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Optimal Taxation in a Stochastic Economy
A Cobb-Douglas Example
P.
Diamond, J. Helms, and J. Mirrlees*
Number 217
March 1978
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Optimal Taxation in a Stochastic Economy:
A Cobb-Douglas Example
P.
Diamond, J. Helms, and J. Mirrlees*
Number 217
March 1978
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
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http://www.archive.org/details/optimaltaxationiOOdiam
Optimal Taxation in a Stochastic Economy:
A Cobb-Douglas Example
P.
I.
Diamond, J. Helms, and J. Mirrlees*
Introduction
The presence of uncertainty about the future is a pervasive fact
that has made an extremely limited appearance in the analysis of optimal
taxation.
As a start to understanding the ways in which stochastic
economies differ from those more commonly analyzed, we have calculated
equilibria in a number of simple economies.
Almost all of these economies
are populated by individuals who maximize the expected value of a CobbDouglas utility function of first- and second-period consumption and
labor.
The individual uncertainty concerns the ability to work in the
second period.
We consider economies with individuals who all have the
same skill in the first period, who have one of two skill levels, and who
are spread continuously over an interval of skills.
In contrast we also
examine determinate economies which have the same ex post possibilities.
Into these economies we introduce linear first- and second-period
earnings taxes to finance a poll subsidy.
We also introduce both flat-
and wage-related pensions with earnings tests and calculate first-best
allocations.
Given the substitutability between first- and second-period
labor and consumption which marks the Cobb-Douglas utility function, we
find the risks are not of very great importance in the economy.
(This
is highlighted by consideration of a fixed coefficients example which
Research assistance by Y. Balcer and financial assistance from the
National Science Foundation are gratefully acknowledged.
If we had only a single period, the model would be equivalent to a
many-person certainty economy.
,
is the same as the Cobb-Douglas example under certainty.)
Because of
moral hazard (labor disincentive) problems, linear taxation is of limited
value in providing insurance.
In economies where workers have the same
ex ante skills, nonlinear taxation (as with a pension plan plus retirement
test) does noticeably better.
The presence of a range of ex ante skills in the economy introduces
the need to redistribute as well as to provide insurance.
The relative
importance of these depends on the range of skill variation relative to the
range of uncertainty about the length of working life.
When we consider
the case in which the range of skills is much larger (on the grounds that
with very short working lives disability programs generally attempt to
verify incapacity rather than simply to pay benefits to all nonworkers)
the need for redistribution becomes the dominant factor in the design of
the tax system.
The value of adding. a pension system to an economy with
optimal linear taxation depends on the extent to which adjustment of the
linear tax system is permitted for those potentially eligible for benefits
under the public pension plan.
Reducing earnings taxation enhances the
ability of a pension plan to increase social welfare.
II.
Consumer Choice
When an individual is subjected to uncertainty regarding the length of
his working life, he would like to insure this risk (assuming that he is
risk averse)
.
If private companies do not offer this insurance, the gov-
ernment can provide a form of insurance through the tax and social insurance systems.
In an economy of individuals who are
(
ex ante) identical,
the government can tax future income (which is subject to random variations
for a given individual) and give each individual the expected value of his
tax payments (provided, as we assume, that there is no aggregate risk)
an economy of individuals who differ ex ante as well as ex post
,
In
.
a govern-
ment tax scheme can both redistribute income and cushion uncertainty.
To
analyze these two pieces separately, we start by asking what size insurance
policy the individual would wish to purchase or, equivalently, what tax
rate would maximize expected utility in an economy of identical consumers.
The consumer problem is described as the choice of first- and secondperiod consumption and labor, with uncertainty about length of working
life modeled in the following way:
in the first period the individual
has a marginal product n for each unit of labor worked.
We assume that
with probability 1-p the worker will have the same marginal product in
the second period, but that with probability p the consumer will be
unable to work at all.
Although second-period decisions are allowed to
be contingent on whether the consumer is able to work in that period,
first-period decisions must be made under uncertainty.
In the current paper we assume that individuals make choices consistent
with the correct optimization of their lifetime utility functions.
None-
theless, it must be recognized that individual myopia, misperceptions,
or poor planning constitute important justifications for social insurance
programs.
A subsequent paper will deal with this issue explicitly.
We assume that no direct attempt is made to avoid the moral hazard
problem of individuals having zero earnings because they choose not to
work rather than having no earning ability.
In particular, we allow the
government to observe total earnings, but not wage rates or hours worked,
and no test is made of disability as a condition for providing benefits.
2
Thus the tax on future earnings provides a disincentive to work in addition
to providing insurance against variability in skill.
Since future labor
supply decisions are being distorted, optimal tax policy will, in general,
require taxing first- as well as second-period labor income.
3
We take an individual's wage rate and the return to savings to be
fixed in terms of the consumption good.
We also assume that work and
consumption choices in the first period in no way affect the marginal
product of labor in the second period or the probability distribution of
the length of working life.
Thus, specifically, we exclude both the possi-
bility of investment in human capital and occupational hazards which
might be associated with first-period work.
not unrelated, however.
The two periods of life are
An individual's savings is a carry-over from the
first to the second period.
Also, since a social insurance system differs
from annual income taxation by relating benefits to past earnings, an
Cf. Kesselman (1976).
2
This form of model more closely resembles retirement than disability
where governments attempt to have medical examinations.
(See Diamond and
Taxation of savings will also be generally desirable.
Mirrlees (1971).) We do not pursue this line of inquiry, examining only
labor income taxation.
individual
'
s
earnings history will be a carry-over when we consider a
model of social insurance.
Below, we will also analyze a case where
preferences are not additively separable over time.
This introduces a
further element of interdependence between decisions in the two periods.
Individuals are characterized by two parameters
product of labor ("skill")
,
which is n in period one,
—
4
the marginal
and the probability,
1-p, that the worker will have the same marginal product in the second
period.
product.
The only alternative we allow in period two is a zero marginal
Let us denote by
x..
,
x_
,
and x~
the individual's level of
consumption in the first period, in the second period if he is lucky (i.e.,
has the same skill), and in the second period if he is unlucky (i.e., has
no earning ability)
.
We denote by y 1 and y„
the amount of work done in
the first period and in the second period if he is lucky.
Work is measured
as the ratio of hours worked to total hours available in the relevant
period; y
and y„
are thus bounded between zero and one.
malized such that earnings are ny 1 and ny_
Wages are nor-
in the two periods, and con-
sumption is measured in terms of consumption expenditures.
In the simulation results presented below, we begin by adopting
a particular form of the Cobb-Douglas utility function, so that expected
utility can be written as
Stern (1976, p. 127) has noted that in the standard one-period nonstochastic model the assumption of a fixed wage does not necessarily imply
Rather,
that the economy is characterized by constant returns to scale.
the wage rate can be regarded as the marginal product of labor in the
neighborhood of the optimum, with the government receiving any profits
as lump-sum income.
However, in our model the level of production will
differ between periods, and we will make comparisons with alternative
economies with different production levels altogether. Hence, the production technology in our model must be regarded as exhibiting fixed wages.
Note that we have implicitly made the assumption that the disutility
of the health or other problem which is responsible for cutting short
the individual's working life is additively separable from the remainder
of the utility function and can thus be ignored in the optimisation.
,
u = ln(x
1
+ ln(l-
)
yi
)
+ (1-p) ln(x
2L
)
(1)
+
p ln(x
2u
)
+ (1-p) ln(l-y
2L )
A Cobb-Douglas formulation was used by Fair (1971) and for the simulation
experiments in Mirrlees (1971).
