Debunking Economics Part 25

Weaknesses One of this school's great strengths is also a weakness. Unlike the neocla.s.sical or Marxian schools, they do not have a 'theory of value' they have nothing to compare to the theory of utility maximization, or even the labor theory of value.

This is certainly a strength when one considers how these theories of value have led these rival schools up intellectual garden paths. However, at the same time it means that post-Keynesians lack a methodological consistency: they are more united by what they oppose than by what they have in common though there are many common threads.

This lack of a theory of value makes it difficult for post-Keynesians to explain why their approach is superior to fledgling students of economics, who have yet to confront any of the intellectual conundrums which afflict neocla.s.sical economics (and they also have difficulty communicating with radical students who are attracted to Marxism). One must normally become disenchanted with mainstream or Marxian economics before one can become a post-Keynesian. That is perhaps too tortuous a path to rely upon, if this school ever hopes to gain the ascendancy in economics.

A final problem is that, despite their rejection of neocla.s.sical economics, they tend to also use static logic in their a.n.a.lysis even though their building blocks might be, for instance, markup pricing rather than 'marginal cost equals marginal revenue.' This lack of appreciation of how different dynamic a.n.a.lysis is from static is not universal among post-Keynesian authors, but it is widespread enough to be a problem. However, it must be said that younger members of the post-Keynesian school are much more comfortable with dynamic a.n.a.lysis than are some of its older members.

The post-Keynesian scorecard Despite the lack of an agreed methodological foundation, the post-Keynesians are easily the most coherent alternative school of economic thought today. They are also likely to gain substantial credence in the event of a financial crisis, given their explicitly monetary approach to economics.

Sraffian economics No prizes for guessing which economist provided the major inspiration to this particular group of economists. Sraffa's Production of Commodities by Means of Commodities became the icon for these economists. As well as applying it to critique other schools notably neocla.s.sical economics and Marxism they attempted to turn it into a means to a.n.a.lyze the real economy.

Strengths There is no doubt that Sraffa's a.n.a.lysis const.i.tuted the most detailed and careful a.n.a.lysis of the mechanics of production in the history of economics. Not for him any simple abstractions, such as the neocla.s.sicals' 'factors of production,' or even Marx's 'industry sectors': his model a.n.a.lyzed the interrelations of production at the level of the individual commodity.

This study turned up many subtleties that escaped other schools of economics: the dependence of the 'quant.i.ty of capital' on the profit rate, rather than vice versa, the phenomenon of reswitching, etc. No other school of economics matches the Sraffians on this insistence of a.s.sumption-free rigor. Well, almost a.s.sumption-free rigor.

Weaknesses The one a.s.sumption Sraffians do make is that the economy can be a.n.a.lyzed using static tools. As a result, even though the proper treatment of time was an essential component of Sraffa's critique of neocla.s.sical economics, modern Sraffian economics makes no use of time or dynamics. Ian Steedman gave the pithiest explanation of why Sraffians a.n.a.lysis ignores dynamics. It is because '"static" a.n.a.lysis does not "ignore" time. To the contrary, that a.n.a.lysis allows enough time for changes in prime costs, markups, etc., to have their full effects' (Steedman 1992).

This proposition can be true only if the long-run position of an economy is an equilibrium one: if, in other words, the economy has just one equilibrium, and it is stable. As Chapter 9 showed, this is highly unlikely to be the case. A market economy is likely to have multiple equilibria, and they are all likely to be unstable. The Sraffian position is thus ignorant of modern dynamic a.n.a.lysis.

Sraffians also have one other flaw: they pay too little attention to Piero Sraffa.

Some post-Keynesians are fond of pointing out how pedantic Sraffa was, and therefore how important was the subt.i.tle to his magnum opus. Sitting beneath the t.i.tle of The Production of Commodities by Means of Commodities was the caveat 'Prelude to a critique of economic theory.'

In other words, these economists argue that Sraffa's method was intended solely to provide a means to critique other economic theories: it was never meant to provide a basis for an economic theory in itself.

Sraffa's 1926 paper provides support for this position. When discussing how the firm should be modeled, Sraffa put great stress upon the issues of importance to 'business men': the necessity and expense of marketing a non-h.o.m.ogeneous product to a market of non-h.o.m.ogeneous consumers, the cost and dangers of credit as a major force limiting firm size, etc. The concepts Sraffa discusses here can be considered only with extreme difficulty in the framework of his 1960 book (check the web link Alternatives/Sraffa for a relevant extract from Sraffa's 1926 paper).

