debt in a model of tax competition

Regional Science and Urban Economics 21 (1991) 371-392. North-Holland

Debt in a model of tax competition*

Richard Jensen and Eugenia Froedge Toma

Uniuersity oj Kentucky, Lexington, KY 40502-0034

Received December 1990, Iinal version received May 1991

This paper develops a two-jurisdiction, two-period model of tax competition where jurisdictions can finance government spending not only with current tax revenues but also with debt. Jurisdictions will have an incentive to issue debt because it reduces the standard tax competition problem of underprovision of government goods. Because the underprovision problem becomes more severe in the period when debt is retired, the net effects on deadweight loss from tax competition cannot be signed.

1. Introduction

In recent years, there has been a growing literature in the area of interjurisdictional tax competition. Generally, tax competition can occur in two basic forms: different types of property (such as business versus residential) can be taxed at different rates, or the average tax rate on all property may be lowered by competition [Wilson (1985, 1986)]. In the latter model, jurisdictions have an incentive to lower tax rates to attract a larger share of a fixed stock of capital. Because each jurisdiction ignores the effects of the loss of capital in other jurisdictions, the general conclusion of tax competition models is that tax rates will be below their optimal levels and there will be an underprovision of government goods.

For the most part, the tax competition models assume a single tax instrument applied to a fixed stock of a mobile factor of production.’ This paper introduces the possibility that jurisdictions possess alternative financing schemes to cover the cost of producing government goods. Not only can jurisdictions raise revenues through taxation, but they also can issue debt.

*We are grateful for financial support from the National Science Foundation (Grant No. RII-8610671) and the Commonwealth of Kentucky through the EPSCoR program. We are indebted to David E. Wildasin and John D. Wilson for their extensive comments and suggestions on numerous occasions while this work was in progress. We thank William Hoyt, Mark Toma, and seminar participants at the Microeconomics Workshop at the University of Kentucky for helpful comments. We also thank Vickie Ford for preparing the manuscript and diagrams. We remain responsible for errors and omissions.

‘Bucovetsky and Wilson (1991) and Hoyt (1991) provide an exception in that they analyze more than a single tax instrument.

016~462/91/$03.50 0 1991-Elsevier Science Publishers B.V. All rights reserved

372 R. Jensen and E.F. Toma, Debt in a model qf tax competition

Traditionally, as much as one-half the capital expenditures of state and local governments have been financed with borrowed funds. Although there are legal restrictions on the use of debt in the form of debt limitations, annual or biennial balanced budget requirements, and even constitutional limits in some states, debt plays an important role at all levels of government.

To analyze the implications of debt financing, we develop a two- jurisdiction, two-period model where jurisdictions can finance government spending with current tax revenues and debt. By developing our model within a tax competition framework, we provide an explanation for why decentralized governments issue debt. Tax competition motivates jurisdictions either to issue debt or to run surpluses. Ceteris paribus, issuing debt commits a jurisdiction to a higher tax rate in period two, when the debt must be retired. If the other jurisdiction’s best reply to this is to increase its tax rate, then both (identical) jurisdictions issue debt in the subgame perfect equilibrium. Compared to the equilibrium with no debt (or surplus), provision of the public good and utility are larger in period one, but smaller in period two. As a result, the public good is underprovided in period two, but need not be in period one. It is possible, at least theoretically, that the public good is optimally provided or even overprovided in period one.

Accumulating a surplus, ceteris paribus, commits a jurisdiction to a lower tax rate in period two, when the surplus is spent. However, if the other jurisdiction’s best reply to this is to increase its tax rate, then both jurisdictions run a surplus in the subgame perfect equilibrium. Compared to the equilibrium with no surplus (or debt), provision of the public good and utility are larger in period two, but smaller in period one. The public good is underprovided in period one, but in period two it can be underprovided, optimally provided, or overprovided. Hence, in our model the standard tax competition problem of underprovision of government services does not necessarily hold in the period in which debt is issued or the period in which a surplus is spent.

2. The basic model

To focus on the effects of allowing jurisdictions to finance government services through the issue of debt, we shall generalize a standard, static model commonly used in this literature [see Wilson (1985, 1986), Wildasin (1988)]. In this model there are two identical jurisdictions, or localities, indexed by i = 1,2 and two time periods, indexed by z = 1,2. Each jurisdiction has a single resident (or equivalently, an arbitrary number of identical residents). There is one homogeneous private good which is produced in each jurisdiction at each date T, denoted xir. Production of the private good requires using a single variable factor and some locationally-fixed factor(s). This variable factor is perfectly mobile across jurisdictions and, consistent

R. Jensen and E.F. Toma, Debt in a model of tax competition 313

with previous models in this literature, we shall call this factor capital. In our framework, it is not critical that the mobile factor be capital; instead, our analysis only requires that some factor be perfectly mobile across jurisdictions. The production function in i is fi(kJ, where ki, is capital employed in i at T and f:>O> f::‘. The total capital stock in the economy at each date is exogenously given and equal to the quantity, K.

