# Tax complexity with heterogeneous voters.

1. INTRODUCTIONThe issue of tax complexity is a crucial one in the policy debate about tax design and reforms. A recent trend towards simplification has emerged, at least in developed countries, such as, for instance, Spain, Germany, The United Kingdom and Italy. Between 2000 and 2005 most of the OECD countries cut the number of brackets with the exception of Canada, Portugal and the U.S1 that all added one.

The experiment of a simplified tax code started in Estonia, the first country in Europe to introduce in 1994 a flat tax on personal and corporate income at a single uniform rate of 26%. Others followed: first Latvia in 1995 and Lithuania in 1994 with a tax rate of respectively 25% and 33%, Russia in 2001 with a rate of 13% on personal income; three years later Slovakia imposed a uniform 19% rate on personal and corporate income and Ukraine a 13% tax rate. Finally in 2005 Georgia and Romania introduced a flat tax rate of 12% and 16%.

Lawyers, economists and accountants have addressed the issue of complexity at length, but in a qualitative way. An initial major difficulty concerns the definition of tax complexity: a broad definition relates tax complexity of a tax scheme to the number of its tax brackets, as well as the number and extension of deductions, allowances, tax credits and special cases (see on this point Hettich and Winer, 1988; Warskett, Winer and Hettich, 1998). But the consensus is not obvious. As Slemrod (2005) emphasizes "Many argue that much of the complexity arises from the use of the income tax system as a vehicle for numerous social policies that are unrelated to raising revenue but that piggyback on the income tax system as an administratively or politically convenient way to deliver these policies. Many say that it is the constant change in the tax law, rather than the state of the law at any one time, that breeds complexity. Finally some argue that spelling out the tax ramifications of as many as possible situations reduces uncertainty and therefore complexity, with others taking the opposite positions" (p.281).

A difficult definition implies that measuring tax complexity is a challenging task. For the personal income2 an appropriate indicator of tax complexity should include the number of tax rates, together with the number of allowances, exemptions, deductions and tax credits. There are no quantitative analyses concerning what engenders tax complexity even though the complexity of the tax system has been debated for years in the European countries (see Bernardi and Profeta, 2004). The debate around tax complexity is centered on two questions: What are the advantages of a complex tax code? And, what are the advantages of a simple one? A complex tax scheme allows to better and more appropriately differentiate individuals, thus increasing their support at elections. The main advantage of a simple tax code instead relies on administration and compliance costs: a simplified scheme is easier to administer. Estimates for the United States whose tax regime is by no means the world's most burdensome, put the costs of compliance, administration and enforcement between 10% and 20% of revenue collected (The Economist, April 16, 2005). Once tax codes have degenerated to the extent they have in most industrialized countries, laden with so many breaks, deductions, allowances and tax expenditures that substantially changed their original shape, even the pretence of any interior logic disappears.

We currently observe worldwide a common declared intention of increasing simplicity of the income tax, mainly through a reduction of the number of brackets. However, de facto the implemented changes do not really seem to lead to this objective of simplicity, since deductions, exemptions, special cases multiply. Italy represents an interesting case. In recent years the issue of tax complexity has been a crucial one in the policy debate about tax design and a trend towards simplification has emerged. There is a wide, although not unanimous, consensus that the Italian income tax is very complex and that such complexity is not simply justified by its economic feasibility, but there is no widely accepted story about it and certainly no empirical evidence concerning why the income tax system is so complex. Changes in the structure of the personal income tax in the period 1974-2005 are meant to reduce tax complexity, mainly through the reduction of the number of brackets. However, deductions, allowances etc. have increased during our sample period. We perform a simple regression exercise on Italian data in the period 1974-2005, which suggests the existence of a trade-off in the determination of tax complexity, measured by the number of tax brackets, deductions and allowances: on one side it positively depends on the degree of heterogeneity of individuals, while on the other side it negatively depends on the cost of having more cases. In other words, a more complex tax structure is required by a more heterogeneous population to meet individuals' preferences, while it may be costly. We also find that ideological factors play a crucial role, with the poor (and thus governments representing closer their preferences) requiring more complexity.

