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9 6 2 1
Marginal Tax Rates
and Income Inequality
in a Life-cycle Model
by David Altig and
Charles T. Carlstrom
FEDERAL RESERVE BANK
OF CLEVELAND
Working Paper 9621
MARGINAL TAX RATES AND INCOME INEQUALITY IN A LIFE-CYCLE
MODEL
by David Altig and Charles T. Carlstrom
David Altig is assistant vice president for economic
research at the Federal Reserve Bank of Cleveland, and
Charles T. Carlstrom is an economist there. The authors
thank seminar participants at Arizona State University,
Bowling Green State University, the Federal Reserve
System Committee on Macroeconomics, the University of
Pennsylvania, and the University of Pittsburgh for many
useful comments. They are especially indebted to two
anonymous referees for their extensive suggestions for
improving this paper.
Working papers of the Federal Reserve Bank of Cleveland
are preliminary materials circulated to stimulate discussion
and critical comment. The views stated herein are those of
the authors and are not necessarily those of the Federal
Reserve Bank of Cleveland or of the Board of Governors of
the Federal Reserve System.
Federal Reserve Bank of Cleveland working papers are
distributed for the purpose of promoting discussion of
research in progress. These papers may not have been
subject to the formal editorial review accorded official
Federal Reserve Bank of Cleveland publications.
Working papers are now available electronically through
the Cleveland Fed’s home page on the World Wide Web:
http://www.clev.frb.org.
December 1996
Abstract
In this paper, we perform computational counterfactual experiments to examine the
quantitative impact of marginal tax rates on the distribution of income. Our methodology
builds on previous simulation models developed by Auerbach and Kotlikoff and Fullerton
and Rogers, and uses an algorithm that allows us to examine marginal tax rate structures
in their literal form. We find that distortions associated with particular marginal tax rate
structures have sizable effects on income inequality in a reasonably quantified life-cycle
setting: In our baseline experiments, the change in steady-state income inequality under
1989 U.S. income tax rates vis-à-vis 1984 rates is about half as large as the change
actually seen in the data over those two years, when measured in terms of a monetary
metric derived from Gini coefficients.
I.
Introduction
This paper is motivated by two well-known observations. The …rst is that, by
almost any measure, income inequality in the United States increased substantially
throughout the 1980s. The second is that over the same period of time there were
several substantial changes in the U.S. personal income tax code, notably major
changes in the structure of marginal tax rates.
With these observations in mind, we ask a straightforward question: What is
the plausible magnitude of the e¤ect on the distribution of income from distortions
created by changing marginal tax rates. More speci…cally, in this article we take
the marginal tax rate changes indicated by the Tax Reform Act of 1986 (TRA86),
and examine the e¤ect on the distribution of income arising from endogenous labor
supply and saving responses in a reasonably quanti…ed general equilibrium model
with life-cycle consumers.
In our opinion, the question we ask has been incompletely addressed – and, hence,
our study is warranted – due to the focus of previous research on other, undeniably
important, aspects of the changing income distribution. First, much recent empirical
research has concentrated on wage income, and in particular on the growing disparity
between returns to high-skill versus low-skill labor. (For an extensive review, see
Levy and Murname [1992].) As documented by Karoly (1994), however, wage-income
inequality accounts for only part of the recent trend in overall income inequality.
Second, studies that explicitly consider changes driven by the tax code have focused
on the top of the income distribution (Feenberg and Poterba [1993] and Slemrod
[1993], for instance), or on changes in taxable, rather than gross, income (Feldstein
[1995], for instance). In both cases, the data is limited in its ability to distinguish
the behavioral responses of total labor supply and saving from “income sheltering”
activities (for example, a change in the manner in which wealth is accumulated that
is not associated with a change the aggregate level of household saving).
We add to the picture drawn by these studies by providing an analysis in the
spirit of the computational experiment described by Kydland and Prescott (1996).
We begin with an explicit benchmark representation of the U.S. economy, calibrated
to match key features of both the macroeconomy and the income distribution under
a post-TRA86 marginal tax rate structure (1989, speci…cally). We then consider a
counterfactual economy in which the only exogenous di¤erence from the benchmark is
the replacement of the TRA86 rate structure with one taken from a pre-TRA86 year
(1984, speci…cally). Our experiments involve no change in the relative (pre-tax) price
of di¤erent types of labor, or in the opportunities for sheltering income by altering
the form in which income is received. Thus, we examine the quantitative impact on
inequality arising solely from behavioral responses associated with labor supply and
saving decisions.
Our analysis builds on the computable general equilibrium framework pioneered
by Auerbach and Kotliko¤ (1987). In particular, we adopt their basic simulation
approach, with two key extensions. First, we follow the recent work of Fullerton and
1
Rogers (1993) and allow for multiple types of life-cycle agents, each type distinguished
by its exogenous labor e¢ciency pro…le, and hence by its maximum attainable wealth
from human capital. Second, we use a simple algorithm for solving Auerbach-Kotliko¤
models with discrete marginal-rate structures. This allows us to introduce tax rate
schedules in their literal form.
The bottom line of our study is that the distortions associated with particular
marginal tax rate structures have sizable e¤ects on income inequality in a reasonably
quanti…ed life-cycle setting. In our baseline experiments, the change in steady state
inequality under the 1989 rates vis-a-vis 1984 rates is about half as large as the change
found in the data over those two years (when measured in terms of a monetary metric
derived from Gini coe¢cients). The increasing inequality in the model arises in the
distribution of both wage- and nonwage-income, despite the fact that, by construction,
relative wages are constant across the tax regimes. However, relative to its share of
total income, increasing inequality in the distribution nonwage income contributes
disproportionately to the change in the Gini across the two steady states. We conclude
from these results that endogenous labor supply and saving responses to changes in
rate structures like those seen in the 1980s can play a signi…cant role in explaining
observed trends in income inequality.
II.
The Model Economy
A.
Basic Demographic Structure
Our model economy is populated by sequences of distinct cohorts that are distinguished by a) date of birth; b) lifetime labor productivity endowments; and c) their
preferences for leaving bequests. Each generation born at a speci…c date contains
13 di¤erent agent types, indexed by j, each with di¤erent exogenous labor e¢ciency
pro…les and distinct utility weights concerning transfers of wealth to their children.
All j-types of a given generation are identical. With the exception of size, successive
generations of j -type agents are identical. All agents live for 80 periods with certainty,
and each j -type generation is 1 + n times larger than its predecessor.
At “age” 21, each j -type cohort gives “birth” to a cohort of the same type. Given
our assumption about the exogenous rate of population growth, every newborn cohort
is (1+ n)20 times the size of its “parent” cohort. Also at age 21, each cohort begins its
economic life, and begins working, saving, and consuming. We assume that children
neither consume nor work prior to separating from the parent households at age 21.
Throughout, then, we will think of an “agent” as being a household, exclusive of
children.
2
B.
Preferences and Household Budget Constraints
Each j-type agent beginning its economic life at calendar date T chooses perfectforesight consumption paths (c), leisure paths (l), and intergenerational transfers (b)
to maximize a time-separable utility function of the general form
UTj =
80
X
j
j
¯ s¡21 u(cjs;T +s¡21 ; ls;T
+s¡21 ; bs;T +s¡21 );
(1)
s=21
where ui > 0, uii < 0, limi!1 = 0, limi!0 = 1, and ui is the partial derivative of
the function u(¢) with respect to argument i. The form of the utility function and
the subjective time-discount factor, ¯, are assumed to be the same for all agents. We
require that ¯ > 0, but need not impose the condition that ¯ < 1.
Letting ajs;t be capital holdings for type j agents, of age s; at time t, maximization
of (1) is subject to a sequence of budget constraints given by
j
j
j
j
j
ajs;t = (1 + rt )(ajs¡1;t¡1 + gs;t
) + ws;t
(1 ¡ ls;t
) + zs;t
¡ T (¹
ys;t
) ¡ Nbjs;t :
(2)
The variable rt represents the return to saving, which is held in the form of claims
to physical capital. We assume that aggregate wage payments at time t, wt, are
distributed according to the e¢ciency levels of individual labor units (i.e. each agent
receives the value of its marginal product in exchange for providing labor services).
Furthermore, an individual’s labor e¢ciency level is an exogenous function of their
j
age and type, and is denoted by "js . Thus, ws;t
= "js wt .
