Interest Rate Unequal To Marginal Product Of Capital (Part 1 Of 4)

"Then Buddha said: ... But tell me, Subhuti, do you really believe that having only one homogeneous capital good will permit you to derive a rate of profit purely from the technical relationship between homogeneous capital and output?

Subhuti replied: Thus it is said in some venerable books.

Buddha said: Revere them, Subhuti, but trust them not. Suppose you do get the value of the marginal product of capital in terms of output of consumer goods. In what units will it be expressed? Physical units of additional consumer goods per unit of additional homogeneous capital. But the rate of profit is a pure number. Surely you will need something more in going from the first to the second to reflect the relative price of the capital good vis-a-vis the consumer good. But the equilibrium price of capital in units of consumer goods depends on the rate of profit used for discounting, and a variation of the rate of profit can involve a variation of the value of the same physical capital in units of consumer goods. This difficulty is not eliminated by having one homogeneous good." -- Amartya Sen (1974). "On Some Debates in Capital Theory", Economica V. 41 (August)

1.0 Introduction
This essay demonstrates that the existence of price Wicksell effects can lead to the inequality of the marginal product of capital and the interest rate. The equality being challenged here should be understood as it is used in macroeconomic models with aggregate production functions. That is, macroeconomic modeling with aggregate production functions is inadequately grounded in microeconomic theory. I conclude with some rather far-reaching possibilities.

The length of my exposition here results from my attempting to clarify several points of confusion exhibited by economists responding on Usenet to previous versions. This argument is well-established in the literature [1]. I suggest that those who think this argument mistaken should take a look at some of my references. If my argument were mistaken, demonstrating the mistake would be worthy of a paper.

I claim this argument is not about index number problems or the aggregation of capital [2]. I also do not see how it relates to the aggregation of production functions. Those who believe otherwise are encouraged to be explicit about the connections. Perhaps, the question, from a neoclassical perspective, is how the services of capital goods are related to the quantity of "waiting" they supposedly represent.

Footnotes
[1] Elements of this argument can be found in Joan Robinson's 1953-1954 article "The Production Function and the Theory of Capital," Review of Economic Studies, 1953-4 and Geoff Harcourt (1972), Some Cambridge Controversies in the Theory of Capital. The closest formulation to my argument is in the following papers:

• Amit Bhaduri (1966). "The Concept of the Marginal Productivity of Capital and the Wicksell Effect," Oxford Economic Papers, XVIII: 284-288
• Amit Bhaduri (1969). "On the Significance of Recent Controversies on Capital Theory: A Marxian View," Economic Journal, LXXIX: 532-539

But the argument was also in Sraffa. See, for example:

• Piero Sraffa (1962). "Production of Commodities: A Comment," Economic Journal, V. LXXII (June): 477-9.

Here is one expositor of Sraffian economics:

"Now a major problem existed because capital, unlike either labor or land, is a produced means of production and cannot be measured unambiguously in purely physical terms: the amount of capital can be measured only in value terms. The problem was to establish the idea of a market for capital, the quantity of which could be expressed independently of the price of its service (i.e. the rate of profit)... The basic deficiency with this approach is in its treatment of capital, which cannot be measured independently of the rate of profit. As observed above, the value of capital, like that of all produced commodities, depends on the rate of profit, or interest." -- J. E. Woods (1990). The Production of Commodities: An Introduction to Sraffa, Humanities Press International: 306-307

[2] The divergence between the marginal product of capital and the rate of interest

"is attributable to the fact that it is impossible to find an invariant unit in which to measure the social quantity of capital.

To put the matter another way, we may say that a change in the supply of capital - arising, for example, from new voluntary savings - alters the units in which all the previously existing capital is measured; and it is therefore incorrect to say that the supply of capital as a whole has increased by the amount of the voluntary saving. It is important to emphasize that this problem of measuring the quantity of capital is not an index-number problem. There are, to be sure, numerous index-number problems of the greatest complexity in the theory of capital. But the problem to which I now refer would exist even in the simplest economy in which all output consisted of a single type of consumer's good and firms were exactly alike." -- L. A. Metzler (1950). "The Rate of Interest and the Marginal Product of Capital," Journal of Political Economy, Vol. 53: 284-306

Metzler provides a brief literature review of awareness of this problem going back to Wicksell. My analysis is closest to his comments on Knight's capital theory, though I think my presentation is clearer.

Do Patents And Copyrights Spur Innovation?

An expert on United States law on intellectual property tells me a National Academy of Sciences study found empirical evidence suggesting the answer is "Yes" to the question in the post title. Presumably, he was referring more to the former than the latter of:

• Patents in the Knowledge-Based Economy (2003)
• A Patent System For The 21st Century(2004)

I've downloaded the executive summaries, but I doubt I will purchase these reports.

Apparently if I go longer between posts, commentators will have more to say on each one.

Break In Blogging

I will be on travel for a week or more starting, I guess, Saturday. And I don't expect to blog in that time.

I feel I am tardy in acknowledging comments. Given this tardiness, I'm more than happy when commenters are discussing topics among themselves.

Does Studying Mainstream Economics Make You A Bad Person?

(I had the title and the next three paragraphs written before Radek's comments today.)

