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Optimal income taxation: an example with a U-shaped
pattern of optimal marginal tax rates
Peter Diamond
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
Optimal income taxation: an example with a U-shaped
pattern of optimal marginal tax rates
Peter Diamond
94-14
April 1994
M.I.T.
JUL
LIBRARIES
1 2 1994
RECEIVED
April 15,
1994
Optimal income taxation: an example with a U-shaped pattern of
optimal marginal tax rates
Peter Diamond 1
In the presence of
Income effects complicate tax analysis.
distorting taxes, income effects imply that lump-sum taxes have
efficiency effects. Thus it is instructive to examine the optimal
income tax in the Mirrlees (1971) model for the case of no income
effects 2 the case where everyone has the utility function
,
(1)
u(x,y) = x + v(l-y)
and v is
where x is consumption, y is labor (in percentage terms)
assumed to be strictly concave. 3 The only difference assumed
across individuals is a difference in skills, with an individual of
The optimal income
skill n having a marginal product egual to n.
taxation problem is the maximization of the integral over the
population of a concave function of utilities, subject to an
aggregate budget constraint and subject to the constraint that
individuals optimize in their choice of labor supply given the
relationship between work and after-tax income.
,
If, above some income level, the distribution of skills is the
exponential distribution and the utility-of-leisure function, v(l-y)
Other
is the logarithm, then the marginal tax rate is increasing.
combinations of utility function and skill distribution yielding the
same conclusion are also given, including the combination of a
constant elasticity labor supply and the Pareto distribution of
skills
With either a logarithmic utility-of-leisure or a constant
elasticity of labor supply, the marginal tax rate is falling where
the density of skills is rising and the income level is high enough
to want to transfer resources from that skill level to the
government budget. Combining this result with that above, we
have examples with a U-shaped pattern of optimal marginal taxes
I am grateful to Jon Gruber for suggesting the inclusion of data
in this paper and providing me with the figures presented below,
1
and to Jay Wilson for helpful suggestions.
I am also grateful to
the National Science Foundation for research support.
2 This assumption is probably more appropriate at high income
levels, since people at the top of the income distribution are likely
to leave large estates and may not adjust their earnings to the
exact level of estate.
3 For a discussion of this case with a constant elasticity of labor
supply, see Atkinson (1990).
The complementary case where
utility is linear in leisure has been studied; see Lollivier and
Rochet (1983)
-1-
This argument
if the density of skills first rises and then falls.
does not refer to the lowest skill levels, only to the levels of skills
from which it is desired to transfer income.
The argument for a U-shaped pattern of marginal tax rates
assumes that there is no upper bound to skill levels. As is well
known, in the presence of an upper bound the marginal rate on
the highest skill should be zero. 4 This analysis shows that the
shape of the distribution near the top is essential for determining
the pattern of taxation near the top. There is no need for tax
rates to decline slowly toward zero as one approaches the absolute
top of the skill distribution.
The absence of income effects also allows an intuitive grasp
Increasing
of the factors that determine the optimal tax structure.
some
skill
level
involves
marginal
tax
rate
affecting
an
the
increase in the deadweight burden for people at this skill level.
Thus, the optimal marginal tax rate at some income level depends
on the elasticity of labor supply at that income level, since this is
important for marginal distortions.
Increasing the marginal tax
rate also transfers income from all individuals with higher skills
to the government, without changing the distortions of their labor
supplies.
The weights on these two elements depend on the ratio
of individuals with skills above this level to individuals with
skills at this level. This intuition is displayed in the equations
below, where the first order condition for the optimal income tax
is written with a product of these three terms.
Model
We assume that a person of type n has a marginal product n
with the distribution of skills written as F(n). Thus, the ratio of
people with higher skills to those with given skills is (l-F(n) /f (n)
where f is the density associated with the distribution F.
Moreover, it is assumed that the social welfare function is the
integral of a strictly concave function of utility, G(u)
with utility
written in the form given in equation (1) and G independent of n.
Thus the objective function to be maximized is
)
,
/c(x(n) + v(l-y(n)))f (n)dn
so
with (x(n)
y(n)) chosen optimally by individuals of type n given
the tax function, T, that determines individual budget constraints,
(2)
,
(3)
x(n)
= ny(n)
-
T(ny(n))
To examine the optimal tax, we start with the optimal tax
formula for the case of the additive utility function, equation (21)
in Mirrlees, specialized for the utility function linear in
consumption.
