Review of Economic Dynamics 3, 365᎐395 Ž2000.
doi:10.1006rredy.2000.0098, available online at http:rrwww.idealibrary.com on
Optimal Intertemporal Consumption
under Uncertainty1
Gary Chamberlain
Department of Economics, Har¨ ard Uni¨ ersity, Cambridge, Massachusetts 02138
and
Charles A. Wilson 2
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 consumer
who maximizes the discounted sum of utilities subject to a sequence of budget
constraints where both the interest rate and his income are stochastic. We show
that if the income and interest rate processes are sufficiently stochastic and the
long run average rate of interest is greater than or equal to the discount rate, then
consumption eventually grows without bound with probability one. We also establish conditions under which the borrowing constraints must be binding and examine how the income process affects the optimal consumption program. Journal of
Economic Literature Classification Number: D91. 䊚 2000 Academic Press
Key Words: uncertainty; consumption; permanent income hypothesis.
1. INTRODUCTION
We shall consider the following problem. In each period t, a consumer
receives income x t . After receiving his income, he must decide how much
to consume in that period, c t , and how much to save for future consumption. His savings earn a gross rate of return, rtq1 , so that the value of his
1
Support from the National Science Foundation and the Alfred P. Sloan Foundation is
gratefully 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: Charles.Wilson@nyu.edu.
365
1094-2025r00 $35.00
Copyright 䊚 2000 by Academic Press
All rights of reproduction in any form reserved.
366
CHAMBERLAIN AND WILSON
assets at the beginning of period t q 1 Žafter he has received income x tq1 .
is given by the relation a tq1 s rtq1Ž a t y c t . q x tq1. The consumer’s problem is to choose a consumption plan to maximize the expectation of
Ý⬁ts0 uŽ c t . ␤ t subject to a budget constraint, where u is an increasing,
strictly concave function. His only source of uncertainty is that x t and rt
follow some stochastic process. The question we address is: What happens
to the levels of c t and a t as t goes to infinity?
The motivation of the paper lies in the work of several authors who
attempt to formalize the permanent income hypothesis of Friedman Ž1957..
The first of these papers, Yaari Ž1976. and Schechtman Ž1976., considers
the case where utility is not discounted Ž ␤ s 1., the interest rate is zero
Ž rt s 1., and the horizon is finite; i.e., the consumer maximizes ÝTts0 uŽ c t ..
Assuming that income x t is identically and independently distributed and
imposing only a solvency constraint on lifetime consumption, Yaari shows
that as T ª ⬁, the optimal consumption plan requires c Tt , the consumption
in period t with horizon T, to converge with probability one to Ew x t x for
all t. Schechtman tightens the solvency constraint to forbid borrowing and
obtains the weaker result that, as both T and t go to infinity, c Tt converges
to Ew x t x with probability one. He also establishes that a t converges to
infinity. Bewley Ž1977. retains the restriction on borrowing but considers a
more general case where both x t and u t follow a stationary stochastic
process. He also allows for discounting Ž ␤ - 1.. He obtains the analogous
result that uXt Ž c Tt . converges to a constant as T ª ⬁, t ª ⬁, and ␤ ª 1, so
that, asymptotically, the consumer becomes insulated from risk. Some of
the implications of this result are developed in Bewley Ž1980a, 1980b..
In this paper we investigate the asymptotic properties of the optimal
consumption program for the infinite horizon model when income and
interest rates are stochastic and the consumer discounts a bounded utility
function. We put no additional restrictions on the form of the stochastic
process Žsuch as stationarity ., and we allow for arbitrary restrictions on the
permissible level of borrowing in each period. Under these assumptions,
we show that if the discount rate is smaller than the long run average rate
of interest, then c t will converge to infinity almost surely. When the long
run interest rate is equal to the discount rate, the asymptotic properties of
c t depend on how stochastic the income stream is. If the income stream is
certain, c t is nondecreasing and converges to the supremum of the
maximum sustainable consumption level starting from the minimum permissible level of wealth in any given period. However, if the income stream
is suitably stochastic, c t must converge to infinity almost surely. This result
extends some of the results of Sotomayor Ž1984., who examines the case
where income is identically and independently distributed Žbut allows for
utility to be unbounded..
CONSUMPTION UNDER UNCERTAINTY
367
Our analysis is based on the following observation. For simplicity,
suppose that the interest factor is a constant, ␥ . Let ¨ Ž a t , z t . be the
expected discounted utility from following the optimal policy, given asset
level a t and information state z t at time t. Then the first order conditions
for utility maximization require ¨ qŽ a t , z t . G ␤␥ Ew ¨ qŽ a tq1 , z tq1 .< z t x, where
¨ q denotes the right-hand derivative of ¨ with respect to a t . When
␤␥ G 1, this implies that ¨ qŽ a t , z t . is a supermartingale. By a theorem of
Doob Ž1953., this implies a finite random variable d⬁ such that ¨ qŽ a t , z t .
converges to d⬁ with probability 1. If ␤␥ ) 1, ¨ q Ž a t , z t . G ␤␥
Ew ¨ qŽ a tq1 , z tq1 .< z t x implies ¨ qŽ a t , z t . ª 0 from which it is easy to show
that c t must converge to infinity. If ␤␥ s 1, this argument can still be used
to show that c t converges with probability 1 to some extended random
variable c⬁ , but in this case, c⬁ need not equal infinity. In fact, as we show
in Theorem 3, if x t is not stochastic, c⬁ is generally not infinite. However,
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 w b, b q ␧ x and a ␶ - ⬁ such that there is
positive probability that c␶ g w b, b q ␧ x, and, with probability greater than
1 y ␧ , c t g w b, b q ␧ x for all t G ␶ whenever c␶ g w b, b q ␧ x. But if
c ( c␶ for t G ␶ , then a t will diverge to "⬁ unless Ž␥rŽ␥ y 1.. c␶ s a␶ q
Ý⬁js ␶q1 x j␥ ␶yj. We can show that it is not optimal to have c ( c␶ if a t ª ⬁.
So conditional on c␶ ( b, it follows that Ý⬁js ␶q1 x j␥ ␶yj ( Ž␥rŽ␥ y 1.. b y
a␶ . But since a␶ is known at time ␶ , this implies that the conditional
variance of Ý⬁js ␶q1 x j␥ ␶yj can be made arbitrarily small for ␧ sufficiently
small. Therefore, unless there is perfect foresight, savings must diverge.
We conclude that if the income stream is suitably stochastic, then c t
converges to infinity with probability 1. In the body of the paper, these
arguments are generalized to allow for a stochastic interest rate. In
particular, if the discount rate is equal to the long run average rate of
interest, then c t grows without bound if there is sufficient uncertainty in
the joint distribution of income and interest rates. This case arises in the
treatment of the optimum quantity of money by Bewley Ž1980c, 1983..
We see that with discounting and a positive interest rate, the only
counterpart to the Yaari᎐Schechtman᎐Bewley result is a zero limit for
marginal utility. If ␤␥ G 1, then consumption does converge, but it converges to bliss, not the expected value of discounted income.3 We should
also note that although both Schechtman and Bewley use the convergence
theorem for supermartingales, their argument requires either the law of
3
An example of Schechtman and Escudero Ž1977. shows that convergence to bliss with
probability 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.
368
CHAMBERLAIN AND WILSON
large numbers or the ergodic theorem, which in turn requires a stationarity
assumption. By itself, however, the supermartingale theorem requires no
additional assumptions. It is the stochastic generalization of the result that
a bounded, monotone sequence converges to a finite limit. Consequently,
when we eliminate the need for a law of large numbers by introducing
discounting, we are able to considerably weaken our assumptions on the
stochastic process and still obtain convergence of c t , even with a positive
interest rate.
If the underlying process is stationary, there are alternative conditions
on the interest rate sequence which ensure that consumption converges to
infinity. In Subsection 4.4, we show that if the interest rate sequence is
stationary and ergodic, then consumption will grow without bound if
Ewlog ␤ rt x ) 0. If Ewlog ␤ rt x s 0 and an additional condition is satisfied
which guarantees that the interest rate has sufficient variance and its
dependence on the past dies out sufficiently fast, then some subsequence
of consumption will grow without bound. In both cases, the result follows
from the fact that ␤ t Ł tjs1 r j Žor some subsequence . converges to infinity
with probability 1.
Section 5 deals with some implications of the budget constraint. Rather
than restrict ourselves to a no borrowing constraint at the outset, we allow
the lower bound on borrowing in any period to be an arbitrary function of
the consumer’s information in that period. We then note that by redefining income and wealth, the original problem is equivalent to a problem
where income is nonnegative in each period and no borrowing is permitted. Making this translation explicit emphasizes that these assumptions are
not really substantive restrictions on the model, but are merely a simpler
representation of a more general model. In particular, they are consistent
with the possibility that the only constraint on borrowing is an intertemporal budget constraint. With this interpretation, we establish some conditions under which the borrowing constraint is never binding as well as
conditions under which it must be binding in some periods. We also show
how the optimal program changes as the borrowing constraints are relaxed.
In the final section of the paper, we show how some of our results can
be extended to the case where the consumer chooses a portfolio of several
risky assets. The main complication introduced by this extension is that the
interest rate becomes endogenous.
2. ASSUMPTIONS AND NOTATION
n
⺢ is the real line, and ⺢ n is n-dimensional Euclidean space. ⺢q and ⺢q
refer to the corresponding subsets of nonnegative elements. Nonnegative
integers are denoted by t, ␶ , i, j. Other lower-case Roman letters generally
CONSUMPTION UNDER UNCERTAINTY
369
refer to random variables or their realizations, and lower-case Greek
letters Žparticularly, ␧ , ␦ , and ␣ . generally refer to real numbers.
Let z ' Ž . . . , zy1 , z 0 , z1 , . . . . be a stochastic process with transition
probabilities pŽ dz t < z ty1 . used to define conditional expectation, where
z t g ⺢ n and z t ' Ž z ty1, z t . ' Ž . . . zy1 , z 0 , z1 , . . . , z t .. Interpret z t as new
information the consumer receives at time t and z t as the information
state at time t. Let Z t Ž z 0 . s z t : z jt s z j0 , j F 04 be the set of states at
time t which are consistent with state z 0 . Except for Section 4.4 where we
consider the case where z is stationary, we will take z 0 as fixed.
We assume that the income of the consumer at time t G 0 is a continuous function x t : Z t Ž z 0 . ª ⺢.4 At each time t G 1, the consumer may
borrow and lend between time t y 1 and time t at interest factor rt which
is assumed to be a positive, continuous function rt : Z t Ž z 0 . ª ⺢q. For
j
t G 0, let R t t ' 1. For j ) t G 0, let R t j ' Ł kstq1
r k denote the interest
y1
factor between periods t and j, and let R t j denote its inverse.