The Sraffian scorecard Though the Sraffian school was fairly influential up until 2000, there have been few developments in it since, certainly in comparison to the growth in post-Keynesian economics since that date.

Complexity theory and Econophysics Complexity theory is not so much a school of thought in economics as a group of economists who apply what is popularly known as 'chaos theory' to economic issues. Since the first edition of this book, there has also been an enormous growth in the number of physicists taking an active interest in economics and finance, and this new school of 'Econophysics' has largely subsumed the complexity theory approach.

The concept of chaos itself was first discovered in 1899 by the French mathematician Henri Poincare. However, knowledge of it languished until the mid-1960s because it could not be fully explored until after the invention of computers. Chaotic models of necessity cannot be understood simply by writing down the equations which represent them: instead, they must be simulated, and their properties a.n.a.lyzed numerically. This was simply not possible before the advent of computers.

An essential aspect of complexity is the existence of nonlinear relations.h.i.+ps between elements of a system, and the apparent ability of complex systems to 'self-organize.' The Lorenz model, noted in Chapter 8, has both these attributes: the nonlinear relations between displacement and temperature lead to behavior which on the surface is 'chaotic,' but behind which lies the beautiful organizing force of Lorenz's 'strange attractor.' Complexity theorists argue that the economy demonstrates similar attributes, and these are what give rise to the cycles which are a self-evident aspect of real-world economies.

Econophysics substantially adds to the contribution made by the early proponents of complexity in economics such as Richard Goodwin (Goodwin 1990, 1991), Benoit Mandelbrot (Mandelbrot 1971, 2005), Hans-Walter Lorenz (Lorenz 1987a, 1987b, 1989), Paul Ormerod (Ormerod 1997, 2001, 2004); Ormerod and Heineike (2009), Carl Chiarella (Chiarella and Flaschel 2000, Chiarella, Dieci et al. 2002, Chiarella et al. 2003) and myself, among many others by bringing both the techniques and the empirical mindset of physicists to bear upon economic data.

Over the last century, physicists have developed a vast array of techniques to interpret the equally vast range of physical processes encountered in everything from fluid dynamics to the behavior of subatomic particles. Their approach has been fundamentally empirical, and devoid of any a priori a.s.sumption that physical processes occur in equilibrium and the concept of equilibrium itself is far more richly specified.

These techniques have enabled Econophysicists to make substantial progress in understanding how finance markets in particular actually operate, with a range of models that accurately capture the 'fat tails' that bedevil a.s.set price data and lie well outside the predictive capacity of neocla.s.sically inspired models.

Strengths It is extremely difficult to work in complexity theory and not understand dynamics. Though some neocla.s.sical dabblers occasionally attempt to use equilibrium thinking in so-called chaotic models, in the main pract.i.tioners in this camp are extremely well versed in dynamics.

They are also normally very competent in mathematics; far more so, not only than other alternative schools of economics, but also than most neocla.s.sicals. Many complexity theorists in economics started out doing PhDs in physics, biology, or mathematics itself, and later delve into economics out of curiosity.

This alone has meant that complexity theorists have had a significant impact upon the profession. While they rarely indulge in direct attacks upon neocla.s.sical economics per se, neocla.s.sical economists are aware that they are quite capable of doing so if provoked. This technical superiority over neocla.s.sical economists has taken the mathematical 'big stick' out of the hands of neocla.s.sicals. This has been taken to another level by Econophysicists, whose training in mathematics and computing is far more rigorous and extensive than that undertaken by economists.

Weaknesses Though many complexity economists are inclined to a post-Keynesian perspective on economics, in general they lack a full appreciation of the history of economic thought. For this reason, they will often generate models which combine incompatible streams in economics. Concepts such as IS-LM and rational expectations often crop up in complexity or Econophysics models of the economy, with the authors rarely being aware of the origins of these 'tools.'

While Econophysics has developed a very rich and empirically based a.n.a.lysis of financial markets to date, and their statistical a.n.a.lysis here involving concepts like Power Law distributions and Tsallis-statistics is far more accurate than neocla.s.sical models, success here has led to neglect of the 'econo' part of the developing discipline's name: at present it could more accurately be called 'Finaphysics' than 'Econophysics.'

Econophysicists also occasionally succ.u.mb to the temptation to introduce one of the strongest weapons in their a.r.s.enal, which I believe has no place in economics: 'conservation laws.' These apply where some fundamental aspect of a system such as, for example, the amount of ma.s.s and energy in the universe is not altered by the physical processes that apply to it, though its distribution and nature may alter. This condition that 'the change in the amount of X equals zero' has been the source of many of the greatest advances in physics, including the derivation of the theory of relativity.

