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"k
Slrrple Model of Qprimum
Life-Cvcle
CoTnsuTr.-ption
With Earnings Uncerrainrv
Eyran ShestLinski
Kunber A09
Januar\' 1986
January
1
9B6
A Simple Model of OptiTnum Life-Cycle Consumption
¥ith Earnings Uncertainty
ty
.
Eytan Sheshinski*
Massachusetts Institute of Technology
and
The Hebrew
Tfrii
versity, Jerusalem
Support from USE Grant Ho. SES-8105184 is gratefully acknowledged.
am indebted to Peter Diamond for a helpful discussion.
I
Abstract
Ve analyze the optimuic conBucption path of an individual who maximizes
expected discounted utility over an infinite horizon, when savings earn
fixed return and wage earnings follow a two-state random process.
It
assumed that no insurance is available against earnings uncertainty.
a
is
Ve
show that optimum consumption always changes disco ntinuously with the level
of earnings.
The optimum consui:ption path is characterized for the case
where the subjective discount rate is equal to the interest rate.
Vhile
under certainty' consumption is constant in this case, it is shown that here
consumption strictly increases or decreases over time.
Specifically, when
the 'Hazard-Rate' non-decreases with time, consumption approaches infinity
or zero,
respectively.
The discontinuity in consumption,
smaller than the difference in earnings across states.
however, is always
Comparison to the
full insurance case when earnings can be annuitized is also provided.
These qualitative results may explain some recent findings (Hall
(19S2)) concerning the volatility of consumption and point to the
appropriate life-cycle modelling in the absence of earnings insurance
(Diamond-Kirrlees (19S2)).
.
.
1
.
Introduction
We consider
a
consumer vho earns
a
wage income that may change
stochastically over time vhile savings earn
chooses
a
fixed return.
The consumer
consumption path that maximizes expected discounted utility over
a
an infinite horizon.
Oar objective is to characterize the optimum
consumption path, focusing on the dependence of consumption on future
earnings and on the corresponding relation between changes in consumption
and changes in actual earnings.
Clearly,
the relation between consumption
and current earnings depends on the nature of the underlying stochastic
nrocess
Vhen earnings over-time are identicallj' distributed random variables
(as assumed,
for example, in laari (1976) and Schechtman (1576)) or, more
generally, if earnings follow a stationary stochastic process (as in Bewley
(1977)»
(1980)),
changes in earnings are regarded as transitory and hence do
not convey information vith regard to fuxure earnings.
asymptotic convergence results
-
of the expected optimum consumption path in
the absence of discounting and, more generally,
of expected marginal
which were taken as the appropriate analogue of the 'permanenx-
utilit;,—
income'
This explains the
(Friedman (l957)) concept under stochastic conditions.
we adopt a different approach.
contain a large permanent component.
disability
(
Diamond -Kirrlees (19S2))—
Changes in earnings are assumed xo
Obvious applications are to cases of
or occupational positions,
events have significant long-term effects.
In fact,
we make the extreme
assumption that once earnings change this change is permanent.
quently,
earning.
where
Conse-
consumption changes discontinuouslj- (and non-negligibljO
^-'i'th
,
-2-
In general,
consumption at any point in time aepends on the level of
assets and on expectations concerning future earnings.
Thus,
even when
actual earnings do not change and the rate of interest is equal to the
subjective rate of time preference (which, under certainty, yields a
constant optimum consumption path), optimum consumption changes over time.
Specifically, when earnings may fall at some future date, savings are
positive and consumption rises over time until the occurance of the fall in
earnings.
Afterwards savings seize the consumption remains constant.
opposite pattern is found when earnings are expected to rise.
The
Optimum
consumptior in the initial state increases indefinitely in the former case
and approaches zero in the latter case.
interpretation of the
At any time,
'
permenent-income
optimum consumption
There is therefore no obvious
'
m
concept in this context.
the initial state is bounded
between total income - i.e., the return on assets plus wage earnings - when
earnings are high and total income when earnings are low.
Thus, the
magnitude of the discontinuit3' in consumption as the level of earning
changes is bounded by the difference in earnings.
