# A Mathematical Revisit of Modeling the Majority Voting on Fixed-Income Quadratic Taxations

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## Abstract

Analyzing voting on income taxation usually implies mathematically cumbersome models. Moreover, a majority voting winner does not usually exist in such setups. Therefore, it is important to mathematically describe those cases in which a majority winner exists, at least for the basic models of voting on income taxation. We provide a complete mathematical description of those income distribution functions for which a majority winning tax exists (or does not exist), in the quadratic taxation model à la Roemer (1999), with tax schedules that are not necessarily purely redistributive. As an intermediate step, we identify by the corner method what are the most preferred taxes of the individuals, when taxation is not purely redistributive. Finally, we prove that for both purely and nonpurely redistributive quadratic taxations, the sufficient inequality condition of De Donder and Hindriks (2004) on the income distribution functions, for the existence of a Condorcet winner, can be relaxed to a broader condition.

## Keywords

Income Distribution Median Voter Condorcet Winner Feasibility Condition Vote Game## 1. Introduction

One important question that the positive theory of income taxation tries to answer is why marginal-rate progressive tax schedules are preponderant in democracies. An heuristic argument commonly invoked to explain this stylized fact resides in the observation that in general, the number of relatively poor (self-interest) voters exceeds that of richer ones. Nevertheless, mathematically formalizing the argument is not an easy task and the literature is rather inconclusive in this respect.

One very important difficulty which arises when studying these issues is that usually the existence of a majority winner (i.e., Condorcet winner) is not guaranteed. Voting games over redistributive tax schedules lack in general the existence of a static equilibrium (see Marhuenda and Ortuño-Ortin [1], Hindriks [2], De Donder and Hindriks [3]). The seminal papers of Romer [4, 5] and Roberts [6] consider only flat rate taxes in order to make use of the median voter theorem, after imposing some natural additional restrictions. However, the overrestrictive assumption of linear tax schemes does not provide the framework to investigate important issues like the high prevalence of marginal-rate progressive taxations in democracies. Therefore, many authors study the basic problem of voting on income taxations in terms of larger classes of tax functions.

Gouveia and Oliver [7] work with two-bracket piecewise linear functions, Cukierman and Meltzer [8] and Roemer [9] study quadratic tax functions, while Carbonell and Klor [10] consider a representative democracy model that allows for the class of all piecewise linear tax schedules. Marhuenda and Ortuño-Ortin [11] allow for the class of all concave or convex tax functions, proving by Jensen's inequality that for income distributions with the median below the mean income, any concave tax scheme receives less popular support than any convex tax scheme.

Carbonell and Ok [12] provide a two-party voting game in which each party (whose objective is to win the elections) proposes tax schemes from an unrestricted set of admissible functions and the voters selfishly vote for the tax that taxes them less. Establishing the existence of mixed equilibria, they identify certain cases in which marginal-rate progressive taxes are chosen almost surely by the political parties. However, Carbonell and Ok [12] find that if the tax policy space is not artificially constrained, the support of at least one equilibrium cannot be obtained within the set of marginal-rate progressive taxes. This result is in the same line with the one of Klor [13], who shows that a majority of poor voters does not necessarily imply progressive taxation for a more general policy space than the one in Marhuenda and Ortuño-Ortin [11].

Although it is hard to find an economically meaningful way of restricting the admissible set of income tax functions, the literature on voting over income taxes which are chosen from restricted policy spaces provides useful and powerful insights into the general problem. In particular, the quadratic model was very much used in the literature to generate interesting results. Cukierman and Meltzer [8] analyze the conditions under which the median voter's most preferred tax policy is a majority winner, in quadratic distortionary tax environments. Roemer [9] uses the quadratic taxation framework to define a different solution concept than the majority winner, based on the need to reach an intraparty agreement between the "opportunists" and the "militants" of the parties. In the same setup of fixed income (i.e., income not distorted by taxes) and quadratic taxations as the one in Roemer [9], Hindriks [2] establishes the inevitable vote cycling theorem.

