JOURNAL
OF ECQNOMlC
THEORY
16, 151-l 66 (1977)
Some Results on “An
Income
Fluctuaticx
Pr
JACK SCHECHTMAN
Uafversity
Federal
of Rio de Janeiro/IkfUFRJ,+
Rio de Janeiro,
Brazil
Rio de Janeiro,
Brazil
AND
VERA L. S. ESCUDERO
Wniversity
Fedeval
of Rio de JaneirojCOPPE,
Received May 18, 1976; revised October 21, 1976
A consumer at each period, given the income available, y, has to decide how
much to consume and save. If he consumes c > 0 units he gets u(c) units of
satisfaction or utility, and if x = y - c > 0 is the amount saved then the available income in the next period is PX + w*, where wB is a random variable, and
I’ is an interest factor that is assumed to be known with certainty. Infinite time
horizon problems are considered, and it is shown tbat if 0 < 6r < 1, where
0 < 6 < 1 is a discount factor, then the limiting policy is optimal. Questions
about the behavior of the stock level, such as bouudness, are considered, and an
example is given that shows that the stock level might converge almost surely
to infinity. Finally an economic explanation is given.
In this paper we consider the following situation. A consumer has a t
period planning horizon, and at each period he must decide how much to
consume and save, in order to maximize the total expected utility accumulated
in the t periods. If y is the available income at the kth period, and. 0 < x < y
is the amount saved at that period, then the available income in the next
period is YX + 02, where 03 is a random variable and r is an interest factor
* This research has been partially supported by the Ofhce of Naval Research under
Contract NOOOI4-76-C-0134 and the National Science Foundation under Grants %X7515566 and MPS74-21222 with the University of California. The U. S. Government’s
right to retain a nonexclusive royalty-free license in and to copyright covering this paper
is acknowledged.
T Now at the Instituto de Matematica Pura e Aplicada IMPA:‘CNPq, Rio de Janeiro,
Rrazii.
Copyright
Ail rights
Q 1977 by Academic
Press, Inc.
of reproduction
in any form reserved.
:ssK
00:2-0531
152
SCHECHTMAN
AND ESCUDERO
that is assumed to be known with certainty. If c = y - k units are consumed
in period k, this produces 8%(c) units of satisfaction or utility to the
consumer, 0 < 6 < 1 is a discount factor. It is assumed that u(c) is strictly
concave, increasing, and differentiable and that the random variables wk
are i.i.d.
In Section 1 we describe the model and state the general results that are
going to be used; they are basically the concepts of competitive prices and
policies for the stochastic case.
In Section 2 we consider the deterministic case, i.e., w = a. It is shown that
the limiting policy is optimal when 0 < 6r < 1, and the behavior of the
consumption, except for the infinite time horizon problem, is studied. In
Section 3 we consider the stochastic case; i.e., w is a nondegenerated random
variable. For the case 0 < 8 < 1, r = 1, it is shown that the limiting policy
is optimal and also that the level of income is bounded above. For 0 < 6 < 1,
6r < 1, and Y > 1, it is also shown that the limiting policy is optimal, but it
is not necessarily true that the stock level will always stay in an interval
[0, j?]. An example is given for which the stock level converges; almost
surely, to infinity. Sufficient conditions are obtained in order to assure the
existence of a bound for the stock level.
We should also mention that some of the results of this paper were
presented in the First Brazilian Symposium in Probability held at Rio de
Janeiro in 1974.
1. THE MODEL
AND SOME GENERAL
RESULTS
Finite Time Horizon ProbIem
A consumer is faced with the following situation. He has a t period planning
horizon, and at each period he must decide how much to consume and save,
in order to maximize the total expected utility accumulated in t periods. If y
is the available income at the kth period, and 0 < x < y is the amount saved
at period k, then the available income in the next period is rx + wk, where wk
is a random variable and r is an interest factor that is assumed to be known
with certainty. If c = y - x units are consumed in period k, then this
produces 8%(c) units of satisfaction or utility to the consumer, and 0 < 6 < 1
is a discount factor. We wiI1 assume that u(c) is strictly concave, increasing,
and differentiable and that the random variables mk are i.i.d. The range of w
is an interval [a, A], A < + co and Prob[w = a] > 0. More formally our
problem is maximize
AK
INCOME
FLUCTUATION
PROiBLEM
153
subject to the stochastic constraints
C7G +
7.X7c-l
xlc = yk,
+ wk-l = yk,
co + XQ= yQ,
CT = yT,
for 1 < k < T.
cL > 0, 9 3 0 for 1 3 k < a, and y” is the initiai income available.
