Does tax competition really promote growth?
 Marko Koethenbuerger^{a}^{,
}^{,
},
 Ben Lockwood^{b}^{,
}
 ^{a} Department of Economics, University of Copenhagen,
Øster Farimagsgade 5, DK 1353 Copenhagen, Denmark
 ^{b} Department of Economics, University of Warwick,
Coventry CV4 7AL, UK
 Received 30 March 2008. Accepted 15 March 2009. Available online 15
September 2009.
Abstract
This paper considers the relationship between tax competition and growth in
an endogenous growth model where there are stochastic shocks to productivity,
and capital taxes fund a public good which may be for final consumption or an
infrastructure input. Absent stochastic shocks, decentralized tax setting (two
or more jurisdictions) maximizes the rate of growth, as the constant returns to
scale present with endogenous growth implies “extreme” tax competition.
Stochastic shocks imply that households face a portfolio choice problem, which
dampens down tax competition and may raise taxes above the centralized level.
Growth can be lower with decentralization. Our results also predict a negative
relationship between output volatility and growth with
decentralization.
Keywords
 Tax competition;
 Uncertainty;
 Stochastic growth
JEL classification
1. Introduction
The link between fiscal decentralization and
economic growth is increasingly attracting the attention of economists. In
particular, a growing body of empirical research is investigating the links
between measures of fiscal decentralization and growth, both at the country and
subnational level. Overall, the evidence is mixed. In particular, crosscountry
studies, which generally use similar measures of fiscal decentralization, can
find positive or negative effects, depending on the precise measure of
decentralization, sample, estimation method, etc. (
[Davoodi and Zou, 1998],
[Wooler and Phillips, 1998],
[Zhang and Zou, 1998],
[Iimi, 2005] and [Thornton,
2007]). More recently, two studies on US data have found more robust
evidence that fiscal decentralization increases growth (
[Akai and Sakata, 2002] and [Stansel,
2005]). For example, Stansel
(2005), in a study of growth over 30 years in 314 US metropolitan
areas, has found that the degree of fractionalization (the number of county
governments per million population in a metropolitan area) significantly
increases growth.
On the theoretical side, explanation of the
mechanisms linking fiscal decentralization and growth are thin on the ground.
Two mechanisms have been studied. First, as shown by Hatfield
(2006), tax competition will quite generally raise the posttax
return on capital, thus increasing the return to savings, and thus growth, in an
endogenous growth model. Second, Brueckner
(2006) shows that centralization, if it imposes uniform public good
provision across regions, can lower the rate of savings and thus growth,
although this mechanism appears to require differences in the mix of young and
old across fiscal jurisdictions.
This paper makes a contribution to understanding of
the tax competition mechanism. We set up a multijurisdiction endogenous growth
model which combines two empirically relevant motives for investing outside the
jurisdiction, namely rateofreturn arbitrage and portfolio diversification. To
create demand for portfolio diversification we assume that there are independent
stochastic shocks to production in each of n regions. The
diversification motive is absent in standard tax competition models which only
take rateofreturn differentials as the driving force for investing abroad. The
rateofreturn motive accounts for the wellknown positive fiscal externality in
tax competition: a higher capital tax rate causes capital to flow to other
jurisdictions and expands the tax base therein. The consequence is that taxes
will be set at an inefficiently low level.
The existence of output shocks generates stochastic
returns to capital invested by a household in each region. So, if taxes are not
too different, the household located in one region will want to invest some of
its accumulated capital in all regions. This in turn generates a
negative fiscal externality: an increase in the capital tax in any “foreign”
region reduces the return on capital invested in that region. Specifically,
because the “home” household will not wish to withdraw all of its savings from
the foreign region in response to higher taxes, in order to maintain a
diversified portfolio, its interest income will go down. The key point is that
this negative “rateofreturn” externality offsets the usual positive fiscal
externality arising from mobile capital (i.e. that an increase in the foreign
region's tax leads to a capital outflow from the foreign region to the home
region). This implies that when the second externality dominates, taxation under
decentralization will be higher, and growth lower, than with
centralization. In the simple AK version of our growth model, in which a
consumption public good is financed out of a tax on capital, analytical results
show that this occurs when: (i) the number of regions is small and (ii) the
variance of the shock is sufficiently high.
We then modify the AK model to the Barro
(1990) model of infrastructure growth in which we allow the
production technology to be stochastic. This has the consequence that the
pretax rate of return to capital in a region becomes more variable as the tax
and thus the amount of infrastructure good in that region increases. This
specification introduces a third type of tax externality, not seen in the
literature on fiscal competition so far, which we call the
riskexposure externality. Specifically, a higher tax increases the
riskiness of investment and thus the riskbearing of nonresidents. The
riskexposure externality is negative and, thus, counteracts the tendency of
taxes to race to the bottom in fiscal competition. As to growth, we find,
consistently with Alesina
and Rodrik (1994), that absent stochastic shocks, centralization
yields a tax rate which is too high to be growth maximizing, while
decentralization yields a tax rate which is growth maximizing. By a continuity
argument, growth is higher under decentralized government when the variance of
the shock is small. But, as the variance of the shock increases, centralization
may generate higher growth than decentralization, as in the consumption
public good case.