De Donder and Hindriks [14] introduce preferences for leisure in the quadratic taxation model and study the voting process over tax schedules using other political equilibria than the Condorcet winner. For the quadratic model with fixed income, De Donder and Hindriks [3] show that incentive constraints result in the policy set to be closed and that individuals all have corner solutions over this set. They also provide a necessary and sufficient condition on the income distribution such that a Condorcet winner exists. Moreover, for income distributions with the median less than the mean, if a majority winner exists then it involves maximum progressivity.

This paper provides a complete description of those income distribution functions for which a majority winning tax exists (or does not exist), when the quadratic taxation model is not purely redistributive. For reasons of completeness, the analysis is not limited only to right skewed income distributions (which are empirically predominant), but there has been also considered the case of the left skewed income distributions. We also identify what are the most preferred taxes of the individuals (and the corresponding income groups they can be classified in, based on the preferred policies), when taxation has more than a purely redistributive purpose. Moreover, we show in this paper that the sufficient condition of De Donder and Hindriks [3], imposed on the income distribution functions in order to insure the existence of a Condorcet winner, can be relaxed to a broader condition.

The paper is organized as follows. Section 2 presents the model. Section 3 states and proves the results. Section 4 discusses and draws the conclusions.

## 2. The Model

The economy consists of a large number of individuals who differ in their (fixed) income. Each individual is characterized by his/her income Open image in new window . The income distribution can be described by a continuous function Open image in new window , differentiable almost everywhere and strictly increasing on the interval Open image in new window . Each individual with income Open image in new window has strictly increasing preferences on the set of its possible net incomes. For any Lebesque measurable set Open image in new window , the associated Lebesque-Stieltjes probability measure induced by Open image in new window is denoted by Open image in new window and it is defined as Open image in new window .

For better comprehensibility of the text, any parameter calculated based on the distribution Open image in new window is denoted using the letter Open image in new window (e.g., the mean is Open image in new window , the median is denoted by Open image in new window , the noncentered moment of second order is Open image in new window , and the variance of the income distribution is Open image in new window ), while Open image in new window refers to a random income in the interval Open image in new window .

The fixed amount Open image in new window should be collected through means of a tax imposed on the agents. When Open image in new window , the tax is purely redistributive. It is assumed that there is no tax evasion, and there are no distortions induced by the taxation system in the economy (i.e., the income is fixed), respectively. The set of admissible tax functions satisfies certain conditions. For a given Open image in new window and Open image in new window , Open image in new window denotes the set of all functions Open image in new window such that (without the second and third conditions below, we would have a resource redistribution problem like in Grandmont [15], which is known not to have a Condorcet winner; see at the end of this section the definition for a majority winner.)

(1) Open image in new window , for all Open image in new window ;

(2) Open image in new window , for all Open image in new window ;

(3) Open image in new window , for all Open image in new window ;

*progressive*(

*regressive*) if and only if Open image in new window is convex (concave). In the following, we consider only quadratic taxes of the form Open image in new window , Open image in new window . (The analysis also includes the case of linear tax schedules, when the coefficient " Open image in new window " takes the zero value.) We restrict our analysis to Open image in new window , the set of quadratic tax functions that satisfy the feasibility conditions (1)–(4). It can be easily proved that conditions (1) to (4) restrict the set of quadratic feasible taxes to functions of the form Open image in new window , Open image in new window , which satisfy the following conditions:

Note that for every given distribution Open image in new window and feasible Open image in new window , to every tax Open image in new window from Open image in new window , it corresponds one and only one element Open image in new window in the feasible area (FA), and vice versa. Thus, the set of feasible quadratic tax policies Open image in new window can be illustrated as follows (the intervals for Open image in new window are mathematically well defined due to the inequality Open image in new window , more specifically from Open image in new window .)

(i)The case Open image in new window is represented in Figure 1(a).

- (iii)
The case Open image in new window is represented in Figure 1(c).

