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\begin{document}

\title{French Favored Redistribution Derived From Surveys:\\
a Political Assessment of Optimal Tax Theory}

\author{Adrien Fabre\thanks{École Normale Supérieure, Paris School of Economics \textemdash{}
Université Paris 1. adrien.fabre@psemail.eu}}
\maketitle
\begin{abstract}
An original method of estimating desired income tax rates is presented,
which is in turn used to assess the political acceptability of the
optimal tax theory. Two surveys have been conducted in 2016 to quantify
French preferences for income redistribution. In the first survey,
respondents picked their preferred values for parameters which were
used to determine the shapes of redistributions. These parameters
included the proportion of winners and losers from a tax reform, and
the minimum guaranteed income. Using different algorithms, several
redistributions were derived from the interpolation of the median
choices of each parameter (50\% of winners, 10\% of losers and a monthly
demogrant of 800¤). They resulted in transfers from high to low incomes
of one tenth of the national disposable income. In the second survey,
a majority of respondents agreed on implementing these redistributions.
These results are in line with previous literature and robust to alternative
specifications. Interestingly, the average desired redistribution
corresponds closely to the shape of the optimal taxation derived from
an utilitarian criterion. This allowed to show that this redistribution
fails to obtain a significant majority support (contrarily to others),
despite its good reception in a setting inhibiting self-interest.
Finally, this study provides evidence that French citizens support
a more direct democratic procedure to define the income tax rates,
and proposes a way to do so.

\bigskip{}
\bigskip{}
\end{abstract}
\begin{flushleft}
\textbf{\small{}Keywords:}{\small{} Preferences for redistribution,
Desired tax, Income tax rates, Income distribution, France.}
\par\end{flushleft}{\small \par}

\begin{flushleft}
\textbf{\small{}JEL classification:}{\small{} D31, D63, H21, P16.}
\par\end{flushleft}{\small \par}

\bigskip{}
\bigskip{}
\bigskip{}
\begin{center}
\textbf{Acknowledgments}
\par\end{center}

I am deeply indebted towards Étienne Lehmann, Laurence Jacquet and
Bruno Van der Linden for providing me their code.

I would like to thank Michel Forsé and Thomas Piketty for their insightful
comments and for the supervision of my master's thesis, from which
this work originates.

I am grateful to Stefanie Stantcheva, Michael I. Norton and the Harvard
Business School for allowing me to use their \emph{Qualtrics }account
to code the questionnaires; and to Matthew Weinzierl for letting me
reproduce some of his figures.

I am also very thankful to all those who have given me precious advice,
notably Malo Huard.

\pagebreak{}

\section*{Introduction}

``What distribution of incomes would best conciliates fairness and
efficiency?'' This question is addressed by the theories of optimal
taxation and redistributive justice, and resolved by assuming a normative
criterion (see e.g. \citet{PikettyEtAl2013}, \citet{FleurbaeyEtAl2016}).
Using the correspondence between ethical criteria and tax schedules,
the so-called inverse-optimum literature reverse-engineers tax systems
to reveal the implicit criteria that would make current income tax
rates optimal (see e.g. \citet{BourguignonEtAl2012,BargainEtAl2013,Hendren2014,ChangEtAl2017}).
At variance with the notion of optimality, this work is built upon
extensive survey evidence which shows that current level of inequalities
rarely reflects the majority's preference. The problematic then becomes:
``What redistribution would (a majority of) people support?'', and
the political acceptability of redistributive reforms is investigated.
This allows in turn to assess empirically the political reception
of the main criteria used in the optimal taxation literature (the
Rawlsian and utilitarian ones). They appear to be well evaluated but
to lack from political acceptability, given the (relatively) high
proportion of disadvantaged people they imply. Indeed, the most widely
approved reform entails a redistribution of similar extent to those
derived from the theory but concentrates the burden to fewer people
at the top. Thanks to an original methodology, this redistribution
is certainly the most precise and favored one to have been elicited
from surveys until now.

A growing literature uses surveys to exhibit determinants for preferences
over income distribution, quantify the inequality aversion, measure
its evolution and compare inclinations in different countries.\footnote{See \citet{Fabre2016} for a thorough review of this literature.}
A general finding emerging from international surveys such as the
2009 \emph{\href{https://dbk.gesis.org/dbksearch/download.asp?file=ZA5400_cdb.pdf}{International Social Survey Program}
}(ISSP) is that, in almost every country, a majority of people supports
that it is ``the government\textquoteright s responsibility to reduce
income differences''. Indeed, over the 40 surveyed countries, such
a proposition fails to obtain a majority only in New Zealand (42\%)
and in the United States (US, 33\%): overall, only 14\% of the respondents
disagree with the statement while 72\% agree (among non-missing answers).
Despite a growing interest for preferences over distribution, only
a few studies focus on determining the desired extent of inequality
or on quantifying the \emph{preferred distribution }(or redistribution)
of a country. Among them, \citet{Singhal2008} and \citet{ForseEtAl2014}
quantify desired rates for the income tax at four levels of income
(respectively for seven OECD countries using ISSP 1996, and for France
using \emph{\href{http://dynegal.org}{Dynegal}}\textemdash a 2014
survey). Both results seem in line with current income tax rates.
Nonetheless, the view that respondents do not want a more progressive
tax system seem contradicted by other results, including in these
surveys: for example, in \emph{\href{http://www.dynegal.org/sites/default/files/brochure_resultats_tns_version_def.pdf}{Dynegal}},
60\% agree for an increase of taxes on the wealthiest even if they
may flee from France, 51\% stand for a range between minimum and maximum
incomes of 1 to 10 or below, and 72\% find the French tax system unfair
or very unfair. However, this discrepancy can be explained by at least
two factors other than mere inconsistency. Firstly, when asked directly
for the appropriate level of income tax rates, respondents may have
restricted their answers to non-negative values because of their representation
of what a tax is: this zero lower bound surely pushed up answers of
rates for low and middle-income. Secondly, respondents may roughly
agree with tax rates in the income range tested in the survey\footnote{The income range stretched from 1100¤ to 12000¤ per month in Dynegal,
and exhibited comparable values in ISSP 1996.} while desiring a more redistributive policy outside these bounds.
This suggests that a survey should be administered with more degrees
of freedom to define income tax rates, allowing for more than four
different levels of income and for negative tax rates. This is precisely
what \citet{Weinzierl2014} undertook in his own survey, by asking
American people to rank different tax systems (graphically presented).
Weinzierl also finds that preferred income tax rates in the US roughly
correspond to the actual tax schedule. This is less surprising than
for the 8 countries examined by Singhal, Forsé and Parodi, because
Americans do not believe that it's the government responsibility to
reduce income differences. Indeed, 51\% of American respondents \emph{disagree}
or \emph{strongly disagree} with that idea, whereas 51\% of French
respondents \emph{strongly agree} with it (among non-missing answers,
in \href{https://dbk.gesis.org/dbksearch/download.asp?file=ZA5400_cdb.pdf}{ISSP 2009}). 

While Weinzierl was interested in eliciting the criteria that Americans
implicitly use to judge a tax system they broadly accept, another
research question emerges in the French context since surveys like
\emph{\href{http://bdq.reseau-quetelet.cnrs.fr/fr/Details_d_une_enquete/1279}{Perceptions des Inégalités et Sentiments de Justice}
}(PISJ, 2009) has shown that \href{http://nesstar.cmh.ens.fr/webview/index.jsp?v=2&submode=abstract&study=http\%3A\%2F\%2Fnesstar.cmh.ens.fr\%2Fobj\%2FfStudy\%2Flil-0731&mode=documentation&top=yes}{60\%{} are favorable}
(strongly or not) to ``an increase of taxes in order to redistribute
the surplus to the least fortunate'' and that an overwhelming \href{http://nesstar.cmh.ens.fr/webview/index.jsp?v=2&submode=abstract&study=http\%3A\%2F\%2Fnesstar.cmh.ens.fr\%2Fobj\%2FfStudy\%2Flil-0731&mode=documentation&top=yes}{89\%{} of French people}
agree (strongly or not) that ``differences between high and low incomes
should be reduced'' in their country. Knowing that the French favor
a redistributive reform, arises the question: which reform do they
want? What redistribution(s) would satisfy their desire for a reform
while still obtaining a majority support? Using a methodology specially
conceived to address this problem of political economy, two surveys
were conducted in 2016 whose results are consistent with the French
endorsement of an additional redistribution. Not only the first survey
was helpful to investigate in detail preferences for the tax system,
but it allowed to define the shape of three redistributions\emph{
}directly from the median answers to key parameters of a reform. The
relevance of this original procedure was demonstrated throughout the
second survey, as respondents have clearly expressed their support
for the income tax reforms derived from the answers of the first survey. 

The surveys will be presented in section 1. Their results will be
exposed in section 2. A rich discussion will ensue in section 3, which
will show the robustness and the weaknesses of this method, study
the reception of the utilitarian and Rawlsian criteria, and pave the
way for a more direct inclusion of the citizens' preferences in the
shaping of tax schedules.

\section{Presentation of the Surveys}

\subsection{Data Collection (and Data Cleansing)\label{subsec:Data-Collection-(and}}

The surveys were conducted in September and October 2016 on two separate
representative samples of a thousand of French adults each. The respondents
were picked by the company \emph{Bilendi} from the 700,000 persons
of the \emph{Access Panel On-line Bilendi} (which is continuously
filtered to improve its quality), and remunerated through gift points
tantamount to 3 euros. The method of quotas insured \emph{a priori}
representativeness according to five socio-demographic characteristics:
age (7 brackets), sex, socio-professional category (8), size of town
(5) and region (9, solely in metropolitan France), while an \emph{a
posteriori }adjustment accounted for the over-representation of highly
educated people.\footnote{See subsection \ref{subsec:Robustness-Check} for a robustness check
with non-adjusted data, and \href{http://adrien-fabre.com/sondage/doc_methode.php\#_r}{on-line documentation}
for more details on the sampling method (in French). All sources are
available here: \href{http://adrien-fabre.com/documents.php\#sondages}{adrien-fabre.com/documents.php\#{}sondages}
(see \emph{Données complètes} for ready-to-use dataset); a \href{http://adrien-fabre.com/sondage/doc_var.php}{codebook}
and the \href{http://adrien-fabre.com/sondage/doc_q.php}{questionnaires}
are also provided.} As the surveys included several graphs and interactive animations,
they have been administered on-line, on each respondent's computer.
Although results from computer-based surveys may differ significantly
from face-to-face or telephone interviews, as pointed out by \citet{Parodi2014},
it is unclear which method of administration is less biased (as compared
to voters' preferences). The attrition rate (measuring the proportion
of respondents who started but did not complete the survey) was respectively
10\% and 30\% in the first and second survey.\footnote{The higher figure in the second survey is due to a bug in the beginning
of its implementation, which did not affect the results (because only
complete answers were retained).} The response time was \emph{a priori }estimated to 20 minutes, and
opportunist respondents who answered in less than 9 minutes were screened
out. \emph{A posteriori}, the median response time for accepted respondents
in each survey was respectively 18.4 and 20.6 min. In order to spot
inattentive respondents, a test of quality of the responses was inserted.
It consisted in adding a choice labeled ``\emph{Please tick `Slightly
decrease' (test of quality of your answer)}'' inside a matrix with
multiples questions and multiple choices (themselves ranging from
``Strongly increase'' to ``Strongly decrease''). Between 16\%
and 24\% failed this test, depending on the survey. Still, it would
be excessive to consider all these respondents as phony or lacking
of seriousness. Indeed, as I submitted personally the questionnaires
to a dozen of people before launching the surveys, I noticed that
some people did not understand this test and responded ``\emph{I
don't know}'' or ``\emph{Indifferent}'' (the choice that looked
the most neutral), like the majority of those who failed this test.
Finally, the final (or \emph{restricted}) sample was constituted from
the original (or \emph{augmented}) sample after the elimination of
the respondents who did not complete the questionnaire, who responded
too quickly, who failed the test of quality or whose quota was already
full. The broader \emph{augmented} sample is used as an alternative
one to test the robustness of the results in subsection \ref{subsec:Robustness-Check}.

\subsection{Survey Questions and Methodology}

The surveys contained many questions on fiscal and political preferences
that are outside the scope of this article. However, their main questions
were dedicated to the study presented here, and were designed to infer
a precise\emph{ }redistribution supported by a majority of French
people. In this subsection, I will detail in turn the source and variable
used to plot the current distribution, the procedures chosen to derive
a redistribution from a few key parameters, and the redistributions
derived from the typical answers to the first survey, hereafter referred
to as the \emph{proposed redistributions}.

\subsubsection{Sources and Variables}

All the data used to plot the distributions is taken from the \emph{Enquête
sur les Revenus Socio-Fiscaux }(ERFS 2012) produced by \emph{INSEE
}(the French national statistics bureau). The standard variable of
the \emph{ERFS} which allows to present income inequalities at the
household level\textemdash arguably the most relevant level to consider
them\textemdash is the equivalised disposable income (or \emph{niveau
de vie}), which equals the disposable income of the household divided
by its number of consumption units.\footnote{Following \href{http://ec.europa.eu/eurostat/statistics-explained/index.php/Glossary:Equivalised_disposable_income}{Eurostat},
the \emph{INSEE} defines the number of consumption units by summing
different weights for each household member: 1 for the household head,
0.5 for each additional person aged 14 or more, and 0.3 for each child
below the age of 14.} This variable has been used in a question where respondents had to
grade different income distributions between -2 and +2 (see subsection
\ref{subsec:Evaluations-of-Distributions:}). However, a variable
at the individual level was preferred for the main questions. In effect,
a redistribution derived from their previous answers was proposed
to each respondent in the first survey, as well as an interactive
graph where s-he could fine-tune their preferred redistribution. The
algorithms which computed interactively the displayed redistribution
from a change in the parameters featured a simple constraint on the
aggregate income: the latter was assumed to stay constant throughout
the redistribution.\footnote{Behavioral responses were taken into account by another mean.}
Indeed, the constraint on the aggregate income ought to be simple
to optimize the loading time and the treatment of the data. Additionally,
at variance with household variables, individual incomes simply sum
up to the aggregate (national) income. Because of this computational
ease, and as an individual variable can still be satisfactory to study
income inequalities, the \emph{individual disposable income }was adopted.
It was defined for the occasion, by imputing the disposable income
to the adult members of the household. Non-contributive social benefits
were imputed to the least contributor(s) of the household, while other
incomes were imputed to their respective entitled person.\footnote{When some income was not attributable to a peculiar individual by
this method (which was the case for capital income), it was allocated
pro rata according to the contribution of each adult in the household
(excluding non-contributive benefits).} 

\subsubsection{Parameters Determining the Redistribution}

Defining a redistribution consists in deriving a \emph{future }distribution
from a \emph{current }distribution, using an algorithm fed with some
parameters. In order to limit the number of such parameters, four
strategic points have been chosen on the cumulative density function
of incomes, through which the future distribution will pass (when
possible). These points correspond to the bounds of the income distribution:
the demogrant and the maximum income, and to the crossing points with
the current distribution: the quantile of adults advantaged and disadvantaged
by the reform (between which the current and future distribution coincide).
The persons advantaged (respectively disadvantaged) by the redistribution
are assumed to have the lowest (resp. the highest) incomes, which
restricts our set of allowed redistributions to redistributive ones
(in accordance with previous insights on French preferences). Furthermore,
it is worth noting that the maximum income plays here little role,
in the sense that in practice, its value does not affect the shape
of the redistribution, so that the results are similar to a situation
with no maximum income (which is equivalent in the algorithm to assigning
an arbitrarily high value to this parameter). 

