The nexus between perceptions of inequality and preferences for redistribution

This paper shows that perceptions of inequality are a key factor in the formation of preferences for redistribution and thereby in the determination of the equilibrium redistribution level. We build on the novel stylized facts provided by the survey experimental literature on perceptions of income inequality, highlighting that agents incorrectly estimate the shape of the income distribution because of limited information. Agents with income above the mean believe they are poorer than they actually are, and agents with income below the mean believe themselves to be richer. We revisit the standard framework on the political economy of redistribution and extend it in two ways. First, we introduce a more general two-sided inequality aversion. Second, we incorporate perceptions of income inequality, modeled by assuming that agents form expectations on the income level of the richest and the poorest in society. We show analytically that the equilibrium redistribution level is crucially determined by the interplay between the information treatment correcting the bias in perceptions of inequality and fairness considerations specified by the degree of inequality aversion. By doing this, we add (biased) perceptions of inequality to the list of potential factors explaining why, notwithstanding high inequality, an increase in the desire for redistribution has not been observed in many countries.


Introduction
What determines individuals' preferences for redistribution? To answer this question, many scholars have investigated the role income inequality plays in the subject matter: does a high level of income inequality stimulate a higher demand for redistribution? Thus far, a consensus on the relationship between these two dimensions has not been reached in the literature.
Lately, the focus of the quest for the key determinants of redistributive preferences has shifted away from the actual inequality level to its perception. Individuals' perceptions of the level of macroeconomic and distributional variables can shape attitudes and modify preferences and voting choices. In turn, voting choices play a decisive role in the design of public policies in democratic societies. Understanding (and eventually altering) perceptions can therefore be crucial to achieving different equilibrium policy outcomes.
The shift in focus is primarily motivated by the fact that preferences for redistribution have indeed proven to correlate more with perceptions of inequality than with the actual inequality level (Gimpelson and Treisman 2018;Choi 2019). This paper investigates theoretically how perceptions of inequality influence both redistributive preferences and the resulting voting outcome in terms of taxation and redistribution.
Assume that there is a population of agents that care about income inequality 1 and that do not have perfect information about their own income ranking and income distribution in general. This assumption implies that preferences for redistribution will be biased, and as a consequence, agents will vote for a level of income taxation and equilibrium redistribution that is inconsistent with real preferences 2 .
An increasing number of surveys and experimental papers (Cowell and Cruces 2004;Cruces et al. 2013;Gimpelson and Treisman 2018;Hoy and Mager 2019;Karadja et al. 2017;Kuhn 2011) have attempted to estimate individuals' bias in the perception of income inequality. Kuhn (2011) and Cruces et al. (2013) design country-specific surveys for Switzerland and Argentina, respectively, whereas Gimpelson and Treisman (2018) and Hoy and Mager (2019) present evidence from 9 cross-national surveys and a large-scale 10country survey, respectively. Considering that a multi-country study allows us to externally validate results with higher accuracy and precision, the evidence provided by the latter studies will feature prominently in our analysis.
Although studies from the literature differ somewhat in their approaches and results, a consensus seems to emerge: agents incorrectly estimate the shape of the income distribution due to limited information. Hence, due to a perception bias in the level of income inequality, individuals' preferences for redistribution cannot be deemed to reflect real preferences. More interestingly, there is an emerging consensus within the literature on the sign of this perception bias. This sign depends on whether an agent underestimates or rather overestimates her position in the income ranking (relative income inequality). For instance, Cruces et al. (2013) show that the sign of the estimation bias depends on whether an agent belongs 1 Several studies in the economic literature have tested and accepted the hypothesis that fairness considerations are among the determinants of preferences for redistribution. See, e.g., the social survey data by Fong (2001), the empirical results in Isaksson and Lindskog (2009), and the results from the large international survey conducted by Corneo and Grüner (2002). 2 The issue of perceptions of income inequality and their effect on the desire for redistribution was also analyzed in Pellicer et al. (2019). However, the researchers assume that inequality is to some degree inevitable, and the desire for redistribution policy aimed at curbing inequality drops as a consequence. In the framework of this paper, redistribution policy, instead, effectively reduces inequality, making it a valuable policy choice. to a high-income or a low-income reference group. If an agent belongs to a high-income reference group (e.g., above the mean income), she tends to overestimate the income of the rest of the population. As a consequence, she underestimates her own income ranking, believing herself to be poorer than she actually is. On the other hand, an agent belonging to a lowincome reference group tends to believe herself to be relatively richer than she actually is. Hoy and Mager (2019) also confirm the evidence whereby all agents believe themselves to belong to the middle class, the so-called middle-class bias.