The borrowing constraint of the consumer at time t G 0 is a continuous
function, k t : Z t Ž z 0 . ª ⺢ that satisfies an intertemporal consistency constraint,
P Ž rt k ty1 q x t G k t , t G 0 < z 0 . s 1,
Ž 1.
where r 0 ky1 ' 0. Let x ' Ž x 0 , x 1 , . . . . and k ' Ž k 0 , k 1 , . . . .. Interpret k t as
the minimum amount of wealth, measured in terms of period t consumption, that the individual may hold at the end of period t Žafter he receives
his period t income and spends his period t consumption.. Equation Ž1.
requires that the lower bound on current net wealth be consistent with the
borrowing constraint the consumer will face in the next period. It implies
that current wealth can never be so low that it may become impossible for
the individual to satisfy his borrowing constraint in the next period even if
nothing is consumed in the current period. Defining r 0 ky1 ' 0 is simply a
convention which implies that the consumer’s initial wealth is derived
solely from his period 0 income.
Since the consumer’s decision at each information state depends only on
what he knows at that state, a consumption program, c ' Ž c 0 , c1 , . . . ., is a
sequence of Borel measurable functions, c t : Z t Ž z 0 . ª ⺢q, t G 0. Let u:
⺢qª ⺢ be an increasing, continuous, strictly concave function with 0 F u
F M ' sup c ª⬁ uŽ c . - ⬁. For any consumption program c, uŽ c t . is the
undiscounted utility to the consumer from his consumption at time t. We
suppose the consumer discounts utility by a factor ␤ Ž0 - ␤ - 1. in each
period. 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 a
variable as one of the coordinates of z t .
370
CHAMBERLAIN AND WILSON
consumption program c to maximize EwÝ⬁js0 uŽ c j . ␤ j < z 0 x subject to
P ŽÝtjs0 Ž x j y c j . R jt G k t , t G 0 < z 0 . s 1.
Some of our results require some regularity assumptions on the transition probabilities. We assume that for any z 0 the Feller property holds: if
f : ⺢ n ª ⺢ is a bounded continuous function, then g Ž z t . s H f Ž w . pŽ dw < z t .
is a continuous function from Z t Ž z 0 . to ⺢. This condition combined with
the requirement that x t , rt , and k t be continuous functions is shown in
Theorem A.1 Žof the Appendix. to guarantee that the consumer’s problem
has a solution.
Before proceeding with our analysis, it will be useful to redefine our
income variable so that the problem may be reduced to a simpler form. As
stated, the problem of the consumer is to choose a consumption program
that maximizes the discounted sum of utility subject to a Žpossibly random.
borrowing constraint. Aside from technical considerations, there are no
restrictions on either the income stream or the borrowing constraint other
than the requirement that the borrowing constraint satisfy Eq. Ž1.. For
instance, if we assume that x t is nonnegative and we wish to prohibit
borrowing, we may set k t s 0 for all t. Alternatively, if we wish to impose
only an intertemporal solvency constraint, the appropriate constraint is
< t.
45
k t Ž z t . ' inf ␬ g ⺢: P Ž ␬ q Ý⬁jstq1 x j Ry1
t j G 0 z s 1 . However, we also
allow for intermediate cases as well as cases where lim t ª⬁ k t Ry1
0 t / 0.
Whatever additional assumptions we impose on x t and k t , however, we
may always translate the variables so that the problem is equivalent to one
in which income is nonnegative and borrowing is not permitted. The key is
to redefine income at time t to be the increase in available purchasing
power the consumer receives at time t. This is the difference between x t ,
the income he receives in period t, and k t y rt k ty1 , the change in the
minimum allowable wealth from period t y 1 to period t, measured in
units of income at time t. Letting ˆ
x t ' x t y k t q rt k ty1 and letting ˆ
kt ' 0
for all t G 0, Eq. Ž1. may be stated as P Ž ˆ
x t G 0, t G 0 < z 0 . s 1, and the
< 0.
borrowing constraint may be stated as P ŽÝtjs0 Ž ˆ
x j y c j . Ry1
0 j G 0, t G 0 z
s 1. Note that since x t , k t , and k ty1 are all continuous functions of z t , x̂ t
is also a continuous function of z t. Consequently, the technical requirements and information restrictions on ˆ
x t and ˆ
k t are satisfied. Unless we
indicate otherwise, we shall assume for the remainder of the paper that
this translation has already been made so that we may restrict attention to
the case x t G 0 and k t s 0. However, it is occasionally useful to emphasize
the interpretation of our results in the original framework.
5
To ensure the existence of some feasible consumption program consistent with ky1 s 0,
< 0 . s 1, and to ensure that yk t - ⬁ so that an optimal
we then require P ŽÝ⬁js 0 x j Ry1
0j G 0 z
t
< t.
consumption program exists, we then require P ŽÝ⬁js t x j Ry1
t j - ⬁ z ) 0 for all z .
371
CONSUMPTION UNDER UNCERTAINTY
To establish the results that follow, it is convenient to work with the
value function of future discounted utility defined over the level of
accumulated assets. For any state z t and wealth level a G 0, define
¨ Ž a, z t . ' max E
c
⬁
Ý u Ž c j . ␤ jyt < z t
jst
subject to
P a y ct q
ž
␶
< t
Ý Ž x j y c j . Ry1
t j G 0, ␶ G t z
jstq1
/
s1
to denote the expected utility of the consumer starting at time t with
information z t , given purchasing power a at time t. Following the arguments of Blackwell Ž1965., Strauch Ž1966., and Maitra Ž1968., we establish
as Theorem A.1 in the Appendix that ¨ Ž a, z t . s max 0 F c F a uŽ c . q
␤ Ew ¨ ŽŽ a y c . rtq1 q x tq1 , z tq1 .< z t x4.6
With the exception of Subsection 4.4, which uses the stationary assumptions, we shall assume the initial information state z 0 is fixed so that
unconditional expectations should be understood to be conditional on z 0 .
It is also frequently convenient to suppress explicit reference to the state
z t , so that for a given state z t , we sometimes write ¨ Ž a, z t . ' ¨ t Ž a. and
regard ¨ t Ž a. as a random variable. Also, to simplify the statement of some
of the theorems, we sometimes state properties of the optimal program
as if they hold for all z t , although we prove our results on a set of probability 1.
3. PRELIMINARY RESULTS
The existence and uniqueness of a solution to the consumer’s maximization problem is established in the Appendix as Theorem A.1. We also
establish there that ¨ t is a bounded, strictly concave function. Therefore,
right- and left-hand derivatives exist. For any function f : w0, ⬁x ª ⺢, let
fqŽ x . denote the right-hand derivative of f Ž x ., let fyŽ x . denote its
left-hand derivative when x ) 0, and let fqŽ 0. s lim x x 0 fqŽ x .. Let c* s
Ž cU0 , cU1 , . . . . denote the consumption program which solves the consumer’s
6
The result is actually established for the more general model with many assets discussed
in Section 6.
372
CHAMBERLAIN AND WILSON
maximization problem, and let a* ' Ž aU0 , aU1 , . . . . denote the corresponding
sequence of random variables representing the level of the consumer’s
wealth in each period after current income has been received but before
consumption has been spent. The wealth sequence a* is defined recurŽ
.
sively by aU0 ' x 0 and aUt ' rt Ž aUty1 y cUty1 . q x t s x t q Ýty1
js0 x j y c j R jt ,
for t ) 0.
With these definitions, we may state some conditions which an optimal
program must satisfy. The proof is quite standard and will be omitted.
Ž U.
qŽ cUt ., ␤ Ew rtq1¨ q
Ž U .< t x4 . Žb. SupLEMMA 1. Ža. ¨ q
t a t s max u
tq1 a tq1 z
U
pose a t ) 0. Then
U
¨y
t Ž at .
¡u
y
~
Ž cUt .
if aUt y cUt s 0
U
y
t
s ␤ E rtq1¨ tq1 Ž a tq1 . < z
¢min u
y
U
< t
Ž cUt . , ␤ E rtq1¨ y
tq1 Ž a tq1 . z
if cUt s 0
4
if 0 - aUt y cUt - aUt .
For t G 0, let ␪ t ' ␤ t R 0 t . If the level of consumption in period 0 is
equal to the level of consumption in period t, then ␪ t measures the rate at
which the consumer can transform discounted utility in period t for utility
Ž U . is a supermartingale.
in period 0. Condition Ža. establishes that ␪ t¨ q
t at
Our first theorem uses the supermartingale convergence theorem to
Ž U . must converge to some random variable e⬁ . The
establish that ␪ t¨ q
t at
Ž U . must be
only problem is that to apply the convergence theorem, ¨ q
t at
Ž .
finite which is not true if x 0 s 0 and ¨ q
0 0 s ⬁. This difficulty can be
avoided if we assume that the discounted value of the entire income
stream is always positive. In this case, there is a stopping time ␶ such that
aU␶ ) 0, which allows us to apply the convergence theorem to the sequence
Ž ␪␶ ¨␶qŽ aU␶ ..y1␪␶qt¨␶qqt Ž aU␶qt ..
THEOREM 1.
If P Žx s 0. s 0, then there is a Ž real-¨ alued . random
Ž U.
.
¨ ariable e⬁ such that P Žlim t ª⬁ ␪ t¨ q
t a t s e⬁ s 1.
Proof. P Žx s 0. s 0 implies a stopping time ␶ such that x␶ ) 0. ThereŽ U . is finite. Let
fore, aU␶ G x␶ ) 0 and the concavity of ¨␶ imply that ¨ q
t at
U y1
U
q
q
d t ' Ž ␪␶ ¨␶ Ž a␶ .. ␪␶qt¨␶qt Ž a␶qt . for t G 0. Since ␶ is a stopping time,
Lemma 1 implies that Ž d 0 , d1 , . . . . is a nonnegative supermartingale ŽMeyer,
1966, p. 66.. But since d 0 s 1, there is a random variable d⬁ with
CONSUMPTION UNDER UNCERTAINTY
373
Ew d⬁ F 1x such that P Žlim t ª⬁ d t s d⬁ . s 1 ŽDoob, 1953, p. 324.. Then if
.
Ž U.
we define e⬁ ' ␪␶ ¨␶qŽ aU␶ . d⬁ , it follows that P Žlim t ª⬁ ␪ t¨ q
t a t s e⬁ s 1.
In the next section, we use Theorem 1 to establish some conditions
under which consumption must converge to infinity. To establish these
conditions we use the fact that whenever assets grow without bound,
consumption also grows without bound. This result is established as
Lemma 2.
cUt
LEMMA 2.
G ␣ 1.
For any ␣ 1 ) 0, there is an ␣ 2 ) 0 such that aUt G ␣ 2 implies
M
Proof. From Lemma A.2, 0 F ¨ t Ž aUt . - 1 y
␤ . Therefore, the concavity
U q U
U
M
Ž
of ¨ t implies a t ¨ t Ž a t . F ¨ t Ž a t . y ¨ t Ž0. F 1 y
␤ . Choose ␣ 2 s Mr 1 y
U
qŽ
Ž
.
.