Let us now detail the questions asked in the survey about these parameters.
For all these questions, the respondents had to type their answers
in an entry field (see Appendix \ref{sec:Survey-Screen-Shots} for
screen shots). Also, a different variant of each question (among two
to four) was randomly allocated to each respondent. Finally, the values
of the parameters were computed from the weighted augmented sample
of the first survey, in order to gain in the precision of the estimates.
The values obtained then and reported in this subsection are much
the same as those from the restricted sample of the first survey as
well as those from the restricted samples of both surveys combined,
which constitute the final results that are presented in subsequent
figures. The summary statistics of the variables for the different
samples are presented in Appendix \ref{sec:Summary-Satistics}. As
for the few questions further mentioned that did not enter in the
computation of the parameters, all statistics are given from the final
results.

\begin{figure}[H]
\caption{Key parameters of a redistribution: the demogrant and the bounds where
current and future distribution coincide}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/Reform parameters\string".eps}}}\selectlanguage{english}%
\end{figure}

\bigskip{}

\paragraph{The demogrant }

Four variants of the question have been asked, which allow to assess
the importance of the phrasing in such questions and help understand
the expectations of the population over the welfare state. The wording
that seems to better correspond to the demogrant is ``\emph{What
should be the amount of welfare }{[}aides de l'État{]} \emph{for those
who have no income?}'': over the 64\% of non-missing answers, the
median is 800¤/month (see Appendix \ref{sec:Results} for the density
of answers and Appendix \ref{sec:Survey-Screen-Shots} for the phrasing
of questions).\footnote{The median for the final sample (combined both surveys) is slightly
lower: 738¤/month.} Other phrasings (the amount of a basic income, the minimum amount
guaranteed to all, or how much the state should insure to all) have
resulted in much higher median answers (1000, 1200 and 1400¤/month,
respectively), probably because the respondents interpreted them as
pertaining to an ideal situation unreachable immediately, or thought
of other mechanisms than social benefits, such as full-employment,
to improve the most modest incomes. Indeed, only the answers to the
former phrasing are compatible with other results, as can be shown
with two pieces of evidence. Firstly, the desired trade-off in the
reduction of inequalities clearly leans towards an increase in the
minimum wage rather than in an increase in welfare benefits (65\%
vs. 7\% of answers).\footnote{This question was asked in the second survey, the remaining answers
fan out as follows: 13\% do not wish to reduce inequalities, and 16\%
choose ``\emph{I don't know, I don't want to answer}''.} Secondly, a majority wishes to keep the \emph{Revenu de Solidarité
Active }(RSA socle) to its current level (37\% of answers) or below
(31\%).\footnote{This result contrasts with the data from \citet{Piketty2003} (the
only previous study which directly asked for the desired amount for
the RSA), which showed that the median person desired an increase
of the RSA by 20\% in 1999 (since, it has been increased by 9\% in
real terms). However, a median desired amount of (minimal) social
aid of 750¤/month observed in the final sample is consistent with
the findings of a 2015 \href{http://drees.social-sante.gouv.fr/IMG/pdf/principaux_enseignements_barometre_2015.pdf}{survey}
from the DREES, which shows that 55\% are favorable to an increase
of the RSA, knowing that it is between 500¤ and 760¤/month (taking
housing benefits into account)\textemdash while 75\% were in favor
of an increase in 2009. Three reasons explain the lower support for
an increase of the RSA in the recent surveys: its revaluation since
2012, a shift in views since the recession, and the framing of the
question (people tend to be more supportive when the survey indicates
the amount of the RSA, because they would not estimate it well otherwise,
and/or because the survey omits housing benefits\textemdash which
is the case in \citet{Piketty2003}).} RSA is a welfare benefit of 535¤/month for French people with no
(or very low) income, which easily results in a demogrant of 785¤/month,
once combined with \href{http://www.igas.gouv.fr/IMG/pdf/2014-149R.pdf}{housing benefit}
(which depends on the rent, the geographical area and the household
structure, and averaged at \href{http://www.developpement-durable.gouv.fr/IMG/pdf/Le_compte_du_logement_2014.pdf}{239¤/month}
in 2013) and \emph{Prime de Noël }(a Christmas grant of 152¤). After
accounting for imperceivable in kind benefits (such as the gratuity
of public transportation in the Paris area for RSA recipients), the
French demogrant can be rounded up to 800¤/month, which is in line
with the dominant preference to keep the demogrant at its current
level. 

\begin{figure}[H]
\caption{\textbf{Demogrant}: desired amounts depending on the phrasing of the
question (in ¤/month)}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.6\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/demogrant honest\string".eps}}}\selectlanguage{english}%
\end{figure}
\bigskip{}

\paragraph{The maximum income\label{par:The-maximum-income}}

Three variants were tested to determine the desired maximum income
for France (if any): straightforwardly, the desired amount for the
legal maximum income (hereafter \emph{legal max}); in addition, the
same question primed with an argument on the counter-productivity
of taxing too much the richest people (\emph{legal max + anti-tax
argument}); and finally, the maximum income in an ideal society (\emph{ideal
max}).\footnote{The phrasing of the variants were respectively: ``In your opinion,
what is the maximal income that should be legally established in France?'';
the same question followed by ``It is worth noting that above a certain
threshold, the taxation of the richest if often counter-productive,
because they move to another country or reduce their activity to avoid
the tax increase''; and ``If France was an ideal society, what would
be the highest income? Let us precise that this question does not
ask whether a legal ceiling ought to be established on French income:
it simply amounts to inquire what would be the highest income in a
society with the appropriate level of inequalities.'' The respondents
then had the possibility to choose the absence of limit, or not to
answer, along the entry field.} Median answers of these variants were always finite: respectively
100,000¤/month for legal max (44\% stated that they did not wish any
ceiling on incomes, and their answer were counted as \emph{infinity}
while computing the median answer), 20,000¤/month for legal max +
anti-tax argument (36\% for \emph{infinity}), and 15,000¤/month for
ideal max (16\% for \emph{infinity}). Interestingly, the priming had
the opposite effect than expected: in effect, the anti-tax argument
induced highly significantly lower answers for the (logarithm of)
desired maximum income.\footnote{See Appendix \ref{sec:Regressions} for the results of all regressions.}
This result may be a manifestation of the \emph{boomerang effect}:
indeed, \citet{HovlandEtAl1953} showed that when someone is pressured
to make a certain choice, psychological reactance (theorized by \citet{Brehm1966})
can cause her or him to resist this pressure by adopting an opposite
alternative. 

The simple phrasing (\emph{legal max}) was logically chosen to set
the value of the parameter, to its median answer of 100,000¤/month.
It is worth noting that the final median answer to the simple phrasing
(when taking due account of both surveys) proved somewhat higher:
it is 250,000¤/month. But in any case, as already mentioned, such
difference in the value of this parameter have very little impact
on the shape of the graph presented to the respondent. Finally, these
results are mostly relevant for themselves: it is interesting to learn
that French are rather in favor of a ceiling on incomes.

\begin{figure}[H]
\caption{\textbf{Maximum income}: desired amounts depending on the phrasing
of the question (in ¤/month)}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.6\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/income_max_miss honest\string".eps}}}\selectlanguage{english}%
\end{figure}
\bigskip{}

\paragraph{The proportion of dis/advantaged by the reform}

The last two parameters were given by two similar questions, which
asked for the preferred proportion of persons to advantage, or to
disadvantage, through a redistribution of incomes from the richest
to the poorest.\footnote{\citet{Fabre2016} relied on parameters expressed in absolute terms
(e.g. the desired income for an unskilled worker) rather than in relative
terms (such as the proportion to advantage) to infer desired redistribution.
The former method does not allow to internalize the constraint on
aggregate income on the answers, thereby burying the trade-off between
advantaging the poor and disadvantaging the rich. Hence, parameters
expressed in relative terms have been chosen.} A \href{http://adrien-fabre.com/sondage/Fiscalite\%20des\%20francais.html\#QuestionText_q36035863_FR}{slider}
was provided graphically to help the respondents, which indicated
the income corresponding to each percentile of the distribution. The
answers are slightly different in the second survey, where the distribution
was given in terms of equivalised household disposable income instead
of individual disposable income, indicating that respondents take
into account both the proportion and the absolute level of income
in their choice.\footnote{This is consistent with a similar finding in \citet{SaezEtAl2016}.
See Appendix \ref{sec:Regressions} for the results and Appendix \ref{sec:Survey-Screen-Shots}
to see the slider.} That being said, the medians are in both cases 50\% and 10\% for
the proportion of persons to advantage and disadvantage, respectively.
They correspond to individual (resp. equivalised household) monthly
incomes of 1450¤ (resp. 1700¤) and 2950¤ (resp. 3150¤).

\begin{figure}[H]
\caption{\textbf{Dis/advantage}: preferred percentage of French people to dis/advantage
through a redistributive reform of the income tax}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.7\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/dis_advantage_miss honest\string".eps}}}\selectlanguage{english}%
\end{figure}
\bigskip{}

\subsubsection{Redistributions Derived From Survey Answers\label{subsec:Redistributions-Derived-From}}

Two algorithms were used to derive different redistributions using
the parameters aforementioned, they are explained in Appendix \ref{sec:Algorithms-Used},
and one can interactively play with them \href{http://adrien-fabre.com/sondage/Fiscalite\%20des\%20francais.html\#q21976492}{here}\footnote{\emph{Dis/av}: \href{http://adrien-fabre.com/sondage/Politique\%20des\%20francais.html\#q19183287}{http://adrien-fabre.com/sondage/Politique\%{}20des\%{}20francais.html\#{}q19183287}}
and \href{http://adrien-fabre.com/sondage/Politique\%20des\%20francais.html\#q19183287}{there}.\footnote{\emph{Demogrant}: \href{http://adrien-fabre.com/sondage/Fiscalite\%20des\%20francais.html\#q21976492}{http://adrien-fabre.com/sondage/Fiscalite\%{}20des\%{}20francais.html\#{}q21976492}}
These algorithms, as compared to others that have been imagined (relying
e.g. on the Lorenz curve, on the desired minimum wage or on a non-parametric
graphical definition), were preferred because of the simplicity of
the questions which they rely upon. Finally, they require an additional
parameter, named \emph{Extent}, adjusted using known median preferences,
which corresponds to the magnitude (coded between 0 and 10) of the
transfer from rich to poor, other things equal.

\paragraph{Algorithm \emph{Demogrant\label{par:Algorithm-Demogrant}}}

The first algorithm uses a unique \emph{neutral point} instead of
a range of quantiles between which the current and future distributions
coincide. To determine this particular quantile which splits the population
between advantaged and disadvantaged people (making everyone somewhat
affected by the reform), I took the median answer to both variants
combined (advantage and disadvantage). The neutral point obtained
was 77\%, corresponding to an individual income of 2150¤/month. Besides,
the parameter \emph{Extent} was chosen so as to suit the median desire
for a slight increase of the minimum wage observed in this survey.
If one considers that the minimum wage should correspond to the ``minimum
income below which one cannot make a decent living''\footnote{\selectlanguage{french}%
The original phrasing (in French) was: ``le revenu minimum mensuel
net en dessous duquel on ne peut s'en sortir sans difficultés importantes''.\selectlanguage{english}%
}, its median desired value was quantified in another survey (PISJ)
to be 1360¤/month. Hence, the minimum wage (net of taxes and subsidies),
amounting to 1280¤/month before the reform, was brought to 1360¤/month
by setting \emph{Extent }to 3.5. Finally, applying the algorithm with
these parameters produced the \emph{demogrant median} redistribution,
presented in Figure \ref{fig:Median-demogrant-proposed}.

