Krishna Bharadwaj, A Sraffian Economist

A web site devoted to freeing Binayak Sen reprints an article from The Telegraph-Calcutta. This article draws a parallel between Sudha Bharadwaj and Binayek Sen.

I don't know anything about B. Sen or S. Bharadwaj. But if Arundathi Roy and Amartya Sen are protesting your arrest, I assume you should be freed. The People's Union for Civil Liberties, at first glance, sounds like a fine organization.

Apparently Sudha Bharadwaj is the daughter of Krishna Bharadwaj. I happen to have read her review of Sraffa's book - or at least the extracts that Harcourt and Laing (1971) reprint. As I recall from somewhere, she took a couple of years to write this review. When given Sraffa's book to review, she felt obligated to reread Adam Smith and David Ricardo. This was a perceptive understanding.

I first became aware of Krishna Bharadwaj's work, though, by stumbling upon her 1989 collection of essays. These are reprinted from such journals as Australian Economic Papers, the Cambridge Journal of Economics, and others. I found these essays quite good. I later read her 1978 lecture and the 1990 conference volume she co-edited with Bertram Schefold. Until the last few years, that conference seems to have been the most thorough assessment of Sraffa's contributions - not that economists such as Pierangelo Garegnani and Paul Samuelson could agree. She also has some applied work which I haven't read.

References

• Krishna Bharadwaj (1963) "Value Through Exogenous Distribution", Economic Weekly (Bombay), 24 August: 1450-1454
• Krishna Bharadwaj (1978) Classical Political Economy and the Rise to Dominance of Supply and Demand Theories, Orient Longman
• Krishna Bharadwaj (1989) Themes in Value and Distribution: Classical Theory Reappraised, Unwin Hyman
• Krishna Bharadwaj and Bertram Schefold (editors) (1990) Essays on Piero Sraffa: Critical Perspectives on the Revival of Classical Theory, Unwin Hyman
• G. C. Harcourt and N. F. Laing (editors) (1971) Capital and Growth, Penguin

Reswitching With Smooth Production Functions

I cite authority:

"Something precious I gained from Robinson's work and that of her colleagues working in the Sraffian tradition. As I have described elsewhere, prior to 1952 when Joan began her last phase of capital research, I operated under an important misapprehension concerning the curvature properties of a general Fisher-von Neumann technology.

What I learned from Joan Robinson was more than she taught. I learned, not that the general differentiable neoclassical model was special and wrong but that a general neoclassical technology does not necessarily involve a higher steady-state output when the interest rate is lower. I had thought that such a property generalized from the simplest one-sector Ramsey-Solow parable to the most general Fisher case. That was a subtle error and, even before the 1960 Sraffa book on input-output, Joan Robinson's 1956 explorations in Accumulation of Capital alerted me to the subtle complexities of general neoclassicism.

These complexities have naught to do with finiteness of the number of alternative activities, and naught to do with the phenomenon in which, to produce a good like steel you need directly or indirectly to use steel itself as an input. In other words, what is wrong and special in the simplest neoclassical or Austrian parables can be completely divorced from the basic critique of marginalism that Sraffa was ultimately aiming at when he began in the 1920s to compose his classic: Sraffa (1960). To drive home this fundamental truth, I shall illustrate with the most general Wicksell-Austrian case that involves time-phasing of labor with no production of any good by means of itself as a raw material.

As in the 1893-1906 works of Knut Wicksell, translated in Wicksell (1934, Volume I), let corn now be producible by combining labor yesterday, labor day-before-yesterday, etc):

Q_t = f(L_t-1, L_t-2, ..., L_t-T) = f(L)                                                     (1)

Q = f(L₁, L₂, ..., L_T) in steady states                                              (2)

    = L₁ f(1, L₂/L₁, ..., L_T/L₁) 1st^o-homogeneous and concave      (3)

    = L₁ (df(L)/dL₁) + ... + L_T (df(L)/dL_T), Euler's theorem            (4)

df/dL_j = f_j(L), d²f/(dL_i dL_j) = f_ij(L) exist for L ≥0                         (5)

f_j > 0, (z₁, ..., z_T)[f_ij(L)](z₁, ..., z_T)' < 0 for z_j ≠ b L_j > 0                (6)

Nothing could be more neoclassical than (1)-(6). If it obtained in the real world, a Sraffian critique could not get off the ground.

Yet it can involve (a) the qualitative phenomena much like 'reswitching', (b) so-called perverse 'Wicksell effects', (c) a locus between steady-state per capita consumption and the interest rate, a(i, c) locus, which is not necessarily monotonically negative once we get away from very low i rates. This cannot happen for the 2-period case where T = 2. But for T ≥ 3, all these 'pathologies' can occur, and there is really nothing pathological about them. No matter how much they occur, the marginal productivity doctrine does directly apply here to the general equilibrium solution of the problem of the distribution of income.

Remarks. What eternal verities do always obtain, even when corners in the technology make derivatives [dQ_j/dL_j, dQ_j/dQ_ij] be somewhat undefined? Always, it remains true:

(a) To go from an initial sub-golden-rule steady state to a maintainable golden-rule steady state of maximal per capita consumption, must involve for society a transient sacrifice of current consumptions ('waiting' or 'abstinence' a la Senior, Böhm, and Fisher!).

(b) For non-joint-product systems, there is a steady-state trade-off frontier between the interest rate and the real-wage (expressed in terms of any good).

This monotone relation between (W/P_j, i) was obscurely glimpsed by Thunen and other classicists and by Wicksell and other neoclassicists. But the factor-price trade-off frontier did not explicitly surface in the modern literature until 1953, as in R. Sheppard (1953), P. Samuelson (1953), and D. Champernowne (1954). One can prove it to be well-behaved for (1)-(3), or any convex-technology case, by modern duality theory. Before Robinson (1956), I wrongly took for granted that a similar monotone-decreasing relation between ( i, Q/(L₁ + ... + L_T) ) must also follow from mere concavity - just as does the relation -d²C_t+1/(dC_t)² = di/dC_t) > 0. But this blythe expectation is simply wrong! I refer readers to my summing up on reswitching: Samuelson (1966).

I realize that there are many economists who tired of Robinson's repeated critiques of capital theory as tedious and sterile naggings. I cannot agree. Beyond the effect of rallying the spirits of economists disliking the market order, these Robinson-Sraffa-Pasinetti-Garegnani contributions deepen our understanding of how a time-phased competitive microsystem works." -- Paul A. Samuelson (1989) "Remembering Joan" in Joan Robinson and Modern Economic Theory (ed. by George R. Feiwel), New York University Press.