The coordinates of the vertices of the above polygons are easily obtained by elementary computations and are given by

*progressive taxations*: Open image in new window , Open image in new window , Open image in new window ,

*regressive taxations*: Open image in new window , Open image in new window , Open image in new window ,

*no taxation*: Open image in new window , *confiscation policy*: Open image in new window .

Figure 1 presents the feasibility areas for different cases of the collected amount Open image in new window . These areas are determined by the (FA) conditions as follows: the first two conditions determine the interior and the sides of the Open image in new window parallelogram. The third condition is the tax revenue requirement constraint, graphically identified by the half-plane situated above the line Open image in new window .

For the case depicted in Figure 1(a), the tax Open image in new window is the most progressive, Open image in new window is the most regressive, and Open image in new window and Open image in new window are out of the feasible area (FA). If Open image in new window then Open image in new window is not feasible, while if Open image in new window then Open image in new window . For the case depicted in Figure 1(b), the tax policy Open image in new window is the most progressive, Open image in new window is the most regressive, and the tax policies Open image in new window , Open image in new window , Open image in new window , Open image in new window are not feasible. For the case depicted in Figure 1(c), the tax schedule Open image in new window is the most progressive, Open image in new window is the most regressive, and the tax policies Open image in new window , Open image in new window , Open image in new window , Open image in new window are not feasible.

A *majority* (or *Condorcet*) *winning tax policy* is a pair Open image in new window in the feasible set such that is preferred by a majority of individuals to any other feasible pair Open image in new window in Open image in new window . An equivalent definition used in our proofs is the following: a tax function is a majority winner if and only if there is no *objection* to it (given Open image in new window , a tax policy Open image in new window is an objection to Open image in new window if Open image in new window ). We denote by Open image in new window the set of all objections to the taxation function Open image in new window . Therefore, the above definitions for Open image in new window being a Condorcet winner are equivalent to the condition Open image in new window .

If Open image in new window , by considering Open image in new window and Open image in new window , De Donder and Hindriks [3] defined the low middle and large income groups which are obtained based on the three intervals Open image in new window and Open image in new window divide Open image in new window . Note as well that Open image in new window . Nevertheless, the other two fixed values of the income are important for the analysis that follows. Those values are Open image in new window . In the same spirit as the interpretation offered by De Donder and Hindriks [3], the voters Open image in new window are poor with relatively high income, and Open image in new window are rich voters with relatively low income, respectively. As one can see in the section of results, these values will play an important role for stating the necessary and sufficient conditions for the existence of a Condorcet winner in the described environment.

## 3. Results

In order to identify the majority winning tax policies (if any), the first step is to characterize the tax policies Open image in new window that are objections to a given tax policy Open image in new window . Therefore, we need first to determine the sign of the function Open image in new window on the interval Open image in new window and then to find the Lebesgue measure Open image in new window of the set Open image in new window on which the difference function is negative. The following lemma presents the way in which the two roots of the quadratic function Open image in new window vary. Since the difference function is Open image in new window , Open image in new window , then it is sufficient to study the sign of the following quadratic function: Open image in new window , Open image in new window , Open image in new window . (We will analyze in the lemma only the case when Open image in new window ; the case Open image in new window will be discussed separately each time when it occurs in our discussion.)

Lemma 3.1.

The behavior of Open image in new window and Open image in new window .

0 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

0 | |||||||||||

Proof of Lemma 3.1.

The discriminant of Open image in new window can be written as Open image in new window ; hence Open image in new window has two real roots. For each Open image in new window , we will denote by Open image in new window and by Open image in new window the smallest and, respectively the largest of the roots. After short computations, we get Open image in new window and Open image in new window .

The behavior of the roots as functions of Open image in new window can be elementary studied by computing their derivatives and the limits at the endpoints of Open image in new window . Since Open image in new window , for each Open image in new window and Open image in new window , for each Open image in new window , then Open image in new window and Open image in new window are increasing functions of Open image in new window . The limits of the functions Open image in new window and Open image in new window at the endpoints of the definition domain are Open image in new window , Open image in new window , Open image in new window , Open image in new window .

