Mathematical models for sustainable wealth redistribution

120 Pages·2017·2.17 MB·English

Mathematical models
for sustainable wealth redistribution
JC van der Walt
Thesis presented in partial fulfilment of the requirements for the degree of
Master of (Industrial) Engineering
in the Faculty of Engineering at Stellenbosch University
Supervisor: Prof JH van Vuuren December 2017
Stellenbosch University https://scholar.sun.ac.za
Declaration
By submitting this thesis electronically, I declare that the entirety of the work contained therein
ismyown,originalwork,thatIamthesoleauthorthereof(savetotheextentexplicitlyotherwise
stated), that reproduction and publication thereof by Stellenbosch University will not infringe
any third party rights and that I have not previously in its entirety or in part submitted it for
obtaining any qualification.
Date: December 1, 2017
Copyright ' 2017 Stellenbosch University
All rights reserved
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Abstract
Economic inequality has increased in most large free-market economies during the last century
and it has been suggested that this phenomenon is an inherent feature of free-market activities.
It seems self-evident, however, that a continual rise in economic inequality is unsustainable. In
fact,severeeconomicinequalityhashistoricallybeenassociatedwithnegativeeffectssuchaspoor
economicgrowth, severefinancialrecessionsorevenviolentrevolutions. Wealthredistributionis
present in every form of government, although the extent thereof varies, and existing theoretical
justifications of redistributive actions usually rely heavily on utility theory.
Most economic postulates related to inequality are empirically inspired and defended, but be-
cause of the vast variety of possible economic contexts in which they may prevail, many of these
claims are disputed. One example is the so-called Robin Hood paradox, which asserts that the
extent of wealth redistribution is less in more unequal societies, where it is needed most, than in
more economically equal societies. Another is the Kuznets curve, which predicts that the extent
of inequality in a developing economy will follow an inverted ‘u’ curve as a result of development
over time.
The implications of increasing relative inequality over time as an inherent feature of wealth
growth are investigated in the presence of wealth redistribution. Very simple mathematical
model abstractions are employed to shed light on the possible evolution over time of wealth
distribution in the context of very basic assumptions, since such behaviour may perhaps then
also be inferred in more complicated settings.
Assuming increasing per capita wealth growth-rate functions is one way of capturing increasing
relative inequality over time, the very simplest case being linearly increasing per capita wealth
growth-rate functions, which are considered in this thesis. Two examples of redistribution dy-
namicsareinvestigated. Oneexamplemimicsdiffusion-likeeffectsoftrickle-downredistribution,
while the other represents a conservative, linear-tax transfer scheme.
Itisestablishedanalyticallywithinthecontextoftheaforementionedmathematicalmodelsthat
increases in economic inequality can always be limited by means of sufficient redistribution. It is
also demonstrated that the Robin Hood paradox may follow from very simple assumptions. It is
furthermoreillustratedthatfluctuatingbehaviourintheevolutionovertimeofwealthinequality
canevenmanifestitselfintheabsenceoftime-dependentprocesses, andhencethatexplanations
of such trends which merely assume time-dependent underlying processes might be of dubious
value. Examples of analytical formulations of theoretical justifications for redistributive actions,
independent of utility theory, are finally also provided.
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Uittreksel
Ekonomiese ongelykheid het gedurende die laaste eeu in die meeste groot vryemark ekonomie¨e
toegeneem en daar is al voorgestel dat hierdie verskynsel ’n inherente eienskap van vryemark
aktiwiteite is. Dit blyk egter voor die hand liggend te wees dat ’n voortdurende toename in
ekonomiese ongelykheid onvolhoubaar is. Ernstige ekonomiese ongelykheid het trouens histo-
ries hand aan hand gegaan met newe-effekte soos stadige ekonomiese groei, ernstige finansi¨ele
resessies en selfs gewelddadige revolusies. Die herverdeling van rykdom is in elke vorm van
regering teenwoordig, alhoewel die mate daarvan varieer, en bestaande teoretiese regverdigings
vir herverdelingsaksies berus gewoonlik swaar op nutsteorie.
Die meeste ekonomiese postulate wat te make het met ongelykheid is empiries-ge¨ınspireer en
word ook sodanig verdedig, maar baie van hierdie bewerings word as gevolg van die groot
verskeidenheidmoontlikeekonomiesekontekstewaarinhulmagvoorkom, betwis. Eenvoorbeeld
hiervan is die sogenaamde Robin Hood-paradoks waarvolgens die mate van rykdom-herverdeling
minder is in meer ongelyke gemeenskappe, waar dit juis m´e´er benodig word, as in ekonomies
meer gelyke gemeenskappe. Nog ’n voorbeeld is Kuznets se kromme waarvolgens voorspel word
dat die mate van ongelykheid in ’n ontwikkelende ekonomie ’n omgekeerde ‘u’ kromme sal volg
soos ontwikkeling oor tyd geskied.
Die gevolge van toenemende relatiewe ongelykheid oor tyd as ’n inherente kenmerk van toene-
menderykdom,indieteenwoordigheidvanrykdom-herverdeling,wordinhierdietesisondersoek.
Baie eenvoudige wiskundige modelabstraksies word ingespan om lig te werp op die moontlike
evolusie oor tyd van die verdeling van rykdom in die konteks van baie basiese aannames, aange-
sien sodanige gedrag dan ook moontlik in meer ingewikkelde kontekste afleibaar is.
Die aanname van toenemende per kapita groeitempo-funksies is een manier waarop die toe-
name in relatiewe ongelykheid oor tyd vasgevang kan word. Die heel eenvoudigste geval hier-
van is lineˆer-toenemende per kapita groeitempo-funksies, wat in hierdie tesis oorweeg word.
Twee voorbeelde van herverdelingsdinamika word ondersoek. Een voorbeeld boots die diffusie-
verwante gedrag van deursyferingsherverdeling na, terwyl die ander ’n konserwatiewe, lineˆere
belasting-oordragskema is.
Daarwordbinnediekonteksvandiebogenoemdewiskundigemodelleanalitiesvasgesteldattoe-
names in ekonomiese ongelykheid altyd deur genoegsame herverdeling beperk kan word. Daar
word ook aangetoon dat die Robin Hood-paradoks die gevolg van baie eenvoudige aannames
mag wees. Verder word daar gedemonstreer dat wisselende gedrag in die evolusie van rykdom-
ongelykheidoortydselfsindieafwesigheidvantyd-afhanklikeprosessemagvoorkomengevolglik
dat verklarings van sulke tendense waarin onderliggende tyd-afhanklike prosesse bloot aange-
neem word, van twyfelagtige waarde mag wees. Voorbeelde van analitiese formule-rings vir die
teoretiese regverdiging van herverdelingsaksies word laastens ook buite die konteks van nuts-
teorie gegee.
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Acknowledgements
The author wishes to acknowledge the following people and institutions for their various
contributions towards the completion of this work:
(cid:136) Professor van Vuuren for his unwavering support, meticulous feedback and friendship.
(cid:136) The National Research Foundation for funding this research.
(cid:136) The SUnORE research group for the provision of office space, computing facilities and
thoroughly enjoyable social events.
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Table of Contents
Abstract iii
Uittreksel v
Acknowledgements vii
List of Reserved Symbols xiii
List of Figures xv
List of Tables xvii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Scope delimitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Thesis organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
I Literature review 7
2 Economic inequality 9
2.1 Inequality and social justice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Trends in economic inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 The redistribution of wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 The notion of optimal redistribution . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Metrics of economic inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Existing models of wealth distribution . . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
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