Does tax competition really promote growth?

  • Marko Koethenbuergera, Corresponding author contact information, E-mail the corresponding author,
  • Ben Lockwoodb, E-mail the corresponding author
  • 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

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

  • H77;
  • E62;
  • F43

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 sub-national level. Overall, the evidence is mixed. In particular, cross-country 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 post-tax 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 multi-jurisdiction endogenous growth model which combines two empirically relevant motives for investing outside the jurisdiction, namely rate-of-return 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 rate-of-return differentials as the driving force for investing abroad. The rate-of-return motive accounts for the well-known 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 “rate-of-return” 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 pre-tax 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 risk-exposure externality. Specifically, a higher tax increases the riskiness of investment and thus the risk-bearing of non-residents. The risk-exposure 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 AK-type growth model without stochastic shocks, the firm's demand for capital is perfectly elastic at the tax-inclusive 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 micro-founded 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 generally2 in Barro's (1990) model of infrastructure growth: that is, there is a tax-inclusive 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 pre-tax price of capital. As there is an infrastructure public good funded by the capital tax, this price-maximizing 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 set-up

3.1. The model

We work with a dynamic stochastic version of the Zodrow–Mieszkowski (1986) model, where regional governments use source-based 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

dyi=ki(dt+dzi),
where ki is the capital stock at t, and zi(t) is a Wiener process,6 i.e. over the period (t,t+dt), the change dzizi(t+dt)-zi(t) is Normally distributed with variance View the MathML source and zero mean, and non-overlapping increments are stochastically independent. Also, zi(t),zj(t) are independent for all ij.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+dzi, is equal to the cost of capital, which comprises the rental price of capital (expressed in differential form), dri, and the tax on the use of capital, View the MathML source. Thus, the rental price of capital in region i follows the process

(1)
View the MathML source

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:

(2)
View the MathML source
where each of the two sub-utility functions has the same fixed degree of relative risk-aversion equal to unity. At each t, the household in region i has an endowment of the private good (wealth) wi, which it can rent to any firm. The evolution of wi over time is described in detail below.

Finally, in region i, the public good is wholly financed by a source-based tax on capital, i.e. gi=τiki. This is without loss of generality, as there is no wage income in the model, so our analysis would go through if τi were re-specified 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 strategies8 in this game. These are strategies where τi is restricted to depend only on the state of the economy, the vector of wealth variables w=(w1,…,wn). Then, in this class of strategies, we focus on symmetric perfect equilibrium (Dockner et al., 2000). This requires that the τi are mutual best responses conditional on any w, and that in equilibrium, τi=τ and thus wi=w, for all i. Finally, we guess initially that equilibrium strategies are state-independent, i.e. τi 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 (σi≡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 τi 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 sij as the share of wealth, wi, rented to a firm in region j, wealth of the consumer located in i evolves according to

(3)
View the MathML source
where jsijdrj is the overall return on wealth over the interval (t,t+dt). Combining (1) and (3), we get
(4)
View the MathML source
where μi=∑jsij(1-τj) is the deterministic part (average) rate of return on wealth. Thus, household in region i chooses ci,sij to maximize
(5)
View the MathML source
subject to (2) and (4) and View the MathML source. Note, in maximizing expected utility the household perceives public consumption gi to be given, although possibly time-varying, and also takes τi as fixed and time-invariant.

The solution to this problem is well-known (e.g., Jones and Manuelli, 2005) and easily stated. First, the optimal consumption rule is simply

(6)
ci=ρwi.

Second, the portfolio shares sij,ji, are determined by the n-1 first-order conditions with respect to the sij

(7)
View the MathML source
To get intuition, consider the two-region case. Then, (7) solve for regions 1 and 2 to give
(8)
View the MathML source
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 τi-τj and (ii) the lower the relative uncertainty of investing abroad, i.e. View the MathML source. In the sequel, we simplify the analysis by assuming σi=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 two-region case (n=2), thinking of region 1 as the home region. We trace the effects of a change in τ2 on home welfare. From (8), using σi=σ, the portfolio allocation rule for home and foreign households is

(9)
View the MathML source
Moreover, by definition, the capital employed in the home region is
(10)
k1=(1-s12)w1+s21w2.
Finally, in the home region, the household optimal accumulation of capital follows the rule (4), given also (6);
View the MathML source
By inspection of (4), (9) and (10), we can identify two fiscal externalities in the model.

