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Treaty of Rome (articles 48 and 51), the U.S. Constitution, the Canadian. Charter of Rights and Freedom (Constitution Act of 1982)...). Building on .... Section. 3 derives the symmetric equilibria of this game. The asymmetric equilibria are derived in ...... recent results ", Canadian Journal of Economics, 15, 613-33. 3] Caplin, A.
MOBILITY AND REDISTRIBUTIVE POLITICS Jean HINDRIKS University of Exeter March 1999

Abstract.

There is a widespread concern that a greater mobility of individuals can

undermine any attempt to redistribute income at the local level. In this paper we derive the equilibrium level of redistribution when both the rich and the poor are mobile (although in di erent degrees) and when each jurisdiction chooses its redistributive policy by majority voting. This leads to a fundamental interaction whereby the policy choices of jurisdictions determine who they attract, and who they attract determines their policy choices. Our main ndings are twofold. First, we show that a greater mobility of the poor can increase the equilibrium amount of redistribution. Second, some jurisdictions can be stuck in equilibrium on the \wrong" side of their La er curve. The reason is that the poor are in a majority in these jurisdictions and they oppose to a potentially Pareto improving tax reduction because it would attract the rich and shift the majority.

JEL classi cation. C72, D71, H71, R51. Keywords. Majority voting, mobility, redistribution. Mailing address: Jean Hindriks, Department of Economics, University of Exeter,

EX4 4PU, Exeter, UK. [email protected]. I am grateful to Tim Besley, Philippe De Donder, Dennis Epple, Francois Maniquet, Gordon Myers and Fred Schroyen for stimulating discussions and suggestions on earlier versions of this work. I have also bene ted from the comments of participants at seminars at the CORE, the University of East-Anglia, the University of Essex, and the University of Namur. Financial support from the European Commission under contract no. ERBFMBICT971968 is gratefully acknowledged. I retain responsability for errors and views. 

1 Introduction The conventional wisdom suggests that in the presence of free mobility, any local government that seeks to redistribute income would be driven to bankruptcy because it would face out-migration of those who are supposed to give (the rich) and in-migration of those who are supposed to recieve (the poor).1 In North America, the fact that some more generous states have become a destination resort for welfare recipients is currently the subject of an important debate in North America ( see The Vancouver Sun, August 1995). The existing literature on the subject provides only three solutions to this problem.2 One is to base income taxation on a nationality, rather than residence, principle. The second, called the scal federalism solution, is to design corrective schemes at the federal level that aim to internalize the scal externalities that the local jurisdictions in ict to each other in setting their redistributive policies. Examples of such corrective schemes are federal matching grants to the payments to the poor made by the local jurisdictions and federal transfers to local jurisdictions according to the number of poor residents (see Boadway and Flatters, 1982). The implementation of these mechanisms is obviously problematic and has led some authors to propose a third solution.3 This third solution, building on the seminal work of Myers (1990), challenges the need of a central government intervention. The idea is that there exist Nash-equilibrium inter-jurisdictional transfers that can 1 See Mott (1992) for a review of the empirical literature on the scally-induced

migrations. See also Kirchgassner and Pommerehne (1996) for an empirical study of labour mobility in Switzerland. 2 See Cremer et al. (1995) for a recent overview and additional references to the literature. 3 See Piketty (1996) for a critical assessment of the scal federalism solution.

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reach the same outcome as a central government intervention. The great advantage of this solution is its implementability, however it leads to eciency only if all individuals are identical.4 This paper starts from the facts that all these solutions are rarely observed in real world and that at the same time we do not observe huge migration

ows between jurisdictions although their redistributive policies may differ substantially. On the other hand, the principles of free movement and equal treatment are very commonly observed within most countries (e.g. the Treaty of Rome (articles 48 and 51), the U.S. Constitution, the Canadian Charter of Rights and Freedom (Constitution Act of 1982)...). Building on these three observations, we argue that redistribution at a local level is possible without central government intervention or inter-jurisdictional transfers, provided that each jurisdiction abides de facto to the free movement and equal treatment principles. The redistribution problem in fact arises when the local-governments limit the right of poor people to migrate or their access to redistributive bene ts ; and we show that this incentive may be real under majority rule. Our main departure from previous work on the subject is that (i) we allow for di erent degrees of mobility among the rich and the poor, and (ii) we assume that redistributive policies within jurisdictions are chosen by majority voting. In doing so we reach two novel and surprising conclusions. First, we nd that improving the mobility of the poor is bene cial to redistribution when the poor are many enough to form a majority in each jurisdiction..5 The logic behind this result is that the poor are less vulnerable to the mo4 I am grateful to Gordon Myers for this remark. 5 This contrasts sharply with Wildasin (1991) who shows that the mobility of the poor

is harmful to redistribution if the rich are immobile.

