Functions For Fashion

How might a concern with being up on current fashions in mainstream economics models help perpetuate the current sociology of economics?

Economists could study various economies in various regions and at various stages of development, economic history, the history of economics, epistemology and methodology, various alternative theories (Austrian, Feminist, Institutionalist, Marxist, Post Keynesian, ...), and other social sciences. Studying varieties of mathematical models takes up their time and provides an excuse for remaining ignorant of so much.

If you want to argue against mainstream economics, a mainstream economist can dismiss you as ignorant of some model variation and as attacking a strawperson. Furthermore, this dismissal could be "justified" by just checking whether you have a degree from a small number of schools, and, if you do, just mocking you as not having fully learned what they are teaching. Thus, your time can be taken up with argument about whether you know what you are talking about. The mainstream economist never need get to the point of engaging a critique.

With all these varieties of models, surely one will do better than another in some specific historical circumstance or when applied to some specific time series. But likely another will do better in a different circumstance. Thus, one need never empirically assess mainstream economics as a whole in some prominent field or empirically compare and contrast a mainstream theory to a non-mainstream theory.

Economics Of The Steady State

A steady state is characterized by the economy having a constant rate of growth. I here select a number of expositions of analyses of steady state I have made on this blog:

• The Harrod-Domar model of warranted and natural rates of growth.
• Karl Marx's volume 2 model of simple and expanded reproduction.
• An extension of the Kahn-Kaldor-Pasinetti-Robinson macroeconomic model of income distribution.
• Explanations of aspects of the non-substitution theorem.
• Neoclassical Overlapping Generations (OLG) models with intertemporal utility maximization.

Consider the hypothesis that a consumer's decision to save should be viewed as a choice between current consumption and future consumption so as to maximize an utility function. Except in the last case above, I have no need of that hypothesis. Alternative theories of value and distribution exist. (Does Nick Rowe imagine that I am in his intended audience for this post?)

Krugman Confused About His Profession, Preferences

Paul Krugman has explained on several occasions that he was inspired to become an economist by Isaac Asimov's Foundation triology. The collapse of a galaxy-wide civilization into an interim dark age, before the revival of a second galactic empire, provides the background setting of the novels. The organizing conceit is that Hari Seldon, the expert founder of the discipline of psychohistory, has figured out how to set up initial conditions such that the intervening and unpleasant dark ages will last for only a millennium, instead of 30 millennia. In Asimov's telling, psychohistory is an explicitly mathematical discipline.

It turns out that Asimov was not the only science fiction author in that era writing about mathematical psychology:

"So? The greatest mathematical psychologist of our time, a man who always wrote his own ticket even to retiring when it suited him..." -- Robert A. Heinlein (1958).

Heinlein's character has written a book titled, On the Statistical Interpretation of Imperfect Data, and his colleagues are at the Institute for Advanced Study, in Princeton, New Jersey.

When people referred to "Mathematical psychology" during the 1950s, what were they talking about? I suggest that they had in mind cybernetics, as invented by Norbert Wiener, and later developments. For example, consider the 1960 book, Developments in Mathematical Psychology, with contributions from R. Duncan Luce, Robert R. Bush, and J. C. R. Licklider.

Many have built on this work over the last half-century, in a variety of disciplinary settings. A few years ago, one could find the label Command, Control, Communications, and Intelligence (C³I) used to refer to much of this work. Departments and ministries of defense provided quite a bit of funding for research in these areas. And if you want to find current research on these topics, you could do worse than read such journals as IEEE Transactions on Communications, IEEE Transactions on Control Systems Technology, and IEEE Transactions on Signal Processing. These journals are all put out by the Institute of Electrical and Electronics Engineers (IEEE).

Thus, Paul Krugman wanted to be a electrical engineer, although he does not know it.

References

• Isaac Asimov (1953). Second Foundation. [I happen to have this book in the trilogy handy.]
• Robert R. Bush (1960). "A Survey of Mathematical Learning Theory", in (ed. by R. D. Luce), The Free Press of Glencoe, Illinois.
• Robert A. Heinlein (1958). Have Space Suit-Will Travel.
• J. C. R. Licklider (1960). "Quasi-Linear Operator Models in the Study of Manual Tracking", in (ed. by R. D. Luce), The Free Press of Glencoe, Illinois.
• R. Duncan Luce (1960). "The Theory of Selective Information and Some of Its Behavioral Applications, in (ed. by R. D. Luce), The Free Press of Glencoe, Illinois.
• Norbert Wiener (1948). Cybernetics: Or Control and Communication in the Animal and The Machine.