No such equivalent concepts exist in economics, which is more akin to biology than physics in this respect: biological populations fluctuate, and there is no law requiring the ma.s.s of biological ent.i.ties to remain constant, for example. Consequently economics belongs to the cla.s.s of dynamical systems known as 'dissipative,' rather than 'conservative.'

A concern that conservation laws were being introduced into areas where they did not belong for example, the a.n.a.lysis of money (Patriarca, Chakraborti et al. 2004; Ding, Xi et al. 2006) or the distribution of wealth led me to contribute to a paper that was critical of recent developments in Econophysics (Gallegati, Keen et al. 2006). However, over time I expect developments like these to dissipate, given the innately empirical focus of physicists.

The complexity scorecard Complexity theory and Econophysics are among the 'glamour' areas of science in general today, and this affects economics, even given its relative isolation from the scientific mainstream. The techniques which complexity modelers in economics employ are thus 'refertilizing' economics with concepts from other disciplines. The economic fixation upon equilibrium appears quaint to these mathematically literate economists, and this alone may significantly undermine the hold which static thinking has on economics.

If statics were to die, then inevitably so too would neocla.s.sical economics, since its way of thinking is unsustainable in dynamics. So Econophysics may be a harbinger of real change in economics, after sixty years of effective ossification.

Evolutionary economics Evolutionary economics draws its inspiration from the theory of evolution. In this, it has much in common with the majority of physical sciences, which in recent years have started to apply the concept of evolution so much so that it has been proposed that Darwinism is the 'universal' basis of science (Nightingale 2000).2 In all sciences, the basic building blocks of the evolutionary way of thinking are diversity, the environment, and adaptation. Diversity gives a range of possible 'solutions' to the challenges thrown up by the environment. The environment interacts with these diverse forms to favor some over others and the environment itself may be altered by feedback between it and these newly emergent species (Levins and Lewontin 1986). Adaptation occurs at the systemic level through the differential survival of some of these diverse forms: while some die out, others prosper, and thus their characteristics are pa.s.sed on more strongly to subsequent generations.

The economic equivalents of diversity are the heterogeneity of consumers, and the variety of commodities; the equivalents of adaptation are new product development, and the consequent endogenous alteration of consumer tastes; the equivalent of the environment is the economy itself, which is endogenously created by the actions of myriad individuals, social groups and corporations.

Strengths It is undeniable that the economy is an evolutionary system with the one embellishment that change in economics is often purposeful, as opposed to the random nature of variation in the environment (though of course, purposive change can fail to achieve its intended ends).3 This self-evident fact was the basis for Thorstein Veblen's query, over a century ago, of 'Why is economics not an evolutionary science?'

Manifestly it should be, and this alone should be a major factor in the rise of evolutionary thinking in economics.

Weaknesses One problem with evolutionary systems is that, effectively, everything can change. Economists, on the other hand, have been wedded to the notion of 'ceteris paribus' ('all other things remaining equal') as a way of being able to impose some order on the apparent chaos of the market.

Ceteris paribus is of course an illusion, but the illusion often seems preferable to reality when it appears that fully acknowledging reality forces one to abandon structure.

This, of course, is not correct: evolutionary modeling still has structure, as shown by the advances made in genetics and many other areas where evolutionary thinking rules. However, economists are thrown back upon a.n.a.logy here, since in economic systems there is no comparable ent.i.ty to the gene, nor to the processes of biological interaction.

The difficulty evolutionary economists face is developing a.n.a.lytic tools which are consistent with evolution, and yet which still enable meaningful statements to be made about economic issues. Generally these have to include computer simulation, but unfortunately economists receive no training in computer programming. Fortunately, many students arrive at university with these skills already, and programming tools for evolutionary modeling such as NetLogo (ccl.northwestern.edu/netlogo/) and Repast (repast.sourceforge.net/) are far more accessible than their predecessors of even a decade ago.

The evolutionary scorecard Evolutionary economics is still in its infancy, and a lot of its time is spent defining basic philosophical concepts at one extreme, and developing computer-based evolutionary economic models at the other. If it can coalesce into a coherent school of thought with effective a.n.a.lytic tools, then it could at last make economics what, one century ago, Veblen knew it should be: an evolutionary science.

W(h)ither economics?