If V,(y) is the maximum expected utility that we can get when y is the
income available and we have t periods to go, then the usual dynamic programming formulation leads us to the functional equations:
It can be shown that V6(y) inherits the basic properties of u(c), i,e., V,(y) is
increasing, strictly concave, and differentiable.
The derivatives of V,(y) will be denoted by p,(y) and the solution of (1.1)
will be denoted by (ct( y), x,(y)).
The following properties of (ct( y), x,(y)) and pl( y) will be stated without
proof (for proofs see [3]):
(1.2) pt(y) is a positive decreasing function, p,(y) > ptU1(y) for ally and
all t, and if U’(C) is a convex function then pt( y) is also a convex tknction for
all t.
(1.3) c,(y), x,(y) are nondecreasing continuous functions and ci(y) <
ct-,t~9 txd~> >, xt&))
for all t.
(,1.4) The functions ct( y), x,(y) satisfy the following relations:
(I .5) pt( y) >, &Ep,-,(rx,( y) + W) with equality if x&y) > 0;
(1.6) pt(y) > u’(c,(y)) with equality if et(y) > 0.
Notation. If w E u (a constant), then the corres~o~d~~g prices and
policies will be denoted by p,“(y) and (c,~(Y), x,“(y)).
Now we will state a theorem that permits us to compare p&y) and consewen~iy c,(Y> with P?(Y) with PP(Y)>, P~~(Y~?(Y)),
and P*~(Y)(c~~~Y)~
where ~3is the Ew. A proof of it can be seen in [3].
THEOREM
1.1.
The following relations are true:
(i) pi(y) <p,“(y) for all t, and consequentZy et(y) 3 Ct”(y);
(ii) pt( y) > p/(y)f~r
all t and consequently c,(y) < c/(y);
(iii) if u’(c) is a convex firnction thea pt( y) >, p;O(y) and ~Q~seqLie~t~~
c,(y) < cha(y), where 6 = Ew.
154
SCHECHTMAN
AND
ESCUDERO
Infinite Time Horizon Problems
Now the consumer objective is to maximize
subject to the stochastic constraints
&
+
Xt
=
&t-1
+
for t > 1,
(O-1
co -+ x0 = y”,
ct 2 0,
xt 3 0
for t >, 0.
DEFINITION.
A policy is a pair of functions (c(y), x(y)) such that c(y) > 0,
x(y) > 0, c(y) + x(y) = y. If y is the stock of capital available then c(y) > 0
tells us how much to consume and x(y) > 0 how much to save.
Given a policy (c(y), x(y)) the stock of capital available at period t will be
denoted by yt (observe the superscript).
For some cases the series E[CE, S”u(c”)] might diverge and in order to
compare the return of policies we will introduce the following definition; for
a more detailed discussion see [l].
DEFINITION.
A policy (c(y), x(y)) is said to overtake another policy
(6(y), a(y)) if starting from the same initial stock yl, the policies (c(y), x(y)),
(6(y), k(y)) yield (stochastic) consumptions paths et and P that satisfy
I
> 0
for all T > some To .
A policy is said to be optimal if it overtakes all other policies.
Now we will state without a proof a theorem that gives sufficient conditions for the optimality of a policy (c(y), x(y)). A proof of it for a similar
problem can be seen in [2].
DEFINITION.
THEOREM
1.2. If there exists a positive and nor&creasing function p(y),
and a policy (c(y), x(y)) that satisfies the following conditions, then the policy
(c(y), x(y)) is an optimalpolicy:
(1.7) p(y) 2 u’(c( y)) with equality if c( y) > 0;
(1.8)
p(y) 3 SrEp(rx(y)
(1.9)
limt-,
+ W) with equality $x(y)
> 0;
EGtp( ye) yt = 0.
1 Observe the superscript ct indicates the consumption
from the first period.
at the tth period, now counting
AN
INCOME
FLUCTUATION
155
PROBLEM
In general it is not easy to find an optimal policy to infinite time horizon
problems. Natural candidates are of course the limiting policies c(y) =
lim,-, et(y) and x(yj = Em,-, x,(y) which are continuous nondecreasing
functions. If bounds for pt(y) can be found, limits in (1.5) and (1.6) can be
taken and they yield
and
where p(y) = Bim,,, pt( y>. Hence whenever a bound for plj y) can be foun
to show that the limiting policy is optimal it will sufhce to show that lim,,,
E@p( y”) y* = 0, where yt is the stock available at the tth period when we use
the limiting policy.