Building on this experimental consensus, this paper's objective is to analyze the effect of individuals' perceptions of inequality on the equilibrium redistribution level in an economy. To this end, we revisit the standard Meltzer and Richard (1981) (MR) political economy framework by building on the work of Galasso (2003). To the best of our knowledge, Galasso (2003) is the first to introduce altruistic preferences in an MR model by assuming that fair agents are one-sided inequality-averse. We generalize the contribution of Galasso (2003) in two ways. First, we introduce two-sided inequality aversion to make the model sufficiently flexible to incorporate the previously discussed findings of the experimental literature. Second, we incorporate perceptions of income inequality into the framework.
Such a novel theoretical framework allows us to address the following questions: does the median voter demand more or less redistribution whenever she receives information (regarded as a treatment, as in Duflo and Saez (2003) and Chetty and Saez (2013)) on the actual shape of the income distribution? To what extent do the results depend on the degree of inequality aversion? Is it the inequality aversion with respect to the top incomes in society or with respect to the poorest in society that plays a larger role?
The main contribution of this paper is to show analytically that perceptions of inequality indeed play a decisive role in the determination of the equilibrium redistribution level. In addition, we show that the degree of inequality aversion of the fair agents crucially determines the magnitude of the effect of (biased) perceptions on the equilibrium redistribution level. In response to the information treatment (a mean-preserving spread in the income distribution) that corrects the perception bias, fair agents with income below the mean support a higher tax rate and demand greater equilibrium redistribution. In turn, the higher taxation and redistribution level will presumably increase the post-tax disposable income of poor agents and reduce the absolute level of inequality in society.
Apparently, this result does not deviate from the conventional framework of the political economy of redistribution (Benabou and Ok 2001;Meltzer and Richard 1981;Piketty 1995) 3 , in which the rational median voter with income below the mean supports a positive level of taxation and redistribution, while agents with income above the mean prefer no redistribution. However, the mechanisms driving the result here differ substantially. In the conventional MR model, fairness considerations are totally absent, and agents are led only by selfish motives. In the framework of this paper, instead, it is both fairness considerations and selfish motives that jointly drive the result.
In a nutshell, perceptions of inequality play a decisive role in the determination of the equilibrium redistribution level through the interplay with fairness considerations specified in the two-sided equality function. Without fairness considerations, the role of the perceptions of inequality would be undermined and not properly understood. We believe that these results can help understand various mechanisms at work behind real-world outcomes of voting on redistribution issues, now that better data and estimates of income inequality are reaching the larger public (hence, gradually reducing perception biases), and the attention given to inequality by policymakers is increasing.
The paper proceeds as follows. Section 2 introduces the modeling approach and constitutes the core part of the study, while Section 3 discusses the results. Section 4 summarizes and concludes the paper.

The economic environment
We design the economic environment by building on the framework developed by Galasso (2003). Such a framework is itself an adaptation of the standard MR model with the introduction of fairness (i.e., a fraction of the population values fairness). The framework of Galasso (2003) is a natural starting point for generalizing fairness considerations by assuming inequality aversion towards both ends of the distribution.
Consider a unitary mass of individuals who differ from each other only with respect to their working ability given by e. These abilities are distributed on the support [e,ē] ⊂ R + according to the cumulative distribution function F (.). As in Galasso (2003), we assume the distribution of abilities to be skewed, and therefore, the median ability e M is below the mean abilityẽ. An individual's income y is a function of her ability and her amount of labor supplied n(e) and can thus be written as the following simplified production function: y(e) = en(e).
Suppose that a fraction λ ∈ [0, 1] of all agents in the population is of the selfish type and value consumption c and leisure l according to a standard quasi-linear utility function U(c, l) = c + u(l) that is twice-differentiable and concave. A fraction 1 − λ of the population is, on the other hand, composed of fair agents that value the degree of equality in the economy as well. The equality function, which measures the aversion to inequality, is given by E. As in Galasso (2003), we assume that the fairness considerations summarized by E do not enter the economic optimization problem. This choice leads to a higher degree of analytical tractability, and it also implies that an individual's choice as to the labor supply is not influenced by how much the single agent dislikes inequality in society. In summary, the utility function of fair agents can therefore be written as U(c, l) = c + u(l) + E.
