Do Angeletos And La'O Know What They Are Talking About?

Some comic gives some stupid and vile economists an opportunity to comment on some of Cosma Shalizi's ideas [1]. (I find particularly stupid the commentator that cites Amazon, a Web 2.0 exemplar, for an example of agents interacting only through prices.) Eventually, one respondent says that George-Marios Angeletos and Jennifer La'O show how to embed animal spirits in Dynamic Stochastic General Equilibrium (DSGE) models. I'm not sure which paper they are talking about, but I did download their paper Sentiments (with accompanying slides.) I do not object to how Angeletos and La'O model shocks, although in this setting I don’t see why I should care.

I was quickly stopped in reading this paper at the first footnote:

"By 'mainstream' we mean the prototypical RBC and New-Keynesian models, as well as the more recent DSGE models. This excludes models with multiple equilibria or irrational agents, which we discuss in due course."

How are models with multiple equilibria non-mainstream? They continue in this vein throughout their paper:

"In our model, agents are fully rational; preferences and technologies are standard; markets are Walrasian; there are no nominal frictions, no externalities, and no non-convexities; the equilibrium is unique; and there is no room for correlation devices or lotteries. In these respects, our theory has squarely neoclassical foundations."

"This extrinsic uncertainty has very similar flavor as the one encountered in models with multiple equilibria: it captures the self-fulfilling nature of short-run fluctuations. Importantly, though, it does not rest on the severe externalities, non-convexities and missing markets that are most often needed to sustain multiple equilibria-nor does it come with the usual difficulties in conducting policy analysis."

It is my understanding that the Sonnenschein-Mantel-Debreu theorem is proved in a Walrasian model with no externalities, non-convexities, or missing markets. The excitement over the theorem comes from being derived with the same sort of assumptions that are used in Debreu's Theory of Value to derive the existence of an equilibrium. So I find it hard to believe the authors know what they are talking about when they suggest multiple equilibria are non-Walrasian or non-neoclassical.

By the way, I happen to have available a model with multiple equilibria. Figures 1 and 2 below illustrate. I'd like somebody to point out to me how agents in this model are not fully rational; how preferences and technologies are non-standard; or where nominal frictions, externalities, and non-convexities exist in this model. I suppose one can say some markets do not exist in an overlapping generations model. Agents cannot buy commodities or sell their labor before they are born or after they are dead. I did not think such an assumption made a model heterodox or non-neoclassical.

Figure 1: Equilibrium Interest Rates as a Function of One Parameter in the Utility Function

Figure 2: Equilibrium Wages as a Function of Another Parameter

[1] Cosma Shalizi is on a team of three that has recently received a grant awarded by the Institute for New Economic Thinking.

A Slight Illness - The Doctor Jests, A King Today - Tomorrow He Is Dead

Has the global financial crisis discredited Dynamic Stochastic General Equilibrium (DSGE) models? It seems to me that that may be so, but I wonder if new criticisms of DSGE models have been put forth. It seems to me the fundamental flaws of such models have been unaddressed for decades. Cosma Shalizi describes these models in a critical way. Joseph Stiglitz is not impressed with them either:

"It is hard for non-economists to understand how peculiar the predominant macroeconomic models were. Many assumed demand had to equal supply - and that meant there could be no unemployment. (Right now a lot of people are just enjoying an extra dose of leisure; why they are unhappy is a matter for psychiatry, not economics.) Many used "representative agent models" - all individuals were assumed to be identical, and this meant there could be no meaningful financial markets (who would be lending money to whom?). Information asymmetries, the cornerstone of modern economics, also had no place: they could arise only if individuals suffered from acute schizophrenia, an assumption incompatible with another of the favoured assumptions, full rationality." -- Joseph Stiglitz

As I understand it, Smets and Wouters (2007) is an example of a DSGE model widely approved of by mainstream economists. Sbordone et al. (2010) is a recent presentation of an introductory DSGE. One can see that the output of these models is a set of stochastic processes meant to model certain time series available in empirical data. Nominal interest rates, (real) income, the inflation rate, the volume of one-period government bonds, employment, and nominal wages are all examples of such time series. The input into such models is another set of stochastic processes. These inputs are given names that suggest they are random terms in functions characterizing either government entities - e.g., monetary policy shock - or agents in microeconomic models. Examples of the latter kind of names are a household discount rate shock, productivity shock, markup shock, and firm discount rate shock. Stochastic processes are specified by parameters of certain probability distributions. As one can see from the names of these inputs, the agents are supposed to be optimizing, including across time. A story, expressed in mathematics and supposedly of microeconomic equilibrium, connects the inputs to the outputs in the model. That is, the DSGE models are supposed to have microfoundations.

But they do not have microfoundations. I look for a number of mistakes in such models:

• Are inputs into production function measured in numeraire units? (The numeraire is often taken to be a basket of consumer goods.) Joan Robinson (1953-54) explains why measuring the quantity of capital in production functions in numeraire units is an error. Notice this is not solely a question of the aggregation of capital. A model can have a continuum of capital goods, yet still exhibit this mistake.
• Are representative agents used? Kirman (1992) explains why the use of representative agents is unfounded.
• Is money modeled? Frank Hahn (1965) explains why money does not matter in General Equilibrium models, even though it does seem to matter for actually existing capitalist economies. Mainstream economists have a couple of strategies for introducing money in an ad hoc way into DSGE models. But I am not convinced the typical modeler has ever managed to address Hahn's point.
• Is the possibility of multiple equilibria taken seriously? Is it demonstrated that non-equilibrium dynamic processes converge to the modeled equilibrium? Richard Goodwin (1990) illustrates what a macroeconomics looks like that, in contrast to typical DSGE models, takes dynamics seriously. Kirman (1989) shows that ignoring muliple equilibria and stability issues was demonstrated to be unfounded by the Sonnenschein-Mantel-Debreu results. Shiller (1978) long ago raised the issues of multiple equilibria and convergence. Shiller was critiquing the tradition out of which DSGE models evolved.

I haven't read much in the literature of DSGE models. It seems to me, however, that these issues are routinely ignored by many mainstream economists. Perhaps a wider range of macroeconomic models should be considered by serious researchers.

References

• Ricardo J. Caballero (2010) "Macroeconomics after the Crisis: Time to Deal with the Pretense-of-Knowledge Syndrome", MIT Dept. of Economics, Working Paper 10-16 (September 27)
• Richard M. Goodwin (1990) Chaotic Economic Dynamics, Oxford University Press.
• Robert J. Gordon (2010) "Is Modern Macro or 1978-era Macro More Relevant to the Understanding of the Current Economic Crisis".
• Frank H. Hahn (1965) "On Some Problems of Proving the Existence of an Equilibrium in a Monetary Economy", in The Theory of Interest Rates (Ed. by F. H. Hahn and F. Brechling), Macmillan.
• Frank Hahn and Robert Solow (1995) A Critical Essay on Modern Macroeconomic Theory, MIT Press.
• Kevin D. Hoover (1988) The New Classical Macroeconomics, Basil Blackwell.
• J. E. King (2010) "Microfoundations for Macroeconomics? The Pre-History of a Dogma, 1936-1975", HETSA 2010, (June)
• Alan Kirman (1989) "The Intrinsic Limits of Modern Economic Theory: The Emperor has No Clothes", Economic Journal, V. 99, N. 35: Supplement: Conference Papers: pp. 126-139.
• Alan P. Kirman (1992) "Whom or What Does the Representative Individual Represent?", Journal of Economic Perspectives, V. 6, N. 2 (Spring): pp. 117-136.
• Joan Robinson (1953-54) "The Production Function and the Theory of Capital", Review of Economic Studies, V. 21, N. 55: pp. 81-106.
• Argia M. Sbordone, Andrea Tambalotti, Krishna Rao, and Kieran Walsh (2010) "Policy Analysis Using DSGE Models: An Introduction", Federal Reserve Bank of NY Economic Policy Review.
• Robert J. Shiller (1978) "Rational Expectations and the Dynamic Structure of Macroeconomic Models: A Critical Review", Journal of Monetary Economics, V. 4: pp. 1-44.
• Frank Smets and Rafael Wouters (2007) "Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach", American Economic Review, V. 97, N. 3: pp. 586-606.
• Alessandro Vercelli (1991) Methodological Foundations of Macroeconomics: Keynes & Lucas, Cambridge University Press.

