Existence of Nash Equilibrium for Tax Competition among a Large Number of Jurisdictions

S. Bucovetsky Arts/Economics York University Toronto Ont. M3J 1P3 CANADA email : [email protected]

June 2003

abstract : In models of tax competition, payoffs to the countries setting taxes need not be quasi–concave in their tax rates, even if preferences and technology are both convex. The existence of Nash equilibria in pure strategies is therefore not easily assured. However, assuming a continuum of agents helps convexify matters, as is well–known. Here it is shown that a pure strategy Nash equilibrium must exist if there is a continuum of countries. It is also shown that this result guarantees the existence of an approximate equilibrium if the relatively size of each country is sufficiently small.

1

Introduction

The relatively simple model of capital tax competition due to Wilson ( 1986 ) , and to Zodrow and Mieszkowski ( 1986 ) has given rise to quite a large literature 1 . In most of this literature, local ( or other lower–level ) governments are assumed to set source–based taxes non–cooperatively. Under a variety of assumptions, considerable insight has been gained about the properties of Nash equilibria when jurisdictions compete in this manner for a mobile tax base. Since the strategic variables are the jurisdictions’ capital tax rates 2 , Nash equilibrium is characterized by each jurisdiction maximizing some payoff function with respect to its own tax rate, taking the tax rates of other jurisdictions as given. The properties of the Nash equilibrium are given by the first–order conditions for the jurisdictions’ maximization problems. Relatively little attention has been paid to the technical aspects of this game : whether the second–order conditions for maximization hold, and whether a Nash equilibrium to the game exists. By changing the model slightly, so that the jurisdictions’ governments acted as revenue–maximizing Leviathans, Laussel and le Breton ( 1998 ) provided a detailed characterization of conditions which ensured the existence of a ( pure strategy ) Nash equilibrium. More recently, in an as yet unpublished paper, Bayindir–Upmann and Ziad look at the Wilson–Zodrow–Mieszkowski model with a finite number of identical jurisdictions. By weakening slightly the requirement for a jurisdiction’s best response, they are able to provide conditions on the production technology which ensure the existence of a symmetric equilibrium. Another unpublished paper, Iritani and Fuji ( 2002 ), provides an

1

2

surveyed in Wilson ( 1999 )

typically : Wildasin ( 1989 ) and Figuieres et al ( 2001 ), for example, consider the implications of

public expenditure levels being the strategic variables

2

example in which a pure strategy Nash equilibrium does not exist.

3

The purpose of this note is to make another small contribution to this literature. I show that there must exist a Nash equilibrium in pure strategies to the tax competition game, if all the jurisdictions are sufficiently small. This result does not require any assumptions about the technology, such as those in Bayindir–Upmann and Ziad. It will hold even if the payoff of a jurisdiction is not convex in its own tax rate. Thus unlike Laussel and le Breton and Bayindir–Upmann and Ziad, which provide conditions implying the continuity of jurisdictions’ reaction functions, this paper establishes that a Nash equilibrium will exist even if the reaction functions have discontinuities. The equilibrium may not be symmetric. However, it is crucial that the jurisdictions be small, in the sense that no single jurisdiction has any influence on the overall “national” net return to capital. Wilson, and Zodrow and Mieszkowski both made such an assumption, although much of the subsequent literature has not. The main result in section 1, takes this smallness quite literally by assuming that there is a continuum of jurisdictions, each of them of measure zero. Section 2 contains an attempt to apply this result to a more realistic setting, with a finite number of jurisdictions. In this case, I do not ( and cannot ) show the existence of a pure strategy Nash equilibrium, no matter how large the number of jurisdictions. However, if the number of jurisdictions is large than the equilibrium derived in section 1 becomes an arbitrarily close approximation of an equilibrium. That is, if the relative size of each jurisdiction is sufficiently small, then there exists a vector of tax rates such that each jurisdiction’s choice of tax rate is arbitrarily close to being a best response to the others’. Since Bayindir–Upmann and Ziad show that there must be a Nash equilibrium when there are two identical jurisdictions, my

3

Rothstein ( 2002 ) as well proves existence of equilibrium in this sort of model. However, the payoff

functions are taken as primitives ( and assumed quasi–concave ) rather than being derived from production and utility functions.

3

results are in some sense complementary. When the number of jurisdictions is very small, or very large, existence of Nash equilibrium is not a major problem. Not surprisingly, perhaps, the leap of faith taken in most of the literature, of simply assuming the existence of a Nash equilibrium, seems quite valid.

1. A Continuum of Jurisdictions

Assuming a continuum of jurisdictions makes the “smallness” of each jurisdiction explicit. Let a jurisdiction’s “type” be a sort of dummy variable : the types are distributed uniformly on the interval [0, 1], but each jurisdiction is identical in every respect. A jurisdiction of type z

4

is inhabited by a single agent, who supplies a fixed quantity

¯ The of labour inelastically. All agents are endowed with the same quantity of capital k. function k : [0, 1] → R+ represents the capital employed per worker in a jurisdiction of type k, so that total employment of capital in all jurisdictions is

Z

1

k(z)dz 0

( if k(z) is integrable ), and the market–clearing condition for capital is

R

¯ k(z) = k.

Output is produced in each jurisdiction, using capital and labour as inputs, under perfect competition. If R(z) denotes the gross return to capital in a jurisdiction of type z, then the perfect competition implies that f 0 (k(z)) = R(z), where f (k(z)) is a strictly concave production function, representing the value of output produced per worker if the capital employed per worker is k(t). Since all jurisdictions are identical, they all have the same production function f (·). Since f (·) is assumed strictly concave, this marginal productivity relation can be inverted, so as to define a “demand for capital” function φ : R+ → R+

4

t will stand for “tax”, so z

will index types

4

implicitly by : f 0 [φ(R)] ≡ R

(1)

for any R ∈ R+ . It then follows from the implicit function theorem that φ0 (R) = (f 00 [φ(R)])−1 < 0

(2)

for any R > 0. The market–clearing condition for capital can now be written 1

Z

φ(R(z))di = k¯

(3)

0

People derive utility from two goods : a private good x and a publicly provided good g. The two goods are perfect substitutes in production. The publicly provided good must be provided by the jurisdiction’s government, and can be financed only by a source–based tax on capital employed in the jurisdiction, so that g(z) = t(z)k(z)

(4)

where t(z) is the unit tax rate on capital in the jurisdiction. Mobility of capital means that the net return to capital is the same in each jurisdiction. Let r denote this return, so that R(z) = r + t(z)

(5)

in each jurisdiction. Private consumption equals the resident’s disposable income : the value of output produced in the jurisdiction, minus the gross return to capital employed there, plus the net return to the resident’s capital endowment : x(z) = f (k(z)) − R(z)k(z) + rk¯

(6)

x(z) = f [φ(R(z))] − R(z)φ(R(z)) + rk¯

(7)

which can be written

5

The resident chooses her jurisdiction’s tax rate t(z) so as to maximize her utility u(x(z), g(z)). With a continuum of jurisdictions, no jurisdiction’s unilateral tax changes have any effects on the “national” net return to capital r. That return is defined, as a function of the tax rates in all the jurisdictions, through the capital market clearing condition eqrefcapeq1, which can also be written Z

