Home > Uncategorized > Arrow-Debreu and the Bourbaki illusion of rigour

Arrow-Debreu and the Bourbaki illusion of rigour

from Lars Syll

By the time that we have arrived at the peak first climbed by Arrow and Debreu, the central question boils down to something rather simple. We can phrase the question in the context of an exchange economy, but producers can be, and are, incorporated in the model. There is a rather arid economic environment referred to as a purely competitive market in which individuals receive signals as to the prices of all goods. All the individuals have preferences over all bundles of goods. They also have endowments or incomes defined by the prices of the goods, and this determines what is feasible for them, and the set of feasible bundles constitutes their budget set.bourbaki Choosing the best commodity bundle within their budget set determines their demand at each price vector. Under what assumptions on the preferences will there be at least one price vector that clears all markets, that is, an equilibrium? Put alternatively, can we find a price vector for which the excess demand for each good is zero? The question as to whether a mechanism exists to drive prices to the equilibrium has become secondary, and Herb Scarf’s famous example (1960) had already dealt that discussion a blow.

The warning bell was sounded by such authors as Donald Saari and Carl Simon (1978), whose work gave an indication, but one that has been somewhat overlooked, as to why the stability problem was basically unsolvable in the context of the general equilibrium model. The most destructive results were, of course, already there, those of Hugo Sonnenschein (1974), Rolf Mantel (1974), and Debreu (1974) himself. But those results show the model’s weakness, not where that weakness comes from. Nevertheless, the damage was done. What is particularly interesting about that episode is that it was scholars of the highest reputation in mathematical economics who brought the edifice down. This was not a revolt of the lower classes of economists complaining about the irrelevance of formalism in economics; this was a palace revolution.

Alan Kirman

Mainstream theoretical economics is still under the spell of the Bourbaki tradition in mathematics. Theoretical rigour is everything. Studying real-world economies and empirical corrobation/falsification of theories and models nothing. Separating questions of logic and empirical validity may — of course — help economists to focus on producing rigorous and elegant mathematical theorems that people like Lucas and Sargent consider as “progress in economic thinking.” To most other people, not being concerned with empirical evidence and model validation is a sign of social science becoming totally useless and irrelevant. Economic theories building on known to be ridiculously artificial assumptions without an explicit relationship with the real world is a dead end. That’s probably also the reason why Neo-Walrasian general equilibrium analysis today (at least outside Chicago) is considered a total waste of time. In the trade-off between relevance and rigour, priority should always be on the former when it comes to social science. The only thing followers of the Bourbaki tradition within economics — like Karl Menger, John von Neumann, Gerard Debreu, Robert Lucas and Thomas Sargent — has given us are irrelevant model abstractions with no bridges to real-world economies. It’s difficult to find a more poignant example of a total waste of time in science.

Applying an axiomatic hypothetico-deductive system to the real world can only be done by means of a mapping, which creates a model for the axiomatic system. These mappings then lead to assertions about the real world which require empirical verification. These assertions (which are proposed scientific laws) can NEVER be proven in the sense that mathematical theorems can be proven …

hqdefaultThe scientific method arose as a rejection of the axiomatic method used by the Greeks for scientific methodology. It was this rejection of axiomatics and logical certainty in favour of empirical and observational approach which led to dramatic progress in science. However, this did involve giving up the certainties of mathematical argumentation and learning to live with the uncertainties of induction. Economists need to do the same – abandon current methodology borrowed from science and develop a new methodology suited for the study of human beings and societies.

Asad Zaman

Models may help us think through problems. But we should never forget that the formalism we use in our models is not self-evidently transportable to a largely unknown and uncertain reality. The tragedy with mainstream economic theory is that it thinks that the logic and mathematics used are sufficient for dealing with our real-world problems. They are not! Model deductions based on questionable assumptions can never be anything but pure exercises in hypothetical reasoning. And that kind of reasoning cannot establish the truth value of facts. Never has. Never will.

  1. Yoshinori Shiozawa
    July 29, 2019 at 7:17 pm

    Alan Kirman’s parable is superb and deep. Sonnenschein-Mantel-Debreu was a palace revolution. It does not change class structure. General Equilibrium theory remained the highest priests in economics. Shouting Hurrah to a palace revolution is a silly comedy of the populace.

    If one wants a real revolution, it is necessary to change the people’s mind.

    Arrow and Hahn refers to what we can call Adam Smith problem. The aura of General Equilibrium theory lies in the fact that Arrow, Debrue and Hahn had solved the Adam Smith problem. They explain its significance in this sentence:

    What ever the source of the concept [my note: equilibrium], the notion that a social system moved by independent actions in pursuit of different values is consistent with a final coherent state of balance, and one in which the outcomes may be quite different from those intended by the agents, is surely the most important intellectual contribution that economic thought has made to the general understanding of social processes. (Arrow and Hahn, General Competitive Analysis, 1971, p.1)

    However, Arrow and Debreu’s GE model requires that agents have an infinite sight and complete rationality. Morioka, Taniguchi and I have solved the Adam Smith problem showing how a large economic system as big as a nation or the world works with agents with bounded rationality and narrow range of sight. In other words, we show how real economy works.

