Bracketology Combinatorics

This is combinatorics, not economics. I suppose this sort of math is useful for political science in calculating power indices.

Consider a single-elimination tournament with 2ⁿ teams entered. Ignoring the play-in games, n is six for the NCAA basketball tournament. Some tournaments have "bys" on the first round, and I am ignoring that aspects of those tournaments too.

Let f(n) be the number of ways of filling out your brackets for the tournament. The problem considered here is to calculate f(6).

Since 2ⁿ teams are in the tournament, 2^{n - 1} games are played in the first round. Since there are two possible victors for each game, there are 2^{(2n - 1)} possible ways of filling out your brackets for the first round. The total number of ways of filling out your brackets is then:

f(n) = 2^{(2n - 1)} f(n - 1)

One could guess a closed-form solution for this difference equation and prove it correct by mathematical induction. If I recall correctly, generating functions are useful in finding the solution to such recurrence relations. But, since I am only interested in a few terms, I adopt the brute force calculations shown in Table 1.

Table 1: Tournament Counts

n	Number of Teams	2(2n - 1)	Ways of Filling Out Brackets
1	2	2	2 = 21 = 100.301
2	4	4	8 = 23 = 100.903
3	8	16 = 24	128 = 27 = 102.11
4	16	256 = 28	215 = 104.515
5	32	216	231 = 109.33
6	64	232	263 = 1019.0

Suppose, by some luck of the draw, you correctly chose the 32 winners of all the first round games. There would still be over one billion (10⁹) possible ways of filling out the remainder of the brackets. Even if everybody in the world filled out a bracket for the full tournament, the overwhelming majority of possible brackets would still be unfilled.

Lo and Mueller's Need for More Scholarship

Consider Andrew W. Lo and Mark T. Mueller's draft paper "WARNING: Physics Envy May Be Hazardous To Your Wealth!", to appear in the Journal of Investment Management. They argue that economists' "physics envy" has led to "a false sense of mathematical precision in some cases", and they illustrate their argument by pointing to Paul Samuelson. They distinguish between uncertainty and risk and offer a checklist to assess the degree of uncertainty in your decision-making environment. They mention chaotic dynamics.

I find their references lacking. They don't reference Philip Mirowski, particularly his book More Heat Than Light in which he considers Samuelson. They do reference Frank Knight's Risk, Uncertainty, and Profit, but not Keynes. They do not reference G. L. S. Shackle and his role on the development of scenario planning. No reference to Joan Robinson appears. I think her 1974 paper "History vs. Equilibrium", with its distinction between logical time and historical time, is particularly apropos. Paul Davidson's 1982 JPKE paper, "Rational Expectations: A Fallacious Foundation for Studying Crucial Decision-Making Processes" is also of some importance, with its emphasis on the mainstream economist's assumption of ergodicity. J. Barkley Rosser, Jr., with his treatment of insights from complex dynamics, is also unreferenced.

I don't see why one would want to read Lo and Mueller until they engage some of this literature on their point.

Me, Elsewhere

At NEP-DGE Blog, I commented on some post about some dynamic general equilibrium model explaining the distributions of wealth and income:

"Chapter 13 of Cockshott, Cottrell, Michaelson, Wright, and Yakovenko’s Classical Econophysics (Routledge, 2009) explains the distribution of income and wealth to some extent. They have social classes and are interested in statistical equilibrium, as in thermodynamics. I don’t know why one should care about what [one] can do in the failed neoclassical paradigm."

Over at Crooked Timber, I mentioned some books that I think were influential for me:

"I guess the Lord of the Rings is the book I’ve read the most times.

I read the Bible from cover to cover once at an early age.

One friend in college had a couple of serious books of physics. So, if I was going to spout off on politics, I ought to read some serious books on economics. The two I found in a used book store were Keynes’ General Theory (which I reread several times) and Von Neumann and Morgenstern’s Theory of Games. Part of the influence of these is to show me I can read original research, whether I understand it or not. I’ve read a number of books others have listed, but one can say that that’s a consequence of this lesson.

Somewhere I came across a reference to Joan Robinson as “the english Galbraith”. I had liked Galbraith, so I read her. I read a lot of her collections and then Sraffa’s Production of Commodities, as well as secondary literature such as Geoff Harcourt’s book, Some Cambridge Controversies. The lesson here is that almost everything economics professors were teaching me as an undergraduate had been shown to be mostly nonsense decades before.

Somewhere in here I read Schumpeter’s History and Hayek’s Individualism and Economic Order. Basically, I read Hayek before I found out right-wingers cite him without reading him. Why wouldn’t a leftist who has also read Orwell accept that Stalinist central planning couldn’t be expected to work well?