The private good serves as the numeraire, and can be either consumed or used as an input to produce a local public good. This implies that units can be chosen so that public goods provision in i at each date r can be measured in terms of units of the private good, or simply local public expenditure at T, which we denote gi,. The utility of the resident in i at z is denoted uir = ui(xir,gi,). We assume the utility function has positive marginal utilities. Denote the marginal rate of substitution in jurisdiction i at period 7 by Mi, = [(&,/3giJ/(&,/8xi7)]. We also assume that 8M,,/dx, > 0 and ZM,,/dg,, < 0, which implies that both goods are normal and that uir is strictly quasi-concave.

Each jurisdiction’s government can raise revenue in each period by imposing a unit tax on capital employed in that jurisdiction. In each period, capital moves across jurisdictions until the after-tax rate of return on capital is equalized. Let ti, be jurisdiction i’s tax at date 7 and p, be the after-tax rate of return at r. Because capital is paid its marginal product, the equilibrium conditions for the capital market to clear at 7 are

klr+kzr= K. (lb)

These equations implicitly define the capital employments and net return to capital as functions of the taxes and capital stock. Because the production functions and capital stock are the same in each period, these functions are the same in each period. They can be written as ki,=q(tlrrt2J, and ~~=~(ti~, tZr), where K is suppressed for notational convenience. It is straightforward to show that capital employed in a jurisdiction is decreasing in its own tax and increasing in the other jurisdiction’s tax, while the after- tax return to capital is decreasing in both taxes [i.e., for each 7,

&,/&,,<O<&C~/&~, for i, j= 1,2 (i# j) and ZJ~JC%~,<O for both i]. At this point our model departs from previous analyses. We allow the

government to finance its expenditures not only with tax revenues but also by issuing debt. All borrowing takes the form of a government bond that is bought by the resident of the jurisdiction (i.e., no money creation is allowed) in period one. The debt can be held for one period only and must repaid with interest in period two. Under these assumptions, if di is the amount

374 R. Jensen and E.F. Tomu, Debt in a model of tux competition

borrowed by jurisdiction i in period one, the budget constraints facing the government in i for periods one and two are:

gil =tilkil -tdi, (2)

gi, = tizki, -( 1 + r)dj, (3)

where Y is the rate of interest paid on the government bond. Eqs. (2) and (3) define the period one and two constraints on public good production. In the first period, public expenditures are equal to the capital tax revenues plus any deficit spending. In period two, expenditures on production of the public good must be reduced by the principal and interest payments on the debt.

The amount of the private good consumed by the resident of juristiction i at T depends on both net income and saving (i.e., lending to the government) at that date. In our framework. net income is

where Oi is jurisdiction i’s share of capital. The resident’s net income consists of returns to the fixed factor plus net capital income received. Because the private good is the numeraire, the amount of it consumed by i in period one is equal to net income, y,i, less any purchase of government bonds, .si; or,

xi1 =yi, psi. (5)

In period two, consumption of the private good is increased by the amount of the previous government bond purchase plus interest earned;

Xiz = _Vi, + ( 1 + r)Si. (6)

We assume each government’s objective is to maximize its resident’s total utility from both periods by choosing a tax and level of debt in period one and a tax in period two (where the feasible sets of taxes and debts for each jurisdiction are non-empty, compact, and convex intervals of the real line). The amount of government spending will adjust to the tax rate and debt chosen by the government. Total utility in i is

(7)

Because the resident chooses si to maximize his welfare, equilibrium in the government bond market requires that

t?u,/dx,, =(l +r)(au,/ax,,),


au2/ax2, =(l +r)(au21ax22), (8b)

Sl +s, =d, +d,. (84

Conditions (8a) and (8b) describe the residents’ marginal utility of private good consumption across the two periods and (8~) specifies that the total savings by the residents must be equal to the total borrowing by the governments. The conditions (8a) and (8b) implicitly define residents’ savings as a function of the interest rate, net income, and government spending, or

(9)

Given (8~) the equilibrium interest rate can, in turn, be written as a function of total debt, net income, and government spending. Substituting from the capital market, the interest rate can be written as a function of all debt and taxes, or

The signs of as,/&, &/6’d,, and &/at, are, in general, ambiguous. As is standard, we model the problem as a game between the governments.

Because there are two periods, the equilibrium concept we employ is subgame perfection. This concept requires that the strategies employed at any stage constitute a Nash equilibrium for the remainder of the game. We solve for the equilibrium of the game in which each government maximizes w conditional on Nash equilibrium behavior in period two. We, therefore, must consider the period-two problem first.

