EFTA00625129.pdf

Why Money Trickles Up Geoff Willis gwillis@econodynamics.org The right of Geoffrey Michael Willis to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. EFTA00625129 0.0 Abstract This paper combines ideas from classical economics and modern finance with Lotka-Volterra models, and also the general Lotka-Volterra models of Levy & Solomon to provide straightforward explanations of a number of economic phenomena. Using a simple and realistic economic formulation, the distributions of both wealth and income are fully explained. Both the power tail and the log-normal like body are fully captured. It is of note that the full distribution, including the power law tail, is created via the use of absolutely identical agents. It is further demonstrated that a simple scheme of compulsory saving could eliminate poverty at little cost to the taxpayer. Such a scheme is discussed in detail and shown to be practical. Using similar simple techniques, a second model of corporate earnings is constructed that produces a power law distribution of company size by capitalisation. A third model is produced to model the prices of commodities such as copper. Including a delay to capital installation; normal for capital intensive industries, produces the typical cycle of short- term spikes and collapses seen in commodity prices. The fourth model combines ideas from the first three models to produce a simple Lotka-Volterra macroeconomic model. This basic model generates endogenous boom and bust business cycles of the sort described by Minsky and Austrian economists. From this model an exact formula for the Bowley ratio; the ratio of returns to labour to total returns, is derived. This formula is also derived trivially algebraically. This derivation is extended to a model including debt, and it suggests that excessive debt can be economically dangerous and also directly increases income inequality. Other models are proposed with financial and non-financial sectors and also two economies trading with each other. There is a brief discussion of the role of the state and monetary systems in such economies. The second part of the paper discusses the various background theoretical ideas on which the models are built. This includes a discussion of the mathematics of chaotic systems, statistical mechanical systems, and systems in a dynamic equilibrium of maximum entropy production. There is discussion of the concept of intrinsic value, and why it holds despite the apparent substantial changes of prices in real life economies. In particular there are discussions of the roles of liquidity and parallels in the fields of market-microstructure and post-Keynesian pricing theory. 2 EFTA00625130 0. Zeroth Section 0.0 Abstract 0.1 Contents 0.2 Introduction 0.3 Structure of Paper Part A — Some Models Part A.I — Heavy Duty Models 1. Wealth & Income Models 1.1 Wealth & Income Data — Empirical Information 1.2 Lotka-Volterra and General Lotka-Volterra Systems 1.3 Wealth & Income Models - Modelling 1.4 Wealth & Income Models - Results 1.5 Wealth & Income Models - Discussion 1.6 Enter Sir Bowley - Labour and Capital 1.7 Modifying Wealth and Income Distributions 1.8 A Virtual 40 Acres 1.9 Wealth & Income Distributions - Loose Ends 2. Companies Models 2.1 Companies Models - Background 2.2 Companies Models - Modelling 2.3 Companies Models - Results 2.4 Companies Models - Discussion 3. Commodity models 3.1 Commodity models - Background 3.2 Commodity models - Modelling 3.3 Commodity models - Results 3.4 Commodity models - Discussion 4. Minsky goes Austrian a la Goodwin — Macroeconomic Models 4.1 Macroeconomic Models - Background 4.2 Macroeconomic Models - Modelling 4.3 Macroeconomic Models - Results 4.4 Macroeconomic Models - Discussion 4.5 A Present for Philip Mirowski? — A Bowley-Polonius Macroeconomic Model EFTA00625131  Part A.II - Speculative Building 4.6 Unconstrained Bowley Macroeconomic Models 4.7 A State of Grace 4.8 Nirvana Postponed 4.9 Bowley Squared 4.10 Siamese Bowley - Mutual Suicide Pacts 4.11 Where Angels Fear to Tread - Governments & Money 4.12 Why Money Trickles Up Part B - Some Theory 5. Theory Introduction Part B.I — Mathematics 6. Dynamics 6.1 Drive My Car 6.2 Counting the Bodies - Mathematics and Equilibrium 6.3 Chaos in Practice — Housing in the UK 6.4 Low Frequency / Tobin Trading 6.5 Ending the Chaos 7. Entropy 7.1 Many Body Mathematics 7.2 Statistical Mechanics and Entropy 7.3 Maximum Entropy Production 7.4 The Statistical Mechanics of Flow Systems Part &II — Economic Foundations 8. Value 8.1 The Source of Value 8.2 On the Conservation of Value 8.2.1 Liquidity 8.2.2 On the Price of Shares 9. Supply and Demand 9.1 Pricing 9.2 An Aside on Continuous Double Auctions 9.3 Supply — On the Scarcity of Scarcity, or the Production of Machines by Means of Machines EFTA00625132  9.4 Demand Part &III — The Logic of Science 10. The Social Architecture of Capitalism 11. The Logic of Science Part C — Appendices 12. History and Acknowledgements 13. Further Reading 14. Programmes 15. References 16. Figures 5 EFTA00625133 0.2 Introduction "The produce of the earth -- all that is derived from its surface by the united application of labour, machinery, and capital, is divided among three classes of the community; namely, the proprietor of the land, the owner of the stock or capital necessary for its cultivation, and the labourers by whose industry it is cultivated To determine the laws which regulate this distribution, is the principal problem in Political Economy..." On The Principles of Political Economy and Taxation - David Ricardo [Ricardo 1817] "We began with an assertion that economic inequality is a persistent and pressing problem; this assertion may be regarded by many people as tendentious. Differences in economic status - it might be argued - are a