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Remembering Krishna Bharadwaj
K Vela Velupillai
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Omkarnath (2005) has documented quite handsomely3 the approval with which even the legendary silences of Sraffa – especially with regard to comments on his own contributions – were broken by the elegance and competence of Krishna Bharadwaj’s maiden foray into
On this, the 20th anniversary of the untimely death of Krishna Bharadwaj, this article tries to reconsider some theoretical aspects of her fundamental contributions to capital theory by showing how applicably relevant they are in modern contexts. Krishna Bharadwaj had an admirable mastery of Srafﬁan methodology and remained loyal to that tradition in a most enlightened manner. Her theoretical contributions to Srafﬁan scholarship enhanced and enlarged the frontiers of applicable capital theory. One particular application of two Bharadwaj Theorems is also considered in this article.
K Vela Velupillai (kvelupillai@gmail.com) is at the department of economics/ASSRU, University of Trento, Italy.
1 A Preamble on Anniversaries, Realism and Parables
[Krishna Bharadwaj] ‘did not then [in 1962] know of the more fundamental critique of economic theory heralded by Piero Sraffa’s work’ (Bharadwaj 1992: 38). This came in 1962 when ‘Sachin Chowdhury [sic], the editor of The Economic Weekly [which subsequently became The Economic and Political Weekly], …. drew out of his drawer [Sraffa’s] slim volume … [which Krishna …] agreed to review … in a month or so!’ (ibid: 39).
– Harcourt 199394: 301; italics in the original.
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the world of PCMC. I shall not traverse these welltrodden noble paths anew. I aim to use the traditional intellectual excuse of anniversaries to try to point out a neglected aspect of Sraffa’s concerns on anchoring serious theory on observational realism, without idle construction of nonrelevant parables that do not aid understanding reality but obfuscate and obscure the distinction between relevant abstractions and irrelevant approximations to a nonobservable reality.
For this purpose I invoke two Bharadwaj Theorems, in the next section, which are then used as a springboard to compute an index number in a computationally efﬁcient manner to compare two productions systems, characterising two alternative economies. Neither Sraffa, nor Krishna Bharadwaj, indulged in “abstractionmongering” in “ahistorical” intellectual exercises, without anchoring their theories in relevant realism, free of parables.
Even in the context of the deepest issues of the theory of value, Sraffa’s anchoring – and Krishna Bharadwaj’s, too
– was in the observable entities of production, and not pseudometaphysical psychological bases of socalled rational
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behaviour. In a note written in summer 1928, as Heinz Kurz has recently reminded us (Kurz 2011), Sraffa emphasises (italics added):
The question asked of the theory of value is the following: Given (from experience) the prices of all commodities ..., ﬁnd a set of conditions that will make these prices appear to be necessary. This means, given the unknowns, ﬁnd the equations (i e, the constants)… (Sraffa Papers D3/12/9: 65).
The immediate implication is that the economic analyst faces a Diophantine Decision Problem (as I have argued in Velupillai 2005). The “given (from experience) prices” cannot be other than – at best – rational valued. The same applies to the “constants” (deﬁned by technology). Such problems are naturally algorithmic – i e, procedural. It is this aspect that is highlighted in the key result I have called Bharadwaj’s Theorem I, in the next section.
Bharadwaj noted, in the concluding lines of her review of PCMC (Bharadwaj 1963: 1454; italics added):
Written in an unusually compact style and embellished with chiselled logic, [PCMC] bears the imprint of sustained reﬂ ection. Unmistakably, this is the work of a master written with authority and insight. ….
[T]ime has dealt kindly with Sraffa’s contribution. It is as relevant today as it was when conceived [over seven and a half decades ago].
No one understood better than Krishna Bharadwaj that the “chiselled logic” embodied in the theoretical propositions of PCMC, proved with impeccably faultless mathematical reasoning, albeit unconventional,4 were motivated by intensely applicable and observational relevance.