All agents choose consumption and leisure (facing the usual trade-off between leisure l and supply of labor n limited by the total amount of disposable time) to solve the economic maximization problem. Individuals pay a payroll tax on their income from labor at tax rate τ equal for all (no means-testing), and receive an equal transfer T . Their budget constraint is thus c = (1 − τ )en(e) + T , with the balanced budget of the government given by T = τ en(e)dF (e). The maximization problem yields the usual result, i.e., (1 − τ )e = u l (l * ). In turn, one can make substitutions to determine the optimal amount of labor supply and of income y * = en * . Galasso (2003) shows that the introduction of a fair agent type in a standard MR framework produces, ceteris paribus, an increase in the equilibrium tax rate and redistribution level, which is induced by fairness considerations. The type of fairness Galasso (2003) introduces is one-sided inequality aversion, i.e., the fair agent compares the absolute distance between her income from that of the poorest in society 4 .

Voting game with a two-sided equity function
From now on, we intend to deviate from the framework in Galasso (2003). We believe that a theoretical framework seeking to incorporate the middle-class bias observed in the survey experimental literature (high-income agents underestimating and low-income agents overestimating their own income rank) benefits by the introduction of a double-sided inequality averse fair agent. Hence, we formulate a double-sided equality function E of the following form: where β =ỹ * −y * y * −y * , γ =ȳ * −ỹ * y * −y * , and β + γ = 1. The coefficients β and γ incorporate fair individuals' concern related to a specific side of the distribution (either the top or the bottom), depending on the society's mean incomẽ y * , here taken as an indirect measure of the society's wellbeing.
In other words, if the society's mean incomeỹ * is closer to the top of the distribution, fair agents are relatively more concerned with the income of the poor. Such a mechanism mathematically results in a low γ and a high β. On the other hand, whenever the society's mean incomeỹ * is closer to the bottom of the distribution, fair agents are more concerned with the income of the rich, reflected by a high γ and a low β.
The features of the median-voter equilibrium that results from the voting game with the introduction of the above equality function E are explained as follows. Recall the general economic setting of Section 2.1. All agents make their voting decisions maximizing their indirect utility, and preferences are single-peaked as in MR and Galasso (2003).
Precisely, selfish agents maximize the utility function V S (τ ) = (1 − τ )y * + T + u(l * ), whereas fair agents maximize the same utility function, however augmented by fairness considerations: The following first-order condition is obtained by fair agents: with ∂ỹ * ∂τ < 0 5 . The first-order condition of the selfish agent can easily be obtained by setting E = 0. Solving for τ , it can be shown that a selfish agent with ability e chooses the tax rate whereas a fair agent with ability e chooses the tax rate Given that β + γ = 1, any combinations of value taken by the pair [β, γ ] lowers the numerator of Eq. 4 with respect to Eq. 3 and consequently leads to τ F > τ S . This result clearly shows that a fair agent supports a higher level of redistribution than that supported by the selfish agent of the same ability, due to the effect of fairness considerations. Note that for any agent, the higher the income y * is, the lower the willingness to support a positive redistribution policy (this is due to purely selfish motives). However, fairness considerations enter the first-order condition of fair agents. For a fair agent to support a positive tax rate, her income y * should be lower than mean incomeỹ * augmented by the terms introducing fairness considerations. Even with income y * slightly above the meanỹ * , the fair agent will still vote for a positive tax rate due to the effect of inequality aversion.
For clarity, we combine the two types of agents within a single expression (a mathematical correspondence) that gives us, for any given positive level of tax rate τ , the corresponding mass of voters willing to support it. The correspondence between the income of a selfish agent and that of a fair agent choosing to vote for the same non-negative tax rate is given by: Hence, we are able to plot the masses of selfish and fair agents on the same diagram, as shown in Fig. 1, ranked by the selfish voters' income, as done in Galasso (2003): Figure 1 shows that a higher tax rate τ is supported by low-income agents, while a lower τ is voted for by high-income agents, consistent with the standard MR model and with Galasso (2003). The correspondence plotted in Fig. 1 is also useful for visualizing the (self- Fig. 1 Selfish Median (SM) voter equilibrium. Note: this graph plots the masses of selfish and fair agents on the same diagram, ranked by the selfish voters' income. It shows that a higher tax rate τ is supported by lowincome agents, while a lower τ is voted for by high-income agents, consistent with the standard MR model and with Galasso (2003) ish) median voter equilibrium (y SM ) of this voting game, which can be determined through the following expression: in which λ is the fraction of selfish voters. A couple of remarks as regards a comparison between our framework and that of Galasso (2003) should be inserted here. It is easy to appreciate that the median voter's income (y SM ) in the context of double-sided inequality aversion is lower than in the context of one-sided inequality aversion. In addition, in Galasso (2003) the minimum income fair agents can earn corresponds to the minimum income of selfish agents voting for the same tax rate. This result can be obtained by simply substituting y F for y in the correspondence equation of Galasso (2003) 6 . However, once we perform the same operation with Eq. 5, which establishes a relationship between the incomes of selfish and fair agents choosing the same tax rate in the context of double-sided inequality aversion, we obtain a lower minimum income for fair agents than in the previous scenario. Specifically, this income equals y − γȳ. The maximum income of fair agents both in the context of single-and double-sided inequality aversion will however be the same. Therefore, the median of the fair agents' voting distribution will be lower in a double-sided inequality aversion context than in a single-sided one. In summary, by applying equity considerations on both sides of the income distribution, the median voter will be willing to support a higher level of equilibrium redistribution.