In this section we will consider the case w EZ a :> 0, 0 < 6 -=z1, r Z I, and
6r < I. We will obtain bounds for the prices p,(y), study the behavior of the
income level when we follow the limiting policy, and finally it will be shown
that the limiting policy is optimal.
Proof.
Suppose not, i.e., x,(a) > 0. Hence from (15) and (1.6) we get
Pt(Y> = ~~Pt-,k%(.Y~ + 4
< Pt-~(xt(Y) + 4
G P&(Y)
So, a: >, uxt( yj + a, by the monotonicity
LEMMA
2.2.
+ 4
by (1.2).
ofp,( y), which is a contradiction.
g-0 4 6 < 1, Y > I, 6r < 1 then c,(y)
b [(Y -
1)/r] yfor
nli
Y.
Proof.
e proof depends on the value that x*(y) assumes:
(a) xt( y) = 0. Then ct( y) = y and consequently ct( y) 3 [(P - 1)/r] y.
(b) x&j > 0. From the optimality condition we get pt(yj = 8p.p,-,
(vxt(yj $ a). Now, from (1.2), the monotonidty
of pi(y) and the fact that
Sr < 1 we get pi(y) < &m,(y)) and consequently we get y/r > x,(y) and
C*(Y) 2 Kf- - Wl Y. a
156
SCHECHTMAN
AND ESCUDERO
From Lemmas 2.1 and 2.2 it follows that p,(y) I u’[(Y - 1)/r] a}. So, the
limiting policy satisfies the relations
with equality if x(y) > 0,
P(Y) 3 kPb(Y)
+ 4
C(Y) > 0.
P(Y) = U’(4Y>>5
(2.0
(2.2)
We should mention at this point that if we assume that u’(0) < co, we
could drop the assumption that a > 0, and bounds for the prices would then
be u’(O).
The next theorem tells us how the stock level behaves when we follow the
limiting policy.
THEOREM 2.3. For the limiting policy yt+l < yt (c(y”+l)) 5 c(yt), and if
y” 2is the initial stock, then there exist a T( y”) such that yt = afor all t > T( y”).
Proof. First we will show that y $+l < yt. If yt > a and x( yt) = 0, then
Yt+l = a and so yt+l < yt. If yt > a and x( yt) > 0 then from (2.1) we get
P(Y”) = arp(ytfl)
< P( Yt+l)*
So, yt > yt+l by the monotonicity of p( y). If yt = a from Lemma 2.1 we get
that x(y”) = 0 and hence that yt+l = a, so ytS1 < yt.
To show the existence of T( y”) we will argue by contradiction, i.e., suppose
that yt > a for all t, hence x( yt) > 0 for all t. So, from (2.1) we get
P(Y”) = @Myt)
for all t,
but 0 < 6r < 1 and p(y) < u’(a) and consequently
all t. 1
(2.3)
(2.3) cannot hold for
From the previous theorem it follows that the sequences yt converge to a
in a finite number of periods.
The behavior of the sequence of capital (y”>, consumption {ct>, and saving
(xt) can be represented by the following graphs:
2 We will assume without loss of generality that v0 > a.
AN INCOME FLUCTUATION
THECREM
2.4.
The limiting
policy
PROBLEM
157
is optimal.
Proof. It suffices to show that lim,,, Pp( yt) yt = . From Lemma 2.2
and Theorem 2,3 we have that p( y’) < d([(r - 1)/r] a] nd yt < y* for all t.
ence 0 < 8p(yt) yt < GW{[r - 1)/r] a} y”, for all t. So, lim,,,
8p(yt) yt = 0, since 0 < 8 -=c1.
3. STOCHASTIC CASE, I. E.,w Is ANONDEGENERATED
~A~~O~~AR~A~LE
In this section we will study the stochastic case in which w E [a: A]? A < a
is a nondegenerated random variable, and Prob[w = a] = 01 > 0.