2 Two Bharadwaj Theorems
Methodologically, [Sraffa] explicitly states that his immediate concern is the properties of the system which do not depend upon change. Erroneously interpreting this as invariance to change or changelessness, some have regarded Sraffa’s analysis as restricted to a stationary equilibrium or, when extended, to signify steady states. There have been repeated charges about the ahistoricty of the exercise, which appears as merely abstractionmongering (Bharadwaj 1989: 321, italics added).
Neither Sraffa nor Bharadwaj stated any of their results or propositions in terms of the formal notion of a theorem, although that is what they should have been called, given the pseudomathematical jargon of economic theoretical practice. Not calling them theorems has led various economists, with only a modicum of formal mathematical training, to carp and ﬁnd lacunae in Sraffa’s rigorously derived demonstrations of his exact, impeccably unambiguous, propositions. Even worse misunderstandings have resulted in Sraffa avoiding the use of the word “proof” to indicate the rigorous procedures with which he demonstrates the validity of his propositions (i e, theorems).5
Krishna Bharadwaj followed Sraffa’s noble example and did not refer to her results and propositions, particularly in Bharadwaj (1970) as theorems; nor did she allude to her procedural demonstrations as proofs. In what can only be termed a truly prescient and remarkable footnote, Krishna Bharadwaj pointed out (ibid, footnote 13, p 415; bold italics added):
Analytically there is no loss of generality involved in a procedure of successive consideration of production systems using a different method of production for only one of the commodities common to them as, given all possible systems of production, it could not lead to any different outermost boundary of wageproﬁ t curves. Incidentally, it would be noted that whatever be the number of commodities produced by different methods in the two systems the maximum number of switching possibilities would still be equal to the total number of distinct (without double counting) basics in the two systems together.
In the language of conventional mathematical economics there is (at least) one formal theorem in this observation – that which gave the paper its title “The Maximum Number of Switches between Two Production Systems”. More importantly, there is also a clear hint on stating, as a theorem, a result on the uniqueness of the “outermost boundary of the wageproﬁt curves”, but that is not all. There is also a clear statement of a way to use – a mode of constructing a procedure – this unique outermost boundary of the wageproﬁt curves in comparative studies between production systems, algorithmically. The precise computational complexity of the procedure is, of course, not mentioned, but such things did not exercise the mind or the pen of many economists then, and they do not do so even now, despite much hype about computable general equilibrium theory (even in developmental contexts) and computational economics.
The above observations are summarised in three numbered statements, in the concluding Section iv of this classic paper, simply referred to in terms of the phrase, “To sum up”. They encapsulate, again in the jargon of the more formal, if less justiﬁed, mathematical economist two absolutely fundamental theorems, both of which have important applicable richness. To state them more formally, giving them the completely justiﬁ ed preﬁ x “Bharadwaj”:
Bharadwaj’s Theorem I (ibid: 42324): At a switch point the adjacent production systems differ in the method of production for only one of the commodities common to them. The maximum number of switching possibilities between two such systems is equal to the number of distinct (i e, without double counting) commodities entering, directly or indirectly, into the two alternative
methods which respectively characterise the two systems. Bharadwaj’s Theorem II (ibid: 424): The choice of the value unit does not affect
the maximum number of switching possibilities. If I succumb to the temptation of pseudomathematical practice, then I would add what those who indulge in such mumbo jumbo call a “Remark”:
Remark The economic content of Bharadwaj’s Theorem II is simply that the choice of the numeraire does not affect the content of Bharadwaj’s Theorem I.
As for Bharadwaj’s Theorem I, it may be useful to recall Sraffa’s remark (sic!) on the different way orthodox, marginal, theory interprets and utilises the existence of switch points, as clearly brought out in Kurz (op cit: p 4):
The characteristic feature of switchpoints is that both distributive variables, wages, w, and the rate of proﬁ ts, r, are rigidly ﬁ xed and are the same in both systems. Sraffa found this marginalist presupposition unacceptable. In a note written on 15 December 1943 he stressed that:
[T]he socalled determination is due to circumstances, which exist, not in the real world of actual production, but only
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in the world of imagination and possibilities: they are not intrinsic to the [actual] system and other levels [of w and r] cannot be ‘inconsistent’ with it. As far as the real, existing, system described by the equations is concerned, any levels [of w and r] are consistent with it (Sraffa Papers D3/12/35: 43(2)).
