If'-'
working paper
department
of economics
el of Social Insurans
Variable Retiremei
Diamond
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
A Model of Social Insuran
Variable Retireme
A
P. A, Diamond
and J. A.
Number 210
Department of Economics, M.I.T,
**
Nuffield College, Oxford
Octob
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1
.
Introduction
No-one knows what work he will be capable of in the future.
Uncertainty about earning ability in the last years of life is
particularly great.
The burden of this risk to the individual is
eased both by private insurance and by the tax and social insurance
system.
Complete relief from risk is not available, because neither
private insurance nor public arrangements distinguish fully between
low income by choice and low income by necessity.
might defeat itself through moral hazard.
Full insurance
For this reason, the design
of optimal social insurance is an interesting and difficult problem.
In this paper, we consider a simple form of earning ability risk, and
2
study optimal insurance for a population of identical individuals."
We suppose that everyone lives for the same length of time, and
that at any date an individual has either full earning capability or
none:
there is no partial loss of earning ability.
strikes randomly.
Loss of ability
The government is assumed unable to distinguish
those who cannot work from those who merely choose not to.
Individuals are assumed to maximize their expected utility, and to
.
>
be willing, without concern for the truth, to claim inability to
work when it suits them.
Three models will be used, with one period, with two periods,
and with continuous time.
The one-period model allows us to develop
simply the basic condition determining whether or not there is a moral
hazard problem.
For the case where there is a problem, we relate the
size of the optimal insurance plan to characteristics of the utility
function.
With the two-period model, we demonstrate an additional
aspect of the moral hazard problem, that taxation of alternative
- 2 -
commodities can increase expected utility.
For our model, we find
that, under plausible assumptions, an untaxed individual would try to
save too much.
Thus the optimal social insurance plan needs to be
supplemented by an interest income tax.
In the continuous-time model,
we examine the optimal consumption path while working, and the optimal
relationship between pension and date of retirement.
Even in the
absence of utility-discounting, and of a positive return to saving,
we find that it is optimal for consumption to increase with age, and
for retirement benefit to increase with retirement age.
This leads
us to suggest a way of incorporating these conclusions into the benefit
structure of the U.S. Social Security system.
2.
3
The One-Period Model
Either the individual is capable of work or he is not.
probability
that he can work.
He knows the
He is an expected-utility
maximizer, his utility being specified by three utility functions
u. (c)
•
utility of consumption
c
:-
when working,
UjCc)
utility when able to work, but not working,
u,(c)
utility when unable to work.
When working, he produces one unit of output.
Assuming nonlinear
taxation and one type of individual, the lack of continuous adjust-
ment of labour supply is not critical.
On the assumption that work is unpleasant in Che aggregate, we
have
U2(c) > u
(c)
for all
c
(1)
-
3
We also assume that work plus consumption is preferable to no work:
u,(l)
U2(0)
>
(2)
There are no private insurance markets.
The government pools risks
well enough to he able to plan to distribute consumption, conditional
on work, in such
way that the expected value of consumption equals
a
the expected value of output tor any individual.
The government being unable to tell why an individual does not
work,
its policy is entirely described by
person who
is
working, and
c«,
c., consumption for a
consumption for one who
is
not.
This
policy provides the individual with expected utility
6u (c.)
6u_(c^)
-
(I
+
(1
if he works
9)u,(Cp)
- 0)u-(c^J
if
when he can
he does not work
Thus he is willing to work if and only if
Uj(Cj) - u^ic^)
It
is
as well to suppose
individual works
if
(3)
that, even in the case of indifference,
he is willing and able to do so.
the
We shall
comment on this assumption below.
When individuals who can, work, the aggregate resource
constraint
is
ecj
since the output of
a
+
(1
wcrker
is
-
0)C2 -
1.
(4)
e
Otherwise,
c_
= 0.
The situation can be portr.ived in a diagram, with
Cj
on the axes.
This
is
done in Figures
1
c
and
and 2, where the plane
- 4 -
is divided
into Lwo regions by the curve
assumption (2), the point
=
c
u
=
u^(c„).
By
lies below this curve.
c„ =
1,
(c.)
Everywhere on and below the curve, the individual works if he can;
The feasible region is the shaded
and above it, he does not work.
Three indifference curves,
area plus the origin-
Ii^i*
^?^9»
are drawn, along which expected utility is constant given the
I_I.,
private decision on whether to work.
curve
= u^
u.
It will be noticed
that the
lies entirely below the forty-five degree line:
this
expresses assumption (1).
Evidently there are two cases to consider, which are shown in
the two figures.
In the first,
the line with equation
indifference curve
0c
+
(I
ihe
-
indifference curve
9)c„ =
6.
is
tangent to
Since the slope of the
is
"l^^l^
6
1
and the slope of the budget
(5)
- 3 u^Cc^)
line is
- 6),
- 9/(1
the optimum in this
case is given by
uj(c*)
=
u^(c*)
Gc* +
and
(I
-
0)c*
=9
(6)
In this case we have the familiar description of a full optimum.
The
moral hazard constraint is ineffective, as can be seen from the
diagram:
in the full optimum,
If at
we have
Figure
u
2
people are willing to work if they can.
the allocation described by
<
u-,,
(6)
the situation is different, and must be as shown in
Here the optimum is given by the intersection of the
budget constraint and the curve
u
= u
:
F,louri
F,
1
Z
- 5 -
Uj(c*)
=
u^fc*)
For this case to occur,
and
6c* +
(1
-
6)0* =
(7)
6
the indifference curve at this
intersection
This will
point must slope down less steeply than the budget line.
be the case if
i
u'
on the contrary,
case applies.
>
u!
all points of the curve
at
u'
at all
u'
= u„.
u
If,
points of the curve, the first
For convenience, ue state all this as a formal
theorem.
Theorem
for all
If
1
and
x
Uj(x) = u^(y)
the optimum is given by
=
Uj(x)
y
implies
u'(x) - u'(y)
(7).
for all
If
implies
Uj(y)
the optimum is given by
x
(8)
and
y
u'(x) - u'(y)
(9)
(6).
The conditions in the theorem take on a more interesting shape
if we
make the further assumption that the loss in the ability to
work has an additive effect on utility:
u^(r)
=
u^(c) - b,
b
-
(10)
Since in this case marginal utility of consumption is the same in
the two states,
u
(x)
=
condition
u„(y)
It seems to us
implies
that
than its contrary.
additively separ.jblc
(8)
can be stated as
u'(:) - u'(y),
for all
x
and
(II)
y
this condition will much more often be relevant
For
in
example,
it
follows when utility is
consumption and labour, and labour
is
disliked
rioLire
3
r
- 6 -
3
.
Comparative StJtirs under
By means of
iht-
dirtgrain,
al
Mt ^
we can
i
Ilaza'rd
xaiTiine
the v/ay in which the optii
:inium
changes when probabilities change, when the disutility of labour is
changed, and when the utility of consumption is changed.
When
increases, the budget line becomes steeper, rotating
6
clockwise about the fixed point
present,
and
c
so that
is
c.
u„ - u
i.e.
increased and
c
,
,
if moral hazard
Thus,
is
Similarly an increase in the
both incr-jase.
c^
disutility of work,
(1,0).
moves the curve
= u
u
downwards,
decreased.
A change in the utility of '.cnsumption can also be analysed by
considering how
affects the curve
it
u
For example, we can
= u„.
show that an increase in risk-aversion can lead to a reduction in the
extent of insurance, which may bt contrary to some people's intuition.
Suppose utility and marginal utility at
remain fixed, both with
c
and without work, and that risk-aversion increases,
decreased for
Therefore the
and in particular for
c^,
^
r.
cur'.'e
=
u.
u„
c
optimal
is decreased:
claimed.
utility at
4.
thereby
(See Figure 3.)
.
c
is
increased and the
the extent of insurance is reduced, as
A diiteitnt kind of ir^-rease in risk-aversion can have the
opposite effect
3
is
shifted to the right in the neigh-
Consequently the optimal
bourhood of
c
i?
c
u.
the curve
For
example, by preserving utility and marginal
while increasing, risk-aversion.
(
u,
=
u
The Two-Period
is
By analogy to Figure
shiftec; up.
