Regional Science and Urban Economics 21 (1991) 333-350. North-Holland

Tax competition with two tax instruments*

Sam Bucovetsky

York University, North York, Ont., Canada M3J IP3

John Douglas Wilson

Indiana University, Bloomington, IN 47405, USA

Received September 1990, final version received May 1991

Models of tax competition among local or regional governments typically assume that only a source-based tax on capital income is available. This paper presents a model where a wage tax is also available and shows that small regions choose not to tax capital income, given that it can only be taxed on a source basis. However, the model also yields the conclusion from previous tax competition models that local public goods are underprovided. In contrast, government use of the available tax instruments is etlicient when both source- and residence-based capital taxes are available, even in the absence of wage taxation.

1. Introduction

The recent literature on tax competition among local governments demon- strates that inefficiently low tax rates and levels of local public good provision are the result of competition for scarce capital [see Wilson (1986) and Zodrow and Mieszkowski (1986)]. In these models, each local govern- ment’s public expenditures must be financed by a tax levied on capital income earned within its boundaries, i.e., a source-based capital income tax. However, other forms of taxation are clearly preferable to a tax of this type, which distorts investment location decisions. In particular, taxes on the labor income or worldwide capital income earned by the region’s residents are nondistortionary, because the models in this literature assume that capital ownership and the supply of labor within each region are fixed. It is also known that the desirability of a source-based capital tax disappears in

*Much of the research reported in this paper was completed while the first author was a member of the economics department at the University of Western Ontario. Comments by the referees, David Wildasin and seminar participants at the University of Kentucky are appreciated.

016&0462/91/$03.50 0 l991-Elsevier Science Publishers B.V. All rights reserved

334 S. Bucovetsky and J.D. Wilson, Tax competition with t~‘o ta.u instruments

models with a system of optimal commodity taxes. This result is related to the Diamond-Mirrlees (1971) conclusion that aggregate production efficiency is desirable in the presence of optimal commodity taxation. In Gordon’s model (1986) where labor supply and savings are subject to choice, commodity taxation consists of taxes on wage income and the worldwide capital income of the region’s residents, i.e., a residence-based tax on capital income. By extending his single-region analysis to consider the equilibrium for a system of many identical regions, it can be shown that the availability of these two taxes eliminates any tendency for regional governments to underprovide public goods.

Since optimal commodity taxation is a strong assumption, an interesting issue is whether the decentralized decision-making by regional governments continues to be efficient in cases where there do exist taxes other than a source-based capital income tax, but not a full complement of commodity taxes. The issue here is not whether the source-based tax is utilized in the absence of optimal commodity taxation, but whether decentralized decision- making produces a constrained Pareto optimum, where the constraint is on the set of available tax instruments.

The present paper analyzes this issue within the simplified framework of Gordon’s paper, where the relevant private commodities are present con- sumption, future consumption, and labor; and optimal commodity taxation consists of a tax on wage income and a residence-based tax on capital income. We allow regions to possess market power but assume that all residents of all regions are identical. This enables us to restrict attention to symmetric equilibria, in which each region’s net capital exports are zero. The incentives to use the tax system to manipulate the terms of trade are thereby eliminated.

Two main cases are considered. First, it is assumed that, in addition to a source-based tax on capital income, regional governments can tax only wage income. This case is particularly relevant for the interpretation of regions as countries, since administrative difficulties often make it relatively easy to evade domestic taxes on foreign-source capital income. In the absence of this residence-based tax, it turns out that regional governments continue to underprovide public goods, given the available methods of taxation. More- over, they finance these goods using inefficiently low rates of taxation on capital income relative to wage income. Thus, the main implications of single-period models continue to hold even when wage taxation is introduced.

These results may appear to conflict with Razin and Sadka’s (1989) conclusion that ‘. tax competition is a constrained optimum, relative to the set of tax instruments that is available’, if small countries ‘cannot tax their residents on income from capital invested in the rest of the world’. This conclusion holds even though the countries do not employ any source-based

S. Bucooetsky and J.D. Wilson, Tax competition with two tax instruments 335

taxes on capital. In the present paper, countries also rely solely on wage taxation, if they are small enough relative to the rest of the world. However, the wage tax is shown to be inefticiencly low. The apparent conflict in results is resolved by observing that Razin and Sadka consider only two small countries that face a fixed world interest rate determined in the rest of the world. In our model, the tax policy for every country in the world is determined endogenously.