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bility of the rich when they are able to chase them. Our second result is that some jurisdictions can choose in equilibrium a tax rate on the downward sloping side of their La er curve. These jurisdictions where the poor are in a majority oppose to a potentially Pareto-improving tax reduction because it would attract the rich and shift the majority to the rich. As a result the rich will reduce further the tax rate in the next period. This case of political failure is reminiscent of Besley and Coate (1998), although in their model the political failure is produced by political uncertainty and not free mobility. In both cases, however, it is the fears of a change in the majority that prevent the realisation of potentially Pareto improving policy changes (namely, a public investment in their model and a tax reduction in our model). 6 We develop the minimal framework useful to establish these results. The purpose of this minimalist modelling approach is to make the logic behind the results suciently clear to convince the reader of their validity in more general environments. Throughout we shall adopt the policy-based approach to scal competition games. This is the approach of a Nash equilibrium which takes the policies of other jurisdictions as xed, by contrast with the membership-based equilibrium which takes the memberships of jurisdictions as xed (see Caplin and Nalebu , 1997, for a comparison between the two 6 It is worth noting that this ineciency result di ers from the cycling problem which

implies that a majority choice equilibrium does not exist. Indeed in our model, the majority choice equilibrium exists but is not Pareto optimal. This ineciency result must also be contrasted with the ineciencies resulting from scal/congestion externalities which can essentially be resolved by a coordination between jurisdictions. Indeed, such a coordination is pointless to solve the ineciency identi ed in this paper which arises from the vulnerability of the current majority to policy changes.

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approaches). So in the membership-based approach (like Westho , 1977 and Epple, Filimon and Romer, 1984), all jurisdictions ignore the migration e ects of their policy choices while in the policy-based approach (like Epple and Romer, 1991), they are fully aware of these migration e ects. We suppose a xed number of jurisdictions.7 Individuals di er both in their income and their preference for jurisdiction. Each jurisdiction abides to the free mobility and equal treatment principles and chooses its redistributive policy according to the majority rule. Redistributive policies in each jurisdiction transfer ressources from their rich residents to their poor residents. Both the rich and the poor are (imperfectly) mobile.8 To keep the analysis tractable, we abstract from the production side and assume that individuals have xed income (no incentive e ect). When jurisdictions have di erent redistributive policies, moving can bring higher incomes to some individuals. But we also all understand that changing location involves non-pecuniary costs such as the time spent on retraining, job search, search for housing, some psychological costs of separation from family and friends, and so on. To formalize these non-pecuniary cost of moving, we use a spatial competition model a la Hotelling in which individuals have various degrees of attachment to location.ffootnoteMansoorian and Myers (1993,1997) make the same assumption in a di erent model to show the eciency of inter-jurisdictionnal transfers. In their model all indi7 In contrast, Greenberg and Weber (1986), Konishi, Lebreton and Weber (1996), Ray

and Vohra (1996) endogenize the number of jurisdictions and so shed light on the important issue of the formation of a federation of jurisdictions. 8 In contrast Wildasin (1991,1994)considers that the rich are immobile. At the other extreme, Epple and Romer (1991) assume that moving is costless which leads to perfect strati cation in response to di erentials redistributive policies. In this paper we assume that moving is costly

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viduals have the same income and so there is no issue of redistribution. A spatial competition model captures quite well the reality that jurisdictions have some degree of monopoly over their members (say, the native population); a slight increase in the tax charged will not result in the jurisdiction losing all its rich members. Because individuals value di erent jurisdictions di erently there will be some individuals for whom it will not pay to move to another jurisdiction and the jurisdiction thus has some degree of monopoly over these individuals. The paper is organised as follows. Section 2 provides the description of our scal competition game and the de nition of the equilibrium. Section 3 derives the symmetric equilibria of this game. The asymmetric equilibria are derived in Section 4. Section 5 concludes the paper.

2 The Fiscal Competition Game There are two exogenously given jurisdictions, called the domestic and the foreign jurisdictions for sake of de niteness9 . We refer to the foreign jurisdiction by the use of the superscript (*). The two jurisdictions are symmetric in a sense we shall make precise shortly. There is a large set N of individuals with di erent income levels and di erent preferences for jurisdiction. For simplicity we consider only two income levels, normalized to 0 and 1. The number of poor individuals (i.e., those with income equal to zero) is n1, and the number of rich individuals is n2. The relative number of rich 9 The assumption of two jurisdiction is not essential. What is really restrictive is the

assumption of a xed number of jurisdictions, which rules out the important issues of making and breaking jurisdictions. In that respect, we view our approach as an indispensable prelude to the more ambitious analysis of the endogenous formation of jurisdictions (see Ray and Vohra, 1997).

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individuals is denoted by   nn12 . We describe the preference for jurisdiction by a single taste parameter x 2 [0; 1]. Those with low x prefer the domestic jurisdiction, those with high x prefer the foreign jurisdiction and those in the middle are indi erent. We further make the simplying (but innocuous) assumption that x is uniformly distributed within each class (i.e., rich and poor). Both jurisdictions impose a tax T and T  (with 0  T; T   1) on their rich residents, and pay a transfer B and B  to their poor residents; and each individual freely joins the jurisdiction that maximises his utility. Note that this redistribution policy can also be thought of as the net e ect of any kind of redistributive policies (e.g. redistributive public employment, free education and free medical care). Since the population is large, no individual believes that his location decision will in uence the policy outcome. Each jurisdiction is constrained to have a balanced budget, so the set of feasible tax-transfer policies that each jurisdiction can a ord to o er depends on who they attract; that is, the number of rich individuals (or the contributors) and the number of poor individuals (or the bene ciaries) . Furthermore, who they attract depends on their policy choices z = (T; B ), z  = (T  ; B ). Formally, each jurisdiction selects a feasible policy

z 2 Z (S (z; z)) z 2 Z (S (z; z)) where fS (z; z ); S (z; z )g is the partition of the population between the two jurisdictions, that will result from their policy choices z; z  and Z is the set of tax-transfer policies that break even given the composition of the jurisdiction. 6

Individuals care only about their net income and their location. The payo of a rich individual with preference x 2 [0; 1] is

