On income fluctuations and capital gains

JOURNAL

OF ECONOMIC

32, 14-35

THEORY

On Income

(1984)

Fluctuations

and Capital

Gains

MARILDA A. DE OLIVEIRA SOTOMAYOR

Pontifiia

Universidade

Received

Departamento de Matema’tica, Catdlica do Rio de Janeiro, Rua Marquh 225~CEP 22.453, Rio de Janeiro, Brasil April

28, 1982; revised

December

de Sdo Vicente,

13, 1982

We consider the infinite time horizon problem of asymptotically maximizing the expected accumulated discounted utility in a one-good production economy. The available capita1 in a given period is given by the production of the previous period plus a random variable. The product of the discount and interest factors is either (1) greater than or (2) equal to one. Under (1) the optimal policy exists under certain conditions and always under (2). The optimal capital sequence almost surely goes to infinity. Under (1) with conditions on the utility one almost surely reaches a capital level above which the sequence is increasing. Journal ofEconomic Literature Classification Number: 213.

INTRODUCTION This paper considers the following situation: A consumer is faced with an infinite time horizon problem and in each period he must decide how much to consume and how much to save in order to maximize asymptotically the expected accumulated and discounted utility. If the amount consumed in the tth period is et, the consumer gets 6’u(c’) units of satisfaction or utility, where 0 < 6 < 1 is the discounting factor and u is the utility function. If xL is the corresponding amount saved, with xf + cf = yf being the available capital at the beginning of the tth period, then the capital at the beginning of the next period will be yf+’ = TX’ + wt, where r > 1 is an interest factor known with certainty and w’ is a random variable. We will consider that _U is increasing strictly concave and differentiable and the random variables wt are equally distributed independent and nonnegative with values in [a, A] with a > 0, A < co. Our problem can be formulated as follows: The aim of the consumer is to find a sequence (et, x’) such that if (F’, 2’) is another sequence of consumptions and savings then, the first sequence overtakes the second, that is,

E c 6f[u(c’) - u(2)] > 0,

for all T > To,

t=0

14 0022-053 Copyright All rights

l/84 $3.00 0 1984 by Academic Press, Inc. of reproduction in any form reserved.

for some T,,

ON INCOME

FLUCTUATIONS

AND

CAPITAL

GAINS

15

subject to the stochastic constraints

ct+xt=yt,

ct> 0,

xt> 0,

y’+1= rx”+ w’, t: cl,..., T

and E is the expected value operator. The stationary characteristics of the model lead us to assume that the decision of how much to consume and how much to save is function only of the present capital and not of the time period considered. This paper has four parts. In Section 1 we formulate the economic model using the dynamic programing recursive equation method and we state the general results that are going. to be used in the next section. If 6r < 1, it was shown in [l] (for r < 1) and in [2] (for r > 1 and U’ with asymptotic exponent) that there exists a level of capital V, such that the optimal sequence of capitals, { JJ’},, decreases to Y; passes it with probability one and stays between 1 and JX In Section 2, we show that if 6r > 1, and U’ has an asymptotic exponent and if the consumer follows the optimal policy there is a level of capital, j, above which the capital is always increasing (Theorem 2.6). Furthermore, with probability one this level of capital is reached and the capital becomes arbitrarily large when t tends to co (Theorems 2.7 and 2.8). The relevance of this result is that if the capital of the consumer is bigger than 9 he is guaranteed independently of the uncertainty a minimum consumption for his whole life. The behavior of the sequence of capitals in these two cases can be represented by Fig. 1. We show by an example, that the level F does not need always exist when U’ does not have an asymptotic exponent. Furthermore, we assert that if the optimal policy exists then the optimal sequence of capitals becomes arbitrarily large with probability one when t tends to co. This assertion corrects an erroneous result of [7], where an attempt was done to prove, through an example, for 6r > 1, that by an appropriate choice

FIGURE

642/32/I-2

1

16

MARILDA

A. DE

OLIVEIRA

SOTOMAYOR

of the random variable w and of its distribution the optimal sequence of capitals {y’}, is bounded almost surely. In Section 3, we find utility functions which admit a linear optimal policy, characterizing them in the deterministic case, and we study some of them considered of special interest for economists (Sections 3.1 and 3.2). This implies the existence of different classes of utility functions which have a linear optimal consumption, in the case w = a >, 0, which corrects the mistaken assertion of [8], according to which, in the case w = 0, the only utility functions with this property are y’ (A > 0) and lg y. Still in the case 6r > 1 and w a random vriable, and using a result of comparison between the derivatives of the utility functions, we get necessary conditions and also sufficient conditions for the existence of the optimal policy. In Section 4, we study the case in which 6r = 1. We show there that the limit policy is optimal, that the prices are strictly decreasing and the optimal sequence of capitals tends to CXIwith probability one when t tends to 00. This assertion also corrects the erroneous result of [7] for the case 6r = 1. Nevertheless there is no level of capital 7 starting from which the optimal capital sequence is increasing. In the deterministic case the optimal policy for any utility function is given by x(y)==,

r

= 0, c(y)=

(r-

l)Y -l-a r

=Y,

’