First, increasing τ2 implies s21 up and s12 down from optimal portfolio choice, implying k1 up. This is the well-known 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 τ2 has other two related effects. First, it directly affects home welfare by lowering μ1=1-(1-s12)τ1-s12τ2, the deterministic part of the return on w1. Second, it lowers μ2, and thus w2 and thus k1. We call this the negative rate of return externality. Note that the size of this externality is measured by s12,s21: specifically, this externality only operates in equilibrium if s12,s21>0. But, the portfolio allocation rule ensures that at symmetric equilibrium, s12=s21=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. sij=1/n. Then, from (4), an important simplification is that every wealth level wi follows the same process, and so wi=w, implying a single state variable. Again from (4), this state variable evolves according to

(11)
View the MathML source
Moreover, again as sij=1/n, capital employed in region i is equal to ki=w. Using all this information in addition to the consumption rule (6), instantaneous utility in region i is
(12)
View the MathML source
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 chooses11τ to maximize
View the MathML source
subject to (11). Setting up the Bellman equation, deriving the first-order 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

View the MathML source
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 τj,ji. Then, invoking symmetry, decentralized equilibrium is characterized by the solution to this problem when τi=τ, for all i. But, it is difficult to solve for government i's problem given general taxes View the MathML source. So, we assume that all regions ji set the same tax τ*, and solve for region i's best response to τ*, say τi. Then, imposing τi=τ* gives a condition for the Nash equilibrium tax.

We proceed as follows. First, let View the MathML source, be the share of wealth invested by the households in region i in any region ji. Note that from (7), and View the MathML source, these share must all be the same. Indeed, evaluating (7) at σi=σ, sij=s, and View the MathML source, and solving, we get

(13)
View the MathML source

Next, let s*=sji, ji, View the MathML source be the shares of wealth invested by the households in region ji in region i and the n-2 regions kji, respectively. Again from (7), View the MathML source solve the two simultaneous equations

(14)
View the MathML source
(15)
View the MathML source
We are only interested in s*. Solving (14) and (15), we get
(16)
View the MathML source

Now let w be the wealth of region i, and w* be the average wealth of regions ji. Then, by definition,

(17)
k=(1-(n-1)s)w+(n-1)s*w*.
Then, using the government budget constraint g=τk, and the consumption rule c=ρw, region i's instantaneous payoff can be written
(18)
lnci+βlngi=κ+lnw+βlnk+βlnτ.

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), σi=σ, and the definition of s:

(19)
View the MathML source
where
View the MathML source
Second, the process for some wj is
View the MathML source
As
View the MathML source
tedious but straightforward calculation gives
(20)
View the MathML source
So, the problem for the government of region i is to choose τ to maximize
View the MathML source
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 closed-form solution for the equilibrium τ, we must guess the correct form of the value function12V(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

(21)
View the MathML source
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,

(22)
View the MathML source
where in the last line we used w=w*=k. So, the semi-elasticity
View the MathML source
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 τd 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(n-1)/σ2 which is simply ρβ times the semi-elasticity of the tax base with respect to τd.

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 ((n-1)/n)(1+β).

As the externalities have opposite effects on τd, we expect that the effect of n on τd is not monotonic, and this is confirmed by the following result:

Proposition 3. AssumeView the MathML source. Then there is a critical value ofView the MathML source, such thatτdis higher thanτcforView the MathML source, and lower thanτcforView the MathML source. In particular, τd→0asn→∞.

Proof. The denominator of (21), ρβ(n-1)/σ2+(1+β)/n, is a convex function of n, with a minimum at View the MathML source. Thus, τd must be a quasi-concave function of n with a maximum at n=n*. Note, τc=τd for n=1. So, as n*≥2 by assumption, there must be an integer View the MathML source for which τc<τd. Also, τd→0 as n→∞ while τc>0, n. Thus, there always exists a value of View the MathML source with τcτd if View the MathML source. □

Unlike the traditional tax competition model, the effect of n on τd 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 (τc<τd), whereas for n large, the mobile tax externality dominates (τc>τd).

Fig. 1 illustrates the tax rate differential (assuming β=1 and ρ=0.1). The thick line is the centralized tax rate τc which is independent of uncertainty. The thin lines depict the decentralized tax rates τd for different levels of uncertainty. For instance, for σ2=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:

View the MathML source
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 1-τ-ρ as the average growth rate.

