Discussion Paper
No. 0134
Internationally Mobile Factors of Production
and Economic Policy in an Integrated
Regional Union of States
Perry Shapiro
Jeffrey Petchey
August 2001
Adelaide University
Adelaide 5005 Australia
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CIES DISCUSSION PAPER 0134
Internationally Mobile Factors of
Production and Economic Policy in an
Integrated Regional Union of States
Perry Shapiro
Jeffrey Petchey
Centre for International Economic Studies
University of Adelaide
[email protected]
August 2001
Internationally Mobile Factors of Production and Economic Policy in an
Integrated Regional Union of States
Perry Shapiro1, Jeffrey Petchey2
Abstract
There is a general class of model used to examine the expenditure and tax strategies
adopted towards mobile factors by states that are members of regional unions (eg.,
federations, confederations or common markets). We develop a variant of this model,
extended to allow for imperfect factor mobility between the region and the world, and
characterize a game between states in which tax and expenditure policies are the
strategies. It is shown that a Nash equilibrium to this game exists (though there may be
multiple equilibria). We also argue that, unlike previous results, states have an incentive
to levy non-zero taxes, or subsidies, on the migrant factor (with the precise mix of
strategies depending on the degree of factor mobility). Generally, the strategies adopted
lead to an inefficient supply of the factor to the region, as well as an inefficient
distribution across member states.
Keywords: regional unions, tax competition, economic integration, common factor
market, labor and capital mobility, externalities, taxes, subsidies, public goods.
JEL Classifications: H21, H41, F15, F20.
1
Department of Economics, University of California, Santa Barbara, California, and Department of
Economics, University of Melbourne: Email: [email protected]: Phone: (international) 61 3 9621
2718.
2 (Corresponding author): Federalism Research Program, School of Economics and Finance, Curtin
University of Technology, GPO Box U 1987, Perth, Western Australia, 6845, Australia: Email:
[email protected]: Phone (international) 61 8 92667408.
1.
Introduction
Regional unions of states include existing federations and emerging
confederations such as the European Union. Both forms of union are characterized by
rules that prohibit member states from impeding the movement of factors of production
between member states. In other words, they have common internal factor markets.
Usually, member states have retained sufficient sovereignty over public expenditure and
tax instruments to allow them to influence, through their policy choices, the distribution
of mobile factors within the common market.
The efficiency of policy choices made by states within these unions has been
examined extensively using a “fiscal competition” model that, while varying according to
the particular application, has certain general features. The model supposes that there are
two states, each endowed with a concave production technology with at least two factors
of production, identical immobile citizens (workers) and a second factor (capital or labor)
that is mobile between states but whose supply to the regional union is fixed. In other
words, the union is assumed to have a closed common factor market. States levy a tax on
the mobile factor, and sometimes citizens as well, to provide a public good which
benefits the immobile citizens. The mobile factor migrates between states to satisfy an
equal net return condition. Competitive factor markets are assumed so that each factor
receives a monetary reward equal to the value of its marginal product. The mobile factor
may, or may not, be owned by domestic citizens (absentee ownership can be allowed).
States choose their policies to maximize a payoff function (often the utility function of
the immobile citizens) subject to budget constraints and the mobile factor equal return
condition. The structure implies interdependence between the policy choices of the states
that is modeled by adopting the Nash equilibrium concept. The policies chosen by states
are commonly shown to be inefficient.
Two papers that employ aspects of this general structure are the ones by Wildasin
(1991a) and Wellisch and Wildasin (1996). The first of these supposes a decentralized
economy of two states each with two types of labor; one rich immobile group and a group
of poor mobile workers who migrate between states to equate per capita utility. Since the
total supply of mobile workers is fixed the model is one of a closed common labor
market.
The rich control the public choice process so that the interests of each
jurisdiction are those of the rich class. Further, the rich are altruistic towards the poor
and hence each jurisdiction chooses some level of income redistribution in favor of the
mobile poor. It is argued that if sub-national governments choose this redistribution the
resulting Nash equilibrium is inefficient: redistribution creates fiscal externalities that are
ignored by competing jurisdictions.
Various coordination options and central
intervention are then examined and shown to be efficiency enhancing.
Wellisch and Wildasin (1996) extend this framework in at least two important
ways. First, they incorporate internationally mobile capital which migrates between
states and the world to satisfy a condition that its net rate of return should equal some
given world return. Second, the authors ‘open’ the common labor market by allowing for
the possibility of international migration of the mobile group.
This was a major
innovation to the general model of common markets within regional unions since, as
noted, until then the efficiency of inter-state competition had been examined within the
context of models with closed common factor markets. Wellisch and Wildasin again
characterize an inefficient Nash equilibrium in which competing sub-national
jurisdictions choose their policies for some exogenously determined and arbitrary number
of international migrants. It is shown that the optimal tax/subsidy on mobile capital is
zero. This mirrors the result in Oates and Schwab (1988). The authors also examine how
the Nash policies are influenced by changes in the arbitrary levels of international
migration. They find that more migration always raises social welfare if migrants are net
taxpayers and lowers social welfare if the migrants are a net fiscal burden.
A game in which states directly choose the level of immigration is also
considered.
Rather than model the game explicitly, Wellisch and Wildasin draw
conclusions about its possible features by referring to their results on the Nash
equilibrium obtained using the assumption of some exogenously given level of migration.
They conclude that if immigrants are net fiscal burdens the Nash equilibrium is one in
which there is no international migration. If migrants are net fiscal contributors some
2
positive level of immigration is chosen by jurisdictions. Finally, a "corrected" Nash
equilibrium is examined in which a central government provides matching transfers to
each jurisdiction (the policy instrument is the matching rate) that take account of the
fiscal externalities and achieve an efficient outcome. The authors also note that there is
no general existence proof for this class of model though one can construct examples,
using specific functional forms, where existence and uniqueness are assured3.
Braid (1996) considers a similar model with the addition of an ad valorem,
rather than a lump-sum tax, on the mobile factor.
He examines only the
symmetric Nash equilibrium, one in which all jurisdictions are the same.
His
analysis
examine
production
the
is
exploits
predicted
perfectly,
the
similarity
outcome.
and
He
freely,
in
equilibrium
also
mobile
assumes
policy
that
the
internationally.
choices
to
factor
of
It
allows
him, as it has previous analysts, to assume a constant return to the mobile
factor of production.
We develop a model of a regional union in the spirit of the general model
discussed above. The model is of a two-state region with free migration of a mobile
factor between states. An innovation of the work is to incorporate a separate mobile
factor supply condition that also allows the total supply of the mobile factor to the region
to vary depending on the payoff it receives within the region relative to the world payoff.
Unlike previous studies, we allow this migration to be imperfect due to migration
frictions. Therefore, in equilibrium it is possible for the regional payoff to differ from the
world payoff. The migration friction is captured through a parameter , ε, that we take to
be a measure of the level of ‘integration’ between the region and the world.