The first survey included a question where the respondents could adjust
the parameters of the first algorithm using sliders, in order to propose
their own preferred reform. At the beginning of this interactive process,
the two first sliders were set to their previous answers, while the
slider for \emph{Extent} was set at its middle value of 5. The median
answers at this question (to which 70\% of the sample responded) were
1000¤/month for the demogrant, 56\% for the proportion of advantaged,
and 5 for \emph{Extent}.\textbf{}\footnote{Only 5 respondents over 1007 filled the entry fields with the initial
value of the parameters, suggesting that those who responded to this
question did it sincerely and carefully.\\
Another version of the question used the other algorithm, \emph{Dis/av},
and its results are not presented because they are similar to those
of the previous subsection: the median answers to advantage and disadvantage
are the same, and the median for \emph{Extent }is only slightly lower,
between 7 and 8.} One could have taken this median value for \emph{Extent }as an alternative
to compute the previous proposed reform (it would have resulted in
a more redistributive proposition). However, the results of this question
were used in a different way: an \emph{average }proposed reform was
computed by averaging all the reforms proposed by the respondents:
it is shown in Figure \ref{fig:Average-proposed-reform}.

\begin{figure}[H]
\caption{The \emph{demogrant median} proposed reform\label{fig:Median-demogrant-proposed}
(Yes/No/PNR: 42/38/20\%).}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/reform demogrant median 2\string".eps}}}\selectlanguage{english}%
\end{figure}

\bigskip{}

\begin{figure}[H]
\caption{The \emph{average} proposed reform\label{fig:Average-proposed-reform}
(Yes/No/PNR: 39/37/24\%).}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/reform average 2\string".eps}}}\selectlanguage{english}%
\end{figure}

\paragraph{Algorithm \emph{Dis/av}}

The second algorithm uses both the preferred proportion of advantaged
and disadvantaged people as parameters. The value of \emph{Extent
}(8.57) was chosen in such a way that the demogrant would equal its
median preferred value. Therefore, the main difference between the
\emph{demogrant median} proposed reform (described above) and the
\emph{median} proposed reform (this is how this one is named) lies
in lower middle of the distribution, where the former is less generous,
and at the top, where the latter concentrates more the burden of the
new transfer. The shape of the \emph{median} proposed reform is shown
in Figure \ref{fig:Median-proposed-reform}.

\begin{figure}[H]
\caption{The \emph{median} proposed reform\label{fig:Median-proposed-reform}
(Yes/No/PNR: 52/26/22\%).}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/reform median 2\string".eps}}}\selectlanguage{english}%
\end{figure}

Admittedly, the three first proposed reforms did not address behavioral
responses. Indeed, including a modeling of behavioral responses in
the algorithm would have been too computationally intensive for an
interactive program. In order to overcome this shortcoming, an additional
parameter was added to the algorithms, which controls the variation
in aggregate (disposable) income through the reform. Therefore, this
parameter can be interpreted in two ways: as a proxy for behavioral
responses or as a change in the government's budget. By setting this
parameter to 5\% of national (disposable) income, a last reform was
proposed, named \emph{distortionary median} proposed reform (see Figure
\ref{fig:Median-distortionary-reform-1}). The value of 5\% roughly
corresponds to the aggregate loss of consumption in redistributions
of similar magnitude (see subsection \ref{subsec:A-Surprising-Link})\footnote{5\% is also the median answer to another question of this survey:
``What proportion of the income of rich countries should be transferred
to poor countries?''. However, interpreting this value as the desired
increase in development aid would wipe out the intended capture of
behavioral responses; thus, an interpretation in terms of a proxy
for distortions should be preferred.} Removing 5\% of national income while disadvantaging only the richest
10\% led to cap all incomes at 3000¤/month, which was obviously not
in accordance with the desired maximum income. Hence, the proportion
of disadvantaged people was raised to 12\% for this redistribution.
Also, the budget was too tight to procure a decent demogrant, so one
could not impede it to decrease, even after increasing the \emph{Extent}
to 9.\footnote{Setting \emph{Extent }to its maximal value of 10 would have resulted
in either an overwhelmingly low maximum income or an even higher proportion
of disadvantaged people, so 9 was chosen as an acceptable trade-off.} 

\begin{figure}[H]
\caption{The \emph{distortionary median} proposed reform\label{fig:Median-distortionary-reform-1}
(Yes/No/PNR: 46/28/27\%).}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/reform distortionary median 2\string".eps}}}\selectlanguage{english}%
\end{figure}

\subsubsection{Approval and Evaluation of the Proposed Distributions}

The question inquiring about the approval of proposed distributions
was phrased as follows: 
\begin{quote}
Imagine a tax reform with the following characteristics: the actual
income distribution of French people (in red) would be replaced by
a more egalitarian distribution (in green); this reform would establish
a basic income guaranteed to all of {[}demogrant{]}¤/month, would
disadvantage the richest {[}disadvantage{]}\% as compared to current
situation but would advantage the poorest {[}advantage{]}\%; it would
entail a transfer of {[}computed{]}\% of GDP from the richest to the
poorest, as compared to current situation. 

Your after-tax income is estimated at {[}computed{]}¤/month. If this
reform were taking place, your after-tax income would become {[}computed{]}¤/month.

Would you approve such a reform?
\end{quote}
%
The proposed reform was displayed below this text, followed by three
options: \emph{Yes}, \emph{No} and \emph{PNR (I don't know, I don't
want to answer)}. The characteristics (square brackets in the above
text) for each proposed redistribution are presented in Annex \ref{sec:Results};
as a matter of fact the transfer was $\ensuremath{10.5\pm1.2}\%$
of GDP in each case. In addition to the four redistributions described
above that were presented in the second survey, the approval of a
\emph{personalized }redistribution was inquired in the first survey.
This redistribution was derived from each respondent's previous answers
using the \emph{Demogrant }algorithm.

Furthermore, a question asked the respondents to grade in $\left\llbracket -2;+2\right\rrbracket $
(from\emph{ I don't like... }to \emph{I like} \emph{this repartition})
the distributions of equivalised household disposable income presented
in Figure \ref{fig:Distributions-of-equivalised}, as well as the
\emph{personalized }redistribution. Each household's equivalised income
was retrieved from its members' individual disposable income and its
number of consumption units.\footnote{A dataset of 5000 households was extracted from the original 128000
individuals database to make computations on each respondent's computer.
Several random extractions were drawn, and the one that fitted the
best the original distribution was retained.} As one can see in Annex \ref{sec:Survey-Screen-Shots}, the respondents
were guided to properly understand the distributions and the question,
although the distributions were not labeled. Besides, the derivation
of distributions from the theory of optimal taxation was carried out
using the model and source code of \citet{JacquetEtAl2013}\footnote{This model computes the optimal nonlinear tax using an elasticity
of labor supply of 0.25 to account for the equity-efficiency trade-off,
in a manner similar to the seminal \citet{Saez2001}. I am indebted
to Étienne Lehmann for having graciously provided me the code.}, adapted to French data with no extensive margin.

\begin{figure}[H]
\caption{Distributions of equivalised disposable income presented for grading,
in ¤/month\label{fig:Distributions-of-equivalised}}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{}\subfloat[\foreignlanguage{english}{\emph{Demogrant median}}]{\selectlanguage{english}%

\includegraphics[width=0.32\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/nivvie mediane rdb int\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{}\subfloat[\foreignlanguage{english}{\emph{Median}}]{\selectlanguage{english}%

\includegraphics[width=0.32\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/nivvie mediane int\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{}\subfloat[\foreignlanguage{english}{\emph{Egalitarian}}]{\selectlanguage{english}%

\includegraphics[width=0.32\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/nivvie egalitaire int\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{}}

\makebox[\textwidth][c]{\foreignlanguage{english}{}\subfloat[\foreignlanguage{english}{\emph{Utilitarian optimal taxation}}]{\selectlanguage{english}%

\includegraphics[width=0.32\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/nivvie utilitarien int\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{}\subfloat[\foreignlanguage{english}{\emph{Rawlsian optimal taxation}}]{\selectlanguage{english}%

\includegraphics[width=0.32\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/nivvie rawlsien int\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{}\subfloat[\foreignlanguage{english}{\emph{Actual}}]{\selectlanguage{english}%

\includegraphics[width=0.32\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/nivvie actuel int\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{}}\selectlanguage{english}%
\end{figure}

\section{Main Results}

\subsection{Majority Adhesions to Proposed Redistributions\label{subsec:Robustness-Check}}

Of the five redistributions proposed to the respondents, each obtained
more approvals than disapprovals (see Table \ref{tab:adhesions}).
One of them, the \emph{median} proposed reform, obtained a majority
of approvals taking account of non-answers, and two-thirds of approvals
when excluding these \emph{PNR} answers.

\begin{table}[H]
\caption{\label{tab:adhesions}Rate of approval of different proposed redistribution
(in \%). 95\% confidence intervals are reported inside square brackets.}

\selectlanguage{french}%
\makebox[\textwidth][c]{%
\begin{tabular}{cc>{\centering}p{2.5cm}>{\centering}p{2.5cm}cc}
\noalign{\vskip-0.5cm}
\selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
 & \foreignlanguage{english}{} & \foreignlanguage{english}{} & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
\tabularnewline
\hline 
\hline 
\noalign{\vskip0.2cm}
\selectlanguage{english}%
\textbf{Proposed reform}\selectlanguage{french}%
 & \selectlanguage{english}%
\textbf{\emph{Demogrant median}}\selectlanguage{french}%
 & \foreignlanguage{english}{\textbf{\emph{Average}}} & \foreignlanguage{english}{\textbf{\emph{Median}}} & \selectlanguage{english}%
\textbf{\emph{Distortionary median}}\selectlanguage{french}%
 & \selectlanguage{english}%
\textbf{\emph{Personalized}}\selectlanguage{french}%
\tabularnewline
\selectlanguage{english}%
\emph{Number of respondents}\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{488}\selectlanguage{french}%
 & \foreignlanguage{english}{\emph{509}} & \foreignlanguage{english}{\emph{505}} & \selectlanguage{english}%
\emph{492}\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{1007}\selectlanguage{french}%
\tabularnewline[0.1cm]
\hline 
\noalign{\vskip0.1cm}
\selectlanguage{english}%
\textbf{Yes}\selectlanguage{french}%
 & \selectlanguage{english}%
\textbf{42}\selectlanguage{french}%
 & \foreignlanguage{english}{\textbf{39}} & \foreignlanguage{english}{\textbf{52}} & \selectlanguage{english}%
\textbf{46}\selectlanguage{french}%
 & \selectlanguage{english}%
\textbf{50}\selectlanguage{french}%
\tabularnewline
\selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{{[}37.9; 46.5{]}}\selectlanguage{french}%
 & \foreignlanguage{english}{\emph{{[}35.1; 43.7{]}}} & \foreignlanguage{english}{\emph{{[}47.8; 56,4{]}}} & \selectlanguage{english}%
\emph{{[}41.1; 50.0{]}}\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{{[}46.4; 52.6{]}}\selectlanguage{french}%
\tabularnewline[0.1cm]
\noalign{\vskip0.1cm}
\selectlanguage{english}%
\textbf{No}\selectlanguage{french}%
 & \selectlanguage{english}%
38\selectlanguage{french}%
 & \foreignlanguage{english}{37} & \foreignlanguage{english}{26} & \selectlanguage{english}%
28\selectlanguage{french}%
 & \selectlanguage{english}%
28\selectlanguage{french}%
\tabularnewline
\selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{{[}34.1; 42.6{]}}\selectlanguage{french}%
 & \foreignlanguage{english}{\emph{{[}33.0; 41.4{]}}} & \foreignlanguage{english}{\emph{{[}22.5; 30.1{]}}} & \selectlanguage{english}%
\emph{{[}23.7; 31.7{]}}\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{{[}25.6; 31.1{]}}\selectlanguage{french}%
\tabularnewline[0.1cm]
\noalign{\vskip0.1cm}
\selectlanguage{english}%
\textbf{PNR}\selectlanguage{french}%
 & \selectlanguage{english}%
20\selectlanguage{french}%
 & \foreignlanguage{english}{24} & \foreignlanguage{english}{22} & \selectlanguage{english}%
27\selectlanguage{french}%
 & \selectlanguage{english}%
22\selectlanguage{french}%
\tabularnewline
\selectlanguage{english}%
(People Not Responding)\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{{[}16.4; 23.3{]}}\selectlanguage{french}%
 & \foreignlanguage{english}{\emph{{[}20.1; 27.5{]}}} & \foreignlanguage{english}{\emph{{[}18.4; 25.5{]}}} & \selectlanguage{english}%
\emph{{[}23.2; 31.1{]}}\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{{[}19.8; 24.9{]}}\selectlanguage{french}%
\tabularnewline[0.1cm]
\hline 
\noalign{\vskip0.1cm}
\selectlanguage{english}%
\textbf{Yes, excluding PNR}\selectlanguage{french}%
 & \selectlanguage{english}%
\textbf{52}\selectlanguage{french}%
 & \foreignlanguage{english}{\textbf{51}} & \foreignlanguage{english}{\textbf{67}} & \selectlanguage{english}%
\textbf{62}\selectlanguage{french}%
 & \selectlanguage{english}%
\textbf{64}\selectlanguage{french}%
\tabularnewline
\selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{{[}47.6; 57.3{]}}\selectlanguage{french}%
 & \foreignlanguage{english}{\emph{{[}46.4; 56.4{]}}} & \foreignlanguage{english}{\emph{{[}61.9; 71.0{]}}} & \selectlanguage{english}%
\emph{{[}57.1; 67.2{]}}\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{{[}60.2; 66.9{]}}\selectlanguage{french}%
\tabularnewline[0.1cm]
\hline 
\hline 
\selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
 & \foreignlanguage{english}{} & \foreignlanguage{english}{} & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
\tabularnewline[-0.5cm]
\end{tabular}}\selectlanguage{english}%
\end{table}

Overall, the most successful algorithm is \emph{Dis/av}. It revealed
a redistribution favored by a clear majority of respondents. In addition,
even with a national income diminished of 5\% (which proxies the distortions),
this algorithm outperforms the others. Nevertheless, the \emph{median
}redistribution cannot be considered as the \emph{preferred }redistribution
of French people: one can still administer a new survey, with a new
algorithm, and find a more favored redistribution. In that sense,
this method of using a survey to reveal favored redistribution only
provides an heuristic solution to the elicitation of the most accepted
reform. Its strength lies instead in its success in exhibiting politically
palatable redistributions. That being said, if successive governments
were to use this property at regular intervals to shape a redistribution
from survey answers and put it on referendum, the iterative process
would likely converge towards a politically stable distribution: i.e.
one for which no redistribution would be preferred.\footnote{From a theoretical point of view, if each citizen can rank distributions
in a non-evolving pre-order, the democratic process converges to the
Smith set, i.e. the smallest non-empty set of distributions such that
each distribution defeats every distribution outside the set in a
pairwise comparison. Then, the iterations could diverge if and only
if the Smith set contains a cycle, e.g. three distributions \emph{A},
\emph{B}, \emph{C}, with \emph{A}$\prec$\emph{B}, \emph{B}$\prec$\emph{C}
and \emph{C}$\prec$\emph{A}, where \emph{X}$\prec$\emph{Y} denotes
that a majority approves the reform from \emph{X} to \emph{Y}.} 

\subsection{Evaluations of Distributions: Triumph of the Optimal Taxation Theory\label{subsec:Evaluations-of-Distributions:}}

One may be worried that the process just described could lead to the
\emph{tyranny of a majority}, hence would not entail a most desirable
distribution. To address this issue, one can enrich the voting procedure
and refine the criterion of acceptation of a redistribution. For example,
taking ground on the theory developed by \citet{BalinskiEtAl2007},
one can ask respondents to grade the current and proposed redistributions
between -2 and +2, and use this information in a more sophisticated
rule to select the most accepted redistribution. Balinski and Laraki
demonstrate that the most desirable aggregation of preferences consists
in electing the proposition with the highest grade at the \emph{k}-th
quantile in its grades' distribution\footnote{\label{fn:To-resolve-ties,}To resolve ties, the propositions are
ranked according to the following formula (as long as the grading
scale is made up of unitary intervals): $g+\mathbf{1}_{p\cdot q\neq0}\left(\frac{p}{p+q}-\left(1-k\right)\right)+\mathbf{1}_{p=0}\left(1-k-q\right)+\mathbf{1}_{q=0}\left(p-\left(1-k\right)\right)$,
where\emph{ g }is the grade at the \emph{k}-th quantile and \emph{p
}(resp. \emph{q}) is the proportion of grades strictly above (resp.
strictly below) \emph{g.}} (this notably overcomes Arrow's impossibility theorem and minimizes
manipulability). Whereas they argue in favor of majority judgment,
which relies on the the median (\emph{$k=\nicefrac{1}{2}$}), in our
case one could prefer to use a lower percentile, say the first quartile,
so that least satisfied people get more influence on the final decision.
Combining several criteria would further hamper a possible \emph{tyrannical
reform of a majority} by favoring the status quo. For example, one
could require that for a reform to pass, the new distribution would
need to win the majority judgment, to be better graded than the incumbent
distribution by the first quartile, and to obtain a \emph{strict}
majority of approvals (i.e. a majority even when non-answers are included).
Using the evaluation of actual and proposed distributions by the respondents
between -2 and +2, one can assess if this more conservative \emph{three
steps} process would lead to a redistributive reform or not. 