Experimental evidence on the topic suggests a disquieting affirmative answer. Specifically, I refer to "Does Studying Economics Inhibit Cooperation", by Robert H. Frank, Thomas Gilovich, and Dennis T. Regan (Journal of Economic Perspectives, V. 7, N. 2 (Spring 1993): 159-171)

I believe that more up-to-date work exists in this vein. I was able to quickly locate a reference to "Does Studying Economics Discourage Cooperation? Watch What We Do Not What We Say or How We Play", by Yetzer, Goldfarb, and Poppen (Journal of Economic Perspectives, V. 10, N. 1 (1996): 177-186). (I haven't read this.)

But look at the URL for that copy of the Frank, Gilovich, and Regan paper. Why should Richard Stallman want more people to know of their findings? What does this have to do with open-source and free (as in "freedom") software?

Perhaps if I browsed around the proceedings of one of the Wizard of OS conferences, I would see some connection between advocating open source and being anti-mainstream economics. Given the interests of at least one of my readers, I want to note that Lawrence Lessig is the keynote speaker for this year's conference, which just ended.

At SSRN And Mankiw's Blog

I have finally completed the first draft of my paper Some Fallacies of Austrian Economics. I wonder if the historical documentation is too extensive for it to be accepted by some journal somewhere. On the other hand, I suppose some Austrians might like some text between equations.

I have been amused by watching commentators on Greg Mankiw's blog be unable to register that some might doubt that minimum wages above the (?) market-clearing wage cause unemployment.

Where Do Correct Ideas Come From? Do They Drop From The Skies?

Have the ideas that mainstream economists teach and apply won out against alternatives in intellectual debate? Perhaps modern economics has been shaped by interventions with a non-cognitive basis.

Academics have one well-established method for writing books. They can try out chapters as articles, thereby seeing what reviewers find convincing and where reviewers think the argument needs strengthening. Maybe Fred Lee is working on a book about the sort of interventions mentioned above and counter-hegemonic movements within academic economics:

• Lee, Frederick S. (2000a). "Conference of Socialist Economists and the Emergence of Heterodox Economics in Post-War Britain", Capital and Class, V. 75: 15-39
• Lee, Frederick S. (2000b) "The Organizational History of Post Keynesian Economics in America, 1971-1995", Journal of Post Keynesian Economics, V. 23, N. 1 (Fall): 141-162
• Lee, Frederick S. (2002). "Mutual Aid and the Making of Heterodox Economics in Postwar America: a Post Keynesian View", History of Economics Review: 45-62
• Lee, Frederick S. (2004a). "History and Identity: The Case of Radical Economics and Radical Economists, 1945-70", Review of Radical Political Economics, V. 36, N. 2 (Sprong): 177-195
• Lee, Frederick S. (2004b). "To Be a Heterodox Economist: The Contested Landscape of American Economics, 1960s and 1970s", Journal of Economic Issues, V. XXXVIII, N. 3 (September): 747-763
• Lee, Frederick S. and Sandra Harley (1998). "Peer Review, the Research Assessment Exercise and the Demise of Non-Mainstream Economics", Capital and Class: 23-51
• Mata, Tiago and Frederick S. Lee (2006). "Making Visible What is Hidden: The Role of Life Histories in Writing the History of Heterodox Economics", 19th Annual Conference of the History of Economic Thought Society of Australia, Ballarat, Victoria (4-7 July)
• Tymoigne, Eric and Frederick S. Lee (2003-4). "Post Keynesian Economics Since 1936: A History of a Promise that Bounced?", Journal of Post Keynesian Economics, V. 26, N. 2 (Winter): 273-287

Contradiction In Arrow-Debreu Model Of Intertemporal Equilibrium?

This post looks at just one problem with the Arrow-Debreu model.

The data of the Arrow-Debreu model of intertemporal equilibrium include the initial quantities, that is, the endowments of all commodities. This specification includes the distribution of these endowments, a statement of who owns what. Some of these endowments are unproduced natural resources which yield services over future time periods in the model. Other endowments can be conceptualized as capital goods, as produced means of production.

In the model, markets clear once, at the beginning of time. The commodities traded in markets include promises to deliver specified commodities at specified dates in the future, perhaps contingent on the state of the world. In the full model, one can purchase now a contract to have delivered any specificied commodity at any date in the future. So market-clearing in this model implies that plans are pre-coordinated. No room exists for expectations about the future to be disappointed.

Where do the capital goods in existence at the beginning of time in the Arrow-Debreu model come from? Presumably, they were manufactured in a prior time period not formally modeled. Some entrepreneurs, with plans and expectations held before the start of time, were directing production toward the time in the model. The initial capital goods embody those plans and expectations. But, in the model, the relative quantities of those goods is arbitrary. For example, an endowment can be in excess supply, with an equilibrium price of zero. This arbitrariness seems to imply that those unmodeled past expectations and plans can be mistaken.

So the Arrow-Debreu model describes a world in which agents have made mistakes in the past and some plans made in the past have been toppled. But the plans and the expectations made now will be found correct forever in the future. It is a model in which agents must both have correct expectations and in which they can have mistaken expectations.

You can find in the literature comments on Joan Robinson's puzzlement at being given the Arrow-Debreu of intertemporal equilibrium as the answer to her question about what is the general neoclassical theory of value. She complained that she wanted to have a model that she could make stand up before she knocked it down. And she would ask, if the economy was not in equilibrium yesterday, why would you expect it to be in equilibrium tomorrow?

