optimal intertemporal consumption under uncertainty · 366 chamberlain and wilson assets at the...

Ž .Review of Economic Dynamics 3, 365!395 2000doi:10.1006!redy.2000.0098, available online at http:!!www.idealibrary.com on

Optimal Intertemporal Consumptionunder Uncertainty1

Gary Chamberlain

Department of Economics, Har"ard Uni"ersity, Cambridge, Massachusetts 02138

and

Charles A. Wilson2

Department of Economics, New York Uni"ersity, 269 Mercer Street, New York,New York 10003

Received November 30, 1999

We analyze the optimal consumption program of an infinitely lived consumerwho maximizes the discounted sum of utilities subject to a sequence of budgetconstraints where both the interest rate and his income are stochastic. We showthat if the income and interest rate processes are sufficiently stochastic and thelong run average rate of interest is greater than or equal to the discount rate, thenconsumption eventually grows without bound with probability one. We also estab-lish conditions under which the borrowing constraints must be binding and exam-ine how the income process affects the optimal consumption program. Journal ofEconomic Literature Classification Number: D91. ! 2000 Academic Press

Key Words: uncertainty; consumption; permanent income hypothesis.

1. INTRODUCTION

We shall consider the following problem. In each period t, a consumerreceives income x . After receiving his income, he must decide how muchtto consume in that period, c , and how much to save for future consump-ttion. His savings earn a gross rate of return, r , so that the value of hist#1

1Support from the National Science Foundation and the Alfred P. Sloan Foundation isgratefully acknowledged. We have benefitted from discussions with William Brock, Ed Green,Jose Scheinkman, and Itzak Zilcha.´

2 To whom correspondence should be addressed. E-mail: [email protected].

3651094-2025!00 $35.00

Copyright ! 2000 by Academic PressAll rights of reproduction in any form reserved.

CHAMBERLAIN AND WILSON366

Ž .assets at the beginning of period t # 1 after he has received income xt#1Ž .is given by the relation a $ r a % c # x . The consumer’s prob-t#1 t#1 t t t#1

lem is to choose a consumption plan to maximize the expectation of" Ž . tÝ u c # subject to a budget constraint, where u is an increasing,t$ 0 t

strictly concave function. His only source of uncertainty is that x and rt tfollow some stochastic process. The question we address is: What happensto the levels of c and a as t goes to infinity?t t

The motivation of the paper lies in the work of several authors whoŽ .attempt to formalize the permanent income hypothesis of Friedman 1957 .

Ž . Ž .The first of these papers, Yaari 1976 and Schechtman 1976 , considersŽ .the case where utility is not discounted # $ 1 , the interest rate is zero

Ž . T Ž .r $ 1 , and the horizon is finite; i.e., the consumer maximizes Ý u c .t t$ 0 tAssuming that income x is identically and independently distributed andtimposing only a solvency constraint on lifetime consumption, Yaari showsthat as T ! ", the optimal consumption plan requires cT, the consumptiont

& 'in period t with horizon T , to converge with probability one to E x fortall t. Schechtman tightens the solvency constraint to forbid borrowing andobtains the weaker result that, as both T and t go to infinity, cT convergest

& 'to E x with probability one. He also establishes that a converges tot tŽ .infinity. Bewley 1977 retains the restriction on borrowing but considers a

more general case where both x and u follow a stationary stochastict tŽ .process. He also allows for discounting # ! 1 . He obtains the analogous

" Ž T .result that u c converges to a constant as T ! ", t ! ", and # ! 1, sot tthat, asymptotically, the consumer becomes insulated from risk. Some of

Ž .the implications of this result are developed in Bewley 1980a, 1980b .In this paper we investigate the asymptotic properties of the optimal

consumption program for the infinite horizon model when income andinterest rates are stochastic and the consumer discounts a bounded utilityfunction. We put no additional restrictions on the form of the stochastic

Ž .process such as stationarity , and we allow for arbitrary restrictions on thepermissible level of borrowing in each period. Under these assumptions,we show that if the discount rate is smaller than the long run average rateof interest, then c will converge to infinity almost surely. When the longtrun interest rate is equal to the discount rate, the asymptotic properties ofc depend on how stochastic the income stream is. If the income stream istcertain, c is nondecreasing and converges to the supremum of thetmaximum sustainable consumption level starting from the minimum per-missible level of wealth in any given period. However, if the income streamis suitably stochastic, c must converge to infinity almost surely. This resultt

Ž .extends some of the results of Sotomayor 1984 , who examines the caseŽwhere income is identically and independently distributed but allows for

.utility to be unbounded .

CONSUMPTION UNDER UNCERTAINTY 367

Our analysis is based on the following observation. For simplicity,Ž t.suppose that the interest factor is a constant, $ . Let " a , z be thet

expected discounted utility from following the optimal policy, given assetlevel a and information state z t at time t. Then the first order conditionst

#Ž t. & #Ž t#1. # t 'for utility maximization require " a , z ( #$ E " a , z z , wheret t#1"# denotes the right-hand derivative of " with respect to a . Whent

#Ž t.#$ ( 1, this implies that " a , z is a supermartingale. By a theorem oftŽ . #Ž t.Doob 1953 , this implies a finite random variable d such that " a , z" t

#Ž t.converges to d with probability 1. If #$ $ 1, " a , z ( #$" t& #Ž t#1. # t ' #Ž t.E " a , z z implies " a , z ! 0 from which it is easy to showt#1 t

that c must converge to infinity. If #$ $ 1, this argument can still be usedtto show that c converges with probability 1 to some extended randomtvariable c , but in this case, c need not equal infinity. In fact, as we show" "

in Theorem 3, if x is not stochastic, c is generally not infinite. However,t "

the following argument shows that c must be infinite if the discounted"

value of future income is suitably stochastic.Suppose there is positive probability that c ! ". Then we can choose"

& 'an arbitrarily small interval b, b # % and a & ! " such that there is& 'positive probability that c ) b, b # % , and, with probability greater than&

& ' & '1 % % , c ) b, b # % for all t ( & whenever c ) b, b # % . But ift &Ž Ž ..c * c for t ( & , then a will diverge to %" unless $! $ % 1 c $ a #& t & &

Ý" x $ &%j. We can show that it is not optimal to have c * c if a ! ".j$ &#1 j & t" &%j Ž Ž ..So conditional on c * b, it follows that Ý x $ * $! $ % 1 b %& j$ &#1 j

a . But since a is known at time & , this implies that the conditional& &

variance of Ý" x $ &%j can be made arbitrarily small for % sufficientlyj$ &#1 jsmall. Therefore, unless there is perfect foresight, savings must diverge.We conclude that if the income stream is suitably stochastic, then ctconverges to infinity with probability 1. In the body of the paper, thesearguments are generalized to allow for a stochastic interest rate. Inparticular, if the discount rate is equal to the long run average rate ofinterest, then c grows without bound if there is sufficient uncertainty intthe joint distribution of income and interest rates. This case arises in the

Ž .treatment of the optimum quantity of money by Bewley 1980c, 1983 .We see that with discounting and a positive interest rate, the only

counterpart to the Yaari!Schechtman!Bewley result is a zero limit formarginal utility. If #$ ( 1, then consumption does converge, but it con-verges to bliss, not the expected value of discounted income.3 We shouldalso note that although both Schechtman and Bewley use the convergencetheorem for supermartingales, their argument requires either the law of

3 Ž .An example of Schechtman and Escudero 1977 shows that convergence to bliss withprobability 1 can occur when #$ ! 1 and income is independent and identically distributed.They also provide a condition on the utility function that rules this out.


large numbers or the ergodic theorem, which in turn requires a stationarityassumption. By itself, however, the supermartingale theorem requires noadditional assumptions. It is the stochastic generalization of the result thata bounded, monotone sequence converges to a finite limit. Consequently,when we eliminate the need for a law of large numbers by introducingdiscounting, we are able to considerably weaken our assumptions on thestochastic process and still obtain convergence of c , even with a positivetinterest rate.

If the underlying process is stationary, there are alternative conditionson the interest rate sequence which ensure that consumption converges toinfinity. In Subsection 4.4, we show that if the interest rate sequence isstationary and ergodic, then consumption will grow without bound if

& ' & 'E log # r $ 0. If E log # r $ 0 and an additional condition is satisfiedt twhich guarantees that the interest rate has sufficient variance and itsdependence on the past dies out sufficiently fast, then some subsequenceof consumption will grow without bound. In both cases, the result follows

t t Ž .from the fact that # Ł r or some subsequence converges to infinityj$ 1 jwith probability 1.

Section 5 deals with some implications of the budget constraint. Ratherthan restrict ourselves to a no borrowing constraint at the outset, we allowthe lower bound on borrowing in any period to be an arbitrary function ofthe consumer’s information in that period. We then note that by redefin-ing income and wealth, the original problem is equivalent to a problemwhere income is nonnegative in each period and no borrowing is permit-ted. Making this translation explicit emphasizes that these assumptions arenot really substantive restrictions on the model, but are merely a simplerrepresentation of a more general model. In particular, they are consistentwith the possibility that the only constraint on borrowing is an intertempo-ral budget constraint. With this interpretation, we establish some condi-tions under which the borrowing constraint is never binding as well asconditions under which it must be binding in some periods. We also showhow the optimal program changes as the borrowing constraints are re-laxed.

In the final section of the paper, we show how some of our results canbe extended to the case where the consumer chooses a portfolio of severalrisky assets. The main complication introduced by this extension is that theinterest rate becomes endogenous.

2. ASSUMPTIONS AND NOTATION

! is the real line, and ! n is n-dimensional Euclidean space. ! and ! n# #

refer to the corresponding subsets of nonnegative elements. Nonnegativeintegers are denoted by t, & , i, j. Other lower-case Roman letters generally


refer to random variables or their realizations, and lower-case GreekŽ .letters particularly, % , ', and ( generally refer to real numbers.

Ž .Let z & . . . , z , z , z , . . . be a stochastic process with transition%1 0 1Ž # t%1.probabilities p dz z used to define conditional expectation, wheret

n t Ž t%1 . Ž .z ) ! and z & z , z & . . . z , z , z , . . . , z . Interpret z as newt t %1 0 1 t tinformation the consumer receives at time t and z t as the information

tŽ 0. $ t t 0 4state at time t. Let Z z $ z : z $ z , j + 0 be the set of states atj j

time t which are consistent with state z 0. Except for Section 4.4 where weconsider the case where z is stationary, we will take z 0 as fixed.