U2(z; z; x) = (1 ? T ) ? d2x in the domestic jurisdiction U2(z; z; x) = (1 ? T ) ? d2(1 ? x) in the foreign jurisdiction; where the parameter d2 > 0 measures the intensity of the preference for location among the rich (or their degree of \attachment to home "). So, a higher degree of attachment is akin to lower mobility. We shall assume that this attachment is not too high (d2 < 1) so that jurisdictions are indeed competing to attract the rich. Each rich individual with preference x such that U2(z; z ; x)  U2(z; z ; x) joins the domestic jurisdiction. The payo of a poor individual with preference x 2 [0; 1] is

U1(z; z; x) = B ? d1x in the domestic jurisdiction U1(z; z; x) = B ? d1(1 ? x) in the foreign jurisdiction; where d1 > 0 measures the intensity of preference for location among the poor. Note that their degree of \attachment " (and thus their degree of mobility) can be di erent from the rich. Let d = dd12 denote the relative degree of attachment of the rich or, equivalently, the relative degree of mobility of the poor. So if d < 1 the rich are less attached to location and thus more mobile than the poor, and vice versa. Each poor individual with x such that U1(z; z ; x)  U1 (z; z ; x) joins the domestic jurisdiction. Given these individual payo functions, any policy con guration (z; z ) induces the following partition of the population between the two jurisdictions

S (z; z) = fx 2 [0; 1] : Ui(z; z; x)  Ui(z; z; x)(i = 1; 2)g S (z; z) = fx 2 [0; 1] : Ui(z; z; x) < Ui(z; z; x)(i = 1; 2)g: 7

We emphasize that this linear structure of preferences permits a simple characterization of equilibria but that it is not essential for our main results. Note also that the model is general enough to accomodate with an arbitrarily low mobility of the poor (by setting d1 suciently high). We consider that each jurisdiction takes the policy of the other as given when making its own policy choice and that it is fully aware of the migration e ects of its policy choice. This is the policy-based approach (by opposition to the membership-based approach). In this policy-based approach, each jurisdiction selects a feasible policy according to the exogenously given decision rule, taking the policy of the other jurisdiction as given and anticipating correctly the migration ows that will result. Equilibrium is a xed-point in which no individual wishes to switch juridiction and no jurisdiction wishes to switch policy. There are di erent possible formulations possible of this equilbrium depending on whether or not budget balance is required out-ofequilibrium. If not, then we have the following de nition:

De nition 1 a policy outcome z; z is a (pure strategy) Nash equilibrium i

z = D(S (z; z)jz) z = D(S (z; z)jz)

with z 2 Z (S (z; z )) with z  2 Z (S (z; z ));

where D is the exogenous decision rule, common to both jurisdictions, which maps their respective membership into the set of feasible policies, taking as given the policy choice of the other jurisdiction.

This policy-based equilibrium has however been criticized by Caplin and 8

Nalebu (1997) on the grounds that it assumes, rather naively, that each jurisdiction can commit to a policy which is made infeasible (as a result of the migration ows)by the policy change of the other jurisdiction. This is the approach adopted in particular by Epple and Romer (1991). In this paper we shall adopt an alternative concept of equilibrium which implies a lower degree of commitment on the part of jurisdictions in the sense that the jurisdiction deviating from the equilibrium outcome anticipates that the other jurisdiction will change its policy in response to maintain budget balance. More speci cally, we shall assume that each jurisdiction chooses its tax rate taking as given the tax rate of the other jurisdiction and adjusts its transfer level according to the migration ows in order to maintain budget balance.10 We denote by B (T; T ) and B  (T; T ) the transfer levels in both jurisdictions that result from the tax pair (T; T ). Substituting these transfer functions into the payo functions, we obtain that for each pair (T; T ) and for each class i (with i = 1; 2), there exists xi (T; T ) 2 [0; 1] such that all individuals in class i with preference x  xi (T; T ) join the domestic jurisdiction and all individuals in class i with preference x > xi (T; T ) join the foreign jurisdiction. Hence, S (T; T ) = fx 2 [0; 1] : x  xi (T; T ); (i = 1; 2)g and S (T; T ) = fx 2 [0; 1] : x > xi (T; T ); (i = 1; 2)g. Since x is uniformly distributed over [0; 1] among each class i (i = 1; 2), xi (T; T ) is also the percentage of individuals in class i who join the domestic jurisdiction and 1 ? xi (T; T ) is the percentage of individuals in class i who join the foreign jurisdiction. Therefore, the buget balance requirement in each jursidiction 10 This is in e ect the assumption implicitely made in most of the scal competition

literature.

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reduces to and,

T; T ) B(T; T ) = T  xx2((T; T )

(1)

x2(T; T ) : B (T; T ) = T   11 ? ? x (T; T )

(2)

1

1

We then adopt in the sequel the following de nition of an equilibrium : De nition 2 a policy outcome z; z is a (pure strategy) Nash equilibrium in tax rates i

T = D2(S (T; T )jT ) T  = D2(S (T; T )jT )

with B = B (T; T  )

with B  = B  (T; T );

where D2 is the majority voting decision rule, common to both jurisdictions, which maps their respective membership into the set of feasible tax rates . Note the interaction involved in this equilibrium. Constrained to break even, the tax-transfer policies each jurisdiction can a ord to o er depend on who they attract, and who they attract determines by majority rule what they choose to o er. An equilibrium is a xed-point in which no individual whishes to move given the policy choices and no jurisdiction wishes to change its policy (according to the majority rule) given the residential choices. Note also that we are here modelling dynamic features in a reduced-form static framework. Indeed, we are facing a dynamic game in which the current majority chooses the present policy which then determines the future majority. However these dynamic features can be subsumed in our reduced-form static game to a stationary state in which no one has an active incentive to deviate from his current strategy being far-sighted enough to recalculate correctly the resulting change in the majority that such action has. So, if the poor