In other words, the orthodox – i e, marginalist – mode of analysing a given system of production is, counterfactually, to postulate the existence of hypothetically “adjacent” – i e, “marginally different from” – production systems. The important qualiﬁcation by Sraffa is that such orthodox counterfactual reasoning is valid “only in the world of imagination and possibilities: they are not intrinsic to the [actual] system”, i e, in the world of parables and have nothing to do with realism.
Bharadwaj’s Theorem I is entirely consistent with Sraffa’s strictures and has nothing to do with counterfactual, hypothetical, “adjacent” production systems.
3 Tractably Computing WageProﬁ t Frontiers
If one measures labour and land by heads or acres the result has a deﬁ nite meaning, subject to a margin of error: the margin is wide, but it is a question of degree. On the other hand if you measure capital in tons the result is purely and simply nonsense. How many tons is, e g, a railway tunnel?6 (Sraffa (1936); italics added.)
Some 20 years ago I noted7:
Production structures carry with them natural prices corresponding to particular analytical assumption about the economics of the production system. What is needed is a device for extracting these prices from the observed data of a functioning economy.
Now, 20 – or so – years later, Stefano Zambelli has developed an algorithm to compute the index I constructed, based on wageproﬁt curves, to compare production systems, to measure technological progress. The connection with the Bharadwaj Theorems is the relevant point in the implementation of the algorithms. Brieﬂy, socalled “bruteforce” algorithms are of exponential time complexity, as the methods of production, for any production system, is increased in number. Using the two Bharadwaj Theorems, the exponentialtime computational complexity can be reduced to polynomialtime complexity in increasing the number of methods of production.
Outline of the BruteForce Algorithm
(Zambelli and Fredhom 2010):
The proverbial “curse of dimensionality” enters copiously in the above algorithm because, for each rate of proﬁ t, all possible combinations of methods of production have to be evaluated. For example, for N countries, each with n production sectors, the total number of wageproﬁt frontiers are Nn. In the applications in Zambelli and Fredholm (op cit), N = 64 and n = 23, which implies Nn = 6423  3.5*1041. A standard computer, running one whole year, must evaluate 1.1*1034 wageproﬁt frontiers per second. No computing facilities existing in standard format or at “normal” institutions have the capacity to do any such computing today – or in the reasonable future.
This is where the Bharadwaj Theorems enter and help dissipate the “curse of dimenionality”: if the computational complexity measure of a bruteforce algorithm is given by S, then utilising the Bharadwaj Theorems this measure can be reduced to the order of S1/n.
I shall not enter into technical details of the algorithm that has been devised, using the Bharadwaj Theorems, in this paper (the interested reader can see the details in Velupillai, 1994 and Zambelli and Fredholm 2010). The point I wish to make, in this homage to the memory of Krishna Bharadwaj, is the following. Theoretical results, entirely motivated by empirical anchorings, devised by Krishna Bharadwaj, based on the framework developed by Sraffa in PCMC, suggested an algorithmic procedure for computing wageproﬁ t frontiers. These, in turn, were the basis on which I constructed an index, which now can be computed with measurable computational efﬁciency, as a result of Zambelli’s understanding of the relevance of the Bharadwaj Theorems for this purpose.
Surely, this is an effective counterexample to the pointless and ceaseless accusations of abstractionmongering and ahistoricity of the Srafﬁ an framework? And it was achieved by a combination of theoretical ingenuity on the part of Krishna Bharadwaj, intuitive knowledge of the procedural nature of the demonstrations of propositions in PCMC by Bharadwaj, felicitously combined with Zambelli’s desire to circumvent the neoclassical counterfactual fudge of productivity comparisons using untenable production functions, having as arguments nonmeasurable inputs.