fact of life; they are no more a 'problem' than are biological differences amongst people, or within and amongst other species for that matter. Furthermore, some economists and social philosophers see economic inequality, along with unfettered competition, as essential parts of a mechanism that provides the best prospects for continuous economic progress and the eventual elimination of poverty throughout the world. These arguments will not do. There are several reasons why they will not do However there is a more basic but powerful reason for rejecting the argument that dismisses economic inequality as part of the natural order of things. This has to do with the scale and structure ofinequality " Economic Inequality and Income Distribution — DG Champernowne [Champernowne & Cowell 1998] "Few if any economists seem to have realized the possibilities that such invariants hold for the future of our science. In particular, nobody seems to have realized that the hunt for, and the interpretation of, invariants of this type might lay the foundations for an entirely novel type of theory." Schumpeter (1949, p. 155), discussing the Pareto law — via [Gabaix 2009] This paper introduces some mathematical and simulation models and supports these models with various theoretical ideas from economics, mathematics, physics and ecology. The models use basic economic variables to give straightforward explanations of the distributions of wealth, income and company sizes in human societies. The models also explain the source of macroeconomic business cycles, including bubble and crash behaviour. The models give simple formulae for wealth distributions, and also for the Bowley ratio; the ratio of returns to labour and capital. Usefully, the models also provide simple effective methods for eliminating poverty without using tax and welfare. The theoretical ideas provide a framework for extending this modelling approach systematically across economics. 6 EFTA00625134 The models were produced firstly by taking basic ideas from classical economics and basic finance. These ideas where then combined with the mathematics of chaotic systems and dynamic statistical mechanics, in a process that I think can be well summed up as 'econodynamics' as it parallels the approaches of thermodynamics, and ultimately demonstrates that economics is in fact a subset of thermodynamics. This makes the process sound planned. It wasn't. It was a process of common sense and good luck combined with a lot of background reading. It was suggested to me in 2006 That the generalised Lotka-Volterra (GLV) distribution might provide a good fit for income data. The suggestion proved to be prescient. The fit to real data proved to be better than that for other previously proposed distributions. At this point, in 2006, I used my limited knowledge of economics to propose two alternative models that might fit the simplest economically appropriate terms into two different generating equations that produce the (GLV). I passed these ideas forward to a number of physicists. The history of this is expanded in section 12. After that, nothing very much happened for three years. This was for three main reasons. Firstly, I didn't understand the detailed mathematics, or indeed have a strong feel for the generalised Lotka-Volterra model. Secondly, my computer programming, and modelling skills are woeful. Thirdly, the academics that I wrote to had no interest in my ideas. In 2009/2010 I was able to make progress on the first two items above, and in early 2010 I was able, with assistance from George Vogiatzis and Maria Chli, to produce a GLV distribution of wealth from a simulation programme with just nine lines of code, that included only a population of identical individuals, and just the variables of individual wealth (or capital), a single uniform profit rate and a single uniform (but stochastic) consumption (or saving) rate. This simple model reproduced a complex reality with a parsimony found rarely even in pure physics. After a brief pause, the rest of the modelling, research and writing of this paper was carried out between the beginning of May 2010 and the end of March 2011. This was done in something of a rush, without financial support or academic assistance; and I would therefore ask forbearance for the rough and ready nature of the paper. From the first wealth-based model, and with greater knowledge of finance and economics; models for income, companies, commodities and finally macroeconomics dropped out naturally and straightforwardly. The models are certainly in need of more rigorous calibration, but they appear to work well. The wealth and income models appear to be powerful, both in their simplicity and universality, and also in their ability to advise future action for reducing poverty. The macroeconomic models are interesting, as even in these initial simple models, they give outcomes that accord closely with the qualitative descriptions of business and credit cycles in the work of Minsky and the Austrian school of economics. These descriptions describe well the actual behaviour of economies in bubbles and crashes from the Roman land speculation of 33AD through tulipomania and the South Sea bubble up to the recent credit crunch. Part A of this paper goes through these various models in detail, discussing also the background and consequences of the models. 