␤ . uqŽ ␣ 1 .. Then if aUt G ␣ 2 , Lemma 1 implies uq Ž cUt . F ¨ q
a
F
¨
t
t
t ␣2
F MrŽ1 y ␤ . ␣ 2 s uq Ž ␣ 1 .. The concavity of u then implies cUt G ␣ 1.
With these results in hand, we are prepared to examine the limiting
behavior of the optimal consumption sequence.
4. CONVERGENCE THEOREMS
Ž U . must
Recall that Theorem 1 establishes that the function ␪ t¨ q
t at
converge to some random variable e⬁ . In this section we demonstrate that
the implications of this result for the limiting behavior of the optimal
consumption sequence depend primarily on the limiting value of ␪ t .
Roughly, our results may be summarized as follows. If lim t ª⬁ ␪ t s ⬁, the
optimal consumption sequence of the consumer must grow without bound,
regardless of the properties of the income and interest rate sequences. If
the limiting value Žor values. of ␪ t is bounded above and away from zero
and the income stream is suitably stochastic, then consumption still grows
without bound. However, if the income sequence is not stochastic, then the
consumption sequence generally converges to a finite limit.
These results may also be interpreted in terms of the relationship
between the rate at which the consumer discounts future utility and the
long run rate of interest. In Section 2, we defined rt to be the one period
j
interest factor between period t y 1 and period t and R t j ' Ł kstq1
r k to
be the accumulated interest factor between periods t and j. Therefore,
jyt .
R1rŽ
represents the a¨ erage interest factor between periods t and j. If
tj
t
lim t ª⬁ R1r
0 t exists, we call it the long run interest factor; otherwise, we say
that the long run interest rate fluctuates. Now consider the meaning of
t
1r t
␪ t ' ␤ t R 0 t . Rewriting the expression, we obtain ␤ R1r
. Therefore,
0 t s ␪t
374
CHAMBERLAIN AND WILSON
the long run rate of interest is equal to the discount rate if and only if
lim t ª⬁ ␪ t1r t s 1. Clearly, if the limiting values of ␪ t are positive and finite,
then 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 rate
of interest and the rate at which the consumer discounts utility. If the long
run rate of interest is greater than the discount rate, then consumption
grows without bound as long as the consumer earns a positive income in
some period. If the long run rate of interest is equal to the discount factor,
then consumption generally converges to infinity only if there is sufficient
uncertainty in either the income or interest rate sequences.
All of our results are established for the case where both the income
and interest rate sequences may be stochastic. In many cases, however, the
intuition behind our results and the meaning of the assumptions become
more apparent when only the income sequence is stochastic. Throughout
this section, therefore, we will frequently focus the discussion on the
special case where the interest rate is a constant. We conclude the section
with an interpretation of our results in the context of a stationary distribution of interest rates.
We turn first to the case where the long run interest rate exceeds the
rate at which the consumer discounts utility.
4.1. Lim ␪ t s ⬁
Our main result for the case where the long run rate of interest exceeds
the discount rate is summarized in the following theorem.
THEOREM 2. Suppose P Žx s 0. s 0. Then Ži. P Žlim sup t ª⬁ ␪ t s ⬁. s 1
implies P Žlim sup t ª⬁ cUt s ⬁. s 1, and Žii. P Žlim t ª⬁ ␪ t s ⬁. s 1 implies
P Žlim cUt s ⬁. s 1.
Proof. Theorem 1 implies that if P Žlim sup t ª⬁ ␪ t s ⬁. s 1 and P Žx s
Ž U.
.
0. s 0, then P Žlim inf t ª⬁¨ q
t a t s 0 s 1. Lemma 1 then implies that
U
q
P Žlim inf t ª⬁ u Ž c t . s 0. s 1, and therefore, that P Žlim sup t ª⬁ cUt s ⬁. s
1. This proves part Ži.. The proof of part Žii. is similar.
Roughly, Theorem 2 says that consumption grows without bound so long
as the long run rate of interest always exceeds the discount rate. If the
long run interest rate fluctuates, but some subsequence always exceeds the
discount rate, then some subsequence of consumption grows without
bound. When the interest rate is constant, the theorem may be restated as
follows.
CONSUMPTION UNDER UNCERTAINTY
375
COROLLARY 1. Suppose that rt s ␥ for t G 1. If P Žx s 0. s 0 and
␤␥ ) 1, then P Žlim t ª⬁ cUt s ⬁. s 1.
4.2. The Case of Certainty: ␤␥ s 1
For the remainder of this section, we are concerned primarily with the
case where the long run interest rate is equal to the rate at which the
consumer discounts future utility. To underline the importance of the role
of uncertainty in the main results which follow, we first concentrate on the
case where the income sequence is nonstochastic and the interest rate in
each period is equal to the rate at which the consumer discounts utility.
Again denote the constant interest factor by ␥ .
When ␤␥ s 1, the first-order conditions for utility maximization imply
that the consumer tries to equalize consumption in each period. As long as
such a program does not violate the borrowing constraint, this condition
characterizes the solution. If such a program does violate the borrowing
constraint, however, the consumer must choose a consumption stream with
unequal levels of consumption over time. Nevertheless, consumption never
decreases over time. Otherwise, by saving more in an earlier period and
consuming more later, lifetime utility can be increased. This observation
implies that the consumption level approaches a limit Žpossibly equal to ⬁.,
which we characterize in the next theorem.
Define yt ' ŽŽ␥ y 1.r␥ .Ý⬁jst x j␥ tyj to be the supremum of those consumption levels that can be sustained indefinitely when we consider only
the borrowing constraints in the distant future, given that the borrowing
constraint is binding in period t y 1. Our next theorem states that cUt
converges to the supremum of these maximum sustainable consumption
levels.
THEOREM 3.
sup t yt .
If x is not stochastic, then ␤␥ s 1 implies lim t ª⬁ cUt s
Proof. We show first that for t G 1, either Ža. cUty1 s cUt , or Žb. cUty1 U
c t and cUty1 s aUty1 Žin which case, aUt s x t .. Suppose cUty1 - aUty1. If
cUty1 s 0, then Lemma 1 and the strict concavity of u and ¨ ty1 imply
q Ž U .
qŽ .
Ž .
¨q
0 , which violates Lemma A.3. So suppose 0 ty1 0 ) ¨ ty1 a ty1 G u
U
U
Ž U .
c ty1 - a ty1. Then, on one hand, Lemma 1 implies uy Ž cUty1 . G ¨ y
ty1 a ty1
U
U
U
U
U
qŽ
qŽ
Ž
.
.
G ¨q
c t ., and therefore, 0 F c ty1 F c t . On the
ty1 a ty1 G ¨ t a t G u
y Ž U .
Ž U .
other hand, Lemma 1 also implies uqŽ cUty1 . F ¨ q
ty1 a ty1 F ¨ ty1 a ty1 F
yŽ U .
Ž U.
¨y
c t , and therefore, cUt F cUty1.
t at F u
Let c ' lim t ª⬁ cUt and let y ' sup t yt . We show first that c F y. Suppose
not and let t be the smallest ␶ G 0 such that c␶U ) y. Then, condition Žb.
376
CHAMBERLAIN AND WILSON
above implies aUt s x t . From the definition of aUt , it then follows that
ty1
Ý Ž x j y cUj . ␥yj s 0.
Ž 2.
js0
Conditions Ža. and Žb. also imply that c␶U is nondecreasing in t. Therefore,
cUj ) yt for all j G t, so that ŽŽ␥ y 1.r␥ .Ý⬁jst cUj ␥ tyj G cUt ) yt s ŽŽ␥ y
1.r␥ .Ý⬁jst x j␥ tyj, which implies a ␶ sufficiently large such that
␶
Ý Ž x j y cUj . ␥ tyj - 0.
Ž 3.
jst
Combining Ž2. and Ž3. then yields Ý␶js0 Ž x j y cUj .␥yj - 0, violating the
budget constraint of the consumer’s maximization problem.
To show that c G y, again assume the contrary. Then there is a yt such
that cUj - yt for all j G 0, which implies
⬁
⬁
jst
jst
Ý cUj ␥ tyj - Ý x j␥ tyj .
Ž 4.
But if cUt is feasible, then
ty1
Ý Ž x j y cUj . ␥yj G 0.
Ž 5.
js0
It follows from Ž4. and Ž5. that there is an ␧ ) 0 and ˆ
␶ G t such that, for
all ␶ G ˆ
␶ , Ý␶js0 Ž x j y cUj .␥yj ) ␧ . But then ˆ
c, defined by ˆ
c j s cUj for j / ˆ
␶
U
and ˆ
c␶ˆ s c␶ˆ q ␧ , also satisfies the consumer’s budget constraint. But then
Ý⬁js0 uŽ ˆ
c j . ␤ j ) Ý⬁js0 uŽ cUj . ␤ j implies that c* is not an optimal consumption program.
For our purposes the main implication of this theorem is that when the
income stream is certain, the consumption sequence generally converges
to a finite limit. Consumption grows without bound only if the discounted
value of future income is not bounded. However, the theorem also has a
noteworthy implication in the context of the original formulation of the
model.
Suppose that x is an arbitrary income sequence and k is an intertemporally consistent sequence of borrowing constraints from which we derive
ˆx t ' x t y k t q ␥ k ty1 G 0. Then, yt ' ŽŽ␥ y 1.r␥ .Ý⬁jst ˆx j␥ tyj s ŽŽ␥ y
1.r␥ .lim ␶ ª⬁ŽÝ␶jst x j␥ tyj y ␥ ty ␶ k␶ . q Ž␥ y 1. k ty1. Now suppose we start
with a sequence of borrowing constraints which generates an optimal
consumption program c 1. Then if the k 1 constraint is replaced by a tighter
CONSUMPTION UNDER UNCERTAINTY
377
constraint, k 2 G k 1 Ži.e., k t2 G k 1t for all t ., but which allows for the same
discounted value of consumption Ži.e., lim ␶ ª⬁␥y␶ Ž k␶1 y k␶2 . s 0., then the
tighter constraint k 2 generates a limiting value of consumption at least as
large as did the k 1 constraint Ži.e., lim t ª⬁ c t2 G lim t ª⬁ c 1t .. An even stronger
result can be established about the behavior of the sequence of accumulated wealth in each period, but this is left to Section 5, where the
stochastic case is treated as well.
4.3. The Stochastic Case
As noted above, the analysis of the certainty case yields little insight into
the limiting behavior of consumption when we introduce uncertainty into
either the income sequence or the interest rate sequence. In this subsection, we formulate a general ‘‘uncertainty’’ condition under which we show
that consumption grows without bound even when the long run interest
rate equals the discount rate.
The meaning of our uncertainty condition is most transparent when the
interest factor is fixed at ␥ s ␤1 . In this case, we require:
Condition ŽU␥ .. There is an ␧ ) 0 such that for any ␣ g ⺢,
P ␣F
ž
⬁
Ý x j␥ tyj F ␣ q ␧ < z t
jst
/
-1y␧
for all z t , t G 0.
Condition ŽU␥ . says that starting at any information state, there is a
fixed probability that the discounted value of future income lies outside
any sufficiently small range. The key implication of this condition is when
consumption stays within a sufficiently small range in each period, assets
must diverge with some fixed probability from any information state.