Money Missing From Mainstream Economics

"In practice, of course, the purists were unable to deliver, and the new tricks involve the 'modern macroeconomists' in ad hoc assumptions of their own that are at least as objectionable as the Keynesian macroeconomic generalizations that [Michael] Wickens objected to. We have already encountered one example, the 'Gorman preferences' needed to make the representative agent at least minimally plausible... Two others are equally incredible. The first is the 'no-bankruptcies' assumption in Walrasian models and the related 'No Ponzi' conditon that is imposed on D[ynamic] S[tochastic] G[eneral] E[quilibrium] models. This eliminates the possibility of default, and hence the fear of default (since these are agents with rational expectations, who know the correct model, and hence know that there is no possibility of default), and hence the need for money, since if your promise to pay is 'as good as gold', it would be pointless for me to demand gold (or any other form of money) from you. Money would be at most a unit of account, but never a store of value. The second is the unobtrusive postulate of 'complete financial markets', smuggled into Michael Woodford's Interest and Prices (Woodford 2003, p. 64), which means that all possible future states of the world are known, probabilistically, and can be insured against: this eliminates uncertainty, and hence the need for finance..." -- J. E. King, The Microfoundations Delusion: Metaphor and Dogma in the History of Macroeconomics (2012: p. 228)

Can General Equilibrium Theory find a role for money? Consider Frank Hahn's "On some problems of proving the existence of an equilibrium in a monetary economy" (1965). He considered the question posed above to be an unmet challenge at the time. Hahn did not think, for example, Don Patinkin's attempt to justify the use of money through the inconvenience of indirect transactions and through transactions cost fit comfortably in a General Equilibrium model. Hahn wanted to find a model in which the existence of money was essential, in some sense, in which an equilibrium with money differed from one without.

mainstream macroeconomists claim to base their approach on General Equilibrium Theory. Experts on GE (for example, Alan Kirman) have been saying for decades that this claim is dubious. The critiques that I am most aware of are based on price theory.

These modern macroeconomists claim to have available models incorporating money and finance. This availability does not mean that they are unwilling to deploy models without money in some contexts for some purposes. As far as I am aware, Woodford is widely cited and widely respected in current mainstream monetary economics.

I have wondered how and if mainstream macroeconomists address the problems highlighted by Hahn in incorporating money in General Equilibrium Theory. But I have not wondered enough to read much. If I take King as an authority, I can spare myself the trouble of establishing that mainstream macroeconomists are basically confused about their own monetary theory. Is this a sound conclusion?

(Colin Rogers is an expert on how the Cambridge Capital Controversies can be used to critique Wicksellian theories of money and of the natural rate of interest. Woodford and Rogers had an exchange of views in the Cambridge Journal of Economics a number of years ago.)

Nick Rowe Knavery

So much nonsense packed into so few words:

"But the lefty Sraffian model has only labour and time as inputs." -- Nick Rowe

I do not know what politics has to do with it. I'd like to see some evidence of the political beliefs of, say, Neri Salvadori. When Sraffa states, in the preface of Production of Commodities By Means of Commodities, that "Others have ... independently taken up [similar] points of view," I think he includes John Von Neumann. Von Neumann wanted to wage an atomic war against the Soviet Union. I guess Rowe must think Von Neumann was a Trotskyite.

The Sraffa model, of course, includes non-produced inputs other than labor. "Land" is the title of Chapter 11 of Sraffa's book. I have attempted to explain some of the points in this chapter here and here. Heinz Kurz, Neri Salvadori, and Bertram Schefold are just some of the economists who have contributed to the literature building on Sraffa's analysis of land.

To address another Rowe misconception, Sraffians have also analyzed heterogeneous labor. In fact, "Heterogeneous Labor" is the title of Chapter 7 of Ian Steedman's book, Marx After Sraffa. I have an example with heterogeneous labor here.

The bit about time being an input in Sraffa's model is apparently some convex combination of a lie and begging the question. Sraffa explicitly states, in the title of his book, that he is considering production processes with inputs of commodities. These commodities include seed corn, iron, pigs, and so on. Although Sraffa assumes a yearly cycle of production for convenience, time is explicitly not an input in Sraffa's model. In fact, Sraffa can be said to have proven that "capital" cannot be reduced to time. In the case of joint production without land, the technique cannot even be reduced to labor inputs applied over time. (By the way, in the first chapter in her Essays in the Theory of Economic Growth, Joan Robinson explicitly analyzes an economy in which robots are produced by robots.)

Does anybody expect Rowe to acknowledge that he has no concern whatsoever over whether what he says is true or even makes sense?

Krugman Promoting Zombie Horror, Not SF Futures

Paul Krugman has been writing about robots lately. He explicitly cites J. R. Hicks' incoherent and mistaken 1932 book, The Theory of Wages. This is a classic statement of the neoclassical theory of factor substitution, of the choice of technique in allocating scarce factors among alternative uses.

If I want to analyze the adoption of new technology, I turn to:

• David Ricardo's chapter, "On Machinery", in the third edition of his book.
• The Von Neumann model of growth, which can be read as a model in which robots produce robots.
• The Harrod-Domar model of the warranted and natural rate of growth, along with the definition of Harrod-neutral and biased technological change.
• Joan Robinson's models of metallic ages.
• Kaldor's growth models of various vintages and his definition of the technical progress function.

I am in agreement with Krugman on the importance of Hicks' book in the development of the neoclassical canon. And I recognize the existence of a problem in empirically distinguishing between the choice of technique and the adoption of new technology in Kaldor's model(s) of economic growth.

Update: Matias Verengo has two posts on this topic.

Anti-Reductionism: An Example?

"There are three good reasons to think that reduction will fail on any likely development of social sciences: (1) multiple realizations of social events are likely; (2) individual actions have indefinitely many social descriptions depending on context; and (3) any workable individualist social theory will in all likelihood presuppose social facts. Each of these claims, if true, rules out reduction as defined here." -- Harold Kincaid (quoted in J. E. King (2012)).