7 EFTA00625135 The agents in the initial models were identical, and painfully simple in their behaviour. They worked for money, saved some of their money, spent some of their money, and received interest on the money accumulated in their bank accounts. Because of this the agents had no utility or behavioural functions of the sort commonly used in agent-based economic modelling. As such the models had no initial underlying references to neoclassical economics, or for that matter behavioural economics. There simply was no need for neoclassicism or behaviouralism. As the modelling progressed, somewhat to my surprise, and, in fact to my embarrassment, it became clear that the models were modelling the economics of the classical economists; the economics of Smith, Ricardo, Marx, von Neumann (unmodified) and Sraffa. With hindsight this turned out to be a consequence of the second of the two original models I had proposed in 2006. In this model wealth is implicitly conserved in exchange, but created in production and destroyed in consumption. Ultimately total wealth is conserved in the long term. This model denies the premises of neoclassicism, and adopts an updated form of classical economics. Despite the rejection of neoclassicism, the models work. Classical economics works. Where the classical economists were undoubtedly wrong was in their belief in the labour theory of value. They were however absolutely correct in the belief that value was intrinsic, and embodied in the goods bought, sold and stored as units of wealth. Once intrinsic wealth, and so the conservation of wealth is recast and accepted, building economic models becomes surprisingly easy. The re-acceptance of intrinsic wealth; and so the abandonment of neoclassicism, is clearly controversial. Given the wild gyrations of the prices of shares, commodities, house prices, art works and other economic goods, it may also seem very silly. Because of this a significant section of part B of this paper discusses these issues in detail, and the economic and finance background in general. The other main aim of part B of this paper is to introduce the ideas of chaotic systems, statistical mechanics and entropy to those that are unfamiliar with them. Partly because of these theoretical discussions this paper is somewhat longer than I initially expected. This is mainly because I have aimed the paper at a much larger audience than is normal for an academic paper. In my experience there are many people with a basic mathematical background, both inside and outside academia, who are interested in economics. This includes engineers, biologists and chemists as well as physicists and mathematicians. I have therefore written the paper at a level that should be relatively easy to follow for those with first year undergraduate mathematics (or the equivalent of a UK A-level in maths). Although the numbers are much smaller, I believe there is also a significant minority of economists, especially younger economists, who are acutely aware that the theory and mathematical tools of economics are simply not adequate for modelling real world economies. This paper is also aimed at these economists. 8 EFTA00625136 I would not be particularly surprised if every single model in this paper has to be reworked to make them describe real world economies. It may even be the case that many of the models have to be superseded. This would be annoying but not tragic, but is beside the point. The main point of this paper is the power of the mathematical tools. The two main tools used in this paper are chaotic differential equation systems and statistical mechanics. In both cases these tools are used in systems that are away from what are normally considered equilibrium positions. It is these tools that allow the production of simple effective economic models, and it is these tools that economists need in order to make progress. Comparative statics may be intellectually satisfying and neat to draw on a blackboard, but it doesn't work in dynamic multi-body systems. For a dynamic system you need dynamic differential equation models. For systems with large numbers of interacting bodies you need statistical mechanics and entropy. Although a minority of economists have toyed with chaos theory, and many economists claim to use 'dynamic' models, I have only encountered one economist; Steve Keen, who truly 'gets' dynamic modelling in the way that most physicists, engineers and mathematical modellers use dynamic modelling. Indeed the macroeconomic model in this paper shares many ideas with, and certainly the approaches of, Steve Keen who has used dynamical mathematical models to follow the ideas of Goodwin, Minsky and others; and who has used the Lotka-Volterra dynamics in particular. Although Keen's models are certainly heterodox he is almost unique in being an economic theoretician who predicted the credit crunch accurately and in depth. While other economists predicted the credit crunch, almost all the others who did so did this from an analysis of repeating patterns of economic history. That is, they could spot a bubble when they saw one. Steve Keen is unusual in being a theoretical economist who is able to model bubbles with a degree of precision. The use of statistical mechanics in economics is even more frustrating. Merton, Black and Scholes cherry-picked the diffusion equation from thermodynamics while completely ignoring its statistical mechanical roots and derivation. They then sledge-hammered it into working in a neoclassical framework. Tragically, a couple of generations of physicists working in finance have not only accepted this, but they have built more and more baroque models on these flimsy foundations. The trouble with Black-Scholes is that it works very well, except when it doesn't. This basic flaw has been pointed out from Mandlebrot onwards, to date with no notice taken. This is most frustrating. If physicists were doing their jobs properly, finance would be one of the simplest most boring parts of economics. The only economist I have encountered who truly 'gets' statistical mechanics is Duncan Foley. He is uniquely an economist who has fully realised not only the faults with the mathematics used by most economists, but also dedicated considerable effort to applying the correct mathematics, statistical mechanics, to economics. Although primarily modelled in a static environment, Foley's work is profoundly insightful, and demonstrates very clearly how statistical mechanical approaches are more powerful than utility based approaches, and how statistical mechanics approaches naturally lead to the market failures seen in real economies. Despite this visionary insight he has ploughed a somewhat lonely furrow, with the relevant work largely ignored by economists, and more embarrassingly also by physicists. 