.'lodel
With two periods, we assume that everyone can work in the first
-
7
-
period, and that an itidividual is able to work in period two with
probability
fdcts are known to the government.
rh'3se
want to use nhis model
to
Wu
slial
1
discuss the significance of private saving
behaviour, but ior the present, ihe government is assumed to offer
individuals pairs of first and second period consumptions:
if
work is done in period two,
every case
a
The budget constraint
c
being assumed
tVTO
J
- e)c
(i
t
is
a
The
=
I
+ 0R
(12)
I
aggregate output
unity, and that there
now in
is
iserved for derivatives.
1
ih^it
periods.
when everyone works is
constant interest factor
siiuation on
)
taken to be
is
R[ec
*
.
i
Utility
if not.
)
.c
be denoted by superscript instead
funciii-'n will
subscripts being
of subscript,
to portray
,c
function oi consumption in the
particular utility
it
ic
(c
R.
In order
budget constraint
two-dimensional diagram, we use the
when everyone works who is able
{^2) to eliminate one of the variables'.
It is most
convenient to use
:
the
a
he variables
where
z
o
- c,
= c,
z
and
c
(13)
is the return to working, 1 - z
the tax on labour income.
c
-^
=
2
Since consur.ipt u.n levels have
so
e
*
^—
R
-
9(i
-
I
c.
-
-
(1
(U)
e)z
c
t.'
-
(15)
z)
be nonnegative,
long as everycne wciks when le can,
and
c
it
follows that,
z
are constraind
o
by three ILneai
;
ni:qua
1
1
t
ies
,
t
- 0,
c
-
0,
and
c„ - 0.
These define
triangle in
a
However, people
,z)-space.
(c
willing to work when they can only if
art;
>
1
u
,c
(c
-
)
u
2
(c
If
-
z
0,
- c
c
has disutility,
implies that
increases
2
u
,
.
1
,
(c^,c^)
^o''2^
Thus the inequality
(t^,c,).
-0-1
u
"
>
'
and decreases
c,
(16)
)
(16)
Notice also that, since an increase in
0-
'>
z
z
so that, maintaining our assumption that work
,
.
,c
o
1
c-.
satisfies (16) and
(16)
o
(c
,z)
if
they can is the one bounded by
z'
satisfies
,z')
(c
l
I
>
z
v/henever
The region in which people work
z.
ABC
in Figure 4.
If
is
,z)
(c
not in this region, eilher there is no feasible allocation with
these values of
and
c
In the latter case,
Gu^Cc
,c
)
+
-
(1
o'^2'
V
=
m
z,
or
c
^
o
e)u-^(c
no-one works in the second period.
Rr^
,c,^).
Max [9u^
*
=
and expected utility is
1,
!.et
-
(I
e)u-^
Rc^
+
c
:
o
=
(17)
i]
/
be the maximum utility obtainable without work in the second period
The indifference curves drjwn in the diagran. are the curves of
constant expected utility
work
1
6u
*
the full optimum - ibe point
attainable utility
is
F
greater
same argument as in Section
unattainable
,x
O
where
u^
3
on the assumption that
is not
v
is
m
,
so
feasible; and the maximum
that moral hazard is
The
done in the second period.
optimum
sKows chat the full
is
if
1
(x
2
-
th.^n
present in the optimum, and worV
9)u
The case sho\vn is one for which
done in the second period
is
u
-
(1
)
I
-
u
2
(X
o
,N,)
_
<
1
ir.ilics
u,(x
10 ,x,
)
-
1
is the partial derivative with respect to
3
u
,x^)
(x
I
O
x..
i.
(l:
c
/'':
<;):('<:;•
4-
-z
[
I Cii^^i
c
S"
-
- 9 -
We concentrate on the case wiiore moral hazard
there
work in the second perir-d at the optimum.
is
optimum the indifference curve
small increase in
z
Then at the
tangent to the curve
is
AB.
An increase in
reduce utility.
v;culd
present and
is
Also a
may
c
o
either increase or decrease utility, depending on the sign of the
We shall discuss conditions for
slope of the curves at the optimum.
this in a moment
The interesting point is that
increase expected utility while
change in
a
is
z
that would
c
held constant represents
a
savings opportunity that the individual would wish to avail himself
of if he were allowed
change in
happen
brings about
c
if
it
as can be seen from
for,
to;
the
i
hanges in
and
c
and
(14)
a
(15),
that would
c^
were an act of pri\Mtc saving or dissaving.
Thus in
the opliKium cannot be obtained without preventing access to
general,
market, or, alternatively,
the capita]
imposing an appropriate tax
or subsidy on wealth.
It
is
most likely that the
hazard constraint will have
2
If
u
u
12
the slope of
the curve
representing the moral
negative slope at the point of tangency.
a
c
tends to zero,
then the curve
AB
only where
=
c
=
z
0.
Since
z
c
>
is
.
at the point
predominantly negative.
is
also zero, i.e.
B,
We can also
rather plausible local cndiiion for the optimum to appear as
in Figure U.
For
differ only by
Theorem
AB
.
can hit the line
on the vertical axis, with
a
urve
tends to minus infinity as
= u
find
i
2
a
this rurther analysis
constant,
Assume that
assume that
2
u
i.e., have the same derivatives.
3
u
v.'o
^
=
u'
- b.
and
3
u
10 -
-
1
u
(i)
(x
O
,x,)
= u
2
(x
I
O
<
2
u,(x ,x,) - u„(x ,x
]
implies
,x„)
/
o
1
o
I
I
)
I
Then at the optimum, individuals want to save more,
Our claim is that at the optimum a reduction
Proof
increase expected utility,
curve
u
12
= u
m
..12
.
is an increasing function of
3
9,1
(u
dc
c
would
being held constant; i.e. that the
z
has a negative slope at the optimum.
increasing function of
c
Since
we have to show that it
z,
u
- u
is an
.
- u 2,)
I
= u
- u
o
21.1
- ^(u
K
o
o
I
- u
2.
)
2
Here numerical subscripts denote differentiation with respect to the
indicated consumption level.
We shall argue that, at the optimum,
2
1
u
1
=
e_£
.
(1
u
- e)-|-
(19)
"
Given- this equation,
u
than
2
—
cc
u
?
3
-^^
1
- u")
(u
is
positive if
—r
is
larger
1
o
(and conversely).
u
The latter conditions is
precisely (ii)
"2
It remains to derive
optimum we have
From Figure
4
a
(19)
given the as,-,umption that at the
moral hazard problem (which follows from (i)).
we know that at
the moral hazard constraint.
D
an indifference curve is tangent to
Equating these two slopes we have
-
11
IN
oRl
1
o,'
-
6(1
o^.
/,
12-
2
,1
oR^
11,
e}(u[ - u^)
oR2
- e)uj
(1
12-
,2
oRl
+
&u^
Crossmult iplying, we obtain (19), which completes the proof.
The first condition in the theorem is the one that we have
earlier claimed to be plausible, and which
is
pretty much required
The second condition
for moral hazard to be present in the optimum.
says that, at equal utilities whether working or not, the individual
Our conclusion is
has a greater incentive to save u'hen not working.
that it is normal for a tax on saving to be associated with the
optimal social insurance policy.
Suppose now that saving is
means that optimisation
is
,il
lowed, and is not taxed.
further constrained by
a
This
saving equilibrium
condition, that the derivative uf expected utility with respect to
is
c
Thus the condition is represented by the locus of
zero.
o
points where the indifference curves have vertical slope, the curve
in Figure 5.
FE
As one moves right along this curve, utility, and
insurancL
the size of the social
diminish,
programjT.e,
If
individuals made their savings plans under the assumption that they
would work in the future, analysis xvould be completed by combining
12
the
u
savings constraint with the moral hazard constraint
,c,)
(c
o
=
u
(c
o
i
,C-,)
z
and
the 'iptimum would be at
E,
where the two
constraints intersect.
However if the individual plans savings and
future work at the same
tim.e,
Max [eu'(c
-
s,c,
o
g
-
Max [eu^(c
+
s)
v-.
