Marginal Income Tax Reforms and Welfare IX EcuentrodeEconomiaP¶ ublica,Vigo2002
Maria Cubel University of Barcelona and University of York Facultat d'Economiques Departament d'Hisenda Publica 690, Av. Diagonal, 08034, Barcelona, Spain Email:
[email protected] October 29, 2001
Abstract Is it possible to increase overall welfare through a spatial di®erentiation of the income tax ? In this paper we explore the possibility for a marginal tax reform based on the di®erentiation of a proportional tax among jurisdictions. We use a methodology developed by Yitzhaki and Slemrod (1991) to analyse Dalton-improving tax reforms applied to commodity taxation. We adapt this methodology replacing commodities by incomes belonging to households who live in two di®erent regions A and B, and we de¯ne welfare improving dominance conditions associated to a marginal di®erentiation of the tax rate in every region of a uniform proportional income tax. Migration movements and the e®ect of the tax reform on taxpayer's taxable income are considered in the analysis. Numerical simulations using income tax data for Spain illustrate the possible impact of the suggested tax reform on overall welfare. JEL Numbers D63, H24, H73, R12. Keywords: income tax, regions, welfare.
1
Introduction
In this paper we explore the possibility for a revenue-neutral marginal tax reform based on the di®erentiation of a proportional income tax among jurisdictions taking into account the possible migration and behavioural e®ects of the reform. To simplify the analysis we consider only two jurisdictions to which we will refer as region A and region B. We use the term regions arbitrarily so the analysis could be perfectly well applied to any other administrative division like municipalities, EU regions (NUTS), countries, etc.,... We adapt a methodology developed by Yitzhaki and Slemrod (1991, hence - forth referred to as Y-S)1 to analyse Dalton-improving tax reforms applied to commodity taxation. The objective of this methodology is to evaluate the e®ect of a marginal tax reform on overall social welfare allowing for a wide range of social welfare functions (SWFs). We apply the same methodology replacing commodities by incomes belonging to households who live in A and B, and we analyse the welfare improving 1
Extensions of this approach can be found in Yitzhaki and Thirsk (1990), Mayshar and Yitzhaki (1995), Mayshar and Yitzhaki (1996),Yitzhaki and Lewis (1996).
1
necessary conditions associated to a marginal di®erentiation of the tax rate in every region of a uniform proportional income tax. The main advantage of this methodology is that it allows us to bring in plausible migration and changes of behaviour resulting from the reform. The simplest framework is to consider only two regions A and B. The suggested tax reform involves a small reduction in the tax rate of region A, (dtA < 0), and a small increase in the tax rate of region B, (dtB > 0), in such a way that total revenue does not change2 and we take into account the possible migration and behavioural e®ects of the reform on taxpayer's income. It is convenient to consider revenue-neutral tax reforms since we can then ignore the issue of the optimal size of government activity. As a result of the reform one would expect people to move from region B, where there is a marginal tax rate increase, to region A, where there is a marginal tax rate decrease, however, a priori we can not discard the possibility of migration from region B to region A. To analyse the possible migration patterns if any, we assume that migration is costly and that when household h moves from one region to the other she receives a monetary compensation, z h . We should think of this monetary compensation z h as a way of discriminating between mobile and immobile workers. Furthermore, we assume that only skilled workers are able to obtain this monetary compensation and consequently only skilled workers could migrate3 . Only when, after the proportional tax, z h exactly covers the moving cost, ch , and some conditions are respected, could migration appear. We also consider the special case of z h = 0 and we prove that in this scenario migration never occurs. The particular case of costless migration is also considered. In the following sections we describe the work of Y-S from which we borrow the methodology we utilise. In section 3 we describe the analytical framework. In section 4 we develop the model and de¯ne the necessary and su±cient conditions under which the suggested tax reform is welfare improving. In section 5 we introduce some special cases which allow to obtain further results. Section 6 presents some numerical simulations. And ¯nally section 7 is dedicated to the discussion and conclusions.
2
Methodological background
Y-S develop a smart methodology to identify revenue-neutral marginal commodity-tax reforms that will bene¯t welfare considering a wide range of SWFs. This methodology is based on an extension of the criterion of second-degree stochastic dominance4 and its objective is precisely to avoid any particular speci¯cation of the SWF and therefore to avoid any particular value judgments. The only needed requirement is the existence of an a priori ordinal ranking of households by their marginal social desert (from relatively less deserving "poor" households to relatively more deserving "rich" households). But it does not require a cardinal measure 2
We could also consider that both the tax rate in region A and the tax rate in region B would change in the same direction without modifying total revenue, in other words, that both tax rates would increase or decrease without modifying total revenue. This would mean that regions are on opposite parts of their La®er curves, an unlikely scenario which we shall subsequently neglect. 3 That only skilled workers are able to migrate is a common assumption in models of tax competition , however, the opposite assumption has also been explored. See, for example, Wildasin (1992), Cremer et al. (1996) Wildasin (1998) and Wilson (1999) for a complete discussion of factor mobility and many references. 4 This methodology was originally developed in the ¯nance literature. There has been special interest in developing rules for ranking assets according to their expected utility assuming only that the marginal utility of income is nonnegative and nonincreasing. Further information can be found in, for example, Hadar and Russell (1969), Hanoch and Levy (1969).
2
of the degree by which one household is more deserving than another. Furthermore it can be applied to any a priori ordinal ranking of households. The tax reform that Y-S propose is a marginal modi¯cation of the tax rates of pairs of commodities such that the tax increase in one commodity is used to subsidize the tax decrease in another commodity holding total revenue from commodity taxation constant. In doing that they are producing implicit transfers from households who consume the commodities with an increase in their taxes to households who consume the subsidized commodities. Following Dalton (1920) these transfers should reduce inequality and be welfare improving provided that there is not reranking of households as a result of the transfers. Because Y-S are considering a marginal tax reform, transfers should be small enough not to reverse the ranking of households. Therefore it may be possible to ¯nd out which commodities should be subsidized and which commodities should have their taxes augmented such that welfare increases for all additive concave SWFs. If this is possible then speci¯c recommendations for tax reforms can be made. However, if such commodities can not be found then no recommendations can be made in the absence of further information about the SWF. Y-S develop an elegant and simple method to distinguish such pairs of commodities in the case that they might exist. Y-S assume an additive SWF from which the only information known it is that it has positive and declining social evaluation of the marginal utility of income. Households may have di®erent utility functions, however, when two households have the same income it is assumed that in the eyes of society they have equal welfare. Households are ordered by nondecreasing income and the variable used to evaluate the welfare implications of the reform is the household's share in total expenditure for the selected pair of commodities. Their method ¯nally reduces to the comparison of (scaled) concentration curves5 . E±ciency implications of the reform result in shifts of the concentration curves. They demonstrate that if the (scaled) concentration curve of one commodity dominates the (scaled) concentration curve of another commodity this means that social welfare can be improved by reducing the tax rate on the ¯rst commodity and increasing the tax rate on the other commodity. However, when the concentration curves of any two commodities intersect then nothing can be said and a Dalton-improving reform involving this pair of commodities does not exist. Additional insight is given by relating these marginal conditional stochastic dominance rules to a decomposition of the Gini coe±cient. The marginal conditional stochastic dominance rules do not o®er a complete ordering of all commodity tax reforms. In order to obtain a complete ordering Y-S look for necessary conditions for welfare dominance using a decomposition of the Gini coe±cient. This procedure makes marginal conditional stochastic dominance rules more operational since it reduces the number of concentration curves which should be analysed, and the procedure it is reduced to a familiar regression program. However, the Gini coe±cient implicitly imposes a speci¯c welfare function with a speci¯c weighting scheme. They themselves, suggest as an alternative to the Gini coe±cient the use of the extended 5
The concentration curve is a diagram similar to the Lorenz curve. On the horizontal axis the cumulative percentage of households ordered from poorest to richest appears, whilst the vertical axis describes the cumulative percentage of the total expenditure on a speci¯c commodity that is spent by the families whose incomes are less than or equal to the speci¯ed income level. The concentration curve, like the Lorenz curve, passes through the origin, but, unlike the Lorenz curve, it need not always be convex. Its curvature depends on the structure of the income elasticity of the commodity, if the income elasticity is negative (positive) the concentration curve is convex (concave) to the origin.
3
Gini coe±cient, which allows for di®erent weighting schemes given the possibility of choosing di®erent values for the inequality aversion parameter. Finally they present an empirical application of their methodology using Israeli data from the Survey of Family Expenditure (1979/1980) and using the extended Gini coe±cient. Extensions of this approach can be found in Yitzhaki and Thirsk, (1990) with an illustration using data from de Cote d'Ivoire, in Mayshar and Yitzhaki, (1995) using data on excise taxes in the UK, and in Yitzhaki and Lewis, (1996) with an empirical application to the energy sector of Indonesia. More interesting is the more sophisticated derivation of this approach which appears in Mayshar and Yitzhaki, (1996) where the criterion for Dalton improving reforms is extended to a two-dimensional criterion taking into consideration di®erences in abilities and needs when ranking households by social desert. Hence it is assumed that there exist social approval for transfers from the more able to the less able and from the less needy households to more needy ones. In all those works the authors emphasize that the exact properties of SWFs are unknown and consequently it is extremely valuable to have a methodology which enables us to make judgments about potential tax reforms that depend only on relatively noncontroversial characteristics of the SWF. Therefore, Dalton improving tax reforms are a very useful alternative to social welfare-based tax reforms because even if the amount of required data is the same at least the reliance on data is weaker under Dalton improving reforms (Yitzhaki and Lewis, (1996), p.541). Thinking of further applications of this methodology, Yitzhaki and Lewis, (1996) point out that it would be possible to apply this kind of analysis approach to di®erent collectives of people. For example, Dalton improving tax reforms which could bene¯t both rural and urban populations or, alternatively, for several subgroups of the population, such as di®erentsize households, see Mayshar and Yitzhaki, (1996). We have decided to choose regions as our criterion to classify households and we apply a modi¯ed version of this methodology considering income tax di®erentiation instead of commodity tax di®erentiation. We are aware of the political problems to implement such a tax reform, however, we should not forget that we are only contemplating a marginal tax reform and therefore the actual di®erentiation of the income tax is very small. Nonetheless, we still think it is valuable to explore whether it is possible to de¯ne stochastic conditions under which welfare improvement may occur. If this would be possible then we could maybe all agree to introduce a small degree of horizontal inequity in favor of vertical equity if the ¯nal result would be an increase in overall social welfare. This is what we try to explore in the subsequent sections.
