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Working Paper Series
Growth Effects of Progressive Taxes
WP 01-09
Wenli Li
Board of Governors of the Federal
Reserve Systems
Pierre-Daniel Sarte
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Growth E¤ects of Progressive Taxes¤
Wenli Li
Board of Governors of the Federal Reserve System
Pierre-Daniel Sartey
Federal Reserve Bank of Richmond
November 2001
Federal Reserve Bank of Richmond Working Paper 01-09
Abstract
Criticisms of endogenous growth models with ‡at rate taxes have highlighted two
features that are not substantiated by the data. These models generally imply: (1) that
economic growth must fall with the share of government expenditures in output across
countries, and (2) that one-time shifts in marginal tax rates should instantenously lead
to similar shifts in output growth. In contrast, we show that allowing for heterogeneous
households and progressive taxes into otherwise conventional linear growth models
radically changes these predictions. In particular, economic growth does not have to
fall, and may even increase, with the share of government expenditures in output across
countries. Moreover, discrete permanent shifts in tax policy now lead to protracted
transitions between balanced growth paths. Both of these …ndings hold whether or not
government expenditures are thought to be productive and better conform to available
empirical evidence.
JEL Classi…cation: E13, O23
Keywords: Economic Growth, Progressive Taxation, Heterogeneous Households
¤
The views expressed in this paper are solely those of the authors and do not necessarily represent those
of the Federal Reserve Bank of Richmond, the Board of Governors, or the Federal Reserve System.
y
We wish to thank especially Robert King for a number of helpful suggestions. We also thank B. Ravikumar, Darrel Cohen, and participants at the 2001 SED meeting in Stockholm for their comments.
1
The evidence that tax rates matter for growth is disturbingly fragile. This empirical fragility
contrasts sharply with the robustness of the theoretical predictions: most models predict that
income and investment taxes are detrimental to growth.
William Easterly and Sergio Rebelo, (1993)
1
Introduction
A central tenet of early endogenous growth models was that cross-country di¤erences in
average growth rates could be explained by variations in government policy. In contrast
to the older neoclassical framework, where long-run growth was exogenously determined by
the rate of technical progress, these models predicted that permanent variations in tax rates
would give rise to di¤erent steady-state growth rates. However, many of the cross-country
studies that followed were unable to con…rm this prediction and shed doubt on the validity
of the endogenous growth framework.1 Jones (1995), and Stokey and Rebelo (1995) further
argued that U.S. time series data was at odds with the implications of linear growth models.
On the basis of these models, the dramatic increase in income taxation which took place
in the early 1940s would have been expected to contemporaneously decrease the U.S. per
capita growth rate. This did not appear to be the case. Thus, both cross-country studies
and time series empirical evidence have suggested that long-run growth is independent of
…scal policy. In this paper, we present a case against this conclusion. Speci…cally, we argue
that relaxing two important assumptions of the early endogenous growth framework, namely
‡at rate taxes and a representative agent, helps reconcile theory and data.
One of the more pressing problems intrinsic to any cross-country investigation of the
growth e¤ects of taxation is that marginal tax rates are not easily observable. Since the data
on individual income and taxes required to compute these rates is only available for a small
set of economies, one is forced to choose an appropriate proxy. Under the assumption that
taxes are proportional, these proxies typically involve using some measure of tax revenue (or
government expenditures) as a fraction of GDP, or regressing this revenue on its tax base
(see Koester and Kormendi [1989]). If the government does not play a productive role, then
endogenous growth models directly imply that per capita output growth should be decreasing
in these ratios across economies. In contrast, cross-country data suggest that economic
growth either does not change much or even increases with the share of tax revenue in GDP.
In this paper, we show that this cross-sectional …nding is consistent with standard endogenous
growth models once progressive taxes are allowed and the income and wealth distributions are
1
See, for instance, Levine and Renelt (1992), Levine and Zervos (1993).
2
nondegenerate. Furthermore, this result holds whether or not public expenditures contribute
to private production.
There is little disagreement that tax policy is not only nonlinear but also varies substantially across economies (see Sicat and Virmani [1988]). The extent to which country-speci…c
tax systems are considered progressive, however, is generally sensitive to the choice or progressivity index.2 That being said, as a tax code becomes more progressive, one expects
marginal tax rates for the wealthy to increase relative to the poor. This notion has two
important consequences that help explain why economic growth may increase, rather than
fall, with the ratio of tax revenue to GDP.
First, richer agents have less incentive to accumulate wealth, both physical and human,
and may ultimately have lower pre-tax earnings in equilibrium. In developing economies,
it is also common for these agents to spend substantial resources escaping taxation. In the
end, as the degree of tax progressivity increases, it is not clear that the share of tax revenue
in GDP should rise and, in fact, it may even decline. Second, because more progressive
tax systems are generally more distortional (see Sarte [1997], and Castañeda, Diaz-Gimenez,
and Rios-Rull [1999]), they are also more likely to be associated with lower economic growth
ceteris paribus. Put together, these two observations imply that variations in progressivity
across economies cause output growth and the share of tax revenue (or public expenditures)
in GDP either to have little correlation or to move together.
The explicit modeling of non-linear taxes also has important dynamic implications. Consider, for instance, a closed economy where all factors of production are reproducible and the
production technology is linear. This is the well-known Ak framework. When the marginal
tax rate increases with income, the after-tax rate of interest is no longer invariant to changes
in the composite capital good . Therefore, contrary to the original framework, a change in
tax policy will induce some transitional dynamics as the economy moves from one balanced
growth path to another (see Yamarik [2000]). Interestingly, in Rebelo’s (1991) original linear
growth model, transitional dynamics stemming from progressive taxation play very little role
as the growth e¤ects of a tax reform occur largely on impact. However, when government
spending is allowed a productive role as in Barro (1990), the initial adverse e¤ects of a tax
increase on growth are small, and may even be reversed. In addition, the transition to a
new balanced growth path may now be quite protracted which, consistent with Jones (1995),
would make it di¢cult to identify the growth e¤ects of tax changes in time series data.
This paper is organized as follows. In section 2, we brie‡y review previous cross-country
2
Silber (1994) works out theoretical cases in which popular indices, such as Kakwani’s as well as Musgrave
and Thin’s, rank tax system di¤erently. Bishop, Formby, and Zheng (1998) show that these cases are
empirically relevant.
3
evidence on …scal policy and economic growth. Section 3 introduces our modeling of progressive taxes which stays constant across the di¤erent frameworks we consider. Section 4
revisits Rebelo’s (1991) original endogenous growth model with progressive taxes and heterogenous households. In section 5, we allow government expenditures to play a productive
role along the lines of Barro (1990), and Glomm and Ravikumar (1994, 1997). Section 6
o¤ers concluding remarks.