(I have changed some of the symbols above.) I've noted before comments from Samuelson in papers that have made claims much the same as above.

Students at Schools With Interesting Economists

E. Roy Weintraub and Edwin Burmeister are two Duke economists I find worth reading. Here are some Duke students:

Duke, Quaterfinals at Ithaca, 18 May 2008

Below are some Notre Dame students, except for the upper left. Thos are Syracuse University students. I did not ask any Notre Dame fans what they thought of their administration's shameful treatment of some of their economists or talk about an on-line petition.

Notre Dame, Quaterfinals at Ithaca, 18 May 2008

I do not have any photos of U-Mass, Amherst, students, although I did go to Syracuse's last home game of the regular season.

Days Late And A Penny Short

I find from Crooked Timber another interesting blog - Nancy Folbre's Care Talk. I don't know how much, if any, of Folbre's work I've read. She is quite prominent, if I understand correctly, as a developer of feminist economics.

Elsewhere

Some feminists have started blogging on economics: Kathy G. and Allison. Kathy G. doesn't seem to draw on Feminist Economics. I don't know about Allison.

I recently stumbled on the blog of an economist at Cambridge, UK.

As I understand it, this blog is from Edward Nell's son. I might as well give a quote from Edward Nell:

"Joan Robinson started the capital theory/production function controversies in the 1950s. After Sraffa's book in 1960 the next decades saw major battles in the journals, battles which resulted in conclusions widely held today: to wit, the technical errors are conceded, but their significance is contested. This has a practical meaning: open any major journal at random today, and there will be marginal products, aggregate production functions, et hoc genus omnia - with no hint that any technical error is involved. The critique is simply ignored. It can't be answered, but it is held to be unimportant.

The neo-Ricardian project initially aimed at reviving the Classical approach. The idea, it seemed was to develop an alternative economics, a science of economic phenomena grounded on different principles...

...The original idea was to move toward a complete reconstruction of economics, on a revived and revised form of the Classical approach, not merely critism of neo-Classical arguments, nor clarification of Classical arguments. The approach would be different: it would be sound theory, but theory based on a realistic account of institutions and history. Furthermore, such analyses could be expected to lead to new, useful, and progressive formulations of policy. That was also the hope of the summer school in Trieste.

What has emerged must be considered disappointing. A Classical 'general equilibrium' theory has been worked out, together with a critique of neo-Classical [economics] - but there has been no development of a new economics. To be sure, there are a few scattered articles on a number of ... topics. But besides the critical work and the development of price theory, the important and widely recognized work has centered on the History of Economic Thought." -- E. J. Nell (1998), The General Theory of Transformational Growth: Keynes After Sraffa, Cambridge University Press

I am aware that I have only here on this blog touched on the potential of Sraffa's work.

One of the seven bloggers here is Tiago Mata.

Tim Lee has a review of Math You Can't Use, a book by Ben Klemen objecting to software patents. Matthew Yglesias's reacts. Some of Matt's commenter's bring up another book, Patent Failure, by Bessen and Meuer. I've found a post from another blog on Matt's post and another reaction from Matt. By the way, when considering the desirability of software patents, one might distinguish between bad patents in an area of technology and the (un)desirability in principle of having patents in an area. As I understand, software patents differ from copyrights in that they impose a burden on developers of doing searches of already established intellectual property, while copyrights don't.

Robert Murphy On Sraffa: In Error

Some discussion with Peter Boettke has inspired me to point out some technical mistakes in Robert Murphy's on-line comments on Sraffa and reswitching.

I begin with Murphy's comments on reswitching. He looks at Samuelson's example in Samuelson's "Summing Up" article. Murphy implicitly suggests that reswitching is only possible in models in which a finite number of techniques are available:

"What Samuelson has done is simply invent a fictitious world in which there are only two ways of producing a particular good... Böhm-Bawerk felt that [his] story was accurate, because at any given time there are more technically efficient but very time-consuming processes 'on the shelf' that are unprofitable at the market rate of interest, but would become profitable at lower rates."

But reswitching is possible when a continuum of techniques lie along the so-called factor price frontier. That is, the possibility of reswitching is consistent with the existence of an uncountably infinite number of techiques. It is also consistent, of course, with the existence of only a countably infinite number and only a finite number of techniques.

Murphy also writes an equally informed comment on Sraffa's book, The Production of Commodities by Means of Commodities. I will adopt Austrian - in fact, Misian - terminology. Sraffa compares prices in Evenly Rotating Economies (EREs) in which the same commodities are produced with the same inputs. Under Sraffa's assumptions in the first part of his book, the construction of the so-called factor price frontier is perfectly valid mathematically. Murphy notes that Sraffa does not model utility-maximization and states that if utility maximization is introduced into the model, the location on the frontier becomes determined uniquely:

"Sraffa's techniques leave no room for the individual members of society to influence the methods of production that end up being used (whether or not there is a surplus), ultimately because there are no individuals in Sraffa's models... However, if we also require that the market rate of interest reflects the subjective premium placed by consumers on present versus future consumption—a feature lacking in Sraffa's aggregate models—then this will eliminate the multiplicity of equilibrium rates of interest."

But Murphy is, again, mathematically incorrect. Multiple equilibrium rates of interest can arise in an ERE model with utility maximization, including intertemporally.

One might look outside a model of an ERE. Murphy suggests he wants to consider models of an approach to an ERE:

"Sraffa's method of determining equilibrium prices in a surplus economy already assumes that the system has settled down at the optimum level of production in all possible lines."

The Arrow-Debreu model of intertemporal equilibrium, despite all its problems, is sufficient for my point here. In such a model of an economy not in an ERE, the equilibrium rate of interest at any point in time for loans of a given length is also not necessarily unique. Not only can multiple equilbrium rates of interest arise, so can a continuum of equilibrium interest rates, if the technology is modeled as discrete.

Why might Murphy be inclined to insist on mathematical error? Consider his statements:

"Sraffa derives results that depict a tradeoff between the real wage and rate of profits. In particular, Sraffa's analysis suggests that in a developed economy, the proportion of the 'surplus' that goes to the workers versus the capitalists is arbitrary, and not at all 'determined' by technological or economic facts... Although he was wrong to condemn interest as an unnecessary and exploitive institution, Sraffa was perfectly correct to criticize the conventional, mainstream justification of the capitalists' income."