Suppose one models the economy as starting with an arbitrary set of capital goods, that is, with a short run model. Perhaps part of the data should then be the expectations, some of which are being disappointed, with which those capital goods were manufactured.

• Gerard Debreu, Theory of Value: An Axiomatic Analysis of Economic Equilibrium, Yale University Press, 1959.

Sraffa or Marx?

I think I'll demonstrate that a range of views exists on the relation of the ideas of Sraffa and the Sraffians to the ideas of Marx. I think Ben Fine is expressing a fairly traditional Marxist view:

"Volume III of Capital addresses the distribution of surplus value but not in the simple sense of who gets how much. Note, however, that even this superficial interpretation presupposes, correctly in line with Marx's method, that the surplus value has to be produced before it is distributed. If, though, the distribution is simply interpreted as a cake-division exercise, as in the (neo-)Ricardian (or Sraffian) interpretations, then the concepts of surplus value and profit collapse and the former simply servers as a superfluous accounting exercise. In contrast, Marx deals with distribution of surplus value as a refinement of value. The results of the previous Volumes are brought together and used to develop more complex and concrete categories in terms of the economic processes by which production and exchange are integrated." -- Ben Fine (2001)

I am amused by Alan Freeman (who is the co-inventor of a new mathematical economics interpretation of Marx):

"Variant b of step 7: the Value System is primary
We know values are primary because of all Marx's qualitative arguments concerning the nature of exchange, because of a wealth of empirical evidence, and because of many philosophical and socio-political arguments on the role of human labour. Let us therefore take the 'primary causal' role of value as an axiom. Let us postulate that, against substantial evidence from the texts, Marx unconditionally asserted that the value of every commodity is determined without the mediation of money.

This leads to a veritable garden of forking paths. We can discern at least the following variants

Variant 7b.I: philosophico-mystical
The determination of price by value takes place behind our backs. It is part of the internal workings of the capitalist system which are ever so mysterious and can only be understood by reciting das Kapital six times before breakfast and joining my group. There is no such thing as the transformation problem and it doesn't matter that the figures don't add up, but you wouldn't understand that because you are a bourgeois revisionist.

Variant 7b.II: pseudo-dialectical
The determination of prices takes place as the Sraffians describe it, and the determination of values takes place as Marx describes it. This can only be understood by reciting das Kapital twelve times before breakfast and joining my study circle. It is true that the figures don't add up, but that is because capital is inherently contradictory, and you should learn to live with it. You can't understand that because you haven't read Hegel.

Variant 7b.III: fake materialist
As Marx explains, the forces of production determine everything. This as Plekhanov explains is the basis of historical materialism. What Marx meant by the determination of value by labour time was the determination of value by technology as you will realize if you read Sraffa and buy my newspaper. The figures do add up. You do not understand this because you are not a worker." -- Alan Freeman (1996)

This is one Sraffian view:

"Therefore, to grasp the essence of the Sraffian revolution it is necessary to define more clearly Sraffa's position in regard to the classical and Marxian approach. The point is that a 'classical-Marxian approach' does not exist, if it is true that Marx considered himself to be a critic of Ricardo and Smith. In other words, was Sraffa a 'neo-Ricardian', as some orthodox Marxists and some orthodox neoclassical economists have argued? Or was he a Marxist, as many of his followers maintained?

The level of abstraction of Sraffa's model is defined, albeit with the usual cryptic conciseness, by Sraffa himself, in the subtitle of his book: Prelude to a Critique of Economic Theory. It is the same as the first chapter of Capital, which is entitled 'The commodity', and which is the real prelude to the Critique of Political Economy. In that chapter, Marx tackled the analysis of the commodity and its value, and laid down the theoretical bases of all his successive work; he attacked the 'vulgar economists', whom he accused of looking for the explanation of the value of goods in exchange relations; and he linked himself explicitly to Ricardo in looking for it, instead in the production sphere.

However, even though Marx had clearly pointed out that value is a social phenomenon, Ricardo's influence induced him to determine the value of goods by ignoring the capitalist form in which they are produced. Value, according to the theory set out in the chapter on 'Commodities', depends solely on the quantity of labour employed in its production, its social determination being reduced to the way in which society allocates working activity among the various industries. In other words, value is not influenced by the way in which production is socially structured, nor by the way in which the social classes face each other in the production sphere. Thus, for example, the value of the social product does not depend on the way in which the product is distributed - a clear reminder of the Ricardian claim to measure value by making it independent from the variations in distribution. It is obvious that, in order to reach such a result, Marx had to rely on a high level of abstraction, and it is almost paradoxical that a book on 'capital' opens with a chapter in which capital is ignored. Now, it is almost as if Sraffa had rewritten that chapter, trying to reach the maximum level of abstraction, from returns to scale, growth, disequilibrium adjustments, even from the specific institutional structure to which the type of capitalist set-up may conform historically - exactly as in the chapter on 'Commodities'. However, he made it clear once and for all that the only thing which it is impossible to ignore in determining the value of commodities produced in a capitalist economy, is the fact that they are produced in capitalist conditions: that it is meaningless to abstract from the wage, as Marx did in that chapter. In other words, Sraffa took Marx seriously in his treatment of value as a social phenomenon - so seriously that he distanced himself from Ricardo more than Marx did himself.

