Optimal Intertemporal Consumption under Uncertainty 1

Transcription

1 Review of Economic Dynamics 3, doi:101006redy , available online a hp:wwwidealibrarycom on Opimal Ineremporal Consumpion under Uncerainy 1 Gary Chamberlain Deparmen of Economics, Harard Uniersiy, Cambridge, Massachuses and Charles A Wilson 2 Deparmen of Economics, New York Uniersiy, 269 Mercer Sree, New York, New York Received November 30, 1999 We analyze he opimal consumpion program of an infiniely lived consumer who maximizes he discouned sum of uiliies subjec o a sequence of budge consrains where boh he ineres rae and his income are sochasic We show ha if he income and ineres rae processes are sufficienly sochasic and he long run average rae of ineres is greaer han or equal o he discoun rae, hen consumpion evenually grows wihou bound wih probabiliy one We also esablish condiions under which he borrowing consrains mus be binding and examine how he income process affecs he opimal consumpion program Journal of Economic Lieraure Classificaion Number: D Academic Press Key Words: uncerainy; consumpion; permanen income hypohesis 1 INTRODUCTION We shall consider he following problem In each period, a consumer receives income x Afer receiving his income, he mus decide how much o consume in ha period, c, and how much o save for fuure consump- ion His savings earn a gross rae of reurn, r, so ha he value of his 1 1 Suppor from he Naional Science Foundaion and he Alfred P Sloan Foundaion is graefully acknowledged We have benefied from discussions wih William Brock, Ed Green, Jose Scheinkman, and Izak Zilcha 2 To whom correspondence should be addressed CharlesWilson@nyuedu $3500 Copyrigh 2000 by Academic Press All righs of reproducion in any form reserved

2 366 CHAMBERLAIN AND WILSON asses a he beginning of period 1 afer he has received income x 1 is given by he relaion a r a c 1 1 x 1 The consumer s prob- lem is o choose a consumpion plan o maximize he expecaion of Ý0 uc subjec o a budge consrain, where u is an increasing, sricly concave funcion His only source of uncerainy is ha x and r follow some sochasic process The quesion we address is: Wha happens o he levels of c and a as goes o infiniy? The moivaion of he paper lies in he work of several auhors who aemp o formalize he permanen income hypohesis of Friedman 1957 The firs of hese papers, Yaari 1976 and Schechman 1976, considers he case where uiliy is no discouned 1, he ineres rae is zero T r 1, and he horizon is finie; ie, he consumer maximizes Ý uc 0 Assuming ha income x is idenically and independenly disribued and imposing only a solvency consrain on lifeime consumpion, Yaari shows ha as T, he opimal consumpion plan requires c T, he consumpion in period wih horizon T, o converge wih probabiliy one o Ex for all Schechman ighens he solvency consrain o forbid borrowing and obains he weaker resul ha, as boh T and go o infiniy, c T converges o Ex wih probabiliy one He also esablishes ha a converges o infiniy Bewley 1977 reains he resricion on borrowing bu considers a more general case where boh x and u follow a saionary sochasic process He also allows for discouning 1 He obains he analogous resul ha u c T converges o a consan as T,, and 1, so ha, asympoically, he consumer becomes insulaed from risk Some of he implicaions of his resul are developed in Bewley 1980a, 1980b In his paper we invesigae he asympoic properies of he opimal consumpion program for he infinie horizon model when income and ineres raes are sochasic and he consumer discouns a bounded uiliy funcion We pu no addiional resricions on he form of he sochasic process such as saionariy, and we allow for arbirary resricions on he permissible level of borrowing in each period Under hese assumpions, we show ha if he discoun rae is smaller han he long run average rae of ineres, hen c will converge o infiniy almos surely When he long run ineres rae is equal o he discoun rae, he asympoic properies of c depend on how sochasic he income sream is If he income sream is cerain, c is nondecreasing and converges o he supremum of he maximum susainable consumpion level saring from he minimum permissible level of wealh in any given period However, if he income sream is suiably sochasic, c mus converge o infiniy almos surely This resul exends some of he resuls of Soomayor 1984, who examines he case where income is idenically and independenly disribued bu allows for uiliy o be unbounded

3 CONSUMPTION UNDER UNCERTAINTY 367 Our analysis is based on he following observaion For simpliciy, suppose ha he ineres facor is a consan, Le a, z be he expeced discouned uiliy from following he opimal policy, given asse level a and informaion sae z a ime Then he firs order condiions 1 for uiliy maximizaion require a, z E a, z z 1, where denoes he righ-hand derivaive of wih respec o a When 1, his implies ha a, z is a supermaringale By a heorem of Doob 1953, his implies a finie random variable d such ha a, z converges o d wih probabiliy 1 If 1, a, z 1 E a, z z implies a, z 1 0 from which i is easy o show ha c mus converge o infiniy If 1, his argumen can sill be used o show ha c converges wih probabiliy 1 o some exended random variable c, bu in his case, c need no equal infiniy In fac, as we show in Theorem 3, if x is no sochasic, c is generally no infinie However, he following argumen shows ha c mus be infinie if he discouned value of fuure income is suiably sochasic Suppose here is posiive probabiliy ha c Then we can choose an arbirarily small inerval b, b and a such ha here is posiive probabiliy ha c b, b, and, wih probabiliy greaer han 1, c b, b for all whenever c b, b Bu if c c for, hen a will diverge o unless 1 c a Ý j1 x j j We can show ha i is no opimal o have c c if a j So condiional on c b, i follows ha Ý x 1 j1 j b a Bu since a is known a ime, his implies ha he condiional variance of Ý j1 x j j can be made arbirarily small for sufficienly small Therefore, unless here is perfec foresigh, savings mus diverge We conclude ha if he income sream is suiably sochasic, hen c converges o infiniy wih probabiliy 1 In he body of he paper, hese argumens are generalized o allow for a sochasic ineres rae In paricular, if he discoun rae is equal o he long run average rae of ineres, hen c grows wihou bound if here is sufficien uncerainy in he join disribuion of income and ineres raes This case arises in he reamen of he opimum quaniy of money by Bewley 1980c, 1983 We see ha wih discouning and a posiive ineres rae, he only counerpar o he YaariSchechmanBewley resul is a zero limi for marginal uiliy If 1, hen consumpion does converge, bu i converges o bliss, no he expeced value of discouned income 3 We should also noe ha alhough boh Schechman and Bewley use he convergence heorem for supermaringales, heir argumen requires eiher he law of 3 An example of Schechman and Escudero 1977 shows ha convergence o bliss wih probabiliy 1 can occur when 1 and income is independen and idenically disribued They also provide a condiion on he uiliy funcion ha rules his ou