1

φ(r + t(z))dz = k¯

(8)

0

If r is given, then the factor mobility condition R(z) = r + t(z) implies that a jurisdiction’s choice variable can be regarded as its gross return to capital R(z), rather than its tax rate t(z). re–phrasing a jurisdiction’s choice problem in terms of R(z) rather than t(z) will prove helpful in establishing the existence of equilibrium in this continuum model. The resident of each type of jurisdiction z chooses her jurisdiction;s gross return to capital R(z) so as to maximize u(x(z), g(z)), subject to equations (7) and to g(z) = (R(z) − r)φ(R(z))

(9)

Since all types of jurisdiction have the same endowments, and the same tastes and technol¯ (R − r)φ(R)] if the “national” ogy, any jurisdiction’s payoff will be u[f (φ(R) − Rφ(R) + rk, net return to capital is r, and if the jurisdiction chooses a gross return to capital R. Let R∗ (r) be the set of R’s which maximize this payoff. In general, R∗ (r) may have many elements. However, the optimal R∗ (r)’s must be strictly increasing, in the following sense : LEMMA 1 : If r0 > r, and if R ∈ R∗ (r), R0 ∈ R∗ (r0 ), then R0 ≥ R. PROOF : Let x(R, r) and g(R, r) denote private and public consumption respectively, when the national net return to capital is r and when the gross return to capital in the jurisdiction is R. From equation (7), x(R00 , r) − x(R0 , r) = f [φ(R00 )] − f [φ(R0 )] − R00 φ(R00 ) + R0 φ(R0 ) 6

(10)

which is independent of r. ∂g(R, r) = −φ(R) ∂r

(11)

so that if R00 > R0 , then ∂ [g(R00 , r) − g(R0 , r] = ∂r

Z

R00

R0

∂2x dR = − ∂R∂r

Z

R00

φ0 (R)dR > 0

(12)

R0

implying that g(R00 , r) − g(R0 , r) is strictly increasing in r. Suppose then that r0 > r, that R ∈ R∗ (r) and that R0 < R. Then u[x(R, r0 ), g(R, r0 )] − u[x(R0 , r0 ), g(R0 r0 )] > u[x(R, r), g(R, r)] − u[x(R0 , r), g(R0 , r)] ≥ 0 so that R0 cannot be in R∗ (r0 ), proving the lemma.

•

An equilibrium to the game played by the continuum of jurisdictions can now be defined in the usual manner.

DEFINITION : The national net return to capital r, and the ( integrable ) gross return to capital function R : [0, 1] → R+ constitute an equilibrium if R1 i 0 φ(R(z))dz = k¯ ii R(z) ∈ R∗ (r) for almost every z

The first condition in the above definition is the market–clearing condition for capital. The second condition says that each jurisdiction’s choice is one of its best responses to the national net return r. A symmetric equilibrium is an equilibrium in which almost every jurisdiction chooses the same gross return R(z) ( which also implies almost every jurisdiction is choosing the same tax rate t(z) = R(z) − r.

LEMMA 2 : A symmetric equilibrium will exist if and only if there exists some t such that ¯ ∈ R∗ [f 0 (k) ¯ − t] f 0 (k) 7

(13)

¯ − t, so that k¯ = φ(r + t). PROOF : Suppose that there exists such a t. Set r = f 0 (k) Then if each R(z) equals f 0 (bark, the two conditions for equililibrium are satisfied. Conversely, if an equilibrium is symmetric, then almost every jurisdiction chooses the same gross return to capital R, so that condition i in the definition of equilibrium ¯ or f 0 (k) ¯ = R. Condiiton ii then requires that f 0 (k) ¯ ∈ R∗ (r) implies that φ(R) = k, for some national net return to capital r, and condition (13) will be satisfied by setting ¯ − r. t ≡ f 0 (k)

•

It has been assumed here that the supply of capital is fixed. This fixity means that jurisdictions might actually want to set their tax rates so high that capital’s net return is negative. This would be the case if demand for the publicly provided output is very strong, relative to capital’s share of income. This phenomenon will be ruled out by assumption.

5

ASSUMPTION 1 : ug (x, 0)/ux (x, 0) > 1 for any x > 0. ASSUMPTION 2 : The values of k which maximize u[f (k) − f 0 (k)k, f 0 (k)k] are all ¯ strictly less than k.

Assumption 1 ensures that the jurisdictions want to set a positive tax rate, so that R > r for any R ∈ R∗ (r). Assumption 2 ensures that jurisdictions want tax rates less than f 0 (k ∗ ) when r = 0, so that R < f 0 (k ∗ ) when R ∈ R∗ (0). To look for a symmetric Nash equilibrium, suppose first that R∗ (r) is single–valued for all r ≥ 0. A third assumption is needed, and it is also relatively weak.

5

For an analysis of existence of equilibrium when this sort of assumption is not made,

see Rothstein ( 2002 ).

8

ASSUMPTION 3 :

ug ux (g, x)

is non–decreasing in x, and decreasing in g.

This assumption would be satisfied, for instance, if both goods were normal. Then lemma 1 can be strengthened to LEMMA 3 : Under Assumption 3, R∗ (r) must be strictly increasing. That is, If r0 > r, and if R ∈ R∗ (r), R0 ∈ R∗ (r0 ), then R0 > R. PROOF : If R was in both R∗ (r) and R∗ (r0 ), with r0 > r, then Lemma 1 implies that R∗ would have to be single–valued over the interval (r, r0 ). If R∗ (r) is single–valued, then the unique optimal R for the net return r must satisfy the first order condition for the maximization of u(x, g) subject to conditions (7) and (9). That first–order condition is −φ(R) +

ug [x(R, r), g(R, r)](φ(R) + (R − r)φ0 (R)) = 0 ux

(14)

Increases in r, holding R constant, must increase x(R, r) and decrease g(R, r), and so increase ug /ux , if Assumption 3 holds. If (14) holds, then the coefficient φ(R)+(R−r)φ0 (R) must be positive. Therefore, increases in r must increase the left side of equation (14), so that dR∗ /dr > 0 if R∗ is single–valued.

•

In general, the best response correspondence R∗ (r) need not be single–valued. If it is multiple valued, it need not be convex. ¯ ∈ R∗ (r) for some From Lemma 2, there will exist a symmetric equilibrium if f 0 (k) non–negative r. if R∗ (r) were single–valued, and if it were continuous, then a symmetric ¯ and if R∗ (r) > k) ¯ for large enough r. Assumptions equilibrium would exist if R∗ (0) < f 0 (k), 1 and 2 ensure these properties hold. The correspondence is single–valued almost everywhere. It is also continuous when it is single–valued. These two properties follow from the fact that any monotonic function of a single variable is continuous almost everywhere, and has at most a countable number of 9

jump discontinuities.6 R∗ (r) is not a function ; but R+ (r), defined as the maximal element in R∗ (r), is. It also follows, from the continuity of u[x(R, r), g(R, r)] in R and r, that R∗ (r) must be multiple–valued whenever R+ (r) jumps.