^
have a differeni condition:
(1
- e)u-^(c
-
s,c„
o
1
-
s,c„
'
s)
+
(1
-
e)u^(c
*
s)]
2
-
s,c„
+
s)]
-
This is
12
-
when savings are optimal assuming work,
a more stringent cond it ifir since the ability to adjust savings
increases the utility available to those planning not to work..
is shown in Figure 5 as
FG
with the optimum occurring at
model is explored in more detail in another paper.)
when individuals are free to
H.
This
(This
Notice that,
without constraint or taxation, it
save-
Q
is still optimal
to be on the boundary of the feasible set.
One aspect of these solutioiis should be further noted.
they are on the boundary of the feasible set,
fact indifferent ^'hether to work in period
individuals are in
Similarly in
or not.
2
Since
the one-period model they are indifferent between working and not
working.
If
this
indifference
w.,
re reflected
in
random choice rather
than an implausible adherence to government wishes,
there would,
though the government could adopt
strictly speaking, be no optimum,
policies arbitrarily close to thf ones we have discussed under which
individuals would certainly wish to work
it is not
inappropriate to assunn
,
if
That is why
they could.
as we have done,
ment can choose when the consumri' is indifferent.
that
the
govern-
Of course a fully
satisfactory treatment of these matters would have to model the
costs of implementing choices explicitly, and that would take us
fo
fara field.
The chief conclusion of this section is that a government
concerned about social insurance would normally want to discourage
private saving, for example by taxing it; and that,
do so,
the extent of
the
social
diminished.
This exemplifies
situations,
that the provider
a
o]
if
it could not
insurance programme would be
general feature of moral hazard
insurance would usually want to
control trade in related commodities.
For example, fire insurance
-
-
13
companies should want their clients to buy fire extinguishers; and
they like to see high taxes on
5.
t'lie
consumption of tobacco.
The Continuous-Time Model
We turn to
model in which the
a
etirement date
i
is
continuously
variable, while maintaining the assumption that labour choice at each
This model
moment is disctetc.
to ask two further
.illows us
questions: how consumption shouM be made to vary with age when
working, and how the retirement benefit should depend upon the age
Since the moral hazard constraints can no longer be
of retirement.
pictured in
a
diagram, the mathenat ics of the model is more
complicated than for the previous models, but we shall use the
earlier analysis to guide us to the solution.
We make the follov;ing assumptions.
output per unit period would be unity.
zero.
Utility
taken to be
is
taneous utility.
Ivhen
the
tlie
everyone were working,
If
The real
interest rate is
undiscountcd integral of instan-
indi'.idual
is
working, his utility
is
according
u.(c), when not working, his utility is
u (c )
u^(c)
or
All three functions are strictly concave.
to whether he is able to work cr not. /To simplify the analysis, ve
assume from the outset that
u„ (c)
Saving
is
= u. (c)
controlled by
and
u.,
b,
+
differ by
a
a
Consequently,
function of age,
c
consumption when retired can be a function both of age
age at retirement,
becom.es unable
to
r,
c^Ct.r').
work is
a
The date
constant:
b - 0.
government.
th^i
consumption when working can be
u-
s
(t);
t
at which an
and
and of
individual
random variable, with density function
f
-
and distribution function
and
f(s)
for
'^
<
<
-
The length of life is denoted by
F.
s
14
T,
There are no atoms in the distribu-
T.
tion.
If
s,
the age of retirement is
r
and the age of disability
is
utility is
r
T
s
/uj (C|(t))dt
/u^{c,(t,r))dt
+
o
"
r
-
s
T
r
=
/u^(C2(t,r))dt
+
/U|(t)dt
+
/u,(t,r)dt
-
b(T - s),
"
o
(20)
r
where we have used (19), and introduced the notation
U|(t) = U|(Cj(t)),
U2(t,r)
=
U2(c2(t,r))
Neither individual nor government can affect the last part of
(20)-
What we are interested in is
or
r
v(r) =
/uj(t)dt
+
T
/u2(t,r)dt
(21)
and expected utility can be taken to be
r
V(Cj,c,,r)
=
,*"
/v(s)f(s)ds
v(r)[]
+
-
F(r)]
(22)
o
since the individual has to retire at the date of disability
he has not already done so.
In
'22),
individual decides to retire if he
is
r
is
s
it
che age ac which the
still able to work at that
date.
(22)
v,
expresses the maximand for our problem as an average of
the expected value of
frequently
m
v(Min I's.r)).
our analysis, and
it
is
Tliis
average occurs
convenient to have e special
,
,
15 -
For any integrable function
notation for it-
/h(s)f(s)ds
=
(h)
J
+
h(r)[l
-
h, we define
F(r)]
(23)
For later reference we note the following
Properties of
(i)
J
(ii)
J
is a positive
J
h
is a
(iv)
h
If
is
is
and
(i)
-
we let
r'
-^
dif f erentiabl
(iv)
J,(h)-J^(h)=
'^
Thus
,
(h)
^
r
where
h
and only if
(h)
=
h'(r)[l - F(r)]
(24)
follow immediately from (23).
/ [h(s)
-
h(r)]f(s)ds
To prove
(ii)
that
[h(r') - h(r)] /f (s)ds
+
(25)
r'
(h)
J
h
if
h
is
monotonically increasing.
The case
r
is
monotonically decreasing
From (25)
constant.
a constant,
it
also follows that
To complete the proof of
for all
dif f erentiable, and
constant.
if
r
'^
J
r
e
and calculate from (23)
r,
r
constant.
dr^r
Proofs
r.
monotonically increasing (or decreasing).
constant function of
is a
(h)
linear functional for each
mono ton ica 1 ly increasing (or decreasing) in
is
(h)
if
(iii)
J
r.
(24)
is
J
proved similarly.
(h)
(iii),
is
constant if
suppose that
From (23) it follows that
then implies
h'
= 0,
i.e.
h(r)
h
J
h
(h)
is
is a
is
= c,
16 -
-
The net resource cost of an individual who retires at
becomes unable to work at
s
is
T
r
z(r) =
/cj(t)dt
requires
a
/c2(t,r)dt
+
o
recalling that
and
r
- r
(26)
r
worker produces unit output.
If the
government
units of output for its own purposes, the aggregate
A
resource constraint is
J
(z)
+
A -
(27)
We are
We can now state formally the problem to be addressed.
to
Maximize
J
subject to (27) and
(v)
J|.(v)
- J^(v),
all
t
- r
(28)
This last constraint says that individuals do not prefer to retire
before
r.
Since the government can set consumption levels equal
to zero for anyone who retires after a date it chooses, we need not
be concerned about individual plans to work too long.
effective constraint on government is that it choose
functions
c.
and
In the full,
c„
so that
(27)
and
(28)
Thus the
r,
and the
hold.
first-best optimum, the government is constrained
only by (27), and marginal utilities of consumption should be
equated in all circumstances;
c,(t)=c°.
If
C2(t,r) - c°;
under these circumstances we had
with
u.(c
)
u| (c°)
-
- u^ (c")
(29)
u„(c„), there would
- 17 -
be no moral hazard problem, for, as one can see from (21),
would be
nondecreasing function of
a
and
r,
J
To see this consider the derivatives of
J
consequently
(v)
nondecreasing function of t.
there is no moral hazard problem.
In the case where /
the optimal value of
J
Using property (iv) of
r.
^^(v)
=
^^(2)
Thus
TT—J
with
u,(c
dr r
II
)
= u„ (c
L
i.
c
)
= {c° - c° -
and
;
,
{u,(c°) - U2(c°)}{l
unless
>
implies
= ul
u;
(v)
J
- c^,
r
= T,
»
^—J
or
r
1}{1
normally
with
(z)
we have
-
F(r)}
- F(r)}
except
in the borderline case
r
(z)
normally, e.g. when
-
A -
or if
a
r
r
respect to
is
r
and
(v)
v(r)
0,
for
r
-
A
Cannot
Thus it
(c° - c°-l){/sfds + r(l - F(r)]}.
i.
J
o '
in these cases be optimal to have
r < T, since an increase
in
would increase utility without increasing the resource cost.
= J
r
r
(z)
= c°T +
z
\
The interesting case is where
u
- u„.
u!
= u'
inconsistent with
is
Accordingly we assume from now on that
U|(Cj) = \i^{<i^
implies
uj(cj) -U2(c2)
Before proceeding with the analysis, we note
feature of the optimum.