Elementary computations give us the following results: Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window . Due to the previous computations, the behavior of the functions Open image in new window and Open image in new window is as presented in Table 1.

The purely redistributive tax policies that individuals prefer are described in De Donder and Hindriks [3]; all individuals in the same income class prefer the same policy. The low income group prefers confiscation policy (represented by the point Open image in new window in the feasible region (FA) when Open image in new window ), the middle income class prefers the maximum progressivity (represented by the point Open image in new window if Open image in new window ), and no taxation (the point Open image in new window ) is preferred by the high income group. The next lemma shows how this simple description changes when the tax schedules are not purely redistributive. A sketch of the proof is provided after stating the result and further details are available upon request.

Lemma 3.2.

The preferred tax for an individual with the income Open image in new window is

(1)the case Open image in new window (Figure 1(a)):

(1a) Open image in new window for Open image in new window (for the income Open image in new window , the individual is indifferent between the taxes on the segment Open image in new window ),

(1b) Open image in new window for Open image in new window (for the income Open image in new window , the individual is indifferent between the taxes on the segment Open image in new window ),

(1c) Open image in new window for Open image in new window (for the income Open image in new window , the individual is indifferent between the taxes on the segment Open image in new window ),

(1d) Open image in new window for Open image in new window ,

(2)the case Open image in new window (Figure 1(b)):

(2a) Open image in new window for Open image in new window (for the income Open image in new window , the individual is indifferent between the taxes on the segment Open image in new window ),

(2b) Open image in new window for Open image in new window (for the income Open image in new window , the individual is indifferent between the taxes on the segment Open image in new window ),

(2c) similar to (1d),

(3)the case Open image in new window (Figure 1(c)):

(3a) similar to (2a),

(3b) Open image in new window for Open image in new window (for the income Open image in new window , the individual is indifferent between the taxes on the segment Open image in new window ),

(3c) Open image in new window for Open image in new window .

Proof of Lemma 3.2.

An individual with income Open image in new window prefers the tax Open image in new window for which the difference Open image in new window is maximum. Hence, we have to solve the following linear programming problem: determine the maximum of the function Open image in new window , subject to the constraints Open image in new window , Open image in new window , Open image in new window , Open image in new window . The problem can be elementary solved by using the corner method.

Irrespective of the amount Open image in new window that should be collected, the low income group prefers the tax policy that equalizes the posttax income. The middle income group prefers the most progressive tax policy. The high income group is divided in a lower part and an upper one by the value Open image in new window . The upper part always prefers a regressive taxation when Open image in new window (in fact, for high values of the amount to be collected, this income group prefers the most regressive tax schedule—see Lemma 3.2(3c)) above). The lower part of the high income group usually behaves as the middle income group, except for the case of low levels of Open image in new window . Even in such a case (see Lemma 3.2(1c)), the lower part of the high income group prefers a progressive taxation instead of a regressive one. These observations motivate a possible redefinition of the middle income group from Open image in new window to Open image in new window . However, in order to have clear comparisons between the results in De Donder and Hindriks [3] and our results, we consider Open image in new window as the lower part of the high income group, while the interval Open image in new window keeps its interpretation of middle income class.

Having Lemmas 3.1 and 3.2 at hand, we are in the position to provide a complete description of the cases in which there is a majority winning tax, or when there is not. The next proposition can be immediately obtained from the lemmas and it is a first step to provide such a description.

Proposition 3.3.

The following assertions hold.

(1)If Open image in new window , then for each Open image in new window the tax policy Open image in new window is a majority winner (a Condorcet winner).

(2)If Open image in new window , then for each Open image in new window the tax policy Open image in new window is a majority winner (a Condorcet winner).

(3)If Open image in new window and Open image in new window , then the tax policy Open image in new window is a majority winner (a Condorcet winner).

- (1)
Let Open image in new window be defined by Open image in new window . In order to prove that under the conditions imposed by the hypothesis the function Open image in new window is a majority winner, it is sufficient to show there is no objection to it. Suppose by contrary that there exists Open image in new window . Then Open image in new window , Open image in new window satisfies the feasibility conditions (FA).