We assume that the income of the consumer at time t ( 0 is a continu-tŽ 0. 4ous function x : Z z ! !. At each time t ( 1, the consumer mayt

borrow and lend between time t % 1 and time t at interest factor r whichttŽ 0.is assumed to be a positive, continuous function r : Z z ! ! . Fort #

t ( 0, let R & 1. For j $ t ( 0, let R & Ł j r denote the interestt t t j k$ t#1 kfactor between periods t and j, and let R%1 denote its inverse.t j

The borrowing constraint of the consumer at time t ( 0 is a continuoustŽ .function, k : Z z ! ! that satisfies an intertemporal consistency con-t 0

straint,

# 0P r k # x ( k , t ( 0 z $ 1, 1Ž .Ž .t t%1 t t

Ž . Ž .where r k & 0. Let x & x , x , . . . and k & k , k , . . . . Interpret k as0 %1 0 1 0 1 tthe minimum amount of wealth, measured in terms of period t consump-

Žtion, that the individual may hold at the end of period t after he receives. Ž .his period t income and spends his period t consumption . Equation 1

requires that the lower bound on current net wealth be consistent with theborrowing constraint the consumer will face in the next period. It impliesthat current wealth can never be so low that it may become impossible forthe individual to satisfy his borrowing constraint in the next period even ifnothing is consumed in the current period. Defining r k & 0 is simply a0 %1convention which implies that the consumer’s initial wealth is derivedsolely from his period 0 income.

Since the consumer’s decision at each information state depends only onŽ .what he knows at that state, a consumption program, c & c , c , . . . , is a0 1

tŽ 0.sequence of Borel measurable functions, c : Z z ! ! , t ( 0. Let u:t #! ! ! be an increasing, continuous, strictly concave function with 0 + u#

Ž . Ž .+ M & sup u c ! ". For any consumption program c, u c is thec!" tundiscounted utility to the consumer from his consumption at time t. We

Ž .suppose the consumer discounts utility by a factor # 0 ! # ! 1 in eachperiod. Therefore, at state z 0, the problem of the consumer is to choose a

4 The continuity requirement is without loss of generality since we may always include avariable as one of the coordinates of z .t


& " Ž . j # 0 'consumption program c to maximize E Ý u c # z subject toj$ 0 jŽ t Ž . # 0.P Ý x % c R ( k , t ( 0 z $ 1.j$ 0 j j jt tSome of our results require some regularity assumptions on the transi-

tion probabilities. We assume that for any z 0 the Feller property holds: ifn Ž t. Ž . Ž # t.f : ! ! ! is a bounded continuous function, then g z $ H f w p dw z

tŽ 0.is a continuous function from Z z to !. This condition combined withthe requirement that x , r , and k be continuous functions is shown int t t

Ž .Theorem A.1 of the Appendix to guarantee that the consumer’s problemhas a solution.

Before proceeding with our analysis, it will be useful to redefine ourincome variable so that the problem may be reduced to a simpler form. Asstated, the problem of the consumer is to choose a consumption program

Ž .that maximizes the discounted sum of utility subject to a possibly randomborrowing constraint. Aside from technical considerations, there are norestrictions on either the income stream or the borrowing constraint other

Ž .than the requirement that the borrowing constraint satisfy Eq. 1 . Forinstance, if we assume that x is nonnegative and we wish to prohibittborrowing, we may set k $ 0 for all t. Alternatively, if we wish to imposetonly an intertemporal solvency constraint, the appropriate constraint isŽ t. $ Ž " %1 # t. 4 5k z & inf ) ) !: P ) # Ý x R ( 0 z $ 1 . However, we alsot j$ t#1 j t j

allow for intermediate cases as well as cases where lim k R%1 ' 0.t !" t 0 tWhatever additional assumptions we impose on x and k , however, wet t

may always translate the variables so that the problem is equivalent to onein which income is nonnegative and borrowing is not permitted. The key isto redefine income at time t to be the increase in available purchasingpower the consumer receives at time t. This is the difference between x ,tthe income he receives in period t, and k % r k , the change in thet t t%1minimum allowable wealth from period t % 1 to period t, measured in

ˆunits of income at time t. Letting x & x % k # r k and letting k & 0t̂ t t t t%1 tŽ . Ž # 0.for all t ( 0, Eq. 1 may be stated as P x ( 0, t ( 0 z $ 1, and thet̂

Ž t Ž . %1 # 0.borrowing constraint may be stated as P Ý x % c R ( 0, t ( 0 zˆj$ 0 j j 0 j$ 1. Note that since x , k , and k are all continuous functions of z t, x̂t t t%1 tis also a continuous function of z t. Consequently, the technical require-

ˆments and information restrictions on x and k are satisfied. Unless wet̂ tindicate otherwise, we shall assume for the remainder of the paper thatthis translation has already been made so that we may restrict attention tothe case x ( 0 and k $ 0. However, it is occasionally useful to emphasizet tthe interpretation of our results in the original framework.

5To ensure the existence of some feasible consumption program consistent with k $ 0,%1Ž " %1 # 0 .we then require P Ý x R ( 0 z $ 1, and to ensure that %k ! " so that an optimalj$ 0 j 0 j t

Ž " %1 # t . tconsumption program exists, we then require P Ý x R ! " z $ 0 for all z .j$ t j t j


To establish the results that follow, it is convenient to work with thevalue function of future discounted utility defined over the level ofaccumulated assets. For any state z t and wealth level a ( 0, define

"t j%t t#" a, z & max E u c # zŽ . Ž .Ý j

c j$ t

subject to

&%1 t#P a % c # x % c R ( 0, & ( t z $ 1Ž .Ýt j j t jž /

j$ t#1

to denote the expected utility of the consumer starting at time t withinformation z t, given purchasing power a at time t. Following the argu-

Ž . Ž . Ž .ments of Blackwell 1965 , Strauch 1966 , and Maitra 1968 , we establishŽ t. $Ž .as Theorem A.1 in the Appendix that " a, z $ max u c #0 + c+ a

& ŽŽ . t#1. # t '4 6#E " a % c r # x , z z .t#1 t#1With the exception of Subsection 4.4, which uses the stationary assump-

tions, we shall assume the initial information state z 0 is fixed so thatunconditional expectations should be understood to be conditional on z 0.It is also frequently convenient to suppress explicit reference to the state

t t Ž t. Ž .z , so that for a given state z , we sometimes write " a, z & " a andtŽ .regard " a as a random variable. Also, to simplify the statement of somet

of the theorems, we sometimes state properties of the optimal programas if they hold for all z t, although we prove our results on a set of prob-ability 1.

3. PRELIMINARY RESULTS

The existence and uniqueness of a solution to the consumer’s maximiza-tion problem is established in the Appendix as Theorem A.1. We alsoestablish there that " is a bounded, strictly concave function. Therefore,t

& 'right- and left-hand derivatives exist. For any function f : 0, " ! !, let#Ž . Ž . %Ž .f x denote the right-hand derivative of f x , let f x denote its

#Ž . #Ž .left-hand derivative when x $ 0, and let f 0 $ lim f x . Let c* $x " 0Ž ( ( .c , c , . . . denote the consumption program which solves the consumer’s0 1

6 The result is actually established for the more general model with many assets discussedin Section 6.


Ž ( ( .maximization problem, and let a* & a , a , . . . denote the corresponding0 1sequence of random variables representing the level of the consumer’swealth in each period after current income has been received but beforeconsumption has been spent. The wealth sequence a* is defined recur-

( ( Ž ( ( . t%1 Ž .sively by a & x and a & r a % c # x $ x # Ý x % c R ,0 0 t t t%1 t%1 t t j$ 0 j j jtfor t $ 0.

With these definitions, we may state some conditions which an optimalprogram must satisfy. The proof is quite standard and will be omitted.

Ž . #Ž ( . $#Ž ( . & # Ž ( . # t '4 Ž .LEMMA 1. a " a $ max u c , #E r " a z . b Sup-t t t t#1 t#1 t#1pose a( $ 0. Thent

"% a(Ž .t t

, % ( ( (u c if a % c $ 0Ž .t t t( (% t- ##E r " a z if c $ 0Ž .$ t#1 t#1 t#1 t

( ( ( ( (% % t. #min u c , #E r " a z if 0 ! a % c ! a .Ž . Ž .$ 4t t#1 t#1 t#1 t t t

For t ( 0, let * & # tR . If the level of consumption in period 0 ist 0 tequal to the level of consumption in period t, then * measures the rate attwhich the consumer can transform discounted utility in period t for utility

Ž . #Ž ( .in period 0. Condition a establishes that * " a is a supermartingale.t t tOur first theorem uses the supermartingale convergence theorem to

#Ž ( .establish that * " a must converge to some random variable e . Thet t t "#Ž ( .only problem is that to apply the convergence theorem, " a must bet t

#Ž .finite which is not true if x $ 0 and " 0 $ ". This difficulty can be0 0avoided if we assume that the discounted value of the entire incomestream is always positive. In this case, there is a stopping time & such thata( $ 0, which allows us to apply the convergence theorem to the sequence&Ž #Ž ( ..%1 # Ž ( .* " a * " a .& & & &#t &#t &#t

Ž . Ž .THEOREM 1. If P x $ 0 $ 0, then there is a real-"alued randomŽ #Ž ( . ."ariable e such that P lim * " a $ e $ 1." t !" t t t "

Ž .Proof. P x $ 0 $ 0 implies a stopping time & such that x $ 0. There-&( #Ž ( .fore, a ( x $ 0 and the concavity of " imply that " a is finite. Let& & & t t

Ž #Ž ( ..%1 # Ž ( .d & * " a * " a for t ( 0. Since & is a stopping time,t & & & &#t &#t &#tŽ . ŽLemma 1 implies that d , d , . . . is a nonnegative supermartingale Meyer,0 1

.1966, p. 66 . But since d $ 1, there is a random variable d with0 "


& ' Ž . Ž .E d + 1 such that P lim d $ d $ 1 Doob, 1953, p. 324 . Then if" t !" t "#Ž ( . Ž #Ž ( . .we define e & * " a d , it follows that P lim * " a $ e $ 1." & & & " t !" t t t "

In the next section, we use Theorem 1 to establish some conditionsunder which consumption must converge to infinity. To establish theseconditions we use the fact that whenever assets grow without bound,consumption also grows without bound. This result is established asLemma 2.

LEMMA 2. For any ( $ 0, there is an ( $ 0 such that a( ( ( implies1 2 t 2c( ( ( .t 1

( MŽ .Proof. From Lemma A.2, 0 + " a ! . Therefore, the concavityt t 1 % #( ( ( M#Ž . Ž . Ž . Žof " implies a " a + " a % " 0 + . Choose ( $ M! 1 %t t t t t t t 21 % #

. #Ž . ( #Ž ( . #Ž ( . #Ž .# u ( . Then if a ( ( , Lemma 1 implies u c + " a + " (1 t 2 t t t t 2(#Ž . Ž .+ M! 1 % # ( $ u ( . The concavity of u then implies c ( ( .2 1 t 1

With these results in hand, we are prepared to examine the limitingbehavior of the optimal consumption sequence.