9 EFTA00625137 Because chaos and statistical mechanics are unfamiliar in economics, I have spent some effort in both the modelling sections and the theory sections in explaining how the models work in detail, how these concepts work in general, and why these mathematical approaches are not just relevant but essential for building mathematical models in economics. This extra explanation for less mathematical scientists and economists may mean that the paper is over-explained and repetitive for many physicists and mathematicians. For this I can only offer my apologies. However, even for physicists some of the background material in the discussions on entropy contains novel and powerful ideas regarding non-equilibrium thermodynamic systems. This is taken from recent work in the physics of planetary ecology and appears not to have percolated into the general physics community despite appearing to have general applicability. The ideas of Paltridge, Lorenz, Dewar and others, along with the mathematical techniques of Levy & Solomon, may not be familiar to many physicists, and I believe may be very powerful in the analysis of complex 'out of equilibrium' systems in general. In fact, although I was trained as a physicist, I am not much of a mathematician, and by emotional inclination I am more of an engineer. My skills lie mostly in seeing connections between different existing ideas and being able to bolt them together in effective and sometimes simpler ways. Part of the reason for the length of this paper is that I have taken a lot of ideas from a lot of different fields, mainly from classical economics, finance, physics, mathematics and ecology, and fitted them together in new ways. I wish to explain this bolting together in detail, partly because very few people will be familiar with all the bits I have cherry-picked, but also I suspect that my initial bolting together may be less than ideal, and may need reworking and improving. I feel I should also apologise in advance for a certain amount of impatience displayed in my writing towards traditional economics. From an economics point of view the paper gets more controversial as it goes along. It also gets increasingly less polite with regard to the theories of neoclassical economics. In the last two years I have read a lot of economics and finance, a significant proportion of which was not profoundly insightful. Unfortunately, reading standard economics books to find out how real economies work is a little like reading astrology books to find out how planetary systems work. Generally I have found the most useful economic ideas in finance or heterodox economics, areas which are not usually well known to physicists, or indeed many economists. These ideas include recent research in market microstructure, liquidity, post-Keynesian pricing theory as well as the work of Foley, Keen, Smithers, Shiller, Cooper, Pettis, Pepper & Oliver, Mehrling, Lyons and others. Neoclassical economics, while forming an intellectually beautiful framework, has proved of limited use to me as a source of knowledge. Partly this is because the mathematics used, comparative statics, is simply inappropriate. Partly it is because some of the core suppositions used to build the framework; such as diminishing returns and the importance of investment and saving, are trivially refutable. 10 EFTA00625138 The only defence I can make for my impoliteness is a very poor one; that I am considerably more polite than others. If any of my comments regarding neoclassical economics cause offence, I advise you to read the work of Steve Keen and Phillip Mirowski with some caution. Both are trained economists who have the mathematical and historical skills to realise the inappropriateness of neoclassicism. Their writing has the polemical edge of a once devout Christian who has recently discovered that the parish priest has been in an intimate liaison with his wife for the last fifteen years. Finally I would like to comment on the work of Ian Wright, Makoto Nirei & Wataru Souma and others. Throughout this paper comparisons are made to the work of Ian Wright who describes simulated economic models in two notable papers [Wright 2005, 2009]. Wright's models are significantly different to my own, most notably in not involving a financial sector. Also, unlike the present paper, Wright takes a 'black box' and 'zero intelligence' approach to modelling which eschews formal fitting of the models to mathematical equations. Despite these profound differences, at a deeper level Wright's models share fundamental similarities with my own, sharing the basic conservation of value of the classical economists, as well as using a dynamic, stochastic, statistical mechanical approach. More significantly, the models are striking in the similarities of their outputs to my own work. Also it is important to note that Wright's models have a richness in some areas, such as unemployment which are missing from my own models. In relevant sections I discuss detailed differences and similarities between the models of Wright and myself. In two papers Souma & Nirei [Souma & Nirei 2005, 2007] build a highly mathematical model that produces a power tail and an exponential distribution for income. Their