To read:

• Wynne Godley and Marc Lavoie (2007) Monetary Economics: An Integrated Approach to Credit, Money, Income, Production and Wealth, Palgrave MacMillan.

James Joyce On Identity Economics

I think that if one looked, one would be able to find in lots of depictions in literature of multiple selves. Here's an example:

"...he had heard about him the constant voices of his father and of his masters, urging him to be a gentleman above all things and urging him to be a good catholic above all things. These voices had now come to be hollow-sounding in his ears. When the gymnasium had been opened he had heard another voice urging him to be strong and manly and healthy and when the movement towards national revival had begun to be felt in the college yet another voice had bidden him to be true to his country and help to raise up her language and tradition. In the profane world, as he foresaw, a worldly voice would bid him raise up his father's fallen state by his labours and, meanwhile, the voice of his school comrades urged him to be a decent fellow, to shield others from blame or to beg them off and to do his best to get free days for the school. And it was the din of all these hollow-sounding voices that made him halt irresolutely in the pursuit of phantoms." -- James Joyce, A Portrait of the Artist as a Young Man

Does how artists depict human beings carry any weight for how economists choose to portray agent's choices? Should it?

Economic & Political Weekly

Every once in a while, I notice popular and semi-popular journals in which ideas I find interesting are discussed. As I understand it, Economic & Political Weekly is an Indian journal. It was named Economic Weekly when it published Krishna Bharadwaj's 1963 review of Sraffa's book. I cannot speak to Indian politics. But I find two articles of interest in the 16 October number.

Ghose (2010) builds on Arthur Lewis's model of development. Lewis depicts undeveloped economies as exhibiting a kind of dualism in which capitalist and traditional (subsistence) sectors coexist. The traditional sector experiences disguised underemployment and can provide an infinite supply at labor at the going wage. Lewis mentions farmerss, casual workers, petty traders, domestic and commercial retainers, and woman in the household as sources of such a labor supply. Economic development is a matter of structural change in which the capitalist sector expands and replaces the traditional sector. My impression is that this distinction between structural change and a mere quantitative expansion used to distinguish the economics of development and of growth. Ghose suggests this perspective has been lost in development economics.

Sinha (2010) provides an appreciation of Sraffa's book. Sinha does not see Sraffa's prices of production as dynamically stable limit points of some sort of gravitational process governing market prices. Nor does he think an internal critique of neoclassical economics based on reswitching was central to Sraffa's project. He emphasizes Sraffa's standard commodity and likes Joan Robinson's reading of Sraffa as generalizing Ricardo's corn economy, in which the rate of profits is a physically specified ratio independent of relative prices. This reading resembles Bharadwaj's.

• Krishna Bharadwaj (1963) "Value through Exogenous Distribution", Economic Weekly (24 August): pp. 1450-1454. (Republished in Capital and Growth (edited by G. C. Harcourt and N. F. Laing), Penguin (1973))
• Ajit K. Ghose (2010) "Reinventing Development Economics", V. XLV, N. 42 (16 October): pp. 41-50.
• Ajit Sinha (2010) "Celebrating Fifty Years of Sraffa's Production of Commodities by Means of Commodities", V. XLV, N. 42 (16 October): pp. 61-65.

Bad Teaching By DeLong?

Brad DeLong has kindly made his lecture notes on General Equilibrium available online. I think he explicitly only asserts that equilibria exist (under certain conditions), not that any are necessarily stable. But I do not see how any student reading his notes cannot come to that conclusion. He never explicitly states that economists have found that General Equilibrium Theory imposes basically no limit on dynamics - this is an implication of the Sonnenschein-Mantel-Debreu results. I also don't care for DeLong's treatment here of Karl Marx, who was not writing about the allocation of given resources. (I think that a formalization of Marx's notion of prices of production is more like a Von Neumann ray without requiring labor markets to clear.)

Not being a teacher, I'm willing to entertain discussion of simplifications in teaching beginners. I can see why Joan Robinson thought that General Equilibrium Theory doesn't stand up long enough to be knocked down. The complete model postulates that enough markets exist such that you can trade any commodity for any other commodity across all time periods and all states of nature. This is wildly non-descriptive of any actual capitalist economies. But I can see how the introductory teacher might not get to point where this objection makes any sense.

West, East, And South

Peter Nobel, who is descended from Alfred Nobel, has come out against the Nobel Prize in economics. I want to focus on this part of his statement:

"With no knowledge of economics, I have no opinions about the individual economics prize winners. But something must be wrong when all economics prizes except two were given to Western economists, whose research and conclusions are based on the course of events there, and under their influence."

My question is which two prize winners does Nobel have in mind? I think Leonid Kantorovich, the Soviet co-inventer of Linear Programming, is obviously one. Is the other Amartya Sen? I think one might contrast the West with both Eastern block countries and the global South. If one counts the South as non-West, then Peter Nobel's count is not quite correct. One would also have to count Arthur Lewis, for his contributions to development economics.

Papers To Read

I seem to be very slow to either read these or write up a detailed explanation:

• Francis M. Bator (1958) "The Anatomy of Market Failure", Quarterly Journal of Economics, V. 72, N. 3 (Aug): 351-379. John Cassidy takes this paper as the authoritative definition in his book How Markets Fail: The Logic of Economic Calamities.
• Arindrajit Dube, T. William Lester, and Michael Reich (2008) "Minimum Wage Effects Across State Borders: Estimates Using Contiguous Counties", forthcoming in the Review of Economics and Statistics. Generalizes the natural experiment approach of Card and Krueger to look at all cross-state local differences in minimum wages in the United States between 1990 and 2006. They find no adverse employment effects from higher minimum wages in the ranges examined. You can watch a video interview with Dube here. (The Wikipedia page on minimum wages also lists meta-analyses, by Stanley and by Doucouliagos & Stanley, more recent than Card and Krueger's meta-analysis.)
• Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou (2009) "The Complexity of Computing a Nash Equilibrium", Communications of the ACM, V. 52, No. 2: pp. 89-97. Defines a complexity class between P and NP and proves that computing a Nash equilibrium is in that class. Thus, if P ≠ NP, Nash equilibria cannot be computed in polynomial time for arbitrary games. In other words, computing a Nash equilibrium in general is infeasible in practice. (Tim Roughgarden's "Algorithmic Game Theory" (Communications of the ACM, V. 53, No. 7 (Jul. 2010)) and Yoav Shoham's "Computer Science and Game Theory" (Communications of the ACM, V. 51, No. 5 (Aug. 2008)) are survey articles.)
• Colin F. Camerer (2006) "Wanting, Liking, and Learning: Neuroscience and Paternalism", University of Chicago Law Review, V. 73 (Winter). Argues that three neural subsystems in our brains process "wanting", "liking", and "learning" separately. I don't think this is quite what Ian Steedman and Ulrich Krause mean by a Faustian agent, but it seems to be related.

P.S. Commentator Emil Bakhum lists some objections to Sraffa's analysis from Alfred Muller. I do not agree that these objections correctly characterize Sraffa's analysis, a point to which I may return. I think I would like a more complete reference, although I might not be able to read it if it is in german.

A "Nobel" Prize For Epicycles In Labor Economics?

Peter Diamond, Dale Mortensen, and Christopher Pissarides won the "Nobel" prize in economics this year. I do not have Bill Mitchell’s expertise on search theory and labor economics. Nevertheless, I thought I’d record my reactions.

As I understand it, Diamond, Mortensen, and Pissarides think labor would be described by by the interactions of well-behaved supply and demand functions for labor if it were not for the heterogeneity of workers and jobs and the time to form matches between them. The orthodox theory would be wrong even if workers and jobs were homogeneous. So I find puzzling why I should approve of this year’s award. I think my opinion is consistent with some of Alessandro Roncaglia’s observations of trends in mainstream economics.

David Ruccio and Richard McIntyre & Michael Hillard (Hat tip to Nick Kraft) have been critical of this year’s award. Paul Krugman has praised it. Doubtless, one could find more praise in the blogosphere.

Grandpa's Giant Pancake

Ingredients

1/2 cup Flour
3 teaspoons baking powder
3 eggs
1/2 cup milk

(This is my uncle's recipe. My father uses 1 cup, 1 teaspoon, 1 egg, and 1 cup, respectively.)

1) Mix and stir. Pour in large, greased fry pan.

2) Cook on medium high heat until bubbles burst. Flip and cook other side.

My father eats this with lots of butter, not maple syrup. Makes 2 or 3 pancakes.