The
parameter can vary from zero (a completely isolated region) to infinity (a fully integrated
region). Using this basic structure, we are able to develop a game with policies as
strategies and characterize a Nash equilibrium in which the total supply of the mobile
factor to the region, and its regional distribution, are determined by the competitive tax
and expenditure policies of the states.
3
3
Some examples are provided in Wildasin (1991b).
The first contribution of the paper, though not a direct consequence of the
inclusion of the mobile factor supply function, is to provide a general existence proof for
a Nash equilibrium. As noted above, no proof of global existence has been developed for
this class of models. The existence result is an example of the well-known Gale MasColell Theorem (1975). It proceeds by transforming the game with policies as strategies
into a game with the desired mobile factor supplies as strategies.
Existence of
equilibrium for this “dual”game is assured by the application of well-known fixed point
results. Every choice of mobile factor supply is associated with a “best” feasible policy.
Thus, the existence of an equilibrium in the dual game ensures the existence of an
equilibrium in the “primal” game, the one in which the strategies are the states’ tax and
expenditure policies.
The second result is to show that states have an incentive to choose a non-zero
mobile factor tax/subsidy in equilibrium. This is in contrast to the findings in Wellisch
and Wildasin, and Oates and Schwab. The non-zero tax/subsidy is a consequence of two
factors. The first, that is independent of the degree of integration, is an externality
generated by the migrating factor. It tends to make the tax positive or negative (a
subsidy) depending on the sign of the externality. The second, which is dependent on the
degree of integration as captured by ε, we call a ‘hiring cartel’ effect. If ε is less than
infinite (less than full integration) states have some monopoly power over the mobile
factor and the hiring cartel effect has a positive impact on the tax/subsidy. States will
wish to use their monopoly power over the mobile factor to tax it and redistribute in favor
of immobile domestic labor. Whether states levy a tax or subsidy depends on the
interaction between the externality and the hiring cartel effects.
Therefore, the zero mobile factor tax result in previous papers is a special case in
our model where the externality is zero and the regional common factor market is fully
integrated with the world factor market. Allowing for migration frictions across the
common external border, and externalities, one derives the result that states will levy nonzero mobile factor taxes/subsidies.
4
The third result is to show that the magnitude of the tax/subsidy depends upon the
value of ε , the integration parameter. Specifically, higher degrees of integration imply a
smaller hiring cartel effect. As states are exposed to more international competition for
the mobile factor, their monopoly power diminishes. In the limit, where there is full
integration, the tax/subsidy is determined only by the externality.
Finally, we are able to generate some new insights into the efficiency of interstate competition within a regional union. It is important to note here that our treatment
of efficiency is somewhat different to the standard one. The usual approach, as discussed
in Wellisch (2000), has been to examine whether public goods are under or over
provided by states that tax a mobile factor. Rather than pursue this issue, we choose to
examine how the tax/subsidy policy of the states affects efficiency in the regional
distribution and total regional supply of the mobile factor. It is shown that if ε is less
than infinite, the monoply power of the states, exercised through the (non-zero)
tax/subsidy, creates a distortion in the distribution of the mobile factor within the region,
and inefficiency in its total supply. The presence of this distortion depends on the states
choosing a non-zero tax/subsidy, and this, in itself, is a result of the adoption of our more
general factor migration condition with imperfect mobility. In a sense then, we identify a
new source of inefficiency associated with policy competition between states. However,
in the special case where ε is infinite (full integration), the monopoly power of the states
disappears (the tax reflects only the externality) and the Nash equilibrium results in an
efficient distribution of the mobile factor and total supply.
The paper is organized as follows. Section 2 develops the basic set up of the
model. Section 3 examines the nature of the Nash equilibrium. It provides the general
proof of existence and characterizes the (non-zero) equilibrium choice of the tax/subsidy.
Section 4 examines the effect of greater integration on the tax/subsidy policies adopted by
states. In Section 5, the efficiency of these policies is examined while conclusions are
presented in Section 6.
5
2.
The Set Up of the Model
Consider a regional union (a federation or confederation) of two independent
states i =1,2 each with a population of immobile residents with such high attachment to
place that they never migrate between states, or between the region and the rest of the
world4. These residents own the land, fixed capital and means of production (firms) in
state i, and they each supply one unit of labor. Hence, they can be thought of as domestic
workers, capitalists or landowners. From now on we refer to them, for presentational
convenience, as homefolks, by virtue of their "stay at home" mentality.
There is also a generic mobile factor of production in state i, which can be thought
of as capital, or labor (e.g. guest or itinerant workers). Unlike the homefolks, this factor
migrates between states, as well as between the region and the rest of the world. The
quantity of the factor present in state i is denoted as n i . Since the number of homefolks
(and their labor supply), the quantity of land and the amount of (fixed) capital are given,
the strictly quasi concave production function of state i can be written as
fi (n i )
(1)
where fi' (n i ) > 0 and −∞ < fi'' (n i ) ≤ 0 .
Each unit of the mobile factor receives a return, w i , which is equal to the value of
its marginal product, fi' (n i ) . Implicit here is the assumption that there are many firms in
each state, and hence, that the market for the mobile factor is perfectly competitive. The
homefolks in state i receive the residual from production, which might include rent from
any of the fixed factors, or the return to their labor input (this return can be aggregated
with rents when homefolk labor is inelastically supplied). The total residual is
R i (n i ) = fi (n i ) − w i (n i )n i .
(2)
The mobile factor has a strictly quasi concave continuous payoff function,
p ( xi , qi ) ,
4
6
The results generalize to k states.
(3)
where qi is a local public good provided by state i for the benefit of the mobile factor,
x i = w i (n i ) − t i is the net monetary reward accruing to each unit of the mobile factor and
t i is a lump sum tax levied on each unit of the mobile factor. In the case of capital, qi
can be thought of as economic infrastructure offered by state i (e.g. port and rail
facilities). For migrant labor, qi might include the provision of services such as health,
education and welfare. The tax, which can be negative (a subsidy), zero or positive, is a
generic instrument designed to capture the policies that states use to impose monetary
costs on migrating factors, or provide them with subsidies.
Homefolks and the mobile factor contribute to the cost of providing the public
good. The per unit contribution of the mobile factor in state i is via the tax/subsidy
discussed above.
If the tax is positive then the mobile factor makes a financial
contribution to the public good but when the tax is negative (a subsidy) the mobile factor
benefits from the public good and is provided with a monetary subsidy. The contribution
of the homefolks in state i is
Ti = q i − t i n i .
(4)
Therefore, the total monetary reward to the homefolks in state i, denoted as
X i (n i ) , is equal to the residual together with the contribution from the mobile factor
(which is positive, zero or negative) less expenditure on the public good,
X i (n i ) = R i (n i ) + n i t i − qi .
(5)
There are multiple consequences from the inward migration of the mobile factor
into any particular state. For example, the homefolks are favored by an increase in the
economic residual, since R i (n i ) is increasing in n i . However, there may be broader
external benefits and costs from in and out migration of the mobile factor. In the case of
migrating labor, immigrants may bring with them cultural ideas that can enhance the
quality of life in the host country and skills that add to overall productivity. But the
cultural differences can be annoyances to the homefolks and immigration can also cause
increased congestion.