So, Proposition 3 indicates that when the number of regions is less (greater) than View the MathML source, growth will be lower (higher) under decentralization than centralization. From Fig. 1, when σ=0.5, for example, View the MathML source. Intuitively, it is only when View the MathML source 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 set-up 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

dy=g1-αkα(dt+dz).
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 pre-tax 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.
(23)
View the MathML source
using the budget constraint g=τk. So, the wage is non-random. The pre-tax rate of return on capital over the period (t,t+dt) is thus determined residually:
(24)
View the MathML source
using the budget constraint g=τk. So, the post-tax rate of return thus follows the process
(25)
View the MathML source
where r(τ)≡ατ1-α-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

(26)
View the MathML source
subject to the stochastic wealth equation which is, using (25)
(27)
View the MathML source
Unlike the consumption good case, this problem is non-standard, and so for completeness, we provide a solution in the Appendix. Moreover, in deriving the solution, we suppose that the household believes13 that kw, a belief which is true in equilibrium. The consumption and portfolio allocation rule for the home region are
(28)
View the MathML source
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
View the MathML source
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 pre-tax 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 rate-of-return externality identified in the AK-model, 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 post-tax return to capital over an interval dt is View the MathML source and is thus increasing in the tax rate. Since each region also hosts capital from non-residents, this specification introduces a second negative externality, i.e. a higher tax increases the riskiness of investment and thus the risk-bearing of non-residents. We call this externality the risk-exposure 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

(29)
View the MathML source
where we have also used k=w. So, under centralization, the government maximizes
View the MathML source
subject to (29). The Bellman equation is
(30)
View the MathML source
where View the MathML source. The first-order condition w.r.t. τ in (30) is
View the MathML source

Now, we guess V(w,w)=A+Blnw. Then, the first-order condition becomes, cancelling B,

(31)
View the MathML source
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)=ατ1-α-τ is a strictly concave function with a unique maximum at View the MathML source. 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=(1-α)2τ-α>0, i.e. the tax has a positive effect on the wage, r<0. Thus, the tax rate is too high to be growth-maximizing. 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 post-tax return on capital, View the MathML source, 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 σ2 increases by selecting a lower tax rate. So, in principle, τC 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

View the MathML source
It also understands that capital allocations are
(32)
View the MathML source
and portfolio shares are
(33)
View the MathML source
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:
(34)
View the MathML source

Again, we set up the Bellman equation and guess the functional form of the value function in order to derive the first-order condition determining τ. All these steps are dealt with in detail in the Appendix, and the resulting first-order condition, evaluated at symmetric equilibrium, is

(35)
View the MathML source
Finally, using (32) and (33), we can calculate:
View the MathML source
Thus, in symmetric equilibrium
(36)
View the MathML source
Comparing (35) to the centralized case, (31), one observes three differences:

First, due to the mobile tax base externality, we have the term θ(kτ/k) in the FOC. kτ/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 View the MathML source rather than View the MathML source 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=(1-α)2τ-α>0, but with centralization, r<0, and as r is single-peaked, this in turn implies that the tax is too high to be growth-maximizing.

Next, combining (35) and (36), plus θ=(1-α)τ1-α, we see that (35) reduces to

(37)
View the MathML source
So, it is clear that as σ2→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. IfView the MathML source, 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 τC and τD, appealing to Proposition 4 and a continuity argument we know that τC>τD 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 semi-elasticity 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 kτ is independent of ρ from (36), and as (31) and (35) are otherwise independent of ρ, then τD,τC are also independent of ρ. This is in contrast to the consumption public good case where the trade-off between higher future growth and more public consumption is decided in favor of more public consumption when ρ rises. With infrastructure spending τD,τC 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 rC=r(τC) and rD=r(τD) vary as σ increases. From Fig. 2τC>τD. Since the growth rate is hump-shaped 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 hump-shaped 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 uncertainty15 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 growth-curve 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 hump-shaped 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 iso-elastic utility, we can still solve the household savings and portfolio choice problem in closed form, but we cannot get a closed-form 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 tax-sensitivity of capital by choosing a higher tax rate.17 This suggests that there is a tendency for low-variance 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 decision-making. 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

View the MathML source
Differentiating the RHS w.r.t. τ and setting the result equal to zero, and assuming V(w)=A+ψlnw, we have
(38)
View the MathML source
To derive ψ we rewrite the Bellman equation, using (38) and V(w)=A+ψlnw, as
View the MathML source
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

(39)
View the MathML source
(40)
View the MathML source
Noting that
View the MathML source
one can compute from (39) and (40) that
View the MathML source
So, assuming a value function V(w,w*), the Bellman equation for the government problem can be written
(41)
View the MathML source
where k=((1-(n-1)s)w+(n-1)s*w*). Taking into account the effect of τ on s and s* and, thus, on View the MathML source, and ξ, we have
(42)
View the MathML source
(43)
View the MathML source
(44)
View the MathML source
So, using (41), (42), (43) and (44), the FOC for the tax is
(45)
View the MathML source
where
(46)
View the MathML source
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
View the MathML source
at symmetric equilibrium we can rewrite (45) as
(47)
View the MathML source
Next, note that in symmetric equilibrium, w=w*, and assume V(w,w)≡A+ψlnw. Then, at symmetric equilibrium,
Vww+Vw*w*=ψ,Vwww2+Vw*w*(w*)2+2Vww*ww*=-ψ,
and consequently, (47) can be rewritten as
(48)
View the MathML source
It just remains to solve for ψ. Note also at symmetric equilibrium that View the MathML source. Using all these facts, the Bellman Eq. (41) reduces to
(49)
View the MathML source
So, by inspection, ψ=(1+β)/ρ. So, from (48) and ψ=(1+β)/ρ, we obtain the expression for τdin 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