LEMMA 4 : R∗ (r0 ) has at least two elements, if the graph of R∗ (r) jumps discontinuously at r0 . PROOF : If R∗ (r) jumps discontinuously at r0 , then so does R+ (r). But since R+ (r) has only a countable number of jump discontinuities, it is continuous in some intervals (r0 − , r0 ) and (r0 , r0 + ), if it jumps at r0 . That means that there is a sequence R+ (r) approaching r0 from above, such that u[x(R+ (r), r), g(R+ (r), r)] ≥ u[x(R, r), g(R, r)] for any R. By continuity, then, this sequence reaches some limit R+ (r0 ), with u[x(R+ (r0 ), r0 ), g(R+ (r0 ), r0 )] ≥ u[x(R, r0 ), g(R, r0 )] for any R. Similarly, there is a sequence of r’s approaching r0 from below, and some R− (r0 ) which is the limit of the ( singleton ) elements of R∗ (r) as r approaches r0 from below, with u[x(R− (r0 ), r0 ), g(R− (r0 ), r0 )] ≥ u[x(R, r0 ), g(R, r0 )] for any R. So both R+ (r0 ) and R− (r0 ) are in R∗ (r0 ). Since there is a jump in R∗ (r) at r = r0 , R+ (r0 ) 6= R− (r0 ).

•

Thus the possibility exists of a jump in the graph of R∗ (r) as in figure 1, in which the ¯ is in the interior of the jump, and is not graph jumps discontinuously at some r. If f 0 (k) in R∗ (r) at the jump, there is no symmetric Nash equilibrium. For example, in figure 1, ∗ ¯ < R∗ (ˆ (ˆ r) < f 0 (k) R∗ (ˆ r) has two values, with R− + r ), where the subscripts − and + denote

the two elements in R∗ (r). In such a case, it may be clear from the figure that there can be no symmetric equilibrium. Lemmas 1 and 3 imply the graph of R∗ (r) can cross the

6

cf Devinatz ( 1968 ), Proposition 2.3.5

10

¯ at most once, and the graph jumps instead of crossing. horizontal line R = f 0 (k)

Formally, let rˆ be the unique r at which this crossing occurs. ¯ ∀R ∈ R∗ (r)} rˆ ≡ sup{r|R(r) < f 0 (k) Then ¯ ∈ R∗ (ˆ LEMMA 5 : There will be a symmetric equilibrium if and only if f 0 (k) r). ¯ ∈ R∗ (ˆ ¯ − rˆ, and Lemma 2 applies. PROOF : If f 0 (k) r), then let t ≡ f 0 (k) On the other hand, if there is a symmetric equilibrium, Lemma 2 implies the existence 11

¯ − t]. Then the definition above implies of a common tax rate t such that f 0 (k¯ ∈ R∗ [f 0 (k) ¯ − t = rˆ. that f 0 (k)

•

However, there will always be an equilibrium, even if not symmetric.

PROPOSITION 1 : Under Assumptions 1 and 2, there always exists an equilibrium to the tax competition game played by a continuum of jurisdictions. ¯ ∈ R∗ (ˆ PROOF : If f 0 (k) r), then there is a symmetric equilibrium, in which every ¯ − rˆ. jurisdiction chooses a tax rate of f 0 (k) ¯ ∈ If f 0 (k) / R∗ (ˆ r), then continuity of u[x(R, r), g(R, r)] in R and r implies that there ∗ ∗ must be at least two optimal values for R in R∗ (ˆ r), R∗ − (ˆ r) and R+ (ˆ r), with R− (ˆ r)

0, and that g(f 0 (k)) ¯ satisfying (22). of g(t) implies that there must be at least one t∗ in (0, f 0 (k)) ¯ As t increases, x decreases and g increases, along the line x + g = f (k). Quasi– ¯ concavity of preferences implies therefore that ug /ux falls as t rises. Since f 00 < 0, t/f 00 (k) must decrease as t increases.Therefore, if [k¯ +

N − 1 t∗ ¯ ]>0 N f 00 (k)

(23)

then g 0 (t) < 0. But (23) must hold if (22) does. Therefore g 0 (t) < 0 whenever g(t) = 0, so that there is exactly 1 value for t∗ satisfying the first–order condition (22) for a symmetric Nash •

equilibrium.

Since the left side of (22) is decreasing in N ,and since g 0 (t) < 0 at t = t∗ , it follows that 8

it is presented in Bayindir–Upmann and Ziad ( 2001 ), for example

15

LEMMA 7 : The t∗ satisfying the first–order conditions for a symmetric Nash equilibrium decreases with the number N of jurisdictions.

Lemma 7 is also not a new observation ; it underlies the result that increased fragmentation worsens tax competition. It implies that t∗ approaches some limit t∞ as N grows large. This limiting tax rate is the solution to (22) as N → ∞, that is the solution to t∞ ug ∞¯ ∞¯ ¯ ¯ (f (k) − t k, t k)[k + 00 ¯ ] = 0 ux f (k)

(24)

This helps relate the finite tax competition model to the continuum model of the previous section. LEMMA 8 : Let t∞ be defined by equation (24). In the continuum model of section ¯ − t∞ . Then R = f 0 (k) ¯ satisfies the first–order condition (14) to 1, suppose that r = f 0 (k) the maximization problem faced by jurisdictions in the continuum model. PROOF : Since φ(·) is the inverse function to f 0 (·), therefore φ0 (R) = 1/[f 00 (φ(R))]. ¯ − t∞ , and if R = f 0 (k), ¯ then equations (24) and (14) are the same. If r = f 0 (k)

•

Lemma 8 connects the equilibria of the continuum model with those of the finite model. What it implies is PROPOSITION 3 : If the best reaction R∗ (r) in the continuum model is single–valued ¯ , so that there is a symmetric Nash equilibrium to the finite at rˆ, then there is some N ¯ or more. model, when the number of jurisdictions N is N PROOF : In the continuum model, choosing R and choosing t are the same problem. ¯ (R − r)φ(R) if and only if a tax rate A gross return R maximizes u(f (φ(R)) − Rφ(R) + rk, ¯ tφ(r + t)). If the best reaction is t = R − r maximizes u(f (φ(r + t)) − (r + t)φ(r + t) + rk, ¯ is strictly preferred to any other R when r = rˆ, which single–valued at rˆ, then R = f 0 (k) is the same as t∞ being strictly the best tax rate when r = rˆ. 16

Theefore, if R∗ (r) is single–valued at r = rˆ r, then t = t∞ will be the unique maximizer ¯ tφ(ˆ of u(f (φ(ˆ r + t)) − (r + t)φ(ˆ r + t) + rk, r + t)). Further, if all jurisdictions ( except a set of measure zero ) chose tax rates t(z) = t∞ ¯ − t∞ = rˆ. in the continuum model, then the national net return would equal f 0 (k) Let π(t; t0 , N ) denote the payoff to a jurisdiction in the finite model, if it chooses a tax rate t, and every one of the N − 1 other jurisdictions choose a tax rate t0 . The definition of t∞ , and the assumption of the hypothesis, imply lim π(t∞ ; t∞ , N ) > π(t; t∞ , N )

N →∞

for any tax rate t 6= t∞ . Since t∗ → t∞ as N → ∞, then continuity of the payoffs implies that π(t∗ (N ); t∗ (N ), N ) > π(t; t∗ (N ), N )