Weird Science III

I guess this is part of a series in which I describe oddities upon which I have stumbled. Here I focus on two phenomena in the measurement of gravity. Perhaps the theory of general relativity is wrong. (The link goes to an explanation of a different problem in physics). Of course, much more prosaic explanations are possible.

Maurice Allais, the recently dead "Nobel" laureate in economics, experimented with a paraconical pendulum during the 1950s. He discovered the Allais effect, which is a variation in the behavior of a pendulum during an eclipse. The plane of the pendulum rotated approximately ten degrees during the eclipse and then returned to the previously pattern. A number of scientists have tried to replicate this and similar effects with various experimental equipment during various eclipses. Some succeeded in replication and some failed. Allais’ explanation, apparently, was to revive the 19th century concept of the aether and argue that space is anisotropic.

Pioneers 10 and 11 were launched in 1972 and 1973, respectively. NASA was still in communication with them after they had passed beyond the orbit of Pluto and were more than 20 Astronomical Units away from the sun. (An AU is the average distance from the Earth to the sun.) Pioneers 10 and 11 have an anomalous acceleration towards the sun of an order of magnitude of 10^-7 centimeters per square seconds. In other words, as they move away from the sun, they are very slowly slowing down more than can be accounted for under the current (relativistic) understanding of gravity. Pioneers 10 and 11 are moving away from the sun at a rate of approximately 12 kilometers per second. (It dawned on me while writing the above that I am no longer sure how many planets are in our solar system.)

References

• Maurice Allais (1986). "The Concepts of Surplus and Loss and the Reformulation of the Theories of Stable General Equilibrium and Maximum Efficiency", in Foundations of Economics (edited by Mauro Baranzini and Roberto Scazzieri), Basil Blackwell.
• John D. Anderson et al. (1998) "Indication, from Pioneer 10/11, Galileo, and Ulysses Data, of an Apparent Anomalous, Weak, Long-Range Acceleration", arXiv:gr-qc/9808081v2 (1 October)
• John D. Anderson et al. (2005) "Study of the Anomalous Acceleration of Pioneer 10 and 11", arXiv:gr-qc/0104064v5 (10 March)
• Gibor Basri and Michael E. Brown (2006) "Planetesimals to Brown Dwarfs: What is a Planet?", Ann. Rev. Earth Planet. Sci., V. 34: pp. 193-216
• M. E. Brown, C. A. Trujillo, and D. L. Rabinowitz (2005) "Discovery of a Planetary-Sized Object in the Scattered Kuiper Belt", The Astrophysical Journal, 635: L97-L100, (10 December)
• Chris P. Duif (2004) "A Review of Conventional Explanations of Anomalous Observations during Solar Eclipses", arXiv:gr-qc/0408023 v5 (31 December)

Stephen Williamson, Again Is He A Fool Or A Knave?

Stephen Williamson writes some ignorant balderdash about heterodox economics. At least two of his commentators recognize the quality of his remarks.

If Williamson had a clue, he would know some scholars distinguish between the heterodox/orthodox distinction and the mainstream/nonmainstream distinction. Following Davis's taxonomy, I would classify North and new institutionalist economics as "mainstream heterodox". I do not think of either Paul Krugman or James Tobin as non-mainstream.

Non-mainstream heterodox economics do not all reject or dislike mathematics. Williamson doesn't seem to know about the existence of sraffians. Of course, he doesn't list feminist economics as a target of his ill-informed calumny. I think most heterodox economists would agree that one can make true and insightful statements about economics without using mathematics. One capable of a moment's reflection can see that that doesn't imply no mathematics can be be useful for economics. Williamson seems to think that the point of using mathematics is to seperate oneself from the hoi polloi. It is no legimate criticism of a paper that a graduate student can write out the model in twenty minutes. The questions one should ask oneself, in this case, are "Does this paper provide empirical and policy guidance for understanding some aspect of Japan's economy?" and "Is this paper original?"

Consider such journals as the American Economics Review or the Journal of Political Economy. And consider the Cambridge Journal of Economics, Metroeconomica, or the Review of Political Economy. The first set contains examples of mainstream economics. The second set of journals publish non-mainstream heterodox economists. An outsider will find journals in both set contain a mixture of natural language and mathematics. They both have both theoretical and empirical papers. The academics are not all from one university, they argue with one another, and they seem to have a variety of sources of funding. Clearly, non-mainstream heterodox economists are not "fringe" in the same sense that most are who argue for astrology, creationism, or a flat earth theory.

Some Reports

I haven't had time to read these:

• 2010 Ibrahim Index of African Governance: Summary (More here and here)
• Shadow Banking, a Federal Reserve Bank of New York Staff Report (h/t Kasey Dufresne)
• Findings Regarding the Market Events of May 6, 2010, a report by the staffs of the Commodity Futures Trading Commission (CFTC) and the Securities and Exchange Commission (SEC) on the flash crash (press release) (h/t Rajiv Sethi)
• Symantec's "W32.Stuxnet Dossier" and the Industrial Control Systems Cyber Emergency Response Team's "ICS-CERT Advisory ICSA-10-272-01 - Primary Stuxnet Indicators"

Inside Job

National Public Radio broadcast an interview last evening with Charles Ferguson, the director of the soon-to-be-released documentary, Inside Job. It is apparently about recent financial shenanigans on Wall Street and the proximate causes of the global financial crisis. The director seemed to be struck about how unlikely it is that top financiers would be heading off to prison. He also was astonished about the pervasive advocacy and lack of ethics he found among economists. He mentioned economists writing articles to promote some industry and not disclosing funding, serving on corporate boards, and acting as expert witnesses in court cases.

NPR played a clip where Glenn Hubbard, the dean of the Columbia University School of Business, a former chair of the Council of Economic Advisors, and a Visiting Scholar (if you can call it that) at the American Enterprise Institute, basically threw Ferguson out of his office. Hubbard apparently did not want to talk about his outside sources of income.

Update: In the comments, Tomboktu links to the NPR piece. Elsewhere, Chris Bertram links to a piece by Charles Ferguson in the Chronicle of Higher Education.

Mainstream Economists: When The Storm Is Past The Ocean Is Flat Again

Did economists predict the possibility of the global economic crisis before it occurred? Did they describe sources of instability as they were building up? I think the following three papers are good for exploring these questions:

• Dirk J. Bezemer (2009) "'No One Saw This Coming': Understanding Financial Crisis Through Accounting Models".
• James K. Galbraith (2009) "Who Are These Economists, Anyway?", Thought and Action.
• Andrea Imperia and Vincenzo Maffeo (2009) "As if Nothing Were Going to Happen: A Search in Vain for Warnings about the Current Crisis in Economic Journals with the Highest Impact Factors" 13th Conference of the Research Network Macroeconomics and Macroeconomic Policies: The World Economy in Crisis - The Return of Keynesianism, Berlin (30-31 October.

The answer I get from these articles is mainly negative for orthodox economics. Robert Shiller receives praise. He could be said to be a mainstream economist. But, as I understand it, his analysis was based on behavioral economics and the rejection of the Efficient Market Hypothesis. The empirical evidence suggests macroeconomists should expand research following Wynne Godley's stock-flow consistent models. Imperia and Maffeo point to those who argued that financial innovation was leading to increased debt, an attempt to compensate for reduced aggregate demand resulting from increased income inequality.

For A New Robinson And Eatwell

Michael Hirsh recently ("Our Best Minds Are Failing Us", Newsweek, September 16, 2010) lamented how unwilling mainstream economists are to change their thinking in light of the events of the last few years. Brian Milner, a columnist for The Globe and Mail has added to the chorus of complaints.

Perhaps an opportunity now exists for a new introductory textbook in economics that differs fairly comprehensively from mainstream textbooks. Never mind Colander’s approach of modifying his textbooks at most 15% from the previous editions so that mainstream economists will not reject it. Maybe some enterprising heterdox economist should write an uncompromising introduction to economics that is also up-to-date on current events. Years ago, I listed some textbooks. More textbooks have become available since then, for example, G. C. Harcourt’s The Structure of Post-Keynesian Economics: The Core Contributions of the Pioneers. I am not sure that Luigi L. Pasinetti’s Keynes and the Cambridge Keynesians: A ‘Revolution in Economics’ to be Accomplished counts as a textbook. I’m sure I’m leaving much out.