This,
(and other aspects
of the model) could provide a testable hypothesis to the volatility of
consumption question
2.
The
analj'-zed
03'
Hall (1978) and Hall and Mishkin (19S2).
I'jodel
An individual may earn a wage of w
respectively.
individual is
Time,
m
t,
state
or w
,
labeled state
is taken to be continuous.
0,
earning v
.
For
a
At time t = 0,
1
the
small dt, there is a
probability of f(t)dt that in the interval (t,t+dt)
occur.
and state
a
switch to state
1
Once such a switch occurs the individual stays forever in state
will
1.
It is assumed that there is no pcssibilitj' for insurance against such event.
Tnere is, however, a perfect capital market vith a fixed return,
savings.
r,
on
The individual's objective is to caxir-ize his expected discounted
utility over an infinite hcrizon.
Once state
1
occurs there is no further uncertainty.
Hence, as in
standard djiiamic programming, we solve the individual's problem backwards.
Suppose that state
1
The individual's problem is then to
occurs at time t.
choose the consumption path in state
1,
c
(E,t)
>_
0,
for all
s
>
t,
that
maximizes
/
u(c^(s,t))e-'^^^-^^^ds
(1)
subject to the intertemporal budget constraint on assets, a(s).
a(s) = ra(s) + v^
-
c^(s,t)
s
,
^
t,
(2)
for a given initial a(t), where the utility function, u(c), is assumed to be
strictly increasing and concave, bounded above, twice conti.nuously
diff erentiable for all
c
>
0,
with u'(0) =
subjective time-preference rate,
Tne boundedness of u ensures that
6,
==.
The interest rate,
r,
and the
are assumed to be positive constants.
(l)
is finite for any- feasible consumption
path satisfying (2).
The Euler first-order conditions for a narimum of (l) subject to (2)
are
(r-6) u'(c^(s,t))
^1^^'"^=-
u'Hc.ls,t))
'
^^^
^5)
.
.
-4-
and the transversality condition lim a(s;e
=
where
C,
c
QS
1
integrating (2) and taking
f
c
J
.
,
(s,t)e
s
-r(s-t)
I
tne latter is equivalent to
",
-*
/^s
ds = a(t) + v
-r(s-t).
r
as
e
J
,
V
(4)
,
t
t
i.e., the present discounted value of cons'usption equals the present
discounted value of future wage earnings plus initial assets.
(5)
and
determine the optinum
(4)
c
denoted
,
1
c"
(B,t) = ^^(t) h(s-t),
—
as r - 6
where h(0)
=
c
^
,
which can be written
(
1
by (3),
and,
for all v
Optinum consumption at the onset of state
0.
Conditions
1,
c
>_
0,
(t),
h'(v)
is
determined by (4),
^M)
=|{ra(t)-w
1
Uj
(5)
)
1
CD
vhere
4;
=
rj
h(v-t)e
"
"
dv
is
Denote by U(a(t)) the maxinized value of (')i
.A
U(a(t)) =
/
It is easy to verify that
U'(a(t))
=
,
^
N
-6(s-t )
u(^^(s,t)e-"^^-^^ds
(3)
u'(^Jt))
.
(6)
implies that
.
(7)
-
Erpected discounted utility, V, is now given by
V =
where
c
(t)
>
/
is consumption in state
at time
(s)
and F(t) is the
t
Tne intertemporal budget constraint in
cumulative distribution of f(t).
state
U(a(t))f(t)je"^*dt
*
(t))Tl-F(t))
[u(c
is
a(t) = ra(t)
+
v^ - c^(t)
t
,
The individual's objective is to choose
to (9)>
a
i"or
given a(0)
3/
0.—
>
c
2
(t)
(9)
.
so as to maximize
(8)
subject
The Euler first-order condition, using (7),
IS
>.'(*,(t)) - u'(c„(t))
(r-6) u'( = „(t))
where
Q:(t)
=
f ( t)/(
1
probability of state
-F( t)
1
)
,
termed the
'
Hazard Rate
occuring at time t.
xransversality condition Ixm a(t)e
^
=
0.
