Consensus Income Distribution

TL;DR: In this paper, the authors consider the optimal redistribution of a given population's income, and find that the social planner's aversion to inequality, embedded in an isoelastic social welfare function indexed by a parameter alpha, or the individuals' concern at having a low relative income, indexed by the parameter beta in a utility function that is a convex combination of (absolute) income and low relative incomes.

Abstract: In determining the optimal redistribution of a given population's income, we ask which factor is more important: the social planner's aversion to inequality, embedded in an isoelastic social welfare function indexed by a parameter alpha, or the individuals' concern at having a low relative income, indexed by a parameter beta in a utility function that is a convex combination of (absolute) income and low relative income. Assuming that the redistribution comes at a cost (because only a fraction of a taxed income can be transferred), we find that there exists a critical level of beta below which different isoelastic social planners choose different optimal allocations of incomes. However, if beta is above that critical level, all isoelastic social planners choose the same allocation of incomes because they then find that an equal distribution of incomes maximizes social welfare regardless of the magnitude of alpha.

Summary

Consensus income distribution

• Bonn, November 2016 The CENTER FOR DEVELOPMENT RESEARCH (ZEF) was established in 1995 as an international, interdisciplinary research institute at the University of Bonn.
• Research and teaching at ZEF address political, economic and ecological development problems.
• ZEF – Discussion Papers on Development Policy are intended to stimulate discussion among researchers, practitioners and policy makers on current and emerging development issues.
• The papers mostly reflect work in progress.
• Chiara Kofol is the Managing Editor of the series.

The author[s]:

• Oded Stark, Center for Development Research (ZEF), University of Bonn.
• Marcin Jakubek, Institute of Economics, Polish Academy of Sciences.
• The authors analysis unravels an interesting distinction between the social planners’ aversion to inequality (represented by the parameter in the isoelastic social welfare function) and the individuals’ concern at having a low relative income (represented by the parameter in the individuals’ utility functions).
• From a comparison of Lemma 1 with Lemma 2, the authors see that in the presence of a deadweight loss of tax and transfer, the optimal choices of isoelastic social planners (including a utilitarian social planner and a Bernoulli-Nash social planner) differ from the choice of a Rawlsian social planner; only the latter chooses to distribute incomes equally.

Figures

Figure 1: The optimal tax *t as a function of for three different values of : 0 (dashed line), *0.9 (dotted line), and * (solid line), for ( ) ln( 1)i if x x , 1,2i , 1 4e , 2 13e , and 1/ 2 .

Full text

ZEF-Discussion Papers on
Development Policy No. 227
Oded Stark, Fryderyk Falniowski, and Marcin Jakubek
Consensus income distribution
Bonn, November 2016

The CENTER FOR DEVELOPMENT RESEARCH (ZEF) was established in 1995 as an international,
interdisciplinary research institute at the University of Bonn. Research and teaching at ZEF
address political, economic and ecological development problems. ZEF closely cooperates
with national and international partners in research and development organizations. For
information, see: www.zef.de.
ZEF – Discussion Papers on Development Policy are intended to stimulate discussion among
researchers, practitioners and policy makers on current and emerging development issues.
Each paper has been exposed to an internal discussion within the Center for Development
Research (ZEF) and an external review. The papers mostly reflect work in progress. The
Editorial Committee of the ZEF – DISCUSSION PAPERS ON DEVELOPMENT POLICY includes
Joachim von Braun (Chair), Christian Borgemeister, and Eva Youkhana. Chiara Kofol is the
Managing Editor of the series.
Oded Stark, Fryderyk Falniowski, and Marcin Jakubek, Consensus income distribution, ZEF
– Discussion Papers on Development Policy No. 227, Center for Development Research,
Bonn, November 2016, pp. 29.
ISSN: 1436-9931
Published by:
Zentrum für Entwicklungsforschung (ZEF)
Center for Development Research
Walter-Flex-Straße 3
D – 53113 Bonn
Germany
Phone: +49-228-73-1861
Fax: +49-228-73-1869
E-Mail: zef@uni-bonn.de
www.zef.de
The author[s]:
Oded Stark, Center for Development Research (ZEF), University of Bonn. Contact:
ostark@uni-bonn.de
Fryderyk Falniowski, Cracow University of Economics. Contact:
fryderyk.falniowski@uek.krakow.pl
Marcin Jakubek, Institute of Economics, Polish Academy of Sciences. Contact:
mjak@mjak.org