To pinpoint the tax assumptions responsible for public good under- provision, we next consider the case where both source- and residence-based taxes on capital income are available, but not taxes on wage income. For this case, it turns out that the equilibrium behavior of regional governments is efficient, given the available tax instruments. Specifically, regions make efficient use of both source- and residence-based taxes to finance public good provision, and they choose the efficient level of public good provision. In contrast to the Diamond-Mirrlees production etliciency result, this efftciency result is obtained even though regions do not have access to an optimal commodity tax system, which would generally include a tax on labor. In contrast to the results of Bucovetsky (1991), Hoyt (1991), and Wilson (1991), which are discussed later, the equilibrium tax and expenditure policies are independent of regional size under the constant-returns-to-scale assumptions in the model.

Thus, the overall conclusion is that the absence of residence-based taxes on capital income, not taxes on wage income, is responsible for the under- provision of public goods. This distinction does not arise in the single-period models of tax competition. There the assumed lixity of both the economy’s total capital stock and each region’s labor supply insures that a residence- based capital income tax and a wage income tax are both lump-sum taxes. Wilson (1986, p. 308) observes that the introduction of lump-sum taxes restores efficiency in regional government behavior.

The plan of the paper is as follows. After introducing the model in section 2, we examine tax competition without residence-based capital income taxation in section 3. Tax competition without wage income taxation is then investigated in section 4. Section 5 contains concluding remarks.

2. The model

We consider a system of a fixed number of identical regions, each containing the same number of residents. (Size differences are considered in section 4.) Each region’s population is normalized to equal one. To concentrate on the efficiency issues associated with capital mobility, indivi- duals are assumed to be immobile among regions, but capital is perfectly mobile.

We wish to provide as simple a model as possible in which savings

336 S. Bucouetsky and J.D. Wilson, Tax competition with two tax instruments

decisions are endogenous. We thus assume that the economy lasts for two periods. This sort of model provides a tractable framework for presenting intertemporal decision problems, as evidenced by the classic work of Irving Fisher. The two-period framework is utilized in the recent papers on capital taxation by Gordon (1986) and Gordon and Varian (1989).

The time structure of the model is as follows. The first period has two stages. In the initial stage, governments choose their future tax rates non- cooperatively. It is assumed that governments can commit to a tax policy. In the second stage, each region’s identical residents choose how to divide a given endowment between current consumption and savings. Once the second period arrives, these residents choose how much labor to supply, and competitive firms produce a homogeneous output from labor and capital. This output is used for private and public consumption. Of course, residents forecast correctly the market-clearing wages and returns to capital when they make their first-period savings decisions. In the second period as well, each region’s government collects its tax revenue and uses the proceeds to supply a public good, which is distributed uniformly across the region’s residents.

Turning to details, we assume that each individual possesses a well- behaved utility function that is separable between the public good and private commodities:

where ci is private good consumption in period i, n is labor supply in the second period, g is the public good supply, and v( .) is the ‘private utility’ function. A resident chooses his labor and private good levels to maximize utility, subject to a standard lifetime budget constraint:

s.t. c,=(l +p)(e--i)+wn,

where

p =after-tax domestic interest rate, w = after-tax wage rate, e = initial resource endowment.

(2)

In the usual way, this problem defines the indirect private utility function, v*(w,p), consumption functions, ci(w,p), and the labor supply function, n(o,p). Savings is then s(~,p)=e-c,(w,p). For some of the analysis, it will be useful to work with compensated demand and supply functions, denoted cf(w,p, u) and n”(o,p,u) (the ‘s’ stands for Slutsky). It is assumed throughout

S. Bucovetsky and J.D. Wilson, Tax competition with two tax instruments 337

the analysis that first-period consumption is a normal good and a Hicksian substitute with leisure.