'
,
is the conditional
in addition,
we have the
integrating (9), letting
t
-
the latter is ecuivalent to
/
CQ(t)e"'^dt
=
a(0) + v^
/
e"''dt
,
(ll)
=°,
-6-
vhich equates the present discounted value of consuicption to tnat of wage
earnings plus initial assets at time
2l
the optinum consunption path
Conditions (5),.00) and (ll) fully determine c\(t) for all
by c\(t).
t
I>enote
0.
0.
Uncertainty is spurious in two obvious cases, in both of which
consuEption is equal to its level under certainty:
(1)
^'
^'i~.~
the same
.
Observe that in this case conditions (4) and (11) are
•
and hence setting
(at t=0),
=
(t)
c
for all
(t)
c
also satisfies
t
conditions (3) and (10).
F(t)
(2)
=0,
t
_<
<
in which the onset of state
Rate',
t
1
and F(t)
,
=
1,
t
<_
This is the case
=>.
<
is certain to occur at time t
is then identically zero for all
Q:(t),
t
t
Setting
and (3) have the same functional form.
The
.
'Hazard-
>_
C
Hence, equations (10)
c
(t)
=
c
(t)
=
c(t), is
seen to solve (3) and (IO) subject to the budget constraint
'0
-^4-
/ c(t)e"'''dt
=
V
/
e~' "dt + v
/ e"^ 'dt
.
(12)
t^
Me shall henceforth assume that v„
^ v\
1
zero for all
"We
t
>
—
and that aft) is not identically
0.
can now address the question of discontinuous changes in the optimum
cons;imption path when the state changes.
we shall prove that,
consumDtion is never continuous:
Theorem
-^
1
:
c
(t)
=
c
(t)
for all
t
_>
is impossible
generically,
By ccritradition
Proof:
c
=
(t)
Suppose that
.
c
(t)
=
where, by (5), c,(0) =-{ra(o)+w
c^,(0)h(t),
V
I
1
substituting c^(0)h(t) for-^^(t) in
is impossible since w
*
(11
)
C
>
7ner.
However,
).
1
yields c^^CG}
-(ra(0)
=
v^),
*
whici
the difference between the uncertainty and
certainty case, we shall focus on tne case where
(10),
t
'•
^%
To further characterize
known,
for all
(t)
c
r -
6
=
C.
the
It is well-
that under certainty optinuia consumption is constant in this
case.
3.
Optinun Consumption Vhen r
h(t) =
In this case,
c
=
(t)
ra(t) + w
ft)
c
=
,
1
for all
-
5
=
for all
t
_>_
C.
and hence
t
=
1
.
By (5)r
differentiating, using (9),
r(v -w.-^ (t)*c\(t))
1
4;
(13)
.
I
Condition (IO) now becomes
u"(^.(t))
•
,
c„(t) = °
-
u'(c^^(t))
2
^
(
(u)
),(,)
u"(^(x))
Equations (13)-(14) are two differential equations that, together with the
transversality condition,
(II),
determine the o-c-imum Dath (c (t),
we have not yet made any specific assumption concerning the
it£te',
ci(t),
as a function of time.
for the onset of state
Formally,
1
'We
c,
(t)).
'Haiiard
shall assume that the probability
does not decrease the longer state
prevails.
-6-
a(t)
non-decreases for all
>
If for some t^
Q:(t)
0,
>
for all
=•
vith certainty before or at
t
t
t
t
>
0.
>
(15)
this means that state
,
occurs
1
The optinum path in this case can easily be
.
Condition (15) is satisfied by the
inferred from the following analysis-
most common distribution functions, such as the exponential (Poisson).
Ve can now provide a characterization of the optimum consumption path.
Theorem
2
Under (15),
:
if v^ _ v^
(a)
then v^ -
>
strictly increase for all
v^
with lim
t,
c^iz) - ^^(t)
>
^^^
=
C(-)(*)
(t)
c
and
>
=
c^(t))
(c'Q(t),
=>;
•^-*<o
if v^ - v^
(b)
then v^
<
strictly decrease for all
- v^
CQ(t) - c'^(t)
<
A
,
,
A
s
with lim (c^(t) -
t,
(t))
c
and ic^it)
<
=
w
^^(t))
,
- w
•i--l.CS
Proof
Consider the case w
:
for all
t 2. 0*
which
(t
c
(t
<_
)
Then, by
0.
for all
t
>
t^.
