Games without Borders with Many Players
S. Bucovetsky
Economics, LAPS, York University
Toronto ON M3J 1P3 Canada
October 20, 2016
Abstract
If agents find all jurisdictions — other than their home jurisdiction —
to be equally good substitutes, then competition to attract these agents
becomes very sharp. This note extends the Kanbur–Keen model to more
than 2 jurisdictions, under this assumption that all outside jurisdictions
are equally good substitutes. Under this assumption, no Nash equilibrium
exists in pure strategies if there are more than 2 jurisdictions.
Keywords tax competition, Nash equilibrium
JEL Classification H73, H77
1
Introduction
The Kanbur–Keen (1993) [henceforth KK] model of commodity tax competition is a wonderfully tractable and intuitive model of fiscal competition. The
key features of this model1 are that individual agents make an all–or–nothing
decision whether to shift their activities to a lower–tax jurisdiction, and that
the cost of that shift differs among agents.
The cost of shifting can be interpreted as a travel cost, so that commodity taxation among contiguous jurisdictions can be explained. It also can be
interpreted as a metaphor2 for other costs of shifting activities to lower tax
liabilities.
In this note, I take the latter interpretation, that there is no explicit spatial
structure, but agents vary in their costs of shifting activities.
For example, consider the choice between purchasing an item in a bricks–
and–mortar store near one’s residence, and purchasing the item online. Distance
does not matter very much for online purchases3 . Both the delivery costs,
and the time lag in receiving the item, seem minimally related to the distance
1 which distinguish it from the Wilson–Zodrow–Mieszkowki–Wildasin model of competition
for mobile capital
2 see particularly Keen and Konrad (2013)
3 within a federation such as the EU or the United States
1
from the vendor. Shopping online is costly. There is the time cost of finding
vendors on the internet, the cost of waiting for delivery, the disutility of a more
complicated process for refunds or repairs. These costs differ across people,
depending on their impatience, their degree of technological sophistication, and
the subjective probability they attach to a product failure. But these costs do
not seem to vary with distance from the vendor.
The costs also do not vary much with the jurisdiction in which the vendor
is located.
An implication of these claims is that the notion of a “contiguous jurisdiction” may not apply in many interpretations of the Kanbur–Keen model.
There have been several extensions of the Keen–Kanbur model to more than
two jurisdictions.4 But these extensions use a spatial interpretation : they are
based on an explicit locational model, in which costs of shifting are the travel
cost to the nearest competing jurisdiction. In a one–dimenional model, that
means that each jurisdiction is competing with one or two contiguous jurisdictions : there may be many jurisdictions, but jurisdiction i’s tax base depends
only on the tax rates of its one or two neighbors.5
Here I assume that all outside jurisdictions are equally attractive, should an
agent choose to shift. That is, an agent from country i can pick the lowest–cost
alternative to her home country. There is a cost to exercising this option, and
the cost varies among agents. But every agent in every country who exercises
this option will pick the same country to which to shift her transaction — the
lowest–cost country.
What this specification implies is very sharp Bertrand competition for agents
who choose to shift. If country i’s tax rate is above the lowest of the other
countries’ tax rates, then it will attract no cross–border commerce. If it lowers
its tax rate by 2, then it attracts all of the cross–border commerce.
As in in many other models of Bertrand competition, this sharp competition
for cross–border commerce destroys the stability of the model. The point of this
note is to demonstrate that there can be no Nash equilibrium6 in pure strategies,
when there are 3 or more identical (ex ante) jurisdictions, and when the lowest–
tax jurisdiction attracts all the world’s cross–border commerce.
2
The Model
There are N identical countries. Each resident of each country buys 1 unit of
the good, either in her home country or from the cheapest of the other N − 1
countries. Each country chooses (non–cooperatively) a destination–based unit
4 see
Ohsawa (1999) and Agrawal (2015)
in the Hotelling model, the possibility exists that a non–contiguous jurisdiction’s tax
rates are so low that some customers are willing to travel more than the length of the neighboring country to shop there. But in equilibrium, tax differences will not be that large.