approach also builds ultimately on the work of Solomon & Levy. However their approach is substantially more complex than my own. Their models do however share a number of similarities to my own models. Firstly, the models of Souma & Nirei use consumption as the negative balancing term in their model in a manner almost identical to the role of consumption in my own model. Secondly, their models ascribe a strong positive economic role to capital as a source of wealth, however this is ascribed to the process of capital growth, not the dividends, interest, rent, etc that is used in my own models. Both Wright's work and that of Souma & Nirei predate this paper. Their work also predates my original models produced in 2006. Given the process by which I came to produce the models below, I believe I did so independently of Wright, Souma & Nirei. However, I would be very foolish to discount that possibility that I was subconsciously influenced by these authors, and so I do not discount this. It is certainly clear to me that Wright, Souma & Nirei have made very substantial inroads in the same directions as my own research, and that if I had not had lucky breaks in advancing my own research, then one or other of them would have produced the models below within the near future. Given that the work of Wright, Souma & Nirei predates my own, and so gives rise to questions of originality, I have included a brief history of the gestation of the present paper in section 12, History and Acknowledgements. With regard to precedence, I would like to note that the general approach for the macroeconomic models in section 4 were partly inspired by the work of Steve Keen, though the 11 EFTA00625139 models themselves grew straight out of my company and commodity models; and ultimately out of my income models. More importantly, not a word of this paper would have been written without the work of Levy & Solomon and their GLV models. Manipulation of the GLV is beyond my mathematical ability. Although Levy & Solomon's economic explanations are naive, their gut feeling of the applicability of the GLV to economics in particular, and complex systems in general, was correct. I believe their work is of profound general importance. In later sections of this paper I quote extensively from the work of Ian Wright, Duncan Foley and Steve Keen, as their explanations of the importance of statistical mechanics and chaos in economics are difficult to improve on. 0.3 Structure of the Paper Part A of this paper discusses a number of economic models in detail, Part A.I discusses a number of straightforward models giving results that easily accord with the real world and also with the models of Ian Wright. Part A.II discusses models that are more speculative. Part B discusses the background mathematics, physics and economics underlying the models in Part A. The mathematics and physics is discussed in Part B.I, the economics in part B.II, the conclusions are in part B.III. Finally, Part C gives appendices. Within Part A; section 1 discusses income and wealth distributions; section 1.1 gives a brief review of empirical information known about wealth and income distributions while section 1.2 gives background information on the Lotka-Volterra and General Lotka-Volterra models. Sections 1.3 to 1.5 gives details of the models, their outputs and a discussion of these outputs. Section 1.6 discusses the effects that changing the ratio of waged income to earnings from capital has on wealth and income distributions. Sections 1.7 and 1.8 discuss effective, low-cost options for modifying wealth and income distributions and so eliminating poverty. Finally, section 1.9 looks at some unexplained but potentially important issues within wealth and income distribution. Sections 2.1 to 2.4 go through the background, creation and discussion of a model that creates power law distributions in company sizes. Sections 3.1 to 3.4 use ideas from section 2, and also the consequences of the delays inherent in installing physical capital, to generate the cyclical spiking behaviour typical of commodity prices. Sections 4.1 to 4.4 combine the ideas from sections 1, 2 and 3 to provide a basic macroeconomic model of a full, isolated economy. It is demonstrated that even a very basic model can endogenously generate cyclical boom and bust business cycles of the sort described by Minsky and Austrian economists. 12 EFTA00625140 In section 4.5 it is demonstrated that an exact formulation for the Bowley ratio; the ratio of returns to labour to total returns, can easily be derived from the basic macroeconomic model above, or indeed from first principles in a few lines of basic algebra. In section 4.6 and 4.7 the above modelling is extended into an economy with debt. From this a more complex, though still simple, formulation for the Bowley ratio is derived. This formulation suggests that excessive debt can be economically dangerous and also directly increases income inequality. The more general consequences of the Bowley ratio for society are discussed in more depth in section 4.8. In section 4.9 two macroeconomic models are arranged in tandem to discuss an isolated economy with a financial sector in addition to an ordinary non-financial sector. In section 4.10 two macroeconomic models are discussed in parallel as a model of two national economies trading with each other. To conclude Part A, section 4.11 introduces the role of the state and monetary economics, while section 4.12 briefly reviews the salient outcomes of the modelling for social equity. In Part B, section 6.1 discusses the differences between static and dynamic systems, while section 6.2 looks at the chaotic mathematics of differential equation systems. Examples of how this knowledge could be applied to housing markets is discussed in section 6.3, while