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Growth, Income Inequality, and Fiscal Policy: What are the Relevant Tradeoffs?*
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Abstract:
We develop an endogenous growth model with elastic labor supply, in which agents
differ in their initial endowments of physical capital. In this context, the growth rate
and the distribution of income are jointly determined. We then examine the
distributional impact of different ways of financing an investment subsidy. Policies
aimed at increasing the growth rate result in a more unequal distribution of pre-tax
income, consistent with the positive correlation between income inequality and
growth observed in the recent empirical literature. However, there is no conflict
between efficiency and equity if inequality is measured in terms of the distribution of
JEL Classification : E62, O17, O40. Keywords: endogenous growth, income inequality, fiscal policy. 1. Introduction
The last decade has seen a revival of interest in the relationship between income inequality
and growth. This research has employed a range of theoretical frameworks and yielded conflicting
empirical results. While earlier evidence suggested a negative tradeoff between growth and
inequality, more recent studies have tended to support a positive relationship; see e.g. Barro (2000),
Forbes (2000), and Lundberg and Squire (2003). Virtually all of this debate has ignored the role of
distortionary taxes. But once their presence is acknowledged, it becomes evident that policymakers
may face two potential tradeoffs, in that growth-enhancing policies may have conflicting effects on
the pre-tax and post-tax distributions of income. Of these two measures, it would seem that the
latter is in fact the more relevant as a guide to policy. Indeed, one can go even further. Since
presumably, one is ultimately interested in the effect of fiscal policy on economic welfare, a further
potential tradeoff – one between growth and welfare inequality itself – naturally arises.
In this paper we analyze the effects of fiscal policy on these various growth-inequality
tradeoffs. To do so we develop an endogenous growth model with elastic labor supply and agents
who differ in their initial endowments of physical capital. In this framework the equilibrium growth
rate and the distribution of income are jointly determined. The key mechanism generating the
endogenous distribution of income is the positive equilibrium relationship we derive between agents’
relative wealth (capital) and their relative leisure. This relationship has a very simple intuition.
Wealthier agents have a lower marginal utility of wealth. They therefore choose to work less and to
enjoy more leisure, and given their relative capital endowments, this generates an equilibrium
This role played by endogenous labor supply is analogous to its role in other models of
capital accumulation and growth, where it provides the crucial mechanism whereby demand shocks,
such as government consumption expenditure, stimulate capital accumulation. The key factor is the
wealth effect and the impact this has on the labor-leisure choice. This mechanism is also central to
empirical models of labor supply based on intertemporal optimization; see e.g. MaCurdy (1981).
There is substantial empirical evidence documenting this negative relationship between
wealth and labor supply. Holtz-Eakin, Joulfaian, and Rosen (1993) find evidence to support the
view that large inheritances decrease labor force participation. Cheng and French (2000) and
Coronado and Perozek (2003) use data from the stock market boom of the 1990s to study the effects
of wealth on labor supply and retirement, finding a substantial negative effect on participation.
Algan, Chéron, Hairault, and Langot (2003) use French data to analyze the effect of wealth on labor
market transitions, and find a significant wealth effect on the extensive margin of labor supply.
Assigning such a central role to the adjustment in labor supply relates our analysis to another
body of literature. The widening gap between working hours in the United States and Europe has
recently sparked a debate about the causes and effects of differences in labor supply; see Prescott
(2004) and Alesina et al. (2005). This literature has largely focused on whether taxes have driven
these differences, and on the impact of labor supply on growth. However, little attention has been
paid to the distributional implications of an endogenous labor supply. Our analysis can therefore be
viewed as extending this discussion to focus on this important, but neglected, aspect. The central
role of the labor-leisure tradeoff in our model, whereby policies that lead to a higher labor supply
also tend to increase inequality is consistent with the positive correlation between average hours
worked in a country and the Gini coefficient of income reported by Alesina et al. (2005) for OECD
As is well-known, the AK model laissez-faire implies a sub-optimally low equilibrium
growth rate, and some form of stimulus to investment is desirable. We therefore introduce a direct
investment subsidy and compare the growth and distributional consequences of financing this
subsidy by either a tax on capital income, on labor income, or on consumption. Changes in tax rates
are shown to have a substantial effect on the supply of labor, in line with recent empirical evidence;
see Cardia, Kozhaya, and Ruge-Murcia (2003), as well as Prescott (2004). Our results highlight the
sharply contrasting effects of these three different modes of finance.
Two essential results emerge. First, we find that policies that enhance the growth rate are
most frequently associated with greater pre-tax income inequality. This is because growth is
fostered by policies that increase the return to capital, and since capital is more unequally distributed
than is labor, higher returns to capital translate into greater income inequality. The positive
correlation between growth and income inequality, and the fact that both variables are jointly
determined, are consistent with the recent empirical findings of Barro (2000), Forbes (2000), and
Lundberg and Squire (2003). However, our analysis also indicates that such policies tend to reduce
welfare inequality, thus suggesting that (gross) income inequality is a poor proxy for the assessment
of the effects of policy on the distribution of welfare. Second, because some policies tend to have
opposite effects on the pre-tax (gross) and post-tax gross (net) distributions of income, it is possible
to induce faster growth in conjunction with a more equal distribution of disposable income.