I had read a lot of commentary – I particularly like Harrington’s The Twilight of Capitalism – before reading Marx with understanding. I actually read Theories of Surplus Value before the first volume of Capital.

I found some works of economic history eye-opening – maybe Braudel’s Capitalism and Material Life, Hobsbawm’s The Age of Revolution: 1789-1848, or Polanyi’s The Great Transformation.

I’m not sure about what were the earliest works in philosophy that I think I might have understood somewhat – probably some Russell, Kuhn’s Structure of Scientific Revolutions, or Popper’s The Open Society and It’s Enemies. Wittgenstein’s Philosophical Investigations is on my list of books I’ve read multiple times."

With this exercise, you will see books on others lists that you maybe should have put on yours. Then there are all the books I haven't yet read, have unjustly forgot, or never understood in the first place. I'll refrain from commenting on any other comments on Crooked Timber, but I will note that young Matt Zeitlin includes Rorty's Achieving Our Country - a good book - on his list.

James Galbraith "In Defense of Deficits"

James Galbraith speaks up against "one of the great misinformation campaigns of all time":

"To put things crudely, there are two ways to get the increase in total spending that we call 'economic growth.' One way is for government to spend. The other is for banks to lend. Leaving aside short-term adjustments like increased net exports or financial innovation, that's basically all there is. Governments and banks are the two entities with the power to create something from nothing. If total spending power is to grow, one or the other of these two great financial motors--public deficits or private loans--has to be in action." -- James K. Galbraith, "In Defense of Deficits", The Nation, 4 March 2010

Historians and Philosophers on Empirical Failures of Neoclassical Economics

Empirical evidence went against neoclassical economics in the following three cases:

• Empirical studies and surveys of businessmen found that they followed a full cost policy, not marginalism
• Behavioral economists have accumulated a body of experimental evidence, including preference-reversals and violations of transitivity, that people are not utility maximizers.
• David Card and Alan Krueger found that increased minimum wages did not decrease employment.

These incidents present data for philosophers, historians, and sociologists of economics. They can explore how mainstream economists reacted to these empirical findings. And three have done just this. Daniel Hausman and Philippe Mongin compare and contrast the reactions to full cost pricing and preference reversals. Tim Leonard compares and contrasts the reaction of mainstream economists to their findings on full cost pricing and on the minimum wage. In keeping with current trends, these articles are descriptive, not prescriptive. That is, they try to understand the positions of participants without passing judgement.

References

• Daniel M. Hausman and Philippe Mongin (1998) "Economists' Responses to Anomalies: Full-Cost Pricing versus Preference Reveresals" [Should this link work?]
• Thomas C. Leonard (2000) "The Very Idea of Applying Economics: The Modern Minimum-Wage Controversyand Its Antecedents, History of Political Economy, V. 32, Annual Supplement: 117-144

Kaldor's Model of Industry and Agriculture

In comments, an anonymous poster asks:

Is the Kaldorian model on industrial and General productivity in an economy applicable in understanding economic development in LDC's like Zambia?

I don't know anything about Zambia.

As I understand it, Kaldor developed that model for the world as a whole. Thirlwall applied that model to a Less Developed Country in a 1986 paper.

In the model, labor is originally not scarce - there is disguised unemployment in agriculture. Industrial production, unlike in agriculture, experiences increasing returns. If wages are low in agriculture, there would be more savings to finance expansion of industry. But there might not be the demand for industrial products. Demand for industrial products is increased by a relatively low price of industrial products, as compared to the price of agricultural products. Demand for industrial commodities might also be for exports. A dynamic equilibirum arises in the model in which a steady state of growth is achieved and the terms of trade between agriculture and industry are specified. The model is supposed to capture certain stylized facts and exhibit a certain complementarity between agriculture and industry. It is also supposed suggest different possibilities, such as the possibility of economic development from favorable terms of trade for agriculture at an initial stage and export-oriented growth at a later stage.

Later essays in Thirlwall's book in which his essays was reprinted treat Africa, particularly the Sudan. This empirical work treats some other considerations than those included in the mathematical outlined verbally above. I don't know current thinking about these issues, although I question the thinking that had been dominant in the Internation Monetary Fund, that is, the "Washington consensus".