applications to share markets are discussed in section 6.4. A general overview of the control of chaotic systems is given in section 6.5. Section 7.1 discusses the theory; 'statistical mechanics', which is necessary for applying to situations with many independent bodies; while section 7.2 discusses how this leads to the concept of entropy. Section 7.3 discusses how systems normally considered to be out of equilibrium can in fact be considered to be in a dynamic equilibrium that is characterised as being in a state of maximum entropy production. Section 7.4 discusses possible ways that the statistical mechanics of maximum entropy production systems might be tackled. Moving back to economics; in section 8.1 it is discussed how an intrinsic measure of value can be related to the entropy discussed in section 7 via the concept of 'humanly useful negentropy'. Section 8.2 discusses the many serious criticisms of a concept of intrinsic value in general, with a discussion of the role of liquidity in particular. Section 9.1 looks at theories of supply and pricing, the non-existence of diminishing returns in production, and the similarities between the market-microstructure analysis and post-Keynesian pricing theory. Section 9.3 looks for, and fails to find, sources of scarcity, while section 9.4 discusses the characteristics of demand. In section 10 both the theory and modelling is reviewed and arranged together as a coherent whole, this is followed by brief conclusions in section 11. Sections 12 to 16 are appendices in Part C. Section 12 gives a history of the gestation of this paper and an opportunity to thank those that have assisted in its formation. 13 EFTA00625141 Section 13 gives a reading list for those interested in learning more about the background maths and economics in the paper. Section 14 gives details of the Matlab and Excel programmes used to generate the models in Part A of the paper. Sections 15 and 16 give the references and figures respectively. 14 EFTA00625142  Part A — Some Models Section A.I — Heavy Duty Models 1. Wealth & Income Models 1.1 Wealth & Income Data — Empirical Information "Endogeneity of distribution Neoclassical economics approaches the problem of distribution by positing a given and exogenous distribution of ownership of resources. The competitive market equilibrium then determines the relative value of each agent's endowment (essentially as rents). I think there are problems looming up with this aspect of theory as well. One reason to doubt the durability of the assumption of an exogenous distribution of ownership of resources is that income and wealth distributions exhibit empirical regularities that are as stable as any other economic relationships. I think there is an important scientific payoff in models that explain the size distributions of wealth and income as endogenous outcomes of market interactions." Duncan K. Foley [Foley 1990] Within theoretical economics, the study of income and wealth distributions is something of a backwater. As stated by Foley above, neo-classical economics starts from given exogenous distributions of wealth and then looks at the ensuing exchange processes. Utility theory assumes that entrepreneurs and labourers are fairly rewarded for their efforts and risk appetite. The search for deeper endogenous explanations within mainstream economics has been minimal. This is puzzling, because, as Foley states, it has been clear for a century that income distributions show very fixed uniformities. Vilfredo Pareto first showed in 1896 that income distributions followed the power law distribution that now bears his name [Pareto 1896]. Pareto studied income in Britain, Prussia, Saxony, Ireland, Italy and Peru. At the time of his study Britain and Prussia were strongly industrialised countries, while Ireland, Italy and Peru were still agricultural producers. Despite the differences between these economies, Pareto discovered that the income of wealthy individuals varied as a power law in all cases. Extensive research since has shown that this relationship is universal across all countries, and that not only is a power law present for high income individuals, but the gradient of the power law is similar in all the different countries. Typical graphs of income distribution are shown below. This is data for 2002 from the UK, and is an unusually good data set [ONS 2003]. Figure 1.1.1 here 15 EFTA00625143 Figure 1.1.1 above shows a probability density function. A probability distribution function (pdf) is basically a glorified histogram or bar chart. Along the x-axis are bands of wage. The y-axis shows the number of people in each wage band. As can be seen this shape has a large bulge towards the left-hand side, with a peak at about £300 per week. To the right hand side there is a long tail showing smaller and smaller numbers of people with higher and higher earnings. Also included in this chart is a log-normal distribution fitted to the curve. The log-normal distribution is the curve that economists normally fit to income distributions (or pretty much anything else that catches their attention). On these scales the log-normal appears to give a very good fit to the data. However there are problems with this. Figure 1.1.2 here Figure 1.1.2 above shows the same data, but this time with the y-axis transformed into a log scale. Although the log-normal gives a very good fit for the first two thirds of the graph, somewhere around a weekly wage level of £900 the data points move off higher than the log- normal fit. The log-normal fit cannot describe the income of high-earners well. Figure 1.1.3 here Figure 1.1.3 above shows the same data but organised in a different manner. This is a 'cumulative density function' or cdf. In this graph the wealth is still plotted along the x-axis, but this time