Elsewhere

I have added the blog of some economists at the University of Missouri-Kansas City to my blogroll. That blog is more policy-oriented than this. Bill Mitchell blogs from Australia, also more about policy than I do. Grupo Lujan-Circus seems like a blog of interest to me, but I can read only the names. The same remark applies to the blog of the Italian Association for the History of Political Economy.

Occasionally I stumble across curious articles in Wikipedia. The one on Surplus economics references Paul Baron and Paul Sweezy. It doesn't describe their ideas very well, and could do with some reference to Sraffa too. The entry on Newtonian time in economics seems to have been written by Austrian fanboys who, typically, know about neither Joan Robinson's distinction between logical and historical time nor Paul Davidson's attack on the Austrian school.

Current Events Foreseen

First Instance

"[Condolences]
G.O.P.Y.T.

A letter from Ronald Reagan to Michael Jackson, dated 1 February, 1984, five days after the singer's hair was set afire by pyrotechnics during the filming of a Pepsi commercial...

Dear Michael,

I was pleased to learn that you were not seriously hurt in your recent accident. I know from experience that these things can happen on the set - no matter how much caution is exercised. All over America, millions of people look up to you as an example. Your deep faith in God and adherence to traditional values are an inspiration to all of us, especially young people searching fro something real to believe in. You've gained quite a number of fans along the road since "I Want You Back", and Nancy and I are among them. Keep up the good work, Michael. We're very happy for you.

Sincerely,
Ronald Reagan" -- Harper's Magazine (June 2009)

Second Instance
This was almost certainly not written last week:

"[James Hansen] said that he was thinking of attending another deomonstration soon, in West Virginia coal country." -- Elizabeth Kolbert, "The Catastrophist", The New Yorker (June 29, 2009): 39-45

And here we have some news:

"SUNDIAL -- Coal miners confronted environmental protesters June 23 during a sometimes tense standoff at a focal point in the battle over mountaintop mining -- a protest that attracted one of the nation's foremost experts on global warming.

NASA climate scientist James Hansen was among the protesters, and West Virginia State Police arrested him during a planned act of civil disobedience. While upstaged in the media spotlight by actress Daryl Hannah, who also was arrested, it was Hansen's presence at the rally that drew widespread interest in the event from the environmental community..." -- Walt Williams

Suppressed Empirical Results

"An overwhelming majority of the entrepreneurs thought that a price based on full average cost (including a conventional allowance for profit) was the 'right' price, the one which 'ought' to be charged...

...the procedure can be not unfairly generalized as follows: prime (or 'direct') cost per unit is taken as the base, a percentage addition is made to cover overheads (or 'oncost' or 'indirect' cost), and a further conventional addition (frequently 10 per cent.) is made for profit. Selling costs commonly and interest on capital rarely are included in overheads; when not so included they are allowed for in addition for profits." -- R. L. Hall and C. J. Hitch, "Price Theory and Business Behavior", Oxford Economic Papers, (May 1939): 12-45

These findings motivated Milton Friedman in his badly-argued work on methodology.

Not Nostradamus

Brad DeLong still wants the attempted extrapolation of trends to be equivalent to prophecy. So I thought I'd point out another nineteenth century writer:

"There are at the present time two great nations in the world, which started from different points, but seem to tend towards the same end. I allude to the Russians and the Americans. Both of them have grown up unnoticed; and while the attention of mankind was directed elsewhere, they have suddenly placed themselves in the front rank among the nations, and the world learned their existence and their greatness at almost the same time.

All other nations seem to have nearly reached their natural limits, and they have only to maintain their power; but these are still in the act of growth. All others have stopped, or continue to advance with extreme difficulty; these alone are proceeding with ease and celerity along a path to which no limit can be perceived. The American struggles against the obstacles that nature opposes to him; the adversaries of the Russian are men. The former combats the wilderness and savage life; the latter, civilization with all its arms. The conquests of the American are therefore gained by the plowshare; those of the Russian by the sword. The Anglo-American relies upon personal interest to accomplish his ends and gives free scope to the unguided strength and common sense of the people; the Russian centers all the authority of society in a single arm. The principal instrument of the former is freedom; of the latter, servitude. Their starting point is different and their courses are not the same; yet each seems marked out by the will of Heaven to sway the destinities of half the globe." -- Alexis de Tocqueville, Democracy in America, V. 1, last page

By His Bootstraps

1.0 Introduction
One can hold savings in various forms of assets. In effect, savings is a time machine for transferring purchasing power into the future. A debt, when purchased - that is, a bond - is one such asset in which one can store savings. The relationships between bonds of various maturities and the existence of well-developed markets in which to trade bonds allows the determination of interest rates without relying on the theory of time preference, a theory which is a lot of utter hogwash anyhow.

The point of this post is to explain how, under certain institutions for selling second-hand debt, a relatively stable long-term interest rate can be maintained by beliefs in its stability. No need arises to call on the forces of thrift and productivity. This post might even be relevant to current events in the USA.

2.0 Institutions Providing a Setting in Which a Second Decision Must be Made
The model I outline here is based on Keynes' account of the two decisions a saver must make:

"The psychological time-preferences of an individual require two distinct sets of decisions to carry them out completely. The first ... determines for each individual how much of his income he will consume and how much he will reserve in some form of command over future consumption. But this decision having been made, there is a further decision which awaits him, namely, in what form he will hold the command over future consumption which he has reserved, whether out of his current income or from previous savings." -- J. M. Keynes, The General Theory of Employment, Interest and Money (1936): p. 166

Assume that the debts of the best quality available for purchase consist of Treasury bills (T-bills) that mature in three months, T-bills that mature in a year, and Treasury notes (T-notes) that mature in 10 years. These are all available in the U.S.A., along with T-bills, T-notes, and T-bonds of other maturities. In this exposition, I abstract from the existence of these other maturities. By including debts of these three maturities, the model incorporates the decision to hold money, assets that pay the short-term interest rate, or assets that pay the long-term interest rate.