Once a man has retired,
a
(30)
convenient
there is no
advantage from inefficient intertemporal allocation of his consumption.
Incentives to work depend on expected utility as a function
of the retirement date, so the cost of providing expected utility
conditional on retirement should be minimized.
C2(t,r) = C2(r)
Thus
(31)
T.
18 -
This means that
and
v
z
can be written
r
/U|(t)dt
v(r) =
+
U2(r)(T - r)
(32)
o
r
z(r) =
/cj(t)dt
+
c^CrXT
- r)
- r
(33)
o
The bad feature of the maximization problem we have to deal
with is a lack of the concavity conditions that would render
necessary conditions for the optimum also sufficient.
The moral
hazard constraints (28) cannot be expected to be concave in any of
the control variables
u-
c.,
c„,
as the control variables,
or
But if we regard
r.
the maximand, and the constraints
(28), are linear, and the resource constraint
is convex in them.
(27)
Thus we have a well-behaved programming problem if
given.
and
u
r
is
taken as
We shall first derive sufficient conditions for the optimum
conditional upon a specified value of
and afterwards derive a
r;
necessary condition for optimality with respect to
r.
The earlier models yielded solutions in which the individual
was indifferent whether to work or not.
expect that in the present model,
about the age of retirement.
the individual will be indifferent
We first consider the best path with
this property and then show that it is
is a
constant independent of
independent of
r,
t,
then
by property (iii).
/uj(t)dt
Correspondingly, one can
+
indeed optimal.
v(r)
is a
If
J
(v)
constant
We write
U2(r)(T - r) = V
(3A)
o
Given that
v
is constant, we ask what is
the optimal combinati on
.
-
of
and
u.
maximizes
Let
It is the one that for
u„.
the given value of
or equivalent ly for given
v,
and
G.
G
be
:
/Gj(u,)dt
+
v
minimizes
he inverse functions of
C|(t) ^^GjCujCt))
u^Cc), so that
2(r) «
19 -
-
o
/ujdt])(T
and
u.(c)
- r)
(z)
(z)
c^Ct) =G2(u2(t)).
and
G^ij-^iv
J
J
Then
- r
(35)
o
and it follows that
'Gj(U|(t)) - GjCu^Cr))
Consequently for
5u (t) ^r^^^
"
(t
- r)
(t
> r)
t < r
^fSl^'^^ " g2(s)]f(s)ds +
[g,(t) - g2(r)][l
- F(r)],
(37)
where we have introduced the notations
gj(t) - G|(U|(t)) = I/u|(c,(t))
(38)
g^Ct) - G'Cu^Ct)) = l/u^(c2(t))
For cost-minimization, the derivative (37) should vanish for all
t;
r
/ (g,(t) - g2(s)]f(s)ds +
[gj(t) - g2(r)][I
and using a
•
Differentiating with respect to
(1
- F(t)]^,(t) o
- F(r)]
(39)
for a time derivative
c,
we obtain
[g^(t) - g^(c)]f(t)
which can be written equivalently as
= 0.
(40)
\
)r
\
Xq, =
a
/
V/
\
A
.
//
/
^
f \Ciur e
6
/
.-'-'U,='^2
n
-/
/
"
-^
- 20
i'-f<"i^;ry<r7 t))
Also, setting
=
t
in
r
"
"[uj(cj(t))
u^^c^Cr.))]^^''^
(39), we have
g,(r) = g2(r)
(Al)
or, equivalently,
u|(c,(r))
=
u^Cc^Cr))
We have shown that if it is true that there is indifference
about retirement age at the optimum, then
and
(Al)
constant, and (AO)
is
We shall show that this is indeed
hold.
to do this we need
I,t
v
to know a little more about
satisfies two differential equations,
differentiating (3A) with respect
(T - t)u2(t)
to
the optimum,
and
the proposed solution,
(40), and one we obtain by
r:
= U2(t:)
- Uj(t)
(42)
In Figure 6, we have a phase diagram for this pair of differential
equations, in
,c„)-space.
(c
functions of
c
and
c^.
functions of
c.
and
c^
and
u
and
g
g„
u„
are of course increasing
are also increasing
respectively; since
is equal to
g
i/u;(c,).
From (40),
c_
u.
c.
is
is stationary when
= u„
stationary when
u
~
^y
lies below the curve
the two curves, both
and
c.
g.
g
= g,;
By assumption
= g^
(i.e.
= r.
Therefore both
throughout the solution.
c
and
the curve
(30),
u!
=
c„
g
Between
u').
are increasing with
c„
says that the solution we propose has to hit the
t
while from (42)
=
t.
g„
(41)
curve at
are increasing functions of
t
.
- 21 -
Theorem
3
Assume that
implies
= u„
u
T -
dt
'^
^^1
g,
r.
' ^2^1
= gj
at
- F
r
the policies so defined are optimal for
J (z) + A = 0,
given
t
f
th..
and
If
u^ - u,
dUj
dt
- u'.
u!
If the optimum for a given
Assuming that
u
(0)
= u-(0)
stated properties exists, for any
= -«,
r
exists, it is unique.
r
a
solution with the
and
between
T
such that
r
/
o
(i.e.
(1
- F(s)]ds > A
(43)
such that positive consumption is feasible).
We first proved this theorem by introducing Lagrange multipliers
for the constraints, and following the standard method of proving
sufficiency of the first-order conditions for constrained maximization problems.
here,
This method is rather involved.
The proof we give
though less clearly motivated in that Lagrange multipliers are
not introduced explicitly,
is
some preliminary results.
The policies satisfying the conditions of
briefer.
the theorem are denoted by asterisks.
We begin by establishing
The outcome is compared with
alternative policies satisfying the constraints of the maximization
problem.
*
Lemma
J
(g^v)
>
- g
*
(0)J
(v)
for any
v
satisfying
J
(v)
<
- J
(v)
.
22 -
*
Since
increasing function of
is an
c
t,
is an increasing
g
1
function of
and we have
t,
/g,(t)[J^(v)
-
J^(v)]dt -
(4A)
o
Now
^*
/gjJ^(v)dt
=
-
[gj(r)
g,(0)]J^(v)
(45)
and
^ *
r ^
fk]\iv)dt=
r ^
t
/g,(t) /v(s)f(s)dsdt
O
/gj(t)v(t)[l - F(t)]dt
+
o
'
o
T
=
r
/[g*(r)
-
g*(t)]v(t)f(t)dt
/[g*(t)
+
o
.
-
g*(t)]v(t)f(t)dt
o
by reversing the order of integration in the first integral, and
using (40) in the second integral.
This last expression simplifies
to
/g*Jj.dt = g*(r)/vfdt -
since
Lemma
=
g^fr)
Combining
("44),
gj (r)
2
Jj-Cg*)
(45),
/g2vfdt
and
(46)
=
g*(r)J^(v) - J^(g*v)
we obtain Lemma
(^6)
1.
= 8*^°)
The second and third conditions of the theorem are equivalent to the
vanishing of (37) for all
result.
t.
Putting
t
=
0,
we obtain the stated
- 23 -
Proof of the Theorem
We now show that any feasible path does not have higher expected
utility.
From the definition of
we have
z
/[Gj(Uj) - Gj(u*)]dt
z(s) - z*(s) =
+
[G^Cu^Cs)) - G2(u*(s))](T -
s)
o
Since
G
and
G_
are convex functions of their arguments, we can
use the inequality for convex functions to obtain
g
z(s) - z*(s) -
/g*(t)(Uj(t)
-
u*(t))dt + g*(s)(u2(s) - u*(s))(T -
s)
o
Using the definition of
v,
v (s), we
(32), and the constancy of
can write this as
g
z(s) - z*(s) -
/(g*(t) - g2(s))(uj(t) - u*(t))dt
o
*
T
/ ££Jll(u
o
9U|(t)
=
(t)
+
g2(s)(v(s) - V*)
-
u*(t))dt
+
(A7)
g*(s)(v(s) - V*)
'
The second step follows from the expression for the derivative,
We now wish to apply the linear operator
(36).