One of the interesting predictions of our model
concerns the relationship between the variance of stochastic shocks and growth.
With a public consumption good, we show analytically that growth is (at least
weakly) decreasing in the variance of the output shock. With a public
infrastructure good, and fiscal decentralization, simulation results indicate a
negative relationship between growth and the variance of output shock. This is
consistent with the macroeconomic evidence (see Ramey
and Ramey, 1995), although of course there are other mechanisms
linking output shocks and growth (Jones
and Manuelli, 2005).
Finally, the question arises as to why we need
stochastic shocks and portfolio diversification as the mechanism for generating
a countervailing negative externality in our endogenous growth model. After all,
there are various other mechanisms (see Section
2) that tend to raise taxes above the socially optimal level in
static tax competition models. The answer is the following. In the AKtype
growth model without stochastic shocks, the firm's demand for capital is
perfectly elastic at the taxinclusive price of capital. This in turn, under the
standard assumption of perfect mobility of capital across regions (also made in
this paper) implies “extreme” or Bertrand tax competition: each jurisdiction can
undercut the others by a fraction and capture all the capital in the economy
(Hatfield,
2006). This extreme competition dominates other mechanisms (such as
tax exporting) which tend to raise taxes. Stochastic shocks, by contrast, have
two effects. As already pointed out, they generate the countervailing
rate of return externality. But also, they weaken the tax undercutting
incentive; a small cut in tax will now only lead to a small capital inflow.
So, our view is that stochastic shocks are the only means by which a
microfounded model can generate higher taxes and lower growth under
decentralization.
The rest of the paper is organized as follows.
Section
2 discusses the related literature. Section
3 introduces the model, solves for equilibrium conditional on fixed
government policy, and identifies the fiscal externalities at work in the model.
Section
4 contains the main results. Section
5 modifies the model to include infrastructure public goods. Section
6 concludes.
2. Related literature
The static tax competition literature emphasizes
several mechanisms which can offset, or even dominate, the basic positive mobile
tax base externality. For example, foreign ownership of fixed factors can lead
to a tax exporting incentive to raise taxes (Huizinga
and Nielsen, 1997). Or, if countries are asymmetric, capital
importers wish to set higher taxes in order to lower the cost of capital (
[Bucovetsky, 1991] and [Wilson,
1991]). Neither of these mechanisms apply here. Fixed factors (if present)
are owned by domestic residents, and since the model is symmetric and we confine
attention to symmetric equilibria, net trade in capital is zero in
equilibrium.
In a more closely related contribution, Lee
(2004) studies the impact of stochastic output shocks on tax
competition in the usual static model of mobile capital. However, in his model,
as the number of regions is assumed large, investors can be sure of a certain
return on capital, and only face uncertain wage income. Thus, the negative
externality arising though portfolio choice which is studied here does not arise
in his model; his focus is on the fact that capital taxes can provide insurance
against random fluctuations in wages, and thus will be used even a fixed factor
can be taxed directly.^{1}
Related literature on economic growth and public
finance is as follows. We have already noted that in the AK model, absent
uncertainty, with decentralization, Bertrand tax competition will drive taxes
down to zero and maximize growth. Hatfield
(2006) has shown that a similar conclusion holds very generally^{2}
in Barro's
(1990) model of infrastructure growth: that is, there is a
taxinclusive price of capital at which firms are willing to employ any level of
the capital input. So, the allocation of capital is determined by households,
who move capital to the region with the highest price. Thus, again there is
extreme or Bertrand tax competition: jurisdictions compete to set taxes to
achieve the highest pretax price of capital. As there is an infrastructure
public good funded by the capital tax, this pricemaximizing tax is not zero,
but strictly positive. Nevertheless, the conclusion is the same as in the AK
model: under decentralization, the rate of return on savings, and thus the
growth rate is maximized. Our paper thus shows that this otherwise quite general
result does not survive the introduction of productivity shocks.
Second, Lejour
and Verbon (1997) address the issue of (de)centralization and growth
in a deterministic AK growth model. Capital is made imperfectly mobile between
regions by the assumption of an ad hoc convex mobility cost of moving capital
between regions. Moreover, built into the cost function is an ad hoc “preference
for diversification”.^{3}
But, an explicit analysis of portfolio diversification is lacking in their
analysis.
Third, Wildasin
(2003) and Becker
and Rauscher (2007) study dynamic tax competition models in a
deterministic environment where there is a convex cost of adjusting the physical
capital stock over time, but where the household's financial capital is
perfectly mobile across jurisdictions.^{4}
Wildasin shows that in the steady state, the optimal tax set by a small
jurisdiction is increasing in the adjustment cost parameter; i.e. taxes are
higher, the less mobile over time is the physical capital stock. Becker
and Rauscher (2007) extend this result to an endogenous
infrastructure growth version of Wildasin's model. Thus, in their model, growth
is unambiguously lower, the less mobile over time is the physical capital stock.