However, the restrictiveness of this
formulation is well known; in particular, a unitary elasticity of sub-
stitution between labor and leisure is implausibly high.
Stern (1976)
for example, estimates this elasticity to be on the order of 0.4.
More-
over, only with an elasticity of substitution which is less than one can
a backward-bending supply curve for labor be observed.
and Feldstein (1973)
,
Kesselman (1976)
as well as Stern, use constant elasticity of sub-
stitution (CES) utility functions; in general, the resultant optimal
tax rates are quite sensitive to the elasticity of substitution.
For
the case at hand, however, we do not generalize to the CES utility function,
largely for computational reasons.
The consumer faces linear income taxation in each period and a poll
subsidy.
Since there is a perfect capital market, the timing of the
payment of the subsidy is not essential.
paid in the first period.
We model the subsidy as being
He is thus subject to two budget constraints,
one for each state of nature:
X
X
2U
2L
= I(S +
= I(S +
= x
2u
^"V
11
?! " *i>
^"V"?!
+ d-t )ny
2
" x
i>
+ (1_t )ny
2
2L
(2)
.
2L
In these equations s is the poll subsidy which each individual receives
in period one, t. is the rate of proportional income taxation in period i,
For computations of optimal social security without uncertainty, see
Sheshinski (1977).
.
and I is one plus the interest rate which the consumer faces.
Consumption
when the consumer is unlucky equals the poll subsidy plus compounded savings.
Consumption when the consumer is lucky exceeds consumption when
he is unlucky by net second-period earnings.
Demand equations are de-
rived from the maximization of (1) subject to (2)
.
III. Stochastic Economy with Identical Individuals
We first consider an economy of identical consumers, in which societal
well-being can be measured by the expected utility of a representative
consumer.
Rather than using expected utility as the objective function,
however, we report on a monotone transform of expected utility, w = exp(%E(u))
In this way, social welfare is measured as the level of consumption which
would give the same expected utility, provided that there were no work
and equal consumption in each period.
Assuming that the risks to which
the consumers are subjected are uncorrelated (and that we have a continuum
of consumers) so that we have a societal realization of the probability
distribution, the government chooses tax rates and the poll subsidy to
maximize social welfare (the representative consumer's transformed expected
utility function) subject to consumer demand equations and the budget (or,
equivalently, resource) constraint
Is = It
+ (l-p)t y
+ IG,
2 2L
lYl
(3)
where G is the lump sum grant provided from outside the system or (if
negative) the government revenue requirements.
In this economy, maximizing
welfare is equivalent to asking what linear taxes a single consumer would
subject himself to.
of n.
2
When G is zero the optimal tax rates are independent
Thus the desired level of insurance is independent of skill when
each consumer is on a breakeven basis.
An iterative search procedure is employed to calculate the optimal
values for
s,
t
,
and
t
.
Specifically, we conduct a grid search in
This measurement of welfare parallels that of Atkinson (1970)
o
Individual labor supply functions are homogeneous of degree zero in n
and s and demand functions are homogeneous of degree one, as is the exponentiated expected utility function. Net government expenditures are homogeneous
Thus proportional changes in n and G induce
of degree one in n, G, and s.
in
optimal
s and no change in optimal t.
proportional
change
a
(t
,t„)
For each gridpoint it is necessary to calculate the optimal
space.
subsidy level, subject of course to the government budget constraint.
Fortunately, since utility is increasing in s for fixed tax rates and,
with positive tax rates, tax revenue is decreasing in
a superior good)
,
government budget.
s
(leisure being
there is a unique (optimal) subsidy which balances the
This is illustrated in Figure 1 for a particular
set of parameter values with equal tax rates in the two periods.
The
frontier of the feasible set gives the optimal subsidy given the tax
rates, and the point of tangency of the consumer's indifference curve
and the frontier fixes the optimal tax rates.
Figure
curves in
(
2 is a
t
1
»t
2
)
sketch of a representative consumer's indifference
space when p = 1/3, n = .5, G = 0, and
I
= 1.25.
Calculating the best tax rate to two decimal places, the optimum occurs
at t
= .04, t_ = .17.
4
Thus, just as low tax rates have been found to
be desirable in the Cobb-Douglas case for a one-period economy with optimal
redistribution, we have found that low rates are optimal when the goal
is to provide pure insurance in our stochastic economy.
Note that social
welfare as a function of the tax rates is quite flat in the vicinity of
the optimum.
In fact, the introduction of optimal linear taxes increases
social welfare by only 0.6 percent, and raising the taxes to twice their
optimal levels results in a level of social welfare just 1.2 percent
below the optimum.
Social welfare does, however, fall off rather sharply
3
To facilitate comparison of results presented throughout this paper
for various economies, we present results principally for economies
where the mean skill level is .5, p = 1/3, and I = 1.25 for a government
which has no outside revenue requirements or sources.
At lower interest rates taxes tend to be relatively higher in period
The ratio of output in period two to output in period one increases,
total
output and the qualitative nature of the results remain essenbut
tially unchanged.
two.
10
.40n
.36n
•
32n
.28n
.24n
.20n
.16n
.12n
,08n
.OAn
r
.1
Figure
1.
.2
.3
.4
.5
.6
.7
.8
.9
Consumer indifference curves and feasible tax/subsidy
combinations for a stochastic economy of identical
individuals with t. - t-, p <* 1/3, I = 1.25.
=
t
11
t2
075
.1073
.1066
.1052
.1027
.0988
.0925
.0828
.0674
.0425
.8
075
.1073
.1066
.1052
.1027
.0988
.0925
0828
.0692
.0472
.7
1075
.1073
.1066
.1052
.1027
0988
.0942
0872
V0737
.0490
.6
0991
\.0905
.0755
.0503
.5
1014
.0919
.0764
.0514
1023
.0925
.0769
.0522
1026
.0926
.0771
.0527
1023
.0924
.0771
.0530
1018
.0920
.0769
.0532
.1010
.0913
.0765
.0531
.9
i
\
/
1166
.1164
.1156
.1141/
.1115/
.1073
7
Figure
2.
Social indifference curves in (t^j.tj)
space for an economy of identical
consumers.
Numbers given are social
welfare levels for n = .5, p = 1/3,
G = 0.
.8
.9
12
for tax rates in excess of those which maximize gross government revenue,
as is suggested by Figure 1.
The fact that the optimum occurs where tax rates (and so deadweight
burdens) are still quite low suggests that the moral hazard problem makes
the insurance gain from linear taxation relatively small.
To demonstrate
that this is indeed so, we consider the analogous first-best maximization
problem where the government can distinguish those who are able to work.
In the first-best economy we can consider the government as selecting
the levels of consumption and labor to maximize social welfare subject
only to the resource constraint
I Xl
+ (l-p)x
2L
+ px
2u
= Iny
1
+ (l-p)ny
2L
+ IG.
(This constraint is equivalent to the budget constraint,
(4)
(3).)
The
first-best optimum, the second-best optimum, and the solution to the
consumer problem without government intervention are presented in Table
1.
Note that optimal linear taxation captures only 13.9 percent of the
potential gains from perfect insurance.
From the method of measuring
social welfare, we can interpret this figure in terms of economies with
equally distributed consumption.
We have also calculated another measure
by dividing the welfare difference by the marginal utility of resources
in the hands of the government at the point of no government intervention.