But none of these claims, including about exploitation, are made in Sraffa's book.

Contrasting Views On Sraffa's Mathematics

"...Sraffa's prices produce questions, besides whatever else, about the mathematics of his arguments." --S. N. Afriat (2008) "Sraffa's Prices", Sraffa or an Alternative Economics (ed. by G. Chiodi and L. Ditta), Palgrave Macmillan.

Here are two perspectives:

"I think that a very important difference exists between: (i) the process through which a mathematical result is reached, and (ii) a rigorous proof of the result. ... Regarding (i) I mean a sequence of mental objects: examples that appear to contain all of what is essential, graphical tools providing proofs that are only valid for dimensions two or three, incomplete proofs that appear as 'almost' correct, auxiliary constructions that show what is not immediately visible in the problem..."

...We know that all the results contained in Production of Commodities, Part I, can be restated in the language of standard mathematics (matrix theory, eigenvalues, eigenvectors, Perron-Frobenius Theorem, etc.) and rigorously proved. My opinion ... is that Sraffa's presentation is closer to the process that I have indicated by (i) in the Introduction, than to formal proofs. In some cases Sraffa's arguments are defective or insufficient, in others they introduce useless complications." --Marco Lippi (2008) "Some Observations on Sraffa and Mathematical Proofs with an Appendix on Sraffa's Convergence Algorithm", Sraffa or an Alternative Economics (ed. by G. Chiodi and L. Ditta), Palgrave Macmillan.

Lippi's position that Sraffa's mathematics contains defects is strengthed by his demonstration of a bug in Sraffa's algorithm for the construction of the standard commodity.

Is Velupilla in disagreement:

"From a purely mathematical point of view, PCC lacks nothing. The concerns in PCC are the solvability of equations systems and, whenever existence or uniqueness proofs are considered, they are either spelled out in completeness, albeit from a non-formal, non-classical point of view or detailed hints are given, usually in the form of examples, to complete the necessary proofs in required generalities. Pure laziness, inertia and ignorance of alternative traditions in mathematical philosophy have caused untold mischief and created an industry of re-casting and distorting PCC, a work of aesthetic purity and mathematical elegance, into a trivial application, to a large extent, of linear algebra." --Kumaraswamy Velupillai (2008) "Sraffa's Mathematics in Non-Classical Mathematical Modes", Sraffa or an Alternative Economics (ed. by G. Chiodi and L. Ditta), Palgrave Macmillan.

Velupilla is severely critical of the use of Perron-Fobenius theorems in the recasting of Sraffa's theory, when Sraffa essentially gave a constructive proof in demonstrating the existence of the standard commodity.

Firms Run By The Power-Mad

"In a recent biography Eleanor Dulles reports on her experience in a New York hairnet factory circa 1920. 'The owner of the factory never came out there, he just sat in New York and took the money ... The manager was a very sharp type. I told him I could increase production, so I worked out an incentive scheme whereby for a 50 percent increase in production they could make 30 to 40 percent more in wages ... The girls really began to put out. They got very much interested in their work, and the good ones were soon earning 16 dollars and more a week.'

To her astonishment, the manager didn't like it.

'"I'm not going to have those girls thinking they are good," he said. "I'm going to get rid of the good girls. I didn't pay them to get above themselves."'

'He deliberately slowed down supplies and made things awkward for the smarter girls, so they just lost spirit and left.'" -- Harvey Leibenstein (1981) "Microeconomics and X-Efficiency Theory: If There Is No Crisis, There Ought to Be", in The Crisis in Economic Theory (ed. by D. Bell and I. Kristol), Basic Books

Two Problems, One Mathematics (2 of 2)

4.0 Mathematical Notes
4.1 Questions of Existence and Uniqueness
Sections 2 and 3 of the first part present two problems in which the following system of linear equations is derived:

p^T A = p^T

The elements of A are all non-negative, and each row sums to unity. For a non-trivial solution to exist, unity must be an eigenvalue of A. In a physically-meaningful solution, a corresponding left-hand eigenvector must have non-negative entries, with at least some being strictly positive. Furthermore, we would like the solution to be unique, up to a multiple. (In the economics case, multiplying prices by a constant corresponds to a change in the numeraire.) As a matter of fact, the problems as stated do not yet guarantee uniqueness.

It is easy to show that unity is an eigenvalue for right-hand eigenvectors of A. Let e be the n-element column vector where all elements are unity. Since the rows of A all add up to unity, the following equation must hold:

A e = e

So unity is an eigenvalue of A. (This proof relies on the property that the set of eigenvalues for left-hand eigenvectors of A is the same as the set of eigenvalues for right-hand eigenvectors of A.) Non-negativity and uniqueness, when it obtains are less obvious.

4.2 Irreducible Matrices
The left-hand eigenvector P^T corresponding to the eigenvalue unity contains all positive elements if A is irreducible. Furthermore, if A is irreducible, the left-hand eigenvector P^T is unique, up to a multiple. A matrix is irreducible, obviously, if it is not reducible. To explain this, I need to define what it means for a matrix to be reducible.

Suppose A is transformed by interchanging a pair of rows and then interchanging the corresponding columns. Any permutation of rows and columns can be performed by repeating this operation for an appropriate sequence of pairs of row and column indices. In the economics case, such a sequence of operations corresponds to selecting a different ordering of the industries in which to express A. In the case of page ranks, such a sequence consists in taking a different ordering for the (unranked) pages. In both cases, the ordering is arbitrary, so no problem arises here.

The non-negative matrix A is reducible if there exists such a sequence of operations that transform A into the block structure form:

A1,1	A1,2
0	A2,2

where A_1,1 is a square non-negative irreducible matrix.

I think the meaning of reducibility in the two problems is suggested under the special case where:

• All the elements of A_1,2 are zero, and
• A_2,2 is irreducible (as well as A_1,1)

The economics problem would then correspond to two non-trading islands, each in a self-replacing state with no surplus. The web pages would consist of two islands of web pages, in which links can be used to get from any one page on an island to any other page on that island, but with no path between these islands of pages.

Unity would be a repeated eigenvalue for a reducible A. One solution vector P^T has strictly positive prices for the industries corresponding to A_1,1 and zero prices for the remaining industries. The other solution has zero prices for the industries corresponding to A_1,1 and strictly positive prices corresponding to industries for A_2,2. It seems reasonable to me to assume in the economics model one is considering a single economy. I don't see why in the page rank case, some set of pages cannot be partially isolated in some sense from the remaining pages. A page ranking algorithm needs to address this possibility.