.asc, a(t)
for all
t
>
t
In fact, E(t)
.
- c^0^(t) +
w„
0 - w\
-1
imtlies that
0.
>
'We
first show that -p,(~)
Suppose, xo the contrary, that there exists
- c
)
- v
c"
c
-
(t)
(t)
<
c^
(t)
=
for all t
>
and
c^
<_
(t)
t
<
>
0,
2
t
,
c^
Since
0.
fact, lim
c
(t)
=
c
0,
(t)
>
fcr
>_
and c'^(t)
<
>
c, (t)
>
which, by (13) and (14),
(t)
<
(t
)
thereafter.
But
1
then there could not exist a time
>
(t)
fcr if not.
0-
1
^(t-)
(t)
c
t
a
c
ra(t) + w^- c^(t) = v^ - w^- c^Ct) +
fcr some
>
(U),
(13) and
-
t
fcr which
c
is strictly decreasing with t,
because,
Dy (14)
and
(15),
-
c
(t
)
<_
0.
Thus,
it has a lir:it.
at any positive
^
(t).
In
.
-y-
c^
(t)
IE bounded awav from 0.
<
-
V
(t)
c
A
given that lie
IJov,
for all
>_
t
,
,
=
(t;
c
>_
\.
there exists a
C,
a(t)e-^^
-rt
lim a(t)e
>
0,
we now prove that v
- v
>
and a(t
>
)
C,
the contrary,
By (13) and
(13),
(w^-^^(s))e~^^ds
/
c
(t)
A
,
c.(t) decreases with t.
A
.
-
(tj
c
^
t
t^
(16;
,
.
.
for all
(t)
c
t_^ 2.
and c,(t)
>
a(t^)e
*
violating the transversality condition.
that there exists some
(u),
such that
t^,
>
Since
.
t
=
t
there exists
Hence,
>_
Suppose,
0.
^ for which c_(t) -
for all
<
t
t
>
t
(t)
2. ^V~''*'i
*
Furthermore, from
.
T(>
a
c
to
t
)
such that
1
(T)
=
u'(0)
=
c
This path, however, cannot be optimal because, by assumption,
0.
Consider next the case v
c
(t)
-c
(t)
for all t.
<
v
-
we first show that
0.
<
to the contrary,
SurtJOse,
that there exists a
I
t
>
c
(t)
such that
c
and
(t)
>
c
(t
)
c,(t
-
)
for all t
<
It
0.
>_
>
t
follows from (13) and (14) that
Furthermore, from (13)t
.
(t)
c
is
1
1
Hence, there exists a T(> t.) such that c,(T) = 0, which
decreasing with t.
I
cannot be optima.1
ijow
we show that w
there exists a t.
U
—>
(14),
c_(t)
w
<
c
(t) -
c
Ul —
such that w. -
•
and
-
w^
>
(tj
for all
OU
t
>_
c^(t^) - c,(t^).
lO
C
Suppose that
Then, by (15)
•
<
and
c
(t)
>
for all
t
>_
t
.
Furthermore,
.
-10-
lirc
t
(>_
and lie
(t)
=
t
such that a(t
c
)
°°
=
(t)
c
>
)
G.
Since
c
(t)
and
c
(t)
<
w
ra(t)
=
for all
before, this violates the transversality condition
to prove that lin
Finallj',
lim (c (t) -
(t))
c
w
>
- V
,
c
(t)
then
c
-
c
(t)
+
t
w
t
>_
tnere exists a
,
As was shown
.
lim a(t)e
v
-rt
=
0.
observe that if
(t)
=
<
is bounded away from 0,
-
v
,
which
t-«°
inplies that there exists
this cannot be optimal
Theoren
consumption
finite T for which
a
m
state
=
Since u'(0)
C.
=
°°,
li
proves that consunption in state
2
(T)
c
A
1,
c
\
.