However, the proof of our main result requires a fixed probability that
assets diverge whenever the consumption program keeps the marginal
utility of consumption approximately constant. Consequently, when we
allow for a stochastic interest rate, the uncertainty condition requires a
slight reformulation that uses the utility function to restrict the relation
between the income and interest rate sequences. Define the ‘‘inverse’’ of
uq by hŽ0. ' ⬁ and, for y ) 0, hŽ y . ' inf c g ⺢q: uqŽ c . F y4 .
Condition ŽU.. There is an ␧ ) 0 such that for any ␣ g ⺢ and any
␾ ) 0,
ž
P ␣F
for all z t , t G 0.
⬁
Ý
jst
␾
ž ž //
xj y h
␪j
< t -1y␧
Ry1
tj F ␣ q ␧ z
/
378
CHAMBERLAIN AND WILSON
Suppose the consumption program were chosen so that, in any information state, an increase in the present value of consumption by one unit
generates an increase in discounted utility equal to ␾ . Then the consumption level in each information state would be determined by the relation
cUt s hŽ ␾r␪ t ., and Condition ŽU. would imply a nontrivial probability that
the present value of current assets plus future income is bounded away
from the present value of current plus future consumption, starting at any
state z t. Roughly speaking, the condition says that the actual income
stream is stochastic relative to the hypothetical income stream required to
make the marginal discounted utility of a ‘‘present value’’ unit of consumption equal in all information states. When the interest factor is constant,
Condition ŽU␥ . implies Condition ŽU..
Our results are based on Lemma 4 below. Roughly, it says that if ␪ t is
bounded away from zero and infinity Žthe long run interest rate is equal to
the discount rate., then Condition ŽU. implies that the marginal utility of
Ž U . must converge to 0. The argument goes as follows. Suppose
assets ¨ q
t at
qŽ U .
¨ t a t does not converge to zero. Then we may choose ␶ - ⬁ and an
arbitrarily small interval w b, b q ␧ x with b ) 0 such that Ži. there is
positive probability that ␪␶ ¨␶qŽ aU␶ . g w b, b q ␧ x, and Žii. if ␪␶ ¨␶qŽ aU␶ . g w b, b
x
Ž U. w
q ␧ x, then, with probability greater than 1 y ␧ , ␪ t¨ q
t a t g b, b q ␧ for
all t G ␶ . Now consider the Žnonnull. set of paths for which ␪␶ ¨␶qŽ aU␶ . g
w b, b q ␧ x for all t G ␶ . Since the first-order conditions for utility maxiŽ U .., Condition ŽU. implies that whenever
mization imply that cUt s hŽ ¨ q
t at
qŽ U .
␪ t¨ t a t g w b, b q ␧ x for all t G ␶ , aUt must diverge with probability at
least ␧ . But if aUt diverges, Lemma 2 implies that cUt must converge to
Ž U . converges to
infinity, which if ␪ t is bounded above, implies that ␪ t¨ q
t at
0. This contradiction establishes the result.
We proceed now with the formal analysis. To use Theorem 1 we must
first establish that Condition ŽU. implies a nonzero income stream.
LEMMA 3. Condition ŽU. implies P Žx s 0. s 0.
Proof. We show first that Condition ŽU. implies that P Žx s 0 < z t . - 1
y ␧ for all z t. For this, it is sufficient to show that for any z t we may
< t.
choose ␾ sufficiently large so that P ŽÝ⬁jst hŽ ␾r␪ j . Ry1
t j - ␧ z s 1. Since
yŽ .
u is concave and bounded between 0 and M, we have cu c - M for all
c g ⺢qr 04 . By definition, uyŽ hŽ ␾r␪ j .. G ␾r␪ j . Therefore, Ž ␾r␪ j .
Ž
.ŽŽ ␾r␪ j . ␤ jyt . implies Ý⬁jst hŽ ␾r␪ j .
hŽ ␾r␪ j . F M. Then Ry1
t j ' ␪ tr␾
⬁
jyt
y1
R t j s Ž ␪ tr␾ .Ý jst Ž ␾r␪ j . hŽ ␾r␪ j . ␤
F Ž ␪ tr␾ .Ž MrŽ1 y ␤ ... Setting ␾ )
␪ t Mr␧ Ž1 y ␤ . establishes the result.
All that remains is to show that P Žx s 0 < z t . - 1 y ␧ for all z t implies
P Žx s 0. s 0. We establish this by contradiction. Suppose P Žx s 0. ) 0.
Let A t s x t s 04 and A⬁ s x s 04 . Then P Ž A t . ) 0 for all t G 0, and
A j x A⬁ . Then, since P Žx s 0 < z t . - 1 y ␧ for all z t and A t is measurable
379
CONSUMPTION UNDER UNCERTAINTY
z t , we have P Ž A j < A t . ª P Ž A⬁ < A t . - 1 y ␧ . Therefore, there is an increasing sequence Ž␶ Ž t .. such that P Ž A␶ Ž tq1. < A␶ Ž t . . - 1 y ␧ which implies
P Žx s 0. s Ł⬁ts0 P Ž A␶ Ž tq1. < A␶ Ž t . . P Ž A␶ Ž0. . F lim t ª⬁Ž1 y ␧ . t P Ž A tŽ0. . s 0.
LEMMA 4. Suppose there is an ␧ ) 0 for which Condition ŽU. is satisfied
Ž U.
.
and P Ž ␧ - ␪ t - 1␧ , t G 0. s 1. Then P Žlim t ª⬁ ␪ t¨ q
t a t s 0 s 1.
Proof. We will suppose the lemma is false and show that this implies a
violation of Condition ŽU..
Our first step is to find an interval w ␾ , ␾ q ␦ x and a time ␶ such that w ␾ ,
␾ q ␦ x contains ␪␶ ¨␶qŽ aU␶ . with positive probability and, in the event that
U
Ž U. w
x
␪␶ ¨␶qŽ aU␶ . g w ␾ , ␾ q ␦ x, it is also true that ␪ t¨ q
t a t g ␾ , ␾ q ␦ and c t
is near hŽ ␾r␪ t . with probability close to 1 for all t G ␶ . Theorem 1 and
Ž U.
.
Lemma 3 imply a random variable e⬁ such that P Žlim t ª⬁ ␪ t¨ q
t a t s e⬁
s 1. If P Ž e⬁ s 0. - 1, then there is a ␺ ) 0 such that for any ␦ ) 0,
P Ž e⬁ g w ␺ , ␺ q ␦ x. ) 0. Let ␩ ' ␧ 3 Ž1 y ␤ .r2. Then, since h is uniformly
continuous on any positive interval bounded away from zero, we may
choose ␾ and ␦ , 0 - ␾ - ␺ - ␾ q ␦ , so that Ži. P Ž e⬁ g w ␾ , ␾ q ␦ x. ) 0,
Žii. P Ž e⬁ s ␾ . s P Ž e⬁ s ␾ q ␦ . s 0, and Žiii. < hŽ ␾r␪ t . y hŽŽ ␾ q ␦ .r␪ t .<
- ␩ for ␧ - ␪ t - 1␧ . Define B ' e⬁ g w ␾ , ␾ q ␦ x4 , and for ␶ G 0, define
Ž U.
A␶ ' ␪␶ ¨␶qŽ aU␶ . g w ␾ , ␾ q ␦ x4 and B␶ ' < cUt y hŽ ␾r␪ t .< - ␩ , ␪ t¨ q
t at g
w ␾ , ␾ q ␦ x, t G ␶ 4 . Then lim ␶ ª⬁ P Ž A␶ . s P Ž B . ) 0. Also, Lemma 1 imŽ U .. s 0, t G 0. s 1. Therefore, P Ž ␧ - ␪ t - 1␧ . s 1
plies P Ž cUt y hŽ ¨ q
t at
and the continuity of h imply P Žlim t ª⬁Ž cUt y hŽ ␧⬁ r␪ t .. s 0 < B . s 1. Then,
since the monotonicity of h implies P Ž hŽ ␾r␪ t . G hŽ e⬁ r␪ t . G hŽ ␾ q ␦ .< B .
s 1, we have lim ␶ ª⬁ P Ž B␶ . s P Ž B .. Consequently, we may choose ␶ - ⬁
such that P Ž B␶ . ) Ž1 y ␧ . P Ž A␶ . ) 0.
We show next that if the marginal utility of consumption c j is always
equated to ␾r␪ j , then the variation in the present value of the limiting
level of assets is less than ␧2 on B␶ . The monotonicity of h and the
construction of B␶ imply P Ž c␶U - hŽ ␾␧ . q ␩ , t G ␶ < B␶ . s 1. Therefore,
letting ␣ 2 ' hŽ ␾␧ . q ␩ , Lemma 2 implies an ␣ 1 - ⬁ such that P Ž0 F aUt
y cUt F ␣ 1 , t G ␶ < B␶ . s 1. Then, since P Ž ␧ - ␪ t - 1␧ , t G 0. s 1 implies
jy ␶
P Ž Ry1
r␪ j F ␧y2␤ jy ␶ , j G ␶ . s 1, it follows that P Žlim t ª⬁Ž aUt
␶ j ' ␪␶ ␤
U
y1
y c t . R␶ t s 0 < B␶ . s 1. Therefore,
Ž aUt y cUt . Ry1
␶t
s aU␶ y c␶U q
s aU␶ y x␶ q
t
Ý Ž x j y cUj . Ry1
␶j
js ␶ q1
t
Ý
js ␶
␾
ž ž //
xj y h
␪j
Ry1
␶j q
t
Ý
js ␶
␾
y cUj Ry1
␶j
žž / /
h
␪j
380
CHAMBERLAIN AND WILSON
implies
P aU␶ y x␶ q
⬁
␾
ž ž //
ž
Ýž ž /
/
/
⬁
q
h
js ␶
Ý
js ␶
␾
␪j
xj y h
␪j
Ry1
␶j
Ž 6.
y cUj Ry1
␶ j s 0 B␶ s 1.
jy ␶
But from P Ž Ry1
r␪ j F ␧y2␤ jy ␶ , j G ␶ . s 1 and the definition
␶ j ' ␪␶ ␤
of B␶ it follows that
⬁
P
␾
y cUj Ry1
␶j -
ž žž / /
Ý
js ␶
h
␪j
␩
Ž1 y ␤ . ␧
2
'
␧
2
/
B␶ s 1.
Ž 7.
/
Ž 8.
Combining Ž6. and Ž7. then yields
P
ž
aU␶ y x␶ q
⬁
Ý
js ␶
␾
ž ž //
xj y h
␪j
Ry1
␶j -
␧
2
B␶ s 1.
We now use these results to show that Condition ŽU. must be violated.
Let ␣ s x␶ y aU␶ y ␧2 . Since B␶ ; A␶ and P Ž B␶ . ) Ž1 y ␧ . P Ž A␶ ., it fol<
< .
lows from Ž8. that P Ž ␣ - <Ý⬁js ␶ Ž x j y hŽ ␾r␪ j .. Ry1
␶ j - ␣ q ␧ A␶ ) 1 y ␧ .