I might as well mention a condition for an interesting generalization of the economics problem. Let A be a non-negative, reducible matrix with no row sums that exceed unity. Suppose the maximum eigenvalue of A_1,1 exceeds the maximum eigenvalue of A_2,2. Then A is a Sraffa matrix. I'm not sure if the definition of a Sraffa matrix requires some of the elements of A_1,2 to be non-negative so that this input-output matrix hangs together to describe a single economy. Some such condition makes sense to me for an analysis of an economy with a surplus.

4.3 Perron-Frobenius Theorems
I state a theorem, or rather, a combination of eight theorems:

Theorem: Let A be an irreducible non-negative nxn matrix. Then:

1. λ_m, the maximum eigenvalue of the matrix A is bounded below by the minimimum row-sum of A and is bounded above by the maximum row-sum of A.
2. The maximum eigenvalue of A is a continuous, increasing function of the elements of A.
3. Let μ = 1/ν be strictly positive. If μ > λ_m, then all the elements of the matrices (μ I - A)^-1 and (I - ν A)^-1 are strictly positive.
4. Any eigenvalue α of A is bounded above in modulus by the maximum eigenvalue of A:

|α| ≤ λ_m

5. The maximum eigenvalue of A is associated with a left-hand eigenvector p^T whose elements are strictly positive:

p^T A = λ_m p^T

p_i > 0, for i = 1, 2, ..., n,

6. The maximum eigenvalue of A is associated with a right-hand eigenvector q whose elements are strictly positive:

A q = λ_m q

q_i > 0, for i = 1, 2, ..., n,

7. To each eigenvalue α of A different from the maximum eigenvalue λ_m there corresponds a non-zero left-hand eigenvector which has at least one negative component.
8. To each eigenvalue α of A different from the maximum eigenvalue λ_m there corresponds a non-zero right-hand eigenvector which has at least one negative component.

I deliberately included more Perron-Frobenius theorems above than I need for this problem. Perron-Forbenius theorems of a slightly different form have also been stated for reducible matrices.

Anyways, from the first condition, one sees that unity is the maximum eigenvalue for an irreducible A in both the economics and page rank problems. From the fifth conditon, it follows that there exists a set of strictly positive prices, in the economics case, or of strictly positive page ranks in the other case. And by the seventh condition, I guess, this is an unique solution (up to a multiple of the eigenvector).

5.0 References

• Kurt Bryan and Tanya Leise, "The $25,000,000,000 Eigenvector: The Linear Algebra Behind Google"
• Luigi L. Pasinetti (1977) Lectures on the Theory of Production, Columbia University Press
• Piero Sraffa (1960) Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, Cambridge University Press

Two Problems, One Mathematics (1 of 2)

"Besicovitch insists that I publish; the fact that I was able to forsee interesting mathematical results shows that there must be be something in the theory." -- Piero Sraffa (Diary entry, 31 May 1958)

1.0 Introduction
In this post, I derive the same equation for two completely different problems. One is an economics model. The other is a simplified presentation of how Google might automatically calculate page ranks to determine the order in which web pages are presented to a user on completion of his search. I could have complicated my exposition by considering a third problem: the steady state probability distribution in a Markov chain.

2.0 Prices for Simple Reproduction
Consider an economy in which n commodities are produced. Each commodity is produced in a process in which it is the only output. In other words, no joint production, such as of wool and mutton, occurs in this economy. n processes are in use, each producing one of the n commodities, and all commodities are produced by one of these processes. Each production process requires a year to complete and uses up all its inputs.

Let a_i,j denote the quantity of the ith commodity used in the production of the jth commodity. Quantities are measured in normalized units, such that the output of each process is one unit of the respective commodity. The nxn matrix A is the Leontief input-output matrix of interindustry quantity flows for this economy. Each element of A is non-negative.

Assume that this economy is undergoing simple reproduction. That is, the output of each process is exactly equal to the total inputs of that commodity used across all processes. (If it helps, one might think of the inputs to each process as including the commodities consumed by the workers operating that process. Labor inputs are not shown in the representation of this economy being considered here.) Anyways, this assumption implies that the sum for each row in A is unity.

Suppose each process is operated by a separate firm. The firm own ats the end of each year a single (normalized) unit of a single commodity. For the firm to continue in operation, it must trade this commodity for an appropriate amount of each of its inputs in all-around markets. Let p^T denote the row vector of the prices in these markets. The condition that the economy continue in operation implies the following equation for prices:

p^T A = p^T

This characterization of prices is a non-neoclassical idea. Markets have not been modeled here as including any sort of maximization process. Nor have these prices been presented as a (stable?) limit point of some sort of dynamic process. Sraffa describes these prices as follows:

"There is a unique set of exchange-values which if adopted by the market restores the original distribution of the products and makes it possible for the process to be repeated; such values spring directly from the methods of production." -- P. Sraffa (1960)

3.0 Google Page Ranks
I now consider a re-definition of all of my symbols. Suppose n web pages have been identified, perhaps by a web-crawler. We want to rank these pages in some way.

These web pages contain links, including to one another. In ranking them, perhaps a page in which a high proportion of the links on other pages goes to that page should have a high rank. But ratios of the proportion of links on other pages that go to that page should be weighted by the ranks of those other pages. These ideas can be formalized.

Let m_i,j be the number of links on the ith web page to the jth web page, for i unequal to j. Let m_i,i be zero. Let m_i be the total number of links on the ith page, excluding links to itself and to pages outside the pages being ranked.

a_i,j = m_i,j/m_i, i = 1, 2, ..., n; j = 1, 2, ..., n

I have now defined a nxn matrix A, where each element is the proportion of the links on a page within a web that go to another specified page in that web. Each element in A is non-negative, and each row adds up to unity. Also, the principal diagonal of A is zero, although that property is not used in the following mathematics.

Let p^T denote the row vector of page ranks. Page ranks satisfy the following system of equations:

p^T A = p^T

In the next part, I consider conditions under which a solution exists and is unique.

5.0 References

• Kurt Bryan and Tanya Leise, "The $25,000,000,000 Eigenvector: The Linear Algebra Behind Google"
• Piero Sraffa (1960) Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, Cambridge University Press

Letters From Soros

Last month, I noted resemblances between Soros' concept of "reflexivity" and Davidson's use of non-ergodicity to formalize the notion of a model economy set in historical time. Davidson drew this point to Soros' attention over a decade ago. Soros has commented on this resemblance.