In describing three-month T-bills as money, I again follow Keynes:

"...we can draw the line between 'money' and 'debts' at whatever point is most convenient for handling a particular problem. For example, we can treat as money any command over general purchasing power which the owner has not parted with for a period in excess of three months, and as debt what cannot be recovered for a longer period than this; or we can substitute for 'three months' one month or three days or three hours or any other period; or we can exclude from money whatever is not legal tender on the spot. It is often convenient to include in money time-deposits with banks and, occasionally, even such instruments as (e.g.) treasury bills." -- J. M. Keynes, The General Theory of Employment, Interest and Money (1936): p. 167

Suppose, contrary to fact, that the short term interest rate, r, was known to be constant for the next ten years, where 100 r is stated as an annual percentage. Then the long term interest rate would be established in the market at the start of the year as 100 [(1 + r)¹⁰ - 1] percent for 10 years, and the interest rate on money would be 100 [(1 + r)^1/4 - 1] percent for three months. A higher price on a bond corresponds to a lower interest rate. For example, the price of a T-bill with a face value of $1000 to be paid in a year is 1000/(1 + r) dollars.

3.0 The Individual
In this model, federal authorities set the interest rate on money. The short term interest rate provides a market consensus on monetary policy is likely to be over the next year. If the annual interest rate embodied in the price of one-year T-bills is higher than the annualized interest rate on money, the market price of T-bills is predicting a tightening of monetary policy. The individual allocates his savings partly on his opinion of this consensus. If he thinks, for example, that the monetary authority is not going to tighten that much, he would sell three-month T-bills and buy one-year T-bills, so as to make a profit from speculation when the price of the latter rises.

The individual, one assumes, has some idea of what is a normal long-term interest rate. He expects that over a long enough period, the federal authority's monetary policy will average out, thereby achieving this normal rate. The individual expects the price of T-notes to eventually rise when the current long-term interest rate is above that normal long-term rate and to fall when the current long-term rate is below that normal rate. Here, too, the possibility for speculative gains influences the individual in his allocation of his savings between T-notes and T-bills.

3.0 Markets
Consider a range of the price of T-notes. For a high enough price, those who are bears on this market (who expect the long term interest rate to rise) would dominate the bulls (who expect the long term interest rate to fall). More would be selling than buying, and the price would fall. The opposite is true for a low enough price. The equilibrium price at an instant of time balances bulls and bears:

"In the Treatise [Keynes] pictures the Bulls and Bears of the gilt-edged market going into and out of bonds as they individually come to think that the next price movement will be up or down. In this speculative market the price of bonds and thus their yield, the interest rate, can only settle if opinion is divided, so that those who wish to sell for fear of a fall find their offers matched by the bids of those who wish to buy in hope of a rise. It is thus, as Keynes says, a variety of opinion in the gilt-edged market which gives stability to the interest rate and some control over it to the monetary authorities." -- G. L. S. Shackle, "Simplicity in Keynes's Theory of Money and Employment", The South African Journal of Economics, v. 51, n. 3 (1983): 357-367

Elsewhere Shackle talks about equilibrium in such a speculative market as inherently restless.

4.0 Conclusions and a Policy Implication
I suppose one could express the above model in mathematics, if one were so inclined. One might start with some distribution of agents' beliefs about the conventional long term interest rate, and allow each agent to slowly update their view, maybe with the addition of random noise. (One might draw on Shackle's "The Bounds of Unknowledge" (in Beyond Positive Economics (ed. by J. Wiseman) Macmillan, 1983) in specifying this updating.) And the agents would decide on the distribution of their savings based on their views. Maybe the model should have more types of assets. One would want a model in which a diversity of opinion is maintained among agents, and in which time series for stock equilibria exhibit hysteresis and non-ergodicity. It wouldn't surprise me if somebody has already published such a model.

Keynes had something to say about policy based on this sort of analysis:

"Thus a monetary policy which strikes public opinion as being experimental in character or easily liable to change may fail in its objective of greatly reducing the long-term rate of interest... The same policy, on the other hand, may prove easily successful if it appeals to public opinion as being reasonable and practicable and in the public interest, rooted in strong conviction, and promoted by an authority unlikely to be superseded." -- J. M. Keynes, The General Theory of Employment, Interest and Money (1936): p. 203

Weird Science

I'm not sure I had heard of the Voynich Manuscript before this xkcd cartoon. But it did remind me of a couple of other artifacts whose existence I think extremely curious:

• The Antikythera Mechanism is a mechanical computer for calculating astronomical positions, and it dates from classical Greece.
• Piri Reis map dates, maybe, from before the Europeans had explored the New World, but yet shows the coast of South America and Anartica under the ice sheet.
• The Nazca lines are a series of drawings on a plateau in Peru dating from before Columbus and that make the most sense when viewed from the air.

Backstop Technologies And Natural Resources

1.0 Inroduction
Patch writes:

"I just finished reading Pasinetti's

Lectures on the Theory of Production (New York: Colombia University Press and London: The Macmillan Press Ltd., 1977.).

It is a readable introduction of production price models and it gave me an idea of what actually is the difference between prices optimizing the use of endowments and production prices. But then again I found some problems and I thought I might ask you before I send an email to Pasinetti himself.

The most important problem is, how do the production models work if you don't have an endowment? If you need commodity A to produce commodity B AND VICE VERSA, you can't start off producing at all because you don't have the necessary A to produce B and vice versa. The second problem maybe already is solved but I don't know: Can you use these models to say anything about natural resources? These resources are 'given', but you don't have to use them up in the first period so you still need an optimization I think."

Luigi Pasinetti, I think, has an ability to distill his ideas and express them remarkably clearly. His Lectures is a great textbook, and I could easily imagine Pasinetti getting a "Nobel" prize. Perhaps Sraffa's abstractions can be combined with Joan Robinson's emphasis on historical time to provide insight into actually existing capitalist economies. If so, such an advance might be based on Pasinetti's work, especially on structural economic dynamics.

I think both of Patch's questions are good questions, and I am going to decompose each into two.

2.0 Proportions and Endowments
One can think of the first as asking one of the following:

1. What happens if the means of production are not in the proportions needed for steady growth at the start of a production period?
2. What happens if none of some input needed for a cost-minimizing technique exists at the start of the production period?

An orthodox economist might want to analyze the first question by talking about intertemporal equilibria, the turnpike theorem, or tranverses between long period equilibria. However:

"5. Thus, if Sraffa's propositions are to be contained in an 'orthodoxy', which as [Frank] Hahn states has the endowment of factors among its data, such a datum should be present also in Production of Commodities. But how can Hahn trace it since that datum is not there?