We can then compare the additional resources needed to achieve the same
welfare gain with the present discounted value of output in the economy.
Below we will consider piecewise-linear taxation, with the moral hazard
problem still present.
This Lagrangian is relatively constant over the range of budget differences we are considering.
13
No
Government
Intervention
Optimal
Linear
Taxes
First-best
Optimum
189
186
.209
,326
.291
.261
,152
,167
261
622
.612
.582
.348
299
,477
311
.306
,291
Output per capita
in period two
.116
,100
159
Present value of
total output
,404
,386
,418
Aggregate firstperiod savings
per capita
,122
,120
082
,1174
1218
2L
2U
2L
Output per capita
in period one
w
Table
1167
1.
Description of the stochastic economy of
identical individuals with p = 1/3, n = .5,
I
= 1.25, and G = 0.
14
In the case at hand, the benefits of perfect insurance are equal to the
increase in welfare that would result from an increase in the government's
budget (G) amounting to 3.5 percent of total output, while the gains
available in the second-best economy are equivalent to a 0.49 percent
increase in output.
It is interesting to compare resource allocations in the three
economies.
Introducing perfect insurance decreases both first-period
output and aggregate savings.
The latter decrease is most striking, with
aggregate savings falling by one-third when first-best insurance is introduced into the economy with no government intervention.
In the second
period the first-best optimum has more work than in the absence of insurance, but the second-best economy has less work.
The great fall in
savings in the first-best economy (compared to no intervention) makes work
more valuable and sharply increases work done in the second period,
stated alternatively, without government intervention Insurance is pro-
vided by large savings.
When the lucky state occurs, the high level of
accumulated wealth yields a smaller incentive for work.
In the second-best
economy, savings fall only slightly relative to no intervention and labor
supply is discouraged by the tax rate.
rises, in the second period.
Thus output falls, rather than
15
IV.
Determinate Economy with Identically Skilled Individuals
We have examined the taxes to which a single individual would choose
To highlight the effects of the stochastic nature
to subject himself.
of the length of working life, we now analyze an economy which has the
same ex post structure as the stochastic economy, but in which every
individual knows whether he will be lucky or unlucky.
That is, all in-
dividuals have n equal to .5 but 1/3 have zero marginal product in the
second period and 2/3 have the same wage as in period one.
The lucky
consumers thus maximize
u
= ln(x
L
1L
+ ln(l-y
)
1L
)
+ ln(x
2L
)
+ ln(l-y
2L )
(5)
subject to the single budget constraint
X
2L
= I(S +
(
1-t
i>
ny
l
- x
+ (1-t )ny
iL )
2
2L
(6)
while the unlucky consumers maximize
Uu = ln( X;LU ) + ln(l-y 1TJ ) + ln(x 2u )
(7)
d-t^ny^
(8)
subject to
x
2u
= I(s
+
- x
1TJ
)
If the fraction p of the consumers are unlucky,
the social choice problem
is to maximize the social welfare function
v = exp OsCd-p)^ + puy))
(9)
subject to the budget constraint
s
= It ((l-p)y
1
lL
+ Pyiu ) +
t
2
(l-p)y
2L
+
IG.
(10)
16
Given the additive structure of preferences, the first-best optimum
for this determinate economy is the same as that in the stochastic economy
considered above.
The second-best economies are different, however.
The average of the first-period labor supply functions of the lucky and
the unlucky does not equal the labor supply of the individual subject to
uncertainty.
Similarly, average savings decisions are different.
As a
consequence of savings differences between the determinate lucky and those
in the stochastic economy who turn out to be lucky, labor supply functions in the second period are also different.
Thus the constraints
imposed on a planner by the use of markets and the freedom of individual
choice are different.
In the absence of government intervention it is clear that an economy
with the stochastic structure we have assumed cannot have a higher level
of social welfare than the corresponding determinate economy.
It is
interesting to note that this need not be the case at given tax rates
when the government is intervening.
g
Because of the presence of un-
certainty and risk aversion, a given (t ,t_) tax pair may raise more
revenue (i.e., allow a greater lump-sum subsidy) in the stochastic economy
than in the analogous determinate economy.
Since risk aversion leads
the stochastic economy to have generally higher (lower) output levels in
the first (second) period, this will generally happen when
relative to t„, and not in the neighborhood of the optimum.
t..
is large
However,
within the region where this does happen there may be tax combinations
= x.
= y^
and y..
The stochastic problem requires x
insurance/redistribution satisfies these constraints.
and perfect
Q
This corresponds to the potential for welfare gain when the government
introduces uncertainty in Stiglitz (1976) and Weiss (1976).
17
for which the stochastic economy will have higher social welfare than
will its determinate counterpart, as can be seen from Figure
3.
In our
calculations, the determinate economy has a higher level of second-best
optimal welfare.
Comparing stochastic and determinate economies, we will consider,
in turn, differences in optimal tax rates, consumer behavior, and social
welfare.
In contrast to the stochastic economy, which has optimal taxes
of 4 and 17 percent for our illustrative parameter set, the determinate
economy has a tax of 8 percent in period two and no tax in period one.
The absence of uncertainty leaves social welfare even less sensitive to
changes in the tax rates:
lowering the taxes to zero or doubling them
results in a loss in social welfare which amounts to only 0.14 percent.
And even raising the taxes to
of less than 1 percent.
t
= t
= 0. 2 brings about a welfare loss
More revealing is a comparison of individual
behavior in the two economies.
Table 1, which summarizes behavior in the stochastic economy,
can be compared directly to Table
determinate economy.
2,
which describes the corresponding
In the absence of government intervention, the
unlucky in the determinate economy work more in the first period (.667)
than do the lucky (.550), due to differing anticipations regarding second-
period earning ability.
In the stochastic economy anticipations and
therefore first-period labor supply are necessarily the same for all
individuals; however, first-period production is greater in the presence
of uncertainty.
Similarly, the lucky consume more in the first period
(.225) than do the unlucky (.167) in the determinate economy, but average
consumption is smaller in the stochastic economy.
After the introduction
of optimal linear taxes, the effect of income variability is lessened in
both economies.
18
"2
o
>— optimum for
• ^ J
^
•
<—*
stochastic
economy
optimum for
determinate
economy
O
o
o
<g>
O
o
o
s>
®
o
o
o
o
®
s>
o
<8>
<8>
o
.6
.7
.8
o
o
o
.1
.2
.3
.4
3.
8
o
o
Figure
o
.5
Comparison of the stochastic economy with n = .5, p
1/3,
I = 1.25, and G •
with the analogous determinate economy.
O
-
grid point where tax revenue raised in the stochastic economy
exceeds that raised in the determinate economy.
^)
-
grid point where the stochastic economy has a higher level of
social welfare than does the determinate economy.
.9
19
No
Government
Intervention
Optimal
Linear
Taxes
First-best
Optimum
225
219
209
.167
,170
209
.281
.274
261
208
,212
261
.550
.562
582
.667
.661
582
.438
.404
,477
Output per capita
in period one
.294
.297
291
Output per capita
in period two
.146
,135
159
Present value of
total output
411
,405
,418
Aggregate firstperiod savings
per capita
.089
095
082
1L
1U
V
x
2L
2U
'1L
1U
2L
w
Table
1199
2.
.1200
1218
Description of the determinate economy of individuals
with skill level n = .5, 2/3 of whom are lucky and
1/3 of whom are unlucky, with I = 1.25 and G = 0.