(25)

for any other t 6= t∗ (N ), if N is large enough. But (25) says that t∗ (N ) is a best reaction of a jurisdiction if all the other jurisdictions choose tax rates of t∗ (N ), proving the Proposition. •

Unfortunately, the hypothesis of Proposition 3, that the best reaction correspondence R∗ (r) be single–valued at rˆ, has not been expressed in terms of the primitives of the model. Equation (24) defines what the equilibrium tax rate must be if there is a symmetric Nash equilibrium in the continuum model. But ( under assumptions 1 and 2 ) there always is a unique t∞ satisfying (24). The hypothesis of Proposition 3 is that t∞ be the unique best tax rate for a jurisdiction when all the other jurisdictions set tax rates of t∞ . Local properties of the production and utility functions are not sufficient to guarantee that this condition holds. Conditions can be provided which imply that the second–order condition for a maximum holds at t = t∞ ; these are provided in Bayindir–Upmann and Ziad. 17

However, as the authors note, these conditions imply only that t∞ is the best tax rate among all tax rates in some small interval (t∞ − , t∞ + ). Even if t∞ is the best tax rate, it may be one of several best tax rates. Although the ¯ is one of three elements in R∗ (ˆ situation depicted in figure 2, in which R = f 0 (k) r) is a very unlikely one, it is possible.

If the hypothesis of Proposition 3 does not hold, then there may not be a Nash equilibrium to the tax competition played by a finite number N of jurisdictions, even when N is large. However, there is an approximate equilibrium, approximate in the sense that the payoff each jurisdiction gets can be made arbitrarily close to the maximum possible, 18

given the other jurisdictions’ tax rates. For Nash equilibria which may not be symmetric, the definition of the payoff function π(t; t0 , N ) will be extended. Let ψ(t; t1 , t2 , m, N ) be the payoff to a jurisdiction if it chooses the tax rate t, when a proportion m of the N other jurisdictions choose the tax rate t1 , a proportion 1 − m of the jurisdictions choose a tax rate t2 , and when there are N jurisdictions in total.

9

Let

ψ(t; t1 , t2 , m, ∞) be the limit of this payoff as N → ∞. So, if all the other jurisdictions levied tax rates of t1 and t2 , then t∗ would be jurisdiction i’s best response if it maximized ψ(t; t1 , t2 , m, N ) over all tax rates t. Here Proposition 1 can be applied in order to derive an approximate equilibrium in which at most two distinct tax rates are chosen, which is why the function ψ is defined as it is.

PROPOSITION 4 : Under assumptions 1 and 2, there is an approximate Nash equi¯ () such that M < N librium in the following sense. For any small > 0,there is some N jurisdictions choose a tax rate t1 , N − M choose a tax rate t2 , M −1 M −1 , N − 1) > max ψ(t; t1 , t2 , , N) − t N −1 N −1

(26)

M M , N − 1) > max ψ(t; t1 t2 , , N − 1) − t N −1 N −1

(27)

ψ(t1 ; t1 , t2 , and ψ(t2 ; t1 , t2 ,

PROOF : If the hypotheses of Proposition 3 hold, then there is an exact Nash equilibrium, in which each jurisdiction sets the tax rate t∗ (N ). In this case t1 = t2 = t∗ (N ), 9

so that

π(t; t0 , N ) ≡ ψ(t; t0 t0 , m, N ) 19

M can be any number less than N , conditions (26) and (27) are the same, and both will hold with = 0 for large enough N . ¯ were in R∗ (ˆ If R∗ (ˆ r) were multiple valued, but f 0 (k) r), then t∞ would be one of the tax rates which maximized any jurisdiction’s payoff when r = rˆ. That is the same as ψ(t∞ ; t∞ , t∞ , m, ∞) = max ψ(t; t∞ , t∞ , m, ∞) t

for any 0 < m < 1. Continuity of ψ then implies that condition (26) ( which is the same as condition (27) ) holds when N is large enough. ¯ ∈ R∗ (ˆ Finally suppose that f 0 (k) r). From the construction of rˆ, there must be a jump ¯ and R− < f 0 (k) ¯ are both in R∗ (ˆ in R∗ (r) at r = rˆ such that R+ > f 0 (k) r). Proposition 1 demonstrated that in this case, there exists a Nash equilibrium to the continuum model in which a proportion m of the jurisdictions choose a tax rate t− ≡ R− − rˆ and a proportion 1 − m choose t+ ≡ R+ − rˆ. Since this is a Nash equilibrium to the continuum model, it would be a Nash equilibrium as well if there were a countable infinity of jurisdictions, so that ψ(t− ; t− , t+ , m, ∞) = max ψ(t; t− , t+ , m, ∞) t

ψ(t+ ; t− , t+ , m, ∞) = max ψ(t; t− , t+ , m, ∞) t

For any proportion m, if the number N of jurisdictions is large enough, then there is some M such that M/N is arbitrarily close to m. Continuity of ψ then implies that conditions •

(26) and (27) hold, completing the proof.

Continuity also implies that the underprovision result in the continuum model must apply in the finite model, if N is sufficiently large. That is, if N is large enough, ug /ux must exceed 1 in any ( exact, pure strategy ) Nash equilibrium of the tax competition game with a finite number of jurisdictions. In this section, it was assumed that all jurisdictions were identical. That assumption could be relaxed slightly by allowing for differences in size. That is, suppose there are N 20

jurisdictions, each with the identical technology, each containing identical residents, but with different numbers of residents. Let si denote the share or the population in jurisdiction i, and s the largest of these shares. Even when the si ’s differ among jurisdictions, all of the results in this section will apply equally well in the limit as s → 0. As long as no jurisdiction’s influence on the national net return to capital is large, the outcome of the continuum model is a good approximation of the Nash equilibrium.

21

References

[1] T. Bayindir-Upmann and A. Ziad. Existence of equilibria in a basic tax-competition model. working paper, University of Bielefeld, 2001. [2] A. Devinatz. Advanced Calculus. Holt Rinehart Winston, 1968. [3] C. Figuieres, J. Hindriks, and G. Myles. Revenue sharing versus expenditure sharing. working paper 01/15, CORE, 2001. [4] J. Iritani and Y. Fujii. A note on non–existence of equilibria in tax competition models. unpublished note, Graduate School of Economics, Kobe University, July 2002. [5] D. Laussel and M. Le Breton. Existence of nash equilibria in fiscal competition models. Regional Science and Urban Economics, 28:283–296, 1998. [6] P. Rothstein. Discontinuous payoffs, shared resources, and games of fiscal competition : Existence of pure strategy nash equilibrium. working paper, Department of Economics, Washington University in St. Louis, August 2002. [7] D. Wildasin. Interjurisdictional capital mobility : Fiscal externality and a corrective subsidy. Journal of Urban Economics, 25:193–212, 1989. [8] J. Wilson. A theory of interregional tax competition. Journal of Urban Economics, 19:296–315, 1986. [9] J. Wilson. Theories of tax competition. National Tax Journal, 52:269–304, 1999. [10] G Zodrow and P. Mieszkowski. Pigou, tiebout, property taxation and the underprovision of local public goods. Journal of Urban Economics, 19:356–370, 1986.