3
Analytical framework
Consider for the simplicity of the analysis that a country is composed of only two regions A and B. Overall social welfare may be expressed according to an individualistic social welfare function. All that is known about this social welfare function is that the social evaluation of the marginal utility of income is positive and declining. Formally, the welfare function takes the form, W [v 1 (y 1 ); ::::; v N (y N ); p1 ; ::::; pm ]
4
(1)
where y h is the disposable income of household h, the p's are prices of n goods, and v h is the indirect utility, of household h. We assume that all households face, of m goods, the same vector of prices p independently of the region they live in. Hence the welfare of every household depends only on her level of income, considering income in a broad sense. Furthermore, we assume that regardless of the speci¯c form of the utility function, the social evaluation of the marginal utility of income (marginal welfare), denoted by ¯, is only a function of income y h . @¯(yh ) @W @vh Hence ¯(y h ) = ( @v < 0. This means that, h )( @y h ) > 0 and we further assume that @yh two households with the same disposable income have also the same social evaluation of their @vr @W @v s utilities, that is, for any two households r and s with y r = ys then ( @W @vr )( @yr ) = ( @vs )( @y s ). The Government implements a proportional income tax uniform in all the territory. The total revenue collected from this proportional income tax is,
R=t
N X
xh = tX
(2)
h=1
where t is the uniform tax rate, xh is the personal income of household h, N is total number of households and X is total taxable income. Now suppose that the government is considering a marginal tax reform based on the spatial di®erentiation of the proportional tax rate t. Consider that the government decides a small reduction in the tax rate of region A, (dtA < 0), and a small increase in the tax rate of region B, (dtB > 0), in such a way that total revenue does not change. It is convenient to consider a revenue-neutral reform since then we can omit the issue of the optimal size of government activity. We should take into account that this kind of tax di®erentiation might well provoke migration movements from one region to the other. We would expect people to move from region B, where there is a marginal tax rate increase, to region A, where there is a marginal tax rate decrease. However, there are many factors that may discourage households from moving. Following King (1984), the more relevant ones are family and friends liaisons , job commitments, cost of moving which includes removal costs, the costs of searching for new accommodation, the possible costs of selling and purchasing new properties in the case of owner-occupiers, the advantages of living in an area well known, or in other words, the cost of losing a familiar environment, the cost of losing a speci¯c landscape and climate, and ¯nally he also mentions institutional rigidities such as zoning regulation, speci¯c social policies of local governments, etc... To this long list of factors we should also add language di®erentials between jurisdictions besides it could also be included in the group of factors associated with the environment. The smaller and the more homogenous the jurisdictions are the most likely migration is to occur since the costs of moving are reduced. Therefore, one would expect to ¯nd more migration between municipalities than between regions and countries. In order to analyse the possible migration patterns resulting from the reform, if any, we assume that wages are uniform along all the territory, which of course is a restrictive simpli¯cation of reality. Then we also assume that when individuals move from one region to the other their incomes change since they receive a certain amount of money in compensation, z h . After this we contemplate as well the special case of no income di®erentials as a result of migration among regions. Individual incomes would also remain the same after migration when considering the case of commuting for working reasons. If households become commuters changing their 5
region of domicile while they keep their original jobs consequently their wages do not change. Obviously, the commuting phenomenon is more likely to occur the shorter is the distance between the places they live and they work in since for shorter distances the transport cost and the non-monetary cost (environmental di®erences, friends, family, etc.,....) of moving are smaller. Therefore, it is expected that commuting, as in the case of migration, would more likely appear between municipalities of the same region and in the bordering areas between regions. We should prove in the following section that under the methodology we are applying commuting for working reasons can never occur since the marginal tax bene¯ts are not important enough to compensate the cost of moving. Only when we would consider costless migration that situation would be possible. Commuting because of change in the work place is not considered in the analysis, instead we have assumed that when a household moves to work in another region it also changes its residence to that region. This is because we are considering a tax based on the region of domicile obviously the case of a tax based on job location could also be explored, then we could contemplate the case of changing the job location for ¯scal reasons while keeping the domicile. Therefore we should admit that commuting phenomena are not properly explored in this work. Regarding migration, only when households migrate and change their jobs receiving a monetary compensation equal to the cost of moving, might migration appear as a result of the tax reform. We prove that this is a necessary but not su±cient condition for migration.
3.1
General scenario: individual incomes change with migration
We assume that, when households move from their region of domicile to another one, they receive a monetary compensation, which we name z h , and that this is taxable. Thus, if household h would move from region i to region j, before the tax reform, she would receive an amount of money z h in such a way that her disposable income, previously xh (1 ¡ t), would then be (xh + z h )(1 ¡ t). Households living in any region would decide to move to another region, as a result of the tax reform, if and only if the disposable income that they would obtain in the new region, after migration, is greater than the disposable income that they would obtain in their original region plus the moving cost. The disposable income of household h after the proportional uniform tax is de¯ned by h y = (1¡t)xh regardless the region of residence, and the disposable income after the marginal tax reform when household h does not migrate from region i is de¯ned by, yh + [
@y h @xh ]dti = [xh + ( )dti ][1 ¡ t ¡ dti ] @t @t
(3)
where i = A and B. If before the tax reform household h would have decided to migrate from region i to region j, however, she would have received the additional compensation z h and, therefore, her disposable income would have become (1 ¡ t)(xh + z h ) = y h + z h (1 ¡ t). Hence, the disposable income after the marginal tax reform when household h migrates because of the reform is, yh + [
@y h @xh ]dtj + z h (1 ¡ t ¡ dtj ) = [xh + z h + ( )dtj ][1 ¡ t ¡ dtj ] @t @t 6
(4)
noting that the term in dt2j can be neglected as second order. Then, household h living in region i would move to a di®erent region j as a result of the tax reform if and only if, comparing (3) and (4)
(
@y h )(dtj ¡ dti ) ¡ z h dtj + z h (1 ¡ t) ¡ ch > 0 @t
(5)
where ch is the cost that household h has to bear for migrating from region i to region j. Let us now consider the case of a household living in region B who is thinking of moving to region A. The case of households living in region A and wanting to migrate to region B can be developed in a similar way, see on. Household h living in region B would be willing to move to region A, if and only if, from (5),
(
@y h )(dtA ¡ dtB ) ¡ z h dtA + z h (1 ¡ t) ¡ ch > 0 @t
(6)
Taking into account that dtA and dtB are in¯nitesimal, migration is possible if and only if the additional compensation received after the uniform tax, z h (1 ¡ t), is equal or greater than the cost of moving ch : However, if z h (1 ¡ t) > ch , households would have already decided to move regardless of the tax reform that we are considering. When z h (1 ¡ t) is strictly greater than ch , rational households would migrate anyway without needing ¯scal incentives. In the special case of having equal net of tax compensation z(1 ¡ t) and cost of moving c for everybody, a permutation of a certain subset of the populations between regions could occur. Theorem 1 The contemplated tax reform would induce household h to migrate from B to A if and only if the net of tax compensation z h (1 ¡ t) is equal to the cost of moving ch and the following condition is respected 6 , (
@y h )(dtA ¡ dtB ) ¡ z h dtA > 0 @t
Alternatively the expression in theorem 1 can be written as,
[
@y h @y h ¡ z h ]dtA > ( )dtB @t @t
(7)
where the left hand side is the gain from moving from region B to region A, and the right hand side is the gain from staying in region B (no migration). Our conclusion is thus, that when individuals receive a monetary compensation z h because of migration, a marginal tax reform which increases the tax rate in one region and decreases the tax rate in the other region, will induce migration in the obvious direction if and only if this compensation exactly covers without exceeding the moving cost (this is, z h (1 ¡ t) = ch ) and condition (7) is respected. This means that only when individuals were indi®erent about 6
h
If zh is not taxable (4), (5) and (6) change. Hence migration occurs if and only if zh = ch and ( @y )(dtA ¡ @t h
dtB ) > 0, ie., if and only if ( @y ) < 0. @t
7
migration before the tax reform was implemented, could they decide to migrate as a result of the tax reform. Everybody who migrates receives a compensation z h (1 ¡ t). If z h (1 ¡ t) > ch , household h already would have migrated regardless of the tax reform. If z h (1 ¡ t) = ch , only if condition (7) is respected would they migrate. In this latest case, the tax reform is the reason which induces individuals to migrate. Finally, if z h (1 ¡ t) < ch , individual h would not migrate since the e®ect of the tax reform is not strong enough to compensate for the cost of moving ch . We should understand the monetary compensation z h as a selecting mechanism which indicates households who are mobile and can migrate between regions. Having z h tax free would produce equivalent results to consider ch = 0 and divide the population arbitrarily between mobile and immobile households. For example, we could consider that only skilled workers are mobile which is equivalent to consider that only skilled workers can obtain the monetary compensation z h and hence become mobile. The extreme cases appear when we consider ch = 0 and z h = 0 which means that all households are perfectly mobile In that h case, migration from i to j would occur whenever ( @y @t )(dtj ¡ dti ) > 0: The other extreme case is to assume ch > 0 and z h = 0 which implies that households are completely immobile (no migration case). Analysing now the necessary but not su±cient condition for migration expressed by (7) we can go further and clarify under which requirements migration can take place. Hence, we ¯nd out that to have migration from region B to region A one of the following conditions is necessary although not always su±cient (recall that dtA < 0 and dtB > 0): h
h
@L a) ( @y @t ) < 0. This is consistent with having the labour supply (L) upward sloping ( @t < @xh @xh 0; xh = f(L; m); @L h > 0 (m being non labour income) and therefore, @t < 0); but, h it is also consistent with the case of backwards bending labour supply ( @L @t > 0, and h 7 therefore, @x @t > 0) . However, in this latest case the following additional condition is h h also required: (1 ¡ t) @x @t < x : This condition is also su±cient.
b) 0
( )dtA @t @t
7
The slope of the labour supply depends on the shape of the utility function . We would just mention some major cases: the Cobb-Douglas utility function and the utility function with constant marginal utility of leisure which are well known for having no labour e®ect when considering a proportional income tax. And, the LES and CES utility functions which show positive partial derivatives of labour with respect to the proportional tax rate.
8
This case is more intriguing since we would expect intuitively that no migration occurs from region A to region B. Therefore, it is important to clarify if there could exist migration in that direction. The necessary but not always su±cient conditions to have migration from region A to region B can be written as follows: (since in theorem 2, dtA < 0 and dtB > 0 ). h
h h a) 0 < z h < @y @t . Alternatively, this condition can be expressed as follows: 0 < z + x < h (1 ¡ t) @x @t This condition is also su±cient.
b) 0
0 everything crystallizes and it is easy to see that the necessary condition in theorem 1, [¡xh (1+½hx;(1¡t) )¡z h ]dtA > [¡xh (1+½hx;(1¡t) )]dtB , is always true and that the necessary condition of theorem 2, [¡xh (1 + ½hx;(1¡t) ) ¡ z h ]dtB > [¡xh (1 + ½hx;(1¡t) )]dtA , can never possibly occur. In other words, only people who were indi®erent between migrating or not migrating before the tax reform and who live in region B might migrate induced by the reform. Thus migration from region B to region A could possibly occur. However, migration because of ¯scal reasons from region A to region B de¯nitely can never exist. This simpli¯es the analysis considerably and supports the intuitively expected result.9 Up to now we have considered that migration is costly (ch > 0), however if we assume instead that ch = 0, and z h = 0 too then migration from region i to j would take place whenever h h ( @y @t )(dtj ¡ dti ) > 0. Under the assumption of ½x;(1¡t) > 0 this condition is respected only for migration from B to A. Furthermore, since now migration has no cost all the population from B would move to A for ¯scal reasons which is a very unrealistic situation. Therefore, we should not explore further the case of costless migration for the overall population. However, we could think of the whole population divided between mobile and immobile households in function of their income or their skills at work and this is precisely the role that the monetary compensation z h plays in the analysis. In order to analyse the impact of the considered tax reform, we impose that the revenue collected before and after the tax reform should be the same. The equal revenue constraint can be expressed as follows,
@XAA @XBA )]dtA + [XBA + t( )]dtA + @t @t X @XBB [XBB + t( zh = 0 )]dtB + (t + dtA ) @t
dR = [XAA + t(
(8)
h2BA
8
Estimations of the elasticity of taxable income with respect to t or (1 ¡ t) are o®ered for the whole population and in some studies are also desagregated for intervals of income. Here we are considering ½hx;(1¡t) and the main thing that we are concern about is its sign. However, later on we consider ½hx;t instead and we also assume for simpli¯cation in some sections that ½hx;t = ½x;t 8 h: h 9 Also if we consider the case ( @x @t ) = 0, theorems 1 and 2 show that there could only exist migration from region B to region A.