We are now well into the economic crisis that I antic.i.p.ated in the first edition of this book in 2000, and which I (and a handful of other non-neocla.s.sical economists) had actively warned of since late 2005. The public backlash against neocla.s.sical economics that I expected this would cause has also occurred, with one-time supporters like The Times of London's economic columnist Anatole Kaletsky now openly attacking it: These are just a few examples of the creative thinking that has started again in economics after 20 years of stagnation. But the academic establishment, discredited though it is by the present crisis, will fight hard against new ideas. The outcome of this battle does not just matter to academic economists. Without a better understanding of economics, financial crises will keep recurring and faith in capitalism and free markets will surely erode. Changes in regulation are not sufficient after this financial crisis it is time for a revolution in economic thought. (Kaletsky 2009) Now that the need for 'a revolution in economic thought' is more widely acknowledged, the question is, how to achieve it?

I have no faith in the capacity of academic economics to reform itself. The historic record on this front is evidence enough: Keynes's challenge was a.s.similated and emasculated within a year of it being made by Hicks's IS-LM misinterpretation, and within thirty years all semblance of the change Keynes wished to cause had been eliminated. The misinterpretation of Fisher's debt-deflation hypothesis dismembered the one other substantive challenge to the neocla.s.sical equilibrium, non-monetary mindset. Consequently, the neocla.s.sical orthodoxy that dominated academic economics prior to this crisis was even more extreme, virulent and intolerant of alternative approaches than that which Keynes and Fisher tried to challenge during the Great Depression.

Though there have been some signs of contrition and realization that the core of neocla.s.sical economics may not be the perfect jewel they once believed it was, the overwhelming reaction of neocla.s.sical economists to this crisis has been to maintain business as usual. I attended the 2011 American Economic a.s.sociation meeting in Denver this year, at which there was a session on 'the 50th Anniversary of Rational Expectations.' What should have been a wake was in fact a celebration, and when one of its proponents was asked what economics would be like in fifty years, he was adamant that 'rational expectations' would still be at the heart of macroeconomic modeling.

Not if I can help it. If that fate does eventuate, then there will be another financial crisis right around the corner, and another rebel will have to try to bring about real change where I and my colleagues in reform Edward Fullbrook, Paul Ormerod, Michael Hudson and many others will have failed.

Change, if it is to come now rather than later, will have to be driven by outside influences: by journalists and influential commentators like Kaletsky who now realize how barren the neocla.s.sical approach is; by a public far better informed via the Internet about the weaknesses of conventional economic thinking than was the public of the 1930s; by intellectuals from other disciplines who have long questioned the merits of neocla.s.sical theory and can no longer be rebuffed when the global economy wallows in a crisis that neocla.s.sical economics completely failed to antic.i.p.ate; and by new students who, again via the Internet, now know that there are other ways to think about economics.

There are some encouraging signs today, though only time will tell if these lead to the change that economics desperately needs: * The PAECON ('Protest against Autistic ECONomics') movement that began in France with the rebellion of a group of young economics students has since sp.a.w.ned an international movement, with both a network that unites the many academic opponents of neocla.s.sical economics (www.paecon.net/PAEmovementindex1.htm), a publicly accessible journal, the Real-World Economics Review (www.paecon.net/PAEReview/), and an active blog (rwer.wordpress.com/).

* George Soros has put some of his substantial wealth behind the Inst.i.tute for New Economic Thinking (INET, ineteconomics.org/), in an effort to redress the effective exclusion of non-neocla.s.sical researchers from official funding channels like, to be parochial, the Australian Research Council (which, to be personal, has turned down my applications for funding to develop models of debt deflation nine times since 1996).