(50)
View the MathML source
Assume the value function for this problem takes the form V(w)=A+Blnw. In this case the Bellman equation is
View the MathML source
where
View the MathML source
is the variance of wealth. The first-order conditions for c and s are
(51)
View the MathML source
(52)
B(r*-r-σ2[s(τ2(1-α)+(τ*)2(1-α))-τ2(1-α)])=0,
and the Bellman equation becomes
View the MathML source
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

View the MathML source
View the MathML source
View the MathML source
ξ=[(1-s)s*τ2(1-α)+s(1-s*)(τ*)2(1-α)]σ2.
Evaluated at τ=τ*, the FOC for the tax from the Bellman equation is
View the MathML source
where we already used the fact that
View the MathML source
Now, guess V(w,w*)=A+Blnw, i.e. the value function is independent of w*. Then,
View the MathML source
Thus, the FOC becomes, cancelling B and using k=w in symmetric equilibrium:
View the MathML source
as required.

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Corresponding author contact information
Corresponding author. Tel.: +45 35324417; fax: +45 35323000.
1

See also Wrede (1999) for an analysis of fiscal externalities in dynamic economies. But, his paper abstracts from portfolio diversification and endogenous growth.

2

This holds even when labor supply is endogenous and is taxable, when there are consumption and infrastructure goods, and where agents are different within regions (with respect to their endowments of capital) and decisions are made by majority voting.

3

Formally, if s is the share of capital invested outside the home region, the mobility cost is c(s)=-vs+μs2/2. Given this specification, it is possible to establish that for v large enough relative to μ, the equilibrium tax on capital can be higher under decentralization, and thus growth lower. Unfortunately, the v and μ parameters are almost impossible to relate to the fundamentals of the stochastic process (generating a desire for diversification) and to interpret empirically.

4

This is clear from the fact that in these models, there is a single “world” rate of return on household savings. Thus, household financial capital is distinct from the physical capital employed by firms. Physical capital is in fact, completely immobile between jurisdictions.

5

Becker and Rauscher claim that the adjustment cost parameter (b in their model) is a measure of inter-jurisdictional capital mobility, but this does not seem to be very plausible.

6

The assumption that random shocks take the form of Brownian motion is standard in stochastic growth models. It describes a situation where productivity is subject to frequent small changes; see Turnovsky (2000) for more discussion.

7

We assume that total factor productivity is deterministic (and normalized at unity)—see Turnovsky (2000), Kenc (2004), and Jones and Manuelli (2005). An alternative representation of the production technology is to assume that total factor productivity follows a geometric Brownian motion—see, e.g., Waelde (2009) on the virtue of this approach.

8

Without the Markov restriction, there would be a very large number of equilibria; e.g., those sustained by punishment strategies.

9

We cannot avoid this conclusion by assuming decreasing returns, i.e. y=f(k), with f(k)<0, as then we are back to a Solow-type growth model, where taxes do not affect growth (in the long run).

10

Strictly speaking, a zero equilibrium tax is only possible if the marginal utility of the public good at zero provision is finite, which is not the case with logarithmic utility, i.e. (2). A precise statement would be that there is no equilibrium with strictly positive taxes in this case.

11

In principle, τ can be any function of w, the state variable. But, given that V(w) is linear in View the MathML source, it is computed in the proof of Proposition 1 that the optimal τ is independent of w. This is not due to the special log form for utility—it can be computed that τ is independent of w also in the more general case where utilities from private and public consumption are iso-elastic (details available on request).

12

Note, the value function depends on the regions’ tax rate choices through private wealth.

13

This assumption is slightly different than rational expectations, as when ττ*, kw in general. However, we cannot solve the household problem in closed form under fully rational expectations when ττ*.

14

In fact, invoking the implicit function theorem we find dτ/dσ<0 if the share of output accruing to capital α≤0.5.

15

This may imply a zero or even a negative net return to investors. The result is reminiscent of Alesina and Rodrik (1994).

16

Details of these calculations are available on request.

17

See Eq. (8).

18

In fact, reiterating the steps involved in deriving τc in Section 4, the uniform central tax rate with heterogenous variances coincides with the central equilibrium tax rate τc in Proposition 1. A formal proof of the result is available upon request.