6. Hahn attempts to do so when he introduces a simplified model of 'neoclassical theory' involving two commodities - wheat and barley - each of which is both a capital good and a consumption good. He asserts that, in order to obtain Sraffa's uniform rate of return on the two capital goods, the initial endowments of wheat and barley must bear some particular proportion to each other, because 'It cannot be part of the doctrine of Sraffa that you are uninterested in whether there is enough ... wheat and barley to meet demand" (Hahn, 1982, p. 365; emphasis added). In other words, Hahn finds it inconceivable that the simple answer to the question he raises in that passage might be:

Yes, it can, and it is, part of the doctrine of Sraffa's that distribution and normal prices do not require the initial endowments of wheat and barley for their determination.

Clearly, Hahn has begged his question here. He has assumed, and not proved, that what constitutes a characteristic feature of 'neoclassical' theory by his own definition of it - the determining role of factor endowments - is present in Sraffa's analysis. To that extent he has assumed what he intended to prove, namely that Sraffa's analysis is a 'special case' of neoclassical theory.

In fact, as he tells us in his Preface, Sraffa's standpoint is that 'of the old classical economists from Adam Smith to Ricardo', and, as we shall presently see, this standpoint does not attribute a determining role to 'the endowment of agents'. Therefore, his analysis can hardly be a 'special case' of the 'neoclassical' theory, which does admittedly rest on such a role." - P. Garegnani, "Sraffa" Classical versus Marginalist Analysis", in Essays on Piero Sraffa: Critical Perspectives on the Revival of Classical Theory (Edited by K. Bharadwaj and B. Schefold), Unwin Hyman (1990).

The second question is addressed by postulating the existence of a backstop technology in which, say, B can be produced without the existence of any A. I think this accords well with experience. Maybe no industry in the USA today produces without inputs of computer software and hardware, at least indirectly. Yet clearly this situation has come about without computer software and hardware being produced forever in the past.

3.0 Natural Resources
I think of natural resources as falling into two kinds:

1. Resources, like land in classical economics, that are given in quantity, cannot be manufactured, and provide services over a production period without ever suffering any diminution.
2. Resources, like oil, that are used up over time and cannot be replaced.

Land is usefully thought of as a case of joint production. Many economists, including Pasinetti himself, have analyzed joint production. I think of Christian Bidard, Bertram Schefold, and Heinz Kurz & Neri Salvadori. I even have an example treating land in the theory of joint production.

Although issues exist in extending Sraffa's results for single production to the theory of joint production, I think how to treat exhaustible resources in the long-period method is more problematic. As I understand it, Kurz and Salvadori have treated exhaustible resourses in what they call the guano model.

Two Tabs

• Matthew Yglesias (and informed commentators) on the microfoundations of macroeconomics
• Robert Nadeau on neoclassical economic theory for the Encyclopedia of Earth. (Recall this.)

Bolsheviks Versus Keynes

Anti-intellectuals and stupid people have long been conflating all variants of Keynesianism with communism.

In arguing against this conflation, Keynesians of some sort like to point out Keynes' explicit remarks on Marx:

"How can I accept a doctrine which sets up as its bible, above and beyond criticism, an obsolete text-book which I know to be not only scientifically erroneous but without interest or application for the modern world? How can I adopt a creed which, preferring the mud to the fish, exalts the boorish proletariat above the bourgeois and the intelligentsia who, with whatever faults, are the quality in life and surely carry the seeds of all human advancement? Even if we need a religion, how can we find it in the turbid rubbish of the red bookshop? It is hard for an educated, decent, intelligent son of Western Europe to find his ideals here, unless he has first suffered some strange and horrid process of conversion which has changed all his values." -- John Maynard Keynes, "A Short View of Russia" (1925) in Essays in Persuasion

Equally of interest, perhaps, is the abuse that Soviet Russia heaped on Keynes.

The following are some of the first comments in Soviet literature about Keynes's General Theory:

"One should not say that the conclusions of Keynes are a novelty. Analogous repair patches on capitalism were offered by reformers of various creeds in the USA and other countries long ago. Economic reality has proved the unadaptability of all these prescriptions for the saving of the capitalist system a countless number of times..." -- L. Freiman, "Bezrabotitsa V Kapitalisticheskikh Stranakh", Planovoe Khozyaistvo (1938)

"This theory is one of the many attempts of bourgeois science to adapt its apologetics to demands of the rule of capital at the present stage of development." -- A. Arutinyan, "Garvardskie Ekonomisty I Burzhuaznoe Konyunturovedenie, Problemy Ekonomiki (1940)

"Through the right-wing socialists, Keynesian ideas penetrate the working class. Keynesianism sows among the workers the harmful illusion of the possibility of overcoming crises and unemployment within the framework of capitalist society. The fight against Keynesianism is one of the most important tasks of the ideological work of Marxist economic science." -- I. G. Blyumin, from the Introduction to the first (1948) translation of The General Theory of Employment, Interest and Money

Here are some extracts from a thesis accepted by the Institute of Economics of the Academy of Sciences of the USSR:

"The chatter of the possibility of liquidating unemployment and economic crises in capitalism serves Keynes only as a smoke screen with the help of which he attempts to hide the real content of his especially anti-people's program directed pointedly against the working class. The unmasking of the conscious distortions by Keynes of the role of the contemporary bourgeois state is the most important requisite in explaining the real content of the Keynesian program of the transition to the so-called 'regulated' capitalism, which in itself is nothing other than a new charlatan project for the notorious 'organized capitalism'...

...The socialist system of the Soviet Union is the best refutation of all the Keynesian apologetics of capitalism and is proof that the only method of liquidating crises, unemployment, and all the other inevitable fellow travelers of capitalism is the destruction of the latter." -- V. S. Volodin (1950)

The above abuse was compiled and translated from the russian by Carl B. Turner (An Analysis of Soviet Views on John Maynard Keynes, Duke University Press, 1969).

This story about the antipathy between Keynes and Marxists is complicated by the existence of Michal Kalecki and of Paul Baran and Paul Sweezy.

An Experiment Protocol

1.0 Introduction
The point of the experiment described here is to offer empirical evidence for the importance of the distinction between uncertainty and risk, as put forth by Frank Knight and by John Maynard Keynes. People are not "rational", as "rationality" is defined by neoclassical economists.

