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UNCERTAINTY AND OPTIMAL SOCIAL SECURITY
SYSTEMS
Eytan Sheshinski and Yoram Weiss
Number 225
October 1978
^•'evi/ev'
{
JUL
25
1979
UNCERTAINTY AND OPTIMAL SOCIAL SECURITY SYSTEMS
by
Eytan Sheshinski and
Yoram Weiss
Introduction
1.
The
work
effects of social security have
of an overlapping generations
can be summarized quite simply.
been discussed within the frame-
model by Barro (1974).
If
His analysis
social security is fully funded
i.e.
,
contributions to the system are invested at the market rate of interest,
is a
perfect substitute for private savings.
it
Consequently, a forced
increase in social security will reduce private savings by an equal amoiint.
Consumption, bequests and aggregate savings will be unaffected.
security is financed on a pay -as- you -go basis, i.e.
If
social
taxes on the cur-
rently working population are used to finance benefits to the retired population,
it
is a perfect substitute for private
Hence, a forced
bequests.
increase in social security will reduce bequests by an equal amount.
Consumption, private savings and aggregate savings will be unaffected.
In
such models the optimal level of social security
In this
istic context.
paper we analyze social security
The duration
of life is
in a
is
clearly undetermined.
somewhat more real-
assumed uncertain and
the annuity
3je
The Hebrew University of Jerusalem and Tel-Aviv University. This
research was partially supported by National Science Foundation Grant
SOC-00587.
1689150
-2-
aspect of social security benefits is incorporated explicitly.
We
use a
simplified two-period version of Yaari's model (1965) to determine the
demand
for annuities.
of annuities
Optimal social security
demanded by a representative
hypothesis is that
such insurance
all
is
is defined
individual.
as the amount
Our working
provided publicly.
Although in
principle the private market could also satisfy this demand, in practice
the public supply appears to be dominant.
2
The demand for social security depends upon the mode
Under a
fully funded
so as to maximize
system, demand
its
future generations.
is
of financing.
determined by each generation
lifetime utility, taking into account the welfare of
A pay-as-you-go system
the yoimger generation to the old.
Demand
is
involves transfers from
therefore defined from the
point of view of the old, taking into account the social security taxes
paid by the
young.
Under an actuarially
fair, fully funded social security
perfect insurance against life uncertainty is feasible.
individual will prefer this option which enables
utility of
bequests across states of nature.
him
system,
The representative
to equate the
Consequently,
at the
marginal
optimum
private savings are reserved for bequests while social security benefits
are used solely for consumption.
The same pattern appears
in
an opti-
mal pay-as-you-go system.
A
steady-state is defined as a path along which all the choice vari-
ables, in per-capita terms,
remain
fixed.
There
is a
unique steady-
-3-
state path associated with each
system.
shown
is
It
mode
of financing the social security-
exogeneous variables,
that for any values of the
these steady-state paths are equivalent in the following sense:
(a)
Consumption, aggregate savings and individual
utility
are the
sanae in both systems;
(b)
sum
the
of private savings
under a fully-funded system
is
and social security contributions
equal to private savings under a pay-as-
you-go system and
product of the population growth
(c) the
ra' ?
and the level of social
security taxes under a pay-as-you-go systemi is equal to the product of
the rate of return and the level of social security taxes under a fully-
funded system.
As an immediate implication, when
the rate of return
exceeds the population growth rate then the optimal level
of social
security under a pay-as-you-go system exceeds the one under a fully-
funded system.
ments
in the
We
These conclusions are made consistent through adjust-
optimal level of bequests.
analyze the effects of two demographic changes which were the
subject of recent discussion:
crease
in the birth rate.
expectancy
is
shown
an increase in
life
expectancy and a de-
Under certain conditions, an increase
in life
to increase the optimal level of social security, to
decrease private savings and to increase aggregate savings.
These
results are expected since, with a given income, an increase in
life
expectancy calls for a decrease in the flow of lifetime consumption
but to an increase in total consumption during the retirement period,
which
is
financed by social security benefits.
The implications
of
-4-
a decrease in the birth rate are unambiguous:
bequests, private and
aggregate savings decrease, while the optimal level of social security
increases.
The reason for these results
is that a
reduction in the birth
rate and, correspondingly, in the size of optimal bequests, increases
consumption during retirement and hence social security benefits.
The
above results apply to the long-run, comparing steady-states, and to the
short-run, with given initial endowments.