9
P h where Xij = h2ij x refers to total pre-reform gross income accruing to people who live i before the reform and in region j after the tax reform, t is the tax rate and P in region h z is the summation of monetary compensations received by migrants who move form h2BA region B to region A10 . After some manipulation, expression (8) can be written as follows, @XAP @XBP )]dtA + [XBP + t( )]dtB @t @t X zh = 0 +(t + dtA )
dR = [XAP + t(
(9)
h2BA
where XiP is total pre-reform gross income of those living in region i after the tax reform ( that is, XiP = Xii + Xji )11 , thus the subscript iP refers to new population (in region i) after the tax reform and therefore after migration occurs12 . Because we are considering a marginal tax reform the number of migrants should be in¯nitesimal. Let us refer to households migrating from i to j as ni . Hence, ni would tend to zero when dtA and dtB tend to zero, because the reason for migrating was the tax rate di®erential and now this di®erential has disappeared. When dtA and dtB tend to zero, households come back to their pre-reform stage, where they were indi®erent between migrating or staying in their original region. Therefore, they will not migrate and ni tends to zero. ni Let us now de¯ne "i as follows, "i = ( N ) : ( dtt i ), where Ni is total population in region i i before the tax reform and ni is the (in¯nitesimal) number of migrants from region i to region j. The parameter "i could be understood as an elasticity measuring the percentage migration response to the percentage change in the rate of the proportional tax. Hence we know that "A = 0 but we do not know the value of "B . Now, isolating t from "i and substituting it in (9) we obtain,
dR = [XAP (1 + ½xAP ;t ) + (1 + SA )
X
z h ]dtA + [XBP (1 + ½xBP ;t )]dtB
(10)
h2BA
where SA = dttA < 0 and ½xiP ;t = ( XtiP )( @X@tiP ) 13 . Taking into consideration that we are only contemplating a marginal tax di®erentiation, nB would tend to zero when dtA and dtB tend to zero, and then SA would tend to minus in¯nity. For small ¯nite changes (rather than in¯nitesimal ones) SA is large and negative and nB is small and ¯nite. Then manipulating equation (10) the following expression is derived: 10
Because there is no migration from region A to region B the equal revenue constraint is simpli¯ed. The complete expression for it in the case of migration among both regions would be: dR = [XAA + t( @X@AA )]dtA + P P t [XBA +t( @X@tBA )]dtA +[XBB +t( @X@tBB )]dtB +[XAB +t( @X@tAB )]dtB +(t+dtB ) h2AB z h +(t+dtA ) h2BA zh = 0 11 Recall that XAB = 0 since there only exist migration form B to A. Hence XBP = XBB and XAP = XAA + XBA . 12 When zh = z 8h, equation (9) becomes, dR = [XAP +t( @X@tAP )]dtA +[XBP +t( @X@tBP )]dtB +(t+dtA )nB z = 0, where nB is the number of households migrating from region B to region A (recall that migration from region A to region B can never happen). 13 Again when z h = z 8h equation (10) is also simpli¯ed , dR = [XAP (1 + ½xAP ;t ) + (1 + SA )znB ]dtA + [XBP (1 + ½xBP ;t)]dtB = 0:
10
dR = XAP dtA ®A + XBP dtB ®B = 0
(11)
P where ®i = (1 + ½xiP ;t ) + (1 + Si )( X1 ) h2ji z h for i = A; B and i 6= j. Recall nA = 0 iP P and therefore h2AB z h = 0 and ®B = (1 + ½X ;t ). The interpretation of the parameter BP ®i is then a bit obscure. We could understand ®i as a sort of marginal e±ciency cost of the suggested tax reform. Manipulating for ®i , this parameter can be written as: P theh expression t 1 14 . Hence, the ¯rst term in ®i includes the elasticity ®i = (1+½xiP ;t )+(1+ dti )( X ) h2ji z iP
of the tax base with respect to the tax rate in every region, [½xiP ;t = ( XtiP )( @X@tiP )]. This direct e®ect of the tax reform on the regional tax base encapsulates the e®ect that changes in the rate of the proportional income tax may produce on taxpayer's incomes. As Auten and Carroll (1999) point out taxpayer's incomes are in°uenced by many factors: modi¯cations of taxpayer's labour supply, change in the form of compensation from cash taxable compensation to fringe bene¯ts, savings and investment patterns, realization of expenditures which receive favourable tax treatment (charity, home mortgage, etc.,...), compliance with the tax law and also factors beyond the household's control such as businesses cycles, changes in interest rates, demographic changes and the general evolution of the economy. The second term in ®i refers to the proportion that the sum of compensations received by those individuals moving to region i represents over total new income in region i, plus the tax collection associated with this proportion divided by the tax ¯ di®erential (dti ). This terms only appears in ®A and ¯ rate ¯ t ¯ it is negative provided that 1 < ¯ dtA ¯. In any case, it is important to have in mind that the number of people migrating as a result of the considered tax reform must be small since nB is only an in¯nitesimal value. The same kind of analysis applied to di®erential commodity taxation gives a similarlystructured analysis with a di®erent interpretation of ®i . Yitzhaki and Slemrod (1991) interpret ®s as the revenue e®ect of a change in the tax rate of commodity "s". Likewise, Wildasin (1984) and Mayshar (1991) interpret s as the marginal social cost of raising one dollar of revenue by taxing the sth commodity15 . Isolating dtA , equation (11) can be expressed as,
dtA = ¡(
XBP )®BA dtB XAP
14
(12)
Is not possible to determine the limit of ®i as dti tends to zero, however, it is easy to check that ®i dti tends to zero as dti tends to zero and therefore dR also tends to zero. 15 Wildasin (1984) and Mayshar (1991) both analyse the marginal social cost (MSC) of public good provision ¯nanced by distortionary taxes. Wildasin takes into account the e®ect of incremental good provision in the demand for taxed goods. He contemplates two excludible benchmarks: First, the ordinary demand of taxed good and the public good are independent and therefore, spending has no repercussion on the amount of collected tax revenue. And second, the compensated demand of the taxed good and the public good are independent and therefore, the marginal revenue is rebated as a lump-sum transfer to taxpayers. He shows that the true MSC of public funds is dependent on those assumptions. This clari¯es the apparent contradiction between the results of Pigou (1947) and Browning (1976) who defend that MSC >1 and Atkinson and Stern (1974) who obtain the opposite result, MSC 0 is: ¯1 + dttA ¯ < P iP zh . This is only respected for certain h2BA values of dtA since dtA is an in¯nitesimal number. In the case that this condition is reversed we would be facing a di®erent tax reform where dtA and dtB would both be positive. As we mentioned in the introduction we are not contemplating explicitly this kind of reform. 16
12
dvh = (
dv h = (
@v h h h XBP )[x ' ]®BA ( )dtB @y h XAP
@v h h h XBP )[x ' + z h ]®BA ( )dtB h @y XAP
f or h 2 AA
for h 2 BA
(17)
(18)
h
) and dtB are both positive, household h obtains a bene¯t from the tax reform Since ( @v @yh when the expression in square brackets in the equation among (15)-(18) corresponding to her category (AA; AB; BB; BA) is positive as well. If the expression in square brackets would be nonnegative for all h, then the tax reform would be a Pareto improving tax reform. This is a very demanding condition, however, we can use a methodology based on rules of seconddegree stochastic dominance, which allow some individuals to lose due to the reform, provided that those losses are compensated with the gains of the rest of individuals and then overall welfare improves. The change in overall welfare as a result of the suggested tax reform can be expressed as follows:
dW =
N X @W ( h )dv h @v
(19)
h=1
Now substituting into (19) from (15), (16) and (17), classifying households by their ¯nal h region of residence [AP = AA [ BA and BP = BB 17 ], and recalling that ¯(y h ) = ( @W )( @v ) @vh @y h expression (19) becomes,
dW = XBP dtB f®BA
X
h2AP
¯(y h )'h (
X xh zh xh + h )¡ ¯(y h )'h ( )g XAP ' XAP XBP
(20)
h2BP
h
Where the term inside the brackets 'hzX derives from the fact that the monetary comAP h pensation z is not tax free. Let households be ordered non-descendingly by taxable income before the tax reform, Ph h h x eh h h x , h = 1:::N , and let CAP ( N ) = k=1 ' ( XAP + 'hzX ) be the concentration curve for AP eh = xh eh = 0 if h 2 B and x income of household k after the reform in region A where x h if h 2 A; and where z = 0 if household k stayed in region A before and after the tax > 0 if household k migrated from B to A because of the reform. Likewise, reform, and z hP h h let CBP ( N ) = hk=1 'h ( XxbBP ) be the concentration curve for income of household k after the bh = 0 if h 2 A and x bh = xh if h 2 B. It is worthwhile to notice reform in region B where x that the ranking from poor to rich in every region is related to the overall ranking h in such a way so that we produce arti¯cial distributions of income for each region which reach the size 17 After the tax reform population in region B is designated by BP = BB [ AB: Since we have accepted that there is no migration from region A to region B, AB = ; and therefore, BP = BB.
13
of the overall population, replicating cumulated incomes when is needed18 . Hence expression (20) can be written as follows,
dW
= XBP dtB f®BA ¡
N X
N X
¯(y h )[C AP (
h=1
¯(y h )[C BP (
h=1
h h¡1 ) ¡ C AP ( )] N N
(21)
h h¡1 ) ¡ C BP ( )] N N
Applying combinatoric theory to equation (21) we obtain the following expression:
= XBP dtB f®BA
dW
N X h [¯(y h ) ¡ ¯(y h+1 )]CAP ( ) N
(22)
h=1
N X h ¡ [¯(yh ) ¡ ¯(y h+1 )]CBP ( ) N h=1
and then simplifying, N X h h dW = XBP dtB f [¯(y h ) ¡ ¯(y h+1 )][®BA CAP ( ) ¡ CBP ( )]g N N
(23)
h=1
where [¯(y h ) ¡ ¯(y h+1 )] is positive since by an earlier assumption ¯(y h ) is positive and @¯(yh ) decreasing with income, @yh < 0. And consequently, dW ¸ 0; for all W satisfying this restriction, if and only if, ®BA CAP ( Nh ) ¸ CBP ( Nh ) 8 h. If we consider uniform elasticity of taxable income, i.e., ½hx;t = ½x;t 8 h and, therefore, 'h = ' 8 h then dW becomes, N X h h dW = XBP dtB f [¯(y h ) ¡ ¯(y h+1 )][®BA CAP ( ) ¡ CBP ( )]g N N
(24)
h=1
Ph
x zh h eh k=1 ( XAP + 'XAP ) and CBP ( N ) = h) < 0; if and only if, ¯(y h ) > 0 @¯(y @yh
where CAP ( Nh ) = '
Ph
x bh k=1 XBP : ®BA CAP ( Nh )
'
As before, dW ¸ 0,
¸ CBP ( Nh ) 8 h. for all W satisfying To consider homogenous elasticity of taxable income is a useful assumption when producing simulations of the tax reform as we will see later on. 18
This means that the concentration curves for income in every region present °at intervals wherever we have replicated cumulated values in order to meet the overall ranking.