In considering the period-two problem, first observe that the levels of debt, period-one taxes, savings and the interest rate are exogenously given in period two, so each government’s problem is to choose a tax ti, to maximize ui2 subject to the modified budget constraint [eq. (3)]. Also note that yi, and gi, depend on (t 12, t,,) through the functions rci and p. Because si and r are exogenous going into period two, utility in i at r = 2 can be written as

V2(t 121 t22, si,di,r)=ui(yi2(t,2,t22)+(1 +r)si,ti2ki2(tl2,t22)-(1 +r)dJ. (11)

The conditions defining the period-two Nash equilibrium in taxes are simply the first-order conditions

(12)

376 R. Jensen and E.F. Toma, Debt in a model of tax competition

for each i= 1,2. It follows from (11) that these equations define equilibrium period-two taxes as functions of d,, d,, sl, s2 and r. Hence, we can write period-two equilibrium taxes as &(dl, d,, sI, s2, r).

Under the assumption of identical jurisdictions, 8,K = K/2. With equal taxes, we have ki, = kj, = K/2, and therefore L7yi2/L7tiz = - ki, = - K/2. Substi- tuting into eq. (12)

(13)

Given the period-two first-order conditions we now turn to period one. Observe from eqs. (2) and (5) that the amount of each good consumed in i in period one, xi1 and g,i, depends on si and di as well as on period-one taxes t,, and r,,. Similarly from eqs. (3) and (6) utility in period two also depends on si and r because of their effects on period-two taxes. Lifetime utility can then be written as

(14)

Eqs. (11) and (14) can be used to calculate the first-order conditions for the jurisdictions. The game at this stage consists of each jurisdiction choosing a tax rate, t,i, and a level of debt, di, to maximize total utility. Note that the consumer’s first-order conditions for si imply that 2W/t3si=0. The governments’ first-order conditions, therefore, contain no derivatives with respect to si, so we have

and

dJQUi, __$ql+r)+ ddi &;I 12

(16)

Using our earlier definition of the marginal rate of substitution between the public and private goods and substituting &,/ax,, =(&,/~x,,)( 1 + r) from the consumer’s first-order condition for saving, we can rewrite eq. (16) as


M. 11 (17)

For a vector of debts and tax rates (tT1, d:, t:,, t:,,dt, tT2) to be a subgame perfect equilibrium, it is necessary that all first-order conditions be simulta- neously satisfied.

3. Subgame perfect equilibrium behavior

3.1. Period-two behavior

To solve for the subgame perfect equilibrium, we begin with period two. To begin, we determine the equilibrium taxes which are determined by the following system of equations:

(1W

where t$(dl,d2,sl,sz,r) is the value of ti, that solves (13) for given d,, d,, sl,

s2, and r. To determine the effects of period-one taxes and debt level on period-two equilibrium taxes, first substitute the interest rate function in (18~) into the saving functions in (18b). Then substitute these plus the interest rate function into (18a). This gives the two equations in two unknowns which determine the period-two equilibrium taxes as functions of period-one taxes and level of debt, t~Z(tll,dl,t21,d2) and t:2(tll,dl,t21,d2), and similarly for the interest rate, r(t,,, d,,t,,,d,). Given the complexity of this

system of equations, and the ambiguity of the effects of taxes and debt levels on savings and the interest rate, it is convenient for both analytical and expositional purposes to assume that the common utility function takes a particular form.2

Assumption 1. The utility function takes the form

(19)

where u’ > 0, u” < 0, h’(0) > 1, h’ > 0, and h” < 0.

‘We are grateful to David E. Wildasin and John D. Wilson for suggesting this form of utility function, and for providing us with Lemma 1 and its proof.


That u and h(g) increase at a decreasing rate, of course, merely guarantees positive but diminishing marginal utility. Because the marginal rate of substitution with this utility function is Mi,=h’(gi,), the condition h’(O) > 1 guarantees that the resident in each jurisdiction will consume the public good. This form of utility function is useful because it allows us to show that, in equilibrium, period-one taxes do not affect the interest rate, savings, or period-two taxes (and so period-two taxes do not affect the interest rate or savings).

Lemma 1. Under Assumption 1. ?rJ8ti, =8tj,/ati, =0 where both derivutives

are evaluated at the Nash equilibrium.

Proof. We show that all the required equilibrium conditions are satisfied when

C?r 8tjz _ ‘?ti, = o -=- _~ clti, 2ti, 2ti, .

(20)

Substituting these equalities into the first-order condition in eq. (15) we get

aui aYil I aui agil -0,

sxi, ati, &il atil

Given Assumption 1, this means that

The assumed utility function and the equalities of (22) also give

(21)

(22)

(23)

The first-order condition for si under Assumption 1 is

u’(xil + hkil)) ~(1 +r)u’(xiz + hki2)). (24)

Given (22) and (23) and the equalities in (20), the conditions in eq. (24) can hold after a tax change only if

as,/&,, =o. (25)


The equalities of eq. (20) also imply C%~/&~, =O. Because the debt levels are held fixed, it follows that capital-market equilibrium is maintained: s1 +s, =d, +d,. To conclude, eq. (20) satisfies all the required equilibrium conditions. Q.E.D.