Kincaid is arguing against strong methodological individualism. To help explicate his first reason, I want to consider a different domain, biology, and some literature crossing over between biology and computer science.

I suppose I ought to first state the reductionist theory I want to oppose: The theory of evolution can be reduced to the biochemistry of DNA.

A large body of literature explores the logical structure of reproduction, including reproduction with mutations, independently of consideration of the structure of DNA. I think of Von Neumann's work on celluar automata, which, despite the publication date of the work Burks edited, pre-dates the discovery of the molecular structure of DNA. As I understand it, Von Neumann described a structure in which a part simultaneously functions as a blueprint for the next generation and as a component that is duplicated in reproduction.

Von Neumann described a celluar automaton with many states, but his logic can be implemented in a particular celluar automata with only two states, namely Conway's Game of Life. Plausibility arguments that this celluar automata can be used to form a universal computer were available before Paul Rendell implemented a Turing machine in the game of life. Basically, one can identify mechanisms for implementing a memory and an array of gates (for example, AND, OR, and NOT). The gates apply to bits flowing across a wire, in some sense. Jacob Aron has create another interesting pattern in the game of life relevant to my thesis, namely a self-replicating creature.

Some researchers have also explored the role of mutations in self-replicating automata. As I understand it, they typically assume the existence of an assembly language for a virtual machine. One can imagine small programs being executed in parallel in some sort of common memory. One needs some way of introducing random changes in some of the instructions over time cycles and a way of rewarding successful programs with, say, more energy, in some sense.

The different artificially alive creatures in these simulations do not reside in separate protected memories. They have the capability of overwriting one another and resisting such overwriting. Some have even arranged tournaments, called core wars, in these simulations. In some sense, the literature I am referencing includes some bits of recreational mathematics.

I have never seen much more than what the literature says in the little bit of exploration of the above I have done. I did once write an implementation of Conway's Game of Life in which the rules were configurable. I was able to create crystal-like growth, but nothing as interesting as in the original game.

I have pointed to some work exploring a logic of reproduction above the level of the biochemistry of DNA. DNA is one means of instantiating this logic. I have not pointed out any other non-virtual mechanisms for instantiating this logic. I do not know if mitochondrial DNA differs sufficiently from regular DNA to count. Silicon-based life forms on other planets is a standard trope in science fiction. Apparently, Reaves et al. (2012) show the supposed discovery of arsenic-based life forms in certain California lakes has not worked out. But does this anti-reductionist argument require the actual existence of another instantiation, or merely the demonstration of the possible existence of one?

References

• J. E. King (2102). The Microfoundations Delusion: Metaphor and Dogma in the History of Macroeconomics.
• Lenski, Richard E., Charles Ofra, Robert T. Pennock, and Christoph Adaml (2003). The Evolutionary Origin of Complex Features, Nature, V. 423 (May): pp. 139-144.
• Poundstone, William (1984). The Recursive Universe. William Morrow.
• M. L. Reaves et al. (2012). Absence of detectable arsenate in DNA from arsenate-grown GFAJ-1 cells.
• Thearling, Kurt and Thomas S. Ray. Evolving Multi-Cellular Artificial Life.
• Von Neumann, John (1966). Theory of Self-Reproducing Automata (ed. by A. W. Burks).

A Keen Defense

My major point in this post is to draw attention to the existence of Kapeller and Pühringer (2010).

Steve Keen's book, Debunking Economics, is mainly a compilation of well-established criticisms of textbook economics. He attempts as popular a presentation as the material will permit. These criticisms, in my opinion, leave textbook economics, both microeconomics and macroeconomics, in tatters¹.

Keen, in the first edition, also offered his own original criticism of the textbook theory of the firm under perfect competition. You can find various brouhahas on the internet over Keen's remarks. Between editions of his book, Keen has published, with others, a series of papers developing his criticism².

Some have asserted theories of perfect competition in models with a continuum of agents provide a defense of the textbook theory. As Kapeller and Pühringer point out, this is a change of subject. A non sequitur should not be a considered an adequate defense of the textbook model. Furthermore, the primary developer of models with a continuum of agents presents his approach as inconsistent with the textbook theory:

"Though writers on economic equilibrium have traditionally assumed perfect competition, they have, paradoxically, adopted a mathematical model that does not fit this assumption. Indeed, the influence of an individual participant on the economy cannot be mathematically negligible, as long as there are only finitely many participants. Thus, a mathematically model appropriate to the intuitive notion of perfect competition must contain infinitely many participants. We submit that the most natural model for this purpose contains a continuum of participants." -- Robert Aumann (1964).

It seems to me only four possibilities are open here:

1. Aumann is not talking, in his critical remarks, about the model of perfect competition taught in almost any intermediate microeconomics textbook.
2. Aumann is mistaken.
3. The economists who write and teach the self-contradictory textbook model are deliberately teaching self-contradictory models to their students.
4. The economists who write and teach the self-contradictory textbook model are ignorant.

I think only the third and fourth options are credible³. I can understand the difficulty of writing and teaching in an intellectually bankrupt discipline.