4 368 CHAMBERLAIN AND WILSON large numbers or he ergodic heorem, which in urn requires a saionariy assumpion By iself, however, he supermaringale heorem requires no addiional assumpions I is he sochasic generalizaion of he resul ha a bounded, monoone sequence converges o a finie limi Consequenly, when we eliminae he need for a law of large numbers by inroducing discouning, we are able o considerably weaken our assumpions on he sochasic process and sill obain convergence of c, even wih a posiive ineres rae If he underlying process is saionary, here are alernaive condiions on he ineres rae sequence which ensure ha consumpion converges o infiniy In Subsecion 44, we show ha if he ineres rae sequence is saionary and ergodic, hen consumpion will grow wihou bound if Elog r 0 If Elog r 0 and an addiional condiion is saisfied which guaranees ha he ineres rae has sufficien variance and is dependence on he pas dies ou sufficienly fas, hen some subsequence of consumpion will grow wihou bound In boh cases, he resul follows from he fac ha Ł r or some subsequence j1 j converges o infiniy wih probabiliy 1 Secion 5 deals wih some implicaions of he budge consrain Raher han resric ourselves o a no borrowing consrain a he ouse, we allow he lower bound on borrowing in any period o be an arbirary funcion of he consumer s informaion in ha period We hen noe ha by redefining income and wealh, he original problem is equivalen o a problem where income is nonnegaive in each period and no borrowing is permied Making his ranslaion explici emphasizes ha hese assumpions are no really subsanive resricions on he model, bu are merely a simpler represenaion of a more general model In paricular, hey are consisen wih he possibiliy ha he only consrain on borrowing is an ineremporal budge consrain Wih his inerpreaion, we esablish some condiions under which he borrowing consrain is never binding as well as condiions under which i mus be binding in some periods We also show how he opimal program changes as he borrowing consrains are relaxed In he final secion of he paper, we show how some of our resuls can be exended o he case where he consumer chooses a porfolio of several risky asses The main complicaion inroduced by his exension is ha he ineres rae becomes endogenous 2 ASSUMPTIONS AND NOTATION is he real line, and n is n-dimensional Euclidean space and n refer o he corresponding subses of nonnegaive elemens Nonnegaive inegers are denoed by,, i, j Oher lower-case Roman leers generally

5 CONSUMPTION UNDER UNCERTAINTY 369 refer o random variables or heir realizaions, and lower-case Greek leers paricularly,,, and generally refer o real numbers Le z, z, z, z, be a sochasic process wih ransiion 1 probabiliies pdz z used o define condiional expecaion, where n 1 z and z z, z z, z, z,, z Inerpre z as new informaion he consumer receives a ime and z as he informaion 0 0 sae a ime Le Z z z : z z, j 04 j j be he se of saes a ime which are consisen wih sae z 0 Excep for Secion 44 where we consider he case where z is saionary, we will ake z 0 as fixed We assume ha he income of he consumer a ime 0 is a coninu 0 4 ous funcion x : Z z A each ime 1, he consumer may borrow and lend beween ime 1 and ime a ineres facor r which is assumed o be a posiive, coninuous funcion r : Z z 0 For 0, le R 1 For j 0, le Rj Ł k1 j rk denoe he ineres facor beween periods and j, and le R 1 j denoe is inverse The borrowing consrain of he consumer a ime 0 is a coninuous funcion, k : Z z 0 ha saisfies an ineremporal consisency con- srain, 0 1 P r k x k, 0 z 1, 1 where r k 0 Le x x, x, andk k, k, Inerpre k as he minimum amoun of wealh, measured in erms of period consumpion, ha he individual may hold a he end of period afer he receives his period income and spends his period consumpion Equaion 1 requires ha he lower bound on curren ne wealh be consisen wih he borrowing consrain he consumer will face in he nex period I implies ha curren wealh can never be so low ha i may become impossible for he individual o saisfy his borrowing consrain in he nex period even if nohing is consumed in he curren period Defining r0 k1 0 is simply a convenion which implies ha he consumer s iniial wealh is derived solely from his period 0 income Since he consumer s decision a each informaion sae depends only on wha he knows a ha sae, a consumpion program, c c, c, 0 1, is a sequence of Borel measurable funcions, c : Z z 0, 0 Le u: be an increasing, coninuous, sricly concave funcion wih 0 u M sup uc For any consumpion program c, uc c is he undiscouned uiliy o he consumer from his consumpion a ime We suppose he consumer discouns uiliy by a facor 0 1 in each period Therefore, a sae z 0, he problem of he consumer is o choose a 4 The coninuiy requiremen is wihou loss of generaliy since we may always include a variable as one of he coordinaes of z

6 370 CHAMBERLAIN AND WILSON consumpion program c o maximize E Ý uc j z 0 j0 j subjec o P Ý x c R k, 0 z 0 j0 j j j 1 Some of our resuls require some regulariy assumpions on he ransiion probabiliies We assume ha for any z 0 he Feller propery holds: if n f: is a bounded coninuous funcion, hen g z H f w pdw z is a coninuous funcion from Z z 0 o This condiion combined wih he requiremen ha x, r, and k be coninuous funcions is shown in Theorem A1 of he Appendix o guaranee ha he consumer s problem has a soluion Before proceeding wih our analysis, i will be useful o redefine our income variable so ha he problem may be reduced o a simpler form As saed, he problem of he consumer is o choose a consumpion program ha maximizes he discouned sum of uiliy subjec o a possibly random borrowing consrain Aside from echnical consideraions, here are no resricions on eiher he income sream or he borrowing consrain oher han he requiremen ha he borrowing consrain saisfy Eq 1 For insance, if we assume ha x is nonnegaive and we wish o prohibi borrowing, we may se k 0 for all Alernaively, if we wish o impose only an ineremporal solvency consrain, he appropriae consrain is 1 k 4 z inf : P Ý j1 xr j j 0 z 1 5 However, we also allow for inermediae cases as well as cases where lim kr Whaever addiional assumpions we impose on x and k, however, we may always ranslae he variables so ha he problem is equivalen o one in which income is nonnegaive and borrowing is no permied The key is o redefine income a ime o be he increase in available purchasing power he consumer receives a ime This is he difference beween x, he income he receives in period, and k rk, he change in he 1 minimum allowable wealh from period 1 o period, measured in unis of income a ime Leing x x k rk and leing ˆ ˆ 1 k 0 for all 0, Eq 1 may be saed as Px 0, 0 z 0 ˆ 1, and he 1 borrowing consrain may be saed as P Ý ˆx c R 0, 0 z 0 j0 j j 0 j 1 Noe ha since x, k, and k1 are all coninuous funcions of z, xˆ is also a coninuous funcion of z Consequenly, he echnical requiremens and informaion resricions on x and ˆ ˆ k are saisfied Unless we indicae oherwise, we shall assume for he remainder of he paper ha his ranslaion has already been made so ha we may resric aenion o he case x 0 and k 0 However, i is occasionally useful o emphasize he inerpreaion of our resuls in he original framework 5 To ensure he exisence of some feasible consumpion program consisen wih k1 0, 1 we hen require P Ý xr 0 z 0 j0 j 0 j 1, and o ensure ha k so ha an opimal 1 consumpion program exiss, we hen require P Ý xr z 0 for all z j j j