20
These differences in first-period labor supply and savings functions,
which are attributable to uncertainty, result in the creation of different
implicit maximization problems for consumers in the second period.
The
second period presents a simple one-period, two-commodity maximization
problem with a certain tax rate t„ and a lump-sum subsidy equal to firstperiod savings, multiplied by the interest factor.
Without government
taxation, the individual who has been subjected to uncertainty has saved
so much (.122)
to reduce his risk that he chooses to work only .347,
while the (lucky) man who has not been subjected to risk has saved much
less (.050) in anticipation of working .437 in the second period.
Sim-
ilarly, a comparison of the lucky in the two economies reveals that the man
subject to risk has lower first-period and higher second-period consumption, while for the unlucky man the reverse is true.
The same general
relationships hold after the introduction of the linear income tax.
But,
in addition, the disincentives of taxation result in a decrease in total
output (especially in the stochastic economy where output declines by 4.4
percent) and a shift from second-period production to first-period production.
This shift in production results in an increase in aggregate
savings from the introduction of optimal linear taxes.
In contrast, the
first-best economy has less savings, having more production in the second
period and less in the first.
These differences in behavior which we have noted result in smaller
differences in social welfare.
The magnitude of the difference is partially
a consequence of the Cobb-Douglas utility function which allows a great
deal of substitution.
Nonetheless, welfare comparisons between the two
economies do show noticeable differences from the presence of uncertainty
in the economy.
Without government intervention, social welfare falls
21
only 1.5 percent short of the first-best optimum in the determinate economy
a gap equivalent to 1.3 percent of total output.
economy the gap is almost three times as large.
9
In the stochastic
And while linear taxation
is relatively ineffectual in curtailing the welfare loss in the stochastic
economy
—
14 percent of the potential (first-best) gains are actually
realized given linear taxation, it leads to even less improvement
8
percent of a much lower potential gain
—
—
in the determinate economy.
Thus, insuring the length of working life and redistributing between
individuals with different working lives have noticeable differences.
An alternative way of thinking about the differences between stochastic
and determinate economies, as described in Tables 1 and 2, is to consider
changes in information, holding constant the extent of government intervention.
Thus, when there is no government intervention, an innovation
which permitted the lucky and the unlucky to identify themselves before
making first-period decisions would increase social welfare by 2.7 percent,
which is 63 percent of the total improvement from introducing information
and the first-best allocation.
This welfare increase is accompanied by a
1.7 percent increase in the present discounted value of output.
With
output going down in the first period and up in the second period, aggregate savings fall by 27 percent.
Q
As noted above, the comparison with total output comes from dividing the
welfare difference by the marginal value of resources to the economy.
—
22
V.
Stochastic Economy with Diverse Skills
We proceed now to a discussion of an economy in which there is un-
certainty about the length of working life and different consumers have
different first-period skill levels.
Thus we introduce distributional as
well as insurance considerations into the model.
In this section and the
next we will consider economies with just two skill classes.
Then we will
introduce a continuum of skills to examine cross-section patterns.
In both
cases we will consider economies where the mean skill level is .5.
Consumer demand functions are determined as in the stochastic economy
specified above, and the social choice problem is one of choosing tax
rates and the subsidy to maximize social welfare,
w = exp(JsJV(n;
,t
t
]L
2
,s)dF(n))
(11)
subject to the budget constraint
Is =
Itj/y^n;
t
1
,s)dF(n) +
,t
2
(12)
(l-p)t /y
2
2L
(n;
t 1> t
,s)dF(n) + IG
2
where F(n) is a distribution function giving the fraction of consumers
in the economy with skill levels no higher than n, v(n;
t
,t„,s)
is the
maximum expected utility which a consumer with skill n can attain when
taxes and the subsidy are as indicated, G is the per capita grant to the
system, and y n and y„
of identical consumers,
tax rates.
are the labor supply functions.
As in the case
scale changes in n and G do not affect the optimal
Since the lucky workers have approximately twice the poten-
tial income of the unlucky, we start with the case where the highly skilled
have about twice the marginal product of the lesser skilled.
Very short
23
working lives tend to be covered by disability insurance which attempts
to evaluate loss of earning ability and so mitigate the moral hazard
problem.
Thus the model with a diversity of skills may be more representa-
tive of public tax transfer programs which do not attempt to measure
ability.
Consider the economy where half the population has skill .607,
while the other half has skill .393.
This choice of parameters at which
to begin analysis was made to balance the importance of redistribution
and insurance, in a sense to be described below.
We shall also examine
optimal taxes as we vary the skills, preserving the mean skill level.
In Table 3 we describe the economy in the absence of government intervention.
Everyone works the same amount in the first period, with the more
highly skilled having proportionately higher consumption than the lesser
skilled.
The differences between the lucky and the unlucky are the same as
those in the one-consumer stochastic economy considered above.
The introduction of linear taxation gives the social welfare grid
shown in Figure
4.
The optimal taxes of .13 and .20 in the two periods can
be compared with the taxes an individual would choose for himself of .04
and .17.
Thus the need for redistribution as well as insurance raises the
desirable tax level.
To get one sense of the relative importance of in-
surance provision and redistribution in the 1.1 percent increase in social
welfare, we can examine the economy in which each skill class is taxed
separately at the same tax rates, so that each individual receives as a
lump-sum subsidy the expected value of his tax payments.
In Table 4 we
give the characteristics of the second-best economies in contrast to those
of an economy with no government intervention and the first-best optima.
Provision of insurance alone yields 54 percent of the social gain from the
provision of both insurance and redistribution by optimal linear taxes.
24
Low-skilled
Unlucky
Low-skilled
Lucky
High-skilled
Unlucky
High-skilled
Lucky
Average
No Government Intervention
x
.149
.149
.230
.230
.189
.622
.622
.622
.622
.622
.120
.256
.185
.396
.268
.0
.348
.0
.348
.232
l
y
l
x
2
y2
-4.997
E(u)
.0822
exp(' 5E(u))
-4.667
.0970
-4.130
-3.797
.1268
.1498
w = .113
Optimal Linear Taxation
x
x
y
1
x
2
y2
.145
.145
.215
.215
.575
.575
.593
.593
.132
.223
.190
.338
.0
.290
.0
.303
-4.812
E(u)
exp(^E(u))
.0902
4.630
.0988
-4.097
.1289
.5
.198
-3.882
.1436
w = .1152
First-best Optimum
x
.209
.209
.209
.209
.209
y1
.468
.468
.656
.656
.5<
x
.261
.261
.261
.261
.0
.336
.0
.570
2
y2
-3.540
E(u)
exp(^E(u))
.1703
Table
3.
3.949
.1388
-3.976
.1370
.452
-4.820
.0898
Description of the stochastic economy in which half
of the individuals have n = .393 and the other half
have n = .607, with p = 1/3, I = 1.25, and G = 0.
w = .1242
25
1050
.1053
.1051
.1041
.1020
.0984
.0925
.0803
,.0677
.0433
1050
1053
.1051
.1041
.1020
.0984
.0925
0830
.0699
.0499
1050
.1053
.1051
.1041
.1020
.0987
.0946
\.0875
.0741
.0529
1139
.1141
.1138/ .1126
.1104
.1065
.1004
.0910
.0764
.0530
\
\
.2
Figure
4.
.3
.7
Social indifference curves in (tpt2) space
for the stochastic economy in which half of
the individuals have n - .393 and the other
Numbers given are social
half have n = .607.
welfare levels for p = 1/3, I = 1.25, G = 0.