Update: Nick Rowe illustrates the willingness of some economists to teach nonsense to students: "To the individual farmer, who sees only a tiny slice of the whole demand curve, because even a 100% change in his output will cause only a tiny percentage change in total output, it will look perfectly flat." Note that in his post, even when considering the limiting case, he never considers the existence of a continuum of producers.

Update 2: Steve Keen, in the Business Spectator, re-iterates his critique of the incorrect neoclassical textbook theory of perfect competition. Tim Worstall lies to readers of Forbes. It is not true that "everyone subscribes" to the "usual basics of economics" that Keen debunks. It is not true that the bulk of Keen's book is about his "breakthroughs in showing us all the errors of our ways." One can accept almost all of Keen's demonstration of the mendacity of neoclassical textbooks without accepting any claim that Keen says is original with him. In fact, Keen notes that Stigler showed that perfectly competitive firms that are not systematically mistaken, if they produce a positive, non-infinitesimal quantity in equilibrium, will not produce at a level of output where the market price, Marginal Revenue, and Marginal Cost are all equated.

Footnotes

1. Some areas of economics, such as game theory or the recent popularity of instrumental variables and experiments (natural and otherwise), remain unaddressed by Keen.
2. Kapeller and Pühringer point to a 2008 paper published by Anglin in Physica A as a peer-reviewed response.
3. The existence of the downward-sloping part of the U-shaped average cost curve for the textbook firm hardly seems compatible with the existence of an infinite number of firms, each producing a quantity of zero units of an homogeneous good.

References

• Aumann, Robert J. (1964). Markets with a Continuum of Traders, Econometrica, V. 32, No. 1-2.
• Kapeller, Jacob and Stephan Pühringer (2010). The Internal Consistency of Perfect Competition, The Journal of Philosophical Economics, V. III, No. 2: pp. 134-152.

A Barefoot Bum Critiques Propertarianism

"Larry, the Barefoot Bum" has been refuting propertarianism¹. Crudely stated, his thesis is that if you believe taxation is unjust because it implies the initiation of force (coercion by the state), you cannot coherently also defend private property.

1. To me, what was traditionally called "libertarianism" is anarchism, that is, a kind of communism.

Widespread Incompetence In Economics

Should one try to engage with those putting forth positions refuted decades ago? Maybe one's time would be better spent on looking at arguments that come closer to reflecting the state of the art, even when those arguments are not backed up by external funding from vicious reactionaries.

Suppose the economy were a complex dynamic system. Those who have investigated this idea have long ago shown that, even if all wages and prices were perfectly flexible, no tendency need exist for the economy to tend towards an equilibrium in which all markets, including the labor market, clear. Some just do not know:

"Given the position [Casey Mulligan is] trying to defend, these are the best arguments available. And that position is widely shared, not only by economists much more famous than Mulligan but by lots of governments and policymakers. Most mainstream opponents of Keynesianism are committed, one way or another, to the view that persistent high unemployment must be caused by problems in labour markets. But it's much easier to talk in vague general terms about rigidities and structural imbalances than to present an operational explanation for the sustained high US unemployment of the last four years. Mulligan at least makes the attempt, which is more than most of the New Classical/Chicago/Real Business Cycle school have done, and necessary if there is to be any progress in the debate." -- John Quiggin

"If (Casey) Mulligan were an isolated crank, I'd ignore him. But he’s endorsed by people like Tyler Cowen who should know better. And, as I said in the opening para, most of the freshwater crowd and quite a few people who were once "New Keynesians" believe or go along with this stuff." -- John Quiggin

The Historical Failure of Neoclassical Economics

"Both classical and marginalist economics provided accounts of the long-period (uniform rate of profit) theory of value and distribution, but whereas a classical economist could take the real wage as a datum for the purpose of such analysis (whatever the implicit ‘background’ theory of wages might be), the marginalist economist had to ‘close the system’ in some other manner. In effect, since ‘resource supplies’ were often taken as given, this meant that the ‘the supply of capital’ had to be taken as given, in one way or another. Just how the given supply of capital was to be represented was an issue which led to considerable heterogeneity amongst even those marginalist economists who shared the long-period method of analysis with the classical economists and with each other. That heterogeneity cannot be entered into here (see Kurz and Salvadori, 1995: 427-43) but it is now widely recognized that each version of such traditional long-period marginalist theory of value and distribution encountered insoluble problems (ibid: 443-8)." -- Ian Steedman (1998).

1.0 Introduction

The ‘neoclassical’ revolution is conventionally dated to the 1870s, with the works of William Stanley Jevons, Leon Walras, and Carl Menger. Neoclassical economists, from the 1870s to the 1930s, tried to develop neoclassical theory:

• To include production, including production with previously produced means of production, within the scope of the theory.
• To extend supply and demand-based reasoning to all runs, including the long run.

All long-run neoclassical models produced in this period failed; they were logically self-contradictory.

2.0 The Endowment of Capital in Early Neoclassical Theories

In long-run theory, relative spot prices are stationary in equilibrium¹. In a long-run equilibrium, entrepreneurs have correctly anticipated effective demand, and the size of plants have been adapted to this demand. Furthermore, plants are being operated at their ideal capacity for which they have been designed². These early economists can be said to have specified the given endowment of capital in two ways:

1. As a vector of physical quantities of heterogeneous produced goods to be used as inputs in production.
2. As a homogeneous quantity, given in value, but free to change its form.