As usual, I don't claim much originality except, maybe, in details. Daniel Ellsberg described the experiment below, as well as another. He references Chipman as having conducted experiments much like these. (Although Ellsberg's paper is oft cited and has been republished, Daniel Ellsberg is probably best known for having leaked The Pentagon Papers to the New York Times and others. Nixon's "plumbers" illegally broke into and searched Ellsberg's psychiatrist's office.)

2.0 The Protocol
The experimenter shows the test subject two urns, urn I and urn II. The test subject is shown that urn 1 is empty. The experimenter truthfully assures the test subject that urn II contains 8 balls, with some or none of them red and the remainder black. The test subject sees the experimented put one red and one black ball in urn II. The experimenter also puts in five red and five black balls in urn I in the test subject's presence. The urns are shaken.

So the test subject knows that urn number I contains 5 red and 5 black balls. Urn number II contains 10 balls. All are either red or black. At least one is black, and at least one is red.

The experimenter flips two coins so as to offer a gamble to the test subject. The coin flipping ensures the probability of offering each gamble is one in four. The gambles are described to the test subject:

• Gamble A: You pay $5 for a draw from urn number I. You choose before the draw whether to play red or black. If a ball is drawn of your color, you receive a payout of $10.
• Gamble B: You pay $5 for a draw from urn number II. You choose before the draw whether to play red or black. If a ball is drawn of your color, you receive a payout of $10.
• Gamble C: You pay $5. You choose urn number I or urn number II. A ball is drawn from the urn you selected. If the ball is red, you receive $10.
• Gamble D: You pay $5. You choose urn number I or urn number II. A ball is drawn from the urn you selected. If the ball is black, you receive $10.

Each test subject goes exactly once, and no test subject is able to observe previous plays by other test subjects (so urn number II cannot be sampled by a test subject).

The hypothesis is that in gambles A and B, statistically equal numbers of people will choose each color, while in gambles C and D, people will prefer to choose urn nmber I.

3.0 To Do

• Demonstrate mathematically that no assignments of probability in urn number II are compatible with the hypothetical behavior.
• Decide on a sample size. Perhaps a sequential test can be defined in which the sample size is not known beforehand.
• Read Craig and Tversky (1995) and Chipman (1960). Where else is Ellsberg referenced?

References

• J. S. Chipman, "Stochastic Choice and Subjective Probability", in Decisions, Values and Groups (edited by D. Willner), Pergamon Press (1960)
• Daniel Ellsberg, "Risk, Ambiguity, and the Savage Axioms", Quarterly Journal of Economics, V. 75, N. 4 (Nov. 1961): 643-669
• Craig R. Fox and Amos Tversky, "Ambiguity Aversion and Comparative Ignorance", Quarterly Journal of Economics, V. 110, N. 3 (1995): 585-603

Friedman Blinded Me With Science

Scientists aspire to develop theories that observations can potentially demonstrate to be wrong. Here I examine whether this aspiration can possibly be achieved when economics is practiced in keeping with one of two views on methodology, the deductive-nomological or the instrumental view. I get the argument below from Donald P. Green and Ian Shapiro, Pathologies of Rational Choice Theory: A Critique of Applications in Political Science (Yale University Press, 1994).

Consider the covering law model, also known as the deductive-nomological view of scientific methodology. In this view, scientists formulate universal laws, in some sense. In an application of a scientific law, the hypotheses or antecedents are asserted to be true. That is, the statement of scientific law is conjoined with initial conditions. One then checks that the consequent holds. If observation is inconsistent with the consequent and one is sure that the initial conditions are true, the law is refuted.

Milton Friedman advocates instrumentalism, in which the assumptions of a scientific theory are false. (Actually, his famous essay, "The Methodology of Positive Economics", is so incoherent, Friedman can be interpreted as advocating almost any methodology you care to name. But let's stick with a widely argued view.) In Friedman's view the antecedents are always false in a significant theory:

"Truly important and significant hypotheses will be found to have 'assumptions' that are wildly inaccurate descriptive representations of reality, and, in general, the more significant the theory, the more unrealistic the assumptions" -- Milton Friedman

Thus, if one holds that economic theories state covering laws and that economists are and should be instrumentalists, economic theories cannot be refuted by observation. The logical implications of false antecedents need not be true.

Can economics be a science if it is practiced in keeping with Friedman's strictures?

Capital Is Dead Labor, That, Vampire-Like, Only Lives By Sucking Living Labor

1.0 Introduction
Stupidity about Marx is never-ending. So I thought I would put up a post about Marx as a mathematical economist. This is exposition of unoriginal ideas. To amuse myself, I didn't review Sraffa or any other author when writing this.

2.0 The Technology
Consider an economy in which n commodities are produced, each in a separate industry. The technique in use is represented by the nxn Leontief matrix A and the n-element row vector a₀ of labor inputs. A column, a_.,j, in A and an element of a_0,j represent an industry. The ith, jth element of A is the quantity of the ith commodity input per unit output of the jth industry. Quantities are here measured in physical units (e.g., bushels corn per ton steel). The jth element of the row vector of labor inputs, a_0,j, is the person-years of labor services hired in the jth industry per unit output.

By assumption, all industries require at least some positive amount of labor to produce their outputs. All industries produce their outputs in a year, and they consume all their inputs in producing their output. This is a model of circulating capital alone; no fixed capital (e.g., long-lasting machines) appears in the model.

Assume that the economy is viable, that is, some levels of operation exist for the industries such that there is a surplus product available, after replacing used-up means of production, when industries are operated at that level. For simplicity, assume all commodities are basic. In other words, every commodity enters either directly or indirectly into the production of every other industry. Presumably, steel enters directly into the production of automobiles. Iron would enter indirectly into the production of automobiles through its use in the production of steel.

No choice of technique occurs in the model.

3.0 Quantity Flows

3.1 Labor Values
Let q be an n-element column vector, where each entry is the gross output of that industry. Each entry is measured in the corresponding physical units (tons, bushels, kilograms, etc.). Let y be an n-element column vector of net outputs. Gross and net outputs are related like so:

y = q - A q

Or:

q = (I - A)^-1 y

where I is the identity matrix. The existence of the inverse follows from viability. If industries were operated at levels to produce the gross outputs, the net output would be available for consumption or accumulation after replacing exactly the inputs consumed in production.