Finally,
we consider
the level of social security.
compensated by a decrease
aggregate savings.
in
Starting at the optimum, an increase in the
level of a fully-funded social security
ally
imposed change
the short-rim effects of an
system
is
shown
to be only parti-
in private savings, thereby increasing
Full compensation occurs only in the absence of
uncertainty, which is the case discussed by Barro (1974).
tional assumptions concerning risk-aversion, the
Under addi-
same conclusions
apply when the initial level of social security exceeds the optimal level.
An imposed increase
in the level of a social security
system based on
pay-as-you-go will increase aggregate savings and bequests.
However,
under uncertainty, the increase in bequests does not fully compensate
for the increased taxes on the younger generation.
A nonoptimal
of bequests,
states
and
level of social security leads to a
generated by lifetime uncertainty.
the
distribution
Consequently, steady-
long-run effects of a change in social security should
be discussed in terms of stable distributions.
such an analysis.
random
We
have not included here
-5-
The only type
of uncertainty considered in this
the duration of life.
The inclusion
as health and future wage income,
we have reached.
It
of other uncertain
may change some
paper relates to
elements, such
of the conclusions that
will be important in these extensions to specify
which decisions by the individual are made ex ante and which ex post
.
-6-
2.
Optimal Social Security Systems
We
shall analyze a
model
in
which
all annuities
the public sector through a social security
determination of
its
Two methods
are supplied by
system and focus on the
optimal level.
of financing will be considered:
a fully-funded
system and a system based on a pay-as-you-go principle.
Our
on
life
point of departure
from
the previous literature is the
uncertainty, which implies that social security is not a perfect
substitute for private savings.
We
extend the standard model of overlap-
ping generations (Samuelson [1958] and
uncertain lifetime.
periods:
Diamond
[1965]
)
to include
Life of a representative individual is divided into two
a working period of fixed duration and a retirement period
whose length
Utility
emphasis
is
random.
Wages
in
each period are assumed fixed.
depends upon own consumption
in
each period, the length of
and the expected utility of the next generation.
life
Each generation may
affect the welfare of the next one through the transfer of bequests.
Two
types of assets are available:
an annuity which yields a given
return for the duration of retirement and regular savings.
certain lifetime, the rate of return on annuities is random.
Due
to the un-
We assume
that the rate of return on savings is certain.
Let c„ and c^ be the consumption flows in the first and second periods,
respectively, of generation
i.
Let
B be
the level of per-capita bequests
-7-
left
by generation
to generation
i
retirement period which
The distribution
is actually
assumed
of 9 is
i
+
1
The fraction
.
of the potential
to be the
same
^
1.
for all generations.
The budget constraint for a representative member
i
^
realized is denoted by 0,
of generation
is
_
-w
i
c_
+
,
T-,i-l
B
-a-s
i
i
(2.1)
GB^
where w
the
= Rs^ + (R'a^-c^j)0
is first-period
amount
of savings,
R
earnings, a the investment in annuities and s
the return on savings, R' the benefit-
investment ratio on annuities and
assumed
that w, R, R' and
G
G
the
number
are fixed.
of children.
It
Note also that there
is
is
no
reinvestment of funds during the second period.
The determination
security system.
of R'
depends on the method
In an actuarially fair , fully-funded
of financing the social
system, expected
benefits are equal to the return on the investment in the system.
R' =
(2.2)
That
is.
^
6
where
6 is the
expected length of the retirement period in the population.
With a pay-as-you-go system, expected benefits to retirees are
equal to social security taxes collected from the working population, i.e.
(2.2')
R' =
3
e
.
,
-8-
Equations
(2. 2)
social security
and
(2. 2')
assume
that risk pooling is feasible.
The
system can therefore offer a nonrandom benefit rate to
each individual.
We
shall first analyze a funded system.
Funded Social Security
A.
Since a funded system does not involve intergenerational transfers,
its
optimality can be analyzed
from
For a given optimal policy
the point of view of a single generation.
of the next generation,
indirect utility depends on its initial
the expected utility of generation
the level of bequests.
V^
(2.3)
i
We assume
=u(cb +
endowment.
We
each generation's
can therefore write
as a function of its consumption and
the following additive form:
E[v(c!,,0) + Gh(B^)]
e
where
u{c-.) is
is the
evaluation of the next generation's representative individual
first-period utility, v(c. ,G) second-period utility and h(B
indirect utility.