The following letter has an Open Society Institute letterhead:

February 28, 1997

Professor Paul Davidson
Holly Chair of Excellence in Political Economy
The University of Tennessee Knoxville
College of Business Administration
Department of Economics
Stokely Management Center
Knoxville, Tennessee 37996-0550

Dear Professor Davidson,

Thank you for sending me your book Economics for a Civilized Society. I found your comments on Samuelson's ergodic hypothesis very pertinent.

Yours Sincerely,

George Soros

From the 15-21 March 1997 issue of The Economist:

Sir - In "Palindrome repents" (January 25th) you accuse me of ignorance of economic theory. In particular, you say that my "claim that economics is inherently flawed on some deep epistemological level is just embarrassing." Is it?

Economics aspires to the status of a hard science. Specifically, it seeks to establish universally valid laws similar to 19th-century physics. For this purpose it relies on the concept of equilibrium, similar to the resting place of the pendulum, which is the same irrespective of any temporary perturbation. Paul Samuelson, an economist, called this the "ergodic hypothesis" and considered it indispensable to making economics a hard science.

The trouble is that economics cannot be made into a hard science, because of the reflexive interaction between the participants' thinking and the actual state of affairs. The interaction does not have a determinate outcome, because the outcome is contingent on the participants' expectations, and the participants' decisions do not merely passively discount the future but also actively help to shape it. There is a two-way feedback mechanism that does not lead to a predetermined resting place, but keeps a historical process in motion. Economic theory can protect the false analogy with 19th-century physics only by eliminating reflexivity. It does so by assuming demand and supply as independently given. The result is an axiomatic system that has little relevance to the real world.

You are correct to claim that, in practice, economists have learnt this, in order to deal with the real world. Alan Greenspan's recent Humphrey-Hawkins testimony is a brilliant exercise in reflexivity. But the theory has never been discarded and it serves as the scientific underpinning for the prevailing belief in the magic of the marketplace.

You are also right to claim that markets do not reign supreme; but you cannot deny that there is a powerful body of opinion that passionately believes that they should. You are plain wrong in asserting that I do not know the "big difference" between laisser-faire and totalitarian ideologies. I stated it explicitly in my Atlantic Monthly article and have been guided by it in my philanthropic activities. I can tolerate personal attacks but I must object when they are used to obfuscate valid arguments.

New York
George Soros

Will Notre Dame Be Serious In Teaching Economics?

Apparently some members of the Department of Economics and Policy Studies would still like to teach classes and hire colleagues. If Notre Dame is serious about educating their students, shouldn't they be taught of the existence of the full range of views on economics? After all Notre Dame has some excellent scholars, including some great historians of economics. (I'm not sure that Esther-Mirjam Sent is still at Notre Dame.)

An on-line petition has been put up in support of these wild ideas. (I haven't yet signed it.)

Hat tip to shagan at daily Kos.

Update: I have now signed the petition. Christopher Hayes comments.

Einstein in Sweezy's Mag

Albert Einstein had an article in the inaugural issue of Monthly Review. I don't know if there is a story here. Did Paul Sweezy, the publisher of the magazine ask Einstein for this article? Did Einstein merely answer a call for papers?

When I first found out about existence of this article, I expected Einstein to merely put forward comments about how we should all try to get along. (I'm never quite sure if I understand the word "Bien-pensant". I know enough French to translate it, but I think I've stumbled on the concept here.) But when I read Einstein's article I found out that he accepts Marx as having made scientific claims:

"The owner of the means of the means of production is in a position to purchase the labor power of the worker. By using the means of production, the worker produces new goods which become the property of the capitalist. The essential point about this process is the relation between what the worker produces and what he is paid, both measured in terms of real value. Insofar as the labor contract is 'free', what the worker receives is determined not by the real value of the goods he produces, but by his minimum needs and by the capitalists' requirements for labor power in relation to the number of workers competing for jobs. It is important to understand that even in theory the payment of the worker is not determined by the value of his product." -- Albert Einstein (1949) "Why Socialism?", Monthly Review (May), as reprinted in Out of My Later Years

A Perspective on the Coase "Theorem"

"'Coase's Theorem' says that once we determine who can legally do what, externalities will vanish as actors pay one another off. This methodology - either the victims bribe the perpetrator to stop the distasteful behavior (when the perpetrator has the legal right to do it) or the perpetrator bribes the victim to put up with the distasteful outcome (when the victims have a right not to suffer the pain) - is considered an epochal innovation in American economics and jurisprudence. Even without understanding all the details and context, this tells us a great deal about our intellectual culture since, in fact, Coase's methodology doesn't work.

1. Bargains made with only two parties, perpetrator and victim, will rarely if ever be efficient.
2. Whenever the perpetrator is a corporation and the victim a far-flung collection of disparate individuals, the latter is at such a disadvantage in seeking recompense that without intervention correction rarely takes place at all.
3. Coase's methodology provides a 'bully incentive' for anyone able to muster the courts on their behalf. Suppose Z owns a train. If we have the legal right to spew sparks, we should create the most spark-spewing train we can. If we have the right to be noisy, we should create a decibel demon. Whenever anyone with the means to pursue legal claims has a right that could cause others pain, they should threaten to exercise it and then extort bribery payments to desist. In the absence of a specific legal prohibition, a bully walking down the street should spit on everyone, claiming that he will only suffer the pains of stopping if his vitims bribe him to do so.

Not surprisingly, Coase's 'extort the victim,' 'empower the powerful' analysis was welcome as soon as economists could understand its utility to elites able to make legal claims here, there, and everywhere, thereby profitably spitting this way and that with impunity.

Why the hoopla for Coase? First, Coase's Theorem, better named the Bully Theorem, reinstates markets and privatization as optimal, even providing an argument against unwanted government regulation. Second, it elevates 'property rights' to the defining position in jurisprudence and establishes a logic for dealing with externalities perfectly suited to co-optation by capital." -- Michael Albert (November 1991)

Soros on Historical Time

The New York Times a couple of weeks ago profiled George Soros, the billionaire financial speculator, philanthropist, and student of Karl Popper's ideas. According to this profile, Soros would like to have an impact on the discipline of economics:

"Now in his eighth decade, [Soros] yearns to be remembered not only as a great trader but also as a great thinker. The market theory he has promoted for two decades and espoused most of his life - something he calls 'reflexivity' - is still dismissed by many economists. The idea is that people's biases and actions can affect the direction of the underlying economy, undermining the conventional theory that markets tend toward some sort of equilibrium." -- Louise Story (2008)

Stiglitz is quoted as saying that Soros might become successful at his goal:

"But Joseph E. Stiglitz, a professor at Columbia who won the Nobel for economics in 2001, said Mr. Soros might still meet success. 'With a slightly different vocabulary these ideas, I think, are going to become more and more part of the center,' said Mr. Stiglitz, a longtime friend of Mr. Soros." -- Louise Story (2008)

I suppose one could debate about whether mainstream economics is as open to these ideas as Stiglitz suggests.