.8
.9
26
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27
We can approach the question of the relative importance of insurance
and redistribution in a different way by considering redistribution
which does not insure length of working life.
to tax income only in the first period.
One way to do this is
In the stochastic economy lucky
and unlucky individuals both work the same amount in the first period.
Thus the tax (and transfer) redistribute between skill classes but fall
equally on the lucky and on the unlucky.
period income is
9
percent.
The optimal tax rate on first-
(The economy is described in Table 4.)
This
yields only 23 percent of the potential gain from linear taxation.
We can consider a similar division of the gains of moving to the
first-best economy.
Providing insurance without redistribution can be
accomplished by optimizing separately in economies with just the higher
skilled and just the lower skilled, redistributing optimally between
the lucky and the unlucky.
We can redistribute without insuring by having
an optimal lump-sum transfer between a higher skilled and lower skilled
person and having no other intervention.
each of these moves equally important.
This economy was selected to make
It is interesting to note that with
linear taxation, the bulk of the savings decrease comes from redistribution.
With first-best tax instruments, the bulk of the much larger decline
in savings comes from the provision of insurance.
In Table 5 we relate optimal taxes to the skill mix in the economy.
The optimal tax rates are steady where it is optimal to have the lower
skilled individual not work.
This occurs when the ratio of the higher
skill level to the lower skill level exceeds approximately 4:1.
If we optimized second-period taxation, with the first-period tax set
equal to zero, social welfare would be .1148 and the tax rate would
be .8 percent.
28
Determinate Economy
Stochastic Economy
no
no
intervention
Ski
1
1
.5,
1
optima
w
Levels
w
1
1
inear taxes
intervention
optimal
w
+
w
l
.5
.1
167
.
1
inear taxes
+
f
2
.
1
174
.04
.
r?
.1199
.1200
.00
.08
.1195
.02
.09
.45,
.55
.1161
.1
169
.06
.17
.1193
.40,
.60
.1143
.1
155
.12
.20
.
175
.1
179
.08
.13
.35,
.65
.1113
.
135
.19
.23
.1:1.44
.1
157
.16
.
.30,
.70
.1069
.1112
.27
.27
.1099
.1 131
.25
.24
.25,
.75
.
1010
.1090
.34
.31
.1038
.1
107
.33
.31
.20,
.80
.0933
.1075
.41
.37
.0959
.
1090
.40
.37
.15,
.85
.0833
.1079
.56
.50
.0856
.1094
.56
.52
.10,
.90
.0700
.
1
143
.56
.50
.0719
.1
158
.56
.52
.05,
.95
.0508
.
1206
.56
.50
.0523
.1223
.56
.52
1
1
18
1
Table
5.
Social welfare with and without government intervention for stochastic
and determinate economies with various skill level pairs; p = 1/3,
= 1.25, G = 0.
I
29
VI.
Determinate Economy with Diverse Skills
Paralleling the analysis above we can consider a determinate economy
where individuals differ in both skill and length of working life.
We
can inquire into the relative importance of redistribution across these
two dimensions.
For ease of reference, we will inaccurately refer to
redistribution across individuals with different lengths of working life
as providing insurance.
In Table 6 we describe the determinate economy
under different tax regimes.
Comparing Table 6 with Table 4, we see that
the provision of information about length of working life raises social
welfare by 2.7 percent in the absence of any government intervention.
We also see that there are smaller gains to be had from tax interventions
in the determinate economy.
Table 5 reveals that optimal tax rates differ
less between the stochastic and determinate economies when there is
more ex ante inequality of skill in the population.
As above we have considered different partial optimization plans.
To
redistribute across skills without redistributing between those with different lengths of working life we have given different subsidies to those
with different lengths of working life so that the budget is balanced
within each group.
If the tax is levied only in the first period the
optimal rate is 8 percent.
Allowing taxation in both periods gives optimal
rates of 10 percent and 8 percent in the two periods.
A similar breakdown of the total gain is available for first-best
instruments.
Redistributing across skills but balancing the budget
within each group of different working lives yields a social welfare of
.1223.
Redistributing across different lengths of working life but balancing
the budget for each skill class yields a social welfare level of .1190.
s
•
^
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31
These welfare levels lie In the interval of .1171 for no intervention and
.1242 for full optimization.
It is interesting that savings fall when the
government intervenes, with the decrease being much larger when there is
just redistribution.
Comparing savings levels in Tables 4 and 6 one sees
that individual uncertainty plays a major role in the determination of
aggregate savings.
32
VII. Economies with a Continuum of Skills
To pursue comparative statics, we might allow both the level of
skill and the expected length of working life to vary continuously in
the population.
We have not considered this second dimension of variation.
Instead, we turn now to an approximation to a continuous skill distribution
with an economy with 100 skill classes uniformly spread between
and 1.
Each individual continues to have a probability of 2/3 of being able to
work in the second period.
In Figure 5 we show the social indifference contours in (t ,t_)
space when p = 1/3, G = 0, and
I =
1.25.
Redistributive factors are strong
enough in this population to have optimal taxes of .42 and .37, whereas
a single consumer would only subject himself to 4 and 17 percent tax
rates.
This compares closely with the two-skill economy where the lower
skill group has n = .20.
In Figure 6 we show the cross-section pattern of the expected present
value of pretax earnings and consumption.
(Expected utilities and equivalent
uniform consumption levels are shown in Figures 7a and 7b.)
In the
absence of government intervention consumption and earnings coincide in
lifetime terms as shown by the dotted straight line.
In contrast, in
the first-best economy, higher skilled individuals work more than lower
skilled individuals, but pay larger lump-sum taxes.
Thus while everyone
enjoys the same consumption, those with higher skill levels have lower
utility.
In contrast to the economy without intervention, optimal linear
taxes (which consist of a positive poll subsidy and positive income tax
rates) discourage work
—
with everyone having lower expected work.
There
Our objective here is to understand the properties of this stochastic
problem rather than to simulate the distribution of skills in the economy.
First-best calculations are performed using a continuous distribution
rather than a discrete approximation.
33
.0794
.0895
.0952
.0986
.1000
.0993
.0958
.0794
.0895
.0952
0986
.1000
.0993
.0958
0794
.0895
.0952
.0986
.1001
.1012
.0794
.0901
.0975/ .1026^^.1057
.1065
092
0957V .1010
\1042
.1054
Xl044
/
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.0753
.0531
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.0794
.0587
.1001
.0951
.0834
.0609
.1043
\.0978
.0847
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0850
.0621
.0846
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0840
.0614
0830
.0607
.0819
.0598
.0806
.0588
.1053/
.1007
'
.0974
.0933
.5
Figure
5.
Social indifference curves in (t^,t 2 ) space
for the stochastic economy of individuals
with uniformly distributed skill levels.
Numbers given are social welfare levels for
p = 1/3. I - 1.25, G = 0.
\
J
.
34
1.1
expected
present
value of
pretax
earnings;
/
1.0
/
no Intervention:
consumption
and earnings
.9
/
expected
present
value of
consumption
.8
/
.7
first-best optimum:
earnings
consumption
.5
.A
optimal linear taxation:
consumption
earnings
.3
.3
Figure 6.
.A
.7
.8
.9
Consumption and earnings profiles without government
intervention, with optimal linear taxation, and at
the first-best optimum.
