Subjciv Discouning in an Exchang Economy Erzo G. J. Lumr Univrsiy of Minnsoa, Fdral Rsrv Bank of Minnapolis, and Cnr for Economic Policy Rsarch Thomas Marioi London School of Economics, Univrsié d Toulous, and Cnr for Economic Policy Rsarch This papr dscribs h quilibrium of a discr-im xchang conomy in which consumrs wih arbirary subjciv discoun facors and homohic priod uiliy funcions follow linar Markov consumpion and porfolio sragis. Explici xprssions ar givn for sa prics and consumpion-walh raios. W provid an analyically convnin coninuous-im approximaion and show how subjciv ras of im prfrnc affc risk-fr ras bu no insananous risk-rurn rad-offs. Hyprbolic discoun facors can b a sourc of rurn volailiy, bu hy canno b usd o addrss ass pricing puzzls rlad o high-frquncy Sharp raios. I. Inroducion Many xplici dynamic quilibrium modls in macroconomics and financ ar basd on h assumpion ha consumrs hav im- and sa- W hank h paricipans in numrous sminars and h dior, Frnando Alvarz, for hlpful commns. Th Financial Marks Group and Sunory-Toyoa Inrnaional Cnr for Economics and Rlad Disciplins a h London School of Economics providd financial suppor. An arlir vrsion of his papr circulad as Cnr for Economic Policy Rsarch Discussion Papr no. 2503. Th viws xprssd hrin ar hos of h auhors and no ncssarily hos of h Fdral Rsrv Bank of Minnapolis or h Fdral Rsrv Sysm. [Journal of Poliical Economy, 2003, vol. 111, no. 5] 2003 by Th Univrsiy of Chicago. All righs rsrvd. 0022-3808/2003/11105-0005$10.00 959
960 journal of poliical conomy sparabl prfrncs and ha hy discoun fuur uiliis a a consan ra. This implis ha consumr choics ar dynamically consisn. Psychologiss hav qusiond h validiy of h assumpion of xponnial discouning on h basis of xprimnal vidnc (Chung and Hrrnsin 1967; Ainsli 1975; Kirby and Hrrnsin 1995). Ths sudis suggs insad ha subjciv discoun funcions ar approximaly hyprbolic. According o his liraur, vns in h nar fuur nd o b discound a a highr ra han vns ha occur in h long run. This cras a conflic bwn an individual agn s prfrncs a diffrn poins in im: h cours of acion prfrrd oday by a hyprbolic agn dos no coincid wih h on h knows h would lik o implmn omorrow. As a rsul, slf-conrol and h dgr o which agns ar abl o commi o fuur choics bcom cnral issus for dcision making. If consumrs can prfcly commi o a squnc of consumpion choics, hn sandard consumr hory applis, whhr subjciv discoun funcions ar gomric or no. Considr, for xampl, an xchang conomy wih im-invarian priod uiliy funcions, consan aggrga ndowmns, and da 0 marks for consumpion a all fuur das and in all fuur sas. In his conomy, h rm srucur of inrs ras coincids wih h rm srucur of h rprsnaiv consumr s ras of im prfrnc. As Lownsin and Prlc (1992) hav suggsd, h xprimnal vidnc would hn lad on o xpc highr yilds on shor-mauriy bonds han on long-mauriy bonds. Empirical sudis of h rm srucur of inrs ras indica ha, on avrag, h opposi is ru. In his papr, w assum insad ha consumrs hav no mans hrough which hy can commi o fuur consumpion choics. Undr his assumpion, w xamin an xchang conomy wih a squnc of marks. In vry priod, consumrs can rad in a compl s of onpriod sa-coningn claims on consumpion (and possibly in som long-livd scuriis as wll). Ths conracs can b prfcly nforcd, so ha consumrs ar allowd o borrow up o h prsn valu of hir ndowmns. In conras, hr ar no nforcabl conracs ha allow consumrs o commi in advanc o a paricular squnc of consumpion choics. Following Sroz (1956), Pollak (1968), and many ohrs sinc, w ak individual consumpion and porfolio choics o b h oucom of an inraprsonal gam in which h sam individual consumr is rprsnd by a diffrn playr a vry da. 1 All consumrs ar as- 1 Byond h consumpion-savings problm, h mulipl slvs mhodology has bn applid o a broad s of slf-conrol issus. S,.g., O Donoghu and Rabin (1999, 2001), Carrillo and Marioi (2000), and Bnabou and Tirol (2002).
subjciv discouning 961 sumd o hav h sam consan rlaiv risk avrsion prfrncs. Endowmn procsss may diffr across consumrs. Th only rsricion w impos on subjciv discoun facors is ha h uiliy of aggrga ndowmns is fini. W considr compiiv quilibria in which consumrs follow inraprsonal sragis ha giv ris o consumpion and porfolio choics ha ar proporional o walh. Ths sragis ar rquird o consiu a Markov prfc quilibrium of h inraprsonal gam. If hr is a long-livd ass in posiiv n supply, hn hr is on and only on such compiiv quilibrium. Th fac ha individual bhavior is h oucom of a gam implis ha sandard drivaions of a ransvrsaliy condiion do no apply. W provid condiions on prfrncs and ndowmns ha nvrhlss rul ou bubbls on long-livd asss. Ths condiions also nsur h xisnc of quilibrium in h absnc of a long-livd ass. W obain xplici xprssions for h consumpion and porfolio sragis as wll as for h quilibrium prics of saconingn claims. In wo spcial cass, quilibrium prics will b xacly as hough consumrs discoun uiliy gomrically, vn whn hy do no. This happns whn ndowmns ar such ha xpcd uiliy growh is consan or whn priod uiliy funcions ar logarihmic. In boh cass, agns consum a consan fracion of walh in ach priod, irrspciv of h shap of h subjciv discoun funcion. This obsrvaional quivalnc rsul gnralizs a rsul of Barro (1999) o h cas of an conomy wih uncrainy. To gain mor insigh ino h propris of quilibrium sa prics, w xamin a limi conomy obaind by ling h lngh of a priod go o zro. In his limi conomy, h sandard consumpion capial ass pricing modl (C-CAPM) of Brdn (1979) applis, irrspciv of how consumrs discoun uiliy: insananous risk prmia ar givn by insananous covariancs of rurns wih marginal uiliy. Th risk-fr ra procss, howvr, is affcd by im inconsisncy. I is h sum of wo rms. On is h xpcd insananous growh ra of marginal uiliy, as is h cas whn consumrs ar im-consisn. Th ohr is a wighd avrag of consumr subjciv ras of im prfrnc a diffrn horizons. Th wighs ar proporional o discound xpcd fuur priod uiliis. This scond rm inroducs a nw sourc of risk-fr ra dynamics: variaion in xpcd fuur ndowmn growh shifs wighs across subjciv discoun ras a diffrn horizons, and his in urn affcs h currn risk-fr ra. Th fac ha Brdn s formula for insananous risk prmia coninus o hold implis ha h high-frquncy Sharp raios of rurns in an conomy wih nongomric subjciv discoun funcions ar h sam as hos in an conomy wih im-consisn consumrs. Many
962 journal of poliical conomy modls of prfrncs ar no abl o gnra h high Sharp raios found in many daa ss (Hansn and Jagannahan 1991). This is a puzzl ha hrfor canno b addrssd using only nonsandard assumpions abou subjciv ras of im prfrnc. Alhough raios of risk prmia ovr rurn volailiis do no dpnd on how consumrs discoun uiliy, h rurn volailiis hmslvs ypically do. In paricular, w giv condiions undr which crain forms of hyprbolic discouning can incras h volailiy of aggrga walh. Inuiivly, h high subjciv discoun ras for narby uiliis implid by hyprbolic discoun funcions mus b machd by low subjciv discoun ras for uiliis in h disan fuur, or ls inrs ras would b oo high compard o wha is obsrvd in h daa. Ths low long-run subjciv discoun ras can mak walh mor snsiiv o informaion abou long-run ndowmn growh ras. Rlad liraur. On of h subjciv discoun funcions w considr is h quasi-hyprbolic discoun funcion inroducd by Phlps and Pollak (1968) in a modl of imprfc inrgnraional alruism. I was lar usd by Laibson (1994) o capur h qualiaiv faurs of hyprbolic discouning for an individual consumr. Laibson (1997) shows ha a parially illiquid ass may b usd as a commimn dvic by consumrs wih im-inconsisn prfrncs. Harris and Laibson (2001) sudy h dynamic choics of a quasi-hyprbolic consumr facing a consan risk-fr inrs ra and subjc o borrowing consrains. Thy driv an Eulr quaion ha dpnds no only on h lvl of consumpion a wo conscuiv das bu also on h marginal propnsiy o consum ou of walh. Krusll and Smih (1999) considr an conomy wih quasi-hyprbolic discoun facors and argu ha consumrs mus hav ngaiv ras of im prfrnc for narby uiliis in ordr o accoun for h low lvl of inrs ras obsrvd in U.S. daa. In conras, w incorpora h high shor-run ras of im prfrnc suggsd by h psychological liraur and mphasiz ha on can us low long-run ras of im prfrnc o accoun for h low risk-fr ras found in h daa. Oulin of h papr. Th conomy is dscribd in Scion II. In Scion III, w analyz h inraprsonal gam facd by a ypical consumr and driv compiiv quilibrium prics. Scion IV drivs xprssions for inrs ras and risk prmia in a limiing conomy obaind by ling h lngh of a priod go o zro. In Scion V, w xamin h combind implicaions of hyprbolic discouning and monoon dynamics in xpcd uiliy for h dynamics of h risk-fr ra and h volailiy of aggrga walh. Scion VI conains concluding rmarks.