Each
of these functions is
time and to satisfy, for any
assumptions.
9,
Note that both
assumed
to be invariant over
the usual monotonicity and strict concavity
c^
the fixed flow of consumption, and 9, its
,
duration, affect second period's utility.
All the variables, except
i.e.
,
prior to the realization of
assume
that
)
B
9.
,
are chosen by the individual ex ante
To ensure an
interior solution,
we
,
u'(0) = v^(O,0) = h'{0) =
(2.4)
—
v. =
where
-^
The maximization
.
=o,
for any 0,
of (2. 3) subject to (2. 1) yields the
following first-order conditions (F.O.C.):
(2.5)
1^
= -u'(cjj) + RE[h'(B')] =
(2.6)
H-
= -u'(cjj)+5LE[0h'(B^)] =
1^
= E[vj(c'j,0)-0h'(B')] =
(2.7)
9
The objective function can be shown
variables s, a and c^
,
,
a
,
c^
responding c^ and
Due
sidered.
and
to
First,
B >
By
.
(2. 1)
i
3
by
we obtain the cor-
.
Furthermore, a
implies, by
*
suppose that
=0
s
Cov[h'(B),e] >
9.
and
9V
^— ^0,
However, by
0.
Due
(2. 1), that
that a
It
and
s
either a
must be
or s
strictly positive.
then follows from (2.
(2. 1),
Ra' -
c'j
>
Second,
is positive.
5) - (2. 6)
and hence B
that
is in-
to risk aversion, h"(B) < 0, the above covariance
must therefore be negative, which
ment shows
B
unique.
(2. 5) - (2. 7) is
assumption (2.4), only interior solutions need to be con-
for all 0.
creasing in
of
in the
follows directly from these conditions that c^ > 0, c^ >
It
B >
B
concave
shall denote the solution for each
which are functions
),
to be globally strictly
and thus the solution to
For following reference we
(s
.
>
0.
is
a contradiction.
A
similar argu-
-10-
The condition
Cov[0,h'(B)] = E[0h'(B)]
(2.8)
which follows from
h" <
0, that
result.
B
(2. 5)
-
9E[h'(B)] =
and (2.6) implies, since
B
monotone
is
in 6
and
This, of course, is a well-known
is constant for all d.
Annuities enable the individual to transfer consumption across
states of nature and
h'(B), in all states.
c/
(2.9)
=
it
is
optimal to equate the marginal utility of bequests,
Further,
3
a
^
it
follows
and
GB
from
^
=
(2. 1) that
Rs
\
e
That
is, annuities
are used exclusively for consumption during retirement
while all savings are reserved for bequests.
bequest motive, h' (B) =
Clearly, in the absence of a
0, the individual's portfolio
would contain only
annuities.
Substituting (2.9) into (2.3), the objective function
V' =
(2.10)
which
is to
u(cjj)
+
E[v(3a\
e)]
+Gh(^)
be maximized subject to the constraint
The problem
is
,
c„+a+s =w + B
thus reduced to a problem of choice under certainty.
This does not imply, of course, that uncertainty
On
becomes
is
redimdant in our model.
the contrary, in the absence of life uncertainty the optimal level of
social security is indeterminate, being a perfect substitute with private
savings.
mum
However, the achievement
facilitates the
of
complete insurance
comparative statics analysis.
at the opti-
—
-11-
Two
types of comparative statics analyses are of interest.
short-run analysis for generation
w+B
satisfying
B
=
fixed.
In both
w
R
and
B
which holds
initial
endowments,
i.e.
,
Second, a long-run steady-state analysis along a path
constant.
,
i
One, a
,
which implies that
all the
choice variables remain
cases we restrict ourselves to partial equilibrium, holding
constant.
We
shall discuss here the steady-state effects of two
changes which were the subject of recent discussion:
demographic
an increase in
life
expectancy and a decrease in the birth rate.
Recalling that HB^) = F(V*^'*"^(B^)), where
of utility for generation i+1 and
F
is a
V*^"*"^ is
the optimal level
monotone increasing function, we
can rewrite condition (2.5) in recursive form
-u'(cjj) + RF'(V*'"^^)u'(cj)"*'^) = 0.