About a decade ago is the first time Soros put these ideas into print, as far as I aware. Here is his then definition of reflexivity:

"In the case of scientists, there is only a one-way connection between statements and facts. The facts about the natural world are independent of the statements that scientists make about them... If a statement corresponds to the facts, it is true; if not, it is false. Not so in the case of thinking participants. There is a two-way connection. On the one hand, participants seek to understand the situation in which they participate. They seek to form a picture that corresponds to reality. I call this the passive or cognitive function. On the other hand, they seek to make an impact, to mold reality to their desires. I call this the active or participating function. When both functions are active at the same time, I call the situation reflexive...

...When both functions are at work at the same time, they may interfere with each other. Through the participating function, people may influence the situation that is supposed to serve as an independent variable for the cognitive function. Consequently, the participants' understanding cannot qualify as objective knowledge...

...Our expectations about future events do not wait for the events themselves; they may change at any time, altering the outcome. That is what happens in financial markets all the time... But reflexivity is not confined to financial markets; it is present in every historical process. Indeed, it is reflexivity that makes a process truly historical...

A truly historical event does not just change the world; it changes our understanding of the world - and that new understanding, in turn, has a new an unpredictable impact on how the world works."-- George Soros (1998: 6-8)

And Soros uses these ideas to criticize the use of equilibrium models in economics.

To me, Soros is expressing the same concept that Post Keynesians call historical time. Post Keynesians reject neoclassical economics. In the failed neoclassical approach, the economy tends towards an economic equilibrium pre-determined by the objective data of tastes, technology, and endowments. Both Soros and the Post Keynesians, with their emphases on expectations, question the objectivity of the data. Today, I turn to Jan Kregel for a statement of the Post Keynesian position:

"There can be no tendency to equilibrium based on a relation between expectations and the objective data of what the consumer will demand and the price he will pay which describes the conditions of equilibrium because the incomes available to consumers will be determined ultimately by the very decisions taken by entrepreneurs on the basis of these expectations.

The post Keynesian approach is thus influenced by Keynes' insistence that the level of output and employment cannot be considered as objective data determining the conditions of equilibrium because they will be endogenously determined by entrepreneurs' decisions... Keynes is concerned with the role of expectations in the coordination of individual production plans in a society consisting of several independent producers whose expectations determine the means available to satisfy an uncertain multiplicity of future demands. Expectations themselves determine the objective facts of the conditions of equilibrium... The problem is not whether the objective data necessary to achieve equilibrium will be reflected in subjective data available to the individual, but the very definition of the objective data. Indeed, even its objectivity is questioned." -- Jan Kregel (1986).

And these ideas accompany an concern with processes set in history. (I have previously mentioned Paul Davidson's use of the concept of non-ergodicity in a formalization of the idea of a process set in history.)

Soros and Post Keynesians like Davidson draw similar practical conclusions from developments of these ideas. The financial system, including internationally, can be a source of economic instability. We need to design new international institutions and conventions to govern finance. The system that has evolved since Richard Nixon abolished the Bretton Woods system doesn't work, as international economic crisis succeeds international economic crisis.

References

• J. A. Kregel (1986). "Conceptions of Equilibrium: The Logic of Choice and the Logic of Production", in Subjectivism, Intelligibility, and Economic Understanding: Essays in Honor of Ludwig M. Lachmann on his Eightieth Birthday (ed. by Israel M. Kirzner), New York University Press.
• George Soros (1998). The Crisis of Global Capitalism: Open Society Endangered, Public Affairs
• Louise Story (2008). "The Face of a Prophet: Soros Craves Respect for His Theories, Not Just His Money", New York Times (11 April) Business, p. 1

Foundations of Probability: No Decided Opinion

Over at Good Math, Bad Math, Mark Chu-Carroll has brought up the disagreement between frequentists and Bayesians. Here's a highly technical and theoretical argument, conducted with a great deal of vitriol, and perhaps with practical consequences. You might think this is the sort of thing I would have a decided opinion on. Yet I don't, perhaps because I don't know enough about Bayesianism in practice.

I have sometimes applied Neyman-Pearson hypothesis testing, sometimes in the context of the design of experiments. I think it useful to have a decision rule before looking at the data, but I try not to get hung up on ontological commitments. I am not sure that my practice is compatible only with some position on these arguments. I generally explain the math with frequentist arguments.

I did have some thoughts on the comments. I wondered whether more schools should be distinguished than Mark or his commentators have done. The first comment supports what might be called a formalist or axiomatic approach: Probability is a mathematical theory of certain measures on sigma algebras. Somewhere in the comments, E. T. Jaynes is mentioned. Is Jaynes' entropy-maximization approach co-extensive with Bayesianism? Somewhere else, personalism is mentioned. Is this also a synonym of Bayesianism? I think some approaches are about objective properties of propositions. I gather this is Keynes' position is his book on probability. Can the non-frequentist school be further decomposed into objective and subjective branches? Somewhere I seem to vaguely recall a fidicual approach, which I understood even less. Where does this fit? I suppose I also ought to ask where Savage fits in.

I was not aware that those that worry about these sorts of things have tended to swing from frequentists to Bayesians with the growth in computing power. I was taught a frequentist approach, with an acknowledgement of the existence of debate. I'm aware that the growth of computing power has led to greater popularity of bootstrap/jackknife/resampling methods since I received my undergraduate degree. Perhaps that's another topic.

Some comments at Mark's go down what I think is a blind alley - they suggest the Monte Hall Problem is an illustration of the strength of Bayesianism. While Bayes' theorem is useful in calculating the correct solution, I don't see the problem as connected strongly to foundational principles. Perhaps, use of Bayes' theorem emphasizes. that Monte's decision rules must be precisely specified beforehand.