subjciv discouning 963 II. An Exchang Economy A. Prfrncs and Endowmns W considr a discr-im, infini-horizon conomy. Tim is labld by p 0, 1, 2,, and uncrainy is dscribd by a probabiliy spac ndowd wih a nondcrasing filraion {F} p0. For ach, w dno by E[7] h condiional xpcaion opraor wih rspc o F. Throughou, random variabls indxd by ar akn o b F -masurabl. Thr is a singl good availabl for consumpion in vry priod. Th rprsnaiv consumr s nonngaiv ndowmns of his good ar dnod by {n p0 }. On inrpraion is ha h consumr supplis labor inlasically and has accss o a linar chnology ha convrs labor ino consumpion goods. Thr ar also k1 0 unis of a long- livd ass or sock ha producs nonngaiv dividnds {d p0 }. Aggrga ndowmns ar dnod by p n dk 1. W assum ha aggrga ndowmns ar sricly posiiv a all das, wih probabiliy on. Following Plg and Yaari (1973), w viw h rprsnaiv consumr as composd of a squnc of auonomous dcision makrs, indxd by im. W rfr o h dcision makr a da as h da consumr. This da consumr valuas currn and fuur consumpion according o a uiliy funcion [ ] n np0 n n np0 U({c } ) p E d u(c ), (1) whr d0 p 1 and dn 1 0. Ths prfrncs ar im-inconsisn if d n1/dn is no a consan funcion of n. Throughou, w assum ha h priod uiliy funcion is givn by 1g c u(c) p, 1 g for som g 1 0, g ( 1. W ak g p 1 o man u(c) p ln (c). I is asy o modify (1) o incorpora habis or subsisnc lvls of consumpion. B. Exampls of Subjciv Discoun Facors I is convnin o s dn p d(n), whr d is a posiiv discoun funcion dfind on [0, ), and 1 0 is h lngh of a priod. Th discr-im subjciv ra of im prfrnc can b dfind as ln (d n1/d n)/. For smooh discoun funcions, his convrgs o r() p Dd()/d() as gos o zro. Exprimnal sudis by psychologiss (s Ainsli [1975], Kirby and Hrrnsin [1995], and, for a survy, Ainsli [1992]) and conomiss
964 journal of poliical conomy (Thalr 1981) suggs ha ras of im prfrnc nd o dclin as a funcion of h horizon ovr which uiliy is discound. O Donoghu and Rabin (1999, 2001) rfr o his phnomnon as prsn bias. A discoun funcion ha capurs prsn bias is y/z d() p (1 z) xp (r). (2) This combins an xponnial discoun funcion wih h gnralizd hyprbolic discoun funcion for which Lownsin and Prlc (1992) hav suggsd an axiomaic jusificaion. I gnralizs h hyprbolic discoun funcion proposd by Ainsli (1975) o inrpr xprimns ha indica rvrsals ovr im of prfrncs for rwards a diffrn horizons. Th rsuling subjciv discoun ra is givn by r() p r [y/(1 z)]. This convrgs o r y as h horizon gos o zro, and i dclins monoonically o r as h horizon gos o infiniy. Incrasing r incrass h discoun ra a all horizons, and incrasing y raiss h subjciv discoun ra mor a shor horizons han a long horizons. Th paramr z govrns h spd a which h subjciv ra of im prfrnc changs from is shor-run valu r yo is long-run valu r. Ay p 0, hr is no diffrnc bwn shor-run and long-run discoun ras, and on obains h sandard xponnial discoun funcion xp (r). Similarly, ling z go o zro yilds h xponnial discoun funcion xp [(r y)]. A paricularly convnin limiing discoun funcion is obaind by ling h spd paramr z and h discrpancy y bwn shor- and long-run discoun ras go o infiniy a h sam im. Spcifically, ak som d! 1, s y p ln (d )z/ln(z), and l z go o infiniy. Thn d() convrgs o d() p d xp (r) (3) for all 1 0. Th corrsponding discr-im discoun facors can b n wrin as dn p db for som posiiv b and all n 1 0. Ths ar h discoun facors usd by Phlps and Pollak (1968) and mor rcnly by Laibson (1994, 1997), Harris and Laibson (2001), and many ohrs. Laibson (1994) argud ha hs discoun facors can provid a good approximaion o hyprbolic discoun facors and calld his quasihyprbolic discouning. C. Marks Our main assumpion is ha marks ar squnially compl. Onpriod-ahad sa-coningn claims ar radd a vry da and in vry sa. Th cos a da of a porfolio of sa-coningn claims ha dlivrs b unis of consumpion a da 1 is E[p b ]/p unis 1 1 1
subjciv discouning 965 of consumpion, whr {p} p0 is a sricly posiiv squnc of probabiliywighd sa-coningn prics. Socks ar radd a all das a xdividnd prics qual o s unis of consumpion pr uni of sock. W rsric anion o quilibria in which sock prics ar nonngaiv. A da, h da consumr can choos nonngaiv consumpion c, a porfolio of sa-coningn claims b 1, and an amoun of sock k, subjc o h priod budg consrain pc E[p 1b 1] psk p[n b (s d )k 1] (4) and subjc o h prsn-valu borrowing consrain [ ] 1 s s 1 1 1 1 sp1 E p n p [b (s d )k ] 0. (5) Thr ar no coningn claims ousanding a h iniial da, and so b 0 p 0. Th consrain (5) rsrics h porfolio of b1 and k chosn by h da consumr o b such ha walh a h bginning of priod 1 is nonngaiv in vry sa. In viw of h fac ha marks ar compl, and sinc hr ar no consrains on porfolios ohr han ha currn consumpion and nxpriod walh mus rmain nonngaiv, w focus on quilibria in which prics saisfy h no-arbirag condiion: ps p E[p (s d )]. (6) 1 1 1 A violaion of his condiion would imply ha h da consumr could choos o consum an unlimid amoun. 2 Th porfolio choics of h da consumr affc h s of fasibl choics a lar das only hrough hir ffc on da 1 walh. I will b convnin o wri w for h consumr s walh a h bginning of da : [ ] s s 1 sp pw p E p n p[b (s d )k ]. (7) Givn h no-arbirag condiion (6), i hn follows ha h s of budg-fasibl consumpion choics dfind by (4) and (5) is quivaln o h s of squncs {c p0 } ha saisfy, for som squnc {w }, 1 p0 pc E[p w ] pw, 1 1 c, w1 0, (8) 2 For crain priod uiliy funcions u, i may b possibl o consruc quilibria in which such arbirag opporuniis ar no xploid. W shall no considr his possibiliy hr.