But then A␶ measurable z ␶ implies that the set of z ␶ such that P Ž ␣ <Ý⬁js ␶ Ž x j y hŽ ␾r␪ j .. Ry1
<
< ␶ . ) 1 y ␧ has positive probability,
␶j - ␣ q ␧ z
which violates Condition ŽU..
We use this lemma again in Section 5. For the moment, however, it
serves as the basis for the main result of this paper.
THEOREM 4. Suppose there is an ␧ ) 0 such that Condition ŽU. is
satisfied. If P Ž ␧ - ␪ t - 1␧ , t G 0. s 1, then P Žlim t ª⬁ cUt s ⬁. s 1.
Proof. If P Ž ␧ - ␪ t - 1␧ , t G 0. s 1, then Lemma 4 implies
Ž U.
.
P Žlim t ª⬁¨ q
t a t s 0 s 1. The conclusion then follows from Lemma 1.
With sufficient uncertainty in the income and interest rate sequences,
consumption will grow without bound even if the long run rate of interest
is equal to the discount rate. A case in which this equality holds has been
considered by Bewley Ž1980c, 1983. in his treatment of the optimum
quantity of money. There is an asset with a fixed nominal return factor
equal to ␤y1 . The real return is rt s ␤y1 Ž qty1rqt ., where the price Ž qt . of
the consumption good in terms of the asset is uniformly bounded away
CONSUMPTION UNDER UNCERTAINTY
381
from zero and infinity. Consequently, ␪ t s q0rqt satisfies the conditions of
Theorem 4. We conclude that if the uncertainty condition is satisfied, c␶U
converges to infinity.
For the case where the interest rate is constant, Theorem 4 implies the
following corollary.
COROLLARY 2. Suppose rt s ␥ for all t G 0 and suppose ␤␥ s 1. If
Condition ŽU␥ . is satisfied, then P Žlim t ª⬁ cUt s ⬁. s 1.
If income is bounded above, then Condition ŽU␥ . in Corollary 2 can be
replaced by the condition that the conditional variance of discounted
future income is uniformly bounded away from zero; i.e., there is a ␺ ) 0
such that VarŽÝ⬁jst x j␥ tyj < z t . G ␺ for all z t , t G 0. However, if the income
stream is stochastic, but the conditions of Corollary 2 are not satisfied,
there are examples where the limiting level of consumption is finite with
some positive probability.
When contrasted with the outcome in the case of certainty, Corollary 2
is perhaps a surprisingly strong result. Unfortunately, the line of argument
used in the proof does not provide a very convincing economic explanation. 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
sufficiently large domain and Ž x t . is bounded.. But to simply attribute the
result to risk aversion on the grounds that uncertain future returns will
cause risk-averse consumers to save more, given any initial asset level, is
not a completely satisfactory explanation either. In fact, it is a bit misleading. First, that argument only explains why expected accumulated assets
would tend to be larger in the limit. It does not really explain why
consumption should grow without bound. Second, over any finite time
horizon, the argument is not even necessarily correct.
Suppose, for example, that x t s x ty1 ␧ t for t s 1, . . . , T, where the Ž ␧ t .
are identically and independently distributed with Ew ␧ t x s 1 and a compact support in ⺢. Suppose, also, that the consumer’s utility function is
quadratic over a sufficiently large domain. Then if the consumer has a
T-period planning horizon, it can be shown that his optimal consumption
program is to set c t s x t for all t. In particular, mean-preserving spreads of
future income leave current consumption unaffected. Moreover, the expected value of consumption in any period t is just equal to period 0
income. 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 of
consumption in an infinite horizon problem? Although we have developed
other arguments to establish our results, any explanation that we have
been able to devise ultimately appeals in an essential way to the martingale convergence theorems.
382
CHAMBERLAIN AND WILSON
4.4. Limit Theorems When Ž rt . Is Stationary
In this subsection, we assume that the sequence of information states is
generated by a stationary process and establish some restrictions on the
distribution of the single period interest rate that imply the conditions of
Theorem 2. Since the stationarity restriction will be placed on the entire
distribution of histories, we will be explicit about conditioning on z 0 .7
THEOREM 5. Suppose P Žx s 0. s 0. If Ž r 1 , r 2 , . . . . is stationary and
ergodic and Ewlog ␤ r 1 x ) 0, then P Žlim t ª⬁ cUt s ⬁< z 0 . s 1 with probability 1.
Proof. Let ␺ s Ewlog ␤ r 1 x. The Ergodic Theorem implies that with
probability 1, lim t ª⬁ 1t Ýtjs1 log ␤ r j s ␺ ŽDoob, 1953, p. 465.. Therefore,
for almost all z there is a ␶ Žz. such that Ýtjs1 log ␤ r j ) ␺2t for t ) ␶ Žz., or
equivalently, ␪ t ) e ␺ t r2 . Therefore, P Žlim t ª⬁ ␪ t s ⬁. s 1, which implies
that z 0 : P Žlim t ª⬁ ␪ t s ⬁< z 0 . s 14 has probability 1. The desired result
then follows from Theorem 2.
For the case where EwlogŽ ␤ r 1 .x s 0, we need some additional regularity
conditions. For s F t, define ␴ Ž r s , . . . , rt . as the ␴-field generated by
r s , . . . , rt , and define ␴ Ž rt , rtq1 , . . . . as the sigma field generated by
rt , rtq1 , . . . . Consider a nonnegative function ␾ of the positive integers.
The sequence Ž r 1 , r 2 , . . . . is ␾-mixing if for each t, j G 1, A1 g ␴ Ž r 1 , . . . , rt .
and A 2 g ␴ Ž rtqj , rtqjq1 , . . . . together imply that < P Ž A1 l A 2 . y
P Ž A1 . P Ž A 2 .< F ␾ Ž j . P Ž A1 ..
Condition ŽR.. Ži. Ž r 1 , r 2 , . . . . is a stationary, ␾-mixing stochastic process with Ý⬁js1Ž ␾ Ž j ..1r2 - ⬁. Žii. EwŽlog ␤ r 1 . 2 x - ⬁ and
␴ 2 ' E Ž log ␤ r 1 .
2
q2
⬁
Ý E Ž log ␤ r1 . Ž log ␤ r j .
/ 0.
js2
The first part of Condition ŽR. ensures that the dependence between rt
and rtqj dies out sufficiently fast as j ª ⬁. Suppose part Ži. is satisfied and
Ewlog ␤ r 1 x s 0. Then Ý⬁js2 EwŽlog ␤ r 1 .Žlog ␤ r j .x converges absolutely and
␴ 2 s lim T ª⬁ Ew T1 ŽÝTjs1 log ␤ r j . 2 x ŽBillingsley, 1968, Lemmata 1 and 3, pp.
170, 172.. In this case, part Žii. may be satisfied as well if there is sufficient
variability in ␤ t R 0T . However, if P Ž ␤ r 1 s 1. s 1, part Žii. is not satisfied,
and we know from Theorem 4 that the behavior of Ž cUt . may depend upon
7
Theorem A.1 establishes the existence of an optimal program only for a fixed z t. Because
we did not restrict z t to be drawn from a compact set, we are able to show that cUt Ž z t . is a
measurable function of z t only for a fixed tail Že.g., a given z 0 .. Without additional
restrictions we are unable to prove that cUt is a measurable function if all components of z t
are allowed to vary.
383
CONSUMPTION UNDER UNCERTAINTY
whether or not Ž x t . is stochastic. Nevertheless, our next theorem states
that if Condition ŽR. is satisfied and Ewlog ␤ r 1 x s 0, then sup t cUt s ⬁,
without a requirement that Ž x t . be stochastic.
THEOREM 6. Suppose P Žx s 0. s 0 and Ž r 1 , r 2 , . . . . satisfies Condition
ŽR.. Then Ewlog ␤ r 1 x s 0 implies P Žsup t G 0 cUt s ⬁< z 0 . s 1 with probability 1.
Proof. Define St ' Ýtjs1 log ␤ r j Žs log ␪ t . and define Yt Ž ␦ . '
S ? ␦ t @ Ž ␴ 2 t .y1 r2 , where 0 F ␦ F 1 and ? ␦ t @ is the greatest integer less than
or equal to ␦ t. Then the functional central limit theorem implies that the
distribution of the random function Yt converges weakly to Weiner measure ŽBillingsley, 1968, Theorem 20.1., from which it follows that
lim P Ž ␴ 2 t .
tª⬁
ž
y1 r2
max S j F ␣ s 2 Ž 2␲ .
jFt
/
y1r2
␣
yu 2 r2
H0 e
du
Ž 9.
for ␣ G 0 ŽBillingsley, 1968, p. 138, and Eq. Ž10.18., p. 72..
Define ␺ ' P Žsup t G 1 St - ⬁.. We shall assume that ␺ ) 0 and obtain a
contradiction. Choose ␧ ) 0 so that 2Ž2␲ .y1 r2H0␧ expŽyu 2r2. du - ␺2 and,
for t G 1, define A t ' Ž ␴ 2 t .y1r2 max j F t S j F ␧ 4 . Then lim t ª⬁ P Ž A t . - ␺2 .
For t G 1, define Bt ' Ž ␴ 2␶ .y1 r2 max j F ␶ S j F ␧ for all ␶ G t 4 . Then there
is a B such that sup t G 1 S t - ⬁4 ; B and Bt ­ B. Therefore, using Eq. Ž9.,
Bt ; A t implies P Ž A t . G P Ž Bt . ª P Ž B . G ␺ , which contradicts
lim t ª⬁ P Ž A t . - ␺2 . We conclude that P Žsup t G 1 St - ⬁. s 0, or equivalently, P Žsup t G 1 ␪ t s ⬁. s 1. The desired conclusion then follows from
Theorem 1 and Lemma 1.
5. THE BUDGET CONSTRAINT
In this section, we address two questions. First, given an arbitrary
stochastic income sequence, under what conditions should we expect the
borrowing constraint to be binding at some information state? Second,
how does a change in the borrowing constraint Žor 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 is
finite conditional on any information state, we can represent any intertemporal budget constraint as a sequence of one period borrowing constraints.
It may well turn out that none of these constraints are actually binding at
the optimum, and yet the consumer is still constrained to choose a
384
CHAMBERLAIN AND WILSON
consumption program whose present value does not exceed the present
value of the income stream. Our next theorem gives conditions under
which this is the case.
THEOREM 7.
If Ew rtq1 uqŽ x tq1 .< z t x s ⬁ and aUt ) 0, then cUt - aUt .
Proof. Suppose cUt s aUt . Then P Ž cUtq1 F x tq1 < z t . s 1. Lemma 1 then
implies
U
U
q
< t
⬁ ) uy Ž cUt . G ¨ q
t Ž a t . G ␤ E r tq1¨ tq1 Ž a tq1 . z
G ␤ E rtq1 uq Ž cUtq1 . < z t G ␤ E rtq1 uq Ž x tq1 . < z t s ⬁,
a contradiction.