For a type-j agent of a given generation, transfers received from parents are
j
represented by the gs;t
in (2), and the bjs;t denote transfers from parents to each of
their N = (1 + n)20 children. (Recall that all j-type parents give birth to children of
the same type.) Transfers all take the form of exchanges of capital, and are received
by children, with interest, at the beginning of the period after they are accumulated
by parents. We restrict all parental transfers to bequests, so that bjs;t = 0, for s 6= 80,
j
and gs;t
= 0, for s 6= 61. In the steady state, of course, gj = bj , for all j (where we
have dropped the age subscripts for convenience).
j
Finally, zs;t
refers to lump-sum transfers from the government, the function
j
j
represents taxable income. We
T (¹
ys;t) determines income-tax payments, and y¹s;t
de…ne taxable income as
j
y¹s;t
=
rt + ¼t j
j
j
j
(1 ¡ ls;t
) ¡ d(ys;t
);
a + ws;t
1 + ¼t s;t
(3)
where ¼t is the rate of in‡ation between times t and t¡1, and d(¢) speci…es deductions
and exemptions as a function of gross income. The in‡ation rate is included in 3 to
capture the imperfect nature of indexation in the U.S. tax code.1 (Note that this is
consistent with the de…nition of capital income included in adjusted gross income for
1
Indexation in the U.S. personal tax code is e¤ectively done by de‡ating income in a given year
by a lagged value of the gross in‡ation rate. Imperfections in this scheme arise for two reasons.
3
tax purposes, and hence is also consistent with the de…nition implicitly contained in
other empirical studies based on actual tax return data.) We do not include explicit
taxation of bequests.
Throughout we will assume that d(¢) is a linear function of income, with a strictly
positive …rst derivative. Although this assumption is not necessary, it appears to
conform to the historical tax structures we consider in that simple linear regressions
of deductions plus exemptions on income result in high values (over 0:95) of the
resulting R2.
We require that the function T (¢) be continuous, but not everywhere di¤erentiable. In fact, our subsequent analysis is conducted using U.S. personal income-tax
codes that have the familiar discrete step-function marginal rate structures.
C.
Firms and Technology
Output in the model economy is produced by identical competitive …rms using
a neoclassical, constant-returns-to-scale production technology. Letting ° j be the
fraction of j-type agents in each generation, aggregate capital (K) and labor (L) (in
per capita e¢ciency units) are obtained from individual asset and labor supplies as
Kt =
13
X
°
j=1
and
Lt =
13
X
j=1
°j
j
80
X
(1 + n)80¡s¡1 ajs;t¡1 ;
(4)
s=21
80
X
s=21
j
(1 + n)80¡s "js (1 ¡ ls;t
):
(5)
Note that, for simplicity, we have normalized the population so that the total number
of age-80 households is one, for all t.
Written in terms of the capital-labor ratio, ·, the aggregate production technology is
qt = f (·t );
(6)
where qt is output per labor input (measured in e¢ciency units), and f(¢) satis…es the
usual Inada conditions. The competitive aggregate wage and interest rate are given
by
wt = qt ¡ ·t f 0 (¢);
(7)
and
rt = f 0 (¢) ¡ ±;
(8)
where ± is the (constant) depreciation rate of physical capital.
First, de‡ating nominal asset income (approximately (r + ¼)at¡1 ) by a given gross in‡ation rate,
t¡1
1 + ¼, overstates real capital income by an amount equal to ¼a
1+¼ . The second imperfection arises
because the adjustments use lagged, rather than contemporaneous, in‡ation rates. The …rst type of
imperfection is incorporated into equation (3) The second type, of course, disappears in the steady
state, and is not incorporated into (3). Further discussion of this issue can be found in Altig and
Carlstrom (1993).
4
D.
The Government
Our interest in this paper is solely in the distortionary e¤ect of di¤erent incometax rate structures. Consequently, the government in our model has a very simple
role: It raises revenue from income taxes, which it then rebates to the agents from
whom the taxes are collected in the form of lump-sum transfers. The government’s
activities in each period are fully captured by the “rule”
j
j
T (¹
ys;t
) = zs;t
;
(9)
for all t; s, and j. Thus, our simulations correspond to compensated demand experiments, and the e¤ects of government policy arise only from the distortionary impact
of marginal tax rates on household labor supply and saving decisions.
III.
Model Calibration
Our model is calibrated to Internal Revenue Service Statistics of Income data
for the taxable returns of married persons …ling jointly in 1989. In what follows, all
references to income, taxpayers, and so on, should be understood as applying to this
population.
Most of our choices for parameterizing the model are standard. Exceptions involve the special features of our framework, speci…cally the tax codes, multiple agent
structure, and bequest motives. We therefore …rst turn to a discussion of how we
quantify these elements of the model, followed by a discussion of the more familiar
choices for (or strategies for choosing) preference and technology parameters.
A.
Labor E¢ciency Pro…les
The labor e¢ciency pro…les in equations (2) and (5) are of the form
j
j
j 2
+aj3 s3
"js = ea0 +a1 s+a2 s
:
(10)
Values of the a coe¢cients for j-type groups 1 through 12 – numbered in ascending
order of lifetime income – are taken from estimates provided by Fullerton and Rogers
(1993, table 4-11).2
2
Fullerton and Rogers use data from the Michigan Panel Study of Income Dynamics (PSID)
to calculate life-cycle labor e¢ciency endowments for the twelve groups. Focusing on a subsample
that includes only households with stable marital histories over the 1970-1987 period, they …rst …t
a common wage function across all households. Speci…cally, they regress the log of average hourly
earnings for each individual on a cubic in age and the interaction of age and age-squared with sex,
education, and race. The coe¢cients so estimated are used to construct synthetic individual wage
observations outside the 18-year sample, which are then combined with actual wage observations to
calculate lifetime labor income for all households. Finally, individuals are ranked by lifetime income
level and separated into twelve groups according to that ranking, for which distinct wage pro…les
are then estimated.
5
In calibrating the model we aim to match two important elements of the income
distribution: mean income and the percent of income earned by taxpayers with high
income. Unfortunately, we are unable, using the Fullerton-Rogers estimates alone,
to simultaneously match average income and generate incomes for the “rich” that
are su¢ciently high to mimic the actual distribution of income. Our, admittedly ad
hoc, solution to this problem is to add a thirteenth j-type whose members have an
e¢ciency pro…le that is proportional to that of the twelfth group. Speci…cally, the
intercept of the pro…le for this group (i.e. a13
0 ) is chosen so that the top 6 percent
of taxpayers (those with incomes over $100,000 in 1989) earn 26 percent of total
measured income in the benchmark steady state.3
B.
Tax Code
In the benchmark case, we use the actual marginal tax rates applied to personal
income in 1989, which re‡ects the fully phased-in changes introduced by TRA86. Our
experiments intentionally exclude many obvious and important forms of tax avoidance
– Subchapter C …lings and the transformation of compensation into nontaxable bene…ts, for example. However, we do incorporate personal exemptions and deductions
by introducing the function d(¢) in the calculation of taxable income (see equation
[3]), which posits that such adjustments to income are linear in gross income. To estimate this function, we use the Statistics of Income data to calculate average personal
exemptions and deductions by listed income classes, which are then regressed on the
midpoint of each income class.4 The estimated function is then used to convert gross
income to taxable income. (The tax functions are more fully described in …gure 5
and equations [13] and [14], below.)
C.
The Production Technology: Scaling the Model
We adopt a simple Cobb-Douglas production function,
qt = A·µt ;
(11)
where µ is capital’s share in production and A is a scale factor. Our benchmark value
for µ is 0.36, and we assume that physical capital depreciates at a rate of 10 percent
per annum. Both of these choices are motivated by familiar long-run observations on
capital shares in the U.S. economy. (See, for example, the arguments in Cooley and
3
The population distribution of lifetime income cohorts 1 through 12 is obtained from table 4-11 in
0
Fullerton and Rogers (1993). Letting ° j ; j 0 = 1:::12, be the proportion of weighted observations for
0
the 12 groups in the Fullerton and Rogers data, we choose ° 13 = 0:06; and then set ° j = ° j (1 ¡ ° 13 )
for j = 1:::12. Note, however, that this does not necessarily imply that the top 6 percent of wage
earners in a given cross-sectional distribution are all of type 13.
4
Deductions are calculated as a weighted average of total standard deductions and total itemized
deductions in each reported income class, where the weights are determined by the share of income
accounted for by taxpayers taking each particular type of deduction.
6
Prescott [1994].) In addition, we set annual population growth to 1.3 percent, the
postwar U.S. average.
Unlike many similar calibrated simulation exercises, the scale factor A does play
a role in our experiments. In particular, because our study uses actual marginal rate
brackets, it is necessary to scale the model so that generated model incomes can be
sensibly applied to the chosen tax codes. Essentially, we choose A so that the mean
income generated in the model’s benchmark steady state equals the average gross
income of joint …lers in tax year 1989. In practice (as discussed earlier), matching
the data requires that we simultaneously choose the scale factor A and the intercept
of the labor e¢ciency pro…le for the thirteenth j-type group (the “richest” lifetime
income type).
D.