7
Also, while imported capital produces higher income for the
homefolks, and is a benefit to them, industrialization may strain social cohesion and
environmental quality can be compromised by increases in capital intensity.
We capture these benefits and costs with a net externality function, E i (n i ) , where
Ei , the net externality value in state i, is a strictly quasi concave continuous function of
the quantity of the mobile factor present. The homefolks’ additively separable pay-off
function, Pi (n i ) , depends on the monetary reward and the net externality5
Pi (n i ) = X i (n i ) + E i (n i ) .
(6)
As observed in the introductory comments, federations (at least democratic ones)
have common internal markets in which factors of production are free to move between
states without artificial restriction. Confederations often also have common internal
capital markets and, as with the declared aim of the European Union, an emerging
common labor market in which citizenship of one state guarantees region wide
citizenship.
To capture this feature of regions, we assume that the two states share a common
factor market in which the mobile factor moves without restriction between states to
maximize its payoff. As noted earlier, this is a standard way to model common regional
factor markets. The implication is that, for a given regional mobile factor supply, if the
pay-off is higher in state i than j then the mobile factor will move from j to i. This
depresses the return in state i and increases it in state j until payoffs are equal (there are
no regional arbitrage opportunities). Therefore, in equilibrium,
p ( w i (n i ) − t i , q i ) = p ( w j (n j ) − t j , q j ) .
(7)
Another important feature of regions that share a common factor market between
member states, including federations and confederations, is that labor and capital migrate
between the region and the rest of the world. There are various ways to capture this
process. In many studies, including Oates and Schwab, and Wellisch and Wildasin, the
5
We assume that the homefolks derive no benefit from the public good, which is provided purely for the
benefit of the mobile factor. It would also be possible to allow the homefolks to generate externalities for
the migrant factor. However, such an extension would add some complexity without changing the results
in any material way.
8
mobile factor equilibrium is achieved when the payoff is the same internally, as it is in
the rest of the world. Specified as a condition on relative payoffs, if p* is the world
factor payoff, the equilibrium requires p/p* = 1. This equilibrium condition is the result
of assumptions, either explicitly or implicitly made; first, that the mobile factor can move
costlessly across all borders, and second, the mobile factor is homogeneous, both in its
productivity, and regional preferences.
We maintain the productivity homogeneity condition implicit in this approach, but
drop the assumption that the mobile factor is alike in all other ways. While movement
between states, within the region, is assumed costless, migration in and out of the region
may not be costless. Factor owners may have regional preferences. For labor there is a
strong pull towards home. Familiar customs, language, as well as a desire for familial
propinquity, influence the readiness of labor to migrate. Owners of capital also display
an inclination to invest in familiar states. The desire to avoid uncertainties associated
with foreign investment is thought to explain the capital market home bias: investors will
accept a lower return from investments in their own country than those taken abroad.
The market manifestations of a home bias in both capital and labor markets are
equilibria with a difference between regional payoffs and those received in the rest of the
world. The potential for differences in tastes for the region, and/or differences in the ease
of movement in and out of the region is modeled here as a reservation payoff, denoted as
ρ. This is the value of the payoff required by an individual unit of the mobile factor,
relative to the world value, to locate within the region rather than outside it. Since the
international return is fixed, for convenience, and without loss of generality, it is assumed
to be one (p* = 1). With this condition, hence forth, we will be concerned only with the
local equilibrium payoff, p. The reservation value is expressed as a locational choice
condition: a factor is located in the region if the specific reservation value is no larger
than the local payoff: ρ ≤ p. If ρ > p, the factor will either emigrate from or remain out of
the region.
9
It is assumed that ρ is a continuous random variable with a distribution f (ρ) .
The proportion of the population of the mobile factor that either remains, or migrates to,
the region is
p
F(p) = ò f (ρ)dρ
(8)
0
The regional mobile factor supply is thus
n1 + n 2 = NF(p)
(9)
where N is the world supply.
The regional mobile factor supply is potentially a very complicated function of p.
It might depend on the higher order moments of the distribution, as well as its mean. For
that reason, it is convenient for purposes of analysis to choose a one-parameter
distribution. The Weibull distribution is often employed in the analysis of extreme
values. If ρ is distributed as Weibull, the regional mobile factor supply function is
ε
n1 + n 2 = N(1 − e−αp )
(10)
While not strictly an elasticity, in the common use of the term, ε performs much
the same function: the supply response to changes in the relative payoffs, depends on ε.
The response is not proportional to it, as it is with a true elasticity, but it nonetheless
varies positively with ε. We also think of the elasticity parameter ε as a measure of the
degree of integration between the region's market for the mobile factor, and the world
market.
In this scheme, ε = 0 implies no integration, 0 < ε < ∞ implies partial
integration while ε = ∞ indicates full integration (the factor is perfectly mobile between
the region and the world).
In the discussion in Section 4, we will examine two polar cases; ε = 0 and ε = ∞.
For ε = 0, the mobile factor does not move in or out of the region and from (10) it can be
seen that total regional supply of the mobile factor is fixed at N(1 − e − α ) . The parameter
α is determined by the relative size of the regional population. A small value of α is
consistent with the assumption that the region is small compared to the rest of the world
(and vice versa). For ε = ∞, the entire mass of the distribution is concentrated at ρ = 1.
10
This means that if p is smaller than 1, the region is denuded entirely of population, and if
p is larger than one, the entire world resides in the region. In this regard, the parameter ε
is similar to the common supply elasticity. As will be discussed later, when ε = ∞, the
only equilibrium is one in which the regional mobile factor payoff is equal to the given
world payoff (as discussed, this is assumed to be one). This, in turn, implies that the total
regional supply of the mobile factor is fixed at N(1 − e − α ) when ε is infinite.
3.
The Nature of an Equilibrium
We now characterize an equilibrium where it is supposed that the government of
state i is elected by the homefolks who maintain a numerical majority.6 Therefore, the
interests of state i are synonymous with those of the homefolks and qi and t i are chosen
by state i to maximize Pi , as defined by (6). There are two types of policy choice
interdependencies between states. First, through the internal equilibrium condition (7)
the policies of state i affect its own mobile factor supply n i and the mobile factor supply
in state j (i.e. n j ). Since state j's net income and mobile factor externality are functions
of n j this means that state i's policies affect the welfare of the homefolks in state j. Thus,
the model captures the direct effect that independent policies in a regional union of states
have on one another. Second, because state policies also affect the total supply of the
mobile factor through (10) there is an additional source of interdependence.
Interdependence raises the prospect that states will act strategically. To capture
this, we suppose that when making policy choices, states take as given the policies of the
other states (Nash behavioral conjectures). However, each state is assumed to assess
(accurately) the impact of its policies on the supply of the mobile factor within its own
borders, conditional on the (supposed) fixed value of the other state's policies. The state
takes account of these mobile factor supply changes in so far as they affect the well being
6 This might be due to migration outcomes or constitutional provisions that give a voting franchise to the
homefolks (e.g. landowners) only.