1.0
35
E(u)
-3
-4
optimal linear taxation
-5
-6
no intervention
#
-7
.1*
,6
.7
.8
.9
1.0
9
"
-8
#
Figure 7a.
Expected utility profiles without government
intervention, with optimal linear taxation, and
at the first-best optimum.
)
36
exp
(JjE
(u)
.25
A
\
\
.20
\
\
\
optimal linear
\
.15
first-best optimum
.10
.05
"1_ no
.1
.2
Figure 7b.
.3
.4
intervention
.5
.6
.7
.8
.9
1.0
Uniform consumption equivalent of expected
utility profiles without government intervention,
with optimal linear taxation, and at the
first-best optimum.
taxation
37
is a fall in aggregate consumption and a shift in consumption toward those
with lower skills; thus expected consumption goes up for the bottom 22 percent of the skill classes while it goes down for the remaining 78 percent.
Since the diagram gives the expected value calculation, it does not present
the insurance aspects of the change in consumption.
Figures 8 and
9.
This is shown in
From these figures we see that the present discounted
value of consumption increases for 33 percent of the unlucky and for 18
percent of the lucky.
As in the two-skill-class economy we can examine separately the
insurance and redistributive potential in the economy.
dividual in the economy.
Consider an in-
He is making labor and consumption choices to
maximize his utility given that he receives the subsidy and faces the
Given his demands,
tax rates that are optimal for the economy at large.
his tax payments will exceed (fall short of) his poll subsidy by a certain
amount
—
which amount is equal to his contribution to (receipts from)
the redistribution program of the economy at large.
Now suppose instead
that he were to regard this contribution as fixed and that he could then
select any tax rates and subsidy level he desired subject to the constraint
that his tax payments still exceed his chosen subsidy by the amount of
the required contribution.
This problem is of course equivalent to asking
what tax system a single individual would subject himself to when the
grant G is set equal to the receipts from the redistribution program.
Since in the economy of identical individuals the optimal tax rates decline with G, we find here that those with higher (lower) skills would
choose higher (lower) tax rates for insurance purposes only, holding
constant the pattern of redistribution in the economy.
For example, we find that the individuals in the economy discussed
above, facing income taxes of .42 and .37, receive a poll subsidy of .13.
38
/
1.3
/
/
1.2
present
value of
pretax
earnings;
/
/
1.1
/
no intervention:
consumption and
earnings
/
1.0
/
present
value of
consumption
/
/
/
first-best optimum
earnings
.5
optimal linear taxation:
consumption
pre-tax earnings
.3
.1
.2
Figure 8.
.3
.6
.7
.8
.9
1.0
Consumption and earnings profiles without
government intervention, with optimal linear
taxation, and at the first-best optimum for
lucky individuals.
.
39
1.1
present
value of
pretax
earnings;
1.0
.9
present
value of
consumption
.8
/
.7
first-best optimum:
earnings —
consumption
.6
/""
'f
/ L
y
•
no intervention:
consumption
+
and earnings
.5
.4
.3
optimal linear taxation:
consumption
pre-tax earnings
.2
.1
.2
Figure 9.
.4
.5
.6
.7
.8
.9
1.0
Consumption and earnings profiles without
government intervention, with optimal linear
taxation, and at the first-best optimum for
unlucky individuals
40
An individual whose marginal product of labor is .23 would prefer, however,
to subject himself to tax rates of only .02 and
.07 and to have his subsidy
reduced to .10, which would result in his still receiving the same net
transfer from the rest of the economy:
.09.
Without the deadweight burden
from the redistributional system, his consumption and labor supply increase,
and the uniform consumption equivalent of his expected utility (exp(%E(u)))
increases by 6.3 percent.
Similarly, the person with n = .5, whose
contribution to the redistribution program is approximately zero, would
prefer to have his tax rates reduced to .04 and .17 which, as we have noted
above, are the optimal taxes for a single individual when he is on a break-
even basis.
Thus we find that if redistribution were accomplished through
lump-sum taxation rather than through distorting taxes, then individuals
would want considerably reduced tax rates for the provision of insurance.
The relative importance of redistributional as opposed to insurance
considerations in the economy can also be demonstrated by introducing
these two elements separately via restricted linear taxation.
provide insurance without redistribution by optimizing
t-
We can
and t„, returning
to each individual as a subsidy exactly the present discounted value of
his tax payments.
2
Because of the homogeneity properties mentioned above,
the optimal tax rates are equivalent to those found for a single skill
class.
As has been noted, these rates are low
the improvement in social welfare.
—
(See Table 7.)
.04 and
.17
—
as is
This improvement is
equivalent, in the economy with a continuum of skill levels, to an increase
in total output of 0.26 percent, which amounts to only 2.1 percent of the
benefits available from optimal linear taxation.
Note that the poll subsidy will not be the same for individuals with
different skill levels; it will be proportional to n.
41
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On the other hand, we can introduce redistribution without insurance
by not allowing transfers between states of nature; i.e., we can optimize
subject to t„ =
t
This yields a much larger tax (t. = .4) and much
0.
1
larger gains in social welfare (equivalent to a 9.8 percent increase in
output).
Thus, while this is a very restrictive way of redistributing
without insuring, it enables us to gain 79.5 percent of the available
welfare benefits of optimal linear taxation.
3
In the first-best setting,
perfect insurance alone accounts for 6.7 percent of the welfare gains
which are realized by introducing first-best insurance and redistribution
together into the stochastic economy.
On the other hand, redistribution
by itself achieves 95.4 percent of the potential welfare gains in the
first-best economy.
And while optimal linear taxation without redistribu-
tion achieves only 14 percent of the available welfare gain from perfect
insurance, optimal taxation without insurance captures 39 percent of the
(much larger) potential gain from perfect redistribution.
In combination,
fully optimal linear taxation gives 43 percent of the welfare improvement
which is obtained at the full first-best optimum.
Thus we find that the potential gains from providing insurance in
the stochastic economy are small relative to the gains from perfect
(nondis tor ting) redistribution.
Moreover, linear taxation is much less
effective in providing the achievable insurance protection than it is in
redistributing income.
3
Of course, as the inequality in earning ability in the economy is decreased, the relative importance of distributional factors decreases as
But the presence of any sizeable percentage of individuals with
well.
skill near zero does much to insure that distributional considerations
If, for example, the population is distributed uniformly
will predominate.
except that 17 percent of the population is concentrated at n = .005,
redistribution alone achieves 87.5 percent of the welfare gains from
linear taxation while insurance alone accounts for a scant 0.5 percent
of the possible improvement.
,
43
In Table
8
we repeat the analysis for the determinate analog to the
stochastic economy with uniformly distributed skill levels.
Then three
separate social choice problems can again be distinguished:
1)
we can find
the full (second-best) optimum by maximizing welfare with respect to
t„, and s subject to the aggregate resource constraint;
2)
t.
we can insure
without redistributing by returning to each consumer exactly the present
discounted value of his total tax receipts as a lump-sum subsidy; and
3)
we can redistribute without insuring by requiring that the aggregate
resource constraint hold separately for the lucky and the unlucky.
The determinate economy has approximately the same optimal tax rates,
.41 and .38, as are found for the stochastic economy.
presence of uncertainty tax revenues are larger.
insensitive to changes in the tax rates.
4
However, in the
Welfare is again rather
Welfare in the untaxed stochastic
economy falls 38.4 percent short of that in the economy with first-best
redistribution, which is equivalent to 27.2 percent of the output in
the untaxed economy.
