Sunday, June 12, 2016

Boylan and O’Gorman’s “Kaldor on Debreu: The Critique of General Equilibrium Reconsidered”

Thomas A. Boylan and Paschal F. O’Gorman’s paper “Kaldor on Debreu: The Critique of General Equilibrium Reconsidered” (2009) makes rewarding reading.

Boylan and O’Gorman review Kaldor’s work on general equilibrium in, for example, Kaldor (1972) and (1985), in which Kaldor had argued that the repudiation of Walrasian general equilibrium theory, especially as newly expounded in the models of Debreu and others (e.g., see Debreu 1959), was a precondition for any proper and truly empirical economic science.

More than this, Kaldor thought Walrasian general equilibrium theory was the greatest obstacle to the development of economics as a science (Boylan and O’Gorman 2009: 448; Kaldor 1972: 1237).

Boylan and O’Gorman briefly mention Kaldor’s methodological critique of general equilibrium theory, which has been interpreted by Lawson (1989) as within critical realism.

But that is not their concern. They state the fundamental issue on which they focus as follows:
“In this paper we do not propose to revisit these debates, but rather to engage with an aspect of Kaldor’s call for the ‘demolition’ of general equilibrium theory. The aspect of the problem that arguably troubled Kaldor most profoundly arose from the colonization of economics by mathematics in the neo-Walrasian research programme that arose after the Second World War. According to Kaldor, economists sought to create a ‘mathematical crystal’ (the expression is borrowed from Heisenberg) a logical system ‘which cannot be further improved or perfected’ (Kaldor, 1985, p. 60).” (Boylan and O’Gorman 2009: 448).
This occurred after World War II and involved the application of Bourbakist formalism by Debreu to Walrasian general equilibrium theory, a process which was connected to the intellectual history of the first half of the 20th century and especially the philosophy of mathematics (Boylan and O’Gorman 2009: 448).

Kaldor himself argued that other Post Keynesians focused too much on criticisms of marginal productivity theory to the detriment of other important critiques of neoclassical theory (Boylan and O’Gorman 2009: 450).

Boylan and O’Gorman then turn to Nicholas Kaldor’s marvellous little monograph Economics without Equilibrium (Armonk, N.Y., 1985).

They point out that Kaldor’s critique extended to a fundamental attack on neoclassical price theory, which, when compared to the real world reality of cost-based mark-up prices, is shown to be a fiction. And real world supply and demand, in contrast to marginalist theory, tends to be equated by use of inventory stock and management of capacity utilisation (for a summary of Kaldor’s Economics without Equilibrium, see here).

General equilibrium (GE) theory was invented by Walras, but found one of its major 20th century developers in the French economist Gérard Debreu’s (1921–2004).

At the time when Kaldor was writing, Gérard Debreu’s Theory of Value: An Axiomatic Analysis of Economic Equilibrium (1959) was a notable exposition of the mathematised Walrasian general equilibrium theory. Debreu had adopted Cantorian set theory in his mathematisation and proof of general equilibrium (Boylan and O’Gorman 2009: 452).

Debreu was influenced by the Bourbaki school of mathematics that upheld mathematics as an autonomous discipline essentially divorced from the real world (Davidson 2009: 170). That Bourbakianism had in turn been influenced by the mathematical formalism of David Hilbert (see more on that here) (Boylan and O’Gorman 2009: 453). An interesting point is that mathematical formalism is anti-realist (which means that anti-realists think human beings invent mathematics or construct it in some sense: mathematical entities are therefore not independent of conscious rational minds capable of understanding and perceiving these concepts).

As Kaldor noted, Debreu’s book does not purport to be an empirical description of reality, as, for instance, in describing how prices are actually formed in the real world (Kaldor 1972: 1237).

GE theory is an example of the aprioristic or deductive method in economics: a formal system in which a set of theorems are logically deduced from certain assumptions (Kaldor 1972: 1237). But there is no real attempt to verify whether the assumptions are true, or whether its system of established equilibrium prices has any explanatory power or even relevance to actual market economies (Kaldor 1972: 1238).

As Boylan and O’Gorman rightly emphasise, the axiomatic and deductive method of mathematised Walrasian general equilibrium theory also begins with outrageously anti-empirical axioms (or starting assumptions), which are either unprovable or blatantly falsified by reality (Boylan and O’Gorman 2009: 450).

In this respect, although Boylan and O’Gorman do not mention this, the deductivist general equilibrium method is a type of apriorism. The ultimate end of apriorism as a method – when combined with the epistemology of Kant – is the insanity of Austrian praxeology, whose inferences are totally immune to any empirical refutation of any kind, as Ludwig von Mises (with a straight face) contended:
“Praxeology is a theoretical and systematic, not a historical, science. Its scope is human action as such, irrespective of all environmental, accidental, and individual circumstances of the concrete acts. Its cognition is purely formal and general without reference to the material content and the particular features of the actual case. It aims at knowledge valid for all instances in which the conditions exactly correspond to those implied in its assumptions and inferences. Its statements and propositions are not derived from experience. They are, like those of logic and mathematics, a priori. They are not subject to verification or falsification on the ground of experience and facts. They are both logically and temporally antecedent to any comprehension of historical facts” (Mises 2008: 32).