There is a discussion of confidence intervals and the correct way of thinking about them from the frequentist perspective. One of my colleagues once suggested to me that constructing a confidence interval is like trying to throw a hat over, say, an apple. The apple is at some position, and you don't know whether it is or is not under the hat after it is thrown. If your hat is a sombero - like a 99% confidence interval - you are more likely to catch the apple. If your hat is a English derby - like a 90% confidence interval - you are less likely to catch the apple. But when you have caught it, you've more precisely specified where the apple is.

I Pity The Fool

Apparently, Philip Mirowski is currently researching how economics can inform science and technology policy. Mirowski (2007) is a step in this enterprise. This paper reviews Warsh (2006). Warsh's book is a popular account of economics organized around the history of the idea of increasing returns. Paul Romer is the hero of Warsh's story. Warsh praises, specifically, Romer (1990).

Mirowksi concludes that neither Solovian growth theory nor so-called new growth theory can legitimately say much for science and technology policy, since they are so defective. Basically, Mirowski agrees with me (for example, look here).

Some quotes from Mirowski follow:

"...the entire book is so lacking in any skeptical perspective that would ideally derive from a deep background familiarity with the history of economics, not to mention some modicum of philosophical humility..., that one has to wonder just what Warsh was thinking.

I want to be clear about this, since I honestly think that David Warsh is the best we have in terms of economics journalism in America."

"The problem is not simply a secondary matter of aggregation, as some suggested back during the capital controversies of the 1970s. The entire motivation behind early production functions was to posit a sharp distinction between factor substitution and something more dynamic and pervasive, which for lack of a better term, is still commonly called technological change."

"Yet, economists and journalists alike extol the new growth theory, and there is no denying its popularity in certain circles. ... Rigorous mathematics and assiduous empiricism had little to do with it."

"I pity the poor student of modern economics, trying to make some sense of what can only appear to the outsider as cryptic oracular pronouncements emitted from people who claim to be experts in the nature and validity of knowledge. But when you get your news from Jon Stewart, your history from Paul Krugman, and your research facts from Wikipedia, maybe the nature of knowledge has itself changed.

References

• Mirowski, Philip (2007) "Review Essay: Did the (Returns to) Scales Fall From Their Eyes?", Journal of the History of Economic Thought, V. 29, N. 4 (Dec): 481-494
• Romer, Paul M. (1990) "Endogenous Technological Change", Journal of Political Economy V. 98, N. 5 (Oct): S71-S102.
• Warsh, David (2006) Knowledge and the Wealth of Nations: A Story of Economic Discovery, Norton

A Reswitching Example With Fixed Capital

1.0 Introduction
This post presents an example of reswitching with fixed capital. This is Example 5 in Chapter 9 of J. E. Woods, The Production of Commodities: An Introduction To Sraffa (Humanities Press, 1990). Fixed capital is a special case of joint production. I often analyze the choice of technique in circulating capital models by constructing the so-called factor-price frontier as the outer envelope of factor-price curves for each technique. This method of analysis does not generalize to general models of joint production. I confine myself in this post to methods of analysis that apply generally to joint production. This example illustrates the reswitching of techniques. A manifestation of reswitching in this example is the non-monotonic dependence of the economic life of machinery on the rate of profits.

2.0 Technology
The example is of a vertically-integrated industry producing a net output of corn with inputs of corn, machines, and labor. Machines can last for two yearly cycles of production, while corn inputs are entirely used up each year in production. Firms have available the three Constant-Returns-to-Scale (CRS) processes shown in Figure 1. New machines are produced by the first process, while corn is produced by both of the two remaining processes. The third process uses a one-year old machine as an input. The one-year old machine is produced jointly with corn by the second process.

Table 1: The Technology

	Machine Industry	Corn Industry
Inputs
Labor	1/10 Person-Year	43/40 Person-Year	1 Person-Year
Corn	1/16 Bushel	1/16 Bushel	1/4 Bushel
New Machines		1	0
Old Machines		0	1
Outputs
Corn	0 Bushels	1 Bushel	1 Bushel
New Machines	1	0	0
Old Machines	0	1	0

3.0 Quantity Flows
A choice of technique arises over the economic life of the machine. A firm might choose to discard the machine after the first year and never use the third process, whether in conjunction with the other processes or not. (I assume that machines can be freely disposed of after both one and two years of operation.)

The proportions in which processes are operated varies, depending on how many years the machine is used. Table 2 shows the first two process being used to produced a net output of one bushel corn. Notice that when these processes are operated in parallel new machines are simultaneously produced by the first process and used up by the second process. (I suppose I could scale up the processes in Tables 2 and 3 by 91 so that an integral number of machines is used in each technique and the net output of both is the same quantity of bushels of corn.)

Table 2: Use Of Machine For One Year

	Machine Industry	Corn Industry
Inputs
Labor	4/35 Person-Year	43/35 Person-Year
Corn	1/14 Bushel	1/14 Bushel
New Machines	0	8/7
Outputs
Corn	0 Bushels	8/7 Bushel
New Machines	8/7	0

Table 3 shows how all three processes must be scaled when machines are used for two years and the net output is one bushel corn. Tables 2 and 3 can be used to calculate the flow of labor inputs and capital goods required to produce a net output of one bushel corn for each technique. Table 4 exhibits the result of this calculation. If machines are junked after one year, 2/455 person-years more labor is hired per bushel corn net, as compared to machines being junked after two years.

Table 3: Use Of Machine For Two Years

	Machine Industry	Corn Industry
Inputs
Labor	4/65 Person-Year	43/65 Person-Year	8/13 Person-Year
Corn	1/26 Bushel	1/26 Bushel	2/13 Bushel
New Machines	0	8/13	0
Old Machines	0	0	8/13
Outputs
Corn	0 Bushels	8/13 Bushel	8/13 Bushel
New Machines	8/13	0	0
Old Machines	0	8/13	0

Table 4: Inputs Per Net Output of 1 Bushel Corn

	Years Machine Is Operated
	1 Year	2 Years
Labor	47/35 Person-Years	87/65 Person-Years
Capital	1/7 Bushel, 8/7 New Machines	3/13 Bushel, 8/13 New Machines, 8/13 Old Machines

4.0 Prices and the Choice of Technique
Firms choose the technique - that is how long machines are used - based on profitability. Hence, an analysis of the choice of technique requires an analysis of prices.