(2.5')
Hence,
i
i+1
in a steady-state, c^ = c^
,
we have F'
ficient condition for a unique steady-state is that
of
1
= h'j- = p-
.
when marginal
A
suf-
utilities
consumption are equalized across generations, the following "selfish-
ness" condition
is satisfied:
u"
(2.11)
In
-
Rh" > 0.
subsequent discussion we shall assume that condition
(2. 11)
holds.
Using (2.9), the F.O.C.
e is a
(2. 5) -(2. 7)
random variable independent
of
and rewriting
with zero mean,
4
6 = 9
we
+
t.
obtain
where
12-
da
(jg
1
/
A
V
R
G
^^ = ^h'u"(
(2.13)
A
where
= -
|^
h"u"
J\
E[vJ
J^ +
ii-^
(u"( 1-|) +
-
E[^12^^
\/ ^[^ll^^l
^^)
\
E[vj]
/
l)
§^E[vjJ<0.
Similar to standard models of savings under imcertainty, the effect
of a
spread preserving
shift
depends upon whether the relative change_in
E[v^^e
E[v^Jcj
+ -rprr i~
—
^ri
^ L^
E V
expected marginal utility of future consumption,
1
]^
larger or smaller than one.
utility directly
An increase
It
a-nd
is only
[
—
j^
is
J
expected marginal
and indirectly through the decrease in the flow of social
security benefits during retirement.
Vj2 >
in 6 affects
J
thus the
sum
of the
It
is
reasonable to assume that
above two expressions will be positive.
with stronger conditions, however, that one obtains the
"intuitive"
—^
outcome
savings, i.e.
,
a
+s
,
>
and
—^
<
0.
also depends on the
The
same
effect on aggregate
condition.
The source
of the ambiguity concerning the optimal level of social security is quite
clear.
For given labor inputs and income, an increase
in life
expectancy
calls for a reduction in the flow of second-period consumption, c^.
How-
ever, expected total consumption, 0c., which is provided by social
may
security taxes,
increase or decrease.
The implications
A
reduction in
G
private savings.
of a
decrease
in the birth rate
are unambiguous.
leads to a decrease in the size of bequests and hence in
Per-capita bequests will, however, increase and so
-13-
will second-period
security.
consumption and thus the optimal level
While a
and
change
s
in opposite directions, their
i.e., aggregate savings, will decrease.
%r
dG
(2.14)
=
-^(
^^2
V
^^•^^^
dG
^
(2.16)
^
*
^^2
Rs
=
*
Specifically,
IV
}
2
dG
g
^„j,^^
2
^ "
^ u
j<0
*
•^^
R_s_ e[v,
J(u^^-Rh^O >
^^
The short-run
effects of changes in Q and in
d(a
(2.17)
+5
)
^
sum,
>
9u"h"-^E[v,J(u"-Rh"))
-
*
dc
of social
0.
AG0
identical to the long-run effects presented above.
between the short-run and long-run effects holds
G
are qualitatively
The usual relation
in this case:
long-run elasticities, across steady-states, exceed
(in
the
absolute values)
the short-run elasticities.
B.
Pay-As-You-Go System
A pay-as-you-go system
An optimal
social security
involves transfers across generations.
must weigh
the welfare of the retired
recipients against that of working population.
planner's point of view is identical in this
generation at each point of time.
We assume
reject with
that the social
that of the older
At a point in time when the population
-14-
consists of generations
the older generation
and i+
i
1,
the
problem solved by a member
of
is
i
Max E[v{cpe) + Gh(B^-a^)]
(2.18)
i
i
a ,c^
subject to
GB^
(2.19)
= Rs^ +
(^
-
c^J 9
6
where
s^ is
predetermined.
The F.O.C. are given by
E[h'(B^-a^)(9-9)] =
(2.20)
and
E[v^(c^j,0)-0h'(B^-a^)] = 0.
(2.21)
We
shall denote the solution to (2, 20) and (2. 21) by (a
depends on
s
.
The nature
,
c^)
which
of the optimal solution is similar to the one
obtained in the case of the funded system; namely, social security
benefits are used solely for consumption.
level of private savings and are thus
•
(2.22)
—
-^
Ga^ = c\9,
B^ =
Bequests are adjusted to the
nonrandom:
^
R
^
.
The long-run, steady-state effects can be analyzed with the additional restriction that a
is
time invariant, which implies that
other choice variables are also constant.
conditions (2. 20) -(2. 21)
mines the optimal level
we also have
of savings
all the
Accordingly, in addition to
condition (2. 23) which deter-
15-
-u'(Cq) + RE[h'(B-a)] =
(2.23)
where c„
w
=
denoted by
+
a
-^
The endogeneous steady-state level
- s.
of s is
s.