* The 'blogosphere,' a phenomenon that has arisen since the first edition of Debunking Economics was published, now allows a plethora of commentators to take pot-shots at conventional thinking on economics. I list my favorites (in no particular order) below; while I don't agree with everything published by these commentators, I agree with a lot, and they are doing serious good in letting people know that economics need serious reform: * Yves Smith at Naked Capitalism: www.nakedcapitalism.com/ * David Hirst at Planet Wall Street (website currently not available) * Dan Denning at www.dailyreckoning.com.au/author/dan/ * Max Keiser at The Keiser Report: maxkeiser.com/ * Mish Shedlock at MISH'S Global Economic Trend a.n.a.lysis: globaleconomica.n.a.lysis.blogspot.com/ * Chris Martenson: www.chrismartenson.com/ * Doug Noland at The PrudentBear: www.prudentbear.com/index.php/commentary/creditbubblebulletin * Harry Dent at www.hsdent.com/ * Edward Harrison at Credit Writedowns: www.creditwritedowns.com/ * Zero Hedge: www.zerohedge.com/ * The Automatic Earth: theautomaticearth.blogspot.com/ * The Levy Inst.i.tute's program: www.levyinst.i.tute.org/ and its blog The Multiplier Effect: www.multiplier-effect.org/ * The University of Missouri Kansas City Economics Department's blog New Economic Perspectives: neweconomicperspectives.blogspot.com/ * The Inst.i.tute for New Economic Thinking's (ineteconomics.org/) blog The Money View: ineteconomics.org/blog * Bill Mitch.e.l.l's Billy Blog: bilbo.economicoutlook.net/blog/ * Michael Hudson one of the Bezemer 12 who foresaw and warned of the 2007 financial crisis, and a leading contributor to the academic literature on the origins on money at michael-hudson.com/ * And my own Steve Keen's Debt.w.a.tch: www.debtdeflation.com/blogs/ Much more than this is needed, however.

Within universities, I would like to see other departments start to offer courses on economics using their methodologies, rather than that of neocla.s.sical economics. Here I believe it is possible to use the ideology of neocla.s.sical economics against it. Neocla.s.sical economists are vehement opponents of monopolies, and yet in the past economics departments have jealously and destructively protected their monopoly on the word 'economics.' However, the empirical failure of neocla.s.sical economics in predicting the Great Recession, and the paucity of alternative approaches within economics departments, is a good reason to remove this monopoly from them. I would be especially pleased to see engineering departments start to offer courses on a Systems Engineering Approach to Economics.

New students of economics can also do their bit. Don't let lecturers get away with teaching the same old stuff during the Great Recession that they taught before. Challenge them about why they exclude money and debt from their macro models, why they pretend to model dynamic processes using comparative statics, and so on. Make a nuisance of yourself and organize with your fellow students to get a voice in designing the curriculum. This is how I began on my path thirty-eight years ago, and it is even more necessary now than it was then and fortunately, there are much better resources to guide you about what an alternative curriculum should include.

Go beyond the standard curriculum too, to learn the skills you will need to be a twenty-first-century economist, rather than a not-yet-extinct fossil from the nineteenth century. Do basic courses in mathematics (calculus, algebra and differential equations), computer programming, history and sociology, rather than the additional fare neocla.s.sical economists prescribe. If you're really lucky, and you have an engineering department that teaches system dynamics (see en.wikipedia.org/wiki/System_dynamics), do those courses. Download and become familiar with programs like QED (www.debtdeflation.com/blogs/qed/), Vensim (vensim.com/), NetLogo, and build your own dynamic models, working from the leads I've given in this book.

Ultimately, I have faith in humanity's ultimate capacity to develop a realistic theoretical perspective on how a complex monetary market economy functions, and to leave behind the neat, plausible and wrong creation that is neocla.s.sical economics.

Whether my faith on this front proves justified or delusional is not up to me, but to you.

NOTES.

Preface.

1 This is actually an application of the 'theory of the second best' (Lancaster and Lipsey 1956). Briefly, Lancaster and Lipsey showed that any single step towards what economics describes as the ideal situation could reduce welfare, if more than one step was required to move from the present situation to the ideal.

Chapter 1.

1 Though somewhat later than I had antic.i.p.ated, since the continued growth of the Subprime Bubble (and Federal Reserve interventions) had papered over the DotCom downturn.

2 There will be resistance aplenty too from government departments, and the bureaucracies of central banks, where promotion has come to those who have held the economic faith.

Chapter 2.

1 The other sub-group calls itself 'New Cla.s.sical.' As I explain in Chapter 10, neither of these subgroups bears any resemblance to either Keynes or the Cla.s.sical School of economic thought. But their battle publicized in the press as a battle between 'Keynesians' and the rest has confused many members of the public into believing that the dominant school of thought in economics at the time of the crisis was 'Keynesian economics.' Nothing could be farther from the truth if Krugman, Woodford and other self-described 'New Keynesians' are Keynesian, then because I can say 'quack,' I am a duck.

2 Bill White, the research director at the Bank of International Settlements, was a notable exception here since he was a proponent of the non-neocla.s.sical 'Financial Instability Hypothesis.'