J
first term on the right hand side vanishes since (37)
all
(47).
is
t
T
""
to
o
^
au,(t)
Thus we have
T
'
'
o
/
it
\
''\9u*(t)/
'
'
The
zero for
.
- 24
J/g2)v
J^(z) - J^(z*) - J^(g*v) -
- g*(0)[J^(v)
by Lemmas
1
and
2
J
(z)
+
(A8)
A
<
Now
- v]
A -
0,
and
J
(z
)
+
A
Therefore (48) implies
= 0.
that
J
- V,
(v)
.
r
expected utility
i.e.,
is
I
no greater on the alternative path than on
the one satisfying the conditions of the theorem.
This proves the
sufficiency part of the theorem.
We next prove the existence part of the theorem.
Refer to the
diagram.
Since marginal utilities go to infinity as consumption goes
to zero,
g
and
g
then go to zero, and the
g
= g.
curve
Similarly the curve
u
= u
passes
passes through the origin.
through the origin.
The value of
g,(r)
determines the point on the curve
at which the solution path ends, and therefore
choosing
the whole path.
small enough, aggregate consumption,
g,(t)
be made as small as we please, uniformly for
J
(z(t)
+
t)
=
(t)
choosing
+
r
- rF(r)
/[I - F(t)]dt
=
z(t) + t, can
- r.
Thus
Therefore
g,(r)
if
(43)
holds,
J
(z)
(z)
+
+
A
can be made negative by
small enough.
J
r
g,(r)
on integrating by
O
^
We now have to show that
choosing
t
By
r
/tf(t)dt
O
parts.
-
= g„
can be made as small as we please.
r
J
g
large enough.
A
can be made positive by
By equation
(40), we have
- 25 -
g
- -
since
to
g,/g~ ~
r, we
and
-^ log
1
- F
[1
- F(t)],
decreases with
Integrating from
t.
obtain
g,(0) - g,(r)[l - F(r)]
Therefore by choosing
- r,
-
J
without limit.
(z)
large enough,
c
(0)
,
and all
c
(t)
can be made as large as we wish, thus increasing
for
t
g,(r)
(49)
The proof of existence is complete.
The rather restrictive condition about utility functions
introduced to prove existence is by no means necessary.
As will be
evident from the proof, the condition was introduced merely to exclude
cases with zero consumption, and their attendant corner conditions.
Throughout the argument,
r
has been taken to be less than
T.
The form of the differential equations makes it clear that the case
r
«»
T
would need careful handling; and the above existence proof
would not go through at all.
most plausible cases,
Fortunately, it turns out that, in the
the optimal value of
r
is
less than
T.
This will emerge as a by-product of our discussion of the optimization of
r.
We have shown that, for a given value of
r
< T,
an optimum exists.
u,'(c
(r))
=
and for a given
The optimum is given by a solution of our
differential equations which at time
satisfying
A,
u'(c
(r)).
r
takes values
c,(r),
c„(r)
Denote utility arising from the
26 -
-
(40)-J42)
the solution to /
c
J
by
(z)
Z(r,x).
formal proof, that
and
= x
(r)
V(r,x), and denote the value of
by
We first note, without taking the space for
and
V
are dif f erentiable functions of
Z
X.
for given
If,
Theorem
3
Z(r,x) + A = 0, it follows from
satisfies
x
r,
that the solution path so defined yields an optimum.
utility is strictly concave, the optimum for given
Therefore the equation
Z(r,x)
A
+
Z(r,x(r))
for the set of
Lemma
3
'
x(r)
r
is a
(50)
is unique.
r
has a unique solution, and
=
A =
+
(50)
which are feasible.
continuous function for
implies that
x(r)
is
T.
r <
To prove this, we must establish that when
r - r < T,
If
x(r) by the equation
we can define a function
by
r
is
r
restricted
From the
bounded.
differential equations we have the inequalities
^1
dt
Integrating from
g,(0)
>
to
1
- F'
T -
dt
t
we deduce that
r,
g,(r)[l - F(r)],
U2(0)
>
U2(r)(T
- r)
(51)
- g,(r)[I
Also
- F(;)]
-
are related by
lower bounds on all
It follows
c
that equation
=
u!
(t),
'r)
c^(0) - c^Ct) - c^Cr); and
C|(0) - c'j(t) - Cj(r),
c.(r) = X
U2(r)(T -
C2(r)
Therefore we have upper and
u'.
which are increasing functions of
(50)
consider a convergent sequence
^
imposes an upper bound on
r
-^
V
and
r
<
o
r(< T)
.
Since
Now
x.
r
V
- r
x.
- 27 -
eventually,
Z(r ,x
being
° continuous, we have
and equal to
x(r
and has a limit point,
is bounded,
{x(r )}
o
o
)
continuous function of
r,
and that, as
is a dif f erentiable function of
derivative.
3,
x
o
V(r) = V(r,x(r))
is
unique
r
is a
varies continuously,
r
We next establish that
all vary continuously.
c.(t)
Thus
0.
Z
.
This proves the lemma.
).
The lemma implies further that
optimal
A =
+
x
V(r)
and obtain a formula (66) for its
The argument used is similar lo that employed in Theorem
where we expressed
z(s)
the functions
in terms of
and
u
v.
In the present case, we must identify dependence on the retirement
date
r,
We have
and this will be done by superfixes.
s
/c[(t)dt
=
z'^(s)
+
C2(s)(T
-
s)
-
(52)
s
o
with
c^(t) = G|(Uj(t))
:2(s)
=
Gi7T-7lV(r)
\
/u[(t)dt}j.
-
o
Since the resource constraint is fixed,
\,(z^')
for any
r,
r'
<
and eventually let
- J^iz"")
=
(53)
We shall manipulate the left hand side of
T.
r'
Since the function
->•
(53)
r.
u
is
so chosen as
J^(z^) - J^(z)
to
minimize
J
(z
),
(54)
28 -
where
/c^(c)dL
=
:(s)
+
Gj~--{V(r)
/u[ (t)dt})(T
-
-
s)
-
By the mean value theorem, we have
z(s) - z'^'cs) =
where
T
g„
J
V
o
to
lies between
(r,r',s)
as
(s)
G'f^r^{V^(r,r',s)
(55),
/ u|
V(r)
and
'
(t)dt }j{V (r) - V(r')}
V(r').
- r'; and
r
and use
r
r
g^(s)
to
(5A)
(53),
- {J
(z'^')
r
r
as
r'
-*
Apply the operator
r.
:
(g^)
+
£,(r,r')}{V(r)
z
-
V(r')}
(56)
1
e(r,r')-*Oasr'-*r.
where
Reversing the roles of
(g„)
is continuous
J^Az"") -
where £2(^»^')
-^
in
Jj.Cz'')
as
and
r
r', and using the fact
i\(^&l) *e2(T,r')]{V{v) - V(r')}
-
r'
that
we also have
r,
-
(57)
r.
Applying the mean value theorem to the left hand side of (56),
recalling property (iv) of
J
that
,
(42)
=
=
c[(s) - C2(s)
H^(s).
=
h'(r)[l - F(r)],
and
and using the tact, derived frc,n/(52),
^"(s)
(h)
ir—J
'
'
+
that
g2(s)[u2(s) - U|(s)] -
1
(58)
say,
l-F(r)
we obtain
J^,(z''')
- J
(^'^^
= H'''(r)(r'
- r).
(55)
here tends to
G'
/
J .(z'"') - J
J
-
(59)
- 29 -
with
between
r
and
r
By the continuity of consumption
r'.
levels both with respect to time and with respect to the terminal
date, we see that
H
r
'
r
^
(f)
as
H (r)
-
r'
imply that
(59)
{H'(r) + n,(r,r')}(r' - r) -
where both
and
e.
r
H (r)
and
and
G-
(60) and
- r)
,
(^^.(82)
J/g^)
(g
"i
to
r'
V,r') - V(r)
<
Letting
0.
=•
)
* e,
{V(r') - V(r))/(r' - r)
is equal
-*
H
r
(r)/J
as
r
(g,).
'
''
r'
r,
r'
<
-^
r
)
{V(r) - V(r')}
(60)
.
also obrain from (60),
r.
'
{V(r
) }
)
-
V(r')}
Now consider
(62)
tends to
r
from below.
is dif f erentiable,
r'
>
(61)
r.