For the deterministic case we have shown that in a finite number of periods,
when we follow the limiting policy, the stock level reaches the level a. The
analog for the stochastic case would be a renewal process. This is in fact true
whenever 0 < 6 < 1 and I” = 1, the stock { yt) evolves in an interval (a, j7j
and a is a renewal point. But, for 0 < 6 < 1, 61~< 1, r* > 1 this result is not
necessarily true, we will give an example in which (~“1 -+a.s. cx:.
(a) Tl~eGaseO<6~1,~=1
Will show that the limiting policy is optimal,
ounds for the prices pt( y).
Proof.
but first it is necessary to fin
From Theorem 1.1 and Lemma 2.1 we get
Pt(Yl < PtYY) < u’(a)
for all y 2 CI
(3.1)
and all f, consequently et(y) 2 eta(y) > a.
THEOREM
Pro@
3.2.
The limiting
First the limiting
policy
is optimal.
policy satisfies the conditions
P(Y) > @P(X(Yl i
P(Y) = I’),
a>,
(3.2)
(3.3)
which is obtained from the optimality conditions (1.5) and (1.6) after taking
the limits and using Lemma 3.1. To prove that the limiting policy is optimal
it suffices to show that lim,,, EG*p( y”) yt = 0. From Lemma 3,1 we have that
p(y) < u’(a). Now as w < A we have that yt < rA + y”. So,
15x
THEOREM
SCHECHTMAN
3.3.
AND
ESCUDERO
There exists a j such that for ally > j7 x(y)
f A < y.
ProoJ
First we will show that there is a 7 such that x(j) f A < j7. By
contradiction suppose not, i.e., for all y, x(y) + A > y; hence, c(y) < A
for all y. So, lim,,, c(y) = A4 and
lim p(y) = u’(M) > 0,
y-m
since M 3 c(~l) > 0 for y > 0.
(3.4)
From (3.2) and (3.3) we have
P(Y) = WMY)
+ w> < Mx(.Y) f a).
(3.5)
Now, taking the limit in (3.5), using (3.4) and the fact that lim,,, x(y) = cc
we get that A4 < 8M which is a contradiction since 0 < 8 < 1. The theorem
now follows from the fact that c(y) is a nondecreasing function which implies
that c( I>) 3 A for all y 3 jj and consequently that x(y) + A < y for all
Y2.P.
1
From the previous theorem it follows that the process evolves in an interval
[a, J], For the deterministic case we were able to show that {y”] converges in
a finite number of periods to a (whenever the process reaches the stock level
a we will say that the process is in state a; for the stochastic case we will show
that the process { y”} is a delayed renewal process, in which state a is a renewal
point, i.e., each time the process reaches state a it starts all over again.
Let X0 be the number of periods that it takes to reach for the first time
state a. And let Xj be the number of periods that elapses between the (j - 1)th
time we reach state a and the jth time we reach it again. The random variables
x, ) Xl ). .. , xj ). .. are independent and X1 , X, ,..., Xj , are identically distributed. Hence the process Xj is a delayed renewal process. To show that it is
nonterminating
we need to show that EXi < 03. In order to show that Xj
has finite expectation we will consider an artificial process that is described
below.
Consider a process with T’(j) = T(j) + 1 states, where T(y) is the number
of periods that it takes to reach the state a when we assume that w = a and
we follow the optimal policy (see Theorem 2.3); jj is an upper level for which
yt < J for all t, which existence we have shown before. Now returning to the
stochastic case, at each period we observe the value of w. If w = a and we
are at state 1 cj < T’(y) then we move to statej - 1 and if we are at state 1
we stay at that state. If w + a and we are at state 1 < j < T’(y) we move to
state T’(J). The following picture describes the process, where the numbers
in the arrows are the transition probabilities. If z is the random variable that
represents the number of periods that the process takes to reach the state 1
having started from state T’(J), it can be shown that
1 +
EZ =
01 +
. .. +
aTW)
&Ybl
2
AK iNCOME FLUCTUATION
t
PROBLEM
159
?+I
and it is easy to see that X, < Z and so consequently we wilt have that
EXj<03.
Using a well-known result from the renewal theory (for instance, see
[4, p. 3651, we can state the following theorem.
THEOREM
has a limiting
3.4. Zf w is a denumevable random aariabk, then the process ( y”>
distribution.