Another assumption of the model that is worth discussing is the fact that the distribution of fair agents across the income spectrum is the same as that of the selfish agents. This distribution is represented by the cumulative distribution function F (y). If we relaxed this assumption, and therefore assumed that the cumulative distribution of the fair agents, F F , is different from that of selfish agents, F S , then the income of the median voter would be defined as follows: To grasp the potential differences between this latter scenario, in which selfish and fair agents are differently distributed across the income spectrum, and the previous one, in which their distributions are the same, let us consider the following example. Suppose to know the mean and the variance of the two distributions F F and F S . Let us denote the two means E F and E S and the two variances V F and V S , where F and S refer to fair and selfish agents, as usual. Assume the two variances to be the same, V S = V F , but the mean of the fair agents be higher than the mean of the selfish agents, E S < E F . In this scenario it is difficult to conclude whether the resulting income of the median voter would be higher, or lower than that in the previous scenario, in which both typologies of agents shared the same income distribution. In fact, a higher mean implies both a potential higher median, but also a lower minimum income.
Recall now that the lowest income of the fair agents, once expressed in terms of selfish agent's income, equals y − γȳ, whilst the highest income remains the same. An increase in the mean therefore increases γ , and hence reduces the lowest income, as well as the median. When we assume the two distributions to have same means E S = E F , but different variances, the resulting effect on the median voter is also ambiguous, since both extremes of the distribution y andȳ move away from each other.
To sum up, when we allow the two groups of agents to be differently distributed across the income ladder, it is not straightforward to draw any clear conclusion about the impact this difference may have on the income of the median voter and, hence, on its preferences for redistribution.

Perceptions of inequality
In this subsection, we introduce perceptions of inequality in our theoretical framework. Let us first introduce the type of perception we will be dealing with. To the best of our knowledge, only survey-based measures of relative inequality have been proposed in the literature (Choi 2019;Engelhardt and Wagener 2014;Powdthavee 2009;Schalembier 2015). In contrast, in this paper, we introduce a simple analytical formulation of perceptions of inequality, aiming at reproducing the consensus from the survey experimental literature within the theoretical framework constructed in Sections 2.1 and 2.2.
We assume for simplicity that the mean incomeỹ * in the society is public knowledge. Although this assumption has been discussed critically in the survey literature on biased perceptions of inequality (Bublitz 2016), we do not regard it as controversial, especially if we consider that mean income in the society is often communicated to the general public by national statistical agencies.
The agents also know with certainty their own income y * . We also assume that to solve their economic maximization problem, the agents only need to know the income level of the richest and the poorest in society, i.e., the extreme ends of the distribution. We assume that all individuals form their perceptions concerning the extreme ends of the distribution in the same manner. This assumption will be discussed and partially relaxed in the Section 2.5.
Agents form a perception of the highest income in society, which we denote byȳ * P , and of the lowest income, denoted by y * P . Furthermore, we defineȳ * P and y * P as follows: with ζ, η ∈ R and ζ, η = 0 7 . ζ, η are societal perception biases, in the sense that being poor or rich does not play a role for their magnitude. Such a simple formalization of the perceived incomes of the richest and the poorest in society (which will be relaxed in Section 2.5) will allow us to perform a tractable analysis of all possible cases of perceptions playing a role in the voting outcome.
Recall that, although all agents form perceptions of inequality, perceptions only matter in the determination of tax rate for fair agents as they are the ones for which perceptions enter in the utility function. Hence, after the introduction of perceptions the equality function of the fair agents specified in Eq. 1 becomes where β P =ỹ * −y * P y * P −y * P and γ P =ȳ * P −ỹ * y * P −y * P , with β P + γ P = 1. It is of utmost importance to notice that, even though individuals' perceptions are modeled in absolute terms (i.e. perception of the level of income of the richest and of the poorest in society), these perceptions are, then, partially converted by the model into relative perceptions. Such conversion is made possible thanks to the coefficients β P and γ P , which are measures of the perceived relative distance of the poorest and richest income from the population mean, respectively. When β P > γ P individuals are more concerned to the bottom of the distribution. When β P < γ P individuals are more concerned to the top of the distribution. Therefore, out of the interaction between β P (or γ P ) and (y * − y * P ) (or (ȳ * P − y * )), our framework embeds both relative and absolute inequality concerns.