7 CONSUMPTION UNDER UNCERTAINTY 371 To esablish he resuls ha follow, i is convenien o work wih he value funcion of fuure discouned uiliy defined over he level of accumulaed asses For any sae z and wealh level a 0, define j Ý j c j a, z max E u c z subjec o 1 Ý j j j ž / j1 P a c x c R 0, z 1 o denoe he expeced uiliy of he consumer saring a ime wih informaion z, given purchasing power a a ime Following he argumens of Blackwell 1965, Srauch 1966, and Maira 1968, we esablish as Theorem A1 in he Appendix ha a, z max uc 0 c a 1 E a cr 4 1 x 1, z z 6 Wih he excepion of Subsecion 44, which uses he saionary assumpions, we shall assume he iniial informaion sae z 0 is fixed so ha uncondiional expecaions should be undersood o be condiional on z 0 I is also frequenly convenien o suppress explici reference o he sae z, so ha for a given sae z, we someimes wrie a, z a and regard a as a random variable Also, o simplify he saemen of some of he heorems, we someimes sae properies of he opimal program as if hey hold for all z, alhough we prove our resuls on a se of probabiliy 1 3 PRELIMINARY RESULTS The exisence and uniqueness of a soluion o he consumer s maximizaion problem is esablished in he Appendix as Theorem A1 We also esablish here ha is a bounded, sricly concave funcion Therefore, righ- and lef-hand derivaives exis For any funcion f: 0,, le f x denoe he righ-hand derivaive of f x, le f x denoe is lef-hand derivaive when x 0, and le f 0 lim f x x 0 Le c* c, c, denoe he consumpion program which solves he consumer s The resul is acually esablished for he more general model wih many asses discussed in Secion 6

8 372 CHAMBERLAIN AND WILSON maximizaion problem, and le a* a, a, 0 1 denoe he corresponding sequence of random variables represening he level of he consumer s wealh in each period afer curren income has been received bu before consumpion has been spen The wealh sequence a* is defined recur- 1 sively by a x and a r a c x x Ý x c j0 j j R j, for 0 Wih hese definiions, we may sae some condiions which an opimal program mus saisfy The proof is quie sandard and will be omied LEMMA 1 a a max u c, E r a z 4 b Sup- pose a 0 Then a u c if a c 0 E r a z if c 0 4 min u c, E r a z if 0 a c a For 0, le R 0 If he level of consumpion in period 0 is equal o he level of consumpion in period, hen measures he rae a which he consumer can ransform discouned uiliy in period for uiliy in period 0 Condiion a esablishes ha a is a supermaringale Our firs heorem uses he supermaringale convergence heorem o esablish ha a mus converge o some random variable e The only problem is ha o apply he convergence heorem, a mus be finie which is no rue if x 0 and 0 This difficuly can be 0 0 avoided if we assume ha he discouned value of he enire income sream is always posiive In his case, here is a sopping ime such ha a 0, which allows us o apply he convergence heorem o he sequence a 1 a THEOREM 1 If P x 0 0, hen here is a real-alued random ariable e such ha Plim a e 1 Proof P x 0 0 implies a sopping ime such ha x 0 There- fore, a x 0 and he concaviy of imply ha a is finie Le d a 1 a for 0 Since is a sopping ime, Lemma 1 implies ha d, d, is a nonnegaive supermaringale 0 1 Meyer, 1966, p 66 Bu since d 1, here is a random variable d wih 0

9 CONSUMPTION UNDER UNCERTAINTY 373 Ed 1 such ha P lim d d 1 Doob, 1953, p 324 Then if we define e a d, i follows ha Plim a e 1 In he nex secion, we use Theorem 1 o esablish some condiions under which consumpion mus converge o infiniy To esablish hese condiions we use he fac ha whenever asses grow wihou bound, consumpion also grows wihou bound This resul is esablished as Lemma 2 LEMMA 2 For any 0, here is an 0 such ha a implies c 1 M Proof From Lemma A2, 0 a 1 Therefore, he concaviy M of implies a a a 0 Choose M u Then if a, Lemma 1 implies u c a M 1 u The concaviy of u hen implies c Wih hese resuls in hand, we are prepared o examine he limiing behavior of he opimal consumpion sequence 4 CONVERGENCE THEOREMS Recall ha Theorem 1 esablishes ha he funcion a mus converge o some random variable e In his secion we demonsrae ha he implicaions of his resul for he limiing behavior of he opimal consumpion sequence depend primarily on he limiing value of Roughly, our resuls may be summarized as follows If lim, he opimal consumpion sequence of he consumer mus grow wihou bound, regardless of he properies of he income and ineres rae sequences If he limiing value or values of is bounded above and away from zero and he income sream is suiably sochasic, hen consumpion sill grows wihou bound However, if he income sequence is no sochasic, hen he consumpion sequence generally converges o a finie limi These resuls may also be inerpreed in erms of he relaionship beween he rae a which he consumer discouns fuure uiliy and he long run rae of ineres In Secion 2, we defined r o be he one period ineres facor beween period 1 and period and Rj Ł k1 j rk o be he accumulaed ineres facor beween periods and j Therefore, R 1 j j represens he aerage ineres facor beween periods and j If lim R 1 exiss, we call i he long run ineres facor; oherwise, we say 0 ha he long run ineres rae flucuaes Now consider he meaning of R Rewriing he expression, we obain R 1 1 Therefore, 0 0