However, these papers do not calculate the outcome under centralized tax
setting, and thus do not explicitly evaluate the effect of tax competition on
growth.^{5}
Finally, the paper is also related to an existing
literature on public finance in models of stochastic growth, where fiscal policy
rules are taken to be fixed (
[Turnovsky, 2000] and [Kenc,
2004]). By contrast, in this paper, taxes are optimized by governments. So,
this paper is the only one, to our knowledge, that studies endogenous tax policy
in a stochastic growth model. For endogenous tax policy in a deterministic
growth model, see Philippopoulos
(2003) and Philippopoulos
and Park (2003). But these papers do not deal with the issue of
capital mobility and its impact on tax policy.
3. The setup
3.1. The model
We work with a dynamic stochastic version of the
Zodrow–Mieszkowski
(1986) model, where regional governments use sourcebased capital
taxes to finance the provision of a public good, and which also has elements of
the portfolio choice model of
[Merton, 1969] and [Merton,
1971]. The economy evolves in continuous time: t∈[0,∞).
There are n>1 regions, i=1,…,n.
There is one firm in each region, which produces output from capital according
to a constant returns production function. Expressed in differential form, the
change in output in region i over the interval (t,t+dt)
is
where
k_{i}
is the capital stock at
t, and
z_{i}(t)
is a Wiener process,
^{6}
i.e. over the period
(t,t+dt),
the change
dz_{i}≡z_{i}(t+dt)z_{i}(t)
is Normally distributed with variance
and zero mean, and nonoverlapping increments are
stochastically independent. Also,
z_{i}(t),z_{j}(t)
are independent for all
i≠j.
^{7}
Firms in region i are willing to operate
at any scale iff, over any interval (t,t+dt),
the change in output per unit of capital dt+dz_{i},
is equal to the cost of capital, which comprises the rental price of capital
(expressed in differential form), dr_{i},
and the tax on the use of capital, . Thus, the rental price of capital in region
i follows the process
Each region is populated by a number of identical
infinitely lived households, and the population in each region is normalized to
1. In region i, each household has a flow of utility from private
consumption good and a public good:
where each of the two
subutility functions has the same fixed degree of relative riskaversion equal
to unity. At each
t, the household in region
i has an
endowment of the private good (wealth)
w_{i},
which it can rent to any firm. The evolution of
w_{i}
over time is described in detail below.
Finally, in region i, the public good is
wholly financed by a sourcebased tax on capital, i.e. g_{i}=τ_{i}k_{i}.
This is without loss of generality, as there is no wage income in the model, so
our analysis would go through if τ_{i}
were respecified as an output tax. The tax is chosen by a benevolent
government, that maximizes the sum of utilities of the households in its
jurisdiction.The details of the two fiscal regimes are described in more detail
below.
With decentralization, the n governments
play a stochastic differential game. As is usual in the literature on dynamic
tax models, we focus on Markov strategies^{8}
in this game. These are strategies where τ_{i}
is restricted to depend only on the state of the economy, the vector of wealth
variables w=(w_{1},…,w_{n}).
Then, in this class of strategies, we focus on symmetric perfect equilibrium
(Dockner
et al., 2000). This requires that the τ_{i}
are mutual best responses conditional on any w, and that in
equilibrium, τ_{i}=τ
and thus w_{i}=w,
for all i. Finally, we guess initially that equilibrium strategies are
stateindependent, i.e. τ_{i}
is independent of w, and we verify that this is true in equilibrium.
This greatly simplifies the analysis by ensuring that the household faces a
relatively simple savings and portfolio design problem.
Finally, we can now see more formally why in the
absence of uncertainty (σ_{i}≡0),
taxes and public good provision would be zero under decentralization. Without
uncertainty, the households in any region will simply allocate its capital to
the region where the return on savings is highest. Given the negative
relationship between r and τ
in (1),
it is then clear that all capital will flow into region(s) where τ_{i}
is lowest. This in turn implies that without mobility costs,^{9}
there will be a “race to the bottom” in capital taxes, with the only possible
equilibrium tax being zero.^{10}
3.2. Solving the household problem
Households solve a portfolio allocation/savings
problem under uncertainty. Denoting s_{ij}
as the share of wealth, w_{i},
rented to a firm in region j, wealth of the consumer located in
i evolves according to
where
∑_{j}s_{ij}dr_{j}
is the overall return on wealth over the interval
(t,t+dt).
Combining
(1) and
(3),
we get
where
μ_{i}=∑_{j}s_{ij}(1τ_{j})
is the deterministic part (average) rate of return on wealth. Thus, household in
region
i chooses
c_{i},s_{ij}
to maximize
subject to
(2) and
(4)
and
. Note, in maximizing expected utility the
household perceives public consumption
g_{i}
to be given, although possibly timevarying, and also takes
τ_{i}
as fixed and timeinvariant.
The solution to this problem is wellknown (e.g.,
Jones
and Manuelli, 2005) and easily stated. First, the optimal consumption
rule is simply
Second, the portfolio shares s_{ij},j≠i,
are determined by the n1
firstorder conditions with respect to the s_{ij}
To get intuition,
consider the tworegion case. Then,
(7)
solve for regions 1 and 2 to give
The portfolio rule is
simple and intuitive; invest more “abroad” (i.e. outside the home region
i): (i) the higher the difference in average returns as measured by
τ_{i}τ_{j}
and (ii) the lower the relative uncertainty of investing abroad, i.e.