The technology for output production in a given region exhibits constant returns to scale and is described by a well-behaved production function, f(k,n), which relates the region’s output gross of investment to the level of investment, k, and labor supply, n. Normalizing the price of output to equal one, we have the familiar profit-maximization condition,

fk(k, n) = 1 + r and f,( k, n) = w,

where subscripts k and n denote partial derivative, r is the before-tax domestic interest rate, and w is the before-tax wage rate. The requirement that profits equal zero in equilibrium defines a negatively-sloped factor-price frontier,

w = w(r); w’(r) = - k/n < 0. (4)

Either of the first-order conditions in (3) then defines k/n as a decreasing function of r:

k/n=y(r); y’(r)<O. (5)

Three tax instruments are considered in this paper, residence- and source-based taxes on capital income and a tax on wage income. For convenience, we work with the tax rates defined in unit terms. With R

denoting the world return on capital, we have:

source-based capital tax rate: t=r-R; residence-based capital tax rate: r = R - p; wage tax rate: T=w-co.

In words, the source-based tax inserts a wedge between the cost of capital to domestic firms and the world return on capital, whereas the residence-based tax places a wedge between this world return and the after-tax return residents earn on their income from savings. The next section assumes that regions cannot use a residence-based tax on capital income (r =O), and section 4 assumes that the wage income is not taxed (T=O). In each case, regional governments play a Nash game in tax rates, with each choosing the

338 S. Bucovetsky and J.D. Wilson, Tax competition with two tax instruments

tax rates that maximize the representative resident’s utility, subject to a government budget constraint that relates the public good supply to the tax rates.’ We consider only symmetric equilibria, where all regions choose the same tax rates in equilibrium.

3. Tax competition without residence-based capital taxation

We now consider the case in which each regional government can impose taxes on the wage income and capital income earned within its borders but is unable to tax income earned by residents outside of its borders. The crucial implication of this constraint is that the after-tax rate of return facing residents is the world return: p=R. A single government’s maximization problem may then be stated as follows:

(P.1) max u[o*(w(R + t) - T, R),g], 1, 7‘39. R

s.t. g=[ty(R+t)+T].n(w(R+t)-T,R), (6)

R = R( t, T). (7)

Eq. (6) is the government budget constraint, requiring that public good expenditures equal tax revenue from capita1 yn and labor n (the unit cost of g is normalized to equal one). Eq. (7) relates the equilibrium world return on capital to the single region’s tax policy, given that all other regions’ tax rates are fixed at their equilibrium levels. The function R(t, T) is implicitly defined by the requirement that net exports of capital, summed over all regions, add up to zero:

b(R,t,T)+(N-l)b(R,t*,T*)=O, (8)

where b(R, t, T) denotes the given region’s net exports of capita1 under world return R and tax rates t and T, N is the number of regions, and t* and T* are the equilibrium tax rates. The region’s net exports equal savings by its residents minus the demand for investment by its firms:

b(R,t,T)=e-c,(w(R+t)-T,R)-y(R+t).n(w(R+t)-T,R). (9)

‘The results presented in section 3 can also be proved for other choices of the strategic variables, consisting of the public good supply and one of the two tax rates, provided we are willing to assume that any single region faces an upward-sloping supply curve for capital imports. Such a relation necessarily exists under the current assumptions. The results in section 4 hold even without this assumption. See Wildasin (1988) for a comparison of the equilibrium under different choices of strategic variables in single-period tax competition models. Wildasin (1991) endogenizes the choice between strategic variables.

S. Bucovetsky and J.D. Wilson, Tax competition with two tax instruments 339

At the symmetric Nash equilibrium, where one region’s tax policy is identical to that in any other region, implicit differentiation of (9) gives the derivatives

8Rlat = -( l/N)(b,/b,) and L?RjaT= -( l/N)(b,/b,), (10)

both of which will later be shown to be negative. The main result in this section is that both equilibrium tax rates are

inefficiently low, at least in the case where the labor supply curve is not negatively sloped. To prove this result, we first derive a preliminary result about the equilibrium tax rates:

Proposition I. The equilibrium tux rates on wage income and capital income

are both positive, with the tax rate on capital income converging to zero as the

number of regions becomes infinite.