Recollect that
g
(62)
^
^2
implies that the limit ot
r
from above exists, and
i
'
'-
r
tends to the same limit
that
and that
-LilL
(r), defined
H
"2
We have therefore established
V'(r) = -
We now express
*
A similar argument wiih
as
tends to
"'('^^
jy,)
{V(r') - V(r)]/(r' - r)
way.
'
* £2 ^r ,r
establishes that
r'
)
,,r
'
t
J
r
V
"'^"^^
since
'
,
imply
(61)
,r
-
(r ,r
e
r, we
tend to zero as
r\
+
r', and using the continuity of
and
r
with respect to
(g„)
+ n2(r,r')}(r'
{H'^(r)
where
r
J
{\(gl)
tend to zero as
n,
Reversing the roles of
V(r)
Therefore (56) and
r.
+
(r)
=
g^(r).
by (58),
(63)
in
a
slightly different
Then we can write
- 30 -
= {c^(r)
H^'Cr)
-
= {- h,(r)
g^(r)u[(r)
+
h^Cr) -
-
c^Cr) +
- F(r)]
l}[l
F(r)]
-
1}(1
-
g''^(v)u^^{r)
(64)
where we define
h.(r) = gj(r)uj(r) - c^(r),
r =
1,2
u.(c^)
1
r
1
- c.
evaluated ac
r
(65)
1]
(66)
u.(c.)
Thus we have, finally, the formula
V(r)
=
'
"/^^^ h ,(r)
-
We must now consider the sign of
g
= g
,
h,(r)
h
-
+
h^
+
along the curve
I
since that curve contains the locus of end-points for
r-optimal paths as
r
A straightforward calculation yields
varies.
dh.
-—
From this it follows that, as
^(h,
- u.
- h^)
u
<
u„
on the curve
The inequality (68)
(c
,c
)
on the curve
which therefore
(c
(r),c
(r))
= u,
to
=
- U2
(68)
0,
g
.
implies that there is ac most one point
g
V'(r) =
lies
g
varies,
= g^
g = g,
<
since
(67)
the
=
g
at which
if
c^(r) = c.,
h
left of that point
- h,
i
=
+
=
I
1,2.
V'(r)
0,
and at
If
>
0;
if
it
lies
.
- 31
V (r)
to the right
<
*
*
exists
for which
T
for
If
4
exists
a
(r
.
=
i
ul(0) = «,
- a
when
Consider first how
Since
defines
r
u
(°°)
I
= 0;
and if there
-^
= u'
(69)
u|(c,) = u^(c2)
1,
behaves as
h.(c.)
11
Let
0.
u!
^
hj(Cj) - h2(c2) = -
number.
this
1,2,
such that
c^
c,,
1
g
=
such that
>
then there exists
equivalently,
i
,
.
1,2,
u^ ~ u
Proof.
*
= c.
)
We have not argued that there is at most one value
satisfying C (r) = c
r
Lemma
c
-
*
r
.
<
r
the optimal policy.
of
there exists such a point, and if there
If
0.
c.
-»
(70)
0,
or,
1
be an arbitrarily small positive
e
is concave,'
u.
1
+
u. (c.)
u.'
(c.) (e - c.)
- u. (e)
Therefore
u.(c.)
>
->
^(O
- e
as
c
->
.
1
It follows
that if
u.
is
1
while if
u.(0) - 0,
it
1
negative for small
is also
Lim h.(c.)
11
c.,
*
1
Lim h
true that
Lim [hj(Cj) - h
(c
)]
11
.
(c
=
.
)
= 0.
=
0;
Thus
(71)
g-^o
Now consider what happens as
since
Lim ul
=
0,
g
-^
<».
c
,
Fron (68),
c
-+
(69),
°°;
or,
equivalently,
and the assumption that
- 32 -
g
->
- h
h
°°,
decreases without limit:
Lim [hjCcj)
-
-
=
h^ic^)]
~
(72)
g-«o
since
~ h„
h
continuous decreasing function of
is a
hold, a solution of
(72),
Lemma
(70) must
exist, and
tlie
g,
and
(71)
lemma is proved.
Let
5
_ (T - t)f(t)
A^^^
<l>(t) _ F(t)
,
.
^^^)
,
be bounded as
t
-^
T.
x(r)
If
Z(r,x(r))
A
and
T
/
o
<
feasible),
[1
- F(t)]dt
Let
A
some
M
defined, by
A = 0,
(so that positive consumption is
then
x(r)
Proof
+
is
as
°o
^-
r
- T
(7A)
be a positive number, and consider paths optimal for
with
c,(r) - M
On any optimum path,
number
a,
g
dependent on
> g
,
u„; and there exists a positive
<
u.
but independent of
M
<
of these paths and for all
Max {g|
If
this were not so,
have
g
-
g
and
(75)
t,
-
g^,
-
t
r,
such that for all
<
- r,
Ut - Uj}
-
a
(76)
then by compactness and continuity, we should
for some r and t
u„ - u / which is impossible by assumption.
From (76) and the differential equations satisfied by the
- 33 -
optimal path,
follows that
it
+ u^)
^(g,
=
-
(g,
$
"i^r^
^)
^ a Min (-pi-^,
Since
^"2 "
&2^Y^*
(77)
defined in (73) is bounded, there exists
(T - t)f (t)
- F(t)
1
and therefore (77)
d
<
^^
,
V
>
2'
a
N
f
- F
1
.-l^log
Integrating from
to
t
Therefore if
r
g^(t)
as
->
-F(t)]
~
log
[1
- F(r)]
-
^
log
F(t)]
-
[1
and the corresponding r-optimal paths all
->
T
••
u^(t)
-<
-
^o
- T, and
r
[1
we obtain
r
g,(t) + U2(t) - g|(r) + U2(r) +
c^(t), C2(t)
such that
1
implies
dt'^1
satisfy (75),
N >
Thus
t < T.
for any fixed
T
- F(t)]dt.
J^(z) -> - / [
It
i
o
follows that for all sufficiently large
r,
inconsistent with (75).
->•
x(r)
"»
a
weak one, automatically satisfied if
zero limit as
result of Lemma
T.
-*
t
5
+
(z)
A =
is
as asserted.
should be noted that the assumption that
It
rather
Therefore
J
(p
f(t)
is bounded
is
tends to a non-
would have to be very pathological for the
f
to fail.
When the assumptions of these Lemmas are satisfied, it follows
that the optimal
r
is
less than
T.
For when
x(r)
>
c,
,
.
.
- 3A -
V (r)
0, and this must be
<
V
Since
is
value of
true for all sufficiently large
evidently continuous at
T
T,
r.
cannot be the optimal
r
Summarizing the above analysis, we have
Theorem
Under the assumptions of Theorem
4
3,
assumption (69), and
the assumption that marginal utilities tend to zero as
consumption levels tend to infinity, there exists an optimal
The optimal path or paths is Identified by the
unique (c-,C2) satisfying the additional condition
<
/
T.
r
h^Cr) - h,(r) =
where
h
are defined
h„
,
It should be recognized that
that optimal
less than
is
r
quite plausible-
It
T
in
(78)
1
(65).
the condition
is
states that the marginal disutility of labour
bounded away from zero for large consumption.
T
is
to
infinity as
t
used to prove
excessively strong, though
is
the optimal
(69)
r,
it
will be optimal to have
tends to
T.
In cases where
c
and
tend
c
Such cases are therefore rather
peculiar
6
.
Features of Solutions
Several aspects of the optimum emerge from the general analysis and
call for explicit notice.
In the first place,
the consumer is indifferent about
since
v
is
constant,
the date of his retirement.
In
effect, he is assumed to retire when the government wishes him to.
A small deviation from the optimum would make the consumer strictly
prefer
r
to any other
retirement date.
But the government wishes
35 -
consumers to retire before the end of life (and could set consumpThis latter
tion for later retirers to zero to achieve this).
achievable
is
situation where the
feature contrasts with the /full optimum'in which, normally, everyone
who can work does.
Surprisingly, that curious feature of the real
world, the compulsory recirement age, may have some optimality
properties.