966 journal of poliical conomy wih iniial walh w 0 dfind by (7). No ha h sochasic procss of ndowmns { p0 } affcs h consumr s s of fasibl consumpion choics only via iniial walh, in conras o modls wih borrowing consrains ha ar ighr han (5), such as in Harris and Laibson (2001). For an individual consumr, hr is hrfor no commimn valu o changing his or hr ndowmn procss using, say, pnsion invsmns ha ar no dircly radabl. D. Inraprsonal and Compiiv Equilibrium In any priod, h consumr can in no way rsric his fuur acions ohr han hrough h amoun of walh ransfrrd o h nx priod. Bhavior of h infinily livd consumr is hrfor no h oucom of a singl uiliy maximizaion, bu of a sragic inracion among h squnc of da consumrs who mak choics a succssiv das. This rsuls in an inraprsonal gam in which ach da consumr chooss his currn consumpion c and nx-priod sa-coningn walh w 1, aking as givn a squnc of prics {p, s } p0 and h sragis of h da s consumrs for all s 1. A sragy for h da consumr in h inraprsonal gam is a mapping (C, W 1) ha spcifis, for any hisory h of h gam up o da, a consumpion lvl c p C (h ) and amouns of sa-coningn walh w1 p W 1(h ) such ha h budg consrain (8) is saisfid givn w. Th hisory hconsiss of all vns obsrvd by h da consumr, including h ralizaions of ndowmns and prics, as wll as pas consumpion and walh choics. Givn a pric squnc {p, s } p0, an inraprsonal quilibrium is a subgam-prfc quilibrium of h inraprsonal gam playd by h squnc of da consumrs. A compiiv quilibrium of h rprsnaiv agn conomy is givn by a sragy profil {(C, W )} 1 p0 in h inraprsonal gam and a pric squnc {p, s } p0 such ha (i) {p, s } p0 saisfis h arbirag condiion (6); (ii) h prsn valu of ndowmns is fini a prics {p} p0; (iii) {(C, W 1)} p0 is an inraprsonal quilibrium a prics {p, s p0 } ; and (iv) goods, sa-coningn claims, and sock marks clar a vry da and in vry sa: C (h ) p, [ ] 1 1 1 s s 1 1 1 1 sp1 p W (h ) p E p n p (s d )k, whr h is h da hisory of an inraprsonal quilibrium pah im- plid by {(C, W 1)} p0. Th pric-aking assumpion nails ha ach da consumr uss
subjciv discouning 967 h prvailing prics whn valuaing h payoff of a dviaion from h inraprsonal quilibrium. Marks nd no clar following a dviaion in h inraprsonal gam. III. Equilibrium Bcaus consumrs ar assumd o b infinily livd, h inraprsonal gam ha drmins bhavior may hav many subgam-prfc quilibria. W focus hr on Markov prfc quilibria in which h consumpion and porfolio choics of h consumr a any da dpnd only on currn walh and on xognous variabls. 3 This ruls ou boosrap sragis ha migh ohrwis b usd o miiga h consumr s slf-conrol problm (s Laibson 1994). A. Inraprsonal Equilibrium Fix prics, and considr a Markov prfc quilibrium for h inraprsonal gam. Wih sligh abus of noaion, l C (w) and W 1(w) dno h consumpion and walh choics of a da consumr who sars wih da walh w. Givn hs Markov sragis, l C,n(w) b h implid lvl of consumpion chosn by h da n consumr in h subgam in which h da consumr sars wih da walh w. Dfin [ ] [ ] np0 n,n np0 F(w) p E d u(c (w)), V(w) p E d u(c (w)). (9) n1,n Th currn valu F(w) is h xpcd uiliy for h da consumr in h subgam in which h da consumr sars wih walh w. Th coninuaion valu V(w) is h uiliy prcivd by h da 1 consumr for his sam subgam. In any Markov quilibrium, C (w) and W 1(w) mus b a bs rspons for h da consumr whn his iniial walh is w. Tha is, w mus hav F(w) p max {u(c ) E[V (w )] : pc E[p w ] pw} (10) 1 1 1 1 c,w1 0 for all possibl lvls of iniial da walh w. 4 3 This is also h prspciv adopd by Phlps and Pollak (1968) and Harris and Laibson (2001). 4 This quaion is rlad o h quasi-bllman quaion dvlopd by Harris and Laibson (2001) for quasi-hyprbolic subjciv discoun funcions.
968 journal of poliical conomy Saring from iniial walh w0 dfind by (7), wri w1 p W 1(w ) and c p C (w ) for all. Undr rgulariy condiions ha will b vrifid blow, h squnc of da firs-ordr and nvlop condiions for (10) can hn b wrin as and p1 DV 1(w 1) p (11) p Du(c ) DF(w ) p Du(c ). (12) Combining h firs-ordr condiion a da wih h nvlop condiion a da 1, w obain p1 DV 1(w 1) Du(c 1) p. (13) p DF 1(w 1) Du(c ) n Clarly, if dn p b, hn (9) implis V 1(w 1) p bf 1(w 1), and (13) yilds h sandard firs-ordr condiion. Mor gnrally, (13) shows ha h sandard gomric subjciv discoun facor mus b rplacd by a raio of marginal uiliis of walh basd on h coninuaion valu funcion V1 and nx priod s currn valu funcion F 1. Th homohiciy of prfrncs suggss a Markov prfc quilibrium of h inraprsonal gam ha is linar in walh: C (w) p fw, W (w) p w w (14) 1 1 for som im- and sa-dpndn cofficins f and w1 ha do no dpnd on walh. Th sragis (14) imply ha F(w) and V(w) as 1g dfind in (9) ar boh proporional o w /(1 g). Th maximizaion in (10) hrfor implis a da bs rspons ha is linar in da walh, and h righ-hand sid of (10) will again b proporional o 1g w /(1 g). Th Markov sragis (14) consiu a subgam-prfc quilibrium of h inraprsonal gam if h cofficins {f, w 1} p0 ar such ha (10) is saisfid a vry da and in vry sa for F and consrucd from hs cofficins according o (9). V 1 B. Compiiv Equilibrium Excp for h spcial cas of logarihmic prfrncs, i is hard o xplicily solv for a Markov quilibrium of h inraprsonal gam a arbirary prics. Bu o consruc a compiiv quilibrium, w nd only o solv for an inraprsonal quilibrium a mark-claring prics. Equilibrium sa prics. Th homohiciy of h priod uiliy funcion implis ha F(w)/[C (w)] and V(w)/[C (w)] dpnd only on w 1g 1g via h consumpion growh ras C (w)/c (w). Bu h linariy of h,n
subjciv discouning 969 sragis (14) implis ha C,n(w)/C (w) dos no dpnd on w, and mark claring rquirs ha his raio b qual o n/. Thrfor, a mark-claring prics, h valu funcions (9) can b wrin as 1g 1g G(fw) D(fw) F(w) p, V(w) p, (15) 1 g 1 g whr h cofficins Gand D ar givn by 1g n G p E [ d n( ) ], np0 1g n n1( ) np0 [ ] D p E d. (16) Th nvlop condiion (12) combind wih c p fw and h xprs- sion for F(w) in (15) yilds f p 1/G. Toghr wih w1 p w1 w and h fac ha mark claring rquirs ha c 1/c p 1/, w hn obain h following cofficins for h Markov sragis: 1 1G1 f p, w1 p. (17) G G This xprsss h cofficins of h linar Markov sragis (14) in rms of h ndowmn procss of h conomy. To consruc an xplici formula for quilibrium sa prics, w can us (13), (15), (16), and mark claring o obain ( ) g p1 D1 1 p. (18) p G 1 In any quilibrium in which consumrs follow Markov sragis ha ar linar in walh, sa prics mus hav his form. Rmaining quilibrium condiions. Clarly, h consrucion lading up o (17) and (18) rquirs ha G and D b fini. Tha is, uiliy of h aggrga ndowmn procss mus b fini a vry da and in vry sa. If his is h cas, hn h sragis dfind by (14) and (17) ar such ha consumpion grows a h sam ra as h aggrga ndowmns. Th only rmaining rquirmn for a compiiv quilibrium is ha h lvls of h consumpion and ndowmn procsss b h sam as wll. By iraing on (8) and using h mark-claring rquirmn T p w T/G T, on can vrify ha iniial walh mus b qual o h prsn valu of h aggrga ndowmns plus limtr E 0[p T TG T]. Similarly, by iraing on (6) and using h dfiniion of walh (7), on obains ha iniial walh mus also b qual o h prsn valu of h aggrga