2
Fiscal Policy and Economic Growth in the CrossSection
Under the assumptions of proportional taxes and a representative agent, endogenous growth
models typically predict a negative correlation between growth, °, and the ratio of public
spending to GDP, G=Y . This negative correlation re‡ects the distortional e¤ects of taxation
in that , with proportional taxes, G=Y = ¿ . This prediction has long been a hallmark of the
endogenous growth literature. However, empirical cross-country growth studies, notably by
Levine and Renelt (1992), and Levine and Zervos (1993), have not been able to con…rm this
negative correlation.
Figures (1a) and (1b) illustrate this notion. Figure 1, panel (a) plots average per-capita
growth rates versus taxes on income, pro…ts, and capital gains as a fraction of GDP across
74 countries over the period 1976-1997. Figure 1, panel (b) illustrates the link between percapita growth rates and the ratio of government expenditures to GDP for the same set of
countries. The data is obtained from the World Development Indicators published by the
World Bank in 2000. If anything, the link between per-capita growth rates and the relative
size of public expenditures is increasing. To the extent that issues such as tax evasion are an
important consideration in developing economies, Figures 1, panels (c) and (d) illustrate the
same relationships as in Figures (1a) and (1b) for OECD countries only. In both cases, the
data fail to establish a negative link between the relative size of government and per-capita
output growth.3
On a less equivocal note, Tanzi and Zee (2000) argue that the relative size of government
is actually higher for richer countries. They …nd “that for the period 1985-1987, the average
total tax level in developing countries was about 17:5 percent of GDP. ... In contrast, the
average total tax level in OECD countries in the same period was more than twice as high
(36:6 percent of GDP), although there was signi…cant variance across the OECD subcountry
3
Levine and Renelt (1992) argue that this result continues to hold even when a wide range of conditioning
variables are taken into account, including intial income.
4
groups. Essentially all of the foregoing comparative observations are equally applicable to
the tax revenue data for the period 1995-1997.”
To account for these cross-sectional relationships, the next sections explore the growth
e¤ects of progressive taxes in two prototypical endogenous growth models augmented to include a non degenerate distribution of income. These models, one …rst formulated by Barro
(1990), and the other by Rebelo (1991), account for two polar assumptions regarding the use
of public expenditures. At one extreme, in Rebelo’s (1991) two-sector framework, government spending does not play a productive role. At the other extreme, in the environment
envisioned by Barro (1990), all tax revenue serves to …nance public services that enter as
an input into private production. In both cases, we show that long-run growth can increase
with the ratio of tax revenue to GDP as in the cross-section. In essence, the fact that taxes
are progressive drives a wedge between the average marginal tax rate and the ratio of tax
revenue to GDP; the distortional e¤ects of higher marginal tax rates remain, but cannot be
captured empirically with the latter ratio. Contrary to the original models, we also show
that changes in tax policy now induce distinct short-run and long-run e¤ects on economic
growth. In Barro’s model, the adverse e¤ects of an increase in taxes are reversed initially,
and there exists a long transition from one balanced growth path to another. This …nding
helps explain why the growth e¤ects of changes in tax policy may be di¢cult to identify in
time series data.
3
Progressive Taxation
We begin by describing the modeling of tax policy, which is common across the frameworks
we consider. The government balances its budget at each point in time and chooses a tax code
summarized by the tax rate, ¿ (y=Y ), where y denotes household income and Y is aggregate
income. Thus, the tax rate which applies to a given household depends only on its standing
in the economy. This modeling assumption ensures that not all households eventually face
the highest marginal tax rate simply as a result of economic growth. In other words, for the
purpose of this paper, we abstract from tax drift considerations.4 In the analysis below, we
further assume that the government sets ¿ (y=Y ) according to the following tax schedule,
4
³ y ´Á
y
; with 0 · ³ < 1; Á > 0;
¿( ) = ³
Y
Y
(1)
This phenomenon is also known as “bracket creep”. As part of the Cato Institute’s policy recommendations to the 106th U.S. Congress, Moore (1999) suggests that “real income bracket creep should be ended
by indexing tax brackets for in‡ation plus real income growth. ... In 1998, for example, worker incomes rose
by a respectable 6 percent, but tax receipts were up 10 percent. The primary culprit is real bracket creep.”
5
where, similarly to Lansing and Guo (1998), the parameters ³ and Á determine the level
and the slope of the tax schedule respectively. When Á > 0, households with higher taxable
income are subject to higher tax rates, and the more common case of proportional taxes
corresponds to Á = 0, ¿ (y=Y ) = ³. In making decisions about how much to consume and
invest, households will take into account the particular way in which the tax schedule a¤ects
their earnings.
Given the tax rate in (1), ¿ (y=Y )y represents the total amount of taxes paid by a household with income y. Because we wish to draw the implications of progressivity on economic
growth, it is helpful to distinguish between average and marginal tax rates. In this case, as
taxable income changes, total taxes paid evolve according to
³ y ´Á
@ [¿ (y=Y )y]
y
;
= ¿ m ( ) = (1 + Á)³
@y
Y
Y
(2)
where ¿ m (y=Y ) is the tax rate applied to the last dollar earned. The average tax rate is
simply ¿ (y=Y ).
While there exists no single “appropriate” way to de…ne the degree of progressivity of a
tax schedule, one of the more widely used de…nitions is expressed in terms of the ratio of
the marginal to the average tax rate. Speci…cally, a tax schedule is said to be progressive
whenever the marginal rate exceeds the average rate at all levels of income.5 In our set-up,
equations (1) and (2) imply that ¿ m (y=Y )=¿ (y=Y ) is simply 1 + Á, so that the parameter
Á captures the degree of progressivity in the tax code. In the limit, where Á = 0, the
tax schedule is “‡at” and ¿ m (y=Y ) = ¿ (y=Y ). Other methods of measuring progressivity
involve the use of indices and attempt in part to capture the degree of tax burden borne by
households at di¤erent income levels.6 A potential problem here is that the distribution of
pre-tax income is endogenous and, therefore, expected to vary in response to changes in the
tax schedule. To the degree that one is concerned with e¤ective progressivity, Creedy (1999)
writes that “the tax structure alone is insu¢cient to judge progressivity because the overall
e¤ect of a tax structure on the distribution of tax payments and the inequality of net income
cannot be assessed independently of the form of the distribution of pre-tax income.” In the
models below, we shall see that as Á increases, Kakwani’s index suggests a less than perfect
correlation between statutory and e¤ective progressivity.
While we have chosen to summarize the tax code by the speci…cation in (1) for simplicity,
it can be di¢cult in practice to gauge the degree of statutory progressivity of a given tax
schedule. Even absent tax drift, such calculations would involve sifting through the tax code
5
See Musgrave and Musgrave (1989). Another way to de…ne progressivity is to require that the average
tax rate be increasing over all income ranges, which is also satis…ed in our framework.
6
See, for instance, Kakwani (1977) and Suits (1977).