if

y>a

if

y
if

y>a

if

y
and therefore the optimal sequence of capitals stays constant and equal to y”, if y* > a or equal to a, starting from t = 1, if y* < a. The case where 6 = 1 and r = 1 was studied by J. Schechtman in [6]. There, he also proved that the optimal sequence of capitals almost surely tends to co. We consider _a and _A such that P(w E [A - &,A]) > 0 and P(w E [a, a + s]) > 0 for all s > 0. In some proofs we will suppose P(w = A) > 0 and P(w = a) > 0, but these conditions are not necessary and the same results can be obtained without them. Throughout this work we will use the terms “increasing” and “decreasing” to denote, respectively, “non-decreasing” and “non-increasing,” reserving the word “strictly” to mean exactly that. Thus “increasing” means the value does not decrease (but could be constant) and “strictly increasing” means

ON

INCOME

FLUCTUATIONS

AND

TABLE

Existence of an Optimal Policy 0<6<1,r>l

CAPITAL

GAINS

17

I

Asymptotic Behavior of the Optimal Sequence of Capitals, ( y’ j If u’ E E, 3p such that VJJ> ~4 rx(y) + A < y, and Ppt*;y’E [a,y],vt>t*)= (a) If u’ E E, 3jsuch that

6~ < I The limit policy is optimal V_u. 6r> 1 (a)Ifu’hasan asymptotic exponent # 0 or u is bounded above, then the limit policy is optimal. (b) If limy-rm u’(y) # 0 .Jan optimal policy. 6r = 1 The limit policy is optimal Vu.

Vy>.!,m(y)+a>yand P(3t*;y’ > j, v’t > t*)

(b) P(lim i*E,.v’=

1.

= 1.

aJ)= 1. v_u.

39 and 35 P (lim ,+my’ = co) = 1, vu.

the value does not decrease and is not constant on any interval. We will use the notation lgy to denote the natural logarithm. We can describe the main results of this paper according to Table 1, where E is the class of functions with asymptotic exponent.

1.

THE

MODEL

Finite Time Horizon Problem A consumer has T periods to go. In each period he must decide how much to consume and how much to save in order to maximize the total expected utility accumulated in T periods. More formaly, the problem is to maximize

subject to the stochastic constraints c’+x’=y’ Yt+ l = rx.x’+ Wf, c==y*

for

O
ct > 0, xt > 0,

for

0
a,

18

MARILDA

A. DE OLIVEIRA SOTOMAYOR

where y” is the initial capital available, c’ and x’ are the corresponding amounts consumed and saved at the beginning of the tth period, y’ is the capital available in the tth period, 0 < 6 < 1 is a discount factor, r is an interest factor and w’ is a random variable. We will assume that u is strictly concave, increasing and differentiable, and the random variables {w’} are equally distributed, independent, with range in an interval [a, A], A < co and a > 0. If V,(y) is the maximum expected utility that we can get when y is the capital available and we have t periods to go, then the Dynamic Programming Formulation of the problem leads us to the following functional equations: V,(Y) = $:zxy

{u(c) + GEV,-,(rx

+ w)},

for all t > 0

c>o,x>o

(l-2)

V,(Y) = ,‘2x”=“, u(c)c>o,x>o

It can be shown that V,(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 (1.2) will be denoted by {c,(y), xt( y)}. The proofs of the following properties of (c,(y), x,(y)) and p,(y) can be found in (3). (a) p,(y) > 0, decreasing in y and increasing in t. (b) C,(Y), xt(v>, continuous and increasing in y. ct( y) is decreasing in t and x,(y) is increasing in t. (c) pt( y) > Jr@- ,(rxt( y) + w), with equality if xt( y) > 0. (l-3) (d) pr( y) > u’(c~( y)), with equality if ct( y) > 0. (l-4) If w E 1 (a constant), we will denote the corresponding prices and policies by P:(Y) and (C:(Y), X:(Y)>. The proof of the following results can be found in [3]. PROPOSITION

1.1.

(i> P,(Y) QJXY>, f or all t and consequently cy(y) >

CXY>.

(ii)

p,(y) > pf ( y), fir all t and consequently c,(y) < c;‘(y).

Infinite Time Horizon Problem

The consumer objective is to maximize asymptotically E 5 6’u(c’), t=0

19

ONINCOMEFLUCTUATIONSANDCAPITALGAINS

where

cf>o, Xf> 0, Cf+xf=yf, Yf+l=

TX’ + wf,

t : o,..., T.

1.1. A (stationary) policy is a pair of functions (c(y), x(y))

DEFINITION

such that

C(Y)> 0,

X(Y) > 0,

C(Y) + X(Y) = Y.

1.2. (c(y), x(y)) is an optimal policy if for any other policy 2(Y)>> st ar t ing from the same initial stock,

DEFINITION

(a>,

E + G’[p(c(y’)) - UF(~‘)] > 0,

for all T > some r,,

f=O

where y’ and j’ are the corresponding available capitals in period t using the policies (C(Y), X(Y)) and (F(Y), .f(v>>, respectively. That is, (c(y), x(y)) overtakes all the other policies. DEFINITION 1.3. A policy (c(y), x(y)) is said to be competitive exists a decreasing and positive function p(y) such that

(i) (ii)

c(y) maximizes u’(c) -p(y) c, c > 0; x(y) maximizes GEp(tx(y) + w)(Tx + w) -p(y),

if there

x, x > 0.