the x-axis is also a log scale. This time the y-axis shows the proportion of people who earn more than the wage on the x-axis. In figure 1.1.3 about 10% of people, a proportion of 0.1, earn more than £755 per week. It can be seen that the curve has a curved section on the left-hand side, and a straight line section on the right-hand side. This straight section is the 'power-tail' of the distribution. This section of the data obeys a 'power-law' as described by Pareto 100 years ago. The work of Pareto gives a remarkable result. An industrial manufacturing society and an agrarian society have very different economic systems and societal structures. Intuitively it seems reasonable to assume that income would be distributed differently in such different societies. What the data is saying is that none of the following have an effect on the shape of income distribution in a country: • Whether wealth is owned as industrial capital or agricultural land • Whether wealth is owned directly or via a stock market • What sort of education system a country has 16 EFTA00625144  • What sort of justice system a country has • Natural endowments of agricultural land or mineral wealth • And so on with many other social and economic factors Intuitively it seems reasonable that any or all of the above would affect income distribution, in practice none of them do. Income distributions are controlled by much deeper and basic processes in economics. The big unexpected conclusion from the data of Pareto and others is the existence of the power tail itself. Traditional economics holds that individuals are fairly rewarded for their abilities, a power tail distribution does not fit these assumptions. Human abilities are usually distributed normally, or sometimes log-normally. The earning ability of an individual human being is made up of the combination of many different personal skills. Logically, following the central limit theorem, it would be reasonable to expect that the distribution of income would be a normal or log-normal distribution. A power law distribution however is very much more skewed than even a log-normal distribution, so it is not obvious why individual skills should be overcompensated with a power law distribution. While Pareto noted the existence of a power tail in the distribution, it should be noted that more recently various authors have suggested that there may be two or even three power tail regions, with a separation between the 'rich' and 'super-rich', see for example [Borges 2002, Clementi & Gallegati 2005b, Souma, Nirei & Souma 2007]. While the income earned by the people in the power tail of income distribution may account for approximately 50% of total earnings, the Pareto distribution actually only applies to the top 10%-20% of earners. The other 80%-90% of middle class and poorer people are accounted for by a different 'body' of the distribution. Going back to the linear-linear graph in figure 1.1.1 it can be seen that, between incomes of £100 and £900 per week, there is a characteristic bulge or hump of individuals, with a skew in the hump towards the right hand side. In the days since Pareto the distribution of income for the main 80%-90% of individuals in this bulge has also been investigated in detail. The distribution of income for this main group of individuals shows the characteristic skewed humped shape similar to that of the log-normal distribution, though many other distributions have been proposed. These include the gamma, Weibull, beta, Singh-Maddala, and Dagum. The last two both being members of the Dagum family of distributions. Bandourian, McDonald & Turley [Bandourain et al 2002] give an extensive overview of all the above distributions, as well as other variations of the general beta class of distributions. They carry out a review of which of these distributions give best fits to the extensive data in the Luxembourg Income Study. In all they analyse the fit of eleven probability distributions to twenty-three different countries. They conclude that the Weibull, Dagum and general-beta2 distributions are the best fits to the data depending on the number of parameters used. 17 EFTA00625145 For more information, readers are referred to 'Statistical Size Distributions in Economics and Actuarial Sciences' [Kleiber & Kotz 2003] for a more general overview of probability distributions in economics, and also to Atkinson and Bourguignon [Atkinson & Bourguignon 2000] for a very detailed discussion of income data and theory in general. The author has analysed a particularly good set of income data from the UK tax system, one example is shown in figures 1.1.1-3 above. This data suggests that a Maxwell-Boltzmann distribution also provides a very good fit to the main body of the income data that is equal to that of the log-normal distribution [Willis & Mimkes 2005]. The reasons for the split between the income earned by the top 10% and the main body 90% has been studied in more detail by Clementi and Gallegati [Clementi & Gallegati 2005a] using data from the US, UK, Germany and Italy. This shows strong economic regularities in the data. In general it appears that the income gained by individuals in the power tail comes primarily from income gained from capital such as interest payments, dividends, rent or ownership of small businesses. Meanwhile the income for the 90% of people in the main body of the distribution is primarily derived from wages. These conclusions are important, and will be returned to in the models below. This view is supported, though only by suggestion, by one intriguing high quality income data set. This data set comes from the United States and is from a 1992 survey giving proportions of workers earning particular wages in manufacturing and service industries. The ultimate source of the data is the US