Leon Walras adopted the first approach, even though his fully developed model contained a market for value capital. His approach comes to grief on the need to simultaneously assume a uniform rate of profit in all markets for produced goods, to equate supply prices and market prices of capital goods, and to impose the condition that capital goods are being produced in proportions that will allow the economy to be reproduced (perhaps on an expanded scale).

The second approach can be further subdivided. In the first subdivision, Jevons and Eugen von Böhm-Bawerk, for example, took the homogeneous stuff of which capital consists to be a fund of subsistence goods to maintain the workers while they labored. More capital somehow represents a longer period of production. These economists incorrectly thought that a meaningful physical measure of this period of production could be defined independently of prices and that a lower interest rate would necessarily encourage entrepreneurs to extend this period, given the available technology.

In the second subdivision, this homogeneous stuff consists of the value of a heterogeneous quantity of capital goods. This ignores price Wicksell effects, the variation of the value of a given collection of physical quantities of capital goods with distribution and prices. Knut Wicksell was both a prominent proponent of this approach and an early economist to realize why it does not work.

3.0 Later Developments

From around the end of the 1920s to the 1960s, neoclassical economists abandoned the long-run to concentrate on the refinement of certain general, logically consistent, although empirically empty, theories. I refer to the works of Erik Lindahl, J. R. Hicks, Friedrich Hayek, and Gerard Debreu and Kenneth Arrow on intertemporal and temporary equilibrium³. Towards the end of this second period, economists returned to the elaboration of long run theories of stationary and steady states. In the logically consistent multisectoral models in this trend, capital is not taken as given, either as a value quantity nor as a vector of physical quantities. Rather, the quantity of capital, in both senses, is found by solving the model⁴.

4.0 Conclusion

Economists have developed a logically consistent and empirically applicable theory of classical ‘natural prices’ (also known as Marxian ‘prices of production)’. As I and others have repetitively demonstrated, such prices are inconsistent with supply and demand-based reasoning. Since the endowment of means of production is not taken as given in such theories, these theories are not about the allocation of given resources among alternative ends.

Over, the last century economists have extensively explored the logic of models in which given resources are allocated among alternative ends. Although such models might be of use to a central planner, they seem to be unable to describe prices in actually existing capitalist economies.

The development of these claims have been available in the scholarly literature for about a third of a century. They have not been refuted. Most mainstream economists just ignore this collapse of neoclassical economics, in their teaching, in their applied work, in policy advice, and in their research.

Footnotes

1. Long run equilibrium is compatible with slow, secular changes, such as improvements in technology and in the composition of output.
2. Neoclassical economists, of course, did not claim that an actual economy would ever be in such a long-run equilibrium. The model was developed as an aid to analyze tendencies to equilibrium thought to be in existence at any given moment of time.
3. I have seen some claim Irving Fisher as a forerunner for these theories.
4. Many of the recent mainstream developers of models of endogenous growth (such as, Paul Romer) seem to be ignorant of this fact.

Elsewhere

• A blog for a close and critical reading of Mas-Colell, Whinston, and Green. MWG is the most dominant introductory graduate microeconomics textbook.
• Alan Kirman calls for a paradigm shift in economics. (Hat tip to Lars Syll.)
• A NOAA FAQ: Why don't we try to destroy tropical cyclones by nuking them?

What Is Mathematics - And Sraffa

An Unsurveyable Rule For Generating A Real Number In Binary Format

Noah Smith offers a definition: "Mathematics is the manipulation of the symbols of a language according to explicit, syntactical rules." ("Unlearning Economics" has also recently written on mathematics in economics). To me, the manipulation of meaningless symbols is a powerful form of reasoning. Taking this definition as is, I think two questions can be raised here:

• What is the interest that mathematicians find in these rules and these symbols in the historical circumstances current at the time?
• What does it mean to follow a rule?

Ludwig Wittgenstein is the philosopher most known, I think, for raising the question of what it means to follow a rule. Any summary of his views will be controversial, but I suppose one can fairly say that he adopted an anthropological point of view, at least for some purposes. Describing how to follow a rule by another rule raises the prospect of an infinite regression. Rather, one might show how people do actually follow a rule, how these uses and practices work pragmatically in some form of life. I find it difficult to see how such description conveys the logical must, so to speak, of many rules. But Wittgenstein was alive to this difficulty. He notes that a judge does not seem to treat a statute book as a manual of anthropology.

Furthermore, Wittgenstein spent quite some time in elaborating how these ideas relate to the philosophy of mathematics. His views on the foundations of mathematics seems to have been constructivist and included questioning whether mathematics needs a foundation. Wittgenstein has frequently been labeled an anti-foundationalist. From this viewpoint, one might question whether existence proofs that do not specify how to construct the relevant object can be reformulated. And one even ends up doubting the meaningfulness of defining the real numbers as, say, any set isomorphic to a set of certain equivalence classes of Cauchy-convergent sequences of rational numbers. The use of the notion of infinity remains, I guess, as a standard topic in the philosophy of mathematics.