But I’m not sure the packaging on most of these is what I’d like to see tried. The textbook I have in mind should be fairly thick, have various boxed asides, and have problem sets after every chapter. (The problem sets could include essays questions and have less numerical examples than is common.)

Chicken Rice Pilaf

Ingredients

2 Tablespoon olive oil
2 smoked sausage, bratwurst, italian sausage, andouille, or other flavorful link sausage (optional)
1 onion
2 ribs celery
1 large green or red pepper
1 bay leaf
1 teaspoon thyme
1 teaspoon tumeric (optional)
6 chicken thighs, skin removed if desired (could be boneless) or 2 x 1 1/2 pounds bone-in, skinless chicken breasts
2 cups uncooked rice
3 to 4 cups canned or home-made chicken broth

1) Heat vegtable or olive oil over medium-high heat in a wide, deep pot (e.g., a Dutch oven). Cut sausage in half lengthwise, then in half lengthwise again. Dice into small pieces and add to pot.

2) Peel and dice onion, adding it to the pot as you do. Dice celery (including leaves) and pepper, adding them to the pot. (Based on what I've seen, one could also add a diced carrot.)

3) When all vegtables are added, cook about 5 minutes so they soften a little. Add bay leaf, thyme, tumeric if desired chicken (e.g., whole thighs) and rice. Add broth plus water to equal 4 cups. Season with a little salt and pepper.

4) Bring to a boil, cover, reduce heat to simmer and cook 30 or 40 minutes, or until chicken is cooked and rice is tender. (I'm thinking of trying it with frozen peas added with about 20 minutes left.)

Makes approximately 6 servings.

Elsewhere

• Bruce McFarling is blogging at Daily Kos. I like his take on the (US) Social Security non-crisis.
• Paul Davidson is now on the the Advisory Board for the Institute for New Economic Thinking.
• David Hendy, another member of INEF Advisory Board, has co-written a paper, "On the Mathematical Basis of Inter-Temporal Optimization", which argues that rational expectations is inapplicable to the sort of non-stationary time series that often arise in economics.
• I stumbled upon Daniel Gay's blog the other day. He has a book, Reflexivity and Development economics: Methodology, Policy and Practice, which sounds interesting from a Post Keynesian perspective.
• I've added Tony Myatt and Rod Hill's blog to my blog roll. They have a book, the Economics Anti-Textbook, too. (I'm not going to try to read either of these books soon.)
• "Magpie" has some comments on a couple of my old posts.

Nonergodicity And The Butterfly Effect

1.0 Introduction
Cosma Shalizi says, "It is not true that ergodicity is incompatible with sensitive dependence on initial conditions." This poses some questions for me: Can I give an example of a non-ergodic process that also exhibits sensitive sensitive dependence on initial conditions? Can I give an example of an ergodic process that exhibits sensitive dependence on initail conditions?

This post answers the former question. I consider Newton's method for finding roots of unity in the complex plane. The latter question is probably more important for Cosma's assertion. For now, I cite the Lorenz equations as an example of an ergodic process with the desired sensitive dependence.

Cosma's assertion, "It is not true that non-stationarity is a sufficient condition for non-ergodicity," directly contradicts Paul Davidson. I do not address that contradiction here.

2.0 Newton's Method
Newton's method is an algorithm for finding the zeros of a function. In this post, I illustrate the method with the function:

F(z) = z³ - 1,

where z is a complex number. A complex number can be written as a two-element vector:

z = [x, y]^T = x + jy

where j is the square root of negative one. (I've been hanging around electrical engineers.) Likewise, one can consider the function F as a vector of two elements:

F(z) = [f₁(z), f₂(z)]^T

The first component maps the real and imaginary parts of the argument to the real part of the function value:

f₁(z) = x³ - 3xy² - 1

The second component maps to the imaginary component of the function value:

f₂(z) = y (3x² - y²)

Newton's method is for numerically finding a solution to the following equation:

F(z) = 0

In my case, one is searching for the cube roots of unity. The method is an iterative method. An initial guess is refined until successive guesses are close enough together that one is willing to accept that the method has converged. A guess is refined by taking a linear approximation to the function at the guess. That guess is refined by solving for the zero of that linear approximation. The zero is the next iteration.

The derivative of a function, when evaluated at the current iterate, provides the linear approximation. The Jacobian is the two-dimensional equivalent of the derivative. The Jacobian, J, is a matrix with the following elements:

J_{i, 1}([x, y]^T) = df_i([x, y]^T)/dx, i = 1, 2.

J_{i, 2}([x, y]^T) = df_i([x, y]^T)/dy, i = 1, 2.

Newton's method is specified by the following iterative equation:

z_{n + 1} = z_n - J^-1(z_n) * F(z_n)

3.0 Numeric Explorations
Figure 1 shows a coloring of the plane based on the application of Newton's method. Each point in the plane can be selected as an initial point. The method is applied, and the point is colored according to which of the three cube roots of unity to which the method converges. Figure 2 shows an enlargement of the region around the indicated point in the northeast of Figure 1. Notice the fractal nature of the regions of convergence.

Figure 1: Fractal Basins of Attraction for Newton's Method

Figure 2: An Enlargement of These Fractal Basins

To exhibit sensitive conditions on initial conditions, I wanted to find nearby points whose trajectory diverges under this dynamical process. Table 1 lists six points selected from Figure 2. They fall into three groups, depending on which root they converge to. I claim that one can find at least three distinct points such that each pair is as close as one wants that each converge to a seperate root.

Table 1: Limit Points for Newton's Method

Initial Guess	Limit Point
0.3899 + j 0.6871	1
0.3938 + j 0.6780	(-1/2) - j (31/2)/2
0.3986 + j 0.6811	(-1/2) + j (31/2)/2
0.4010 + j 0.6868	1
0.3980 + j 0.6908	(-1/2) - j (31/2)/2
0.3943 + j0.6949	(-1/2) + j (31/2)/2

Figure 3 and 4 display the trajectories of the six points selected for Table 1. Apparently the function is very shallow in this region. I had not realized before these explorations that these sorts of trajectories go so far from the origin before returning to converge to a root on the unit circle.

Figure 3: Real Part of Some Time Series From Newton's Method

Figure 4: Imaginary Part of Some Time Series From Newton's Method

4.0 Conclusions
Cosma provides this definition, among others, of an ergodic process:

"A ... process is ergodic when ... (almost) all trajectories generated by an ergodic process belong to a single invariant set, and they all wander from every part of that set to every other part..."

This definition is appropriate for both deterministic and stochastic processes.

The three roots of unity constitute the non-wandering (invariant) set for the dynamical system created by the above application of Newton's method. A trajectory that has converged to one of the roots does not wander to any other root. So the process is non-ergodic. Yet which root a process converges to is crucially dependent on the initial conditions. A small variation in the initial conditions leads to a long-term divergence in trajectories. This is especially evident because of the fractal structure of the basins of attraction of the three roots.

I think of the above as close to recreational mathematics. I have not tied the above example into any economics model. Common neoclassical models, such as the Arrow-Debreu model of general equilibrium, fail to tie dynamics down. I find it difficult to see how one who has absorbed this fact and understands the mathematics of dynamical systems can find credible much orthodox teaching in economics.

Faustian Agents

"Two souls, alas, do dwell within this breast. The one is ever parting from the other" -– Goethe

"He [i.e., Dickens] told me that all the good simple people in his novels, Little Nell, even the holy simpletons like Barnaby Rudge [Slater comments parenthetically that this must have been Dostoevsky's description, not Dickens' -- indeed] are what he wanted to have been, and his villains were what he was (or rather, what he found in himself), his cruelty, his attacks of causeless enmity towards those who were helpless and looked to him for comfort, his shrinking from those whom he ought to love, being used up in what he wrote. There were two people in him, he told me: one who feels as he ought to feel and one who feels the opposite. From the one who feels the opposite I make my evil characters, from the one who feels as a man ought to feel I try to live my life. Only two people? I asked." -- Fyodor Dostoevsky

I have previously described agents that assess an action by ranking outcomes among a number of incommensurable dimensions. By Arrow's impossibility theorem, such an agent in general cannot have a single aggregate ranking of the outcome of actions. I was able to list all best choices for my simple example. That is, for each menu, I listed best choices, with ties being possible. (By the way, a budget constraint is a menu.) If one wants to generalize this approach, one would need to specify methods for specifying best choices when listing all possible menus by hand becomes impractical. Pairwise voting is not a good idea, since the results depend on the voting order in which pairs are compared. Furthermore, one would not want to specify one such method, but allow for many different possibilities. Ulrich Krause has done this. He calls the method for choosing out of these rankings of different aspects an agent's "character". As I understand it, he allows for these rankings to change, based on the agents experience. And so he ends up with a formal model of opinion dynamics. I don't know if or how this relates to Akerlof's identity dynamics, but, I think, that would be an interesting question to explore. References