Despite the fact that increased growth is compatible with lower welfare inequality, no policy
dominates in both dimensions. Tradeoffs therefore still exist among these three modes of finance,
and the policy maker needs to weigh these carefully in evaluating the consequences for growth and
distribution. Thus, while all three are growth enhancing, consumption tax-financing is superior from
a growth perspective, capital income tax-financing is superior from the standpoint of reducing
welfare inequality, while wage income tax-financing may actually exacerbate welfare inequality.
Overall, the analysis provides support for the use of either a tax on capital income or a tax on
consumption to finance a subsidy on investment, in that both policies increase the growth rate and
reduce inequality in post-tax income and welfare. An even more attractive policy consists of
adopting a consumption tax together with an equal-in-magnitude wage subsidy to finance the
investment subsidy, since this does not distort the labor-leisure choice.
The paper contributes to the recent literature on the relationship between income distribution
and growth.1 It is close to Bertola (1993), who also examines how policies directed at increasing the
growth rate affect the distribution of consumption, although his assumption of a constant labor
supply implies that the distribution of income is independent of policy choices. It is also related to
Bénabou (2002), where the tradeoffs engendered by different fiscal polices are examined. Bénabou
considers a model with risky human capital investment, and compares direct income redistribution
with redistributive education finance. He finds that the latter is preferable in terms of growth, but
inferior from an insurance point of view. We propose an alternative scenario, but share the
conclusion that even if it is possible to enhance both growth and equity, different policies affect
these two objectives to different extents, and hence a careful analysis of policy options is necessary.
Our approach has two main limitations. First, the assumption that agents differ only in their
initial endowments of capital coupled with an AK technology implies that there are no income
dynamics. While this is restrictive, it has the compensating advantage of enabling us to examine the
distributional consequences of fiscal policy analytically.2 The other is that we are ignoring other
important elements central to the growth-income inequality relationship, most notably human capital
and education. These aspects are emphasized by Galor and Zeira (1993) and Viaene and Zilcha
(2003), among others. We choose to focus on a different source of income differences, the role of the
return to capital, which has been largely ignored by the recent literature. The argument that the
behavior of capital returns is essential to understanding distributional differences has, however, been
emphasized by Atkinson (2003) and is supported by recent empirical evidence for the OECD (see
The paper is organized as follows. Sections 2 and 3 present the structure of the model and
derive the macroeconomic equilibrium. Section 4 employs the framework to address the impact of
taxation on both growth and inequality. Section 5 supplements our theoretical analysis with some
numerical simulations, used to illustrate some of the distributional implications of the various
policies. Section 6 concludes, while technical details are provided in the appendix.
The Model Description of the decentralized economy
Firms shall be indexed by j. We assume that the representative firm produces output in
accordance with the Cobb-Douglas production function3
where Kj denotes the individual firm’s capital stock, L denotes the individual firm’s employment of
labor, K is the average stock of capital in the economy, so that L K measures the efficiency units of
labor employed by the firm. The production function exhibits constant returns to scale in the private
factors -- labor and the private capital stock.
All firms face identical production conditions. Hence they will all choose the same level of
employment and capital stock. That is, K = K and L = L for all j, where L is the average
economy-wide level of employment. Furthermore, we assume that the aggregate labor market
clears, so that L = 1− l , where l is the average leisure time. The economy-wide capital stock yields
an externality such that in equilibrium the aggregate (average) production function is linear in the
aggregate capital stock, as in Romer (1986), namely
Y = AL K ≡ Ω(L)K
Ω L) ≡ AL and ∂Ω / L
We assume that the wage rate, ω , and the return to capital, r, are determined by their
respective marginal physical products. Differentiating the production function and given that firms
are identical, we derive equilibrium factor prices as a function of leisure time,
These expressions imply that the equilibrium return to capital is independent of the stock of capital
while the wage rate is proportional to the average stock of capital, and therefore grows with the
economy. 4 In addition, ∂r / ∂l < 0 and ∂w / ∂l > 0 , reflecting the fact that more employment (less
leisure) raises the productivity of capital but lowers that of labor.
There is a mass 1 of infinitely-lived agents in the economy. Consumers are indexed by i and
are identical in all respects except for their initial endowments of capital, Ki0. Since the economy is
growing, we are interested in the share of individual i in the total stock of capital, ki, defined as
k ≡ K K . Relative capital has a distribution function G(k ) , mean ∑ k = 1, and variance 2
All agents are endowed with a unit of time that can be allocated either to leisure, l or to
work, 1− l ≡ L . A typical consumer maximizes expected lifetime utility, assumed to be a function
of both consumption and the amount of leisure time, in accordance with the isoelastic utility function
/( − γ ) equals the intertemporal elasticity of substitution.5 The preponderance of
empirical evidence suggests that this is relatively small, certainly well below unity, so that we shall
assume γ < 0. The parameter η represents the elasticity of leisure in utility. This maximization is
subject to the agent’s capital accumulation constraint
(1− s)K = (1−τ )rK + (1−τ )(1− l )wK − (1+τ )C
where s denotes a subsidy to investment in physical capital, while τ , τ , and τ denote the tax rates
on capital income, labor income, and consumption, respectively. With the equilibrium wage rate
being tied to the aggregate capital stock, we observe from (4) that the individual’s rate of capital
accumulation depends on the aggregate stock of capital, which the individual takes as given.
The objective of this paper is to examine the various trade-offs faced by the policymaker.
The macroeconomic equilibrium therefore needs to take explicit account of the constraints to which
the government is subject. Given the four policy instruments defined above, the government
balances the public budget at each instant t in accordance with the constraint
sK = τ C +τ rK +τ (1− l)wK (5)
where C denotes aggregate consumption.