4
Sadka (1976), Seade (1977)
-2-
(4)
p(n
-
v')f = [(v'-yv")/n][/
(p-G')dF].
where p is the Lagrange multiplier on the government's budget
constraint and the functions v and G are evaluated at the
appropriate labor supplies. We note that p is equal to the average
of G' over the entire population since consumption can be
transferred one-for-one between the government and the
Thus, the
households, provided the tranfers are equal per capita.
integral in (4) is first increasing and then decreasing as a function
of n.
It is convenient to use the marginal condition for individual
choice,
(5)
V
= (l-T')n,
where T" is the marginal tax rate, and the definition of the
elasticity of labor supply
(6)
e = -v'/yv".
Then, we can write
(7)
(4)
as
T'/(l-T') = [(e- 1 +l)/n][/(p-G')dF]/[pf].
-A
Multiplying and dividing (7) by (1-F) to turn the integral into an
average term, we can rewrite the first order condition as
(8)
Using
(9)
T'/(l-T') = [(e-i+lJ/nH
(5)
J
(p-G')dF/p(l-F)][(l-F)/f].
again, we can write it as
T'/(1"T')
2
=
[(v'-yv")/(v')
2
][f (p-G' dF/p 1-F)
)
(
]
[
(
1-F) /f
]
Rising Marginal Tax Rates
We are now prepared to examine the pattern of marginal tax
rates given by these optimality conditions, considering the levels
of skills at which labor supply is positive.
Equation (8) shows the
three terms mentioned above. The second term reflects the gain
from transferring resources from workers with skill levels above n
to the government, averaged over the people paying the higher
taxes.
The first term reflects the distortion associated with such a
change, recognizing the fact that a one dollar increase in taxes
paid requires a smaller increase in tax rates the larger the level of
skill at the income level at which taxes are increased.
The third
term gives the ratio of individuals above this skill level to
individuals at this skill level.
If, above some income level, the distribution of skills is the
exponential distribution, then the third term, the ratio (l-F)/f, is a
constant, independent of skill level.
If the utility-of-leisure v(l-y)
is the logarithm, then, where labor supply is positive, the first
-3-
is equal to 1 independent of skill level.
(v'-yv") / (v' 2
term in (9),
With these two assumptions, the marginal tax rate will vary with
This yields an increasing marginal tax rate
the remaining term.
since the social marginal utility of income is decreasing with
income
]
[
,
Thus, over the range where the distribution of skills is
exponential, marginal tax rates increase if the utility-of -leisure
function is logarithmic. The same conclusion would hold for a
distribution that goes to zero more slowly than the exponential, so
that the third term in (9) was increasing. 5 In addition, the result
follows for any utility function that showed the first term
increasing as one moved up the skill distribution (and so the level
of labor)
Another combination of assumptions yielding the same
from the
conclusion can be found by moving n, in equation (8)
first term to the third term. Then, we have rising marginal rates
where the elasticity of labor supply is constant or falling and (1The latter holds for the Pareto
F) /nf is constant or rising. 6
distribution, where the density is proportional to l/n 1+a for a>0.
,
Asymptotic Marginal Rates
With a known finite top to the skills distribution, the optimal
marginal tax rate should be zero at the top of the income
This need not imply that rates decline until very
distribution.
close to the top. 7 Thus it has been natural to consider the
behavior of the optimal marginal tax rate satisfying the conditions
above as skills rise without limit.
If the welfare weight of those
at the top tends to zero as skill rises without limit, then the tax
rate tends to the revenue maximizing rate.
This can be derived
from the equations above after noting that if G' goes to zero as n
rises without limit, then the middle term in (9) goes to 1.
If the
utility-of-leisure function is logarithmic, then the first term in (9)
is also equal to 1 and the tax rate converges to the solution to
(10)
T'/(1"T')
2
= (l-F)/f.
If the distribution of skills is exponential toward the top, (l-F)/f
is equal to 1/b, where b is the coefficient in the exponential
distribution.