22

S. Bucovetsky Arts/Economics York University Toronto Ont. M3J 1P3 CANADA email : [email protected]

June 2003

abstract : In models of tax competition, payoffs to the countries setting taxes need not be quasi–concave in their tax rates, even if preferences and technology are both convex. The existence of Nash equilibria in pure strategies is therefore not easily assured. However, assuming a continuum of agents helps convexify matters, as is well–known. Here it is shown that a pure strategy Nash equilibrium must exist if there is a continuum of countries. It is also shown that this result guarantees the existence of an approximate equilibrium if the relatively size of each country is sufficiently small.

1

Introduction

The relatively simple model of capital tax competition due to Wilson ( 1986 ) , and to Zodrow and Mieszkowski ( 1986 ) has given rise to quite a large literature 1 . In most of this literature, local ( or other lower–level ) governments are assumed to set source–based taxes non–cooperatively. Under a variety of assumptions, considerable insight has been gained about the properties of Nash equilibria when jurisdictions compete in this manner for a mobile tax base. Since the strategic variables are the jurisdictions’ capital tax rates 2 , Nash equilibrium is characterized by each jurisdiction maximizing some payoff function with respect to its own tax rate, taking the tax rates of other jurisdictions as given. The properties of the Nash equilibrium are given by the first–order conditions for the jurisdictions’ maximization problems. Relatively little attention has been paid to the technical aspects of this game : whether the second–order conditions for maximization hold, and whether a Nash equilibrium to the game exists. By changing the model slightly, so that the jurisdictions’ governments acted as revenue–maximizing Leviathans, Laussel and le Breton ( 1998 ) provided a detailed characterization of conditions which ensured the existence of a ( pure strategy ) Nash equilibrium. More recently, in an as yet unpublished paper, Bayindir–Upmann and Ziad look at the Wilson–Zodrow–Mieszkowski model with a finite number of identical jurisdictions. By weakening slightly the requirement for a jurisdiction’s best response, they are able to provide conditions on the production technology which ensure the existence of a symmetric equilibrium. Another unpublished paper, Iritani and Fuji ( 2002 ), provides an

1

2

surveyed in Wilson ( 1999 )

typically : Wildasin ( 1989 ) and Figuieres et al ( 2001 ), for example, consider the implications of

public expenditure levels being the strategic variables

2

example in which a pure strategy Nash equilibrium does not exist.

3

The purpose of this note is to make another small contribution to this literature. I show that there must exist a Nash equilibrium in pure strategies to the tax competition game, if all the jurisdictions are sufficiently small. This result does not require any assumptions about the technology, such as those in Bayindir–Upmann and Ziad. It will hold even if the payoff of a jurisdiction is not convex in its own tax rate. Thus unlike Laussel and le Breton and Bayindir–Upmann and Ziad, which provide conditions implying the continuity of jurisdictions’ reaction functions, this paper establishes that a Nash equilibrium will exist even if the reaction functions have discontinuities. The equilibrium may not be symmetric. However, it is crucial that the jurisdictions be small, in the sense that no single jurisdiction has any influence on the overall “national” net return to capital. Wilson, and Zodrow and Mieszkowski both made such an assumption, although much of the subsequent literature has not. The main result in section 1, takes this smallness quite literally by assuming that there is a continuum of jurisdictions, each of them of measure zero. Section 2 contains an attempt to apply this result to a more realistic setting, with a finite number of jurisdictions. In this case, I do not ( and cannot ) show the existence of a pure strategy Nash equilibrium, no matter how large the number of jurisdictions. However, if the number of jurisdictions is large than the equilibrium derived in section 1 becomes an arbitrarily close approximation of an equilibrium. That is, if the relative size of each jurisdiction is sufficiently small, then there exists a vector of tax rates such that each jurisdiction’s choice of tax rate is arbitrarily close to being a best response to the others’. Since Bayindir–Upmann and Ziad show that there must be a Nash equilibrium when there are two identical jurisdictions, my

3

Rothstein ( 2002 ) as well proves existence of equilibrium in this sort of model. However, the payoff

functions are taken as primitives ( and assumed quasi–concave ) rather than being derived from production and utility functions.

3

results are in some sense complementary. When the number of jurisdictions is very small, or very large, existence of Nash equilibrium is not a major problem. Not surprisingly, perhaps, the leap of faith taken in most of the literature, of simply assuming the existence of a Nash equilibrium, seems quite valid.

1. A Continuum of Jurisdictions

Assuming a continuum of jurisdictions makes the “smallness” of each jurisdiction explicit. Let a jurisdiction’s “type” be a sort of dummy variable : the types are distributed uniformly on the interval [0, 1], but each jurisdiction is identical in every respect. A jurisdiction of type z

4

is inhabited by a single agent, who supplies a fixed quantity

¯ The of labour inelastically. All agents are endowed with the same quantity of capital k. function k : [0, 1] → R+ represents the capital employed per worker in a jurisdiction of type k, so that total employment of capital in all jurisdictions is

Z

1

k(z)dz 0

( if k(z) is integrable ), and the market–clearing condition for capital is

R

¯ k(z) = k.

Output is produced in each jurisdiction, using capital and labour as inputs, under perfect competition. If R(z) denotes the gross return to capital in a jurisdiction of type z, then the perfect competition implies that f 0 (k(z)) = R(z), where f (k(z)) is a strictly concave production function, representing the value of output produced per worker if the capital employed per worker is k(t). Since all jurisdictions are identical, they all have the same production function f (·). Since f (·) is assumed strictly concave, this marginal productivity relation can be inverted, so as to define a “demand for capital” function φ : R+ → R+

4

t will stand for “tax”, so z

will index types

4

implicitly by : f 0 [φ(R)] ≡ R

(1)

for any R ∈ R+ . It then follows from the implicit function theorem that φ0 (R) = (f 00 [φ(R)])−1 < 0

(2)

for any R > 0. The market–clearing condition for capital can now be written 1

Z

φ(R(z))di = k¯

(3)

0

People derive utility from two goods : a private good x and a publicly provided good g. The two goods are perfect substitutes in production. The publicly provided good must be provided by the jurisdiction’s government, and can be financed only by a source–based tax on capital employed in the jurisdiction, so that g(z) = t(z)k(z)

(4)

where t(z) is the unit tax rate on capital in the jurisdiction. Mobility of capital means that the net return to capital is the same in each jurisdiction. Let r denote this return, so that R(z) = r + t(z)

(5)

in each jurisdiction. Private consumption equals the resident’s disposable income : the value of output produced in the jurisdiction, minus the gross return to capital employed there, plus the net return to the resident’s capital endowment : x(z) = f (k(z)) − R(z)k(z) + rk¯

(6)

x(z) = f [φ(R(z))] − R(z)φ(R(z)) + rk¯

(7)

which can be written

5

The resident chooses her jurisdiction’s tax rate t(z) so as to maximize her utility u(x(z), g(z)). With a continuum of jurisdictions, no jurisdiction’s unilateral tax changes have any effects on the “national” net return to capital r. That return is defined, as a function of the tax rates in all the jurisdictions, through the capital market clearing condition eqrefcapeq1, which can also be written Z