The following results can be found in 13, 4). PROPOSITION

1.2.

If (C(Y), x(y); P(Y)> is a competitive policy, then

P(Y) > U’(C(Y>>,

with equality ifc(y)

p(y) > GrEp(rx(y) + w), PROPOSITION

>0

with equality $x(y)

> 0.

1.3. u (c(y), x(y); p(y)) is competitive and lim E#p( y”) y’ = 0,

t-m

then it is optimal. PROPOSITION

competitive.

1.4.

Zf

(c(y),x(y))

is an optimal policy

then it is

20

MARILDA A. DE OLIVEIRA SOTOMAYOR DEFINITION

1.4.

The policy (c(y), x(y)), where

is called the limit policy, and consists of continuous increasing functions. If lim,+, P,(u) =P(y), in order to prove that the limit policy is optimal it suffices to show that lim,,, E#p(y’) y’ = 0, since the conditions of competitivity are easily verified.

2. SOME RESULTS ON A CAPITAL FLUCTUATION PROBLEM:

0<6<

l,r>

I,&>

1

In this section we will assume that _ubelongs to the class of functions for which the limit policy is optimal. This happens, for instance, when _u is bounded or _u’ has asymptotic exponent ~0. (See Theorems 3.3.2 and 3.3.3.) In the deterministic case, that is, where w = a > 0, we have the following. THEOREM 2.1. Under the conditions increasing and Em,,, y’ = co. Proof.

From the competitivity

P(Y) > h+(y)

of the previous

section

yf

is

of the optimal policy it follows that

+ a) > p(rx(y> + a).

P-1)

Since p is decreasing we find from (1.1) and (2.1) that yf is increasing. Suppose lim,,, yt = K < 00. Then, from (2.1) it follows that lim p( y’) > Jr I”;*

t-tm

a contradiction.

p(y’) > 0,

1

In the stochastic case, i.e., w is a non-degenerate random variable, we have the following result. THEOREM 2.2. A sufficient condition for the existence of a F such that rx(y) + a > y for all y >/T is that . Ep(rx(Y) + W) l ‘1-” p(rx(y) + a = .

ONINCOME

FLUCTUATIONS

AND CAPITALGAINS

21

ProoJ: Given an E > 0, there exists a F such that

<

P(Y)

Vy>y=.

Grp(rx( y) f a) ’

The rest follows analogously to [2, Theorem 3.X]. PROPOSITION

lim,,,

2.3. If

(c(y), x(y)

is

an

optimal

policy

then

P(Y) = 0.

ProoJ Suppose lim,,,p(y) = K > 0. From (1.6) and the monotonicity of p( y) it follows that K > 6r K, a contradiction. 2.4. = co.

PROPOSITION

lim,,,c(y)

If

(c(y),

x(y))

is

arz

optimal

policy

thefi

ProoJ: Suppose lim, ‘co c(y) = K < co. Now, consider M > K and A > such that Qf)

- U(C(Y>> > A > 0,

bJ*

@.2)

c(Y)I, for ally > M.

(2.3)

From the competitivity of c(y) we have Of)

- MY>> < P(Y)M

-

From (2.2), (2.3) and Proposition 2.3 we have

o
lim ,+xAY)

1 2.5. If

(c(y),

x(y))

is

an

optimal

policy

then

= 0.

Proof. Suppose lim,,,x(y)= k < co. From Grp(rK -t A) > 0, contradicting Proposition 2.3. a

(1.6) we get p(y)>

DEFINITION 2.1. The exponent of the function f a.t the point x is e,(x) = Igf(x)/lg x and the asymptotic exponent e, off is lim,,, eJx>. Iff(x) is a positive function, then the following fact can be shown. If x > x’ and a < e,. < L?,then for x large

22

MARILDA

A. DE OLIVEIRA SOTOMAYOR

THEOREM 2.6. If g’(y) has an asymptotic exponent then there exists a F such that rx( y) + a > y, for all y > j?

ProoJ:

Eu’W-M + w)>> u’(c(rx(v> +-4)) W(rW +a>>/ W(rW +a)> > c(rx(v> +4 ’

( c(rx(y)

+a)

“+_ 1 ’

for y large. Now, using the results of Propositions 2.4 and 2.5, the rest follows analogously to [2, Theorem 3.91. I The following question arises: What is the chance of an individual who follows the optimal policy to reach the critical capital level -? of Theorems 2:2 and 2.6? Theorem 2.7 says that the optimal sequence of capitals ( yf}t reaches F with probability one. Using Theorem 2.7 and Lemma 2.1 it is immediate that, with probability one, y’ tends to co when t tends to co. To prove Theorem 2.7 we need the following lemma: LEMMA

2.1. Let K > 0. Then P(3t; y’ > K) = 1.

ProoJ Call T the minimum time needed to reach K with initial stock _O in the deterministic case when w E A. We can consider P(w = A) > 0. Set to= min{t; w’=A,..., w’+~=A}. Now, using the fact that to+1 > rxA(yfa) +A =yy+l. By induction we find that X(Y) > XAY) we get Y ‘OST > y2+T > K and hence P(3t; y’ > K) = 1. Y THEOREM

2.7. Let $ be a level of capital such that for all y > F we have

Y < 4~) + a. Then P(3t;yf > Jq = 1. An example of the behavior of the optimal sequenceof capitals { yt}r is given by Fig. 2. We can now prove a result which gives us the asymptotic behavior of the optimal sequenceof capitals, which corrects an erroneus result of [7] for the case 6r> 1.