Proof. This result is proved by combining the tirst-order conditions for the equilibrium t and T. These first-order conditions may be obtained by substituting constraints (6) and (7) into the objective function and differen- tiating the resulting expression with respect to t and T. To simplify these derivatives, we use the observation that the region’s savings-labor ratio, s/n, satisfies

s/n = y = - w’(r), (11)

which follow from (4) and the equilibrium condition, b = s - yn = 0. We also use Roy’s identity,

v’=n and v*=s w P ’ (12)

which, with (1 l), implies that the price changes, dw = w’(r) and dp = 1, have a zero first-order impact on utility:

vz w'( r) + v,* = 0. (13)

Employing (11)+13), the first-order conditions for t and T may then be written as

-yn+MRS{[yn-yy(ty+T)n,+tny’]

+(aR/at)[(ty+ T)(n,-yn,)+ try’]} =0 (14)

and

340 S. Bucovetsky and J.D. Wilson, Tax competition with two tax instruments

-n+MRS{[n-(ty+T)n,]

+(dR/ZT)[(tjl+ 7-)(n,-yz,,)+tny’]) =O, (15)

where the arguments of functions have been omitted for brevity, and MRS is the marginal rate of substitution between the public good and private income.

The derivatives of R in these first-order conditions are obtained by differentiating (9) and using (10):

and

(16)

(17)

By substituting (16) into (14), and (17) into (15) and then subtracting the product of (15) and y from (14) we obtain a single first-order condition:

t[Nb, + y’n] = [ty + T] [ya,, - n,]. (18)

Eq. (18) is now manipulated into a form that can be used to sign the two tax rates. Note first from (11) and (12) that the price changes, do=? and dp= - 1, leave utility unchanged. Thus, we have the following relations between uncompensated and compensated derivatives:

yn, - no = yt~o ~ n” P (19)

and

ycrCD-( IP=yc;<O-C;p. (20)

Using these two equalities, (9) can be differentiated to obtain

b,=yc”,,-c”;, -fn+y[pL-$1. (21)

Eqs. (19) and (21) may then be substituted into (18) to get our final optimality condition:

All terms multiplying the tax rates in (22) are positive. In particular, the compensated demand derivatives, - c”lp and n”, are always positive, while the assumption that leisure and first-period consumption are Hicksian substitutes

S. Bucovetsky and J.D. Wilson, Tax competition with two tax instruments 341

implies that cS,, and -no are both positive. Finally, (5) implies that -fn is positive. Thus, (22) implies that both t and T have the same sign, which must be positive to satisfy the government budget constraint.

Finally, observe from (22) that t/T must go to zero as N goes to infinity. This completes the proof.

The absence of capital taxation in an economy with many regions is intuitively reasonable. With many regions, each region faces an infinitely elastic supply of capital at the equilibrium world return. In contrast, the supply elasticity for labor is finite. The inverse elasiticity rule from optimal tax theory then suggests that only labor should be taxed to meet a government’s revenue needs. Note also that this result is also related to the Diamond-Mirrlees finding that there should be aggregate production efficiency when commodity taxes are optimal. Here the result is that a small country should not distort the allocation of capital across regions. But this result holds despite the absence of an optimal commodity tax system for the region, which would generally include a residence-based tax on capital

income. This brings us to the issue of whether the tax instruments available to the

region are set efftciently from the viewpoint of the entire economy. The next proposition shows that each tax rate is inefficiently low in equilibrium, at least for the case of a non-backward-bending labor supply curve, in the sense that a central planner could improve utility by instructing each region to raise its tax rate.

Proposition 2. Assume that n,(o, p) 20. Starting from the Nash equilibrium, utility can be increased in each region, if every region raises t by the same small amount, or if every region raises T by the same small amount, with g also rising to balance each region’s government budget.