1

φ(r + t(z))dz = k¯

(8)

0

If r is given, then the factor mobility condition R(z) = r + t(z) implies that a jurisdiction’s choice variable can be regarded as its gross return to capital R(z), rather than its tax rate t(z). re–phrasing a jurisdiction’s choice problem in terms of R(z) rather than t(z) will prove helpful in establishing the existence of equilibrium in this continuum model. The resident of each type of jurisdiction z chooses her jurisdiction;s gross return to capital R(z) so as to maximize u(x(z), g(z)), subject to equations (7) and to g(z) = (R(z) − r)φ(R(z))

(9)

Since all types of jurisdiction have the same endowments, and the same tastes and technol¯ (R − r)φ(R)] if the “national” ogy, any jurisdiction’s payoff will be u[f (φ(R) − Rφ(R) + rk, net return to capital is r, and if the jurisdiction chooses a gross return to capital R. Let R∗ (r) be the set of R’s which maximize this payoff. In general, R∗ (r) may have many elements. However, the optimal R∗ (r)’s must be strictly increasing, in the following sense : LEMMA 1 : If r0 > r, and if R ∈ R∗ (r), R0 ∈ R∗ (r0 ), then R0 ≥ R. PROOF : Let x(R, r) and g(R, r) denote private and public consumption respectively, when the national net return to capital is r and when the gross return to capital in the jurisdiction is R. From equation (7), x(R00 , r) − x(R0 , r) = f [φ(R00 )] − f [φ(R0 )] − R00 φ(R00 ) + R0 φ(R0 ) 6

(10)

which is independent of r. ∂g(R, r) = −φ(R) ∂r

(11)

so that if R00 > R0 , then ∂ [g(R00 , r) − g(R0 , r] = ∂r

Z

R00

R0

∂2x dR = − ∂R∂r

Z

R00

φ0 (R)dR > 0

(12)

R0

implying that g(R00 , r) − g(R0 , r) is strictly increasing in r. Suppose then that r0 > r, that R ∈ R∗ (r) and that R0 < R. Then u[x(R, r0 ), g(R, r0 )] − u[x(R0 , r0 ), g(R0 r0 )] > u[x(R, r), g(R, r)] − u[x(R0 , r), g(R0 , r)] ≥ 0 so that R0 cannot be in R∗ (r0 ), proving the lemma.

•

An equilibrium to the game played by the continuum of jurisdictions can now be defined in the usual manner.

DEFINITION : The national net return to capital r, and the ( integrable ) gross return to capital function R : [0, 1] → R+ constitute an equilibrium if R1 i 0 φ(R(z))dz = k¯ ii R(z) ∈ R∗ (r) for almost every z

The first condition in the above definition is the market–clearing condition for capital. The second condition says that each jurisdiction’s choice is one of its best responses to the national net return r. A symmetric equilibrium is an equilibrium in which almost every jurisdiction chooses the same gross return R(z) ( which also implies almost every jurisdiction is choosing the same tax rate t(z) = R(z) − r.

LEMMA 2 : A symmetric equilibrium will exist if and only if there exists some t such that ¯ ∈ R∗ [f 0 (k) ¯ − t] f 0 (k) 7

(13)

¯ − t, so that k¯ = φ(r + t). PROOF : Suppose that there exists such a t. Set r = f 0 (k) Then if each R(z) equals f 0 (bark, the two conditions for equililibrium are satisfied. Conversely, if an equilibrium is symmetric, then almost every jurisdiction chooses the same gross return to capital R, so that condition i in the definition of equilibrium ¯ or f 0 (k) ¯ = R. Condiiton ii then requires that f 0 (k) ¯ ∈ R∗ (r) implies that φ(R) = k, for some national net return to capital r, and condition (13) will be satisfied by setting ¯ − r. t ≡ f 0 (k)

•

It has been assumed here that the supply of capital is fixed. This fixity means that jurisdictions might actually want to set their tax rates so high that capital’s net return is negative. This would be the case if demand for the publicly provided output is very strong, relative to capital’s share of income. This phenomenon will be ruled out by assumption.

5

ASSUMPTION 1 : ug (x, 0)/ux (x, 0) > 1 for any x > 0. ASSUMPTION 2 : The values of k which maximize u[f (k) − f 0 (k)k, f 0 (k)k] are all ¯ strictly less than k.

Assumption 1 ensures that the jurisdictions want to set a positive tax rate, so that R > r for any R ∈ R∗ (r). Assumption 2 ensures that jurisdictions want tax rates less than f 0 (k ∗ ) when r = 0, so that R < f 0 (k ∗ ) when R ∈ R∗ (0). To look for a symmetric Nash equilibrium, suppose first that R∗ (r) is single–valued for all r ≥ 0. A third assumption is needed, and it is also relatively weak.

5

For an analysis of existence of equilibrium when this sort of assumption is not made,

see Rothstein ( 2002 ).

8

ASSUMPTION 3 :

ug ux (g, x)

is non–decreasing in x, and decreasing in g.

This assumption would be satisfied, for instance, if both goods were normal. Then lemma 1 can be strengthened to LEMMA 3 : Under Assumption 3, R∗ (r) must be strictly increasing. That is, If r0 > r, and if R ∈ R∗ (r), R0 ∈ R∗ (r0 ), then R0 > R. PROOF : If R was in both R∗ (r) and R∗ (r0 ), with r0 > r, then Lemma 1 implies that R∗ would have to be single–valued over the interval (r, r0 ). If R∗ (r) is single–valued, then the unique optimal R for the net return r must satisfy the first order condition for the maximization of u(x, g) subject to conditions (7) and (9). That first–order condition is −φ(R) +

ug [x(R, r), g(R, r)](φ(R) + (R − r)φ0 (R)) = 0 ux

(14)

Increases in r, holding R constant, must increase x(R, r) and decrease g(R, r), and so increase ug /ux , if Assumption 3 holds. If (14) holds, then the coefficient φ(R)+(R−r)φ0 (R) must be positive. Therefore, increases in r must increase the left side of equation (14), so that dR∗ /dr > 0 if R∗ is single–valued.

•

In general, the best response correspondence R∗ (r) need not be single–valued. If it is multiple valued, it need not be convex. ¯ ∈ R∗ (r) for some From Lemma 2, there will exist a symmetric equilibrium if f 0 (k) non–negative r. if R∗ (r) were single–valued, and if it were continuous, then a symmetric ¯ and if R∗ (r) > k) ¯ for large enough r. Assumptions equilibrium would exist if R∗ (0) < f 0 (k), 1 and 2 ensure these properties hold. The correspondence is single–valued almost everywhere. It is also continuous when it is single–valued. These two properties follow from the fact that any monotonic function of a single variable is continuous almost everywhere, and has at most a countable number of 9

jump discontinuities.6 R∗ (r) is not a function ; but R+ (r), defined as the maximal element in R∗ (r), is. It also follows, from the continuity of u[x(R, r), g(R, r)] in R and r, that R∗ (r) must be multiple–valued whenever R+ (r) jumps.