11
of the state's homefolks. In other words, each state within the region pursues its own selfinterest and ignores the broader regional effects of its policies7.
3.1
Existence of Equilibrium
Since it makes little sense to analyze the nature of a non-existent outcome, we
first examine the question of the existence of a Nash equilibrium of the non-cooperative
game implicit in the states’ policy choices. The difficulty we see for the problem as
stated is that the objective functions of the states may not be concave in the strategies.
Therefore, while there may be an optimum for both states, conditional on its neighbor’s
policy choices, the set of optima may be neither continuous nor convex. While this
potential problem does not rule out existence, we are not able to prove that, in general, or
for our particular problem, an equilibrium exists in the game described with tax/subsidy
and expenditure policies as strategies.
However, we find that we can prove the existence of a Nash equilibrium in a
game that is the “dual” of the game in policy strategies. The duality is such that the
existence of equilibrium of the dual game, so described, implies existence of a Nash
equilibrium of the “primal” game in which the strategies are tax/subsidy and public good
provision policies.
We start with the recognition that states, while they can make choices from the
entire set of feasible strategies, will choose only those that are least expensive and
achieve a desired supply of the mobile factor. While there may be many combinations of
taxes and public good levels that will induce a chosen ni, states will choose only that
policy combination that has the smallest cost. We show, in what follows, that state
specific mobile factor supply is a strictly quasi-concave function of the policy variables.
This being the case, it follows that there is a unique minimum cost combination of public
7
The impact that one state’s choice of policy has on another state’s welfare is often termed a fiscal
externality (see Wildasin (1988)). Our model set up implies that there is both tax/subsidy and policy
competition between states. Wildasin (1988) characterizes equilibria in a fiscal competition model where
states have one strategic variable, either public expenditure (a Z equilibrium), or a tax (a T equilibrium).
Further discussion of fiscal competition models with two strategic variables is provided in Wildasin and
Wilson (1991) and Wildasin (1991b).
12
policies that will produce a give ni response, given the desired level of mobile factor in
the neighbor state, n-i. The dual game is the one with desired mobile factor supplies as
strategies. We show what restrictions are necessary to insure that the state objective
functions are strictly quasi-concave in ni. Since the set of feasible n’s is convex, it
follows that a Nash equilibrium of the dual game with n’s as strategies, exists. Since
there is a unique mapping from desired n to state policies, it follows that an equilibrium
in the primal policy game exists as well.
In the model structure above, individual states have two policy instruments – a
tax, t, and a public good, q8.
which case it is a subsidy.
As stated, the tax can be either positive or negative, in
For the discussion of existence, we find it more natural to
think of the monetary instrument as a subsidy, s. When s is positive, it enhances the
mobile factor’s payoff and, when negative, it diminishes it.
There are three fundamental elements of the model developed above:
(i)
Mobile factor payoff is a strictly quasi concave continuous function of its income
and the public good.
Income is the wage, a declining function of mobile factor
population, plus the state subsidy,
p i = p( w i ( n i ) + s i , q i ) .
(ii)
(11)
A state’s mobile factor supply is an increasing function of its own factor payoff.
The internal migration equilibrium condition (7), can be used to derive state-specific
mobile factor supply as a function of its policies, conditional on its neighbor’s policies,
n i = n i (s i , q i | s − i , q − i ) .
(iii)
(12)
The payoff to a state is the sum of two strictly quasi concave functions of n i
Π i (n i ) = R i (n i ) + E i (n i )
(13)
less its expenditure on the public good, q, and its subsidy payout (tax collected), s.
8 The proof that follows is restricted to but one, constant marginal cost, non-monetary policy. It would
apply to a vector of publicly supplied goods and services, with constant or rising marginal cost, that are
either uncrowded or crowded.
13
We start our development of the dual game by showing that n i is a strictly
concave function of si and qi . From (11), one can show that pi is a strictly quasiconcave continuous function of si and qi for a constant value of n i 9.
p i = v i (s i , q i , n i | s − i , q − i )
(14)
We will use the concavity of vi (.) with constant n i to prove that the function ni( ) is,
itself, strictly quasi concave. For every value of ni , define its upper contour set as
UC i ( n | s −i , q −i ) ≡ {s i , q i | n i (s i , q i | s −i , q −i ) ≥ n} .
Lemma
The set UCi(n|s-i,q-i) is convex.
Proof: Choose policy values s 0i , q 0i and s1i , q1i on the boundary of UCi( n | s-i , q-i).
Define s λi , q λi = λ(s 0i , q 0i ) + (1 − λ )(s1i , q1i ), 0 < λ < 1 .
Because ni is monotonically
increasing in v i (si0 , q i0 , n | s − i , q −i ) = v i (s1i , q1i , n | s − i , q − i ) , and because vi (.) is strictly quasi
concave, v i (si0 , q i0 , n | s − i , q −i ) < v i (siλ , q iλ , n | s − i , q − i ) . The policy choice ( s λi , q λi ) is on the
boundary of an upper contour set of value of n greater than n . Thus, UCi(n|s-i,q-i) is
convex. //
Corollary: ni(si,qi|s-i,q-i) is a strictly quasi-concave function of si,qi.
Proof: This follows immediately from the convexity of its upper contour set.//
9
This can be shown as follows (where the i subscripts are deleted). For a constant n = ~
n , choose any two
+ s , q ) = p(w(n)
+ s , q ) . For 0 ≤ λ ≤ 1 , define
pairs of values, s q , and s q , such that p(w(n)
λ
λ
0
0
1
1
n ) + s , q ) . The strict quasi concavity of the mobile factor
w( ~
n ) + s , q = λ( w( ~
n ) + s , q ) + (1 − λ )( w ( ~
λ λ
~
payoff function implies that p( w ( n ) + s , q ) ≤ p( w ( ~
n ) + s 0 , q 0 ) . Thus, for fixed n the function p(.) is
0
0
1 1
strictly quasi concave in s and q.
14
0
0
1
1
The analysis above is characterized in Fig. 1. The pi function is drawn for a fixed
value of mobile factor supply of n i = n . p i is the mobile factor payoff that induces a
mobile factor supply of n . The line p i p i represents the combinations of si and qi (given
n ) that yield the (constant) payoff p i . The points si0qi0 and s1i q1i are points on the “iso-n”
line with the convex combination siλ q iλ represented by the dashed line between.
Fig. 1: Mobile Factor Payoff for n i = n
pi
p i
p i
si0qi0
s1i q1i
qi
si
0
For a given n i , the cost or price of each unit of the monetary subsidy is just $1.
Similarly, the cost or price of a unit of the public good is a constant amount, in this case,
$1. However, since a single unit of the public good benefits all units of the mobile factor
15
equally, its cost per unit of mobile factor is, 1/ n i .10 Thus, the total cost of a particular
policy choice with a factor supply of n i is (si +
qi
)n i . In si , qi space, these costs are
ni
represented by a linear iso-cost line that has a slope of −n i .
