Dynamic Capital Tax Competition Miltiadis Makris Department of Economics, University of Exeter, CMPO, University of Bristol, and EMOP, Athens University of Economics and Business November 7, 2005
Abstract We re-examine the view that capital taxes are too low when capital is mobile across tax jurisdictions. We do so by emphasising two previously neglected implications of noncooperative capital tax setting in a fully dynamic environment. First, in a truly-dynamic environment, the returns to the non-capital factors of production and thereby disposable income and savings depend on predetermined capital stock. Thus, capital-market clearing interest rates depend also on past rates and taxes. Hence, taxes affect future interest rates, and thereby future consumption levels, capital stocks and tax revenues abroad. Second, in an overlapping generations economy, the effect of capital taxes on the current income of different generations is not the same across generations. The reason is that for old generations capital taxes affect the returns to their savings, while for younger generations capital taxes affect their non-capital income. This implies that capital taxes may produce an externality through affecting current consumption levels abroad. These two horizontal externalities may lead, ceteris paribus, to too high national capital taxes, and may more than offset the usual effects of tax competition. In this case, and contrary to conventional wisdom, national capital taxes will be too high.
Keywords: Tax Competition, Dynamic Taxation, Capital Taxation. JEL Classification Numbers: H25, H77, F0
Correspondence: Dr. Miltiadis Makris, Department of Economics, University of Exeter, Streatham Court, Rennes Drive, Exeter EX44PU, EMAIL:
[email protected].
1
1
Introduction
Mobility of tax bases between tax jurisdictions gives rise to horizontal externalities that tend to leave regional/state/national taxes too low.0 This standard wisdom is expressed forcefully in the well-established models of Zodrow and Mieszkowski (1986) and Wilson (1986) (ZMW hereafter): competition for mobile capital leads to too low source-based capital taxes, as regional taxes have a positive effect (through the world interest rate) on capital tax bases abroad.1 The analysis in ZMW has since been enriched in various directions to provide instances in which source-based capital taxes may be too high.2 These instances include:3 trade in capital- and labour-intensive goods (Wilson, 1987), large capital-importing countries (De Pater and Myers, 1994), large foreign ownership of immobile factors (Huizinga and Nielsen, 1997), competition for amenities (Noiset, 1995, and Wooders et. al., 2001), commonality of the capital tax base between states and federal governments (Keen and Kotsogiannis, 2002), government failure (Edwards and Keen, 1996), sharing of a common currency (Makris, 2005), political economy considerations (Fuest and Huber, 2001, Kessler et al, 2002, Grazzini and van Ypersele, 2003, and Lockwood and Makris, 2005). In addition, there is a growing body of empirical evidence that higher capital mobility has not clearly led to cuts in corporate tax rates, at least for OECD countries. In particular, recent 0
I would like to thank Daniel Becker, Sebastien Mitraille, and participants at seminars in Exeter and at the
IEB Workshop on Fiscal Federalism in Barcelona, 6-7 June 2005, ”Decentralization, Governance and Economic Growth” for useful comments and discussions prior to this draft. The usual disclaimer applies. 1
Low capital taxes may not be an exclusive characteristic of open economies, that compete for capital. As
Chamley (1986) and Judd (1985) emphasise, even when a closed economy is considered, in the long-run the accumulated distortions on capital and labour, through the effect of capital taxes on interest rates and wages, dominate the distortions on labour, that arise from the effect of labour income taxes on labour supply. So, the optimal steady-state capital tax in a closed economy is zero. 2
For some excellent recent surveys see Wilson (1999) and Wilson and Wildasin (2004).
3
On a related topic, Kehoe (1989), by building on the capital levy problem discussed, for instance, in Fischer
(1980), has shown that an attempt to coordinate to higher taxes may not prove beneficial even if tax competition leads to a ‘race to the bottom’, as long as governments cannot pre-commit to their tax policies. In such an environment, an anticipation of coordination to higher taxes after savings have taken place will lead to low savings and hence low aggregate capital stock in the first place. On another related topic, Bucovetsky and Wilson (1991) show that if tax authorities can deploy a savings, as well as a capital, tax then the resulting policy mix is efficient, and hence there is no scope for coordination.
2
work by Hallerberg and Basinger (1998, 2001), Devereux, Lockwood, and Redoano (2002), Garrett (1998), Quinn (1997) Rodrik (1997), Swank and Steinmo (2002)) find rather mixed effects4 of relaxation of capital controls on corporate tax rates. Nevertheless, the message of the basic ZMW model and anecdotal evidence of capital taxes in OECD countries seem to drive many of the debates for tax coordination. For instance, OECD (1998) calls for countries to refrain from harmful tax competition. This paper re-examines the view that capital taxes are too low when capital is mobile across tax jurisdictions. It does so, by emphasising two previously neglected implications of non-cooperative capital tax setting in a dynamic environment. First, capital taxes, give rise to an horizontal externality by affecting future interest rates. This effect of capital taxes arise because in a truly dynamic environment (a) the returns to non-capital factors and thereby disposable income depend on current capital, and (b) endogenous savings depend also on current disposable income. Thus, endogenous savings depend also on current capital and hence current interest rates and taxes. As in equilibrium current savings determine future capital stocks, we thus have that current taxes affect, both directly and indirectly - through the current interest rate, future market-clearing interest rates. These relations are completely ignored by the received literature, where the models are either static (in which case savings are exogenous) or two-period ones (where the first-period disposable income is fixed).5 In fact, the above externality can not be captured in a static or a two-period model; in such a model current taxes affect only the current market-clearing interest rate.6 4
Specifically, Devereux, Lockwood, and Redoano (2002) is probably the most comprehensive, as it allows
for four different measures of exchange controls, and studies not only statutory rates of corporate tax, but also effective marginal and average rates, for almost all OECD countries, and allows for strategic interaction in corporate tax setting between countries. It finds that depending on the choice of measure of capital controls and corporate tax rates, a unilateral or multilateral liberalisation of controls may lower or raise corporate taxes. This is broadly consistent with the findings of Quinn (1997), Rodrik (1997), Garrett (1998) and Swank and Steinmo (2002) who simply find that capital controls have no significant effect. 5
See, for instance, the reviews by Wilson (1999) and Wilson and Wildasin (2004).
6
Our paper is related to Philippopoulos and Economides (2001) and Park and Philippopoulos (2001) in that in
these papers the non—cooperative tax-setting in a dynamic environment is also studied. In the first one, however, there is no capital mobility, and the tax externality arises because of public good spillover effects. In the latter one, capital is mobile and local public goods are also provided by governments. However, in that work interest rates are exogenously fixed (as the model is an AK growth one) and hence the externalities we emphasise here
3
Second, in an overlapping generations economy with positive supply of capital, the effect of capital taxes on the current income of different generations is not the same across generations. The reason is that for old generations capital taxes affect the returns to their savings, while for younger generations capital taxes affect their non-capital income. This implies that capital taxes do produce an externality through affecting current consumption levels abroad. This holds even if countries are symmetric, and non-capital income is not taxed.7 The tax externalities we highlight in this paper may lead, ceteris paribus, to too high national capital taxes, and may more than offset the usual effects of tax competition. In this case, and contrary to conventional wisdom, national capital taxes will be too high. So, the main result of this paper is that tax coordination, by means of a small multilateral increase in the are absent. Also, the focus is on optimal union membership. Our work is also related to Correia (1996) in that dynamic taxation in a small open economy is analysed. However, in that paper, foreign governments are passive and the rest of the world is assumed to be in a steady-state. Also, and more importantly, the world interest rate is assumed to be fixed and time-invariant. Thus, the horizontal tax externalities we are concerned here with are not present in Correia (1996). A fixed and time-invariant world interest rate is also assumed in Wildasin (2003). The focus in both Correia (1996) and Wildasin (2003) is on whether capital should be taxed, and externalities are not explicitly analysed. Another related work is that of Palomba (2004). There, however, the analysis is positive: taxes are not endogenous and it is the effects on growth of tax changes that are investigated. Also, once comparative statics become complicated, the focus switches on an example where savings are constant. Our work is also related to another strand of research, with a more macroeconomic orientation. In particular, Roeger et.al. (2002), Klein et.al. (2003) and Mendoza and Tesar (2003) calibrate two-economy dynamic models where capital is taxed. In these papers the world interest rate is endogenous, and therefore horizontal tax externalities are potentially present. Yet in all these papers public consumption is exogenously given and thereby the horizontal tax externality emphasised in ZMW, and re-visited here, is not present. In fact, in these papers, the externalities that emerge from the use of capital taxes work exclusively through the endogenous adjustment of the rest of the taxes, with the latter taking place in order to maintain fiscal solvency. In all these papers, the emerged externalities are not explicitly analysed. Klein et.al. (2003) assume that both residence-based and source-based capital taxes are used, and focus on whether time-consistent capital taxes in U.S.A differ from those in Europe. In Roeger et.al. (2002), the main aim is to quantify possible gains from tax cooperation. However, source-based capital taxes are not used and consumers differ with respect to their unemployment status. In Mendoza and Tesar (2003), source-based capital taxes which can be pre-committed upon are deployed, and the main aim is to quantify the possible gains from tax cooperation. Nevertheless, in contrast to our work, tax policies there, as well as in Roeger et.al. (2002), are time-invariant. 7
As the analysis of the non-dynamic models in Persson and Tabellini (1992), De Pater and Myers (1994),
Keen and Kotsogiannis (2002) make clear, if countries are symmetric, and non-capital income is not taxed then capital taxes do not produce a consumption externality.