• K. J. Arrow (1963) Social Choice and Individual Values (2nd. Edition), John Wiley & Sons.
• Ulrich Krause (2010) "Collective Dynamics of Faustian Agents", in Economic Theory and Economic Thought: Essays in Honour of Ian Steedman (ed. by J. Vint, J. S. Metcalfe, H. D. Kurz, N. Salvadori, and P. Samuelson), Routledge.
• S. Abu Turab Rizvi (2001) "Preference Formation and the Axioms of Choice", Review of Political Economy, V. 13, N. 2: pp. 141-159.
• A. K. Sen (1969) "Quasi-Transitivity, Rational Choice and Collective Decisions", Review of Economic Studies, V. 36, N. 3 (July): pp. 381-393.
• A. K. Sen (1970) "The Impossibility of a Paretian Liberal", Journal of Political Economy, V. 78, N. 1 (Jan.-Feb.): pp. 152-157.

To read:

• G. A. Akerlof and R. E. Kranton (2010) Identity Economics: How Our Identities Shape Our Work, Wages, and Well-Being, Princeton University Press.
• J. B. Davis (2003) The Theory of the Individual in Economics: Identity and Value, Routledge.
• A. Kirman and M. Teschl (2004) "On the Emergence of Economic Identity" Revue de Philosphie Économique, V. 9, N. 1: pp. 59-86
• U. Krause (2009) "Compromise, Consensus and the Iteration of Means", Elemente der Mathematik, V. 64: pp. 1-8
• I. Steedman and U. Krause (1986) "Goethe's Faust, Arrow's Possibility Theorem and the Individual Decision-Taker" in The Multiple Self: Studies in Rationality and Social Change (ed. by J. Elster), Cambridge University Press.

Stephen Williamson, Fool or Knave?

Stephen Williamson quotes Narayana Kocherlakota, apparently a very stupid person:

"Kocherlakota says this...:

'But over the long run, money is, as we economists like to say, neutral. This means that no matter what the inflation rate is and no matter what the FOMC does, the real return on safe short-term investments averages about 1-2 percent over the long run.'

Again, uncontroversial." -- Stephen Willaimson

This, of course, is false. Communities of economists exist who set their theories in historical time and dispute that money is neutral in any run. I prefer to point to Post Keynesians, but Austrian School economists satisfy these criteria also. Furthermore, economists within such schools surpassed mainstream economists in the current historical conjuncture by having pointed out the possibility of the global financial crisis before its occurrence.

I think economists should strive not to tell untruths abouts what economists believe.

Why Income Inequality Leads To Recessionary Conditions

1.0 Introduction
Apparently, some have been discussing whether the gross increased inequality in the USA is connected with the depressionary conditions we are in. So I thought I would climb on my bicycle and do some arithmetic.

I take it as a stylized fact that an increase in inequality is associated with an increase in the average and marginal propensity to save.

There's something called the Harrod-Domar model of growth. I'm not sure I've ever read Domar. I've certainly read more of Harrod than I have of Domar. So in the sequel, I refer exclusively to Harrod.

Harrod defined three rates of growth: the actual rate, the warranted rate, and the natural rate. Increased inequality can result in the warranted rate exceeding the natural rate. Since the warranted rate is unstable and the actual rate cannot long exceed the natural rate, increased inequality is likely to lead to the actual rate of growth falling below and away from the warranted rate, that is, to depressions.

2.0 Harrod's Model
Harrod's model is fairly simple, but it raises deep questions.

2.1 The Actual Rate
Along a steady state growth path, the ratio, v, of the value of capital to the value of net income is constant:

v = K/Y,

where K is the value of the capital stock, and Y is the value of net income. v is known as the capital-output ratio. Thus:

dY/dt = (1/v) dK/dt

Investment, I, is defined to be the change in the value of capital with time. Hence,

(1/Y) dY/dt = (1/v) (I/Y)

The left-hand-side of of the above equation is, by definition, the rate of growth, g, of the economy. The equality of investment and savings is an accounting definition in a model with no foreign trade and no government. Therefore,

g = (1/v) (S/Y)

Define the savings rate, s:

s = S/Y

Then, a steady state growth ratio is the ratio of the savings rate to the capital-output ratio:

g = s/v

That is, the (actual) rate of growth is the quotient of the savings rate and the capital-output ratio.

2.2 The Warranted Rate
Suppose the savings rate and the capital-output ratio are as desired by income recipients (consumers) and firms, respectively. This defines Harrod's warranted rate of growth:

g_w = s_d/v_d

where the subscripts on the right hand side stand for "desired". The warranted rate of growth is being achieved when expectations are being realized and current actions are not setting up forces to disturb current expectations.

The warranted rate of growth extends Keynes' analysis to the long period. Consider the stability of a warranted growth path. If the actual rate of growth exceeds the warranted rate, capacity will be utilized at a greater rate than firms expected. They will increase investment faster than the warranted rate, and the rate of growth will deviate from the warranted rate even more. Likewise, if the actual rate falls below the warranted rate, firms will cut back on investment since the plans upon which their investment was made are not being realized. Hence, the warranted rate is unstable.

Harrod suggested that this instability of the warranted rate is more like an inverted flat-bottomed bowl than a knife-edge.

2.3 The Natural Rate
Suppose the labor force is initially fully employed. Let n be the rate of growth of the labor force:

n = (dL/dt)/L

Define the value of output produced per employed worker:

f = Y/L

Harrod-neutral technical change occurs when the value of output per worker grows at a constant rate, m, while the rate of profit stays unchanged:

m = (1/f) df/dt

Harrod-neutral technical progress implies that the productivity of labor is growing at the same rate in all industries.

Anyways, the following equation follows:

dY/dt = f dL/dt + L df/dt

Some algebra yields:

(1/Y) (dY/dt) = ( 1/L) (dL/dt) + (1/f) (df/dt)

The left hand side of the above equation is the rate of growth that keeps the labor force fully employed (or a constant percentage unemployed). Harrod calls this the natural rate of growth. Hence, assuming Harrod-neutral technological progress, the natural rate of growth is the sum of the rate of growth of the labor force and the rate of growth of labor productivity.

g_n = n + m

3.0 Conclusions
Notice that the determinants of the warranted rate of growth - the savings rate and the desired capital-output ratio - are taken as exogeneous constants. The determinants of the natural rate of growth - the growth of the labor force and Harrod-neutral technological progress - are also given. Hence, the warranted and natural rates can only be equal by a fluke.

Solow, following up on some work by Pivlin, suggested that the desired equality between the warranted and natural rates can be brought about by considering the capital-output ratio as a well-behaved function of the rate of interest. Divergences between the two rates can be corrected by variations in the distribution of income. This approach of neoclassical macroeconomics is exemplified in Solow's eponymous growth model, but it has been shown to be not well-founded in the Cambridge Capital Controversy.

If the warranted rate is below the natural rate, a moderate increase in the saving rate is desirable if the economy is exhibiting boom-like conditions. This would bring the warranted rate towards the actual rate of growth while still keeping it below the natural rate of growth.

Notice that when the warranted rate exceeds the natural rate, the economy must sometime fall below the warranted rate. The natural rate sets a limit which the economy cannot long exceed. Because of the instability of the warranted rate, such an economy will experience frequent and perhaps prolonged recessionary conditions. Since increased savings intensify the discrepancies between the warranted and natural growth rates under these conditions, increased savings intensify the frequency and severity of recessions. That is, increased inequality can intensify the frequency and severity of recessions.

References

• A. Asimakopulos (1991) Keynes's General Theory and Accumulation, Cambridge.
  1991
• Roy F. Harrod (1948) Towards a Dynamic Economics, Macmillan.
• Joan Robinson (1962) Essays in the Theory of Economic Growth, Macmillan.

Barnett's Fried Apples

Ingredients

4 Tablespoons butter
1 #2 can sliced apples or 2 1/2 cups fresh apple
1/8 teaspoon salt
1/4 cup sugar
Cinnamon to taste

1) Peel and core apples.