It seems one of my favorite economists, Piero Sraffa, was an important stimulus in Wittgenstein's development of these views. Sraffa has been said to have led Wittgenstein to see the importance of an anthropological point of view. Sraffa's masterpiece, The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory, is written in a unique style, not less in the presentation of the mathematics underlying the economics in the book. Sraffa frequently provides outlines of algorithms for constructive existence proofs, maybe most famously for the Standard Commodity. So Sraffa and Wittgenstein might be said to have shared a certain attitude to the philosophy of mathematics, although I do not expect to ever see oral discussions on this topic to be well documented. Sraffa's book can also be said to address only a limited range of topics in economics. An earlier statement of his seems to suggest that he thought room should exist in economics for non-formal treatment of some topics:

"The causes of the preference shown by any group of buyers for a particular firm are of the most diverse nature, and may range from long custom, personal acquaintance, confidence in the quality of the product, proximity, knowledge of particular requirements and the possibility of obtaining credit, to the reputation of a trademark, or sign, or a name with high traditions, or to such special features of modelling or design in the product as - without constituting it a distinct commodity intended for the satisfaction of particular needs - have for their principal purpose that of distinguishing it from the products of other firms. What these and the many other possible reasons for preference have in common is that they are expressed in a willingness (which may frequently be dictated by necessity) on the part of the group of buyers who constitute a firm's clientele to pay, if necessary, something extra in order to obtain the goods from a particular firm rather than from any other." -- Piero Sraffa (1926). "The Laws of Returns Under Competitive Conditions", Economic Journal (Dec.): pp. 544-545.

Whatever you think of the speculations in this post, I think some conclusions are nearly inarguable. Advocates and opponents of the use of mathematics in economics do not neatly divide between mainstream and non-mainstream economists. In particular, one important non-mainstream economist, Piero Sraffa, demonstrated one approach to mathematical economics, while still being aware of the limits to formalism in economics. Furthermore, any comprehensive scholarly study of the philosophy of mathematics will necessarily look at his work as long as Wittgenstein's later views are considered germane to such scholarship.

A Student's Recollection

More than two decades ago, I took a course in intermediate microeconomics. The textbook was R. Robert Russell and Maurice Wilkinson's Microeconomics: A Synthesis of Modern and Neoclassical Theory. "Modern", in this case, refers to the use of set theory terminology, linear programming, and proofs like those in an introductory real analysis class. In contrast, "Neoclassical" refers to the use of continuously differentiable functions. In any case, the substance of the theory - which is only one possible theory - is unaffected. (I would not have been clear on this at the time.)

One day, our professor was teaching us about oligopoly and the theory of the kinked demand curve. And, in response to a question, the professor said something like, "This is a theory I actually believe". Yet, in the rest of the classes, when he was teaching us to manipulate utility functions or production functions or to take Lagrangians or whatever, he never expressed an opinion of the empirical applicability of what he was teaching us.

I also recall that our professor made a special effort to teach us input-output analysis one week. This topic was not in the textbook, if I recall correctly. But Leontief was coming to give a lecture (not to our class, but in a big lecture hall, that is, CC308). And our professor wanted us prepared. As it was, Leontief's lecture did not concern the details of input-output analysis, but the complaint that most of then contemporary economics was unconcerned with empirical results. Most economists did not even cast their theory in a form where it could be connected up to empirical data that one might collect.

References

• Wassily Leontief (1982). Academic Economics, Science, New Series, V. 217, N0. 4555 (9 July): pp. 104-107.
• Wassily Leonteif (1983). Academic Economics Continued, Science, New Series, V. 219, No. 4587 (25 February): 904.
• R. Robert Russell and Maurice Wilkinson (1979). Microeconomics: A Synthesis of Modern and Neoclassical Theory, John Wiley & Sons.

A Simple, But Perverse, Neoclassical Model

Figure 1: Less Plentiful Supply of Capital Lowers the Interest Rate

1.0 Introduction

I claim that capital reversing can be a source of instability and interesting dynamics in neoclassical models. I am interested in, for example, the convergence or not of equilibrium paths in models of intertemporal and temporary equilibrium to steady states, but not in tâtonnement dynamics. The ill-behaved nature of many neoclassical models is a challenge in demonstrating this claim.

This post is a start on revisiting these issues. I here outline a simple model of overlapping generations with a simple production model that cannot exhibit reswitching, capital reversing, or even price Wicksell effects. Yet, in this model, a greater willingness among the households to save is associated with a higher interest rate. This is inconsistent with the supposedly intuitive stories told in outdated and exploded neoclassical textbooks.

2.0 The Model

The model describes an economy in which a single commodity, corn, is produced. In this model, corn functions as both the consumption good and as the only capital good. In production, all (seed) corn is used up in producing the harvest; that is, all capital is circulating capital. For my purposes in this post, I want to consider an economy in a stationary state.

The point of these assumptions is not to describe any actually existing capitalist economy. Rather, the point is to demonstrate that neoclassical theory does not justify conclusions commonly made. I suppose you can say that these types of models raise the following empirical question: why do mainstream economists continue to teach, both in the classroom and in policy work, conclusions long exposed as nonsense by their own theory?

2.1 Utility-Maximizing Agents

Suppose the population consists of overlapping generations, as in Figure 2. Each generation lives for two years. In a given year, all members of the generation born at the start of that year work a full year. They are paid their wages at the end of the year. Out of their wages, they consume some and they save the remainder at the going interest rate. They are retired during the second year of their life. At the end of their second year, they consume the remainder of their income and die.