The evidence of this trade-off motivates our analysis. We build a model which predicts the emergence of a trade-off in the determination of tax complexity of the personal income tax and analyzes its main features. Taxpayers heterogeneous by income are grouped in different tax brackets. We abstract from oth er individual economic characteristics (marital status, number of children, age, physical handicaps etc.) which may determine a different tax treatment (3). Following a political economy approach, we argue that economic and political considerations are interrelated in explaining the complexity of the income tax that we observe in most democratic countries and the trend towards a simplified structure. In a democracy the political competition requires complexity, i.e. many tax rates which allow to carefully discriminate among heterogeneous voters (Hettich and Winer, 1988; Warskett et al., 1998) and thus to maximize the support that each party expects to receive in the next election. In the limit each person should be taxed with a different tax rate, to better respond to his preferences. However, when the system becomes well developed, public expenditures and taxation raise and this requires an increase of complexity of regimes and raises the administration costs, which make such a complicated tax structure too expensive, and force to reduce complexity and to group the individuals together (4). Thus, the government decides to create rate brackets to group individuals. Then, it has to decide how to establish the politically optimal number of brackets, how to assign individuals to these brackets in a manner consistent with its political objectives and how to choose the rate of taxation applied to each group.

This choice is intrinsically multidimensional. A standard median voter approach is not appropriate to deal with it, since Nash equilibria of a majoritarian voting game may fail to exist when the issue space is multidimensional (Persson and Tabellini, 2000). We therefore introduce a simple probabilistic voting model (Profeta, 2002). Differently from a median voter approach, probabilistic voting has the advantage of explicitly consider the role of heterogeneity of voters. The model delivers the existence of a trade-off in the choice of the number of brackets: on one side decreasing the number of brackets implies a larger loss in expected support, since it is no longer possible to equalize marginal political costs of taxation across individuals. This loss is larger the larger the political power of the individuals which are affected by the grouping. On the other side decreasing the number of brackets implies lower administration costs, and thus higher revenue and the possibility to spend more on public goods, which can be converted into additional support (5).

As in Warskett et al. (1998), which formalizes an earlier intuition of Hettich and Winer (1988), our model predicts that political competition requires tax complexity. We make a step further by making explicit, starting from the Italian evidence, how this result depends on the existence of a trade-off between costs and benefits of tax complexity. Our model also delivers interesting and new predictions for the evolution of tax complexity. In particular, a well-developed system may be at a point where the costs are so high that the need to simplify is predominant.

The paper is organized as follows. Next section provides evidence on the Italian personal income tax which motivates our analysis. Section 3 develops a probabilistic voting model of tax complexity. Section 4 concludes.

2. PERSONAL INCOME TAX COMPLEXITY: THE CASE OF ITALY

This section provides evidence on the existence of a trade-off between benefits and costs of the Italian personal income tax complexity from 1974, year of a radical reform, to 2005: a more complex structure is beneficial for a heterogeneous population, while it is administratively costly.

The personal income tax (IRPEF) represents about one fourth of total Italian fiscal revenue and more than one third of tax revenue (see Gastaldi and Liberati, 2004). It can be considered the main instrument to achieve redistribution, in the form of vertical equity, through its progressive scheme and in the form of horizontal equity, through the qualitative discrimination of incomes and allowances schemes. The structure of the personal income taxation has been modified several times in the last decades (1974-2005). A series of changes of different magnitude has characterized the last decades, such as the variation of tax brackets, number of tax rates and the increase of tax allowances in the form of tax credits for type of income and for family charges. These changes have been justified both by efficiency and equity reasons, but it is also well known that electoral reasons have always deeply motivated them (see Profeta, 2007). The level of complexity of IRPEF has significantly evolved over time towards a reduction of the number of tax rates (from 32 tax rates in the period 1973-82, to 9 in the period 1983-88, 7 in the period 1989-97, 5 in the period 1989-2004 and 4 in 2005) while the number of allowances, exemptions, deduc tions and tax credits instead has increased during our sample period, especially in the 1990s and 2000s.

We focus on the relation between the degree of complexity of the personal income tax and, respectively, the heterogeneity of individuals (as the benefit-side of a potential trade-off) and the enforcement cost (as the cost-side). There are inherent difficulties of measurement of these variables that may be at the origin of the lack of empirical investigations in the literature. We now explain how we choose to measure each of these three variables.

We proxy the degree of complexity of the personal income tax in each year by a composite indicator represented by the number of the tax rates multiplied by the number of allowances, deductions, exemptions and tax credits, or, alternatively, by the sum of all these items.