Figure 2 presents the feasibility areas for the coefficients Open image in new window and Open image in new window of the functions Open image in new window , which occur in the proofs of the Propositions 3.3 and 3.4. The feasibility areas are determined in a similar way as for the Open image in new window -feasible taxes: a parallelogram is separated by the line generated by the budget constraint condition.

We denote by Open image in new window , Open image in new window , where by Open image in new window and Open image in new window we mean Open image in new window and Open image in new window , respectively. From the feasibility conditions (FA) for the tax function Open image in new window , we obtain that the coefficients Open image in new window and Open image in new window must satisfy Open image in new window , Open image in new window , and Open image in new window . The feasible area for the coefficients Open image in new window and Open image in new window can be represented as it is shown in the Figure 2(a).

If Open image in new window , then Open image in new window and Open image in new window since Open image in new window .

If Open image in new window and Open image in new window , then Open image in new window and the roots of Open image in new window satisfy the inequalities Open image in new window and Open image in new window (see Lemma 3.1). In this case Open image in new window (see Table 2, line 2).

If Open image in new window and Open image in new window , then Open image in new window and the roots of Open image in new window satisfy the inequalities Open image in new window and Open image in new window (see Lemma 3.1). In this case Open image in new window (see Table 2, line 3).

- (2)
We will prove that there is no objection to the tax policy Open image in new window given by Open image in new window . Suppose, by contrary that there exists Open image in new window . Let Open image in new window be the tax policy given by Open image in new window and let Open image in new window , Open image in new window , Open image in new window where by Open image in new window and Open image in new window we mean Open image in new window and Open image in new window , respectively. The feasibility conditions for Open image in new window conduct to the following conditions on the coefficients Open image in new window and Open image in new window : Open image in new window . The feasible area for the coefficients Open image in new window and Open image in new window can be represented as it is shown in the Figure 2(b).

If Open image in new window , then Open image in new window and Open image in new window .

If Open image in new window , then Open image in new window and the roots of Open image in new window satisfy the inequalities Open image in new window and Open image in new window . In this case Open image in new window (see Table 2, line 4).

If Open image in new window then Open image in new window and the roots of Open image in new window satisfy the inequalities Open image in new window and Open image in new window . In this case Open image in new window (see Table 2, line 5).

- (3)
We will prove that in this case there is no objection to the tax policy Open image in new window given by Open image in new window . Suppose by contrary that there exists Open image in new window . Let Open image in new window be the tax policy given by Open image in new window . The feasibility area for the coefficients Open image in new window and Open image in new window is presented in Figure 2(c).

If Open image in new window , then Open image in new window and Open image in new window .

If Open image in new window , then Open image in new window , Open image in new window , Open image in new window and Open image in new window (see Table 2, line 6).

If Open image in new window , then Open image in new window , Open image in new window , Open image in new window and Open image in new window (see Table 2, line 5).

The sign of the function Open image in new window .

1 | 0 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | + | + | + | + | + | + | + | 0 | − | − | − | − | − | |

3 | + | + | + | + | + | 0 | − | − | − | − | − | − | − | |

4 | − | − | − | − | − | − | − | − | − | 0 | + | + | + | |

5 | − | − | − | − | − | 0 | + | + | + | + | + | + | + | |

6 | − | − | − | − | − | − | − | 0 | + | + | + | + | + | |

7 | + | + | + | + | + | + | + | + | + | 0 | − | − | − | |

8 | − | 0 | + | + | + | + | + | + | + | + | + | 0 | − | |

9 | − | − | − | 0 | + | + | + | + | + | + | + | + | + |

Note that if Open image in new window , the result from Proposition 3.3(1) was first obtained by De Donder and Hindriks [3] (see Proposition 1(a) in that paper). Proposition 3.3(1) is a generalization: it states that for every feasible value of Open image in new window , if a majority of individuals is in the low income group, then the voting outcome will determine that all individuals are equal in the posttax income. The second and third parts of the proposition have no empirical relevance since there is overwhelming evidence ruling out negatively skewed income distributions. However, these parts are reported for the purpose of completeness, such that Proposition 3.3 and the next three form together a knit result. (In fact, the results from the last two parts of Proposition 3.3 are very logical; e.g., the second part states that an existing majority of individuals in the upper part of the high income class will induce as a voting outcome the regressive tax system preferred by all the individuals with income in that subclass.)