4 Brief Reﬂections of Krishna Bharadwaj’s Srafﬁ an Methodology
[T]he review, ‘Value through Exogenous Distribution’, appeared in August 1963. In order to write it, she followed a demanding intellectual pilgrim’s progress, taking the same journey that Sraffa himself had taken over the 30 to 40 years prior to the publication of the book. She read, as he did, Smith, Ricardo, Malthus, Marx, Mill, Jevons, Marshall, Walras, Wicksell. The result was her outstanding review article …
– Harcourt, op cit, p 301. I believe the Srafﬁan basis of Krishna Bharadwaj’s scientiﬁc methodology is most clearly evident in Bharadwaj (1970). In particular, in her mode of discussing the nature of the economically motivated difference between the Srafﬁ an categories of basic and nonbasic commodities and the mathematical distinction between decomposable and indecomposable matrices. This difference, when obfuscated by illdigested mathematics, grafted on to an economic theory and its categories without serious anchoring in observable relevance, grounded in the traditions of economic thought, leads to the kind of appeals made to “aggregation” by Peter Newman (op cit) and Levhari’s (1965) hasty conclusions. Her concluding remarks on this double distinction are worth their economic weight in gold (ibid: 423):
The classiﬁcation of commodities into basics and nonbasics in a given system uses more of
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the available information about the system than does the classiﬁcation of the system as decomposable or indecomposable. The additional information incorporated in the former distinction is essential for the discussion of switching possibilities between two systems.
It is not surprising, then, that it was Krishna Bharadwaj who derived the important two theorems that enabled us to construct an algorithm and compute efﬁ ciently an intractable measure and use it for the eminently empirical purpose of comparing the productive cap abilities of two economic systems.
Krishna Bharadwaj had an admirably complete command of the traditions of economic thought that underpinned much of contemporary economic theory. She, like Sraffa, may not have had command over the formal side of mathematics, but had an unsurpassed intuition to guide her towards an understanding of the way economic theory had to be developed to serve in the interpretation of the evolution of economic institutions – and how they might be shaped to further humane ends, in their future development.
[Krishna Bharadwaj, whose Srafﬁ an Scholarship was supreme, died on 8 March, 20 years ago. Krishna Bharadwaj began her remarkable journey towards a mastery of Sraffa (1960) in 1962, a half a century ago. My own studies of Sraffa began 10 years later, in 1972, but it was only in 1992, the year Krishna died, that I was able to develop an applicable index, based on Sraffa’s magnum opus. This particular saga was completed by Stefano Zambelli, who constructed an algorithm to use my index, based on a deep and abstract result in Bharadwaj (1970). I am indebted to my colleague and friend (of 30 years), Stefano Zambelli, whose own adherence to Srafﬁ an methods has been a beacon of light in my eternal struggles with Production of Commodities by Means of Commodities.]
Notes
1 Guglielmo Chiodi drew my attention to a handwritten letter by Sraffa to Gareganani, dated 22 June 1962, referring to Newman’s review as “È tipico effetto della troppa (e troppo poco digeribili) matematica”, which in my ‘free translation’ would read: “[Newman’s review is] a typical outcome of too much (and little digestible) mathematics”.
2 Sraffa, in the above letter to Garegnani, refers to this review in blunt terms as: “‘Harrod è una perdita di tempo’ (again, a ‘free translation’ by me would render it as: ‘Harrod is a waste of time.’”).
3 Curiously, the one issue of EPW inaccessible via JSTOR is Vol 27, # 12 (1992), which is supposed to contain Deena Khatkhate’s ‘Letter to the Editor’ referring to Sraffa’s own letter to Sachin Chaudhuri stating that the Bharadwaj review of PCMC “was one of the three best reviews” of his book (Omkarnath, op cit, p 460). Omkarnath has kindly made available to me a copy of this Letter to the Editor of EPW, a perusal of which leaves the “mystery” of the other two “best reviews” unresolved, however.