The benefit side of the income tax complexity is measured by the Gini pre-tax income inequality index, to capture the idea that the higher the degree of income differentiation the higher the income tax complexity. However we should take into account that the ideology of governments is also important: the political influence of the poor increases the degree of complexity, because the rich oppose less to an increase of the tax rate on them, as required by the poor, if there is more differentiation (see Profeta, 2007). Therefore, leftist governments which take more into account the interests of the poor tend to increase the number of brackets, special regimes and complexity in general. We measure the partisan element of government coalitions ideology with a dummy variable (LEFT) that takes the value 1 if there is a left-wing domination in both government and parliament and if left-wing or centre parties make up between 33.3% and 66% of the government; 0 otherwise. A positive sign is expected on the coefficient of LEFT. To include this ideological component into our measure of the benefit side of the income tax complexity, we use an interactive variable (Gini*LEFT) that captures the impact that the polarization of individuals (groups) preferences in terms of income and ideology have on the number of tax rates. We expect a positive sign of this variable: the political influence of the low- and middleincome groups together with the left-wing governments increases the degree of complexity of the personal income tax.

As for the cost-side of income tax complexity, we should look at total resources for collecting taxes, which are the sum of the tax collection administration's budget, the value of time and money spent by taxpayers and any costs incurred by third parties to the collection process (Slemrod, 2005). Unfortunately, data on the cost of collecting taxes are not available for Italy in any of these components. In order to overcome this problem, we resort to a conceptually less rigorous but more easily measurable indicator of the cost of income tax complexity: the number of lines (COMP) or, alternatively, the number of boxes (COMPR) (6) to be fulfilled by the taxpayer in the income tax form of the year before the considered one (7). This measure represents visible aspects of the income tax system that is likely to be correlated with the compliance and enforcement costs of the tax (see on this point Moody, 2002; Slemrod, 2005) even though it is an imperfect measure. For example, not all lines on a tax form have similar implications in terms of complexity; some may apply to only a small fraction of the taxpayers. Also, some income tax forms contain lines regarding non-income tax programs such as homestead credits for property tax. But, in spite of these shortcomings, this measure is worthy to be used to capture the cost dimension of the personal income tax.

Finally, we introduce some political control variables. The first one captures the degree of political competition among individuals (groups) measured by a government fragmentation index, i.e. the power dispersion in the government coalitions. This variable (FRAG) takes value 1 in the presence of a coalition government and 0 otherwise, that is in the presence of a single-party majoritarian government and a minority government, as in Roubini and Sachs (1989). More competition among parties in the government might result in higher complexity: coalition governments tend to meet the support of the voters by giving more weight to the benefits of increasing the number of tax rates and neglecting the administrative costs. The second one is the election variable (ELE), consisting in a dummy that takes the value of 1 in the year of the election and 0 otherwise. To capture the idea that proximity to elections may have impact on complexity, we use the election variable of the previous year, and we expect that this variable would affect positively the degree of complexity, since the policy makers are inclined to favor the heterogeneity of voters preferences. The third one is NGOV, which is a dummy variable capturing a change of government in the considered year. This is an indicator of instability of governments, which we expect to be positively related with tax complexity.

The political data come from Ministero dell'Interno (various years). Data on GINI index are from Brandolini (1999) for the period 1974-1995 and from Bank of Italy for the years 1998-2005. Information about the number of tax brackets, allowances, deduc tions and tax credits come from Bosi and Guerra, 2007; Gastaldi and Liberati, 2004; personal income tax forms for the years 1974-2005.

Table 1 shows the regression results: our measure of the benefit-side of tax complexity (GINI*LEFT) is significantly and positively related to our index of tax complexity (the product of the number of the tax rates multiplied by the number of allowances, deductions, exemptions and tax credits), while our measure of the cost-side (the number of boxes of the tax form of the previous year, COMPR (-1)) is significantly and negatively related to it. Political control variables have the expected sign: the complexity of the income tax is higher when fragmentation of the governments is higher, the proximity of the elections is closer and the government is more instable. Notice however that only fragmentation turns out to be significant.

Considering the measurement problems that we have discussed above and the limited number of observations available, these results has to be taken with some caution. They mainly provide sketchy evidence that will motivate our next study, rather than a rigorous econometric analysis. They are however robust to different specifications, for instance the results do not qualitatively change when using as complexity index the sum, rather than the product, of the number of tax rates, tax allowances, deductions, exemptions and tax credits and as cost variable the number of lines, rather than boxes, of the tax form.