• A. P. Thirlwall (1986) "A General Model of Growth and Development on Kaldorian Lines", Oxford Economic Papers (July) (Reprinted in The Economics of Growth and Development: Selected Essays of A. P. Thirlwall, Edward Elgar (1995))

Anti-Intellectualism Among Mainstream Economists

I find these comments to be anti-intellectual:

• John Quiggin rejects the Austrian school of economics on the ground that partisans of that school discuss political philosophy and the epistemology and methodology of economics.
• Roberto Perotti critizes Post Keynesians and neo-Ricardians on the grounds that they don't spend their time exclusively constructing formal models and estimating correlations. (I used Google's translation feature. Sergio Cesaratto answers from a Sraffian perspective.)
• Commentators at Mark Thoma reject discussions about what Adam Smith wrote.

I thought the point of scholarship was to attempt to make true statements. If somebody makes an untrue statement about what Keynes or Adam Smith said, one should correct them. This is not to say that that the fact that Keynes or Smith advocated something or other is a justification for policy. I think a historically accurate representation of an old text entails quite a bit of contextualization in terms of its time. To apply policy conclusions to our time would require recontextualization in contemporary terms, as well as empirical work.

I would think different scholars, even within a discipline, would find different questions of interest. Some economists argue for a supposed freedom to choose. Shouldn't some then be legitimately allowed to explore old texts or methodology or whatever? If Thomas Kuhn was somewhat correct, wouldn't one expect more discussion about methodology when the defining paradigm in a field has so obviously broken down, as today among mainstream economists?

On Sraffa, Elsewhere

Alex M. Thomas has begun a series of posts about Piero Sraffa's work in economics.

Survey of Utility Theory?

1.0 Introduction
I think utility theory has a canonical textbook presentation. Many variations seem to exist. In some, the additional structure is imposed on the (commodity?) space over which agents choose. In others, more basic assumptions are made from which preferences can be derived under certain special cases.

I'd like to know if there are any surveys to read over these variations. I'm not insisting on something critical. And, given the dryness of the subject matter, I might not put such a survey on top of my queue. As can be seen below, I'm not sure of the field that would be demarcated by such a surveys. But literature surveys, in some sense, construct their object.

2.0 Textbook Treatment
Consider a space of n commodities. Each element of the space is a vector x = (x₁, x₂, ..., x_n). Under the usual interpretation, x_i is the quantity of the ith commodity.

An agent is modeled as having a preference relation, ≤, over the space of commodities. A typical question is what assumptions must hold for a utility function to exist. A utility function u(x) exists if, for all x and y in the space of commodities:

x ≤ y if and only if u(x) ≤ u(y)

Typically, the preference relation is taken to be a total order, that is, complete, reflexive, and transitive. A preference relation is complete if, for all x and y in the space of commodities,

x ≤ y or y ≤ x

A preference relation is reflexive if, for all x in the space of commodities

x ≤ x

A preference relation is transitive if, for all x, y, and z in the space of commodities,

if x ≤ y and y ≤ z then x ≤ z

If the quantities of commodities fall along a continuum, a preference relation being a total order is not sufficient for a utility function to exist. Lexicographic preferences are an example of a preference relation for which a utility function does not exist. A continuity assumption rules out this case. This assumption is that for all x in the space of commodities, the sets {y | y ≤ x} and {z | x ≤ z} of commodities not preferred to x and commodities x is not preferred to, respectively, are closed.

Theorem: If a preference relation is a total order and is continuous in the above sense, then a utility function exists.

The utility function is only defined up to a monotonically increasing transformation. In other words, utility is ordinal. Typical exercises are to show certain properties of utility functions, such as ratios of marginal utilities (du/dx_i)/(du/dx_j), are invariant over the set of such transformations.

3.0 Probability
Von Neumann and Morgenstern generalized the commodity space to include vectors of the form: (p₁, x⁽¹⁾; p₂, x⁽²⁾; ..., p_m, x^(m)), where:

p₁ + p₂ + ... + p_m = 1

A commodity, in this sense, is a lottery. Each superscripted commodity vector x⁽ⁱ⁾ is associated with a probability p_i that it will be chosen.

Von Neumann and Morgenstern defined a new set of axioms to go along with their redefined commodity space. One implication is that for any two elements x and y in the commodity space, the linear combination (p, x; (1 - p), y) is also in the space. They obtain that a utility function exists, and it acts like mathematical expectation:

u(p₁, x⁽¹⁾; p₂, x⁽²⁾; ..., p_m, x^(m)) = p₁ u(x⁽¹⁾) + p₂ u(x⁽²⁾) + ... + p_m u(x^(m))

Under Von Neumann and Morgenstern's approach, utility functions are only defined up to affine transformations. That is, they are cardinal. In other words, they attain an interval measurement scale level. The utility for a lottery depends only on the probabilities and the resulting outcomes. It does not depend on how many spins of the wheel or roll of the dice are needed to decide between otherwise equivalent lotteries. Gambling is assumed to have no utility or disutility.