Acknowledgements
We are indebted to two referees for thoughtful comments and constructive suggestions.
Marcin Jakubek gratefully acknowledges the support of the National Science Centre, Poland,
grant 2014/13/B/HS4/01644.

Abstract
In determining the optimal redistribution of a given population’s income, we ask which
factor is more important: the social planner’s aversion to inequality, embedded in an
isoelastic social welfare function indexed by a parameter alpha, or the individuals’ concern
at having a low relative income, indexed by a parameter beta in a utility function that is a
convex combination of (absolute) income and low relative income. Assuming that the
redistribution comes at a cost (because only a fraction of a taxed income can be transferred),
we find that there exists a critical level of beta below which different isoelastic social
planners choose different optimal allocations of incomes. However, if beta is above that
critical level, all isoelastic social planners choose the same allocation of incomes because
they then find that an equal distribution of incomes maximizes social welfare regardless of
the magnitude of alpha.
Keywords: Maximization of social welfare, Isoelastic social welfare functions, Deadweight
loss of tax and transfer, Concern at having a low relative income, Social planners’
aversion to inequality
JEL Codes: D31, D60, D63, H21, I38

1
1. Introduction
The fundamental tension between different social planners with regard to the income
allocation rule under a deadweight loss of tax and transfer is easily understood, and has
been alluded to for many years. For example, Tullock (1975) and Sen (1982) already grappled
with the assumptions or conditions necessary to render equal division the optimal
distributional rule for a given total income. However, neither of them enlisted individuals’
concern at having a low relative income as a conciliator. In this note we bring together under
the same isoelastic roof all the pivotal social planners, we incorporate a deadweight loss of
tax and transfer, we display the received tension between the different social planners, and
we ask what strength of the individuals’ concern at having a low relative income will cause
all the social planners to choose the same - equal - distribution of income.
1
The class of isoelastic social welfare functions (Atkinson, 1970) enables us to
represent the varying degrees of the social planners’ aversion to inequality in the
population’s distribution of income as special cases. Due to its appealing axiomatic
foundation and flexibility in embracing basic equality criteria,
2
the function has become a
popular measure of social welfare in a variety of fields, ranging from optimal taxation
(Atkinson and Stiglitz, 1976; Stern, 1976; Slemrod et al., 1994) to health economics (Abasolo
and Tsuchiya, 2004, and references cited therein) and environmental economics (Shiell,
2003).
Our aim is to uncover a condition under which all the pivotal “isoelastic social
planners” - a utilitarian, a Rawlsian, a Bernoulli-Nash, or any planner “in-between” - will
come up with the same optimal income distribution when a tax and transfer procedure is
subject to a deadweight loss. We obtain a strong congruence result: when the individuals’
utility functions exhibit a sufficiently high concern at having a low relative income, the
optimal tax policies of all the social planners align: this unanimity holds for the entire class of
isoelastic social welfare functions with a parameter of inequality aversion,