(b) Case 0 < 6 < 1, )’ > 1,6v < 1
First we have from Theorem 1.1 and Lemma 2.2 that p,(y) d p,“(y) <
u’{[(r - 1)/r] 4’) 2 u’([(r - l>/ Y1)a f or all y >, a and all t. Hence, the limiting
policy satisfies
(3.6)
P(Y) 3 of%@+)
+ 44
(3.7)
P(Y) = e4.v>>.
THEOREMS 3.5.
The limiting policy is optimal.
ProoJ Tt suffices to show that lim,,,, ,Wp(yt) yt = 0. Now, yt < rt
{ y” + [(r - 1)/r] A) andp(yt) < ~‘{[(r - 1)/r] a), so
Taking the limit and using the fact that Sr < 1, we get the desired result.
When 0 < 6 < 1, r = 1, we were able to show that there exists a jj such that
x(y) t A G y for all y > J. We could expect this result holds for I’ > 1,
6u < 1. The following example will show that in some cBses we coul
-:y”> -B.S. co.
Consider the following problem expressed in its dynamic ~ro~ramrn~ng
formulation.
PROBLEM
1.
V,(y) = Max(--eee + GEV~-,(~x t- CO)),
c t x = t’,
c > 0,
x 20,
V,(y) = -e-u.
160
SCHECHTMAN AND ESCUDERO
To find an explicit solution of Problem I is “very difficult,” and as a metter
of fact, we are not going to solve it. We will find the solution of another
problem, and then show that for the limiting policy the stock level {ytj of
Problem I converges almost sure to +co.
Consider the following problem, in its dynamic programming formulation.
PROBLEM II.
Vt(y) = Max{+-”
f SEV,&x
$ w)),
c+x=y,
V,(y) = -e-“.
Observe that now we do not impose the condition of nonnegativeness on
c and x. This fact will allow us to find an explicit solution of Problem II.
This solution is given in Appendix A, and the limiting policy is
r-l
C(Y) = 7
Y - ---&f
x(Y)=;+
log
8rE[e-W(+l)lr],
(3.9)
(3.10)
log 8rE[e-w(T-1)/T].
The solution of Problem II, its prices, and the stock level will be denoted by
MY), 3Y)l, F(Y), and {jtj and the corresponding ones for Problem I by
MY>, x(AL MY% and in”>.
From (3.10) we get the following relation for the stock level
Y-t+l
=
jt
+
[r/(r
l)] log GrE[e-[(V-l)/rlW] + w;
-
i.e., {j?} is a random walk in which the random variable is
w’
=
w
+
[r/(r
-
log
l)]
BrE[e-[(T-l)/rlw].
Now, if we take Prob[w = 0] = 1 - 8 and Prob[w = A > 0] = 8 we will
have that
Ecu'= eA + -&
i0g
6r
+
i0gp
-
e+
ee-A+lq
-&
for a given 0 if we choose A sufficiently large, and we will have that Ew' > 0
and so {Jt} -a.S. + cc. Now, we will show that Jt < Y* for all t, but first we
need a lemma, whose proof is given in Appendix B.
LEMMA
3.6.
3~) < 4~)
THEOREM
for y 2 0.
3.7.
jt < yt
for
all
t, if y”
=
j”.
AN
1NCOME
FLUCTUATION
PROBLEM
161
Proo$ The proof will be done by induction. First, it is true for f = 0
since y” = j?. N ow assume that it is true for t, i.e., j? < yt. Xfjt” < 0 then
j”+l < yt+l since yt 2 0 for all t. if jP+l > 0, we have
jw
by monotonicity
we get
= rX(jF) + cot < rx(j7”) + LLIt
of .Z(.) and the induction hypothesis. Now, using Lemma 3.6
Y-t+1 = I”qyy + o)t
< rqy) + 03
< rx(yt) + c0* = yt+l.
From the previous results we could state that if u(c) = -ET-~ and for a
given 8 A is sufficiently large, then (y”] -fse6. foe.
Since in general it is not necessarily true that there exists a jj such that
yt E 10, 71 for all t, it would be interesting to &ad conditions under which
yt E 10, j] for some J.
3.8. A suj5kient condition for the existence of a p such that
-t A < y for ally >, 7 is that
THEOREM
rxly)
(3.11)
ProoJ
From (3.11) we have that given an E > 0 there exists a 3 such that
ewe,
P(Y)
<Ii-E
Wrx(y)
t A)
and
P(Y)
p(r-4.d + 4
< Sr + E&.