The iso cost line is displayed in Fig. 2, together with the convex iso-n curve from
Fig. 1. The point of tangency between the two yields the combination of policies that
achieves n i = n at minimum cost.
Fig. 2: Iso Cost Line and Iso n Curve
qi
q*i
n i = n
0
s*i
si
The strict quasi concavity of ni(si,qi|s-i,q-i) means that there is a unique minimum
cost policy choice for every value of ni. The efficient input combination (the inputs are
the subsidy and public good policies), and thus the minimum cost, depends on the desired
factor supply, n-i, of the neighbor. The minimum cost is then a function of the desired
choice of each state
10
If q is not produced at a constant marginal cost, and it is crowded, its per unit factor price would be
MC(q)/B(n) where B(n) represents crowding.
16
C i = C i ( n i | n −i ) .
(15)
We have now transformed the primal game with policy choices as strategies to its
dual in which the quantity of the mobile factor is the strategic variable for state i. In a
sense, the transformation restricts the original strategy set of policy variables to be only
those that produce the target level of factor supply at minimum cost. This is a plausible
restriction since it is always in a state’s interests to make cost minimizing choices in order
to maximize the payoff to citizens. The set of possible choices of n , Ω , is convex and
bounded. Where N is the world mobile factor population, Ω = {n ∈ R | n ∈ [0, N]} .
In the dual game, the objective of each state is a function of n i alone
Pi ( n i | n −1 ) = Π i ( n i ) − C i ( n i | n −i ) .
(16)
A Nash equilibrium, (n*i , n *− i ) ∈ Ω × Ω , is defined in the usual way, namely,
Pi (n *i | n*− i ) ≥ Pi (n i | n *− i ) ∀n i ∈ Ω ∀i .
(17)
Theorem (Existence): If Ci(ni , n-i) is a convex function of ni, a Nash equilibrium to the
policy game exists.
Proof: Since Π i (.) is strictly quasi concave, the convexity of Ci(.) implies that each
state’s objective function Pi(ni|n-i) is continuous and quasi concave. This game is played
by the choice of n from a compact convex set to maximize a strictly quasi-concave
objective function. This is sufficient for the existence of a Nash equilibrium11 [see Rosen
(1965), Border (1985), p. 90, Mas-Colell, et.al. (1995), p. 253].
It is well known that with constant prices, the cost function is convex if ni(.) is
strictly quasi concave. In the case here, even though n i (.) is strictly concave in si and
qi , the price of the public good declines with n i . Thus, as a state moves from an
equilibrium at q*i ,s*i (Fig. 2) and achieves a higher value of n i , the slope of the iso-cost
11
17
This Theorem is an example of the well-known Gale and Mas- Colell (1975, 79) Theorem.
line changes in such a way that reduces the relative price of using the public good versus
the subsidy. As the size of the mobile factor population increases, there will be a
tendency for states to substitute public goods for subsidies, or decreasing taxes. With a
sufficient mobile factor response to increases in public good supply, the cost function
may not be convex.
Nevertheless, a Nash equilibrium may still exist, even if the cost function is not
convex. Concavity of the objective function is a sufficient, not a necessary condition for
existence. Furthermore, the objective function Π i (n i ) can be concave in mobile factor
supply if the concavity of Ri(.) outweighs the non-concavity of –Ci(.),
that is if
R ' ' > C' ' for all n.
3.2
Uniqueness of Equilibrium
We have not been able to demonstrate uniqueness in general. However, there is
one value of the integration parameter, ε , where it is possible to show that the
equilibrium is unique. This is the case where ε is infinite (full integration). In this
instance, equilibrium imposes an additional constraint on states in the sense that the
mobile factor must receive a monetary return that is exactly equal to the world payoff
(assumed to be one). If the regional return to the mobile factor is less than the world
return, the entire factor supply will leave. If the regional return is higher, the entire world
supply will flow into the region. Thus, as long as the state production functions exhibit
decreasing returns to scale, it is impossible for a state to sustain a mobile factor return
that is higher or lower than the world value if ε is infinite.
Since there is only one mobile factor supply in each state consistent with the
world payoff, there can only be a single Nash equilibrium if ε is infinite. Interestingly,
this is the case found in the existing literature. However, for the more general case where
ε is less than infinite, and the mobile factor is imperfectly mobile between the region and
the world, there may be more than one Nash equilibrium.
Thus, what we have shown is existence and uniqueness for the more usual model
where the mobile factor migrates to equate its regional return with the world return, and
18
existence and possible multiple equilibria for the more general case of imperfect
integration where the world and regional payoffs diverge. An implication is that it would
be extremely difficult to undertake comparative static analysis, except in the polar case of
ε = ∞ where we are guaranteed to have both existence and uniqueness.
3.3
The Tax/Subsidy
As noted in the Introduction, a major focus of the paper is to examine whether
states within regional unions will adopt non-zero tax/subsidy policies towards a factor of
production that is perfectly mobile within the region, but imperfectly mobile between the
region and the outside world. For this part of our analysis, we find it more convenient to
continue developing the model in terms of its primal version, where state i chooses
t i and qi to maximize (6), conditional on t j and q j .
Development of the model in this way requires knowledge of the mobile factor
supply responses to changes in t i and qi . Since our intention is to examine the Nash
equilibrium tax/subsidy, rather than the public good, which we have shown in the
discussion of existence to be provided in equilibrium at minimum cost, we need only
derive the mobile factor supply responses to changes in t i . They can be found by
differentiating (7) and (10) with respect to t i ,
æ ∂n i ö
−p x j w ö ç ∂t i ÷
÷=p æ 1 ö,
֍
xi ç
÷
1 ÷ø ç ∂n j ÷
è −b ø
çç ∂t ÷÷
è i ø
æ p xi w
ç
ç1 − bp x w i'
i
è
'
i
'
j
(19)
ε
where b = N(αεp ε −1e − αp ) ≥ 0.
In the primal version of the policy game, state i chooses its policies, t i and qi , to
maximize (6), subject to its neighbor’s policy choices, t j , q j . With this set up, there are
two necessary conditions, one for t i and one for qi , that must be satisfied at any Nash
equilibrium.
Since our focus is on the tax/subsidy, we only require the necessary
condition for t i , namely,
19
( t i − w i' n i + E i' )
∂n i
+ ni = 0
∂t i
(20)
From (19), the mobile factor response to a change in state i's tax is
∂n i p x i
=
(1 − bp x j w 'j ) < 0 i ≠ j
∂t i
∆
where ∆ = p xi w i' + p x j w 'j − bp xi p x j w i' w 'j < 0 .
(21)
Substituting (21) into (20) yields an
expression for the tax/subsidy that is satisfied in any Nash equilibrium, in either the dual
or primal versions of the game,
*
i
'
i
t = −E (n i ) − n i (
p x j w 'j
p xi [1 − bp x j w 'j ]
).
(22)
The first point to note is that, unlike the result obtained in the papers discussed in
the Introduction, in general the equilibrium tax/subsidy adopted by state i is non-zero.