. In the sequel, we simplify the analysis by
assuming
σ_{i}=σ,
i=1,…,n.
3.3. Fiscal externalities
In this section, we identify the fiscal
externalities at work in the model. For clarity, we focus on the tworegion case
(n=2),
thinking of region 1 as the home region. We trace the effects of a change in
τ_{2}
on home welfare. From (8),
using σ_{i}=σ,
the portfolio allocation rule for home and foreign households is
Moreover, by
definition, the capital employed in the home region is
Finally,
in the home region, the household optimal accumulation of capital follows the
rule
(4),
given also
(6);
By inspection of
(4),
(9) and
(10),
we can identify two fiscal externalities in the model.
First, increasing τ_{2}
implies s_{21}
up and s_{12}
down from optimal portfolio choice, implying k_{1}
up. This is the wellknown positive capital mobility externality: an
increase in the foreign tax causes a capital outflow, benefitting the home
region. Note that it is measured (inversely) by σ;
the higher σ,
the weaker is this externality, as the higher σ,
the stronger the incentive for the household to maintain a balanced
portfolio.
Second, an increase in τ_{2}
has other two related effects. First, it directly affects home welfare by
lowering μ_{1}=1(1s_{12})τ_{1}s_{12}τ_{2},
the deterministic part of the return on w_{1}.
Second, it lowers μ_{2},
and thus w_{2}
and thus k_{1}.
We call this the negative rate of return externality. Note that the
size of this externality is measured by s_{12},s_{21}:
specifically, this externality only operates in equilibrium if s_{12},s_{21}>0.
But, the portfolio allocation rule ensures that at symmetric equilibrium, s_{12}=s_{21}=0.5.
4. Equilibrium taxes
4.1. Centralization
Following much of the literature in fiscal
federalism (e.g.,
[Lockwood, 2002] and [Besley
and Coate, 2003]), we assume one of the defining characteristics of
centralization is that taxes are set uniformly, i.e. at the same level across
regions. Such uniformity is widely observed in practice. So, from (7),
with capital taxes being identical in all regions, each households invests an
equal share of the savings in each region, i.e. s_{ij}=1/n.
Then, from (4),
an important simplification is that every wealth level w_{i}
follows the same process, and so w_{i}=w,
implying a single state variable. Again from (4),
this state variable evolves according to
Moreover, again as
s_{ij}=1/n,
capital employed in region
i is equal to
k_{i}=w.
Using all this information in addition to the consumption rule
(6),
instantaneous utility in region
i is
So, households in all
regions have the same instantaneous utility
(12),
and this must therefore also be the objective of government. Formally, the
central government chooses
^{11}τ
to maximize
subject to
(11).
Setting up the Bellman equation, deriving the firstorder condition, and
guessing that the value function
V(w)
is linear in
lnw,
we can show (see the Appendix):
Proposition 1.
The centralized equilibrium tax rate is
The tax rate is
increasing in the preference for the public good,
β,
and the rate of discount,
ρ.
The comparative static results are intuitive.
First, the higher the marginal valuation of the public good, the higher the tax.
Second, the less the household values future growth, i.e. the higher ρ,
the more it is willing to increase the tax rate to fund the public good now, at
the expense of future growth in the tax base. Note finally that the centralized
tax is independent of the number of regions.
4.2. Decentralization
Here, the government in region i maximizes
the expected discounted utility of the household in region i—given
formally below—subject to the state Eq. (4),
and arbitrary taxes τ_{j},j≠i.
Then, invoking symmetry, decentralized equilibrium is characterized by the
solution to this problem when τ_{i}=τ,
for all i. But, it is difficult to solve for government i's
problem given general taxes . So, we assume that all regions j≠i
set the same tax τ^{*},
and solve for region i's best response to τ^{*},
say τ_{i}.
Then, imposing τ_{i}=τ^{*}
gives a condition for the Nash equilibrium tax.
We proceed as follows. First, let , be the share of wealth invested by the
households in region i in any region j≠i.
Note that from (7),
and , these share must all be the same. Indeed,
evaluating (7)
at σ_{i}=σ,
s_{ij}=s,
and , and solving, we get
Next, let s^{*}=s_{ji},
j≠i,
be the shares of wealth invested by the
households in region j≠i
in region i and the n2
regions k≠j≠i,
respectively. Again from (7),
solve the two simultaneous equations
We are only interested
in
s^{*}.
Solving
(14) and
(15),
we get
Now let w be the wealth of region
i, and w^{*}
be the average wealth of regions j≠i.
Then, by definition,
Then,
using the government budget constraint
g=τk,
and the consumption rule
c=ρw,
region
i's instantaneous payoff can be written
So, it is clear from (18) and (17)
that there are two state variables in the problem, w and w^{*}.
These follow the following processes. First, from (4),
given also (6),
σ_{i}=σ,
and the definition of s:
where
Second, the process for
some
w_{j}
is
As
tedious but
straightforward calculation gives
So, the problem for the
government of region
i is to choose
τ
to maximize
subject to
(19) and
(20),
and the portfolio allocation rules
(13) and
(16).
Our approach to this problem follows Turnovsky
(2000). We first write down the Bellman equation defining the value
function for the home region V(w,w^{*}),
and thus characterizing the optimal choice of τ,
given τ^{*}.