Removing uncertainty reduces this gap slightly
to an equivalent of 26.0 percent of output, a much smaller change from
the provision of information than in the economy with a smaller spread
in skills.
Introducing taxation for redistributive purposes only into
the determinate economy
leads to the realization of approximately the same
percentage of the potential gains from first-best redistribution as in
the case of the stochastic economy
—
32 percent as compared to 36 percent.
In the economy in which individuals have different marginal products
in period one, as in the economy of identical consumers, there are some
values of t^ and t£ for which social welfare is higher in the stochastic
economy than in the economy which is identical but for the absence of
uncertainty.
To maintain comparability, we require that t2 =
as well as in the stochastic economy.
in the determinate,
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45
For many of the policy configurations savings are much higher when there
is individual uncertainty.
Thus, while the impact of uncertainty is
small relative to distributional considerations, it does play a significant
role in the optimization problem, even if insurance is not to be provided
at all.
46
VIII. Results for a More Risk Averse Consumer
The Cobb-Douglas model, being intertemporally additive, shows no
particular gain from giving consumption to those who have been accustomed
to high consumption.
But maintaining income during retirement at some
appropriate fraction of preretirement earnings is a common goal of social
insurance systems.
To see the effects of incorporation of this idea into
the specification of the utility function, we consider a polar case of
interdependence between the utility of consumption in the two periods.
Suppose that consumers desire consumption in the two periods to be
in some exact ratio, and that consumption in one period which is in
excess of the level that would be desired given consumption in the other
period provides no added utility.
u = 2(l-p) ln(min( Xl ,
We define expected utility to be
Sx^)) +
2p ln(min(x
r
Ox^))
(13)
+
where
6
InU-yp +
(1-p) ln(l-y
2L
)
- ln(6)
Geometrically,
is the inverse of the desired "replacement ratio."
indifference curves between first- and second-period consumption are right
To make our results comparable to those discussed above for the
angles.
Cobb-Douglas utility function, we will take as a value of
I
.
8
the value
Then individual behavior under certainty and the first-best optimum
will be the same as those described above for the Cobb-Douglas utility
function.
First, we ask how effectively we can insure such a consumer against
loss of skill in the second period.
Optimal taxes for the provision of
insurance alone are even smaller than in the Cobb-Douglas case
and .08
—
—
.01
and only 0.5 percent of the welfare gain that would result
47
from perfect insurance is realized.
This is shown in Table
The same
9.
risk aversion that makes insurance so desirable (the gains from firstbest Insurance are 2.7 times as great in the present case as they are
when consumers have Cobb-Douglas utility functions) leads to a work disincentive problem in the second period which almost totally precludes
the consumer from being insurable.
In contrast, recall that optimal linear
taxes give the Cobb-Douglas consumer 13.7 percent of the gain in social
welfare that would come from first-best insurance.
The role of uncertainty in affecting both savings and second-period
labor supply is very marked.
In the absence of government intervention,
second-period output is only 3.9 percent of first-period output.
analogous determinate economy the ratio is 50 percent.
In the
Savings in the
uncertain economy is 78 percent higher than in the determinate analog.
The
introduction of optimal linear taxes practically eliminates second-period
output and results in a small increase in savings.
Moving to the first-
best economy nearly halves aggregate savings and raises second-period
output to approximately half of first-period output.
Turning to the economy of consumers with uniformly distributed skill
levels, we again consider the benefits from pure insurance and from pure
redistribution.
In Table 10 we show the calculations for the economy
in which consumers have the utility function which we have been considering.
Fully optimal taxes in the economy are large
—
.43 and .23
—
but even
more so than in the Cobb-Douglas case it is redistribution that is responsible for the high tax rates as well as for the gains in social welfare.
Pure insurance yields only 0.2 percent of the gains from optimal
linear taxation, whereas pure redistribution yields 96.4 percent of the
welfare gains.
(For the Cobb-Douglas consumer these gains were 2.1 percent
48
No
Optimal
Linear
Taxes
Government
Intervention
First-best
Optimum
.174
170
209
.217
213
261
198
202
261
664
662
582
.038
.023
477
Output per capita
in period one
332
331
291
Output per capita
in period two
.013
008
159
Present value of
total output
.342
.337
,418
Aggregate firstperiod savings
per capita
158
.161
082
.1078
1218
2L
^2U
'2L
w
,1077
Table
9.
Description of the stochastic economy of
identical individuals with the utility function
described in this section, with p = 1/3, n = .5,
I
= 1.25,
and G =
0.
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50
and 79.5 percent, respectively.)
With optimal linear taxes, 10 percent of
the consumers do not work at all and an additional 23 percent of the lucky
do not work in period two.
(In the Cobb-Douglas economy these numbers are
11 percent and 3 percent.)
And expected output in period two represents
only 5 percent of the present value of expected total output (as compared
to 29 percent in the Cobb-Douglas economy)
.
Despite the greater need for
insurance here than in the Cobb-Douglas case, second-period taxes are much
lower due to the moral hazard problem.
(When
t..
= .4 and t„ is greater
than .41, there is no production at all in period two.)
Thus for a consumer who is extremely risk averse in this sense,
the benefits from first-best insurance are large, the benefits realized
from second-best insurance are small, and the moral hazard problem is
highly significant.
51
IX.
Pensions with an Earnings Test
The linear tax system has been found to be surprisingly ineffective
in providing the consumer with insurance against the contingency of loss
of earning ability in the second period.
We therefore are led to ex-
amine how much more effectively simple nonlinear taxation can transfer
income between states of nature.
We shall consider two ways of introducing
nonlinearity into second-period income taxation.
One way parallels nega-
tive income taxation for the elderly or a universal pension with an earnings
test by allowing a higher marginal tax rate for low incomes.
Secondly we
will parallel social security by giving a second-period benefit which
depends (linearly) on first-period earnings.
linearly with second-period earnings.
This benefit is reduced
Thus, again, we will have a larger
marginal tax rate on low than on high earnings, with the break point determined by the size of social security benefits and the rate of decline of
benefits with earnings.
Since either this universal pension system or this
wage-related pension system add two more parameters into the calculation,
we have not attempted to consider both at once in the calculations, although we present the model with both systems present.
Suppose a payroll tax is levied on earnings (and is merged with
the income tax in this model), and benefits, paid in the second period,
consist of a minimum benefit of a plus lb dollars for every dollar of
earnings in period one.
In addition, there may be a retirement test and
c dollars of benefits are withheld for each dollar of second-period
Because social welfare is not concave in the tax parameters, it would
have been prohibitively expensive to search the six-dimensional (t.. .t-.s.a.b.c)
space to find the full optimum.
.
,
:
52
earnings.
Total benefits are thus the bracketed amounts in the revised
consumer budget constraints
X
2U
= I ^S +
+
X
2L
(
i^
ny
~ x
i
i^
(14)
yi ]
1~ t
+ blny
1
n
~ x )
1) y1
x
- cny
2L ] +
+ (l-t 2 )ny
2L
.
refers to the maximum of the amount in brackets and zero.
]
[
[a
1_t
+ bln
= I (s +
+
where
[a
(
Note that benefits are constrained to be nonnegative.
Thus an unlucky
person consumes accumulated savings plus the social dividend plus benefits
from the pension system.