10 374 CHAMBERLAIN AND WILSON he long run rae of ineres is equal o he discoun rae if and only if lim 1 1 Clearly, if he limiing values of are posiive and finie, hen he long run ineres rae exiss and is equal o he discoun rae Alhough his is no a necessary condiion, we will someimes blur his disincion for exposiional convenience We may ranslae our conclusions as follows The limiing behavior of consumpion depends primarily on he relaion beween he long run rae of ineres and he rae a which he consumer discouns uiliy If he long run rae of ineres is greaer han he discoun rae, hen consumpion grows wihou bound as long as he consumer earns a posiive income in some period If he long run rae of ineres is equal o he discoun facor, hen consumpion generally converges o infiniy only if here is sufficien uncerainy in eiher he income or ineres rae sequences All of our resuls are esablished for he case where boh he income and ineres rae sequences may be sochasic In many cases, however, he inuiion behind our resuls and he meaning of he assumpions become more apparen when only he income sequence is sochasic Throughou his secion, herefore, we will frequenly focus he discussion on he special case where he ineres rae is a consan We conclude he secion wih an inerpreaion of our resuls in he conex of a saionary disribuion of ineres raes We urn firs o he case where he long run ineres rae exceeds he rae a which he consumer discouns uiliy 41 Lim Our main resul for he case where he long run rae of ineres exceeds he discoun rae is summarized in he following heorem THEOREM 2 Suppose P x 0 0 Then i P lim sup 1 implies Plim sup c 1, and ii P lim 1 implies Plim c 1 Proof Theorem 1 implies ha if P lim sup 1 and P x 0 0, hen Plim inf a 0 1 Lemma 1 hen implies ha P lim inf u c 0 1, and herefore, ha Plim sup c 1 This proves par i The proof of par ii is similar Roughly, Theorem 2 says ha consumpion grows wihou bound so long as he long run rae of ineres always exceeds he discoun rae If he long run ineres rae flucuaes, bu some subsequence always exceeds he discoun rae, hen some subsequence of consumpion grows wihou bound When he ineres rae is consan, he heorem may be resaed as follows

11 CONSUMPTION UNDER UNCERTAINTY 375 COROLLARY 1 Suppose ha r for 1 If P x 0 0 and 1, hen Plim c 1 42 The Case of Cerainy: 1 For he remainder of his secion, we are concerned primarily wih he case where he long run ineres rae is equal o he rae a which he consumer discouns fuure uiliy To underline he imporance of he role of uncerainy in he main resuls which follow, we firs concenrae on he case where he income sequence is nonsochasic and he ineres rae in each period is equal o he rae a which he consumer discouns uiliy Again denoe he consan ineres facor by When 1, he firs-order condiions for uiliy maximizaion imply ha he consumer ries o equalize consumpion in each period As long as such a program does no violae he borrowing consrain, his condiion characerizes he soluion If such a program does violae he borrowing consrain, however, he consumer mus choose a consumpion sream wih unequal levels of consumpion over ime Neverheless, consumpion never decreases over ime Oherwise, by saving more in an earlier period and consuming more laer, lifeime uiliy can be increased This observaion implies ha he consumpion level approaches a limi possibly equal o, which we characerize in he nex heorem j Define y 1 Ý jx j o be he supremum of hose con- sumpion levels ha can be susained indefiniely when we consider only he borrowing consrains in he disan fuure, given ha he borrowing consrain is binding in period 1 Our nex heorem saes ha c converges o he supremum of hese maximum susainable consumpion levels THEOREM 3 If x is no sochasic, hen 1 implies lim c supy Proof We show firs ha for 1, eiher a c c,or b 1 c1 c and c a in which case, a x 1 1 Suppose c1 a 1 If c 1 0, hen Lemma 1 and he sric concaviy of u and 1 imply 0 a u , which violaes Lemma A3 So suppose 0 c a Then, on one hand, Lemma 1 implies u c a a a u c 1 1, and herefore, 0 c1 c On he oher hand, Lemma 1 also implies u c a a a u c, and herefore, c c 1 Le c lim c and le y supy We show firs ha c y Suppose no and le be he smalles 0 such ha c y Then, condiion b

12 376 CHAMBERLAIN AND WILSON above implies a x From he definiion of a, i hen follows ha 1 Ý x j c j j 0 2 j0 Condiions a and b also imply ha c is nondecreasing in Therefore, j c y for all j, so ha 1 Ý c c y j j j j 1 Ý x, which implies a sufficienly large such ha j j Ý x j c j j 0 3 j Combining 2 and 3 hen yields Ý j0 x j cj j 0, violaing he budge consrain of he consumer s maximizaion problem To show ha c y, again assume he conrary Then here is a y such ha c j y for all j 0, which implies Bu if c is feasible, hen j j Ý j Ý j j j c x 4 1 Ý x j c j j 0 5 j0 I follows from 4 and 5 ha here is an 0 and ˆ such ha, for j all ˆ, Ý j0 x j cj Bu hen ˆc, defined by ˆc j cj for j ˆ and c ˆ c ˆ ˆ, also saisfies he consumer s budge consrain Bu hen j j Ý j0 uc ˆj Ý j0 ucj implies ha c* is no an opimal consumpion program For our purposes he main implicaion of his heorem is ha when he income sream is cerain, he consumpion sequence generally converges o a finie limi Consumpion grows wihou bound only if he discouned value of fuure income is no bounded However, he heorem also has a noeworhy implicaion in he conex of he original formulaion of he model Suppose ha x is an arbirary income sequence and k is an ineremporally consisen sequence of borrowing consrains from which we derive j ˆx x k k 0 Then, y 1 Ý x 1 jˆj j 1 lim Ý x k 1 j j k 1 Now suppose we sar wih a sequence of borrowing consrains which generaes an opimal consumpion program c 1 Then if he k 1 consrain is replaced by a igher

13 CONSUMPTION UNDER UNCERTAINTY consrain, k k ie, k k for all, bu which allows for he same 1 discouned value of consumpion ie, lim k k 2 0, hen he igher consrain k 2 generaes a limiing value of consumpion a leas as 1 2 large as did he k consrain ie, lim c lim c 1 An even sronger resul can be esablished abou he behavior of he sequence of accumulaed wealh in each period, bu his is lef o Secion 5, where he sochasic case is reaed as well 43 The Sochasic Case As noed above, he analysis of he cerainy case yields lile insigh ino he limiing behavior of consumpion when we inroduce uncerainy ino eiher he income sequence or he ineres rae sequence In his subsecion, we formulae a general uncerainy condiion under which we show ha consumpion grows wihou bound even when he long run ineres rae equals he discoun rae The meaning of our uncerainy condiion is mos ransparen when he 1 ineres facor is fixed a In his case, we require: Condiion U There is an 0 such ha for any, j ž Ý j / j P x z 1 for all z, 0 Condiion U says ha saring a any informaion sae, here is a fixed probabiliy ha he discouned value of fuure income lies ouside any sufficienly small range The key implicaion of his condiion is when consumpion says wihin a sufficienly small range in each period, asses mus diverge wih some fixed probabiliy from any informaion sae However, he proof of our main resul requires a fixed probabiliy ha asses diverge whenever he consumpion program keeps he marginal uiliy of consumpion approximaely consan Consequenly, when we allow for a sochasic ineres rae, he uncerainy condiion requires a sligh reformulaion ha uses he uiliy funcion o resric he relaion beween he income and ineres rae sequences Define he inverse of u by h 0 and, for y 0, h y infc : u c y 4 Condiion U There is an 0 such ha for any and any 0, for all z, 0 ž ž // ž / P x h R z 1 1 Ý j j j j