Hence, under Assumption 1 each jurisdiction’s first-order condition in period two reduces to

(26)

Note well that Zi depends only on t,,, t,,, d,, and d, under this assumption. Hence, it follows that period-two equilibrium taxes, if they exist, will depend only on the level of debt and period-one taxes, or t$= t&(tll,d,, t,,,d,). To ensure the existence of equilibrium taxes we make the following assumption.

Assumption 2. For each jurisdiction i, Z,(t,,,t,,,d,,d,) is decreasing in ti2, or dzijati, ~0.

This implies that, for any tax levied by jurisdiction j and any level of debt, jurisdiction i’s period-two utility is strictly concave in ti2. This also ensures the existence of uniquely defined best reply tax functions in period two. It is worth noting that concavity is a stronger assumption than we need for existence (i.e., simply assuming the existence of a unique best reply tax for jurisdiction i given any tj2, d,, and d, is sufficient for existence of equilibrium, and this requires only that jurisdiction i’s period-two utility be strictly quasi-concave in its tax). We need the stronger assumption of concavity in order to determine the comparative static effects of a change in debt on period-two equilibrium taxes, which are critical to our analysis (see proof of Proposition 1).

Jurisdiction i’s best reply Qi(tj2,dl,d2) is the tax that maximizes its period-two utility, given any period-two tax levied by jurisdiction j and any levels of debt. The best replies are defined implicitly by

Z~(@~(t22,dl,d2),t2z,d,,d,)=0 and Z,(t,,,@,(t,,,d,,d,),d,,d,)=O.

Holding debts constant, we can graph these best reply functions in tax space. The equilibrium period-two taxes are then given by the intersection of these best replies (see figs. 1 and 2). Unfortunately, the slopes of these best replies

R.S.U E. C

380 R. Jensen and E.F. Toma, Debt in a model qftax competition

I

t :2(d, , d,) t 12

Fig. 1

t:2 (d,,d,) t 12

Fig. 2

cannot, in general, be determined. That is, jurisdiction i’s best reply function is positively (negatively) sloped when an increase in jurisdiction j’s tax


increases (decreases) the marginal utility of the resident in jurisdiction i. We therefore conduct the analysis for both positively and negatively sloped best replies. The period-two equilibrium is shown for positively sloped best replies in fig. 1 and for negatively sloped ones in fig. 2.

As we noted above, what is critical for our analysis is the effect of a change in debt on period-two equilibrium taxes. Indeed, the essence of subgame perfection in this analysis is that each jurisdiction recognizes the effect of its choice of debt in period one on equilibrium taxes in period two [recall the term &j*,/d& in eq. (16)]. To determine this effect, we first consider the effect of a change in debt on each jurisdiction’s period-two best reply tax function.

Lemma 2. If d, =d, =O, an increase in debt by jurisdiction i increases its

period-two best reply tax function, but leaves jurisdiction j’s period-two best

reply tax function unchanged. That is, &Di/adi>O for all tj2 and &Dj/adi=O for

all ti,.

Proof. We demonstrate the result for a change in d, (the proof for a change in d, is symmetric because jurisdictions are identical). It follows from Assumption 2 that the sign of a@,/adl is given by the sign of Z,/ad, and the sign of &D2/adl is given by the sign of Z,/dd,, where

and

(27)

(28)

The result then follows from the fact that under Assumption 1, (12) implies agiz/&, >O at any point on a jurisdiction’s best reply tax function. Q.E.D.

To see what this result says, suppose we begin in an equilibrium where neither jurisdiction has issued debt. Then a marginal increase in debt by jurisdiction 1 shifts @i out, so jurisdiction l’s best reply tax is larger for any given t,,. However, this marginal increase in debt has no effect on jurisdiction 2’s best reply function. The intuition for this is simple. Ceteris paribus, a change in d, changes the amount of the debt repayment, and so changes public good consumption in period two. In jurisdiction 1 this change has two components, as can be seen from the term in square brackets in eq.


Fig. 3

(27). First, for a given interest rate, a change in d, changes the amount that must be repaid (this effect is given by the term 1 + r). Second, a change in d, also changes the interest rate [this effect is given by the term (dr/Jd,)d,]. Hence, the issuing of debt by jurisdiction 1 (compared to no debt) commits it to a higher best reply tax in period two, when it must repay the debt. However, a change in d, affects the amount of the debt repayment in jurisdiction 2 only through its effect on the interest rate [see eq. (28)]. Because jurisdiction 2 has no debt to repay, this commitment by jurisdiction 1 has no effect on jurisdiction 2’s best reply tax in period two. Note well that this result holds regardless of the slope of the best replies. An increase in d, from 0 to d>O shifts jurisdiction l’s best reply to the right, as shown in fig. 3 for the case of positively sloped best replies, and in fig. 4 for the case of negatively sloped best replies.