4
capital tax, may not lead to a welfare improvement relative to the non-cooperative outcome. The organisation of the paper is the following. Next Section presents the basic model, while Section 3 investigates whether non-cooperative capital taxes are too low or too high. Section 4 discusses some extensions to the basic model. Section 5 concludes and points to directions of future research.
2
The Model
To present the basic argument in the simplest possible manner, we deploy here a stylised model which abstracts from many features of empirical reality. How our basic argument might be modified in a more general framework is discussed in Section 4. Our framework is the standard capital taxation model of ZMW, appropriately modified to incorporate full dynamics in capital accumulation. We do so, by deploying the well-known Samuelson economy with production. There are n > 1 symmetric countries, each populated by two-period overlapping generations. Taxes and public spending in each country are set by the national government. Let subscript t = 0, .., ∞ and j = 1, ..., n denote period t and country j respectively. There is a single composite and traded good, and no uncertainty. There is also a world market for capital. It is assumed that there are no transaction costs or restrictions in trading in this market. That is, there is perfect capital mobility. Let ρt denote the real interest rate in this market in period t. To capture the basic workings in place, and build our main intuition, we deploy here the simplest possible model for our purpose. Specifically, the basic model postulates that the single good is produced in each and every jurisdiction by means of combining capital and a fixed factor, like land. The case of capital being combined, for production purposes, with an endogenous but immobile across jurisdictions factor is discussed in Section 4. Each and every government possesses a per-unit tax on capital employed domestically. In addition, public spending takes the form of public good provision. Assume, for the time being, that governments do not issue public debt, i.e. they do not enter the capital market. Public debt is introduced in Section 4. Expressed in real terms, denote with gt,j the level of public good. Moreover, denote with τ t,j and kt,j the tax on and the level of capital. We also assume that governments do not possess an unrestricted lump-sum tax. So, in the basic model
5
tax authorities do not tax the fixed factor. The case of governments taxing the income from the fixed factor at a rate θt > 0, which is less than the unrestricted optimum one, is discussed in Section 4. One reason for governments facing restriction in their ability to use lump-sum taxes is that administratively feasible forms of such taxes would not be politically feasible. A typical example here is the poll tax in Great Britain imposed by Margaret Thatcher, which is largely viewed as one of the reasons for her having been driven out of office.8 We also assume that a tax on savings is not available. This assumption is motivated from the fact that in practice it is difficult to tax capital income on a residence basis, due to administrative and tax compliance problems associated with taxing foreign-source income.9 The government’s budget constraint is gt,j = τ t,j kt,j .
(1)
Governments are assumed to be benevolent: they choose national policies {gt,j , tt,j }∞ t=0 to maximise total intertemporal welfare. Note that in our basic model, due to the immobile factor being fixed and the unavailability of lump-sum taxes, we effectively abstain from labour income taxes. However, in all OECD countries, a big part of government revenue derive from taxes on wage income. Thus, in Section 4 we also discuss how our basic result is affected by the availability of such taxes. We turn to the description of the private sector in the typical country. Private production in period t in country j takes place by means of a production function f (kt,j ) with the standard properties f (0) = 0, f 0 > 0, f 00 < 0. Capital kt,j is bought in the capital market, and does not depreciate after its use. Rents, i.e. payments to the fixed factor, are thus given by f (kt,j )−(ρt + τ t,j )kt,j , and the demand for capital follows from the standard profit-maximisation condition f 0 (ktj ) = ρt + τ tj .
(2)
So, capital is a decreasing function of the gross rate of interest ρt + τ t,j , kt,j = k(ρt + τ t,j ) with k 0 = 1/f 00 (k). Also, equilibrium returns to the immobile factor, rt,j are a decreasing function of the gross interest rate: rt,j = r(ρt + τ t,j ) with r0 = −k and r(ρt + τ t,j ) ≡ f (k(ρt + τ t,j )) − f 0 (k(ρt + τ t,j ))k(ρt + τ t,j ). 8
See for instance Wilson (1999).
9
For a model where the degree of information sharing between tax authorities is endogenously determined to
be zero, which in turn implies that residents do not, in effect, face a tax on their capital income upon repatriation, i.e. a tax on their savings, see Makris (2003).
6
Let Nt be the size of generation t. Let a superscript denote the period an individual is born. In period t the typical agent allocates her income from the fixed factor between current consumption ctt and savings st+1 . In period t + 1, the agent born in period t consumes her return from savings. Notice at this point that if young agents were not owning any of the immobile factor there would be zero supply of capital. Thus, a necessary condition for the existence of an equilibrium with positive capital is that the firm is partly owned be young citizens. Assume that this is indeed the case. In fact, to simplify exposition, let us assume for the time being that in each period the ownership rights of the firm are transferred from old to young citizens. Specifically, assume that the firm is equally owned by, and only by, young citizens. As we show in Section 4, allowing for senior citizens to own a part of the firm would not affect qualitatively our main results. If the firm is owned only by young citizens, the intertemporal budget constraint of an individual born in period t in country j is, ctt+1,j = (1 + ρt+1 )(
r(ρt + τ t,j ) − ctt,j ) Nt
(3)
We postulate the following preferences V (ctt,j , ctt+1,j )Γ(gt,j , gt+1,j ),
(4)
where Γ(0, 0) = 0 and Γ1 > 0, Γ2 > 0, Γ11 < 0, Γ22 < 0. We10 also assume that V (0, 0) = 0, V1 > 0, V2 > 0, V11 < 0, and V22 < 0. It follows that welfare maximisation for given interest rate ρ and policies, taking into account the budget constraint (3), gives consumption functions ctt,j = c1 (ρt+1 , c2 (ρt+1 ,
rt,j Nt ),
and a savings function st+1,j = s(ρt+1 ,
rt,j Nt ).
rt,j Nt ),
ctt+1,j =
In more detail, in accordance with
the majority of the existing literature, let us assume that the second order sufficient condition is satisfied, that consumption is a normal good and that first- and second-period consumptions are gross substitutes. We thereby have that s1 > 0 and, more importantly, s2 > 0, i.e. savings are increasing with the interest rate and the first-period income. Crucially, for our purposes, the latter, in conjuction with the endogeneity of non-capital income, implies that income and savings in period t depend also on the gross interest rate that clears the current capital market, ρt + τ t . In particular, this dependence is negative. This dependence will be 10
Numbered subscripts are used to denote partial derivatives.
7
further discussed below, when equilibrium in the capital market is investigated. Notice, here, that in a static model savings are exogenous, see for instance, Zodrow and Mieszkowski (1986). In a two-period model first-period income rt,j and, thereby, savings st+1,j are independent of the current gross interest rate ρt + τ t,j , see, for instance, Bucovetsky and Wilson (1991) and Keen and Kotsogiannis (2002).11 Given the above, the value function of typical agent born in period t in country j is U t ≡ U (ρt+1 , ρt , τ t,j , gt,j , gt+1,j ) ≡ V (c1 (ρt+1 ,
(5)
r(ρt + τ t,j ) r(ρ + τ t,j ) r(ρt + τ t,j ) ), (1 + ρt+1 )( t − c1 (ρt+1 , )))Γ(gt,j , gt+1,j ), Nt Nt Nt
with - due to the envelope theorem: U1t /Γt = V2t st+1,j ,
(6)
U2t /Γt = U3t /Γt = −(1 + ρt+1 )V2t kt,j /Nt .