ONINCOME

FLUCTUATIONS

FIGURE

THEOREM

2.8. P(lim,,,

AND CAPITAL

GAINS

23

2

y’ = co) = 1.

ProoJ: Define Z, = -p(v”). Then, from (1.6) it follows that EZ,+, > Zr and the process {Z,} is a semi-Martingale with converging means and hence it converges almost surely to a random variable -M. Denote by < a generic element of the sample space on which all our random variables are defined. Let B = {<; lim,,, p(y’(C)) = M(r)}. We have that P(B) = 1. Let A = np=, A, = n;= I {<; 3 such that ~‘(5) > X}. From Lemma 2.1 it follows that P(A) = 1. Let t,(r) = min{t;y’([) > K), for all K., Let C = B PIA. Then P(C) = 1. If < E C, lim,,,

ytkc5)(r) = co and lim,+,p(JjLX’“‘(r))

= 0.

(lr)

Since lim,,,p(y’(<) exists we have that lim,+,p(y’(<) = 0. Suppose that lim t+av’(~) does not exist; then, the sequenceJI’(~) oscillates. That is, there exists a K(5) and a sub-sequence{ y’“‘“‘(l)}n of {JJ’(~)} such that #n(“‘(c) < K(t), for all n. Then, p(y’“‘“‘(<)) >p(K(<)), for all pz, which is impossible, since lim p( y’(t)) = 0. t-to3 Hence, lim,,,y’(r)

exists and by (*) for all 5 E C:

and consequently P(,“:y’= +

co)=

1.

I

Remarks. (1) If u’(v) is a convex function then the process {u’} is a semi-Martingale.

24

MARILDA

A. DE

OLIVEIRA

SOTOMAYOR

In fact, if u’(y) is convex we have that p(y) is convex, too (see [3]). Then, using Jensen’s inequality we get

P(Y’>> ~r&drx( y’) + w’) > p(Ewt(rx(y’) + w’)) =p(&Yf+l), where E,, is the expectation operator with respect to w’. Hence yf for all t, that is, E{ y’+’ / y’} > yf. (2) If u’(y) is a convex function, then lim, loo Ey’ = co. Using exists and cannot be finite, we see that lim,,, Ey’ P(lim,,, yf = do) = 1. We will now see an example where u’(y) hasn’t an asymptotic and depending on the distribution of w a level j may or may not that for all y > 7, rx(y) + a > y. Consider u(y) = -eey. Case 1. Take 6r = l/Eeeaw > 1,

< EwtyL+‘, Remark 1 because exponent, exist such

a = r - l/r.

It is easy to verify that (c(y) = ay, x(y) = y/r) is optimal. In this case y’ = co, just as in the deterministic case. rx( y) + a = y + a > y, Vy and lim,,,

Case 2. First we are going to find the limit policy for any w. Using (1.2) we get V,(y) = maxOGcGY{ePC - Be-r(y-c) . N,}, where N, = EepW. If y > (l/r) lg(6r. N,) then 0 < cl(y)
Cl(Y)= L-y-~-~. Thus V,(y)

=

-,-(r/l+r)Y

.

By induction we can show that the solution in (0, y) for a t-periods problem is given by 1 a)=

1 +r,‘;..

. k[W)

+rf f+(f-l)r+“‘+r*-’

for Y> YN,

y-

l+r+a.S+rf’ . N,

. ,;+t’...

N;+‘+“‘++‘],

ONINCOMEFLUCTUATIONSAND

CAPITALGAINS

25

where YN = lg(6r) tt(t-l)r+...+r’-‘/r~

. ,;/ri

. . ~jq t ’ . t rf -‘/rf,

where N

=

Ee-('k-'/l

t .

trk-1)w

K

We have that

and

,im I@, . ,;+r . . . N;+‘f”‘+‘I-’ t-02 1 +r+ *a’ +r’

=-LlgEe-“-l”‘“.

Then, r-l

- z&

lim c,(y) =ry t+cc

lg Ee-“-f”/““’

--=

lg 6r r--l

c(y)

and Yttl=$

+ __r-l

lg &.Ee-(‘-i/“W

+ wt

r

A++*

Ig 6r - Ew + w’.