970 journal of poliical conomy ndowmns plus limtr E 0[p Ts T]k 1. Our consrucion yilds an qui- librium if hs wo limi rms ar qual. If h sock is in posiiv n supply, his qualiy can b guarand by assigning a high nough valu o s 0 ; in paricular, hr may hav o b a bubbl on h sock. On h ohr hand, if k 1 p 0, hn an assumpion is ndd o nsur ha E[p 0 T TG T] gos o zro as T gs larg. Using (18), on can vrify ha his is quivaln o rquiring ha [ ] T 1g D 0 ( ) Tr p1 G1 1 lim E p 0. (19) This is no somhing ha follows simply from assuming ha uiliy of h aggrga ndowmns is fini. Eihr of h following wo assumpions is sufficin o nsur ha (19) holds. Condiion A. Thr is an a 1 0 such ha ag (0, 1] a all das and in all sas. Condiion B. Thr is a b 1 0 such ha d n1/dn b for all n 0 and such ha 1g n n [ ( ) ] np0 E b! (20) a vry da and in vry sa. Th proof of h following proposiion is givn in Appndix A. 5 Proposiion 1. If hr is a long-livd ass in posiiv n supply or if a las on of condiions A and B holds, hn hr is an quilibrium in which consumpion and porfolio choics ar proporional o walh. Th sragis ar givn by (14), (16), and (17), and quilibrium sa prics ar givn by (18). I is possibl o consruc xampls in which hr mus b a bubbl on h long-livd ass. 6 In such xampls, h consumpion-walh raio gos o zro fas nough so ha consumrs do no xhaus hir prsnvalu budg consrains. This is consisn wih quilibrium only if hr is a long-livd ass ha rads abov h prsn valu of is dividnds. For im-consisn consumrs, singl-agn opimizaion implis ha consumrs xhaus hir prsn-valu budg consrains, and his is h argumn ha is radiionally usd o driv a ransvrsaliy condiion ha can hn in urn b usd o rul ou bubbls on long-livd asss (s Schinkman 1977; Brock 1979; Obsfld and Rogoff 1983). Hr, h rquirmn ha consumrs play a subgam-prfc quilib- 5 Th proposiion dos no covr h cas of quasi-hyprbolic prfrncs wih d 1 1, bu a spara proof can b givn in his cas. 6 An xampl is an conomy wih (1)/(1g) and n 1 p (!) d p (b /n!). This yilds a n posiiv limi in (19) if 1 0, vn hough xpcd uiliy is fini.
subjciv discouning 971 rium dos no guaran ha hy xhaus hir prsn-valu budg consrains, and so a ransvrsaliy condiion nd no apply. Th possibl nd for condiion A or B or bubbls is a consqunc of our focus on Markov sragis. If w allow for non-markov sragis, hn hr is always a compiiv quilibrium in which hr ar no bubbls on long-livd asss. Th quilibrium consumpion and porfolio choics ar affin funcions of walh, and h formula for sa prics (18) coninus o hold. Logarihmic prfrncs. Th drivaions abov do no apply dircly o h cas of g p 1, alhough (16) (18) do hold. For logarihmic prf- rncs, h inraprsonal gam can b solvd xplicily. Th soluion is f and whr p 1 b w1 p bp/p 1, b p 1 (1/ np0 d n), givn any sa prics for which walh is fini. On can us his o solv for quilibrium sa prics, vn whn consumrs ar hrognous in rms of hir subjciv discoun funcions. Th raio of consumpion (walh) o aggrga consumpion (walh) convrgs o on for h consumr who is h mos pain on avrag (h on wih h highs b). In an conomy wih boh im-consisn and im-inconsisn consumrs, his could asily b a im-inconsisn consumr wih high subjciv ras of im prfrnc a narby horizons as long as his consumr s long-run subjciv ras of im prfrnc ar sufficinly low. C. Implicaions Effciv subjciv discoun facors. As can b sn from h quaion for sa prics (18), h wo variabls ha drmin sa prics in his conomy ar ndowmn growh and h raio D 1/G 1. If consumrs discoun gomrically, h raio D 1/G1 is consan and qual o h subjciv discoun facor b. For gnral subjciv discoun facors, D 1/G1 is an ffciv subjciv discoun facor ha can b xprssd as a wighd avrag of d /d : whr h wighs q n,1 D G n1 1 1 n n,1( np0 ) dn1 p q, (21) d ar givn by n 1g E 1[d n( 1n/ 1) ] n,1 1g q p. E [ d ( / ) ] 1 n 1n 1 np0 Ths wighs ar proporional o h xpcd uiliy of da 1 n consumpion from h prspciv of h da 1 consumr. In h
972 journal of poliical conomy spcial cas of quasi-hyprbolic discouning, d p 1 and d p d b n 0 n for all n 1, his yilds h gnralizd Eulr quaion p p 1 ) g 1 p [f1 db (1 f 1 )b] ( (22) of Harris and Laibson (2001). Mor gnrally, considr h propris of (21) if subjciv discoun ras ar rlaivly high a narby horizons and low a disan horizons. Th discoun ras implid by D 1/G1 will hn dpnd on h iming of ndowmn growh. If g 1 1, high arly ndowmn growh lowrs h wighs on d n1/dn for small valus of n, and his lowrs h discoun ra implid by D 1/G 1. If h sam amoun of ndowmn growh is dlayd, mor wigh is pu on d n1/dn for small valus of n. Dlayd growh hrfor incrass h discoun ra implid by D 1/G 1. Idnificaion. Th disinguishing characrisic of nongomric discouning is h fac ha h usual gomric subjciv discoun facor is rplacd by D 1/G1 in (18). This implis ha i is no possibl o diffrnia nongomric discoun facors from gomric ons if D 1/G1 happns o b consan. This will b h cas if h condiionally 1g xpcd uiliy raio E[( 1/ ) ] is consan. For any subjciv discoun funcion, on can hn consruc an alrnaiv conomy wih a gomric subjciv discoun facor givn by b p D 1/G 1. Sa prics will b h sam in boh conomis. I is no difficul o vrify ha h walh-consumpion raio in h alrnaiv conomy is again qual o a sum of xpcd uiliy raios, discound using h gomric subjciv discoun facor b. Thus consumpion-walh raios canno b usd o idnify propris of h subjciv discoun funcion ihr. As an xampl, on can ak indpndnly and idnically disribud ndowmn growh and an informaion srucur {F} p0 such ha a any da nohing is known abou fuur ndowmn growh. Alrnaivly, on can ak prfrncs o b logarihmic. If g p 1, hn G 1 and D1 ar simply sums of subjciv discoun facors, and hus D 1/G1 is obviously consan across im. 7 IV. A Coninuous-Tim Approximaion In h coninuous-im vrsion of h C-CAPM sudid by Brdn (1979), insananous xpcd rurns in xcss of h risk-fr ra ar qual o h insananous covarianc of rurns wih marginal uiliy. Th risk-fr ra is qual o a consan subjciv ra of im prf- 7 Barro (1999) obsrvs ha in h sandard drminisic Cass-Koopmans growh modl, on canno infr from daa whhr consumrs discoun gomrically or no if h conomy is in sady sa or if prfrncs ar logarihmic. S also Laibson (1996).