(7)
Equilibrium in the market for capital in period t + 1, t ≥ 0, is given by X
k(ρt+1 + τ t+1,j ) = Nt
j
X
s(ρt+1 ,
j
r(ρt + τ t,j ) ). Nt
(8)
To understand this note that the left hand side corresponds to total demand for capital by firms. Total supply on the other hand consists of private savings. In effect, this condition is equivalent to the equilibrium condition that total demand for the single traded good in period t + 1 equals total supply.12 Let the equilibrium interest rate in period t + 1 be given by ρt+1 = ρ(ρt , ~τ t , ~τ t+1 ) where ~τ ≡ {τ 1 , ..., τ n }. Note that in a symmetric equilibrium we have τ j = τ and sj = s for any j = 1, ..., n, and hence k(ρt+1 + τ t+1 ) = Nt s(ρt+1 , 11
r(ρt + τ t ) ). Nt
(9)
The reason for the latter is straightforward. Even in a two-period model, the inherited aggregate supply of
capital, which affects the first-period payments to the internationally immobile factor, is exogenously fixed. 12
To see this add the t + 1 − period private and public budget constraints for some country j to get Nt ctt+1,j
+ Nt+1 ct+1 t+1,j + gt+1,j = (1 + ρt+1 )Nt st+1,j + rt+1,j − Nt+1 st+2,j + τ t+1,j kt+1,j . After using the definition for rents and summing over j one has
P
j
(Nt ctt+1,j + Nt+1 ct+1 t+1,j + gt+1,j ) =
P
j
((1 + ρt+1 )(Nt st+1,j − kt+1,j ) +
kt+1,j + f (kt+1,j ) − Nt+1 st+2,j ). Since capital is an intermediate good, supplied by individuals to firms, total demand for the single traded good in period t + 1 is supply is
P
j
P
j
(Nt ctt+1,j + Nt+1 ct+1 t+1,j + gt+1,j + Nt+1 st+2,j ). Total
( kt+1,j + f(kt+1,j )). So, demand equals supply if and only if
8
P
j
(Nt st+1,j − kt+1,j ) = 0.
Let the symmetric equilibrium interest rate in period t + 1, t ≥ 0, be given by ρt+1 = p(ρt , τ t , τ t+1 ). We have that p3 ∈ (−1, 0) and p1 = p2 > 0. ¯ it is pre-determined Notice, however, that in period 0 the supply of capital is fixed, at k: by the savings, when they were young, of period−0 senior citizens. So, the symmetric equilib¯ rium interest rate in period 0, as perceived by the governments, is given by ρ0 ≡ pZ (τ 0 , k). ¯ This interest rate is determined implicitly by the market-clearing condition k(ρ0 + τ 0 ) = k, ¯ where k¯ is the given supply of capital in period t = 0 in each region. Clearly, pZ 1 (τ 0 , k) = −1. Given the symmetry of our model we focus on symmetric equilibria. To capture noncooperative tax-setting, we also focus on situations where, given history, fiscal authorities hold Nash conjectures against each other when policy is chosen. We are now ready to investigate the equilibrium national tax policies.
3
Capital Taxation
Our aim is to investigate how the predictions of the ZMW model, which came to be the workhorse in the capital tax competition literature, are modified once one takes into account a fully-dynamic environment. So, as in ZMW, we assume policy pre-commitment. That is, we assume that tax authorities have a commitment mechanism that enables then to announce the whole path of taxes {τ t,j }t=∞ t=0 at time 0 and abide by such an announcement when the time comes to administer the taxes. The investigation of dynamic tax competition in the absence of pre-commitment is out of the scope of the present paper, and is left for future research. We also assume that countries are small open economies. That is, we assume that when tax authorities choose their policies non-cooperatively they take the path {ρt }t=∞ t=0 as given and outside their control. Relaxing this assumption would not affect qualitatively our results - conditional on the existence of equilibrium.13 So, in what follows, each government j chooses in period t = 0 the tax-path {τ t,j }∞ t=0 taking into account the reaction of the private sector and the path of the interest rates {ρt }t=∞ t=0 . 13
The reason is simple. As one can see easily after some trivial derivations, the effect on the t + 1−period
equilibrium interest rate ρ(ρt , ~τ t , ~τ t+1 ) of a marginal change in the capital tax τ t+1,j , evaluated at a symmetric equilibrium, is equal to p3 (ρt , τ t , τ t+1 )/n. That is, tax authorities take into account only 1/nth of the total effect on the symmetric equilibrium interest rate of their tax choices. So the horizontal externalities we highlight here are still present, though somewhat dampened.
9
Notice that the small open economy assumption renders redundant the requirement that governments take as given the tax-paths chosen by the other governments. The reason is that agents’ indirect utilities are independent of foreign taxes for given interest rates. Drop the subscript j until further notice. Define with W (ρt+1, ρt , τ t , τ t+1 ) ≡ Nt U (ρt+1 , ρt , τ t , τ t k(ρt + τ t ), τ t+1 k(ρt+1 + τ t+1 )), for t ≥ 0, the total welfare of consumers born in period as a function of the current and next-period’s interest rates and capital taxes. Similarly, for senior citizens ˆ (ρ0 , ρ−1 , τ −1 , τ 0 ) ≡ N−1 U (ρ0 , ρ−1 , τ −1 , g−1 , τ 0 k(ρ0 + τ 0 ))). in period 0, define W The objective function of the typical tax authorities can then be written as: ˆ (ρ0 , ρ−1 , τ −1 , τ 0 ) + W
∞ X
W (ρs+1 , ρs , τ s , τ s+1 ).
(10)
s=0
When choosing the tax of period t ≥ 0 the typical government takes as given the interest rate and the domestic taxes in the previous and next periods and the current interest rate. Assuming that W (ρκ , ρκ−1 , τ κ−1 , τ κ ) + W (ρκ+1 , ρκ , τ κ , τ κ+1 ) is strictly concave with respect to τ κ , for any {ρκ−1 , ρκ , ρκ+1 , τ κ−1 , τ κ+1 }, κ ≥ 0, the typical capital tax τ t+1 , t ≥ 0, is given at an interior solution by
∂[W (ρt+1 ,ρt ,τ t ,τ t+1 )+W (ρt+2 ,ρt+1 ,τ t+1 ,τ t+2 )] ∂τ t+1
= 0, which evaluated at the
symmetric equilibrium gives: Et+1 = 0.
(11)
t+1 t+1 0 Here Et+1 ≡ [Nt V t Γt2 + Nt+1 V t+1 Γt+1 Γ kt+1 1 ]{kt+1 + τ t+1 k (ρt+1 + τ t+1 )} − (1 + ρt+2 )V2
is the marginal effect of a change in the capital tax τ t+1 on the welfare of generations t and t + 1, for given interest rates and past and future taxes. To understand this condition, note, first, that here the public good in period t + 1 is consumed by both the young and old agents. So the total marginal utility, i.e. the marginal utility of the “typical household”, from public good provision in period t + 1 is Nt V t Γt2 + t+1 t+1 Γ kt+1 represents the negative effect on Nt+1 V t+1 Γt+1 1 . Also, the term − (1 + ρt+2 ) V2
the welfare of generation t + 1 due to a tax-induced decrease in the disposable income of young agents in period t + 1. Concerning the typical capital tax τ 0 , we have that it is given by the above condition after setting, with some abuse of notation, t = −1. We are now ready to analyse the efficiency of the typical regional policy. In particular, we investigate whether a coordinated change in regional taxes is welfare improving or not. In doing so, notice that we postulate a high valuation for public good, and so that the equilibrium taxes are strictly positive.
10
To start with, note that at a symmetric equilibrium the welfare of the typical agent born in period t ≥ 0 is W o (ρt , τ t , τ t+1 )/Nt ≡ U (p(ρt , τ t , τ t+1 ), ρt , τ t , τ t k(ρt + τ t ), τ t+1 k(p(ρt , τ t , τ t+1 ) + τ t+1 )). So, a marginal increase in the (announced) symmetric capital tax τ t+1 , t ≥ 0, while holding all future taxes fixed, results in:14 ∂
P∞
s=0 W
o (ρ
s , τ s , τ s+1 )
∂τ t+1
= Zt+1 p3 (ρt , τ t , τ t+1 ) + Et+1 + +[p1 (ρt+1 , τ t+1 , τ t+2 )p3 (ρt , τ t , τ t+1 ) + p2 (ρt+1 , τ t+1 , τ t+2 )]∆t+1 . Here, ∆t+1 ≡
P∞
s=t+2 Z
s Qs−1 κ=t+2 p1 (ρκ , τ κ , τ κ+1 )}.