Let us illustrate this aspect by means of a stylized numerical example. Suppose individual i perceives her income y i be three times higher than the lowest income (y i = 3 × y P ) and two thirds the highest income (y i = 2 3 ×ȳ P ). Suppose, now, that β P < γ P , and specifically that β P = 1 3 and γ P = 2 3 . 8 It can be noticed that, under this scenario, an absolute concern to the bottom of the distribution (y i − y P = 2 3 y i >ȳ P − y i = 1 3 y i ) may be contrasted by a relative concern about the top (β P < γ P ), thus resulting in an equal impact on E P of the two types inequality aversion. This example allows us appreciating the role that both absolute and relative inequality concerns play in the formation of perceptions of inequality.
Performing the same optimization procedure as in Section 2.2 results in the following optimal lump sum tax rate for the fair agent with ability e in the context of perceptions of inequality: The effect of correcting the societal bias in perceptions of inequality can therefore be pinned down through a before-and-after comparison of the above Eq. 11 with the tax rate resulting from the elimination of the perception bias for terms y * P andȳ * P (i.e., when (η, ζ ) tends to [0, 0]). The entire set of possible scenarios that can arise is summarized in Fig. 2.
According to the values of perception biases η and ζ , we identify 12 scenarios that describe all possible realizations of the societal bias determining perceptions of inequality. These 12 scenarios depend on the relationship between η and ζ in terms of both their magnitudes and signs. Any one of these scenarios affects the equality function E (P ) in a different manner. In the first tree of the diagram, the magnitude of the perception biases is equal with respect to both ends of the distribution (|η| = |ζ |). In the remaining two trees of the diagram, instead, the bias is either stronger when it comes to estimating the income of the poorest in society (the cases in which |η| > |ζ |) or, on the contrary, is stronger when it comes to estimating the income of the richest in society (whenever |η| < |ζ |).
In conclusion, this subsection prepared the ground for the analysis of the effect that the elimination of perception biases (which happens when the pair (η, ζ ) tends to [0, 0]) can have on the equilibrium tax rate chosen by the median voter.

Effect of perceptions on redistribution
This subsection pins down the effect of perceptions of inequality on the equilibrium redistribution level. A real-world example of an information "treatment" or "shock" that corrects the societal perception bias can, e.g., be a pre-election television debate on inequality and redistribution, in which precise information regarding the income of the poorest and the Fig. 2 Tree diagram with magnitude and sign of perception biases η, ζ . Note: the above tree-diagram identifies 12 scenarios describing all possible realizations of the societal bias determining perceptions of inequality. These 12 scenarios depend on the relationship between η and ζ in terms of both their magnitudes and signs. Any one of these scenarios affects the equality function E (P ) in a different manner. In the first tree of the diagram, the magnitude of the perception biases is equal with respect to both ends of the distribution (|η| = |ζ |). In the remaining two trees of the diagram, instead, the bias is either stronger when it comes to estimating the income of the poorest in society (the cases in which |η| > |ζ |) or, on the contrary, is stronger when it comes to estimating the income of the richest in society (whenever |η| < |ζ |) richest in society (i.e., the absolute level of income inequality in society at that time) is disclosed to the general public.
More importantly, we intend to build on the results and consensus emerging from the experimental literature on perceptions of inequality. To this end, recall the insights from the multi-country surveys of perceptions of inequality mentioned in the introduction. Gimpelson and Treisman (2018) present an overview of the results from nine large cross-national surveys that point in the same direction: agents know neither the shape of the distribution of income in their country nor their own position along the income rank. This phenomenon can be justified either by missing information or by bounded rationality arguments. Based on this striking stylized fact, Gimpelson and Treisman (2018) state that the common assumption that rational economic agents accurately observe the level of income inequality in society should be disregarded in future studies. In other words, Gimpelson and Treisman (2018) advocate that a more in-depth analysis of perceptions of inequality (including how they are formed) should be included in forthcoming political economy models of redistribution, which is justified by the fact that survey evidence shows that preferences for redistribution happen to be more strongly correlated with perceptions of inequality rather than with the actual inequality level. This corresponds to the framework of this paper, which uses as a starting point the fact that agents do not know the absolute distance in income from the richest to the poorest in society. More precisely, the perceived income of the richest in society is different from their real income,ȳ P =ȳ, and the same occurs for the poorest in society, y P = y, due to a systematic societal perception bias that we denote by parameters ζ, η = 0.