subjciv discouning 973 rnc plus h xpcd insananous growh ra of marginal uiliy. In his scion w dscrib how Brdn s rsuls chang whn consumrs ar im-inconsisn. This will allow us o highligh propris of shor-horizon rurns in an conomy wih im-inconsisn consumrs ha ar no apparn from h discr-im sa pric formula (18). A. Limi Propris of Discr-Tim Economis Suppos ha hr xiss an undrlying coninuous-im ndowmn procss { 0 } ha volvs according o a diffusion l d p [m (x )d j(x )dw ], (23) whr and {W } 0 {x } 0 is a vcor of indpndn sandard Brownian moions, is a vcor of sa variabls ha saisfis l dx p m x(x )d j x(x )dw. (24) Suppos also ha h subjciv discoun funcion d is dfind for all [0, ) and normalizd so ha d(0) p 1. Givn h ndowmn procss { 0 } and h discoun funcion d, w consruc a squnc of discr-im conomis as follows. For any priod lngh 1 0, considr a discr-im conomy wih a squnc of subjciv discoun facors dn p d(n) and priod n ndowmns givn by n, n p 0, 1, 2,. For any and posiiv, dfin 1g G() p E d(n) n [ ( ) ], np0 1g n ( ) np0 [ ] D() p E d((n 1)). (25) Whn is an ingr mulipl of, hs dfiniions corrspond o hos givn in (16). In quilibrium, h raio of consumpion pr uni of im ovr walh in his discr-im conomy is qual o 1/[G()], for p 0,, 2,, as in (17). From (18), h on-priod-ahad sa prics ar givn by ( ) g p () D () p, (26) p() G () again for p 0,, 2,. W now consruc a coninuous-im sa pric procss from (26) and xamin h propris of is sampl pahs as gos o zro. W g adop h normalizaion p 0() p 0. Dfin p() o b h produc of p 0() and h squnc of on-priod-ahad sa prics (26) up o
974 journal of poliical conomy priod [/] 1, whr [/] is h ingr par of /. A somwha indirc bu rvaling way o wri his is [/] 1 G n() D n() g [ ( { [ ] }) ] [/] np1 G n() p() p xp ln 1. (27) For vry 1 0, his dfins a coninuous-im procss indxd by ha coincids wih (26) for any qual o an ingr mulipl of. Proposiion 2. Undr rgulariy condiions, h sa prics {p()} 0 for a discr-im conomy wih priod lngh convrg as gos o zro, wih probabiliy on, o sa prics givn by whr and F(x v ) g 0 G(x v) {p} 0 p p xp dv, (28) [ ] 1g [ v ] [0,) F(x ) p E dd(v) (29) ( ) 1g ( v ) 0 G(x ) p E d(v) dv. (30) [ ] Appndix B givs a prcis samn and proof of his rsul. Th xprssions (28) (30) ar as migh b xpcd givn (25) (27). In paricular, no ha G() can b inrprd as an ingral of discound fuur xpcd uiliy raios agains im ha convrgs o G(x ) and ha h diffrnc G() D() can b inrprd as an ingral of fuur xpcd uiliy raios agains d ha convrgs o F(x ) as gos o zro. Effciv subjciv discoun ras. Th raio F(x )/G(x ) will b rfrrd o as h ffciv subjciv discoun ra. I rplacs h consan subjciv discoun ra ha would appar in (28) if consumrs discound uiliy gomrically. If d is sufficinly smooh so ha on can wri [ ] 0 d() p xp r(v)dv, hn h ffciv subjciv discoun ra simplifis o a wighd avrag of h subjciv discoun ras r(v) : F(x ) p r(v)q(x, v)dv, (31) G(x ) 0
subjciv discouning 975 whr h wighs q(x, v) ar givn by 1g E[d(v)( v/ ) ] 1g 0 v q(x, v) p, E[ d(v)( / ) dv] as in (21). Of cours, (31) will b consan if d is xponnial. Alrnaivly, considr h quasi-hyprbolic discoun funcion (3). This is clarly nonsmooh: an arbirarily small posiiv dlay rsuls in a discr drop in h discoun funcion. This drop shows up in h ffciv subjciv discoun ra via F(x ) 1 d p r. (32) G(x ) G(x ) Th associad walh-consumpion raio can b wrin as 1g ( v ) 0 G(x ) p d E xp (rv) dv. (33) [ ] As xpcd, (32) and (33) imply ha h ffciv subjciv discoun ra is incrasing in r and h insananous discoun 1 d. Rmark. Proposiion 2 dscribs h limiing propris of a squnc of discr-im conomis as h priod lngh gos o zro. I is also possibl o analyz h coninuous-im conomy dircly. Following Barro s (1999) analysis of h drminisic Cass-Koopmans growh modl, on way o do his is o assum ha consumrs can commi o a paricular consumpion sragy for a shor priod of im and hn l his commimn priod go o zro. In Lumr and Marioi (2000b), w prov ha his also yilds (28) whn h discoun funcion for an conomy wih a commimn priod of lngh is givn by d () p 1 if [0, ) and d () p d() if [, ). If d () p d() for all insad, hn any disconinuiy of d a zro would no b rflcd in h limi, and h domain of ingraion in (29) would b (0, ) insad of [0, ). B. Inrs Ras and Risk Prmia L R b h da cumulaiv rurn on som ass. Tha is, on uni of consumpion invsd a da yilds R T/R unis of consumpion a da T 1 if h ass is hld from o T and any dividnds ar rinvsd. Suppos ha w hav an ass wih cumulaiv rurns ha saisfy l dr p R [m (x )d j (x )dw ]. R R
976 journal of poliical conomy Th sa prics (28) form an Io procss ha can b wrin as l dp p p[r(x )d l(x )dw ] for som r(x ) and l(x ) ha ar givn blow. To avoid arbirag op- poruniis, cumulaiv rurns mus saisfy pr p E[p TR T] for all T. Thus p R has no drif, and an applicaion of Io s lmma hrfor implis h wll-known rlaion l m R(x ) r(x ) p j R(x ) l(x ). (34) If j R(x ) p 0, hn h cumulaiv rurn R is insananously risk-fr, and so r(x ) can b inrprd as h risk-fr ra. Th cofficin l(x ) is usually rfrrd o as h mark pric of risk. Using (28) and Io s lmma, w obain and F(x ) 1 l r(x ) p gm (x ) 2g(1 g)j(x ) j(x ) (35) G(x ) l(x ) p gj(x ). (36) Equaions (34) (36) show how h quilibrium risk-fr ra and risk prmia ar drmind by prfrncs and ndowmns. Th ky implicaion of (35) and (36) is ha h shap of h subjciv discoun funcion d influncs sa prics only via h risk-fr ra. In urn, h risk-fr ra is affcd by h shap of d only hrough is ffc on h ffciv subjciv discoun ra F(x )/G(x ). Th ohr drminans of h risk-fr ra ar h usual inrmporal subsiuion and prcauionary savings ffcs rprsnd by h scond and hird rms in (35). Th dynamics of h risk-fr ra will dpnd on h inracion of hs sandard ffcs wih h variaion in h ffciv subjciv discoun ra ha ariss whn discouning is no xponnial. By conras, h shap of h subjciv discoun funcion d has no ffc on h mark pric of risk l(x ). Rcall ha h insananous Sharp raio for h cumulaiv rurn is dfind by From (34), his is qual o R m R(x ) r(x ) [j (x ) j (x )] l 1/2 R R l j R(x ) l(x ) l 1/2 R R [j (x ) j (x )] As poind ou by Hansn and Jagannahan (1991), h absolu valu l 1/2 of his raio is a lowr bound for [l(x ) l(x )] and hus, bcaus of l 1/2 (36), for g[j(x ) j(x )]. Givn svral rurns procsss, on can..