Now, choosing a convenient distribution for w such that e-““/Ee-“W > Sr, where a = r - l/r, it follows that r-x(u) + a < y, Vy and there is no u under the conditions of Theorem 2.2. Now set w” = (r/r - 1) lg 6r - Ew + d. Ew’ = (r/r - 1) lg Jr > 0. Hence, from (I’) it follows that lim,,, y’ = co with probability one. fl

3. EXISTENCE

OF AN OPTIMAL

POLICY

Our objective in this section is to study the existence of an optimal policy. We will show that if u(y) is bounded or u’(u) has an asymptotic exponent different from zero, then the limit policy will be optimal. We will arrive at this result indirectly, by studying the linearity of the optimal policy and comparing u’(y) with the derivatives of some special functions. 3.1. Linearity of the Optimal Policy

(A) Deterministic case: w 3 a > 0. The next propositions

characterize the

26

MARILDA

class of utility w=a>o.

functions

A.DE OLIVEIRA

with optimal

3.1. Let (c(y), x(y)) 1)/r, then

PROPOSITION

0
u’(y) = C(Sry

1 (d [m+l

SOTOMAYOR

linear consumption

in the case

be an optimal policy. If c(y) = ay,

(y+sJ],

forally>O,

where m = r(1 - a), l/mn - aa/(m - 1) < y < l/m”-’ integer and

(3.1.1)

- aa/(m - 1) with n

(3.1.2) strictly decreasing and continuous. Proof. If c(y) = ay, we have for u’(Z) = &u’(Zm + au). Define g(Z) = u’(Z $ aa/( 1 - m)). g(s) = C(&)+l fj[m’-’ . s], n integer,

all y > 0, p(y) = u’(ay). Then

g(s/m) = or g(s).

Hence Thus

1 -&s<-n-1 3 m m where (3.1.2) holds. Thus (3.1.1) follows.

#

This result corrects the mistaken assertion of [8] according to which, in case w =_0, y”(L < 0) and Ig y are the unique ones. PROPOSITION 3.1.2. If u(y) is strictly concave, increasing, differentiable and satisfies the functional equation

u’(ay) = 6r u’(ayr(1 - a) + au), O
for all

y > 0,

r-l r

then

(c(Y) = ah x(y) = y(1 - a)> is optimal. Furthermore, if u’(y) is also a solution of u’(j3y) = 6r u’vym), for all y > 0, then c(y) = j3y and p = a.

ON INCOME FLUCTUATIONS

AND CAPITALGAINS

2-i

ProoJ Setting p(y) = u’(oy), we have p(y) = Grp(rx(y) + a). Hence, the policy with c(y) = ay is competitive. A solution of (3.1.3) satisfies (3.1.1). Given y’ = m(y” + a/(m - 1)) - a/(m - 1), then if l/am” - u/(m - 1) < yf < P/anT1 - a/(m - I), we have n < 1 - t - lg(ay’ + aa/(m - l))/lg m. In this case, from p(yf)yf < C(&)n yf, we deduce that lim,,, G’p(y”)y’ = 0. By Proposition 1.5 the policy is optimal. To show the uniqueness it suffices to use the fact that u’ is injective. PROPOSITION

3.1.3.

Let (c(y), x(y)

be an optimal policy. If c(y) = ay?

where a = (r - 1)/r, then u’(y) = C/(C%)~-’ 4 [y - (n - 1) au],

(3.1.4)

with II integer and (n - 1) aa < y < n (ua), n E Z’ with #: [Q, au] + qq0) = 1, C(aa) = l/ ar, strictly decreasing and continuous. ProoJ: It suffices to see that the solution u’(y) = Jr u’(y + aa) is given by (3.1.4). a

of the functional

equation

PROPOSITION 3.1.4. If u( y) is strictly concave, increasing and a solution of the functional equation

(3.1.5)

u’(y) = dr u’( y + au),

for all y, where a = (r - 1)/r, then the policy (y(r - 1)/r, y/r) is optimal and it is the unique linear one. ProoJ Setting p(y) = u’(cry) of (3.1.5) satisfies (3.1.4), Using that lim,,, 6’p( y’) y’ = 0, and The uniqueness follows from the (B) Stochastic case, i.e., w E [a, A 1,a > 0. PROPOSITION

O
3.1.5.

w is a non-degenerate

If (c(y), x(y))

random

variable

is an optimal policy and c(y) = ay,

then

u’(y)=6rEu’(ym+aw), Proof.

the given policy is competitive. A solution that yf = y” + ta and u’(y’) < C we deduce by Proposition 1.3, the policy is optimal. fact of _u’ is injective. 1

firaZZy>Oandm=r(l-a).

(3.1.6)

Obvious, since c(y) > 0 for y > 0.

PROPOSITION

3.1.6.

If u(y) is a solution of the integral equation (3.1.6),

28

MARILDA

A. DE

OLIVEIRA

where 0 < a < (r - 1)/r and u is strictly tiable, then the policy

SOTOMAYOR

concave, increasing

and differen-

(c(y)= ay,x(y)=Y(l - a> is optimal and it is the unique linear one. ProoJ:

u’( 1) = C [(m”-‘-

Setting p(y) = u’(ay), the given policy is competitive. Now, set get u’(ay) < C/(6r)n-1, if m”-’ + and by induction l)/(m - l)] a4 < ay < m” + [(m” - l)/(m - l)] CA.