Figure \ref{fig:Evaluation-of-different} shows these raw results
while Table \ref{tab:Aggregate-evaluation-of} presents statistics
from their aggregation. While the totally egalitarian distribution
is the only one that is less appreciated than the current one; the
distributions that obtain the best grades under most criteria are
derived from the theory of optimal taxation, using either a Rawlsian
or a utilitarian criterion. This demonstrates the acceptability of
this theory: not only it produces solid justification for a formula
of income tax rates, but it also outperforms competing propositions
under popular evaluation. Moreover, it does so irrespective of the
ethical criterion chosen, which does not significantly change the
average grade. One may regret that these winning distributions were
not tested for approval in the survey: the only two that can be reviewed
with respect to the three steps process proposed in the previous paragraph
are the \emph{median }and \emph{demogrant median} redistributions.
Interestingly, the \emph{demogrant median }obtains slightly better
results at the evaluation although the \emph{median }reform gets the
highest rate of approval. This may be due to the exposition of the
current distribution along with the proposed one in the approval question,
which revealed to the respondents the higher proportion of disadvantaged
people in the \emph{demogrant median }than in the \emph{median }reform.
Indeed, the spread between rates of approval among those who are disadvantaged
by the \emph{demogrant median} but not by the \emph{median }reform
is 9\% higher than among the whole sample (even though the low number
of observations prevents a statistical significance). As the \emph{median
}reform is the only one to obtain a strict majority of approvals,
it is the only one tested that meets the third criterion of the three
steps process. As is shown in Table \ref{tab:Aggregate-evaluation-of},
it also fulfills the two first steps. Therefore, this proposed reform
is a robust candidate for a redistribution of French incomes. Finally,
it is worth noticing that the respondents tend to better grade redistributions
derived from an external source (be it a theory or the aggregation
of preferences) than from their own figures: it suggests that in this
particular case, collective wisdom overcomes personal intelligence. 

\begin{figure}[H]
\caption{Evaluation of different distributions of income\label{fig:Evaluation-of-different}}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.6\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/notations_en honest\string".eps}}}\selectlanguage{english}%
\end{figure}

\begin{table}[H]
\caption{Evaluation statistics of different income distribution.\label{tab:Aggregate-evaluation-of}
The distributions in bold were tested in the second survey (997 respondents)
while the others were tested in the first one (1007 respondents),
except for the \emph{actual} distribution, which was tested in both.
See note \ref{fn:To-resolve-ties,} for the definition of the score's
formula and \citet{BalinskiEtAl2007} for that of the majority gauge.}

\selectlanguage{french}%
\makebox[\textwidth][c]{ 

\begin{tabular}{cccccccc}
\noalign{\vskip-0.5cm}
\multirow{1}{*}[0.1cm]{\selectlanguage{english}%
\selectlanguage{french}%
} & \multicolumn{5}{c}{\selectlanguage{english}%
\selectlanguage{french}%
} & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
\tabularnewline
\hline 
\hline 
\noalign{\vskip0.1cm}
\selectlanguage{english}%
\selectlanguage{french}%
 & \multicolumn{5}{c}{\selectlanguage{english}%
Score from \emph{k}-th quantile in grades' distribution\selectlanguage{french}%
} & \selectlanguage{english}%
Majority \selectlanguage{french}%
 & \selectlanguage{english}%
Average \selectlanguage{french}%
\tabularnewline[0.2cm]
\selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\emph{k} = 0.15\selectlanguage{french}%
 & \selectlanguage{english}%
0.25\selectlanguage{french}%
 & \selectlanguage{english}%
0.35\selectlanguage{french}%
 & \selectlanguage{english}%
0.45\selectlanguage{french}%
 & \selectlanguage{english}%
0.5\selectlanguage{french}%
 & \selectlanguage{english}%
gauge\selectlanguage{french}%
 & \selectlanguage{english}%
grade\selectlanguage{french}%
\tabularnewline[0.2cm]
\hline 
\noalign{\vskip0.2cm}
\selectlanguage{english}%
\emph{Rawlsian optimum}\selectlanguage{french}%
 & \selectlanguage{english}%
-1.019\selectlanguage{french}%
 & \selectlanguage{english}%
-0.919\selectlanguage{french}%
 & \selectlanguage{english}%
-0.054\selectlanguage{french}%
 & \selectlanguage{english}%
0.046\selectlanguage{french}%
 & \selectlanguage{english}%
0.096\selectlanguage{french}%
 & \selectlanguage{english}%
0+\selectlanguage{french}%
 & \selectlanguage{english}%
0.237\selectlanguage{french}%
\tabularnewline[0.2cm]
\selectlanguage{english}%
\emph{Utilitarian optimum}\selectlanguage{french}%
 & \selectlanguage{english}%
-1.003\selectlanguage{french}%
 & \selectlanguage{english}%
-0.903\selectlanguage{french}%
 & \selectlanguage{english}%
-0.803\selectlanguage{french}%
 & \selectlanguage{english}%
0\selectlanguage{french}%
 & \selectlanguage{english}%
0.050\selectlanguage{french}%
 & \selectlanguage{english}%
0+\selectlanguage{french}%
 & \selectlanguage{english}%
0.108\selectlanguage{french}%
\tabularnewline[0.2cm]
\selectlanguage{english}%
\textbf{\emph{Demogrant median}}\selectlanguage{french}%
 & \selectlanguage{english}%
-2.032\selectlanguage{french}%
 & \selectlanguage{english}%
-0.977\selectlanguage{french}%
 & \selectlanguage{english}%
-0.877\selectlanguage{french}%
 & \selectlanguage{english}%
-0.019\selectlanguage{french}%
 & \selectlanguage{english}%
0.031\selectlanguage{french}%
 & \selectlanguage{english}%
0+\selectlanguage{french}%
 & \selectlanguage{english}%
0.043\selectlanguage{french}%
\tabularnewline[0.2cm]
\selectlanguage{english}%
\textbf{\emph{Median}}\selectlanguage{french}%
 & \selectlanguage{english}%
-2.031\selectlanguage{french}%
 & \selectlanguage{english}%
-0.991\selectlanguage{french}%
 & \selectlanguage{english}%
-0.891\selectlanguage{french}%
 & \selectlanguage{english}%
-0.081\selectlanguage{french}%
 & \selectlanguage{english}%
-0.031\selectlanguage{french}%
 & \selectlanguage{english}%
0 -\selectlanguage{french}%
 & \selectlanguage{english}%
-0.104\selectlanguage{french}%
\tabularnewline[0.2cm]
\selectlanguage{english}%
\emph{Personalized }\selectlanguage{french}%
 & \selectlanguage{english}%
-2.068\selectlanguage{french}%
 & \selectlanguage{english}%
-1.078\selectlanguage{french}%
 & \selectlanguage{english}%
-0.978\selectlanguage{french}%
 & \selectlanguage{english}%
-0.878\selectlanguage{french}%
 & \selectlanguage{english}%
-0.828\selectlanguage{french}%
 & \selectlanguage{english}%
-1+\selectlanguage{french}%
 & \selectlanguage{english}%
-0.403\selectlanguage{french}%
\tabularnewline[0.2cm]
\selectlanguage{english}%
\textbf{\emph{Actual}}\selectlanguage{french}%
 & \selectlanguage{english}%
-2.258\selectlanguage{french}%
 & \selectlanguage{english}%
-2.158\selectlanguage{french}%
 & \selectlanguage{english}%
-2.058\selectlanguage{french}%
 & \selectlanguage{english}%
-1.083\selectlanguage{french}%
 & \selectlanguage{english}%
-1.033\selectlanguage{french}%
 & \selectlanguage{english}%
-1 -\selectlanguage{french}%
 & \selectlanguage{english}%
-0.782\selectlanguage{french}%
\tabularnewline[0.2cm]
\selectlanguage{english}%
\emph{Egalitarian}\selectlanguage{french}%
 & \selectlanguage{english}%
-2.325\selectlanguage{french}%
 & \selectlanguage{english}%
-2.225\selectlanguage{french}%
 & \selectlanguage{english}%
-2.125\selectlanguage{french}%
 & \selectlanguage{english}%
-2.025\selectlanguage{french}%
 & \selectlanguage{english}%
-1.062\selectlanguage{french}%
 & \selectlanguage{english}%
-1 -\selectlanguage{french}%
 & \selectlanguage{english}%
-0.752\selectlanguage{french}%
\tabularnewline[0.2cm]
\hline 
\hline 
\selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
\tabularnewline[-0.5cm]
\end{tabular}}\selectlanguage{english}%
\end{table}

\subsection{Socio-demographic Determinants and Other Correlations}

The correlations found between a redistributive taste and socio-demographic
characteristics are in line with existing literature (see \citet{Fabre2016}
for a review). In particular, the negative relationship between income
and preferences for redistribution is retrieved: on average, an additional
thousand euros in monthly income is associated with a 4.5\% lower
rate of approval when controlling for political leaning (6.1\% without
this control). That being said, as one can see in Table \ref{tab:Socio-demographic-determinants-f},
preferences remain mainly idiosyncratic: $R^{2}=0.06$ at best, while
age, gender and education are not significantly correlated with the
approval of a redistribution.\footnote{Although the highest degree is negatively correlated with a significance
below the 10\% level when it is the only explanatory variable, this
significance vanishes when one includes the income in the regression.} Unsurprisingly, the variable that explains the highest part of the
variance is the political leaning. 

A variable \emph{disadvantaged} was constructed for each variant of
the reform proposed: it is a dummy taking the unitary value when the
individual disposable income of a respondent indicates that s-he will
be disadvantaged by the reform. Being disadvantaged by a reform is
associated with a significantly lower propensity to approve it, although
this does not capture all the effects associated with a higher income.
Whatever the reform, the rate of 55\% of approval separates the categories
of incomes advantaged and disadvantaged by the reform (see Appendix
\ref{sec:Summary-Satistics}). 

Interestingly, although wealth is negatively correlated with the approval
of a redistributive reform, future wealth (which is estimated on a
scale from 0 to 6 using the answers on current wealth and expected
bequest) is positively correlated. This might be interpreted as follows:
those who are about to inherit but are not wealthy yet may be more
aware of the extent of inequalities than those who do not expect to
become wealthy, hence they may be more keen to redistribution; while
wealthy people simply express their self-interest. 

\begin{table}[H]
\caption{\label{tab:Socio-demographic-determinants-f}Socio-demographic determinants
for approval of redistributive reforms (OLS)}

\selectlanguage{french}%
\noindent\begin{minipage}[t]{1\columnwidth}%
\selectlanguage{english}%
\centering  \begin{tabular}{@{\extracolsep{5pt}}lccc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]  \\[-1.8ex] & \multicolumn{3}{c}{Approval of the reform} \\  \\[-1.8ex] & (1) & (2) & (3)\\  \hline \\[-1.8ex]   Constant & 0.675$^{***}$ & 0.508$^{***}$ &  \\    & (0.041) & (0.036) &  \\    Variant: Average & $-$0.154$^{***}$ &  & 0.404$^{***}$ \\    & (0.040) &  & (0.100) \\    Variant: Median &  & 0.062$^{*}$ & 0.495$^{***}$ \\    &  & (0.036) & (0.108) \\    Variant: Distortionary & $-$0.086$^{**}$ & 0.025 & 0.463$^{***}$ \\    & (0.041) & (0.037) & (0.105) \\    Variant: Demogrant & $-$0.180$^{***}$ & 0.025 & 0.349$^{***}$ \\    & (0.039) & (0.035) & (0.104) \\    Income (k¤ per month)\footnote{ The variable used is the individual disposable income capped at 4500¤/month. This trimming concerns the top 5\% of the distribution and helps getting relevant coefficients for incomes, given that the effect vanishes above 4500¤/month. } & $-$0.045$^{***}$ &  & 0.091 \\    & (0.013) &  & (0.075) \\    Income$^2$ &  &  & $-$0.020 \\    &  &  & (0.012) \\    Left - Right leaning (-2 to +2) & $-$0.078$^{***}$ &  & $-$0.087$^{***}$ \\    & (0.012) &  & (0.014) \\    Left - Right $^2$ &  &  & 0.001 \\    &  &  & (0.013) \\    Disadvantaged &  & $-$0.125$^{***}$ & $-$0.072 \\    &  & (0.031) & (0.053) \\    Misunderstanding of graphics &  & $-$0.090$^{***}$ & 0.022 \\    &  & (0.024) & (0.032) \\    Wealth (0 to 6) &  & $-$0.019$^{*}$ & $-$0.016 \\    &  & (0.010) & (0.014) \\    Future wealth (0 to 6) &  & 0.023$^{***}$ & 0.017$^{*}$ \\    &  & (0.007) & (0.009) \\    Age (1 to 8) &  &  & $-$0.005 \\    &  &  & (0.010) \\    Gender: female &  &  & $-$0.019 \\    &  &  & (0.032) \\    Highest degree (0 to 6) &  &  & 0.003 \\    &  &  & (0.009) \\   \hline \\[-1.8ex]  Observations & 1,146 & 1,658 & 1,048 \\  R$^{2}$ & 0.064 & 0.032 & 0.501 \\  \hline  \hline \\[-1.8ex]  \textit{Note:}  & \multicolumn{3}{r}{$^{*}$p$<$0.1; $^{**}$p$<$0.05; $^{***}$p$<$0.01} \\  \end{tabular} \selectlanguage{french}%
%
\end{minipage}\selectlanguage{english}%
\end{table}

Besides, it is worth noticing that the 56\% who claimed to have understood
the graphical questions without trouble were more prone to approve
the reform by 9 percentage points (with a high significance). However,
this correlation is explained by the political leaning: while people
who situate themselves at the extreme-left are more likely to approve
a redistribution than right-wing people by more than 20 percentage
points, they are also half as likely to misunderstand the graphics
as compared to every other political orientations (see Appendix \ref{sec:Summary-Satistics}).

