Mathematical models for sustainable wealth redistribution

Abstract

ENGLISH 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, severe economic inequality has historically been associated with negative effects such as poor economic growth, severe financial recessions or even violent revolutions. Wealth redistribution is 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 because 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 dynamics are investigated. One example mimics diffusion-like effects of trickle-down redistribution, while the other represents a conservative, linear-tax transfer scheme. It is established analytically within the context of the aforementioned mathematical models that 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 furthermore illustrated that fluctuating behaviour in the evolution over time of wealth inequality can even manifest itself in the absence of time-dependent processes, and hence that explanations 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.

AFRIKAANSE OPSOMMING: Ekonomiese ongelykheid het gedurende die laaste eeu in die meeste groot vryemark ekonomieë 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 histories hand aan hand gegaan met newe-effekte soos stadige ekonomiese groei, ernstige finansiële 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-geïnspireer en word ook sodanig verdedig, maar baie van hierdie bewerings word as gevolg van die groot verskeidenheid moontlike ekonomiese kontekste waarin hul mag voorkom, betwis. Een voorbeeld hiervan is die sogenaamde Robin Hood-paradoks waarvolgens die mate van rykdom-herverdeling minder is in meer ongelyke gemeenskappe, waar dit juis méér 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 toenemende rykdom, in die teenwoordigheid van rykdom-herverdeling, word in hierdie tesis ondersoek. 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, aangesien sodanige gedrag dan ook moontlik in meer ingewikkelde kontekste a eibaar is. Die aanname van toenemende per kapita groeitempo-funksies is een manier waarop die toename in relatiewe ongelykheid oor tyd vasgevang kan word. Die heel eenvoudigste geval hiervan is lineêr-toenemende per kapita groeitempo-funksies, wat in hierdie tesis oorweeg word. Twee voorbeelde van herverdelingsdinamika word ondersoek. Een voorbeeld boots die diffusieverwante gedrag van deursyferingsherverdeling na, terwyl die ander 'n konserwatiewe, lineêre belasting-oordragskema is. Daar word binne die konteks van die bogenoemde wiskundige modelle analities vasgestel dat toenames 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 rykdomongelykheid oor tyd selfs in die afwesigheid van tyd-afhanklike prosesse mag voorkom en gevolglik dat verklarings van sulke tendense waarin onderliggende tyd-afhanklike prosesse bloot aangeneem word, van twyfelagtige waarde mag wees. Voorbeelde van analitiese formule-rings vir die teoretiese regverdiging van herverdelingsaksies word laastens ook buite die konteks van nutsteorie gegee.