Preferences
We specialize the utility function in (1) to
0
1
j
1¡¾
(ls;T
(bj )1¡¾b
+s¡21 )
j
j
s¡21 @
A + ¯ 59 ½j N
UT =
¯
ln(cs;T +s¡21) + ®
:
1¡¾
1 ¡ ¾b
s=21
80
X
(12)
In (12), ® is the utility weight on leisure, ¾ is the inverse of the elasticity of leisure
with respect to wages, and ¾ b is the inverse of the elasticity of bequests with respect
to consumption. The parameter ½j is j-type speci…c utility weight placed on bequests
left to children.
This formulation of preferences is chosen for two reasons. First, it is consistent
with the existence of balanced growth, in that average hours are invariant to positive
rates of labor-augmenting technical progress (see King, Plosser, and Rebelo [1988]).5
Second, the speci…cation in equation (12) has the property that, for given marginal
tax rates, the capital-labor ratio is invariant to the scale factor A in (11). Both of
these features follow from the fact that changes in the level of wages have o¤setting
income and substitution e¤ects on individual labor supply decisions.
Given the functional form in equation (12), the parameters ® and ¾ are, respectively, set to 0:6 and 4. These choices imply that steady state hours worked by the
“average” worker is about 30 percent of their total time endowment (which we take
to be 16 hours per day) and an average intertemporal labor supply elasticity of about
5
We have, however, chosen not to to embed exogenous labor-augmenting technical progress into
the model. While it is certainly easy, by a suitable change of variables, to solve the model with such
an extension, the notion of a steady state becomes somewhat slippery when we contemplate both
economic growth and the type of progressive tax systems that we are considering. In particular,
in a growing economy with an unchanging tax code, all taxpayers face the highest marginal tax
rate in the long run. Although we make an allowance for growth in our benchmark experiment, the
steady-state comparisons in our model implicitly assume that the relevant tax codes are indexed to
real growth.
7
0.5.6
Finally, we set ¾ b to 2.0 and, for j = 1; :::12, choose the ½j so that each groups
endogenous bequest choices match the levels reported in table 3-8 of Fullerton and
Rogers (1993).7 For cohort 13 we assume that bequests represent the same proportion
of peak income as for cohort 12, and choose ½13 to match this value.
E.
Solving the Model
Before proceeding to a discussion of the benchmark equilibrium and the results
of our experiments, we will brie‡y outline our solution algorithm. A more detailed
description is provided in the appendix.
To calibrate the benchmark steady state, we use a modi…cation of the iterative
approach suggested by Auerbach and Kotliko¤ (1987).
1. Conjecture values for ½j (the utility bequest weights); j = 1; :::13.
2. Conjecture values for A (the scale factor in the production function speci…ed in
equation [11]) and a13
0 (the intercept of the labor e¢ciency pro…le for type-13
agents).
3. Conjecture values for the aggregate capital stock and labor supply, which in
turn imply guesses for the aggregate wage and interest rate from equations (7)
and (8).
4. Conjecture marginal tax rates ¿ js for each age s and type j.
5. For given values of r, w, and the ¿ js , calculate life-cycle consumption and leisure
paths for each j-type. In this stage of the procedure, corner conditions for
leisure are resolved.
6
Most empirical studies provide estimates of the intertemporal elasticity of supply in labor (´),
rather than leisure as in our preference speci…cation in equation (12). Because leisure accounts for
approximately two-thirds of the labor endowment, on average ´ ¼ 2¾. However, because the labor
supply elasticity is not constant for this preference speci…cation, we verify that the simulated data
is consistent with a value of about 0.5 by constructing an arti…cial panel of length 20 years for
agents with initial ages from 25 to 44, and then estimate the compensated intertemporal elasticity
of substitution in labor using the approach outlined in MaCurdy (1981). (We thank an anonymous
referee for suggesting this approach.) Doing so yielded the estimate ´^ = 0:51, a result very close to
the theoretical average. Although at the high end of MaCurdy’s estimates for male labor supply,
Rogerson and Rupert (1991) and Rogerson, Rupert, and Wright (1996) argue that conventional
analyses of male labor supply tend to understate the true degree of intertemporal substitution.
Furthermore, our analysis takes the household as the relevant decision making unit, and there
is broad agreement that the elasticity is higher for women than for men (see Killingsworth and
Heckman [1986]).
7
For calibration purposes the bequest data in Fullerton and Rogers were converted to 1989 dollars.
There is, in general, little evidence to guide us in the choice ¾ b : The e¤ect of choosing an alternative
value for ¾b is explored in section VI.
8
6. Update tax rates by applying calculated income levels to the relevant tax rate
schedule. Note that, because of the step-function characteristic of the rate
schedule, some agents may be at “kinks” in their budget surfaces. These conditions are resolved in this stage of the solution procedure.
7. Repeat steps (4) and (5) until consumption and leisure choices are consistent
with the assumed tax rates used in calculating them.
8. Use equations (4) and (5) to aggregate asset and labor supplies over all ages and
j-types to …nd the values K 0 and L0 implied by the individual decisions for given
factor prices. Update guesses for capital and labor and repeat steps (4)-(7) until
the updated values K 0 and L0 are consistent with individual consumption and
leisure choices.
9. Update values for variables in step 2 and repeat steps 3 through 8 until average
gross income and the share of income earned by the top 6 percent of the income
distribution match the appropriate values in the data. (See the discussion in
section III.C, above.)
10. Update values for the ½j and repeat steps 2 through 9 until the endogenously
chosen bequests of each j-type matches the appropriate values in the data. (See
the discussion in section III.D, above.)
The solution procedure is identical for cases other than the benchmark steady
state, except that the calibrated values of the ½j , A, and the intercept for the labor
e¢ciency pro…le for type-13 agents are …xed, eliminating steps 1, 2, 9, and 10. A
schematic of the algorithm is shown in …gure 1.
F.
The Benchmark Equilibrium
We set the discount parameter ¯ in equation (12) to 0.99 so that, given our other
parameter choices, the benchmark steady-state pre-tax real interest rate is about 3
percent and the capital-output ratio is about 2.8 (values which are reasonably close to
their respective historical averages in the U.S. data8 ). In addition to matching these
standard aggregate variables, our calibration exercise is designed to deliver congruence
of the model’s tax- and income-distribution characteristics with those derived from
8
See Siegal (1992), who reports average real interest rates over the 1800-1990 period. We note, for
the record, that the rates appear to di¤er signi…cantly across subperiods. Speci…cally, real returns
to long-term bonds averaged 1.46 percent over the period from 1889-1978, but 5.76 percent outside
that interval.
The capital measure from the data refers to the constant-constant net stock of reproducible tangible wealth reported in the January 1992 Survey of Current Business, and includes consumer durables
and government capital. With this de…nition of K, the capital-output ratio in the United States
averaged about 2.8 for the period from 1959 through 1990.
9
the 1989 tax-return data. Some of the relevant comparisons along these dimensions
are reported in table 1.
As indicated by the Gini coe¢cients, the distribution of income in the model
is somewhat more equal than that exhibited by the data. This results in large part
from the fact that, even after the addition of the thirteenth lifetime-income type
described in III.A, the framework under represents taxpayer incomes at the highest
end of the income distribution. Although our simulations closely match the fraction
of income earned by the top 5 percent of income earners – not surprising given that
our calibration approach essentially engineers this outcome – the share of the top 1
percent is less than half that found in the data. This di¤erence is clearly seen in the
Lorenz curves plotted in …gure 2.
A more detailed view of the model’s distributional features is shown in …gures 3
and 4, which compare actual and simulated distributions of gross income and taxes
paid according to the income classes reported in the Statistics of Income. The most
signi…cant discrepancy between the model and the data is the concentration of mass
in the $40,000 to $50,000 range, a di¤erence that is especially pronounced for gross
income.9
As can also be seen in table 1, our benchmark model does quite well at matching
the broad characteristics of the actual 1989 tax distribution: Simulated average and
average marginal tax rates are virtually identical to the values derived from the data.
Furthermore, the simulated distribution of income among the di¤erent marginal rate
brackets is very close to that obtained from actual tax returns.
IV.
The Quantitative E¤ects of Changes in Marginal Tax
Rates
Our computational experiment begins with the benchmark steady state calibrated to 1989 data, as described in the previous section. We then consider and
compare an alternative economy, identical in every way to the benchmark except
that the 1989 tax structure used to calibrate the model is replaced with a rate structure taken from the personal income tax code for 1984. We focus particularly on how
the distinct rate structures a¤ect overall income inequality.
The rate structures we consider are depicted in …gure 5. The rates are taken
from schedule Y for married persons …ling jointly in tax years 1984 and 1989, but
for two adjustments. First, the 1984 schedules are adjusted for real income growth
between the two years. (Note that, subject to the caveats noted in footnote 1, the
tax schedule in 1989 included an adjustment for in‡ation between the two years.)