Consumer optimization
The consumer’s formal optimization problem is to choose her rate of consumption, leisure,
and rate of capital accumulation to maximize (3) subject to the accumulation equation (4). The
corresponding first-order conditions are
where λ is agent i’s shadow value of capital, together with the transversality condition
These optimality conditions are standard. Together with the individual’s accumulation
equation (4), they yield the individual saving and leisure decisions. In the Appendix we show that
the economy is always on its balanced growth path.6 The two key relationships are
The first equation gives the rate of consumption growth of the individual, while the second
yields the individual’s leisure choice. Equation (7) asserts that in equilibrium the rates of growth of
consumption and capital are equal and the same for all individuals, being equal to the tax-adjusted
return to capital less the rate of time preference all multiplied by the intertemporal elasticity of
substitution. Observe that the return to capital depends upon aggregate leisure, which as we show
below, is jointly determined in conjunction with the growth rate. Furthermore, the capital stock of all
agents grows at the same rate, implying that at any point in time, the share of agent i, k , remains
equal to her initial share k , say. That is, the relative wealth position of agents, k , is unchanging
Equation (8) represents the “relative labor supply” function. Note that the individual relative
labor supply is linear in the agent’s capital, implying that the distribution of capital does not affect
aggregate leisure. The macroeconomic equilibrium, including both the growth rate and aggregate
labor supply (leisure), is therefore independent of the distribution of wealth. Equation (8) is also the
crucial mechanism that equates growth rates across individuals. In (A.10b) in the Appendix we show
that the transversality condition (6d) implies
so that (8) yields a positive equilibrium relationship between relative wealth and leisure. This
relationship provides the link between the agent’s initial relative endowment of capital and the
equilibrium distribution of income. Wealthier agents have a lower marginal utility of wealth. They
therefore choose to supply less labor and to “buy” more leisure. In effect, they compensate for their
larger capital endowment, and the higher growth rate it would support, by providing less labor,
thereby having an exactly offsetting effect on the growth rate.
This role that the elasticity of labor supply is playing in the determination of income
distribution is analogous to the role it plays in other similar growth models. For example,
government consumption expenditure will stimulate capital accumulation in the Ramsey model, and
growth in the Romer model, if and only if labor is supplied elastically. In both cases the underlying
2.3 Macroeconomic equilibrium
With the economy always being on its balanced growth path, the key aggregate equilibrium
relationships can be summarized by the following equations:
Aggregate consumption-capital ratio
Recalling the definitions of r(l) , )
w l , and Ω(l), these equations jointly determine the
aggregate consumption-capital ratio, C K , the average leisure time, l, the average (common)
growth rate, ψ , and one of the fiscal instruments given the other three policy parameters. Given l,
(8) determines the individual leisure time, l , as a function of k , while the individual consumption-
capital ratio can be derived by dividing (6a) by (6b) and expressed as
Substituting (10b) into (10c), and recalling (2a) and (2b), the macroeconomic equilibrium of
the economy can be summarized by the following pair of equations that jointly determine the
equilibrium mean growth rate, ψ , and average leisure time, l:
(1− α )Ω(l)(1−τ ) /(1− s) − β
The first equation describes the relationship between ψ and l that ensures the equality
between the risk-adjusted rate of return to capital and return to consumption. The second describes
the combinations of the mean growth rate and leisure that ensure product market equilibrium holds.
We shall focus our attention on solutions that are not only viable, in the sense of satisfying the
transversality condition, but also generate positive equilibrium growth. From (11b) and (9), the
equilibrium solution for l must therefore lie within the range:
The laissez-faire economy
Setting the tax rates and the subsidy to zero, the equilibrium mean growth rate and leisure in
the laissez-faire economy are determined by the following pair of equations:
These RR and PP locuses are depicted in Figure 1. First, note that equation PP is always
decreasing in l, reflecting the fact that more leisure time reduces output, thus increasing the
consumption-output ratio and having an adverse effect on the growth rate of capital. In addition, the
RR curve is also decreasing in l. Intuitively, a higher fraction of time devoted to leisure reduces the
productivity of capital, requiring a fall in the return to consumption. This is obtained if the growth of
the marginal utility of consumption rises, that is, if the balanced growth rate falls. Both schedules
are concave, and sufficient conditions for a unique equilibrium to exist, at point Q, say, are
which are certainly met if γ ≤ 0 , and hold under much weaker conditions as well.
The Distribution of Income and Welfare
We now consider the relative income of agent i, having capital stock K . Her gross income
is Y = rK + wK (1− l ) , while average economy-wide income is Y = rK + wK (1− l) . Using equation
(8) to substitute for the individual’s labor supply, we can write her relative income, y ≡ Y Y , as
y (l, k ) = k +
y (l, k ) −1 = ρ (l)(k −1), where ρ (l) ≡ 1−
emphasizes that the distribution of income depends upon two factors, the
initial (unchanging) distribution of capital, and the equilibrium allocation of time between labor and
leisure, insofar as this determines factor rewards. The net effect of an increase in initial wealth on
the relative income of agent i is given by ρ(l). As long as the laissez-faire equilibrium is one of
positive growth, it is straightforward to show that7
Thus relative income is strictly increasing in k , indicating that although richer individuals choose a
lower supply of labor, this effect is insufficiently strong to offset the impact of their higher capital
income. Consequently, the standard deviation of income across the agents, σ , is less than the
(unchanging) variability of capital, σ . To see this more intuitively, note that the relative labor
supply function, equation (8), implies that the standard deviation of labor supplies can be expressed
l −η /(1+η) lies between 0 and 1 (1+η) ,
indicating that labor supplies are less unequally distributed than are capital endowments, thus
reducing the variability of income across agents relative to that of their underlying capital
The standard deviation of relative income, σ , serves as an analytically convenient measure
of (gross) income inequality. The DD locus in the lower panel of Fig. 1 illustrates the relationship
between σ and the standard deviation of capital endowments, σ , namely
Given σ , σ is a decreasing and concave function of aggregate leisure time. This is because as
leisure increases (and labor supply declines) the wage rate rises and the return to capital falls,
compressing the range of income differences between the wealthy with large endowments of capital
and the less well endowed. Thus, having determined the equilibrium allocation of labor, denoted by
Q, from the upper panels in Fig. 1, (16) determines the corresponding unique equilibrium variability
of income across agents, denoted by the point M.