Proposition I. In any stable period-two equilibrium with no debt (d, =d2 =O), jurisdiction j’s tax is increasing in jurisdiction i’s debt if best reply tax functions are positively sloped, but decreasing in jurisdiction i’s debt if best reply tax functions are negatively sloped.

Proof. It follows from eq. (13) that

g=[ -(z) (g+(z) (fg/H, (29)

R. Jensen and E.F. Toma, Debt in a model of tax competition

* * t,,(O,O) t,2 (d,O)

Fig. 4

where

383

(30)

Note that Qi is positively (negatively) sloped if and only if i3Zi/8tj, s-0 (8Z,/i%,, CO). Also note that with identical jurisdictions the equilibrium is stable if and only if (a~,/at,,)(a~,/at,,)<l. This implies that H>O in any stable equilibrium. It then follows from the proof of Lemma 2 that &j*,/adi >O if best replies are positively sloped, and &F2j8di ~0 if best replies are negatively sloped. Q.E.D.

As shown in Lemma 2, if we begin in a no-debt equilibrium, then an increase in d, increases jurisdiction l’s period-two best reply function, but has no effect on jurisdiction 2’s period-two best reply function. Hence, the effect of the increase in d, on jurisdiction 2’s period-two equilibrium tax is completely determined by the slope of its best reply function in the no debt equilibrium. If it is positively sloped, as in fig. 3, then jurisdiction 2’s period- two equilibrium tax must increase because its best reply to jurisdiction l’s commitment to a higher period-two tax is to increase its tax. Conversely, if best replies are negatively sloped, as in fig. 4, then jurisdiction 2’s period-two

384 R. Jensen and E.F. Toma, Debt in a model oi tax competition

equilibrium tax must decrease because its best reply to jurisdiction l’s commitment to a higher period-two tax is to decrease its tax.

Of course, there is no a priori reason to expect best reply functions to be either positively or negatively sloped everywhere (as we have drawn them in the figures). That is, it is entirely possible that Qi could be positively sloped for some values of tj2, and negatively sloped for other values of fj2. For this reason, it is worthwhile to note that the results in Proposition 1 do not require that best reply functions be positively or negatively sloped everywhere. Instead, they require only that best replies be positively or negatively sloped in a neighbourhood of the equilibrium which would occur in period two in the absence of debt (d, = d, =O).

We conclude our analysis of the period-two equilibrium by considering the level of public good provision and its relationship to debt issued in period one. Ceteris paribus, because debt must be repaid from tax revenue in period two, an increase in debt reduces the public good level in period two. Although an increase in the period-two tax will increase the public good level, the forces of tax competition prevent this increase from completely offsetting the original decrease due to the debt increase.

Proposition 2. In a symmetric subgame perfect equilibrium, the public good level in period two decreases as the common debt level chosen by euch

jurisdiction increases.

Proof. In a symmetric equilibrium, both jurisdictions have the same debt levels and the same period-two taxes and public good levels. Denote the common debt level by d, the common tax by f,, and the common public good level by g, = t,(K/2)-(1 +r)d. The equilibrium condition is then eq. (26) evaluated at these common levels, which can be written in the form of a marginal benefit, marginal cost rule as

h’(r,($)-(l+r)d)=MC(i,), (31)

where MC(t,) = [ 1 + (2/K)tz(&iz/&i2)] I does not depend on the common debt level or the common public good level. If d increases, then the left-hand side of (31) increases because h”<O, but the right-hand side remains the same. Hence, equality in (31) (and equilibrium) can be restored only if t, increases. However, one can show that MC(t,) is increasing in t, [see Wilson (1986)], so this increase in t, also increases MC(t,). This implies that equilibrium is restored at a lower public good level than that before the increase in debt. Q.E.D.


If debt is nonnegative in equilibrium, then there is an underprovision of the public good in period two for essentially the same reason as in the static tax competition argument. If there is no debt, of course, then the argument is exactly the same. With identical jurisdictions, taxing the perfectly mobile factor to provide the public good introduces a distortion which results in consumption at a point on the production constraint below the first-best public good level [see Zodrow and Mieszkowski (1986) for a graphical representation]. With positive debt, it follows from Proposition 2 that the underprovision becomes even more severe.

Proposition 3. In a symmetric subgame perfect equilibrium with nonnegative debt, the public good is underprovided in period two.

Proof. The first-best public good level must satisfy h’(g,) = 1 and g, +x2 = f(K/2), where x2 is the common private good level in period two and f is the common production function. From the proof of Proposition 2, the period-two equilibrium condition is given by (31), where MC(t,) is increasing in t,. Because MC(O) = 1, it follows that MC(t,)$l if and only if t, $0, and so in equilibrium h’ (g2) k$l if and only if t, $0. Suppose t, =O, so the left-hand side of (31) is h’( -(l +r)d) and the right-hand side is MC(O) = 1. Because h” < 0, we have h’( -( 1 + r)d) 2 h’(0) > 1 = MC(O) for d 2 0. Hence, at t, =O, the left-hand side of (31) is greater than the right-hand side. And because MC’ ~0, restoring equality in (31) requires increasing t,. Hence, the symmetric equilibrium must have t, >O, which implies h’(g,) > 1 and an underprovision of the public good in period two. Q.E.D.