Moreover, Zκ = [V2κ−1 Γκ−1 −(1+ρκ+1 )V2κ Γκ ]kκ
+[Nκ−1 V κ−1 Γ2κ−1 + Nκ V κ Γκ1 ]τ κ k0 (ρκ + τ κ ) is the overall effect, through private and public consumptions, of a marginal change in the interest rate ρκ on the welfare of generations κ−1 and κ. Furthermore,
Qs−1
κ=t+2 p1 (ρκ , τ κ , τ κ+1 ) = p1 (ρt+2 , τ t+2 , τ t+3 ) p1 (ρt+3 , τ t+3 , τ t+4 )...p1 (ρs−1 , τ s−1 , τ s )
for any s ≥ t + 4, That is,
Qs−1
Qt+1
κ=t+2 p1 (ρκ , τ κ , τ κ+1 )
κ=t+2 p1 (ρκ , τ κ , τ κ+1 )
≡ 1,
Qt+2
κ=t+2 p1 (ρκ , τ κ , τ κ+1 )
≡ p1 (ρt+2 , τ t+2 , τ t+3 ).
is the (compounded) effect on ρs , s ≥ t + 2, of a marginal
change in ρt+2 . To understand the above equation, recall first that Et+1 is the direct welfare effect on generations t and t + 1 of a change in the typical tax τ t+1 . Zκ , on the other hand, represents the welfare effect on generations κ − 1 and κ of a change in the interest rate ρκ , with the latter being the result of a coordinated change in all capital taxes of period t + 1. The first difference with the canonical capital tax competition model lies in Zt+1 . In the static and two-period models with symmetric countries and no taxes on immobile factors, Zt+1 consists only of the effect that arises from changes in tax revenues. The reason is that the welfare effects from changing the relative price of consumption and from changing the returns to the immobile factor cancel each other out.15 In our OLG set-up, however, these welfare effects fall upon different generations. The reason is that ρt+1 is the rate of return to savings for generation t, and rt+1 are the rents appropriated by generation t + 1. In other words, the supply of the capital stock of period t+1 is endogenously provided by the agents born in period t, while the firm in period t + 1 is owned by the young agents in period t + 1. So, inducing a 14
Because countries are symmetric there is no reason from an efficiency point of view to distort the international
allocation of capital. So, Pareto efficient capital taxes are uniform, and thereby there is no equilibrium trade in capital. 15
See for instance Persson and Tabellini (1992), De Pater and Myers (1994), Keen and Kotsogiannis (2002).
11
decrease in the interest rate ρt+1 by means of an increase in the typical capital tax τ t+1 leads to lower total welfare in period t + 1 if V2t Γt > (1 + ρt+2 )V2t+1 Γt+1 , and vice versa. On the other hand, a change in the interest rate of period t + 1 has no effect on the welfare of the “typical household” through affecting private consumption only if V2t Γt = (1 + ρt+2 )V2t+1 Γt+1 . Note that optimality of savings implies that V1t+1 Γt+1 = (1 + ρt+2 )V2t+1 Γt+1 . So, if consumption by older and by younger members of a household were perfect substitutes, which is the implicit assumption of static and two-period models, i.e. if V1t+1 Γt+1 = V2t Γt , then capital taxes would indeed have no welfare effect through private consumption. Nevertheless, the more important differences between a truly-dynamic set-up and a static or a two-period model stem directly from the following properties of interest rate determination. The interest rate that clears the capital market depends on the current capital tax as well as on savings. In addition, in a truly-dynamic set-up, savings depend on anticipated real interest rate as well as on the non-capital income when young and thereby on the past capital’s user-cost. Accordingly, we have, first, that there are future savings (made by future generations), the returns of which depend on the current interest rate ρt+1 and tax τ t+1 . Second, future levels of capital and thereby public good provision do depend on the current interest rate and capital tax. Finally, there are endogenous future savings which do depend recursively, through their dependence on disposable income, on the current interest rate and the capital tax. All these effects, which have been neglected by the received literature of capital tax competition, are captured by the last term in the above condition. After using (11), the above equation can be re-written as ∂
P∞
s=0 W
o (ρ , τ , τ s s s+1 )
(12)
∂τ t+1
= {Zt+1 p3 (ρt , τ t , τ t+1 ) + +[p1 (ρt+1 , τ t+1 , τ t+2 )p3 (ρt , τ t , τ t+1 ) + p2 (ρt+1 , τ t+1 , τ t+2 )]∆t+1 } ≡ Wτot+1 , A coordinated increase in symmetric capital taxes τ t+1 is welfare improving if and only if Wτot+1 > 0. If Wτot+1 > 0 then the net tax externality is positive, and the non-cooperative equilibrium is characterised by under-taxation of capital. If, on the other hand, Wτot+1 < 0 then the net tax externality is negative, and the non-cooperative equilibrium capital taxes are too high. We clearly have that the direction of inefficiency in the equilibrium capital tax depends on the balance between the terms in the curly brackets. 12
0 The term [Nt V t Γt2 + Nt+1 V t+1 Γt+1 1 ]τ t+1 k (ρt+1 + τ t+1 )p3 (ρt , τ t , τ t+1 ) > 0 represents the
positive horizontal externality that arises from the effect on period-t + 1 capital tax revenues abroad of tax-induced changes in the current real interest rate. To see this, note that the marginal effect on capital tax revenues of a change in price of capital is τ k0 < 0. As interest rates decrease with capital taxes, i.e. p3 < 0, we have that an increase in the capital tax leads to an increase in tax revenues. As tax revenues are valued, i.e. Γ1 > 0 and Γ2 > 0, this constitutes a positive externality, which leaves, ceteris paribus, taxes too low. Note that the extend of this externality is positively related to the responsiveness of capital to the current before-tax real interest rate ρ + τ . This is the standard horizontal externality that arises due to the mobility of capital, and has been emphasised in the ZMW model. We call this the tax-competition effect. The term [V2t Γt − (1 + ρt+2 )V2t+1 Γt+1 ]kt+1 p3 (ρt , τ t , τ t+1 ) reflects the externality that arises from the effect on period−t + 1 foreign consumption of tax-induced changes in the t + 1 − period price of capital. Specifically, a marginal increase in the gross interest rate of period t + 1 increases the returns from the savings of generation t. It also decreases non-capital income of generation t + 1 (and hence consumption). As we have mentioned also above, these two effects do not necessarily cancel each other out, as it is the case in the received literature when countries are symmetric and non-capital income is not taxed. The reason is that each effect is born by different generations. In fact, the net effect is ambiguous, and it depends on the relative marginal utilities of consumption when old of generations t and t + 1. In more detail, as p3 < 0, this externality is positive, and hence reinforces the tax-competition effect, if V2t Γt /V2t+1 Γt+1 < (1 + ρt+2 ), i.e. if the marginal rate of substitution between consumption when old by generations t and t + 1 is lower than the marginal rate of transformation.16 Notice that in equilibrium the shadow value of transfers from parents to children has the sign t − V2t Γt . So,17 if population is non-decreasing and “forward interof V2t+1 Γt+1 (1 + ρt+2 ) NNt+1 16
Note that in a steady-state with positive interest rates, if it exists, consumption by senior citizens, consump-
tion by young citizens, public good provision and interest rates are constant, and hence V2t Γt = V2t+1 Γt+1 . So, this externality does reinforce the tax competition effect. However, in the presence of growth, consumption will not be constant within age-groups, even along a balanced-growth path. In the present model, growth can be sustained if output is given by Af (k), where A is a technology parameter, and there is technological growth. 17
Under transfers from parents to children the budget constraint of the typical agent born in period t is ctt+1 =
rt + bt (1 + ρt+1 )[ N t
that
Nt−1 Nt
Nt−1 Nt
− ctt ] − bt+1 , where bv is the transfer given by the typical agent born in period v − 1. Note
is the number of children of the typical agent born in period t − 1. Under such transfers the wealth
13