Since aUt G x t , the following corollary follows immediately.
COROLLARY 3.
If Ew rtq1 uqŽ x tq1 .< z t x s ⬁ and x t ) 0, then cUt - aUt .
Theorem 7 and its corollary say that whenever the expected increment
in disposable income in the following period is sufficiently small so that
the expected marginal utility from consuming out of that increment would
be infinite, the consumer chooses to consume less than his current wealth
in the current period in order to pass some of his wealth to the next
period. If we allow the expected marginal utility of future income to be
finite, however, the borrowing constraint may well be binding, at least
occasionally. This will obviously be the case if income received in each
period is growing at a sufficiently high rate over time so that the consumer
wants to transfer future income to present consumption. But if the income
stream is suitably stochastic, a much weaker set of conditions guarantees
that the budget constraint is sometimes binding.
Our next theorem may be summarized as follows. Suppose the single
period interest rate never exceeds the discount rate and the marginal
utility of consuming from current income alone is bounded above. Then if
the long run rate of interest is less than the discount rate or if the income
sequence is suitably stochastic, there is a positive probability that the
borrowing constraint is binding in at least one period.
THEOREM 8. Suppose P Ž ␤ rt F 1, t G 1. s 1 and P Ž uqŽ x t . - ␣ , t G 0.
s 1 for some ␣ g ⺢. If either Ži. P Žlim t ª⬁ ␪ t s 0. s 1 or Žii. there is an
␧ ) 0 for which Condition ŽU. is satisfied and P Ž ␪ t ) ␧ , t G 0. s 1, then
P Ž cUt - aUt , t G 0. - 1.
Proof. We establish the theorem by contradiction. Suppose P Ž0 F cUt
Ž U.
- aUt , t G 0. s 1. Then, by induction, Lemma 1 implies ¨ y
0 a0 F
U
U
y
y
Ew ␤ r 1¨ 1 Ž a1 .x F Ew ␪ t¨ t Ž a t .x for all t G 1. We will establish that
385
CONSUMPTION UNDER UNCERTAINTY
Ž U .x s 0 and therefore, ¨ y
Ž U.
lim t ª⬁ Ew ␪ t¨ y
t at
0 a 0 s 0, violating the monotonicity and concavity of ¨ 0 .
Ž U .x s 0, we first show P Ž ¨ y
Ž U.
.
To show that lim t ª⬁ Ew ␪ t¨ y
t at
t at - ␣ 1 s 1
U
y
for some ␣ 1 g ⺢, and then show P Žlim t ª⬁ ␪ t¨ t Ž a t . s 0. s 1. Since P Ž ␤ rt
F 1, t G 0. s 1 implies P Ž0 F ␪ tq1 F ␪ t F 1, t G 1. s 1, the desired result then follows from the Dominated Convergence Theorem.
Ž U.
.
Ž qŽ x t . - ␣ , t G 0.
To establish P Ž ¨ y
t a t - ␣ 1 s 1, note first that P u
qŽ .
s 1 implies either u 0 - ␣ or P Ž x t ) ␾ , t G 0. s 1 for some ␾ ) 0. If
Ž U.
uqŽ 0. - ␣ , then Lemma A.3 and the strict concavity of ¨ imply P Ž ¨ y
t at
qŽ .
qŽ .
- ¨ t 0 s u 0 - ␣ . s 1, in which case we let ␣ 1 ' ␣ . If P Ž x t ) ␾ , t
Ž U.
G 0. s 1, then P Ž aUt ) ␾ , t G 0. s 1, and so Lemma 1 implies P Ž ¨ y
t at
y
yŽ .
F u ␾ , t G 0. s 1, in which case we let ␣ 1 ' u Ž ␾ ..
Ž U.
.
Ž.
To establish P Žlim t ª⬁ ␪ t¨ y
t a t s 0 s 1, we consider Conditions i
and Žii. separately. If Condition Ži. holds, then the desired property follows
immediately. If Condition Žii. holds, then Lemmata 1 and 4 imply
U
U
q
P lim uq
t Ž c t . F lim ¨ t Ž a t . F
ž
tª⬁
tª⬁
1
␧
U
lim t ª⬁ ␪ t¨ q
t Ž a t . s 0 s 1,
/
and so P Žlim t ª⬁ cUt s ⬁. s 1. Another application of Lemma 1 then yields
U
U
U
y
y
P lim ␪ t¨ y
t Ž a t . F lim ␪ t u Ž c t . F lim u Ž c t . s 0 s 1.
ž
tª⬁
tª⬁
tª⬁
/
If the long run rate of interest is less than the discount rate, the
conclusion of Theorem 8 is not particularly surprising. In this case, the
marginal return to consuming a fixed level of consumption goes to zero
with time. Consequently, if the marginal utility to consuming current
income is bounded, then the borrowing constraint must eventually be
binding. The theorem is less intuitive for the case where the long run rate
of interest is equal to the discount rate Ž ␤␥ s 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 lower
bound at least once before converging to infinity. However, Theorem 4
implies that this happens only a finite number of times.
5.2. Comparati¨ e Dynamics
Suppose the original sequence of borrowing constraints is replaced with
another sequence which allows at least as much borrowing in any information state. The following theorem states that at any information state the
optimal accumulated wealth under the new sequence of borrowing con-
386
CHAMBERLAIN AND WILSON
straints is no higher than the optimal accumulated wealth under the
original sequence of borrowing constraints.
THEOREM 9. Let a 0 and a1 be the sequence of accumulated assets
associated with the solutions c 0 and c 1 corresponding to constraints k 0 and
k 1, respecti¨ ely. Then P Ž k t0 G k 1t , t G 0. s 1 implies P Ž a0t G a1t , t G 0. s 1.
Proof. Let S0 and S1 be the metric spaces defined in the Appendix
corresponding to k 0 and k 1, respectively, and let T0 and T1 be the
corresponding T operators. Note that S0 ; S1 so that T1 is defined on S0 .
Let ¨ 0 and ¨ 1 be the corresponding fixed points of T0 and T1. We shall
show that ¨ 0q Ž a, z t . ) ¨ 1y Ž a q ␧ , z t . for all Ž a, z t . g S0 and ␧ ) 0.
Let f 1 represent ¨ 1 restricted to S0 and let f nq 1 ' T0n f 1, where T0n is
the operator T0 applied n times. For Ž a, z t . g S0 , define
C1 Ž a, z t .
' arg
max
0FcFayk 1t Ž z t .
uŽ c . q ␤ E
¨ 1 Ž Ž a y c . rtq1 q x tq1 , z tq1 . < z t
4,
and for n G 2 define
Cn Ž a, z t .
' arg
max
0FcFayk t0 Ž z t .
uŽ c . q ␤ E
f ny 1 Ž Ž a y c . rtq1 q x tq1 , z tq1 . < z t
4.
Fix Ž a, z t . g S0 and ␧ ) 0. Let c1 ' C1Ž a q ␧ , z t . and c 2 ' C2 Ž a, z t ..
Then either c1 ) c2 G 0 or a q ␧ y c1 ) a y c 2 . Therefore, using the
standard envelope arguments exploited in Lemma 1, the strict concavity of
u and ¨ implies that either
¨ 1y Ž a q ␧ , z t . F uy Ž c1 . - uq Ž c 2 . F f 2q Ž a, z t .
Ž 10 .
or
¨ 1y Ž a q ␧ , z t . F ␤ E rtq1¨ 1y Ž Ž a q ␧ y c1 . rtq1 q x tq1 , z tq1 . < z t
- ␤ E rtq1¨ 1q Ž Ž a y c 2 . rtq1 q x tq1 , z tq1 . < z t F f 2q Ž a, z t . .
In either case, we have shown that ¨ 1y Ž a⬘, z t . - f 2q Ž a, z t . for any
Ž a, z t ., Ž a⬘, z t . g S0 , a⬘ ) a.
Now suppose f nq Ž a, z t . ) f Ž ny1.y Ž a⬘, z t . for any Ž a, z t ., Ž a⬘, z t . g S0
with a⬘ ) a. Fix Ž a, z t . g S0 and ␧ ) 0 and define c n ' CnŽ a q ␧ , z t . and
c nq 1 ' Cnq1Ž a, z t .. Then either c n ) c nq1 G 0 or a q ␧ y c n ) a y c nq1.
CONSUMPTION UNDER UNCERTAINTY
387
Then the envelope argument implies either
f ny Ž a q ␧ , z t . F uy Ž c n . - uq Ž c nq1 . F f Ž nq1.q Ž a, z t .
Ž 11 .
or
f ny Ž a q ␧ , z t . F ␤ E rtq1 f Ž ny1.y Ž Ž a q ␧ y c n . rtq1 q x tq1 , z tq1 . < z t
- ␤ E rtq1 f nq Ž Ž a y c nq1 . rtq1 q x tq1 , z tq1 . < z t
Ž 12 .
F f Ž nq1.q Ž a, z t . .
In either case, we have shown that f ny Ž a⬘, z t . - f Ž nq1.q Ž a, z t . for any
Ž a, z t ., Ž a⬘, z t . g S0 , a⬘ ) a.
Now let Ž ␧ n . be a strictly decreasing sequence such that ␧ 1 s ␧ and
lim nª⬁ ␧ n s ␧2 . Then for n G 2, we have just shown that ¨ 1y Ž a q ␧ , z t . f 1y Ž a q ␧ 2 , z t . - ⭈⭈⭈ - f Ž ny1.y Ž a q ␧ n , z t . - f ny Ž a q ␧2 , z t . for all
Ž a, z t . g S0 . Since T0 is a contraction mapping and T0¨ 0 s ¨ 0 , it follows
that f n Ž a, z t . ª ¨ 0 Ž a, z t . as n ª ⬁. Therefore, Lemma A.4 implies
¨ 0q Ž a, z t . ) ¨ 0y Ž a q ␧2 , z t . G lim sup n f ny Ž a q ␧2 , z t . G ¨ 1y Ž a q ␧ , z t .
for any Ž a, z t . g S0 and ␧ ) 0.
To show that a1 F a 0 , we again argue by induction. Fix any realization of
z s Ž . . . , zy1 , z 0 , z1 , . . . .. By definition, a00 Ž z 0 . s a10 Ž z 0 .. Suppose a0t Ž z t .
y a1t Ž z t . G 0. We will show that a0tq1Ž z tq1 . y a1tq1Ž z tq1 . G 0. Note first,
that for any w g ⺢ N,
a0tq1 Ž z t , w . y a1tq1 Ž z t , w .
s rtq1 Ž z t , w . Ž Ž a0t Ž z t . y a1t Ž z t . . y Ž c t0 Ž z t . y c 1t Ž z t . . . .
Ž 13 .
Therefore, a0tq1Ž z tq1 . - a1tq1Ž z tq1 . implies c t0 Ž z t . ) c 1t Ž z t . and a0tq1Ž z t , w .