Given

mf(y’+A)-- m~l
+ A/m - 1) -A/m

- l]

and

t
K = k[(‘/’

+ A/(m - ‘))/(y’ k m

+ ‘/cm - ‘>>I

3

from which it follows that lim,,, GfEp(yt) yf = 0. By Proposition 1.3 the given policy is optimal. The uniqueness follows from the fact of _u’ is injective. PROPOSITION

3.1.7.

If u(y) is a solution of the integral equation

U’(Y) = 6r Eu’( y + aw),for ally > 0,

(3.1.7)

where a = (r - 1)/r and u is strictly concave, increasing and differentiable, then the policy

(c(y)= as x(Y)= YU- w>> is optimal, and is the unique linear optimal one. Proof. Setting p(y) = u’(ay), the given policy is competitive. To show that lim t+co 6’Ep( y”) y’ = 0, set u’( 1) = C and proceed analogously to the proof of Proposition 3.1.6. 1 3.2. Some Special Utility Functions

In this section we will study some utility functions which seem of special interest for economists. The following three propositions can be proved by showing that u(y) satisfies (3.1.3) with 0 < a < (r - l)/ r and then using Proposition 3.1.2.

29

ONINCOMEFLUCTUATIONSANDCAPITALGAINS

w E a >, 0, u(y) = C(y + aa/(m - l))‘, for 3.2.1. rf C < 0, m = r( 1 - a), 1 ( 0, a = 1 - (&)‘l’-*/r then the policy (c(y) = ayy,x(y) = y( 1 - a)) is optimal, and is the unique linear one. PROPOSITION

I))“, PROPOSITION 3.2.2. In the case w sz a > 0, if u(y)=C(y+/Za/(mfor 0 < /z < 1, C < 0, m = r( 1 - a), a = 1 - (&)"' -‘/r and supposing 6r’ < 1, then (c(y) = cry, x(y) = y(1 -a)) is an optimal policy, and is the unique linear one. PROPOSRION 3.2.3. If w s a and u(y) = C lg( y + au/m - I), c > 0, a = 1 - 6, m = r(l - a), then (c(y) = ay, x(y) = y(l - a) is an optimal policy and it is the unique linear one. COROLLARY

3.2.4.

For w = 0 the functions

u(Y)=ckYT

c>o

U(Y) = wt

A
U(Y) = cya,

O
(1) (2) l,C
1

(3)

admit a linear optimal policy, (c(y) = cry,x(y) = y( 1 - a)) for a= l-6

in (1)

and a = 1 - (“)“‘-’ Y PROPOSITION

6r = l/E(e -““$

in (2) and (3).

3.2.5. Let u(y) = 1 -e-” with w E [a, A]. Then, if so c(y) = ay, with a = r - l/r, is an optimal consumption.

Proof u(y) is a solution of (3.1.7) for 6r = l/E(e-“W). follows from Proposition 3.1.7. I

The resuit

3.3. Existence of an Optimal Policy 3.3.1. Let u and u* be two utility functions with u’
Proof By induction on t : for t = 0, pO(y) < p$( y). Supposing it true for t - 1 we are going to prove it for t. If by contradiction, p,(y) > p:(y), for By definition sf competitivity some y, then p,(y) > u’(c:( y)). u’(c) -p!(y) < 0 for all c > cl(y) > 0 and if cf(y) > 0 we 21~0 have u’(c) -p,(y) > 0, for all c < c,(y). Hence c:(y) > c,(y). Now, p,(y) < Jr EpF-, (TX: ( y) + w) < p:(y), a contradiction.

30

MARILDA LEMMA

then lim,,,

A.DE OLIVEIRA SOTOMAYOR

3.3.1. If u’(y) has an asymptotic exponent different from zero, u’(y) = 0.

Proof lim,,, u’(y)=K, contradiction. 1

for

some K.

If

KfO,

then e,,=O,

a

LEMMA 3.3.2. If u’(y) has an asymptotic exponent dtfirent from zero, then for all number A, 0 < 1 ( -e,,, there exists an MA > 0 such that for all Y > Ma, U’(Y) < l/Y+

Proof Set 0 < .A < -e,, . Then, given an E= -e,, - A, there exists an M > 0 such that lg u’(y)/lgy < -A, for all y > M, from which u’(y) < l/y”, for all y > M. I LEMMA 3.3.3. Let u(y) = C lgy, C > 0. Let ti E [a,A] variable. Then the limit policy is optimal.

be a random

Proof We know that p,(y) 1. Then the limit policy is optimal in the stochastic case, w E [a, A], a > 0.

Proof. In the case w E 0 we can calculate, using (1.2), that P”(Y) = C/(av)” w here C= C(l -A) and CI= 1 - (6r)1’A/r. Now, set p(y) = y = 0. By lim,,, P,(Y) < c/7(wY and using that A > 1 get lim,,,p(y) Theorem 2.6 {y’} is increasing starting from a certain t and hence lim,, 6’ Ep( y’) yf = 0, and the limit policy is optimal, 1 LEMMA 3.3.5. Set u(y) = Cy’-‘, C > 0, 0 < /z < 1, and 6r’-’ the limit policy is optimal for w E [a, A], a > 0.