14 378 CHAMBERLAIN AND WILSON Suppose he consumpion program were chosen so ha, in any informaion sae, an increase in he presen value of consumpion by one uni generaes an increase in discouned uiliy equal o Then he consumpion level in each informaion sae would be deermined by he relaion c h, and Condiion U would imply a nonrivial probabiliy ha he presen value of curren asses plus fuure income is bounded away from he presen value of curren plus fuure consumpion, saring a any sae z Roughly speaking, he condiion says ha he acual income sream is sochasic relaive o he hypoheical income sream required o make he marginal discouned uiliy of a presen value uni of consumpion equal in all informaion saes When he ineres facor is consan, Condiion U implies Condiion U Our resuls are based on Lemma 4 below Roughly, i says ha if is bounded away from zero and infiniy he long run ineres rae is equal o he discoun rae, hen Condiion U implies ha he marginal uiliy of asses a mus converge o 0 The argumen goes as follows Suppose a does no converge o zero Then we may choose and an arbirarily small inerval b, b wih b 0 such ha i here is posiive probabiliy ha a b, b, and ii if a b, b, hen, wih probabiliy greaer han 1, a b, b for all Now consider he nonnull se of pahs for which a b, b for all Since he firs-order condiions for uiliy maxi- mizaion imply ha c h a, Condiion U implies ha whenever a b, b for all, a mus diverge wih probabiliy a leas Bu if a diverges, Lemma 2 implies ha c mus converge o infiniy, which if is bounded above, implies ha a converges o 0 This conradicion esablishes he resul We proceed now wih he formal analysis To use Theorem 1 we mus firs esablish ha Condiion U implies a nonzero income sream LEMMA 3 Condiion U implies P x 0 0 Proof We show firs ha Condiion U implies ha Px 0 z 1 for all z For his, i is sufficien o show ha for any z we may 1 choose sufficienly large so ha P Ý h R z j j j 1 Since u is concave and bounded beween 0 and M, we have cu c M for all c 0 4 By definiion, u h Therefore, j j j 1 j h M Then R implies Ý h j j j j j 1 j R Ý h M 1 j j j j Seing M 1 esablishes he resul All ha remains is o show ha P x 0 z 1 for all z implies P x 0 0 We esablish his by conradicion Suppose P x 0 0 Le A x 04 and A x 04 Then P A 0 for all 0, and A A Then, since P x 0 z 1 for all z and A is measurable j

15 CONSUMPTION UNDER UNCERTAINTY 379 z, we have P A A PA A j 1 Therefore, here is an in- creasing sequence such ha P A A 1 1 which implies P x 0 Ł P A A P A lim 1 P A LEMMA 4 Suppose here is an 0 for which Condiion U is saisfied 1 and P, 0 1 Then Plim a 0 1 Proof We will suppose he lemma is false and show ha his implies a violaion of Condiion U Our firs sep is o find an inerval, and a ime such ha, conains a wih posiive probabiliy and, in he even ha a,, i is also rue ha a, and c is near h wih probabiliy close o 1 for all Theorem 1 and Lemma 3 imply a random variable e such ha Plim a e 1 If P e 0 1, hen here is a 0 such ha for any 0, Pe, 0 Le Then, since h is uniformly coninuous on any posiive inerval bounded away from zero, we may choose and, 0, so ha i Pe, 0, ii P e P e 0, and iii h h 1 for Define B e, 4, and for 0, define A a, 4 and B c h, a,, 4 Then lim P A P B 0 Also, Lemma 1 im- 1 plies Pc h a 0, 0 1 Therefore, P 1 and he coninuiy of h imply Plim c h 0 B 1 Then, since he monooniciy of h implies Ph he h B 1, we have lim P B P B Consequenly, we may choose such ha P B 1 P A 0 We show nex ha if he marginal uiliy of consumpion cj is always equaed o j, hen he variaion in he presen value of he limiing level of asses is less han 2 on B The monooniciy of h and he consrucion of B imply Pc h, B 1 Therefore, leing h, Lemma 2 implies an such ha P a 1 c, B 1 Then, since P, implies 1 j 2 j P R, j 1, i follows ha Plim a j j 1 c R 0 B 1 Therefore, a c R 1 1 Ý j j j j1 1 1 Ý j j Ý j j ž / ž / j j j j a c x c R ž / ž / a x x h R h c R

16 380 CHAMBERLAIN AND WILSON implies ž Ý ž ž // ž ž / / 1 / P a x x h R 1 j j j j 6 Ý j j j h c R 0 B 1 1 j 2 j Bu from P R, j j j 1 and he definiion of B i follows ha ž ž / j / ž / 1 P Ý h cj R j B j Combining 6 and 7 hen yields ž ž / j / ž / 1 P a x Ý x j h R j B j We now use hese resuls o show ha Condiion U mus be violaed Le x a Since B A and P B 1 P A 2, i fol- 1 lows from 8 ha P Ý x h R A j j j j 1 Bu hen A measurable z implies ha he se of z such ha P 1 Ý x h R z j j j j 1 has posiive probabiliy, which violaes Condiion U We use his lemma again in Secion 5 For he momen, however, i serves as he basis for he main resul of his paper THEOREM 4 Suppose here is an 0 such ha Condiion U is 1 saisfied If P, 0 1, hen P lim c 1 1 Proof If P, 0 1, hen Lemma 4 implies Plim a 0 1 The conclusion hen follows from Lemma 1 Wih sufficien uncerainy in he income and ineres rae sequences, consumpion will grow wihou bound even if he long run rae of ineres is equal o he discoun rae A case in which his equaliy holds has been considered by Bewley 1980c, 1983 in his reamen of he opimum quaniy of money There is an asse wih a fixed nominal reurn facor 1 1 equal o The real reurn is r q q, where he price q 1 of he consumpion good in erms of he asse is uniformly bounded away