We then evaluate this Bellman equation at τ=τ^{*}.
To get a closedform solution for the equilibrium τ,
we must guess the correct form of the value function^{12}V(w,w^{*}),
which we are able to do, using the fact that at a symmetric equilibrium (s=s^{*}=1/n)w=w^{*},
from (19) and (20).
Specifically, we guess that V(w,w)
is linear in lnw.
All these steps are dealt with in detail in the Appendix, and the end result is:
Proposition 2.
The decentralized equilibrium tax rate is
The tax rate is
increasing in the preference for the public good,
β,
the rate of discount,
ρ,
and the size of the output shock,
σ.
This formula and the comparative statics are
intuitive. First, as under centralization, the higher the marginal valuation of
the public good, the higher the tax and, the less the household values future
growth (higher ρ),
the more it is willing to increase the tax rate to fund the public good now.
Second, the higher σ,
the weaker the response of the capital stock in any region to a change in the
tax rate in that region and so the smaller the mobile tax base externality, thus
increasing the equilibrium tax. Intuitively, the smaller the variance, the more
willing are investors to move their wealth between regions in response to tax
differences, thus increasing the mobility of the tax base.
More formally note from (17),
in symmetric equilibrium,
where in the last line
we used
w=w^{*}=k.
So, the semielasticity
is clearly decreasing
in
σ.
Note also that the more regions, the bigger this elasticity. This is a similar
result to those for the standard static model of tax competition. Note finally
this effect is operative only when
n>1;
with no fiscal competition (
n=1),
the size of stochastic shocks makes no difference to
τ.
Last, we turn to a comparison of taxes under
decentralization and centralization and how they relate to the number of regions
n. Inspection of (21)
shows that n affects the denominator of τ_{d}
in two places, corresponding to the two different externalities identified
above. First, a higher n increases the tax base elasticity, as already
remarked; this is measured by the term ρβ(n1)/σ^{2}
which is simply ρβ
times the semielasticity of the tax base with respect to τ_{d}.
Second, an increase in n increases the
rate of return externality, corresponding to the term (1/n)(1+β).
Intuitively, any resident of region i only invests 1/n
of his wealth at home in equilibrium. So, the government of region i
ignores the negative effect of the tax in region i on the rate of
return to investors in all the other regions, measured by ((n1)/n)(1+β).
As the externalities have opposite effects on τ_{d},
we expect that the effect of n on τ_{d}
is not monotonic, and this is confirmed by the following result:
Proof. The denominator
of (21),
ρβ(n1)/σ^{2}+(1+β)/n,
is a convex function of n, with a minimum at . Thus, τ_{d}
must be a quasiconcave function of n with a maximum at n=n^{*}.
Note, τ_{c}=τ_{d}
for n=1.
So, as n^{*}≥2
by assumption, there must be an integer for which τ_{c}<τ_{d}.
Also, τ_{d}→0
as n→∞
while τ_{c}>0,
∀n.
Thus, there always exists a value of with τ_{c}⪌τ_{d}
if . □
Unlike the traditional tax competition model, the
effect of n on τ_{d}
is generally ambiguous. This is of course, because an increase in n
increases both the mobile tax base and rate of return externalities, which have
opposite signs. So, for n small, the tax base externality is small and
so the rate of return externality dominates (τ_{c}<τ_{d}),
whereas for n large, the mobile tax externality dominates (τ_{c}>τ_{d}).
Fig.
1 illustrates the tax rate differential (assuming β=1
and ρ=0.1).
The thick line is the centralized tax rate τ_{c}
which is independent of uncertainty. The thin lines depict the decentralized tax
rates τ_{d}
for different levels of uncertainty. For instance, for σ^{2}=0.5
the rate of return externality dominates the mobile tax externality for n∈{2,3,4}.
The final step in our analysis is to relate taxes
and growth. First, at any instant, output must be divided between private and
public consumption, i.e. y=c+g.
Next, from the government budget constraint and (6),
we have in symmetric equilibrium that y=ρw+τk=(ρ+τ)w.
So, the growth of output is just proportional to that of wealth. Finally, from
(20)
and the definition of dz^{*},
in symmetric equilibrium:
Growth has a stochastic
and deterministic component, and only the latter is affected by taxes, being
decreasing in the tax. In what follows, we refer to
1τρ
as the average growth rate.
So, Proposition
3 indicates that when the number of regions is less (greater) than
, growth will be lower (higher) under
decentralization than centralization. From Fig.
1, when σ=0.5,
for example, . Intuitively, it is only when that the mobile tax base externality dominates
and thus growth under decentralization is higher.
One further implication of the model concerns the
relationship between the variance of stochastic shocks and growth. Under
decentralization growth is decreasing in the variance of the output shock. This
is consistent with the macroeconomic evidence (see Ramey
and Ramey, 1995).
5. Infrastructure public goods
5.1. The model
We now modify the baseline setup by allowing the
government to spend on a public infrastructure good rather than consumption
good. For tractability, we assume two regions, unstarred (home) and starred
(foreign). We will focus on the home region. Output follows the process
We
assume that the returns to
g accrue to a third factor of production
(labour) which is immobile across all regions and is fixed at 1 in each region.