Figure 2: Lifespans of Overlapping Generations

Furthermore, assume that each generation consists of a single individual, also known as an agent. Furthermore, suppose all generations are identically characterized by the following Cobb-Douglas utility function:

U(c₀, c₁) = (c₀)^γ(c₁)^{(1 - γ)}

where c₀ is the bushels corn the agent consumes at the end of the first year of their life, c₁ is the bushels corn consumed at the end of the second year, and

0 < γ < 1

A higher value of γ indicates a lesser willingness to defer consumption and a smaller supply of savings. Let w be the wage, and r the interest rate. Under these assumptions, the agent born in each generation solves the following utility-maximization problem:

Given w, r

Choose c₀, c₁

To Maximize U(c₀, c₁)

Such that c₀(1 + r) + c₁ = w(1 + r)

c_i ≥ 0; i = 0, 1.

The constraint states that the total value of consumption, evaluated at a single point in time, equals the income of the agent, also evaluated at the same point in time. The solution to this mathematical programming problem is:

c₀ = γ w

c₁ = (1 - γ) w(1 + r)

S = (1 - γ) w

where S is the bushels corn saved at the end of each year.

2.2 Production

For simplicity, I assume a Leontief, fixed coefficients production function. Let L be the person-years of labor employed during the year, K be the bushels corn used as capital during the year, and q be the bushels corn produced during the year. The production function is:

q = min( L/a₀, K/a₁)

where:

a₀ > 0

0 < a₁ < (1/2)

(Productivity has to exceed a certain threshold for an equilibrium to exist in this model.)

Only consider cases where both constraints bind. In a stationary state, the corn available at the end of the year is divided up into a₁/a₀ bushels to use as capital next year and (1 - a₁)/a₀ corn to consume, per person-year employed.

Given this technology, the wage-rate of profits frontier is easily expressed:

a₁(1 + r) + a₀w = 1

Hence, one can solve for the wage as a function of the interest rate and the coefficients of production:

w = [(1 - a₁)/a₀] - (a₁/a₀)r

When the interest rate is zero, the wage is (1 - a₁)/a₀, that is, the total surplus of corn, after subtracting the seed corn needed to sustain production at the same level. When the wage is zero, the interest rate is (1 - a₁)/a₁.

2.3 Equilibrium

This model is completed by assuming that the households want to hold the capital stock at the end of every year. since only one generation is saving for retirement at the end of this year, this equilibrium condition is:

S = a₁/a₀

I might as well make an aside on marginal productivity. In models in which the firms choose the cost-minimizing technique, marginal productivity conditions are used to specify the coefficients of production. The price of each commodity used as a capital good is equal, in equilibrium, to the present value of the marginal product of that commodity. In models in which the technology is specified as a set of fixed-coefficient techniques, the value of marginal product, as I understand it, is an interval in which left-hand and right-hand derivatives enter. In any case, since prices and the quantities of capital goods are both found by solving the model, one cannot say that the (rental) price of a capital good is determined by its marginal product. Furthermore, wages are not determined by the marginal product of labor. A fortiori, the rate of profits is not determined by the marginal product of finance capital, even if one can concoct some equation involving the return on capital, some measure of the value of capital goods, and its marginal product.

Anyways, one can solve the above model to find the following closed-form expression for the interest rate in a stationary state:

r = [(1 - a₁)/a₁] - [1/(1 - γ)]

Figure 1 above graphs this function. And one can see that, in this model, a stationary state in which households are less willing to save is associated with a lower interest rate. If the interest rate were the price of capital and prices were indices of relative scarcities, this example could not be created. But equilibrium prices are not scarcity indices and neoclassical economics, as taught by most university professors, is nonsensical poppycock.

3.0 Conclusion

This post has presented a simple neoclassical model, a limit point, in some sense, of the kind of model that neoclassical economists advocated as a resolution of the Cambridge Capital Controversies. And this simple model shows that much of mainstream teaching and policy work is theoretically unfounded, by their own logic.

Nick Rowe Teaching Miasma Theory Of Plague...

...and other outdated blatherskite:

"An increase in desired saving will only affect the rate of interest slowly, over time, as the greater flow of investment slowly increases the stock of capital and reduces MPK [Marginal Product of Capital]."

"If people want to save more, the rate of interest will fall, the price of capital goods will rise, and there will be a movement along the PPF as existing resources move away from producing consumption goods towards producing investment goods." -- Nick Rowe

(Some have tried to explain.)

Reproducing Civil Society

"Civil Society - an association of members as self-subsistent individuals in a universality which, because of their self-subsistence, is only abstract. Their association is brought about by their needs, by the legal system - the means to security of person and property - and by an external organization for attaining their particular and common interests." -- G. W. F. Hegel, The Philosophy of Right

"...in the case of the most advanced States, ...civil society has become a very complex structure and one which is resistant to the catastrophic incursions of the immediate economic element (crises, depressions, etc.). The superstructures of civil society are like the trench systems of modern warfare. ... In Russia the State was everything, civil society was primordial and gelatinous; in the West, there was a proper relation between State and civil society, and when the State trembled a sturdy structure of civil society was at once revealed. The State was only an outer ditch, behind which there stood a powerful system of fortresses and earthworks: more or less numerous from one State to the next, it goes without saying - but this precisely necessitated an accurate reconnaissance of each individual country." -- Antonio Gramsci, Prison Notebooks

There exist at least two approaches to economics:

• One focused on the allocation of given scarce resources among alternative ends.
• One focused on the conditions for the reproduction of society.