Leonard Savage develops axioms of probability concurrently with axioms of utility theory in his personalistic approach to probability and statistics. I'm not sure how much the survey I would like would go into approaches to probability, even if probability is important to decision theory. The same comment applies to game theory.

4.0 Attributes and Needs
Some see commodities as being chosen as an indirect means to choose something more abstract. As I understand it, Kevin Lancaster depicts a commodity as a bundle of attributes. Different commodities can have some attributes in common. A choice of an element in the space of commodities can then be related to an element in a space of commodity attributes.

The early Austrian school economists thought of goods as being desired for the satisfactions of wants. Water, for example, can be used to water your lawn, to satisfy a pet's thirst, or to drink yourself. One can imagine ranking wants in disparate categories. I am thinking of the triangular tables in Chapter III of Carl Menger's Principles of Economics, in Book III, Part A, Chapter III of Eugen von Böhm-Bawerk's Positive Theory of Capital, and in Chapter IV of William Smart's An Introduction to the Theory of Value. The tables are triangular because the most pressing want in one category typically is less pressing than the most pressing want in another category. An element in the space of commodities corresponds to the set of wants that the agent would choose to satisfy with the quantities of commodities specified by that element.

This mapping from quantities of commodities to sets of wants leads to a redefinition of marginal utility, which one might as well designate by a new name - marginal use. The marginal use of a quantity of commodity is, roughly, the different wants that would be added, with a set union, to the set of wants satisfied by the the given quantities of commodities with that additional quantity of the given commodity. McCulloch shows that a ranking of wants in different categories can arise such that a measure does not exist for the space of sets of wants. (A measure in this sense is a technical term in mathematics, typically taught in courses in analysis or advanced courses in the theory of probability.) He argues that the Austrian theory of the marginal use is thus ordinal. Surprisingly, his argument implies that the law of diminishing marginal utility does not require utility to be measured on a cardinal scale.

I haven't read Ian Steedman's work on consumption, but I think I'll mention it here.

5.0 Choices from Menus
Another generalization of the textbook treatment is to examine how a preference relation can be built out of a more fundamental structure. Imagine the agent is presented with a menu, where a menu is a nonempty set of elements of the commodity space. The agent is assumed to have a choice function, which maps each menu to the set of best choices, in some sense, in that menu. The agent is not postulated to rank either the elements not chosen for a given menu or the elements in the choice set.

A question: what constraints need to be put on choices out of menus such that preferences exist? Since a choice function can be constructed for which no preference function exists, some such constraints exist. I previously noted literature drawing on the logical structure of social choice theory in this context. Alan Isaac emphasizes temporal and menu independence in his overview of abstract choice theory.

6.0 Experimental Economics
I am emphasizing theory. A literature exists on experiments, many of which have falsified the textbook treatment of economics.

7.0 Computatibility, Conservation Laws, Etc.
Some of the above extensions of the textbook treatment seem to postulate some sort of structure within the agent's mind. Computers provide an arguable metaphor of mental processes, and some literature applies the theory of computability to economics. Gerald Kramer, for example, shows that no finite automaton can maximize utility in the simplest setting. I gather others have shown that the textbook treatment postulates that each agent's computation powers exceed those of a Turing machine, that agents compute functions that are, in fact, noncomputable. I turn to Kumaraswamy Velupillai's work for insights into computability, constructive mathematics, and economics. Philip Mirowski is always entertaining. One might also mention the literature on Herbert Simon's notion of satisficing

8.0 Conclusion
This post is a brief overview of some of what would be treated in a survey of variations and approaches to utility theory. Apparently, the notion of economic man can be complicated.

An Incomplete List of References

• Colin F. Camerer (2007) "Neuroeconomics: Using Neuroscience to Make Economic Predictions", Economic Journal, V. 117 (March): C26-C42.
• Alan G. Isaac (1998) "The Structure of Neoclassical Consumer Theory"
• Daniel Kahneman and Amos Tversky (1979) "Prospect Theory: An Analysis of Decision under Risk" Econometrica, V. 47, N. 2 (March): pp. 263-292
• Gerald H. Kramer () "An Impossibility Result Concerning the Theory of Decision-Making", Cowles Foundation Paper 274
• Kevin J. Lancaster (1966) "A New Approach to Consumer Theory", Journal of Political Economy, V. 75: pp. 132-157.
• J. Huston McCulloch (1977) "The Austrian Theory of the Marginal Use and of Ordinal Marginal Utility", Journal of Economics, V. 37, N. 3-4: pp. 249-280.
• Judea Pearl (1988) Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann
• Leonard J. Savage (1954, 1972) The Foundations of Statistics, Dover Publications
• Chris Starmer (1999) "Experimental Economics: Hard Science or Wasteful Tinkering?" Economic Journal, V. 109 (February): pp. F5-F15
• Ian Steedman (2001) Consumption Takes Time: Implications for Economic Theory, Routledge
• S. Abu Turab Rizvi (2001) "Preference Formation and the Axioms of Choice", Review of Political Economy, V. 13, N. 12 (Nov.): pp. 141-159
• John Von Neumann and Oskar Morgenstern (1953) Theory of Games and Economic Behavior, Third Edition, Princeton University Press