6
and accounting for various deductions to be netted out of gross income, determining the
income tax rate which applies to net income, and computing the credits deductible from the
resultant tax liability. Sicat and Virmani (1988) manage to work out marginal statutory
tax rates at discrete income levels for a number of low-income and middle-income countries.
Their results show that marginal tax rates on the highest bracket vary anywhere from 30
percent (Burkina Faso) to 95 percent (Tanzania) among the low-income countries alone. In
contrast, the marginal tax rate on the lowest bracket computed for the same set of countries
varies only from 2 percent to 20 percent. Although the authors do not publish estimated
average tax schedules, their …ndings are nevertheless suggestive of signi…cant di¤erences in
statutory progressivity across economies.
In the U.S., the income tax has undergone dramatic changes over the past two decades as
the result of several important pieces of legislation, notably the Economic Recovery Act of
1981 and the Tax Reform Act of 1986. These changes in law, a¤ecting tax deductions, credit
phase-outs, and the bounds on statutory tax brackets, have made it di¢cult to measure
true marginal tax rates. Studies on the subject are limited and probably the best data
can be found in a Congressional Budget O¢ce (2001) report which documents changes in
the e¤ective, or average, federal tax rate faced by households over the period 1979 to 1997,
and includes estimates that embody the 2001 tax law. Interestingly, the study …nds that
“progressivity has increased over the past two decades, primarily because the rate faced by
households with low incomes fell by nearly a third with the expansion, in the 1990s, of the
earned income tax credit.” In the context of our model, we can approximate this statutory
change with an increase in the curvature parameter Á. We should point out that e¤ective
rates overall have also declined, or at most remained constant, between 1979 and 1997.
Therefore a simultaneous reduction in the shift parameter ³ is likely to have occurred during
this period.
4
Long-Run Policy and Long-Run Growth Revisited
This section modi…es Rebelo’s (1991) original linear endogenous growth model to take account of progressive taxes. In order for progressivity to play a redistributive role, we further
introduce a nondegenerate distribution of income and wealth into the environment by assuming that households di¤er in their rates of impatience. We adopt this method of inducing
income heterogeneity mainly for tractability. There exists, however, substantial empirical
work which links di¤erences in earnings and wealth to diverse rates of time preference.7 More
7
See Hausman (1979), Lawrance (1991) for work using the Panel Study of Income Dynamics, Samwick
(1998) for an analysis using the Survey of Consumer Finances, as well as Warner and Pleeter (2001) for an
7
importantly, the results in this paper hinge on the fact that relatively wealthier agents may
have an incentive to reduce their pre-tax earnings relative to aggregate income as the tax
schedule becomes more progressive. In principle, this channel would remain operative in
models where heterogeneity arises from other considerations, such as borrowing constraints
as in Hugget (1993), Ayagari (1994), and Rios-Rull (1995) among others, or permanent
di¤erences in productivity as in Caucutt, Imrohoroglu, and Kumar (2001).
Consider a closed economy populated by a large number of households uniformly distributed on [0; 1]. There are N types of households and each household type is indexed by a
discount factor ¯ j where 0 < ¯ 1 · ¯ 2 ::: · ¯ N < 1. Thus, the most patient households have
discount factor ¯ N . Within each group, the measure of households is given by 1=N. Each
household receives income from previous savings and from 1 unit of an inelastically supplied
nonreproducible factor (which we can think of as land or labor). Income is either saved or
used to purchase consumption goods. Households are allowed to borrow and …nance their
debt out of wage income.
On the supply side, the economy remains exactly as in Rebelo (1991) and consists of two
production sectors. The …rst sector produces investment goods, It , using a fraction 1 ¡ Át of
the available capital stock, Zt , according to the linear technology It = A(1 ¡ Át )Zt . Here, Zt
is to be interpreted as a reproducible composite capital good, which includes both human and
physical capital, and which can be accumulated over time. Speci…cally, Zt+1 = It + (1 ¡ ±)Zt ,
where 0 < ± < 1, is the capital depreciation rate. The second sector combines the remaining
capital stock, Át Zt , with nonreproducible factors to produce consumption goods, Ct . The
quantity of nonreproducible factors is denoted by T and available in …xed supply in each
time period. Consumption goods are produced according to the Cobb-Douglas technology,
Ct = B(Át Zt )® T 1¡® .
1
Each household of type j chooses paths of consumption, fcjt g1
t=0 , and capital, fzjt gt=0 ,
to solve
max
cjt ; zjt+1
1
X
t=0
c1¡¾
t jt
¯j
¡1
, ¾ > 0,
1¡¾
"
(P1)
subject to qt cjt + zjt+1 ¡ (1 ¡ ±)zjt = yjt 1 ¡ ³
where yjt = rt zjt + ut T; Yt =
N
X
j=1
yjt
µ
yjt
Yt
¶Á #
1
,
N
and cjt ; zjt ¸ 0 for all j and t, zj0 > 0 given for all j.
article based on the military drawdown program of the early 1990s.
8
;
(3)
(4)
We denote the price of consumption in terms of the composite capital good by qt . The
variables rt and ut denote the rates of return to capital and nonreproducible factors respectively. In solving their optimal consumption-investment allocation problem, households take
1
the sequence of prices frt g1
t=0 and fut gt=0 as given. Thus, the following Euler equation is
obtained for each household of type j,
("
)
µ
¶¾
µ
¶Á #
qt+1 cjt+1
yjt+1
= ¯j
1 ¡ (1 + Á) ³
rt+1 + 1 ¡ ± , j = 1; :::; N:
(5)
qt
cjt
Yt+1
¡ ¡¾ ¢
In addition, a transversality condition must hold for each household type, limt!1 ¯ tj ctj
=qt
zjt = 0.
Firms make their production decisions to maximize pro…ts and solve, maxÁ; Z; T A(1 ¡
P
Át )Zt + qt B(Át Zt )® T 1¡® ¡ rt Zt ¡ ut T , where Zt = N
j=1 zjt (1=N ), 0 · Át · 1, Zt ¸ 0 for all
t. Optimal …rm behavior implies that
rt = A(1 ¡ Át ) + qt ®BÁ®t Zt®¡1 T 1¡® ;
(6)
qt ®B(Át Zt )®¡1 T 1¡® = A:
(8)
ut = qt (1 ¡ ®)B(Át Zt )® T ¡® ;
(7)
and
Total tax revenues are simply used to …nance government expenditures on goods and
services. The government budget constraint is then given by
µ ¶Á
N
X
yjt
1
Gt =
³
yjt .
Yt
N
j=1
(9)
It can be easily shown that the above set-up implicitly de…nes an economy-wide resource
constraint.
Equilibrium
An equilibrium for this economy is a set of prices frt ; ut ; qt g, t = 0 ; :::; 1, household allo-
cations fcjt ; zjt g, t = 0 ; :::; 1, j = 1 ; :::; N , and …rms’ decision rules fÁt ; Zt ; T g, t = 0 ; :::; 1,
such that given prices and the tax schedule ¿ (:), i) households’ allocation decisions maximize
their lifetime utility, ii) …rms’ decision rules maximize pro…ts, and iii) all markets clear.