It seems possible to conclude that, while on the subject of the determination of profit, Sraffa's theory does not bring to light any basic analytical differences between Ricardo and Marx, on the subject of value his work can be read in only one way: the Prelude to a Critique of Economic Theory seems to us just like a first chapter of Capital which Marx would have written if he had been a little less Ricardian and a little more Marxist." -- Ernesto Screpanti and Stefano Zamagni (2005, p. 445-446)

References

• Fine, Ben (2001). "The Continuing Imperative of Value Theory", Capital and Class, V. 75: 41-52
• Freeman, Alan (1996). "The Psychopathology of Walrasian Marxism", in Marx and Non-Equilibrium Economics (Edited by Alan Freeman and Guglielmo Carchedi), Edward Elgar
• Screpanti, Ernesto and Stefano Zamagni (2005). An Outline of the History of Economic Thought, Second Edition, Oxford University Press

Expertise Devalued At Wikipedia

I have some knowledge about how 20th century English-speaking mathematical economists have treated the labor theory of value. The transformation problem is a central focus of discussions of whether the theory is internally consistent.

I have previously mentioned my dissatisfaction with the evolution of the Wikipedia entry on the Labor Theory of Value. But consider the transformation problem.

At one point, (29 July 2005) the entry was mainly written by Mario Ferretti. Although he describes neither the New Interpretation nor the Temporal single system approach adequately for my taste, I have no doubt about his expertise on linear production models. He's written, for example, "The Neo-Ricardian Critique: An Anniversary Assessment", (University of Rome seminar, 26 October 2004). This is not the post for me to go into where I agree and disagree.

Look at the current entry on the transformation problem. Basically, Ferretti's work has been stripped out.

One could talk about the difficulties of including mathematics in work that will be read by a general audience. And some supposed Marxists are impatient with the transformation problem - I also find some just don't understand the topic. But, once again, the evolution of this article doesn't seem right to me.

I might as well mention an article I'm happier with. I have some disagreements with Radek on General Equilibrium. But overall I think the evolution of this article, primarily with his rewriting, has resulted in a better organized and better referenced article than my previous attempt.

I'm not a registered Wikipedia user. If I were, I would probably vote for Aaron Swartz.

An Ideal Of Discussion

“Socrates: I imagine, Gorgias, that you, too, have taken part in many discussions and have discovered in the course of them this peculiar situation arising: people do not find it easy by an exchange of views to arise at a mutually satisfactory definition for the subjects under discussion, and in this way bring the argument to an agreeable end. Rather, when they disagree on any point, and one declares the other to be guilty of incorrect or vague statements, they grow angry and imagine that everything that is said proceeds from ill will, not from any concern about the matters under discussion. Some of these arguments end most disgracefully, breaking up in mutual vituperation to such an extent that the bystanders are annoyed at themselves for having become auditors of such people. Now why do I say this? Because at the moment you seem to me to be making statements which do not follow from, and are not consistent with, what you first said about rhetoric. I hesitate, therefore, to embark on a refutation in the fear that you may imagine that I am speaking, not with a view to illuminating our subject, but to discredit you. Now if you are the sort of person I am, I shall gladly continue the questions and answers; if not I shall let them go. And what sort of person am I? One of those who are happy to be refuted if they make a false statement, happy also to refute anyone else who may do the same, yet not less happy to be refuted than to refute. For I think the former a greater benefit, in proportion it is of greater benefit to be oneself delivered from the greatest harm than to deliver another. No worse harm, it is true, can befall a man than to hold wrong opinions on the matters now under discussion between us. If, then, you declare yourself to be such a person as I am, let us continue the discussion; but if you think we ought to let it go, let us at once dismiss it and close the interview.” -- Plato, Gorgias (457-458)

Sraffian Triumphalism

This is from the sort of professional literature I read:

"Convincing conditions of sufficient generality which ensure a well-behaved technology have not been proposed. We therefore should not seek for those special assumptions under which neoclassical theory might work but for a different theory of distribution and employment altogether. Keynesian modifications of neoclassical full employment theory are still being discussed by the mainstream economists. I am reminded here of the Ptolemaic world system: if planets do not move in circles although circles are thought to describe their behaviour, epicycles are invented. Why do we not turn to the ellipses straightaway, as Kepler did, forgetting about the circles and the harmonies of spheres, which would mean in our context: why do we not turn to a theory of value which does not presuppose full employment as the natural state?" --Bertram Schefold (1990, p. 383)

"...since both groups of versions of marginalist equilibrium theory - the long-period versions and the neo-Walrasian versions - encounter what appear to be radical and insurmountable difficulties, one must conclude that at present there is no defensible neoclassical theory (in the sense of explanation) of prices and distribution. The onus is on the neoclassicals to show that this is not so. Unless and until they succeed, it seems reasonable to turn to different, non-neoclassical approaches to value and distribution (and employment and growth)." --Fabio Petri (1999, p. 55-56)

"A growing number of contemporary economists recognizes the need to overcome the cowardly protectionism through which the academy endeavors to defend itself from criticism deriving from the facts as well as from the innovations that originate from other social sciences." --Ernesto Screpanti and Stefano Zamagni (2005, p. 513)