The amount of labor hired to produce the net output y is:

L = a₀ q = a₀ (I - A)^-1 y

Suppose net output consisted of only one unit of the commodity produced in the jth industry:

y = e_j

where e_j is the jth column in the identity matrix. The labor value of the jth commodity, that is, the amount of labor hired to produce one more unit of the jth commodity net, is:

v_j = a₀ (I - A)^-1 e_j

Labor values are expressed as an n-element row vector:

v = a₀ (I - A)^-1

Labor values then have a sensible meaning; nothing radical is involved in defining them.

3.2 The Standard Commodity
One might expect any arbitrary basket with a large number of commodities to have both some labor-intensive and some capital-intensive commodities, in some sense. On average, these will approximately cancel in an arbitrary basket. Accordingly, let's assume that gross outputs, net outputs, and the commodity in which wages are paid are all of average capital-intensity, in some sense.

Since both gross and net outputs are of average capital intensity, it seems sensible to assume they are composed of the same proportions, just different in amount:

q* = (1 + 1/R) y*

where the asterisks indicate a basket in standard proportions. R is a strictly positive constant. As a normalization condition, the standard system is assumed to employ a unit amount of labor:

a₀ q* = 1

It follows that the gross output of the standard system is a right-hand eigenvector of the Leontief input-output matrix:

A q*= [1/(1 + R)] q*

The outputs of the standard system are guaranteed to be positive by setting [1/(1 + R)] to the maximum eigenvalue of the Leontief matrix, also known as the Perron-Frobenius root of the Leontief matrix.

The net output of the standard system, y*, is the standard commodity.

3.3 The Rate of Exploitation
The labor embodied in the gross output of the standard system is easily found. One has:

1 = a₀ q* = v(I - A)q* = v q* - v A q*

Or, taking advantage of the fact that the gross output of the standard system is an eigenvector of the Leontief matrix:

1 = [R/(1 + R)] v q*

Hence, the labor value of gross output of the standard system is found as:

v q* = 1 + 1/R

Marx expressed the labor value of gross output as C + V + S. Constant capital C is vAq*, the labor value of the means of production used up in producing the net output. Variable capital V is the labor value of the value added by labor paid out in wages. Surplus value S is the labor value of the remaining net output, which is obtained by the capitalists. Since one person-year is employed, the sum of variable capital and surplus value in the standard system is unity:

V + S = 1

Let w denote the proportion of the net output of the standard system (that is, the standard commodity) that is paid out in wages. Hence:

0 ≤ w ≤ 1

And variable capital is given as:

V = w

It follows that surplus value is now defined:

S = 1 - w

Marx denoted the ratio of surplus value to variable capital as the rate of exploitation:

e = S/V = (1 - w)/w = (1/w) - 1

where e is the rate of exploitation.When the whole value of the net product is paid out to workers as wages, workers are not exploited and the rate of exploitation is zero. The rate of exploitation is otherwise positive, and increases without bound as the wage becomes a lesser proportion of the value of the net product.

4.0 Price Equations
Let p denote a row vector of prices of production. Prices of production permit smooth reproduction in a competitive capitalist economy. They are defined by the condition that the same rate of profits is obtained in each industry:

p A(1 + r) + a₀ w = p

where r is the rate of profits. Since profits are not earned on wages, the workers are paid at the end of the year. Wages are not advanced in this model. Since the same symbol for wages is used in calculating the labor value of quantities in the standard system, the standard commodity is the numeraire. Thus, the price of the standard commodity is unity:

p y* = 1

Recall that the net and gross outputs of the standard system are in proportion. One can thus calculate the price of the gross output of the standard system:

p q* = 1 + 1/R

Postmultiply the price equations by the gross output of the standard system:

p A q* (1 + r) + a₀ q* w = p q*

Or:

p q* [1/(1 + R)] (1 + r) + w = (1 + 1/R)

Or:

(1 + 1/R)[1/(1 + R)] (1 + r) + w = (1 + 1/R)

The rate of profits is an affine function of the wage:

r = R(1 - w)

The above equation can also be expressed as:

w = 1 - r/R

The rate of profits ranges from zero to the maximum R. The maximum rate of profits is obtained when workers live on air, with a wage of zero. A higher wage is associated with a lower rate of profits, with a very simple relationship with this numeraire.

Total wages are a₀ q* w. But, since one person-year is employed in the standard system, total wages are simply w.

Total profits are p A q* r, that is:

p A q* r = [1/(1 + R)] p q* R(1 - w)

Or:

p A q* r = [1/(1 + R)](1 + 1/R)R(1 - w)

Or:

p A q* r = 1 - w

The above is hardly surprising. The ratio of the rate of profits to the maximum rate is an increasing function of the rate of exploitation:

r/R = e/(1 + e)

When the rate of exploitation is zero, the rate of profits in the system of prices of production is also zero. As the rate of exploitation increases without bound, the ratio of the rate of profits to the maximum rate monotonically increases to unity.

5.0 Invariants
The following statements hold, whether the quantities mentioned are evaluated in labor values or in prices of production:

• The gross output of the standard system is valued at 1 + 1/R
• The net output of the standard system is unity
• Variable capital is valued at w
• Surplus value, that is, profits are (1 - w)

Furthermore, the rate of profits is positive if and only if workers are exploited.

This model certainly suggests that market phenomena are a veil over the exploitation inherent in capitalism. And calculations with labor values exhibit that exploitation.

"Capital As Power"

Maybe I'll purchase the book containing this truism:

"It should be noted upfront that economics – or, more precisely, the neoclassical branch of political economy – is not an objective reality. In fact, for the most part it is not even a scientific inquiry into objective reality. Instead, neoclassical political economy is largely an ideology in the service of the powerful. It is the language in which the capitalist ruling class conceives and shapes society. Simultaneously, it is also the tool with which this class conceals its own power and the means with which it persuades others to accept that power." -- Jonathan Nitzan and Shimshon Bichler, Capital As Power: A Study of Order and Creorder (Routledge 2009)

The above is not a novel idea.