Finally, as the approval of the \emph{personalized }reform was inquired,
the comparison of the respondent's (and when appropriate, of her-his
household's) income before and after the reform was randomly displayed
(or not). There was no effect of showing the personal impact of the
reform on one's answer (see  Appendix \ref{sec:Regressions}). This
is why randomizing its display was no longer necessary in the second
survey, where it was always shown. The absence of effect is best explained
by the fact that, with the current and proposed distribution within
sight, the respondents could already infer the impact of the reform
on their income. Indeed, this is in agreement with the finding that
the only significant effect of this treatment lies in its interaction
with a lack of comprehension of the questions containing graphics:
while those who struggled understanding these questions were less
likely by 16 percentage points to take a side on the approval of a
reform when its impact on their own income was displayed, this higher
indecision reached 26 percentage points when it was not shown (see
Appendix \ref{sec:Regressions}). 

\section{Discussion}

\subsection{A Surprising Link With the Theory of Optimal Taxation\label{subsec:A-Surprising-Link}}

This survey has produced a surprising result: as one can see on Figure
\ref{fig:Comparison-between-the}, the \emph{average} proposed redistribution
has a shape very similar to the one derived from utilitarian optimization.\footnote{The only difference between the two lies in the \emph{average} distribution
being almost always above the utilitarian one, because the former
does not take into account behavioral response.} The reason for this sameness is unclear, and it may well be a coincidence.
Still, an interesting finding can be deduced using this resemblance:
the rate of approval of the optimal utilitarian reform. Assuming that
it would equal the one of the \emph{average }reform, it would not
be significantly higher than 50\% even excluding missing answers,
making this redistribution more controversial than the \emph{median
}one. Indeed, the derivation of the optimal utilitarian reform lacks
of political acceptability considerations (in the same manner of averaging
of people's preferences), whereas the construction of the \emph{median
}reform embeds some insight from political theory\textemdash namely,
the median voter theorem\textemdash through its use of median preferred
parameters.

Yet, the optimal utilitarian distribution obtains a good evaluation,
surpassing all proposed reforms (see subsection \ref{subsec:Evaluations-of-Distributions:}).
Interestingly, this shows that in a situation closer to the veil of
ignorance\textemdash because the impact of the reform on their income
was not displayed, people tend to value what is socially optimal (according
to the theory of optimal taxation), while they tend to favor their
self-interest when the latter is made clear. Indeed, excluding missing
answers, 58\% of the respondents disadvantaged by the reform (proxied
by those who have an individual disposable income greater than 1600¤/month)
disapprove the \emph{average} redistribution although the proportions
of each grade in their evaluation of the optimal utilitarian reform
are the same ($\pm1\%$) as in the whole sample (their average grade
is even higher by 0.06); and this effect is exacerbated for smaller
subsets (resulting in 65\% of disapproval and an average grade higher
by 0.14 for those earning more than 3000¤/month). 

\begin{figure}[H]
\caption{\label{fig:Comparison-between-the}Comparison between the \emph{average}
and the optimal utilitarian reforms.}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/reform average utilitarian 2\string".eps}}}\selectlanguage{english}%
\end{figure}

\subsection{Robustness Checks}

There are several ways to prove the robustness of the results exposed
in the last section, both internal (by processing the data differently)
and external (by comparing these findings with previous literature).
Four robustness checks will be tested in turn: including screened
out respondents in the sample, studying the non weighted answers,
controlling for the quality of responses, and comparing the findings
with an earlier estimation of French preferences for income redistribution.
For the sake of concision, I will focus on the approval of the proposed
redistribution\textemdash which is emphasized in this article, but
robustness has also been checked on other variables.

\subsubsection{Using a Broader Sample}

As explained in subsection \ref{subsec:Data-Collection-(and}, a broader
sample is at our disposal. As shown in Table \ref{tab:Answers-to-any},
there is a significantly lower rate of approval in this \emph{augmented
}sample than in our sample of reference. However, this discrepancy
is entirely explained by a higher rate of non-answers: the difference
vanishes when those are excluded. Furthermore, this lower proportion
of non-answers is not surprising because the \emph{restricted }sample
excludes respondents who are likely to choose not to answer: those
who responded too quickly and those who did not complete the survey.
On the contrary, it indicates that the restriction of the \emph{augmented}
sample has been successful.

\begin{table}[H]
\caption{\label{tab:Answers-to-any}Approval of proposed redistributions in
the \emph{augmented }sample}

\makebox[\textwidth][c]{ 
\begin{tabular}{@{\extracolsep{5pt}}lcccc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]  \\[-1.8ex] & Headcount & \multicolumn{2}{c}{Approval (of any reform)} & Non-answer \\  \\[-1.8ex] & (additional) & (2) & (3)\\  \hline \\[-1.8ex]   In \textit{augmented} (constant) & 1,007 & 0.353$^{***}$ & 0.563$^{***}$ & 0.373$^{***}$ \\   & & (0.015) & (0.019) & (0.013) \\    In \textit{restricted} & 1,994 & 0.100$^{***}$ & 0.021 & $-$0.150$^{***}$ \\    & & (0.019) & (0.023) & (0.017) \\   \hline \\[-1.8ex]  Includes non-answers & Yes & Yes & No & Yes \\  \hline \\[-1.8ex]  Observations & 3,001 & 3,001 & 2,231 & 3,001 \\  \hline  \hline \\[-1.8ex]  \textit{Note:}  & \multicolumn{4}{r}{$^{*}$p$<$0.1; $^{**}$p$<$0.05; $^{***}$p$<$0.01} \\  \end{tabular} 
}
\end{table}

\subsubsection{Non Weighted Results}

Despite the adoption of the method of quotas to constitute the main
sample (which did not take highest degrees into account), higher educated
people are largely over-represented in the non-weighted sample: only
24\% do not have the \emph{Baccalauréat}\footnote{The \emph{Baccalauréat }is the examen concluding high-school. Source:
\href{https://www.insee.fr/fr/statistiques/1906701}{insee.fr/fr/statistiques/1906701}}\emph{ }in the sample as opposed to 55\% in the general population.
Hence, the weights bring the distribution of degrees much closer to
the reality.\footnote{The weighted share of respondents without \emph{Baccalauréat }is thus
42\%. This figure is still a bit away from the reality, because weights
have been trimmed in order to be in $\left[\nicefrac{1}{4};4\right]$:
indeed, authorizing too large weights would have hampered the precisions
of estimates.} However, had the results be computed without weights, they could
have changed of one or two percents, but they would not have been
significantly different, as is shown in Table \ref{tab:Effect-of-weighting}.
This is not surprising, as we have seen that approving a reform was
not correlated with the highest degree of the respondent. Finally,
only omitted or unobservable variables could have caused a sample
bias, the main socio-demographic characteristics having been addressed.

\begin{table}[H]
\caption{\label{tab:Effect-of-weighting}Effect of weighting on approval of
a reform}

\makebox[\textwidth][c]{ 
\begin{tabular}{@{\extracolsep{5pt}}lccccc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]   & \multicolumn{5}{c}{Approval of a reform} \\  \cline{2-6}  \\[-1.8ex] & Any reform & Average & Demogrant median & Median & Distortionary median  \\  \\[-1.8ex] & (1) & (2) & (3) & (4) & (5)\\  \hline \\[-1.8ex]   Effect of weighting & $-$0.004 & 0.003 & 0.010 & $-$0.017 & $-$0.012 \\    & (0.007) & (0.015) & (0.015) & (0.015) & (0.015) \\    Constant & 0.452$^{***}$ & 0.418$^{***}$ & 0.383$^{***}$ & 0.539$^{***}$ & 0.467$^{***}$ \\    & (0.011) & (0.022) & (0.022) & (0.022) & (0.023) \\   \hline \\[-1.8ex]  Observations & 1,994 & 509 & 488 & 505 & 492 \\  \hline  \hline \\[-1.8ex]  \textit{Note:}  & \multicolumn{5}{r}{$^{*}$p$<$0.1; $^{**}$p$<$0.05; $^{***}$p$<$0.01} \\  \end{tabular} 
}
\end{table}

\subsubsection{Quality of Responses}

Some variables enable to measure the quality of the responses, which
can be affected voluntarily or not. Indeed, the time of response to
several questions, including those on the approval of the \emph{average
}and \emph{demogrant }reforms, have been recorded: they reveal the
extent of voluntary effort to understand the questions.\textbf{}\footnote{The global response time and the number of clicks for different questions
have also been recorded; they appear to be uncorrelated with our variables
of interest.}\textbf{ }At the end of the surveys, the respondents were asked whether
they had understood the questions with graphics: this shows the level
of (involuntary) understanding. Both variables exhibit the same outcomes,
as one can see in Table \ref{tab:Correlation-between-approval} :
the quality of a response is positively correlated with non-answering,
but it is not significantly correlated with the rate of approval of
a reform, after excluding \emph{PNR} (People Not Responding). This
absence of correlations achieves the validation of internal robustness:
indeed, it demonstrates that the findings are not driven by a potential
lack of seriousness of the responses or a low understanding of the
questions.

\begin{table}[H]
\caption{\label{tab:Correlation-between-approval}Correlation between approval
of a reform and quality of a response (OLS)}

\makebox[\textwidth][c]{ 
\begin{tabular}{@{\extracolsep{5pt}}lcccccc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]  \\[-1.8ex] & \multicolumn{4}{c}{Approval of any reform} & \multicolumn{2}{c}{Non-answer} \\  \\[-1.8ex] & (1) & (2) & (3) & (4) & (5) & (6)\\  \hline \\[-1.8ex]   log of time response (for approval) & 0.083$^{***}$ & 0.027 &  &  & $-$0.128$^{***}$ &  \\    & (0.022) & (0.027) &  &  & (0.017) &  \\    Misunderstanding graphical questions &  &  & $-$0.088$^{***}$ & 0.008 &  & 0.165$^{***}$ \\    &  &  & (0.022) & (0.026) &  & (0.018) \\    Constant & 0.141$^{**}$ & 0.410$^{***}$ & 0.490$^{***}$ & 0.566$^{***}$ & 0.594$^{***}$ & 0.135$^{***}$ \\    & (0.071) & (0.088) & (0.015) & (0.016) & (0.056) & (0.012) \\   \hline \\[-1.8ex]  Includes non-answers & Yes & No & Yes & No & Yes & Yes \\  \hline \\[-1.8ex]  Observations & 995 & 803 & 1,994 & 1,586 & 995 & 1,994 \\  \hline  \hline \\[-1.8ex]  \textit{Note:}  & \multicolumn{6}{r}{$^{*}$p$<$0.1; $^{**}$p$<$0.05; $^{***}$p$<$0.01} \\  \end{tabular}  
}
\end{table}

\subsubsection{Comparison with an Earlier and Indirect Estimation}

In an earlier attempt whose result is shown in Appendix \ref{tab:Characteristics-of-Distributions},
\citet{Fabre2016} derived the median favored redistribution for 43
countries from ISSP surveys, using responses of perceived and desired
incomes of different professions (unskilled worker, doctor, chairman
of a large corporation, etc.). The variability of answers concerning
perceived current incomes provided the median desired income for a
broad range of income, from which was inferred the median desired
redistribution. As the questions used in this earlier estimate were
not originally designed for this purpose, one could not interpret
with certainty their implied desired redistribution in terms of a
preference for a tax reform.\footnote{The desired redistribution estimated in this earlier work has to be
understood in a broad sense, including for example a renegotiation
of the wage scale. See \citet{Fabre2016} for a discussion on the
hypotheses and the method.} Yet, the earlier results proved to be very similar to those revealed
in this paper. The desired Gini coefficient implied by the earlier
estimation of the median reform favored by French people is 0.30,
almost in the range of 0.24-0.29 corresponding to Gini of proposed
redistributions. Similarly, the earlier desired transfer from rich
to poor of 11\% of GNI falls in the range of the new estimations:
9-12\%, and so do other statistics reported in Appendix \ref{sec:Summary-Satistics}:
income shares, the Theil index, the demogrant, the proportions of
advantaged and disadvantaged by the reform... This convergence of
results with two different methods both reinforces the credibility
of the proposed redistributions as favored reforms and validates the
estimations of desired redistributions by country of the earlier approach.

\subsection{Limits of This Work and Future Research\label{subsec:Shortcomings-of-This}}

As a primary attempt to derive the parametrization of a redistributive
reform favored by citizens from a survey, this work can logically
be challenged in some aspects. The following exposition of these weaknesses
will help mapping the future work needed in order to refine the reform
derived from citizens' preferences.

\subsubsection{Choice of the Individual Disposable Income}

The main drawback of the current method is to rely on the individual
disposable income to present the income distribution. Indeed, this
variable exhibits a significant number of very low incomes, including
3.5 percents\textbf{ }of people with no income, which does not adequately
corresponds to the extent of extreme poverty. The reason for this
is twofold: firstly, the dataset provided by INSEE is not fully precise
on the ends of the distribution\footnote{In particular, the ERFS fails to take account of the transfers from
parents to their children living independently.}; secondly, income is arguably better measured at the household level,
because intra-household transfers often make zero-earners well-off.
In effect, only 7\% of people with no income had an household equivalised
income below 10,000¤/year: these zero-earners are most often young
adults or spouses in a typical household. 

The effect of this bias concerning the extent of poverty in the graphics
is unclear. On the one hand, seeing that many people live in misery
may induce a higher rate of approval for a redistribution. On the
other hand, the smaller magnitude of the redistribution that would
have appeared had the reforms be presented in terms of household (equivalised)
income might have also increased its acceptation. In addition, more
households are disadvantaged by the reforms in terms of equivalised
disposable income than suggested from the individual perspective (see
Appendix \ref{sec:Distributions-Characteristics}). The approval rate
might have benefited from the fewer proportion of households disadvantaged,
had the proposed reforms respected the median desired proportion of
\emph{households }to disadvantage (as opposed to the proportion of
individuals).\footnote{The respondents are unlikely to have misjudged the impact of the reform
on their household's income because the comparison before and after
the reform was displayed.} Finally, some respondents may have chosen not to answer because of
doubts concerning the distribution displayed.