Secondly, since the lifetime utility of a man who loses his
ability at
s
is
is borne by the
v(s) - (T - s)b,
sufferer,
the entire cost of ill-health
but this is the only source of difference
among the utilities of different individuals.
If
b
were zero,
everyone would have the same lifetime utility, while the marginal
utility of income consumption in different periods would vary from
individual to individual.
Insurance would be perfect in a naive
sense, not the economist.
sense.
's
Thirdly, it may seem curious that the solution is independent
of
b,
that is, independent of the effect of ill-health.
is that marginal
The reason
utilities are, by assumption, unaffected by the
state of health of the consumer.
difference between
u
and
In the more general
varies with
u.
c
,
case where the
the manner of
dependence does affect the optimum.
Fourthly, it is worth summarizing the basic features of the
optimum that emerge from the diagram:
U2(C2)
^''
u,(c,)
(79)
Thus an individual would always be immediately better off if he
retired.
He is prepared to continue working because of improved
retirement benefits with additional work.
It may be helpful
consider why one would not want
Starting from such a
u^ ~ "i*
to
- 36 -
situation, an increase in
decrease in
at
c
retirement date, financed by a
earlier, increases expected utility from planning
c
transfers consumption to the state
retirement at that date, since
it
with higher marginal utility.
Thus work is not discouraged, and
The argument works at
utility increased.
r,
and therefore also at
earlier dates.
We also have
u^Cc^)
>
uj(C|)
(80)
Thus the individual would prefer more insurance.
the extent of social
Moral hazard keeps
insurance down.
We recall also the basic intertemporal facts, that
c«
and
c
are different iable functions of time, and that
dc
^>0
(81)
-7-^ ^
at
(82)
There is no difficulty in principle about calculating optimal
policies.
First the terminal point of the optimal path has to be
found, by solving simultaneously
u!
= u'
and
h, - h
=
are calculated backwards from there with various values of
each case,
J
(z)
is
a
Paths
.
r.
In
calculated, and we have the solution for that
particular value of the resource constraint.
procedure with
1
simple case.
We illustrate the
- 37 -
Example
= log c,
u.
u„ = a + log c,
=1,
f(s)
It
and
is
readily checked that Theorems
c.,
3
1
and
4
^2
t)—
c
-
(I
- t)C|
The terminal point is given by
Cjdog
= c„
c.
-
Cj
=
1)
-
=
log
—
"l
+
In terms of
apply.
the differential equations in Theorem
(1
so that
T =
a >
3
c
are
a
c
= Cj
- C2
= c„
c.
c^dog
c^
and
a -
+
+
1)
= 0,
I
1/a.
To solve the differential equations, we define
—
^2
*^1
w = log
.
,
m
=
log
(1
- t)
-
log
- r)
(1
In terms of these variables, we have
dw
dm
-r—
=e ""W -w+a-1,
w =
^^^2
—— =c(w-a),
dm
z
I
c.=—
2
a
m
at
at
m
=
=
Solution of these equations gives the optimal path for any value of
r.
For each
r
we then compute
little manipulation,
to be
J
(z), which is
found, after a
- 38 -
J
(z)
=
;
o
r
-
(1
t)(c
„
=
(1
In this case,
+
c„ -
it
a
-log(l-r)
- r)^[ /
then,
is
^^
-^-
l)dc +
Z
\
-„
"^
(c- + c-e
9„,
-
1
De'^^dm
+ -]
relatively easy- to obtain the solution for
different resource constraints, since only one solution of the
differential equations
In table
is
required.
we give solutions for
I
12
a
=
.5,
of the government net subsidy to the scheme,
Table II displays solutions for
of work,
a =
1
,
a
and three values
-A,
.5,
0,
case with much greater disutility
and the same values of the net subsidy.
be noted that in table
I,
c.
-.2.
and
and
are equal to
c„
2
It
should
at
r.
This is a very high figure relative to the wage, which is unity.
rather implausible assumption about the disutility of work is
required to reduce
c
(r)
and
c-(r)
below unity.
But in all
cases people receive consumption during most of working life that
is less
than the wage plus the net subsidy, and only a very small
proportion of the population obtain
a
retirement benefit that is
greater than wage plus net subsidy.
Two other features of the numerical results deserve comment.
In all
to
cases
c
-
c^
increases with
t
until
t
is
very close
r.
The second feature is that in table
when there is
a
I,
very large government deficit
r
is
nearly
(A = -.5).
1,
even
Thus
the direct effect of moral hazard, arising from the consumer's
ability to retire healthy without special penalty, is almost
A
- 39 -
Table
a =
-.5
.3
1.00
1.03
1.05
1.09
.4
1. 13
.5
.6
1
.7
I
.1
.2
.24
.32
.9
.95
.99
.77
.78
.50
.81
.53
.54
.56
.59
.62
.66
.73
.86
1.01
1.02
1.45
1.69
1.92
.8
.5
1.
12
1.34
.64
1
.38
.39
.40
.42
.43
.45
.47
.51
.2
.23
.24
.24
.25
.26
.27
.28
.31
.32
.33
.34
.35
.37
.40
.44
.51
.969
=
.30
.56
.66
.77
1.13
1.46
r =
A
A =
.83
.86
.90
.95
1.17
I
.31
.33
.39
.46
.67
.51
.60
.88
=
.000
(more accurat
.998
r
-
1
;
1
1
.8
X
10
Table 11
a =
_
A_
=
A_ = .2
^1
.87
.1
.2
.3
.4
.90
.93
.96
.99
.59
.63
.69
.75
.85
.5
.6
'2
.49
.52
.54
.57
.7
.8
.91
.30
.33
.35
.37
.40
.44
.50
.59
.78
.96
.51
.65
.9
=
.494
r
==
.17
.18
.19
.20
.31
.95
.99
r
^2
'^l
.28
.30
.32
.34
.36
,39
.44
.61
.66
-72
.80
'
.882
r
!
.21
.23
.25
.29
.35
.49
.76
=
.963
-
40 -
entirely eliminated: hardly anyone retires early.
outcome
Nevertheless the
far from the first-best optimum that would be achievable
is
if individuals could be
prevented from retiring except through ill-
health; for in that case consumption would be
a
constant independent
of age and state of health.
7
.
Private Saving
As in the simpler models, we can ask whether individuals would
attempt to alter their consumption plans
tiiey
if
thought they
could lend or borrow at the zero interest rate which the government
Calculating the gain from savings, we shall show that
faces.
individuals would choose to save.
A consumer working at date
date
t'
wants to lend for repayment at
t
if
t'
u|(C|(t))
<
[uJ(C|(t'))(l - F(t'))
+
/u^(c2(s))f(s)ds]/[l - F(t)]
When the opposite inequality holds, the consumer would choose to
lend for repayment at date
Dividing by
t'
-
t
t'.
and letting
t'
-*
t,
we find that the
consumer wants to save if
^;(c,)
>r^(u;
- u-)
(83)
Only if there were equality here would the individual be content
with the saving being done on his behalf by the government.
Reference to Theorem
3
shows that the optimal policy does not
eliminate incentives to save or dissave; for in the optimum.
- 41
dt u
1
-
- F u
u
Now a simple calculation shows that
,
^"1
1,
,,2,1
^"P ^Tr^-^)
-
—
"P^
^^
,
*^i -^2
,
(85)
"2
i
Applying this inequality to (84), we obtain
^;
>i4t(u; -u')
(86)
This proves that the consumer wants to lend, that is, to save.
The
government can prevent this happening by suitable taxation of saving.
It
is
interesting to enquire what the government should do if
it has chosen,
market.
or is constrained,
to allow a perfect untaxed capital
Then in equilibrium
^u;
-^(ul
=
-
up
(87)
and it is also the case that
uj
«=
u^
at
r.
(88)
This last equality holds because of the possibility of saving for
retirement, or borrowing against retirement benefits from the period
when one knows one will be retired.
We conjecture that the third-best optimum, when private saving
is unconstrained,
is
defined by (87),
the consumer
retiring age.
be
(88)
and the condition that
indifferent about
We have not been able to prove this.
- 42 -
The Magnitude of Transfers
8.
An insurance scheme
said to be actuarially fair if variations in
is
the date of retirement have no effect upon the net discounted
transfers to the individual.