Since taxes also have direct redistributive effects, we must distinguish between the before-tax
and after-tax income distributions. We therefore define the agent’s after-tax relative income as
− ρ (l,τ ,τ )(1− k ) (17a)
ρ summarizes the distribution of after-tax (net) income and is related to the corresponding
before-tax (gross) measure, ρ(l), by
ρ (l,τ ,τ ) = ρ(l) + − ρ l
with the standard deviation of after-tax income given by8
The dispersion of pre-tax income across agents will exceed the post-tax dispersion if and only if
From (17a) and (17b) we see that the income tax rates, τ and τ , both exert two effects on
the after-tax income distribution. First, by influencing the equilibrium supply of labor, l, they
influence gross factor returns, and therefore the before-tax distribution of income, as summarized by
ρ(l) . In addition, they have direct redistributive effects, which are captured by the second term on
the right hand side of (17b): a higher tax on capital income reduces this component, while a higher
tax on labor income raises it. From (14) and (17) we establish the following partial effects:
where the first term in each of these expressions captures the indirect effect of taxation on net
income inequality, and the second term the direct effect. In contrast, an investment subsidy, s, or a
consumption tax, τ , affect after-tax inequality only indirectly, through their impact on labor supply.
Finally, we compute individual welfare. By definition, this equals the value of the
intertemporal utility function (3) evaluated along the equilibrium growth path. Thus, the optimized
level of utility for an agent starting from an initial stock of capital, K , can be expressed as
The welfare of individual i relative to that of the individual with average wealth is then
where the second term has been obtained by substituting for the consumption-capital ratio. Using
equation (8), we can express relative welfare as
Consider now two individuals having relative endowments k > k . Individual 2 will have a
higher mean income. The transversality condition (9) implies that if γ > 0, then their relative
welfares satisfy x(k ) > x(k ) > 0 , while if γ < 0, x(k ) > x(k ) > 0 . However, in the latter case
absolute welfare, as expressed by (19a) is negative. Thus in either case, the better endowed agent
will have the higher absolute level of welfare.
We can now compute a measure of welfare inequality. A natural metric for this is obtained
by applying the following monotonic transformation of relative lifetime utility, enabling us to
express the relative utility of individual i as
= u(k ) = 1+ϕ(l)(k −1) where
From (9), 0 < ϕ(l) < 1, and is an increasing, concave function of l. Welfare inequality, expressed in
terms of equivalent units of capital, can then be measured by the standard deviation of relative utility
It is straightforward to show that in the absence of taxes, and assuming positive growth
so that (gross) income inequality is less the underlying wealth inequality, but exceeds welfare
inequality. Welfare inequality is plotted as UU in the lower part of Fig. 1.10 Having determined the
equilibrium leisure at Q, (20b) yields the corresponding degree of welfare inequality, denoted by N,
which by virtue of (20c) lies above M. From this figure, it is evident that any structural shift or
policy change that causes a change in the equilibrium leisure affects equilibrium gross income
inequality, M, and welfare inequality, N, in conflicting ways.
We are interested in the impact of taxes and subsidies on the following three measures of
inequality: pre-tax income inequality, post-tax income inequality, and welfare inequality.11 Before
examining the impact of specific policies, we summarize how fiscal policy will affect these
distributional measures with the following proposition:
Proposition 1:
Given the initial distribution of capital across agents:
Fiscal policy influences the before-tax (gross) distribution of income and the
distribution of welfare through its effect on the labor supply.
Any fiscal policy that increases (decreases) the supply of labor increases
(decreases) before-tax income inequality and decreases (increases) welfare
A labor income tax and a capital income tax both influence the after-tax (net)
distribution of capital in two ways; each has a direct redistributive impact, in
addition to an indirect one through changes in the labor supply. A
consumption tax or investment subsidy has only the latter effects.
Proposition 1 highlights how, when labor supply is endogenous, income inequality is a poor measure
of welfare inequality. In fact, changes in pre-tax inequality are inversely related with those of
welfare inequality. Is post-tax income inequality therefore a better measure of welfare inequality?
The answer is not necessarily. As we have seen, fiscal policy has two effects on the distribution of
post-tax income, and net income inequality need not move together with welfare inequality.
To see this, note that welfare inequality is a weighted average of consumption inequality and
leisure inequality, both of which will fall as the labor supply increases. A higher labor supply (lower
l) raises the return to capital, increasing the incentives to save, and reducing the consumption/capital
ratio. This effect is stronger for those with greater wealth holdings, so that the distribution of
consumption becomes less unequal. The distribution of leisure also becomes less unequal, (as can
be observed from σ ), and thus the overall effect is to render the distribution of welfare more equal.