We now turn to the description of a balanced growth equilibrium in which all individual
and aggregate variables, expressed in units of the composite capital good, eventually grow
at the same constant rate.8 In what follows,we denote the growth rate of variable X by
° X = Xt+1 =Xt .
8
Note that if individual variables grow at some constant rate, and their aggregate also grows at a constant
rate, then these rates must all be equal.
9
Along a balanced-growth path, Át is constant and the relative price of consumption
increases at rate ° q = ° Z1¡® by equation (8). Given the production technology in the consumption sector, we have that ° C = ° ®Z . Hence, when measured in units of the composite
capital good, aggregate consumption, qt Ct , grows at rate ° qC = ° Z .
P
PN
From (4), we have that Yt = N
j=1 yjt (1=N ) = rt
j=1 zjt (1=N) + ut T = rt Zt + ut T .
® 1¡®
Therefore, Yt = A(1 ¡ Át )Zt + qt B(Át Zt ) T
by equations (6) and (7), and substituting
for qt in this last expression yields Yt = A(1 ¡ Á)Zt + (A=®)ÁZt . It follows that ° Y = ° Z in
the steady state.
Observe also that rt = A using equations (6) and (8). The law of motion for capital
further implies that ° I = ° Z . Thus, we ultimately have that ° qC = ° I = ° Y = ° Z , and it
remains only for us to describe how the growth rate of the composite capital good, ° Z , is
determined in equilibrium.
Because individual and aggregate variables grow at the same rate in the long run, yj =Y
in equation (5) is constant in the steady state. The left-hand side of this equation can
be written as ° q ° ¾cj = ° Z1¡® ° ¾cj and, since ° cj is the same for all j, individual consumption
increases at rate ° ®Z in the long run. In this model, therefore, the balanced growth rate,
° Z , and the relative distribution of income, as summarized by yj =Y for each j, are jointly
determined as a set of N + 1 equations in N + 1 unknowns,
9
82
3
>
>
>
>
=
<6
³ y ´Á 7
j
1¡®(1¡¾)
7
6
A + 1 ¡ ± , j = 1; :::; N ,
°Z
= ¯ j 41 ¡ (1 + Á) ³
>
>
Y }5
>
>
|
{z
;
:
(10)
marginal tax rate
and
N ³
X
yj ´ 1
= 1:
Y N
j=1
(11)
In the appendix, we discuss the conditions under which a solution to this set of equations
exists and is unique. A crucial di¤erence between this model and Rebelo’s (1991) original
framework is that tax reforms a¤ect both economic growth, ° Z , and the distribution of
relative pre-tax earnings, yj =Y , simultaneously. Furthermore, this implies that e¤ective
¡ y ¢Á
marginal tax rates, (1 + Á) ³ Yj , are ultimately endogenous. In the original single agent
set-up with proportional taxes, the formula for long-run growth,
1¡®(1¡¾)
°Z
= ¯ f(1 ¡ ¿ )A + 1 ¡ ±g ;
(12)
with ¿ being the constant marginal tax rate, could not possibly capture any feedback e¤ects
from economic growth to e¤ective tax rates. Easterly and Rebelo (1993) observed that this
feature of represented a serious caveat in the interpretation of their results.
10
Because households’ relative income respond to changes in progressivity in the environment we consider, the direction in which the share of tax revenue in output adjusts is not
immediately clear. Therefore, whatever the growth response, it may have been misleading to
look for evidence of a robust negative relationship between the size of government, as measured by the ratio of government expenditures to GDP, and economic growth. In contrast,
more conventional endogenous growth models with ‡at rate taxes, whereby G = ¿ Y , necessarily predicted a rise in G=Y as ¿ increased, and this rise was unambiguously accompanied
by a fall in the rate of growth by equation (12).
4.1
The Steady State E¤ects of Changes in Progressivity
To understand the importance of progressivity for the cross-sectional relationship linking
growth and taxes, consider the e¤ects of a rise in Á. For simplicity, let us focus on the case
where there are only two household groups, impatient households indexed by ¯ 1 and patient
households with discount rate ¯ 2 > ¯ 1 .
Figure 2, panel (a) illustrates a typical equilibrium where y1 =Y and y2 =Y solve equation
(10), reproduced below as
"
#
1¡®(1¡¾)
³ y ´Á
1 °Z
j
(1 + Á)³
=1¡
¡ (1 ¡ ±) ;
(13)
A
¯j
|
{z Y }
y
¿ m ( Yj )
for impatient and patient households respectively. As expected, impatient households are
relatively poorer in the long run. At the initial equilibrium growth rate, ° Z , equation (11)
P
must also hold so that (1=2) 2j=1 (yj =Y ) = 1.
Suppose that Á increases to Á0 in Figure 2, panel (b), so that the marginal tax rate
increases at all levels of income. Because taxes are progressive, this upward shift entails a
heavier tax burden for the patient households at the initial solutions for y1 =Y and y2 =Y . Fur-
thermore, as a result of this higher marginal tax rate, all households have an incentive to lower
their relative pre-tax earnings to y10 =Y and y20 =Y , and this change is particularly pronounced
for the more patient households. However, at the initial growth rate, ° Z , y10 =Y and y20 =Y no
P
longer represent an equilibrium distribution of relative income since (1=2) 2j=1 (yj0 =Y ) < 1.
Hence, in order to reach the new steady state, the growth rate must fall to ° 0Z in Figure 2,
panel (b), which induces the new distribution y100 =Y and y200 =Y . Ultimately, an increase in
Á has led to a fall in economic growth, slightly higher pre-tax relative income for impatient
P
households, and lower pre-tax income for patient households so that (1=2) 2j=1 (yj00 =Y ) = 1.
The e¤ects of the adjustment mechanism we have just described are less straightforward for the steady state share of government expenditures in GDP. In our framework,
11
P
G=Y is initially given by (1=2) 2j=1 ³(yj =Y )1+Á . In response to the change in tax policy, and given Figure (2b), the relative size of government expenditures is ultimately (1=2)
P2
00
1+Á0
where Á0 > Á, y100 =Y > y1 =Y , and y200 =Y < y2 =Y . The end result of an inj=1 ³(yj =Y )
crease in progressivity, therefore, is ambiguous as patient and impatient households’ relative
earnings move in di¤erent directions. It follows that, while ° Z unambiguously falls in Figure
2b), the relationship between ° Z and G=Y may be much ‡atter than originally suggested
by the early growth literature. In fact, if G=Y also falls as the tax schedule becomes more
progressive, than di¤erences in progressivity across economies would lead to an increasing
relationship between economic growth and the relative size of government. As in Stokey and
Rebelo (1995), the strength of this relationship would depend importantly on the elasticity
of intertemporal substitution and depreciation rates.