"...the absurd aphorisms of a political economy controlled by property have puzzled the most generous minds." -- P.-J. Proudhon

A Neoclassical Response To The Cambridge Capital Controversy

1.0 Introduction
Around 1980, Edwin Burmeister could have justly thought that he was expressing the most prominent neoclassical response to the Cambridge Capital Controversy. He had championed David Champernowne's chain index as a defense of the aggregate neoclassical model, and continued to do so. Nowadays, though, mainstream economists make claims based on the aggregate model apparently in complete ignorance that they had ever been competently challenged:

"However, the damage had been done, and Cambridge, UK, 'declared victory': Levhari was wrong, Samuelson was wrong, Solow was wrong, MIT was wrong and therefore neoclassic economics was wrong. As a result there are some groups of economists who have abandoned neoclassical economics for their own refinements of classical economics. In the United States, on the other hand, mainstream economics goes on as if the controversy had never occurred." -- Edwin Burmeister (2000)

This post illustrates, by means of an example, elements of Burmeister's approach to the neoclassical aggregate model. It is exposition, with next to no criticism.

2.0 Technology
Consider a very simple economy in which a single consumption good, corn, is produced from inputs of labor, iron, and (seed) corn. All production processes in this example require a year to complete. Two production processes are known for producing corn, and two processes are known for producing iron. These processes require inputs to be available at the beginning of the year for each unit output produced and available at the end of the year. Each corn-producing process produces one bushel corn at the scale of operations shown in Table 1. Similarly, each iron-producing process produces one ton iron at the scale shown in Table 1.

Table 1: The CRS Technology

Inputs	Corn Industry		Iron Industry
	A	B	C	D
Labor (Person-Years):	2	3	1	2
Iron (Tons):	3/50	1/10	3/5	1/2
Corn (Bushels):	1/2	1/4	1/2	3/5
Output (Various):	1	1	1	1

Apparently, inputs of iron and corn can be traded off in producing corn outputs. The process that requires more iron also requires more labor. Inputs of iron and corn are also traded off in producing iron. But in iron production, the process requiring a greater quantity of iron input requires less labor.

A technique consists of a process for producing iron and a process for producing corn. Thus, there are four techniques in this example. They are defined in Table 2.

Table 2: Techniques and Processes

Technique	Processes
Alpha	A, C
Beta	A, D
Gamma	B, C
Delta	B, D

3.0 Quantity Flows
Suppose firms have adopted the alpha technique and they produce 20/43 bushels corn with process A and 3/43 tons iron with process C. One can see, from Table 1, that these firms will employ 40/43 person-years in the corn industry and 3/43 person-years in the iron industry - that is, a total of one person-year throughout the economy. Likewise, firms in the corn industry will purchase inputs of 6/215 tons iron, while firms in the iron industry will purchase inputs of 9/215 tons iron in the iron industry, for a total of 3/43 tons iron inputs throughout the economy. The produced iron at the end of the year exactly replaces the iron used as input, leaving a net output of 17/86 bushels corn. (Calculating corn inputs in the two industries is left as an exercise for the reader.)

Since these processes can be equally scaled up to any desired level, I have described a stationary economy on a per person-year basis. Table 3 shows the results of these calculations, as well as similar calculations for the gamma and delta technique. The beta technique is never cost-minimizing and is not shown in Table 3.

Table 3: Quantities Per Person-Year

Technique	Variable	Value
Alpha	Gross Outputs	(3/43 Tons, 20/43 Bushels)
	Capital Goods	(3/43 Tons, 23/86 Bushels)
	Net Output	17/86 Bushels Corn
Gamma	Gross Outputs	(1/13 Tons, 4/13 Bushels)
	Capital Goods	(1/13 Tons, 3/26 Bushels)
	Net Output	5/26 Bushels Corn
Delta	Gross Outputs	(1/17 Tons, 5/17 Bushels)
	Capital Goods	(1/17 Tons, 37/340 Bushels)
	Net Output	63/340 Bushels Corn

4.0 Prices
In a steady state, the same rate of profits is earned on all processes in use. Furthermore, charging that rate of profits on a process not eligible for use results in costs in that process exceeding the revenues. That is, we seek steady state prices corresponding with the cost-minimizing technique.

Suppose the alpha technique is cost minimizing. Prices for the iron-producing process (C) must satisfy the following equation:

(3/5 p_α + 1/2)(1 + r) + w_α = p_α,

where p_α is the price of iron, w_α is the wage, and r is the rate of profits. The wage is paid at the end of the year, and corn is taken as the numeraire (so the price of a bushel corn is unity). Likewise, prices for the corn producing process (A) satisfy the following equation:

(3/50 p_α + 1/2)(1 + r) + 2 w_α = 1

I have specified a system of two equations in three variables. The wage and price of iron can be found as a function of the third variable, that is, the rate of profits. Table 4 displays this solution, as well as the solutions for the corresponding systems of equations for the other three techniques.

Table : Solutions to Price Equations

Technique	Variable	Equation
Alpha	Wage	wα(r) = (27 r2 - 56 r + 17)/[2 (43 - 57 r)]
	Price of Iron	pα(r) = 25 (3 + r)/(43 - 57 r)
Beta	Wage	wβ(r) = (107 r2 - 286 r + 107)/[40 (14 - 11 r)]
	Price of Iron	pβ(r) = 5 (11 +r)/[2 (14 - 11 r)]
Gamma	Wage	wγ(r) = (2 r2 - 13 r + 5)/[2 (13 - 17 r)]
	Price of Iron	pγ(r) = 5(9 + 5 r)/[2 (13 - 17 r)]
Delta	Wage	wδ(r) = (13 r - 7)(r - 9)/[20 (17 - 13 r)]
	Price of Iron	pδ(r) = (33 + 13 r)/(17 - 13 r)

Figure 1 graphs the wage-rate of profits curves for each technique. The cost-minimizing technique corresponds to the curve on the outer envelope. The wage-rate of profits curves for the alpha, gamma, and delta technique comprise the wage-rate of profits frontier. Alpha is cost-minimizing at low rate of profits, delta is cost-minimizing at high rates, and gamma is cost-minimizing at intermediate rates. Notice that for each pair of techniques, the wage-rate of profits curves cross at most once in the first quadrant. There is no reswitching, either on or off the frontier, in this example.

Figure 1: Wage-Rate of Profits Frontier

5.0 Champernowne's Chain Index

The above analysis specifies for each rate of profits (or for each wage) which technique will be adopted by cost minimizing firms. At switch points, linear combinations of techniques are cost-minimizing. The above analysis also determines the price of each capital good (e.g. corn and iron) for each rate of profits, as well as the composition of capital goods used in each technique per person-year. Figure 2 can thus be drawn based on this analysis.