The emergence of a quantifiable trade-off in the determination of the number of tax rates in this simple exercise for the Italian case motivates the rest of the paper. In next section we build a voting model with the intention to capture and characterize the trade-off.

3. A PROBABILISTIC VOTING MODEL

In this section we introduce a probabilistic voting model to analyze the personal income tax complexity. Probabilistic voting is used to overcome a typical problem of median voter models, which are not able to appropriately capture the multidimensional nature of the determination of tax complexity. Probabilistic voting is particularly useful in our context since it allows to deal with an ideological component, which characterizes the political power of different groups in a society and may be relevant to determine the structure of taxation.

3.1 General Futures

The society is composed of H groups of voters of different income (8), denoted byh = 1, ... H. Each group has different size, nh is the number of individuals in group h. Individuals in each group are homogeneous and they have the same utility function, defined by (9):

[U.sup.h] = ln [y.sup.h] (1 - [t.sup.h]) + G

where yh is the before-tax (given exogenously) income of each individual in group h , [t.sup.h] is the average tax rate that he has to pay on income, and G is the level of a public good. (10)

The government collects taxes to finance the public good G. However, collecting taxes involves a cost which depends on the number of tax rates used by the government. More precisely, the government faces the following budget constraint:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

i.e. the level of public good is equal to total revenues minus the administrative cost, and

C = [alpha][T.sup.2]

where a is the unitary cost associated to each tax rate and T is the number of tax rates used by the government. More tax rates imply a larger administrative cost at an increasing rate.

The government has to decide upon a public policy vector q, which is multi-dimensional, and it is defined by the level of tax rates for all brackets and the number of tax rates to apply. This means that we have two extreme cases and several intermediate possible cases for the policy platform:

Case 1. a single tax rate for each income group: T = H, q = ([t.sup.1] [t.sup.2], ...[t.sup.H])

Case 2. a unique tax rate for all income groups: T = 1, q = t = [t.sup.1] = [t.sup.2] = ... [t.sup.H]

Case 3. a combination of a number I [greater than or equal to] 1 of brackets, indexed by i = ... each containing [I.sub.i] groups of individuals(h [member of] [I.sub.i]) and a number of S [greater than or equal to] 1 tax rates applied to a specific group of individuals (h g Ii ,for all i = ).

Case 4. T = I + S, q = ([t.sup.i], [t.sup.h]) where i = 1, ... ,(h [??] [I.sub.i]).

In this case, T is larger than 1, because, to be different from the extreme cases, in this case there exist at least one bracket and one group which is taxed with its specific tax rate, and smaller than H , because there exists at least one bracket where more than one group of individuals are taxed with the same tax rate.

3.2 The political institution

The political game is described according to a probabilistic voting framework a la Lindbeck and Weibull (1987), which in turn builds on probabilistic voting models by Hinich et al. (1972), Coughlin and Nitzan (1981a, 1981b), and Coughlin (1992)11. This framework has also the advantage to deliver solutions when the policy space is inherently multi-dimensional and the simple majoritarian voting model with a median voter equilibrium cannot be applied, since a Nash equilibrium of a majoritarian voting game may fail to exist.

Consider two parties, or candidates, labeled A and B. Before the election takes place, the parties commit to a policy platform, [q.sup.A] and [q.sup.B]. They act simultaneously and do not cooperate. Each party chooses the platform which maximizes its expected number of votes (12). Platforms are chosen when the election outcome is still uncertain. The two parties differ along some other dimension relevant to the voters than the announced policy, which may reflect ideological elements. Voters are heterogeneous with respect to their ideological preferences.

Each party maximizes the expected total number of votes from all groups.

From the probabilistic voting specification (see appendix 6.1) it follows that party A will choose [q.sup.A] such as to maximize the following objective function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [U.sup.h]([q.sup.A] )is the utility of voters in group h under government policy [q.sup.a] ,[U.sup.h] ([q.sup.B]) is the utility of voters in group h under government policy [q.sup.A] and sh is the density of the ideology of group h . This density captures the number of swing voters, i.e. the political influence of group h.

Party B solves a symmetric problem. Parties act simultaneously, taking the choice of the other party as given, and do not cooperate.

Thus, taking qB as given, party A solves the problem at Eq.4 subject to the following budget constraint:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3.3 The probabilistic voting equilibrium

We solve the problem of party A at Eq. 4 subject to the budget constraint at Eq. 5, given [q.sup.B], under the three possible cases of specification of [q.sup.A] .