6 Gabszewicz et al (2016) also show that there may be no Nash equilibrium if the Kanbur–
Keen model is modified slightly (in this case, to add another dimension of heterogeneity
between jurisdictions).
5 As
2
tax on the good. Countries set tax rates non–cooperatively, seeking to maximize
total tax revenues.
There is a cost δs to a consumer of buying outide one’s own country, where
s varies across consumers, and measures the ease with which they can shop
outside their home country. As in KK, δ, which measures the overall height
of barriers to cross–border shopping, is the same for all countries. Also as in
KK, within each country consumers’ ease s of shopping elsewhere is distributed
uniformly.
This structure means that, of the N countries, only those with the lowest
tax rates attract any foreign business. From the perspective of a single country
i, let t∗ denote the lowest of the N − 1 tax rates levied elsewhere. If country i
chooses a tax rate Ti > t∗ , then total tax revenue in the country is
R(Ti , t∗ ) = Ti h[1 −
t∗ − Ti
]
δ
if
Ti > t∗
(1)
exactly as in equation (4a) of KK, where (as in KK) h is the total number of
residents in the country (the same for each country).
Since R(Ti , t∗ ) defined in equation (1) is a concave function of Ti , there is
at most 1 solution to the maximization of R(Ti , t∗ ) subject to Ti ≥ t∗ .
Therefore, if any country i chooses a tax rate Ti which is strictly higher than
the lowest tax rate chosen by the different countries, then any other country j
which chooses a tax rate higher than the lowest must choose the same Ti .
OBSERVATION 1 In any pure–strategy equilibrium, there are at most two
distinct tax rates chosen.
There is no product differentiation here. If any resident chooses to shop
abroad, she will shop in the lowest–tax foreign country, even if that country’s
tax rate is only less than the next–lowest tax rate.
Because of this extreme sensitivity to small tax differences, the usual Bertrand
undercutting would occur. If countries i and j were tied for the lowest tax rate,
then one of the countries would earn strictly higher revenue if it lowered its tax
rate by some small , doubling its sales to foreign customers with an infinitessimal loss in tax revenue per customer.7
Therefore
OBSERVATION 2 There are only two types of pure–strategy Nash equilibrium : (i) all N countries set the same tax rate ; (ii) N − 1 countries set the
same tax rate as each other, and the remaining country sets a lower tax rate.
But, if N > 2 there cannot be an equilibrium in which all countries chose
the same tax rate. Consider the decision of country i. If the remaining N − 1
countries chose the same tax rate T , country i faces exactly the problem faced by
7 In this paragraph, I have assumed that foreign business will be split evenly among the m
countries with the lowest tax rate, if m > 1. This assumption is not needed.
3
one of the 2 countries in KK : the N −1 other countries can be aggregated into a
single country with a tax rate of T . That aggregate country is N −1 times larger
than country i. Therefore, country i’s problem is exactly the maximization
defined in equation (4) of KK, with H = (N − 1)h.
Equation (5) of KK, and figure 2A, show that, if H > h, it is never a best
response for country i to choose a tax rate equal to T , the common tax rate in
the other N − 1 countries.8
So
OBSERVATION 3 The only possible pure–strategy Nash equilibrium when
N > 2 involves N − 1 countries setting a common high tax rate T , and one
country choosing a lower tax rate t.
In particular, the equilibrium tax rates must be those defined in KK for
an asymmetric 2–country world9 , in which the “large” country is N − 1 times
larger than the small country10 :
t=
δN +1
3N −1
(2)
T =
δ 2N − 1
3 N −1
(3)
The tax revenue in each of the N − 1 high–tax countries is T h[1 −
which (from equations (2) and (3)) equals
R=[
2N − 1 2 h
]
N −1 9
T −t
δ ],
(4)
The single low–tax country gets revenues of th[1 + (N − 1) T δ−t ], which equals
r=
(N + 1)2 h
N −1 9
(5)
The single low–tax country must earn higher tax revenue than the N − 1
high–tax countries, if N > 2.