4. CONVERGENCE THEOREMS

#Ž ( .Recall that Theorem 1 establishes that the function * " a mustt t tconverge to some random variable e . In this section we demonstrate that"

the implications of this result for the limiting behavior of the optimalconsumption sequence depend primarily on the limiting value of * .tRoughly, our results may be summarized as follows. If lim * $ ", thet !" toptimal consumption sequence of the consumer must grow without bound,regardless of the properties of the income and interest rate sequences. If

Ž .the limiting value or values of * is bounded above and away from zerotand the income stream is suitably stochastic, then consumption still growswithout bound. However, if the income sequence is not stochastic, then theconsumption sequence generally converges to a finite limit.

These results may also be interpreted in terms of the relationshipbetween the rate at which the consumer discounts future utility and thelong run rate of interest. In Section 2, we defined r to be the one periodtinterest factor between period t % 1 and period t and R & Ł j r tot j k$ t#1 kbe the accumulated interest factor between periods t and j. Therefore,R1!Ž j%t . represents the a"erage interest factor between periods t and j. Ift jlim R1! t exists, we call it the long run interest factor; otherwise, we sayt !" 0 tthat the long run interest rate fluctuates. Now consider the meaning of* & # tR . Rewriting the expression, we obtain #R1! t $ * 1! t. Therefore,t 0 t 0 t t


the long run rate of interest is equal to the discount rate if and only iflim * 1! t $ 1. Clearly, if the limiting values of * are positive and finite,t !" t tthen the long run interest rate exists and is equal to the discount rate.ŽAlthough this is not a necessary condition, we will sometimes blur this

.distinction for expositional convenience.We may translate our conclusions as follows. The limiting behavior of

consumption depends primarily on the relation between the long run rateof interest and the rate at which the consumer discounts utility. If the longrun rate of interest is greater than the discount rate, then consumptiongrows without bound as long as the consumer earns a positive income insome period. If the long run rate of interest is equal to the discount factor,then consumption generally converges to infinity only if there is sufficientuncertainty in either the income or interest rate sequences.

All of our results are established for the case where both the incomeand interest rate sequences may be stochastic. In many cases, however, theintuition behind our results and the meaning of the assumptions becomemore apparent when only the income sequence is stochastic. Throughoutthis section, therefore, we will frequently focus the discussion on thespecial case where the interest rate is a constant. We conclude the sectionwith an interpretation of our results in the context of a stationary distribu-tion of interest rates.

We turn first to the case where the long run interest rate exceeds therate at which the consumer discounts utility.

4.1. Lim * $ "t

Our main result for the case where the long run rate of interest exceedsthe discount rate is summarized in the following theorem.

Ž . Ž . Ž .THEOREM 2. Suppose P x $ 0 $ 0. Then i P lim sup * $ " $ 1t !" tŽ ( . Ž . Ž .implies P lim sup c $ " $ 1, and ii P lim * $ " $ 1 impliest !" t t !" t

Ž ( .P lim c $ " $ 1.t

Ž . ŽProof. Theorem 1 implies that if P lim sup * $ " $ 1 and P x $t !" t. Ž #Ž ( . .0 $ 0, then P lim inf " a $ 0 $ 1. Lemma 1 then implies thatt !" t tŽ #Ž ( . . Ž ( .P lim inf u c $ 0 $ 1, and therefore, that P lim sup c $ " $t !" t t !" t

Ž . Ž .1. This proves part i . The proof of part ii is similar.

Roughly, Theorem 2 says that consumption grows without bound so longas the long run rate of interest always exceeds the discount rate. If thelong run interest rate fluctuates, but some subsequence always exceeds thediscount rate, then some subsequence of consumption grows withoutbound. When the interest rate is constant, the theorem may be restated asfollows.


Ž .COROLLARY 1. Suppose that r $ $ for t ( 1. If P x $ 0 $ 0 andtŽ ( .#$ $ 1, then P lim c $ " $ 1.t !" t

4.2. The Case of Certainty: #$ $ 1

For the remainder of this section, we are concerned primarily with thecase where the long run interest rate is equal to the rate at which theconsumer discounts future utility. To underline the importance of the roleof uncertainty in the main results which follow, we first concentrate on thecase where the income sequence is nonstochastic and the interest rate ineach period is equal to the rate at which the consumer discounts utility.Again denote the constant interest factor by $ .

When #$ $ 1, the first-order conditions for utility maximization implythat the consumer tries to equalize consumption in each period. As long assuch a program does not violate the borrowing constraint, this conditioncharacterizes the solution. If such a program does violate the borrowingconstraint, however, the consumer must choose a consumption stream withunequal levels of consumption over time. Nevertheless, consumption neverdecreases over time. Otherwise, by saving more in an earlier period andconsuming more later, lifetime utility can be increased. This observation

Ž .implies that the consumption level approaches a limit possibly equal to " ,which we characterize in the next theorem.

ŽŽ . . " t%jDefine y & $ % 1 !$ Ý x $ to be the supremum of those con-t j$ t jsumption levels that can be sustained indefinitely when we consider onlythe borrowing constraints in the distant future, given that the borrowingconstraint is binding in period t % 1. Our next theorem states that c(

tconverges to the supremum of these maximum sustainable consumptionlevels.

THEOREM 3. If x is not stochastic, then #$ $ 1 implies lim c( $t !" tsup y .t t

Ž . ( ( Ž . (Proof. We show first that for t ( 1, either a c $ c , or b c !t%1 t t%1( ( ( Ž ( . ( (c and c $ a in which case, a $ x . Suppose c ! a . Ift t%1 t%1 t t t%1 t%1

c( $ 0, then Lemma 1 and the strict concavity of u and " implyt%1 t%1# Ž . # Ž ( . #Ž ." 0 $ " a ( u 0 , which violates Lemma A.3. So suppose 0 !t%1 t%1 t%1( ( %Ž ( . % Ž ( .c ! a . Then, on one hand, Lemma 1 implies u c ( " at%1 t%1 t%1 t%1 t%1

# Ž ( . #Ž ( . #Ž ( . ( (( " a ( " a ( u c , and therefore, 0 + c + c . On thet%1 t%1 t t t t%1 t#Ž ( . # Ž ( . % Ž ( .other hand, Lemma 1 also implies u c + " a + " a +t%1 t%1 t%1 t%1 t%1

%Ž ( . %Ž ( . ( (" a + u c , and therefore, c + c .t t t t t%1(Let c & lim c and let y & sup y . We show first that c + y. Supposet !" t t t

( Ž .not and let t be the smallest & ( 0 such that c $ y. Then, condition b&


above implies a( $ x . From the definition of a( , it then follows thatt t t

t%1( %jx % c $ $ 0. 2Ž . Ž .Ý j j

j$ 0

Ž . Ž . (Conditions a and b also imply that c is nondecreasing in t. Therefore,&( ŽŽ . . " ( t%j ( ŽŽc $ y for all j ( t, so that $ % 1 !$ Ý c $ ( c $ y $ $ %j t j$ t j t t. . " t%j1 !$ Ý x $ , which implies a & sufficiently large such thatj$ t j

&( t%jx % c $ ! 0. 3Ž . Ž .Ý j j

j$ t

Ž . Ž . & Ž ( . %jCombining 2 and 3 then yields Ý x % c $ ! 0, violating thej$ 0 j jbudget constraint of the consumer’s maximization problem.

To show that c ( y, again assume the contrary. Then there is a y suchtthat c( ! y for all j ( 0, which impliesj t

" "( t%j t%jc $ ! x $ . 4Ž .Ý Ýj j

j$ t j$ t

But if c( is feasible, thent

t%1( %jx % c $ ( 0. 5Ž . Ž .Ý j j

j$ 0

Ž . Ž .It follows from 4 and 5 that there is an % $ 0 and & ( t such that, forˆ& Ž ( . %j (all & ( & , Ý x % c $ $ % . But then c, defined by c $ c for j ' &ˆ ˆ ˆ ˆj$ 0 j j j j

and c $ c( # % , also satisfies the consumer’s budget constraint. But then&̂ &ˆ ˆ" Ž . j " Ž ( . jÝ u c # $ Ý u c # implies that c* is not an optimal consump-ˆj$ 0 j j$ 0 j

tion program.

For our purposes the main implication of this theorem is that when theincome stream is certain, the consumption sequence generally convergesto a finite limit. Consumption grows without bound only if the discountedvalue of future income is not bounded. However, the theorem also has anoteworthy implication in the context of the original formulation of themodel.

Suppose that x is an arbitrary income sequence and k is an intertempo-rally consistent sequence of borrowing constraints from which we derive

ŽŽ . . " t%j ŽŽx & x % k # $ k ( 0. Then, y & $ % 1 !$ Ý x $ $ $ %ˆ ˆt t t t%1 t j$ t j. . Ž & t%j t%& . Ž .1 !$ lim Ý x $ % $ k # $ % 1 k . Now suppose we start& !" j$ t j & t%1

with a sequence of borrowing constraints which generates an optimalconsumption program c1. Then if the k1 constraint is replaced by a tighter


2 1 Ž 2 1 .constraint, k ( k i.e., k ( k for all t , but which allows for the samet tŽ %& Ž 1 2 . .discounted value of consumption i.e., lim $ k % k $ 0 , then the& !" & &

tighter constraint k2 generates a limiting value of consumption at least as1 Ž 2 1.large as did the k constraint i.e., lim c ( lim c . An even strongert !" t t !" t

result can be established about the behavior of the sequence of accumu-lated wealth in each period, but this is left to Section 5, where thestochastic case is treated as well.

4.3. The Stochastic Case

As noted above, the analysis of the certainty case yields little insight intothe limiting behavior of consumption when we introduce uncertainty intoeither the income sequence or the interest rate sequence. In this subsec-tion, we formulate a general ‘‘uncertainty’’ condition under which we showthat consumption grows without bound even when the long run interestrate equals the discount rate.

The meaning of our uncertainty condition is most transparent when the1interest factor is fixed at $ $ . In this case, we require:#

Ž .Condition U$ . There is an % $ 0 such that for any ( ) !,"

t% j t#P ( + x $ + ( # % z ! 1 % %Ý jž /j$ t

for all z t, t ( 0.