subjciv discouning 977 ighn his lowr bound by using h porfolio wih h highs Sharp raio. If w can us mans and variancs of monhly rurns o approxima insananous Sharp raios, hn h simas rpord by Hansn and Jagannahan indica ha xrm lvls of risk avrsion ar rquird o rconcil rurn daa and daa on aggrga consumpion in h Unid Sas. Bcaus h mark pric of risk dos no dpnd on h shap of d, his is a puzzl ha canno b undrsood by modifying sandard assumpions abou how consumrs discoun uiliy. I should b mphasizd, hough, ha his dos no man ha risk prmia and rurn volailiis do no dpnd on h spcificaion of h subjciv discoun funcion. Th dividnds of an ass ar discound g by h produc of xp [ 0 F(x s)/g(x s)ds] and, and h pric of h ass dpnds on how boh hs facors corrla wih dividnds. For an imporan xampl, considr h infinily livd ass ha gnras dividnds qual o aggrga ndowmns. Th pric of his ass is simply aggrga walh G(x ). Th rurn on aggrga walh consiss of an insananously risk-fr dividnd yild, 1/G(x ), and risky capial gains ha aris from changs in G(x ). Io s lmma applid o G(x ) hrfor implis ha l DG(x )j x(x ) l l j R(x ) p j(x ) (37) G(x ) whn R is akn o b h rurn on aggrga walh. Bu h walh- 1g consumpion raio G(x ) is simply h xpcd valu of ( v/ ) for v 1 0, discound by d(v). This mans ha DG(x )/G(x ) and hus h volailiy of h rurn on aggrga walh and, by (34), h risk prmium on aggrga walh will dpnd on h shap of h subjciv discoun funcion d. V. Prsn Bias, Inrs Ra Dynamics, and Volailiy Equilibrium prics dpnd on h shap of h subjciv discoun funcion via is impac on h ffciv subjciv discoun ra F(x )/G(x ). W wan o xamin in mor dail how prsn bias affcs his discoun ra, as wll as h volailiy of h consumpion-walh raio 1/G(x ). From (29) and (30) i is clar ha h dpndnc of xpcd uiliy growh on h currn sa and h horizon is going o b imporan. For gnral ndowmn procsss, his dpndnc can b qui complicad, and his maks i hard o drmin h ffcs of changing h shap of h subjciv discoun funcion. In subscion A, w idnify a class of ndowmn procsss for which h inracion bwn uiliy growh and prsn bias can b xamind analyically. Thn in subscion B, w prsn a paramric xampl for which w giv a
978 journal of poliical conomy compl characrizaion of h ffcs of prsn bias on inrs ras and consumpion-walh raios. A. Monoon Uiliy Dynamics Rcall from (29) and (30) ha F(x) and G(x) ar ingrals of h xpcd uiliy raio G(x, ) p E 1g [( ) F x 0 p x 0 ] agains, rspcivly, dd() and d()d. No ha his raio is ngaivly rlad o xpcd uiliy growh if g 1 1. Suppos now ha {x 0 } is a saionary scalar diffusion, and considr h following rsricion on ndowmns and uiliy. Condiion M. Th xpcd uiliy raio saisfis ] DG(x, x ) DG(x, x ) [ 0. G(x, ) Sinc DG(x, x 0)p 0, condiion M implis ha DG(x, x ) has h sam sign for all 1 0. Condiion M hrfor says ha a chang in h sa has an impac on h log of h xpcd uiliy raio ha incrass monoonically as h horizon incrass. Roughly, his propry can aris if a chang in x0 has an immdia impac on 0, small or no ffcs on ndowmns in h long run, and ffcs on inrmdia ha dclin monoonically in. Using condiion M, w can sign h corrlaion bwn h xpcd uiliy raio G(x, ) and h ffciv subjciv discoun ra F(x)/G(x), and w can drmin h ffc on DG(x)/G(x) of crain paramric changs in d. Rcall ha for smooh discoun funcions, F(x)/G(x) is a wighd avrag of h subjciv discoun ra r() p Dd()/d(), wih wighs q(x, ) ha ar proporional o d()g(x, ). W can hrfor wri [ ] [ ] 0 F(x) Dxq(x, ) x G(x) q(x, ) p r() q(x, )d. No ha h righ-hand sid of his quaion can b inrprd as a covarianc. This lads o h following lmma. Lmma 1. Suppos ha d is smooh, wih subjciv discoun ras r() ha ar dcrasing in. Thn condiion M implis ] F(x) DG(x, x ) [ 0. x G(x)
subjciv discouning 979 W say ha subjciv discoun ras xhibi monoon prsn bias whn r() is a dcrasing funcion of. Lmma 1 sas ha h ffciv subjciv discoun ra F(x)/G(x) and h xpcd uiliy raio G(x, ) vary wih x in opposi dircions whn subjciv discoun ras x- hibi monoon prsn bias. For xampl, if g 1 1 and if a high valu of x implis high fuur ndowmn growh, hn a high valu of x will nd o imply a low valu of h xpcd uiliy raio G(x, ). Thus lmma 1 implis ha high xpcd ndowmn growh nds o go oghr wih high ffciv subjciv discoun ras whn g 1 1. Nx, considr varying h subjciv discoun funcion paramrically. Wri d a for a subjciv discoun funcion indxd by som scalar pa- ramr a, and l G a(x) b h corrsponding walh-consumpion raio. Obsrv ha on can wri [ ] p [ ][ ] a a 0 a DG a(x) DG(x, x ) Daq a(x, ) a G(x) G(x, ) q (x, ) q (x, )d, whr h wighs q a(x, ) ar proporional o d a()g(x, ). No again ha h righ-hand sid of his quaion can b inrprd as a covarianc. Lmma 2. Suppos ha d a is smooh, wih subjciv discoun ras r a() ha ar incrasing in a for all. Thn condiion M implis a F Dx G a(x) G(x) a F 0. For xampl, if d a() p xp (a), his simply says ha gomrically discound xpcd uiliy raios bcom mor snsiiv o h sa as on lowrs h discoun ra. For gnral subjciv discoun funcions, his conclusion coninus o hold if on lowrs subjciv discoun ras a all horizons. I should b mphasizd ha condiion M is only a sufficin condiion. Th conclusions of lmmas 1 and 2 apply mor gnrally if ndowmns ar such ha any oscillaions ovr im in DG(x, x )/G(x, ) ar rlaivly small. B. Paramric Exampls Considr h ndowmn procss p xp (h x ), whr x is a squar roo procss: dx p k(m x )d j xdw, (38) for posiiv k, m, and j (s Fllr 1951; Cox, Ingrsoll, and Ross 1985).