The reason for this can be seen if one examines the right hand side of the tax/subsidy
expression. The first part, E i' (n i ) , is the change in the externality resulting from an
increase in the quantity of the mobile factor present in state i. We refer to this from now
on as the "marginal externality value". As discussed, this term can be positive, zero or
negative in sign.
(
The second part of the equilibrium tax/subsidy, the term
)
n i p x j w 'j / p x i (1 − bp x j w 'j ) , we label the “hiring cartel” component (the rationale for this
terminology is given below). We know that b ≥ 0, and that, because of the concavity of
the production function, w 'j is non-positive. Therefore, the hiring cartel part is always
negative. However, when the minus sign in front of the cartel term is taken into account
one can think of the cartel component as non-negative. Thus, the hiring cartel effect
tends to make t *i positive.
The hiring cartel component of the tax/subsidy requires further explanation.
In
this regard, it should be recalled that there are many perfectly competitive firms
producing a homogeneous product in both states. As a consequence, the mobile factor is
paid a return equal to the value of its marginal product. Were there but one firm, or were
20
firms able to form a statewide hiring cartel, homefolks could improve their position, visà-vis the perfectly competitive outcome, by paying the mobile factor something less than
its marginal product. A hiring cartel could exploit its monopoly position to the advantage
of the firm-owning homefolks.
Clearly, there is no cartel in our model.
However, effectively the state
government acts in the place of such a cartel - it acts on behalf of homefolks since it
represents their interests - and uses its taxing power to exploit the potential monopoly
gains and increase the income of the homefolks. The revenue raised from the tax on the
mobile factor is redistributed to homefolks via a reduced monetary contribution to the
public good (see (4)) and direct monetary awards. The result is that firms and homefolks,
through the policies of the government of a state, pay the mobile factor a return that is
less than its marginal product. It is for this reason that we have used the words "hiring
cartel" to describe this part of the tax/subsidy equation: the state essentially acts as a
hiring cartel which has the option of paying the mobile factor something other than its
marginal product.
When will states tax the mobile factor and when will they provide it with a
subsidy? It is clear that the hiring cartel effect always induces state i to tax the mobile
factor, that is, to try to exploit the mobile factor for the advantage of the homefolks. If
the marginal externality value is also negative (in the case of labor this might be so if the
costs of more guest workers outweighs the benefits) then the tendency to exploit the
mobile factor is intensified and the tax is higher than otherwise. However, if the marginal
externality value is positive it will tend to offset the desire to exploit the mobile factor. If
sufficiently large (and positive), the marginal externality value may encourage a state to
provide the mobile factor with a monetary subsidy.
Hence, the model explains the tax/subsidy policies of states as the result of an
interaction between the desire to redistribute from mobile factors for the benefit of
homefolks and the externalities generated by migrating factors.
It also explains
differences in the policies adopted by states; specifically, states will have different
marginal externality values and different hiring cartel effects, and hence will choose to
21
adopt diverse tax/subsidy policies in equilibrium. For example, a state that perceives that
it has a relatively low, though positive, marginal externality value with respect to
migrating guest workers may wish to tax them. This might be a state that already has
relatively large numbers of guest workers and hence is at a point on its externality
function for which the marginal value of extra guest workers is low. Alternatively, a state
that is relatively small in terms of mobile population may have a much higher marginal
externality value and choose to subsidize mobile labor.
4.
Regional Integration and the Tax/Subsidy
We now examine how the degree of integration between the region's factor
market and the world factor market, as captured by the parameter ε, influences the (nonzero) tax/subsidy chosen by a state within a regional union. Since ε also captures the
degree of competition faced by the region as a whole in competing internationally for the
mobile factor, we can also view this as an analysis of how changing international
competition affects the policies of states within the region.
As part of the discussion, it is also possible to consider the impact of withinregion competition (directly between the states). This is an interesting question to ponder
because the proponents of competitive federalism, such as Breton (1984), have argued
that competition between states is desirable in the sense that it reduces the taxing power
of governments. We tackle this issue by examining what we call the “monopoly state” in
which the region consists of only one state.
The analysis is undertaken for two polar cases. In the first, we suppose that the
region is fully isolated from the world ( ε = 0 ), while in the second case we suppose that
the region is fully integrated ( ε = ∞ ). It has not been possible to obtain analytical results
on the intermediate case, where ε is less than infinite, but more than zero, because b is a
function of ε and varies in an indeterminate way as ε changes.
Before discussing each case, it is useful to note that, as with the competitive twostate case, a monopoly state chooses its policies to maximize the payoff to the homefolks,
ε
but faces only one factor market constraint, specifically, n = N(1 − e −αp ) where we drop
22
the i subscript when considering the monopoly state. Since the state has no fully open
border with a neighboring state, there is no internal free migration condition such as (7).
The state faces only international competition for the mobile factor. In this case,
bp x
∂n
.
=−
∂t
1 − bp x w '
(23)
There is also a difference between the tax/subsidy expression when state i is a
monopoly state, and the tax/subsidy expression if it faces competition for the mobile
factor within the region. This can be seen by noting that if a state is a monopoly state its
tax/subsidy is
t * = − E ' (n ) +
n
.
bp x
(24)
Comparing (24) with (22), the equilibrium tax/subsidy for the competitive case, it
is clear that the hiring cartel effect differs, depending on whether a state faces
competition from within the region.
Now consider the polar case where the region is fully integrated with the world
factor market (ε = ∞) and the regional mobile factor payoff is equal to the world payoff.
For the competitive case, we know from the discussion of uniqueness in Section 3.2 that
the Nash equilibrium is unique. One can also show that b = ∞ when ε = ∞ . Hence, the
hiring cartel effect is zero for both the monopoly and competitive state cases and the
tax/subsidy is
t i = − E i' (n i ) .
(25)
Thus, if the region is fully integrated a monopoly state chooses the same
tax/subsidy as a competitive state. The implication is that the competitive federalism
outcome is no different to the monopoly outcome when the region is fully integrated
simply because the monopoly power of the state is completely constrained by foreign
competition for the mobile factor.
Now consider the other polar case, a fully isolated region (ε = 0) where states face
no outside competition for the mobile factor. Here, we have b = 0 and the hiring cartel
23
component of the tax/subsidy expression for the competitive state case is − n i w 'j . The
competitive equilibrium tax/subsidy is
t *i = − Ei (n i ) − n i w 'j .
(26)
Because we do not know whether or not there are multiple Nash equilibria in the
interstate policy game, we are unable to compare the monopoly and competitive state
outcomes in general. However for the polar cases of ε = ∞, we know that the monopoly
and the competitive outcomes are similar. In both cases, the hiring cartel effects go to
zero and the tax/subsidy reflects only the externalities. An unambiguous comparison is
also possible for the fully isolated region (ε = 0). It is clear, from a comparison of (24)
and (26) that for a region of similar size the single state mobile factor tax is higher than
the equilibrium value of the competitive state tax. As ε (and thus b) goes to 0 the
expression in (24) becomes infinitely large. However, a state cannot tax without limit,
and this indicates that the monopoly state tax is fully exploitive: it will equal the full price
w(n) paid to the mobile factor. Furthermore, the monopoly state will supply no public
good.