A lucky person consumes this plus earnings
net of both the income tax and the offset in benefits for second-period
earnings
The government budget constraint becomes, for the economy of identical
individuals
Is + (1-p) [a + blny
1
- cny
]
2L +
+ p(a + blny^
(15)
=
It^
+ (l-P)t y
+
2 2L
IG.
That is, the social dividend plus pension benefits to the lucky and the un-
lucky equal income tax revenue plus the grant provided from outside of
the system.
We now consider separately the two ways to introduce nonlinear taxation.
We start with the economy of identical consumers.
aspects of labor supply response in the second period.
There are two
The individual can
respond to a small change in taxes by a Hmall change In work or by ceasing
work altogether.
The latter constraint will be our sole concern when
consumers are identical.
To analyze this in isolation we will use a zero
53
marginal tax rate on second-period work.
c equal to one.
Thus we set t„ equal to zero and
This simplification will not continue once there are
diverse skills in the economy.
With the single consumer, the optimal
universal pension is an optimal pair (a,t-), while the optimal wagerelated pension is an optimal pair (b,t-).
With a universal pension, the budget constraint is It 1 ny, = pa, and so
the consumer is receiving the expected value of his tax payments as a
subsidy if he is unable to work.
With a wage-related pension,
t..
= pb.
In either case, the conditional subsidy (a or Ibny.) cannot be made any
larger than the value which would leave the worker indifferent to continuing work when lucky.
Nevertheless the two approaches are not equivalent.
If the pension which is perceived as a lump sum is replaced by a plan
in which the pension depends on the individual's earning record, the
worker perceives a further gain from working in period one than was
seen before.
However, this additional work (and additional savings)
will undercut the scheme because it would lead him to stop working when
lucky.
Thus the maximal scheme must have a smaller level of b than would
give the same pension at the level of work previously chosen.
With a
wage-related pension the individual perceives a zero marginal expected
tax on first-period work.
This avoidance of marginal distortions, however,
is achieved at the cost of being constrained to a smaller pension.
is not clear which method of insurance is preferable.
We calculated the
optimal parameter values for the two schemes in an economy with n =
1.25, p = 1/3, and G = 0.
Thus it
The results are shown in Table 11.
.5,
The earnings-
related social security program yields welfare gains which amount to 45
percent of the potential gains from first-best insurance.
I =
The universal
54
Government
Intervention
Optimal
Linear
Taxes
WageRelated
Pension
Flat
Pension
.189
.186
.198
.192
.209
.326
.291
.308
.300
.261
.152
.167
.177
.171
.261
yl
.622
.612
.613
.582
.582
y 2L
.348
.299
.384
.399
.477
output in period
one per capita
.311
.306
.307
.291
.291
output in period
two per capita
.116
.100
.128
133
159
present value of
total output per
capita
,404
.386
.409
398
.418
aggregate firstperiod savings
per capita
122
.120
.109
.099
,082
.04
.053
082
No
x
X
X
l
2L
2U
FirstBest
Optimum
.17
070
b
.159
pension
.061
.070
.1190
.1192
w
,1167
Table 11.
1174
Introducing Social Security into
a Stochastic Economy of identical
individuals with p = 1/3, n = .5,
I = 1.25, G = 0.
1218
55
pension plan fares slightly better, achieving 49 percent of the potential
welfare gains.
Thus in our example avoiding marginal distortions from
taxation is less important than providing a larger conditional subsidy.
Note that either program is a significant Improvement over the performance
of the linear tax system, which was able to capture only 13.7 percent of
the potential insurance gains.
The difference between the social welfare
under linear taxation and under either of these public pension schemes is
equivalent to approximately 1 percent of total output.
In considering the introduction of a public pension plan into the
economy characterized by a diversity of skill levels, we will ask two questions.
First, what gains in welfare can be realized by adding these programs
to the optimal linear tax system; and second, how should the underlying
tax system be modified to maximize the gains from the additional insurance
systems.
We take as our starting point the economy in which the skill parameter
n is uniformly distributed, p = 1/3,
linear taxes are .42 and .37.
I = 1.25,
and G = 0.
The optimal
If we provide a flat pension benefit a
of .015 (equal to 3.8 percent of per capita output) and subject it to an
earnings test which reduces the benefit by 15 cents for every dollar of
second-period earnings (the optimal (a,c) pair given that
t.
and t„
are maintained at .42 and .37), social welfare increases by an amount
which is equivalent to giving the government a grant equal to 0.32 percent
of total output.
(See Table 12.)
Note that the price which must be
paid for giving this subsidy to the poor (both those with low earning
abilities and those who are unlucky) is the additional deadweight burden
of inducing an additional 9 percent of those with second-period skills
to leave the labor force.
However, the upper 55 percent of the skill
5
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57
distribution
—
those who receive no benefits if lucky
affected by the decrease in
magnitude of
to the poor.
c is not
s
to finance the system.
—
are barely
We find that the
nearly as important as having the benefits confined
Indeed, raising c all the way to 1 sacrifices less than
7
percent of the gains from the introduction of the program.
Reducing the second-period tax rate
t
?
increases the gains which
may be achieved by decreasing the tendency to leave the labor force in
period two.
As is shown in Table 12, reducing t~ to .22 allows the optimal
negative income tax to yield benefits equal to an increase in output of
0.56 percent.
This is 75 percent larger than can be achieved by intro-
duction of a flat pension with no change in tax rates.
We find a similar pattern when we introduce a social security system.
Returning to the linear tax optimum of
t-
= .42 and t„ = .37, we choose
social security benefits and the payroll tax by searching over b, c, and
t
1
with pb =
(t-
- .42).
That is, we introduce a first-period payroll
tax which would just finance the pension system if all the lucky worked.
The rest of the financing comes from a decline in s.
At the optimum,
the welfare gain is equal to a 0.43 percent increase in output.
gain can be increased to 0.50 percent by lowering t„ to .32.
This
In both
cases, at the optimal c no one who works in period two receives a social
security benefit.
There is a wide range of values of c that accomplish
this end.
Thus we find that while social insurance and the negative income
tax can increase welfare by a nonnegligible amount, reduction of the tax
rate in period two is called for if benefits are to be maximized.
58
References
Atkinson, A.B., "On the Measurement of Inequality," Journal of Economic
Theory 2 (1970) 244-263.
Diamond, P. and Mirrlees, J., "A Model of Social Insurance with Variable
Retirement," Journal of Public Economics , forthcoming, 1978.
Fair, R.C., "The Optimal Distribution of Income," Quarterly Journal of
Economics 85 (1971) 551-579.
Feldstein, M.S., "On the Optimal Progressivity of the Income Tax," Journal
of Public Economics 2 (1973) 357-376.
Kesselman, J.R., "Egalitarianism of Earnings and Income Taxes," Journal
of Public Economics 5 (1976) 285-301.
Mirrlees, J. A., "An Exploration in the Theory of Optimal Income Taxation,"
Review of Economic Studies 38 (1971) 175-208.
Sheshinski, E. , A Model of Social Security and Retirement Decisions,
unpunlished, 1977.
Stern, N.H., "On the Specification of Models of Optimal Income Taxation,"
Journal of Public Economics 6 (1976) 123-162.
The Case for
Stiglitz, J.E., "Utilitarianism and Horizontal Equity:
for MathInstitute
Taxation,"
Report
Number
Random
Technical
214,
Stanford
University,
1976.
ematical Studies in the Social Sciences,
Weiss, L. , "The Desirability of Cheating Incentives and Randomness in
the Optimal Income Tax, Journal of Political Economy (1976).
,
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