Fiscal Policy and the Relationship between Inequality and Growth
A familiar feature of the Romer (1986) model is that by ignoring the externality associated
with the aggregate capital stock, the decentralized economy generates a sub-optimally low growth
rate. This suggests that by increasing the growth rate, an investment subsidy will move the
equilibrium closer to the social optimum. With heterogeneous agents, two questions arise. First,
how to finance this subsidy if the government is concerned with both average welfare and welfare
inequality. An investment subsidy raises the return to capital and will thus favor those with large
capital holdings. Are there ways in which this reverse redistribution can be avoided? Second, we
want to know the consequences of different policies for the growth-inequality relationship.
Financing an investment subsidy
In this section we investigate these questions in some detail, by considering the effect of
financing an investment subsidy, s, using one of the three distortionary taxes.
Subsidy to investment financed by a tax on capital income
Suppose that the fiscal authority decides to finance the subsidy to investment by imposing a
tax on capital income, alone. The impact of fiscal changes on the equilibrium growth rate and labor
supply (leisure) can be illustrated graphically in Fig. 1. Holding all other taxes constant, an increase
in the subsidy rate, s, shifts the RR schedule upwards, moving the equilibrium Q to the left along the
PP curve, and increasing the growth rate and reducing leisure. An increase in the tax on capital, τ ,
has the opposite effect, shifting the RR schedule downwards, so that the overall effect of this mode
of financing depends upon the size of the capital income tax needed to finance the subsidy.
Setting τ = τ = 0 in the government budget constraint, (10d), the necessary capital income
From equation (11a) we find that the direct effect of the subsidy dominates, so that the RR schedule
shifts upwards, moving the equilibrium Q to the left, increasing the growth rate and reducing leisure.
As a result M and N move to the left along the DD and UU curves, respectively, so that before-tax
income inequality, σ (l) , increases, while welfare inequality, σ (l) , declines. Recalling the net
distribution of income as characterized by (17b), we see that taxing capital income ensures that
ρ (l,τ τ ) < ρ(l) . If the redistributive effect dominates, as our simulations below suggest may
plausibly occur, the after-tax inequality actually declines, relative to the laissez-faire distribution.
Subsidy to investment financed by a tax on wage income
Alternatively, the subsidy may be fully financed by a wage tax
s (1− (α η )(l (1− l)))
α (1−(s η)(l (1−l)))
In this case, both the RR and PP schedules shift up, resulting in a higher growth rate and greater or
lower leisure, depending on the relative shifts. The ambiguous impact on leisure arises because the
wage tax tends to reduce the supply of labor, while the higher growth rate induced by the subsidy
The ambiguous response of labor complicates the impact on the inequality of income. First,
the increase (decrease) in leisure time will reduce (increase) the dispersion of gross incomes, as seen
from (14’). However, the required (positive) wage tax implies taxing the factor that is more equally
distributed, and for any given distribution of gross incomes this raises the variability of net incomes
(see (17b) above). If the policy reduces leisure it would then unambiguously increase pre-tax and
post-tax income inequality. But when leisure increases the two effects work in opposite directions:
there will be a reduction in the variability of gross income, while net income inequality may increase
or decrease as compared to the equilibrium without taxes.12
Subsidy to investment financed by a tax on consumption
As a third example, the subsidy may be financed by setting the consumption tax equal to
((α η)(l (1−l))− s)
ρ (l,τ ,τ ) = ρ(l) . Again both schedules shift upwards, increasing the growth rate.
In this case it can be shown that leisure declines, so that gross income inequality increases. Since the
consumption tax has no direct redistributive effect, the gross and the net distributions of income are
identical and hence net income inequality increases as well.
Proposition 2: Consider the effects of an investment subsidy.
An investment subsidy financed by a capital income tax increases the growth
rate, labor supply, and before-tax income inequality. It reduces welfare
inequality and has an ambiguous effect on after-tax income inequality.
An investment subsidy financed by a labor income tax increases the growth
rate. Although it has an ambiguous effect on labor supply, and therefore on
before-tax income inequality and welfare inequality, it increases after-tax
An investment subsidy financed by a consumption tax increases the growth
rate, labor supply, and both before-tax and after-tax income inequality, while
Table 1 provides formal expressions for the effects of the investment subsidy on growth and
distribution, under the different financing modes. We see that for each of the three methods of
finance there are two effects that may either reinforce or offset each other. All three taxes tend to
reduce the growth rate. However, if used to finance an investment subsidy, the positive effect of the
latter will always dominate and the growth rate will rise.
Proposition 2 highlights the different tradeoffs between inequality and growth generated by
this policy. First, changes in fiscal policy that increase the growth rate will also increase pre-tax
income inequality, in line with the empirical evidence in Forbes (2000) and others. The reason for
this is simply that faster growth induces a greater supply of labor, reducing the wage rate, and raising
the return to capital, thus making the distribution of income more unequal.13 However, this apparent
conflict between efficiency and equity disappears when inequality is measured in terms of utility.
Second, pre-tax and post-tax inequality need not move together. While the indirect effect of
the subsidy is always the same (increasing the growth rate, labor supply, and hence pre-tax
inequality), the direct effect varies, depending upon the tax used to finance it. The consumption tax
has no direct redistributive impact, the labor income tax redistributes toward those with higher
incomes, while the capital income tax redistributes in the reverse direction. These direct effects can
be large enough to dominate the indirect effect and in Section 5, below, we present numerical
examples where pre-tax and post-tax income respond in opposite ways to changes in τ and τ .