14
Remark 1 A marginal tax reform which reduces taxation in region A (dtA < 0), and increases taxation in region B; (dtB > 0); provoking migration from B to A; is welfare improving with respect to the uniform original proportional tax, for every individualistic social welfare function, W [v1 (y 1 ); ::::; v N (y N ); p1 ; ::::; pm ]; in which the social evaluation of the marginal utility of income is positive and a declining function of disposable income, if and only if, R1 : ®BA CAP (
4.1
h h ) ¸ CBP ( ) 8 h N N
Incomes change with migration but z h is tax free
If z h were tax free then migration would take place only when z h = ch and the following h conditions would be respected: ( @y @t )(dtA ¡ dtB ) > 0, for migration from region B to region A h and ( @y @t )(dtB ¡ dtA ) > 0 for migration from region A to region B. Since dtB < 0 and dtA > 0 h
h
@y there is migration from B to A if ( @y @t ) < 0 and migration from A to B if ( @t ) > 0. Following Long (1999) among others, the estimated tax-rate elasticity of taxable income is negative, then h ( @y @t ) < 0 and there is only migration from B to A. Thus, the equal revenue constraint (10) would be: dR = [XAP (1 + ½xAP;t ) + [XBP (1 + ½xBP;t )]dtB = 0. The e®ect of the reform on P P h h overall welfare would be dW = XBP dtB f®BA h2AP ¯(y h )'h ( XxAP )¡ h2BP ¯(y h )'h ( XxBP )g
and the parameter ®BA =
®B ®A
=
1+½x
BP;t
1+½x
.
AP;t
After some mathematical manipulation and using the same notation as in the previous section we ¯nd that the su±cient condition for a welfare improving tax reform when z h is free h) < 0; is that, ®BA CAP ( Nh ) ¸ CBP ( Nh ) 8 h , of tax, for all W satisfying ¯(y h ) > 0 @¯(y h @y
where z h = 0 in CAP ( Nh ): If we further assume that ½xAP ;t = ½xBP ;t = ½x;t then ®BA = 1. Considering that ®BA may encapsulate the marginal e±ciency cost of the reform, then having z h tax free would imply no e±ciency cost as a result of the reform. Simulation calculations for this case can easily be computed.
4.2
Special case: household's income does not change with migration
Let us now assume that z h = 0 which is an interesting simpli¯cation of the more general scenario described in the previous section. If z h = 0 a household's pre-tax income will remain the same regardless of the region she decides to live in. As in the ¯rst scenario, individuals living in any region would decide to move to another region as a result of the tax reform if and only if the disposable income obtained in the new region (after migration) is greater than the disposable income that they would obtain in their original region plus the moving cost. The necessary condition for the existence of migration from region i to region j is now simpli¯ed to,
(
@y h )(dtj ¡ dti ) > ch @t
15
(25)
It is immediate to observe that expression (25) can never hold since (dtj ¡ dti ) is in¯nitesimal while ch is ¯nite. Hence, our conclusion is thus that, when individual gross incomes do not change with migration, a marginal tax reform which increases the tax rate in one region and decreases the tax rate in the other region is not powerful enough to cause migration from one region to the other when we consider that migration is costly (ch > 0). However, in the case that ch = 0 migration would be costless and all the population in B would migrate to A. Hence, we are not considering this case. Thus the equal revenue constraint can be expressed as follows,
dR = XA dtA + t(
@XA @XB )dtA + XB dtB + t( )dtB = 0 @t @t
(26)
Manipulating equation (26) the following expression is derived: 0
0
dR = XA ®A dtA + XB ®B dtB = 0 0
(27) 0
t i where ®i = 1 + ( @X @t )( Xi ), i = A and B. The parameter ®i includes the elasticity of the tax base with respect the tax rate in every region. The e®ect of the tax reform on overall welfare is described by a similar although simpli¯ed expression as the one obtained for the more general case, in which z h > 0.
dW = XB dtB f®0BA
X
¯(yh )'h (
h2A
X xh xh )¡ ¯(y h )'h ( )g XA XB
(28)
h2B
The are two major di®erences between this special case and the general one: on the one 1+½ 0 hand, the value of the parameter ®BA is now ®BA = 1+½xB ;t , and on the other hand and xA ;t much more relevant, there is no migration because of ¯scal reasons and therefore now we deal with the original distribution of pre-tax income of every region. Thus the important issue to focus on in this case is the in°uence of the tax reform on taxable income. We think that this is an interesting result in itself. Using the notation of the previous section and applying combinatoric theory expressions (28 ) can be written in the following manner, N X 0 h h dW = X B dtB f[ [¯(y h ) ¡ ¯(y h+1 )][®BA C A ( ) ¡ C B ( )]g N N
(29)
h=1
P h where CA ( Nh ) = hk=1 'h ( Xxe A ) is the concentration curve for income of household k after eh = xh if h 2 A; and CB ( Nh ) = eh = 0 if h 2 B and x the reform in region A where x Ph h b h x k=1 ' ( XB ) is the concentration curve for income of household k after reform in region B bh = 0 if h 2 A and x bh = xh if h 2 B. where x Again if we consider uniform elasticity of taxable income, i.e., ½hx;t = ½x;t 8 h and, therefore, 'h = ' 8 h then dW becomes,
16
N X h h dW = XB dtB f [¯(y h ) ¡ ¯(y h+1 )][®0BA CA ( ) ¡ CB ( )]g N N
(30)
h=1
where now CA ( Nh ) = '
Ph
x eh k=1 XA
and CB ( Nh ) = '
Ph
x bh k=1 XB
0
and ®BA = 1.
Remark 2 A marginal tax reform which reduces taxation in region A (dtA < 0), an increases taxation in region B; (dtB > 0); without migration e®ects is welfare improving with respect to the uniform original proportional tax, for all individualistic social welfare functions, W [v 1 (y 1 ); ::::; v N (y N ); p1 ; ::::; pm ]; in which the social evaluation of the marginal utility of income is positive and a declining function of disposable income, if and only if, R2 : ®0BA CA (
5
h h ) ¸ CB ( ) 8 h N N
Some particular cases
We have seen that in the general scenario, where people who migrate receive a compensation z h , the tax reform induces some degree of migration from region B to region A. As a consequence of this migration the change in overall welfare after the reform is expressed using the new distribution of population among regions once migration has occurred. The implication of working with new distributions of population is not trivial since it means that a priori we can only test with simulations the welfare e®ect of the suggested tax reform. Before the reform is implemented it is impossible to produce an evaluation of the welfare implications of the considered tax reform in the absence of data or assumptions about migration patterns. Nevertheless, the analysis is still interesting since we can in principle at least check which values of t, ni , ½x;t , dtA and dtB are compatible with welfare improving reforms. In the particular case of having z h = 0 migration does not occur and then the conditions for welfare improvement are simply expressed in terms of the Lorenz curves for the modi¯ed original distributions of income which can be easily calculated. All that is known about the overall social welfare function which is considered in the analysis is that the social evaluation of the marginal utility of disposable income, ¯(y h ); is positive and decreasing with disposable income. In this section, for the sake of simplicity we make additional assumptions about ¯(y h ); in doing so we are introducing explicit value judgements into the analysis. This is not unusual, actually speci¯c welfare weights are being increasingly used in studies of projects both in less developed countries (see for example, Ahmad and Stern,1984,1991; Little and Mirrless, 1974) and in more advanced countries (Deaton, 1977). The welfare weights ¯(y h ) may be speci¯ed in a number of ways. The simplest assumption to make is to consider that ¯(y h ) = ¯ 8 h which is compatible with a symmetric utilitarian welfare function for which distributional considerations are of no social concern. A common method for generating welfare weights is to use some modi¯cation of the utility function introduced by Atkinson (1970): x1¡e 1¡e = b log(x)
Ue (x) = a + b
17
e 6= 1; e ¸ 0 e=1
where e is a constant parameter which refers to inequality aversion19 . Hence ¯ h = U 0 = b( x1 )e . Ahmad and Stern (1984) for example, used this function substituting incomes by total expenditure per capita and Deaton (1977) considering real income instead of monetary income. ¡ ¢e We then de¯ne ¯(xh ) = ³ x1h , where ³ is just a constant and e refers to social inequality aversion. Ahmad and Stern (1991) for example, explored this assumption ¡ 1 ¢e when analysing tax h reforms in developing countries. They chose ³ = 1 so that ¯(x ) = xh . Likewise, they have used other values for the parameter ³. For³ example in their paper on tax reform and Indian ´e h I1 20 indirect taxes (1984) they consider ¯ = I h where I 1 is the expenditure of the poorest
household and I h is the expenditure of household h. That expression of ¯ h is particularly interesting since it compares the marginal social value of a unit of expenditure to individual h relative to a unit to individual 1. Replacing ³ 1 ´eexpenditure by disposable income, y, we could h h de¯ne ¯(y ) in a similar way as ¯(y ) = yyh : The value of the parameter of inequality aversion e is of primary importance since it is responsible for the relative weights that households receive when measuring overall welfare. ¯(y h ) h < y h+1 . The bigger is e the bigger is the social The ratio ¯(y h+1 ) increases with e for y concern about inequality and the bigger are the weights attached to the lower end of the h @¯ h @¯ h 21 distribution relative to those attached to the upper end ( @¯ @e > 0; @e@yh < 0; @yh < 0). The extreme cases are those for e = 0 and e ! 1. For e = 0 ¯(y) = 1 and the social welfare function becomes the traditional pure utilitarian SWF, we say then that society is inequality neutral and cares nothing about inequality. Hence every household receives the same weight what means that the policy maker values an additional unit of income for the poorest individual as equivalent to an additional unit of income for the richer individual. When e ! 1 ¯(y h ) ! 0 and the SWF tends to the maximin of Rawls. Society has extreme inequality aversion to the point that it only cares about the welfare of the poorest individual. Values of e > 2 give very much greater weight to the poorest and values of e ¸ 5 begin to approach the maxi-min³or ´Rawlsian utility³ function. ´e 1 e If we use ¯(y h ) = yyh or ¯(y h ) = y1h we would have to calculate the exact value of ¯(y h ) for every household in order to be able to say anything about the welfare e®ect of the reform. A possible simpli¯cation would be to divide the population by deciles or by intervals of income and then calculate the weights associated to every group. These are very interesting when doing empirical calculations. We have chosen ¯(y h ) = ¯ 8 h; and ³ options ´ 1 ¯(y h ) = yh as interesting cases for which we are able to de¯ne comprehensible dominance 19
The bigger is e the bigger is the society's concern for inequality. Alternatively the bigger is e the bigger is the social perception of the cost of an unequal income distribution and therefore the bigger is the social perception of inequality. We ensure inequality aversion when we consider an egalitarian social welfare function. Concavity implies that the Principle of Transfers is respected. The more concave is the function the bigger is the inequality aversion and the bigger is the bene¯t from any income transfer from rich to poor. See more on this in Lambert ³ 1 ´e (1993). ¡ ¢e ¡ ¢e 20 h ¯ = IIh derives from assuming ¯ 1 = ³ I11 = 1 so that ³ = I 1 . 21 The bigger is the inequality aversion of society the bigger is the inequality captured by the Atkinson index or indeed the extended Gini index. Although de¯nitions of both indexes are di®erent and the inequality aversion parameters are di®erent too. We could consider that the inequality aversion parameter works like a zoom lens the bigger is the aversion to inequality the bigger is the picture we take of inequality and the bigger is the importance that society attaches to inequality and therefore to the well being of the poorer individuals.