The natural way to correct this shortcoming would be to use the (household)
equivalised disposable income as the variable of reference, and to
take account of the family structure to determine the proposed income
tax rates.

Besides, this choice of variable has also influenced the features
of the proposed reforms. As these redistributions take place at the
individual level, they mitigate the favorable treatment of certain
categories of persons in the current tax system due to the marital
and familial quotients. In particular, singles are favored by the
proposed reforms relatively to couples with heterogeneous income,
and this also applies to a lesser extent to low-income families with
children as compared to high-income ones. Unsurprisingly, one finds
a significant higher rate of approval by 7\% for singles (with or
without dependent child). However, this effect vanishes when one controls
for individual disposable income (see Appendix \ref{sec:Regressions})\textemdash indeed,
singles tend to earn less than couples. This absence of peculiarity
for singles indicates that the results are not driven by the structures
of households nor by their unequal interests in the proposed reforms:
in effect, this difference among types of households remains small
relatively to the whole redistribution. That being said, disregarding
households in the proposed reforms goes against preferences of French
people in favor of a marital quotient documented by other questions
in this survey (see Appendix \ref{sec:Results}), and this questions
the assumption of a simplified and individualised favored redistribution.

\subsubsection{Simplification of the Redistribution Sought}

The individualization of the income tax in the spirit of the redistributions
proposed do not correspond to the French tax system. This feature
is one among several others that have been simplified in order to
present a clear proposal of reform to the respondents. Hereafter are
other improvements that could be made to precise a reform of the income
tax:
\begin{itemize}
\item include the amount of the minimum wage as a fourth parameter of the
reform;
\item take into account the number of hours worked;\footnote{Indeed, it may seem unfair that someone working full-time for 2000¤/month
would not be advantaged by the (current) proposed reforms, contrarily
to someone working half-time for the same hourly wage. \citet{SaezEtAl2016}
provide survey evidence on this question.}
\item distinguish different situations from the benchmark redistribution,
such as: capital income, imputed rents, unemployment, retirement,
students, or even gender or profession;\footnote{For the latter, a distinction could be made between civil servants
and people employed by the private sector, or for jobs in a particular
condition, such as farmers or entrepreneurs, etc.}
\item choose the budgetary cost of the reform.\footnote{Each respondent could propose a new budget by lowering or increasing
each public spending (and possibly also revenues), and the answers
could then be averaged to determine the budget of the reform.}
\end{itemize}

\subsubsection{Framing}

In attempts to gather preferences on such a complex issue as the tax
system, the surveyer tends to\emph{ }frame the questions from a peculiar
perspective. Furthermore, the angle through which a proposal is presented
is likely to bias the responses. The practical solution to overcome
this issue is to multiply the surveys\textemdash preferably with different
surveyers\textemdash as well as the number of persons who review them,
so as to average out their different perspectives. Admittedly, the
framing of my survey may have been biased towards redistribution,
because it presented the reform \emph{as} a redistribution, and masked
the level of tax that people already pay. On the other hand, \citet{Weinzierl2014}
might have been biased towards equal sacrifice, because along the
levels of after-tax income for different schedules, their graphics
presented current pre-tax incomes\textemdash rather than current after-tax
distribution\textemdash (see Figure \ref{fig:Example-of-question}).
This reflects the underlying perspective that pre-tax incomes are
deserved, or at least that their distribution is more relevant to
the determination of the post-reform after-tax distribution than the
current after-tax distribution. In this vein, pooling the top 1\%
in the same rectangle veiled the extent of inequalities at the top
while presenting the level of tax for each group emphasized the \emph{sacrifice}
made by tax-payers.

\begin{figure}[H]
\caption{Example of question from \citet{Weinzierl2014}. Transparent boxes
represent pre-tax income while green rectangles stand for after-tax
income.\label{fig:Example-of-question}}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.8\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/weinzierl\string".eps}}}\selectlanguage{english}%
\end{figure}

While many surveys can be questioned relatively to the perspective
which they favor, letting those attempts to harvest people's preferences
perfectible and incomplete, there is a way to reconcile Weinzierl's
tropism with mine. One can present in the same graphic three distributions
instead of two: both current pre-tax and after-tax distributions,
alongside the alternative after-tax distribution (see Figure \ref{fig:Simulation-of-behavioral}).
Doing this would insure that those who feel that more redistribution
is needed could recognize which proposal implies that feature (correcting
for Weinzierl's bias) while showing the amount of existing transfers
would help people realizing the extent of contribution required for
a reform (correcting for my bias).

\subsection{A New Democratic Process to Choose Income Tax Rates}

While French citizens are dissatisfied with the current distribution
of incomes resulting from the tax system, this article provides a
proof of concept for a new democratic process to choose income tax
rates, whose outcome would likely better fit citizens' preferences.
This process would contain several steps:
\begin{enumerate}
\item The parliament would command a survey aimed at determining the features
and parameters of a reform favored by the citizens: the method presented
in this article improved by the remarks of the subsection \ref{subsec:Shortcomings-of-This}
would constitute a good candidate.
\item The national statistics bureau would administer the survey and expose
the results publicly.
\item The reform that best suits respondents' preferences would be put to
a referendum.
\item In case of success of the first steps, the tax reform would be \emph{progressively}
implemented. 
\end{enumerate}
%
~

A redistribution likely to transfer one tenth of income from the richest
to the poorest would entail a substantial restructuring of the economy
through the re-allocation of consumption across sectors (at least,
for the sectors with a high homogeneity in their customers' incomes).
In order to smooth the re-organization of the economy, as well as
to let enough time for rich indebted people to deleverage before their
incomes decrease, any large reform should be staggered over a dozen
of years or so. Not only would this progressive implementation of
a reform be needed to smooth the transition, but it would also help
improving previsions concerning behavioral responses and the associated
definition of tax rates.

Importantly, the process described above should not be carried out
only once; rather, it ought to be repeated every one, two or three
years, in order to readjust dynamically the reform according to changes
in preferences. Indeed, preferences are likely to change as society
would learn about its own behavioral response to the redistribution.

Finally, French people seem favorable to such a process. Indeed, the
respondents of the first survey were asked whether the tax rates should
be determined from a survey and then put on a referendum. Interestingly,
44\% approved the idea, 29\% thought that the opinion of everyone
should be taken into account, but through another method, while only
14\% expressed their satisfaction with the current system (where tax
rates are voted at the Parliament) and 13\% did not answer.

\subsection{Computations of the Income Tax Rates}

The proposed redistributions were until now expressed in terms of
shift in the after-tax distribution. However, were such redistributions
set in place, the social planner would have to infer the new income
tax rates needed to attain them, taking due account of behavioral
responses. I will present hereafter two ways of doing this: an empirical
one that can be used during the process of the reform, and a theoretical
one, which allows to define the income tax rates \emph{ex ante}.

\subsubsection{Empirical and Dynamical Computation}

The first approach is agnostic about the determinants of the behavioral
response $\rho=\Delta z$ that a one-period redistribution triggers
on the pre-tax distribution $z$. Following a change in tax rates
$\Delta T_{t}^{z}=T_{t}^{z}-T_{t-1}^{z}$, the after-tax distribution
also varies, by $\Delta c_{t}=c_{t}-c_{t-1}$. Hence, the subsequent
aggregate change in pre-tax distribution is borne by a change in tax
revenues and in aggregate disposable income: $\int\rho=\int\Delta T+\int\Delta c$.
If the response $\rho$ is well anticipated, the reform can be made
budget neutral (i.e. $\int\Delta T=0$), so that the entire loss due
to the response is absorbed by an adjustment in the objective of redistribution
$\Delta c$. In theory, the social planner could use the first year
of a progressive implementation of the reform to learn $\rho$, and
then adjust its objective of redistribution for the following years
(e.g. to respect budget neutrality). However, the response has no
reason to be linear, and will inevitably change as the reform is adjusted.
Thus, the learning of the response is a permanent process, which can
be described formally for each quantile \emph{q} of the distribution
as an expression of the intended (or \emph{objective of}) redistribution
$\mathbb{\mathbf{\mathbf{E}}}\left[\Delta c\left(q\right)\right]$,
the expected change in tax distribution $\mathbb{\mathbf{\mathbf{E}}}\left[\Delta T\left(q\right)\right]$
and the expected response $\mathbb{\mathbf{\mathbf{E}}}\left[\Delta\rho\left(q\right)\right]$:

\begin{eqnarray*}
z_{t}=z_{t-1}+\rho_{t} & = & c_{t}+T_{t}\\
 & = & c_{t-1}+\mathbb{\mathbf{\mathbf{E}}}_{t-1}\left[\Delta c_{t}\right]+T_{t-1}+\mathbb{\mathbf{\mathbf{E}}}_{t-1}\left[\Delta T_{t}\right]+\rho_{t}-\mathbb{\mathbf{\mathbf{E}}}_{t-1}\left[\rho_{t}\right]
\end{eqnarray*}

Finally, the next-period tax schedule is given by the expectation
of response and the intended redistribution:

\begin{eqnarray*}
T_{t+1}\left(q\right) & = & T_{t}\left(q\right)+\mathbb{\mathbf{\mathbf{E}}}_{t}\left[\Delta T_{t+1}\left(q\right)\right]\\
 & = & T_{t}\left(q\right)+\mathbb{\mathbf{\mathbf{E}}}_{t}\left[\rho_{t+1}\left(q\right)\right]-\mathbb{\mathbf{\mathbf{E}}}_{t}\left[\Delta c_{t+1}\left(q\right)\right]\\
T_{t+1}^{z}\left(z\right) & = & T_{t+1}\left(\mathbb{\mathbf{\mathbf{E}}}_{t}\left[q_{t+1}\left(z\right)\right]\right)
\end{eqnarray*}

In the case of a linear implementation of the reform, i.e. $\mathbb{\mathbf{\mathbf{E}}}_{t-1}\left[\Delta c_{t}\right]=\Delta c_{t-1}$
for $t\in\left\llbracket 1;t_{max}\right\rrbracket $, the response
can simply be expected to be constant: $\mathbb{\mathbf{\mathbf{E}}}_{t-1}\left[\rho_{t}\right]=\rho_{t-1}$,
and the tax schedule is easy to compute.

\subsubsection{Theoretical Approach}

One can also compute the income tax rates by using a more theoretical
model. Such an approach is indeed necessary, at least to estimate
the response during the first year of the reform. Furthermore, this
modeling can be coupled to the algorithm which determines the intended
after-tax distribution, so that behavioral responses are taken into
account in the project of reform.\footnote{The \emph{distortionary median }reform was already computed by an
algorithm that takes as a parameter the expected loss due to the behavioral
response. However, the value chosen for this parameter (5\%) was estimated
separately, and the process of coupling this algorithm to a model
giving the behavioral response (in order to reach convergence in the
expected behavioral response) is yet to be done.} Modeling the behavioral response requires to make assumptions on
their determinants. Following a common practice in the applied literature
(e.g. \citet{Saez2002} or \citet{JacquetEtAl2013}), I assume away
income effects, so that the pre-tax distribution responds solely to
changes in marginal tax rates.\footnote{Other consequences of the reform on the activity would possibly occur,
such as an increase in aggregate demand following the redistribution
due to a higher marginal propensity to consume of the poor; but they
are not modeled.} This behavioral effect is captured by the elasticity of earnings
\emph{z }with respect to the net-of-tax rate $1-T\text{\ensuremath{^{\prime}}}$
: $\zeta_{z}=\frac{1-T\text{\ensuremath{^{\prime}}}}{z}\frac{\partial z}{\partial(1-T\text{\ensuremath{^{\prime}}})}$.\footnote{Let us precise that $T^{\prime}=\frac{\partial T}{\partial z}$.}

Keeping the notations of the previous subsection, one has:

\begin{eqnarray*}
c\left(q,t\right) & = & z\left(q,t\right)-T\left(z\left(q,t\right),t\right)\\
\frac{dc}{dt}\left(q,t\right) & = & \frac{\partial z}{\partial t}\left(q,t\right)-\frac{\partial T}{\partial t}\left(z\left(q,t\right),t\right)-\frac{\partial T}{\partial z}\left(z\left(q,t\right),t\right)\cdot\frac{\partial z}{\partial t}\left(q,t\right)
\end{eqnarray*}

Forgetting indices for more clarity:

\begin{eqnarray}
\frac{dc}{dt} & = & \frac{\partial z}{\partial t}\cdot\left(1-\frac{\partial T}{\partial z}\right)-\frac{\partial T}{\partial t}\label{eq:1}
\end{eqnarray}

Using the definition of $\zeta_{z}$, one obtains a partial differential
equation for \emph{T}:\footnote{I recall that other functions than \emph{T} are known, as well as
the initial condition $T\left(\cdot,0\right)$.}

\begin{equation}
\frac{dc}{dt}=-\zeta_{z}\cdot z\cdot\frac{\partial^{2}T}{\partial t\partial z}-\frac{\partial T}{\partial t}\label{eq:3}
\end{equation}

The discrete version of this model (which has been originally derived
in \citet{Fabre2016}) gives the semi-discrete counterpart of equation
\ref{eq:3}:

\begin{equation}
\Delta c=-z\cdot\zeta_{z}\cdot\Delta T^{\prime}-\Delta T\label{0.5-1}
\end{equation}

Equation \ref{0.5-1} can in turn be solved, as it is an Euler-Cauchy
equation of the variable $\Delta T$; it is tractable when $\zeta_{z}$
is approximated by a linear, step or power function\emph{. }In practice,
the continuous version can be approximated by $N$ iterations of the
discrete version: one has only to define a path $c_{k}\left(q\right)$
with $k\in\left\llbracket 1;N\right\rrbracket $, choosing \emph{N}
large enough so that, for all \emph{k}, $\Delta c_{k}=c_{k}-c_{k-1}$
remains small enough (so that $\Delta z_{k}$ can be considered infinitesimal).
In practice however, the simulations produced almost the same results
for \emph{N}=1 and \emph{N}=10. Finally, the Figure \ref{fig:Simulation-of-behavioral}
presents the results obtained by applying this model to the \emph{median
}reform with a constant elasticity of labor supply, set to a credible
value of 0.3 (see e.g. \citet{GruberEtAl2002,EversEtAl2008,Chetty2012}
for empirical estimates).\textbf{ }As expected, pre-tax incomes decrease
through the effect of the reform, by an average of 7.4\% (while magnitudes
of 5.3-7.6\% are observed for the other proposed reforms). 

\begin{figure}[H]
\caption{\label{fig:Simulation-of-behavioral}Simulation of behavioral responses}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{\includegraphics[width=0.8\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/pre tax variation median 03\string".eps}}}\selectlanguage{english}%
\end{figure}

\section{Conclusion}

While the theory of optimal taxation proves very successful in determining
socially optimal tax schedules,\textbf{ }it usually relies on \emph{ad
hoc} assumptions about the utility function of agents and on external
principles of social justice. Auspiciously a recent work by \citet{SaezEtAl2016}
demonstrates that these premises did not need to remain arbitrary,
by means of introducing generalized social welfare weights which can
be calibrated through surveys. However, a last practical frailty of
this theory needed to be examined: the political acceptability of
its recommendations. Following Saez and Stantcheva in their exploration
of new methods to define a desirable tax schedule, I departed from
the theory and elicited some redistributions that French citizens
would approve, by asking them directly for their parameters. As one
of the reform proposed to the respondents appeared to resemble closely
that resulting from utilitarian optimization, it could be shown that,
even if it was positively appreciated in a context mimicking the veil
of ignorance, this reform failed to obtain a significant majority
approval when the impact on the respondent's income was made clear.
This provided evidence that, for reasons of political acceptability,
tax policies ought to deviate from the theoretical optimum to take
into account the proportion of people disadvantaged by a reform. Combining
this parameter with three others\textemdash the proportion of people
to advantage, the demogrant and the extent of the redistribution\textemdash allowed
to characterize a broad class of reforms with a minimal amount of
information. Eventually, the redistribution defined by the median
desired value for these parameters obtained the support of two-thirds
of expressed answers. This result opened the way to the proposal of
a participatory process that would modify progressively the French
income tax system, associating recurrent surveys to shape iterative
reforms and referendums to validate them. Indeed, the respondents
broadly supported a more democratic procedure to choose the income
tax rates.\footnote{Only 14\% were satisfied with the current (parliamentary) process.}
As a primary attempt to quantify a favored redistribution from survey
answers, this work leaves room for substantial improvements, in particular
to refine the algorithm used to delineate the curves and to include
more dimensions of choice.