The next two propositions are central for the current paper. We start with the second proposition, that provides a necessary condition for a majority winning tax to exist.

Proposition 3.4.

Let Open image in new window be such that Open image in new window .

(1)If Open image in new window , then for each Open image in new window the tax policy Open image in new window is a majority winner (a Condorcet winner).

(2)If Open image in new window , then for each Open image in new window the tax policy Open image in new window is a majority winner (a Condorcet winner).

(3)If Open image in new window , then for each Open image in new window the tax policy Open image in new window is a majority winner (a Condorcet winner).

- (1)
We have to prove that there is no objection to the tax policy Open image in new window given by Open image in new window . Let Open image in new window be a tax policy given by Open image in new window and let Open image in new window , Open image in new window , Open image in new window , where Open image in new window and Open image in new window . The feasibility conditions for Open image in new window determine the following inequalities: Open image in new window , Open image in new window , and Open image in new window . The feasible area for the coefficients Open image in new window and Open image in new window can be represented as it is shown in the Figure 2(d).

If Open image in new window , then Open image in new window and Open image in new window , which is not an objection to the tax function Open image in new window .

If Open image in new window , then Open image in new window . If Open image in new window , then Open image in new window , Open image in new window and Open image in new window (see Table 2, line 7). If Open image in new window , then Open image in new window , Open image in new window and Open image in new window Open image in new window (see Table 2, line 8). If Open image in new window , then Open image in new window , Open image in new window and Open image in new window (see Table 2, line 9).

- (2)
We have to prove that there is no objection to the tax policy given by Open image in new window . Let Open image in new window be a tax policy given by Open image in new window , and let Open image in new window , Open image in new window , Open image in new window , where Open image in new window and Open image in new window . The feasibility conditions for Open image in new window determine the following inequalities: Open image in new window , Open image in new window , and Open image in new window . The feasible area for the coefficients Open image in new window and Open image in new window can be represented as it is shown in Figure 2(e).

If Open image in new window , then Open image in new window and Open image in new window .

If Open image in new window , then Open image in new window and the proof is similar to the correspondent case of the 1st part.

If Open image in new window , then Open image in new window , and after splitting in subcases Open image in new window , Open image in new window , and Open image in new window the proofs are similar to the correspondent cases of the 1st part.

- (3)
We have to prove that there is no objection to the tax policy Open image in new window given by Open image in new window . If Open image in new window is given by Open image in new window and Open image in new window and Open image in new window , then the feasibility area for Open image in new window is given by the conditions: Open image in new window , Open image in new window , and Open image in new window . The feasible area for the coefficients Open image in new window and Open image in new window can be represented as it is shown in the Figure 2(f).

In this case Open image in new window and Open image in new window . After splitting in subcases Open image in new window and Open image in new window , then the proofs are similar to the correspondent parts of the 1st case.

Hence, the tax policy Open image in new window is a majority winner. This completes the proof.

If Open image in new window , the result from Proposition 3.4(1) was first established by De Donder and Hindriks [3] (see Proposition 1(b) in that paper). However, their result was obtained by imposing the more restrictive condition Open image in new window . In other words, for every distribution function Open image in new window such that Open image in new window , we have with certainty that the maximum progressivity tax is the voting outcome, as far as the median voter prefers this policy. This result does not depend on whether the tax is purely redistributive or not. For purely redistributive taxes, a specific proof is provided in Curt, Litan and Filip [16].