2) Melt butter in iron skillet. Add apples, salt, sugar, and cinnamon. (I'm generous with the cinnamon.)

3) Fry until soft, between low and medium heat about 1/2 hour. (Do not fry dry.)

Makes approximately 3 servings. (I like them served with pork chops.)

That You Should Listen To Mainstream Economists...

... seems often to me to be the main point of many mainstream economists these days. I deliberately don't write, "Why you should listen..." Somebody as stupid as Kartik Athreya, a PhD. with the research department of the Federal Reserve Bank of Richmond, appears to be doesn't deal in arguments. I also see this sort of babble in recent posts by Frances Woolley, and Mike Moffat. (See also Nick Rowe's comments to those posts.)

(I, of course, have read papers making points along Colander's line.)

Jeffrey Miron And Propertarian Advocacy Taught At Harvard

Jeffrey Miron teaches EC1017 at Harvard. "A Libertarian Perspective on Economic and Social Policy" is the course title, and PDFs for the lectures are available for download.

Based on the notes for the three lectures I looked at, Miron supposedly derives propertarian policy from intermediate principles (e.g., "efficiency"), with little to no data on relative magnitudes. I don't care for this approach myself, never mind the policy conclusions. He seems to mention no names. The reading list (from Spring 2009) does not include his book (which I haven't read). Perhaps Miron's experience is that Harvard students can be counted on to bring up Rawls, Karl Popper's piecemeal social engineering, Alan Haworth, and even Nozick.

"When Adam Delved And Eve Span, Who Was Then The Gentleman?"

I had associated the title of this post with the 17th century and the period of the English Civil War. I think it occurs somewhere in Christoper Hill's The World Turned Upside Down: Radical Ideas During the English Revolution. Hill's book is an account of Anabaptists, Diggers, Levellers, Muggletonians, the New Model Army, Ranters, and Quakers - a very heady and confusing mix. So I was startled yesterday to read the phrase in Crispin: The Cross of Lead. This is a Newberry-prize winning children's book, by Avi. It is set in England in the 14th century. I think it conveys a good idea of the drudgery and isolation of village life at the time; the seemingly unchangable hierarchy; and the bustle, confusion, and filth of a city before modern plumbing. I also like that Christianity is presented as a form of life, a language that all we see cannot but help using. So is Avi's use of the phrase an anachronism? Hill may reference it, but, if so, the people of his time were harking back to a previous one. Apparently, the phrase is associated with John Ball, the leader of the 1381 Peasants’ Revolt. I know nothing about the Peasants’ Revolt, although Hill does refer to it in one line. But John Ball does appear in Avi’s book. Crispin, our thirteen-year old hero, overhears him conspiring. John Ball says:

"...that no man, or woman either, shall be enslaved, but stand free and equal to one another. That all fees, obligations, and manorial rights be abolished immediately. That land must be given freely to all with a rent of no more than four pennies per acre per year. Unfair taxes must be abolished. Instead of petty tyrants, all laws shall be made by consent of a general commons of all true and righteous men. Above all persons, our lawful king shall truly reign, but no privileged or corrupt parliaments or councilors. The church, as it exists, should be allowed to wither. Corrupt priests and bishops must be expelled from our churches.. In their place will stand true and holy priests who shall have no wealth or rights above the common man..."

Update: I've learned a new vocabulary word: A Jacquerie is a peasants' revolt, named after the French peasants' revolt of 1358.

Infinities Of Infinities

1.0 Introduction

This is mathematics, not economics. It is meant to be an introduction to how abstract mathematicians can be.

2.0 Some Definitions for Set Theory

Two sets are the same size if and only if they can be put into a one-to-one correspondence with each other.

A set S₁ is bigger than the set S₂ if and only if:

• A subset of S₁ can be put into one-to-one correspondence with S₂, and
• S₂ cannot be put into one-to-one correspondence with S₁.

A set is countably infinite if and only if it can be put into one-to-one correspondence with the set of natural numbers N = {0, 1, 2, ...}. (The integers and the rational numbers are both countably infinite.)

The power set P(S) formed from the set S is the set of all subsets of S. For example, the power set for the set {a, b} contains four elements:

P( {a,b} ) = {S | S ⊂ {a, b}.} = { ∅, {a}, {b}, {a, b} }

3.0 A Theorem

Theorem For all sets S, the power set P(S) is bigger than the set S.

Proof: First, show that a subset of P(S) can be put into one-to-one correspondence with S. Consider the set of singletons:

{ {a} | a is an element of S }.

Since each singleton {a} is a subset of S, the set of all singletons is a subset of P(S). And the set of all singletons maps one-to-one to S.

Next, show, by a proof by contradiction, that P(S) cannot be put into one-to-one correspondence with S. Suppose that there exists a one-to-one function f that maps S into P(S).

Notice that, for all a ∈ S, f(a) is a subset of S. For any given a in S, either

a ∈ f(a)

or

a ∉ f(a).

Define the set T to be the set of all elements in S that map under f to a set not containing themselves:

T = { a | a ∈ S and a ∉ f(a)}

Since f is one-to-one and T is a (possibly empty) subset of S, there exists, by hypothesis, an element b in S such that

f(b) = T.

Now consider whether or not

b ∈ T.

Suppose true. But, by the definition of T as the set of elements of S that are not elements of the subset of S that they map to, b cannot be in f(b), that is, T. But, if b is not in f(b), by the definition of T, b must be in T. So either way yields a contradiction. Thus, no such b can exist.

So I have shown that there does not exist an element b in S that maps under f to T. Yet T is in P(S). Thus, f cannot be one-to-one. Which was to be demonstrated.

4.0 Applying the Theorem to the Set of Natural Numbers

An interesting property of the above proof is that it applies to both finite sets and infinite sets. So start with N, the set of natural numbers. N contains an infinite number of elements. But, by the theorem, P(N), the set of all subsets of the natural numbers, is a set containing a bigger infinity. One can go on to form a set of infinite sets, each with a bigger size infinity:

U₀ = { N, P(N), P(P(N)), ..., Pⁿ(N), ...}

(Under the Zermelo Frankel axioms for set theory, the elements of a set do not need to all be of the same "type".) One can repeat the process of forming a sequence of power sets:

U₁ = { U₀, P(U₀), P(P(U₀)), ..., Pⁿ(U₀), ...}.

One can even imagine constructing a power set of all these difference size infinite sets in this sequence of sequences of sets:

P( { U₀, U₁, U₂), ...} )

The definitions of infinite sets need not stop here.

4.0 Conclusion
I don't find the above hard to follow if I think of it as merely a matter of syntactic manipulation of symbols. Do I have a clear idea of these infinities of different size infinities after every point in this sequence of definitions? Does anybody? This is not so clear to me.

Reference

• Paul R. Halmos (1960) Naive Set Theory, Springer Verlag

Onieda-Like Community Near Stroud, In Gloucestershire?

Martin Gardner once received a letter referring to "an Oneida-like community near Stroud, in Gloucestershire". The topic of the letter was something else, on visualizing four-dimensional space. Can anybody provide me with more information on this community?

Full Unemployment

I find amusing the political slogan with which I title this post. We want the engineers to do their job in applying control theory to stepping motors, in creating Artificial Intelligences, in developing techniques of information management, etc. such that nobody need work out of necessity. Maybe in some future day, machinery will produce all we need, including more machinery. When the economic problem is solved:

"Man will be faced with his real, his permanent problem - how to use his freedom from pressing economic cares, how to occupy the leisure, which science and compound interest will have won for him, to live wisely and agreeably and well." -- John Myanard Keynes (1930)

Marx and Engels envision a post-capitalist society:

"Where nobody has one exclusive sphere of activity, but each can become accomplished in any branch he wishes, society regulates the general production and thus makes it possible for me to do one thing today and another tomorrow, to hunt in the morning, fish in the afternoon, rear cattle in the evening, criticize after dinner, just as I have a mind, without ever becoming hunter, fisherman, shepherd or critic. -- Karl Marx (1947, p. 22)

Bruce Sterling (1989) imagines that, in such a world, one will cultivate ones taste for "The Beautiful and the Sublime". At any rate, in this pleasant world of tomorrow, all will be able to devote themselves to great cooking, fostering social relationships, art, or whatever one may choose.