, (defined in (1)
below) spanning from zero (the case of a utilitarian social function) to infinity (the case of a
1
Rich evidence from econometric studies, experimental economics, social psychology, and neuroscience
confirms that individuals routinely engage in, and are affected by, interpersonal comparisons. In particular,
people are dissatisfied when their consumption or income levels are lower than those of others who
constitute their “comparison group.” Studies that recognize such discontent are, among others, Stark and
Taylor (1991), Zizzo and Oswald (2001), Luttmer (2005), Fliessbach et al. (2007), Blanchflower and Oswald
(2008), Takahashi et al. (2009), Stark and Fan (2011), Stark and Hyll (2011), Fan and Stark (2011), Stark et al.
(2012), and Card et al. (2012). Additionally, the comparisons that affect the sense of wellbeing significantly
are those made by looking “up” the hierarchy, whereas the possibility that individuals derive satisfaction from
looking “down” is not supported by studies of this subject. For example, Andolfatto (2002) demonstrates that
individuals are adversely affected by the material wellbeing of others in their reference group when this
wellbeing is far enough below theirs. See also Frey and Stutzer (2002) and Walker and Smith (2002) for a
large body of evidence that supports the “upward comparison” hypothesis.
2
The isoelastic social welfare function satisfies the criteria of unrestricted domain, independence of irrelevant
alternatives, anonymity, separability, and the weak Pareto criterion (Roberts, 1980).

Frequently asked questions

Q1. What are the contributions in this paper?

In this paper, the authors consider the optimal redistribution of a given population 's income and determine which factor is more important: the social planner 's aversion to inequality, embedded in an isoelastic social welfare function indexed by a parameter alpha, or the individuals ' concern at having a low relative income, indexed by the parameter beta in a utility function. 

Q2. What is the way to determine the optimal allocation of income?

Assuming that the redistribution comes at a cost (because only a fraction of a taxed income can be transferred), the authors find that there exists a critical level of beta below which different isoelastic social planners choose different optimal allocations of incomes. 

Q3. What is the effect of inequality aversion on the planning of income?

The authors find that when an “isoelastic social planner” faces a population characterized by an intensity of concern at having a low relative income that is higher than a critical value, the planner will choose to equalize incomes. 

Q4. What is the effect of inequality aversion on the distribution of income?

Slemrod et al. (1994), and others, who show that embedding inequality aversion in the social welfare function suffices to render taxation more progressive, and the distribution of income more equal. 

Q5. What is the only way the social planner could try to improve social welfare?

the only way in which the social planner could try to improve social welfare is to tax the “rich” individual, and make a transfer to the “poor” individual. 

Q6. what is the maximum of ( )f on (, ) e?

On the other hand, because ( )F is a strictly concave function - it is the sum of strictly concave functions ( )u raised to the power 1-a and divided by 1-a - maximized on a closed subset ( , ) e characterized by a concave constraint function,13 then, if a local maximum on ( , ) e exists, then that maximum is also a global maximum on ( , ) e . 

Q7. What is the solution of the problem of a Rawlsian social planner?

the solution of the problem of a Rawlsian social planner, (A1), has to be a transfer such that the post-transfer incomes are all equal and, as shown in Lemma 3, *x is the unique point in ( , ) e such that all the incomes are equal. 

Q8. what is the simplest way to construct a transfer from an individual with income higher than ?

The authors assume that ( , ) arg max ( )RSWF x e x z , where 1,..., )( nz zz is such that 1min{ ,..., }nz z z 1max{ ,..., }nz z , and the authors show that it is possible to construct a transfer from an individual with income higher than z to individual(s) with income z and obtain a ( , )y e such that( ) ( )R RSWF SWFy z . 

Q9. what is the simplest way to determine the total income of a social planner?

the set on which the authors search for the solution of the social planner’s problem is1 1 1( , ) ( , , ) : 0 for all and max{ ,0}, ,0}max{ n nn i i i i ii i x x e x xx i e e x ,namely, the authors search over the set of incomes that can be attained from the initial allocation eby taxing some individuals; the authors thereby obtain the sum 1max{ ,0} ni ii t e x ; and wedistribute t between the remaining individuals such that the transfer amounts to1max{ ,0} ni ii t x e .6 

Q10. What is the solution to the isoelastic social planner’s maximization problem?

The authors next show that if neither (b) nor (c) holds, then the solution of the isoelastic social planner’s maximization problem is an equal division of incomes if and only if (d) holds, that is, (b) (c) (d) (a) .