Under the assumption that income growth (which we are abstracting from) does not
itself directly alter the distribution of income, the adjustment amounts to treating the
9
Some discrepancy would be expected due to the fact that our wage e¢ciency pro…les are estimated from data spanning the period from 1970 to 1987, and are not purely consistent with the
distribution of wage income in 1989.
10
rate schedules as being applicable to given regions of the income distribution. (We
examine the e¤ect of this adjustment in section V.I.)
Second, the schedules depicted in …gure 5 are “e¤ective” rates in the sense that
we have incorporated the impact of the functions used to convert gross income into
taxable income (see the discussion in section III.B). These functions are
Max f$8368 or $7057 + 0:1566¹
yg
(13)
Max f$9805 or $8280 + 0:1555¹
yg
(14)
for 1984, and
for 1989.10 (The …rst term in the brackets represent our implicit “standard deduction.”) Note that the two functions are very similar, implying that, for our experiments, most of the di¤erence between the two rate structures is derived from the
statutory rate schedule.
A.
Aggregate E¤ects
A comparison of the long-run equilibria under the 1989 and 1984 rate structures
is provided in table 2. Compared to the 1989 benchmark, steady state output is about
6 percent lower under the 1984 tax rate structure. Although the long-run equilibrium
levels of both labor and capital inputs fall in the 1984 regime, the decline in the latter
is proportionately larger, so that the steady state capital-labor ratio is reduced by
introducing the earlier year’s rate structure. Thus, the aggregate wage is lower and
the interest rate higher in the 1984 case than in the 1989 case.
Revenues collected by the income tax are nearly 16 percent higher under the
1984 tax code. The average income-tax rate under the 1984 structure is about 17
percent (compared to the benchmark value of about 14 percent), and the average
marginal tax rate is nearly 32 percent (compared to 23-1/2 percent in the benchmark
case). Because the direct wealth e¤ects of the tax di¤erences are o¤set by lump
sum transfers, it is the change in marginal rates that drive the both aggregate and
distributional e¤ects in our model.
B.
Marginal Tax Rates and the Distribution of Income
The bottom panel of table 2 summarizes the distributional consequences of our
experiment. As measured by the Gini coe¢cient, the distribution of income becomes
more unequal under the 1989 schedule, increasing from 0.33 in the 1984 case to 0.36
in the 1989 case. To provide some intuition for the magnitude of the di¤erence in
inequality across the two tax codes, we calculate a measure suggested by Blackburn
(1989) that translates di¤erences between Gini values into a monetary metric. Focusing on the steady state distribution of income obtained with the 1984 rates, our
10
The function in equation (13) incorporates the zero-bracket amount provided for in the 1984
code.
11
application of the Blackburn measure gives the dollar amount that would have to be
transferred from all agents with incomes below the median to all those with abovemedian income in order to generate the same degree of inequality realized in the
steady state under the 1989 rates. In our experiment this amounts to about $3000
(measured in terms of the 1989 price level).
As a frame of reference, it is instructive to compare the results of our counterfactual experiment with actual experience. From 1984 to 1989, the Gini coe¢cients
derived from the Statistics of Income tax data increased from roughly 0.35 to 0.40
(based on an average of the lower and upper bound estimates suggested by Gastwirth
[1972]). In terms of the Blackburn measure, the monetary equivalent to this change is
a transfer of $5,776 from below-median income earners to above-median income earners. Comparing this number to that obtained from the model, our experiment thus
indicates a response in the distribution of income from the change in rate structures
that is just over 50 percent of the change that actually occurred in the data.
The data further reveal that the share of income earned by the top 5 percent of
income earners rose from 19.7 percent in 1984 to 25.7 percent in 1989. In our model
simulations, the corresponding numbers are 19.5 percent and 22.6 percent, roughly
40 percent of the change in the data. The absolute magnitude of the change in the
income share of the top 1 percent of the distribution is much smaller in the model than
in the data, which is not surprising given that our model signi…cantly underestimates
the share earned by “very high” income earners. In the model, the change in the
income share of the 1 percent of the distribution is less than 1 percentage point (from
4.6 percent under the 1984 structure to 5.4 percent under the 1989 structure). This
compares to a change of just over 4-1/2 percentage points in the data (from 8.9
percent in 1984 to 13.5 percent in 1989).
These comparisons with the data are made solely for the purpose of providing
some perspective on the magnitude of the e¤ects predicted by the model. Our conclusion is not that TRA86 explains half the change in the measured Gini coe¢cient
between 1984 and 1989 (or thirty to forty percent of the increase in the mass at the
top of the income distribution). For one thing, we have obviously excluded important
elements of the Act that may have mitigated or exaggerated its overall impact.11 In
comparing the model’s outcome to the data we are attempting to show that, working
solely through the distortionary e¤ect on household saving and labor supply decisions, the impact on inequality from rate structure changes like those implemented
by TRA86 are quantitatively meaningful.
11
It is also true that the model is comparing steady states, while the data is separated by only
…ve years. However, our experiments with transition paths indicate that measures of inequality in
the model converge very rapidly to levels near their steady state values. These results are reported
in an earlier draft of this paper. (Interested readers are referred to Altig and Carlstrom [1995]. The
model in that paper has a preference speci…cation that does not include bequests, but is otherwise
identical the one used in the experiments reported here.)
12
V.
A Further Analysis of the Results
A.
Some Cohort-Speci…c Examples
To aid in exploring these results a bit further, …gure 6 plots the life-cycle pro…les
of marginal tax rates, labor supply, labor income, asset holdings, and asset income for
two speci…c j-types, designated as cohorts 3 and 11. The income of a representative
member of cohort 3 ranges from about $10,000 to $33,000 in the benchmark steady
state. For cohort 11 the range is $42,000 to $98,000.
The panels in …gure 6 clearly reveal a discernible e¤ect of marginal tax rates on
the leisure and saving decisions of households. For both types the life-cycle pattern of
labor supply decisions clearly reveal the distortionary impact of the tax rate schedules,
each exhibiting discrete kinks in the pro…les corresponding to the discrete steps in
tax rates. With both groups marginal tax rates are lower in the 1989 case for most
periods of life, and labor supply is higher. Because the capital-labor ratio is higher
under the 1989 structure, the aggregate wage rate, and consequently labor income,
are also generally higher.
Changes in labor supply are quite di¤erent for the two groups: The cumulative
increase in hours provided by group 11 under the 1989 tax regime relative to 1984 is
better than 77 percent in excess of group 3’s. The greater labor e¢ciency of cohort
11 then magni…es the responses of total wage income in …gure 6. Thus, inequality in
the wage component of income increases across these two types of agents even though
the relative wage received remains the same.
With respect to saving behavior, for cohort 3 asset positions and asset income are,
on average, lower under the 1989 tax code than under the 1984 code. The converse
is true for cohort 11. Thus, as we compare the steady state with 1989 rates to that
under 1984 rates, inequality in the nonwage component of income increases across
these two types of agents because the asset income of cohort 11 increases and that of
cohort 3 decreases.
This particular result – lower nonwage income for cohort 3 under the 1989 rate
structure compared to higher nonwage income for cohort 11 – is driven by the combination of tax rate changes and general equilibrium e¤ects on the after-tax real rate
of interest. Consider …rst the case of type-3 agents. Although the tax rates realized
by this cohort are generally higher under the 1984 structure, the tax rate di¤erences
relative to the 1989 code are, over most of the cohort’s life-cycle, swamped by the
higher 1984 pre-tax interest rate (see …gure 7). As a result of the higher after-tax
interest rates, asset levels and asset income for cohort 3 are thus higher in the general
equilibrium calculated with the 1984 rate structure.
To help reinforce this point, …gure 7 illustrates the e¤ect of a partial equilibrium
experiment in which factor prices are …xed at their steady state values under the 1989
code, and marginal tax rates are …xed at their equilibrium values under the 1984 code.
(Recall that, because the steady state capital-labor ratio is lower for 1984 case, the
pre-tax real interest rate is higher.) With the interest rate …xed at the lower 1989
13
value, after-tax rates for cohort 3 calculated with the 1984 marginal tax rates result in
lower post-tax interest rates than in either of the two general equilibrium cases, and
so also results in a lower life cycle pro…le of asset holdings and a lower asset-income
pro…le.
For the type-11 group, the marginal tax rates are su¢ciently lower under the
1989 code so as to swamp the general equilibrium pre-tax interest rate di¤erences
across the two steady states. As a consequence, and in contrast to the case of cohort
3, after-tax rates over the life-cycle are higher on average in the benchmark case,
which results in higher asset levels and greater asset-income than chosen under the
1984 rate structure.
B.
A Decomposition of the Income Gini into Wage and Nonwage Components
The comparison of the two groups in the previous subsection suggests that, in our
experiment, changes in inequality arise from both labor- and capital-income sources.