Curiously enough, the classical tradition in economics, as exemplified, for example, in Sraffa or Von Neumann, provides tools for analyzing how prices might be formed in a post-scarcity world. For example, Joan Robinson, in her first essay in (Robinson 1962) has a section titled "A model for the future" with a subsection on "The Robots". This is a model of a (maybe impossible) capitalist economy. In my version, all production is carried out in automated factories, and these factories are owned by firms traded on a stock exchange. Everybody owns shares, and the trading of these shares sets up a tendency torwards a uniform rate of profits.

I have described before some formulation of a price system consistent with this institutional set up. For now, I want to describe prices when the managers of each firm in an industry have chosen a process for producing the firm's output. As usual, I assume, for simplicity that all processes require the same time to operate, say, a year. Inputs must be purchased at the beginning of the year, and outputs become available at the end of the year. A reference set of prices satisfies the following system of equations:

p A β = p B

where

• A is a square matrix; a_i,j is the quantity of the ith commodity used as input when the jth process is operated at a unit level.
• B is a square matrix; b_i,j is the quantity of the ith commodity produced as output when the jth process is operated at a unit level.
• p is a row vector of prices; p_i is the price of the ith commodity.
• (β - 1) is the rate of profits.

This formulation allows for robots to last for more than one year. The quantity of a dated robot enters as an input, and the output includes a robot one year older, as well as whatever other outputs are produced by that process. The use of such robots is a special case of the more general model of joint production encapsulated in the above system of equations.

Various conditions must be imposed on the coefficients of production A and B to ensure a solution simultaneously exists for prices and the dual problem of the choice of technique. Von Neumann, in fact, assumes that each commodity is either used as an input or produced as an output in a poisitive amount in each process. Joan Robinson assumes the existence of "some standard physical elements (say, nuts and bolts) that enter into the production both of robots and of salable goods." But I do not want to discuss more of the mathematics in this post.

References

• D. G. Champernowne (1945-1946) "A Note on J. v. Neumann's Article on 'A Model of Economic Equilibrium'", Review of Economic Studies, V. 13, N. 1: pp. 10-18.
• John Maynard Keynes (1930) "Economic Possibilities for our Grandchildren", in Essays in Persuasion, W. W. Norton & Company
• Karl Marx and Frederick Engels (1947) The German Ideology: Parts I & III, International Publishers
• Joan Robinson (1962) Essays in the Theory of Economic Growth, Macmillan.
• Piero Sraffa (1960) , Cambridge University Press.
• J. v. Neumann (1945-1946) "A Model of General Economic Equilibrium", Review of Economic Studies, V. 13, N. 1: pp. 1-9.
• Bruce Sterling (1989) Crystal Express, Ace Books

Nortz's Johnny Cake

Ingredients

1/4 cup sugar
1/3 cup shortening
1 beated egg
1 cup sour milk
1 teaspoon baking soda
1 teaspoon baking powder
1 cup flour
1 1/2 cup cornmeal
1/2 teaspoon salt

1) Mix in above order, stirring thoroughly after adding each ingredient. Bake about 1/2 hour at 400 F.

2) Serve sliced with applesauce or maple syrup.

Makes approximately 10 servings. I usually make a double recipe when making my great-grandmother's Johnny cake.

Phenomenology

"One of the embarrassing dirty little secrets of economics is that there is no such thing as economic theory properly so-called. There is simply no set of foundational bedrock principles on which one can base calculations that illuminate situations in the real world." -- Brad DeLong

My title does not refer to an approach in continental philosophy associated with Husserl and Heidegger. Rather, I refer to a term used in physics and engineering by practitioners who know they are not trying to develop models derived from fundamental laws, but only modeling the phenomena.

I find it of interest that Brad DeLong has recently described economics as phenomenology. A noted "rocket scientist" on Wall Street came to the same conclusion:

"The techniques of physics hardly ever produce more than the most approximate truth in finance because 'true' financial value is itself a suspect notion. In physics, a model is right when it correctly predicts the future trajectories of planets or the existence and properties of new particles, such as Gell-Mann's Omega Minus. In finance, you cannot easily prove a model right by such observation. Data are scarce and, more importantly, markets are arenas of action and reaction, dialectics of thesis, antithesis, and synthesis. People learn from past mistakes and go on to make new ones. What's right in one regime is wrong in the next.

As a result, physicists turned quants don't expect too much from their theories, though many economists naively do. Perhaps this is because physicists, raised on theories capable of superb divination, know the difference between a fundamental theory and a phenomenological toy, useful though the latter may be. Trained economists have never seen a really first-class model. It's not that physics is 'better', but rather that finance is harder. In physics you're playing against God, and He doesn't change his laws very often. When you've checkmated Him, He'll concede. In finance, you're playing against God's creatures, agents who value assets based on their ephemeral opinions. They don't know when they've lost, so they keep trying." -- Emanuel Derman (2004) My Life as a Quant: Reflections on Physics and Finance, John Wiley & Sons.

I think one can read intimations of Soros' reflexitivity or Joan Robinson's historical time in the above quote. Derman is even more direct about a Post Keynesian concept elsewhere:

"Slowly it began to dawn on me that what we faced was not so much risk as uncertainty. Risk is what you bear when you own, for example, 100 shares of Microsoft - you know exactly what those shares are worth because you can sell them in a second at something very close to the last traded price. There is no uncertainty about their current value, only the risk that their value will change in the next instant. But when you own an exotic illiquid option, uncertainty precedes its risk - you don't even know exactly what the option is currently worth because you don't know whether the model you are using is right or wrong. Or, more accurately, you know that the model you are using is both naive and wrong - the only question is how naive and how wrong." -- Emanuel Derman (2004)

Jacob Schwartz (9 January 1930 - 2 March 2009)

Jacob T. Schwartz was a mathematician at the Courant Institute of Mathematical Sciences at New York University. He once gave a series of lectures on mathematical economics, published as Lectures on the Mathematical Method in Analytical Economics, Gordon and Breach (1961). This book, coming out a year after Sraffa's work, seems to very little known. Its findings parallel Sraffa's in many ways:

"Our interest...will...be...in the use of the input-output model as a framework for ...abstract economic analysis." (p. 8)

"If each time labor appears as an input in production we replace this input by the corresponding real wage bill, we come to a hypothetical situation in which the only inputs required for the production of commodities are other (non-labor) commodities. Thus we may, if it is convenient for one or another theoretical purpose, consider our model to refer to a self-enclosed world of material commodities, produced out of each other with no additional input." (p. 10)

"The proper conclusion at this point is that the rate of profit ρ is not successfully determined by the Walrasian theories from considerations of production coefficients, utility functions, and so forth. What our analysis shows, in fact, is that the determination of the rate of profit is not purely a question of economics at all, but is rather a social-political question involving, among other things, union-management relations, pressures, and counterpressures, etc. Thus an initial skepticism about classical equilibrium analysis is justified. What this analysis aims to give us is a set of prices. But all the price-ratios are already determined by a small part of the theory, to wit by the competitive equality of profit rates. All that remains to be determined on the score of prices, is the rate of profit - but, as we have just seen, the Walrasian determination of this rate is questionable... What are determined more successfully are the amounts of production - but this is more a humble matter of consumption habits at given prices than a highly recondite matter of consumption schedules at a variety of hypothetical prices." (pp. 196-197)

"As our analysis in Lecture 16 shows, as long as we assume a fixed scheme of production the Keynesian conclusion that wage cuts by lowering wage-generated commodity demand must lower demand for labor is inescapable. The neoclassical contention thus depends on the possibility of shifts in the production scheme; a conclusion which the neoclassicist would be the first to emphasize, since the whole apparatus of neoclassical theory, revolving about the notion of marginal product accruing to an increment of each input facor, does in fact center on an analysis of variations in production. This means that the equilibrium analysis of Lecture 16 has come to such distinctly Keynesian conclusions as it has only by assuming away the basis for the neoclassicist's argument. At the present point, therefore, we shall attempt to generalize the analysis of Lecture 16 to include the possibility of shifts in the production scheme, hoping to estimate the extent to which such shifts are likely to affect our earlier conclusions." (p. 239)

"We may at this point remark once more that our analysis of prices shows that even in the framework of the present general model [with a choice of technique] price ratios are determined up to a single parameter from the conditions of production. As we have emphasized in the final paragraph of Section 1, Lecture 3, this conclusion constitutes strong presumptive evidence against theories which attempt to tie prices to consumer demand. More generally, we see that by allowing variation in the scheme of production, we in fact introduce no changes in the fixed-matrix Leontief model other than to make the Leontief matrices dependent on the [rate of profits]." (p. 248)

I prefer Sraffa's book partly on the basis of style.