4.2 Policy rankings
The three modes of financing the subsidy can be ranked in terms of their relative impacts on
the growth rate and various inequality measures. These rankings are reported in Table 2, where we
see how they can be sharpened by mildly strengthening the restrictions, as indicated, but consistent
with our simulation results. This table offers an alternative perspective on the tradeoffs involved for
the different modes of financing the investment subsidy.
While, as noted, all methods of financing the subsidy raise the growth rate, consumption tax
financing is the most effective, and capital income tax financing the least so. At the same time,
capital income tax financing has the most adverse effect on gross income inequality, followed by the
consumption tax. Both these cases yield a positive relationship between growth and gross income
inequality, consistent with the recent empirical evidence. By contrast, the labor income tax may
quite plausibly reduce gross income inequality leading to a negative growth-gross income inequality
relationship in that case. Second, the relative rankings of the three modes of finance are precisely
reversed insofar as net income inequality is concerned, being essentially the same as those of welfare
inequality. This suggests that while gross income inequality is a poor indicator, net income
inequality is a good indicator of the relative rankings of welfare inequality.14 Third, while each
mode of financing may be superior in terms of some criterion, it is dominated in others. The table
thus highlights the need for a policy maker wishing to stimulate investment in this way to take
careful account of the tradeoffs along these various dimensions. In particular, capital income tax
financing, although inferior from the standpoint of stimulating growth, may in fact be desirable if the
policymaker attributes sufficient weight to reducing post-tax or welfare inequality.
Proposition 3:
Suppose a policy maker wishes to stimulate investment through
From a growth perspective, financing the subsidy using a consumption tax is
superior to a wage tax, and in turn to a capital income tax, although all are
In terms of welfare inequality, the capital income tax is superior to the
consumption tax, which in turn dominates the wage income tax. These
rankings also apply to their impact on net income inequality, but are reversed
in terms of their impact on gross income inequality.
5. Numerical Examples
To obtain further insights into the growth-income inequality relationship we provide some
numerical examples. To do so we use the following, mostly conventional, parameter values:
Parameter Values
The choice of production elasticity of labor measured in efficiency units implies that 60% of output
accrues to labor. One consequence of the Romer technology, is that whereas this value is realistic in
terms of the labor share of output, it implies an implausibly large externality from aggregate capital
which implies extreme solutions for the first-best fiscal policy, discussed below. The choice of the
scale parameter A = 0.75, is set to yield a plausible value for the equilibrium capital-output ratio.
Turning to the preference parameters, the rate of time preference of 4% is standard, while the
choice of the elasticity on leisure, η = 1.75, is standard in the real business cycle literature, implying
that about 72% of time is devoted to leisure, consistent with empirical evidence. Estimates of the
intertemporal elasticity of substitution are more variable throughout the literature. Our choice 0.33
is well in the range of the empirical evidence, which with few exceptions lies in the range (0,1).
Our measures of income distributions are reported in terms of standard Gini coefficients, as
well as the standard deviation measures employed in our theoretical discussion. This involves
choosing an initial distribution of wealth, which is less straightforward, as data on the distribution of
wealth are difficult to obtain.15 The choice we have made yields a Gini coefficient of income around
33.3%. This is the value observed for net income in Italy in 2000, and compares with 36.8% and
25.2% for the US and Sweden, respectively, in the same year (Luxembourg Income Study, 2005).
The first line of Table 2 reports the benchmark equilibrium in the laissez-faire economy for
our base parameters. We see that 72.5% of time is allocated to leisure, yielding a growth rate of
3.3%, a standard deviation of income inequality of 0.206, and a Gini coefficient of income of 33.3%.
This is a plausible benchmark and l = 0.725 lies in the range [0.636, 0.745], consistent with (12).
The next three rows report the numerical effects of financing a fixed (arbitrary) investment
subsidy of 30% through a capital income, wage tax, or a consumption tax, respectively. The results
illustrate the analytical results obtained above. Table 3 also confirms some of the more ambiguous
effects discussed. For example, it confirms the theoretical possibility that the redistributive effects in
both capital income tax and wage income tax financing dominate, so that post-tax inequality in the
former declines, while in the latter it rises. In addition, the table reports the welfare gains for the
average individual, expressed as equivalent variations in the initial capital stock.16 One interesting
result of these simulations is that financing the subsidy by a tax on capital income is attractive in that
it is the only policy that increases growth, while significantly reducing both welfare inequality and
net-income inequality. Although using a consumption tax generates more growth and a higher
welfare gain for the average agent, the distributions of post-tax income and welfare need not move
The last row of the table considers financing the subsidy using a combination of wage and
consumption taxes. In particular, we set τ = −τ ; that is, these two taxes are optimally set, although
the subsidy is below the first-best level.17 The effect of this policy on the growth rate is stronger
than in the previous three cases, the reason being that this policy does not distort the allocation of
time between labor and leisure. Employing only a wage or a consumption tax tends to reduce the
supply of labor, partially offsetting the effect of the subsidy. When both are used, this effect is
absent. Since setting τ = −τ results in faster growth than using only one tax, this policy generates
larger welfare gains than any of the pure policies. The effect on distribution is quite significant, as
the policy implies taxing consumption and subsidizing and wage incomes, which reduces
substantially post-tax income inequality and welfare inequality.