This observation generalizes to the entire distribution. As reported in table 3, separate
Gini coe¢cients for wage and nonwage income components reveal that inequality in
both labor and asset sources of income is higher under the 1989 rate structure: The
Gini for wage income under the 1989 rate structure is 0.44, compared to 0.36 in the
case of the 1984 structure. For nonwage income, the Ginis are 0.56 versus 0.51 in the
1989 and 1984 cases, respectively.
Using a decomposition suggested by Lerman and Yitzhaki (1985), we …nd that
the rise of inequality in both income sources contribute equally to the change in the
overall Gini.12 However, the share of the total-income Gini accounted for by nonwage
income is only about 25 to 27 percent. Thus, the fact that the change in the nonwageincome distribution accounts for 50 percent of the change in the overall Gini implies
that inequality from the asset-income source contributes disproportionately to the rise
in overall income inequality. This conforms to the decrease in the share of inequality
accounted for by wage income found in the U.S. data over the period from 1980-1990
(see Karoly [1994]).
12
The Lerman-Yitzhaki decomposition yields an expression for the inequality share of income
component k given by Ik = Sk GGk Rk , where Sk is the component’s share of total income, Gk is the
Gini coe¢cient for the component, Rk is the covariance of the component with the distribution of
total income normalized by the covariance of the component with its own distribution (termed the
“Gini correlation”), and G is the Gini for total income. By construction, the component shares of
inequality must sum to one. Under the 1984 rate structure, Iw = 0:754 and Inw = 0:246 (where the
subscripts refer to w and nw refer to wage and nonwage income respectively). Under the 1989 rate
structure the corresponding values are Iw = 0:733 and Inw = 0:267.
14
C.
Inequality in the “After-Tax” Income Distribution
Although lump-sum transfers imply that net tax payments are always zero in
our experiments, it is instructive to examine the distributional impact of the two
rate structures on income net of income-tax payments. We report these calculations
in table 3.13 As expected given the progressive nature of both rate structures, each
of the after-tax Ginis indicate that net-of-income-tax inequality is less than that
of gross income. (Furthermore, as would be anticipated by perusal of …gure 5, the
di¤erence between the pre- and post-tax distributions is greater under the 1984 code.)
Interestingly, the greater inequality associated with 1989 rate structure is ampli…ed by
the comparison of the net-of-income-tax distributions: Relative to the results reported
in table 2, the Blackburn metric is about $4216 for the “after-tax” calculations, or
approximately 40 percent higher than for our earlier calculations. Still, the overall
picture remains consistent with the conclusion in Karoly (1994) that “pre-tax income
inequality dominates any increase in post-tax inequality.”
D.
Inequality in and Life-Cycle Income Distribution
The conclusion that inequality is greater when households face the 1989 rate
structure does not depend on the fact that the Gini coe¢cients are calculated on the
basis of cross-sectional, rather than life-cycle, income: The Gini coe¢cients calculated
from the distribution of wealth (the present value of labor income plus bequests) across
j-types also reveal more inequality under the 1989 code. The rise in the Gini as we
move from the 1984 to the 1989 steady state – from 0.26 in the former to 0.29 in
the latter – is in fact greater in percentage terms than the change obtained from the
cross-sectional calculations.14
Ultimately, however, a more meaningful measure of the distributional impact
across lifetime income groups would relate the tax structure e¤ects to measures of
economic welfare. We turn next, then, to the calculation of utility losses and gains
associated with our computational experiment. For any individual living in a long-run
equilibrium under the 1984 rate structure, we calculate the welfare gain of shifting to
the 1989 structure as the percentage decrease in full wealth that must be taken from an
individual in the 1989 steady state in order to maintain utility at its initial level. (Full
wealth is de…ned as the present value of labor income when the individual’s entire
labor endowment is allocated to market work, plus the present value of bequests.)
Positive numbers thus indicate that lifetime utility is higher under the 1989 rates.
13
We thank an anonymous referee for suggesting these calculations.
As found in the cross-sectional income distribution, the increase in the wealth Gini can be attributed to both wage and nonwage sources. However, the contribution of nonwage wealth (de…ned
as the present value of asset income plus bequests) is more pronounced in the lifetime wealth distribution across j-types. Using the Lerman-Yitzhaki decomposition described above, we …nd that
about 60 percent of the increase in the wealth-Gini across j-types is attributable to the nonwage
source.
14
15
The welfare results for each of the 13 j-types are depicted in …gure 8. The average
(population-weighted) steady-state welfare gain associated with changing from the
1984 to the 1989 rate structure averages 0.9 percent of full wealth.15 In other words,
the average household gains by a shift to the marginal tax rates implied by TRA86.
However, as is clear from …gure 8, this average masks a wide range of outcomes for the
speci…c j-types. The very wealthiest group (type 13) enjoys a welfare gain equivalent
to over 2 percent of its wealth. Cohort 2, on the other hand, su¤ers a loss of about
1/3 of a percentage point.
These welfare results are largely driven by changes in the rate structures per se,
as opposed to general equilibrium e¤ects on factor prices. Figure 8 also shows the
welfare results from the partial equilibrium experiments described in subsection A,
above. In general, both the quantitative and qualitative outcomes are similar to those
obtained from the full general-equilibrium experiments.
At the low end of the lifetime-income scale, there is a reversal in the partial
equilibrium experiments of the welfare losses found in the full general equilibrium
simulations. As a point of reference, refer back to the labor and asset pro…les for
type-3 agents shown in …gures 6 and 7, and recall the dominant e¤ect of the higher
interest rate on the saving behavior of this group in the steady state under the 1984
tax rates.16 In welfare terms, removing the impact of the higher pre-tax interest
rate in the full 1984 general equilibrium simulation is su¢cient to shift a zero change
in lifetime utility across the tax regimes to a small gain from shifting to the 1989
structure.
These observations notwithstanding, the general implication that the welfare consequences are concentrated at the higher end of the wealth distribution remains. Furthermore the average utility gain from shifting to the 1989 structure barely changes,
rising from 0.9 percent of full wealth in the general equilibrium case to 0.95 percent
in the partial equilibrium experiment.
VI.
Some Alternative Experiments
To provide a sense of the robustness of our results, we report a summary of
several alternative experiments in table 4. These experiments are
1. The partial equilibrium experiment described in the previous section.
2. Setting ¾ to 10, implying a labor supply elasticity of about 0.2.
3. Setting ½j = 0, for all j, thus eliminating bequests from the model.
4. Setting ¾ b = 4, implying a bequest elasticity of 0.25.
15
The essential message of our welfare calculations is not modi…ed by including the transition path
analysis. See, again, Altig and Carlstrom (1995).
16
The pattern of e¤ects shown in …gures 6 and 7, and hence the same general set of observations,
apply also to type-1 and type-2 lifetime income groups.
16
5. Eliminating the adjustment of the 1984 rates for real growth between the two
tax years, so that the experiment involves a straight comparison of the two rate
structures as they appeared in their given tax years.
6. Adjusting the intercept of the adjustment function d(¢) in equation (13) so that
the two rate structures generate the same aggregate revenue from the income
tax.
7. Allowing a deduction equal to 60 percent of nonwage income when calculating
taxable income under the 1984 rate structure.17
8. Calculating wage income taxes inclusive of the social security tax. The combined
OASDI rates applicable to employees and employers together were 10.9 percent
in 1984 and 11.8 percent in 1989. As provided for in law, we applied these rates
to wage income levels up to $37,800 for the 1984 case, and up to $48,000 in the
1989 case.
In each of these experiments the model was calibrated as discussed in section III.
As is clear from inspection of table 4, none of our major conclusions are altered by
the types of revisions contemplated in these experiments. In each case the increase
in inequality arising from the 1989 rate structure vis-a-vis the 1984 structure is substantial when measured against the actual change in the distribution of income across
those two years. In addition, there are, on average, welfare gains associated with a
shift to the 1989 structure, with those gains being concentrated at the upper end of
the lifetime income distribution. In general, the rising tide associated with our “tax
reform” does not lift all boats: Typically, there are welfare losses at the lower end of
the lifetime income distribution, although the e¤ects are small.
We emphasize two aspects of these experiments, in particular. First, our results
are quite insensitive to the implied labor supply elasticity. Thus, although the preference parameters chosen for our benchmark experiment maps into elasticities at the
high end of many estimates, these choices do not drive our results. This is not particularly surprising given that 50 percent of the model’s change in income inequality is
due to changes in the distribution of nonwage income. Second, the partial equilibrium
experiments – which hold …xed factor prices – suggest the general robustness of our
results along transition paths between the two steady states (see footnote 11).
17
The 1984 code allowed for taxation of 40% of the excess of long-term capital gains over shortterm capital gains. The percentage of nonwage income subject to tax was, of course, greater than the
percentage implied by our back-of-the-envelope adjustment. Furthermore, in practice the percentage
varied inversely with the income level because higher income taxpayers realized a greater percentage
of their nonwage income in capital gains. Our adjustment in this experiment therefore overstates
the degree of progressivity inherent in the 1984 capital gains provisions.