The theorems attained by correct praxeological reasoning are not only perfectly certain and incontestable, like the correct mathematical theorems. They refer, moreover, with the full rigidity of their apodictic certainty and incontestability to the reality of action as it appears in life and history. Praxeology conveys exact and precise knowledge of real things.” (Mises 2008: 39).
Was the formalist program of Hilbert (as passing into the Bourbaki school of mathematics and Debreu’s mathematised Walrasian general equilibrium theory) any better than the praxeological epistemology of Mises?

That is a tricky question, but at least Mises thought his starting axioms were true of the real world because he thought they were synthetic a priori – whereas mathematical formalism holds its axioms and derived theorems as an “empty or meaningless, purely formal system” (Boylan and O’Gorman 2009: 453).

In other words, this is an analytic a priori system with a purely abstract or imaginary character (even for starting axioms), whose application to the real world is not even actively asserted – let alone proved!!

A case in point was the use of the concept of “Cantorian actual infinity” derived by Debreu from Cantor’s set theory (Boylan and O’Gorman 2009: 455–458). This was used by Debreu to prove his general equilibrium solution, but Cantorian actual infinity is the very definition of a totally abstract, anti-empirical, and imaginary concept. It’s 100% pure mathematics which always, epistemologically speaking, remains an analytic a priori system.

Boylan and O’Gorman conclude:
“… there is no justification for this economic interpretation of Debreu’s ingenious piece of Cantorian pure mathematics. Debreu’s proof does not support this economic interpretation. Debreu’s so-called economic equilibrium exists only in the domain of Cantorian actual infinity, which transcends any process limited to socio-historical time. More precisely, since the method of the proof of existence is inherently non-constructive, i.e. cannot be carried out in a finite number of steps taken one at a time, Debreu’s equilibrium cannot be given either a finite or a potentially infinite interpretation. Debreu’s equilibrium point is merely shown to exist in a non-temporal, actual infinite Platonic domain, which cannot in any finite effective way be realized in the socio-historical world in which economic agents operate. Alternatively, in the language of the
Hilbertian formalist, there is no evidence to support the assumption that the logical possibility, established by Debreu’s proof of existence, could be realized in any socio-economic system where real historical time operates.” (Boylan and O’Gorman 2009: 458).
That is to say, Debreu’s equilibrium solution has validity only in the world of pure mathematics, nowhere else (Boylan and O’Gorman 2009: 459).

Clearly, this won’t do. In fact, the whole mathematical formalist program as spun out by Debreu as the foundation of modern Walrasian general equilibrium theory won’t do either as a method for an empirical science – it belongs in a lunatic asylum.

As Kaldor complained, such “pure theory is not [sc. even] intended to describe reality” (Kaldor 1972: 1238).

It is not often you discover an epistemological method more half-baked than Mises’ praxeology, but there you go.

But, to return to Kaldor, he identified the major unverified and plainly unrealistic assumptions of neoclassical theory as follows:
(1) that producers maximise profits;
(2) the existence of perfect competition;
(3) linear-homogenous and continuously differentiable production functions;
(4) impersonal market relations;
(5) information communicated by market clearing prices; and
(6) perfect knowledge of all relevant prices and perfect foresight. (Kaldor 1972: 1238).
Another delusional assumption is that real economies can approach, or are close to, a state of equilibrium (Kaldor 1972: 1239).

Yet another is that a convergence to an equilibrium state is governed by exogenous forces such as essentially unchanging production patterns over time (Kaldor 1972: 1244). But the reality is that important sectors are subject to increasing returns to scale (Kaldor 1972: 1244), as Kaldor’s teacher the Marshallian Allyn A. Young at the London School of Economics (LSE) had taught him (see Young 1928 and Sraffa 1926).

Once increasing returns to scale are assumed, a dynamic system has endogenous forces that can drive it away from equilibrium.

Kaldor rightly rejects all these neoclassical assumptions as untrue, and concludes that the main theorems of neoclassical theory “cannot possibly hold in reality” (Kaldor 1972: 1240). Boylan and O’Gorman (2009: 454) agree and charge that “Debreu’s Theory of Value, seen as a work aimed at attaining the highest standards of logico-mathematical rigour and precision, is a purely formal uninterpreted system having no connection whatsoever to any branch of reality in general or real economic processes in particular.”

The upshot of all this, Kaldor argued, is that an empirical economic science needs to be built on realistic starting assumptions such as “stylized facts” (Boylan and O’Gorman 2009: 451–452).

The subsequent fate of Walrasian general equilibrium theory was to construct models so bizarrely contrary to the real world that one can only stand in awe at its ability to be disconnected from reality in its insulated madhouse (for example, no real world money in its models!).

But this also has a long history in the West, and can be best described as Platonic mathematical mysticism – an intellectual tradition which thinks that the beautiful and eternal truths of mathematics or geometry reveal the true nature of reality, whereas the world of woe and pain we live in happens to be just an imperfect reflection of the Platonic Realm of the Forms. That is to say, mathematics is truth; everything else is just a pale reflection of it (as pointed out here).