18
conditions for marginal welfare improving reforms of the kind considered in the analysis. We then contemplate these possibilities.
5.1
Inequality neutral symmetric utilitarian welfare function: ¯(yh ) = ¯ 8 h
Let us consider for simplicity that ¯(y h ) = ¯ 8 h which means that social welfare is measured by a symmetric utilitarian welfare function for which distributional considerations are of no social concern since every household receives equal weight in the SWF. ³The´implications of 1 e setting ¯(y h ) = ¯ 8 h are similar to those of considering e = 0 in ¯(y h ) = yyh or in ¯(y h ) = ¡ h ¢¡e y . When e = 0, ¯(y h ) = 1 which is a particular case of ¯(y h ) = ¯ 8 h. When e = 0 we say that society is inequality neutral or has zero inequality aversion. Using the interpretation of the Atkinson inequality index, Atkinson (1970), we would say that when e = 0 no loss of total income would be accepted in exchange for perfect equality. We must analyse the implications of this assumption in both cases: migration case or general case ( z h > 0) and no migration case (z h = 0). 5.1.1
Migration case when ¯(y h ) = ¯ 8 h
When we assume that ¯(y h ) = ¯ 8 h expression (20) from which we can infer the welfare e®ect of the analysed tax reform becomes,
dW = ¯XBP dtB f®BA
X
'h (
h2AP
X xh zh xh + h )¡ 'h ( )g XAP ' XAP XBP
Consider for simplicity that ½hx;t = ½x;t 8 h becomes, dW = '¯XBP dtB f®BA
(31)
h2BP
X
h2AP
(
and consequently, 'h = '
8 h then dW
X xh xh zh + )¡ ( )g XAP 'XAP XBP
(32)
h2BP
hence,
dW = '¯XBP dtB f®BA (1 +
X
h2BA
zh ) ¡ 1g 'XAP
(33)
Where the subscript h 2 BA refers to households who migrate from B to A. The suggested tax reform is welfare improving, if and only if dW ¸ 0 . A su±cient condition for dW ¸ 0 is 1+½ that ®BA ¸ 1. Considering that ®BA = 1+½ +(1+S ) x;t1 P and that (1+SA ) < 0; ®BA zh A X x;t AP Ph2BA is positive (in fact bigger than one) when j1 + SA j X1AP h2BA z h < 1 + ½x;t , and negative when the inequality relationship is reversed. When ®BA is negative the e®ect of the reform 19
is a reduction in overall welfare, thus we are only interested in positive values of ®BA 22 . We know that when ®BA is positive it is also bigger than one, therefore dW ¸ 0 and the suggested taxPreform is welfare improving. However, the necessary condition for dW ¸ 0 is A )' [1 ¡ (1+S ] h2BA z h ¸ 0: The expression in brackets is always positive therefore dW ¸ 0: 1+½ x;t
Remark 3 A marginal tax reform which reduces taxation in region A (dtA < 0), and increases taxation in region B; (dtB > 0); provoking migration among regions is welfare improving with respect to the uniform original proportional tax, for the inequality neutral symmetric utilitarian social welfare function and assuming uniform and negative tax rate elasticity of taxable income, if and only if, R3 : [1 ¡
(1 + SA )' X h ] z ¸0 1 + ½x;t h2BA
and this is always true. If the compensation z h were not taxable then dW = '¯XBP dtB f®BA ¡ 1g. Furthermore, since we are assuming for convenience that ½hx;t = ½x;t 8 h then ®BA would be simpli¯ed to one and therefore, dW = 0. This means that under the assumption of homogeneous tax rate elasticities of taxable income the possibility of having dW > 0 when ¯(y h ) = ¯ emerges from the tax e®ect derived from the monetary compensation z h that allows households to migrate. Only if z h is taxable we could obtain a positive e®ect of the reform on overall welfare with homogeneous elasticities of taxable income with respect to the tax rate. However, if we would drop the assumption of uniform elasticities, ®BA could be di®erent from one when z h is not taxable and therefore the tax reform would not be neutral on overall welfare (dW 6= 0). 5.1.2
No migration case when ¯(y h ) = ¯ 8 h
In¡ the¢ simpler case of no migration the welfare e®ect of the reform is expressed by setting ¯ y h = ¯ 8 h in (28), dW = ¯XB dtB f®0BA
X
'h (
h2A
X xh xh )¡ 'h ( )g XA XB
(34)
h2B
If again we consider that the tax rate elasticity of taxable of income is uniform for the whole population (½hx;t = ½x;t ) and consequently, 'h = ' 8 h hence dW is, dW = '¯XB dtB f®0BA ¡ 1g
(35)
Considering that the estimated values of ½x;t are generally accepted to be negative (' > 0 0) then dW ¸ 0, if and only if, ®BA ¸ 1. Furthermore, when there is no migration the 0 de¯nition of ®BA is simpli¯ed and, under the assumption of uniform elasticity of taxable 0 income with respect to the tax rate, ®BA = 1 which implies that dW = 0. If we would allow 0 for di®erentiation of the elasticity of taxable income with respect to the tax rate then ®BA 22 Negative values of ®BA would also be compatible with dtA and dtB both positive. This is not the kind of reform we are contemplating.
20
6 0. Consequently in the absence of migration we could could be di®erent from one and dW = say that with regard to the welfare e®ect of the reform the assumption of homogeneity of the h @x tax rate elasticity of taxable income brings equivalent results as assuming @x @t = @t = 0 8 h: Remark 4 A marginal tax reform which reduces taxation in region A (dtA < 0), and increases taxation in region B (dtB > 0) without provoking migration among regions does have no impact on overall welfare with respect to the uniform original proportional tax, for the inequality neutral symmetric utilitarian social welfare function and assuming uniform and negative tax rate elasticity of taxable income. This result is not surprising since we are considering a social welfare function which is inequality neutral and therefore it shows indi®erence to transfers. Actually the obtained result is indeed very similar to that one emerging when we consider z h not taxable. The di®erence is that now we deal with the original distributions of income in regions A and B instead of the new distributions after migration.
5.2
Symmetric utilitarian social welfare function with distributional considerations: ¯(yh ) = (y h )¡1
Setting ¯(y h ) = (y h )¡1 is a particular case of the general expression ¯(y h ) = (y h )¡e where e is a constant parameter which refers to society's inequality aversion. In this subsection we evaluate the welfare e®ects of the reform when e = 1. That e = 1 implies that we are considering a symmetric utilitarian welfare function which gives higher weights to households in the lower end of the distribution and lower weights to those in the upper end. Therefore, society takes into account distributional considerations when measuring overall welfare. Setting e = 1 means that society has a moderate inequality aversion. Again we present the result of the analysis for both cases: migration case or general case (z h > 0) and no migration case (z h = 0). 5.2.1
Migration case when ¯(y h ) = (y h )¡1
Let ¯(yh ) = (y h )¡1 hence expression (20) is now,
dW =
X X 1 1 zh 1 'h ( + h h )¡ 'h ( )g XBP dtB f®BA 1¡t XAP ' x XAP XBP h2AP
(36)
h2BP
Then considering homogenous elasticity of taxable income, dW can be written as,
dW =
X ' 1 zh 1 g XBP dtB f®BA ( + )¡ h 1¡t ¹AP 'x XAP ¹BP
(37)
h2BA
After some simple mathematical manipulation and assuming again that the estimated values of ½x;t are negative so that ' > 0 , a su±cient condition to have dW ¸ 0 is thus, 21
®BA ¹BP ¸ ¹AP
(38)
where ¹AP and ¹BP are the means of gross income after migration for region A and region B.
1 P h As before ®BA is positive when j1 + SA j XAP h2BA z < 1 + ½x;t . We know that when ®BA is positive it is also bigger than one, therefore looking at (38) we know that if ¹BP ¸ ¹AP the suggested reform would improve welfare. This ties in with equation (2) since the reform considers a marginal tax increase in region B and a simultaneous marginal tax decrease in region A. However, even if ¹BP < ¹AP the reform could still improve overall welfare provided 1+½ ¸ ¹¹AP : This is indeed a more interesting result. that ®BA = 1+½ +(1+S ) x;t1 P zh BP A X x;t h2BA AP P zh Then including the e®ect of h2BA 'xh X the necessary condition for dW ¸ 0 is, AP
®BA ¸ Where [ '+
' 1 XBA N AP
P
h2BA
zh
¹AP [ ¹BP ' +
'
1 XBA NAP
P
h2BA z
h
]
(39)
] < 1 and XBA is total gross income belonging to households
migrating from B to A and NAP is total number of households is region A after migration. For ¹AP · ¹BP the right hand side of (39) is smaller than one. Hence since we know that when ®BA is positive it is also bigger than one expression (39) holds always whenever ®BA > 0 and consequently the reform increases overall welfare. The case of ¹AP > ¹BP is also compatible with (39) and welfare improving reforms. Remark 5 A marginal tax reform which reduces taxation in region A (dtA < 0); increases taxation in region B (dtB > 0) and implies migration from B to A is welfare superior with respect to the uniform original proportional tax, for the logarithmic utilitarian social welfare function and considering that the elasticity of taxable income with respect to the tax rate is uniform and negative, if and only if, R5 : ®BA ¸
¹AP [ ¹BP ' +
' 1 XBA NAP
P
h2BA z
h
]
Again if the monetary compensation z h is tax free the reform e®ect on welfare does not depend directly on the value of z h . Thus the contemplated reform would improve welfare, if and only if, ¹BP ¸ ¹AP since ®BA = 1 (recall ½hx;t = ½x;t ). 5.2.2
No migration case when ¯(y h ) = (y h )¡1
Setting ¯(y h ) = (y h )¡1 expression (28) becomes, dW =
X 'h X 'h 1 ¡ g XB dtB f®0BA 1¡t XA XB h2A
22
h2B
(40)
For ½hx;t = ½x;t 8 h ('h = ' 8 h) and assuming that the estimated values of ½x;t are negative 0
(' > 0), the necessary condition to have dW ¸ 0 is thus, (recall that ®BA =
½hx;t
1+½x;t 1+½x;t
= 1 since
= ½x;t ):
¹B ¸ ¹A
(41)
Remark 6 A marginal tax reform which reduces taxation in region A (dtA < 0), and increases taxation in region B (dtB > 0) without provoking migration among regions is welfare superior with respect to the uniform original proportional tax, for the logarithmic utilitarian social welfare function and considering that the elasticity of taxable income with respect to the tax rate is uniform and negative, if and only if, R6 : ¹B ¸ ¹A When we measure social welfare with a logarithmic utilitarian social function we ¯nd the interesting result that only information about the means of income by regions is needed to evaluate the impact of the reform on overall welfare. Only if the reform is done in the right direction so that it bene¯ts the region with the smallest mean income overall welfare will increase as a result. When we use a logarithmic utilitarian social function we are giving higher weights to households in the lower end of the distribution and lower weights to those at the top. Hence, it seems reasonable that the necessary condition for a welfare improving reform should be a function of the income means of every region assuring that the reform is done in favour of the poorest, i.e., in favour of the region with the smallest mean income.