17
VII.
Concluding Remarks
Several studies of recent vintage have examined the distortionary impact of tax
rates on individual behavior using a variety of data-based approaches. Although it
is generally di¢cult for these studies to disentangle the channels through which tax
e¤ects alter the income distribution, Slemrod [1995] concludes that “... much of the
high-income behavioral response to TRA86 is probably better characterized as income
shifting, rather than income creation in the sense of increased labor supply.” We
have adopted a strategy employing a computational methodology in order to isolate
reasonable quantitative guesses about the magnitude of “income creation” e¤ects.
In their recent analysis of tax policy and income inequality, Gramlich, Kasten,
and Sammartino (1993) conclude: “Not only did actual tax and transfer changes
not compensate fully [for the dramatic shift in pre-tax, pre-transfer income over the
1980s], they were not even in the proper direction.” Our experiments lead us to a
similar conclusion, but for a much di¤erent reason. Our analysis indicates that behavioral impact of changes in the marginal tax rate structures like those introduced in
TRA86 are likely to have, in and of themselves, contributed substantially to observed
changes in the pre-tax distribution of income. We emphasize that, by construction,
our computational experiments allow only behavioral responses associated with endogenous labor supply and saving choices or, in Slemrod’s words, income creation
activities.
We also …nd that the tax-induced increases in income inequality in our model
correspond to rising “welfare inequality.” The highest lifetime income groups in our
experiments gain both income share and utility from the 1989 structure. The lowest
lifetime income groups, on the other hand, su¤er utility losses in most variants of our
counterfactual simulations, and would prefer living in a steady state with the 1984
rate structure. This occurs despite the fact that aggregate income is higher under
the 1989 structure. Thus, in our experiments, the general equilibrium spillover e¤ects
do not improve the welfare of those at the bottom of the distribution. In fact, we
show that the general equilibrium e¤ects on factor prices actually serve to reduce the
lifetime utility of these groups.
Generally, our results illustrate the substantial e¤ects that real-world marginal
tax rate di¤erentials can be expected to have in reasonably calibrated general equilibrium models. They thus point to the distortionary e¤ects of marginal tax rates per
se as a potentially important element of trends in income inequality in the United
States and elsewhere.
18
VIII.
Research Bibliography
1
Altig, David and Charles T. Carlstrom, “Marginal Tax Rates and Income Inequality:
A Quantitative-Theoretic Analysis,” Working Paper 9508, Federal Reserve Bank
of Cleveland, August 1995.
2
Altig, David and Charles T. Carlstrom, “Using Bracket Creep to Raise Revenue:
A Bad Idea Whose Time Has Passed,” Economic Review, 29, Federal Reserve
Bank of Cleveland, Quarter 2, 1993, 2-11.
3
Cooley, Thomas F. and Edward C. Prescott, “Economic Growth and Business Cycles,” in Frontiers of Business Cycle Research, Thomas F. Cooley, ed., Princeton
University Press: Princeton, N.J., 1995.
4
Feenberg, Daniel R. and James M. Poterba, “Income Inequality and the Incomes of
the Very High-Income Taxpayers: Evidence From Tax Returns,” in Tax Policy
and the Economy, James M. Poterba, ed., 1993.
5
Feldstein, Martin, “The E¤ect of Marginal Tax Rates on Taxable Income: A Panel
Study of the 1986 Tax Reform Act, Journal of Political Economy, 103, June
1995, 551-72.
6
Fullerton, Don and Diane Lim Rogers, Who Bears the Lifetime Tax Burden? The
Brookings Institution: Washington, D.C., 1993.
7
Gastwirth, Joseph L., “The Estimation of the Lorenz Curve and Gini Index,” Review
of Economics and Statistics, 54, 1972, 320-22.
8
Gramlich, Edward M., Richard Kasten, and Frank Sammartino, “Growing Inequality in the 1980s: The Role of Federal taxes and Cash Transfers,” in Uneven
Tides: Rising Inequality in America, S. Danziger and P. Gottschalk, eds., Russel
Sage Foundation: New York, 1993.
9
Karoly, Lynn A., “Trends in Income Inequality: The Impact of, and Implications
for, Tax Policy,” in Tax Progressivity, Joel Slemrod, ed., Cambridge University
Press: New York, 1994, 95-129.
10
Killingsworth and James Heckman, “Female Labor Supply: A Survey,” in Handbook
of Labor Economics, vol. 1, O. Ashenfelter and R. Layard, eds., North-Holland:
New York, 1986.
11
King, Robert G., Charles I. Plosser, and Sergio T. Rebelo, “Production, Growth,
and Business Cycles,” Journal of Monetary Economics, 21, March/May 1988,
195-232.
12
Kydland, Finn E. and Edward C. Prescott, “The Computational Experiment: An
Econometric Tool,” The Journal of Economic Perspectives, 10, Winter 1996, 6986.
19
13
Lerman, Robert I., and Shlomo Yitzhaki, “Income Inequality E¤ects by Income
Source: A New Approach and Applications to the United States,” The Review
of Economics and Statistics, 67, 1985, 151-6.
14
Levy, Frank and Richard J. Murname, “U.S. Earnings Levels and Earnings Inequality: A Review of Recent Trends and Proposed Explanations,” Journal of
Economic Literature, 30, September 1992, 1333-81.
15
Macurdy, Thomas, “An Empirical Model of Labor Supply in a Life-Cycle Setting,”
Journal of Political Economy, 89, 1981, 1059-85.
16
Rogerson, Richard and Peter Rupert, “New Estimates of Intertemporal Substitution: The E¤ect of Corner Solutions for Year-Round Workers,” Journal of
Monetary Economics, 27, April 1991, 255-69.
17
Rogerson, Richard, Peter Rupert, and Randall Wright, “Homework in Labor Economics: Household Production and Intertemporal Substitution,” Working paper,
Federal Reserve Bank of Cleveland, 1996.
18
Siegal, Jeremy J., “The Real Rate of Interest from 1800-1990: A Study of the U.S.
and the U.K.,” Journal of Monetary Economics, 29, April 1992, 227-52.
19
Slemrod, Joel, “Income Creation or Income Shifting? Behavioral Responses to the
Tax Reform Act of 1986,” American Economic Review, 85, May 1995, 175-80.
20
Slemrod, Joel, “On the High-Income La¤er Curve,” Working Paper No. 93-5, O¢ce
of Tax Policy Research, University of Michigan, March 1993.
20
Table 1: Tax and Distributional Characteristics of Benchmark Model
Model
Gini Coe¢cient
Fraction of Income Earned by:
Top 5% of Income Earners
Top 1% of Income Earners
Average Tax Rate
Average Marginal Tax Ratec
Share of Gross Income for
Taxpayers with Marginal Rated :
.15
.28 (Lower)
.33
.28 (Upper)
0.359
Dataa
Lower Boundb : 0.394
Upper Boundb : 0.404
0.226
0.056
0.139
0.235
0.257
0.135
0.136
0.240
0.388
0.352
0.113
0.147
0.345
0.387
0.128
0.140
a. All data pertain to taxable returns of married persons …ling jointly in 1989, and
are taken from Statistics of Income: Individual Tax Returns 1989 and SOI Bulletin,
Spring 1993
b. Upper and lower bounds are calculated using the formulas suggested in Gastwirth
(1972).
c. Average marginal rates are weighted averages, with the weights determined by the
taxpayers’ shares of gross income.
d. Pertains to the population of taxpayers with taxable income.
Sources: Internal Revenue Service; and authors’ calculations.
21
Table 2: Steady State Comparisons of Alternative Marginal Tax Rate
Structures
1984
Aggregate Comparisons
Output (Relative to Benchmark)
Total Income Tax Revenue
(Relative to Benchmark)
Average Tax Rate
Average Marginal Tax Rate
Distributional Comparisons
Gini Coe¢cient (Income)
Blackburn Inequality Measurea
Fraction of Income Earned by:
Top 5% of Income Earners
Top 1% of Income Earners
0:94
1989
(Benchmark)
–
1:16
0:17
0:32
–
0:14
0:24
0.331
$2993
0.359
–
0.195
0.046
0.226
0.054
a. Dollar amount of lump-sum transfers that must be made from all individuals with
income below the median to all individuals with income above the median in order
to maintain the Gini inequality index at its benchmark value. See Blackburn (1989).