3.2. Period-one behavior

With Assumption 1, the period-one first-order condition for taxes [eq. (15)] reduces to

i3W. ‘=u’(Xil + h(gil)) atil

(32)

because &/ati, = &,*,l&,, =0 by Lemma 1. Assumption 1 also allows us to write the first-order condition for debts in a more convenient form. Because jurisdictions are identical, we have

~=u’(Xi, + h(g,2))di[l -h’(gi,)]. (33)


Because dy,,/at,, =0 with identical jurisdictions and akiZ/8tj2 = -(aki2/&,,), we have 3[xi, + h(gi,)]/atj, = -h’ (giz)ti2(aki,/ati,). So from eq. (26),

avl,, atjz = u'(xiZ + 4gi2))

0 t Ch’ki2)- ll. (34)

Because equilibrium in the bond market under Assumption 1 requires that u’(xi, + h(g,r)) =( 1 + r)u’(xi2 +h(gi2)), it follows that eq. (16) can be written as

::=(I +r)U'(Xi2+h(gi2))[h'(gi1)-h'(gi*)l L

As is well-known, the assumptions on consumption and production do not provide information about the curvature properties of I+$. As a result, we must make the following assumption to guarantee the existence of a subgame perfect equilibrium.

Assumption 3. For each jurisdiction i, y(t, 1, d,, t,,,d,) is strictly concave in

(tir,di).

Again, as in the period-two problem, it is not sufficient to assume that each jurisdiction i has a unique best reply tax ti, and debt level di for any given (tjI, dj). This approach could be used to show existence of a symmetric equilibrium if we knew how eqs. (32) and (35) varied with the common period-one tax and debt level. However, this is impossible for the debt condition (35). The reason is that the effect of a change in the common debt level on the period-two equilibrium tax and the interest rate (aty2,1adi and dr/8di) depends on the third derivatives of the utility and production functions. Because there is no a priori reason to make assumptions on these third derivatives, the only alternative is to assume strict concavity.

Proposition 4. If period-two best reply tax functions are positively sloped at the tax rates which would be chosen in the absence of debt (d, = d, = 0), then there exists a symmetric subgame perfect equilibrium with positive debt.

Proof Suppose that there is a prohibition on debt, so di=O for both i. Then the resulting equilibrium must have ti, =ti2, xi1 =xi2, gi, =gi, and si=O for


both i (period two is just a replica of period one). Hence, just as in the standard, static model of tax competition, equilibrium taxes are given by

for i= 1,2 and z= 1,2. It then follows from both i. Moreover, it follows from (34) that

(36)

(32) that 8W,/LV,, =0 if di=O for

aw_ ad, -“‘(xi2 + hki2))Ch’(ITi*)- ll L

(“)($$J>O if ($$)>O. (37) 2

because h’(gJ> 1 by Proposition 3. The result then follows from Assumption 3 [standard theorms, such as Theorem 7.1 in Friedman (1977), guarantee existence if the payoff functions are continuous, bounded, and strictly concave].

Ceteris paribus, an increase in debt increases the public good level in period one, but decreases it, in period two. It is therefore natural to expect the effect of an increase in the common debt level on the public good level in period one to be the opposite of that in period two.

Proposition 5. Suppose both jurisdictions choose Nash equilibrium period-one taxes conditional on an exogenously given common level of debt. Then the

public good level in period one increases as the common debt level increases.

Proof: In a symmetric equilibrium, both jurisdictions have the same debt levels and the same period-one taxes and public goods levels. Denote the common debt level by d, the common tax by t,, the common public good level by g, = t,(K/2)+d. Eq. (32) evaluated at these common levels can be written in the form of a marginal benefit, marginal cost rule as

h(,, ($)+d)=tK(,,,,

where MC(t,)=[l +(2/K)t,(akil/ati,)]-’ does not depend on the common


debt level or the common public good level. If d increases, then the left-hand side of (38) decreases because h” < 0, but the right-hand side remains the same. Hence, equality in (38) can be restored only if t, decreases. However, because MC’ (t,)>O, this decrease in t, also decreases MC(t,). This implies equilibrium is restored at a higher public good level. Q.E.D.

Hence, in equilibrium, debt serves as a substitute for taxation in the provision of the public good in period one. This severs the direct relationship between the tax rate and the public good level in a symmetric equilibrium, so the public good need not be underprovided in period one.

Proposition 6. In a symmetric suhgume perject equilibrium with positive debt, the public good is underprovided in period one if t, >O, optimully provided ij” t, = 0, and overprovided if t, < 0.