Ž .Condition U$ says that starting at any information state, there is afixed probability that the discounted value of future income lies outsideany sufficiently small range. The key implication of this condition is whenconsumption stays within a sufficiently small range in each period, assetsmust diverge with some fixed probability from any information state.However, the proof of our main result requires a fixed probability thatassets diverge whenever the consumption program keeps the marginalutility of consumption approximately constant. Consequently, when weallow for a stochastic interest rate, the uncertainty condition requires aslight reformulation that uses the utility function to restrict the relationbetween the income and interest rate sequences. Define the ‘‘inverse’’ of

# Ž . Ž . $ #Ž . 4u by h 0 & " and, for y $ 0, h y & inf c ) ! : u c + y .#

Ž .Condition U . There is an % $ 0 such that for any ( ) ! and any+ $ 0,

" +%1 t#P ( + x % h R + ( # % z ! 1 % %Ý j t jž /ž /ž /* jj$ t

for all z t, t ( 0.


Suppose the consumption program were chosen so that, in any informa-tion state, an increase in the present value of consumption by one unitgenerates an increase in discounted utility equal to +. Then the consump-tion level in each information state would be determined by the relation

( Ž . Ž .c $ h +!* , and Condition U would imply a nontrivial probability thatt tthe present value of current assets plus future income is bounded awayfrom the present value of current plus future consumption, starting at anystate z t. Roughly speaking, the condition says that the actual incomestream is stochastic relative to the hypothetical income stream required tomake the marginal discounted utility of a ‘‘present value’’ unit of consump-tion equal in all information states. When the interest factor is constant,

Ž . Ž .Condition U$ implies Condition U .Our results are based on Lemma 4 below. Roughly, it says that if * ist

Žbounded away from zero and infinity the long run interest rate is equal to. Ž .the discount rate , then Condition U implies that the marginal utility of

#Ž ( .assets " a must converge to 0. The argument goes as follows. Supposet t#Ž ( ." a does not converge to zero. Then we may choose & ! " and ant t

& ' Ž .arbitrarily small interval b, b # % with b $ 0 such that i there is#Ž ( . & ' Ž . #Ž ( . &positive probability that * " a ) b, b # % , and ii if * " a ) b, b& & & & & &

' #Ž ( . & '# % , then, with probability greater than 1 % % , * " a ) b, b # % fort t tŽ . #Ž ( .all t ( & . Now consider the nonnull set of paths for which * " a )& & &

& 'b, b # % for all t ( & . Since the first-order conditions for utility maxi-( Ž #Ž ( .. Ž .mization imply that c $ h " a , Condition U implies that whenevert t t

#Ž ( . & ' (* " a ) b, b # % for all t ( & , a must diverge with probability att t t tleast % . But if a( diverges, Lemma 2 implies that c( must converge tot t

#Ž ( .infinity, which if * is bounded above, implies that * " a converges tot t t t0. This contradiction establishes the result.

We proceed now with the formal analysis. To use Theorem 1 we mustŽ .first establish that Condition U implies a nonzero income stream.

Ž . Ž .LEMMA 3. Condition U implies P x $ 0 $ 0.

Ž . Ž # t.Proof. We show first that Condition U implies that P x $ 0 z ! 1% % for all z t. For this, it is sufficient to show that for any z t we may

Ž " Ž . %1 # t.choose + sufficiently large so that P Ý h +!* R ! % z $ 1. Sincej$ t j t j%Ž .u is concave and bounded between 0 and M, we have cu c ! M for all

$4 %Ž Ž .. Ž .c ) ! ! 0 . By definition, u h +!* ( +!* . Therefore, +!*# j j jŽ . %1 Ž .ŽŽ . j% t. " Ž .h +!* + M. Then R & * !+ +!* # implies Ý h +!*j t j t j j$ t j%1 Ž . " Ž . Ž . j% t Ž .Ž Ž ..R $ * !+ Ý +!* h +!* # + * !+ M! 1 % # . Setting + $t j t j$ t j j t

Ž .* M!% 1 % # establishes the result.tŽ # t. tAll that remains is to show that P x $ 0 z ! 1 % % for all z implies

Ž . Ž .P x $ 0 $ 0. We establish this by contradiction. Suppose P x $ 0 $ 0.$ t 4 $ 4 Ž .Let A $ x $ 0 and A $ x $ 0 . Then P A $ 0 for all t ( 0, andt " t

Ž # t. tA " A . Then, since P x $ 0 z ! 1 % % for all z and A is measurablej " t


t Ž # . Ž # .z , we have P A A ! P A A ! 1 % % . Therefore, there is an in-j t " tŽ Ž .. Ž # .creasing sequence & t such that P A A ! 1 % % which implies& Ž t#1. & Ž t .

Ž . " Ž # . Ž . Ž . t Ž .P x $ 0 $ Ł P A A P A + lim 1 % % P A $ 0.t$ 0 & Ž t#1. & Ž t . & Ž0. t !" tŽ0.

Ž .LEMMA 4. Suppose there is an % $ 0 for which Condition U is satisfied1 (#Ž . Ž Ž . .and P % ! * ! , t ( 0 $ 1. Then P lim * " a $ 0 $ 1.t t !" t t t%

Proof. We will suppose the lemma is false and show that this implies aŽ .violation of Condition U .

& ' &Our first step is to find an interval +, + # ' and a time & such that +,' #Ž ( .+ # ' contains * " a with positive probability and, in the event that& & &

#Ž ( . & ' #Ž ( . & ' (* " a ) +, + # ' , it is also true that * " a ) +, + # ' and c& & & t t t tŽ .is near h +!* with probability close to 1 for all t ( & . Theorem 1 andt

Ž #Ž ( . .Lemma 3 imply a random variable e such that P lim * " a $ e" t !" t t t "Ž .$ 1. If P e $ 0 ! 1, then there is a , $ 0 such that for any ' $ 0,"

Ž & '. 3Ž .P e ) , , , # ' $ 0. Let - & % 1 % # !2. Then, since h is uniformly"

continuous on any positive interval bounded away from zero, we mayŽ . Ž & '.choose + and ', 0 ! + ! , ! + # ', so that i P e ) +, + # ' $ 0,"

Ž . Ž . Ž . Ž . # Ž . ŽŽ . . #ii P e $ + $ P e $ + # ' $ 0, and iii h +!* % h + # ' !*" " t t1 $ & '4! - for % ! * ! . Define B & e ) +, + # ' , and for & ( 0, definet "%

$ #Ž ( . & '4 $# ( Ž . # #Ž ( .A & * " a ) +, + # ' and B & c % h +!* ! -, * " a )& & & & & t t t t t& ' 4 Ž . Ž .+, + # ' , t ( & . Then lim P A $ P B $ 0. Also, Lemma 1 im-& !" &

( ( 1#Ž Ž Ž .. . Ž .plies P c % h " a $ 0, t ( 0 $ 1. Therefore, P % ! * ! $ 1t t t t %

Ž Ž ( Ž .. # .and the continuity of h imply P lim c % h % !* $ 0 B $ 1. Then,t !" t " tŽ Ž . Ž . Ž . # .since the monotonicity of h implies P h +!* ( h e !* ( h + # ' Bt " t

Ž . Ž .$ 1, we have lim P B $ P B . Consequently, we may choose & ! "& !" &Ž . Ž . Ž .such that P B $ 1 % % P A $ 0.& &

We show next that if the marginal utility of consumption c is alwaysjequated to +!* , then the variation in the present value of the limitingj

%level of assets is less than on B . The monotonicity of h and the&2Ž ( Ž . # .construction of B imply P c ! h +% # -, t ( & B $ 1. Therefore,& & &

Ž . Ž (letting ( & h +% # -, Lemma 2 implies an ( ! " such that P 0 + a2 1 t( 1# . Ž .% c + ( , t ( & B $ 1. Then, since P % ! * ! , t ( 0 $ 1 impliest 1 & t %

Ž %1 j%& %2 j%& . Ž Ž (P R & * # !* + % # , j ( & $ 1, it follows that P lim a& j & j t !" t( . %1 # .% c R $ 0 B $ 1. Therefore,t & t &

a( % c( R%1Ž .t t & t

t( ( ( %1$ a % c # x % c RŽ .Ý& & j j & j

j$ &#1

t t+ +( (%1 %1$ a % x # x % h R # h % c RÝ Ý& & j & j j & jž / ž /ž / ž /* *j jj$ & j$ &


implies

" +( %1P a % x # x % h RÝ& & j & jž /ž /ž * jj$ &

6Ž ." +

( %1# h % c R $ 0 B $ 1.Ý j & j &ž /ž / /* jj$ &

Ž %1 j%& %2 j%& .But from P R & * # !* + % # , j ( & $ 1 and the definition& j & jof B it follows that&

" + - %( %1P h % c R ! & B $ 1. 7Ž .Ý j & j &2ž /ž /ž /* 21 % # %Ž .jj$ &

Ž . Ž .Combining 6 and 7 then yields

" + %( %1P a % x # x % h R ! B $ 1. 8Ž .Ý& & j & j &ž /ž /ž /* 2jj$ &

Ž .We now use these results to show that Condition U must be violated.( % Ž . Ž . Ž .Let ( $ x % a % . Since B / A and P B $ 1 % % P A , it fol-& & & & & &2

Ž . Ž # " Ž Ž .. %1 # # .lows from 8 that P ( ! Ý x % h +!* R ! ( # % A $ 1 % % .j$ & j j & j && & ŽBut then A measurable z implies that the set of z such that P ( !&

# " Ž Ž .. %1 # # & .Ý x % h +!* R ! ( # % z $ 1 % % has positive probability,j$ & j j & jŽ .which violates Condition U .

We use this lemma again in Section 5. For the moment, however, itserves as the basis for the main result of this paper.

Ž .THEOREM 4. Suppose there is an % $ 0 such that Condition U is1 (Ž . Ž .satisfied. If P % ! * ! , t ( 0 $ 1, then P lim c $ " $ 1.t t !" t%

1Ž .Proof. If P % ! * ! , t ( 0 $ 1, then Lemma 4 impliest %(#Ž Ž . .P lim " a $ 0 $ 1. The conclusion then follows from Lemma 1.t !" t t

With sufficient uncertainty in the income and interest rate sequences,consumption will grow without bound even if the long run rate of interestis equal to the discount rate. A case in which this equality holds has been

Ž .considered by Bewley 1980c, 1983 in his treatment of the optimumquantity of money. There is an asset with a fixed nominal return factor

%1 %1Ž . Ž .equal to # . The real return is r $ # q !q , where the price q oft t%1 t tthe consumption good in terms of the asset is uniformly bounded away


from zero and infinity. Consequently, * $ q !q satisfies the conditions oft 0 tTheorem 4. We conclude that if the uncertainty condition is satisfied, c(

&

converges to infinity.For the case where the interest rate is constant, Theorem 4 implies the

following corollary.