"Sraffa ... provided a logically self-consistent solution to the problem of exchange values to which Ricardo - and, following him, Marx - had given an insufficient answer, constituting one of the causes that led to the abandonment of the classical framework and the rise of the marginalist approach; and he showed that to this problem the marginalist approach offered a solution that was only apparently more 'scientific', but that in reality was vitiated in its foundations in so far as the theory of value and distribution is concerned." --Alessandro Roncaglia (2006, p. 460)

References

• Petri, Fabio (1999). "Professor Hahn on the 'neo-Ricardian' Criticism of Neoclassical Economics", in Value, Distribution and Capital: Esays in Honour of Pierangelo Garegnani (edited by Gary Mongiovi and Fabio Petri), Routledge
• Roncaglia, Alessandro (2006). The Wealth of Ideas: A History of Economic Thought, Cambridge University Press
• Schefold, Bertram (1990). "Joint Production, Intertemporal Preferences and Long-Period Equilibrium: A Comment on Bidard", Political Economy, Studies in the Surplus Approach, V. 6: pp. 139-163 (republished in Normal Prices, Technical Change and Accumulation, by Bertram Schefold, Macmillan)
• Screpanti, Ernesto and Stefano Zamagni (2005). An Outline of the History of Economic Thought (Second Edition), Oxford University Press

Disconnect Of Monetary And Price Theory In Neoclassical Economics

Friedrich Hayek, John Maynard Keynes, and Gunnar Myrdal found themselves in the 1930s addressing the same problem in price theory. I refer not to the practical difficulty in accounting for the Great Depression, but the lack of integration between monetary theory and the theory of value and distribution:

Here's Hayek:

"What I complain of is not only that this theory [Fisher's monetarism] in its various forms has unduly usurped the central place in monetary theory, but that the point of view from which it springs is a positive hindrance to further progress. Not the least harmful effect of this particular theory is the present isolation of the theory of money from the main body of general economic theory.

For so long as we use different methods for the explanation of values as they are supposed to exist irrespective of any influence of money, and for the influence of money on prices, it can never be otherwise." (Hayek 1935, p. 2-3)

Here's Keynes:

"So long as economists are concerned with what is called the Theory of Value, they have been accustomed to teach that prices are governed by the conditions of supply and demand; and, in particular, changes in marginal cost and the elasticity of short-period supply have played a prominent part. But when they pass in volume II, or more often in a separate treatise, to the Theory of Money and Prices, we hear no more of these homely but intelligible concepts and move into a world where prices are governed by the quantity of money, by its income-velocity, by the velocity of circulation relative to the volume of transactions, by hoarding, by forced saving, by inflation and deflation et hoc genus omne; and little or no attempt is made to relate these vague phrases to our former notions of the elasticities of supply and demand. If we reflect on what we are being taught and try to rationalize it, in the simpler discussions it seems that the elasticity of supply must have become zero and demand proportional to the quantity of money; whilst in the more sophisticated we are lost in a haze where nothing is clear and everything is possible. We have all of us become used to finding ourselves sometimes on the one side of the moon and sometimes on the other, without knowing what route or journey connects them, related, apparently, after the fashion of our waking and our dreaming lives.

One of the objects of the foregoing chapters has been to escape from this double life and to bring the theory of prices as a whole back to close contact with the theory of value. The division of Economics between the Theory of Value and Distribution on the one hand and the Theory of Money on the other hand is, I think, a false division..." (Keynes 1936, p. 292-293)

And here's Myrdal:

"It is a peculiarity of all systematic treatises on orthodox economic theory that there is no inner connexion and integration of monetary theory with the central theory of prices. Usually the monetary theory is only a rather loose appendix to the theory of price formation. The central economic problems - according to the classical theory, those of production, of barter-exchange and of distribution - are treated, without exception, as problems of exchange value, or, in other words as problems of relative prices. Obviously, by regarding the central economic problems in this way one entirely detaches their fundamental treatment from any monetary considerations." (Myrdal 1939, p. 10)

I put the above quotes into my latest iteration of this paper.

I think that, despite later work by, for example, Don Patinkin or Eugene Fama or with models of overlapping generations, mainstream economists still fail to satisfactorily integrate monetary and price theory. I an influenced in this view by a Frank Hahn paper and comments of various Post Keynesians, such as Paul Davidson.

References

• Hahn, F. H. (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.
• Hayek, F. A. (1935). Prices And Production, Second Edition, London: George Routledge and Sons.
• Keynes, J. M. (1936). The General Theory of Employment Interest and Money, New York: Harcourt, Brace and Co.
• Myrdal, G. (1939). Monetary Equilibrium, New York: Sentry Press.

Two Papers Just Published

Han and Schefold
When I listed some papers that I guess contain empirical or applied evidence of Sraffa effects, Han and Schefold's paper was only in draft form. It has now been published:

• Han, Zonghie and Bertram Schefold (2006). "An Empirical Investigation of Paradoxes: Reswitching and Reverse Capital Deepening in Capital Theory", Cambridge Journal of Economics, V. 30: 737-765.

Walker Reviews Petri
Donald Walker reviews Petri's 2004 book. Walker finds Petri's criticisms of general equilibrium theory, including the Arrow-Debreu model of intertemporal equilibrium well-taken. But he is more skeptical about Petri's proposed solution, adopting a modern classical long period approach. Does anybody know of other reviews of this book?