Following
Turnovsky
(2000) and
Kenc
(2004), we assume that the pretax wage,
a, over the period
(t,t+dt)
is determined at the start of the period and is equal to the expected marginal
product of labor, i.e.
using the budget
constraint
g=τk.
So, the wage is nonrandom. The pretax rate of return on capital over the
period
(t,t+dt)
is thus determined residually:
using the budget
constraint
g=τk.
So, the posttax rate of return thus follows the process
where
r(τ)≡ατ^{1α}τ
plays an important role below.
5.2. Solving the household problem
Now, setting β=0
in (2),
the consumer in the home region maximizes
subject to the
stochastic wealth equation which is, using
(25)
Unlike the consumption
good case, this problem is nonstandard, and so for completeness, we provide a
solution in the Appendix. Moreover, in deriving the solution, we suppose that
the household believes
^{13}
that
k≡w,
a belief which is true in equilibrium. The consumption and portfolio allocation
rule for the home region are
We can now compare
(28) and
(9).
The difference is only in the portfolio allocation rules; for comparison, the
portfolio allocation rule in the consumption good model can be written
in the notation of this
Section. The difference between this formula and
(28),
therefore, is that in
(28),
τ,τ^{*}
affect the rule directly, and not just via their effects on
r,r^{*}.
This is because a higher tax rate in the home region increases the pretax rate
of return on capital in the home region and thereby the riskiness of the
investment from
(25).
5.3. Fiscal externalities
Besides the tax base externality and the
rateofreturn externality identified in the AKmodel, the infrastructure model
exhibits a third type of externality. Infrastructure spending increases the
return to capital and, as a result, from (25),
we see that the stochastic, as well as the deterministic, part of the
return on capital invested in a given region now depends positively on the tax
rate. This is in contrast to the consumption good model, where the tax in a
region only lowered the mean return on capital invested in that region, but did
not affect the variance of returns.
Specifically, from inspection of (25),
we see that a higher tax rate magnifies the exposure of investors to risk, i.e.
the variance of the posttax return to capital over an interval dt
is and is thus increasing in the tax rate. Since
each region also hosts capital from nonresidents, this specification introduces
a second negative externality, i.e. a higher tax increases the riskiness of
investment and thus the riskbearing of nonresidents. We call this externality
the riskexposure externality. We will now characterize the equilibrium
tax policy under (de)centralization and relate it to the externalities.
5.4. Centralization
As in the consumption good case, we assume that
taxes are set uniformly, i.e. τ=τ^{*}.
So, from (28),
each households invests an equal share of the savings in each region, i.e. s=s^{*}=0.5.
Then, from (27),
w=w^{*},
and the single state variable w evolves according to
where we have also used
k=w.
So, under centralization, the government maximizes
subject to
(29).
The Bellman equation is
where
. The firstorder condition w.r.t.
τ
in
(30)
is
Now, we guess V(w,w)=A+Blnw.
Then, the firstorder condition becomes, cancelling B,
To interpret this
condition, it is helpful to note that (following exactly the argument in the
consumption case), the average growth rate in output is
r(τ)ρ,
where
r(τ)=ατ^{1α}τ
is a strictly concave function with a unique maximum at
. So, although
(31)
cannot be solved explicitly for the tax rate, we can see that in the absence of
uncertainty,
r^{′}+θ^{′}=0.
As
θ^{′}=(1α)^{2}τ^{α}>0,
i.e. the tax has a positive effect on the wage,
r^{′}<0.
Thus, the tax rate is too high to be growthmaximizing. This reproduces the
finding of
Alesina
and Rodrik (1994). Note that uncertainty implies
r^{′}+θ^{′}>0,
i.e. it tends to lower
τ.
^{14}
This is because an increase in
τ
raises the variance of the posttax return on capital,
, and thus the riskiness of capital income. In
consequence, the government can counteract the magnified exposure to income risk
when then variance of the output shock
σ^{2}
increases by selecting a lower tax rate. So, in principle,
τ_{C}
could maximize growth when there are stochastic shocks to production.
Note, the relationship between the variance of
stochastic shocks and growth is not unambiguously negative. The potentially
negative relation between the output variance and the tax rate implies that a
higher variance yields higher growth provided the interest rate adjusts
negatively to an increase in taxes.
5.5. Decentralization
Here, the government in the home region maximizes
the expected discounted utility of the home household subject to the state
equations for w and w^{*},
and equations determining k,k^{*},s,s^{*},
taking τ^{*}
as given. Specifically, using (28),
the government in the home region maximizes
It also understands
that capital allocations are
and portfolio shares
are
The two state
variables,
w and
w^{*} follow
(27)
and the counterpart of this equation for the foreign region. For completeness,
we give both state equations:
Again, we set up the Bellman equation and guess the
functional form of the value function in order to derive the firstorder
condition determining τ.
All these steps are dealt with in detail in the Appendix, and the resulting
firstorder condition, evaluated at symmetric equilibrium, is
Finally, using
(32) and
(33),
we can calculate:
Thus, in symmetric
equilibrium
Comparing
(35)
to the centralized case,
(31),
one observes three differences:
 •

First, due to the mobile tax base externality, we have the term θ(k_{τ}/k)
in the FOC. k_{τ}/k
measures the percentage capital outflow due to a 1 percentage point increase
in the tax.