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are in a majority in one jurisdiction they will never vote for a policy that shift the majority to the rich as this can only make them worse o . This completes the description of the scal competition game. The game is symmetric , in the sense that (i) the rich are endowed with the same (gross) income in both jurisdictions, (ii) the distribution of preference for jurisdiction is symmetric, and (iii) both jurisdictions have the same decision rule. Such a symmetry implies that if T = T , then we have B (T; T ) = B (T; T ) and x1(T; T ) = x2(T; T ) = 12 , that is, the rich and the poor are equally divided between the two jurisdictions. In the sequel, we shall interpret this as the no-migration situation and we shall call the natives of the domestic jurisdiction all individuals with preference x  12 . We now derive the equilibria of this game as a function of the size and mobility of each group. We start with the symmetric equilibria.

3 Symmetric Equilibria In a symmetric equilibrium, both jurisdictions choose the same tax rate and by symmetry of the game the population is equally divided between the two jurisdictions. It follows that either the rich or the poor are in a majority in both jurisdictions depending on their relative number in the population. If the rich outnumber the poor in the population (that is,  > 1), then each jurisdictions is controlled by the rich and the resulting tax choices are T = T  = 0. On the other hand, if the poor outnumber the rich in the population (i.e.,   1)then each jurisdiction is controlled by the poor and selects its tax rate so as to maximise the income of its poor residents, anticipating correctly the induced migration ows and their e ects on the 11

transfer level and the possible shift in the majority. We now characterize this equilibrium level of taxation and show that it is increasing with the mobility of the poor. To do that, we need to derive the migration response of each class to a small tax change. First consider the migration response of the rich. Given the pair (T ,T ), with T close to T  , the equilibrium migration is characterized by the marginal individual with preference x = x2 (T; T ) who is indi erent between the two jurisdictions, so x2(T; T ) solves (1 ? T ) ? d2x = (1 ? T  ) ? d2(1 ? x)

(3)

 x2(T; T ) = 21 + T 2d? T

(4)

@x2(T; T )  1 = ? @T 2d2 T =T 

(5)

This yields 2

Therefore all the rich with x  x2(T; T ) go to the domestic jurisdiction and all the rich with x > x2 (T; T ) go to the foreign jurisdiction. Given the uniform distribution, x2 (T; T ) determines also the fraction of the rich who choose to reside in the domestic jurisdiction. Note that x2 (T; T ) = 1=2 for T=T (due to the symmetry of the model) and that x2 (T; T ) is decreasing in T and increasing in T  . This re ects the fact that the rich prefer to join the jurisdiction with a lower tax rate. The migration response of the rich to a small domestic tax change from T = T  is 

Thus, the migration response of the rich to a marginal tax change is decreasing with their degree of attachment to location. 12

We now turn to the migration response of the poor to a small tax change. Given a pair (T ,T ) with T close to T , the utility of the poor individual with preference x is u1 (T; T ; x) = B (T; T ) ? d1 x in the domestic jurisdiction and u1(T; T ; x) = B  (T; T ) ? d1 (1 ? x) in the foreign jurisdiction. The equilibrium migration of the poor is characterized by the marginal individual with preference x = x1 (T; T )who is indi erent between the two jurisdictions. Using (1) and (2), x = x1(T; T ) solves   T x2(T;x T ) ? d1x = T  1 ? x12?(T;x T ) ? d1(1 ? x) (6) Given the uniform distribution, x = x1 (T; T ) determines also the frac-

tion of the poor who join the domestic jurisdiction. Notice that the equilibrium migration of the poor, x1 (T; T ) depends on the equilibrium migration of the rich, x2(T; T ). Rewriting the above expression as    f (x1; T )  2d1x1 +  (1 1??xx2)T ? xx2T ? d1 = 0; 1 1

(7)

and applying the implicit function theorem, we obtain 

  dx1(T; T )  @f ( x ; T ) =@T 1 =? dT @f (x1; T )=@x1 T =T  : T =T 

(8)

where, using the symmetry of the model11 

T T   @x2(T; T )  @f (x1; T )  x2(T; T ) ?   + = ?  @T T =T  x1(T; T )  1 ? x1(T; T ) x1(T; T ) @T T =T   2T ?1 = ? x2(T; T ) ?   x (T; T ) x1(T; T ) 2d2  1 2 T = ?1 + d ; (9) 2 11 Recall that symmetry implies x1 (T; T  ) = x2 (T; T  ) = 1 at T = T  . 2

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and 

    @f (x1; T )  = 2d1 +  (1 ? x2 (T; T ))2 T  + x2 (T; T ) 2 T @x1 T =T  (1 ? x1 (T; T )) (x1(T; T )) = 2d1 + 4T: (10)

Therefore, 



1 ? 2dT2  (11) = 2d + 4T dT 1 T =T  It is worth noting that [dx1(T; T )=dT ]T =T  < 0 for 2T > d2 . The reason is that when the rich are highly mobile (low d2 ), a tax increase induces so many rich to leave that the poor also nd pro table to leave. We are now in a position to charaterise the (symmetric) equilibium tax rates. Anticipating how the rich and the poor respond to any pair of tax rates (T; T ) with T close to T  , the domestic jurisdiction chooses T so as to maximize the income of its poor residents, taking the tax rate of the foreign jurisdiction as given. Formally, T solves 

dx1(T; T ) 