• Walker, Donald A. (2006). "A Review of General Equilibrium, Capital, and Macroeconomics: A Key to Recent Controversies in Equilibrium Theory, by Fabio Petri" History of Political Economy: 562-565.

Classic Cowles Commission Conference On Activity Analysis

Gabriel Mihalache notes that the Cowles Commission has made available for free downloading the 1951 proceedings of Activity Analysis of Production and Allocation. I have never read this before, but I have seen frequent references to this conference. All sorts of interesting people participated. For example, you can find contributions from Dantzig, Georgescu-Roegen, Samuelson, Arrow, Herbert Simon, and Morgenstern. I once had the privilege of working with a professor who, when an undergaduate at Columbia, learned linear algebra in a class taught by Koopmans.

I've previously downloaded various papers available from the Cowles Commission. They sponsored so much important work, I'm not even going to try to mention selections.

If you want to look at dead tree editions, another classic contribution from that milieu is Robert Dorfman, Paul A. Samuelson, and Robert M. Solow's Linear Programming and Economic Analysis.

Luigi L. Pasinetti's Lectures on the Theory of Production is a later and great textbook in the Sraffian tradition that contains something of a response to Dorfman, Samuelson, and Solow. (Amazon has mischaracterized it.)

I Couldn't Make This Up

Right on cue, Greg Mankiw exemplifies the intellectual impoverishment of mainstream teaching in economics.

I hadn't previously read Gabriel Mihalache's blog. I'm inclined to be more tolerant of this nonsense, given this acknowledgement. (Is it clear that the equations in this post are clickable?)

Based on the comments on this post, I may have more readers than the amount of comments suggests.

Foley On Incoherence Of Marginalist Project

Here are two long quotations from Duncan K. Foley, "Value, Distribution and Capital: A Review Essay", Review of Political Economy, V. 13, N. 3, 2001.

"The strongest and most important point that has come out of the classical critique of marginalism and neoclassical economics is its refutation of the capital scarcity theory of the rate of profit.

The notion that the profit rate can be coherently viewed as being determined by a 'marginal product of capital' given by technology and input availabilities is one of the more confused and ideologically muddied chapters in the history of economics. A cost-minimizing firm facing a wage, cost of capital, and prices of capital goods determined by markets will adapt its relative use of labor and capital to those prices. The value of capital goods is a rational and appropriate measure of capital input to the firm when the prices of capital goods are determined by market forces outside the firm's control. When a cost-minimizing firm faces a range of technical methods of production that approximate a smooth continuum of capital-labor ratios, cost-minimization entails setting the marginal value product of each input equal to its price. (There is, of course, vigorous debate in all schools of economic thought over how well the assumptions of a cost-minimizing firm facing market prices for inputs fits the behavior of real capitalist firms.) In this scenario, however, it is the wage and the cost of capital that determine the marginal value products of labor and capital, not the other way around.

The vision of the marginalists and their neoclassical followers was that this uncontroversial theorem of cost minimization could somehow be transformed into a theory of the wage and profit rate, and hence into a theory of distribution. Their hopes of accomplishing this transformation stemmed from their anthropomorphic vision of the economy and its markets as analogous to the allocation of scarce resources by a single decision maker with a well-defined objective function, who, at least under the assumptions of concavity of the objective function and convexity of resource constraints, can impute shadow prices (Lagrange multipliers) to the resources. This program, despite firing the imaginations of many talented economists, has never managed to disentangle the problems of aggregation of disparate preferences, treatment of time and information, definition of resource limitations, and dynamic stability of market clearing that are inherent in carrying it out to arrive at the robust, unified, and transparent account of distribution it sought. Economics owes a particular debt of gratitude (which it shows precious few signs of recognizing, to tell the truth) to Sraffa and his followers for their dogged insistence on bringing to light the ramifying incoherence of this marginalist project...

...The Cambridge capital debate centered initially on one aspect of this tangle of confusions, the neoclassical hope that the value of capital goods would somehow behave like a single scarce input in equilibrium. The interchanges of this debate, including an important paper by Garegnani (1970), showed unambiguously that the neoclassical construction would work in general only under assumptions on production so stringent as to amount to the assumption of a single capital good. Other work related to the Cambridge debate, for example that of Luigi Pasinetti (1974) and Stephen Marglin (1984), also underlined the other side of Sraffa's critique, the necessity of taking some distributional variable, either the wage or profit rate, as the boundary condition in production models, rather than stocks of individual capital goods. This path clarifies the real relationship between input prices and marginal productivities (if indeed they exist), which is that input prices determine marginal productivities through the cost-minimizing choice of technology.

Neoclassical economists have a hard time keeping their minds clear on this point. They are distracted by the fact that it is possible to embed a Sraffian production system in a general equilibrium model with given stocks of inputs and calculate equilibrium prices and rates of return without any reference to the value of capital goods. They read this model as supporting the scarcity theory of profit rates, although it fact it only reproduces its own assumption of the full employment of all inputs except those that have zero prices in equilibrium. The equilibrium allocation can equally well be viewed as one in which given input prices determine cost-minimizing choices of technique that happen to be compatible with arbitrarily given supplies of inputs. In the absence of a compelling dynamic theory that shows how the market might find these prices, which neoclassical theory has pretty well given up hope for, Sraffa's critique carries the day.