If, E < (I - 8r)/&, thenp( y) < p(vx( y) + A) for all y > j and consequently
y 3 Ix(-y) + A for all y 3 j. g
One case that immediately satisfies the Last theorem is the case u(c) =
Kc + g(c), where K > 0 is a constant and g(c) is a strictly concave f~mctisn,
increasing and differentiable.
he last result would be of limited application if we were restricted to the
lash example. But as we will see, using the concept of exponent of a function
(used by Brock and Gale in [l]) we will be able to apply it to a very generat
class of nroblems.
162
SCHECHTMAN
DEFINITION.
AND ESCLJDERO
The exponent of the function f at the point x is
44 = hU(,u),
and the asymptotic exponent e, off is lim,,
ef(x>.
If f(x) is a positive function, then the following fact can be shown (see
[I] for a proof).
If x > X’ and a < e, < a then for x large
(x’/x)” < f(x)/f(x’)
< (x/x’)5
(3.12)
THEOREM 3.9. If u’(c) has an asymptotic exponent ef , then there exists a 7
such that rx( y) + A < y for ally 2 jj.
Proof. If lim x(y) = K, then for y > tK + A, rx(y) + A < y.
lim,,, x(y) = co then the proof is as follows.
4drxb9+ w>G ,drx(Y)+ 4
~~r-4Y)
+ 4 pW(A+ 4
+ a>>( c(rx(u)+ A) p for y large,
- u’(c(rx(y)
u’(4r-Q) + -4 ’ i 4r.U
-t 4 1
If
(3.13)
by (3.12), and using the fact that lim,,, c(y) = ~0 and C(tX(Y) i- A) >
+X(Y)
+ a>.
Now, c(vx(~> + A) = c(rx(y> + a> + h(y)(A - 4, were 0 ,< NY) G 1.
Hence, from (3.13) we get
44rx(y) + w>< cb4.d + 4 + h(y)(A - a> oi
1
p(rKd + 4 ’ (
c(r-4.d + 4
1 + NY)@ - a> *
Xy)
zz=
for y large.
1
i
Now, taking the limit as y + CCwe get @(KC(~) + w)/[p(rx(y) + A)] = 1.
The theorem now follows from Theorem 3.8.
Now, whenever there exists a j such that VX(y) + A < y for all y >, 7, we
will have that the process evolves in an interval [a, y]. As in the case
0 < 6 < 1, r = 1 .it follows also that the process {y”} is a delayed renewal
process, as are its consequences.
In order to give some economic flavor to our result we should first mention
that there is a close relationship between the limiting exponent of the marginal
utility u’(y) and the Arrow-Pratt relative risk avertion measure RR(y) =
-u”(y)/u’( y). In [7, p. 1 lo], it is shown that if RR(y) < R for all y > y. for
some y0 then u’(y)/u’(yJ > (yJy>” for all y b y,, , that is inequality (3.12)
obtained from the existence of the limiting exponent for the marginal utility
also holds. A possible economic explanation of our result could be argued in
AN
INCOME
FLUCTUATION
PROBLEM
153
the following way: At each period the consumer is guided in his choice
between consumption and saving by the values of an extra unit in face of his
actual wealth, y, and his next PX(y) + w, a random variable, that depends on
his next income w. Those values are expressed by p(y) and p(~x + w), the
latter being also a random variable. His uncertainty is about the changes in
the value p(rx t IV). Now, if p(rx + A)/[Ep(rx L A)] is “close” to i, we
could say that the consumer for-sees small variation of p(rx: i W) with
respect to p(rx f A). If we assume that the consumer’s behavior could be
expressed by a utility function whose marginal utility has a limiting exponent
or a uniformly bounded relative risk avertion for large values of y we will
have that hm,,, (p(rx(y) + A)/&+(y)
t w)]) - I (see theorem 3.9) and
if we also take into account the fact that Sr < ! we will have that for sufl%
cientiy large value of y the consumer will behave as in the deterministic case,
i.e., his wealth is going to decrease no matter what happens with his future
income.3 The utility function e-Y does not have a uniformly boun
risk avertion. nor does its marginal utility have a limiting exponent, that is the
varia’bihty of p(rx + w) with respect to p(rx + A) might not decrease to aa
extent that makes him behave as in the “deterministic case.” This is in fact
what our previous example shows, namely, given a binomial distribution for
w if we choose A sufficiently large the sequence of weahhs (yt3 that we get
when we follow the optimal policy converges almost surely to infinity. 8n the
other hand if A is sufficiently small, for instance, A < --log &, it can ‘be
shown that +vt > A implies yt > yt-’ for all t. A reason for this resuh could
be given in a similar way as before. First, for different values of the wealth
~(2:) 7 1’;: the greatest difference in consumption is A, and if A is ‘“small”
this will imply from the continuity of u’(;:) that p(rx + W) do not vary
relatively 9.0p(rx + A) to a great extent.