17 CONSUMPTION UNDER UNCERTAINTY 381 from zero and infiniy Consequenly, q q saisfies he condiions of 0 Theorem 4 We conclude ha if he uncerainy condiion is saisfied, c converges o infiniy For he case where he ineres rae is consan, Theorem 4 implies he following corollary COROLLARY 2 Suppose r for all 0 and suppose 1 If Condiion U is saisfied, hen Plim c 1 If income is bounded above, hen Condiion U in Corollary 2 can be replaced by he condiion ha he condiional variance of discouned fuure income is uniformly bounded away from zero; ie, here is a 0 j such ha Var Ý jx j z for all z, 0 However, if he income sream is sochasic, bu he condiions of Corollary 2 are no saisfied, here are examples where he limiing level of consumpion is finie wih some posiive probabiliy When conrased wih he oucome in he case of cerainy, Corollary 2 is perhaps a surprisingly srong resul Unforunaely, he line of argumen used in he proof does no provide a very convincing economic explanaion Clearly he sric concaviy of he uiliy funcion mus play a role The resul does no hold if, for insance, u is a linear funcion over a sufficienly large domain and x is bounded Bu o simply aribue he resul o risk aversion on he grounds ha uncerain fuure reurns will cause risk-averse consumers o save more, given any iniial asse level, is no a compleely saisfacory explanaion eiher In fac, i is a bi misleading Firs, ha argumen only explains why expeced accumulaed asses would end o be larger in he limi I does no really explain why consumpion should grow wihou bound Second, over any finie ime horizon, he argumen is no even necessarily correc Suppose, for example, ha x x for 1,,T, where he 1 are idenically and independenly disribued wih E 1 and a com- pac suppor in Suppose, also, ha he consumer s uiliy funcion is quadraic over a sufficienly large domain Then if he consumer has a T-period planning horizon, i can be shown ha his opimal consumpion program is o se c x for all In paricular, mean-preserving spreads of fuure income leave curren consumpion unaffeced Moreover, he expeced value of consumpion in any period is jus equal o period 0 income So here is no endency a all for consumpion o rise over ime Why hen does he resul change when we consider he limiing value of consumpion in an infinie horizon problem? Alhough we have developed oher argumens o esablish our resuls, any explanaion ha we have been able o devise ulimaely appeals in an essenial way o he maringale convergence heorems

18 382 CHAMBERLAIN AND WILSON 44 Limi Theorems When r Is Saionary In his subsecion, we assume ha he sequence of informaion saes is generaed by a saionary process and esablish some resricions on he disribuion of he single period ineres rae ha imply he condiions of Theorem 2 Since he saionariy resricion will be placed on he enire disribuion of hisories, we will be explici abou condiioning on z 0 7 THEOREM 5 Suppose P x 0 0 If r, r, 1 2 is saionary and ergodic and Elog r 0, hen Plim c z wih probabiliy 1 Proof Le Elog r 1 The Ergodic Theorem implies ha wih 1 probabiliy 1, lim Ý log r Doob, 1953, p 465 j1 j Therefore, for almos all z here is a z such ha Ý log r for j1 j 2 z,or 2 equivalenly, e Therefore, P lim 1, which implies 0 ha z : Plim z 0 14 has probabiliy 1 The desired resul hen follows from Theorem 2 For he case where Elog r 1 0, we need some addiional regulariy condiions For s, define r s,,r as he -field generaed by r,,r, and define r, r, s 1 as he sigma field generaed by r, r 1, Consider a nonnegaive funcion of he posiive inegers The sequence r, r, is -mixing if for each, j 1, A r,,r and A r, r, ogeher imply ha P A A 2 j j1 1 2 P A P A j P A Condiion R i r, r, 1 2 is a saionary, -mixing sochasic pro- cess wih Ý j 12 ii E log r 2 and j E log r1 2 Ý E log r1 log rj 0 j2 The firs par of Condiion R ensures ha he dependence beween r and r dies ou sufficienly fas as j Suppose par j i is saisfied and E log r 0 Then Ý E log r log r 1 j2 1 j converges absoluely and 2 1 T 2 lim E Ý log r T T j1 j Billingsley, 1968, Lemmaa 1 and 3, pp 170, 172 In his case, par ii may be saisfied as well if here is sufficien variabiliy in R However, if P r 1 1, par ii 0T 1 is no saisfied, and we know from Theorem 4 ha he behavior of c may depend upon 7 Theorem A1 esablishes he exisence of an opimal program only for a fixed z Because we did no resric z o be drawn from a compac se, we are able o show ha c z is a measurable funcion of z only for a fixed ail eg, a given z 0 Wihou addiional resricions we are unable o prove ha c is a measurable funcion if all componens of z are allowed o vary

19 CONSUMPTION UNDER UNCERTAINTY 383 wheher or no x is sochasic Neverheless, our nex heorem saes ha if Condiion R is saisfied and Elog r 1 0, hen supc, wihou a requiremen ha x be sochasic THEOREM 6 Suppose P x 0 0 and r, r, 1 2 saisfies Condiion R Then Elog r 0 implies Psup c z wih probabiliy 1 Proof Define S Ý log r log and define Y j1 j S 2 12, where 0 1 and is he greaes ineger less han or equal o Then he funcional cenral limi heorem implies ha he disribuion of he random funcion Y converges weakly o Weiner mea- sure Billingsley, 1968, Theorem 201, from which i follows ha ž / u 2 j 2 lim P max S 22 e du 9 j 0 for 0 Billingsley, 1968, p 138, and Eq 1018, p 72 Define P sup S 1 We shall assume ha 0 and obain a 12 2 conradicion Choose 0 so ha 2 2 H exp u 2 0 du 2 and, 2 12 for 1, define A max S 4 Then lim P A j j 2 For 1, define B 2 12 max S for all 4 j j Then here is a B such ha sup S 4 B and B B Therefore, using Eq 1 9, B A implies P A P B P B, which conradics lim P A We conclude ha P sup S 2 1 0, or equiva- lenly, P sup 1 1 The desired conclusion hen follows from Theorem 1 and Lemma 1 H 5 THE BUDGET CONSTRAINT In his secion, we address wo quesions Firs, given an arbirary sochasic income sequence, under wha condiions should we expec he borrowing consrain o be binding a some informaion sae? Second, how does a change in he borrowing consrain or equivalenly, a change in he income sequence affec he paern of borrowing? 51 When Is Budge Consrain Someimes Binding? As noed in Secion 2, if he discouned value of he income sequence is finie condiional on any informaion sae, we can represen any ineremporal budge consrain as a sequence of one period borrowing consrains I may well urn ou ha none of hese consrains are acually binding a he opimum, and ye he consumer is sill consrained o choose a