At this point, we …nd it helpful to introduce a simple calibrated example in order to make
matters more concrete. In addition, this numerical example will also help us in computing
the dynamic e¤ects of tax reforms. We shall think of our benchmark as that of an economy
resembling the United States. We set ® to 0:25 so as to generate a long-run investment
share of 21 percent. The capital depreciation rate is set to 5 percent annually, and we
choose the technology shift parameter, A, to yield a 2 percent per capita output growth
rate per year. Households are assumed to have log utility, ¾ = 1. We set the number of
households, N , to 15 and assume that discount rates are equally spaced between ¯ 1 and
¯ N . We then choose ¯ 1 and ¯ N to generate a Gini coe¢cient of income of 0:40, and a net
rate of return to capital of 5:4 percent. This procedure yields ¯ 1 = 0:95 and ¯ N = 0:99.
The idea that the range of discount rates help shape the distribution of relative income
should be clear. To see why time preference parameters also a¤ect the return to capital,
observe that in our model,
the average neti rate of return
to the composite capital good is
´
¡ yj ¢Á
PN ³h
A + 1 ¡ ± which, by equation (10), is simply
given by (1=N) j=1 1 ¡ (1 + Á) ³ Y
¡
¢
P
1¡®(1¡¾)
°Z
(1=N) N
j=1 1=¯ j . The parameter Á is chosen so as to generate a value of 0:1 for
Kakwani’s index.9 Finally, we set ³ to 0:20 which yields a steady state share of government
expenditures in GDP of 22 percent.
9
Kakwani’s index is de…ned as T CI ¡ Gini, where T CI is the Tax Concentration Index and Gini
denotes the Gini coe¢cient of income. The Tax Concentration Index is a measure of the relative tax
burden born by households at di¤erent income levels and, in our framework, is de…ned as T CI =
PN Pj
PN
1 ¡ (2=N 2 ) j=1 i=1 f[¿(yi =Y )yi ] =T g, where T is total tax revenue, (1=N ) i=1 ¿ (yi =Y )yi . In the case of
‡at rate taxes, Kakwani’s index is zero and, given a …xed distribution of income, this index increases with
the degree of progressivity in the tax code. By subtracting the Gini coe¢cient from the Tax Concentration
Index, Kakwani’s index attempts to neutralize the e¤ects of a change in tax policy on the distribution of
pre-tax earnings. For developed economies, calculations of Kakwani’s index range from 0.1 to 0.17 in Bishop,
Formby and Zheng (1998).
12
Figure 3 illustrates the steady state e¤ects generated by varying the degree of statutory progressivity. As expected, output growth decreases as taxes become more progressive.
This fall in economic growth is associated with a rise in the average marginal tax rate,
P
Á
(1=N ) N
j=1 (1 + Á)³(yj =Y ) , in Figure 3, panel (b). There is no question, therefore, that the
distortional e¤ects of higher marginal taxes remain.10 Additionally, because of the endogenous adjustment in the distribution of relative pre-tax income, Figure (3b) also shows that
the share of government expenditures falls. The end result, therefore, is a slightly increasing
relationship between output growth and the size of government as shown in Figure 3, panel
(c). This relationship stands as the model analog to Figure 1, panels (a) through (d). Evidently, Figure (3c) also shows that if accurate cross-country data on e¤ective marginal tax
rates were obtained, one would continue to expect a negative correlation between per capita
output growth rates and average marginal tax rates. Finally, Figure 3, panel (d) emphasizes
the point made by Creedy (1999), namely that more progressive statutory rates do not always translate into more progressive e¤ective rates. In particular, Kakwani’s index does not
increase monotonically with the ratio of the marginal to the average tax rate. Put alternatively, one cannot judge of the progressivity e¤ects of a statutory reform independently of
the induced changes in the distribution of pre-tax earnings.
4.2
Tax Reform and the Dynamics of Economic Growth
The original environment described in Rebelo (1991) did not allow for transitional dynamics.
Consequently, any change in the marginal tax rate would have been instantaneously re‡ected
in a new steady state growth rate. On the basis of this model, therefore, one expects that
evidence of tax reforms would be detectable in time series data, in the U.S. and elsewhere.
However, once progressive taxes are introduced into the environment, equation (3) shows
explicitly that after-tax output exhibits diminishing returns to the composite capital good.
For that reason, changes in tax policy may now induce a long transitional period as the
economy moves from one balanced growth path to another. In addition, if the initial growth
e¤ects of the tax reforms are small, it is not clear that such changes can be easily identi…ed
in time series data.
To study the dynamics induced by a change in tax policy, we must …rst transform our
economy’s variables so as to make them constant in the steady state. This can simply
be achieved by normalizing each variable by the composite capital good, Zt , except for
consumption variables, which we divide by Zt® , and their relative price which we normalize
by Zt1¡® . This transformation de…nes a new set of state-like variables, zjt =Zt , j = 1; :::; N .
10
Caucutt, Imrohoroglu, and Kumar (2000) also emphasize this point in a di¤erent environment.
13
Since our state space is quite large, we choose to linearize the dynamics of our transformed
system around its stationary equilibrium. The resulting set of linearized equations possesses
a continuum of solutions but only one of these is consistent with the transversality condition
for each household type. It is useful to think of the law of motion for the state variables as
St+1 = M St + H"t ;
(14)
d
1
N
b 0
where St = (zd
t =Z t ; :::; zt =Z t ; Át ) , M is the state transition matrix evaluated at the steady
state, "t captures the innovation to policy, summarized by Át , and the ‘hat’ notation stands
for percent deviations from steady state values. The linearized equations also establish a
relationship between the transformed state and control variables.
Figure 4 illustrates the e¤ects of a one-time permanent increase in tax progressivity
calibrated to generate a 1:5 percent fall in output growth in the long run. As conjectured,
the introduction of progressive taxation does imply some transitional dynamics. However,
the striking aspect of the transition from the old to the new balanced growth path is that
most of the adjustment occurs contemporaneously. At the time of the shock, the balanced
growth rate decreases by 1:45 percent. The dashed line depicts the growth e¤ects of a
permanent rise in the marginal tax rate in Rebelo’s (1991) framework. As the inside panel
makes plain, relative to the initial steady state growth rate (i.e. relative to zero percent
deviation from steady state), there is little di¤erence between the original representative
agent formulation with ‡at rate taxes and our model with heterogeneous households and
progressive taxes. Consequently, the notion that signi…cant changes in tax policy should be
contemporaneously re‡ected in output growth rates continues to hold. In particular, Stokey
and Rebelo (1995) stress that, while U.S. per capita growth rates have shown substantial
variation at time, they have never displayed a clear break in their average value.
5
Government Spending in a Simple Model of Endogenous Growth: New Implications
In the analysis thus far, all tax proceeds were spent in a way that a¤ected neither the
marginal utility of private consumption nor the production possibilities of the private sector.