6
Numerical simulations
In this section we explore through simulations the feasibility of the analysed welfare improving tax reforms. Using real ¯scal data for 1990 we contrast the cases developed in subsections 4 and 5. Just as a reminder for the reader, we have de¯ned a tax reform where the income tax is marginally di®erentiated between regions. We have assumed for simplicity that there are only two regions A and B, which initially face a uniform proportional income tax. Then we consider the di®erentiation of the income tax such that the tax rate of region A decreases marginally (dtA < 0) and the tax rate of region B increases marginally too (dtB > 0) holding total revenue constant with respect to the uniform case. We have arbitrarily chosen pairs of regions from a sample of ¯scal data to explore under which circumstances a welfare improving reform would be feasible. Checking ex-ante the conditions for welfare improving tax reforms drawn in sections 4 and 5 we could tell the policy maker whether to increase or to decrease the tax bill in every region. Also and more interesting we can explore the impact of migration on the feasibility and direction of welfare improving tax reforms. How much migration is allowed before reversing the conditions for welfare improving tax reforms ? or, how much migration would it be necessary for a condition to hold ? If for example, let's say that originally a region should have its tax bill increased, could migration from or towards this region change this result ? And if yes, what is the magnitude and the kind of migration movements that would change the result ? This sort of question arises in the analysis, and we deal with them in the following subsections. 23
6.1
Data and Methodology
We use a sub-sample of ¯scal data with approximately 21,000 observations from the Spanish personal income tax for 1990. This sub-sample has been obtained from the Extended Panel data set produced by the Instituto de Estudios Fiscales (IEF thereafter), Ministerio de Economia y Hacienda, Madrid23 . Besides the well known limitations of using ¯scal data this sample is just perfect for our exercise since it allows us to simulate di®erent tax reforms with real data instead of using arti¯cially generated distributions (e.g. lognormal). Likewise, the fact that we use data from 1990 is not of much importance since our objective is only experimental. Spain is not a federal country but administratively it is divided in seventeen partitions named comunidades autonomas, CAs, which have a great individual historical and cultural idiosyncrasy. These CAs enjoy a high level of decentralization specially in the expenditure side of the budget and increasingly in the revenue side as well. We use in our simulations income tax data from some of these 17 partitions selecting the cases that we consider more interesting or illustrative. Hence simulations are presented for arbitrarily chosen pairs of CAs. Of course the success in testing the dominance conditions for welfare improving reforms depends upon the characteristics of the CA. in our sample (mean income and distribution of income), the number and income of migrants and the direction of the tax reform. We choose 4 pairs of CAs. and we tested conditions drawn in sections 4 and 5 in the two possible directions. The chosen pairs are: Andalucia (And) and Catalunya (Cat) where the CA with higher taxable income mean has a more equal distribution of taxable income; Andalucia and Valencia (Val) where the CA with lower taxable income mean (Val) has also a more equal distribution of taxable income; Madrid (Mad) and Catalunya which follow the same pattern as the ¯rst considered pair but after eliminating the 5% of the richest individuals24 and ¯nally Valencia and Catalunya for which lorenz curves of taxable income cross and therefore we can not say which CA has a more equal distribution using the Lorenz dominance criterion of course. For those conditions where migration is involved we employ di®erent assumptions concerning z h and ½hx;t . To be speci¯c, we test those conditions for the combinations resulting from z h = z 8 h with z = 1 and z = 3 million of pesetas (ptas.), and ½hx;t = ½x;t 8 h, ½x;t = ¡0:161 and ½x;t = ¡0:29 which are the minimum and maximum weighted average values of the estimated tax rate taxable income elasticities obtained by Long (1999) when using the income of the groups as weights.25 We also use in our simulations ½x;t = ¡0:1 which is an approximated 23
The Extended Panel is a random and dynamic sample, produced by the IEF since 1982, which comprises over 200,000 cases and o®ers most of the information included in the income tax declaration. This sample is respresentative for the whole population and also for various desagregations, like regions or municipalities. However we should not forget that when employing ¯scal data we are not dealing with real income but with taxable income and therefore we should be careful extracting conclusions since problems of tax evasion, strategic behaviour etc might arise. We thank the IEF, Ministerio de Hacienda, for the provision of the sample used in this paper. 24 Before the elimination of the 5% of the richest declarants in the sample for every CA the lorenz curves of both CAs cross. 25 Long (1999) estimates the tax rate elasticity of taxable income by income groups and calculates the average values of the elasticity using the income and the returns of each group of income as weights. In all cases he obtains negative values of the elasticities which are always increasing with income. The are many authors which have produced estimations of the net of tax elasticity of taxable income, like for example, Auten and Carroll (1999), Feldstein (1995 a, b, c), Feldstein and Feenberg (1995), Gruber and Saez (2000), Saez (1999).
24
and conservative value for the estimated elasticity of labour supply with respect to the tax rate (see, e.g., Tuomala(1990) and more recently Blundell and MaCurdy (1999) for more on labor supply related topics). Even though there is empirical evidence showing that elasticities may vary with income (see e.g., Gruber and Saez (2000), Long (1999), Saez (1999) among many others) it is useful to start with the case of constant elasticities to see how migration movements can determine the direction of a marginal welfare improving tax reform. The case of varying elasticities could be easily simulated but we prefer to keep things simpler reducing the number of parameters brought into the analysis. Likewise, we simulate di®erent degrees of migration when testing the welfare improving conditions. From every C.A. we select all tax payers with medium and high wages and from those subsamples we randomly chose a 5%, 10%, 20% and 50% of the tax payers and we assume that those tax payers are the migrants. We use in our simulations values of t = 0:2 and t = 0:3. and of dtA = ¡0:1t and ¡0:25t . We also use experimentally dtA = 0:5t to see if it produces unexpected results.
6.2
Results
C.A.pairs (B-A) Cat-Val Val-Cat Cat-And And-Cat And-Val Val-And Cat-Mad Mad-Cat
R1 59/72 2/72 57/72 1/72 20/72 23/72 15/72 58/72
R1* 3/4 0 3/4 0 0 0 0 2/4
R5 59/72 11/72 60/72 18/72 53/72 55/72 41/72 62/72
R5* 4/4 0 4/4 0 3/4 0 0 3/4
R2 S F S F x x F S
R6 S F S F S F F S
Table 1: Successful tests ( no. of successful tests over total no. of implemented tests) by pairs of CAs R1* and R5* are conditions for z tax free
Condition R126 is respected in many cases and usually points out a clear direction for the reform. For example, when we choose Val as jurisdiction A and Cat as jurisdiction B condition R1 holds and welfare increases as a result of the reform, in all cases except when ®BA < 0 (13 over 72 tested cases). However, if we change the direction of the reform, taking Val as B and Cat as A, condition R1 is only respected in two cases and the concentration curves for A and B present intersections in 69 cases from the 72 cases which were tested. Therefore, we can say nothing about the e®ect of the reform in those cases. When we test the pair of jurisdictions Mad and Cat, we ¯nd out that the tests are more often successful when Mad = B and Cat = A (58=72 for Mad = B and Cat = A in contrast with 15=72 for Mad = A and Cat = B). Consequently, it seems that the reform is more often welfare improving when we reduce the tax in Cat and we increase it in Mad. We should emphasize For simplicity we use the estimated values presented by Long since they are tax rate elasticities instead of net of tax rate elasticities. 26 When testing condition R1 values of t are needed to simulate modi¯ed distributions of taxable income for every C.A. involved in the tax reform. We did simulations using t = 0:2 and t = 0:3 and we obtained equal results for both parameters. Therefore we only show one set of results for condition R1 (see tables 1 to 8).
25
that the migration percentage is decisive for the success of the tests. Then it emerges that R1 is nearly always respected for migration of 5%, 10% and 20%. However, for migration of 50% we obtain few cases of crossings of the CAs concentration curves and most of the cases where ®BA < 0 (5=72) Changing the direction of the reform such that Mad = A and Cat = B we can say nothing about the e®ect of the reform in most of the tested cases (15/72 successful tests from which half of them emerge for 20% of migration, 10/72 cases for which ®BA < 0; and 47/72 cases for which the concentration curves cross and nothing can be said about the welfare e®ect of the reform). When we test R1 for A=And and B=Cat we ¯nd out that it is always respected for migration of 5% and of 10%. For migration of 20% R1 always holds except for 3=18 cases where ®BA < 0 and for migration of 50% we obtain 9=18 cases where ®BA < 0 and 3=18 cases where the CAs concentration curves cross (total number of successful R1 tests: 57=72). However, when we change the direction of the tax reform, we ¯nd that R1 is only respected in one case over 72 tested cases. For the pair of CA Val and And, we ¯nd out that R1 is slightly more often respected when we take A=And and B=Val (23=72 successful tests when A=And and B=Val and 20=72 successful tests when A=Val and B=And). However, the number of successful tests is very similar regardless the direction of the reform therefore it seems that overall welfare could be improved implementing the reform in either direction. We have also tested the modi¯ed version of condition R1 when we consider that z is tax free (we refer to this new condition as R1*, recall that ®BA = 1 in this case). Hence, we ¯nd out that for the pair of jurisdictions Val(A) and Cat(B) and also for the pair And(A) and Cat(B), the test succeeds for all migration percentages except the highest (50%) for which the concentration curves cross. However, if we change the direction of the reform in both pairs the test fails for every simulated migration percentage since the welfare improving condition is reversed. Therefore, increasing the tax rate of the proportional income tax on Cat and decreasing the tax rate on Val or And improves overall welfare. Taking the pair of jurisdictions Val and And and testing condition R1* in both directions we obtain no successful results for any percentage of simulated migration either because the test fails or because the concentration curves of both jurisdictions cross, we also obtain similar results for Mad(A) and Cat(B). For this pair of jurisdictions, changing the direction of the reform such that Cat =A and Mad = B, gives successful results for migration of 5% and 10%, however, for migration percentages of 20% and 50% the concentration curves cross and nothing can be said about the welfare e®ect of the reform. Consequently, we can only say that the reform is welfare improving when the tax rate of the proportional income tax is decreased in Cat and increased in Mad considering migration of 5% and 10% from Mad to Cat. Condition R2, which refers to the case of no migration, holds for Val(A) and Cat(B), Cat(A) and Mad(B), And(A) and Cat(B) and fails otherwise. For jurisdiction A=Val and jurisdiction B=Cat, we obtain successful results on testing condition R5 for 59 over 72 cases. This result varies considerably when we reverse the direction of the reform and the number of successful tests is now 11 over 72 tested cases. For jurisdiction A=Cat and jurisdiction B=Mad, condition R5 is respected on 62 cases over 72 tested cases. It should be noticed that failings appear for migration of 50% and 20%. However, when A=Mad and B=Cat, R5 is respected in 41 cases over 72. The largest number of failed tests occurs when simulating 20% of migration and regardless of the value of ½x;t With respect to the pair And and Cat we obtain better results for A=And and B=Cat since
26
the number of successful tests is 60 over 72 while for A=Cat and B=And we only obtain 18 successful tests over 72. For the pair Val and And the number of successful tests does not vary very much when we modify the direction of the reform (53=72 for A=Val and B=And and 55=72 for A=And and B=Val). Therefore, as with the previous conditions we do not obtain clear results when analysing Val and And. Condition R5, when we consider z tax free (we refer to this condition as R5*), is respected for the following pairs of jurisdictions: Val(A) and Cat(B) and And(A) and Cat(B) always; Mad(B) and Cat(A) for migration of 5% 10% and 20%; and ¯nally, Val(A) and And(B) for migration percentages of 5, 10 and 20. Finally condition R6 succeeds for: Val(A) and Cat(B), Cat(A) and Mad(B), And(A) and Cat(B) and Val(A) and And(B).
