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 More recently, in an as yet unpublished paper, Bayindir–Upmann and Ziad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 Another unpublished paper, Iritani and Fuji ( 2002 ), provides an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
of labour inelastically. All agents are endowed with the same quantity of capital k̄. The
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) = k̄.
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 = k̄
(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) + rk̄
(6)
x(z) = f [φ(R(z))] − R(z)φ(R(z)) + rk̄
(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 = k̄
(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 technology, any jurisdiction’s payoff will be u[f (φ(R) − Rφ(R) + rk̄, (R − r)φ(R)] if the “national”
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 = k̄
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
f 0 (k̄) ∈ R∗ [f 0 (k̄) − t]
7
(13)
PROOF : Suppose that there exists such a t. Set r = f 0 (k̄) − t, so that k̄ = φ(r + t).
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
implies that φ(R) = k̄, or f 0 (k̄) = R. Condiiton ii then requires that f 0 (k̄) ∈ R∗ (r)
for some national net return to capital r, and condition (13) will be satisfied by setting
t ≡ f 0 (k̄) − r.
•
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 k̄.
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.
From Lemma 2, there will exist a symmetric equilibrium if f 0 (k̄) ∈ R∗ (r) for some
non–negative r. if R∗ (r) were single–valued, and if it were continuous, then a symmetric
equilibrium would exist if R∗ (0) < f 0 (k̄), and if R∗ (r) > k̄) for large enough r. Assumptions
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
graph jumps discontinuously at some r. If f 0 (k̄) is in the interior of the jump, and is not
in R∗ (r) at the jump, there is no symmetric Nash equilibrium. For example, in figure 1,
∗
∗
(r̂), where the subscripts − and + denote
(r̂) < f 0 (k̄) < R+
R∗ (r̂) has two values, with R−
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
horizontal line R = f 0 (k̄) at most once, and the graph jumps instead of crossing.
Formally, let r̂ be the unique r at which this crossing occurs.
r̂ ≡ sup{r|R(r) < f 0 (k̄) ∀R ∈ R∗ (r)}
Then
LEMMA 5 : There will be a symmetric equilibrium if and only if f 0 (k̄) ∈ R∗ (r̂).
PROOF : If f 0 (k̄) ∈ R∗ (r̂), then let t ≡ f 0 (k̄) − r̂, and Lemma 2 applies.
On the other hand, if there is a symmetric equilibrium, Lemma 2 implies the existence
11
of a common tax rate t such that f 0 (k̄ ∈ R∗ [f 0 (k̄) − t]. Then the definition above implies
that f 0 (k̄) − t = r̂.
•
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.
PROOF : If f 0 (k̄) ∈ R∗ (r̂), then there is a symmetric equilibrium, in which every
jurisdiction chooses a tax rate of f 0 (k̄) − r̂.
If f 0 (k̄) ∈
/ R∗ (r̂), 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∗ (r̂), R∗ − (r̂) and R+
(r̂), with R−
(r̂) <
∗
f 0 (k̄) < R+
(r̂). Consider then a situation in which a proportion m of the jurisdictions
∗
∗
choose the tax rate R−
(r̂) − r̂ and a proportion 1 − m choose R+
(r̂) − r̂. By construction
∗
∗
φ[R+
(r̂)] < k̄ < φ[R−
(r̂)]
(15)
so that there is some m, strictly between 0 and 1, such that
∗
∗
mφ[R−
(r̂)] + (1 − m)φ[R+
(r̂)] = k̄
(16)
Equation (16) says that the national net return to capital r̂ will clear the capital market,
∗
if a proportion m of the jurisdictions choose the tax rate R−
(r̂) − r̂ and the remainder
∗
∗
∗
choose the tax rate R+
(r̂). The fact that both R−
(r̂) and R+
(r̂) are in the best response
set R∗ (r̂) means that each of the jurisdictions is willing to choose the tax rate it does.
∗
∗
(r̂) for m < z ≤ 1
So in this case r = r̂, R(z) = R−
(r̂) for 0 ≤ z < m, and R(z) = R+
satisfies both conditions for an equilibrium.
Therefore, if there is no symmetric equilibrium at the net return r̂ at which the graph
of R∗ (r) jumps the horizontal line R = f 0 (k̄), then there will be an asymmetric equilibrium.
•
12
The two cases described in the proof of the theorem need not be mutually exclusive.
It might be the case that R∗ (r̂) had three or more elements, one of which was f 0 (k̄), one
of which was greater than f 0 (k̄), and one of which was less than f 0 (k̄). In such a situation
there would be a symmetric equilibrium, as well as one or more asymmetric equilibria.
However, the monotonicity of R∗ (r) implies
PROPOSITION 2 : If the best reaction correspondence R∗ (r) is single–valued at r̂ 7 ,
then the only equilibrium is a symmetric one.
For any net return r, any R in the best response correspondence R∗ (r) must satisfy
the first–order condition (14). This condition is necessary ( but not sufficient ) for the gross
return R to maximize u[x(R, r), g(R, r)]. Hence any characterization of the equilibrium to
the tax competition game when there is a large number of identical jurisdictions in terms
of the first–order conditions will be correct.
In particular, equation (14) implies that the marginal rate of substitution ug /ux must
exceed 1. This “under–supply” condition must hold in equilibrium, whether the equilibrium is symmetric or not.