LEMMA 4 : R∗ (r0 ) has at least two elements, if the graph of R∗ (r) jumps discontinuously at r0 . PROOF : If R∗ (r) jumps discontinuously at r0 , then so does R+ (r). But since R+ (r) has only a countable number of jump discontinuities, it is continuous in some intervals (r0 − , r0 ) and (r0 , r0 + ), if it jumps at r0 . That means that there is a sequence R+ (r) approaching r0 from above, such that u[x(R+ (r), r), g(R+ (r), r)] ≥ u[x(R, r), g(R, r)] for any R. By continuity, then, this sequence reaches some limit R+ (r0 ), with u[x(R+ (r0 ), r0 ), g(R+ (r0 ), r0 )] ≥ u[x(R, r0 ), g(R, r0 )] for any R. Similarly, there is a sequence of r’s approaching r0 from below, and some R− (r0 ) which is the limit of the ( singleton ) elements of R∗ (r) as r approaches r0 from below, with u[x(R− (r0 ), r0 ), g(R− (r0 ), r0 )] ≥ u[x(R, r0 ), g(R, r0 )] for any R. So both R+ (r0 ) and R− (r0 ) are in R∗ (r0 ). Since there is a jump in R∗ (r) at r = r0 , R+ (r0 ) 6= R− (r0 ).

•

Thus the possibility exists of a jump in the graph of R∗ (r) as in figure 1, in which the ¯ is in the interior of the jump, and is not graph jumps discontinuously at some r. If f 0 (k) in R∗ (r) at the jump, there is no symmetric Nash equilibrium. For example, in figure 1, ∗ ¯ < R∗ (ˆ (ˆ r) < f 0 (k) R∗ (ˆ r) has two values, with R− + r ), where the subscripts − and + denote

the two elements in R∗ (r). In such a case, it may be clear from the figure that there can be no symmetric equilibrium. Lemmas 1 and 3 imply the graph of R∗ (r) can cross the

6

cf Devinatz ( 1968 ), Proposition 2.3.5

10

¯ at most once, and the graph jumps instead of crossing. horizontal line R = f 0 (k)

Formally, let rˆ be the unique r at which this crossing occurs. ¯ ∀R ∈ R∗ (r)} rˆ ≡ sup{r|R(r) < f 0 (k) Then ¯ ∈ R∗ (ˆ LEMMA 5 : There will be a symmetric equilibrium if and only if f 0 (k) r). ¯ ∈ R∗ (ˆ ¯ − rˆ, and Lemma 2 applies. PROOF : If f 0 (k) r), then let t ≡ f 0 (k) On the other hand, if there is a symmetric equilibrium, Lemma 2 implies the existence 11

¯ − t]. Then the definition above implies of a common tax rate t such that f 0 (k¯ ∈ R∗ [f 0 (k) ¯ − t = rˆ. that f 0 (k)

•

However, there will always be an equilibrium, even if not symmetric.

PROPOSITION 1 : Under Assumptions 1 and 2, there always exists an equilibrium to the tax competition game played by a continuum of jurisdictions. ¯ ∈ R∗ (ˆ PROOF : If f 0 (k) r), then there is a symmetric equilibrium, in which every ¯ − rˆ. jurisdiction chooses a tax rate of f 0 (k) ¯ ∈ If f 0 (k) / R∗ (ˆ r), then continuity of u[x(R, r), g(R, r)] in R and r implies that there ∗ ∗ must be at least two optimal values for R in R∗ (ˆ r), R∗ − (ˆ r) and R+ (ˆ r), with R− (ˆ r)

0, and that g(f 0 (k)) ¯ satisfying (22). of g(t) implies that there must be at least one t∗ in (0, f 0 (k)) ¯ As t increases, x decreases and g increases, along the line x + g = f (k). Quasi– ¯ concavity of preferences implies therefore that ug /ux falls as t rises. Since f 00 < 0, t/f 00 (k) must decrease as t increases.Therefore, if [k¯ +

N − 1 t∗ ¯ ]>0 N f 00 (k)

(23)

then g 0 (t) < 0. But (23) must hold if (22) does. Therefore g 0 (t) < 0 whenever g(t) = 0, so that there is exactly 1 value for t∗ satisfying the first–order condition (22) for a symmetric Nash •

equilibrium.

Since the left side of (22) is decreasing in N ,and since g 0 (t) < 0 at t = t∗ , it follows that 8

it is presented in Bayindir–Upmann and Ziad ( 2001 ), for example

15

LEMMA 7 : The t∗ satisfying the first–order conditions for a symmetric Nash equilibrium decreases with the number N of jurisdictions.

Lemma 7 is also not a new observation ; it underlies the result that increased fragmentation worsens tax competition. It implies that t∗ approaches some limit t∞ as N grows large. This limiting tax rate is the solution to (22) as N → ∞, that is the solution to t∞ ug ∞¯ ∞¯ ¯ ¯ (f (k) − t k, t k)[k + 00 ¯ ] = 0 ux f (k)

(24)

This helps relate the finite tax competition model to the continuum model of the previous section. LEMMA 8 : Let t∞ be defined by equation (24). In the continuum model of section ¯ − t∞ . Then R = f 0 (k) ¯ satisfies the first–order condition (14) to 1, suppose that r = f 0 (k) the maximization problem faced by jurisdictions in the continuum model. PROOF : Since φ(·) is the inverse function to f 0 (·), therefore φ0 (R) = 1/[f 00 (φ(R))]. ¯ − t∞ , and if R = f 0 (k), ¯ then equations (24) and (14) are the same. If r = f 0 (k)

•

Lemma 8 connects the equilibria of the continuum model with those of the finite model. What it implies is PROPOSITION 3 : If the best reaction R∗ (r) in the continuum model is single–valued ¯ , so that there is a symmetric Nash equilibrium to the finite at rˆ, then there is some N ¯ or more. model, when the number of jurisdictions N is N PROOF : In the continuum model, choosing R and choosing t are the same problem. ¯ (R − r)φ(R) if and only if a tax rate A gross return R maximizes u(f (φ(R)) − Rφ(R) + rk, ¯ tφ(r + t)). If the best reaction is t = R − r maximizes u(f (φ(r + t)) − (r + t)φ(r + t) + rk, ¯ is strictly preferred to any other R when r = rˆ, which single–valued at rˆ, then R = f 0 (k) is the same as t∞ being strictly the best tax rate when r = rˆ. 16

Theefore, if R∗ (r) is single–valued at r = rˆ r, then t = t∞ will be the unique maximizer ¯ tφ(ˆ of u(f (φ(ˆ r + t)) − (r + t)φ(ˆ r + t) + rk, r + t)). Further, if all jurisdictions ( except a set of measure zero ) chose tax rates t(z) = t∞ ¯ − t∞ = rˆ. in the continuum model, then the national net return would equal f 0 (k) Let π(t; t0 , N ) denote the payoff to a jurisdiction in the finite model, if it chooses a tax rate t, and every one of the N − 1 other jurisdictions choose a tax rate t0 . The definition of t∞ , and the assumption of the hypothesis, imply lim π(t∞ ; t∞ , N ) > π(t; t∞ , N )

N →∞

for any tax rate t 6= t∞ . Since t∗ → t∞ as N → ∞, then continuity of the payoffs implies that π(t∗ (N ); t∗ (N ), N ) > π(t; t∗ (N ), N )