TABLES 2 to 12 HERE
6.3
Parameter ®BA , monetary compensation z h and welfare e®ect of the reform
Simulations of the suggested tax reform have been presented for four values of migration: 5%,10%, 20% and 50% of the subpopulation of workers with high and medium wage. We have noticed that when using values of 20% and of 50% of migration and mainly a tax rate variation of 10% (dtA = ¡0:1t) the parameter ®BA turns sometimes negative which implies that the su±cient condition for a welfare improving reform is not respected. When the number of migrants increases the value of the total monetary compensation (nB z) also increases (recall that we have assumed z uniform for all households in our simulations), thus keeping the rest of parameters constant this increases the likelihood of ®BA being negative. A question of how big the parameter z might be before ®BA becomes negative arises. From the revenue equal constraint (12) we obtain the de¯nition of ®BA which imposes a limitation on the value of z since we need ®BA to be positive in order that the reform favours welfare. Hence ®BA is positive only if,
z
0 is di®erent for every pair of jurisdictions since the total taxable income of the recipient jurisdiction (XAP ) changes for each pair and also varies with migration. The absolute value of the elasticity of taxable income (½x;t ) is relevant as well since the bigger it is the larger is the negative e®ect on ®BA of increasing migration and increasing z. As an example, we show the obtained limit values of z before ®BA becomes negative for Val and Cat when migration °ows from Val (B) to Cat (A). For a tax rate variation of 10% (dtA = ¡0:1t) and 50% of migration z should be smaller than approximately: 2.87 million of ptas. for a tax rate elasticity of taxable income equal to -0.29, 3.39 million of ptas. for an elasticity of -0.161and 3.64 million of ptas. for an elasticity of -0.1. For migration of 20% the limit values of z are larger: 6.72 million of ptas. for an elasticity of taxable income equal to
27
-0.29, 7.94 million of ptas. for an elasticity of -0.161and 8.52 million of ptas. for an elasticity of -0.1. The limit values of z increase progressively when migration diminishes. Ceteris Paribus for larger tax rate variations the limit values of z are increased. For example, taking 50% of migration and dtA = ¡0:25t we obtain: 8.62 million of ptas. for ½x;t = ¡0:29, 10.18 million of ptas. for ½x;t = ¡0:161 and 10.92 million of ptas. for ½x;t = ¡0:1. From (42) it is obvious that when the jurisdiction receiving migrants has lower overall taxable income the limit values of z are smaller and consequently the parameter ®BA is more likely to be negative. For example, when we simulate the tax reform such that now migration °ows from Cat (B) to Val (A) the results change dramatically with respect to the previous example and the maximum values of z consistent with ®BA < 0 are smaller. As illustration, for dtA = ¡0:1t and 50% of migration the maximum values of z are: 767,573 ptas. for ½x;t = ¡0:29, 907,033 ptas. for ½x;t = ¡0:161 and 972,979 for ½x;t = ¡0:1. This implies that for any given value of z and percentage of migration a tax reform which diminishes the income tax rate of the jurisdiction with the smallest taxable income of the pair is more likely to produce negative values of ®BA . Also it implies that, given a certain percentage of migration, the value of the monetary compensation z compatible with ®BA > 0 and a potential welfare improvement should be smaller when the jurisdiction receiving migrants has lower total taxable income. Alternatively, we could also say that for a given value of z migration should be smaller in order that ®BA > 0. Up to now we have mainly analysed the e®ect of z and migration on the sign of ®BA , we should emphasize that the value of dtA is also important. From expression (42) we can calculate the minimum absolute values of dtA consistent with ®BA > 0, holding constant the rest of parameters. As an example, ¯ for ¯ Valencia (B) and Catalunya (A), and for ½x;t = ¡0:161 ¯ dtA ¯ the lower limit absolute values of ¯ t ¯ are: 0.011, 0.021, 0.040 and 0.090 for 5%, 10%, 20% and 50% of migration respectively and for z = 3 million of ptas.; and 0.004, 0.007, 0.014 and 0.032 for 5%, 10%, 20% and 50% of migration respectively and for z = 1 million of ptas. We should also point out that these limit ¯ ¯values diminish as the absolute value of ½x;t also diminishes. ¯ ¯ Hence, the larger the ratio ¯ dttA ¯ (or the smaller the absolute value of dtA ) and the larger the absolute value of ½x;t and also the larger the value z the more likely is to have ®BA < 0 and a reduction of welfare. This is interesting since it means that very small tax di®erentiation when migration is big is only compatible with small values of z. This is coherent with the fact that migration is induced by tax di®erentiation hence if tax di®erentiation is small migration should be small too to allow for big values of z. Furthermore, if we understand z as a proxy of the moving cost c hence small tax di®erentiation and big migration is only compatible with small moving cost and this e®ect is reinforced the smaller the total gros income of the recipient jurisdiction is.
7
Discussion and concluding remarks
In this paper we analysed the feasibility of an equal-yield marginal di®erentiation of a proportional income tax between pairs of jurisdictions. We used a methodology developed by Yitzhaki and Slemrod (1991) to study Dalton-improving tax reforms on commodity taxation. To simplify the analysis we considered only two regions A and B. The suggested tax reform involves a small reduction in the tax rate of region A, (dtA < 0), and a small increase in the 28
tax rate of region B, (dtB > 0), in such a way that total revenue does not change and we take into account the possible migration and e®ects of the reform on taxpayer's taxable income. It is convenient to consider revenue-neutral tax reforms since we can ignore the issue of the optimal size of government activity. As a result of the reform one would expect people to move from region B, where there is a marginal tax rate increase, to region A, where there is a marginal tax rate decrease. We proved that under the assumption of negative elasticity of taxable income with respect to the tax rate this is always true. To include migration into the analysis we assumed that when household h moves from one region to the other she receives a monetary compensation, z h . Only when, after the proportional tax, z h exactly covers the moving cost, ch , and some conditions are respected, could migration appear. We also considered the special case of z h = 0 and we proved that in this scenario migration never occurs. We de¯ned dominance conditions for welfare improving reforms for the case of migration (or general case) and for the case of no migration. We also explored some di®erent scenarios by simplifying the de¯nition of the SWF considered. To be speci¯c we explored the implications of two types of SWF's: utilitarian symmetric function with no distributional considerations where all households receive the same weight and utilitarian symmetric SWF with a declining weighting scheme in favour of the poor. Dominance conditions were de¯ned for both scenarios in section 5. Numerical simulations using income tax data for Spain and for 1990 were presented. We used as jurisdictions the Spanish administrative division on CA taking selected pairs of CA to check conditions drawn in sections 4 and 5 (R1, R2, R5 and R6). Condition R1 was often respected for every tested pair of jurisdictions and for every combination of parameters. For any of the selected pairs we did not obtain any case for which the dominance condition was reversed although the concentration curves showed crossings in some cases and therefore no conclusion about the welfare e®ect of the reform could be drawn for those cases. We found out that the main issue on testing R1 is related to ®BA which sometimes turned to be negative implying that the su±cient condition for welfare improving tax reforms is not respected. Generally, we obtained ®BA < 0 mainly for high levels of migration (50% and 20%). However, when ®BA > 0; R1 was highly respected and sometimes regardless of the direction of the reform. We presume that the fact of including the monetary compensation z in the new distribution of income after migration might be responsible for this result. However, we should also emphasize that the test of R1 de¯nitely showed a higher number of successful results when the reform was implemented in the direction which favours the jurisdiction in every pair with the smallest taxable mean income before the tax reform. Hence, we can say that the reform is more likely welfare improving when it is implemented in favour of the poorest jurisdiction. And we understand by the poorest jurisdiction as that one with the smallest taxable income mean in every pair and before the tax reform. The only exception to this result is Val and And. For this pair of CA the number of successful tests is very similar regardless of the direction of the reform. To analyse the in°uence of the parameter z on the success of the tests we thought that it would be useful to test condition R1 when z is tax free (R1*). In that case, we are not adding extra money to the original incomes and the distribution of every jurisdiction is only modi¯ed because of migration. On testing R1*, results pointed out a de¯nite direction of the reform for the pairs: Cat and And, Cat and Val and Cat and Mad and we obtained similar
29
suggestions concerning the direction of the reform as when testing condition R1. Tests of R1* only worked successfully when the jurisdiction with the smallest pre-reform taxable mean income was the one having its tax reduced. We should also mention that the test produced better results when we simulated smaller levels of migration (5%, 10% and even 20% of tax payers with high and medium wage). For the pair of jurisdictions Val and And, results were not so clear. As a conclusion we should say that when z is tax free the result of the tests were highly sensitive to migration and we could not suggest any kind of reform provided that enough migration took place. It is also worthwhile to mention that results from testing R1* are always coherent with those obtained from testing R1. Therefore, although R1 can be respected regardless of the direction of the reform there is always a larger number of successful tests in one of the possible directions of the reform and this direction is exactly the one for which condition R1* is respected. In the case that condition R1* does not hold for any direction (case of Val and And) we also obtained no clear results when testing R1. We also tested welfare improving conditions when no migration was involved and we obtained tidier results since the number of parameters coming into play is smaller. Hence, when there exist a clear dominance relationship between concentration curves of taxable income, the reform improves welfare in only one direction. Furthermore, surprisingly we found out that there is a certain connection among the obtained results from testing conditions R1, R1* and R2. If the reform was implemented in the direction suggested by R2 we also obtained better results when testing R1 and R1*. Likewise, when condition R2 was not respected we did not obtain clear results from testing R1 and R1* (case of Val and And). Finally, some simpli¯cations referring to the weighting scheme of the social welfare function were also explored. First of all, we considered an utilitarian symmetric SWF where all households receive the same weight in overall welfare. For this case we found out that when there exists migration the reform is always welfare improving (see remark R3), however, when there is no migration the reform can not possibly increase welfare since we are considering a social welfare function which is inequality neutral and therefore it is indi®erent to transfers (see remark R4). Secondly, we measured welfare using a logarithmic utilitarian SWF which presents a declining weighting scheme in favour of the poor. In the migration scenario the e®ect on welfare from the reform is expressed by condition R5. We tested this condition and a its modi¯ed version under the assumption of z not taxable (R5*) and we obtained successful results in many cases and clear recommendations for welfare improving reforms. For R5 we obtain similar recommendations as when testing R1: the reform is more likely welfare improving when it is implemented in favour of the jurisdiction with the smallest taxable mean income before the reform in every pair. Again, the exception to this rule was the pair Val and And. Testing R5* results agreed with that rule and for all pairs of jurisdictions. In the simpli¯ed scenario of no migration the e®ect of the reform, using the logarithmic utilitarian SWF, is encapsulated just by a relationship between the means of gross income for every jurisdiction in favour of the jurisdiction with the smallest mean (see remark R6). Although it seems quite parsimonious to use only information about the means of income of every jurisdiction to evaluate the welfare e®ect of the reform, it is interesting to see that such a simple criterion could in principle be applied. Actually results from testing every considered condition (R1, R1*, R2, R5 and R5*) pointed out that a marginal reform as the one suggested in this paper is more likely to improve overall welfare when is implemented in favour of the jurisdiction with the smallest taxable mean income in every pair of jurisdictions before the
30
reform which is exactly the message of condition R6. We also noticed that this rule is more likely to be respected the larger the di®erence between means in the pair. We provided dominance conditions for welfare improving reforms taking into consideration migration between jurisdictions and behavioural responses from taxpayers when facing tax changes. The analysis brings interesting results although it also presents many limitations that we should consider for future research. For example it does not take into account market labour considerations, consumer maximization problems and public goods provision. It o®ers however, a static analysis of the possible e®ects, in the short run, of a particular sort of tax reform which besides introducing horizontal inequity might also improve overall welfare under certain circumstances. Future research should also be addressed to introduce dynamic analysis and some sort of ¯scal competition among jurisdictions.