The first is the approach of the so-called neoclassical theory, and the second is the approach of classical political economy.

I like to write about price theory, a field in which one can formulate certain definite quantitative relationships. But the investigation of conditions that facilitate the reproduction of society can extend well outside of price theory and even of economics, as the above Gramsci quote suggests. I suggest the following are examples of components of civil society, whatever your definition: churches, labor unions, charities, civic groups, professional societies, and athletic clubs.

I am not at all sure that anybody has drawn attention in the literature to this commonality between the works of Gramsci and Sraffa: both analyzed the conditions for the reproduction of society, one concentrating on political theory and the other on price theory.

Your Opinion Does Not Matter

Figure 1: Bottom Decile Irrelevant To Policy

Figure 2: Top Decile Much More Influential on Policy Than Median

These striking figures are from Martin Gilens (2005). Gilens looks at 1,781 questions from opinion surveys given between 1981 and 2002 and soliciting opinions on policy changes that could be implemented by some combination of the president and the Congress. He codes the question based on whether the policy change was implemented in the four years after the survey was given. As I understand it, for each question he performs a regression based on opinions and income. This allows him to analyze the consistency, for each income percentile, of opinions and policy outcome.

Gilens then looks at questions where people at different income percentiles differ in opinion by at least 8% for his scale. He finds 887 such questions for the 10th and 90th percentile, and 498 questions for the 50th and 90th percentile. As you can see, poor people at the 10th percentile have virtually no impact on policy outcomes, and middle income people at the 50th percentile have only slight impact. Empirically, the United States is a plutocracy.

• Martin Gilens (2005). "Inequality and Democratic Responsiveness", , V. 69, N. 5: pp. 778-796.

Production Takes Time

In his book¹, Piero Sraffa presented the elements of an economic theory that takes into consideration the empirical fact stated in my post title.

Suppose capitalists make decisions on what and how much to produce. They base these decisions partly on what they can expect to sell. As a consequence of these decisions, they have created a stock of capital goods, a set of interindustry flows, and a production of the surplus of commodities.

Suppose one takes the share of labor in this surplus as a datum², at least for the purposes of a theory of value. Questions could then arise: for what set of prices would these decisions of the capitalists be justified³? For what prices would capitalists be willing to hold the produced capital goods? Sraffa provides an answer. The solution of his price equations⁴ are the desired prices of production.

In giving this answer, one is not claiming that these prices will prevail in an economy. The question of whether or not they will prevail is known as the realization problem⁵. An analysis of consumer demand enters in in addressing this problem.

I do not consider myself to be original in anything above⁶, other than, perhaps, in exposition. And this theory has widespread empirical application, both in the form of Leontief's Input-Output analysis and through Keynesian economics.

Footnotes

1. Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, (1960).
2. One could make other decisions on the givens. One might take expectations about the decisions of the monetary authority as a given. Or one could append the Cambridge equation to this model.
3. Keynes examines a closely related issue on pages 48-50 of The General Theory of Employment, Interest, and Money. Anyway, Sraffa and Keynes's theories share, at least, a family resemblance in that decisions on quantities temporally precede the determination of prices.
4. Since Sraffa presents a system of simultaneous equations, one cannot competently describe his theory as one of uni-directional causation.
5. The realization problem is addressed by Keynes' theory of effective demand.
6. See, for example, Alessandro Roncaglia's Sraffa and the Theory of Prices (1978).

Eigenvectors Generalized?

I have occasionally mentioned some of the mathematics that many find useful in reading Sraffa¹. Here I want to raise a question. Consider the following two problem statements:

Problem 1: Given a vector space V and a linear function A mapping that vector space into itself, find a vector v in V such that the image of v under A is merely the original vector lengthened or shortened. In other words:

A(v) = λ v.

Problem 2: Given a vector space V and two linear functions, A and B, mapping that vector space into itself, find a vector v in V such that the images of v under A and B are two vectors, where one such image is the other vector lengthened or shortened.In other words:

A(v) = λ B(v).

The second problem statement is, in some sense, a generalization² of the first. I know of lots of theory for analyzing the first problem and many application areas³ unrelated to economic models of circulating capital. I do not know of any literature on the second problem outside of mathematical economics and the analysis of joint production. Likewise, I have a name, eigenvector, for a solution to the first problem. But I have no such name for a solution to the second. Where, if anywhere, can one find literature on the second problem and other application areas motivating its application?

Maybe this is a question for math overflow.

Footnotes

1. In such discussions, I usually do not worry about whether or not the mathematics on which I draw is constructive. Arguably, Sraffa insisted that his proofs be constructive. This topic should be of interest to Wittgenstein scholars.
2. Singular values are one generalization of eigenvalues. My question relates to another generalization.
3. Since I stated the problem so as not to be limited to finite-dimensional vector spaces, one such application area is Fourier analysis.