980 journal of poliical conomy I can b vrifid ha condiion M is saisfid for hs ndowmns. 8 Sinc x is nonngaiv and man-rvring, his xampl implis ha logarihmic ndowmns flucua blow som linar rnd. Th xampl can b xndd o allow for a sochasic rnd whil prsrving condiion M. For high valus of x, ndowmns ar far blow rnd. Expcd ndowmn growh is hn high and uncrain. For g 1 1, his implis a low xpcd uiliy raio G(x, ). W now considr h ffcs of prsn bias using wo alrnaiv subjciv discoun funcions. In boh cass, w adjus h subjciv discoun funcion so ha h risk-fr ra rmains unchangd on avrag. Vry high shor-run subjciv discoun ras hav bn suggsd in h liraur. Only if long-run subjciv discoun ras ar corrspondingly low will h implid risk-fr ra b in h rang obsrvd in mos daa ss. Quasi-hyprbolic discouning. Rcall ha h ffciv subjciv discoun ra for h quasi-hyprbolic discoun funcion is incrasing in boh h long-run discoun ra r and h insananous discoun 1 d. Making consumrs prsn-biasd amouns o incrasing h dis- coun 1 d. By (33), his has no ffc on DG(x )/G(x ). Bu o kp h risk-fr ra h sam on avrag, w hav o lowr r. Lmma 2 applid o d r() p xp (r) implis ha his maks h walh-con- sumpion raio mor volail. A highr valu of x implis highr fuur ndowmn growh. Suppos g 1 1, so ha G(x, ) and hrfor G(x) ar dcrasing in x. This mans ha h walh-consumpion raio and currn consumpion mov oghr. Th addd volailiy of h walh-consumpion raio ha coms abou from making consumrs prsn-biasd hrfor incrass h risk prmium on aggrga walh by (34) and (37). Considr h risk-fr ra. Obsrv ha G(x) is dcrasing in x whn g 1 1. Equaion (32) hn implis ha h ffciv subjciv discoun ra F(x)/G(x) is incrasing in x whn h prsn bias discoun 1 d is posiiv. Highr fuur ndowmn growh hrfor maks con- sumrs mor impain. I follows ha h ffciv subjciv discoun 2 ra and h inrmporal subsiuion ffc gk{[1 (j /2k)]x m} ar posiivly corrlad. Th prcauionary savings ffc is givn by 2 g(1 g)j x/2 and hus is ngaivly corrlad wih boh h ffciv subjciv discoun ra and h inrmporal subsiuion ffc. If h prcauionary savings ffc dominas h inrmporal subsiuion 8 This follows from h fac ha DG(x, x ) k 1 1 p xa[(1 B ) (1 B) ], G(x, ) 2 whr A p 2k/j and B p (1 g)/(1 g A).
subjciv discouning 981 ffc, hn making consumrs prsn-biasd will rduc h variabiliy of h risk-fr ra. Gnralizd hyprbolic discouning. Clarly, h discoun funcion (2) xhibis monoon prsn bias. Thrfor, lmma 1 applis, and h ffciv subjciv discoun ra will mov oghr wih h inrmporal subsiuion ffc if g 1 1. W hrfor obain h sam co- movmns among h hr drminans of h risk-fr ra as in h cas of quasi-hyprbolic prfrncs. No ha Dd()/d() is incrasing in r and y and dcrasing in z (sinc a highr valu of z spds up h ransiion from r y o r). To mak consumrs mor prsn-biasd whil kping h lvl of h risk-fr ra h sam, on can simulanously incras y and z and lowr r. Lmma 2 implis ha h incras in y nds o mak h walh-consumpion raio lss volail, whras h incras in z and h dcras in r will mak i mor volail. This suggss an ambiguous ffc of prsn bias. Bu if w ak z o b larg and y on h sam ordr as z/ln(z), hn h hyprbolic discoun funcion is wll approx- imad by a quasi-hyprbolic discoun funcion, and our rsuls for h lar should apply. Numrical xprimns confirm ha his is indd h cas. Implicaions. Our xampls suggs ha if ndowmns ar such ha xpcd uiliy raios xhibi monoon dynamics and if subjciv discoun funcions xhibi monoon prsn bias, hn h ffciv subjciv discoun ra nds o mov oghr wih h inrmporal subsiuion rm ha drmins h risk-fr ra. This can mak h risk-fr ra mor or lss volail, dpnding on h magniud of h prcauionary savings ffc. A h sam im, hs xampls indica ha h walh-consumpion raio bcoms unambiguously mor volail as a rsul of prsn bias. In U.S. daa, aggrga consumpion growh dos no sm o xhibi much prdicabiliy, and his implis ha h ffciv impainc is going o b clos o consan. Nvrhlss, hr ar wo ways in which our rsuls can ponially shd ligh on ass pricing puzzls. Firs, paricipaion in financial marks may b limid, and h consumpion procsss of hos who do paricipa may xhibi significan prdicabiliy. Scond, our rsuls coninu o apply whn h priod uiliy 1g funcion is rplacd by [(c c ) 1]/(1 g), whr c is a subsisnc or habi lvl of consumpion. For xampl, h habi prsisnc modl of Campbll and Cochran (1999) gnras a n-of-habi consumpion procss c c ha has sochasic propris ha ar vry similar o hos of x, whr xis givn by (38). An inrsing qusion for furhr rsarch is whhr prsn bias movs h implicaions of hs modls closr o h daa.
982 journal of poliical conomy VI. Concluding Rmarks In an infini-horizon xchang conomy in which consumrs canno commi o fuur choics and priod uiliis ar xpcd o grow a a consan ra, pric and consumpion daa can b inrprd as rsuling from h opimal choics of consumrs whos subjciv ras of im prfrnc ar consan. A similar obsrvaional quivalnc applis for ssnially arbirary ndowmn procsss if priod uiliy funcions happn o b logarihmic. Infrncs abou h shap of subjciv discoun funcions can b mad whn consumrs can mak irrvrsibl commimns rgarding fuur consumpion lvls, as in Laibson (1997), or whn consumrs fac binding borrowing consrains, as in h buffr sock savings modl analyzd by Harris and Laibson (2001). In his papr, w absrac from mark fricions or ohr commimn dvics. Insad, w xamin h obsrvabl implicaions of alrnaiv assumpions abou subjciv discoun funcions whn hr is srial dpndnc in uiliy. An imporan faur of h coninuous-im approximaion w prsn is h fac ha h insananous mark pric of risk dos no dpnd on h subjciv discoun funcion of consumrs. In ohr words, shor-horizon Sharp raios ar no affcd by how consumrs discoun uiliy. Subjciv ras of im prfrnc do influnc h dynamics of h insananous risk-fr ra. In urn, his affcs how h dividnds of long-livd asss ar discound, and his has implicaions for h volailiy and risk prmia on such asss. For xampl, prsn bias can, undr crain condiions, mak aggrga walh mor volail, vn whn i rducs h volailiy of h risk-fr ra. For h priod uiliy funcions w considr, srial dpndnc in uiliy is h rsul of srial dpndnc in ndowmn growh. Srial dpndnc in uiliy can also b gnrad by habi prsisnc and by h consumpion of durabl goods. If prfrncs ar homohic, linar quilibria of h yp drivd in his papr can again b consrucd. How hs aspcs of consumr prfrncs inrac wih hyprbolic subjciv ras of im prfrnc is an inrsing subjc for furhr rsarch. Linar quilibria can also b consrucd for conomis in which uiliy is no longr im and sa sparabl, as long as prfrncs rmain homohic. In Lumr and Marioi (2000a), w xnd h work of Epsin and Zin (1989, 1991) on non xpcd uiliy o a vry gnral class of nonrcursiv homohic prfrncs.