We also should point out a fact observed early by Pratt [a] that the function
z{‘(y) = 6~ in spite of its nice graph has great relative variations when we
compare va!ues that differ in a relatively small proportion. For instance, if
i’ = 10” and y’ = y + IO-5y
()r y’ = 106 - 10 we have that e-v/e-“’ = P5
which is a very large number!
APPENDIX
A
In this Appendix we will find the solution of the problem:
Maximum
E fj stu(ct)
i=O
3Frorr the fact that lim[p(tx + A)/E’(rx + A)] -+ 1, the optimality condition p(y) ==
GrE’(vx(yf i w) could be replaced by p(y) N (6r f e)p(r.xCy) + A), where E cars be made
arbitrarly small by choosing y arbitrarly large.
164
SCHECHTMAN
subject to ct $ xt = yt,
AND ESCUDERO
O<t<T
Y t+1 = rxt + J
and
XT = 0,
CT = yT,
where
u(c) = -e+,
The dynamic programming
CJ E [a, A] 09 are i.i.d.
formulation
Vt(y) = maxi-e+
leads to the functional equations:
+ GEV,&x
+ at)},
(A.11
X+C=Y,
and
Vi(y) = -e-“,
and
Cl(Y) = Y>
Xl(Y) = 0.
We claim that the solution of (A.l) for any t > 1 has the form
+-1
dJ)
=
-;“A
__---
+
. .. +
1
y - At log 6r - B, ,4
e-2 + ... + 1
Xt(Y) = +1 + ... + 1 y + At log & + Bt .
It is true for t = 1. Now assume that is true for t, then
Bt+l = (4 + log Eexp (-
yt-l it::.
+ 1 w)) x r~~~..‘.“$l
From the recurrence relation, we will have that the limiting
log 6r - -&
4Y.l =
APPENDIX
log
1.
policy is
E[e-(tr-l)l~hJ],
B
In this appendix we will obtain a result that permits us to compare the
optimal policies for the problems:
PROBLEM
I.
Max c @u(c~)
t=o
4 Where the sum in the denominator
ranges over L’s such that t - i > 0.
AN
INCOME
FLlJCTIjATION
165
PROBLEM
Subject to
Yt+1 = rxt + d,
36 + ct = y,
xi > 0,
ct 2 0.
Max i
Gfu(ci)
t=o
Subject to
p1 = rxt f wt,
.xt + et = Yt,
CT = y+,
XT = 0.
The prices and policies for Problem I will be denoted by &y)), (et(y), x5(y)]
and the corresponding ones to Problems II by P&y)% (Et(y), XL(y)).
THEOREM
B1.
FOV y 2
0, X&y)
<
xt(y)
rind
conseqziently
X(y)
<
x(y),
PTS~I The proof will be done by induction. It is true for t = 1 since
x,(y) = 0 and 5&(y) = 0. Now, assume that it is true for t - 1, i.e., .xtll(y) >
Zt-:(y) for y > 0 and by contradiction that xt(y) < .Tt(y). Then y - ei(y) <
y - C,(y) and consequently ctJy) > G&y) and p&y)
< j?-,(y). From
e optimalities conditions we get
I%(Y) = ~Pt-l@xt(Y)
3 J%-lo%-ICY)
i
w>
+ a>
(3.1)
since ptel( .)
is nonincreasing and Y%& y) > x,(y) >, 0. But from the induction hypothesis we have that ctM1(y) < C,,(y) and so, &+(;J) > pttl(y). Then @.I)
implies
PdY> 2 ek-l&G-l(Y)
+ w>
= Pt-l(Y),
which is a contradiction.
4
ACKNOWLEDGMENTS
I would like to thank Professor David Gale for his comments and fzx the opportunity
of being back at Berkeley, and Professor 3. A. Sheinkman for helpfuu2 cowersation. But the
authors are stiil responsible for the errors left.
166
SCHECHTMAN
AND
ESCUDERO
REFERENCES
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