Therefore, it is enough to have a majority formed by individuals between the upper part of the low income group and the lower part of the high income group, in order to obtain support for the highest tax progressivity. Our next example proves that the necessary condition in Proposition 1(b) in De Donder and Hindriks [3] is overrestrictive. We construct a function Open image in new window with Open image in new window , in which case the conclusions of our Proposition 3.4 trivially apply.

Example 3.5.

The mean income Open image in new window , greater than the median income Open image in new window which is equal to Open image in new window . Routine calculations show that Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window .

For this income distribution, our Proposition 3.4 directly applies (while there is no need to check the necessary and sufficient condition of Proposition 3 in De Donder and Hindriks [3]).

The first and third parts of the Proposition 3.4 advocate the idea that the majority winning tax is the most progressive one, when the median voter is part of the middle income group. Regarding the intuition of the result, straightforward and not surprising is the case in which there is a majority within the middle income group (i.e., Open image in new window ); hence, the middle class can afford to minimize its tax, and the burden remains on the rich and the poor.

Not intuitively straightforward is the case in which the middle income group cannot form a majority coalition, but there exists a majority formed by individuals between the upper part of the low income group and the lower part of the high income group (i.e., Open image in new window ). The policy preferred by the middle income group remains the only majority winner because there is disagreement within the low income group, between the upper part and the rest of the group, and within the high income group, between the lower part and the rest, respectively. Although the second part of the proposition implies a left skewed distribution of the income (not existent in practice), the result is in the same line with the first and third parts: as far as the median voter prefers the tax Open image in new window (together with all the individuals in the subclass to which the median voter belongs), then the majority winner must consist of that policy.

The next proposition characterizes the class of income distributions for which the model does not provide a Condorcet winner.

Proposition 3.6.

Let Open image in new window be such that Open image in new window .

- (2)
If Open image in new window , Open image in new window and Open image in new window satisfies the previous conditions (in Proposition 3.6(1)), then there is no majority winner (Condorcet winner).

Proof of Proposition 3.6.

In order to prove the result, it is sufficient for each tax policy Open image in new window to find another tax policy Open image in new window that is an objection to the tax Open image in new window , that is, Open image in new window . Let Open image in new window be a given tax policy. In our attempt to determine an objection to the tax policy Open image in new window , we are looking for a pair Open image in new window such that Open image in new window , Open image in new window and Open image in new window . If we denote by Open image in new window , then Open image in new window and Open image in new window . Since the tax policy defined by Open image in new window must satisfy the feasibility conditions (FA), we have to choose Open image in new window such that Open image in new window , Open image in new window , Open image in new window . Due to the fact that Open image in new window , we have the following inequalities: Open image in new window and Open image in new window . So, only if Open image in new window and Open image in new window , there exists Open image in new window such that the previous conditions are satisfied. Since Open image in new window , by applying Lemma 3.1, we obtain: Open image in new window and in conclusion Open image in new window is an objection for Open image in new window .

It remains to analyze the cases Open image in new window and Open image in new window . Let Open image in new window be a tax policy defined by Open image in new window . In order to determine an objection Open image in new window , we look for values Open image in new window , Open image in new window which satisfy the feasibility conditions (FA) and Open image in new window . Since Open image in new window , then Open image in new window and in consequence Open image in new window . If we denote Open image in new window , then Open image in new window and Open image in new window . After short computations, we observe that we must determine Open image in new window such that Open image in new window and Open image in new window . In the case when Open image in new window , then we can choose Open image in new window which satisfies the previous restrictions. The policy tax Open image in new window is an objection to Open image in new window due to the fact that Open image in new window . In the case when Open image in new window , then the objection tax policy can be determined in a similar manner by choosing Open image in new window , Open image in new window such that Open image in new window . In the case when Open image in new window , the objection function Open image in new window can be also obtained by choosing Open image in new window and Open image in new window which satisfy the feasibility conditions and the equality Open image in new window . If we denote Open image in new window , then Open image in new window and Open image in new window . After short computations, we observe that we must determine Open image in new window such that Open image in new window and Open image in new window . In the case when Open image in new window , then we can choose Open image in new window which satisfies the previous restrictions and we get Open image in new window . Hence, Open image in new window is an objection to the tax policy Open image in new window . If Open image in new window , then Open image in new window and in consequence Open image in new window . For Open image in new window , we have to study the case Open image in new window . In this case the tax policy Open image in new window given by Open image in new window is an objection to Open image in new window . This completes the proof.