Labor Market Flexibility

Some mainstream economists claim that unemployment would be less if labor markets were more flexible and less rigid. In a 1998 paper arguing against this view, Bob Solow explains what this labor market rigidity that so many mainstream economists, especially "freshwater" economists, want to abolish is:

"My first observation is that 'labour-market rigidity' is never defined very precisely or directly in this context, but only be enumeration of tell-tale symptons. Thus a labour market is inflexible if the level of unemployment-insurance benefits is too high or their duration is too long, or if there are too many restrictions on the freedom of employers to fire and to hire, or if the permissible hours of work are too tightly regulated, or if excessively generous compensation for overtime work is mandated, or if trade unions have too much power to protect incumbent workers against competition and to control the flow of work at the site of production, or perhaps if statutory health and safety regulations are too stringent. It seems clear that those who point to labour-market rigidity as the source of high unemployment have something other than simple nominal or real wage rigidity in mind, or so shall I assume." -- Robert M. Solow, "What is Labour-Market Flexibility? What is it Good for?", Proceedings of the British Academy, V. 97

As I understand it, many of these "rigidities" were put in place in the United States context around the time of the New Deal with the cooperation and assistance of Institutionalist economists, a school with some sense of the real world.

Bank for International Settlements (BIS) - Receptive to Heterodoxy?

I never even heard of the Bank for International Settlements (BIS) before a couple years ago. I am vaguely aware of the Basel accords, which I guess they have something to do with.

I have come across two papers - a ridiculously small number of data points - which make me wonder if they are receptive to heterodox economics. William White, the former chief economist of the BIS, criticized last December the direction of research in modern macroeconomics. He thinks macroeconomists should pay more attention to Hyman Minsky and also Austrian Business Cycle Theory. In a current BIS working paper (H/T to D-Squared), Piti Disyatat rejects the loanable funds theory and argues for the theory of an endogenous money supply, if I understand correctly. This paper references, among others, Basil Moore, Thomas Palley, and Randy Wray.

Colin Rogers and Ian Steedman Down Under

Colin Rogers has a review essay of Geoff Tily's Keynes's General Theory, the Rate of Interest and 'Keynesian' Economics: Keynes Betrayed:

"Tily (chapter 4) carefully documents how this Robertson version of 'Keynesian' economics succeeded in squeezing out Keynes' vision. The model behind Robertson’s vision is instantly recognisable by all students of economics today and consists of the idea that there exists a unique long-run equilibrium around which the economy fluctuates during booms and recessions... Apart from the fact that Keynes ... described rigidities as the classical explanation for unemployment, Tily makes it clear that this 'Keynesian' interpretation overlooks entirely that Keynes was concerned with the existence of multiple long-period equilibria in the General Theory. -- Colin Rogers, "Keynes vs the Keynesians: Keynes Rediscovered", History of Economics Review, Winter 2008.

I have never quite understood the idea that Keynesian policy is about helping the economy reach equilibrium quicker. I thought The Arrow-Debreu model of intertemporal equilibria was the canonical neoclassical explanation of prices. That model doesn't allow for out-of-equilibrium behavior in real time. By now, it should be trivial to create models of multiple long-run equilibria. Keynes' explanation for multiple long-run equilibria, I think, had to do with long-run expectations - each state of expectations creates a different (set of?) long run equilibrium.

Ian Steedman reviews another volume in Hayek's collected works:

"Laurence White has been zealous in providing editorial footnotes to Hayek’s text and many of his notes are indeed helpful... Some footnotes seem to be quite unnecessary explications... The simple observation by Hayek, that the choice of production method may change in response to a relative input price change, is made the occasion of a long and irrelevant note on the 'economics of socialism' debate..., while an editorial reference to the 'reswitching controversy' cites a single discussion thereof – one by Leland Yeager! ... It might be felt here that Hayek’s text is being used for the promotion of ‘Austrian’ economics more generally." -- Ian Steedman on "F. A. Hayek. The Pure Theory of Capital. Volume 12 of the Collected Works of F. A. Hayek, History of Economics Review, Summer 2009.