The survey experiments conducted by Hoy and Mager (2019) across 10 countries with over 30000 participants and by Cruces et al. (2013) confirm the result of Gimpelson and Treisman (2018) regarding the fact that agents incorrectly estimate the real inequality level. In addition, Cruces et al. (2013) and Hoy and Mager (2019) provide evidence regarding an interesting and novel stylized fact. Namely, survey evidence shows that a majority of agents tend to believe they are positioned around the middle of the income distribution, regardless of whether they belong to a high or poor reference group (a middle-class bias).
More specifically, agents belonging to a high-income reference group (e.g., above the mean) tend to underestimate their own income rank; hence, they believe that they are relatively poorer than they actually are. In absolute inequality terms, this means that the distance from the income of a rich agent to the income of the richest in society is smaller in reality than in perception. On the other hand, agents belonging to a low-income reference group tend to overestimate their own income rank; hence, they believe they are relatively richer than they actually are. In absolute inequality terms, this means that the distance from the income of a poor agent to the income of the poorest agent in society is smaller in reality than in perception.
We incorporate this middle-class bias in our theoretical framework augmented with perceptions in which a fraction of fair agents is double-sided inequality averse. What will be the effect of the information treatment on the equilibrium redistribution level supported by the median voter?
We start by identifying the case (among those specified in the tree diagram of Section 2.3) for which the income distribution is subject to a mean-preserving spread after the information treatment (when the pair (η, ζ ) tends to [0,0]). In other words, the middle-class bias is identified by the case of the tree diagram for which the richest in society are richer in reality than an agent believes, and the poorest are poorer in reality than an agent believes (with the magnitudes of the biases being equal to ensure the mean-preserving spread feature). This result corresponds to the case given by E 4 in the tree diagram, in which we have |η| = |ζ | with η > 0, ζ < 0.
To find out whether the information treatment will induce the fair agent to support a higher (lower) equilibrium redistribution level, we need to pin down the effect on the fair agent's equality function. The effect on the equality function specified in Eq. 10 is given in turn by: the effects on (i) the coefficients of inequality aversion β and γ , and on (ii) the absolute distances from income of the median voter to the ends of the distribution. Summing up: [E 4 : |η| = |ζ | with η > 0, ζ < 0] Correcting the societal bias in perceptions (hence, going from the equality function of Eq. 10 including perceptions to the function in Eq. 1) has a net effect resulting in a higher tax rate τ F and equilibrium redistribution chosen by the fair agent, which we explain as follows.
The features of the mean-preserving spread leave both coefficients of aversion unchanged after the information treatment: β = β − β P = 0 and γ = γ − γ P = 0. This result implies that the degree of inequality aversion of the fair agent with respect to both ends of the distribution is unchanged.
As to the absolute distance between the fair agent's income and the extreme ends of the distribution, the distance from the highest income (which is nowȳ >ȳ) has increased by exactly as much as the distance from the lowest income in society (which is now y < y). Hence, the two terms that multiply β and γ in Eq. 10 both increase compared to values before the information treatment. All in all, we can conclude that as long as η > 0, ζ < 0 with |η| = |ζ |, the change in the equality function of the fair agents induced by the information treatment is negative in value, due to the increase in the level of absolute inequality caused by the mean-preserving spread: E = E − E P < 0. The effect on the desired tax rate by fair agents τ F is then positive, due to the negative denominator given by ∂ỹ * ∂τ < 0. Figure 3 plots this effect. Fig. 3 Selfish Median (SM) voter equilibrium after correction in perception bias. Note: The graph shows the reduction in the median voter equilibrium (y SM ) after the information treatment correcting the bias in perceptions, and the resulting higher tax rate τ Figure 3 shows the reduction in the median voter equilibrium (y SM ) after the information treatment correcting the bias in perceptions, and the resulting higher tax rate τ . The information treatment results in an increase in the level of absolute inequality in society, and the behavioral response of fair agents with two-sided inequality aversion is that of supporting a higher tax rate and redistribution level. This in turn implies that the median voter of the population comprising both selfish and fair voters will support a higher tax rate and redistribution level after the information treatment (this follows straightforwardly from Eq. 6).

Heterogeneity in perception formation
In the baseline framework of the paper, perceptions are modeled in a way that is homogeneous across all members of the society. We now extend the baseline framework to account for heterogeneity in perception formation. In other words, we allow the bias in perceptions to vary with income.