The next proposition specifies what are the conditions for a majority winner to exist and such that the conclusions of Proposition 3.4 remain valid, even if there cannot be formed a majority by individuals between the upper part of the low income group and the lower part of the high income group.

Proposition 3.7.

Let Open image in new window and Open image in new window . If for each Open image in new window we have Open image in new window , then the conclusions of the Proposition 3.4 are true.

Proof of Proposition 3.7.

Similar to the proof of the Proposition 3.4.

The intuition of this proposition is not straightforward and resides in the same type of arguments invoked by De Donder and Hindriks [3]. The maximum progressivity has majority support due to the disagreement, not only between the lower and upper parts of the high income group, but also among the individuals of the upper part of this income class. Analogous analysis can be performed for the low income class.

To be more specific, for any tax change (represented by the parameter Open image in new window ) involving less progressivity (parameter Open image in new window ) and higher flat tax parameter ( Open image in new window ), there is always some poor with income higher than Open image in new window who do not find the increase in Open image in new window large enough to compensate for the lower Open image in new window . As well, there is always some rich with income lower than Open image in new window who do not find the decrease in Open image in new window large enough to compensate for the increase in Open image in new window . The group in disagreement with the extremes for a given tax change is larger now than in Proposition 3.4, that is, Open image in new window . Therefore the condition on the distribution of income in Proposition 3.7 ensures that the size of the group Open image in new window is large enough to form a majority, for any possible Open image in new window .

## 4. Conclusions

We identified what are the most preferred taxes of the individuals (and the corresponding income groups they can be classified in, based on the preferred policies), for every case of taxation that has more than a purely redistributive purpose (meaning that the tax should collect some positive amount Open image in new window ). In particular, we have proved that if the model departs from the purely redistributive feature, then, at least within the high income class, the preferences differ between the groups Open image in new window and Open image in new window . This fact is essential to understand the result which states that the sufficient condition of De Donder and Hindriks [3] can be relaxed to a broader one.

Indeed, for not very large collected amounts Open image in new window , it is enough to have a majority formed by individuals between the upper part of the low income group and the lower part of the high income group, in order to insure support for the highest tax progressivity (as far as the median voter prefers this policy, that is, Open image in new window ). For Open image in new window , the results in De Donder and Hindriks [3] were obtained around the more restrictive condition Open image in new window . However, the case of purely redistributive taxations can be seen as a limiting case of those situations in which the taxes should collect some positive amounts. Therefore, for Open image in new window and for every distribution function Open image in new window such that Open image in new window , it is also certain that the maximum progressivity is the voting outcome. Example 3.5 shows that the set of the distribution functions with the above property is not empty.

For reasons of completeness, the paper provides an overall description of those income distribution functions for which a majority winning tax exists (or does not exist), when the quadratic taxation model is not purely redistributive. For the same reasons of completeness, the analysis considers both the right skewed income distributions, which are predominant in practice, but also the left skewed ones are analyzed (see the second and third parts of Proposition 3.3, and the second part of Proposition 3.4). We conclude with the idea that, should any political equilibrium different than the Condorcet winner be proposed, it is important to be first tested on those quadratic taxation models without majority winners (both for purely, as well as for nonpurely redistributive taxations). Our work offers a complete mathematical description of this testing set of models.

## Notes

### Acknowledgments

The authors would like to thank Francisco Marhuenda for helpful comments. Cristian Litan aknowledges preliminary discussions with Luis Corchón. Diana Andrada Filip and Paula Curt acknowledge financial support by CNSIS-UEFISCU, project number PNII-IDEI 2366/2008. Cristian Litan acknowledges financial support by CNCSIS-UEFISCSU, project number PN II-RU 415/2010. The usual disclaimer applies.

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