Fred Lee's New History of Heterodox Economics

Frederick Lee has written a book: A History of Heterodox Economics: Challenging the Mainstream in the Twentieth Century. I don't know if I'm going to purchase this Routledge book. I've read papers many of the chapters grew from.

Alex Haley and John Maynard Keynes: Self-Confident Authors

Alex Haley used to hang out at the Savoy, a local restraurant in Rome, NY. While writing Roots, he told the owner, "I am writing a book that is going to change how white people look at black people in America."

While writing The General Theory of Employment, Interest, and Money, Keynes had an argument with George Bernard Shaw about Marx. Keynes wrote to Shaw on 1 January 1935:

"...To understand my state of mind, however, you have to know that I believe myself to be writing a book on economic theory, which will largely revolutionise - not, I suppose, at once but in the course of the next ten years - the way the world thinks about economic problems. When my new theory has been duly assimilated and mixed with politics and feelings and passions, I can't predict what the upshot will be in its effects on actions and affairs. But there will be a great change, and, in particular, the Ricardian foundations of Marxism will be knocked away.

I can't expect you, or anyone else, to believe this at the present stage. But for myself I don't merely hope what I say, - in my own mind I'm quite sure."

I don't accept that Marx built on Say's Law, which is what Keynes refuted.

Notre Dame Faculty Senate Defends Academic Freedom

Maybe Phil Mirowski and his more interesting colleagues will be able to pursue their vocation.

Garrison Orally On Reswitching

The following is from the Question and Answer session for Roger Garrison's recent lecture, "Macroeconomics: The Boom and Bust Cycle" (starting at about 59:29):

Audience member: "Can you reconcile using Hayekian triangles with reswitching and [unclear]?"

Garrison: "Yeah. The question is, 'How do you reconcile the Hayekian triangle with the Cambridge Capital Controversy, which involves capital-reswitching and technique reversing. I got that backwards - isn't it technique reswitching and capital reversing. Let me just take a poll here. How many have heard of the Cambridge Capital Controversy and know about technique reswitching and capital-reversing? Good, OK, good. One guy.

Let me just give you a standing-on-one-foot explanation. Capital theorists in Cambridge - this is Cambridge, England - argued that Böhm-Bawerk didn't have it quite right. His math was too simple, and if you allow for inputs and outputs to be expressed by polynomials, instead of rectangles or linear inputs, you find that the production process could lengthen or shorten with the change in the interest rate. If the interest rate falls, maybe it will shorten, or maybe it will lengthen. And you don't know. You have to look at the math and see how it turns out.

And the way the math turns out, there are only trivial exceptions - very small exceptions to the proposition that a low interest rate encourages long-term investment. You can come up with counterexamples, but even Cambridge capital-theorists, like Joan Robinson - she wrote an article, "The Unimportance of Reswitching" - just the magnitude doesn't amount to anything. If you read Hayek's Pure Theory of Capital, he acknowledges that, yes, yes, you can have funny profiles that seem to run counter to the general proposition that a low interest rate encourages long-term production. But those are very trivial. I've published articles on this. I can give you a link to one. But one thing I observed is that to get numbers where the reswitching is visible to the naked eye, you have to have the interest rate changing from 100% to 50%, or something like that, which itself suggests how minor that distinction is.

There are actually more fundamental criticisms that are based on how the Cambridge capital theorists measure roundaboutness. And the Austrians don't really measure it. They don't need to measure it. It's enough to show you the pictures. You saw the picture of the Research and Development chemist. You saw the picture of the retail clerk. That's what we are talking about, however you describe that mathematically. So I think, economically, that just doesn't work.

There are economists - Post Keynesians are the main ones - who throw up that Cambridge Capital Controversy as a killer of Austrian Business Cycle theory. They see that as the basis for just dismissing Austrian Business Cycle theory, but I think with very little justification."

References:

• Roger M. Garrison (1979) "Comment: Waiting in Vienna", in Time, Uncertainty and Disequilibrium: Exploration of Austrian Themes (ed. by M. J. Rizzo), D. C. Heath and Company
• Roger M. Garrison (2006) "Reflections on Reswitching and Roundaboutness", in Money and the Markets: Essays in Honor of Leland Yeager (ed. by R. Kopl), New York:Routledge.