We aim at comparing the total number of votes that party A can obtain in the different politicoeconomic equilibrium that arise under the three possible cases, which depend on the number of tax rates used. We do not analyze what of the three cases is optimal13.

We define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] the aggregate average

density and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] the aggregate income.

The following proposition summarizes the results on the tax rates.

Proposition:

The equilibrium tax rates are the following:

Case 1. A tax rate for each income group:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Case 2. A unique tax rate for all income groups:

t = 1 -1 1/Y

Case 3. I tax rates for each income bracket and S tax rates specific for income groups not in any bracket

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof: See the Appendix

The proposition shows some interesting results. For a given a [bar.s], the optimal tax rate applied to a group of income (or to different groups) depends negatively on the size and on the political power of the group (or groups) of income to which this tax rate is applied, and positively on its (their) income. In other words, more numerous or more politically powerful groups will pay a lower tax rate, as predicted by standard political economy models. Also, richer groups will pay a higher average tax rate, i.e. the personal income tax will be progressive.

However, in this paper we are mainly interested in comparing the three equilibriums arising in the different cases. To this respect, it is easy to show that the corresponding optimal levels of public good are:

Case 1. G = [bar.y] -1 -[H.sup.2] [alpha]

Case 2. G = y -1 -[alpha]

Case 3. G = [bar.y] -1 - (I + S).sup.2] [alpha].

By combining the above results, the total number of votes collected by party A can be expressed as follows:

Case 1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Case 2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Case 3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3.4 The trade-off

By comparing the total number of votes that party A obtains using a different number of brackets, a trade-off emerges: on one hand a smaller number of tax rates or more brackets imply less administrative costs, which allow a higher level of public good, and thus of the utility of all individuals. On the other hand it also implies a loss in the political support that the party can obtain. In fact, when the party can apply to each income group a specific tax rate, it can apply the tax rate that maximizes the political support from each income group. When, on the contrary, the party decides to apply the same tax rate to different income groups, this tax rate necessarily cannot maximize the political support by all income groups involved.

In the appendix 6.3 we explicitly derive the cost and benefit sides of the trade-off for all cases.

To investigate the determinants of the trade-off, we now focus on a simple case where there are only three income groups, low, middle and high income: [y.sup.1] < [y.sup.2] < [y.sup.3].

Under this specification, the 3 cases become:

Case 1. q = ([t.sup.1] [t.sup.2], [t.sup.3])

Case 2. q = t

a q = ([t.sup.1], [t.sup.3]) where [t.sup.1] is applied to y:and [y.sup.2], [t.sup.3] to [y.sup.3] Case 3.b q = (tl,t2)where [t.sup.1] is applied to [y.sup.1] and [t.sup.2] is applied to [y.sup.2] and [y.sub.3].

It is easy to show that the number of votes in case 1 is larger than the number of votes in case 2 if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the left-hand-side, if positive, is the cost of grouping and the right-hand-side is the benefit from grouping. Since we are comparing here the two extreme cases, the intuition is straightforward: when the groups which are included in the same bracket are more numerous (a higher [n.sup.1] or [n.sup.2] or [n.sup.3] in this case) ore more politically powerful (a higher [S.sup.1] or [S.sup.2] or [S.sup.3]) or poorer (a lower y1 or [y.sup.2] or y3 ) they would push for a higher differentiation, and thus the cost of grouping increases. The benefit of grouping depends instead on the reduction of costs associated with a smaller number of tax brackets.

The comparison between these two extreme cases delivers the more intuitive and interesting result, contained in the following proposition. Results in the same direction can be obtained from the comparison of the other cases (See Appendix for more details).

Proposition:

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] a more numerous, more politically powerful or poorer income group will push stronger for having its specific tax rate, and thus more differentiation. Thus, the loss from grouping depends positively on the size and the political power of the income groups included in the same tax bracket and negatively on their income. The gain from grouping instead depends on the administrative costs which increase when the number of tax rates increases, i.e. for a more complex tax schedule.

Proof: See Appendix 6.4.

Remark: If the three groups have equal size, equal density and equal income, there is no loss from grouping while the benefit is positive, and it is therefore always optimal to apply a unique tax rate.

The results show that the trade-off in tax complexity arises from the heterogeneity of individuals, which crucially shapes the political process. Our probabilistic voting model is a simple framework to capture the role of heterogeneity in the political determination of the tax structure.