References [1] Ahmad, E. and Stern. N. (1984). The theory of reform and Indian indirect taxes. Journal of Public Economics, 25, 259-298. [2] Ahmad, E. and Stern. N. (1991). The Theory and Practice of Tax Reform in Developing Countries. New York: Cambridge University Press. [3] Atkinson, A.B. (1970). On the measurement of inequality. Journal of Economic Theory, vol. 2, pp. 244-263. [4] Atkinson, A.B. and Stern, N. (1974). Pigou, taxation, and public goods. Review of Economic Studies, 41(1), 119-28. [5] Auten, G. and Carroll, R. (1999). The e®ect of income taxes on household income, The Review of Economics and Statistics, 81(4), 681-93. [6] Blundell, R. and MaCurdy, T. (1999). Labor supply: a review of alternative approaches. In O.Ashenfelter and R. Layard (eds.) Handbook of Labor Economics, vol. 3. [7] Browning, E. K. (1976). The marginal cost of public funds. Journal of Political Economy. 84(2), 283-98. [8] Cremer, H., Fourgeaud, V., Leite-MOnteiro, M., Marchand, M. and Pestieau, P. (1996). Mobility and redistribution: a survey, Public Finance / Finance Publiques, 51, 325-52. [9] Cubel, M. and Lambert, P.J. (1997). A regional approach to income tax policy. Economics Discussion Paper No. 97/8, University of York. [10] Dalton, H. (1920). The measurement of the inequality of income. Economic Journal, 30(119), 348-61. [11] Deaton, A. (1977). Equity, e±ciency, and the structure of indirect taxation. Journal of Public Economics, 8, 299-312. [12] Feldstein, M. (1995a). Tax avoidance and the deadweight loss of the income tax. NBER Working Paper, 5055. 31
[13] Feldstein, M. (1995b). The e®ect of marginal tax rates on taxable income: a panel study of the 1986 Tax Reform Act, Journal of Political Economy, 103(3), 551-72. [14] Feldstein, M. (1995c). Behavioral responses to tax rates: evidence from TRA86. NBER Working Paper, 5000. [15] Feldstein, M. and Feenberg, D. (1995). The e®ect of increased tax rates on taxable income and economic e±ciency: a preliminary analysis of the 1993 tax rate increases. NBER Working Paper, 5370. [16] Gruber, J. and Saez. E. (2000). The elasticity of taxable income: evidence and implications. NBER Working Paper, 7512. [17] Hadar, J. and Russell, W.R. (1969). Rules for ordering uncertain prospects. American Economic Review, 59(1), 25-34. [18] Hanoch, G. and Haim, L. (1969). The e±ciency analysis of choices involving risk. Review of Economic Studies, 36(107), 335-46. [19] King, D. N. (1984). Fiscal Tiers: The Economics of Multi-level Government. London: Allen & Unwin. [20] Lambert, P.J. (1993). The Distribution and Redistribution of Income: A Mathematical Analysis. Oxford: Blackwell. [21] Lambert, P.J. (1992). Rich-to-poor income transfers reduce inequality: a generalization. Research on Economic Inequality, 2, 181-90. [22] Little, I.M.D. and Mirrlees, J.A. (1974). Project Appraisal and Planning for Development Countries. London: Heinemann. [23] Long, J. E. (1999). The impact of marginal tax rates on taxable income: evidence from state income tax di®erentials. Southern Economic Journal, 65(4), 855-69. [24] Mayshar, J. (1991). On measuring the marginal cost of funds analytically, American Economic Review, 81(5), 1329-35. [25] Mayshar, J. and Yitzhaki, S. (1996). Dalton-improving tax reform: when households di®er in ability and needs. Journal of Public Economics, 62, 399-412. [26] Mayshar, J. and Yitzhaki, S. (1995). Dalton-Improving indirect tax reform. American Economic Review, 85(4), 793-807. [27] Pigou, A.C. (1947). A Study of Public Finance. 3rd ed. London: Macmillan. [28] Saez, E. (1999). The e®ect of marginal tax rates on income: a panel study of 'bracket creep'. NBER Working Paper, 7367. [29] Tuomala, M. (1990). Optimal Income Tax and Redistribution. Oxford: Clarendon Press. [30] Wildasin, D.E. (1984). On public good provision with distortionary taxation, Economic Enquiry, 22, 227-43. 32
[31] Wildasin, D.E., (1992). Relaxation of barriers to factor mobility and income redistribution, in: P. Pestieau (ed.), Public Finance in a World of Transition, Supplement to Public Finance / Finance Publiques, 47, 216-30. [32] Wildasin , D.E., (1998). Factor mobility and redistributive policy: local and international perspectives in P.B. Sorensen (ed.) Public Finance in a Changing World, London: MacMillan Press, 151-92. [33] Wilson, J.D. (1999). Theories of tax competition. National Tax Journal, 52 (2), 269-304. [34] Yitzhaki, S. and Slemrod, J. (1991). Welfare dominance: an application to commodity taxation. American Economic Review, 81(3), 480-96. [35] Yitzhaki, S. and Lewis, J.D. (1996). Guidelines on searching for a Dalton-Improving tax reform: an illustration with data from Indonesia. The World Bank Economic Review, 10(3), 541-62. [36] Yitzhaki, S. and Thirsk, W. (1990). Welfare dominance and the design of excise taxation in the C^ote d'Ivoire. Journal of Development Economics, 33, 1-18.
33
cat(A), and(B) dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.161 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.29 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.1 dta=0.1t z=3M dta=0.25t dta=0.5t
ATR50 x x x
®BA < 0 x x x x x
®BA < 0 x x x x x
®BA < 0 x x
ATR20 x x x x x x x x x S x x x x x x x x
ATR10 x x x x x x x x x x x x x x x x x x
ATR5 x x x x x x x x x x x x x x x x x x
Table 2: Test of condition R1 with migration from And to Cat. S:successful test, x:crossings on the concentration curves, F:failings
and(A), cat(B) dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.161 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.29 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.1 dta=0.1t z=3M dta=0.25t dta=0.5t
ATR50
®BA < 0 S x
®BA < 0 ®BA < 0 S
®BA < 0 S x
®BA < 0 ®BA < 0 S
®BA < 0 S x
®BA < 0 ®BA < 0 S
ATR20 S S S
®BA < 0 S S S S S
®BA < 0 S S S S S
®BA < 0 S S
ATR10 S S S S S S S S S S S S S S S S S S
Table 3: Test of condition R1 with migration from Cat. to And.
34
ATR5 S S S S S S S S S S S S S S S S S S
val(A), cat(B) dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.161 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.29 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.1 dta=0.1t z=3M dta=0.25t dta=0.5t
ATR50
®BA < 0 S S
®BA < 0 ®BA < 0 S
®BA < 0 S S
®BA < 0 ®BA < 0 S
®BA < 0 S S
®BA < 0 ®BA < 0 S
ATR20 S S S S S S S S
ATR10 S S S S S S S S S
®BA < 0
®BA < 0
S S S S S
S S S S S S S S
®BA < 0
®BA < 0 S S
Table 4: Test of condition R1 with migration from Cat to Val.
cat(A), val(B) dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.161 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.29 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.1 dta=0.1t z=3M dta=0.25t dta=0.5t
ATR50 x x x S x x x x x
®BA < 0 x x x x x S x x
ATR20 x x x x x x x x x x x x x x x x x x
ATR10 x x x x x x x x x x x x x x x x x x
Table 5: Test of condition R1 with migration from Val. to Cat.
35
ATR5 x x x x x x x x x x x x x x x x x x
ATR5 S S S S S S S S S S S S S S S S S S
mad(A), cat(B) dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.161 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.29 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.1 dta=0.1t z=3M dta=0.25t dta=0.5t
ATR50
®BA < 0 x x
®BA < 0 ®BA < 0 x
®BA < 0 x x
®BA < 0 ®BA < 0
ATR20 S x S
®BA < 0 S x S x x
®BA < 0
x S x x
S x S x x
®BA < 0
®BA < 0
S x
S x
ATR10 x x S S x x x x x S x x x x x S x x
ATR5 x x S x x x x x x S x x x x x x x x
Table 6: Test of condition R1 with migration from cat. to mad. cat(A),mad(B) dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.161 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.29 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.1 dta=0.1t z=3M dta=0.25t dta=0.5t
ATR50 S x x
ATR20 S S S
®BA < 0
®BA < 0
S x
S S S S S
®BA < 0 S x
®BA < 0 ®BA < 0 S S x x
®BA < 0 S x
®BA < 0 S S S S S S S S
ATR10 S S S S S S S S S S S S S S S S S S
Table 7: Test of condition R1 with migration from mad. to cat.
36
ATR5 S S S S S S S S S S S S S S S S S S
and(A),val(B) dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.161 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.29 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.1 dta=0.1t z=3M dta=0.25t dta=0.5t
ATR50 S x x
®BA < 0 S x S x x
®BA < 0 S x S x x
®BA < 0 S x
ATR20 S x x S S x S x x S S x S x x S S x
ATR10 x x x S x x S x x S S x x x x S x x
ATR5 x x x S x x x x x S x x x x x S x x
Table 8: Test of condition R1 with migration from val. to and.
val(A), and(B) dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.161 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.29 dta=0.1t z=3M dta=0.25t dta=0.5t dta=0.1t z=1M dta=0.25t dta=0.5t ½ = -0.1 dta=0.1t z=3M dta=0.25t dta=0.5t
ATR50
®BA < 0 x x
®BA < 0 ®BA < 0 x
®BA < 0 x x
®BA < 0 ®BA < 0
ATR20 S x x
®BA < 0 S x S x x
®BA < 0
x S x x
S x S x x
®BA < 0
®BA < 0
S x
S x
ATR10 S x x S S x S x x S S x S x x S S x
Table 9: Test of condition R1 with migration from and. to val.
37
ATR5 x x x S x x x x x S x x x x x S x x
C.A. (A, B) Cat, And And, Cat Cat, Mad Mad, Cat Val, Cat Cat, Val Val, And And, Val
ATR5 F S S F S F x x
ATR10 F S S x S F x x
ATR20 F S x x S F x x
ATR50 F x x F x F F F
Table 10: Condition R1*: R1 when z is tax free. C.A. (A, B) Cat, And And, Cat Cat, Mad Mad, Cat Val, Cat Cat, Val Val, And And, Val
ATR5 F S S F S F S F
ATR10 F S S F S F S F
ATR20 F S S F S F S F
Table 11: Condition R5*: R5 when z is tax free.
C.A. (A, B) Cat, And And, Cat Cat, Mad Mad, Cat Val, Cat Cat, Val Val, And And, Val
R2 F S S F S F x x
R6 F S S F S F S F
Table 12: Conditions R2 and R6.
38
ATR50 F S F F S F F F