20 384 CHAMBERLAIN AND WILSON consumpion program whose presen value does no exceed he presen value of he income sream Our nex heorem gives condiions under which his is he case THEOREM 7 If E r u x z 1 1 and a 0, hen c a Proof Suppose c a Then Pc x z Lemma 1 hen implies u c a E r a z E r u c z E r u x z, a conradicion Since a x, he following corollary follows immediaely COROLLARY 3 If E r u x z and x 0, hen c a 1 1 Theorem 7 and is corollary say ha whenever he expeced incremen in disposable income in he following period is sufficienly small so ha he expeced marginal uiliy from consuming ou of ha incremen would be infinie, he consumer chooses o consume less han his curren wealh in he curren period in order o pass some of his wealh o he nex period If we allow he expeced marginal uiliy of fuure income o be finie, however, he borrowing consrain may well be binding, a leas occasionally This will obviously be he case if income received in each period is growing a a sufficienly high rae over ime so ha he consumer wans o ransfer fuure income o presen consumpion Bu if he income sream is suiably sochasic, a much weaker se of condiions guaranees ha he budge consrain is someimes binding Our nex heorem may be summarized as follows Suppose he single period ineres rae never exceeds he discoun rae and he marginal uiliy of consuming from curren income alone is bounded above Then if he long run rae of ineres is less han he discoun rae or if he income sequence is suiably sochasic, here is a posiive probabiliy ha he borrowing consrain is binding in a leas one period THEOREM 8 Suppose P r 1, 1 1 and Pu x, 0 1 for some If eiher i P lim 0 1 or ii here is an 0 for which Condiion U is saisfied and P, 0 1, hen Pc a, 0 1 Proof We esablish he heorem by conradicion Suppose P0 c a, 0 1 Then, by inducion, Lemma 1 implies a 0 0 E r a E a for all 1 We will esablish ha 1 1 1

21 CONSUMPTION UNDER UNCERTAINTY 385 lim E a 0 and herefore, a 0 0 0, violaing he mono- oniciy and concaviy of 0 To show ha lim E a 0, we firs show P a 1 1 for some, and hen show Plim a 0 1 Since P 1 r 1, 0 1 implies P 0 1, 1 1 1, he desired re- sul hen follows from he Dominaed Convergence Theorem To esablish P a 1, noe firs ha Pu x, implies eiher u 0 or P x, 0 1 for some 0 If u 0, hen Lemma A3 and he sric concaviy of imply P a 0 u 0 1, in which case we le If P 1 x, 0 1, hen Pa, 0 1, and so Lemma 1 implies P a u, 0 1, in which case we le u 1 To esablish Plim a 0 1, we consider Condiions i and ii separaely If Condiion i holds, hen he desired propery follows immediaely If Condiion ii holds, hen Lemmaa 1 and 4 imply ž / 1 P lim u c lim a lim a 0 1, and so Plim c 1 Anoher applicaion of Lemma 1 hen yields ž / P lim a lim u c lim u c 0 1 If he long run rae of ineres is less han he discoun rae, he conclusion of Theorem 8 is no paricularly surprising In his case, he marginal reurn o consuming a fixed level of consumpion goes o zero wih ime Consequenly, if he marginal uiliy o consuming curren income is bounded, hen he borrowing consrain mus evenually be binding The heorem is less inuiive for he case where he long run rae of ineres is equal o he discoun rae 1 for a fixed ineres facor In his case Theorem 4 implies ha consumpion grows wihou bound Evidenly, along almos every pah, he asse level firs falls o is lower bound a leas once before converging o infiniy However, Theorem 4 implies ha his happens only a finie number of imes 52 Comparaie Dynamics Suppose he original sequence of borrowing consrains is replaced wih anoher sequence which allows a leas as much borrowing in any informaion sae The following heorem saes ha a any informaion sae he opimal accumulaed wealh under he new sequence of borrowing con-

22 386 CHAMBERLAIN AND WILSON srains is no higher han he opimal accumulaed wealh under he original sequence of borrowing consrains THEOREM 9 Le a 0 and a 1 be he sequence of accumulaed asses associaed wih he soluions c 0 and c 1 corresponding o consrains k 0 and k, respeciely Then P k k, 0 1 implies Pa 0 a 1, 0 1 Proof Le S and S be he meric spaces defined in he Appendix 0 1 corresponding o k 0 and k 1, respecively, and le T and T be he 0 1 corresponding T operaors Noe ha S S so ha T is defined on S Le 0 and 1 be he corresponding fixed poins of T0 and T 1 We shall 0 1 show ha a, z a, z for all a, z S and 0 Le f 1 represen 1 resriced o S0 and le f n1 T0 n f 1, where T0 n is he operaor T applied n imes For a, z S, define C1 a, z arg max u c E a c r1 x 1, z z, 1 0cak z and for n 2 define Cn a, z 4 n1 1 arg max u c E f a c r1 x 1, z z 0 0cak z Fix a, z S and 0 Le c C a, z and c C a, z Then eiher c1 c2 0 or a c1 a c 2 Therefore, using he sandard envelope argumens exploied in Lemma 1, he sric concaviy of u and implies ha eiher or 1 a, z u c1 u c2 f 2 a, z 10 a, z E r a c r x, z z E r1 a c2 r1 x 1, z z f a, z 1 2 In eiher case, we have shown ha a, z f a, z for any a, z, a, z S 0, a a n n1 Now suppose f a, z f a, z for any a, z, a, z S 0 wih a a Fix a, z S and 0 and define c C a, z 0 n n and c C a, z Then eiher c c 0orac a c n1 n1 n n1 n n1 0

23 CONSUMPTION UNDER UNCERTAINTY 387 Then he envelope argumen implies eiher or f n a, z u c u c f n1 a, z 11 n f a, z E r f a c r x, z z n n1 1 1 n 1 1 n1 E r f a c r x, z z n 1 1 n f n1 a, z n n1 In eiher case, we have shown ha f a, z f a, z for any a, z, a, z S 0, a a Now le n be a sricly decreasing sequence such ha 1 and 1 lim Then for n 2, we have jus shown ha a, z n n 2 1 n1 n f a, z f a, z f a, z 2 n 2 for all 0 0 a, z S 0 Since T0 is a conracion mapping and T0, i follows ha f n a, z 0 a, z as n Therefore, Lemma A4 implies 0 0 n 1 a, z a, z lim sup f a, z a, z 2 n 2 for any a, z S0 and 0 To show ha a 1 a 0, we again argue by inducion Fix any realizaion of z, z, z, z, By definiion, a 0 z 0 a 1 z 0 Suppose a 0 z a z 0 We will show ha a z a z Noe firs, ha for any w N, a 0 z, w a 1 z, w 1 1 r z, w a 0 z a 1 z c 0 z c 1 z Therefore, a z a z implies c z c z and a z, w N a z, w for all w Lemma 1 hen implies 1 E r z, w a z, w, z, w z u c 1 z u c 0 z E r1 z, w a1 z, w, z, w z 0 1 Bu we have jus shown above ha a z, w a z, w 1 1 implies a z, w, z, w a z, w, z, w 1 1 This conradicion com- plees he proof