COROLLARY 2. Suppose r $ $ for all t ( 0 and suppose #$ $ 1. IftŽ . Ž ( .Condition U$ is satisfied, then P lim c $ " $ 1.t !" t

Ž .If income is bounded above, then Condition U$ in Corollary 2 can bereplaced by the condition that the conditional variance of discountedfuture income is uniformly bounded away from zero; i.e., there is a , $ 0

Ž " t%j # t. tsuch that Var Ý x $ z ( , for all z , t ( 0. However, if the incomej$ t jstream is stochastic, but the conditions of Corollary 2 are not satisfied,there are examples where the limiting level of consumption is finite withsome positive probability.

When contrasted with the outcome in the case of certainty, Corollary 2is perhaps a surprisingly strong result. Unfortunately, the line of argumentused in the proof does not provide a very convincing economic explana-tion. Clearly the strict concavity of the utility function must play a role.ŽThe result does not hold if, for instance, u is a linear function over a

Ž . .sufficiently large domain and x is bounded. But to simply attribute thetresult to risk aversion on the grounds that uncertain future returns willcause risk-averse consumers to save more, given any initial asset level, isnot a completely satisfactory explanation either. In fact, it is a bit mislead-ing. First, that argument only explains why expected accumulated assetswould tend to be larger in the limit. It does not really explain whyconsumption should grow without bound. Second, over any finite timehorizon, the argument is not even necessarily correct.

Ž .Suppose, for example, that x $ x % for t $ 1, . . . , T , where the %t t%1 t t& 'are identically and independently distributed with E % $ 1 and a com-t

pact support in !. Suppose, also, that the consumer’s utility function isquadratic over a sufficiently large domain. Then if the consumer has aT-period planning horizon, it can be shown that his optimal consumptionprogram is to set c $ x for all t. In particular, mean-preserving spreads oft tfuture income leave current consumption unaffected. Moreover, the ex-pected value of consumption in any period t is just equal to period 0income. So there is no tendency at all for consumption to rise over time.

Why then does the result change when we consider the limiting value ofconsumption in an infinite horizon problem? Although we have developedother arguments to establish our results, any explanation that we havebeen able to devise ultimately appeals in an essential way to the martin-gale convergence theorems.


Ž .4.4. Limit Theorems When r Is Stationaryt

In this subsection, we assume that the sequence of information states isgenerated by a stationary process and establish some restrictions on thedistribution of the single period interest rate that imply the conditions ofTheorem 2. Since the stationarity restriction will be placed on the entiredistribution of histories, we will be explicit about conditioning on z 0.7

Ž . Ž .THEOREM 5. Suppose P x $ 0 $ 0. If r , r , . . . is stationary and1 2& ' Ž ( # 0.ergodic and E log # r $ 0, then P lim c $ " z $ 1 with prob-1 t !" t

ability 1.

& 'Proof. Let , $ E log # r . The Ergodic Theorem implies that with11 t Ž .probability 1, lim Ý log # r $ , Doob, 1953, p. 465 . Therefore,t !" j$ 1 jt

, ttŽ . Ž .for almost all z there is a & z such that Ý log # r $ for t $ & z , orj$ 1 j 2, t!2 Ž .equivalently, * $ e . Therefore, P lim * $ " $ 1, which impliest t !" t

$ 0 Ž # 0. 4that z : P lim * $ " z $ 1 has probability 1. The desired resultt !" tthen follows from Theorem 2.

& Ž .'For the case where E log # r $ 0, we need some additional regularity1Ž .conditions. For s + t, define . r , . . . , r as the .-field generated bys t

Ž .r , . . . , r , and define . r , r , . . . as the sigma field generated bys t t t#1r , r , . . . . Consider a nonnegative function + of the positive integers.t t#1

Ž . Ž .The sequence r , r , . . . is +-mixing if for each t, j ( 1, A ) . r , . . . , r1 2 1 1 tŽ . # Ž .and A ) . r , r , . . . together imply that P A 0 A %2 t#j t#j#1 1 2

Ž . Ž . # Ž . Ž .P A P A + + j P A .1 2 1

Ž . Ž . Ž .Condition R . i r , r , . . . is a stationary, +-mixing stochastic pro-1 2" Ž Ž ..1!2 Ž . &Ž .2 'cess with Ý + j ! ". ii E log # r ! " andj$ 1 1

"22. & E log # r # 2 E log # r log # r ' 0.Ž . Ž . Ž .Ý1 1 j

j$ 2

Ž .The first part of Condition R ensures that the dependence between rtŽ .and r dies out sufficiently fast as j ! ". Suppose part i is satisfied andt# j

& ' " &Ž .Ž .'E log # r $ 0. Then Ý E log # r log # r converges absolutely and1 j$ 2 1 j12 T 2& Ž . ' Ž. $ lim E Ý log # r Billingsley, 1968, Lemmata 1 and 3, pp.T !" j$ 1 jT

. Ž .170, 172 . In this case, part ii may be satisfied as well if there is sufficientt Ž . Ž .variability in # R . However, if P # r $ 1 $ 1, part ii is not satisfied,0T 1

Ž ( .and we know from Theorem 4 that the behavior of c may depend upont

7Theorem A.1 establishes the existence of an optimal program only for a fixed z t. Because( Ž t .we did not restrict z to be drawn from a compact set, we are able to show that c z is at t

t Ž 0 .measurable function of z only for a fixed tail e.g., a given z . Without additionalrestrictions we are unable to prove that c( is a measurable function if all components of z t

tare allowed to vary.


Ž .whether or not x is stochastic. Nevertheless, our next theorem statestŽ . & ' (that if Condition R is satisfied and E log # r $ 0, then sup c $ ",1 t t

Ž .without a requirement that x be stochastic.t

Ž . Ž .THEOREM 6. Suppose P x $ 0 $ 0 and r , r , . . . satisfies Condition1 2Ž . & ' Ž ( # 0.R . Then E log # r $ 0 implies P sup c $ " z $ 1 with prob-1 t ( 0 tability 1.

t Ž . Ž .Proof. Define S & Ý log # r $ log * and define Y ' &t j$ 1 j t tŽ 2 .%1!2 ) *S . t , where 0 + ' + 1 and ' t is the greatest integer less than) ' t *

or equal to ' t. Then the functional central limit theorem implies that thedistribution of the random function Y converges weakly to Weiner mea-t

Ž .sure Billingsley, 1968, Theorem 20.1 , from which it follows that

( 2%1!2 %1!22 %u !2lim P . t max S + ( $ 2 2/ e du 9Ž . Ž . Ž .Hjž /t!" j+ t 0

Ž Ž . .for ( ( 0 Billingsley, 1968, p. 138, and Eq. 10.18 , p. 72 .Ž .Define , & P sup S ! " . We shall assume that , $ 0 and obtain at ( 1 t

,%1!2 % 2Ž . Ž .contradiction. Choose % $ 0 so that 2 2/ H exp %u !2 du ! and,0 2,2 %1!2$Ž . 4 Ž .for t ( 1, define A & . t max S + % . Then lim P A ! .t j+ t j t !" t 2

$Ž 2 .%1!2 4For t ( 1, define B & . & max S + % for all & ( t . Then theret j+ & j$ 4 Ž .is a B such that sup S ! " / B and B # B. Therefore, using Eq. 9 ,t ( 1 t t

Ž . Ž . Ž .B / A implies P A ( P B ! P B ( , , which contradictst t t t,Ž . Ž .lim P A ! . We conclude that P sup S ! " $ 0, or equiva-t !" t t ( 1 t2

Ž .lently, P sup * $ " $ 1. The desired conclusion then follows fromt ( 1 tTheorem 1 and Lemma 1.

5. THE BUDGET CONSTRAINT

In this section, we address two questions. First, given an arbitrarystochastic income sequence, under what conditions should we expect theborrowing constraint to be binding at some information state? Second,

Žhow does a change in the borrowing constraint or equivalently, a change.in the income sequence affect the pattern of borrowing?

5.1. When Is Budget Constraint Sometimes Binding?

As noted in Section 2, if the discounted value of the income sequence isfinite conditional on any information state, we can represent any intertem-poral budget constraint as a sequence of one period borrowing constraints.It may well turn out that none of these constraints are actually binding atthe optimum, and yet the consumer is still constrained to choose a


consumption program whose present value does not exceed the presentvalue of the income stream. Our next theorem gives conditions underwhich this is the case.

& #Ž . # t ' ( ( (THEOREM 7. If E r u x z $ " and a $ 0, then c ! a .t#1 t#1 t t t

( ( Ž ( # t.Proof. Suppose c $ a . Then P c + x z $ 1. Lemma 1 thent t t#1 t#1implies

( ( (% # # t#" $ u c ( " a ( #E r " a zŽ . Ž . Ž .t t t t#1 t#1 t#1

(# t # t# #( #E r u c z ( #E r u x z $ ",Ž . Ž .t#1 t#1 t#1 t#1

a contradiction.

Since a( ( x , the following corollary follows immediately.t t

& #Ž . # t ' ( (COROLLARY 3. If E r u x z $ " and x $ 0, then c ! a .t#1 t#1 t t t

Theorem 7 and its corollary say that whenever the expected incrementin disposable income in the following period is sufficiently small so thatthe expected marginal utility from consuming out of that increment wouldbe infinite, the consumer chooses to consume less than his current wealthin the current period in order to pass some of his wealth to the nextperiod. If we allow the expected marginal utility of future income to befinite, however, the borrowing constraint may well be binding, at leastoccasionally. This will obviously be the case if income received in eachperiod is growing at a sufficiently high rate over time so that the consumerwants to transfer future income to present consumption. But if the incomestream is suitably stochastic, a much weaker set of conditions guaranteesthat the budget constraint is sometimes binding.

Our next theorem may be summarized as follows. Suppose the singleperiod interest rate never exceeds the discount rate and the marginalutility of consuming from current income alone is bounded above. Then ifthe long run rate of interest is less than the discount rate or if the incomesequence is suitably stochastic, there is a positive probability that theborrowing constraint is binding in at least one period.