We now explore an alternative formulation, …rst suggested by Barro (1990), in which tax
revenue is used to …nance public services that contribute to private production.11 There are
two reasons that lead us to reexamine this case with progressive taxes and heterogenous
households.
11
See also Glomm and Ravikumar (1994), (1997).
14
First, because public spending played a productive role in Barro’s initial set-up, the
relationship between growth and taxes tended to be that of an inverted U, re‡ecting the
fact that higher taxes …nanced a higher level of productive expenditures on the one hand,
and their distortional e¤ects on the other. In our framework, however, the relative size of
government expenditures tends to fall with changes in progressivity in the long-run (Figure
3, panel c). Therefore, as the marginal tax rate rises relative to the average rate, we expect
growth to fall not only because of the distortional e¤ects of taxes but also because government
contributions to private output are lower. Consequently, unlike in Barro’s (1990) article, both
per capita output growth and G=Y would unambiguously decrease in equilibrium so that,
in plotting one variable against the other, the cross-sectional relationships in Figure 1 would
continue to hold.
Second, while more progressive taxes ultimately lower output growth, the idea that taxes
…nance valuable public services opens up the possibility that the initial adverse e¤ects of a
progressivity increase may be small, or even reversed. Note that in Barro’s (1990) original
framework, there are no transitional dynamics. We have seen that an increase in marginal
rates motivated by higher progressivity leads to a gradual endogenous adjustment in households’ pre-tax income. In the long run, this mechanism can lower productive government
expenditures by reducing e¤ective tax revenue. The endogenous adjustment in pre-tax income, however, may be limited at the beginning of the transition, and an increase in marginal
tax rates would simply raise productive public expenditures during this phase. A direct implication is that the short and long-run growth e¤ects of an increase in progressivity may
go in opposite directions; and identifying the impact of tax reforms on economic growth in
time series data may be more subtle than previously anticipated. To illustrate these ideas,
we now turn to a more detailed description of the economic environment.12
The production technology is given by
Yt = AKt® G1¡®
L1¡®
; with 0 < ® < 1;
t
t
(15)
where Kt and Lt stand for aggregate capital and aggregate labor input respectively. As much
as possible, we have attempted to keep the notation in this and the previous section as in
the original papers. We continue to think of Kt as a composite capital good which includes
both human and physical components. It follows that the de…nition of labor in this context
12
In models where the underlying technology for output is of the type AK + H(K), where H(K) satis…es
the properties of a neoclassical production function (see Jones and Manuelli [1990] for example), permanent
changes in tax rates will also induce transition dynamics between balanced growth paths. However, the
steady state implications of these models with a representative agent are typically inconsistent with the
cross-sectional data we reviewed earlier.
15
is that of raw labor and is separate from that which allows for investment in human capital.
Total government purchases at date t are represented by Gt . For the purpose of this analysis,
we shall think of Gt as nonrival and nonexcludable and, therefore, abstract from congestion
considerations.13
As in the section above, we assume that there exists a large number of pro…t-maximizing
…rms that solve, maxKt ; Lt AKt® Gt1¡® L1¡®
¡ rt Kt ¡ wt Lt , where rt and wt denote the rental
t
rate on capital and wages respectively. Pro…t maximization yields
µ ¶1¡®
Gt
rt = ®A
L1¡®
;
t
Kt
and
(16)
µ
¶
Kt ® 1¡®
wt = (1 ¡ ®)A
Gt :
(17)
Lt
The household side of the economy remains essentially as in section 4. We assume
that each household supplies one unit of labor inelastically so that, in equilibrium, Lt =
P
(1=N ) N
j=1 1 = 1. We describe the problem of a type j household as,
max
cjt ; kjt+1
1
X
t=0
c1¡¾
t jt
¯j
¡1
,¾>0
1¡¾
"
(P2)
subject to cjt + kjt+1 ¡ (1 ¡ ±)kjt = yjt 1 ¡ ³
where yjt = rt kjt + wt ; Yt =
N
X
j=1
yjt
µ
yjt
Yt
¶Á #
;
1
,
N
(18)
(19)
and cjt ; kjt ¸ 0 for all j and t, kj0 > 0 given for all j.
1
All households take the sequence of prices frt g1
t=0 and fwt gt=0 as given, and the following
Euler equation obtains,
("
)
µ
¶¾
µ
¶Á #
yjt+1
cjt+1
= ¯j
1 ¡ (1 + Á) ³
rt+1 + 1 ¡ ± , j = 1; :::; N.
cjt
Yt+1
(20)
In addition, the usual transversality condition must also hold for each j, limt!1 ¯ tj c¡¾
tj kjt
= 0. Productive government purchases are …nanced by tax revenue as in equation (9), which
we reproduce below,
13
µ ¶Á
N
X
yjt
1
Gt =
³
yjt .
Yt
N
j=1
(21)
See Barro and Sala-i-Martin (1992) for a discussion of how congestion in public services can eliminate
scale e¤ects in economic growth.
16
The de…nition of equilibrium is analogous to that in section 4. In a manner comparable to
that in the previous section, we now turn to the description of a balanced growth equilibrium
in which all individual and aggregate variables grow at the same constant rate. We denote
this growth rate by °.
Along the balanced growth path, yj =Y is constant for each j. Equation (21) implies that
P
1+Á
the relative size of government expenditures is given by G=Y = (1=N ) N
in
j=1 ³ (yj =Y )
the steady state. Furthermore, in this steady state, wages grow at rate ° while the return
to capital, r, is constant. Speci…cally, r = ®A (G=K)1¡® by equation (16) and, given the
production technology in (15), (G=K)® = A (G=Y ). Combining these two expressions for r
and G=K yields
µ ¶ 1¡®
1
G ®
r = ®A ®
:
(22)
Y
In this model, therefore, an increase in public services relative to GDP unambiguously raises
the marginal product of capital.
In the end, we can think of long-run growth, °, the long-run distribution of relative
pre-tax earnings, yj =Y , and the size of government in the steady state, G=Y , as being
simultaneously determined as a set of N + 2 equations in N + 2 unknowns,
(·
)
µ ¶ 1¡®
³ y ´Á ¸
1
G ®
j
¾
®A ®
° = ¯j
+ 1 ¡ ± , j = 1; :::; N;
(23)
1 ¡ (1 + Á) ³
Y
Y
G X ³ yj ´1+Á 1
=
³
;
Y
Y
N
j=1
N
and
N ³
X
yj ´ 1
= 1:
Y N
j=1
(24)
(25)
In this framework with public expenditures contributing to private production, the longrun e¤ects of changes in tax progressivity cannot be worked out in terms of a simple diagram.
The endogenous adjustment in the distribution of pre-tax earnings now a¤ects economic
growth not only directly, through the income tax rate, but also indirectly through its impact
on the relative size of public infrastructures. Furthermore, unlike the case with ‡at rate
taxes originally explored by Barro (1990), and Glomm and Ravikumar (1994, 1997), these
two channels no longer have to necessarily o¤set each other.