generational” transfers are valued in equilibrium then this externality leads to too low taxes. To understand the remaining term [p1 (ρt+1 , τ t+1 , τ t+2 )p3 (ρt , τ t , τ t+1 )+p2 (ρt+1 , τ t+1 , τ t+2 )]∆t+1 , note first that p1 (ρt+1 , τ t+1 , τ t+2 )p3 (ρt , τ t , τ t+1 ) + p2 (ρt+1 , τ t+1 , τ t+2 ) > 0 is the overall change in interest rate, ρt+2 , due to a marginal change in the typical tax τ t+1 . This overall effect takes into account that in a truly-dynamic environment interest rates depend on previous interest rates and capital taxes. Recall also that ∆t+1 represents the overall welfare effect on generations t + 1, t + 2, ... abroad of a marginal change in ρt+2 . Thus, [p1 (ρt+1 , τ t+1 , τ t+2 )p3 (ρt , τ t , τ t+1 ) + p2 (ρt+1 , τ t+1 , τ t+2 )]∆t+1 represents the externality that arises from the effect of capital taxes on all future interest rates. Call this the dynamic effect of capital taxes. Note that for any t ≥ 0 we have, due to p1 = p2 and p3 + 1 > 0, that p1 (ρt+1 , τ t+1 , τ t+2 )p3 (ρt , τ t , τ t+1 ) + p2 (ρt+1 , τ t+1 , τ t+2 ) > 0 : the direct effect of period−t + 1 capital taxes, with t ≥ 0, dominates the indirect (through ρt+1 ) effect on ρt+2 . Thus, an increase in current tax increases next period’s interest rate, and thereby all future interest rates. Hence if ∆t+1 < 0 the dynamic externality leads, ceteris paribus, to too high taxes! Notice however that our discussion of the sign of Zt+1 implies that ∆t+1 is also of ambiguous sign. So the externality in question can lead, all other things equal, to either an under- or an over-taxation of capital in period t + 1. Interestingly, if Zt+1 and Zκ0 , κ0 ≥ t + 2, have the same signs then the pecuniary externalities in periods t + 1 and κ0 from the use of τ t+1 have opposite direction. The reason is that an increase in τ t+1 leads to a decrease in ρt+1 (p3 < 0) and an increase in ρt+2 and thereby all future interest rates ( p1 (ρt+1 , τ t+1 , τ t+2 ) p3 (ρt , τ t , τ t+1 ) + p2 (ρt+1 , τ t+1 , τ t+2 ) > 0 and p1 > 0). Accordingly, if the externality effect on private consumption, due to the breakdown of households in our OLG economy, counteracts the tax competition effect and is sufficiently strong in periods t + 1 and κ0 so that Zt+1 > 0 and Zκ0 > 0 then the effect of capital taxes τ t+1 on the current interest rate leads, ceteris paribus, to too high taxes, while the effect on the future interest rate ρκ0 leads, all other things equal, to too low taxes. That is, the dynamic effect reinforces the tax competition effect. If, on the other hand, the tax competition effect is sufficiently strong so that Zt+1 < 0 and Zκ0 < 0 then the effect of capital taxes τ t+1 on the current interest rate leads, ceteris paribus, to too low taxes, while the effect on the future rt of an agent, in first-period units, becomes (1 + ρt+1 )[ N + bt t
Nt−1 Nt ]
− bt+1 . So, due to the envelope theorem, a
t for the typical member of generation marginal increase in bt+1 leads to a welfare gain of V2t+1 Γt+1 (1 + ρt+2 ) NNt+1
t + 2 and a welfare loss of V2t Γt for the typical agent born in period t + 1. Thus, the shadow value of transfers from parents to children is as stated in the main text.
14
interest rate ρκ0 leads, all other things equal, to too high taxes. That is, the dynamic effect counteracts the tax competition effect, and might thereby lead to too high taxes. Notice that if savings had been income inelastic then the dynamic effect of capital tax competition would have vanished. This follows directly from the fact that if s2 ≡ 0 then p1 = p2 = 0 : past taxes and interest rates would have no effect on current interest rates and thereby capital stocks. Furthermore, if the returns to the internationally immobile factor and thereby the disposable income of young citizens had been exogenously given we would again have had that p1 = p2 = 0. Concerning the typical capital tax τ 0 , we have that the welfare effect of a coordinated change in τ 0 , keeping all future taxes fixed, is given by the above condition for Wτo after setting, ¯ with some abuse of notation, t = −1 and replacing p3 (ρ−1 , τ −1 , τ 0 ) with pZ 1 (τ 0 , k) = −1. Notice ¯ then that - even if s2 > 0 and r0 > 0, and thereby p1 = p2 > 0 - only Z0 pZ 1 (τ 0 , k) comprises the net benefit of a marginal change in the initial typical capital tax; the dynamic effect of capital taxes vanishes. Thus, τ 0 is inefficiently low if and only if Z0 < 0. To investigate the overall strength of the dynamic effect of capital taxes, let us assume that a steady-state exists.
A steady-state of our economy is defined by Nt = N,
ρt = ρ, τ t = τ , s(ρ, r(ρ + τ )/N ) = k(ρ + τ )/N, ctt =
r(ρ+τ ) N
− s(ρ, r(ρ + τ )/N ) ≡ cy ,
ctt+1 = (1 + ρ)s(ρ, r(ρ + τ )/N ) ≡ co , gt = τ k(ρ + τ ) ≡ g, V1 (cy , co ) = (1 + ρ)V2 (cy , co ) and [k(ρ + τ ) + τ k0 (ρ + τ )]N V (cy , co )[Γ1 (g, g) + Γ2 (g, g)] − (1 + ρ)k(ρ + τ )V2 (cy , co )Γ(g, g) = 0. The two last conditions are the first-order conditions with respect to savings and the capital tax, respectively, from the optimisation problems of the typical agent and the government, respectively, evaluated at a stationary equilibrium. Let Z ≡ −ρV2 (cy , co )Γ(g, g)k(ρ + τ ) + N V (cy , co )[Γ1 (g, g) + Γ2 (g, g)]τ k0 (ρ + τ ) < 0. Z is the overall effect, at the stationary equilibrium, of a marginal change in the current interest rate ρ on the welfare of living generations, keeping future interest rates fixed. Since Z < 0, the temporal net horizontal externality leads, as it is the prediction of the ZMW model, to too low taxes, all other things equal. In this case, however, the dynamic externality leads, ceteris paribus, to too high taxes. In more detail, after defining pµ ≡ pµ (ρ, τ , τ ), µ = 1, 2, 3, we have that at steady-state Wτo /Z = p3 + p1 (1 + p3 )
∞ X
ps1 ,
(13)
s=0
and stationary capital taxes are too low if p3 + p1 (1 + p3 )
P∞
s s=0 p1
< 0, and vice versa.
Interestingly, observe that by using a static or two-period model and identifying Z with 15
the welfare effect on the typical household from a marginal increase in the interest rate, one would arrive at the above condition with p1 ≡ 0 (and pZ 1 = −1 instead of p3 , if the model is ¯ see for instance Keen and Kotsogiannis (2002). Thus, the static, i.e. with fixed savings k); difference of our steady-state analysis with the analysis in static or two-period models boils down to the fact that, in a dynamic framework, it is recognised that a marginal increase in the symmetric-equilibrium (steady-state) capital tax affects current, in a negative manner, and all future, in a positive manner, interest rates. That is, a dynamic framework recognises that the overall, intertemporal, effect on all interest rates of an increase in the stationary capital tax is p3 + p1 (1 + p3 )
P∞
s s=0 p1
and not p3 .
Clearly, then if p3 → −1 taxes are too low, while if p3 → 0 taxes are too high. In particular, capital taxes are too high if the (negative) effect of taxes on current interest rates is P∞
ps1 . ps s=1 1
sufficiently small, i.e. p3 > − 1+Ps=1 ∞
Note that current interest rates are more responsive to
current taxes, the less responsive savings are to the price of capital (i.e. the lower s1 is).18 As −1 < p3 < 0, it follows directly that if p1 → 0 taxes are too low, while if p1 → ∞ taxes are too
high. In general, capital taxes are too high if the (positive) effect of past on current interest rates is sufficiently large, i.e. p1 ≥ p∗1 where p∗1 is defined by
P∞
∗s s=1 p1
p3 = − 1+p . Note that 3
current interest rates are less responsive to past interest rates, the more responsive savings are to the price of capital (i.e. the higher s1 is) and the less responsive savings are too disposable income (i.e. if the lower s2 is).19 Finally, note that the more price-responsive capital is (the higher | k0 | is), the more responsive to current taxes and the less responsive to past interest rates current interest rates are. Thus, to summarise, we have that capital taxes are too high if capital is not very priceelastic and savings are sufficiently high income-elastic, and vice versa The responsiveness, on the other hand, of savings to their returns has an ambiguous effect on the efficiency properties of capital taxes.
4
Extensions
In this Section we discuss some of our assumptions. First we discuss the robustness of our results, while maintaining the assumption that the internationally immobile factor of produc18
Note from the symmetric equilibrium capital market-clearing condition that p3 = [ sk10 − 1]−1 .