(Admittedly the Platonists were mathematical realists, and mathematical formalists are anti-realists, but let us put this arcane point aside.)

If the real world does not work in the way that a neoclassical model predicts, then – for the committed neoclassical theologian – the real world economy must be wrong. If the real world does not conform to the beautiful and eternal truth of Walrasian theory, then it must be evil government intervention, or villainous trade unions, or inflexible wages and prices, or labour market regulations, or price controls, or something – anything that pollutes the truth of the model.

At any rate, Kaldor’s insightful complaint was “equilibrium theory has reached the stage where the pure theorist has successfully (though perhaps inadvertently) demonstrated that the main implications of this theory cannot possibly hold in reality, but has not yet managed to pass his message down the line to the textbook writer and to the classroom” (Kaldor 1972: 1240). Has anything really changed today?

Further Reading
Pilkington, Philip. 2012. “Divine Mathematics – Neoclassical Economics as Spiritual Meditation,” Naked Capitalism, August 21, 2012.

Syll, Lars P. “Axiomatic economics — the Bourbaki-Debreu Delusion,” 21 January, 2016.

Boylan, Thomas A. and Paschal F. O’Gorman. 2009. “Kaldor on Debreu: The Critique of General Equilibrium Reconsidered,” Review of Political Economy 21.3: 447–461.

Davidson, Paul. 2009. John Maynard Keynes (rev. edn.), Palgrave Macmillan, Basingstoke.

Debreu, Gerard. 1959. Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Wiley, New York and London.

Kaldor, Nicholas. 1972. “The Irrelevance of Equilibrium Economics,” Economic Journal 82: 1237–1252.

Kaldor, Nicholas. 1985. Economics without Equilibrium. M.E. Sharpe, Armonk, N.Y.

Lawson, T. 1989. “Abstraction, Tendencies and Stylized Facts: A Realist Approach to Economic Analysis,” Cambridge Journal of Economics 13: 59–78.

Mises, L. von. 2008. Human Action: A Treatise on Economics. The Scholar’s Edition. Mises Institute, Auburn, Ala.

Sraffa, P. 1926. “The Laws of Returns under Competitive Conditions,” Economic Journal 36: 535–550.

Young, A. A. 1928. “Increasing Returns and Economic Progress,” Economic Journal 38: 527–542.


  1. There is an extensive critique of 'neoclassical' economics in the prologue to R. A. Werners book 'New Paradigm in Macroeconoics'

  2. This is all very true stuff. Unfortunately PKEs fall into the same trap of confusing mathematics with reality and have a dreadful tendency to curve fit aggregation formula to empirical data that comes from a system operating under the rules of neoliberalism.

    So they still have no idea about how aggregation happens, nor have they any real idea at the macro level how changing the rules away from neoliberalism will alter the way the system responds at either the individual or aggregate level.

    Simplistic mathematical models, even ones with system dynamics in them like the ones used by Steve Keen, are just not going to be enough to describe human systems.

  3. These quotes indicate that Boylan & O’Gorman don't really understand what they are talking about - the quote, especially after "more precisely" is basically PoMo of the type ridiculed by Gross & Levitt, or Bricmont & Sokal - and I much more sympathetic & generous to the targets of those four authors' critiques than they are. Though it is a slander to both sides, neither side is quite that bad or always that bad - pomo word salad is not the best weapon to denounce neoclassical math salad.

    Wikipedia on Constructive Analysis in particular, the intermediate value theorem is very helpful. Walk before running. A non-constructive existence proof - like that of the intermediate value theorem is just that. What it proves to exist should be, probably will be relevant in some way to the real world object of the theory if the assumptions of the theory are. Ungarbage in, ungarbage out. The real problem is finding / constructing in some sense (something like) what exists in the weak, classical "Cantorian" sense. Nobody sensible nowadays says such existence is meaningless.

    Criticisms of Hilbert & Bourbaki & formalism are nearly as badly informed and unhelpful. Hilbert's philosophical views were far deeper and superior to the usual caricature which was probably due to E. T. Bell putting words into his mouth. But this is because most of what is written on the philosophy of mathematics is just (now highly outdated) chattering (as is most on the philosophy of X that disparages or divorces itself from general philosophy, and the philosophical tradition - which is pretty much most of 20th century philosophy).

    James Franklin's Aristotelian Philosophy of Mathematics book is surely an exception to the snark above - but I haven't had the time to look at it.

  4. Nice article.

    As someone with applied mathematical training, I have no problems with viewing mathematical models as being idealised entities independent of the real world. The question is whether the models have any predictive value (which is tied to the question of falsifiability) when applied to a real world system.

    In practice, mainstream macro is based on non-measured quantities that are only estimated by using models that assume that the postulated economic dynamics are correct (e.g., a Kalman filter). The end result is that mainstream theory will be able to "explain" any observed economic trajectory.

    The upshot is that mainstream theory never need change; it just adds epicycles in order to keep the publication mill going. This is why things are not any different today, nor is there any plausible hope for internal reform.

  5. If I may ask, what is your stance on mathematical realism?