 •

Second, due to the rate of return externality, r^{′}
enters with a weight of only 0.5. The rationale is that half of the total
return to capital goes to foreigners and the effect of tax policy on it is
external to the government.
 •

Third, a higher tax rate increases infrastructure spending and thus the
exposure of investors to the productivity shock. The impact of capital
taxation on risk exposure of foreigners is not recognized (the risk exposure
externality), explaining the weight of rather than in the last term.
Thus, in general the average growth rate of the
economy, which is r(τ)ρ,
may be higher or lower under decentralization relative to centralization. To
begin the comparison, we can obtain an analytical result confirming Hatfield
(2006) when there is no uncertainty. First, by inspection, (31)
reduces to r^{′}=θ^{′}
when σ=0.
As θ^{′}=(1α)^{2}τ^{α}>0,
but with centralization, r^{′}<0,
and as r is singlepeaked, this in turn implies that the tax is too
high to be growthmaximizing.
Next, combining (35) and (36),
plus θ=(1α)τ^{1α},
we see that (35)
reduces to
So, it is clear that as
σ^{2}→0,
(37)
reduces to
r^{′}=0.
Thus, with decentralization, in the absence of productivity shocks the
government sets the tax rate so as to maximize
r and thus the average
growth rate
r(τ)ρ.
This is Hatfield's result. We can summarize as follows:
Proposition 4.
If, so the average growth rate is always higher
under decentralization.
What happens with stochastic shocks? Although we
cannot solve explicitly for the equilibrium tax rates τ_{C}
and τ_{D},
appealing to Proposition
4 and a continuity argument we know that τ_{C}>τ_{D}
also holds for a small enough variance of the production shock. This allows us
to conclude that decentralization yields higher growth when the variance is
small.
The latter finding does, however, not extend to any
size of the variance. First, as see Fig.
2 shows, taxes fall as σ
increases in both fiscal regimes. With centralization, the reason is clear; a
higher tax exposes investors to more risk, and this must be set against the
productivity gain that the infrastructure good provides. This effect is also
present with decentralization, although it only has half the impact, due to the
risk exposure externality referred to above. Even so, it dominates any effect of
σ
on the semielasticity of capital supply, (36).
Generally, as r^{′}>0
for all σ
from (37),
an increase in σ
makes the supply of capital more inelastic from (36),
as in the consumption good case.
Note finally that as k_{τ}
is independent of ρ
from (36),
and as (31) and (35)
are otherwise independent of ρ,
then τ_{D},τ_{C}
are also independent of ρ.
This is in contrast to the consumption public good case where the tradeoff
between higher future growth and more public consumption is decided in favor of
more public consumption when ρ
rises. With infrastructure spending τ_{D},τ_{C}
are independent of ρ
because taxation and infrastructure provision have countervailing effects on
growth which are equated at the margin.
Now we turn to the relationship between
decentralization and growth, recalling that the average growth rate is r(τ)ρ.
Fig.
3 shows how r_{C}=r(τ_{C})
and r_{D}=r(τ_{D})
vary as σ
increases. From Fig.
2τ_{C}>τ_{D}.
Since the growth rate is humpshaped in taxes, the implication of the tax rate
differential for growth can be ambiguous. Specifically, in the absence of
uncertainty a decentralized government engages in Bertrand tax competition with
the consequence of maximizing growth. The decentralized tax rate decreases as
σ
increases and, since the growth rate is humpshaped in the tax rate, growth is
unambiguously decreasing in σ.
A centralized government sets a too high tax rate
to be growth maximizing in the absence of uncertainty^{15}
and lowers it as σ
rises. For sufficiently small values of σ
growth is rising. In fact, for σ≈1.7
the growth rate equals that under decentralization and for σ≈2
the centralized tax rate is growth maximizing. For larger values both a
decentralized and centralized government operate on the upward sloping part of
the growthcurve with centralization yielding higher growth. This is in contrast
to the results of Hatfield
(2006), who assumes a deterministic growth model.
The simulation results indicate that a higher
output variance yield lower growth in contrast to the finding under
centralization. The reason is that decentralization maximizes the rate of return
to investors (i.e. the interest rate) in the absence of shocks and a higher
variance lowers the tax rate. Since the interest rate is humpshaped in taxes,
growth declines as the output variance increases.
6. Conclusions
This paper has considered the relationship between
tax competition and growth in an endogenous growth model where there are
stochastic shocks to productivity, and capital taxes fund a public good which
may be for final consumption or an infrastructure input. Absent stochastic
shocks, decentralized tax setting (two or more jurisdictions) maximizes the rate
of growth, as the constant returns to scale present with endogenous growth
implies Bertrand tax competition. Stochastic shocks imply that households face a
portfolio choice problem. Shocks dampen down tax competition and may raise taxes
above the centralized level when the government provides a public consumption
good. Growth can be lower with decentralization. In the public infrastructure
model shocks may increase the tax base elasticity and the equilibrium
decentralized tax rate may be too low to yield higher growth with
decentralization. Our results may also predict a negative relationship between
output volatility and growth, consistent with the empirical evidence.