19
Note from the symmetric equilibrium capital market-clearing condition that p1 = s2 k [s1 − k0 ]−1 .
16
tion is fixed. Then, we also discuss the robustness of our results to allowing for an endogenous internationally immobile factor of production, like labour.
4.1
Exogenous Non-Capital Factor
When non-capital factors of production are fixed, the main message of our paper - that the dynamic externality counteracts and may outweigh the temporal tax competition effect - is robust. The reason is that the temporal externality works through the negative effect of current taxes on current interest rates, while the dynamic externality works through the positive effect of current taxes on future interest rates. Extending the model, by following closely the extensions in the existing literature of the typical ZMW model, and allowing, for instance, for public debt, other taxes like rent and consumption taxes, foreign ownership, money, multilevelled government e.t.c., would only affect the particular form, and thereby the size and/or sign, of Zκ , κ ≥ 0. The same is true if we allow for partial ownership of the fixed factor on the part of senior citizens. This implies that condition (13) is still relevant, after generally defining Z as the overall welfare effect on the part of living agents in the typical region of a marginal increase in the stationary interest rate for given future interest rates - with the particular form of Z depending on the exact details of the model, like the availability of other taxes, the presence of money e.t.c. So, stationary capital taxes are too low if and only if Z[p3 + p1 (1 + p3 )
P∞
s s=0 p1 ]
> 0.
To put it another way, if some static or two-period model with exogenous labour predicts that the temporal welfare effect of the interest rate is negative and hence capital taxes are too low, then this result extends over to the stationary equilibrium of the OLG version of that model if [p3 + p1 (1 + p3 ) versa.
P∞
s s=0 p1 ]
< 0, otherwise stationary capital taxes tend to be too high, and vice
To put it another way, of some static or two-period model with exogenous labour finds that Z < 0 and hence that taxes are too low, then this result extends to a dynamic environment if [p3 + p1 (1 + p3 )
P∞
s s=0 p1 ]
< 0, otherwise taxes are too high, and vice versa. To demonstrate
the above we examine, in what follows, some indicative extensions. We start with the case of the firm being partly owned by senior citizens. Let 0 < as < 1 denote the proportion of the firm that is owned, in period s, by senior citizens. Disposable income when young decreases by as rs,j /Ns for generation s. At the same time, however, income when old increases by as rs,j /Ns−1 for generation s−1. Following similar steps to the ones in the 17
previous Section one can then easily see that Zs increases by as ks {(1 +ρs+1 )V2s Γs −V2s−1 Γs−1 }. The first part represents the dampening of the effect of a change in the interest rate ρs on consumption in period s + 1 by generation s through the disposable income of young agents in period s. The second part represents the dampening of the effect of the interest rate ρs on consumption in period s by generation s − 1 through the returns to period-s immobile factors. Therefore, if (1 + ρs+1 )V2s Γs > V2s Γs and for s = t + 1, κ with κ > t + 1, the associated temporal externality due to a tax-induced decrease in ρt+1 counteracts the tax competition effect, and the associated dynamic externality due to a tax-induced increase in ρκ reinforces the tax competition effect, and vice versa. Turning to the steady-state, we have, given that at steady-state V2s Γs = V2s−1 Γs−1 and as = as−1 , that increasing the firm ownership of senior citizens reduces, all other things equal, the scope for coordination in the long-run. The reason is that the steady-state temporal welfare effect of interest rates is now higher, and thereby Wτo is higher by akρV2 Γ[p3 + p1 (1 + p3 )
P∞
s s=0 p1 ].
But note that akρV2 Γ has the opposite sign of
−ρV2 (cy , co )Γ(g, g)k(ρ + τ ) + N V (cy , co )[Γ1 (g, g) + Γ2 (g, g)]τ k 0 (ρ + τ ). Thus, by increasing the ownership of senior citizens, the overall externality at steady-state is dampened, all other things equal, regardless of its direction, i.e. regardless of the sign of [p3 + p1 (1 + p3 )
P∞
s 20 s=0 p1 ].
At this point we can bring out another critical aspect of using two-period models to study capital tax competition. Notice first that the model we have been analysing so far can also be interpreted as one where citizens supply labour inelastically. Under such an interpretation, the production function becomes time-dependent, i.e. with some abuse of notation ft (k) ≡ f (k, Ht ) where Ht is total labour employed in period t. Also, rt = ft − kt ft0 becomes the total wages in period t, and at becomes the share of total labour which is supplied by senior citizens. It follows directly that if aκ = 1 had been the case, i.e. if only senior citizens were working (!!), then there would be zero supply of capital and hence no production in period κ, unless young citizens were born with a sufficiently high endowment of the private good - or senior citizens were transferring enough of their income to their children. Suppose then that at = 1, t ≥ 0, and that there are exogenously given endowments/bequests et /Nt received by each and every young citizen of generation t. In this case, our model is in fact an overlapping generations version of the typical barebone two-period model which is used in the received literature. 20
The ‘all other things equal’ refers to the fact that a change in ownership will, in equilibrium, affect also
chosen policies and thereby capital stock, public good provision and marginal utilities.
18
In such an environment, as well as in the standard version of the two-period model, income when young is exogenous and savings depend only on this income and the anticipated interest rate in the next period. Thus, p1 = p2 = 0 and hence the dynamic effect of capital taxes vanishes. Also, there is no externality through the effect of the interest rate ρt+1 on private consumption in period t + 1 by generations t and t + 1. To see this, note first that ρt+1 does not affect consumption of generation t + 1, as income when young is now exogenously given. On the other hand, the effect of ρt+1 on period-t + 1 consumption of the typical member of e
generation t is
r
∂[(1+ρt+1 )( Nt −ctt )+ Nt ] t t ∂ρt+1
= st+1− (kt+1 /Nt ) = 0, .where the last equality follows
from the symmetric capital-market equilibrium. Accordingly, if at = 1 the only externality that survives - even in our dynamic framework - is the one which is emphasised in the ZMW model, and taxes are predicted to be too low. If, on the other hand, at < 1 (and et ≥ 0), then both the externalities we have identified in this paper, i.e. the externalities through the effect of taxes on current private consumption and future interest rates, re-emerge. We believe that a model with (at least some) labour supplied by young citizens is a more realistic, and as we have seen here a certainly richer, framework for the study of capital tax competition. Let us therefore return hereafter to the case of at < 1 and et ≥ 0. In fact, to simplify exposition, and bearing in mind our discussion above of the implications of having at > 0, we focus hereafter on our baseline environment with at = et = 0. The implications of allowing for endogenous labour are discussed shortly after. Before doing so, let us consider next the case of the government issuing public debt. Note that due to perfect capital mobility, agents must earn the same after-tax returns from public and private bonds. Thus, in equilibrium, agents are indifferent between lending firms or governments, as long as total asset holdings equal their planned savings. So, the consumers’ problem and value function remains, in effect, the same, and equilibrium private bond holdings equal savings minus holdings of public debt. Following the convention that dt+1 denotes the debt liabilities of the government at the end of period t, and the beginning of period t + 1, equilibrium in the market for capital in period t + 1 is given by X
[k(ρt+1 + τ t+1,j ) + dt+1,j ] = Nt
j
X j
s(ρt+1 ,
r(ρt + τ t,j ) ). Nt
(14)
Let the equilibrium interest rate in period t+1 be given by ρt+1 = ρ(ρt , ~τ t , ~τ t+1 , d~t+1 ) where d~ ≡ {d1 , ..., dn }. Let also the symmetric equilibrium interest rate be given by ρt+1 = p(ρt , τ t , τ t+1 , dt+1 ), and note that p4 (ρt , τ t , τ t+1 , dt+1 ) > 0. Recall that p1 (ρt+1 , τ t+1 , τ t+2 , dt+2 ) > 0. Thus, one can 19
easily see, following the arguments so far, that public debt policy produces its own pecuniary externalities by affecting positively current and future interest rates.21 These externalities are analogous to the ones that arise by the use of capital taxes, with the difference that for public debt both the temporal and the dynamic externalities are of the same direction. A detailed discussion of these external effects is, however, out of the scope of the present work, and is left for future research.22 To see how the efficiency properties of capital taxes are affected with the introduction of public debt, recall that the consumer’s value function remains the same, and note that the government’s period−t+1 budget constraint becomes gt+1,j = τ t+1 kt+1 +dt+2,j −(1+ρt+1 )dt+1,j . To understand this, note that now, in period t + 1, tax revenue requirements increase by the need to service past debt, i.e. (1 + ρt+1 )dt+1,j , while fiscal revenues can increase by issuing new debt dt+2,j at the end of period t + 1. Following similar steps to the ones in the previous Section one can then easily see that Zs decreases by ds (1 + ρs )[Ns−1 V s−1 Γs−1 + Ns V s Γs1 ]. 2 This term represents the externality that arises from the effect on the servicing of past debt abroad of tax-induced changes in ρs . The static horizontal externality that arises due to the introduction of public debt has also been identified by Jensen and Toma (1991) in a two-period model. Clearly, the temporal effect of positive public debt is that the tax competition effect is reinforced, and taxes tend to be too low. However, the dynamic effect implies that taxes tend to be too high. It follows then, after following similar steps to the ones above when we discussed firm-ownership, that the overall externality at steady-state is reinforced, all other things equal, regardless of its direction. So, the scope for tax coordination in the long-run increases, after the introduction of public debt. Consider now the case of taxable rents at a rate θs > 0, s ≥ 0, which is lower than its unrestricted level. In this case the government’s budget constraint becomes gs,j = τ s,j kj + 21
For completeness, note that the first order condition of government j with respect to end-of-period-t issues of
d d = 0 Here, Et+1 ≡ [Nt−1 V t−1 Γt−1 +Nt V t Γt1 ]− (1+ρt+1 ) [Nt V t Γt2 +Nt+1 V t+1 Γt+1 public debt, i.e. dt+1,j , is Et+1 2 1 ]
is the overall welfare effect on generations t − 1, t, t + 1 from a marginal increase in dt+1 . This effect results from higher public good provision in period t at the cost of lower public good provision in period t + 1 due to the servicing of debt. 22
In fact one can show that, due to p4 > 0 and p1 > 0, steady-state public debt is inefficiently high, for any
given capital tax. For a related discussion, see, for instance, Jensen and Toma (1991), and Beetsma et. al. (2001) and references therein.