However, the tax rates defined by equations (2) and (3) constitute a Nash
equilibrium only if no country wants to deviate unilaterally.
What would happen in one of the large countries were to switch its tax rate
from T to t−?11 If it did so, then it would attract all the cross–border shoppers
from the N − 2 remaining high–tax countries — and no cross–border shoppers
∂r(t,T )
8 Put otherwise, equation (4) in KK implies that
must jump discontinuously upward
∂t
at t = T .
9 Kanbur and Keen (1993), equation (8)
1
10 so that, in equation (8) of KK, θ =
N −1
11 Such a deviation might not be the country’s best strategy. But if it is a better strategy
than the tax rate T , the original set of tax rates cannot be Nash equilibrium. That is, a
necessary condition for the existence of a pure–strategy Nash equilibrium is that undercutting
to t − not be profitable. It may not be a sufficient condition.
4
from the other low–tax country. Therefore, if it deviated to a tax rate just below
t, its revenues would approach th[1 + (N − 2) T δ−t ], or
Rdev =
h
N +1
[N 2 + N + 1]
2
(N − 1)
9
(6)
There can be a Nash equilibrium in pure strategies only if no high–tax country wishes to deviate to a tax rate of t − , only if Rdev ≤ R.
But Rdev > R when N = 3. Moreover, (N − 1)2 [Rdev − R] is an increasing
function of N when N ≥ 3, so that Rdev > R whenever N > 2.
Therefore
Proposition 1 When there are 3 or more identical countries, and when cross–
border shoppers all shop in the cheapest country, there can be no Nash equilibrium in pure strategies to the non–cooperative game played by revenue–maximizing
governments.
3
Asymmetry
The non–existence result does not depend on the assumption that all N countries have the same population.
Observations 1, 2 and 3 did not depend on the assumptions that countries
were equal in size : even with size differences, concavity of revenue functions,
and the temptation to undecut, imply that the only possible pure–strategy Nash
equilibrium is a situation in which one of the countries chooses a lower tax rate,
and in which the remaining N − 1 countries choose the identical higher tax rate.
If countries differ in size, then the high–tax country which will gain the most
from deviation to a tax rate of t− must be the smallest of the high–tax countries
(since it will gain the most revenue, relative to its population, by undercutting
the low–tax country).
So, given the population of the one low–tax country, the configuration of the
other countries which is the least susceptible to deviation by undercutting is
a population distribution in which all of the high–tax countries have the same
population.
Let si be the share of the total population in country i. Consider a population configuration in which country 1 chooses the low tax t, and countries
2, 3, . . . , N choose the high tax T . The argument above shows it is sufficient to
consider population distributions in which s2 = s3 = · · · = sN = (1−s1 )/(N −1)
: if there is no Nash equilibrium (with country 1 choosing the lowest tax) for that
population configuration, then there will be no Nash equilibrium (with country 1 choosing the low tax rate) for any other distribution of the population of
countries 2, 3, . . . , N .
In this configuration, the only possible Nash equilibrium in which country 1
chooses the lowest tax rate is (from equation (8) of KK) one in which country
5
1 chooses
t=
δ
s1
[1 + 2
]
3
1 − s1
(7)
and in which the identical countries 2, 3, . . . , N choose
T =
δ
s1
[2 +
]
3
1 − s1
(8)
As in the previous section, t1 = t and t2 = t3 = · · · = tN = T can be a Nash
equilibrium only of none of the identical high–tax countries wants to deviate to
a tax rate of t − .
Tax revenue for a high–tax country, T [1 − T δ−t ] is
R=
(2 +
s1
2
1−s1 )
9
(9)
If a country did deviate, it would steal from country 1 all the cross–border
shopping revenue from consumers in countries 2, 3, . . . , N , getting total revenue
of
T −t
Rdev = t(1 + (N − 2)
)
(10)
δ
If s1 ≤ 0.5, and N > 2, it must be the case that Rdev > R12 .