Proof: Again the first-best public good level must satisfy h’(gl)= 1 and g, +x1 =f(K/2), where xi is the common private good level in period one. Because MC’(t,) >O and MC(O)= 1, it follows from (38) that in equilibrium h’(g,) $1 if and only if t, $0. Suppose t, =O, so the left-hand side of (38) is h’(d) and the right-hand side is MC(O) = 1. Because d >O, the condition h’(0) > 1 does not imply h’(d) > 1. That is, we may have h’(d) greater than, less than, or equal to 1. If h’(d)> 1, then as in the proof of Proposition 3 we can conclude that t, >O and h’(g,)> 1 in equilibrium, so the public good is underprovided. However, it is possible (at least theoretically) that h’(d) 5 1. If h’(d)= 1, then (38) holds exactly at t, =0 and the public good is optimally provided. If h’(d)< 1, then at t, =0 the left-hand side of (38) is less than the right-hand side. Because h” <O and MC’>O, restoring equality in (38) requires decreasing t,. Hence, the equilibrium now must have t, <O and h’(g,) > 1, and thus overprovision of the public good in period one. Q.E.D.

The equilibrium conditions provide no definitive conclusion regarding the optimality of the public good level in period one. Contrary to the standard conclusions of the tax competition literature, underprovision is not a necessary outcome when jurisdictions have the option of issuing debt. Proposition 6 shows that debt substitutes for taxes in period one, and as a result the public good level may be greater than optimal. If so, the period- one common tax is negative, which means the jurisdictions are subsidizing the use of capital.

Allowing jurisdictions to issue debt in a model of tax competition also has an interesting, and apparently counterintuitive, implication for the equilibrium rate of interest.

Proposition 7. In a symmetric subgame pet-j&t equilibrium with positive debt,


the interest rate must be negative if there is underprovision of the public good in both periods. Moreover, utility is larger in period one than in period two.

Proof. Recall that xi + h(g,) =x2 + h(g,) in a symmetric equilibrium with no debt. Then it follows from Propositions 2 and 5 that x1 + h(g,) > x2 + h(gJ in a symmetric equilibrium with positive debt if the public good is underprovided in both periods. It then follows from Assumption 1 that utility is larger in period one. Moreover, U” < 0 implies u’(x, + h(g,)) < u’(x, + h(g,)), so the bond market equilibrium condition u’(x, + h(g,)) =( 1 + r)u’(x, + h(g2)) holds if and only if r CO. Of course, u’ > 0 and this condition imply r>-1. Q.E.D.

While a negative interest rate is not typical in the economics literature, there is some intuition for why this can occur in a tax competition model with debt. Given the underprovision of government services in the absence of debt, consumers are willing to subsidize provision of the public good in period one. In fact, they are so willing to give up private consumption for the public good today that they drive the interest rate to a negative level, and thereby give up private consumption in the future as well. That is, the negative interest rate essentially serves as a tax on private consumption in the future. Ceteris paribus, issuing debt in period one lessens the underprovision then, but only at the expense of worsening the underprovision in period two. Driving the interest rate to a negative level, and thereby taxing period- two private consumption, alleviates the worsening of the underprovision problem in period two. Because r ~0 only if the marginal utility of the private good is lower in period one, it follows from Assumption 1 that utility is larger in period one.

Finally, it is important to note that a symmetric subgame perfect equilibrium with a surplus (negative debt) is also possible.

Proposition 8. If period-two best reply tax functions are negatively sloped at

the tax rates which would be chosen in the absence of debt (d, =d,=O), then there exists a symmetric subgame perfect equilibrium in which both jurisdictions

accumulate a surplus (debt is negative). Moreover:

(i) The public good is underprovided in period one.

(ii) The public good is underprovided in period two if t, >O, optimally provided if t, = 0, and overprovided if t, =C 0.

(iii) The equilibrium interest rate is positive, tf the public good is underprovided in both periods, in which case utility is larger in period two than in period one.

Proof. If there is a prohibition of debt, then it follows from the proof of Proposition 4 that aB$/&, =0 and the sign of aWJadi is given by the sign of


&$,ladi. Hence, from Proposition 1 we have a~/adi<O when period two best replies are negatively sloped in the no debt equilibrium. The existence of an equilibrium with d<O (a surplus) then follows from Assumption 3. The proofs of statements (i), (ii), and (iii) are analogous to (respectively) the proofs of Propositions 3, 6, and 7. Q.E.D.

What we have essentially shown is that jurisdictions will either issue debt or accumulate a surplus as long as there is any strategic interaction between them. That is, from the proof of Proposition 4 [see especially (35)], we have either debt or surplus in equilibrium as long as &j*Z/adi #O when d, = d, =O, and from Proposition 1, dti*,/i;d,#O as long as period-two best reply tax functions do not have a zero slope when d, = d, = 0. Ceteris paribus, issuing debt is a commitment to a higher tax in period two, while accumulating a surplus is a commitment to a lower tax in period two. Hence, if best replies are positively sloped, a jurisdiction issues debt because this induces the other jurisdiction to increase its tax in period two. However, if best replies are negatively sloped, a jurisdiction does not want to issue debt because this induces the other jurisdiction to lower its tax in period two. Instead, it accumulates a surplus because this is what induces the other jurisdiction to increase its tax in period two if best replies are negatively sloped.