20
θs rs,j , and disposable income when young decreases by θs rs,j /Ns for generation s Following similar steps to the ones in the previous Section one can then easily see that Zs increases by θs ks {(1 + ρs+1 )V2s Γs − [Ns−1 V s−1 Γs−1 + Ns V s Γs1 ]}. This term represents the net externality 2 that arises from the effect on disposable income (and hence private consumption) and rents-tax revenues abroad of tax-induced changes in ρs . This net horizontal externality that arises due to the taxation of the immobile factor has also been identified by Keen and Kotsogiannis (2002) in a two-period model. Clearly, the direction of this net externality depends on the relative marginal valuation of private and public consumptions. By following similar steps to the ones above, when we discussed firm-ownership, one can very easily see that if, at the steady-state, (1 + ρ)V2 Γ > −N V [Γ2 + Γ1 ], then increasing the tax on the internationally immobile factor reduces, all other things equal, the scope for tax coordination in the long-run, and vice versa.
4.2
Endogenous Non-Capital Factor
Next, we consider the implications of the internationally immobile factor being endogenous. In particular, assume that the production function is homogenous of degree one. Let, with a slight abuse of notation, f be the intensive form representation of this technology. Now kt is the period-t capital stock as a proportion of the immobile factor Lt Nt ; that is, period-t capital is equal to kt Lt Nt . Referring, for brevity, to the endogenous immobile factor as labour, rt /Nt is now replaced by the before-tax labour income wt Lt , where wt is the period-t wage rate. Also, θt ≤ 1 is now the labour income tax rate in period t. Let also utility of the typical member of generation t be Φ(Λ − Lt , ctt , ctt+1 )Γ(gt , gt+1 ), where Λ is the time endowment, Λ − Lt is leisure and Φ1 > 0, Φ11 < 0. In this case, the wealth of a t−generation agent in country j, htj , is equal to htj = (1 + ρt+1 )(1 − θt )wt,j Λ and the budget constraint is (1 + ρt+1 )(1 − θt )wt,j (Λ − Lt,j ) + (1 + ρt+1 )ctt,j + ctt+1,j = htj . Assuming that leisure is a normal good, standard consumer theory tells us that period-t labour supply in jurisdiction j is given by a function L(ρt+1 , htj , (1 + ρt+1 )(1 − θt )wt,j ) with L2 < 0 and L3 > 0. Normalise units for clarity so that Λ = 1. Note that the effect on labour supply of changes in the net wage Lω has the sign of [L2 + L3 ](1 + ρ), which captures the balance of the usual income and substitution effects. That is, labour supply is upward sloping if and only if L2 + L3 > 0. Let us denote with Lρ the total effect on labour supply of a marginal increase in the next period’s real interest rate ρ, i.e. Lρ = L1 + [L2 + L3 ](1 − θ)w, and with Lτ the effect on labour supply of a marginal change in the capital
21
tax, i.e. Lτ = −[L2 + L3 ](1 − θ)(1 + ρ)k. Clearly, then, when labour supply is upward sloping, we have that Lτ ≤ 0, while Lρ is ambiguous. One can then see that similar considerations apply to the ones so far, with the only difference that now the interest rate in period t + 1 depends also on the next period ’s interest rate ρt+2 . This follows directly after noting that in a truly dynamic environment endogenous labour supply depends on future interest rates; in particular Lt+1 depends, among others, on ρt+2 . By following the steps in Section 3, one can then see that the counterpart of (13) differs in eight ways.23 First, all terms of Zκ , κ ≥ t + 1, in (13) are multiplied with Lκ Nκ . Second, Zκ , κ ≥ t + 1, in (13) will take into account that ρκ affects negatively the current (upward-sloping) labour supply (through the interest rate’s negative effect on current wages) and thereby public good consumption (through the associated drop in the capital and labour tax bases). Specifically, the capital tax in period κ gives rise to a temporal horizontal externality which is represented + Nκ V κ Γκ1 ] [τ κ kκ + θκ wκ ] Nκ Lκω kκ < by Zκ , with κ ≥ t + 1, being higher by −[Nκ−1 V κ−1 Γκ−1 2 0. Third, Zκ , κ ≥ t + 1, in (13) will also take into account the effect of the tax τ κ , through ρκ , on private and public good consumptions in period κ, due to the taxation of labour income. This gives rise to an externality which is analogous to the one under a tax on rents discussed above. Note that the temporal version of these externalities is analysed in Bucovetsky and Wilson (1991). Note also, after recalling our discussion in the previous sub-section, that the only effect on our analysis so far is on the size and sign of Zκ , κ ≥ t + 1. The following differences, however, are specific to the endogeneity of labour in a dynamic environment, and are not captured by our discussion so far. In fact, the following results are absent in Bucovetsky and Wilson (1991), where Lρ = 0 (as in that model there is no future and, hence, labour income is used only for current consumption). In particular, the fourth difference arises from the fact that an anticipated change in period-κ interest rate ρκ will also affect labour supply and, hence, capital stock in period κ − 1. That is, the welfare effect of an increase in ρκ increases by Zˆκ ≡ [Nκ−2 V κ−2 Γκ−2 + Nκ−1 V κ−1 Γ1κ−1 ] [τ κ−1 kκ−1 + θκ−1 wκ−1 ] Nκ−1 Lκ−1ρ . This 2 23
Allowing for endogenous labour, and endogenous labour income taxes, gives rise to a whole host of issues,
the detailed analysis of which demands, space-wise, a paper of their own. In a companion paper, therefore, we also analyse the efficiency properties of the labour income tax. In addition, we also examine the non-cooperative mix between capital and labour income taxes (in the presence of public debt), re-visiting thereby the lessons taught by Chamley (1986) for the case of a closed economy.