Quantity Flows For Structural Dynamics

1.0 Introduction
This post presents an example of a model of structural economic dynamics. I consider what quantity flows would arise for an economy in which agents make decisions in which the economy smoothly reproduces. The solution for this exercise turns out to be dynamically unstable in the special case I use for illustration. I think this means that, if I solve this special case in a future post for one way of setting out the price system, the solution for prices will be stable. The model presented in this post illustrates the difficult discovery problems that are solved in successful economies.

2.0 Technology
This economy consists of two sectors. In the first sector, labor produces means of production with existing means of production. In the second sector, labor produces means of consumption with existing means of production. (I use steel as as a synecdoche for means of production and corn for means of consumption.) The technique in use in both sectors exhibits Constant Returns to Scale (CRS). Only circulating capital is modeled; the means of production are entirely consumed in producing the output. Table 1 shows the coefficients of production for the technique in use during the t-th year.

Table 1: The Technology

	Steel Industry	Corn Industry
Labor	a0,1(t) person-years	a0,2(t) person-years
Steel	a1,1(t) tons	a1,2(t) tons
Outputs	1 ton steel	1 bushel corn

The technique improves each year. That is, each coefficient of production decreases at a constant rate of 100 c_i,j percent per year:

[a_i,j(t) - a_i,j(t + 1)]/a_i,j(t) = c_i,j

The above difference equation can be solved in closed form. The coefficients of production evolve as:

a_i,j(t) = a_i,j(0) (1 - c_i,j)^t

A more complex formulation might have non-constant percentage rates of decrease in the coeffients of production. For example, the percentage rate of decrease might be larger if the level of output of an industry was larger. Then one would be modeling "learning by doing" or endogenous growth, following in the tradition of Nicholas Kaldor and Kenneth Arrow. (Mainstream economists would cite Paul Romer's confused balderdash.)

3.0 Conditions for Smooth Reproduction
Let q₁(t) and q₂(t) be the tons of steel and the bushels of corn, respectively, produced as output and available at the end of the t-th year. I want to consider the case in which the labor force is always fully employed, the proportions in which output is produced always turns out to be appropriate, and no excess capacity is ever created.

The gross output of corn each year is divided up between the workers and the capitalists and then consumed. The gross outputs of steel and corn in a given year determine, along with the coefficients of production, how much steel should have been produced in the previous year:

q₁(t - 1) = a_1,1(t) q₁(t) + a_1,2(t) q₂(t)

The amount of labor employed in the t-th year is:

L(t) = a_0,1(t) q₁(t) + a_0,2(t) q₂(t),

where L(t) is the person-years of labor employed. In a general formulation, one might model the number of workers growing each year, but with increased productivity being taken partly in the form of decreased working hours per worker. For simplicity, I here model the labor force as a given constant:

L(t) = L^*

The above equations specify a dynamic system. An initial condition needs to be specified for any solution path to be completely determined. I take the initial ratio of employment in the two sectors as a given parameter:

a_0,1(0) q₁(0)/a_0,2(0) q₂(0) = h

The model can be simplified by expressing one quantity flow in terms of other by use of the condition that labor is fully employed. Some algebraic manipulation yields a single difference equation for the output of steel:

q₁(t) = [a_1,2(t) L^* - a_0,2(t) q₁(t - 1)]/d(t),

where

d(t) = [a_0,1(t)a_1,2(t) - a_0,2(t)a_1,1(t)]

If the coefficients of production were constant, the above would be a linear difference equation. If I recall my mathematics correctly, linear systems either blow up; decay to an equilibrium; or, for coefficients meeting an exact balance, generate a constant wave.

4.0 The Solution of a Special Case
I tried a numerical experiment to increase my understanding of this dynamical system. Accordingly, I chose some specific values for the model parameters. Table 2 gives the initial coefficients of production. The difference equation for gross steel outputs is simplified in that the coefficients of production in a sector decrease at the same constant rate. I chose the following rates of decrease:

c_0,1 = c_1,1 = 1/20

c_0,2 = c_1,2 = 1/40

Let the labor force be unity:

L^* = 1

Finally, I carefully specified an initial condition:

a_0,1(0) q₁(0)/a_0,2(0) q₂(0) = 0.22335983

Table 2: The Initial Technology

	Steel Industry	Corn Industry
Labor	a0,1(0) = 1	a0,2(0) = 1
Steel	a1,1(0) = 1/10	a1,2(0) = 1/5
Outputs	1 ton steel	1 bushel corn

One can easily step through the first few years of the solution, thereby obtaining the start of a series for q₁(t) and q₂(t).The solution is dynamically unstable. I carefully chose the initial condition to get six years before the solution blows up. For the first five years, the output of steel grows over 3% and the output of corn grows over 14 1/2%, for a constant labor supply. This set of priorities is the reverse of what was typically achieved in no-longer actually existing socialism. When Imre Nagy, for example, tried to put Hungary on a new course, he was deposed. The distribution of labor, shown in Table 2, is not realistic for a developing capitalistic economy either. In practice, the labor force becomes steadily less concentrated in producing means of consumption and more in producing means of production. Still, I think, this model with a better choice of parameters and perhaps some generalizations can be quite interesting.

Figure 1: Dynamic Distribution of the Labor Force

References

• Karl Marx (1885) Capital, Volume 2
• Luigi L. Pasinetti (1977) Lectures on the Theory of Production, Columbia University Press
• Luigi L. Pasinetti (1983) Structural Change and Economic Growth: A Theoretical Essay on the Dynamics of the Wealth of Nations, Cambridge University Press
• Luigi L. Pasinetti (1993) Structural Economic Dynamics: A Theory of the Consequences of Human Learning, Cambridge University Press

To read:

• Dale W. Jorgenson (1960) "A Dual Stability Theorem", Econometrica, V. 28, N. 4 (October): pp. 892-899

Judt On The Influence Of The Austrian School

I continue to find writers characterizing Austrian school economists as influential.

I think some might quarrel with this description of the influence of the Austrian school on Chicago:

"We are the involuntary heirs to a debate with which most people are altogether unfamiliar. When asked what lies behind the new (old) economic thinking, we can reply that it was the work of Anglo-American economists associated overwhelmingly with the University of Chicago. But if we ask where the 'Chigago boys' got their ideas, we shall find that the greatest influence was exercised by a handful of foreigners, all of them immigrants from central Europe: Ludwig von Mises, Friedrich Hayek, Joseph Schumpeter, Karl Popper, and Peter Drucker." -- Tony Judt (2010) Ill Fares the Land, Penguin Press, pp. 97-98

I don't think differences in details (e.g., aggregation in economic models) adequately refutes Judt's point.

Judt does read, for example, Hayek as more nuanced than some of his followers:

"The intellectual refugees - and especially the economists among them - lived in a condition of endemic resentment toward their uncomprehending hosts. All non-individualist social thought - any argument that rested upon collective categories, common objectives or the notion of social goods, justice, etc. - aroused in them troubling recollections of past upheavals... Men like Hayek or von Mises seemed doomed to professional and cultural marginality. Only when the welfare states whose failure they had so sedulously predicted began to run into difficulties did they once again find an audience for their views: high taxation inhibits growth and efficiency, government regulation stifles initiative and entrepreneuship, the smaller the state the healthier the society and so forth.

Thus when we recapitulate conventional clichés about free markets and western liberties, we are in effect echoing - like light from a fading star - a debate inspired and conducted seventy years ago by men born for the most part in the late 19th century...

It is perhaps worth noting here that even Hayek cannot be held responsible for the ideological simplifications of his acolytes. Like Keynes, he regarded economics as an interpretive science, not amenable to prediction or precision. If planning was wrong for Hayek, this was because it was obliged to base itself on calculations and predictions which were essentially meaningless and thus irrational. Planning was not a moral misstep, much less undesirable on some general principle. It was simply unworkable - and, had he been consistent, Hayek would have acknowledged that much the same applied to 'scientific' theories of the market mechanism...

In the United States, among a younger generation of self-confident econometricians (a sub-discipline of whose bostful scientificity both Hayek and Keynes would have had much to say), the belief that democratic socialism is unachievable and has perverse consequences has become something close to a theology. This creed has attached itself to every effort to increase the role of the state - or the public sector - in the daily lives of American citizens." -- Tony Judt (2010): pp. 102-104