Concluding Comments
The literature on the relationship between growth and inequality has raised new questions
about the impact of fiscal policy, and in particular about whether or not there is a tradeoff between
redistribution and growth. The existing literature has focused on the impact of fiscal policy when
capital markets are imperfect. In this paper we have argued that if the labor supply is endogenous
and agents differ in their initial capital endowments, growth and the distribution of income are
simultaneously determined. As a result, macroeconomic policies aimed at increasing the growth rate
will have distributional implications even in the absence of capital market imperfections.
The key mechanism whereby the initial distribution of capital endowments influences the
distribution of income is through the wealth effect, which implies that wealthier agents supply less
labor, although the resulting distribution of labor supplies is less unequal than that of the capital
endowments. As a result any policy that tends to increase the supply of labor and raise the relative
return to capital raises the return to the factor that is the source of the inequality, causing the
distribution of income to become more unequal.
We have illustrated the effect of policy by examining the distributional consequences of
financing an investment subsidy through a variety of taxes. We consider the impacts of the various
policies on four key variables: the rate of growth, the pre-tax distribution or income, the post-tax
distribution of income, and the distribution of welfare. A number of results emerge. First, we find
that policies that increase the growth rate tend to make the distribution of gross income more
unequal, but that of welfare more equal. Second, it is often the case that fiscal policy has opposite
effects on the distribution of gross and net income. As a result, it is possible to increase the growth
rate and to reduce net income inequality, although gross income inequalities would be exacerbated.
Moreover, net income inequality and welfare inequality need not move together in response to a
policy change. These results highlight the fact that using gross income inequality to assess the
distributional implications of fiscal policy may be misleading when the labor supply is endogenous.
Finally, we conclude with a caveat. While the simple AK model has the advantage of
providing a tractable framework for investigating the growth-inequality relationship and its policy
implications, it also has the limitation that the economy is always on its balanced growth path. It
therefore cannot address issues pertaining to the dynamics of wealth and income distribution. In
current work we find that the essential structure of our analysis carries over to a neoclassical
technology having a diminishing marginal product of capital; see García-Peñalosa and Turnovsky,
(2006b). Specifically, the key relationship linking relative leisure to relative capital continues to
hold and plays the same crucial role. In this case, however, the relationship evolves over time,
thereby generating dynamic time paths for the distributions of wealth and income. However, further
work is needed to determine the effect of fiscal policy on these transitional dynamic time paths.
Effects of introducing investment subsidy under alternative modes of finance
η(1− l) ⎠η(1−l) ⎝ 1−γ ⎠⎦
η(1− l) ⎠η(1−l) ⎝ 1−γ ⎠⎦
η(1− l) ⎠η(1−l) ⎝ 1−γ ⎠⎦
η(1− l) ⎠η(1−l) ⎝ 1−γ ⎠⎦
Relative Rankings of Alternative Modes of Financing Investment Subsidy Gross income inequality Net income inequality Welfare Inequality Alternative Financing of Fixed Investment Subsidy
Fig 1: Equilibrium Growth, Employment, and Income Distribution
Appendix: Derivation of macroeconomic equilibrium
In this appendix we derive the macroeconomic equilibrium, showing that the economy is
always on its balanced growth path. See also García-Peñalosa and Turnovsky (2006a). We begin by
while we may write the individual’s accumulation equation (4) in the form
Taking the time derivative of (6a) and combining with (6c) implies
indicating that each agent, irrespective of her capital endowment, chooses the same growth rate for
her shadow value of capital. Taking the time derivative of (A.1) implies
Considering equations (A.3) and (A.4) for individuals i and k, immediately implies that all agents
will choose the same growth rate for consumption and leisure. This in turn implies that average
consumption, C, and leisure, l, also grow at their respective common growth rates, namely
Now turn to the aggregates. Summing (A.1) over all agents and noting that ∑ k = 1,
∑ l = l, the aggregate economy-wide consumption-capital ratio is
while summing over (A.2) yields the aggregate accumulation equation
⎝ 1− s ⎠ ⎝ 1− s ⎠
The remainder of our derivation is to show that in equilibrium l(t), l (t), k (t) ≡ K (t) K (t)
are constant through time, so that the economy is always on its balanced growth path. To show this
substitute (A.2’), (A.4), and (A.5) into (A.3) expressing it by the following differential equation in l:
Because time is bounded, in steady-state equilibrium l = 0 , with the corresponding stationary level
of l being determined where G(l ) = 0 . Thus the linearized dynamics of l about that point are
(l(t) − l ) (A.7)
It is straightforward to establish that G (
′ l ) > 0 under any plausible conditions, in which case (A.7) is
an unstable differential equation. 18 The only solution consistent with the eventual attainment of
steady state is for l(t) , and therefore for l (t) , to be constant over time.
The next step is to combine (A.2’) and (A.2) to yield the following differential equation in
the relative capital stock, k (t) ≡ K (t) K (t) , namely
This equation describes the potential evolution of the relative wealth (capital), starting from the
initial endowment k . With l , l both constant this is a simple linear equation, the properties of
which will depend the coefficient of k (t) , which we can determine from the transversality
condition. If (6d) holds for all individuals it implies the aggregate condition
With l constant, (A.2’) and (A.3) imply that λ and K both grow at constant rates. It is then
straightforward to show that (A.9) will be met if and only if r ((1−τ ) (1− s) >ψ , i.e. the tax-
adjusted equilibrium return on capital must exceed the equilibrium growth rate. It then follows from
(A.2’) that the transversality condition can be further written in the following two equivalent ways
The first equation asserts that part of income is consumed, while the latter imposes the restriction on
leisure that ensures that this will be the case.