< 1. Then

Proof. We first seethat the limit policy, in the case w = 0, is competitive for p”(y) = c/(ay)“, a = 1 - (&)“‘/r, c= C(l -1). Using that Y’ < m’(y’ + A/@ - I)), m = r(l - a> ad P(Y) = M,,p,(y) < C/My)* we get that Km,,, 6’ Ep(y’) y’ = 0, and the limit policy is optimal. 1 THEOREM 3.3.2. If u’( y) has an asymptotic exponent different from zero then the limit policy is optimal.

Proof. (1) Suppose that e,, < -1. Using Lemma 3.3.2 for A > 1 and Theorem 3.3.1 for u*‘(y) = l/ya we get that p,(y) 1 and the fact that p*(y) = l/(1 - S) y, when )3.= 1, it follows that lim,,, 6’Ep(y’) y’ = 0, and the limit policy is optimal.

ON INCOME FLUCTUATIONS

AND CAPITAL GAINS

31

(2) Let us suppose0 > e,, > -1. Using Lemma 3.3.2 for 0 < nl < 1, such that 6r’-’ < l? and Theorem 3.3.1 for u*‘(y) = l/y’ we get p,(y)
3.3.3.

If u(y)

is bounded above then the limit

policy

is

optimal. Proof. From the concavity and differentiability of u we have that given any y’, for all y > y’ U’(Y) Y < -U(Y’> + 4~) + U’(Y)Y’. Since lim,,, u’(y) = 0 we can write, for a given e > 0 and starting from a certain y^ that u’(y)y < cy’ + (K - u’(y)) = C > 0, where K is such that u(y) 9. Using Theorem 3.3.1 and Lemma 3.3.3 we can set p(y) = lim,,,p,(y) 9. Hence lim,,, fYEp(y’)y’ < lim, +oo6’ max(C/l - &p(a)y^] = 0, which concludes the proof. I COROLLARY 3.3.4. If _uhas absolute risk aversion, R, > K, K > 0, then the limit policy is optimal.

Pro& From R, = -u”(y)/u’(y) > K we can write u”(y)egy f Ku’(y) eKy < 0, that is, u’(y) eKy is strictly decreasing. Using that lim,,, u’(y) eRy= M for some M> 0 and the fact that _uis increasing, we get that u(y) is bounded above and then, by Theorem 3.3.3 the limit policy is optimal. I THEOREM

3.3.5.

If u’(y) y is decreasing then the limit policy is optimal.

ProoJ Given a y*, for all y 2 y* we have that u’(y) < Cly, where c = u’(y*) y*. Repeating the argument used in the proof of Theorem 3.3.3 we get the desired result. m An example of an utility function with a zero asymptotic exponent and no optimal policy. Let K > 0. u(y)= 1 -ee-‘+K.p, Suppose that there exists an optimal policy. Then p(y) > e pcfy) + K, for all y. Hence hm,,, p(y) > K > 0, a contradiction by Proposition 2.3. The following result holds generally, since p(y) > u’(c(y)). PROPQSTION

642132/l-3

3.3.6.

Iflim,,,

uf (y) > 0, them there is no optirna~p~l~~~l.

32

MARILDA

A. DE OLIVEIRA SOTOMAYOR

4. SOME RESULTS ON A CAPITAL FLUCTUATION 0<6< l,r> 1,6r=l

PROBLEM:

(A) Deterministic Case: w E a. LEMMA

4.1.

xy( y) = 0, for all y < a and for all t.

Proof. Suppose that there exists a f such that x:(a) > 0. Then p;(a) < py(rxy(a) + a) and thus a > rxy(a) + a > a, a contradiction. Hence x:(y) = 0 for all y < a. I LEMMA

4.2.

cT(y) > 0 for all y > 0 and for all t.

LEMMA

4.3.

xf( y) > 0 for all y > a and for all t.

LEMMA

4.4. x”(y)

> 0 for all y > a.

The proofs of the three last lemmas are, as that for Lemma 4.1, immediate. THEOREM 4.1. For the limit policy we have yLtl < yi, ‘dt, if y” > a and yi = a, tit > 0, ify” < a.

Proof. THEOREM

It is immediate from Lemma 4.3. 4.2.

1

The limit policy is optimal.

ProoJ It suffices to show that lim,,, #p”(yi) y: = 0, since the limit policy is competitive with p”(y) = lim f+o3pF( y). Using Theorem 4.1 this fact obviously follows. m Now, we will see that there exists only one optimal policy for all utility function. Let x,(y)==,

r

= 0,

if

y>a

if

O
and c

(y)=

(r-

a

l)y+a

r =Y,

’

if

y>a

if

O
ONINCOMEFLUCTUATIONS

ANDCAPITALGAINS

33

SettingP,(Y) = u’(c,(Y>> we have that the given policy is optimal, no matter the utility function. Using Theorem 4.2 and the unicity of the optimal policy when it is the limit policy we get that the given policy is the limit policy and the unique optimal one. (B) Stochastic

Case, i.e., w E [a,A]

is a Non-degenerate Random

Variable THEOREM

4.3.