It would have been a good thing if the Cambridge capital controversy had managed to drive a stake through the heart of the scarcity theory of the rate of profit, but it hasn't. I think this has been because the classical critique tells economists what they shouldn't do (assume full employment and market clearing in order to determine prices) but doesn't tell them very clearly what they should do as an alternative, either at the purely theoretical level or in econometric studies."

Those who have followed this series of posts with attention will understand, to some extent, Foley's statement that "input prices determine marginal productivities through the cost-minimizing choice of technology."

Marginal Productivity Not A Theory Of Distribution (Part 4)

6.0 Conclusions
Let me recap. In the first part, I laid out a model of the economy. In the second part, I derived marginal productivity conditions and noted equations for equilibrium prices. In the third part, I outlined how to construct the factor price frontier, and asserted that that analysis is equivalent to marginal productivity analysis when they both are applicable.

Marginal productivity is not a theory of income distribution. I consider it an analysis, like the construction of the factor price frontier, of the choice of technique.

Consider two nested models of production. In the first, the coefficients of production are fixed. Equilibrium conditions in the production model would consist of the system of equations arising from the condition that no pure economic profit exists. These equilibrium conditions and the choice of the numeraire determine, with one degree of freedom, the wage, the interest rate, and prices. No marginal productivity conditions exist, for interior solutions, in this model.

In the more general model, one allows coefficient of production to vary, instead of taking them as given parameters. Marginal productivity equilibrium conditions arise for each coefficient of production. The model still has a single degree of freedom.

If one took the wage as given from outside, the model shows how firms choose processes in which the present value of the marginal product of labor is equal to the wage. The wage is not determined by the value of the marginal product of labor. Rather, it is more appropriate to say the value of the marginal product of labor is determined by the wage.

Samuelson, however, when calculating the marginal product of labor, noted that the interest rate could be taken as given instead. The factor price frontier shows possible distributions of income, given the possible coefficients of production. But marginal productivity cannot determine a specific location on the frontier. By taking an important distributive variable, the interest rate, as given, an economist, in theory, can determine the location on that frontier and the value of the marginal product of labor. (Notice that in case of reswitching, one cannot even map from the chosen technique, out of the set of all possible techniques, to the wage and the value of the marginal product of labor.)

Interestingly enough, Walras understood that marginal productivity is not a theory of income distribution:

"[The theory of marginal productivity] introduces into the problem of production a system of equations...in which the number of equations is equal to the number of coefficients of production and in which these coefficients are represented as unknowns...[The theory] makes possible a definitive criticism and refutation of the English theory of rent, by showing that the consideration of marginal productivity is relevant to the determination of coefficients of production, but is not relevant to the determination of the price of services." (Leon Walras, Elements of Pure Economics, Appendix III, (Translated by William Jaffe), 1954.)

Occasionally, you may see somebody saying that changes in technology or relative physical productivity are what drives changes in income distribution, at least in theory:

"To a good neoclassical economist, the statement that the relative price of a factor of production--like the labor of the elite top 1% of America's wage and salary distribution--has risen is the same thing as the statement that the relative productivity of that factor of production has risen." -- Brad DeLong

Does DeLong get neoclassical economics correct?

For now, that's all.

Marginal Productivity Not A Theory Of Distribution (Part 3)

5.0 The "Factor Price Frontier"
This post builds on an analysis of marginal productivity in an economic model.

One can rewrite Equation 17 as Equation 20:

(20)

Or:

(21)

The matrix inverse exists for all r between zero and some maximum, inclusive, for each technique if both goods are basic and the economy is viable.

Let e be a column vector denoting the numeraire. Post-multiplying, one obtains:

(22)

Hence, the wage-rate of interest curve for a given technique is:

(23)

This is a downward sloping curve in w-r space, and it cuts both axis.

One can plot the wage-rate of interest curve for each technique. A technique is a set of coefficients of production, a0 and A. The coefficients of production a01, a11, a21, for the wheat industry lie on the unit-isoquant production function for wheat. Likewise, the coefficients of production a02, a12, a22, for the corn industry lie on the unit-isoquant production function for corn. Given a common numeraire used in plotting each curve, these curves fall in the same space, as in this example.

Consider the frontier formed by the outer envelope of all these curves. That is, for a given interest rate, the wage on the factor price frontier is the maximum wage from out of the wages on all the wage-interest rate curves at that rate of interest. Similarly, for a given wage, the factor price frontier shows the maximum rate of interest. And the coefficients of production used in constructing wage-rate of interest curves on the frontier for, say, a given rate of interest can be chosen by competitive profit-maximizing (or cost-minimizing) firms paying the given rate of interest and the equilibrium wage thereby determined.

The coefficients of production found in this analysis of the factor price frontier are the same as those that arise in the marginal productivity analysis. When the production functions are well-behaved and continuously differentiable, no point on the frontier is a switch point, and the optimizing coefficients of production vary continuously with the interest rate. Since the two analyses, of the factor price frontier and of marginal productivity, are equivalent, the wage is equal to the present value of the marginal product of labor for every point on the frontier.