17
5.1
Productive Government Services and the Steady State Implications of Changes in Progressivity
To explore the long-run e¤ects induced by changes in progressivity in this new environment,
we introduce once more a calibrated example similar to the one used in the previous section.
We shall also use this example in computing the transition between di¤erent balanced growth
paths. All the parameters are chosen as in section 4, except for ®, which we now set to 0:45
in order to continue matching a U.S. investment share of approximately 20 percent.
Figure 5 depicts the steady state e¤ects generated by changes in the degree of statutory
progressivity when government expenditures contribute to private output. Contrary to most
other growth models with productive public spending, an upward shift in the tax function
caused by a rise in progressivity, Á, does not lead to an inverted U shape for output growth, °.
Because more patient households choose to lower their pre-tax earnings relative to aggregate
income when faced with higher marginal tax rates, an increase in progressivity reduces
the share of government expenditures in output, G=Y , as shown in Figure 5, panel (b).
By equation (22), this e¤ect directly contributes to reducing the return to investment. In
addition, the distortional e¤ects of higher marginal tax rates remain and, in equilibrium,
output growth falls with increases in progressivity. Note in Figure 5, panel (b) that the
average e¤ective marginal tax rate still increases with Á. Because both output growth and
the relative size of government expenditures fall as the marginal tax rate rises relative to
the average rate, the implied relationship between growth and G=Y is slightly increasing in
Figure 5, panel (c). Hence, as in section 4, this relationship is once more consistent with
Figure 1, panels (a) through (d).
In the end, if there exist important variations in tax progressivity across economies,
whether or not government expenditures contribute to private production is immaterial for
the cross-sectional correlation between economic growth and the ratio of public expenditures
to output. In either case, the upward shift in higher marginal tax rates implied by higher
values of Á lowers output growth and G=Y simultaneously. Should government services
play a productive role, the downward adjustment in G=Y simply further decreases economic
growth.
5.2
Productive Government Services, Tax Reform, and the Dynamics of Economic Growth
We have already remarked that variations in progressivity, while they can explain the crosssectional correlation between output growth and the relative size of government, should also
18
be re‡ected in time series data. In Figure 4, a rise in the marginal tax rate implied by more
progressive taxes was shown to have a large contemporaneous impact on economic growth.
Interestingly, per capita output growth rates in the U.S., and in most OECD countries, do
not show any breaks in their average value.
With government services entering as an input into private production, Figure 6 shows
the dynamic transition between balanced growth paths resulting from a one-time permanent
increase in tax progressivity. As before, the rise in Á is calibrated to generate a 1:5 percent
fall in economic growth in the long run. Note that output growth increases on impact, even
if it is eventually lower in the long run. To see why, observe that as the economy moves to its
new steady state only gradually, the distribution of relative pre-tax income, yj =Y , does not
adjust instantaneously to a change in progressivity. Therefore, as shown in the inside panel
of Figure 6, the immediate e¤ect of an increase in Á is to …nance a higher ratio of public
spending to output, G=Y . Given the model’s assumptions, this increase in the relative size
of government expenditures initially raises the marginal product of capital and, as a result,
economic growth.
In the long run, the more a-uent households eventually have lower pre-tax earnings in
relative terms. As shown in Figure 6, this eventual adjustment implies lower tax revenue
relative to output and, consequently, less public infrastructures and lower economic growth.
The key point here is that the growth e¤ects of tax reform may go in opposite directions
over time. Contrary to standard linear growth frameworks with ‡at rate taxes, a permanent
increase in the marginal tax rate does not necessarily imply a corresponding and permanent
fall in economic growth contemporaneously. In particular, Figure 6 illustrates that the
transition to the new lower balanced growth path may be rather slow, with a half-life of
approximately 40 years.
In order to identify the growth e¤ects of tax policy in the U.S., Stokey and Rebelo (1995)
test for breaks in the average value of per capita output growth. The transitional dynamics
in Figure 6, however, suggest that this strategy may be inappropriate. In our example,
changes in growth rates resulting from the increase in progressivity average close to zero
over the …rst sixty years. Clearly, this does not mean that economic growth is una¤ected by
tax policy. Moreover, the mean change in the share of government spending is 1:65 percent
for the same period. Thus, following a tax reform, it is conceivable for the average ratio
of government expenditures to output to increase while the average growth rate displays
little change. Over the post-war era, U.S. data show exactly this scenario if only in a more
drastic fashion. While the mean rate of economic growth has stayed mainly constant since
the early 1940s, the mean size of government relative to GDP has increased on the order
of 15 percent. In our framework, such a dramatic outcome likely requires a simultaneous
19
increase in the scaling parameter ³ in addition to a change in progressivity. This suggests
that the successive wave of tax reforms in the U.S. since the end of World War II have not
always a¤ected the rich disproportionately relative to the poor.
5.3
Additional Considerations
It is important to emphasize that a discussion of developing economies in terms of the model
in this section, as well as section 4, would have to involve several additional considerations.
In poorer countries, the lack of proper monitoring infrastructure often makes it less costly
for households to underreport income. For instance, a 2000 report from the Associated Press
states that “only a small fraction of Russians …led tax returns in 1999, and many of them
likely fudged their declarations to underreport income.” In and of itself, this phenomenon
only enhances one of the main threads in this paper, namely that some households may …nd
it worthwhile to try and lower their pre-tax income in the face of higher taxes. According to
the same Associated Press report, the “cash-strapped government’s attempt to wrench out
a few more rubles every year through high taxes has back…red, causing rampant evasion.”
This feature of developing economies often calls into question the relevance of taxing income
rather than consumption or imports.
In fact, the composition of tax revenue does di¤er considerably between developed and
developing countries. In particular, the income-consumption tax mix is generally biased towards consumption in the latter economies. That being said, there are at least two reasons
that tend to insure a non-trivial role for income taxes in poorer countries. First, consumption
taxes are commonly regarded as more regressive than income taxes. This is often a concern
in poorer economies and, according to Tanzi and Zee (2000), India and Sri Lanka which
experimented with a graduated tax on consumption 40 years ago soon abandoned it because
of severe di¢culties in implementing it. Second, Tanzi and Zee (2000) also write that “the
rate structure of the personal income tax is often the most convenient and visible policy
instrument for most governments in developing countries to underscore their commitments
to social justice, and hence to gain political support for their policies. It is, therefore, not
surprising to …nd that many developing countries attach great importance to maintaining
some degree of nominal personal income tax rate progressivity by applying many rate brackets, and are reluctant to undertake personal income tax reforms that would suggest any
lessening of such commitments.”
20
6
Conclusion
With the advent of the endogenous growth framework, it became theoretically possible to
address some of the cross-country dispersion in average growth rates in terms of di¤erences in
public policy. Unfortunately, early endogenous growth models, of the type posited by Jones
and Manuelli (1990), or Rebelo (1991), were later shown to be at odds with the data both
in the cross section and in the time series. Above all, these models implied that economic
growth should fall with the size of government spending relative to GDP in the cross section.