\pagebreak{}

\bibliographystyle{plainnat}
\bibliography{\string"/home/adrien/Google Drive/Economie/Articles/Preferences for distribution\string",\string"/home/adrien/Google Drive/Economie/Articles/Optimal Taxation\string",\string"/home/adrien/Google Drive/Economie/Articles/autre\string"}


\appendix
\newpage{}

\part*{Appendix}

\section{Summary Statistics\label{sec:Summary-Satistics}}

\begin{table}[H]
\caption{Summary statistics of reform parameters for different samples}

\makebox[\textwidth][c]{  \setlength\extrarowheight{3pt}
\begin{tabular}{rrrrrrrrrrrr}   \hline \hline & \multicolumn{3}{c}{first sample \textit{restricted} } & \multicolumn{3}{c}{ first sample \textit{augmented} } & \multicolumn{4}{c}{final sample (both surveys)} \\   & median & PNR & obs. & median & PNR & obs. & median & 95\% C.I. & PNR & obs. \\    \hline Advantage & 50 & 175 & 449 & 50.00 & 315 & 645 & 50.00 & [50; 50] & 265 & 621 \\    Disadvantage & 10.0 & 206 & 500 & 10 & 357 & 699 & 10 & [9; 10] & 589 & 1325 \\    Demogrant &  750.00 & 77 & 253 &  800 & 151 & 369 &  738.4 & [700; 800] & 167 & 456 \\    Maximal income & 50k & 44 & 243 & 100k & 92 & 354 & 250k & [100k; Inf] & 130 & 600 \\     \hline \hline \end{tabular} 

}
\end{table}
\bigskip{}
\begin{table}[H]
\caption{\label{fig:Approval-of-a}Approval of a reform by (individual disposable)
income level (in \% and ¤/month.), excluding people not responding
(PNR).\protect \\
The cells in bold correspond to those supposed to be disadvantaged
by the reform.}

\makebox[\textwidth][c]{
\centering \setlength\extrarowheight{3pt} \begin{tabular}{rrrrrr|r}  \hline \hline  & $\leq$ 1200 & 1201 to 1600 & 1601 to 2200 & 2201 to 3000 & $>$ 3000 & Any income \\    \hline PNR (any reform) & 31 & 25 & 26 & 15 & 13 & 23 \\    Approval (any reform) & 75 & 64 & 58 & 56 & 38 & 58 \\\hline     median & 79 & 71 & 61 & 73 & \textbf{53} & 67 \\    distortionary median & 76 & 66 & 67 & 63 & \textbf{35} & 62 \\    demogrant median & 69 & 63 & 58 & \textbf{49} & \textbf{26} & 52 \\    average & 75 & 56 & \textbf{48} & \textbf{42} & \textbf{35} & 51 \\\hline     Number of obs. (average) & 66 & 66 & 86 & 68 & 92 & 396 \\    Number of weighted obs. (average) & 71 & 69 & 81 & 64 & 79 & 384 \\     \hline \hline \end{tabular}
}
\end{table}

\bigskip{}
\begin{table}[H]
\caption{\label{tab:political-leaning}Approval and understanding by political
leaning}

\makebox[\textwidth][c]{
\begin{tabular}{@{\extracolsep{5pt}}lccccccc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]  \\[-1.8ex] & Proportion & Misunderstanding & \multicolumn{3}{c}{Approval among any reform} & \multicolumn{2}{c}{Approval of \textit{median} reform} \\  \\[-1.8ex] & (1) & (2) & (3) & (4) & (5) & (6) & (7)\\  \hline \\[-1.8ex]   Extreme-left & 0.019 & 0.180 & 0.583 & 0.788 & 0.647 & 0.844 & 0.589 \\    Left & 0.218 & 0.384 & 0.565 & 0.691 & 0.686 & 0.769 & 0.742 \\    Center & 0.096 & 0.387 & 0.405 & 0.496 & 0.506 & 0.643 & 0.657 \\    Right & 0.156 & 0.424 & 0.371 & 0.439 & 0.484 & 0.627 & 0.616 \\    Extreme-right & 0.098 & 0.368 & 0.355 & 0.417 & 0.401 & 0.519 & 0.465 \\    Indeterminate & 0.413 & 0.509 & 0.443 & 0.634 & 0.610 & 0.648 & 0.636 \\   \hline \\[-1.8ex]  Includes non-answers & Yes & Yes & Yes & No & No & No & No \\  \textit{Augmented} sample & No & No & No & No & Yes & No & Yes \\  \hline \\[-1.8ex]  Average &  & 0.442 & 0.448 & 0.582 & 0.578 & 0.666 & 0.638 \\[0.5ex]  Observations & 2,004 & 2,004 & 1,994 & 1,586 & 2,231 & 399 & 566 \\  \hline  \hline \\[-1.8ex]  \end{tabular} 
}
\end{table}

\section{Survey Screen Shots\label{sec:Survey-Screen-Shots}}

The questionnaires are available on-line:

first sample: \href{http://adrien-fabre.com/sondage/Fiscalite\%20des\%20francais.html}{adrien-fabre.com/sondage/Politique\%{}20des\%{}20francais.html}

second sample: \href{http://adrien-fabre.com/sondage/Politique\%20des\%20francais.html}{adrien-fabre.com/sondage/Politique\%{}20des\%{}20francais.html}

\begin{figure}[H]
\caption{Disadvantage: \protect \\
``On the occasion of a tax reform which would redistribute income
from the richest to the poorest, \textbf{what proportion of French
people should be disadvantaged }by the reform\textbf{?} What we call
\emph{being disadvantaged by the reform}, is incurring a decrease
in one's after-tax income as compared to the current situation, and
it would concern the richest French.\protect \\
\textbf{The slider below helps you answer the question}: the text
below the slider changes when you shift the slider (by maintaining
the mouse pressed and shifting it on the side). The value of the slider
is not recorder, thus you have to report the value you will have chosen
in the field below.\protect \\
Among French, 20\% earn more than 2450¤/month.\protect \\
A proportion of ... should be disadvantaged (in \%):\protect \\
PNR (Do not know, do not wish to answer)''\protect \\
The slider is on-line: \protect\href{http://adrien-fabre.com/sondage/Fiscalite\%20des\%20francais.html\#QuestionText_q36035863_FR}{adrien-fabre.com/sondage/Fiscalite\%{}20des\%{}20francais.html\#{}QuestionText\_{}q36035863\_{}FR}}

\makebox[\textwidth][c]{\includegraphics[width=0.45\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/screen avantager\string".eps}}
\end{figure}
\bigskip{}
\begin{figure}[H]
\caption{Process of decision for the tax schedule\protect \\
``Currently, tax rates are voted at the Parliament. This is not the
only possible process: for example, it would be possible to determine
the tax rates preferred by citizens in a survey, and then put the
proposal that would emerge from the survey on a referendum. Do you
think that citizens should be consulted in this way to determine income
tax rates?\protect \\
Yes, this is a good idea\protect \\
No, the current system is satisfactory\protect \\
We should better take into account everyone's opinion, but through
another method\protect \\
PNR (Do not know, do not wish to answer)''}

\makebox[\textwidth][c]{\includegraphics[width=0.45\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/screen choix_bareme_impot\string".eps}}
\end{figure}
\begin{figure}[H]
\caption{Evaluations\protect \\
``Three graphs representing the standards of living of French adults,
from the poorest to the richest, are presented \textbf{below one another}.
For example, according to the first graph, the richest 1\% would have
a standard of living of 11700¤ per month (we can read the values of
the graphs by maintaining the mouse over the blue bars). The different
graphs show how the French national income can be distributed among
French, in more or less egalitarian ways. According to your preferences
in terms of inequalities, you can \textbf{grade each graph}, by a
grade between -2 (I don't like this distribution) to +2 (I like this
distribution): you just have to shift the slider next to each graph.
It is greatly recommended to \textbf{scroll until the bottom of the
page to see all graphs before grading them}.''}

\makebox[\textwidth][c]{\includegraphics[width=0.95\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/screen notations\string".eps}}
\end{figure}
\bigskip{}

\begin{figure}[H]
\caption{Legal maximum incomes (only one version was randomly displayed, cf.
sub-subsection \ref{par:The-maximum-income} for the translation)}

\makebox[\textwidth][c]{\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/screen rev_max_ideal\string".eps}}
\end{figure}
\begin{figure}[H]
\makebox[\textwidth][c]{

\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/screen revenu_max\string".eps}}
\end{figure}
\bigskip{}
\begin{figure}[H]
\caption{Demogrant (only one version was randomly displayed)\protect \\
1. What should be the amount of welfare for those with no income?\protect \\
2. What should be the amount for the basic income in France? The basic
income would be a benefit allocated to every adult without any condition
(such as age or activity), in replacement of social minima (notably
RSA {[}\emph{minimum welfare}{]} and APL {[}\emph{housing benefits}{]}).\protect \\
3. What should be the minimal income guaranteed to all French people?\protect \\
4. What is the minimal income that the State should insure to all,
in France?}

\makebox[\textwidth][c]{\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/screen rdb\string".eps}}
\end{figure}
\begin{figure}[H]
\makebox[\textwidth][c]{

\includegraphics[width=0.5\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/screen rdb2\string".eps}}

\end{figure}

\section{Raw Results\label{sec:Results}}

\begin{figure}[H]
\caption{Demogrant}

\makebox[\textwidth][c]{\includegraphics[width=0.65\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/demogrant cdf notitle\string".eps}}
\end{figure}
\bigskip{}
\begin{figure}[H]
\caption{Maximal income}

\makebox[\textwidth][c]{\includegraphics[width=0.65\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/maximal income cdf notitle\string".eps}}
\end{figure}
\bigskip{}

\selectlanguage{french}%
\begin{figure}[H]
\selectlanguage{english}%
\caption{Dis/advantage}
\foreignlanguage{french}{\makebox[\textwidth][c]{\includegraphics[width=0.65\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/dis-advantage cdf notitle\string".eps}}}\selectlanguage{french}%
\end{figure}
\foreignlanguage{english}{\bigskip{}
}

\begin{table}[H]
\selectlanguage{english}%
\caption{Individualisation\protect \\
Percentage of approval among expressed answers under different variants.\protect \\
Two versions of an individualisation of the income tax were tested,
with two variants for each. In the simple version, the question was
phrased either with an emphasis on individualisation: ``The individualisation
consists in determining the income tax of someone only in function
of her or his income, while the marital quotient decreases the tax
rate of married couples. Are you in favor of the individualisation?'';
or with an emphasis on the marital quotient: ``The marital quotient
allows to take into account the spouse's income in the computation
of someone's income tax rate, which decreases the tax rate of married
couples. Are you in favor of the marital quotient?'' This slight
difference of phrasing had a substantial and significant effect on
the answers, suggesting that preferences on this issue are not definitive.\protect \\
The second version proposes a progressive implementation of the reform,
so that ``only couples who got married before the reform would benefit
from the marital quotient''. Overall, compensating for already married
couples did not entail a higher support for individualisation.\label{tab:Individualisation:-percentage-of}\protect \\
}

\selectlanguage{french}%
\makebox[\textwidth][c]{%
\begin{tabular}{c|cc}
\multicolumn{1}{c}{\selectlanguage{english}%
\selectlanguage{french}%
} & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
\tabularnewline[-0.7cm]
\hline 
\hline 
\noalign{\vskip0.1cm}
\selectlanguage{english}%
Emphasis:\selectlanguage{french}%
 & \selectlanguage{english}%
Individualisation\selectlanguage{french}%
 & \selectlanguage{english}%
Marital quotient\selectlanguage{french}%
\tabularnewline[0.1cm]
\hline 
\noalign{\vskip0.1cm}
\selectlanguage{english}%
simple version\selectlanguage{french}%
 & \selectlanguage{english}%
41\selectlanguage{french}%
 & \selectlanguage{english}%
20\selectlanguage{french}%
\tabularnewline[0.1cm]
\noalign{\vskip0.1cm}
\selectlanguage{english}%
progressive version\selectlanguage{french}%
 & \selectlanguage{english}%
29\selectlanguage{french}%
 & \selectlanguage{english}%
36\selectlanguage{french}%
\tabularnewline[0.1cm]
\hline 
\hline 
\multicolumn{1}{c}{\selectlanguage{english}%
\selectlanguage{french}%
} & \selectlanguage{english}%
\selectlanguage{french}%
 & \selectlanguage{english}%
\selectlanguage{french}%
\tabularnewline[-0.7cm]
\end{tabular}