The analyses of the factor price frontier, however, can be used to determine the cost-minimizing coefficients of production when the industry-specific production functions are not smooth. That is, the factor price frontier can be constructed when the set of possible coefficients of production is discrete. In this sense, the analysis of the factor price frontier, inasmuch as it applies to both continuously differentiable production functions and a discrete model of production, is more general than an analysis confined to cases where derivatives exist. (This is not a radical conclusion; post (Second World) war mainstream economists do not confine themselves to calculus-based results; they have long used topological arguments.)

Marginal Productivity Not A Theory Of Distribution (Part 2)

3.0 Marginal Conditions Arising From Profit Maximization

Previously, I outlined a model on an economy in which wheat and corn are produced in yearly production cycles. If the profit-maximizing competitive firm produces wheat, it must solve the following mathematical programming problem:

Given w, P(1), P(2)
Choose a01, a11, a21 to
To Maximize
(6)
Such that

The solution gives the following marginal productivity equations:

(7)

(8)

(9)

In competitive equilibrium, the price of every input is equal to the value of the marginal product for each product.

Pure economic profit must be nonnegative, for if it were negative the firm would have chosen not to produce at all. It works out that pure economic profit cannot be positive either. Equation 10 states that pure economic profit is zero in the wheat-producing industry:

(10)

An analogous set of three marginal productivity equations and a zero economic profit condition arises from analyzing corn-producing firms.

4.0 Interest Rates In Steady States
Consider the following definitions:

(11)

(12)

These are known as the own rates of interest of wheat and corn, respectively.

In a long run equilibrium, the quantities of wheat and corn inputs are not given. Assume they have been adjusted such that relative spot prices of corn, wheat, and labor remain unchanged from year to year. Almost all economists up until the late 1920s, as far as I am aware, thought of this constancy of relative prices, or an equivalent condition, as a defining property of long run equilibrium.

(It was about then that short-run notions of temporary and intertemporal equilibrium were introduced, and the economists' conceptions became more complex. Mistakes were made by the early neoclassicals. Walras thought he could take endowments of produced goods as given, but still consider an equilibrium with steady-state prices. Others mistakenly wanted to take the quantity of "capital" as given, but have its form, in terms of composition of wheat and corn, be endogenously determined.)

Certainly, we need to introduce some condition relating industries. Anyways, If you think about the condition of stationary spot prices, you will see that if no pure economic profits are possible, relative forward and relative spot prices must be equal:

(13)

Otherwise, an agent could buy or sell wheat or corn in the forward market for a year and make a pure economic profit on the spot market at the end of the year.

Therefore:

(14)

Or:

(15)

where r is the common own rate of interest for all commodities. It is not a parameter of some unobservable utility function, although it may become equal to some such parameter in equilibrium in special cases. Such a special case could be created by closing this model with utility-maximization by a infinitely-lived representative agent.

The condition of unchanging relative spot prices allows us to remove P(2) from our equations. We have:

(16)

The condition of no economic profit gives the following system of two equations:

(17)

where I have dropped the time index on the price vector as being no longer needed. The marginal productivity equations become:

(18)

(19)

So we have a system of eight Equations - the two price equations (17), and the six marginal productivity equations (18) and (19). There are ten choice variables to be determined by the model:

• The wage w
• The common own rate of interest r
• Two prices, P = (p1, p2)
• Six production coefficients a0 and A

I assume it's obvious that the above model generalizes to an n good model. There are two degrees of freedom. One is closed by selecting the numeraire.

There is no equation equating r to some mystical, mythical, marginal product of capital.

In the next part, I outline the construction of the so-called factor price frontier.

Marginal Productivity Not A Theory Of Distribution (Part 1)

1.0 Introduction
What is held constant when defining the marginal product of labor? A correct answer, in the context of a neoclassical long period theory, is the interest rate. This is part of the answer Paul Samuelson gave to Joan Robinson in the early 1960s when she visited MIT. My exposition of this answer is indebted to an explanation of Frank Hahn's.

2.0 The Model Economy
Consider an economy that produces wheat and corn in yearly production cycles. At the start of the year, a firm producing wheat buys inputs of labor, corn, and wheat. The production process takes a year. The firm's output of wheat is available at the end of the year. Likewise, a firm buying corn purchases inputs of labor, corn, and wheat at the start of the year and has outputs of corn at the end of the year.

In neoclassical theory, goods are differentiated by their availability at different times. So there will be five prices to consider for a yearly production cycle:

• w, the wage, which I arbitrarily assume here is paid at the beginning of the year.
• P(1) = (p11, p21), the row vector of spot prices of wheat and corn for immediate delivery at the beginning of the year.
• P(2) = (p12, p22), the vector of forward prices of wheat and corn for delivery at the end of the year. For example, p12 is the price that must be paid at the start of the year for a contract in which the buyer will receive a bushel of wheat at the end of the year.

Now consider a competitive firm producing wheat. Assume the firm faces a constant returns to scale production function:

(1)

where Q1 is the output of bushels wheat, L01 is inputs of labor, X11 is inputs of wheat, and X21 is inputs of corn. Since I assume Constant Returns to Scale, I can separate questions of scale and relative proportions:

(2)

where

(3)

(4)

(5)

I make the usual assumption that diminishing marginal returns exist for each input in both production functions.

In the next part, I derive the marginal productivity equations for a competitive wheat-producing firms.