24 388 CHAMBERLAIN AND WILSON I should be clear from our discussion in Secion 2 ha for any change in he se of feasible consumpion programs in response o a change in he borrowing consrain, here is an equivalen change in he income sequence ha generaes he same change in he se of feasible consumpion programs Consequenly, we may reinerpre Theorem 9 as a saemen abou how he opimal consumpion program changes in response o cerain kinds of changes in he income sequence Our nex corollary saes ha if he income sequence changes so ha a any sae z he discouned value of income up o ime has no decreased, hen he discouned value of he consumpion up o ime also does no decrease for any sae z COROLLARY 4 Le c 0 and c 1 be he opimal consumpion programs associaed wih borrowing sequence k and income sequences x 0 and x 1, respeciely Then ž Ý j j 0 j / j ž Ý j j 0 j / j0 P x x R 0, 0 1 implies P c c R 0, Proof Le k k, x x, and define k k 1, k 0, k 1, by k and k Ý j0 x j xj Rj k for 0 By assumpion, x and k joinly saisfy 1 Therefore, x k rk 1 x Ý j0 x j xj Rj k Ý j0 x j xj Rj rk 1 x k rk 1 implies ha x and k do as well Furhermore, he definiions of c 0 and c 1 imply ha hey are also he opimal consumpion programs associaed wih he common income sequence, x, and respecive borrowing consrains, k 0 and k 1 To see his, noe ha for any consumpion program c, we have Ý x 1 c j0 j j Rj k 1 1 Ý j0 x j cj Rj k Therefore, a program c is feasible for x and k if and only if i is feasible for x and k 1 By assumpion c 1 is opimal for x 1 and k Therefore, i is also opimal for x and k 1 Now le a 0 and a 1 denoe he asse sequences associaed wih consumpion programs c 0 and c 1 for income sequence x Then, for i 0, 1, a0 i x0 i 1 and a Ý x c i j0 j j Rj x for 1 The hypohesis of he corollary and he consrucion of k 1 imply k 1 k 0 Therefore, Theorem 9 implies a a Ý c c R R Ý c c R for all 1 j0 j j j 0 j0 j j 0 j 6 MANY ASSETS In his secion, we illusrae how he argumen behind Theorem 2 can be exended o he case where here are several risky asses We show ha o apply he maringale convergence heorem, i is no necessary o acually

25 CONSUMPTION UNDER UNCERTAINTY 389 solve he porfolio problem of he consumer So long as he consumer is able o choose a sequence of marginal ineres raes which, in he long run, exceed he discoun rae, consumpion mus converge o infiniy Suppose he consumer mus choose a porfolio of J risky asses in each period 0 If he invesmen in period in he ih asse is b i measured in erms of period consumpion, hen he payoff a 1is bg i i, 1, 1 where g is a nonnegaive, coninuous funcion from Z z 0 i, 1 o for i 1,, J, and 0 If b b,,b 1 J is he porfolio chosen in period, hen he muli-asse analogue of he budge consrain is a consrain on shor sales which we represen by he requiremen ha b K z We assume ha K is a coninuous correspondence from 0 J J Z z o and ha K z is a nonempy, closed, convex subse of which is bounded from below; ie, here is a funcion m: Z z 0 such J ha K z,, : mz 4 1 J j We also assume ha J K z is comprehensive; ie, if 1 K z and 2, hen 1 2 K z This guaranees ha i is always feasible for he consumer o increase his holding of any asse For any, J, le Ý J j1 j j Le denoe he J-vecor of ones, and define k z infb: b K z 4 Inerpre k as he mini- mum amoun of wealh measured in erms of period consumpion ha he individual may hold a he end of period afer receiving period income and spending period consumpion Le g g,, g 1 J We require ha he curren bounds on shor sales be consisen wih he borrowing consrains he individual will face in he fuure: Pbg1 x1 k1 z 1 for all b K z and all z 0 For any sae z, aconsumpion-porfolio program, c, b, is a sequence 0 0 J of a pair of funcions, c : Z z and b : Z z, 0 For 0 any z Z z and any a k z, le K a, z c, b K z :0cab4 denoe he se of feasible consumpion-porfolio pairs for he consumer, and define j Ý j c, b j a, z max E u c z ž / j subjec o P c j, bj Kj a j, z, j z 1, where a a and aj bj1 gj x j for j Theorem A1 implies ha his problem has a soluion and ha a, z is sricly concave in a k Therefore, righ- and lef-hand derivaives are well defined for a k We le k, z lim a, z a k As in Secion 2, le a a, z, and le c*, b* denoe he soluion o he consumer s problem saring in sae z 0 wih wealh a x 0We

26 390 CHAMBERLAIN AND WILSON shall simplify he maringale argumens by assuming ha x0 k 0 The following condiions are hen implied by an opimal program They are esablished by he same sandard argumens used o esablish Lemma 1 LEMMA 5 i a E g a z k, for k 1,, J, and ii a u c To obain he analogues of Theorems 1 and 2 from Lemma 5, we consider an arbirary marginal porfolio, ie, a porfolio which can be added o he exising porfolio Given our assumpions on K, he only porfolios we migh no be able o consider are hose involving shor sales For his reason, we resric aenion o hose porfolios wihou shor sales We say ha b, b, 0 1 is a posiie porfolio program if each b is a 0 J measurable funcion from Z z ino 0 4 For any posiive porfolio 1 program, we define he sequence of reurn facors by r z 1 1 b z g z bz for 0 1 The nex heorem provides a condiion ha, if saisfied by he reurn sequence generaed by a leas one posiive porfolio, implies ha opimal consumpion converges o infiniy Recall ha R Ł r and 0 j1 j R 0 THEOREM 10 Suppose ha r, r, 1 2 is he reurn sequence for some posiie porfolio program Then, i P lim 1 implies Plim c 1, and ii P sup 1 implies P 0 sup 0 c 1 Proof i Define d 1 and d a a Lemma 5 implies ha a Er a z from which i follows ha d is a nonnegaive supermaringale Therefore, here is a random variable d wih Ed 1 and P lim d d 1 Doob, 1953, p 324 So if, i follows ha a 0, and herefore, from Lemma 5 ha u c 0, from which we conclude ha Plim c 1 ii If along a subsequence, hen, along ha subsequence, a 0 which implies u c 0 and herefore, Psup c 1 We emphasize ha here need be no relaion beween he opimal porfolio and he posiive porfolio program ha generaes he sequence of ineres raes used in Theorem 10 APPENDIX We esablish he exisence of an opimal soluion for he problem presened in Secion 6 where he consumer mus choose an opimal porfolio and consumpion program in each period Recall ha z n