Ž . Ž #Ž . .THEOREM 8. Suppose P # r + 1, t ( 1 $ 1 and P u x ! ( , t ( 0t tŽ . Ž . Ž .$ 1 for some ( ) !. If either i P lim * $ 0 $ 1 or ii there is ant !" t

Ž . Ž .% $ 0 for which Condition U is satisfied and P * $ % , t ( 0 $ 1, thentŽ ( ( .P c ! a , t ( 0 ! 1.t t

Ž (Proof. We establish the theorem by contradiction. Suppose P 0 + ct( . %Ž ( .! a , t ( 0 $ 1. Then, by induction, Lemma 1 implies " a +t 0 0

& %Ž ( .' & %Ž ( .'E # r " a + E * " a for all t ( 1. We will establish that1 1 1 t t t


& %Ž ( .' %Ž ( .lim E * " a $ 0 and therefore, " a $ 0, violating the mono-t !" t t t 0 0tonicity and concavity of " .0

& %Ž ( .' Ž %Ž ( . .To show that lim E * " a $ 0, we first show P " a ! ( $ 1t !" t t t t t 1Ž %Ž ( . . Žfor some ( ) !, and then show P lim * " a $ 0 $ 1. Since P # r1 t !" t t t t

. Ž .+ 1, t ( 0 $ 1 implies P 0 + * + * + 1, t ( 1 $ 1, the desired re-t#1 tsult then follows from the Dominated Convergence Theorem.

Ž %Ž ( . . Ž #Ž . .To establish P " a ! ( $ 1, note first that P u x ! ( , t ( 0t t 1 t#Ž . Ž .$ 1 implies either u 0 ! ( or P x $ +, t ( 0 $ 1 for some + $ 0. Ift

#Ž . Ž %Ž ( .u 0 ! ( , then Lemma A.3 and the strict concavity of " imply P " at t#Ž . #Ž . . Ž! " 0 $ u 0 ! ( $ 1, in which case we let ( & ( . If P x $ +, tt 1 t. Ž ( . Ž %Ž ( .( 0 $ 1, then P a $ +, t ( 0 $ 1, and so Lemma 1 implies P " at t t%Ž . . %Ž .+ u + , t ( 0 $ 1, in which case we let ( & u + .1

Ž %Ž ( . . Ž .To establish P lim * " a $ 0 $ 1, we consider Conditions it !" t t tŽ . Ž .and ii separately. If Condition i holds, then the desired property follows

Ž .immediately. If Condition ii holds, then Lemmata 1 and 4 imply

1( ( (# # #P lim u c + lim " a + lim * " a $ 0 $ 1,Ž . Ž . Ž .t t t t t !" t t tž /%t!" t!"

Ž ( .and so P lim c $ " $ 1. Another application of Lemma 1 then yieldst !" t

P lim * "% a( + lim * u% c( + lim u% c( $ 0 $ 1.Ž . Ž . Ž .ž /t t t t t tt!" t!" t!"

If the long run rate of interest is less than the discount rate, theconclusion of Theorem 8 is not particularly surprising. In this case, themarginal return to consuming a fixed level of consumption goes to zerowith time. Consequently, if the marginal utility to consuming currentincome is bounded, then the borrowing constraint must eventually bebinding. The theorem is less intuitive for the case where the long run rate

Žof interest is equal to the discount rate #$ $ 1 for a fixed interest factor.$ . In this case Theorem 4 implies that consumption grows without bound.

Evidently, along almost every path, the asset level first falls to its lowerbound at least once before converging to infinity. However, Theorem 4implies that this happens only a finite number of times.

5.2. Comparati"e Dynamics

Suppose the original sequence of borrowing constraints is replaced withanother sequence which allows at least as much borrowing in any informa-tion state. The following theorem states that at any information state theoptimal accumulated wealth under the new sequence of borrowing con-


straints is no higher than the optimal accumulated wealth under theoriginal sequence of borrowing constraints.

THEOREM 9. Let a0 and a1 be the sequence of accumulated assetsassociated with the solutions c0 and c1 corresponding to constraints k0 and

1 Ž 0 1 . Ž 0 1 .k , respecti"ely. Then P k ( k , t ( 0 $ 1 implies P a ( a , t ( 0 $ 1.t t t t

Proof. Let S and S be the metric spaces defined in the Appendix0 1corresponding to k0 and k1, respectively, and let T and T be the0 1corresponding T operators. Note that S / S so that T is defined on S .0 1 1 0Let " 0 and " 1 be the corresponding fixed points of T and T . We shall0 1

0#Ž t. 1%Ž t. Ž t.show that " a, z $ " a # % , z for all a, z ) S and % $ 0.0Let f 1 represent " 1 restricted to S and let f n#1 & T n f 1, where T n is0 0 0

Ž t.the operator T applied n times. For a, z ) S , define0 0

C a, z tŽ .1

1 t#1 t#& arg max u c # #E " a % c r # x , z z ,Ž . Ž .$ 4Ž .t#1 t#11 tŽ .0+ c+ a%k zt

and for n ( 2 define

C a, z tŽ .n

n%1 t#1 t#& arg max u c # #E f a % c r # x , z z .Ž . Ž .$ 4Ž .t#1 t#10 tŽ .0+ c+ a%k zt

Ž t. Ž t. Ž t.Fix a, z ) S and % $ 0. Let c & C a # % , z and c & C a, z .0 1 1 2 2Then either c $ c ( 0 or a # % % c $ a % c . Therefore, using the1 2 1 2standard envelope arguments exploited in Lemma 1, the strict concavity ofu and " implies that either

" 1% a # % , z t + u% c ! u# c + f 2# a, z t 10Ž . Ž . Ž . Ž . Ž .1 2

or

1% t 1% t#1 t#" a # % , z + #E r " a # % % c r # x , z zŽ . Ž .Ž .t#1 1 t#1 t#1

1# t#1 t 2# t#! #E r " a % c r # x , z z + f a, z .Ž . Ž .Ž .t#1 2 t#1 t#1

1%Ž t. 2#Ž t.In either case, we have shown that " a0, z ! f a, z for anyŽ t. Ž t.a, z , a0, z ) S , a0 $ a.0

n#Ž t. Žn%1.%Ž t. Ž t. Ž t.Now suppose f a, z $ f a0, z for any a, z , a0, z ) S0Ž t. Ž t.with a0 $ a. Fix a, z ) S and % $ 0 and define c & C a # % , z and0 n n

Ž t.c & C a, z . Then either c $ c ( 0 or a # % % c $ a % c .n#1 n#1 n n#1 n n#1


Then the envelope argument implies either

f n% a # % , z t + u% c ! u# c + f Žn#1.# a, z t 11Ž . Ž . Ž . Ž . Ž .n n#1

or

n% t Žn%1.% t#1 t#f a # % , z + #E r f a # % % c r # x , z zŽ . Ž .Ž .t#1 n t#1 t#1

n# t#1 t#! #E r f a % c r # x , z zŽ .Ž .t#1 n#1 t#1 t#1 12Ž .

+ f Žn#1.# a, z t .Ž .

n%Ž t. Žn#1.#Ž t.In either case, we have shown that f a0, z ! f a, z for anyŽ t. Ž t.a, z , a0, z ) S , a0 $ a.0

Ž .Now let % be a strictly decreasing sequence such that % $ % andn 1% 1% tŽ .lim % $ . Then for n ( 2, we have just shown that " a # % , z !n!" n 2

%1% t Žn%1.% t n% tŽ . Ž . Ž .f a # % , z ! 111 ! f a # % , z ! f a # , z for all2 n 2Ž t. 0 0a, z ) S . Since T is a contraction mapping and T " $ " , it follows0 0 0

nŽ t. 0Ž t.that f a, z ! " a, z as n ! ". Therefore, Lemma A.4 implies% %0# t 0% t n% t 1% tŽ . Ž . Ž . Ž ." a, z $ " a # , z ( lim sup f a # , z ( " a # % , zn2 2

Ž t.for any a, z ) S and % $ 0.0To show that a1 + a0, we again argue by induction. Fix any realization ofŽ . 0Ž 0. 1Ž 0. 0Ž t.z $ . . . , z , z , z , . . . . By definition, a z $ a z . Suppose a z%1 0 1 0 0 t1Ž t. 0 Ž t#1. 1 Ž t#1.% a z ( 0. We will show that a z % a z ( 0. Note first,t t#1 t#1

that for any w ) ! N,

a0 z t , w % a1 z t , wŽ . Ž .t#1 t#113Ž .

$ r z t , w a0 z t % a1 z t % c0 z t % c1 z t .Ž . Ž . Ž . Ž . Ž .Ž . Ž .Ž .t#1 t t t t

0 Ž t#1. 1 Ž t#1. 0Ž t. 1Ž t. 0 Ž t .Therefore, a z ! a z implies c z $ c z and a z , wt#1 t#1 t t t#11 Ž t . N! a z , w for all w ) ! . Lemma 1 then impliest#1

t 1% 1 t t t##E r z , w " a z , w , z , w zŽ . Ž . Ž .Ž .t#1 t#1

( u# c1 z t $ u% c0 z tŽ . Ž .Ž . Ž .t t 14Ž .t 0# 0 t t t#( #E r z , w " a z , w , z , w z .Ž . Ž . Ž .Ž .t#1 t#1

0 Ž t . 1 Ž t .But we have just shown above that a z , w ! a z , w impliest#1 t#10#Ž 0 Ž t . Ž t .. 1%Ž 1 Ž t . Ž t .." a z , w , z , w $ " a z , w , z , w . This contradiction com-t#1 t#1

pletes the proof.


It should be clear from our discussion in Section 2 that for any change inthe set of feasible consumption programs in response to a change in theborrowing constraint, there is an equivalent change in the income se-quence that generates the same change in the set of feasible consumptionprograms. Consequently, we may reinterpret Theorem 9 as a statementabout how the optimal consumption program changes in response tocertain kinds of changes in the income sequence. Our next corollary statesthat if the income sequence changes so that at any state z t the discountedvalue of income up to time t has not decreased, then the discounted valueof the consumption up to time & also does not decrease for any state z&.