Austrian Business Cycle Theory As Uninteresting And Esoteric

I have been unsuccessful in getting my refutation of Austrian Business Cycle Theory accepted for publication in a peer-reviewed journal. My latest publically available version is downloadable from SSRN. I have had a later version rejected a couple of weeks ago. This version's references include Roger Garrison's 2006 "Reflections on Reswitching and Roundaboutness". I hope quoting an extract from a review is acceptable, since some might find this of interest:

"Hayek had washed his hands of the triangles [that ground his theory of the trade cycle] long before the ink was dry on his Pure Theory of Capital... Modern Austrian school economists like Roger Garrison still find the triangle-logic compelling, but I have the impression that, with the possible exception of Leland Yeager’s 1976 Economic Inquiry paper, no systematic attempt has been made to show that Austrian economics is immune to the capital critique; the arguments are intuitive rather than carefully drawn. (Even Yeager’s piece struck me when I read it, many years ago, as largely intuitive.) ...the arguments had never been put forward in a robust way..."

Classical Cross Dual Dynamics

"The actual price at which any commodity is commonly sold is called its market price. It may either be above, or below, or exactly the same with its natural price.

The market price of every particular commodity is regulated by the proportion between the quantity which is actually brought to market, and the demand of those who are willing to pay the natural price of the commodity, or the whole value of the rent, labour, and profit, which must be paid in order to bring it thither. Such people may be called the effectual demanders, and their demand the effectual demand; since it may be sufficient to effectuate the bringing of the commodity to market...

...The natural price, therefore, is, as it were, the central price, to which the prices of all commodities are continually gravitating...

The whole quantity of industry annually employed in order to bring any commodity to market, naturally suits itself in this manner to the effectual demand. It naturally aims at bringing always that precise quantity thither which may be sufficient to supply, and no more than supply this demand." - Adam Smith, The Wealth of Nations, Book I, Chapter VII

1.0 Introduction
Contemporary economists have elaborated Smith's metaphor of the gravitational attraction of market prices to natural prices. Elaborations consist of formal models of cross-dual dynamics.

Systems of equations describing prices and quantitites are dual systems in the post Sraffa/von Neumann tradition. Dynamics are cross dual when changes in prices respond to quantities and changes in quantities respond to prices. In particular, industries expand in which rates of profits are high and contract in industries in which profits are low. And prices fall in industries in which the quantity supplied exceeds the effectual demand. Prices rise in industries in which the quantity supplied is below the effectual demand.

Dupertuis and Sinha (2009) is one immediate impetus for my setting out this model of the reallocation of labor for a given wage. I don't think I thoroughly understand models of cross-dual dynamics. I think they are of interest for exploring the possible dynamics of market prices in competitive capitalist economies. I don't see that they are directly empirically applicable. For that, one needs to worry about markup prices and the degree of utilization of capacity in various industries.

2.0 Natural Prices
The data of our problem are:

• The n x n input-output matrix A, where n is the number of industries and a_{i, j} is the amount of the ith commodity used as input per unit output of the jth industry
• The n-element row vector a₀ of labor inputs, where a_{0, j} is the amount of person-hours hired per unit output in the jth industry
• The money wage w_given
• The composition of net output as expressed in the n-element column vector c_given.

This is a circulating capital model in which all production processes use up their inputs in a year. Labor is assumed to be paid their wages at the end of the year. Only economies capable of producing a surplus product are considered. For simplicity, assume Constant Returns to Scale (CRS) and that every commodity is basic, in Sraffa's sense. This section considers the problem of finding:

• The natural prices, as expressed in the n-element row vector p^*
• The corresponding wages w^*
• The corresponding rate of profits r^*
• The effectual demand as expressed in the n-element column vector of gross quantities q^*
• The n-element column vector of net quantities y^*.

In the system of natural prices, the same rate of profits are made in every industry:

p^* A(1 + r^*) + a₀ w^* = p^*

I take the net output as the numeraire.

p^* y^* = 1

In my formulation here, the wage is taken as a given ratio of the net output:

w^* = w_given

The above equations comprise the price system for natural prices.

Net outputs and gross outputs are related by the following equation:

y^* = q^* - A q^* = (I - A) q^*

where I is the identity matrix. I normalize the units of labor such that one unit is employed throughout the economy:

a₀ q^* = 1

Finally, the net output is assumed to be in the specified proportions. That is, there exists a positive constant k such that

y^* = k c_given

The above systems of equations are sufficient to determine gross and net effectual demands, natural prices, and the distribution of income. (An alternative specification would take the composition of gross output as given, instead of the net output. Perhaps outputs should be in units of Sraffa's standard commodity.)