22
Table 3: Inequality by Income Source
1984
Gini Coe¢cient for Wage Income
Wage Income Share of Inequalitya
(Percent Contribution to Total Change)b
Gini Coe¢cient for Nonwage Income
Nonwage Income Share of Inequalitya
(Percent Contribution to Total Change)b
Gini Coe¢cient for “After-tax” Incomec
(Ratio to Gross Income Gini)
Blackburn Inequality Measure d
for After-Tax Gini
(Ratio to Gross Income Measure)
0.359
0.754
(49.2)
0.513
0.246
(50.8)
0.285
(0.863)
1989
(Benchmark)
0.441
0.733
–
0.563
0.267
–
0.327
(0.910)
$4216
(1.409)
–
–
a: Share of inequality calculated using the Lerman-Yitzhaki (1985) decomposition of
the total-income Gini coe¢cient.
b: Percent of absolute change in total-income Gini accounted for by income source
calculated using the Lerman-Yitzhaki (1985) decomposition.
c: De…ned as income less income-tax payments. The “after-tax” income measure does
not include lump-sum transfers.
d. See notes to table 2.
23
Table 4: Alternative Experimentsa
1984
Benchmark Model
Partial Equilibrium
Lower Leisure Elasticity: ¾ = 10
Higher Bequest Elasticity: ¾b = 4
No Bequests
Rates Unadjusted for Growth
Equal Revenue
40% Capital-Income Exemption
Social Security Taxes Included
Benchmark Model
Partial Equilibrium
Lower Leisure
Elasticity: ¾ = 10
Higher Bequest
Elasticity: ¾b = 4
No Bequests
Rate Unadjusted
for Growth
Equal Revenue
40% Capital-Income
Exemption
Social Security
Taxes Included
0.331
0.333
0.331
0.330
0.331
0.329
0.328
0.335
0.333
1989
(Benchmark)
0.359
0.359
0.359
0.358
0.361
0.358
0.359
0.359
0.360
Blackburn
Measureb
$2993
$2768
$2895
$2975
$3168
$2996
$3294
$2500
$2873
Average
0.90
0.95
Types 1-3
-0.13
0.20
Welfare Gainsc
Types 4-10
0.30
0.40
0.49
-0.12
0.15
0.83
0.93
0.81
-0.11
-0.35
0.31
0.19
1.51
1.38
1.08
0.78
-0.14
-0.16
0.46
0.14
1.71
1.36
0.78
-0.19
0.14
1.38
0.86
-0.09
0.24
1.40
Types 11-13
1.47
1.45
a: See text for explanations of the experiments.
b. See notes to table 2.
c. Entries represent percent of full wealth that must be transferred from individual in
steady state under 1989 marginal tax rate structure in order to maintain utility level
realized in the steady state under 1984 marginal tax rate structure.
24
Technical Appendix to “Marginal Tax Rates and Income Inequality in a
Life-Cycle Model”
IX.
Basic Demographic Structure of Model
² At each time there are 80 overlapping generations.
² Generations correspond to children aged 1-20 and adults aged 21-80.
² Only adults make economic decisions. The presence of children has no e¤ect on
the preferences of parents.
² At age 21, each generation has N = (1 + n)20 children, where n is the exogenous
rate of population growth.
² All generations die with certainty at age 80.
² Each generation consists of 13 di¤erent types that are distinguished by the lifetime wealth cohort to which they belong. The types are numbered in ascending
order from 1 to 13 according to lifetime wealth. The fraction of the total population accounted for by each of the j = 1; :::; 13 types is denoted by ° j .
² The exogenous wage e¢ciency pro…les for types 1-12 are taken from the estimates of Fullerton and Rogers. The pro…le for type 13 is chosen as part of our
calibration procedures.
² All agent-types have distinct preferences over bequests left to children.
X.
The Individual Choice Problem
Preferences
² Suppressing time subscripts for notational simplicity, types j = 1::13 have preferences given by
j
j
U(C ; L ) =
80
X
s=21
¯
(s¡21)
"
ln(cjs )
#
(lj )1¡¾
(bj )1¡¾b
+® s
+ ¯ 59 ½j N
:
1¡¾
1 ¡ ¾b
where ® is a utility weight on leisure, ¾ is the inverse of the intertemporal
elasticity of substitution in leisure, and ¾b is the inverse of the elasticity of
bequests with respect to consumption. The parameter ½j is a j-type speci…c
utility weight placed on bequests left to children.
25
² Utility is maximized over the path of consumption and leisure chosen subject
to the sequence of budget constraints
cjs = "js w(1 ¡ lsj )(1 ¡ ¿ js ) + [1 + r(1 ¡ ¿ js )]ajs¡1 + Tsj + gs ¡ ajs ¡ Nbs ;
where bs = 0 for s 6= 80. The variable gs are bequests received from parents.
Because the bequests are made following the parents’ death, gs = (1 + r)b for
s = 60, and are zero otherwise.
² Assuming that all distortionary tax payments are rebated via lump sum transfers, the wealth constraint for all types is
80
X
(1+r)¡(s¡21) cjs +(1+r)¡(80¡21) Nb
s=21
=
80
X
(1+r)¡(s¡21) "js w(1¡lsj )+(1+r)¡(60¡21) (1+r)b:
s=21
² Leisure cannot exceed the total time endowment, which here is normalized to
one. Thus, we also add the constraint lsj · 1, for all s and j.
First-order conditions
² Dropping the j superscripts for convenience, the FOCs (assuming interior solutions) are
c:
l:
¯ (s¡21) c¡1
s ¡ ¸s = 0
(1)
¯ (s¡21) ®ls¡¾ ¡ ¸s "s ws (1 ¡ ¿ s ) = 0;
(2)
¸s ¡ ¸s+1[1 + r(1 ¡ ¿ s )] = 0
(3)
½N¯ (80¡21) b¡¾b ¡ ¸80 N = 0:
(4)
a:
b:
Solving for Consumption and Leisure Paths
Given values for the aggregate variables r and w, consumption and leisure choices
for types 1-13 can be solved as follows:
² We wish to solve for c21as a function of parameters and aggregate variables.
Ignoring retirement for the moment, …rst use equation (2) to solve for ¸s :
¸s =
¯ (s¡21) ®ls¡¾
"s w(1 ¡ ¿ s )
26
Substituting into (1):
¯ (s¡21) c¡1
s =
¯ (s¡21) ®ls¡¾
"s w(1 ¡ ¿ s )
Solving for ls :
1
(5)
ls = Ás cs¾
h
s)
where Ás = "s w(1¡¿
®
From (3),
i¡ 1
¾
.
(s¡20)
¯ (s¡21) c¡1
[1 + r(1 ¡ ¿ s+1)]c¡1
s = ¯
s+1
=)
(6)
cs+1 = ¯[1 + r(1 ¡ ¿ s+1 )]cs :
To handle the case of retirement, we …rst replace each of the ws in the budget
constraint with ws¤ = ¶s ws , where ¶s = 1 if (1 ¡ ls ) > 0, and ¶s = 0 if (1 ¡ ls ) · 0.
Second, equation (5) is omitted and ls is constrained to equal unity. Substituting
equation (5) into the wealth constraint (and using the parameter ¶s de…ned above)
gives
80
X
(1 + r)¡(s¡21) cs + (1 + r)¡(80¡21) Nb
s=21
80
X
= w
s=21
=)
80
X
(1 + r)¡(s¡21) ¶s "s ¡ w
¡(s¡21)
(1+r)
s=21
80
X
1
(1 + r)¡(s¡21) ¶s "s Ás cs¾ + (1 + r)¡(60¡21) (1 + r)b:
s=21
#
"
80
80
1
X
X
N ¡ (1 + r)19
¡(s¡21)
¡(s¡21)
¾
b
=
w
(1+r)
¶
"
¡w
(1+r)
¶
"
Á
c
:
cs +
s
s
s
s
s
s
(1 + r)59
s=21
s=21
By recursive substitution of equation (6):
c21 (1 +
80
X
(1 + r)¡(s¡21)
s=22
+
"
s
Y
¤i )
i=22
#
19
N ¡ (1 + r)
(1 + r)59
1
¾
+c21
w(¶21"21Á21 +
80
X
(1 + r)¡(s¡21) ¶s "s Ás
s=22
b = w
80
X
(1 + r)¡(s¡21) ¶s "s
s
Y
i=22
(15)
s=21
where ¤i = ¯[1 + r(1 ¡ ¿ s )]: For s = 80,
¸80 = ¯ 59 c¡1
80
From equation (4) we can then obtain
"
1
b=
½c80
27
#¡
1
¾b
:
1
¤i¾
(8)
The implicit equation for c21 thus becomes
c21 (1 +
80
X
(1 + r)¡(s¡21)
s=22
+
"
w
80
X
N ¡ (1 + r)19
(1 + r)59
#"
s
Y
1
¾
w(¶21"21 Á21 +
¤i ) + c21
1
½c80
(1 + r)¡(s¡21) ¶s "s Ás
s=22
i=22
#¡
80
X
1
¾b
s
Y
¤i¾ )
s
Y
¤i¾ )
1
i=22
¡
(1 + r)¡(s¡21) ¶s "s = 0
s=21
Substituting for c80 yields
c21 (1 +
80
X
¡(s¡21)
(1 + r)
s=22
1
¾b
+c21
w
"
80
X
s
Y
1
¾
¤i ) + c21 w(¶21"21 Á21 +
s=22
i=22
19
N ¡ (1 + r)
(1 + r)59
# " #¡
1
½
1
¾b
80
X
80
Y
i=22
¡(s¡21)
(1 + r)
¶s "s Ás
1
i=22
1
¾b
¤i ¡
(1 + r)¡(s¡21) ¶s "s = 0
(16)
s=21
² Given the solution for c21 , the entire path of consumption and leisure can be
derived from equations (5) and (6).