22
effect falls on generations that are alive in period t − 1. Therefore, an anticipated tax-induced reduction in the world interest rate in period t + 1, t ≥ 0, leads to decrease in previous period’s foreign welfare if Lρ > 0, and vice versa. The last four differences arise due to the fact that under endogenous labour a period’s interest rate depends on the previous period’s as well as on the next period’s interest rates, i.e. interest rates are partially forward looking. In more detail, recall that τ t+1 , t ≥ −1, affects positively the period-t + 2 interest rate ρt+2 , and thereby all other future interest rates ρt+3 , ρt+4 .... But when interest interest rates are partially forward looking each of these changes will have an effect on past interest rates. That is, the change in ρκ , κ ≥ 1, will have an effect on all the interest rates of periods κ − 1, κ − 2, ..., 0. The end result will of course depend on the sign of Lρ . We then have the following: The fifth difference is that the counterpart of (13) includes an additional term that captures the fact that an anticipated change in τ t+1 , t ≥ 0, and thereby in ρt+1 , t ≥ 0, will also affect the interest rate in period t, through the associated changes in all the interest rates after period t. To understand this term, notice first that an anticipated marginal change in ρt will in turn lead, through its effect on all previous interest rates, to an overall welfare effect on ˆ t+1 ≡ generations t, t − 1, ..., 0, −1 equal to ∆
Pt
ˆ Qt−1−s π s+κ ], where π s+κ denotes κ=0
s=0 [(Zs + Zs )
the marginal effect on the symmetric equilibrium interest rate in period s + κ of a change in the next period’s interest rate ρs+κ+1 . Thus, the additional term in the counterpart of (13) is ˆ t+1 , where γ t denotes the overall effect on the symmetric equilibrium interest rate equal to γ t ∆ in period t of a change in the capital tax in period t + 1. Sixth, the counterpart of (13) includes also an extra term that captures the fact that an anticipated change in τ t+1 , t ≥ −1, will also affect the interest rate in each and every period κ ≥ t + 1, through the associated changes in all the interest rates after period κ. This term is equal to
P∞
s=t+1 [(Zs
P κ Qκ−1 v κ + Zˆs ) ∞ κ=s+1 [β v=s π ]], where β denotes the marginal effect on the
symmetric equilibrium interest rate in period κ ≥ t + 2 of a change in the capital tax in period t + 1. Note here that β κ takes into account that ρκ−1 depends on ρκ , i.e. that interest rates are partly forward-looking. Let also δ t+1 denote the effect of τ t+1 on the symmetric equilibrium interest rate ρt+1 taking as given the next-period’s interest rate. δ t+1 takes also into account that ρt depends on ρt+1 . We then have that the seventh difference in the counterpart of (13) is that p3 (ρt , τ t , τ t+1 ) and [p2 (ρt+1 , τ t+1 , τ t+2 ) + p3 (ρt , τ t , τ t+1 ) p1 (ρt+1 , τ t+1 , τ t+2 )] 23
Qs−1
κ=t+2 p1 (ρκ , τ κ , τ κ+1 ),
with s ≥ t + 2, are replaced by δ t+1 and β s , respectively.
So, to summarise, with endogenous labour, capital tax τ t+1 , t ≥ −1, is too low if and
only if Wτot+1 > 0 with Wτot+1 = (Zt+1 + Zˆt+1 )δ t+1 + γ t
∞ X
β s (Zs + Zˆs )
(15)
s=t+2
+
t X
[(Zs + Zˆs )
s=0
t−1−s Y
π
s+κ
]+
κ=0
∞ X
[(Zs + Zˆs )
s=t+1
∞ X
[β
κ=s+1
κ
κ−1 Y
π v ]].
v=s
Here, Zκ , is equal to Zκ ≡ (1 − θκ )[V2κ−1 Γκ−1 − (1 + ρκ+1 )V2κ Γκ ]kκ Nκ Lκ +[Nκ−1 V κ−1 Γκ−1 + 2 Nκ V κ Γκ1 ]Nκ {τ κ k 0 (ρκ +τ κ )Lκ −[τ κ kκ +θκ wκ ]Lκω kκ }. In the above, let γ −1 ≡ 0, as at the instant of decision-making 0 the interest rate in period −1 is pre-determined. Note also that, for t ≥ 0, we have γ t = π t [δ t+1 +
P∞
κ=t+2 [β
κ Qκ−1 v v=t+1 π ]].
Clearly, then, the externalities produced by
non-cooperative tax-setting are more complicated now, and the efficiency properties of capital taxation are ambiguous. In fact, note that now if Lρ > 0, and hence Zˆ > 0, then at a stationary equilibrium the welfare effect of an anticipated change in the interest rate of some ˆ may no longer be negative period, while keeping constant all other interest rates, i.e. Z + Z, due to the positive effect of anticipated interest rates on the previous period’s labour supply and, hence, tax revenues.24 Therefore, if current interest rates are still negatively related to current capital taxes, i.e. δ t+1 < 0, then the temporal externality may no longer be positive. That is, even if we had ignored the dynamic externalities, i.e. had set arbitrarily π t = 0 and β κ = 0, κ ≥ t + 2, then capital taxes might still not have been too low, in contrast to the typical tax competition model. The reason is that a non-dynamic model ignores the effect of anticipated changes in future capital taxes on the current labour supply and capital stock. The last difference from the case of inelastic labour is that, crucially, the effects of the anticipated tax τ t+1 on interest rates, and thereby δ t+1 , π κ , κ ≥ 0, and β κ , κ ≥ t + 2, are also ambiguous. Therefore, a prediction that taxes are too low becomes even more problematic. To see the ambiguity in question note that, as it can very easily be shown, in a symmetric ¯ and ρt+1 = pt+1 (ρt , τ t , τ t+1 , θt , θt+1 , ρt+2 ), for t ≥ 0. Suppose equilibrium ρ0 = pZ (τ 0 , θ0 , ρ1 , k) now that capital stocks are decreasing with the interest rate and that savings are increasing with income and their return. It follows that the direct effects of the current capital tax and next-period’s interest rate on the period-0 interest rate can be unambiguously signed as 24
A labour supply which is increasing with future interest rates is one of the fundamental ingredients of the
Real Business Cycle theory. See, for instance, Romer (1996) Ch. 4.
24
∂pZ /∂τ 0 < 0 and ∂pZ /∂ρ1 having the sign of L0ρ . Similarly, one can also see that ∂pt+1 /∂ρt > 0, ∂pt+1 /∂τ t > 0, ∂pt+1 /∂τ t+1 < 0 and ∂pt+1 /∂ρt+2 having the sign of Lt+1ρ . Yet, the overall effects have to take into account that interest rates are also forward looking. As an example, consider the period-1 interest rate and notice that, by means of elimination, we have ρ1 = ¯ τ 0 , τ 1 , θ0 , θ1 , ρ2 ). Clearly, then, the overall effects on period-1 interest rate p1 (pZ (τ 0 , θ0 , ρ1 , k), of past and current capital taxes and next-period’s interest rate depend, if they are well-defined, on the sign of 1 −
5
∂p1 ∂pZ ∂ρ0 ∂ρ1 ,
which is ambiguous if L0ρ > 0.
Conclusions
This paper shows that taxes on mobile capital may not be too low, in a fully dynamic environment. Our explanation is based on a simple ingredient: in a truly dynamic economy income from internationally immobile factors and therefore savings depend on the current capital stock and thereby on the current user-cost of capital. This, in turn, implies, in conjuction with capital-market equilibrium, that capital taxes produce an externality by affecting future interest rates. Second, in an overlapping generations economy with positive supply of capital, the effect of capital taxes on the current income of different generations is not the same across generations. The reason is that for older generations capital taxes affect the returns to their savings, while for younger generations capital taxes affect their non-capital income. This implies that capital taxes do produce an externality through affecting current consumption levels abroad. These two externalities may lead, ceteris paribus, to too high national capital taxes. These additional effects, of capital taxes, may more than offset the usual effects of tax competition. In this case, and contrary to conventional wisdom, non-cooperative capital taxes will be too high. Future research that incorporates, for instance, intergenerational altruism and endogenous bequests, or asymmetric regions will improve our understanding of capital tax competition in a fully dynamic environment. The case of bequests is particularly interesting because in this case savings depend also on the returns to savings of future generations. This in turn implies that the current equilibrium interest rate depends on all (rationally anticipated) future interest rates. In fact this is also the case in a representative agent economy, where intergenerational altruism is perfect. Thus announcing, under pre-commitment, a high tax in the future will have an additional effect on current and short-run interest rates, which in turn will give rise
25
to an additional horizontal externality. The investigation of this externality is left for future research. Finally, another interesting extension would be to investigate the case of amenities, i.e. production-enhancing public spending. For any given public consumption and capital tax, an increase in spending on current amenities has two effects, by affecting production possibilities. First, it affects the current demand for capital for any given price of capital, and hence the current gross interest rate that clears the market for capital. So, if amenities and capital are complements, an increase in public inputs increases the equilibrium current interest rate. Of course, an increase in current interest rates will also lead, as we have seen, to an increase in future interest rates. Second, an increase in amenities affects the current returns to immobile factors of production and hence current savings and future gross interest rates. If amenities increase the returns to immobile factors, higher amenities imply higher savings and hence lower future interest rates. The detailed investigation of these externalities is left for future research.25
6
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