In the following, we shall assume that perception formation is a function of individual's income. There is indeed no reason to believe that a rich and a poor individual have the same perceptions on the income of the richest (or the poorest) in society. A poor individual may have limited information on the earnings, or more generally on the lifestyle of a rich individual, and vice versa.
Perception formation for each agent i is now assumed to take the following behavioral rules:ȳ P i (y; ω, σ, θ) =ω +σ yθ i (12) The three parameters (ω, σ, θ) are behavioral parameters such that ω, θ ∈ R + and σ ∈ R, and should be interpreted as follows. The first term ω represents the societal bias in perception that each agent is subject to. Hence, whenever σ equals zero we return to the baseline framework with perceptions constructed only on common societal bias.
Parameter σ measures the extent to which individuals diverge in their perception formation. When σ is different from zero, the income of the richest (or poorest) is perceived via a combination of societal bias (as in the baseline scenario) and an individual bias that is function of the agent's income. The higher the σ in absolute value, the stronger the divergence in perception between the two income groups.
Finally, parameter θ describes perception heterogeneity within the bottom and the top of the income ladder. When θ is below 1, the perception functions in Eqs. 12 and 13 become concave, hence the variability in perception formation within the bottom half of the distribution turns higher than the variability within the top half. On the contrary, when θ is above 1, the perception functions in Eqs. 12 and 13 become convex and the top half shows more heterogeneity in perception formation than the bottom half. Figure 4 plots examples of the perception functions in Eqs. 12 and 13 when θ = 1.
To make our model mathematically tractable, we have assumed the perception functions in Eqs. 12 and 13 to be differentiable. However, the perception functions in Eqs. 12 and 13 may also be assumed to be piece-wise defined. Such formalization would allow us to describe a society in which full homogeneity in perception formation exists within different segments of the society (or of the income ladder). Assume, for instance, a function that Fig. 4 Heterogeneity in perception formation. Note: On the left-hand (right-hand) side, perception of the income of the richest (poorest) in society is plotted. Parameter σ measures the extent to which individuals diverge in their perception formation. When σ = 0, perceptions are formed regardless of income, as in the baseline model. When σ = 0, the income of the richest (or poorest) is perceived via a combination of societal bias (as in the baseline scenario) and an individual bias that is function of the agent's income. The higher the σ in absolute value, the stronger the divergence in perception between the two income groups. As an example, focus on the diagram on the left-hand side: a negativeσ will imply that the income of the richest in society is correctly perceived by the bottom of the distribution, whilst the richer part of the distribution incurs in a perception bias. On the other hand, a positiveσ will imply that the income of the richest in society is wrongly perceived by the bottom of the distribution (by the extent of the societal bias), whilst the richer part of the distribution incur in a smaller bias takes different values in different fraction of the income ladder, and assume these values to be constant. This formulation would imply different fractions of the population, such as the bottom 10%, to form identical perceptions of the highest or lowest income in society. Or rather, this would entail a process of class-based perception formation in society.
Summing up, we believe that the tractability of the above framework can provide the experimental literature with a formalization of perceptions that could be empirically implemented in surveys. The decomposition of the bias in perceptions of inequality into a societal and an individual component, plus the estimation of the relative importance of these components would contribute to a deeper understanding of mis-perceptions of inequality.

Perception probabilities
Another way of embedding heterogeneity of perception in the model is to write each individual's perception of the gap between her income and the income of the richest in society (Ḡ p ) as well as the gap between her income and the income of the poorest in society (G p ) in the following way: where and¯ are perception probabilities. The perception probabilities can be functions of individuals' income: the higher (lower) an individual's income is, the higher (lower) her ability to perceive the highest income in society; similarly, the lower (higher) an individuals' income is, the higher (lower) her ability to perceive the lowest income in society.
To capture this mechanism, we can define and¯ as two statistically dependent normal distributions of the form ∼ N (0, G + 1) and¯ ∼ N (0,Ḡ + 1). The phenomenon underlying such formalization is that although on average individuals correctly estimate the extremes of the distribution, the farther away their income is from the extreme income they are asked to estimate, the higher the probability they make a mistake in their estimation.
Although the above formalization has the advantage of capturing simply the dynamics at stake, it has the disadvantage of making the model analytically intractable. Indeed, given that the two distributions and¯ are mutually dependent, it is difficult to explicitly derive the distribution of the resulting equity function E in a simple way. However, using numeric computations and simulation techniques, it would be possible to determine the effect of the introduction of these perception probabilities on the equity function E and therefore on the equilibrium redistribution level in the economy.