Department of Labor; Bureau of Statistics, and so the provenance is believed to be of the good quality. Unfortunately, enquiries by the author has failed to reveal the details of the data, such as sample size and collection methodology. The data was collected to give a comparison of the relative quality of employment in the manufacturing and service sectors. Although the sample size for the data is not known, the smoothness of the curves produced suggest that the samples were large, and that the data is of good statistical quality. The data for services is shown in figures 1.1.4 & 1.1.5 below, the data for manufacturing is near identical. Figure 1.1.4 here Figure 1.1.5 here Like the UK data, there appears to be a clear linear section in the central portion of the data on a log-linear scale in figure 1.1.5, indicating an exponential section in the raw data. Again this data can be fitted equally well with a log-normal or a Maxwell-Boltzmann distribution. What is much more interesting is that, beyond this section, the data heads rapidly lower on the logarithmic scale. This means it is heading rapidly to zero on the raw data graph. With these two distributions there is no sign whatsoever of the 'power tail' that is normally found in income distributions. 18 EFTA00625146 It is the belief of the author that the methodology for this US survey restricted the data to 'earned' or 'waged' income, as the interest in the project was in looking at pay in services versus manufacturing industry. It is believed income from assets and investments was not included as this would have been irrelevant to the investigation. This US data set has been included for a further reason, a reason that is subtle; but in the belief of the author, important. Looking back at figure 1.1.1 for the UK income data, there is a very clear offset from zero along the income axis. That is the curve does not start to rise from the income axis until a value of roughly £100 weekly wage. The US data shows an exactly similar offset, with income not rising until a weekly wage of $100. This is important, as the various curves discussed above (log-normal, gamma, Weibull, beta, Singh-Maddala, Dagum, Maxwell-Boltzmann, etc) all normally start at the origin of the axis, point (0,0) with the curve rising immediately from this point. While it is straightforward enough to put an offset in, this is not normally necessary when looking at natural phenomena. In the 1930s Gibrat, an engineer, pioneered work in economics that studied work on proportional growth processes that could produce log-normal or power law distributions depending on the parameters. His work primarily looked at companies, and was the first attempt to apply stochastic processes to produce power law distributions. Following the work of Pareto, the details of income and wealth distributions have rarely been studied in mainstream theoretical economics, a notable and important exception being Champernowne. Champernowne was a highly gifted mathematician who was diverted into economics, he was the first person to bring a statistical mechanical approach to income distribution, and also noted the importance of capital as a major creator of inequality, though his approach concentrated on generational transfers of wealth [Champernowne & Cowell 1998]. Despite the lack of interest within economics, this area has had a profound attraction to those outside the economics profession for many years, a review of this history is provided by Gabaix [Gabaix 2009]. In recent years, the study of income distributions has gone through a small renaissance with new interest in the field shown by physicists with an interest in economics, and has become a significant element of the body of research known as 'econophysics'. Notable papers have been written in this field by Bouchaud & Mezard, Nirei & Souma, Dragulescu & Yakovenko, Chatterjee & Chakrabarti, Slanina, Sinha and many, many, others [Bouchaud & Mezard 2000, Dragulescu & Yakovenko 2001, Nirei & Souma 2007, Souma 2001, Slanina 2004, Sinha 2005]. The majority of these papers follow similar approaches; inherited either from the work of Gibrat, or from gas models in physics. Almost all the above models deal with basic exchange processes, with some sort of asymmetry introduced to produce a power tail. Chatterjee et al 2007, Chatterjee & Chakrabarti 2007 and Sinha 2005 give good reviews of this modelling approach. The approaches above have been the subject of some criticism, even by economists who are otherwise sympathetic to a stochastic approach to economics, but who are concerned that a 19 EFTA00625147 pure exchange process is not appropriate for modelling modern economies [Gallegati et al 2006]. An alternative approach to stochastic modelling has been taken by Moshe Levy, Sorin Solomon, and others [Levy & Solomon 1996]. They have produced work based on the 'General Lotka-Volterra' model. Unsurprisingly, this is a generalised framework of the 'predator-prey' models independently developed for the analysis of population dynamics in biology by two mathematicians/physicists Alfred Lotka and Vito Volterra. A full discussion of the origin and mathematics of GLV distributions is given below in section 1.2. These distributions are interesting for a number of reasons; these include the following: • the fundamental shape of the GLV curve • the quality of the fit to actual data • the appropriateness of the GLV distribution as an economic model Figure 1.1.6 here Figure 1.1.7 here With regard to the fundamental shape of the GLV curve, figures 1.1.6 and 1.1.7 above show plots of the UK income data against the GLV on a linear-linear and log-log plot. The formula for this distribution is given by: P(w) = K(e' r""•)/((w/L)" +"1) (1.1a) and it has three parameters; K is a general scaling parameter, L is a norm