However, full exploitation cannot be an equilibrium outcome for the competitive
state, as long as the marginal externality is not a large negative value. If one state were to
be fully exploitive, its neighbor could induce the entire mobile factor supply into its state
and enjoy the returns from it with a small reduction in its tax, or a small increase in its
supply of the public good. Since the residual, R, is an increasing function of n it will be
in its interest to make the change if the mobile factor-created externality is not negative
and large. The implication is that when ε = 0 there is a difference between a competitive
federalism outcome and a monopoly outcome, namely, the competitive outcome will
involve a lower tax/subsidy on the mobile factor.
This begs the question of what determines the extent to which the competitive
state tax is lower than the monopoly tax? In answering this, one might suspect (rightly,
as it turns out) that the more competitive is the regional factor market, the smaller is the
cartel effect. One can gain some insight here by considering the case of a two state
24
region where both states have the same technology and the equilibrium is one in which
each state has half the total mobile factor supply (which does not change since ε = 0 ).
The value of w ' for each state is also the same in the equilibrium. Since each state has
the same factor supply, the steeper is the wage function (the larger in absolute value is w')
the higher is the equilibrium tax.
At first sight this is curious but it points vividly to the role of intra regional
competition in influencing the tax chosen by states. Specifically, we know that the value
of the mobile factor response to changes in state taxes in the competitive case is
determined by (19).
When ε = b = 0, the response in the competitive case is
∂n i / ∂t i = 1/(w1' + w '2 ) . Clearly, the larger in absolute value is w', the smaller is the
mobile factor loss to a state for any increase in its tax rate. If the marginal value of the
wage function is infinitely steep, a state can increase its factor tax with no change in
factor supply. The limiting case is similar to the fully isolated single state in which the
state possesses a complete monopoly over its own migrating factor. Thus, the magnitude
of the difference between the competitive and monopoly mobile factor tax when ε is zero
depends on the steepness of the wage functions. Since we take the steepness of the wage
function to be a measure of the degree of interdependence, and hence competition
between states, the implication is that greater intra regional competition results in a
smaller difference between the competitive and monopoly mobile factor tax.
5.
Sub-Optimal Supply and Distribution of the Mobile Factor
We now turn to the final part of the paper, the discussion of efficiency. This issue
has been studied extensively in models of regions that levy taxes on mobile factors, for
example, the tax competition literature. As noted in the Introduction, the main issue
addressed is whether the public good is over or under provided relative to an optimal
supply. A comprehensive survey of these ideas, and the main results, can be found in
Chapter 4 of Wellisch (2000).
Rather than follow this path here, we choose to explore a different question: the
interaction between migration frictions, as captured by the ε parameter in the mobile
25
factor supply function, and optimality in the inter-state distribution and total supply of the
mobile factor. This different emphasis allows us to generate some new insights into the
inefficiencies created by competitive state policies (eg., tax competition) which are not
directly related to the over or under provision issue.
5.1
Inter-State Distribution
The first question, whether a given supply of the mobile factor is allocated
efficiently between states, has been studied, though in a somewhat different context to the
one here. The approach has been to suppose that a given (rather than variable) supply of
a factor, such as labor, migrates between the member states of a federation12. The
derived efficiency condition implies that the marginal benefit of an extra unit of the
mobile factor in state i, w i (n i ) − x i (where w i (n i ) is the per unit return to the mobile
factor -its marginal product if competitive factor markets are assumed - and x i is its per
unit monetary reward expressed in terms of private good consumption), must be equal to
the marginal benefit of an extra unit of the mobile factor in state j, w j (n j ) − x j . The wellknown efficiency condition is w i (n i ) − x i = w j (n j ) − x j , that is, marginal benefits must be
equated across states otherwise it is always possible to redistribute the mobile factor and
gain an increase in welfare. Since the tax on the mobile factor is just the difference
between what it contributes to state i, w i (n i ) , and what it receives as a monetary reward,
x i , then the efficiency condition can be written as
ti = t j .
(27)
Thus, in equilibrium competing states must levy the same per unit tax on the
mobile factor for its distribution across states to be efficient. The application of such an
“equating at the margin” idea for our model is more complex. First, there is the mobile
factor externality to consider. The implication of this is that, for any state i, the marginal
12 Key papers in this “fiscal externality” literature include Boadway and Flatters (1982), Myers (1990),
Mansoorian and Myers (1993) and Burbidge and Myers (1994). For an overview see Chapter 7 of
Wellisch.
26
benefit of an extra unit of the mobile factor has two components. One is the change in
the net externality benefits which result from an additional unit of the mobile factor and
the other is the change in the reward to the homefolks. The marginal externality benefit is
simply E i' (n i ) . The change in the reward to the homefolks is just the change in total state
output (the marginal product of the mobile factor) less the net payment to the mobile
factor.
For state i this is fi' (n i ) − x i (where it should be recalled from our earlier
discussion that x i = w i (n i ) − t i is the net monetary reward accruing to each unit of the
mobile factor). As a result, the marginal benefit of an extra unit of the mobile factor in
state i is E i' (n i ) + fi' (n i ) − x i .
Efficiency in the inter-state distribution of a given supply of the mobile factor
requires an equality of this value across states; E i' (n i ) + fi' (n i ) − x i = E 'j (n j ) + f j' (n j ) − x j .
Since the difference between the marginal product and the net monetary reward is equal
to the mobile factor tax, the equating at the margin rule in our model can be expressed as
E i' (n i ) + t i = E 'j (n j ) + t j .
(28)
Whether the tax/subsidy policies of the states in the game characterized in Section
3 meet this condition can be determined by substituting the Nash equilibrium expressions
for t i and t j (from (22)) into (28). This yields
−n j (
p x j w 'j
p x i [1 − bp x j w 'j ]
) = −n i (
p x i w i'
p x j [1 − bp x i w i' ]
).
(29)
For the Nash equilibrium tax/subsidy policies to result in an efficient allocation of
a given quantity of mobile factor within the region requires that the hiring cartel effects
be equal in equilibrium. When ε < ∞ , this requirement is not satisfied, and the Nash
equilibrium tax/subsidy policies of the states create an inefficiency in the inter-state
distribution of the mobile factor. Clearly, the source of the inefficiency is the hiring
cartel effect. The exception is the polar case where ε is infinite. In this case, we know
that the hiring cartel effect is zero and that, therefore, (29) must be satisfied.
27
Thus, if one allows for a divergence in the regional and world payoffs due to
imperfect factor mobility, it is clear that the inter-state mobile factor distribution is
inefficient if the region is not fully integrated with the world. It is only when there is full
integration, the assumption made in the model employed by Oates and Schwab, for
example, that the inter-state mobile factor distribution would be efficient.
This is
because, in such a world, states have no monopoly power over the mobile factor.
However, if there is imperfect integration, states face an incentive to distort the inter-state
mobile factor distribution.