COROLLARY 4. Let c 0 and c1 be the optimal consumption programsassociated with borrowing sequence k and income sequences x 0 and x1,respecti"ely. Then

t1 0 %1P x % x R ( 0, t ( 0 $ 1Ž .Ý j j 0 jž /j$ 0

t1 0 %1implies P c % c R ( 0, t ( 0 $ 1.Ž .Ý j j 0 jž /j$ 0

0 0 1 Ž . 1Proof. Let k & k, x & x , and define k $ k , k , k , . . . by k & 0%1 0 1 %11 t Ž 0 1. 0 1and k & Ý x % x R # k for t ( 0. By assumption, x and k jointlyt j$ 0 j j jt tŽ . 1 1 0 t Ž 0 1. 0satisfy 1 . Therefore, x % k # r k $ x % Ý x % x R % k #t t t t%1 t j$ 0 j j jt t

t%1 Ž 0 1. 0 1 1Ý x % x R # r k $ x % k # r k implies that x and k do asj$ 0 j j jt t t%1 t t t t%1well. Furthermore, the definitions of c 0 and c1 imply that they are also the

Ž .optimal consumption programs associated with the common incomesequence, x, and respective borrowing constraints, k0 and k1. To see this,

t Ž 1 .note that for any consumption program c, we have Ý x % c R % kj$ 0 j j jt tt Ž . 1 1$ Ý x % c R % k . Therefore, a program c is feasible for x and k ifj$ 0 j j jt t

and only if it is feasible for x and k1. By assumption c1 is optimal for x1

and k. Therefore, it is also optimal for x and k1.Now let a0 and a1 denote the asset sequences associated with consump-

tion programs c 0 and c1 for income sequence x. Then, for i $ 0, 1, ai $ x0 0i t%1 Ž i.and a $ Ý x % c R # x for t ( 1. The hypothesis of the corollaryt j$ 0 j j jt t

and the construction of k1 imply k1 + k0. Therefore, Theorem 9 implies0 1 t%1 1 0 t%1 1 0 %1Ž . Ž .0 + a % a $ Ý c % c R $ R Ý c % c R for all t ( 1.t t j$ 0 j j jt 0 t j$ 0 j j 0 j

6. MANY ASSETS

In this section, we illustrate how the argument behind Theorem 2 can beextended to the case where there are several risky assets. We show that toapply the martingale convergence theorem, it is not necessary to actually


solve the portfolio problem of the consumer. So long as the consumer isable to choose a sequence of ‘‘marginal’’ interest rates which, in the longrun, exceed the discount rate, consumption must converge to infinity.

Suppose the consumer must choose a portfolio of J risky assets in eachŽperiod t ( 0. If the investment in period t in the ith asset is b measuredi

.in terms of period t consumption , then the payoff at t # 1 is b g ,i i, t#1t#1Ž 0.where g is a nonnegative, continuous function from Z z to !i, t#1 #

Ž .for i $ 1, . . . , J, and t ( 0. If b & b , . . . , b is the portfolio chosen in1 Jperiod t, then the multi-asset analogue of the budget constraint is aconstraint on short sales which we represent by the requirement that

Ž t.b ) K z . We assume that K is a continuous correspondence fromt ttŽ 0. J Ž t. JZ z to ! and that K z is a nonempty, closed, convex subset of !t

tŽ 0.which is bounded from below; i.e., there is a function m: Z z ! ! suchŽ t. $Ž . J Ž t.4that K z / ( , . . . , ( ) ! : ( ( m z . We also assume thatt 1 J j

Ž t. Ž t. J%K z is comprehensive; i.e., if + ) K z and + ) ! , then + # +t 1 t 2 # 1 2Ž t.) K z . This guarantees that it is always feasible for the consumer tot

increase his holding of any asset.For any ( , # ) ! J, let (# & ÝJ ( # . Let 2 denote the J-vector ofj$ 1 j j

Ž t. $ Ž t.4ones, and define k z & inf b2: b ) K z . Interpret k as the mini-t t tmum amount of wealth measured in terms of period t consumption that

Žthe individual may hold at the end of period t after receiving period t. Ž .income and spending period t consumption . Let g & g , . . . , g . Wet 1 t J t

require that the current bounds on short sales be consistent with theŽborrowing constraints the individual will face in the future: P bg #t#1

# t. Ž t. tx ( k z $ 1 for all b ) K z and all z .t#1 t#1 t0 Ž .For any state z , a consumption-portfolio program, c, b , is a sequence

tŽ 0. tŽ 0. Jof a pair of functions, c : Z z ! ! and b : Z z ! ! , t ( 0. Fort # tt t 0 t tŽ . Ž . Ž . $Ž .any z ) Z z and any a ( k z , let K a, z & c, b ) ! +t t #

Ž t. 4K z : 0 + c + a % b2 denote the set of feasible consumption-portfoliotpairs for the consumer, and define

"t j%t t#" a, z & max E u c # zŽ . Ž .Ý j

c, b j$ t

j t#subject to P c , b ) K a , z , j ( t z $ 1,Ž . Ž .ž /j j j j

where a & a and a & b g # x for j $ t. Theorem A.1 implies that thist j j%1 j jŽ t.problem has a solution and that " a, z is strictly concave in a ( k .t

Therefore, right- and left-hand derivatives are well defined for a $ k . Wet#Ž t. #Ž t.let " k , z & lim " a, z .t a" k t

Ž . Ž t. Ž .As in Section 2, let " a & " a, z , and let c*, b* denote the solutiontto the consumer’s problem starting in state z 0 with wealth a $ x . We0


shall simplify the martingale arguments by assuming that x $ k . The0 0following conditions are then implied by an optimal program. They areestablished by the same standard arguments used to establish Lemma 1.

Ž . #Ž ( . & # Ž ( . # t 'LEMMA 5. i " a ( #E g " a z for k $ 1, . . . , J, andt t k , t#1 t#1 t#1Ž . #Ž ( . #Ž ( .ii " a ( u c .t t t

To obtain the analogues of Theorems 1 and 2 from Lemma 5, weconsider an arbitrary marginal portfolio, i.e., a portfolio which can beadded to the existing portfolio. Given our assumptions on K , the onlytportfolios we might not be able to consider are those involving short sales.For this reason, we restrict attention to those portfolios without short

Ž .sales. We say that b , b , . . . is a positi"e portfolio program if each b is a0 1 ttŽ 0. J $4measurable function from Z z into ! % 0 . For any positive portfolio#

Ž t#1.program, we define the sequence of return factors by r z &t#1Ž t. Ž t#1. Ž t.b z g z !b z 2 for t ( 0.t t#1 tThe next theorem provides a condition that, if satisfied by the return

sequence generated by at least one positive portfolio, implies that optimalconsumption converges to infinity. Recall that R & Ł t r and * &0 t j$ 1 j t# tR .0 t

Ž .THEOREM 10. Suppose that r , r , . . . is the return sequence for some1 2Ž . Ž .positi"e portfolio program. Then, i P lim * $ " $ 1 impliest !" t

Ž ( . Ž . Ž . Ž (P lim c $ " $ 1, and ii P sup * $ " $ 1 implies P sup ct !" t t ( 0 t t ( 0 t.$ " $ 1.

Ž . #Ž ( . #Ž ( .Proof. i Define d & 1 and d & * " a !" a . Lemma 5 im-0 t t t t 0 0#Ž ( . & # Ž ( . # t 'plies that " a ( #E r " a z from which it follows that d is at t t#1 t#1 t#1 t

nonnegative supermartingale. Therefore, there is a random variable d"& ' Ž . Ž .with E d + 1 and P lim d $ d $ 1 Doob, 1953, p. 324 . So if" t !" t "

#Ž ( .* ! ", it follows that " a ! 0, and therefore, from Lemma 5 thatt t t#Ž ( . Ž ( .u c ! 0, from which we conclude that P lim c $ " $ 1.t t !" t

Ž .ii If * ! " along a subsequence, then, along that subsequence,t#Ž ( . #Ž ( . Ž ( ." a ! 0 which implies u c ! 0 and therefore, P sup c $ " $ 1.t t t t t

We emphasize that there need be no relation between the optimalportfolio and the positive portfolio program that generates the sequence ofinterest rates used in Theorem 10.

APPENDIX

We establish the existence of an optimal solution for the problempresented in Section 6 where the consumer must choose an optimalportfolio and consumption program in each period. Recall that z ) ! n

t


0 Ž . tŽ . $ t t Ž 0 t.4 0and for any z $ . . . , z , z , Z z & z : z $ z , z , . . . , z . Fix z .%1 0 0 1Ž . $Ž t. Ž t. tLet S, d denote the metric space defined by S & a, z : a ( k z , zttŽ 0. 4) Z z , t $ 0, 1, . . . , and

d a, z t , a0, z jŽ . Ž .Ž .1 if j ' t

$ " "½ # #max a % a0 , 3 z , z , . . . , 3 z , z if j $ t ,$ 4Ž . Ž .1 1 t t

n , , # Ž . #where 3 is the metric on ! . For any f : S ! !, let f $ sup f s ,s) SŽ .and let D S denote the set of continuous functions f : S ! ! such thatM tŽ .0 + f + and f a, z is nondecreasing and concave in a. Note that1 % #

Ž . , ,D S is a complete metric space under the metric f % g .t tŽ . $Ž . Ž . 4Recall that K a, z & c, b ) ! + K z : 0 + c + a % b2 , where 2t # t

is the n-vector of ones, denotes the set of feasible consumption andportfolio choices for wealth level a in state z t. Define the operator T onŽ .D S by

t t#1 t#Tf a, z & sup u c # #E f bg # x , z z .Ž . Ž .$ 4Ž .t#1 t#1tŽ . Ž .c , b ) K a , zt

Ž . Ž .LEMMA A.1. T is a contraction mapping D S into D S .

Ž . Ž .Proof. We show first that f ) D S implies Tf ) D S . Suppose f )MŽ .D S . It is immediate that 0 + Tf + and that Tf is monotone nonde-1 % #

creasing.Ž .We show next that Tf is continuous. Since d s, s0 ! 1 implies t $ t0, we

suppress the dependence on t and let z denote z t. For w ) ! n, defineŽ . Ž Ž . Ž . Ž .. Ž .m b, z, w & f bg z, w # x z, w , z, w and let h b, z &t#1 t#1Ž . Ž # . Ž . $Ž . Ž .4H m b, z, w p dw z . Then Tf a, z $ sup u c # # h b, z .Žc, b.) K Ža, z .t

Since u is continuous and K is continuous and compact-valued, it followstŽthat Tf is continuous if h is continuous Hildenbrand, 1974, Corollary, p.