Proof. Suppose that every region raises t simultaneously. By optimality condition (14), any given region’s own increase in t has a zero first-order impact on its utility. However, the rise in t in other regions affects the given region’s utility through a change in R. The marginal change in utility from a rise in R is given by the term multiplying 8R/& in (14). With N - 1 other regions raising t, we then have the following utility change in the given region:

(N- l)(MRS)(aR/&)[(ty+ T)(n$-yn:) + tny’], (23)

where use is made of equality (19). The derivative aR/& is given by (16), which has a negative numerator under the assumptions that first-period

342 S. BucourtskJ: and J.D. Wilson. Tax competition with two tax instruments

consumption is a normal good and Hicksian substitute with leisure, and that n,(w,p)zO. Eq. (21) and our previous observations show that the denomina- tor of (16) is positive. Thus, dR/& ~0. Since both tax rates are positive (Proposition 1) and ni<O under the assumption of Hicksian substitutes, the expression in square brackets in (23) is negative. Thus, the whole expression in (23) is positive, as required by the proposition.

By similar manipulations, the utility change from a rise in every region’s T is

(N- l)(MRS)(ZR/(?T)[(ty+ T)(n;-yn;)+tny’],

which is also positive, since (17) and (21) can be used to show that (7R/iT<O

under our assumptions. Q.E.D.

The explanation given for inefficiently low tax rates in the tax competition literature is that each government treats as a cost the capital outflow that occurs when it taxes capital to finance additional public good provision, whereas this outflow is not a cost from the viewpoint of the entire economy, since other regions benefit from the resulting capital inflows. In other words, a rise in a single region’s tax rate creates positive externalities for other regions in the form of capital flows into these regions. In the present model, additional public good provision in a region can be shown to lead to increased exports of capital, regardless of whether it is financed by higher taxes on capital income or higher taxes on wage income. Thus, the standard intuition would seem to also hold in the present model. This cannot be the complete story, however, because Proposition 1 shows that the difference between the social and private cost of capital in a region (i.e., the tax rate t) goes to zero as the number of regions goes to infinity, although (23) and (24) together with (16) and (17) show that the utility gain from a rise in every region’s tax rate stays bounded from below by some positive number. In other words, higher tax rates are beneficial in the many-region case, even though capital inflows or outflows do not directly affect a region’s utility.

This puzzling observation is explained by observing from (23) and (24) that, in the case where t = 0, labor supply changes are solely responsible for the utility gains from higher tax rates. Basically, the capital flows induced by a rise in a single region’s t or T continue to create a positive externality through the interaction of the undistorted capital market with the distorted labor markets. When capital flows out of a region, the equilibrium world return is driven down, raising the wage to maintain zero profits. Through these price changes, a rise in a single region’s t or T leads to a rise in every other region’s labor supply. These other regions benefit from higher labor supplies because the tax on labor keeps the social value of labor above its social opportunity cost. Because of this positive externality, the equilibrium t

S. Bucovetsky and J.D. Wilson, Tax competition with two tax instruments 343

and T continue to be inefficiently low in the many-region case. As described by (23) and (24), if every region raised t or T, utility would rise through the labor supply changes brought about by the resulting fall in R.

Note that the assumptions about consumer preferences used to prove Proposition 2 are only sufficient conditions. In particular, the proposition still holds if the wage elasticity of labor supply is negative but sufficiently small in magnitude. As an empirical matter, this wage elasticity is normally estimated to be about zero for males and positive for females.

Proposition 2 is limited to an analysis of local welfare improvements, rather than a global comparison between equilibrium and Pareto efficient tax rates. The single-tax model of Zodrow and Mieskowski (1986) obtains the strong result that equilibrium tax rates and public good expenditures lie below the Pareto efficient levels, but the existence of two tax instruments in the present model considerably complicates global comparisons. Two inter- esting observations are available, however. First, the equilibrium public good supply is financed with too little reliance on capital taxation. To see this, note first that, if all regions change their tax system so that the equilibrium g is financed by a slightly higher t and lower T, then the resulting welfare change is given by an expression identical to (22) except that N = 1:

TCyn”,-nil-tCycS;.,--~,I, (25)

which, by (22) is positive. This result is intuitively reasonable, since the welfare effects of identical tax changes in all regions should be identical to the welfare effects of the same tax changes in one large region that is closed to capital flows, i.e., the case where N= 1. If we assume that welfare is a concave function of t, given that T falls to balance the budget in each region as t rises, then the positive welfare gain given by (25) implies that the Pareto efficient value of t is greater than the equilibrium value. In fact, if we set (25) equal to zero and use the symmetry of the Slutsky matrix to manipulate the result, then we get the Ramsey-rule result that the optimal commodity tax system should, to a first-order approximation, reduce labor and savings by the same percentages:

CWo+tn;l/n= -[Tc”,,+tc”,,]/s. (26)

Thus, the equilibrium tax system violates the Ramsey rule in the direction of inefficiently low tax rates on capital income. The basic reason for this inefficiency is that a budget-balancing rise in t and fall in T in a single region

344 S. Bucovetsky and J.D. Wilson, Talv competition with two ta.x instruments

b

Fig. 1

creates a positive externality by causing an outflow of capital to other regions, and this externality results in each region financing its public expenditures with an inefficiently low t.

The other global comparison applies to the many-region case, for which Proposition I tells us that only wage income is taxed in equilibrium. Here we can ask: what is the efficient public good supply, given the inefficient financing decisions of local governments? Assuming welfare in the economy is a concave function of the public good level chosen by each region, defined with T balancing the government budgets, Proposition 2 implies that the efficient public good supply lies above the equilibrium level. This is illustrated in fig. 1, in which private utility is on the vertical axis, and the public good supply is on the horizontal axis. Curve ab gives the relation between these two variables when all regions increase their public good supply simultaneously, while a’b’ gives the relation for a single region, holding fixed the tax rates in all other regions. Thus, ab may be called ‘the economy’s consumption possibility frontier’ (CPF), whereas a’b’ is ‘a single region’s CPF’, given the Nash assumption of fixed tax rates. The equilibrium lies at a tangency between u’b’ and an indifference curve, and this tangency is located at a point where the single region’s CPF crosses the economy’s PPF from above. At this point, the economy’s CPF is less steeply sloped than the single region’s CPF, because of the positive externalities associated with higher tax rates. The figure illustrates the assertion in Proposition 2 that utility could be increased in all regions, if they all simultaneously raised their tax rates on labor income above the equilibrium level, thereby moving the

S. Bucovetskp and J.D. Wilson, Tax competition with two tax instruments 345

economy along its CPF. A tangency between the economy’s CPF and an indifference curve would represent the efftcient public good supply.

Finally, observe that the ‘efficient’ public good supply on ab in fig. 1 is actually efftcient only in a ‘third-best’ sense. The first-best public good supply is optimal in the case where lump-sum taxation is available, and the second- best public good supply is optimal when only taxes on wage income and capital income are available, and they are optimally set. The third-best public good supply is optimal under the constraint that the mix of these two taxes used to finance the public good is that chosen under the decentralized decision-making of local governments. Fig. 1 shows that the public good supply actually chosen by these regional governments is not even efficient in this third-best sense.

4. Constrained Pareto efficiency without labor taxation

The previous section ruled out residence-based taxes on capital income but assumed that wage income could be taxed at any desired rate. The result was inefficient regional government behavior. We now reverse the assumptions by assuming that residence-based capital taxation is available, but not wage taxation. The resulting equilibrium turns out to be Pareto efficient, given the available tax instruments. Thus, it is the absence of residence-based taxes on capital income, not taxes on wage income, that is the primary cause of the underprovision of public goods. These residence-based taxes are commonly ignored in single-period models of tax competition, because the absence of savings decisions makes them equivalent to lump-sum taxes, and the goal of the analysis is to explore decentralized government behavior in the absence of lump-sum taxation.