(25)

for any other t 6= t∗ (N ), if N is large enough. But (25) says that t∗ (N ) is a best reaction of a jurisdiction if all the other jurisdictions choose tax rates of t∗ (N ), proving the Proposition. •

Unfortunately, the hypothesis of Proposition 3, that the best reaction correspondence R∗ (r) be single–valued at rˆ, has not been expressed in terms of the primitives of the model. Equation (24) defines what the equilibrium tax rate must be if there is a symmetric Nash equilibrium in the continuum model. But ( under assumptions 1 and 2 ) there always is a unique t∞ satisfying (24). The hypothesis of Proposition 3 is that t∞ be the unique best tax rate for a jurisdiction when all the other jurisdictions set tax rates of t∞ . Local properties of the production and utility functions are not sufficient to guarantee that this condition holds. Conditions can be provided which imply that the second–order condition for a maximum holds at t = t∞ ; these are provided in Bayindir–Upmann and Ziad. 17

However, as the authors note, these conditions imply only that t∞ is the best tax rate among all tax rates in some small interval (t∞ − , t∞ + ). Even if t∞ is the best tax rate, it may be one of several best tax rates. Although the ¯ is one of three elements in R∗ (ˆ situation depicted in figure 2, in which R = f 0 (k) r) is a very unlikely one, it is possible.

If the hypothesis of Proposition 3 does not hold, then there may not be a Nash equilibrium to the tax competition played by a finite number N of jurisdictions, even when N is large. However, there is an approximate equilibrium, approximate in the sense that the payoff each jurisdiction gets can be made arbitrarily close to the maximum possible, 18

given the other jurisdictions’ tax rates. For Nash equilibria which may not be symmetric, the definition of the payoff function π(t; t0 , N ) will be extended. Let ψ(t; t1 , t2 , m, N ) be the payoff to a jurisdiction if it chooses the tax rate t, when a proportion m of the N other jurisdictions choose the tax rate t1 , a proportion 1 − m of the jurisdictions choose a tax rate t2 , and when there are N jurisdictions in total.

9

Let

ψ(t; t1 , t2 , m, ∞) be the limit of this payoff as N → ∞. So, if all the other jurisdictions levied tax rates of t1 and t2 , then t∗ would be jurisdiction i’s best response if it maximized ψ(t; t1 , t2 , m, N ) over all tax rates t. Here Proposition 1 can be applied in order to derive an approximate equilibrium in which at most two distinct tax rates are chosen, which is why the function ψ is defined as it is.

PROPOSITION 4 : Under assumptions 1 and 2, there is an approximate Nash equi¯ () such that M < N librium in the following sense. For any small > 0,there is some N jurisdictions choose a tax rate t1 , N − M choose a tax rate t2 , M −1 M −1 , N − 1) > max ψ(t; t1 , t2 , , N) − t N −1 N −1

(26)

M M , N − 1) > max ψ(t; t1 t2 , , N − 1) − t N −1 N −1

(27)

ψ(t1 ; t1 , t2 , and ψ(t2 ; t1 , t2 ,

PROOF : If the hypotheses of Proposition 3 hold, then there is an exact Nash equilibrium, in which each jurisdiction sets the tax rate t∗ (N ). In this case t1 = t2 = t∗ (N ), 9

so that

π(t; t0 , N ) ≡ ψ(t; t0 t0 , m, N ) 19

M can be any number less than N , conditions (26) and (27) are the same, and both will hold with = 0 for large enough N . ¯ were in R∗ (ˆ If R∗ (ˆ r) were multiple valued, but f 0 (k) r), then t∞ would be one of the tax rates which maximized any jurisdiction’s payoff when r = rˆ. That is the same as ψ(t∞ ; t∞ , t∞ , m, ∞) = max ψ(t; t∞ , t∞ , m, ∞) t

for any 0 < m < 1. Continuity of ψ then implies that condition (26) ( which is the same as condition (27) ) holds when N is large enough. ¯ ∈ R∗ (ˆ Finally suppose that f 0 (k) r). From the construction of rˆ, there must be a jump ¯ and R− < f 0 (k) ¯ are both in R∗ (ˆ in R∗ (r) at r = rˆ such that R+ > f 0 (k) r). Proposition 1 demonstrated that in this case, there exists a Nash equilibrium to the continuum model in which a proportion m of the jurisdictions choose a tax rate t− ≡ R− − rˆ and a proportion 1 − m choose t+ ≡ R+ − rˆ. Since this is a Nash equilibrium to the continuum model, it would be a Nash equilibrium as well if there were a countable infinity of jurisdictions, so that ψ(t− ; t− , t+ , m, ∞) = max ψ(t; t− , t+ , m, ∞) t

ψ(t+ ; t− , t+ , m, ∞) = max ψ(t; t− , t+ , m, ∞) t

For any proportion m, if the number N of jurisdictions is large enough, then there is some M such that M/N is arbitrarily close to m. Continuity of ψ then implies that conditions •

(26) and (27) hold, completing the proof.

Continuity also implies that the underprovision result in the continuum model must apply in the finite model, if N is sufficiently large. That is, if N is large enough, ug /ux must exceed 1 in any ( exact, pure strategy ) Nash equilibrium of the tax competition game with a finite number of jurisdictions. In this section, it was assumed that all jurisdictions were identical. That assumption could be relaxed slightly by allowing for differences in size. That is, suppose there are N 20

jurisdictions, each with the identical technology, each containing identical residents, but with different numbers of residents. Let si denote the share or the population in jurisdiction i, and s the largest of these shares. Even when the si ’s differ among jurisdictions, all of the results in this section will apply equally well in the limit as s → 0. As long as no jurisdiction’s influence on the national net return to capital is large, the outcome of the continuum model is a good approximation of the Nash equilibrium.

21

References

[1] T. Bayindir-Upmann and A. Ziad. Existence of equilibria in a basic tax-competition model. working paper, University of Bielefeld, 2001. [2] A. Devinatz. Advanced Calculus. Holt Rinehart Winston, 1968. [3] C. Figuieres, J. Hindriks, and G. Myles. Revenue sharing versus expenditure sharing. working paper 01/15, CORE, 2001. [4] J. Iritani and Y. Fujii. A note on non–existence of equilibria in tax competition models. unpublished note, Graduate School of Economics, Kobe University, July 2002. [5] D. Laussel and M. Le Breton. Existence of nash equilibria in fiscal competition models. Regional Science and Urban Economics, 28:283–296, 1998. [6] P. Rothstein. Discontinuous payoffs, shared resources, and games of fiscal competition : Existence of pure strategy nash equilibrium. working paper, Department of Economics, Washington University in St. Louis, August 2002. [7] D. Wildasin. Interjurisdictional capital mobility : Fiscal externality and a corrective subsidy. Journal of Urban Economics, 25:193–212, 1989. [8] J. Wilson. A theory of interregional tax competition. Journal of Urban Economics, 19:296–315, 1986. [9] J. Wilson. Theories of tax competition. National Tax Journal, 52:269–304, 1999. [10] G Zodrow and P. Mieszkowski. Pigou, tiebout, property taxation and the underprovision of local public goods. Journal of Urban Economics, 19:356–370, 1986.

22