- a1tq1Ž z t , w . for all w g ⺢ N . Lemma 1 then implies
␤ E rtq1 Ž z t , w . ¨ 1y Ž a1tq1 Ž z t , w . , Ž z t , w . . < z t
G uq Ž c 1t Ž z t . . ) uy Ž c t0 Ž z t . .
Ž 14 .
G ␤ E rtq1 Ž z t , w . ¨ 0q Ž a0tq1 Ž z t , w . , Ž z t , w . . < z t .
But we have just shown above that a0tq1Ž z t , w . - a1tq1Ž z t , w . implies
¨ 0q Ž a0tq1Ž z t , w ., Ž z t , w .. ) ¨ 1y Ž a1tq1Ž z t , w ., Ž z t , w ... This contradiction completes the proof.
388
CHAMBERLAIN AND WILSON
It should be clear from our discussion in Section 2 that for any change in
the set of feasible consumption programs in response to a change in the
borrowing constraint, there is an equivalent change in the income sequence that generates the same change in the set of feasible consumption
programs. Consequently, we may reinterpret Theorem 9 as a statement
about how the optimal consumption program changes in response to
certain kinds of changes in the income sequence. Our next corollary states
that if the income sequence changes so that at any state z t the discounted
value of income up to time t has not decreased, then the discounted value
of the consumption up to time ␶ also does not decrease for any state z ␶ .
COROLLARY 4. Let c 0 and c 1 be the optimal consumption programs
associated with borrowing sequence k and income sequences x 0 and x 1,
respecti¨ ely. Then
t
P
ž
Ý Ž x 1j y x j0 . Ry1
0 j G 0,
js0
/
tG0 s1
t
implies P
ž
Ý Ž c1j y c j0 . Ry1
0 j G 0,
js0
/
t G 0 s 1.
1
Proof. Let k 0 ' k, x ' x 0 , and define k 1 s Ž ky1 , k 0 , k 1 , . . . . by ky1
'0
0
1
0
1
1
t
and k t ' Ý js0 Ž x j y x j . R jt q k t for t G 0. By assumption, x and k jointly
satisfy Ž1.. Therefore, x t y k 1t q rt k 1ty1 s x t0 y Ýtjs0 Ž x j0 y x 1j . R jt y k t0 q
1.
0
1
1
Ž 0
Ýty1
js0 x j y x j R jt q r t k ty1 s x t y k t q r t k ty1 implies that x and k do as
0
1
well. Furthermore, the definitions of c and c imply that they are also the
optimal consumption programs associated with the Žcommon. income
sequence, x, and respective borrowing constraints, k 0 and k 1. To see this,
note that for any consumption program c, we have Ýtjs0 Ž x 1j y c j . R jt y k t
s Ýtjs0 Ž x j y c j . R jt y k 1t . Therefore, a program c is feasible for x 1 and k if
and only if it is feasible for x and k 1. By assumption c 1 is optimal for x 1
and k. Therefore, it is also optimal for x and k 1.
Now let a 0 and a1 denote the asset sequences associated with consumption programs c 0 and c 1 for income sequence x. Then, for i s 0, 1, a0i s x 0
i.
Ž
and a it s Ýty1
js0 x j y c j R jt q x t for t G 1. The hypothesis of the corollary
and the construction of k 1 imply k 1 F k 0 . Therefore, Theorem 9 implies
0.
ty1 Ž 1
0 . y1
Ž 1
0 F a0t y a1t s Ýty1
js0 c j y c j R jt s R 0 t Ý js0 c j y c j R 0 j for all t G 1.
6. MANY ASSETS
In this section, we illustrate how the argument behind Theorem 2 can be
extended to the case where there are several risky assets. We show that to
apply the martingale convergence theorem, it is not necessary to actually
389
CONSUMPTION UNDER UNCERTAINTY
solve the portfolio problem of the consumer. So long as the consumer is
able to choose a sequence of ‘‘marginal’’ interest rates which, in the long
run, exceed the discount rate, consumption must converge to infinity.
Suppose the consumer must choose a portfolio of J risky assets in each
period t G 0. If the investment in period t in the ith asset is bi Žmeasured
in terms of period t consumption., then the payoff at t q 1 is bi g i, tq1 ,
where g i, tq1 is a nonnegative, continuous function from Z tq1 Ž z 0 . to ⺢q
for i s 1, . . . , J, and t G 0. If b ' Ž b1 , . . . , bJ . is the portfolio chosen in
period t, then the multi-asset analogue of the budget constraint is a
constraint on short sales which we represent by the requirement that
b g K t Ž z t .. We assume that K t is a continuous correspondence from
Z t Ž z 0 . to ⺢ J and that K t Ž z t . is a nonempty, closed, convex subset of ⺢ J
which is bounded from below; i.e., there is a function m: Z t Ž z 0 . ª ⺢ such
that K t Ž z t . ; Ž ␣ 1 , . . . , ␣ J . g ⺢ J : ␣ j G mŽ z t .4 . We also assume that
yK t Ž z t . is comprehensive; i.e., if ␾ 1 g K t Ž z t . and ␾ 2 g ⺢qJ , then ␾ 1 q ␾ 2
g K t Ž z t .. This guarantees that it is always feasible for the consumer to
increase his holding of any asset.
For any ␣ , ␤ g ⺢ J, let ␣␤ ' Ý Jjs1 ␣ j ␤ j . Let ␭ denote the J-vector of
ones, and define k t Ž z t . ' inf b␭: b g K t Ž z t .4 . Interpret k t as the minimum amount of wealth measured in terms of period t consumption that
the individual may hold at the end of period t Žafter receiving period t
income and spending period t consumption.. Let g t ' Ž g 1 t , . . . , g J t .. We
require that the current bounds on short sales be consistent with the
borrowing constraints the individual will face in the future: P Ž bg tq1 q
x tq1 G k tq1 < z t . s 1 for all b g K t Ž z t . and all z t.
For any state z 0 , a consumption-portfolio program, Žc, b., is a sequence
of a pair of functions, c t : Z t Ž z 0 . ª ⺢q and bt : Z t Ž z 0 . ª ⺢ J, t G 0. For
any z t g Z t Ž z 0 . and any a G k t Ž z t ., let K t Ž a, z t . ' Ž c, b . g ⺢q=
K t Ž z t .: 0 F c F a y b␭4 denote the set of feasible consumption-portfolio
pairs for the consumer, and define
¨ Ž a, z t . ' max E
c, b
⬁
Ý u Ž c j . ␤ jyt < z t
jst
subject to P Ž c j , bj . g K j Ž a j , z j . , j G t < z t s 1,
ž
/
where a t ' a and a j ' bjy1 g j q x j for j ) t. Theorem A.1 implies that this
problem has a solution and that ¨ Ž a, z t . is strictly concave in a G k t .
Therefore, right- and left-hand derivatives are well defined for a ) k t . We
let ¨ qŽ k t , z t . ' lim ax k t¨ qŽ a, z t ..
As in Section 2, let ¨ t Ž a. ' ¨ Ž a, z t ., and let Žc*, b*. denote the solution
to the consumer’s problem starting in state z 0 with wealth a s x 0 . We
390
CHAMBERLAIN AND WILSON
shall simplify the martingale arguments by assuming that x 0 ) k 0 . The
following conditions are then implied by an optimal program. They are
established by the same standard arguments used to establish Lemma 1.
q Ž U .< t x
Ž U.
w
LEMMA 5. Ži. ¨ q
for k s 1, . . . , J, and
t a t G ␤ E g k, tq1¨ tq1 a tq1 z
U
U
qŽ
Žii. ¨ q
Ž .
c t ..
t at G u
To obtain the analogues of Theorems 1 and 2 from Lemma 5, we
consider an arbitrary marginal portfolio, i.e., a portfolio which can be
added to the existing portfolio. Given our assumptions on K t , the only
portfolios we might not be able to consider are those involving short sales.
For this reason, we restrict attention to those portfolios without short
sales. We say that Ž b 0 , b1 , . . . . is a positi¨ e portfolio program if each bt is a
measurable function from Z t Ž z 0 . into ⺢qJ y 04 . For any positive portfolio
program, we define the sequence of return factors by rtq1Ž z tq1 . '
bt Ž z t . g tq1Ž z tq1 .rbt Ž z t . ␭ for t G 0.
The next theorem provides a condition that, if satisfied by the return
sequence generated by at least one positive portfolio, implies that optimal
consumption converges to infinity. Recall that R 0 t ' Ł tjs1 r j and ␪ t '
␤ tR0 t .
THEOREM 10. Suppose that Ž r 1 , r 2 , . . . . is the return sequence for some
positi¨ e portfolio program. Then, Ži. P Žlim t ª ⬁ ␪ t s ⬁. s 1 implies
P Žlim t ª⬁ cUt s ⬁. s 1, and Žii. P Žsup t G 0 ␪ t s ⬁. s 1 implies P Žsup t G 0 cUt
s ⬁. s 1.
Ž U . qŽ U .
Proof. Ži. Define d 0 ' 1 and d t ' ␪ t¨ q
t a t r¨ 0 a 0 . Lemma 5 imU
q
t
qŽ U .
plies that ¨ t a t G ␤ Ew rtq1¨ tq1Ž a tq1 .< z x from which it follows that d t is a
nonnegative supermartingale. Therefore, there is a random variable d⬁
with Ew d⬁ x F 1 and P Žlim t ª⬁ d t s d⬁ . s 1 ŽDoob, 1953, p. 324.. So if
Ž U.
␪ t ª ⬁, it follows that ¨ q
t a t ª 0, and therefore, from Lemma 5 that
qŽ U .
u c t ª 0, from which we conclude that P Žlim t ª⬁ cUt s ⬁. s 1.
Žii. If ␪ t ª ⬁ along a subsequence, then, along that subsequence,
U
qŽ U .
Ž
.
¨q
a
c t ª 0 and therefore, P Žsup t cUt s ⬁. s 1.
t
t ª 0 which implies u
We emphasize that there need be no relation between the optimal
portfolio and the positive portfolio program that generates the sequence of
interest rates used in Theorem 10.
APPENDIX
We establish the existence of an optimal solution for the problem
presented in Section 6 where the consumer must choose an optimal
portfolio and consumption program in each period. Recall that z t g ⺢ n
391
CONSUMPTION UNDER UNCERTAINTY
and for any z 0 s Ž . . . , zy1 , z 0 ., Z t Ž z 0 . ' z t : z t s Ž z 0 , z1 , . . . , z t .4 . Fix z 0 .
Let Ž S, d . denote the metric space defined by S ' Ž a, z t .: a G k t Ž z t ., z t
g Z t Ž z 0 ., t s 0, 1, . . . 4 , and
d Ž Ž a, z t . , Ž a⬘, z j . .
s
½
1
if j / t
max < a y a⬘ < , ␳ Ž z1 , zX1 . , . . . , ␳ Ž z t , zXt . 4
if j s t ,
where ␳ is the metric on ⺢ n. For any f : S ª ⺢, let 5 f 5 s sup s g S < f Ž s .<,
and let DŽ S . denote the set of continuous functions f : S ª ⺢ such that
M
t.