5.2
Total Regional Supply and Inter-State Distribution
The analysis above was undertaken for a given total supply of the mobile factor
and concentrated on the issue of its inter-state distribution. But a feature of our model is
that it allows for variable regional mobile factor supply. The issue of whether the total
regional supply of the mobile factor is efficient, has received much less attention,
possibly because of prior concentration on models in which there is full integration or
none at all. For example, the fiscal externality literature referred to above, does not
consider this question because it supposes that the supply of the mobile factor within the
region (usually a federation) is fixed. However, with a variable total supply of the mobile
factor, there is not only the issue of the inter-state distribution of the factor to consider,
but also the efficiency of its total supply.
For the total supply of the mobile factor to be efficient also requires that the net
marginal benefit of an additional unit of the mobile factor be equal to zero in each state.
If it is not, then one can always change the supply of the mobile factor and increase the
payoff to the homefolks in state i. But if the net marginal benefit is zero in each state
then (28) must also be satisfied since the net marginal benefits will also be equal across
states. Thus, the following condition must be met in equilibrium if the total supply of the
mobile factor to the region, and its distribution across states, are to be efficient,
E i' (n i ) + t i = 0 = E 'j (n j ) + t j .
28
(30)
Is (30) satisfied in a Nash equilibrium characterized in Section 3? When ε is less
than infinite, we know that t i ≠ − Ei' (n i ) and that t j ≠ − E 'j (n j ) because the hiring cartel
effect is positive. This implies that E i' (n i ) + t i ≠ 0 and E 'j (n j ) + t j ≠ 0 . Therefore, the
total supply of the mobile factor to each state, and the region, is either too high or too
low, relative to the optimal supply. As shown above, its distribution across states is also
inefficient. Again, the hiring cartel effect is the source of the distortion. However, we
know that when ε is infinite the hiring cartel effect is zero and hence that t i = − E i' (n i )
and t j = −E 'j (n j ) . This implies that E i' (n i ) + t i = 0 , E 'j (n j ) + t j = 0 and hence that (30) is
satisfied. Both the total supply and regional distribution of the mobile factor are efficient
in this case13.
Therefore, in a regional union with a variable total factor supply, (30) is the
equating at the margin rule that must be satisfied for both the inter-state distribution and
total supply to be efficient. In general, this condition is not satisfied in equilibrium,
implying that policy competition between states leads to sub optimality in the supply and
distribution of mobile factors, except in the special case of full integration.
What we have is “over” or “under” provision (we cannot in general say which) of
the mobile factor to the region and an inefficient inter-state distribution. This seems to be
a new notion in this literature: that the supply of the mobile factor that results from a
game between states in which each adopts competitive self-serving behavior, leads to a
sub optimal supply of the mobile factor (capital or labor) to the regional union, be it a
federation or confederation. The sub optimality, related to the mobile factor itself, is a
source of inefficiency that is in addition to any inefficiency arising from provision of the
public good, the issue already examined in the literature.
Its identification is a
consequence of exploring a model structure in which monopoly power gives states the
ability to choose non-zero tax/subsidies.
13
29
Recall from Section 3.2 that the equilibrium is also unique in this case.
6.
Conclusion
There is a class of model in which states within regional unions choose policies
directed towards migrating factors of production. We have extended this model by
allowing the total supply of the mobile factor to the region to be determined by a function
that allows the regional mobile factor payoff to differ from the world payoff in
equilibrium (due to imperfect mobility).
We have also modeled the total supply of the
mobile factor to each state, and the region as a whole, as being the result of competitive
optimizing decisions made by member states (ie., total factor supply is endogenous).
This is quite different to the approach of say, Wellisch and Wildasin, who model states
making policy decisions conditional on some given supply of the migrant factor.
The paper contains a number of new results. In particular, we have demonstrated
existence of Nash equilibria by showing that the game in two strategic variables has a
dual game in which each state finds the policy combination that minimizes the cost of
achieving its desired supply of the mobile factor. It has not been possible to show
uniqueness, however, except for the special case where the regional and world factor
markets are fully integrated, and states must offer a payoff to the mobile factor that
equals its world return. In the more general case of imperfect mobility between the
region and the world, there is the possibility of multiple Nash equilibria.
We also show that it is optimal in equilibrium for states to levy non-zero taxes or
subsidies on the mobile factor. As noted, this differs from existing findings where
optimality usually requires a zero tax/subsidy. One reason for the difference is that our
factor supply condition allows states, when there is imperfect mobility between the region
and the world, to have monopoly power over the migrating factor.
This creates
incentives to exploit the factor and redistribute in favor of the domestic residents, who
control the local polity. The non-zero tax/subsidy also leads to sub optimality in the total
supply and inter-state distribution of the mobile factor except in the case of full
integration.
30
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ii
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0133 Rajan, Ramkishen and Graham Bird, Still the weakest link: the domestic
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0132 Rajan, Ramkishen and Bird, Graham, “Banks, maturity mismatches and liquidity
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0130 Francois, Joseph F. “Factor mobility, economic integration and the location of
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0129 Francois, Joseph F. “Flexible Estimation and inference within general
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0128 Rajan, Ramkishen S., "Revisiting the Case for a Tobin Tax Post Asian Crisis: a
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0127 Rajan, Ramkishen S. and Graham Bird, "Regional Arrangements for Providing
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0126 Anderson, Kym and Shunli Yao, "China, GMOs, and world trade in agricultural
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0125 Anderson, Kym, "The Globalization (and Regionalization) of Wine", June 2001.
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iii
(Paper for the National Academies Forum’s Symposium on Food and Drink in
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0124 Rajan, Ramkishen S., "On the Road to Recovery? International Capital Flows
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0123 Chunlai, Chen, and Christopher Findlay., "Patterns of Domestic Grain Flows
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0122 Rajan, Ramkishen S., Rahul Sen and Reza Siregar, "Singapore and the New
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0121 Anderson, Kym, Glyn Wittwer and Nick Berger, "A Model of the World Wine
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0120 Barnes, Michelle, and Shiguang Ma, "Market Efficiency or not? The Behaviour
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0118 Stringer, Randy, "How important are the 'non-traditional' economic roles agriculture
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0117 Bird, Graham, and Ramkishen S. Rajan, "Economic Globalization: How Far and
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0116 Damania, Richard, "Environmental Controls with Corrupt Bureaucrats," April 2001.
0115 Whitley, John, "The Gains and Losses from Agricultural Concentration," April 2001.
0114 Damania, Richard, and E. Barbier, "Lobbying, Trade and Renewable Resource
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0113 Anderson, Kym, " Economy-wide dimensions of trade policy and reform," April
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0112 Tyers, Rod, "European Unemployment and the Asian Emergence: Insights from the
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0111 Harris, Richard G., "The New Economy and the Exchange Rate Regime," March
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0110 Harris, Richard G., "Is there a Case for Exchange Rate Induced Productivity
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0109 Harris, Richard G., "The New Economy, Globalization and Regional Trade
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0108 Rajan, Ramkishen S., "Economic Globalization and Asia: Trade, Finance and
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0107 Chang, Chang Li Lin, Ramkishen S. Rajan, "The Economics and Politics of
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iv