There
simple correspondence between the steady-state
is a
solutions of the two methods of financing the social security system:
a = p- a
(2. 24)
Using relation
(2. 5)
-(2.
7)
and
When R
=
and
(2. 24), it
(2. 21) -(2. 23)
G
security and the
the two
s
= s
+ a
.
can be verified that the system of equations
are equivalent.
systems yield identical optimal levels
same aggregate savings.
When R
>
(
of social
<) G, the optimal
social security tax will be larger (smaller) under a pay-as-you-go system
than under a funded system.
However, bequests adjust so
transfer between generations remains unchanged.
that the net
Consequently, the
steady-state levels of consumption, aggregate savings and utility are
identical under the two systems.
This result has a direct bearing on the Feldstein-Barro controversy
(Feldstein [1974, 1976] and Barro [1974, 1976]) with regard to the effect
of social security on aggregate savings.
when social security
long-run effects.
is
Our discussion indicates
that
optimally chosen, the method of financing has no
The concept
of an optimal level of social security
played no role in the above controversy since lifetime uncertainty has
been disregarded.
The discussion focused on imposed changes
in the
16
level of social security.
We
shall turn to this question in the next
section.
Short-run analysis for the pay-as-you-go system can be applied
to the older generation, for
which private savings are given.
condition (2. 23) need not hold.
The results w.
of social security are qualitatively the
same
r. t. the
Thus,
optimal level
as in the long-run.
•17-
Departures frora the Optimal Level
3.
There are marked differences
of Social Security
in the effects of
an imposed social
security system under conditions of certainty and uncertainty.
certainty, the
If
maximum
Under
level of utility is independent of the level of a.
social security is funded then a is a perfect substitute for s and
if
financed on a pay-as-you-go basis then a is a perfect substitute for B.
Hence any change
None
in a will
be exactly compensated by either
of the other variables will be affected.
Consequently, the
on a as well as
we
initial
shall continue to
resources.
assume
case of a funded and on
This
is
B
-
As
or B.
Under undertainty, there
no perfect substitute for a which, by assumption,
insurance available.
s
maximum
is the
only
form
is
of
level of utility depends
a simplifying assumption, however,
that the utility of bequests depends on
B
in the
a in the case of a pay-as-you-go system.
equivalent to the assumption that each generation is concerned
with the expected wealth, rather than expected utility, of the next generation.
In other
words, each generation disregards the risk aversion of
subsequent generations.
When
a is
imposed
state is considerably
at
an arbitrary level, the definition of a steady-
more complicated than when
When complete insurance
is not feasible,
a is optimally chosen.
bequests become random.
steady state can then be defined as a stable distribution of B.
not attempt to characterize this distribution.
be confined to first-impact effects
the
optimum a
.
We
A
shall
Instead, the analysis will
for a given generation, starting
from
18-
As
we
in the previous section,
shall discuss in turn a funded
system and a pay-as-you-go system.
The maximization problem which we discuss
one in the previous section except that a
Differentiating totally the
F.O.C.
is identical to the
predetermined.
is
(2.5) and (2.7) w. r.
t.
a,
we
obtain
*
^
(3.1)
(
=
^
-Gu"(Cq)E[v^j(Cj,0) + 9V'(B)]
<0
-^E[v^j(c^,0)]E[eh"(B)]
and
dc*
(3.2)
.
,
*
^2 (GE[v
dr^=^"(^
+
V" -^\
+ (E[h"(B)]E[0V'(B)]
(c.,e)]
-^
_
(0E[h"(B)]-E[9h"(B)])
E[0h"(B)]^)
-
|
where
A' = Gu"(cQ)(GE[v^^(c^,0) + E[02h"(B)])
(3.3)
+ R2GE[v^^(c^,0)]E[h"(B)]
+ R^(E[h"(B)]E[0V'(B)]
The
last
term on
Schwartz's inequality.
the
It
R.H.S.
Jc
We
have already seen that
at (a
Jc
,
s
^+
E[0h"(B)]^) > 0.
and (3.3)
of (3. 2)
follows that
-
1
>
if
is positive
Gov
[h"(B),0]
by
^
jje
,
c.)
B
is
nonrandom and thus
0.
•19-
Cov[h"(B),0] = 0, satisfying the above condition.