They also suggested that one-time permanent shifts in tax policy would be associated with
an instantaneous and permanent change in economic growth in the time series.
In this paper, we have attempted to show that allowing for progressive taxes and household heterogeneity in these models considerably changes their predictions. In the economies
presented above, an increase in tax progressivity did lead to lower growth. However, the
endogenous adjustment in the distribution of pre-tax earnings ultimately prevented this policy change from yielding higher tax revenues as a fraction of GDP. When plotted against
each other, both of these results seemed to match well with available cross-country evidence.
Second, the explicit modeling of non-linear taxes meant that a change in tax policy would
induce a gradual adjustment from one balanced growth path to another. Furthermore, in
the case where public spending served as an input into private production, we found that
the short and long-run e¤ects of tax reforms could go in opposite directions. In particular,
within the context of a standard endogenous growth model, a one-time permanent shift in
tax policy did not lead to an instantaneous shift in the average rate of economic growth.
21
Appendix
If (1 + Á)³ < 1, and [A + (1 ¡ ±) ¡ A(1 + Á)³] <
³
¯1
¯N
´
[A + (1 ¡ ±)], then a solution to
the set of equations (10) and (11) exists and is unique.
The …rst restriction places an upper bound on the scale of the marginal tax schedule,
¿ m (y=Y ) = (1 + Á)³(y=Y )Á , and insures that households have su¢cient incentive to invest.
The second restriction on parameters holds when the degree of progressivity in the tax
schedule, as measured by Á, is high enough relative to the spread in discount factors ¯ 1 =¯ N <
1.
Equation (10) can be re-written as
"
# Á1
1¡®(1¡¾)
yj
1
1¡±
°Z
=
+
¡
; j = 1; :::; N:
Y
(1 + Á)³ A(1 + Á)³ ¯ j A(1 + Á)³
(A1)
and, from equation (11), it follows that a solution for ° Z ¸ 0 must solve
# Á1 µ ¶
"
N
1¡®(1¡¾)
X
1¡±
°Z
1
1
+
¡
= 1:
(1
+
Á)³
A(1
+
Á)³
¯
A(1
+
Á)³
N
j
j=1
(A2)
De…ne the left-hand side of (A2) as F (° Z ). There are two cases to consider, namely 1 ¡
®(1 ¡ ¾) ¸ 0 and 1 ¡ ®(1 ¡ ¾) < 0.
Suppose that 1 ¡ ®(1 ¡ ¾) ¸ 0. First, since Á can be greater than 1, the expression inside
the square brackets of equation (A1) cannot be negative. In particular, F (° Z ) is always well
1
de…ned as long as ° Z · ° Z , where ° Z = f¯ 1 [A + (1 ¡ ±)]g 1¡®(1¡¾) . Recall that ¯ 1 is the
discount rate of the most impatient households.
i Á1 ¡ ¢
PN h 1
1
1¡±
Now, when 1 ¡ ®(1 ¡ ¾) ¸ 0, F (0) =
. Therefore, if
j=1 (1+Á)³ + A(1+Á)³
N
1
h
i
P
Á ¡ 1 ¢
1
(1 + Á)³ < 1, then N
> 1 and, for any A ¸ 0, F (0) > 1.
j=1 (1+Á)³
N
1
De…ne ° Z = f¯ N [A + (1 ¡ ±) ¡ A(1 + Á)³]g 1¡®(1¡¾) . Because ¯ N is the discount rate of
the most patient households, when ° Z = ° Z , yj =Y < 1 for j = 1; :::; N ¡ 1 in equation (A1).
Hence, F (° Z ) < 1.
Since F (° Z ) is continuous, by the Intermediate Value Theorem, there exists 0 < ° Z < ° Z
such that F (° Z ) = 1. Furthermore, because F (¢) falls monotonically
with ° Z , this solution
³ ´
is unique. The restriction that [A + (1 ¡ ±) ¡ A(1 + Á)³] <
insures that ¯ N [A + (1 ¡ ±) ¡ A(1 + Á)³]g
particular f¯ N [A + (1 ¡ ±) ¡ A(1 + Á)³]g
Figure 7).
1
1¡®(1¡¾)
1
1¡®(1¡¾)
¯1
¯N
[A + (1 ¡ ±)] above simply
lies in the admissible domain for ° Z , in
1
= ° Z < f¯ 1 [A + (1 ¡ ±)]g 1¡®(1¡¾) = ° Z (see
One can then solve for the distribution of relative income, yj =Y for each j, by simply
using equation (A1). The case where 1 ¡ ®(1 ¡ ¾) < 0 can be worked out in a similar fashion.
22
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25
Figure 1.
(a)
All Countries 1976-1997
(b)
All Countries 1976-1997
60
30
50
Government Expenditures (% GDP)
Taxes on Income, Profits, Captial Gains (% of GDP)
35
25
20
15
10
40
30
20
10
5
0
0
0
2
4
6
8
10
12
0
14
2
4
(c)
OECD Countries 1976-1997
8
10
12
14
(d)
OECD Countries 1976-1997
60
25
50
20
Government Expendtures (% GDP)
Taxes on Income, Profits, Capital Gains (% of GDP)
6
Per Capita GDP Growth (%)
Per Capita GDP Growth (%)
15
10
5
40
30
20
10
0
0
0
2
4
6
8
10
0
2
4
6
Per Capita GDP Growth (%)
Per Capita GDP Growth (%)
26
8
10
Figure 2.
(a)
Marginal Tax Rate
1−
1 γ 1z −α (1−σ )
− (1 − δ )
A β2
1−
1 γ 1z −α (1−σ )
− (1 − δ )
A β1
y
Y
τ m = (1 + φ ) ζ
y1
Y
y2
Y
φ
Relative Income
(b)
y
τ m′ = (1 + φ ′ )ζ
Y
Marginal Tax Rate
1−α (1−σ
1−
1 γ ′z
A β 2
)
− (1 − δ )
1−α (1−σ
z
)
− (1 − δ )
1−α (1−σ
)
− (1 − δ )
1−
1 γ
A β2
1−
1 γ z′
A β 1
1−
1 γ 1z −α (1−σ )
− (1 − δ )
A β1
φ′
y
τ m = (1 + φ )ζ
Y
y1′
Y
y1
Y
y1′′
Y
y ′2
Y
27
y ′2′
Y
y2
Y
Relative Income
φ
Figure 3.
Figure 4.
28
Figure 5.
Figure 6.
29
Figure 7.
F (γz )
F (0 )
1
F (γ z )
0
γ ={β [ A + (1 − δ ) − A(1 + φ )ζ ]}
z
1
1−α (1−σ )
N
30
γ ={β [ A + (1 − δ )]}
1
1−α (1−σ )
z
1
γz
@FedFRASER