2. A Finite Number of Jurisdictions
Suppose now that the nation consists of a finite number N of jurisdictions. The
technology, distribution of endowments, and tastes are all the same as in the continuum
model of section 1. When N < ∞, each jurisdiction’s fiscal policy has an influence on the
national net return to capital r. The same letters will be used to denote variables ( such as
consumption levels and tax rates ), except that they now will be indexed by the finite set
{1, 2, . . . , N }. If t is a vector of tax rates ( not necessarily the same in each jurisdiction ),
7
which is equivalent to the graph being continuous at
13
r̂
then the allocation of capital to the jurisdictions and the net return are now determined
from the system of equations
f 0 (r + ti ) = ki
N
X
i = 1, . . . N
ki = N k̄
(17)
(18)
i=1
( where the subscript i now indexes jurisdictions ). For a given vector of tax rates, the
system (17) and (18) determines the ki ’s and the net return r uniquely. The resident of
jurisdiction i chooses a tax rate ti to maximize u(xi , gi ) subject to
xi = f (ki ) − f 0 (ki )ki + rk̄
(19)
gi = ti ki
(20)
where ki and r both depend on the jurisdiction’s own tax rate ti , through (17) and (18).
The first–order condition for jurisdiction i is
[(k̄ − ki )
ug
∂ki
∂r
− ki ] +
[ki + ti
]=0
∂ti
ux
∂ti
(21)
which is a condition which has been used widely in the literature to analyze the issue of
under–provision of public output under tax competition.
A Nash equilibrium to the simultaneous–move, non–cooperative tax setting game
played by the N jurisdictions is a vector of tax rates (t1 , t2 , . . . , tN ) such that each ti maximizes u(f (ki )−f 0 (ki )ki +rk̄, ti ki ) given the other N −1 tax rates (t1 , t2 , . . . , ti−1 , ti+1 , . . . , tN ),
and given that ki and r are determined by equations (‘egret’) and (18). The Nash equilibrium is symmetric if all jurisdictions choose the same tax rate ti = t∗ in equilibrium.
Equation (21) must hold for each jurisdiction i at any Nash equilibrium.
When all tax rates are equal, so that capital is allocated equally to all jurisdictions,
equations (17) and (18) imply that
∂r
1
=−
∂ti
N
14
∂ki
N −1 1
=
∂ti
N f 00 (k̄)
which simplifies equation (21) into
N − 1 t∗
ug
(f (k̄) − t∗ k̄, t∗ k̄)[k̄ +
]=0
ux
N f 00 (k̄)
(22)
The following result is certainly not new,8 but is certainly important in the analysis of the
existence of equilibrium.
LEMMA 6 : Under assumptions 1 and 2, there is exactly 1 tax rate t∗ in (0, f 0 (k̄))
which satisfies equation (22).
PROOF : Consider the left side of the equation as some function g(t) of the N jurisdictions’ common tax rate t. ( That is, consider the marginal incentive for a jurisdiction
to raise its tax changes, in response to a coordinated change in all the jurisdictions’ tax
rates. ) Assumptions 1 and 2 imply that g(0) > 0, and that g(f 0 (k̄)) < 0, so that continuity
of g(t) implies that there must be at least one t∗ in (0, f 0 (k̄)) satisfying (22).
As t increases, x decreases and g increases, along the line x + g = f (k̄). Quasi–
concavity of preferences implies therefore that ug /ux falls as t rises. Since f 00 < 0, t/f 00 (k̄)
must decrease as t increases.Therefore, if
[k̄ +
N − 1 t∗
]>0
N f 00 (k̄)
(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
∞
∞
]=0
(f (k̄) − t k̄, t k̄)[k̄ + 00
ux
f (k̄)
(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
1, suppose that r = f 0 (k̄) − t∞ . Then R = f 0 (k̄) satisfies the first–order condition (14) to
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))].
If r = f 0 (k̄) − t∞ , and if R = f 0 (k̄), then equations (24) and (14) are the same.
•
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
at r̂, then there is some N̄ , so that there is a symmetric Nash equilibrium to the finite
model, when the number of jurisdictions N is N̄ or more.
PROOF : In the continuum model, choosing R and choosing t are the same problem.
A gross return R maximizes u(f (φ(R)) − Rφ(R) + rk̄, (R − r)φ(R) if and only if a tax rate
t = R − r maximizes u(f (φ(r + t)) − (r + t)φ(r + t) + rk̄, tφ(r + t)). If the best reaction is
single–valued at r̂, then R = f 0 (k̄) is strictly preferred to any other R when r = r̂, which
is the same as t∞ being strictly the best tax rate when r = r̂.
16
Theefore, if R∗ (r) is single–valued at r = rr̂, then t = t∞ will be the unique maximizer
of u(f (φ(r̂ + t)) − (r + t)φ(r̂ + t) + rk̄, tφ(r̂ + t)).
Further, if all jurisdictions ( except a set of measure zero ) chose tax rates t(z) = t∞
in the continuum model, then the national net return would equal f 0 (k̄) − t∞ = r̂.
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 r̂, 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
situation depicted in figure 2, in which R = f 0 (k̄) is one of three elements in R∗ (r̂) 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 equilibrium in the following sense. For any small > 0,there is some N̄ () such that M < 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 .
If R∗ (r̂) were multiple valued, but f 0 (k̄) were in R∗ (r̂), then t∞ would be one of the
tax rates which maximized any jurisdiction’s payoff when r = r̂. 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.
Finally suppose that f 0 (k̄) ∈ R∗ (r̂). From the construction of r̂, there must be a jump
in R∗ (r) at r = r̂ such that R+ > f 0 (k̄) and R− < f 0 (k̄) are both in R∗ (r̂). 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− − r̂ and a proportion
1 − m choose t+ ≡ R+ − r̂.
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