(11)
Thus the tax rate converges to
1-T' = ((b 2 + 4b)
1/2
- b)/2,
The logarithmic utility function has the first term in (9) equal
1.
Varying the value of the constant equalling the first term
generates utility functions klog(l-y) for constants k, utility
functions that do not differ in this implication.
6 Hausman has estimated the elasticity of labor supply by quintile
and finds.
7 This point is also made by the numerical calculations in
Tuomala (1984)
5
to
-4-
For values of (l-F)/f, that is, 1/b, of 5, 10, and 15, (see Tables
and 2, below), the optimal marginal tax rate tends to 64%, 73%,
and 77%.
1
One can do a similar calculation for the Pareto distribution.
Assuming a constant elasticity of labor supply, e, and a Pareto
distribution with coefficient a, so that (l-F)/(nf) equals 1/a, (8)
becomes
(12)
T'/(l-T') = (e _1 +l)/a.
Thus, the the optimal tax rate coverges to
(13)
T'
=
(e
-1
+l)
/
(a+e
-1
+l)
Identifying skill with the wage yields an elasticity based on
adjusting hours of labor supply; identifying skill with an
underlying ability implies a larger elasticity since education is
For values of the elasticity of 0.2 and of a of 0.5,
also variable.
Raising a to 1.5 lowers the
the asymptotic optimal tax rate is 92%.
tax rate to 80%.° Raising the elasticity to 0.5 with a at 1.5 lowers
the tax rate to 67%.
Falling Marginal Rates
It is also interesting to consider the case of a distribution of
skills with a density that first rises and then falls.
For this
analysis we consider only the levels of skills at which G' is less
than p, so that it is desirable to transfer resources away from this
skill level (if it could be done costlessly)
We assume that G'
equals p at a skill level that is below the skill level that is the
mode of the distribution of skills.
It is now convenient to cancel
the two (1-F) terms in equations (8) and (9).
When the density is
rising and p exceeds G' the latter two terms in equations (8) and
Thus with either a constant elasticity of labor
(9) are declining.
supply or a logarithmic utility-of-leisure, the marginal tax rate
must be falling.
.
,
Similarly, one can compare tax rates at two income levels on
either side of the maximum in the density, and such that the
density is equal at the two points.
Provided that G' is less than p
at the lower of the two skill levels being compared, the marginal
tax rate would be higher at the lower income level with either a
constant elasticity of labor supply or a logarithmic utility-ofleisure.
Thus we can see the pattern of tax rates when the density of
skills is first rising and then falling (and such that the workers
8 Feenberg and Poterba (1992) use tax data to estimate the Pareto
coefficients each year from 1951 to 1990 for the distribution of
total incomes for the top 0.5% of the population.
They find
values of a that vary between 0.54 and 1.46.
-5-
with the modal skill work and have G' less than p in equilibrium)
With the logarithmic utility and an exponential distribution of
skills where the density is falling the pattern of marginal tax rates
is U-shaped above the level where G' equals p, with the minimum
of marginal rates occurring at the maximum of the density of
Since there is probably a positive density of skills at the
skills.
level of skills where labor supply is zero, it does not follow from
this argument that marginal tax rates are higher at the bottom of
The same conclusion
the income distribution than at the top.
about a U-shaped pattern holds with a constant elasticity of labor
supply and a Pareto distribution for skills where the distribution
is falling.
Data
While a careful attempt to fit this model to available data is
beyond the scope of this note, it does seem interesting to examine
For this purpose, calculations have
the distribution of wages.
the
March, 1992 CPS.
This survey asked
been done using
individuals for annual earnings in 1991, as well as weeks worked
From these numbers one can
and typical hours per week.
implied
average
wage.
Figure 1 presents the
calculate an
distribution of wages for males aged 21 to 62. 9 In order to have a
visible scale, (and to ignore very low earnings) the figure shows
wage rates from $1 to $100, with intervals of size $2 and is based
Approximately 17% of
on a sample size of approximately 34,000.
the sample report lower wages or no work; less than 0.1% of the
sample report higher wages. The figure shows the single peaked
distribution one would have expected.
Table 1 reports calculations from the same data set. The
columns show the mean wage per cell; the number of observations
per cell, adjusted by cell size in order to be proportional to the
density; the number of observations with higher wages; the ratio
(l-F)/f; and the ratio (l-F)/nf, where n is the wage.