Ž
0 F f F 1y
is nondecreasing and concave in a. Note that
␤ and f a, z
DŽ S . is a complete metric space under the metric 5 f y g 5.
Recall that K t Ž a, z t . ' Ž c, b . g ⺢q= K t Ž z t .: 0 F c F a y b␭4 , where ␭
is the n-vector of ones, denotes the set of feasible consumption and
portfolio choices for wealth level a in state z t. Define the operator T on
DŽ S . by
Tf Ž a, z t . '
u Ž c . q ␤ E f Ž bg tq1 q x tq1 , z tq1 . < z t 4 .
sup
Ž c, b .gK t Ž a, z t .
LEMMA A.1. T is a contraction mapping DŽ S . into DŽ S ..
Proof. We show first that f g DŽ S . implies Tf g DŽ S .. Suppose f g
M
DŽ S .. It is immediate that 0 F Tf F 1 y
␤ and that Tf is monotone nondecreasing.
We show next that Tf is continuous. Since dŽ s, s⬘. - 1 implies t s t⬘, we
suppress the dependence on t and let z denote z t. For w g ⺢ n, define
mŽ b, z, w . ' f Ž bg tq 1Ž z, w . q x tq 1Ž z, w ., Ž z, w .. and let hŽ b, z . '
H mŽ b, z, w . pŽ dw < z .. Then Tf Ž a, z . s supŽ c, b.g K t Ž a, z . uŽ c . q ␤ hŽ b, z .4 .
Since u is continuous and K t is continuous and compact-valued, it follows
that Tf is continuous if h is continuous ŽHildenbrand, 1974, Corollary, p.
30.. To show that h is continuous, note that
< h Ž b, z . y h Ž b⬘, z⬘ . < F
Hm Ž b, z, w . Ž p Ž dw < z . y p Ž dw < z⬘. .
Ž 15 .
q < m Ž b, z, w . y m Ž b⬘, z⬘, w . < p Ž dw < z⬘ . .
H
Define dŽŽ b, z ., Ž b⬘, z⬘.. ' max ␳ 1Ž b, b⬘., ␳ 2 Ž z, z⬘.4 , where ␳ 1 and ␳ 2 are
Euclidean metrics. For fixed Ž b, z ., the Feller property implies that for any
␧ ) 0, there is a ␦ 1 ) 0 such that < H mŽ b, z, w .Ž pŽ dw < z . y pŽ dw < z⬘..< - ␧
for dŽŽ b, z ., Ž b⬘, z⬘.. - ␦ 1. Since Z t Ž z 0 . is a finite-dimensional Euclidean
392
CHAMBERLAIN AND WILSON
space, the Feller property also implies that there is a compact set A ; ⺢ n
and ␦ 2 ) 0 such that dŽŽ b, z ., Ž b⬘, z⬘.. - ␦ 2 implies H⺢ nyA pŽ dw < z⬘.
Ž1 y ␤ . ␧
- M . Since f is uniformly continuous on compact sets, we may
choose ␦ 3 ) 0 so that dŽŽ b, z ., Ž b⬘, z⬘.. - ␦ 3 implies < mŽ b, z, w . y
mŽ b⬘, z⬘, w .< - ␧ for w g A. Therefore, if dŽŽ b, z ., Ž b⬘, z⬘.. - min ␦ 2 , ␦ 34 ,
we have
H< m Ž b, z, w . y m Ž b⬘, z⬘, w . < p Ž dw < z⬘.
-␧
ž
M
1y␤
1y␤
M
Ž 16 .
/
q ␧ s 2␧ .
Combining these results, we have dŽŽ b, z ., Ž b⬘, z⬘.. - min ␦ 1 , ␦ 2 , ␦ 3 4 implies < hŽ b, z . y hŽ b⬘, z⬘.< - 3 ␧ .
All that remains is to show that Tf Ž a, z t . is concave in a. But this
follows since u is concave and hŽ b, z t . is concave in b. This establishes
that Tf g DŽ S ..
To establish that T is a contraction mapping, we may use Blackwell’s
Theorem 5 to show that given f, g g DŽ S ., 5 Tf y Tg 5 F ␤ 5 f y g 5.
LEMMA A.2. There is a unique ¨ g DŽ S . such that T¨ s ¨ .
Proof. Given that T is a contraction and DŽ S . is a complete metric
space, the lemma follows immediately from the contraction mapping fixed
point theorem.
Define h ¨ Ž b, z t . ' H ¨ Ž bg tq1Ž z t , w . q x tq1Ž z t , w ., Ž z t , w .. pŽ dw < z t .. Since
u and h ¨ are continuous functions and K t is a continuous, compact-valued
correspondence, it follows that we can select measurable functions, C and
B, such that for all z t and all a G k Ž z t ., Ž C Ž a, z t ., B Ž a, z t .. g K t Ž a, z t .
and
¨ Ž a, z t . ' u Ž C Ž a, z t . . q ␤ E ¨ Ž B Ž a, z t . g tq1 q x tq1 , z tq1 . < z t
for all Ž a, z t . g S. ŽSee 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 t is convex-valued.
For any z t and any a G k t Ž z t ., define the random variables a* s
U
Ž a t , aUtq1 , . . . ., c* s Ž cUt , cUtq1 , . . . ., and b* s Ž b*t , bUtq1 , . . . . recursively as
follows: aUt ' a, and for j G t, cUj ' C Ž aUj , z j ., bUj ' B Ž aUj , z j ., and aUj '
bUjy1 g j Ž z j . q x j Ž z j .. The next theorem establishes that Žc*, b*. is an optimal consumption-portfolio program for the consumer, and a* is the
corresponding sequence of wealth levels when starting with wealth level a
and state z t.
393
CONSUMPTION UNDER UNCERTAINTY
THEOREM A.1. Ži. Žc*, b*. maximizes EwÝ⬁jst uŽ c j . ␤ jyt < z t x subject to
P ŽŽ c j , bj . g K j Ž a j , z j ., j G t < z t . s 1, where a t ' a and a j ' bUjy1 g j Ž z j . q
x j Ž z j . for j ) t. Žii. For a G k t Ž z t ., ¨ Ž a, z t . s EwÝ⬁jst uŽ cUj . ␤ jyt < z t x, which
is a strictly conca¨ e function of a.
Proof. It follows from Lemma A.2 and Strauch Ž1966, Theorem 5.1b.
that ¨ Ž a, z t . s EwÝ⬁jst uŽ cUt . ␤ jyt < z t x. Condition Ži. then follows from
Blackwell’s Theorem 6. The strict concavity of ¨ follows from the strict
concavity 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 and
income is defined so that x t G 0 and k t s 0 for all t G 0. As in the text, we
let ¨ t Ž a. ' ¨ Ž a, z t ..
LEMMA A.3.
qŽ .
Ž .
P Ž ␤ rt F 1, t G 1. s 1 implies P Ž ¨ q
0 , t G 0. s
t 0 su
1.
Ž .
Proof. Without loss of generality it is sufficient to prove that ¨ q
0 0 s
qŽ .
.
u 0 . Note first that if u 0 s ⬁, the lemma follows immediately from
Lemma 1 of the text. So we may suppose that uqŽ 0. - ⬁. Since Lemma 1
qŽ .
qŽ .
Ž .
Ž .
implies ¨ q
0 , we need to show that ¨ q
0 . Let c 0 and c ␧
0 0 Gu
0 0 Fu
denote the optimal consumption programs for a0 s 0 and a0 s ␧ ) 0,
respectively, and let a 0 and a ␧ denote the corresponding sequence of
wealth levels.
We show first that P Ž a0t F a ␧t , t G 0. s 1. Suppose not, and let t be the
smallest j such that P Ž a0j ) a ␧j . ) 0. For any z t g ⺢ n, we have
qŽ
a ␧t Ž z ty1 , z t . y a0t Ž z ty1 , z t .
␧
␧
s rt Ž z ty1 , z t . Ž Ž a ty1
Ž z ty1 . y a0ty1 Ž z ty1 . . y Ž c ty1
Ž z ty1 .
Ž 17 .
0
yc ty1
Ž z ty1 . . . .
0
Therefore, there is a nonnull set of z ty1 such that a0ty1 F a ␧ty1 , c ty1
0
␧
ty1
0
a ty1 , and P Ž a t ) a t < z . s 1. From this, Lemma 1 and the strict concavity of ¨ t imply
0
y
0 < ty1
¨y
ty1 Ž a ty1 . F ␤ E r t¨ t Ž a t . z
␧ < ty1
␧
- ␤ E r t¨ q
F ¨q
t Ž at . z
ty1 Ž a ty1 . ,
which violates the concavity of ¨ ty1.
Ž 18 .
394
CHAMBERLAIN AND WILSON
0.
t
0 . y1
Ž ␧
Ž ␧
By definition, Ž a ␧tq1 y a0tq1 . Ry1
0, tq1 s a 0 y a 0 y Ý js0 c j y c j R 0 j for
0
␧
any t G 0. Therefore, P Ž a t F a t , t G 0. s 1 implies
t
E
Ý Ž c j␧ y c j0 . Ry1
0j
Ž 19 .
js0
s Ž a0␧ y a00 . y E
␧
y a0tq1 . Ry1
Ž atq1
0, tq1
F ␧.
Furthermore, using Lemma 1 and the strict concavity of u and ¨ t , it also
follows from P Ž a0t F a ␧t , t G 0. s 1 that P Ž c t0 F c t␧ , t G 0. s 1. Then,
t
.
since P Ž ␤ rt F 1, t G 1. s 1 implies P Ž Ry1
0 t G ␤ , t G 0 s 1, it follows
from the concavity of u that
¨ 0 Ž ␧ . y ¨ 0 Ž 0. s E
⬁
Ý Ž u Ž c t␧ . y u Ž c t0 . . ␤ t
ts0
FE
⬁
Ý uq Ž c t0 . Ž c j␧ y c j0 . ␤ t
Ž 20 .
ts0
FE
⬁
Ý uq Ž c t0 . Ž c j␧ y c j0 . Ry1
0t
F ␧ uq Ž 0 . .
ts0
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.
LEMMA A.4. Let D be an open, con¨ ex subset of ⺢ and let Ž f j : D ª
⺢, j G 1. be a sequence of conca¨ e functions such that f j Ž x . ª f 0 Ž x . for all
qŽ .
yŽ .
Ž .
Ž .
x g D. Then fq
x and fy
0 x F lim inf jª⬁ f j
0 x G lim sup jª⬁ f j x .
Ž x .Ž y y x . for x, y g D.
Proof. Since f j is concave, f j Ž y . y f j Ž x . F fq
j
Then f j ª f 0 implies that f 0 is concave and f 0 Ž y . y f 0 Ž x . F
qŽ .
Ž x .Ž y y x ., which implies fq
Ž .
lim inf jª⬁ fq
x . The proof of
j
0 x F lim inf j f j
the second half of the lemma is similar.
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