The limit policy is optimal.

ProojI It suffices to show (1.7) with p(y) = lim,,,p,(y) and so to use Proposition 1.3. r we deduce, respectively, that Ep(y’) > From (1.6) and x(y) 0, ‘dt, and y’< y” + tA, Vt and Vy”, and then that (1.7) holds. THEOREM

4.4. p(y)

is a strictly

decreasing function,

where p(y) =

lim,+,p,(~). Proof. (1) P(Y) = U’(4Y))

Suppose p-‘(M) = [q/3], 0 < a /?-a. Similarly S + A > j?. Since J?(Z)= Ep(z + w), J?is constant in [s, v] and p is non-increasing, p(y) has to be constant in [s + A, v + A]. As before, we now conclude that A - a > s - v = r(j3 - a) and so integrating obtain A - a > r”(j3 - a) for all _n,a contradiction.

(2) If p-I(M)= [ a, co), then, as before, x(y) = y -X for all y > a, contradicting the fact that x(y) Q x0(y) = y/r, r > I, for y > a. THEOREM

4.5.

P(lim,,,yf

= co) = 1.

ProojI As in the proof of Theorem 2.8, we have that Z, = -p( y’) is a semi-Martingale with an almost sure limit. By Theorem 4.4 p- ’ exists and is continuous. Hence y’ =p-I(-Z,) converges almost surely.-Since the set of sample paths for which there are infinitely many _t such that wf = A and wtt ’ = a has probability one we can assumethat the sample paths on which y’ converges also have this property. This fact along with the continuity of x(y) and the relation yff ’ = rx(y’) + W( contradicts a finite limit for yf In case 6r < 1, under certain condictions there exists Y; such that, for all y > Y; rx(y) + A < y, implying yf > yff ’ if yf 27. In case 6r > 1, under certain condictions, too, there exists a level of

34

MARILDAA.DE

OLIVEIRA SOTOMAYOR

capital, F, such that for all y > 7, rx(y) + a > y, and hence y’ < y’+ ’ if y’>y. In our case, 6r = 1, we have that. y
for ally >A.

y>rx(y)+a>Y-(A-a), Although yf --f co when t--f co almost surely, there is no capital level starting from which the optimal sequence of capitals is increasing, since the only way for this to occur is to have y = rx( y) + a, starting from a certain y, which cannot happen, as the following proposition shows. PROPOSITION

4.6.

There is no L such that for all y > F, rx( y) + a = y.

Proof. If there will exist a such F we would have p(rx( y) + a) =p( y) = Ep(rx( y) + w) implying that p(y) = p( $) > 0, for all y > J, a contradiction

by Theorem 4.4. Remark. If u’(y) is convex then { y”} is a semi-Martingale since p(y) is strictly convex and p( y’) > p(Ey’+ ‘). Theorem 4.5 corrects the erroneous result of [7] for the case 6r > 1, where an attempt was done to prove, using u(y) = -e-y, that by an appropriate choice of the random variable w and of its distribution the optimal sequence of capitals { yf}t is bounded almost surely. We found in Section 3 the limit policy for u(y) = -ePy and any w.

x(Y)=:+

--&

lg Ee-(‘-W’)W

lg Ee-

+ (r-

1 r-l

1/r)w

. lg 6r.

----&lg

6r.

ACKNOWLEDGMENTS I would like to express the subject developed here Svetlichny, for his advising de Araujo, for his interest This work is part of my was financially supported Tecnologico) and CAPES of the Brazilian government.

my gratitude to Professor Jack Schechtman, who introduced me to and for his valuable assistence and advising; to Professor George and patient and instructive comments; to Professor Aloi’sio Pessoa and suggestions. thesis presented to the Department of Mathematics of PUC/RJ and by CNPq (Conselho National de Desenvolvimento Cientifico e (Coordenapao de Aperfeicoamento de Pessoal de Nivel Superior)

ON INCOME FLUCTUATIONS

AND CAPITAL GAINS

35

REFERENCES 1. J. L. DOOB, “Stochastic Processes,” pp. 324-325, Wiley, New York 1953. 2. E. KOHLBERG, A model of economic growth with altruism between generations, J. Econ. Theory 13 (1976), I-13. 3. D. LEVHARI, L. J. MIRMAN, AND I. ZILCHA, Capital accumulation under uncertainty, hternal. Econ. Rev. 21 (1980) 661-671. 4. D. LEVHARI AND T. N. SRINIVASAN, Optimal savings under uncertainty, Rev. ECGPZ. Studies 36 (1969), 998-1019. 5. J. SCHECHTMAN, Some applications of competitive prices to dynamic programing problems under uncertainty, ORC 73-5, Operations Research Center, University of California. Berkeley, March 1973. 6. J. SCHECHTMAN, An income fluctuation problem, J. &on. Theory 12 (1976), 218-241. 7. J. SCHECHTMAN, A grain storage problem, with random production, ORC 74-3, Operations Research Center, University of California, Berkeley, January 1977. 8. J. SCHECHTMAN AND V. L. X. ESCUDERO,Some results on an income fluctuation probiem, J. Econ. Theory 16 (1977), 151-166.