4.1 Prices When Machine is Operated One Year
Suppose, to begin with, that the machine is operated for one year only. Prices are such that no pure economic profits can be earned in producing new machines or in operating machines for the first year. Furthermore, costs (inclusive of a charge on capital goods advanced) do not exceed revenues in either of these two processes. These assumptions imply two equalities are met:

(1/16) (1 + r) + (1/10) w_α = p_{0, α}

[(1/16) + p_{0, α}](1 + r) + (43/40) w_α = 1,

where a bushel corn is the numeraire, p_{0, α} is the price of a new machine, w_α is the wage, and r is the rate of profits. These equations embody the implicit assumption that wages are paid after the laborers have been working for a year. The price of a year-old machine is zero.

This is a system of two equations in three unknowns. If the rate of profits is taken as given outside this system, the other two variables can be found as a function of the rate of profits:

w_α(r) = (5/2)(14 - 3 r - r²)/(4 r + 47)

p_{0, α}(r) = (1/16)(103 + 39 r)/(4 r + 47)

The larger is the wage, the smaller is the rate of profits. The maximum rate of profits in this system arises for a wage of zero. That maximum is approximately 253.11%.

The above solution can be used to analyze the profitability of operating the machine for another year. Figure 1 shows the costs and revenues (in bushels corn) for producing a bushel of corn by operating the third process in Table 1. Inputs are valued at the prices given by the above solution. Note that revenues would exceed costs if the rate of profits were below approximately 33%. Likewise, revenues would exceed costs if the rate of profits were above 50% and below the maximum. In these regions, firms would choose to operate the machines for a second year, and the prices would not be equilibrium prices.

Figure 1: Costs and Revenues for Operating Machine for Second Year

On the other hand, the above solution gives equilibrium prices if the rate of profits lies between approximately 33% and 50%. In this region, the costs of operating the machine for a second year exceed the revenues. Firms would only operate the machine for one year, consistently with the hypothesis for this case.

4.2 Prices When Machine is Operated Two Years
The other case arises when prices are consistent with the machine being operated for two years. By assumption, pure economic profits cannot be earned in any of the three processes in the technology. Likewise, costs do not exceed revenues for any of the three processes. These assumptions yield a system of three equations:

(1/16) (1 + r) + (1/10) w_β = p_{0, β}

[(1/16) + p_{0, β}](1 + r) + (43/40) w_β = 1 + p_{1, β}

[(1/4) + p_{1, β}](1 + r) + w_β = 1

where p_{0, β} is the price of a new machine, p_{1, β} is the price of a machine after operating for one year, w_β is the wage, and r is once again the exogeneously specified rate of profits.

This system has one more equation than the system in Section 4.1. But it contains one more variable also. Thus, the wage and the prices of the new and one year-old machine can be found in terms of the rate of profits:

w_β(r) = (5/2)(26 + 7r - 4r² - r³)/(87 + 51r + 4 r²)

p_{0, β}(r) = (1/80)(135 - 166 w_β - 86 w_β r + 50 r - 5 r²)/(1 + r)²

p_{1, β}(r) = (1/4)(3 - 4 w_β - r)/(1 + r)

In this case also, the maximum rate of profits arises when the wage is zero. The maximum rate of profits is approximately 258.77%

The analysis of the choice of technique in this case is based on examining the price of the year-old machine (Figure 2). Only non-negative prices are consistent with equilibria. The price of an used machine becoming negative is a sign that it is cheaper to truncate the use of a machine after it is used for one year. That is, for a rate of profits between approximately 33% and 50%, firms will junk the machine after one year of operation.

Figure 2: Price of One-Year Old Machine

4.3 Choice of Years of Operation
The analysis of the choice of technique is consistent, whether one begins with the price system for machines being used for one or two years. When the rate of profits is below approximately 33%, but non-negative, machines are used for two years. When the rate of profits exceeds 50% and below the maximum, machines are also used for two years. When the rate of profits exceed approximately 33% and are below 50%, machines are discarded after one year of operation.

5.0 Conclusion
Suppose one incorrectly accepts pre-Sraffian neoclassical or Austrian intuition. Then one would believe that a low rate of profits reflects capital, in some sense, being relatively less scarce than labor. And one would expect firms to adopt a more capital-intensive technique at a lower rate of profits. In this context, one would expect the economic lives of machines to be longer at a lower rate of profits. But, as shown by Figure 3, such a belief is incorrect, in general.

Figure 3: Economic Life of Machine versus Rate of Profit

If one had such incorrect pre-Sraffian neoclassical or Austrian intuition, one would likewise expect firms to hire more workers at lower wages, also under conditions of perfect competition. Figure 4 demonstrates this belief is false as well. So much for the idea that wages and employment are determined by the interaction of well-behaved supply and demand functions.

Figure 4: Labor Intensity versus Wage

Madrick on the End of an Age

Jeff Madrick has a Huffington Post column titled, "The End of the Age of Milton Friedman".

Since this is short and popular, one could think up lots of caveats. I'll go first. "Crisis in the 1970s" does not explain why Friedman's ideas became popular. Economists had available another account of stagflation. Something else remains of Friedman's academic contributions other than "the overstated natural rate of unemployment philosophy". I think most working economists would still echo something of Friedman's views on methodology, despite their rejection by specialists in the field.

A Photo of Barkley Rosser

I stumbled upon the following photo accompanying an interview with Albert Tucker (Maurer 1985).



The 3rd man from the left in the first row is the father of Barkley Rosser, Jr.

I have tried reading Rosser (1936), but I did not really understand the proof. As I understand it, Rosser puts some theorems of the time together to alter a theorem of Gödel's so that it's statement seems more natural. It is a very concise paper. Gödel shows that ω-consistency implies the existence of undecidable propositions. Rosser discarded the ω; he showed that consistency implies the existence of undecidable propositions. I guess there can be consistent systems that are not ω-consistent. Consistency is a syntactical property, and it does not require intuitions about universal quantification over all natural numbers.

Definition: A system is ω-inconsistent if and only if there exists a proposition p(n), with free variable n, such that

1. p(0) is provable, p(1) is provable, p(2) is provable, and so on.
2. It is provable that not ( for all n, p(n))

Definition: A system is ω-consistent if and only if it is not ω-inconsistent.

References

• Stephen B. Maurer (1985). "Albert Tucker", in Mathematical People: Profiles and Interviews (Ed. by D. J. Albers and G. L. Alexanderson), Birkhauser
• Barkley Rosser (1936) "Extensions of some Theorems of Gödel and Church", The Journal of Symbolic Logic, 1 (3), (September): 87-91.