One might ask how robust our results are. Two of
our important simplifying assumptions are logarithmic utility of private
consumption, and in the case of the infrastructure model, no taxes on labor. If
logarithmic utility of private consumption is relaxed to isoelastic utility, we
can still solve the household savings and portfolio choice problem in closed
form, but we cannot get a closedform solution for the equilibrium tax, even in
the case of a consumption public good. But the key externalities at work remain
the same, and as a result, it is possible to get higher taxes and lower growth
with decentralization in the public consumption good case.^{16}
As to labor taxation, we conjecture that our main conclusions would be
unaffected in the infrastructure model, as long as the demand for the public
good is high enough so that it is optimal to use a capital tax at the
margin.
A different assumption is that the variance of the
output shock is identical across regions. One may ask how our main finding
extends to regions which differ in their output variance. For instance, in the
consumption good model a region with a lower variance will most likely set a
higher tax rate. The intuition is that investors are willing to earn a lower
rate of return in exchange for a lower variance of the rate of return. The
government will exploit the reduced taxsensitivity of capital by choosing a
higher tax rate.^{17}
This suggests that there is a tendency for lowvariance regions to exhibit lower
growth under decentralization compared to centralization.^{18}
Finally, we should note that the two regimes can be
unambiguously ranked in terms of welfare. To see this, note that the centralized
solution can always replicate tax policy under decentralization. Appealing to a
revealed preference argument, welfare under centralization is weakly higher than
under decentralization. It would be interesting to set up a model in which not
only the growth rate differential but also the welfare differential is
ambiguous; e.g., due to inefficiencies in centralized decisionmaking. We leave
such an exercise to future research.
Acknowledgments
We would like to thank participants at the “New
Perspectives on Fiscal Federalism” Conference in Berlin of October 2007
(especially Enrico Spolaore), at the Royal Economic Society meeting in Warwick
of March 2008, at the CESifo conference “Public Sector Economics” in Munich of
April 2008, at the IIPF meeting in Maastricht of August 2008, at the VfS meeting
in Graz of September 2008, at the CPEG meeting in Toronto of June 2009, and at
seminars at the University of Vienna and University of Copenhagen for their
helpful comments. We are also grateful to an anonymous referee, the editor, Jan
Brueckner, Andreas Haufler, Christian Keuschnigg, and Klaus Wälde, who gave us
numerous helpful suggestions.
Appendix. Proofs of propositions and other
results
Proof of Proposition 1.
There is a single state variable w which follows (11).
So, denoting V(w)
as the value function, the Bellman equation is
Differentiating the RHS
w.r.t.
τ
and setting the result equal to zero, and assuming
V(w)=A+ψlnw,
we have
To derive
ψ
we rewrite the Bellman equation, using
(38)
and
V(w)=A+ψlnw,
as
Thus, equating
coefficients on
lnw,
we see that
ψ=(1+β)/ρ.
Combining with
(38),
the optimal tax rate is
τ=ρβ/1+β
which completes the proof. □
Proof of Proposition 2.
First rewrite the stochastic terms in the state equations as
Noting that
one can compute from
(39) and
(40)
that
So, assuming a value
function
V(w,w^{*}),
the Bellman equation for the government problem can be written
where
k=((1(n1)s)w+(n1)s^{*}w^{*}).
Taking into account the effect of
τ
on
s and
s^{*}
and, thus, on
, and
ξ,
we have
So, using
(41),
(42),
(43) and
(44),
the FOC for the tax is
where
At a symmetric
equilibrium we have
τ^{*}=τ
and so
w^{*}=w=k
and
s=s^{*}=1/n.
Thus, using
(46),
and also noting from
(43) and
(44)
that as
at symmetric
equilibrium we can rewrite
(45)
as
Next, note that in
symmetric equilibrium,
w=w^{*},
and assume
V(w,w)≡A+ψlnw.
Then, at symmetric equilibrium,
and
consequently,
(47)
can be rewritten as
It just remains to
solve for
ψ.
Note also at symmetric equilibrium that
. Using all these facts, the Bellman Eq.
(41)
reduces to
So, by inspection,
ψ=(1+β)/ρ.
So, from
(48)
and
ψ=(1+β)/ρ,
we obtain the expression for
τ_{d}in
Proposition 2. □
A.1. Derivation of the solution to the household problem
in the public infrastructure good case
Using k=w
in (27),
the problem is to maximize (26)
subject to
Assume the value
function for this problem takes the form
V(w)=A+Blnw.
In this case the Bellman equation is
where
is the variance of
wealth. The firstorder conditions for
c and
s are
and
the Bellman equation becomes
So,
B=1/ρ;
using this in
(51) and
(52)
gives
(28).
A.2. Derivation of the FOC (35)
At a symmetric equilibrium, w=w^{*}.
So, the Bellman equation is
Evaluated
at
τ=τ^{*},
the FOC for the tax from the Bellman equation is
where we already used
the fact that
Now, guess
V(w,w^{*})=A+Blnw,
i.e. the value function is independent of
w^{*}.
Then,
Thus, the FOC becomes,
cancelling
B and using
k=w
in symmetric equilibrium:
as required.
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