(tj,
A
0,
c
,
,
(t),
and initial
ere co-monotone, that there is always a jump
in consumption in the direction of the change in earnings upon the onset of
state
and that the absolute size of tne jump is alwaj^s less than the
1
m
difference
earnings in the two states.
Clearly, these are all
empirically testable hypotheses.
The phase diagram for (c
v^
- v^
and in
>
lb
Fig^iire
(t),
c
(t))
is drawn in Figure
fcr the case w^
- v,
(O)
c
^amnle:
=
ra(0) + w
log
Let u(c)
=
tiz)
^(t)
c^Ct)
c^(t)
c
Then equation (14) becomes
l)cr(t)
rhere is no explicit solution for (13) and
w^
1
that in both
L'ote
.
7^—= (7^-
w^-
0.
case
1
1
cases
<
1e for the
>
0,
for large
t
(17)
(17)'
However, for the case
we use Theorem 2 which imulies
that
-
A
c
,
V
,A
(t)/c
(t)
1
~
1
-1
Lenotmg x(t)
c^(t) -
=
(t),
c
and
(13)
(17)
1.
yield
I
x(t)
(r+a(t))x°(t)
=
Further assuiLing that
x(t)
=
^
=
i.e., z(t)
0,
solution is (Figure
A
/
\
c-(t)
=
=
ct(t)
(18)
constant, the only feasible solution is
a
=
X
r(v^-w^)
-
(v-
r+a
-V'
)
Hence the (approrinate)
steadv-state
j-j-
.
'
1
.
2)
\
ra(t} +
r
/
V
r+a
c
+
r+a
V
1
19.
c, (t)
In state 0,
=
ra(t) + V
the relevant variable for consumption is a weighted average of
the wages in state
Rate',
3-
1.
As expected,
the higher the
the higher is the weight of (the lower wage), w
a.,
Fizll
and in state
'Hazard-
•
Insurance
Suppose that earnings can be insured on an actuarially fair basis.
the switch from state
to
1
occurs at time
of earnings is given by
-
Hence, the
[(1-e)
^'')^-Q ^
e
"
\^
]
'human-capital' annuity value is
t,
If
the present discounted value
)
-12-
I
=
where a(r)
[(i_e-^^)wQ
/
/
e
*
When f(t)
=
,*
cie
=
ra(0)
.
raO)
point E in Figure 2.
c*,
while
c
e-'"'w^]f(t)dt
f(t)dt is the
consumption (for the case
=
^
+
6
=
r)
^ [(l-a(r))wQ
^
.
a(r)vj
'Laplace transforiL' of f(t).
is constant,
(l-a(r))v^+ a(r)v^
Optimum
and equal to:
c*,
(21
.
Equation (21) is the same as
(20)
in (19).
c
.-I-w^.-^w^,
Compared xo (19)> therefore,
(22)
c
is always larger than
is initially lower and then higher for large t.
•13-
Footnotes
—
The most general results on asjiiiptotic properties of opticum
stochastic consunption are in Chamberlin and Wilson (1984).
—2/ In
fact,
our two-state earnings nodel is based on the continuous time
version of the Eiamond-Mirrlees (19S2) paper on social insurance.
3/
—
Note that no non-negativity restriction is imposed on a(t,
-u-
References
Bewley, T.
"The Permanent
(1977)
Income Hypothesis
Formulation," Journal of Economic Theory
"The Permanent
(1980)
Bevley, T.
V,'ilson
16,
252-292.
Income Hypothesis and Long Run Economic
Stability," Journal of Economic Theory
Chamberlin, G. and C.
,
A Theoretical
:
(19S4)
,
22
,
377-394--
"Optimal Intertemporal Consumption Under
Uncertainty," C.V. Starr Center for Applied Economics, New York
University, Research Report 84-15Diamond, P. and
Kirrlees (1932) "Social Insurance with Variable
J-
Retirement and Private Savings," Department of Economics, K-I-T'
Friedman, K-
(1957)
"
A
Theory of the Consumption Function ,"
of EconoEH-c Research,
Hall, E-
(National Bureau
New York).
"Stochastic Implications of the Life-Cycle Permanent income
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