Consider first a single region’s maximization problem in the open econ- omy. As before, each region chooses the utility-maximizing tax policy, given that tax rates are fixed at their equilibrium levels in other regions. It is useful, however, to pose a single region’s problem indirectly. In particular, the existence of both source- and residence-based taxes on capital allows us to use as control variables both the cost of capital facing domestic firms, r,

and the after-tax return received by domestic investors, p. The wage rate is then determined by r via the previously defined factor-price frontier, W(I), and net exports of capital are determined by both I and p:

b(r, P) = s(wG% P) -y(r). 449, PI. (27)

The equilibrium world return, R, is related to the given region’s net exports, 6, by the requirement that total net exports, summed over all regions, equal zero:

346 S. Bucovetsky and J.D. Wilson, Tax competition with two tax instruments

h+(N-l)[Is(w(R+t*),R-r*)-~(R+t*)n(w(R+t*),R-r*)]=O, (28)

where the term in square brackets gives the equilibrium net exports of capital in each of the N- 1 other regions, given the equilibrium tax rates, t* and r*. Using the function, R= R(b), as defined by this equality, the government budget constraint for the single given region may be written:

Cr - R(b(r, PHI C.$w(r), d - b(r, dl+ CR(b(r, p)) - pls(W), PI =g. (29)

Here r-R is the source-based tax rate, R-p, is the residence-based tax rate, s is domestic savings, and s-b is the domestic capital this expression, we obtain the following statement maximization problem:

(P.2) max nCu*(w(r), p),gl, ‘,P.9

s.t. (r-p).$w(r)rp)-Cr- R(b(r,pNlb(r,d =g. Compare this problem with the problem facing a

stock. By rearranging of the government’s

(30)

central planner who must choose a common tax and public expenditure policy for each region. Since no capital flows occur between regions with the same tax policies, this is equivalent to the problem a single region would face if its borders were closed to capital flows. Here savings and investment are identical, and there is no difference between residence- and source-based taxes on capital income. Given the absence of taxes on wage income, the government’s problem is then to maximize utility through the appropriate choice of the public good level and tax rate on capital income, subject to the government budget constraint. This problem can be restated by making r, p, and g control variables (so that r-p is the unit tax rate), and adding a constraint that savings equals investment under these three variables:

(P.3) max nCv*(w(r), ~Lfcl, ‘~P.9

s.t. (r-p)s(w(r),p)=g. (31)

b( r, p) = 0. (32)

The main result of this section is that these two problems possess identical solutions when (P.2) is solved with the tax rates in all other regions fixed at their equilibrium levels. Thus, the wasteful tax competition exhibited in the previous section no longer exists. One technical assumption used in the proof is that the first-order conditions for (P.2) uniquely determine the solution. While this assumption need not always hold, due to the possible non- convexities that arise from the second-best nature of the problem, there do not appear to be any important systematic reasons for it to fail. Since

S. Bucovetsky and J.D. Wilson, Tax competition with two tax instruments 347

uniqueness of the solution to (P.2), given the other regions’ tax rates, does not imply uniqueness of the Nash equilibrium, we state the proposition in a way that allows for the possibility of multiple equilibria.

Proposition 3. lj. both source- and residence-based taxes on capital income are available, but not taxes on wage income, then there exists a symmetric equilibrium that is Pareto efficient, given the available tax instruments.

Proof. Letting 3, and p denote the Lagrange multipliers on constraints (31) and (32), the first-order conditions for problem (P.3) are:

u,vtw’(r)+A(r-p)s,w’(r)+/?b,=O; (33)

u,up* + A(r - p)s, + pb, = 0; (34)

u,-A=O. (35)

In contrast, the first-order conditions for problem (P.2) are:

u,v:w’(r)+A(r-p)sww’(r)-A[r-R(O)]b,=O; (36)

u,vp* +i(r-p)s,-A[r-R(O)]b,=O; (37)

IA,-l*=O. (38)

Let (r*,p*,g*, A*,b*) represent a solution to problem (P.3) and the values of the Lagrange multipliers associated with this solution. This vector satisfies (33)(35) and constraints (31) and (32). Then it is clear that (r*,p*,g*,A*) will also satisfy (36)-(38) and constraint (30), if the equilibrium world return, R(O), satisfies

p* = - A*[r* -R(O)]. (39)

In other words, the solution to (P.3) will also be a solution to each region’s problem (P.2) in the symmetric Nash equilibrium, if the equilibrium world return satisfies (39). To be the equilibrium world return, however, R(0) must also satisfy

r*=R(O)+t*; p*=R(O)-s*, (40)

where t* and T* are the tax rates which support the solution to (P.2) [proof: since (r*,p*) satisfies (32), the R that satisfies equilibrium condition (28) with b fixed at zero equals the R(0) in (40)]. By fixing the tax rates in each region

R.S.U.E. B

348 S. Bucouetsky and J.D. Wilson, Tax competition with two tax instruments

i a

Fig. 2

at values which satisfy (40) for the R(0) satisfying (39) we complete the construction of an equilibrium that coincides with the solution to problem (P.3). Q.E.D.