This result is very intuitive: the size of different income groups, the distribution of income, the political power of different income groups and the size of the administrative costs shape the trade-off behind the choice of a complex personal income tax schedule. In particular, the benefit of tax complexity is larger for a more unequal income distribution and for a higher political power of the group which would take advantage from differentiation. But a more complex tax schedule is also more costly. The number of tax rates thus de pends on how the political process balances costs and benefits of tax complexity.

4. POLICY IMPLICATIONS

Our study delivers an interesting policy implication. Even though the flat-rate or broad-based tax seems to be very popular at the moment in the political debate as a way to simplify the income tax regime, in Italy as well as in other industrialized European countries, our model predicts that a flat tax will not be politically feasible, unless the society is characterized by groups of voters rather homogeneous in income, size and political power. When instead voters are heterogeneous, the more numerous, more politically powerful or poorer income groups will push to be taxed according to a specific tax rate which meets their preferences.

The intuition is the following. In a democracy the political competition requires complexity as a way to discriminate among heterogeneous voters and thus to maximize the support that each party expects to receive in the next election. Therefore the structure of income taxation that we observe can be seen as the result of a political process where politicians maximize their expected number of votes. To gain political support, respecting the constraint of administrative costs, a complex tax structure emerges, with grouping individuals in brackets. A trade-off is obviously implicit in the choice of the number of brackets: on one side decreasing the number of brackets implies a larger loss in expected support, since it is no longer possible to equalize marginal political costs or oppositions to taxation across individuals. On the other side decreasing the number of brackets generates lower administration costs, and thus higher revenue and the possibility to spend more on public goods, which can be converted into additional support. In presence of administration costs, information costs may induce an even stronger simplification, if taxpayers self select and choose to earn a reduced in come in order to be eligible for a lower tax rate. Therefore, a well-developed fiscal system may be at a point where costs are so high that the need to simplify is predominant.

Our policy implication is in line with what we observe in European countries. High administrative and information costs are raising the need for a reduction of complexity. The flat-rate or broad-based tax is becoming very popular (UK, Italy). However, in a democracy, political competition requires a combination of a flattax rate (or broad-based) with other provisions (exemptions, deductions, exclusions) aimed at differentiating. Although our model does not explicitly include the role of special provisions, we expect that they will be more extensively introduced in countries with more heterogeneity of voters, in terms of income, political power and size, i.e. when, as suggested by our model, the equilibrium tax structure departs more from a flat-rate system.

5. CONCLUSIONS

We have used a political economy approach to explain the personal income tax complexity. After providing evidence for the Italian case, our probabilistic voting model shows that reducing the number of tax rates of the personal income tax schedule by grouping individuals of different income levels in the same tax bracket involves a trade-off: on one side it reduces the political support by individuals, who necessarily will pay a tax rate different from their preferred one, while on the other side it also reduces the administrative costs of taxation.

Our theoretical analysis has two limits: first, while in the analysis of the case of Italy we allow tax complexity to broadly include the number of tax rates and the number of deductions, allowances and tax credits, in the model we restrict tax complexity to the number of tax rates, abstracting from the role of special provisions. Second, we focus only on the personal income tax neglecting other taxes. It would be interesting to address in a future study what happens to tax complexity when the role of special provisions is explicitly considered, or when more than one tax may change.

Nevertheless our positive analysis is a first step to explain the evolution of tax complexity in modern democratic societies and the recent trend towards simplification. We argue that this is mainly due to the fact that fiscal systems have become so sophisticated that the costs are too high and simplification is required.

Finally, what is the optimal level of complexity of the personal income tax schedule? We have not addressed this issue, although the emergence of the tradeoff that we have explained is a first step to understand which features would characterize the optimal design. A complete normative analysis is left to future studies.

APPENDIX

The Probabilistic Voting Model

Voter j in group h votes for party A if

[U.sup.h([q.sup.A])-Uh([q.sup.B]) + 8 + aj > 0, where [U.sup.h] ([q.sup.A]) is the utility (Eq. 1) of voters in group h under government policy [q.sup.A], [U.sup.h]([q.sup.B]) is the utility of voters in group h under government policy [q.sup.A], and the term ([delta] + [sigma].sup.j][sigma].sup.i] 0) reflects voter j's ideological preferences for party A . This term includes two components: 8, which is common to all voters, and [[sigma].sup.j], which is idiosyncratic. The first component, [delta], reflects the general popularity of party A . We assume that this is a random variable uniformly distributed on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] with expected value equal to 0 and density d. This component represents the source of electoral uncertainty, since it is realized between the announcement of the party platforms and the election. The second component, [[sigma].sup.j], reflects the individual ideology of voter j . Voters are distributed within each group according to a uniform distribution on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], where the density is [s.sup.h], specific to each income group, and the mean is (14) zero .