T; T ) MaxT 2[0;1] B(T; T ) = T xx2((T; T ) 1

(12)

Di erentiating B (T; T ) with respect to T around T = T  and using the symmetry of the model together with (5) and (11), we have 

 ) )  )  @x1 (T; T )  T x ( T; T @x ( T; T Tx ( T; T @B(T; T )  2 2 2 = x (T; T ) + x (T; T ) ? x2(T; T ) @T @T @T 1 1 T =T  T =T  T =T  1!   2 T 1 ? d2 T ?1 ? T = 1 + x (T;   x1(T; T ) 4T + 2d1= 1 T ) 2d2

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T ? 2dT22 T = 1 ? d ? 2T + d = : 2 1

(13)

Simple calculation shows that 

@B(T; T )  > 0 @T T =T  = 0

< 0

: T < d2 1 ? d : T = 1 ?d2d

: T > 1 ?d2d ;

and thus for d2  1 ? d, 

@B(T; T )  >0 @T T =T 

8T 2 [0; 1]

Our rst proposition follows directly

Proposition 3.1 Suppose that each jurisdiction chooses its tax rate by

majority voting. Then (i)if the poor outnumber the rich in the population, (  1), the symmetric Nash equilibrium in tax rates is 8 > < d =1 ? d  =T => 2 : 1

for d2 < 1 ? d for d2  1 ? d (ii) if the rich outnumber the poor, ( > 1), the symmetric Nash equilibrium in tax rates is T = T = 0

T

So, a greater mobility of the rich (that is, a lower d2 ) reduces the equilibrium tax rates. This is not surprising since a greater mobility of the rich increases the tax elasticity of the tax base. More surprisingly, proposition 3.1 suggests that a greater mobility of the poor (higher d) leads to higher 15

taxes and thus more redistribution. Though by no means apparent a priori, this nding that a greater mobility of the poor is bene cial to redistribution has a ready explanation which is best thought of as an envelope result. In equilibrium, when the poor control both jurisdictions, each jurisdiction maximises its transfer level taking the tax rate in the other jurisdiction as given. So a small increase in the tax rate in jurisdiction i has no e ect on its transfer level but induces the rich to migrate in jurisdiction j . The resulting increase in jurisdiction j 's tax base raises its transfer level and thus attracts the poor in this jurisdiction. So any deviation from the equilibrium causes the poor to chase the rich. This reduces the incentive for each jurisdiction to reduce its tax rate and thus leads to higher taxes. We now turn to the derivation of asymmetric equilibria.

4 Asymmetric equilibria We shall focus on asymmetric equilibria resulting from the fact that each jurisdiction is controlled by a di erent class of individuals.12 So in equilibrium the rich are in a majority in only one jurisdiction. Suppose without loss of generality that it is in the foreign jurisdiction. It follows that T  = 0. What is then the best response of the domestic jurisdiction controlled by the poor? To answer this question it is useful to distinguish whether the rich outnumber the poor in the population. 12 As a result we ignore the possibility that two jurisdictions under the control of the

same class (say, the poor) can still choose di erent tax rates. We leave this for future research.

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4.1 The rich outnumber the poor ( > 1) The following proposition states that redistribution is possible under majority voting although the number of rich individuals exceeds the number of poor individuals in the economy. This is because the poor are suciently mobile (i.e. d is high enough) compared to the relative number of rich individuals ( > 1) to form a majority coalition in one jurisdiction.

Proposition 4.1Suppose that each jurisdiction selects its policy by ma-

jority rule and that 1 <  < Minf2 + d; 4g. Then there exist two (pure strategy) Nash equilibria in tax rates: (b) Asymmetric : T = d22 ,T  = 0

The logic behind this proposition is simple. The asymmetric equilibrium (b) is supported by a distinct majority in each jurisdiction. The poor gather in one jurisdiction and impose a positive tax rate on the rich who decide to stay (i.e., those who are the most attached to this jurisdiction). Since  > 1 the other jurisdiction is necessarily under the control of the rich and thus chooses a zero tax rate. This intensi es competition between jurisdictions and explains that the tax rate chosen by the jurisdiction under the control of the poor is lower than the tax rate that would have been charged if the other jurisdiction were also under the control of the poor (leading to tax rates as described in proposition 3.1). In fact the tax rate chosen by the jurisdiction under the control of the poor corresponds to its revenue maximising tax rate, T = d22 which also turns out to be its transfer maximising tax rate