}
\end{table}
\foreignlanguage{english}{\bigskip{}
}
\selectlanguage{english}%

\section{Distributions Characteristics\label{sec:Distributions-Characteristics}}

\begin{table}[H]
\caption{\label{tab:Characteristics-of-Distributions}Characteristics of Distributions
of Income\protect \\
 (\emph{Dis- }and \emph{Adv- }stand for \emph{Dis/advantage }and are
expressed in \%, \emph{Demogrant} is in ¤/month, \emph{Transfer }and
income shares are in proportion of GNI, and \emph{D9/D1} is the inter-decile
ratio)}

\makebox[\textwidth][c]{
\begin{tabular}{rrrrrrrrrrrrr}   \hline \hline  & Gini & Theil & D9/D1 & Bottom50 & Top10 & Top1 & Top0.1 & Demogrant & Dis- & Adv- & Transfer & Extent \\    \hline equivalised disposable & 0.308 & 0.204 & 3.441 & 0.298 & 0.258 & 0.072 & 0.031 & -47 &  &  &  &  \\    actual & 0.434 & 0.454 & 10.326 & 0.224 & 0.335 & 0.128 & 0.076 & 0 &  &  &  &  \\    \textit{demogrant median} & 0.286 & 0.200 & 3.101 & 0.306 & 0.242 & 0.074 & 0.040 & 800 & 23 & 77 & 0.094 & 3.5 \\    \textit{median} & 0.241 & 0.102 & 3.194 & 0.331 & 0.202 & 0.036 & 0.015 & 802 & 10 & 50 & 0.117 & 8.57 \\    \textit{distortionary} & 0.251 & 0.100 & 3.815 & 0.321 & 0.186 & 0.026 & 0.008 & 550 & 12 & 50 & 0.116 & 9 \\    \textit{average} & 0.250 & 0.124 & 2.797 & 0.331 & 0.224 & 0.053 & 0.019 & 859 & 42 & 58 & 0.117 &  \\    earlier median & 0.295 & 0.176 & 5.429 & 0.304 & 0.237 & 0.056 & 0.022 & 550 & 10 & 73 & 0.109 &  \\     \hline \hline \end{tabular}
}
\end{table}
\bigskip{}

\begin{figure}[H]
\caption{Earlier estimation of\emph{ }median desired redistribution, from \citet{Fabre2016}}

\makebox[\textwidth][c]{\includegraphics[width=0.6\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/earlier\string".eps}}
\end{figure}
\bigskip{}
\begin{figure}[H]
\caption{\label{fig:Distribution-housheolds}Distribution of the proposed reforms
in terms of equivalised disposable income\protect \\
The proportions of households disadvantaged by each reform do not
coincide with the points where the reforms curve crosses the current
curve (because the ranking of households is not preserved). They are:
25.4\% (median), 26.9\% (demogrant median), 28.6\% (distortionary
median), 38.8\% (average).}

\makebox[\textwidth][c]{\includegraphics[width=0.6\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/household all reforms\string".eps}}
\end{figure}

\section{Algorithms Used\label{sec:Algorithms-Used}}

The commented pseudo-code of the algorithms used is given \href{http://adrien-fabre.com/sondage/doc_methode.php\#_a1}{on-line},\footnote{\href{http://adrien-fabre.com/sondage/doc_methode.php\#_a1}{http://adrien-fabre.com/sondage/doc\_{}methode.php\#{}\_{}a1}}
along with a pedagogical presentation of the algorithm \emph{Dis/av}
in seven slides (the main ones are presented in Figure \ref{fig:Algorithm-Dis/av}).
Thus, I will only summarize here the key steps of each algorithm.

Both of them start with the \emph{current} distribution of income
as a \emph{working }distribution and make it evolved until the \emph{new
}distribution, using their parameters.

\paragraph{Algorithm \emph{Dis/av}}

It is worth reminding that on a range of income (concerning people
that are not advantaged nor disadvantaged by the reform), both \emph{current}
and \emph{new} distributions coincide. The algorithm proceeds in the
following way:
\begin{enumerate}
\item define the \emph{reference }curve by setting the incomes of advantaged
to their maximum and those of disadvantaged to their minimum, thus
drawing two horizontal lines at each end of the distribution;
\item narrow the gap between \emph{working }and \emph{current} distributions
by a factor \emph{$1-$Extent}$/10$;
\item find the appropriate demogrant and draw a line (straight if possible)
joining the demogrant to the income of the richest advantaged, so
that the reform is budget neutral (absent any behavioral response);
\item decrease incomes by the amount of the loss due the behavioral response
(given as a parameter), by narrowing the gaps between the \emph{working
}distribution and the \emph{current }one (for lowest incomes) or the
income of the poorest disadvantaged (for highest incomes) by a common
factor. 
\end{enumerate}
\begin{figure}[H]
\caption{Algorithm \emph{Dis/av}\label{fig:Algorithm-Dis/av} Figure \ref{fig:3.5-=00201Clinearize=00201D-the}
and \ref{fig:4.-correct-for} result in the \emph{median }and \emph{distortionary
}redistributions.}

\selectlanguage{french}%
\makebox[\textwidth][c]{\foreignlanguage{english}{}\subfloat[\foreignlanguage{english}{1. \emph{reference }curve\label{fig:1.-reference-curve}}]{\selectlanguage{english}%

\includegraphics[width=0.28\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/algo disav 1\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{\qquad{}}\subfloat[\foreignlanguage{english}{2. narrowing the gap}]{\selectlanguage{english}%

\includegraphics[width=0.28\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/algo disav 2\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{\qquad{}}\subfloat[\foreignlanguage{english}{3. adjust for budget neutrality}]{\selectlanguage{english}%

\includegraphics[width=0.28\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/algo disav 3\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{}}

\makebox[\textwidth][c]{\foreignlanguage{english}{}\subfloat[\foreignlanguage{english}{3.5 \textquotedblleft linearize\textquotedblright{} the left end\label{fig:3.5-=00201Clinearize=00201D-the}}]{\selectlanguage{english}%

\includegraphics[width=0.28\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/algo disav 3-5\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{\qquad{}}\subfloat[\foreignlanguage{english}{4. correct for behavioral responses\label{fig:4.-correct-for}}]{\selectlanguage{english}%

\includegraphics[width=0.28\paperwidth]{\string"/home/adrien/Google Drive/Economie/Travail/enquete/images/algo disav 4\string".eps}\selectlanguage{french}%
}\foreignlanguage{english}{}}\selectlanguage{english}%
\end{figure}

\paragraph{Algorithm \emph{Demogrant}}

The algorithm \emph{Demogrant }is quite baroque, because it contains
many adjustments corresponding to cases for which the main steps would
lead to anomalies. For the sake of clarity, I focus on the situation
where the parameters allow not to deviate from the main steps. During
the following sketch of these steps, the \emph{neutral point} designates
the point where \emph{current }and \emph{new }distributions cross.
\begin{enumerate}
\item The \emph{reference }curve passes by the \emph{demogrant}, is parallel
with the \emph{current }curve for the lowest incomes, then is straight
until the neutral point, and coincides with the \emph{current }curve
after the neutral point;\textbf{ }the junction between the straight
line and the parallel part is set where it can be smooth.
\item The deficit \emph{D }is defined as the difference between the integral
of the \emph{current }and the \emph{reference }curve. If possible,
the incomes of the advantaged are decreased by an aggregate amount
of \emph{D$\cdot$$(1-$Extent}$/10)$, and those of disadvantaged
are reduced by the appropriate amount to obtain budget neutrality
(absent any behavioral response). On the left side, the gap between
the \emph{reference }curve and the maximum between the \emph{demogrant
}and the \emph{current }curve is narrowed by a common proportion.
On the right side, the \emph{reference }curve cannot be brought closer
to the neutral income by a common proportion all the way long, because
the maximum income of the \emph{new }curve is constrained by a parameter;
but in practice, this is what happens, except for the very top quantiles
of the distribution, for which a straight line is set.
\end{enumerate}

\section{Regressions\label{sec:Regressions}}

Hereafter are reported all regressions mentioned in the article. When
appropriate, Mood tests of the equality of medians were carried out;
they are not reported because they produced results similar to those
of linear regressions.

\begin{table}[H]
\caption{Effect of an anti-tax argument on the desired maximum income}

\makebox[\textwidth][c]{  
\begin{tabular}{@{\extracolsep{5pt}}lcc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]   & \multicolumn{2}{c}{log$_{10}$ desired maximum income} \\  \cline{2-3}  \\[-1.8ex] & Excluding \textit{infinities} & Setting $ \infty := 10^9 $ \\  \\[-1.8ex] & (1) & (2)\\  \hline \\[-1.8ex]   Anti-tax priming & $-$0.221$^{***}$ & $-$0.751$^{***}$ \\    & (0.071) & (0.201) \\    Constant & 4.195$^{***}$ & 6.424$^{***}$ \\    & (0.043) & (0.113) \\   \hline \\[-1.8ex]  Observations & 392 & 686 \\  \hline  \hline \\[-1.8ex]  \end{tabular} 
}
\end{table}

\bigskip{}
\begin{table}[H]
\caption{Effect of the choice of income variable on the proportion to dis/advantage}

\makebox[\textwidth][c]{  
\begin{tabular}{@{\extracolsep{5pt}}lcc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]  \\[-1.8ex] & Advantage & Disadvantage \\  \\[-1.8ex] & (1) & (2)\\  \hline \\[-1.8ex]   Income expressed at household level & $-$1.705 & 2.500$^{*}$ \\    & (2.909) & (1.354) \\    Constant & 50.102$^{***}$ & 13.308$^{***}$ \\    & (1.430) & (1.047) \\   \hline \\[-1.8ex]  Observations & 356 & 736 \\  \hline  \hline \\[-1.8ex]  \end{tabular} 
}
\end{table}

\bigskip{}
\begin{table}[H]
\caption{Significance of the differences in the average distributions' evaluations
(with the optimal utilitarian as a reference)}

\makebox[\textwidth][c]{ 
\begin{tabular}{@{\extracolsep{5pt}}lc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]  \\[-1.8ex] & Evaluation of distributions \\  \hline \\[-1.8ex]   Optimal utilitarian (Constant) & 0.159$^{***}$ \\    & (0.048) \\    \textit{Median} & $-$0.140$^{**}$ \\    & (0.069) \\    Actual & $-$0.926$^{***}$ \\    & (0.059) \\    \textit{Demogrant median} & $-$0.078 \\    & (0.069) \\    Egalitarian & $-$1.027$^{***}$ \\    & (0.069) \\    Personalized & $-$0.569$^{***}$ \\    & (0.069) \\    Optimal rawlsian & 0.058 \\    & (0.069) \\   \hline \\[-1.8ex]  Observations & 5,883 \\  \hline  \hline \\[-1.8ex]  \end{tabular} 
}
\end{table}

\bigskip{}
\begin{table}[H]
\caption{\label{tab:Effect-of-the-1}Effect of the display of the impact of
the reform on the respondent's income, and understanding}

\makebox[\textwidth][c]{
\begin{tabular}{@{\extracolsep{5pt}}lcccccc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]  \\[-1.8ex] & Approval & \multicolumn{2}{c}{Approval (any reform)} & \multicolumn{3}{c}{Person Not Responding} \\  \\[-1.8ex] & (1) & (2) & (3) & (4) & (5) & (6)\\  \hline \\[-1.8ex]   No display of impact of reform on one's income & $-$0.008 &  &  & 0.013 & $-$0.023 & $-$0.054 \\    & (0.032) &  &  & (0.027) & (0.034) & (0.038) \\    Misunderstanding of graphics &  & $-$0.095$^{***}$ & $-$0.004 &  & 0.262$^{***}$ & 0.310$^{***}$ \\    &  & (0.022) & (0.029) &  & (0.038) & (0.045) \\    Left - Right leaning &  &  & $-$0.074$^{***}$ &  &  & 0.016 \\    &  &  & (0.012) &  &  & (0.013) \\    Misunderstanding \& No display &  &  &  &  & $-$0.103$^{*}$ & $-$0.146$^{**}$ \\    &  &  &  &  & (0.055) & (0.065) \\    Constant & 0.531$^{***}$ & 0.495$^{***}$ & 0.466$^{***}$ & 0.230$^{***}$ & 0.165$^{***}$ & 0.142$^{***}$ \\    & (0.023) & (0.015) & (0.019) & (0.020) & (0.025) & (0.028) \\   \hline \\[-1.8ex]  Observations & 958 & 1,994 & 1,196 & 958 & 958 & 614 \\  R$^{2}$ & 0.0001 & 0.009 & 0.031 & 0.0002 & 0.063 & 0.092 \\  \hline  \hline \\[-1.8ex]  \textit{Note:}  & \multicolumn{6}{r}{$^{*}$p$<$0.1; $^{**}$p$<$0.05; $^{***}$p$<$0.01} \\  \end{tabular} 
}
\end{table}

\bigskip{}

\begin{table}[H]
\caption{\label{tab:singles}Correlation between the approval of a reform and
being single without dependent child}

\noindent\begin{minipage}[t]{1\columnwidth}%
\makebox[\textwidth][c]{
\begin{tabular}{@{\extracolsep{5pt}}lccc}  \\[-1.8ex]\hline  \hline \\[-1.8ex]  \\[-1.8ex] & \multicolumn{3}{c}{Approval (any reform)} \\  \\[-1.8ex] & (1) & (2) & (3)\\  \hline \\[-1.8ex]   Childless single & 0.068$^{***}$ & 0.062$^{**}$ & 0.039 \\    & (0.026) & (0.026) & (0.027) \\    Equivalised household income\footnote{ The variable has been capped at 4500¤/month. This trimming eases the interpretation of the results and concerns 7.6\% (resp. 5.4\%) of the observations in the case of the individual (resp. equivalised) income. } &  & $-$0.046$^{***}$ & $-$0.014 \\    &  & (0.010) & (0.021) \\    Individual disposable income\textsuperscript{$a$} &  &  & $-$0.046$^{**}$ \\    &  &  & (0.020) \\    Constant & 0.436$^{***}$ & 0.521$^{***}$ & 0.571$^{***}$ \\    & (0.013) & (0.022) & (0.027) \\   \hline \\[-1.8ex]  Observations & 1,994 & 1,994 & 1,888 \\  \hline  \hline \\[-1.8ex]  \textit{Note:}  & \multicolumn{3}{r}{$^{*}$p$<$0.1; $^{**}$p$<$0.05; $^{***}$p$<$0.01} \\  \end{tabular} 
}%
\end{minipage}
\end{table}

\end{document}