The optimal social insurance scheme
The net
we have derived is not in this sense actuarially fair.
discounted transfer to an individual retiring at
notation,
Theorem
(see equation
z(s)
For
5
<
s
r,
in our
(33)).
is
z
is,
s
decreasing function of
a
s
when
That is, those forced to retire
social insurance is optimal-
earlier receive larger net transfers from the government.
Proof
Since
z(s)
/C|(t)dt
=
C2(s)(T -
+
s)
- s,
o
z' (s)
=
Cj(s) - C2(s)
By Theorem 4,
r,
z'
=
of
(89)
0,
,
the right hand side of
and negative on the
point.
u^
2
1
'
^
+
u'
c^(s){T -
The situation is shown in Figure
(c
,c„)
obtaining
u..
1,
(89)
I
is
zero at the optimal
7,
where the curve labelled
at which the right hand side
vanishes, is shown sloping downwards and to the right from
differentiate the right hand side of
=
-
(89)
the terminal point of the optimal path.
u
-
curve to the left of the terminal
= u'
which is the locus of
s)
- u
u
- c„
= c
+
-
(u„
-
u
)ul,V(u')
2
.
To justify this, we can
(89) with respect
to
c„,
This is positive above the curve
Consequently points between the curves
u'
=
u'
and
,
- A3
for which
= u
u
is
c
less than
(r)
c
all have
z'(s)
<
0.
Since the optimal path lies entii'ely in this region, the theorem is
proved.
The meaning of this result
is
that those who are unfortunate
enough to suffer disability early in life receive
a
larger net
transfer from the State than those able to work until late in life.
The optimal social
insurance scheme subsidises those who retire
early, though only to the extent compatible with maintaining
incentives to work.
The following result is of interest.
Corollary to Theorem
5
Under the optimal policy, for all
C| (t)
i.e.
-
c^Ct)
<
t
<
- r,
I
the extra consumption obtained by working
is
always less than
the marginal productivity of labour.
Proof
By the Theorem,
We have also shown that
c
9.
-
c„ -
1
<
0,
the right hand side of
u„ > u
(89)
is
on the optimal path.
negative.
Hence
as was claimed.
Social Security in the United States
Despite the level of mathematical complexity, the models we have
considered are very special.
It
would be silly to base policy on
the particular equations we have derived.
the current U.S.
There are two aspects of
publicly provided pensions which it may not be
premature to criticize on the basis of the analysis we have performed,
c,
F'
I
C/if
rt
7
- 44 -
First we shall put the models briefly in perspective.
If
individuals
are saving rationally, the incentive for work comes from the change
in their lifetime budget constraints with additional work.
they
If
are consuming their net wage, it is necessary to consider separately
the incentives which come from current wages for work and from
increased future pensions as
result of additional work.
a
they
If
are following savings rules which differ from both of these models,
the two parts of compensation matter differently, although not
necessarily in the manner we have analysed.
The U.S. population no
doubt contains individuals whose behaviour is describable by
a
wide
variety of models, and not just the fully rational model.
In the model analysed we found that a growing pension benefit
permitted
a
higher pension, relative to the wage, than would be
possible otherwise given the moral hazard problem.
However if a
pension which grows with work done is to serve as an incentive for
work, individuals need to be awaie of the relationship between
pension and work done-
l^ile
it
would be expensive,
a
greater flow
of information from the Social Security administration to individuals
Hearing retirement age about their own pension prospects might well
be worth the cost
-
Under the current U.S. system individuals receive their pension
independent of whether they continue working once they reach age
72.
Before that age there is an earnings limitation on pension
eligibility.
In terms of our notation the pa^inent of benefits
independent of retirement represents
increase in
c.,
a
large and discontinuous
the consumption enjoyed while working.
analysis we found that
a
growing level of
the growth should be continuous,
c
c
In our
was optimal but that
can be increased continuously
- 45 -
either by reducing taxes repeatedly for those continuing to work or
by paying a steadily growing fraction of pension benefits independent
of any retirement test.
15% of benefits might be paid
For example,
ac age 66 independent of retirement and 85% subject to the retirement
The former fraction would grow steadily to
test.
100%
as in individual
ages to 72.
10.
Conclusions
The main results we have obtained may be summarised by highlighting
what they would recommend if they were applicable to the real world.
Many of the results suggest practicable policy changes, and may be
worthy of study in more realistic models.
(1)
The presence of moral hazard implies the desirability of
policies that leave consumers indifferent about the date of
This conclusion might be modified in models where
.different
individuals begin with different abilities, or' probability
retirement.
distributions for the loss of earnings.
(2)
It is an essential adjunct
that capital
to
an opcimal insurance scheme
transactions by consumers be taxed.
cases, this would be
a
In most
positive tax, discouraging saving.
This conclusion might be modified if we allowed for nonrational
saving behaviour.
(3)
Post-retirement benefits show a strong tendency to
increase with the age of retirement.
(4)
Contributions co the insurance scheme should diminish with
age,
sometimes quite rapidly.
eventually.
This
is
They may become negative
not as odd as it seems, for the optimal
-
46 -
allocation can be accomplished alternatively by paying part of
benefits independent of retirement.
The Corollary to Theorem
shows that the additional consumption from working rather than
retiring immediately should never exceed the marginal product.
(5)
There are some indications that contributions diminish
more rapidly, and benefits increase more rapidly, as age
increases.
(6)
Under an optimal scheme, the incidence of retirement for
reasons other than ill-health may be very low, even when the
utility gain from healthy retirement is quite large.
this,
the form of the optimal
Despite
scheme may be very different
from a first-best redistribution between people of different
ages that ignores all incentive considerations.
(7)
An actuarially fair scheme may discourage retirement less
than an optimal scheme.
5
- A7 -
Footnotes
Financial support by the N.S.F, during the preparation of this
1.
paper is gratefully acknowledged.
Computing assistance was provided
by the Computing and Research Support Unit of the Oxford University
Social Studies Faculty.
In another paper, we consider a diverse population, whose
2.
savings opportunities are limited only by linear taxation.
The theoretical effects of social insurance on retirement have
3.
been studied by
4.
Feldstein
(1
974)
and
Sheshinski
(1
977)
.
The model under consideration is equivalent to the standard
maximization of an additive social welfare function.
The particular
distribution of characteristics assumed here has not, to our
knowledge, been investigated previously.
*
5.
For example, by preserving utility and marginal utility at
while increasing risk aversion.
u
= u„
is
By analogy to Figure 3,
c
the curve
shifted up.
equals the inverse of one plus the interest rate.
6.
R
7.
In addition it will be necessary to restate the moral hazard
constraint to allow for the combined choice of savings and labour
plans.
8.
To achieve the optimum at
D
in Figure 4,
the government must
either close the capital market, or introduce nonlinear wealth
taxation to ensure that individuals do not desire to save.
,
,
- 48 -
9.
The analysis of optimal
r
lengthy and rather technical.
is
The impatient reader can turn to Theorem
at the end of this
4
*
section,
The length and delicacy of the
which states the essential results.
argument arises pa'rtly from the difficulty of establishing that
maximum utility
is a dif f erentiable
function of
r
-
if
it
were
not, unpleasant possibilities would arise - and partly from the
difficulty of discussing the case
r
= T
directly.
feature is that, under plausible assumptions,
condition for optimal
r
a
The surprising
simple first-order
can be obtained, and defines optimal
r
uniquely.
10.
We would not expect this result without the assumption that the
marginal utility of consumption of
a
nonworker is independent of his
health.
I
11.
To put it another way, higher retirement benefits are an
incentive to work in every earlier period, the later the date the
more periods in which work
12.
If
is
encouraged.
the utility function with labour
were
y
log
c
+ log
(2 - y)
implying unit labour supply in the absence of lump sum income, the
disutility of work
is
log
(2
- 0)
- log
(2
-
I
)
= log 2 = .6931
This Cobb-Douglas model probably overstates the utility of not
working at all
A9 -
References
Feldstein, M,
1974,
,
Social security, induced retirement and aggregate
capital accumulation, Journal of Political Economy
Sheshinski,
E.
,
1977, A model of social security and retirement
decisions, mimeo.
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