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given that the other jurisdiction chooses a zero tax rate.13 Surprisingly the mobility of the poor has no e ect on this equilibrium tax rate. However, a greater mobility of the poor induces more poor individuals to join the taxing jurisdiction reducing the transfer level in that jurisdiction. In the extreme, if the poor are highly mobile, they would all gather in the taxing jurisdiction. This suggests a potential tension between the poor natives (those with preference x  1=2) and the poor migrants (those with preference x > 1=2) as the migrants depress signi cantly the bene t to the natives. 14 A clear implication of this is that the poor natives may agree with the rich residents to limit the in ows of poor migrants, by supporting a transfer policy that discriminates against the non-natives. This explain why the equal treatment principle may be untenable the facto with an increased mobility of the poor. However, as the next proposition clearly shows, a policy that discriminates against the non-natives in reducing the mobility of the poor (which is akin to a lowering of d) may induce one jurisdiction to overtax its rich residents. This is because the inhibited mobility of the poor requires that the poor natives overtax the rich to keep a majority. As a result, they select a tax rate on the downward sloping side of their La er curve, T > d22 . A striking feature of this is that the poor who are in a majority would resist to a potentially Pareto improving tax reduction (that would attract the rich) in fear of a shift in the majority. 13 Using the fact that x2 (T; 0) = Minf0; 1 ? T g, it is readily seen that the maximisation 2 2d2 of tax revenue T x2 (T; 0) involves T = d22 . 14 As a matter of illustration, in our model when each rich resident transfers T , each poor native receives only T , with  = 1+p11+d . So, the transfer loss due to the poor migrants is increasing with the mobility of the poor (d)

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Proposition 4.2(Political Failure) Suppose that each jurisdiction selects

its policy by majority rule and that Minf2+ d; 4g < . Then the asymmetric Nash equilibrium in tax rates has the features that one jurisdiction selects a tax rate on the downward-sloping side of its La er curve (T > d22 ).

This proposition provides a novel example of political failure in which the fear of a shift in the majority prevents the realisation of a potentially Pareto improving policy change. The asymmetric equilibria in propositions 4.1 and 4.2 are illustrated in gure 2. The condition stated in proposition 4.1 implies that the curve x2 (T; 0) intersects the curve x1 (T; 0) on its upward sloping side and thus that the poor are in a majority at the transfer maximising tax rate (i.e. T = d2=2). As a result, this jurisdiction selects the transfer maximising tax rate (see, equilibrium A). On the other hand, when the condition stated in proposition 4.2 is satis ed, the curve x2 (T; 0) intersects the curve x1 (T; 0) on its downward sloping side so that the poor can keep a majority only by overtaxing the rich (see, equilibrium B). This equilibrium is Pareto dominated since a tax reduction would make the rich better o (even those who are induced to move) and would increase the transfer level making also the poor better o (even those who are induced to move). Notice that this pareto-dominated majority choice arises when the rich individuals are numerous enough and when the poor are not mobile enough. This easily seen on the gure since an increase in  shifts upwards the curve x2(T; 0) while a lower mobility of the poor attens out the curve x1 (T; 0). Insert Figure 2 here

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4.2 The poor outnumber the rich ( < 1) Having more poor individuals than rich individuals in the economy improves the redistribution possibilities under majority voting as they can guarantee themselves a majority in at least one jurisdiction, whatever their relative mobility (d). Moreover, in any symmetric equilibrium (T = T ) the poor and the rich are equally divided between the two jurisdictions and so the poor are in a majority in both jurisdictions. In that case each jurisdiction selects a policy that maximises the income of its poor residents and the equilibrium tax rates are such as given in proposition 3.1. In that symmetric equilibrium the mobility of the poor is thus bene cial to redistribution for the same reason that explained in section 3. The rich however have the possibility, if they are not too few (i.e. if  is not too low) to gather in one jurisdiction so as to form a majority and rule out any redistribution in this jurisdiction. This is more likely to arise if the poor are highly mobile. If the rich are able to gain a majority in one jurisdiction, then the other jurisdiction (where the poor are necessarily in a majority) would select its revenue maximising tax rate. This is the statement of the next proposition.

Proposition 4.3 Suppose p that each jurisdiction selects its policy by ma-

d jority rule and that 1 ? 1+ 3 <  < 1. Then there exist an asymmetric equilibrium in in tax rates de ned as in Proposition 4.1

On the other hand, if the rich are too few, they cannot gain a majority in any jurisdiction. Each jurisdiction is then controlled by the poor and thus will select the tax rate that maximises the income of their poor residents as in Section 3. 20

Proposition 4.4 Suppose p that each jurisdiction selects its policy by ma-

d jority rule and that   1 ? 1+ 3 . Then there is no asymmetric equilibrium in which each jurisdiction is controlled by a distinct majority.

The gure below illustrates the di erent voting equilibria as a function of  and d. Insert Figure 3 here

5 Conclusion The accelerated integration of markets has led to the widespread concern that the associated greater mobility of individuals will make redistribution more dicult. The existing theoretical works on the subject supports by large this concern. However they presume somewhat arbitrarily that either the rich or the poor are immobile. In this paper we consider that both the rich and the poor are mobile (although in di erent degrees) and we analyse local redistribution when jurisdictions are required to balance their budget and choose their tax-transfer policies either to maximise the income of their poor residents (or similarly to minimise income inequality) or by majority rule. This paper reveals two novel results. First, improving the mobility of the poor increases the equilibrium level of redistribution. This is because any departure from the equilibrium tax rates involves the poor chasing after the rich so that a greater mobility of the poor reduces the incentive of each jurisdiction to reduce their tax rate and leads to higher taxes in equilibrium. 21

A second result is that, under majority voting, some jurisdictions may choose in equilibrium a tax rate on the wrong side of their La er curve. The reason is that the poor are in a majority in these jurisdictions and are aware that a tax reduction would attract the rich. As a result they object to any potentially-Pareto improving tax reduction by fears of a shift in the majority. This is a novel example of political failure which is caused by the interplay between mobility and majority voting, and to which there is apparently no solution. Even though these results are derived from a very stylized model, we believe that they suggest to ndings with much broader scope. The rst general nding is that the mobility of the poor (i.e., the net bene ciaries of a policy) mitigates the harmful e ects of the mobility of the rich (i.e. the net contributors to the policy) since it enables the poor to \chase " the rich. In that sense, what is harmful to redistribution is not the mobility of the poor but contrarily the impediments to this mobility (i.e. zoning requirements, immigration policy, ...). The second general nding is that under free mobility (that is, free membership) the majority rule may select Pareto-dominated policies because the current majority would oppose to Pareto-improving policy changes that a ect membership in fears of a loss of political in uence. We believe that this second nding has much relevance in public decision making and may well explain some instances of political inaction.

