, Comparave Dynamc and he Turnpke Propery Economerca, 45, 6-62 Backu, D K, P J Kehoe, and F E Kydland [992]: Inernaonal Real Bune Cycle, Journal of Polcal Economy,, Bewley, T F [98], The Opmum Quany of Money, Model of Moneary Economc, ed J Kareken and N Wallace, Federal Reerve Bank of Mnneapol Duey, P, J Geanakoplo, and M Shuk [988]: Bankrupcy and Effcency n a General Equlrum Model wh Incomplee Marke, Cowle Foundaon Dcuon Paper, Yale Unvery Duffe, D, J Geanakoplo, A Ma-Colell, and A McLennan [994]: Saonary Markov Equlra, Economerca, 62, Duffe, D and W Shafer [985]: Equlrum n Incomplee Marke I: A Bac Model of Generc Exence, Journal of Mahemacal Economc, 4, Duma, B [98]: Two-Peron Dynamc Equlrum n he Capal Marke, Revew of Fnancal Sude, 2: Green, E [987]: Lendn and he Smoohn of Unnurale Income, Conracual Arranemen for Ineremporal Trade, ed E Preco and N Wallace, Unvery of Mnneoa Pre, 3-25 Har, O [975]: On he Opmaly of Equlrum when he Marke Srucure Incomplee, Journal of Economc Theory,, Hayah, F [985]: The Effec of Lqudy Conran on Conumpon: A Cro- Seconal Analy, Quarerly Journal of Economc,, Heaon, J and D Luca [997]: Marke Frcon, Savn Behavor and Porfolo Choce, Macroeconomc Dynamc,, 76-

31 3 Kehoe, P J and F Perr [998]: Inernaonal Bune Cycle wh Endoenou Incomplee Marke, Federal Reerve Bank of Mnneapol Kehoe, T J and D K Levne [993]: De Conraned Ae Marke, Revew of Economc Sude, 6, Kehoe, T J, D K Levne, and M Woodford [992]: The Opmum Quany of Money Reved, Economc Analy of Marke and Game, ed P Daupa, D Gale, O Har and E Makn, MIT Pre, Kocherlakoa, N R [996]: Implcaon of Effcen Rk Sharn whou Commmen, Revew of Economc Sude, 63, Krueer, D and F Perr [998]: The Effec of Redruve Taxe n De Conraned Econome, Federal Reerve Bank of Mnneapol Levne, D K [993]: Tremln Invle Hand Equlrum, Journal of Economc Theory, 59, Levne, D K and W R Zame [996]: De Conran and Equlrum n Infne Horzon Econome wh Incomplee Marke, Journal of Mahemacal Economc, 29,3-3 Manuell, R [986]: Topc n Ineremporal Economc, Unvery of Mnneoa PhD Deraon Marce, A and R Marmon [992]: Communcaon, Commmen, and Growh, Journal of Economc Theory, 58, Preco, E C and R M Townend [985]: Pareo Opma and Compeve Equlra wh Advere Selecon and Moral Hazard, Economerca, 52, 2-45 Radner, R [972]: Exence of Plan, Prce and Prce Expecaon n a Sequence of Marke, Economerca, 4, Schechman, J and V Ecurdero [977]: Some Reul on an Income Flucuaon Prolem, Journal of Economc Theory, 6, 5-66 Schenkman, J A [976]: On Opmal Seady Sae of n-ecor Growh Model When Uly Dcouned, Journal of Economc Theory, 2, -3 Schenkman, J A and L We [986]: Borrown Conran and Areae Economc Acvy, Economerca, 45, 23-45

32 3 Townend, R M [98]: Model of Money wh Spaally Separaed Aen, Model of Moneary Economc, ed J Kareken and N Wallace, Federal Reerve Bank of Mnneapol Zame, W R [993]: Effcency and he Role of Defaul When Secure Marke are Incomplee, Amercan Economc Revew, 83, Zelde, S P [989]: Conumpon and Lqudy Conran: An Emprcal Inveaon, Journal of Polcal Economy, 97,