Figure 2: Value of Capital and the Rate of Profits

Figure 2 shows the effects of both real and price Wicksell effects. The two horizontal lines arise from switch points. At switch points the composition of capital goods varies with the technique, while the rate of profits and the prices of capital goods are fixed. In other words, "real" capital varies in some sense. So the horizontal lines show real Wicksell effects. The curved, non-horizontal, segments display price Wicksell effects. That is, at non-switching points, the composition of capital goods remains invariant, but the prices of capital goods vary. Consequently, the numeraire value of the basket of capital goods varies here also.

Champernowne's chain index (Figure 3) sums up real Wicksell effects alone. Price Wicksell effects are abstracted from. The value of capital goods at the rate of profits of zero is taken in Figure 3 from Figure 2. Horizontal lines are drawn in Figure 3 at the same rates of profits at which they appear in Figure 2. The horizontal lines are also the same length. Vertical lines are drawn between horizontal lines.

Figure 3: Chain Index Value of Capital and the Rate of Profits

Champernowne's chain index only makes sense of the neoclassical parable in this case because all steps in Figure 3 slope down to the right. In other words, for an infinitesimal variation of the rate of profits around a switch point, the capital intensity of the cost-minimizing technique at the lower rate of profits exceeds the capital intensity of the cost-minimizing technique at the higher rate of profits. That is, Burmeister's defense of the neoclassical parable only applies in cases in which real Wicksell effects happen to be always negative:

"It follows, then, that a negative real Wicksell effect is the appropriate concept of 'capital deepening' in a model with many heterogeneous capital goods... Imposing some set of conditions on the technology ... should be sufficient to assure that the real Wicksell effect is always negative. Such conditions would be of interest - especially if they could be empirically tested - since they would validate the qualitative conclusions derived from one-good models often used in macroeconomics without any theoretical justification... Unfortunately, no set of such sufficient conditions is known, but the literature on capital aggregation suggests that they would impose severe restrictions on the technology." -- Edwin Burmeister (1987)

6.0 A Pseudo-Production Function
I finally turn to the aggregate neoclassical production function used in the neoclassical parable:

Y = F(K, L),

where Y is net income, K is capital, and L is labor. Since Constant Returns to Scale are assumed, one can divide through by the labor input:

Y/L = F(K/L, 1)

Or:

y = f(k),

where y is net output per worker and k is capital per worker, in some sense. Figure 4 graphs this function for the example, where Champernowne's chain index is used to measure capital per worker. (If net output consisted of more than the numeraire good, a chain index would be used to measure output also.)

Figure 4: Pseudo-Production Function for the Example

Using this construction, the equilibrium condition that the rate of profits equal the marginal product of capital holds at switch points:

r = f ' (k)

This analysis has accepted that the value of capital goods (that is, the "quantity of capital") depends on the rate of profits. Recall, however, that this analysis only applies to examples in which real Wicksell effects happen to be always negative.

References

• Salvatore Baldone (1984). "From Surrogate to Pseudo Production Functions", Cambridge Journal of Economics, V. 8: 271-288
• Edwin Burmeister (1980) Capital Theory and Dynamics, Cambridge University Press
• Edwin Burmeister (1987) "Wicksell Effects", in The New Palgrave, (ed. by J. Eatwell, M. Milgate, and P. Newman), Macmillan
• Edwin Burmeister (2000). "The Capital Theory Controversy" in Critical Essays on Piero Sraffa's Legacy in Economics (ed. by H. D. Kurz), Cambridge University Press
• D. G. Champernowne (1953-1954). "The Production Function and the Theory of Capital: A Comment", Review of Economic Studies, V. 21: 112-135

Toxic Textbooks

Edward Fullbook, a post autistic economist, is organizing a community to encourage students to protest orthodox economics textbooks.

Reflections On "Sraffian Economics (New Developments)"

Michael Mandler has an article, "Sraffian Economics (New Developments)" in the latest edition of The New Palgrave Dictionary of Economics. I have been trying to read this. (Paul Samuelson's article, "Sraffian Economics", in the original New Palgrave is also heavy going.)

I have previously read Mandler as an anti-Sraffian willing to take the views he opposes seriously. I wonder if he is more positive now. Perhaps he feels that, although Sraffians are mistaken in theory, their mistakes are worthwhile to explore.

That is all subjective on my part, of course. Mandler is explicit on the issues of the indeterminateness of equilibrium and of tâtonnement stability. An indeterminate equilibrium is not merely a case of multiple equilibria. Rather, a continuum of equilibria arise. Perturbations of an equilibrium along this continuum would not set up stable or unstable forces driving the economy back towards or away from the original equilibrium. Rather the economy would just be in another equilibrium. The tâtonnement is a particular kind of exchange process that arises before the beginning of time in the Arrow-Debreu model of intertemporal equilibrium. Mandler argues that Sraffa has failed to demonstrate indeterminateness, and that issues of tâtonnement instability are not essentially connected to Sraffa's model of production; they arise from elements of utility-maximization.

Mandler has certainly been engaged by Sraffians (or vice versa) on exactly these issues. But I'm not sure that I agree that Mandler has picked out the essential points of Sraffa's book. The distribution of income is indeterminate in Sraffa's open model. I do not read Sraffa as claiming this property would still obtain if he closed his model by appending a specification of utility-maximizing consumers, including intertemporally. Rather, I take Sraffa as offering an open model demonstrating non-neoclassical theories of value and distribution can be constructed. If one insists on a closed mathematical model (for example), an empirical issue arises. I think Sraffa did not insist that his model be closed, at least, with elements of a model at the same level of abstractness and generality.

While tâtonnement (in)stability is interesting, I take Sraffian analysis to point towards stability isses elsewhere in, say, the Arrow-Debreu model. One can construct models of spot prices corresponding to the forward prices in the Arrow-Debreu model. These spot prices have their own dynamics that would arise even if spot markets always cleared instantaneously over time. Sraffa's model of production supports an exploration of limit points of this dynamics.

I have constructed examples with bifurcations, pointing to possibilities of complex dynamics in models of temporary or momentary equilibrium. (I don't claim to have a good grasp of the distinction, if any.) One can also show, through an analysis of structural stability, that many of the stories applied economists like to tell are without logical foundation.