22

Appendices A. Proof of Proposition 4.1 The proof of part (a) is immediate as for T = T  the rich and the poor are equally divided between the two jurisdictions and since  > 1 the rich form a majority in both jurisdictions imposing their most preferred policy which involves T = T  = 0. We now prove part (b). We must establish that the pair (T; T  )= ( d22 ; 0) is the unique asymmetric equilibrium if 1 <   Minf2+ d; 4g. Since  > 1, the rich can guarantee themselves a majority in at least one jurisdiction. Suppose (without loss of generality) that it is in the foreign jurisdiction, so T  = 0. Thus for an asymmetric equilibrium we must have T > 0 which requires that the poor are in a majority in the domestic jurisdiction. Given T  = 0, the equilibrium tax rate of the domestic jurisdiction is then the solution of

ArgMaxT 2[0;1]B(T; 0) = T xx21((T;T;0)0) subject to the majority constraint

x1(T; 0)  x2(T; 0)

(A. 1)

To solve this optimisation programme we have rst to determine the migration functions x1 (T; 0) and x2 (T; 0) taking account of the potential corner migration solutions. Given a pair (T; 0), x1 (T; 0) is de ned as

x1(T; 0) = Minf1(T; 0); 1g

(A. 2)

where 1 (T; 0) is characterized by the marginal poor individual x = 1 (T; 0) who is indi erent between the two jurisdictions. That is x = 1 (T; 0) solves 23

0) ? d x = 0 ? d (1 ? x): T x2(T; 1 1 x

(A. 3)

Given a pair (T; 0), x2(T; 0) = Maxf0; 2(T; 0)g where 2 (T; 0) is characterized by the marginal rich individual x = 2 (T; 0) who is indi erent between the two jurisdictions. That is x = 2 (T; 0) solves (1 ? T ) ? d2x = 1 ? d2(1 ? x) Thus we have,

2(T; 0) = 21 ? 2Td2 Therefore,

x2(T; 0) = 21 ? 2Td 2 = 0

:

T 2 [0; d2) : T  d2

(A. 4)

Notice that tax revenue in the domestic jurisdiction are equal to Tx2(T; 0) and thus that the revenue maximising tax rate is T = d22 (ignoring the majority constraint). We now show that the unconstrained optimisation problem of the domestic jurisdiction requires to select the revenue maximising tax rate, T = d22 and that at this solution the majority constraint is satis ed, (i.e.x1 ( d22 ; 0)  x2( d22 ; 0)) if and only if   Minf2+ d; 4g. Two cases must be considered : x1 ( d22 ; 0) < 1 and x1( d22 ; 0) = 1. CASE 1 :x1 ( d22 ; 0) < 1 From (A2), x1(T; 0) = 1 (T; 0) for any T such that 1 (T; 0)  1 with 1(T; 0) given by (A3). Applying the implicit function theorem to condition (A3) gives

24

d1(T; 0) =  x21((T;T;0)0) + 1T(T; 0) @x2@T(T;0) dT T x212((T;t;0)0) + 2d1 x2(T; 0) ? 2Td2 = T x21((T;T;0)0) + 2d11(T; 0) ( 1 ? dT2 ) = d (4 2(T; 0) ? 1) ; 1 1

where the second equality follows from (A4), and the third equality follows from (A3) and (A4). Using the fact that 1 (T; 0)  1=28T 2 [0; 1], it follows that d2 d1(T; 0) > 0 : T < dT 2 d = 0 : T = 22

< 0

:

T > d22 :

(A. 5)

Thus since x1 ( d22 ; 0) = Minf1 ( d22 ; 0); 1g < 1, x1(T; 0) = 1 (T; 0) for all T 2 [0; 1]. Using (A3) this implies that B(T; 0) = 2d1x1(T; 0) ? d1 for all T 2 [0; 1] so that the optimisation problem of the domestic jurisdiction reduces to

MaxT 2[0;1]2d1x1(T; 0) ? d1 subject to (A1). Since x1 (T; 0) = 1 (T; 0) 8T , if follows from (A5) that the unconstrained solution is T = d22 . We now show that at this solution the majority condition (A1) is satis ed. From (A4),x2( d22 ; 0)p= 41 and from (A3) it follows after simple calculation that x1( d22 ; 0) = 1+ 41+d . Hence x1 ( d22 ; 0)  x2( d22 ; 0) p implies 1+ 41+d  4 which is easily seen to be true since by assumption   2 + d. 25

CASE 2 :x1 ( d22 ; 0) = 1 Using (A5) and the fact that x1 (T; 0) = Minf1(T; 0); 1g, it follows that there exists a vector (1; 2)  0 such that x1(T; 0) = 1 for all T 2 [ d22 ? 1 ; d22 +2 ] and x1 (T; 0) = 1 (T; 0) < 1 for all T 2= [ d22 ? 1 ; d22 +2 ]. But from case 1, we know that any T 2= [ d22 ? 1; d22 + 2] cannot be an equilibrium. So, any equilbrium has the features that T 2 [ d22 ? 1; d22 +2 ] and thus x1(T; 0) = 1. This implies that the optimisation problem can be written as

MaxT 2[ d22 ?1 ; d22 +2 ]T x2(T; 0) subject to (A1). Using (A4) the unconstrained solution is T = d22 . From (A4), x2 ( d22 ; 0) = 1 and thus the majority constraint (A1) is satis ed since x1 ( d2 ; 0) = 1 and 4 2 by assumption   4. This completes the proof.