3 The rate of interest the Federal Reserve charges when it loans to a commercial bank.

4 See media.ft.com/cms/3e3b6ca8-7a08-11de-b86f-00144feabdc0.pdf.

5 mpra.ub.uni-muenchen.de/15892/1/MPRA_paper_ 15892.pdf.

6 If you're a neocla.s.sical economist, you're probably offended by this statement and regard it as a parody; if you're a professional from another discipline say, engineering who has not had any previous exposure to economic theory, you probably regard this as hyperbole. In either case, I'd suggest that you hold judgment until you finish this book.

7 The 'Money and Banking' course at Bernanke's alma mater, where he gave this speech, is a case in point. See www.anababus.net/teach/syllabusECO342.pdf.

8 An interesting instance of this is the observation by Mark Thoma on the Economist's View blog on 'What's wrong with modern macroeconomics: comments' (economistsview.typepad.com/economistsview/2009/11/whats-wrong-with-modern-macroeconomics-comments.html), which shows that he was unaware of significant papers that show the foundations of neocla.s.sical theory are unsound research I discuss in the next chapter: 'One thing I learned from it is that I need to read the old papers by Sonnenschein (1972), Mantel (1974), and Debreu (1974) since these papers appear to undermine representative agent models. According to this work, you cannot learn anything about the uniqueness of an equilibrium, whether an equilibrium is stable, or how agents arrive at equilibrium by looking at individual behavior (more precisely, there is no simple relations.h.i.+p between individual behavior and the properties of aggregated variables someone added the axiom of revealed preference doesn't even survive aggregating two heterogeneous agents).'

9 It is also because complexity theory tends to be incompatible with neocla.s.sical economics, since a common property of complex systems is that they have unstable equilibria: see Chapter 9.

10 Nonlinear 'difference' equations also generate chaos, but economics courses normally cover only linear difference equations.

11 By way of balance, I also know that some of what I say about Marxism will be offensive to Marxist economists.

12 Curiously, academic neocla.s.sical economists don't follow this philosophy themselves: they really believe that they are promoting the common good by developing and teaching neocla.s.sical economics.

13 In the first edition, since my target audience didn't have access to academic journals, I decided to make references to academic papers uncluttered by not giving page references. Since I am now catering for an audience that does have access to those journals, all new references in this edition have page references for quotations.

14 Except in one footnote, where the equation concerns meteorology rather than economics, and can easily be skipped. Occasionally, when some proposition in the text is best stated in mathematical form, I have used words rather than mathematical symbols.

Chapter 3.

1 However it is also also empirically impossible, as I discuss in the addendum.

2 Most of Bentham's endeavors in this regard related to devising a scale of punishments that he regarded as just sufficient to discourage the commission of crime.

3 Jevons, one of the three co-founders of neocla.s.sical economics, was justly skeptical that mathematics could treat all behavior. He argued that 'economy does not treat of all human motives. There are motives nearly always present with us, arising from conscience, compa.s.sion, or from some moral or religious source, which economy cannot and does not pretend to treat. These will remain to us as outstanding and disturbing forces; they must be treated, if at all, by other appropriate branches of knowledge' (Jevons 1866). However, subsequent economists have applied this theory to all behavior, including interpersonal relations.

4 Cardinal refers to the ability to attach a precise quant.i.ty, whereas ordinal refers to the ability to rank things in size order, without necessarily being able to ascribe a numeric value to each.

5 As I point out later, the mathematician John von Neumann developed a way that a cardinal measure of utility could be derived, but this was ignored by neocla.s.sical economists (Von Neumann and Morgenstern 1953: 1729).

6 At its base (where, using my 'bananas and biscuits' example, zero bananas and zero biscuits were consumed), its height was zero. Then as you walked in the bananas direction only (eating bananas but no biscuits), the mountain rose, but at an ever-diminis.h.i.+ng rate it was its steepest at its base, because the very first units consumed gave the greatest 'marginal utility.' The same thing applied in the biscuits direction, while there was some path in the combined 'biscuits and bananas' direction that was the steepest of all to begin with.

7 The Wikipedia entry on contours explains how isobars are derived, and actually mentions indifference curves as an example of them in economics: en.wikipedia.org/wiki/Contour_line.

8 Economists a.s.sume that consumers spend all their income. They treat savings, in effect, as a form of consumption only what is 'consumed' are goods in the future.

9 This is different to indirectly altering the consumer's effective income via a change in prices.

10 Hippasus apparently used the geometry of pentagrams to prove the existence of irrational numbers. The proof by contradiction that the square root of two is irrational, though mathematical, is very easy to understand. See footnote 1.