Now returning to (A.8) we see from (A.10b) that the coefficient of k is positive implying
that the only solution consistent with long-run stability and the transversality condition is that the
right hand side of (A.8) be zero, so that k = 0 for all time. Since k reflects capital stocks that
evolve gradually over time, this is accomplished by agents selecting their respective leisure, l , in
accordance with the “relative labor supply” function
Thus (A.10b) implies a positive relationship between relative wealth and leisure. Setting l in
accordance with (A.11) implies k ≡ 0 , so that the relative wealth position of agents, k , is constant
over time. The capital stock of all agents grows at the same rate, so that at any point in time, the
share of agent i, k , remains equal to her initial share k , say. Moreover, it follows from (A.3),
(A.4), and (A.5) that individual and aggregate consumption also grow at the same common rate:
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Footnotes
*Cecilia García-Peñalosa is Chargée de Recherche at CNRS and GREQAM in Marseille, France and
would like to acknowledge the support received from the Institut d’Economie Publique (IDEP),.
Stephen Turnovsky is Castor Professor at the University of Washington and his research was
supported in part by the Castor endowment at the University of Washington. The paper has
benefited from seminar presentations at the Central European University, Erasmus University, the
University of California, Riverside, the University of Kansas, the University of Munich, and the
University of Oregon. The constructive comments of two anonymous referees are also gratefully
1 See, for example, Galor and Zeira (1993), Alesina and Rodrik (1994), Persson and Tabellini
(1994), and Aghion and Bolton (1997), as well as the overview in Aghion, Caroli, and García-
2 This assumption is common to a large part of the literature on distribution and growth, eg. Bertola
(1993), Alesina and Rodrick (1994), Persson and Tabellini (1994). However, in García-Peñalosa
and Turnovsky (2006b) we examine the simultaneous determination of growth and the distribution
of income when there are diminishing returns to capital, in which case both the distribution of wealth
and income exhibit transitional dynamics. See also Caselli and Ventura (2000) for a more general
study of heterogeneity and the dynamics of distribution in growth models.
3We note that the use of the Cobb-Douglas production function, although convenient, is actually less
restrictive than may appear. All of our analytical results continue to hold if we generalize the
production function (1a) to Y = F (L K , K ) where F (.) is homogeneous of degree one in the two
arguments, L K L K and K .
4Intuitively, in a growing economy, with the labor supply fixed, the higher income earned by labor is
reflected in higher returns, whereas with capital growing at the same rate as output, returns to capital
5The restrictions in (3) are required to ensure the concavity of the utility function in its two
6 This is also the case in the representative agent model; see Turnovsky (2000).
7 The fact that ρ(l) < 1 is immediate from its definition in (14’). It is straightforward to show that if
the equilibrium is one of positive growth, (11b) suffices to ensure that ρ(l) > 0 . We should also
note that the policy maker could set tax rates so drive ρ(l) to zero, if he wished to ensure that all
agents have the same pre-tax income. This would require a negative equilibrium growth rate to
offset the differential in the initial capital endowments.
ρ < 0 as a perverse case implying a negative relationship between wealth and
after tax income. It can effectively be ruled out for any plausible tax configuration.
9 In affluent OECD countries, such as US and Canada, pre-tax income inequality typically exceeds
post-tax income inequality (as measured by the Gini coefficients) by about 2-4 percentage points,
reflecting the progressivity of the tax structure characteristic of such economies.
10 The transversality condition (9) implies that the point U lying on the l axis lies to the left of the
11 We may also note that the inequality of current consumption,
C / C − 1 = k c / c − 1 =
12 We can, however, see that when the subsidy rate matches the externality, s = α , 1
ρ = 1 implying that the net income inequality is increased to that of the initial endowment of
13 One can also show that structural changes such as an increase in productivity or a change in the
rate of time preference will generate a positive correlation between inequality and growth; see
García-Peñalosa and Turnovsky (2006a).
14 However, the effect on net income inequality is not necessarily a good indicator of the effect on
welfare inequality in all cases, and example of where these two measures move in opposite ways is
15 We have assumed that distribution of wealth among the 5 quintiles is 0, 0, 1.2%, 12%, 86.8%,
which are consistent with the data. For example, in the US in 1992 the bottom 40% of the
population held 0.4% of total wealth, while the top 20% owed 83.8% of the total; see Wolff (1998).
16We have also evaluated the social benefits of policy in terms of a utilitarian welfare function,
defined as the sum of the welfares over the heterogeneous agents in the economy. In the case that
the distribution of endowments is uniform across agents we find that utilitarian welfare comprises
two components: the utility of the median individual, adjusted by a term that takes account of the
dispersion across agents. Since the welfare comparisons for this function are qualitatively similar to
those obtained by considering just the average individual, we do not report the numerical results in
this case. In general, the welfare level of the median agent overstates the utilitarian level of utility.
17 It is straightforward to demonstrate that in the absence of externalities in consumption or in the
labor market, the first-best combination of the taxes on labor income and consumption must satisfy
/( + τ ) = 1 so as not to distort the corresponding optimality conditions (6a) and (6b). This
relationship, familiar from the representative agent model [see Turnovsky, 2000] extends to the form
18 For example, a uniform tax rate, τ = τ , 0
γ < , together with the transversality condition suffice

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