Each group has neutral voters, called "swing voters", who are indifferent between party A and B. The identity of the swing voters is crucial when a party considers whether to deviate from a common policy announcement, [q.sup.A] = [q.sup.B], or not. Suppose party A decides to decrease taxes of group 1 financed by a budgetbalanced increase of taxes to group 2. Party A expects a gain of votes from group 1 equal to the number of swing voters in group 1, and a loss of votes from group 2 equal to the number of swing voters in group 2. If group 1 has a higher number of swing voters than group 2, this will lead to a net gain of votes. As a consequence, each party tries to attract the more mobile voters. Formally, the swing voter in group h is identified by as.v. where

[[sigma].sup.s.v.] = [U.sup.i] ([q.sup.B] ) - [U.sup.i] ([q.sup.A] ) -[delta]. Voters with [[sigma].sup.h] lower than vote [[sigma].sup.sv] for party B and voters with [[sigma].sup.h] higher than [[sigma].sup.s.v.] vote for party A .

Therefore, the vote share of party A in group h can be expressed by

[[pi].sup.A,h] = [s.sup.h] ([U.sup.h] (q[q.sup.A]) - [U.sup.h] ([q.sup.B] ) + [delta]) + 1/2).

Each party maximizes the expected total number of votes from all groups. Given the definition of [[pi].sup.A,h], the objective function of party A can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Substituting the expression for [[pi].sup.A,h] and given the previous assumptions about the distribution functions, party A will choose [q.sup.A] such as to maximize the following objective function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where ([U.sup.h] (q) is defined at Eq. 1. Proof of proposition 1

* Case 1: T = H, q = ([t.sup.1] [t.sup.2], ... [t.sup.H]) Party A maximization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Substituting the budget constraint into the maximization function, we obtain the first order condition with respect to a generic [t.sup.h]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

from which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

* Case 2: T = 1, q = t = [t.sup.1], [t.sup.2], ... [t.sup.H]

Party A maximization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Substituting the budget constraint into the maximization function, we obtain the first order condition with respect to t :

from which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

* Case 3: 1 < T < H individuals in a generic group [I.sup.i] are taxed with the same tax rate f individuals in a specific group are tax with their specific [t.sup.h].

Party A maximization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Substituting the budget constraint into the maximization function, we obtain the first order condition with respect to [t.sup.i] :

and with respect to [t.sup.h] (h * [I.sub.i]),

from which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The trade-off

We can identify the trade-offs as follows:

* shifting from case (1) to case (2), the cost of grouping depends on

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the benefit of grouping depends on sa(T2 -1).

* shifting from case (1) to case (3), the cost of grouping depends on and the benefit of grouping depends on

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

* shifting from case (3) to case (2), the cost of having only one group depends on

and the benefit depends on

[bar.s] [[alpha]((I + S).sup.2] -1).

Moreover, case 3 implies several alternative possibilities at the same cost, mainly targeting at the bottom versus targeting at the top. A general result is that it is better to target groups more numerous and/or containing more swing voters.

Proof of Proposition 2

* From case 1 to case 2 (Eq. 22):

The interesting case is for a generic h(h = 1,2,3) for which shifting from being taxed with its specific tax rate to a unique tax rate implies a loss

Under the above condition, for a given s and y, it is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. The benefit from

grouping instead is represented by the RHS, which increases with [alpha]. Q.E.D.

We now compare the other cases:

* The number of votes in case 3a is larger than the number of votes in 1 if:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the left-hand-side, if positive, represents the advantage of grouping and the right-hand side the cost. Group 3 is indifferent between case 1 and 3a, because in both cases he will be taxed with his specific tax rate. Group 1 and 2 prefer case1 to case 3b, since they are taxed with their specific tax rate rather than with a unique tax rate. Thus, if they are more numerous or more politically powerful or poorer, the number of votes in case 1 are more likely to be larger than in case 3 a. A similar result applies when comparing 3b with 1.