.30 . To show that h is continuous, note that

# # # #h b , z % h b0, z0 + m b , z , w p dw z % p dw z0Ž . Ž . Ž . Ž . Ž .Ž .H15Ž .

# # ## m b , z , w % m b0, z0, w p dw z0 .Ž . Ž . Ž .HŽŽ . Ž .. $ Ž . Ž .4Define d b, z , b0, z0 & max 3 b, b0 , 3 z, z0 , where 3 and 3 are1 2 1 2

Ž .Euclidean metrics. For fixed b, z , the Feller property implies that for any# Ž .Ž Ž # . Ž # .. #% $ 0, there is a ' $ 0 such that H m b, z, w p dw z % p dw z0 ! %1

t 0ŽŽ . Ž .. Ž .for d b, z , b0, z0 ! ' . Since Z z is a finite-dimensional Euclidean1


space, the Feller property also implies that there is a compact set A / ! n

ŽŽ . Ž .. Ž # .nand ' $ 0 such that d b, z , b0, z0 ! ' implies H p dw z02 2 ! %A1 % # %Ž .! . Since f is uniformly continuous on compact sets, we may

MŽŽ . Ž .. # Ž .choose ' $ 0 so that d b, z , b0, z0 ! ' implies m b, z, w %3 3

Ž . # ŽŽ . Ž .. $ 4m b0, z0, w ! % for w ) A. Therefore, if d b, z , b0, z0 ! min ' , ' ,2 3we have

# # #m b , z , w % m b0, z0, w p dw z0Ž . Ž . Ž .H16Ž .

M 1 % #! % # % $ 2% .ž /1 % # M

ŽŽ . Ž .. $ 4Combining these results, we have d b, z , b0, z0 ! min ' , ' , ' im-1 2 3# Ž . Ž . #plies h b, z % h b0, z0 ! 3% .

Ž t.All that remains is to show that Tf a, z is concave in a. But thisŽ t.follows since u is concave and h b, z is concave in b. This establishes

Ž .that Tf ) D S .To establish that T is a contraction mapping, we may use Blackwell’s

Ž . , , , ,Theorem 5 to show that given f , g ) D S , Tf % Tg + # f % g .

Ž .LEMMA A.2. There is a unique " ) D S such that T" $ " .

Ž .Proof. Given that T is a contraction and D S is a complete metricspace, the lemma follows immediately from the contraction mapping fixedpoint theorem.

Ž t. Ž Ž t . Ž t . Ž t .. Ž # t.Define h b, z & H " bg z , w # x z , w , z , w p dw z . Since" t#1 t#1u and h are continuous functions and K is a continuous, compact-valued" tcorrespondence, it follows that we can select measurable functions, C and

t t t t tŽ . Ž Ž . Ž .. Ž .B, such that for all z and all a ( k z , C a, z , B a, z ) K a, ztand

t t t t#1 t#" a, z & u C a, z # #E " B a, z g # x , z zŽ . Ž . Ž .Ž . Ž .t#1 t#1

Ž t. Žfor all a, z ) S. See Hildenbrand, 1974, Corollary, p. 30, Proposition 1,.p. 22, and Lemma 1, p. 55. Furthermore, C is unique since u is strictly

concave, h is concave in b, and K is convex-valued." tt Ž t.For any z and any a ( k z , define the random variables a* $t

Ž ( ( . Ž ( ( . Ž ( .a , a , . . . , c* $ c , c , . . . , and b* $ b* , b , . . . recursively ast t#1 t t#1 t t#1( ( Ž ( j. ( Ž ( j. (follows: a & a, and for j ( t, c & C a , z , b & B a , z , and a &t j j j j j

( Ž j. Ž j. Ž .b g z # x z . The next theorem establishes that c*, b* is an opti-j%1 j jmal consumption-portfolio program for the consumer, and a* is thecorresponding sequence of wealth levels when starting with wealth level aand state z t.


Ž . Ž . & " Ž . j% t # t 'THEOREM A.1. i c*, b* maximizes E Ý u c # z subject toj$ t j(j t jŽŽ . Ž . # . Ž .P c , b ) K a , z , j ( t z $ 1, where a & a and a & b g z #j j j j t j j%1 j

Ž j. Ž . Ž t. Ž t. & " Ž ( . j% t # t 'x z for j $ t. ii For a ( k z , " a, z $ E Ý u c # z , whichj t j$ t jis a strictly conca"e function of a.

Ž .Proof. It follows from Lemma A.2 and Strauch 1966, Theorem 5.1bŽ t. & " Ž ( . j% t # t ' Ž .that " a, z $ E Ý u c # z . Condition i then follows fromj$ t t

Blackwell’s Theorem 6. The strict concavity of " follows from the strictconcavity of u.

The following results are used in Sections 4 and 5. As in those sections,Lemma A.3 is restricted to the case where there is only one asset andincome is defined so that x ( 0 and k $ 0 for all t ( 0. As in the text, wet t

Ž . Ž t.let " a & " a, z .t

Ž . Ž #Ž . #Ž . .LEMMA A.3. P # r + 1, t ( 1 $ 1 implies P " 0 $ u 0 , t ( 0 $t t1.

#Ž .Proof. Without loss of generality it is sufficient to prove that " 0 $0#Ž . #Ž .u 0 . Note first that if u 0 $ ", the lemma follows immediately from

#Ž .Lemma 1 of the text. So we may suppose that u 0 ! ". Since Lemma 1#Ž . #Ž . #Ž . #Ž . 0 %implies " 0 ( u 0 , we need to show that " 0 + u 0 . Let c and c0 0

denote the optimal consumption programs for a $ 0 and a $ % $ 0,0 0respectively, and let a0 and a% denote the corresponding sequence ofwealth levels.

Ž 0 % .We show first that P a + a , t ( 0 $ 1. Suppose not, and let t be thet tŽ 0 % . nsmallest j such that P a $ a $ 0. For any z ) ! , we havej j t

a% z t%1 , z % a0 z t%1 , zŽ . Ž .t t t t

$ r z t%1 , z a% z t%1 % a0 z t%1 % c% z t%1Ž . Ž . Ž .Ž . Ž . ŽŽt t t%1 t%1 t%1 17Ž .

%c0 z t%1 .Ž . . .t%1

Therefore, there is a nonnull set of z t%1 such that a0 + a% , c0 !t%1 t%1 t%10 Ž 0 % # t%1.a , and P a $ a z $ 1. From this, Lemma 1 and the strict concav-t%1 t t

ity of " implyt

% 0 % 0 t%1#" a + #E r " a zŽ . Ž .t%1 t%1 t t t18Ž .

# % t%1 # %#! #E r " a z + " a ,Ž . Ž .t t t t%1 t%1

which violates the concavity of " .t%1


Ž % 0 . %1 Ž % 0. t Ž % 0. %1By definition, a % a R $ a % a % Ý c % c R fort#1 t#1 0, t#1 0 0 j$ 0 j j 0 jŽ 0 % .any t ( 0. Therefore, P a + a , t ( 0 $ 1 impliest t

t% 0 %1E c % c RŽ .Ý j j 0 j

j$ 0 19Ž .% 0 % 0 %1$ a % a % E a % a R + % .Ž . Ž .0 0 t#1 t#1 0, t#1

Furthermore, using Lemma 1 and the strict concavity of u and " , it alsotŽ 0 % . Ž 0 % .follows from P a + a , t ( 0 $ 1 that P c + c , t ( 0 $ 1. Then,t t t t

Ž . Ž %1 t .since P # r + 1, t ( 1 $ 1 implies P R ( # , t ( 0 $ 1, it followst 0 tfrom the concavity of u that

"% 0 t" % % " 0 $ E u c % u c #Ž . Ž . Ž . Ž .Ž .Ý0 0 t t

t$ 0

"# 0 % 0 t+ E u c c % c #Ž . Ž .Ý t j j

t$ 020Ž .

"# 0 % 0 %1 #+ E u c c % c R + % u 0 .Ž .Ž . Ž .Ý t j j 0 t

t$ 0

Letting % go to zero proves the result.

The arguments behind Lemma A.3 exploit only the strict concavity of u.As long as " is well defined, the conclusions still follow.

ŽLEMMA A.4. Let D be an open, con"ex subset of ! and let f : D !j. Ž . Ž .!, j ( 1 be a sequence of conca"e functions such that f x ! f x for allj 0

#Ž . #Ž . %Ž . %Ž .x ) D. Then f x + lim inf f x and f x ( lim sup f x .0 j!" j 0 j!" j

Ž . Ž . #Ž .Ž .Proof. Since f is concave, f y % f x + f x y % x for x, y ) D.j j j jŽ . Ž .Then f ! f implies that f is concave and f y % f x +j 0 0 0 0

#Ž .Ž . #Ž . #Ž .lim inf f x y % x , which implies f x + lim inf f x . The proof ofj!" j 0 j jthe second half of the lemma is similar.

REFERENCES

Ž .Bewley, T. 1977 . ‘‘The Permanent Income Hypothesis: A Theoretical Formulation,’’ Journalof Economic Theory 16 , 252!292.

Ž .Bewley, T. 1980a . ‘‘The Permanent Income Hypothesis and Long-run Economic Stability,’’Journal of Economic Theory 22, 377!394.

Ž .Bewley, T. 1980b . ‘‘The Permanent Income Hypothesis and Short-run Price Stability,’’Journal of Economic Theory 23, 323!333.


Ž . ŽBewley, T. 1980c . ‘‘The Optimum Quantity of Money,’’ in Models of Monetary Economics J..Karenken and N. Wallace, Eds. , Minneapolis: Federal Reserve Bank.

Ž .Bewley, T. 1983 . ‘‘A Difficulty with the Optimum Quantity of Money,’’ Econometrica 51,1485!1504.

Ž .Billingsley, P. 1968 . Con"ergence of Probability Measures. New York: Wiley.Ž .Blackwell, D. 1965 . ‘‘Discounted Dynamic Programming,’’ Annals of Mathematical Statistics

36 , 226!235.Ž .Doob, J. L. 1953 . Stochastic Processes. New York: Wiley.

Ž .Friedman, M. 1957 . A Theory of the Consumption Function. New York: National Bureau ofEconomic Research.

Ž .Hildenbrand, W. 1974 . Core and Equilibria of a Large Economy. Princeton: Princeton Univ.Press.

Ž .Meyer, P. A. 1966 . Probability and Potentials. Waltham, MA: Blaisdell.Ž .Maitra, A. 1968 . ‘‘Discounted Dynamic Programming on Compact Metric Spaces,’’ Sankhya

Series A 30 , 211!216.Ž .Schechtman, J. 1976 . ‘‘An Income Fluctuation Problem,’’ Journal of Economic Theory 12,

218!241.Ž .Schechtman, J., and Escudero, V. L. S. 1977 . ‘‘Some Results on ‘An Income Fluctuation

Problem’,’’ Journal of Economic Theory 16 , 151!166.Ž .Sotomayor, M. 1984 . ‘‘On Income Fluctuations and Capital Gains,’’ Journal of Economic

Theory 32, 14!35.Ž .Strauch, R. E. 1966 . ‘‘Negative Dynamic Programming,’’ Annals of Mathematical Statistics

37 , 871!890.Ž .Yaari, M. E. 1976 . ‘‘A Law of Large Numbers in the Theory of Consumer’s Choice Under

Uncertainty,’’ Journal of Economic Theory 12, 202!217.

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