Discussion
Recall the main results of Section 2.4. The fair agents with income below the mean react to a mean-preserving increase in inequality (caused by the information treatment) by supporting a higher tax rate and demanding higher equilibrium redistribution, which will presumably increase her post-tax disposable income and reduce the absolute level of inequality in society. This result appears to be consistent with the conventional MR framework of political economy of redistribution. However, in the conventional MR model, fairness considerations were absent, and the median voter was led only by selfish motives. In the framework of this paper, both fairness considerations and selfish motives instead affect the result.
More importantly, the results of Section 2 show that the perceptions of inequality play a decisive role in the determination of the equilibrium redistribution level through the interplay with fairness considerations specified in the equality function. Without the specification of fairness consideration as done in the current framework, the role of perceptions of inequality in determining the level of redistribution would not be properly understood.
There is one additional interpretation of the result of the model in this paper. In the literature, it has been shown that increasing inequality is not necessarily correlated with higher redistribution (Lind 2005). This puzzle has been referred to as the Robin Hood paradox (Lindert (2004), page 15): "History reveals a Robin Hood paradox, in which redistribution from rich to poor is least present when and where it seems most needed". Namely, redistribution is higher in countries in which we observe lower inequality, and vice versa.
In our view, the result of this paper qualifies perceptions of inequality as one of the possible causes that can contribute to explaining the Robin Hood paradox. As soon as the perception bias is resolved, our model indicates that fair agents (and hence the median voter) demand a higher equilibrium redistribution level. In other words, it is precisely the perception bias as to the real level of inequality that could in part explain the paradox of low demand for redistribution in countries with high level of inequality. If agents (with a certain degree of inequality aversion) were provided with full and correct information about the level of inequality in society, they would indeed vote for higher equilibrium redistribution 9 .

Concluding remarks
In this paper, we embedded the findings from the survey experimental literature on perceptions of inequality into a political economy model of redistribution. These findings indicate that biased perceptions of income inequality influence behavioral responses and outcomes of voting on issues of taxation and redistribution. Indirectly, this implies that altering or rather correcting the bias in individuals' perceptions of inequality can have an important effect on the desired level of taxation and redistribution.
More specifically, we introduced perceptions of inequality in a tractable political economy model of redistribution, which closely resembles that proposed by Meltzer and Richard (1981). However, we accounted for fairness considerations similarly to the work of Galasso (2003). Our innovation with respect to the framework of Galasso (2003) is that we extend individuals' fairness to account for a double-sided inequality aversion. Indeed, we argue that double-sided inequality aversion allows us to better incorporate the stylized facts of the experimental literature on perceptions of inequality.
To obtain an estimate of the hypothetical effect of correcting individuals' bias in their perceptions of inequality, we assumed information on the income distribution to be disclosed to agents as a treatment through a public announcement. Consistent with observations highlighted in the experimental literature, the real level of absolute inequality ends up being higher than the perceived level. To recreate this stylized fact in our framework, we hypothesized that information treatment entailed a mean-preserving spread in the distribution of income.
Fair agents with double-sided inequality aversion react to this information treatment by demanding a higher tax rate and redistribution level. In other words, the role of perceptions of inequality is indirectly pinned down by the elimination of biased perceptions and crucially hinges on the degree of double-sided inequality aversion of a fraction of agents in the population. Without fairness considerations (as in the conventional modelsá la Meltzer and Richard (1981)), the role of perceptions of inequality in driving the results would not be decisive.
In our view, the result of this work can also contribute to the broader debate in the economic literature on inequality and redistributive preferences. Lind (2005) points out that the level of redistribution observed in many Western countries does not appear to be inversely related to the degree of income inequality, as one could expect. Lindert (2004) labels this empirical regularity the Robin Hood paradox. Namely, redistribution is higher in countries in which we observe lower inequality, and vice versa. More equal countries continue to redistribute wealth, although they are already relatively more equal (e.g., Nordic countries are often cited in this context), while more unequal countries choose not to redistribute wealth, hence becoming even more unequal.
This paper adds perceptions of inequality to factors that contribute to explaining one side of the Robin Hood paradox. As soon as the fair agents receive the information treatment disclosing that the absolute level of income inequality has increased, our model indicates that voting outcomes imply a higher equilibrium redistribution level. In other words, it is precisely the bias in perception of the level of inequality that could partially explain why we observe low demand for redistribution in countries with relatively higher levels of inequality. If agents with fairness considerations were provided with full and correct information about the level of inequality in society, we would indeed observe higher equilibrium redistribution in more unequal countries.
Funding Open access funding provided by NTNU Norwegian University of Science and Technology (incl St. Olavs Hospital -Trondheim University Hospital).
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