B. Proof of Proposition 4.2 We rst show that T = T  = 0 is an equilibrium outcome. Since  > 1, the symmetry of the model implies that the rich form a majority in both jurisdictions when T = T . Thus T = T  = 0 is the unique symmetric equilibrium outcome supported by a majority of rich residents in each jurisdiction. We now show that if  > Minf2 + d; 4g, there exists a unique asymmetric equilibrium and that at this equilibrium T > d22 . Any asymmetric equilibrium requires that the poor form a majority in one jurisdiction (say the domestic one). Since  > 1, the rich form necessarily a majority in the other jurisdiction and thus T  = 0. Fixing T  = 0, from the proof of proposition 4.1 it is straightforward to verify that

dB(T; 0) > 0 dT

: 26

T < d22

= 0

:

< 0

:

T = d22 T > d2 : 2

(B. 1)

Thus to prove that a jurisdiction with a majority of poor individuals selects T > d22 in equilibrium in response to T  = 0, we proceed in two steps. In step (i) we show that x1(T; 0) < x2(T; 0) for all T  d22 , and then in step (ii) we show that there exists T^ 2 ( d22 ; d2) where T^ is de ned as the lowest T such that x1(T; 0) = x2 (T; 0) . Having shown that, then it is readily seen by using (B1) that in equilibrium T = T^ > d22 . Step (i). Using (A4) gives x2( d22 ; 0) = 14 . Using (A2) and (A3),one nds

p1 + d d 1 + 2 x ( ; 0) = Minf ; 1g: 1 2

4

(B. 2)

It is then straightforward to verify that x1 ( d22 ; 0) < x2( d22 ; 0) since by assumption  > Minf2 + d; 4g. Since from (A2) x2(T; 0) is non-increasing for all T and since from (A5), x1 (T; 0) is non-decreasing for all T  d22 , it follows that x1 (T; 0) < x2(T; 0) for all T  d22 . Step (ii). We must show that there exists T^ 2 ( d22 ; d2) where T^ is the lowest T such that x1 (T; 0) = x2(T; 0). To prove this it suces to show that x1 (T; 0) intersects x2 (T; 0) at least once over the domain T 2 ( d22 ; d2). Inspection of (A4) shows that x2 (T; 0) is continuous in T . On the other hand, it follows from the implicit function theorem that 1 (T; 0) is di erentiable and thus continuous in T . Therefore x1 (T; 0) = Minf1 (T; 0); 1g is also continuous in T . We already know from step (i) that x1( d22 ; 0) < x2( d22 ; 0). Combining (A2) and (A3) together with (A4) we derive that x1 (T; 0) > x2(T; 0) when 27

T approaches d2. Thus by continuity it follows that x1(T; 0) intersects at least once x2(T; 0). This completes the proof.

C. Proof of Proposition 4.3 The proof of part (a) is straightforward since  < 1 implies by the symmetry of the model that for any symmetric equilibrium T = T  , the poor outnumber the rich in each jurisdiction. Thus in any symmetric equilibrium, each jurisdiction chooses its tax rate so as to maximise the income of their poor residents. This leads to the symmetric equilibrium given in proposition 3.1. We now prove part (b). To have an asymmetric equilibrium, the rich must form a majority in one jurisdiction (say, the foreign jurisdiction) and thus T  = 0. Since  < 1, this implies that the poor are in a majority in the domestic jurisdiction. It follows that domestic jurisdiction's equilibrium tax rate T solves MaxT 2[0;1] B (T; 0) subject to x1 (T; 0)  x2(T; 0). From (B1), the unconstrained solution is T = d22 . It remains to check that at this solution, the poor are in a majority in the domestic jurisdiction to support T = d22 and that the rich are in a majority in the foreign jurisdiction to support T  = 0. That is, we must have

x1( d22 ; 0)   x2( d22 ; 0) 1 ? x1 ( d22 ; 0) <  (1 ? x2 ( d22 ; 0)):

(C. 1) (C. 2)

Since  < 1, (C1) is satis ed if (C2) is satis ed. Thus it suces to show that (C2) is satis ed. Using (A4) and (B2), condition (C2) reduces to

p 1 ? Minf 1+ 41+d ; 1g < 43 

It is then straightforward p1+d to verify that this condition is satis ed since by assumption  > 1 ? 3 . This completes the proof. 28

D. Proof of Proposition 4.4 The proof of part (a) is the same as the proof of part (a) in proposition 4.3. To prove part (b) we show that there cannot be an asymmetric equilibp1+ rium if   1 ? 3 d . To have an asymmetric equilibrium, the rich must be in a majority in one jurisdiction. Suppose this is true (say, in the foreign jurisdiction). Then T  = 0 and since  < 1 the poor are in a majority in the domestic jurisdiction. Hence, from the proof of proposition 4.3, we know that the domestic jurisdiction will choose T = d22 and that at this solution p1+d the rich can form a majority in the foreign jurisdiction only if  > 1 ? 3 , which is a contradiction.

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