Therefore, the non–existence result of the previous section holds for any
distribution of population among the n countries.
Corollary 1 When there are 3 or more countries, and when cross–border shoppers all shop in the cheapest country, there can be no Nash equilibrium in pure
strategies to the non–cooperative game played by revenue–maximizing governments.
12 this
inequality is derived in the appendix
6
References
[1] D. Agrawal. The Tax Gradient : Spatial Aspects of Fiscal Competition.
American Economic Journal : Economic Policy, 7(2):1 – 29, 2015.
[2] J. Gabszewicz, O. Tarola, and S. Zanaj. Migration, Wages and Income
Taxes. International Tax and Public Finance, 23(3):434–453, June 2016.
[3] R. Kanbur and M. Keen. Jeux Sans Frontières : Tax Competition and Tax
Coordination when Countries Differ in Size. American Economic Review,
83:877–892, 1993.
[4] M. Keen and K. Konrad. The Theory of International Tax Competition
and Coordination. In A. Auerbach, R. Chetty, M. Feldstein, and E. Saez,
editors, Handbook of Public Economics, volume 5, chapter 5, pages 257 –
328. Elsevier, 2013.
[5] Y. Ohsawa. Cross–Border Shopping and Commodity Tax Competition
among Governments. Regional Science and Urban Economics, 29(1):33 –
51, 1999.
7
4
Appendix : Derivation
Suppose that country 1 is the low–tax country, and that countries 2, 3, . . . , N are
the high–tax countries, with countries 2, 3, . . . , N having identical population,
and with the ratio of country 1’s population to the population of the rest of the
s1
.
world being θ ≡ 1−s
1
Without loss of generality, the population of country 1 can be set equal to
1, and the cross–border shopping cost parameter δ also set equal to 1.
From equations (8) and (10) of KK, the only posible equilibrium in which
country 1 is the low–tax country has country 1 choosing the tax rate
1 + 2θ
3
t1 =
(11)
and all the other countries choosing the same tax rate
2+θ
3
(12)
(2 + θ)2
9
(13)
T =
resulting in revenue per person of
R=
If instead a country deviated to a tax rate of t1 , it would attract all the cross–
border shopping from the remaining N − 2 high tax countries, yielding revenue
of
(1 + 2θ)[3 + (N − 2)(1 − θ)]
Rdev = t1 [1 + (N − 2)(T − t1 )] =
(14)
9
The country will want to deviate if Rdev > R, or if F (θ) > 0, where
F (θ) ≡ (1 + 2θ)[3 + (N − 2)(1 − θ)] − (2 + θ)2
(15)
The function F (θ) is quadratic.
F 0 (θ) = 1 + 2(N − 2)(1 − θ) − 4θ
(16)
so that F 0 (0) > 0 whenever N ≥ 2.
Since F (0) = N − 3, and F (1) = 0, the function F (θ) must be greater than
or equal to 0 for every θ ∈ [0, 1] whenever N ≥ 3.
If countries 2, 3, . . . , N are not identical, then take the smallest of those
countries. Since the other N − 2 high–tax countries are now larger than it, the
payoff per person to that country, should it choose a tax rate of t1 − , will be
strictly higher than t1 [1 + (N − 2)(T − t1 )] — it would be t1 [1 + P (T − t1 )] where
P is the ratio of the population of the other N − 2 high–tax countries to the
population of the deviating country. Therefore, the payoff to deviation by the
smallest high–tax country would be strictly higher than Rdev − R.
8
If country 1, the low–tax country, were strictly larger than all of the other
N − 1 countries together, then θ > 1, so that it is not true that F (θ) > 0.
But when one country is larger than all of the other countries together, the
2–country result of KK applies : there cannot be a Nash equilibrium in which a
country with the majority of the population levies a lower tax than the others.13
13 In this case, the expressions for t , T , R and Rdev would not be valid : equations (11)
1
and (12) would imply t1 > T if θ > 1.
9