XI.
Notes on the Solution Algorithm for Individual Choice
Problem
Note: In what follows, the symbol “'” should be read to mean “is su¢ciently
close to”.
A.
Consumption and Leisure Paths for Given Marginal Tax
Rates
For all j:
1. Set ¶s = 1 for all s. For given factor prices (r and w) and marginal tax rates
(¿ s ), solve for c21 . Use equations (5) and (6) to solve for cs , s = 22; :::; 80, and
ls , s = 21; :::80: Solve for b from equation (8).
2. If (1 ¡ ls ) > 0, then set ¶s = 1, and ls according to equation (5). If (1 ¡ ls ) · 0,
set ¶s = 0, and ls = 1.
3. Recalculate consumption path taking the ¶s calculated from step 2 as given.
Calculate implied ls from equation (5).
28
4. If ls = 1 from step 2 and ls < 1 from step 3, then set value of ls to that calculated
in step 3. Otherwise, keep ls set to 1.
5. Repeat steps 2, 3, and 4 until step 2 yields no cases in which (1 ¡ ls ) < 0.
6. Set a21 = 0 and calculate asset paths (as , s = 22; :::; 80) from the budget
contraints. (Recall that g = (1 + r)b.)
7. Verify that b ¸ 0 and a80 = Nb.
B.
Iterations on Tax Rates
1. Given the consumption and leisure paths described in subsection III.A, de…ne
s¡1
gross income at each age s as ys = "s (1 ¡ ls )w + (r+¼)a
, and taxable income
1+¼
as ys ¡ ds , where ds represents deductions (de…ned below).
2. For k tax brackets, let yek be the maximum amount of income applicable to a
given bracket.
3. Let ¿ s represent the marginal tax rate at age s under which the consumption
and leisure paths described in subsection III.A are calculated. Let ¿ 0s represent
new values of tax rates consistent with the current iteration of consumption and
leisure paths. De…ne ± s as a parameter that controls the updating of tax rates
for the next iteration of consumption and leisure paths. The parameter ±s takes
on the values 0 or ± 0 > 0. De…ne the logical variable tax convergence, which
indicates whether further iterations on tax rates are necessary. Begin iterations
with tax convergence = T RUE.
4. For each age s, determine the appropriate tax rate given ys . [Note: To apply tax
rates, model-generated income is scaled up by the ratio of mean actual income
to mean model income. (See the notes on calibration below.) Income levels are
rescaled downward after determination of appropriate tax rates.]
5. If ys ' ye1 (in which case the individual is at a “kink” in the budget surface),
set ±s = 0. If not, then check if ¿ 1 ' ¿ s . If true, then set ± s = 0 and ¿ s = ¿ 0s .
If not, set ± s = ±0 and tax convergence = F ALSE.
6. For ys > ye1: If ¿ 0s ' ¿ s , then set ± s = 0. If not, loop over the tax brackets to
…nd the maximum k for which y < yek . If ys ' yek (in which case the individual
is at a “kink” in the budget surface), set ± s = 0. If not, set ±s = ±0 and
tax convergence = F ALSE.
7. Update tax rates for next iteration on consumption and leisure paths using the
formula ¿ s = ¿ s + ± s (¿ 0s ¡ ¿ s ).
8. Repeat consumption, leisure, and tax rate iterations until full tax rate procedure
ends with tax convergence = T RUE.
29
C.
Iterations on Aggregate Capital and Labor
1. De…ne the current aggregate capital stock as K =
and the current aggregate labor supply as L =
13
P
j=1
13
P
°j
80
P
s=21
j=1
80
P
j
°
s=21
(1 + n)80¡s¡1as¡1,
(1 + n)80¡s "js (1 ¡ lsj ).
2. Let KOLD and LOLD be the values under which a given set of consumption,
leisure, and tax rate calculations are made. De…ne KN EW and LNEW to be
calculations of capital and labor made following convergence of steps III.A and
III.B.
3. If KNEW ' KOLD and LN EW ' LOLD , then the algorithm has converged to
the …xed point mapping from individual choices to the aggregate equilibrium.
If not, then update capital and labor according to the formulas K = KOLD +
¤(KNEW ¡ KOLD ) and L = LOLD + ¤(LNEW ¡ LOLD ), and repeat steps III.AIII.C until the conditions KNEW ' KOLD and LN EW ' LOLD are satis…ed.
XII.
Calibration Issues
The model will calibrated to individual tax return data from tax year 1991. The
universe of the data is chosen to be taxable returns for married persons …ling jointly.
Given the basic technology and preference parameters, the tax code, and the exogenous labor productivity pro…les for lifetime wealth types 1-12 (from Fullerton and
Rogers), …ve critical sets of parameters remain: the production scale parameter, the
intercept of the wage e¢ciency pro…les for type 13. Conceptually, these parameters
are simultaneously chosen according to the following criteria:
² Given an aggregate production technology speci…ed as y = A·µ , where · is the
aggregate capital-labor ratio, A is chosen so that average income is equal to
$52,511 (AGI divided by total returns for joint …lers with taxable returns).
² The intercept terms of the wage e¢ciency pro…les for type 13 individuals, i.e.
"13
21 , is chosen so that the top 6 percent of income earners account for 26 percent
of total income. (Note: This is equal to the return- and AGI-shares of …lers
with adjusted gross incomes over $100,000.) The slope of the pro…le for type
13 is chosen to be identical to the type-12 estimates provided by Fullerton and
Rogers. Also, we set the share of the population of the type-13 cohorts to be
equal to the return-share of taxpayers with incomes over $100,000.
² Finally, the bequest parameters ½ are chosen so that each j-type agent leaves
the level of bequests implied by table 3-8 of Fullerton and Rogers. (For type-13
agents we assume that bequests represent the same proportion of peak income
as for type-12 agents.)
30
A.
Algorithm for Scaling the Model
1. Prior to initial calculations of consumption, leisure, and tax rates, set "13
21 , ½,
52;511
and A = M ean¡M odel¡Income .
2. De…ne Mean ¡ Model ¡ Income as Y =
14
P
j=1
°j
80
P
s=21
(1 + n)80¡s ysj . Let Y OLD
be the mean income assumed in solving the individual choice problem for any
given iteration, and Y N EW be the new mean income obtained after solving that
problem.
3. If Y N EW ' Y OLD ; then A¤ = 52;511
. If not, then set mean income for the
Y¹OLD
next iteration through the individual choice problem according to the formula
Y¹ = Y OLD + ± Y¹ (Y N EW ¡ Y OLD ).
¤
4. Choosing the human capital parameter "13
21 = " is accomplished by …nding val¤
ues such that, given ½ and A , the income of those earning $100,000 or more
account for approximately 26 percent of total income (see discussion immediately preceding this subsection). Denote the fraction of the model income
accounted for by those earning more than $100,000 by ª. After determining
A¤ from steps 2 and 3, if ª ' 0:279, then set "¤ = "13
21 . If not, de…ne an initial
adjustment factor ± " , and prede…ned (constant) adjustment limits ±" and ¹± "
0
± " +¹
±"
, and repeat steps
(where ±" < ¹± " ). If ª > 0:26, then update "13
21 by ±" =
2
± " +± "
0
13
2 and 3. If ª < 0:26, then update "21 by ±" = 2 , and repeat steps 2 and 3.
Continue iterating in this fashion until the condition ª ' 0:26 is satis…ed.
B.
Algorithm for Solving for the Utility Weights for Bequests
1. Let ^bj denote the target level of total bequests for type-j agents. Choosing the
¤
¤
½j is accomplished by …nding the values such that, given values of A¤ and "13 ,
N bj ' ^bj for all j. After determining A¤ and "¤ from steps 2-4, if Nbj ' ^bj ,
¤
then set ½j = ½. If not, de…ne an initial adjustment factor ± ½ , and prede…ned
(constant) adjustment limits ± ½ and ¹±½ (where ± ½ < ¹± ½ ). If Nbj > ^bj , then
¹
update ½ by ± 0½ = ±½ +2 ±½ , and repeat steps 2-4. If Nbj < ^bj , then update ½ by
± ½ +±
± 0½ = 2 ½ , and repeat steps 2-4. Continue iterating in this fashion until the
condition N bj ' ^bj is satis…ed.
31
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