Hence,
if
the level
happens to be equal to the optimal annuity holding
of social security
ds
level for the individual, then -j— +
1
>
This result means that a
0.
forced increase in the level of social security will reduce private savings
but increase total savings.
Under the additional assumption
aversion in the
utility of
nonincreasing absolute risk
of
bequests, which implies h'"(B)
5^
0,
one can
extend the previous result to levels of social security which exceed the
optimum
By
fied.
level a
This can be shown as follows.
.
9V
— ^
^
<
as
(2. 5)
and
(2. 6),
concavity,
Hence, by
Consider the case a
be decreasing in
is
in
^+
1
>
*
,
when
(2.5) and (2.7) are satis-
Cov(h'(B),e) >
as
h'"(B)
if
for a
in
which a
Under certainty such a
is
..
>
it
must
B
Therefore,
0.
then Cov(h"(B),0) >
0.
Hence, by
'
> a
.
The above result may be interpreted as follows.
experiment
a < a
Since h'(B) is strictly monotone in 6
.
order to satisfy Cov(h'(B),0) <
increasing in 6 and
(3.2),
a
5^
^
a < a
increased and
shift in asset
s
decreases
Consider an
in
equal amount.
composition does not change the
consumption and bequest possibilities and thus this would be the optimal
adjustment.
However, under uncertainty such a change
probability distribution of bequests.
then for a given level of c^
the
,
If
a > a
*
and thus
affects the
Ra
—^
*
- c.
> 0,
the variance of bequests increases while
mean remains unchanged.
The assumption
that h"' >
implies
in
—
20-
this case that the expected
marginal
utility of
bequests increases.
This
Q
when
result is retained
c. adjusts
Another question of interest
so as to satisfy
is
the effect of social security on
Using the F.O.C.
expected bequests.
(2. 7).
(2, 5)
and
we
(2. 7)
find that
*
^mi=^(^^A.p.
\da
da
da
(3.4)
/
(R^GE[v
^
A
'
]
^—
(E[0h"(B)]-eE[h"(B)])
9
Gu"{c^)
+
R
—
{E[d\"{B)]- eE[9h"{B)])
In general, the sign of (3.4) is
B
indeterminate.
However,
at a
,
-- and hence h"(B) -- is independent of 6, which implies that (3.4) is
negative,
dE(B)—
<
-r
Notice that under certainty
0.
-
for all a,
accordance with the observation made previously that the
which
is in
sum
+ a remains constant for changes in
s
dE^B^
—
—-r
a.
Consider now the short-run impact of an increase in social
security payments to generation
generation i+1.
5)
financed by an increased tax on
The only choice variable for generation
savings are predetermined.
(3
i
^^1 _
Differentiating (2, 21) w. r.
i
t.
is c., since
a yields
E[e{e-d)h"]
1
dc.
Evaluated
at
a
,
—>
^da
0, since
h" is constant.
It
follows that
—
21-
dE[B]
1
>
—>
—J-
In contrast to the
0.
case of certainty, the increase in
bequests does not fully compensate for the increased taxes on the younger
generation.
Consider now the effects
the working population.
of the
increase in social security taxes on
Starting at the steady-state values (a, s),
differentiate equations (2. 21) - (2, 23) w.r.t.
The change
a.
we
in first-
period consumption is given by
where
A' =
(u"+—^^
jE[v^jJ +
^
g-"
+
2~
^^
G
G'
o
V^6
= E[9
^i"
9
]
J
-
is
^" the
"^
variance
^.x^..v,^ of
^.
6.
..
even in the absence of uncertainty.
are held fixed.
in the
In the
Notice
.hat .Q
c^, would decrease
.,^..^^ that
This
is
because
initially
endowments
long-run, endowments will vary with bequests and,
case of certainty, c„ will remain invariant.
Aggregate consumption
ation's total consumption, 6c,
in
,
any period
is the
sum
of the older
gener-
and the younger generation's total
consumption, Gc„. Thus, the change in aggregate consumption
is,
by
(3.5) and (3.6),
_
(3.7)
dCn
e-^+G
dC(-,
h"Vg((u"-Rh")(E[vjj]
A^(E[v^J+
'
G
)L
,1^)_rW)_^2^^^^^j,.^^[^^^j
'^)
G
<
22-
As
in a fully
funded system
^
the initial impact of an increase in
social security is to increase aggregate savings.