In order to
have reasonable cell sizes, the wage intervals are first $0.50, but
are expanded above a wage of $2 6.
The table shows sharply
falling values of (l-F)/f through the range where the density is
first rising and then roughly flat, that is, up to a wage of roughly
Beyond this point (l-F)/f is
$13, a little below the mean of $13.7.
roughly constant. This implies a downward trend in (l-F)/nf.
With a longer time horizon than one year, one would consider
education to be an endogenous variable somewhat responsive to
tax incentives.
One might also be interested in the distribution of
skills within a cohort. Thus a further calculation was done by
regressing the log of the wage on education, age and age squared,
and plotting the exponentiated residuals.
Figure 2 shows the
density of skills from this calculation.
In Table 2 are the same
calculations for this distribution as shown in Table 1 for the
No attempt was made to consider both earners in a two-earner
family or wages of single females.
9
-6-
distribution of wages. This distribution shows a fatter tail than
the distribution of wages, with (l-F)/f rising and (l-F)/nf roughly
constant for the top 15% of the skill distribution.
There is not a simple route between the Mirrlees model and
policy implications for annual income taxes levied on families and
covering both capital and labor incomes. The assumption of a
zero income elasticity of labor supply and the limited information
on the pattern of wage elasticities of supply by skill level would
Nevertheless
limit inferences even if there were a simple route.
some
lessons
from
analysis.
The
sharp
are
the
fall in (l-F)/f
there
approach
the
mode
of
the
skill
distribution
from below
skills
as
seems highly relevant, especially if one wants to redistribute from
This finding on the
people near the mode, rather than to them.
shape of an optimal (negative) income tax seems relevant in
thinking about the phaseout of the earned income tax credit, and,
This model confirms the implication of
possibly, welfare reform.
Mirrlees' calculations that the optimality of a zero tax rate at the
highest income level is not a finding that sheds much light on
optimal taxes, especially in the absence of knowledge of exactly
where the top is. That is, if one replaced an unbounded
distribution of skills by a bounded one with the same distribution
up to some level and a (small) atom of skills at the highest level,
the result of rising marginal tax rates continues to hold until the
top itself is reached.
Third, the sensitivity of the pattern of
marginal rates to the measure of skill seems potentially relevant,
although different formulations of skill will be associated with
different estimates of the elasticities of labor supply. Apart from
direct relevance for setting taxes, consideration of this case
clarifies some of the elements that matter for understanding
optimal taxation, particularly the role of the shape of the
distribution of skills.
References
Atkinson, A. B.
1973, How progressive should income tax be?, in
M. Parkin and A. R. Nobay, eds., Essays in Modern Economics
London: Longmans, 90-113.
,
,
Atkinson, A. B.
1990, Public economics and the economic public,
European Economic Review 34, 225-248.
,
,
Feenberg, Daniel R.
and James M. Poterba, 1992, Income
inequality and the incomes of very high income taxpayers:
evidence from tax returns, paper presented at the NBER Tax
Policy and the Economy meeting, November 17, 2992.
,
Lollivier, Stefan, and Jean-Charles Rochet, 1983, Bunching and
Second-Order Conditions: A Note on Optimal Tax Theory, Journal
of Economic Theory 31, 392-400.
.
Mirrlees, J. A., 1971, An exploration in the theory of optimum
income taxation, Review of Economic Studies 38, 175-208.
,
-7-
1976, On income distribution, incentive effects and
Sadka, E.
optimal income taxation, Review of Economic Studies 43, 261-8.
,
,
Seade, J. K.
1977, On the shape of optimal tax schedules, Journal
of Public Economics 7, 203-236.
,
,
Slemrod, Joel, Shlomo Yitzhaki, Joram Mayshar and Michael
Lundholm, 1994, The optimal two-bracket linear income tax,
Journal of Public Economics 53, 269-290.
,
Stern. N. H.
1976, On the specification of models of optimum
income taxation, Journal of Public Economics 6, 123-162.
,
,
Tuomala, Matti, 1984, On the Optimal Income Taxation: Some
Further Numerical Results, Journal of Public Economics 23, 351,
366.
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