3.0 Initial Condititions
The problem in the remaining sections is to define a dynamic process for the quantities produced q(t) and the market prices p(t) for t = 0, 1, 2, ... The initial quantities q(0) and market prices p(0) are givens. For the sake of the argument, I consider a dynamic process in which the amount of labor employed and the value of net output are invariant. So the initial quantities and prices must satisfy the following equations:

a₀ q(0) = 1

p(0) y(0) = p(0)(I - A)q(0) = 1

4.0 Reallocation of Labor
Define r_average(t), the average rate of profits for the economy as a whole at time t:

r_average(t) = [p(t)(I - A - a₀ w_given)q(t)]/[p(t) A q(t)]

The numerator in the expression on the right hand side above is the value of the surplus product remaining in the capitalists' possession after replacing the means of production and paying laborers their wages. The denominator is the value of the capital goods advanced.

Typically, the rate of profits will vary from the average among the industries. The rate of profits for the jth industry at time t is:

r_j(t) = [p_j(t) - p(t) a_{., j} - a_{0, j} w_given]/[p(t) a_{., j}]

where a_{., j} is the jth column of the input-output matrix A.

Define R_average(t) to be the n-element column vector with each element equal to the average rate of profits. Let R(t) be the n-element column vector with each element being the rate of profits for the corresponding sector.

Now dynamics of the quantities of produced commodities can be specified:

q(t + 1) = [1/f₁(t)]{[R(t) - R_average(t)] + q(t)}

Or, in terms of scalars:

q_i(t + 1) = [1/f₁(t)]{[r_i(t) - r_average(t)] + q_i(t)}

where

f₁(t) = 1 + a₀[R(t) - R_average(t)]

The denominator f₁(t) above is a normalization that ensures the quantity of labor employed is always unity. The numerator ensures that the more the rate of profits in a sector exceeds the average, the faster that sector will expand in comparison with other sectors. (An alternative formulation might compare the rate of profits in each industry with the rate of profits r^* in the system of natural prices.)

5.0 Price Changes
Price dynamics are here set out more directly:

p(t + 1) = [1/f₂(t)]{p(t) - [q^T(t) - q^*T]}

where x^T is the transpose of the vector x. In terms of scalars, prices are given by:

p_i(t + 1) = [1/f₂(t)]{p_i(t)- [q_i(t) - q^*_i]}

The time series f₂(t) is defined as follows:

f₂(t) = [q^*T - q^T(t)](I - A)q(t + 1)} + p(t)(I - A)q(t + 1)

The denominator f₂(t) above is, again, a normalization condition. In this case, the normalization ensures the value of the net output is equal to unity. Since the composition of net output typically changes over the course of the process, the real wage varies in terms of any fixed commodity basket. It does remain, however, a given ratio of the net output.

6.0 Conclusion
I have set out above a model of a dynamic process, but without an analysis of its properties. An obvious theorem is that if initial quantities and prices happen to be equal to the effectual demands and natural prices, they will be left unchanged by the dynamics of market adjustments. In other words, the natural system is a stationary point of this dynamic process.

An interesting question is the trajectory of market prices, given an arbitrary starting point. I don't expect the process to necessarily converge to the natural system. At this point, I don't have any numeric examples of limit cycles or chaotic behavior. A failure of local stability doesn't bother me; I have often thought of Sraffa's work as pointing towards the possibility of complex dynamics arising in models of capitalist economies.

Questions of structural stability are of interest as well. Do dynamic properties of the system depend on the level of wages, especially if one introduces into the model a choice of technique? And how do the answers to these questions vary, if at all, with alternative modeling assumptions, some of which I have indicated? I do not know that the literature has reached definitive answers to these questions.

Update (28 January 2010): I have redefined the dynamics above in a way that seems more reasonable to me.

References

• Michel-Stéphane Dupertuis and Ajit Sinha, "A Sraffian Critique of the Classical Notion of Centre of Gravitation", Cambridge Journal of Economics, V. 33 (2009): 1065-1087.

Upcoming URPE Conferences

Chris Pepin informs us that the Union for Radical Political Economics (URPE) will participate in the upcoming Eastern Economics Association (EEA) conference. The conference will be held at the Loews Philadelphia Hotel, February 26 - 28. A program is available. This year is the 50th anniversary of the publication of Sraffa's book.

URPE is also participating in the Left Forum, on March 19-21 at Pace University.

Elsewhere

• Eric Rauchway tells us about the Bretton Woods conference, where John Maynard Keynes showed that some economists could be more useful than dentists.
• David Ruccio reprints a cartoon by B. Deutsch making fun of economists prefering a supposedly elegant theory of the minimum wage to empirical results falsifying the theory.
• Bill Mitchell has a negative view of Greg Mankiw's textbook.