-23-
Pootnotes
1.
See also Feldstein's paper (1974) and the exchange between Barro,
Buchanan and Feldstein (1976).
Samuelson's analysis (1975)
is different
since, following Diamond's discussion of the effects of national debt
(1965), he disregards bequest motives and hence voluntary intergenerational transfers.
2.
In the
U.S., pension funds provide less than
fifty
percent of the
total benefits to retirees.
3.
The Hessian Matrix
of our
problem
2
2
u"+^E[h"]
^
u"+^E[eh"]
2
2
u"+^— E[0V']
u"+^E[0h"]
OG
e
-^E[d\"]
G
OG
-^-E[e\"]
-p^E[0h"]
^
+
E[v^J
li
dG
is negative-definite.
concavity.
-P^E[0h"]
^
0G
^E[A"]
^
The diagonal elements are negative by
The principal minor
of
order
2 is
strict
equal to
2
^-^
[Gu"E[(0-I)\"] + R^(E[0^h"]E[h"]-E[0h"]^)] > 0.
d'^G'^
The second term
in the
above expression
is positive
by Schwartz's
24-
inequality.
The determinant can be reduced
to
2
+^E[vjj) (E[0V]E[h"]
(u"
-
K[Gh"f)
u
J
Gu"E[v
Since 9 is restricted to
4.
_2
^
o-^^ E[(0-0)^h"]
02
+
[O, l]
,
<
the described shift is not strictly
spread preserving.
A
5,
some
special case of
which
,
ElvJ
6.
of
+s
)
Then
standard definition of relative risk
is the
v'(ci)
——
12-'
aversion, and
d(a
v(c.,0) = v(c^)0.
interest is
>
?
:j
'
_
=
1
1.
TT
Hence,
da
—^
> o0,
V-
ds
^^
< 0, and
0,
Expected wealth
is
defined as the expected present discounted value
consumption, including expected bequests.
i^
^OR
In the case of a
GE[B^]
R
^^
By
(3. 1),
i-1
pay-as-you-go system, the definition includes net
bequests, i.e. bequests minus the taxes paid by the next generation.
By (2.1) and (2.2')
i
i
„
'0
,
+
^1
,
-i=r ^
R
GE[B^-a1 _
^R
R
w
^Tji-l
+ B
25-
Notice that
7.
^
if
h'"(B) = 0, i.e.
utility of
bequests
is
quadratic,
*
then
8.
dCj^
-5
—<
da
da
+
1
When
R
—
g
,
c^
i.
—
>
e.
for all a.
increases, the variance would
,
still
increase since
part of social security benefits is left for bequests,
However, mean bequests, due
These two effects combine
to the reduction in savings, decreases.
to increase the expected
marginal
utility of
dc.
bequests.
increases.
When
Our
-j
—<
0, the
variance of
result indicates that the
B increases
former
while the
mean
effect is dominant.
-26-
References
Barro, Robert
J. (1974),
Journal of Political
Barro, Robert
J. (1976),
of Political
"Are Government Bonds Net Wealth,"
Economy
,
82^
(November), 1095-1117.
"Reply to Buchanan and Feldstein," Journal
Economy
,
84 (April), 343-350.
Buchanan, James M. (1976), "Barro on the Ricardian Equivalence
Theorem," Journal
of Political
Economy
Diamond, Peter A. (1965), "National Debt
Model," American Economic Review
in a
,
60^
84 (April), 337-342.
,
Neoclassical Growth
(December), 1126-1150.
Feldstein, Martin S. (1974), "Social Security, Induced Retirement and
Aggregate Capital Accumulation," Journal
of Political
Economy
82 (October), 905-926.
Feldstein, Martin
Security:
S.
(1976), "Perceived Wealth in
A Comment," Journal
of Political
Bond and Social
Economy
,
84
(April), 331-336.
Samuelson, Paul A. (1958), "An Exact Consumiption Loan Model of
Interest with or without the Social Contrivance of
Journal of Political
Economy
,
66
Money,"
(December), 467-482.
,
-2 7-
Samuelson, Paul A. (19 75), "Optimum Social Security
Growth Model," International Econom.ic Review
,
in a
Life-Cycle
16 (October),
538-544.
Yaari,
Menahem
E. (1965), "Uncertain Lifetime, Life Insurance, and
the Theory of the
(April), 137-150.
Consumer," Review
of
Economic Studies
,
32
Date Due
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