Variations in the supply of labor can be modeled by perturbing a parameter in utility functions. An increased supply of labor is modeled by an increased desire for consumption, as opposed to leisure. Nevertheless, the corresponding equilibrium associated with an increased supply of labor, all other parameters held constant, might have a higher wage. The increased supply of labor need not drive the equilibrium wage down.

Likewise, variations in the supply of savings can be modeled by perturbations in a parameter describing intertemporal utility-maximizing. And greater savings can be associated, all other parameters held constant, with a higher equilibrium interest rate.

Relating the structural (in)stability of equilibrium limit points to the dynamics of temporary or momentary equilibria is a challenge to me. I am not sure whether interesting bifurcations are tied to capital-theoretic "paradoxes" such as reswitching and capital-reversing. I think it may depend on details of the model. In one reswitching example, I have found that whether the normal or "perverse" switch is associated with bifurcations depends on whether intertemporal maximizing representative agents are also modeled as choosing between leisure and consumption. Whether the latter choice is included or not seems to flip the result. But perhaps in some model where one has fixed the modeling choice, the existence of interesting dynamic behavior, in some sense, may be tied to the existence of perverse switches.

I may never resolve these theoretical issues to my own satisfaction.

No Thanks, Alessandro. I'm Full.

Alessandro Roncaglia has written a book, Piero Sraffa, for Palgrave's "Great Thinkers in Economics" series. (This is the same series containing Paul Davidson's John Maynard Keynes, available in hardcover and paperback.) I have always enjoyed Roncaglia's take on Sraffa, but I think I'll pass for now. I've seen quite a bit of what he has had to write in the past.

• Alessandro Roncaglia (1978) Sraffa and the Theory of Prices, John Wiley & Sons.
• Alessandro Roncaglia (1979) "The Sraffian Contribution", in A Guide to Post-Keynesian Economics (edited by Alfred S. Eichner), M. E. Sharpe
• Alessandro Roncaglia (2000) Piero Sraffa: His Life, Thought and Cultural Heritage, Routledge
• Alessandro Roncaglia (2001) "Production of Commodities by Means of Commodities Between Criticism and Reconstruction: The Given Quantities Assumption", in Piero Sraffa's Political Economy: A Centenary Estimate (edited by Terenzio Cozzi and Roberto Marchionatti), Routledge
• Alessandro Roncaglia (2005) The Wealth of Ideas: A History of Economic Thought Cambridge University Press

On Austrian Business Cycle Theory, Recently

Brad DeLong offers an empirical criticism based on order of magnitude estimates. For some reason, John Quiggin's blog crashes this browser on this platform. So I look to Mark Thoma to echo Quiggin, who doesn't seem to understand the (failed) concepts. Quiggin doesn't mention Wicksell, the idea of a natural rate of interest, or capital structure, for example. DeLong's post was in response to Roger Garrison. Peter Boettke adds a post. In comments to some of these posts, I link to a recent iteration of my critique, which may have some influence on Roger Koppl.

Is Utility Theory Tautological? An Old Argument

"What does [Jevon's theory] really amount to? In my apprehension to this, and no more - that value depends upon utility, and that utility is whatever effects value. In other words, the name 'utility' is given to the aggregate of unknown conditions which determine the phenomenon, and then the phenomenon is stated to depend upon what this name stands for." -- J. E. Cairnes, Some Leading Principles of Political Economy (1874) (quoted by G. Myrdal in The Politcal Element in the Development of Economic Theory)

"...that value was determined by the conditions which determine it - an announcement, the importance of which, even though presented under the form of abtruse mathematical symbols, I must own myself unable to discern." -- J. E. Cairnes, Some Leading Principles of Political Economy (1874) (quoted by G. Myrdal in The Politcal Element in the Development of Economic Theory)

An Intervention By Kurt Gödel Into Economics

Kurt Gödel attended Karl Menger’s colloquium in Vienna in the 1930s. Sraffians should be interested in this colloquium since Von Neumann, in 1937, presented his classically-inspired economics model to the attendees. (Von Neumann had previously presented an earlier version to a Princeton mathematics seminar.) The later English translation of Von Neumann’s terse article is accompanied by a note from David Champernowne with the following acknowledgement.

"This note is the outcome of conversations with Mr. N. Kaldor, to whom many of the ideas in it are due. I am also indebted to Mr. P. Sraffa of Cambridge and to Mr. Crum of New College, Oxford, for instruction in subjects discussed in this article" -- D. G. Champernowne

Gödel’s published comment, however, was part of a discussion of a marginalist model. Walras’ models of the exchange of several commodities for one another and of production contain equations in which the quantity demanded of each good is a function of prices:

q₁ = d₁(p₁, ..., p_n)

q_n = d_n(p₁, ..., p_n)

Abraham Wald, in 1934, presented to the colloquium a sort of inverse or dual model, building on a paper from Karl Schlesinger. This model contains equations expressing the prices at which the quantities of commodities are demanded:

p₁ = d₁(q₁, ..., q_n)

p_n = d_n(q₁, ..., q_n)

I gather this model also contained inequalities, an important development in the theory of general equilibrium. At any rate, Kurt Gödel commented:

"Actually, for each individual entrepreneur the demand also depends on the price of factors of production. One can formulate an appropriate system of equations and investigate whether it is solvable." -- Kurt Gödel

According to John Dawson, Jr., Gödel’s remark is not well-taken; there is no obvious way to introduce prices of factors of production in this "inverse" model with many consumers. By contrast, when Gödel decided to say something about relevatistic physics, his remarks about rotating universes and world-lines traveling into the past, I guess, challenge physicists even decades later.

I stumbled on Gödel’s remark last week by noticing E. Roy Weintraub (1983) referenced in the first volume of Gödel’s collected works and wondering why that should be. I now see that Weintraub also quotes Gödel's remark, and Dawson is concurring with Weintraub.

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: 10-18
• S. Feferman et al (editors) (1986). Kurt Gödel: Collected Works: Volume 1: Publications 1929-1936, Oxford University Press
• J. v. Neumann (1945-1946). “A Model of Economic Equilibrium”, Review of Economic Studies, V. 13, N. 1: 1-9
• E. Roy Weintraub (1983). “On the Existence of a Competitive Equilibrium: 1930-1954”, Journal of Economic Literature, V. 21, N. 1 (March): 1-39