4. A numerical example

Consider the following example. Let the capital stock be K =2, the common production function be f(k,,) = Ak,, -( l/2)ki, where A is a constant, and the common utility function be u(xi, + h(gi,)) = log (Xir + h(g,,)), where h(g,,) =(gi,)‘12. Then in the symmetric subgame perfect equilibrium: net income is y, = A -(l/2) -t,; consumptions of the private and public goods are x,=y,-d,g,=t,+d, x,=y,+(l+r)d, and g2=t2-(l+r)d; and the interest rate is r = (5t, - 5t2 + 8d)/(4A ~ 5t I - 8d).

Let A =2 (where A is the vertical intercept of the marginal product of capital). If there is no debt, then the equilibrium tax is t,=0.202 in each period 7, the interest rate is r =O, net income is yc= 1.298, consumptions are X, = 1.298 and g, = 0.202, and utility is u(x, + h(g,)) = 0.558. The public good is underprovided because h’(g,) = 1.112. The slope of the common best reply is i?@Jat, = (4 - 1 6tj2 - tj22)/(44 - 20tj, - tT2), which is positive at the equilibrium (&Pi/atj2 =0.018 if tj2 =0.202). However, it need not be positively sloped everywhere (and is in fact negatively sloped for tj2 2 0.25).

If jurisdictions can issue debt, then equilibrium taxes and debt are t, = 0.068, t, =0.338, and d =O. 166, the interest rate is r = -0.009, net incomes are y, = 1.432 and y, = 1.162, and consumptions are x1 = 1.266, g, = 0.234, x2 = 1.327, and g, = 0.173. Utilities are u(xt + h(g, )) = 0.243 and u(x2 +h(g,)) =0.176. The public good is underprovided in each period


because h’(g,)= 1.034 and h’(g,) = 1.200. Hence, taxes and private consumption are lower in period one and higher in period two, while public consumption and utility is higher in period one and lower in period two. Total utility is somewhat lower. We cannot prove this result generally, but it is not surprising. Proposition 4 only shows that, if d, =d, =O, then each jurisdiction has an incentive to issue debt given that the other jurisdiction’s debt is fixed at zero. This does not, of course, imply that total utility will increase for both when both issue debt. That is, at least for this example, allowing the option of debt results in a prisoners’ dilemma type of outcome.

5. Concluding comments

In typical tax competition models, government services are financed through taxation of the mobile factor of production. In addition to its distortionary effects of factor inputs within the taxing jurisdiction, factor taxation distorts the flow of resources across jurisdictions. The resulting effect is that government service levels will be below their first-best level.

Our paper introduces the possibility that governments can finance public services not only through taxes but also through debt financing. We show that as long as there is tax competition among jurisdictions, localities will not choose single period balanced budgets. As a result, the connection between tax rates and the level of public good provision is broken. For example, consider the equilibrium where debt is issued. Observing high taxes does not imply one should also observe high public good levels because in period two taxes are high, but public good provision is low, due to debt retirement. Similarly, observing low taxes does not imply one should also observe low public good levels because in period one taxes are low, but public good provision is high, due to the issuing of debt.

In this paper, we assumed each jurisdiction’s issue of debt would affect the equilibrium rate of interest. An interesting extension of this model would be to assume small competing jurisdictions in an open economy setting. If interest rates were independent of a single jurisdiction’s behavior, issuing debt could be a means for reducing the deadweight loss associated with tax competition.

References

Bucovetsky, Sam and John D. Wilson, 1991, Tax competition with two tax instruments, Regional Science and Urban Economics, this issue.

Friedman, James W., 1977, Oligopoly and the theory of games (North-Holland, Amsterdam). Gordon, Roger H., 1986, Taxation of investment and savings in a world economy, American

Economic Review 76, 1086-l 102. Hoyt, William H., 1991, Competitive jurisdictions, congestion, and the Henry George Theorem:

When should property be taxed instead of land? Regional Science and Urban Economics, this issue.

392 R. Jensen and E.F. Toma, Debt in a model of tux competition

Wildasin, John D., 1988, Nash equilibria in models of tiscal competition, Journal of Public Economics 35, 229-240.

Wilson, John D., 1985, Optima1 property taxation in the presence of interregional capita1 mobility, Journal of Urban Economics 17, 73-89.

Wilson, John D., 1986, A theory of interregional tax competition, Journal of Urban Economics 19, 296315.

Wilson, John D., 1990, Tax competition with interregional difference in factor endowments, Regional Science and Urban Economics, this issue.

Zodrow, George R. and Peter Mieszkowski, 1986, Pigou, property taxation and the underprovision of local public goods, Journal of Urban Economics 19, 356-370.

debt in a model of tax competition

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