Thus, the replacement of the wage income tax with a residence-based tax on capital income eliminates the inefficient forms of decentralized govern- ment behavior described in the previous section. In contrast to fig. 1, this means that the equilibrium lies not only at a point of tangency between an indifference curve and a single region’s CPF, but also at a point of tangency between the same indifference curve and the economy’s CPF, as illustrated in fig. 2. In other words, the two CPFs are tangent at the equilibrium. The reason for this tangency between the two CPFs may be simply explained by returning to problems (P.2) and (P.3) and observing that, if a given (r, p,g) satisfies the constraints to (P.3), then it must also satisfy the constraints to (P.2), since the only real difference in the two sets of constraints is the addition of the zero-net-export constraint in (P.3). Consequently, for any given g, the maximized value of the objective function in (P.2) is at least as high as the maximized value of the objective function in (P.3). Thus, the single region’s CPF must be at least as high as the economy’s CPF. Since the two CPFs intersect at the equilibrium g, they must then be tangent at this point, as illustrated.

Note also that the single region’s CPF in fig. 2 involves efficient financing of the public good, given the available tax instruments. In contrast, the corresponding CPF in fig. 1 is defined for the case of wage taxation only, which is inefficient given the availability of a source-based tax on capital

income. Thus, both public expenditure decisions and financing decisions are efficient under the current assumptions.

Proposition 3 can easily be extended in two directions. First, the proof does not use the separability assumption, which was previously imposed to conduct the analysis in section 3. In other words, we can replace the utility function in eq. (1) with the more genera1 function, u(c,,c,, n,g), and obtain the same results.

Second, Proposition 3 continues to hold when regions possess different population sizes, provided that the public good is a publicly-provided private good in the sense that there are no scale economies in consumption. In this case, larger regions are scaled-up multiples of smaller regions. Differences in regional size then enter the optimization problems for regional governments only through differences in the relation between the world return and per capita exports of capital, R(h). But these differences are irrelevant for the results, since the first-order conditions for each region’s equilibrium strategy [eqs. (36)-(38)] depend only on the equilibrium world return, R(O), not the derivative R’(0). Indeed, these first-order conditions imply not only that the total tax rate on capital, r-p, is independent of regional size, but also that the source- and residence-based components of this tax rate, r-R(O) and R(O)-p, are independent of regional size. Bucovetsky (1991) and Wilson’s (1991) finding that relatively small regions have a competitive advantage in single-period tax competition models does not apply here. Moreover, Hoyt’s (1991) demonstration that an increase in the number of regions leads to greater inefficiency in local government behavior is obviously not applicable here, given the result that efficiency always prevails. Whether these previous results apply to the model in the previous section is a question that we hope to explore in future research.

5. Concluding remarks

We have shown in the last section that opening a region’s borders to capita1 flows unambiguously expands consumption possibilities when both residence- and source-based taxes on capita1 income are available, even in the absence of a tax on wage income. The presence of both taxes allows a regional government to manipulate gross and net returns to capita1 (r and p) independently of the world return (R). With both taxes, the region is effectively able to insulate itself from the capita1 flows that occur in response to another region’s tax and expenditure policy. Thus, we do not have the interregional externalities described in section 3, where the inability to tax capita1 income on the basis of residence results in a region’s p being tied to the world return, R. Without this residence-based tax, a rise in a region’s tax and expenditure levels creates a positive externality by causing a capita1 outflow that drives down the world return. We conclude that it is the

350 S. Bucowtsky and J.D. Wilson, Tax competition with two tax instruments

absence of this residence-based tax, rather than taxes on wage income, that is responsible for the tendency of decentralized decision-making by local governments to produce inefficiently low levels of taxation and public spending.

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