subjciv discouning 983 Appndix A Equilibrium To prov proposiion 1, w procd in wo sps. W firs giv a prcis samn and proof of h firs-ordr condiion (11). Nx, w chck ha condiions A and B imply h xisnc of an quilibrium in which hr is no long-livd ass ha is availabl in posiiv n supply. Proof of h Firs-Ordr Condiion (11) Givn currn walh w 1 0, h da consumr s dcision problm can b wrin as p { 1 1 1 [ 1] } (c,w 1) #L1 p max u(c ) E[A u(w )] : c E w w, (A1) whr L 1 is h s of nonngaiv F1-masurabl random variabls, A 1 1g L 1 is an almos surly posiiv random variabl ha is qual o D 1/G1 in quilibrium, and p 1/p L 1 rprsns h rlaiv pric of nx-priod consumpion. Lmma 3. Suppos ha (A1) has a soluion (c, w 1) a which xpcd discound uiliy is fini. Thn (11) holds almos surly, ha is, p1 A1Du(w 1) p Du(c ) p. Proof. Sinc A 1 is almos surly posiiv and h marginal uiliy Du is infini a zro, c and w1 mus boh b posiiv a h maximum. Furhrmor, givn ha u is sricly incrasing, h budg consrain mus b binding a h maximum. Prurb w1 o w1 p (1 ) w1 for som clos nough o zro and ak cp w E[(p 1/p)w 1]. Clarly, h mapping.e[a 1u((1 ) w 1)] is diffrniabl on Th opimaliy of. (c, w ) hn rquirs ha or, quivalnly, 1 [ ( [ ]) ] p 1 uw (1 )E w E[A u((1 )w )] p 0, p p0 1 1 1 F [[ ] ] p 1 1 1 1 p E Du(c ) A Du(w ) w p 0. (A2) Alrnaivly, considr prurbing walh o w1 p (1 i B)w for som 0 and B F 1. For clos nough o zro, w E[(p 1/p)w 1] is posiiv. Furhrmor, sinc u is incrasing and concav, on has F A 1[u((1 i B)w 1) u(w 1)] 1 B 1 1 F A i w Du(w ) A1w1Du(w 1). Sinc w1du(w 1) p (1 g)u(w 1) and xpcd uiliy is fini, h righ-hand sid of his inqualiy is ingrabl. Th diffrniabiliy of u on oghr wih h dominad convrgnc horm hrfor implis ha h righ d-
984 journal of poliical conomy rivaiv of h mapping.e[a 1u((1 i B)w 1)] a p 0 is wll dfind and is givn by E[A i w Du(w )]. Th opimaliy of (c, w ) hn rquirs ha 1 B 1 1 1 [[ ] ] p 1 1 1 1 B p E Du(c ) A Du(w ) w i 0. (A3) No ha bcaus B is an arbirary lmn of F 1, quaions (A2) and (A3) imply ha [Du(c )(p 1/p) A1Du(w 1)]w 1 p 0 almos surly. Sinc w1 is always posiiv, h rsul follows. Q.E.D. Proof of Proposiion 1 1g By assumpion, G, D, and A 1 p D 1/G1 ar wll dfind and almos surly fini for any 0. By consrucion, h consumpion and walh choics (fw, w1 w ) saisfy h firs-ordr condiion (11) a mark-claring prics g p 1/p p (D 1/G 1)( 1/ ), whr f and w1 ar givn by (17). I hn follows from h concaviy of h objciv funcion in (A1) ha (fw, w1 w ) is opimal from h prspciv of h da consumr. In h x, w hav imposd only ha consumpion choics grow a h sam ra as aggrga ndowmns. Goods mark claring a all das is hrfor quivaln o h condiion w 0 p 0G 0. Iraion on (8) oghr wih goods mark claring implis ha [ ] n n N N N np0 Nr pw p E p lim E [p G ] (A4) 1g for all 0. No ha G and D1 ar rlad via Gp 1 E[( 1/ ) D 1]. Iraing on his idniy, using (18) o limina D 1, on can vrify ha rquiring (A4) a da 0 is in fac quivaln o h goods mark claring condiion w 0p 0G 0. Toghr wih h fac ha sock prics mus b nonngaiv, (6) implis ha ps p E[ sp1 pd s s] pz, whr {z } p0 is a nonngaiv squnc of random variabls ha saisfis p z p E[p z ] (A5) 1 1 for all 0. This says ha h valu of h sock mus b qual o h prsn valu of dividnds plus a nonngaiv bubbl. Th sock mark and marks for coningn claims clar whn k p k1 and b1 p 0 for all. This implis ha, in quilibrium, pw p E[ sp p s s] pzk 1. I hrfor follows from (7) and (A4) ha w hav an quilibrium if and only if, for som squnc {z }, p0 p zk p lim E [p G ]. (A6) 1 N N N Nr If k 1 1 0, hn (A6) uniquly drmins h bubbl procss {z p0 }. No ha h righ-hand sid of (A6) is, by consrucion, a nonngaiv maringal, and hus (A5) holds. If insad k p 0, hn any nonngaiv {z } 1 p0 ha saisfis (A5) will b consisn wih quilibrium providd ha h righ-hand sid of (A6) is zro. Using (18), on can s ha h righ-hand sid of (A6) is zro if and only if (19) holds. Thus, o conclud h proof, w nd only o chck ha
subjciv discouning 985 condiions A and B imply (19). Considr firs condiion A. Using (16), on can wri [ ] [ [ ]] T1 1g D 1 p E0 [ ( ) ( 1 )] p1 G1 1 GT1 T1 1g D E 0 [ ( ) ](1 k) p1 G T 1g T1 1g 1g D D DT T 0 ( ) 0 ( ) T1 ( ) p1 G1 1 p1 G1 1 GT1 T1 E p E E _ 1 T (1 k), from which (19) follows. Undr condiion B w can wri D /G b for any n 0 sinc D /G is an avrag of d n1/dn b. Also, dn b for any n 0, and hrfor n 1g G E[ np0 b ( n/ ) ]. This yilds 1 [[ ( ) ] ] [ ( ) ] n n 0 ( ) T 1g 1g D T T 0 T 0 T p1 G 1 0 1g [ ] npt 0 E G E b G E b. Condiion B implis ha h righ-hand sid of his inqualiy convrgs o zro as T gos o infiniy. Hnc h rsul. Q.E.D. Appndix B Coninuous-Tim Approximaion In his Appndix, {x 0 } dnos a coninuous-im Markov procss dfind on h sam probabiliy spac (Q, F, P) as h ndowmn procss { 0 } and aking N is valus in som sa spac X O. Endowmns { 0 } ar posiiv, and for any x X and 0, G(x, ) is dfind as in h x. Th following assumpions will b mainaind in h rmaindr of his Appndix. Assumpion 1. Th subjciv discoun funcion d : r [0, 1] is nonincras- ing, lf-coninuous, and posiiv on a s of posiiv Lbsgu masur. Morovr, d is ingrabl ovr and d(0) p 1. Assumpion 2. Thr xiss a funcion M : X r ha is boundd on compac subss of X such ha G(x, ) M(x) for all x X and 0. Assumpion 3. Th funcion G(x, 7) is coninuous for vry x X. Morovr, h family of funcions {G(7, )} is quiconinuous a any x X. Sinc d is nonincrasing and ingrabl ovr, lim r d() p 0. Thus d inducs a uniqu probabiliy masur md on h Borl ss of such ha m d([s, )) p d(s) d() for any 1 s 0 (Lang 1993, proposiion X.1.8). By Fubini s horm, w may rwri (29) and (30) as F(x) p [0,) G(x, )dm d() and G(x) p 0d()G(x, )d, rspcivly, for any x X. Assumpions 1 3 nsur ha hs funcions ar wll dfind and fini. Furhrmor, G(x) is posiiv for all x in X sinc