4 I have maintained, ever since I noted it in my review of Pasinetti’s Lectures on the Theory of Production (Velupillai 1980), that every proved proposition in PCMC is mathematically rigorous and all attempts at recasting the formalism employed in the book in terms of linear algebra are unnecessary (Velupillai 2008). In particular, it is not at all necessary to invoke the celebrated results of Perron & Frobenius to “prove” Sraffa’s “theorems”.
5 Two unfortunate examples of this mischief by socalled mathematical economists are those by Burmeister (1968) and Hahn (1982), as I have pointed out in Velupillai (2008).
6 The quote goes on (italics added): If you are not convinced, try it on someone who has not been entirely debauched by economics. Tell your gardener that a farmer has 200 acres or employs 10 men – will he not have a pretty accurate idea of the quantities of land & labour? Now tell him that he employs 500 tons of capital, and he will think you are dotty – (not more so, however, than Sidgwick or Marshall).
7 In a paper prepared for the World Bank, with the collaboration of Stefano Zambelli (Velupillai 1994). The paper had the “tragic” fate of being approved for publication as a World Bank Discussion paper (on the basis of one supportive referee’s report, by Lance Taylor, although heavily “denounced” by the second referee, T N Srinivasan), and then the commitment to publish it in that mode completely ignored!
References
Bharadwaj, Krishna (1963): “Value through Exogenous Distribution”, The Economic Weekly, Vol 24, 24 August, pp 145054.
Bradford, Wylie and G C Harcourt (1997): “Units and Deﬁnitions” in G C Harcourt and P A Riach (ed.), A Second Edition of the General Theory, Volume 1 (London: Routledge).
Burmeister, Edwin (1968): “On a Theorem of Sraffa”, Economica (New Series), Vol 35, No 137, February, pp 8387.
Hahn, Frank H (1982): “The NeoRicardians”, Cambridge Journal of Economics, Vol 6, December, pp 33374.
Harcourt, Geoffrey C (19931994): “Krishna Bharadwaj, August 21, 1935 – March 8, 1992: A Memoir”, Journal of Post Keynesian Economics, Vol 16, No 2, Winter, pp 299311.
Harrod, Roy (1961): “Review of Production of Commodities by Means of Commodities”, Economic Journal, Vol LXXI, pp 78387.
Kurz, Heinz (2011): “Sraffa, Keynes and PostKeynesianism”, Seminar Paper, Trento, October (Forthcoming in Peter Kiesler and Geoff Harcourt (ed.), The Handbook of Post Keynesian Economics (Oxford: Oxford University Press 2012).
Levhari, David (1965): “A Nonsubstitution Theorem and Switching of Techniques”, Quarterly Journal of Economics, Vol LXXIX, No 1, February, pp 98105.
Newman, Peter (1962): “Production of Commodities by Means of Commodities: A Review Article”, Schweizerische Zeitschrift für Volkswirtschaft und Statistik, Vol XCVIII, March, pp 5875.
Omkarnath, G (2005): “ ‘Value through Exogenous Distribution’: A Review Article in 1963”, Economic & Political Weekly, Vol 40, No 5, 29 January4 February, pp 45964.
Samuelson, Paul Anthony (1962): “Parable and Realism in Capital Theory: The Surrogate Production Function”, Review of Economic Studies, Vol 39, No 3, June, pp 193206.
Shaikh, Anwar (2012): “The Empirical Linearity of Sraffa’s Critical OutputCapital Ratios” in Christian Gherke, Neri Salvadori, Ian Steedman and Richard Sturn (ed.), Classical Political Economy and Modern Theory: Essays in honour of Heinz Kurz (London: Routledge).
Sraffa, Piero (1936): “Letter to Joan Robinson”, 27 October 1936, cited in Bradford & Harcourt (1997), p 131.
Velupillai, K Vela (1980): “Review of L L Pasinetti’s ‘Lectures on the Theory of Production’”, Journal of Economic Studies, Vol 7, #1, pp 6465.
Zambelli, Stefano and Thomas Fredholm (2010): “An Algorithmic Measurement of Technological Progress”, ASSRU Discussion Paper 1006, December.
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