* The number of votes in case 3a is larger than the number of votes in case 2 if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since the groups which are included in the same bracket in case 3a are group one and group two while all groups pay the same tax rate in case 2, a higher [n.sup.1] [s.sup.1] or [n.sup.2][s.sup.2] implies a higher cost of grouping, while the benefit from grouping increases with the number of tax rates.

* The number of votes in 3a is larger than the number of votes in 3b if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In this case the cost is the same and we have to compare the relative total benefits.

ACKNOWLEDGEMENTS

We thank Stanley Winer for useful suggestions, an anonymous referee and participants at IIPF 2007 and SIEP 2007 for many comments. All remaining errors are ours.

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(1) For the U.S., see Slemrod (1990).

(2) In our analysis we focus on the personal income tax neglecting the existence of other taxes. Notice however that simplifying one tax may well lead to changes in other taxes that increase the complexity of those taxes, or it may even have an effect on the use of other types of policy instruments, such as regulation. Nevertheless the personal income tax is a major one and the one at the center of the debate on simplification (see for instance the debate on the flatrate tax).

(3) This assumption, justified by tractability reasons, reduces our concept of tax complexity in the theoretical model to the number of tax rates, neglecting the role of special provisions. This is clearly a drawback of our analysis, since deductions, allowances and special cases contribute to define tax complexity, as we also see in the Italian case. The model will however deliver interesting insights.

(4) Complicated rules generate also imperfect enforcement since taxpayers are uncertain as to eligibility when taking a deduction or credit and controls by the financial administration become more difficult (Krause, 2000).

(5) Also, in presence of administration costs, information costs may induce an even stronger simplification, if taxpayers self select and choose to earn a reduced income in order to be eligible for a lower tax rate (if their increase in leisure more than compensate the loss in after-tax earnings).

(6) To make the line-based cost measure comparable through the years, we always include in the numbered count of lines and boxes the name, address and calculations of exemptions and marital status.

(7) To capture a physiological delay between perceptions and tax design, we suggest that more lines in the tax form at year t-1 imply a more complex tax system at year t. The use of a delayed variable is also useful to avoid endogeneity problem between the number of lines in the tax form and our measure of tax complexity.

(8) Exogenously given income is here the only economic source of heterogeneity. A more comprehensive analysis should include other sources to explicitly address the role of special provisions, but at a cost of a reducing tractability.

(9) Quasi-linearity simplifies the model, since the income effects only show up in the linear component, i.e. the public good. It is a common assumption in this kind of redistribution models, see Persson and Tabellini (2000).

(10) For simplicity, and to better focus on the aim of the paper, we do not specify the economic maximization process which is standard in this framework (see Profeta, 2004).

(11) See also Persson and Tabellini (2000) and Profeta (2004 and 2007).

(12) This approach is standard in the literature. Alternatively, the objective of the party can be to maximize the probability of winning, which would leave the results unaffected.

(13) This would be very interesting, but a much more complicated theoretical task, and is out of the scope of this model.

(14) In general, both 8 and a1 may have expected values different from zero, reflecting the across groups difference in average ideology.

CONTACT INFORMATION OF AUTHORS

Galli: Professor, Universita La Sapienza di Roma, Rome, Italy. (emmagalli@hotmail.com)

Profeta: Professor, Universita Bocconi, Milan, Italy (paola.profeta@unibocconi.it)

Emma Galli Universita di Roma, La Sapienza

Paola Profeta Universita Bocconi, Econpubblica and Dondena Research Center

TABLE 1 Regression results (OLS) Dependent variable: Index of complexity = number of the tax rates * number of allowances, deductions, and tax credits Variable Coefficient P-value Constant -96.93 0.3122 (93.36) COMPR(-1) ** -16.38 0.0065 (5.54) GINI x LEFT ** 355.3 0.0036 (106.8) FRAG ** 20.36 0.007 (34.33) ELE(-1) 63.66 0.5601 (21) NGOV 37.84 0.1753 (1.4) [R.sup.2]=0.63; Adj- R2=0.53; S.E.=73 F= 6.4; N.obs. 25 ** significant at 5%

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Author: | Galli, Emma; Profeta, Paola |
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Publication: | Public Finance and Management |

Article Type: | Report |

Geographic Code: | 4EUIT |

Date: | Mar 22, 2009 |

Words: | 8061 |

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