    Please see our new book: Microfoundations of evolutionary economics

    This book presents a new vision on the working of modern economy that is totally different from neoclassical economics. Readers can check it easily because we do not draw on the holy trinity of neoclassical economics: rationality (optimization), equilibrium, and (methodological) individualism.

  2. Yoshinori Shiozawa
    July 30, 2019 at 6:32 am

    This is my comment July 28, 2019 at 3:55 am to

    “A modern economy is a very complicated system.” (Solow in Syll’s citation)

    To understand this complex system, it needs to develop appropriate tools. Some of them may take the form of mathematics. Don’t throw the baby out with the bathwater.

    Those who want to understand this complex system should not spare themselves efforts to obtain the necessary capability. Don’t make all aspirants idle economists.

    • August 4, 2019 at 9:20 am

      I have not (yet?) read your book, but support your comment. As a mathematician it seems to me that there is a lot of what Keynes called ‘pseudo-mathematics’ to be thrown out, but I fail to see how we can make much progress without some mathematics. But which?

      Do you have a view on why so many continue to go so astray in thinking about uncertainty, and how to enlighten them? (E.g., which mathematics of uncertainty do you favour?)

  3. Yoshinori Shiozawa
    July 30, 2019 at 7:24 am

    Lars Syll does not understand (or more exactly is not willing to understand that Bourbaki was not a simple movement for an ideology that Syll summarized as “Theoretical rigor is everything.” Their mathematics was very abstract but this was necessary to to re-organize mathematics in a more integral and intelligible system of knowledge. Bourbaki provides the way to understand the already extremely ramified mathematics at that time and made clear the structure of mathematics in various fields. Mathematicians on the edge are discovering an astonishing connection between fields that have been thought to be very different in nature.

    Lars Syll and Asad Zaman are wrong in thinking that mathematics (axiomatic system is a form of mathematics) is a simple system of tautology. Mathematics is tautology only in the eyes of God. Syll and Zaman are forgetting that people economists and mathematicians including are being whose rational capability is limited. If the economy is a complex and large system, it is not easy to understand the structure and processes of the whole system. Mathematics and axiomatic systems are a tool to analyze and understand this complex system. Don’t be mislead by classical science philosophy before the complexity became topics of scientific investigation.

    The most simple evidence is to give an example. Please read Chapter 2 and Chapter 4 of our book (cited in the first of my comments). Chapter 2 gives a new theorem why price system is stable by a framework totally different from New Keynesian’s price stickiness argument by menu cost. Moreover, the theorem proves that input substitution (which is assumed to be important economics mechanism that supports neoclassical worldview) plays no important role in the price determination. This is simply the total refutation of marginal productivity theory of prices.

    Chapter 4 (by Morioka) is a proof that complex system of input-output correlations can follow the slow movement of final demand. This proves that modern economic system works very differently from what neoclassical economists (and common people educated by them) think. It is not the work of price system that brings demand and supply always nearly equal. On the contrary, it is the firms’ operations to adjust their product inventories at an appropriate level.

    The two results (one in Chapter 2 and one in Chapter 4) are only obtained by long enduring efforts to know the real structure of economic system. They are highly mathematical and Chapter 2 adopts even a sort of axiomatic method. But without using mathematical methods, these important results were impossible. They are mathematical but their contents are strongly anti-neoclassical.

  4. July 30, 2019 at 12:44 pm

    By changing axioms, we can prove anything we like about the “real world” – as long we dont require a match between axioms and reality

    • Yoshinori Shiozawa
      July 30, 2019 at 2:06 pm

      Have you really read our book?

  5. Yoshinori Shiozawa
    August 2, 2019 at 7:52 am

    The trouble with Lars Syll is that he talks as if no heterodox economics exists. I know very well there are many strands or schools among heterodox economics and they all have many defects and weakness. However, if economics can develop beyond actual mainstream economics, the burgeon of new economics must come out from at least one of them. (Of course, it is possible that some of them will be unified as different parts of a more coherent theory.)

    Lars Syll’s talks, it seems to me, as if a new scientific economics emerges once a good philosophy and methodology are established. This is a pure fiction and betrays the history of all sciences and disciplines (natural or social). A science grows very slowly and sinuously. It requires enormous efforts and trials and errors. If Lars Syll wants to be a useful philosopher of economics, he should better pay more efforts on the state of heterodox economics. If he finds some deficiency and flaws, and if he tell those economists who work in those fields, the economists would reflect on their problems and try to find a better way. In a very wide sense, criticizing neoclassical and mainstream economics may be a part of above efforts, but actually it works in a very negative way in this blog site, where almost all readers already recognize there are fatal flaws in neoclassical economics. The next target should be to show a new direction for the construction of a new economics. If not, his efforts enhance disbelief on economics in general including heterodox economics. No science emerges from nothing. A science grows.

    • Craig
      August 2, 2019 at 7:06 pm

      Except for its critique of Lars as an individual, instead of a general critique of economic thinking, this is actually a very good post and nascent recognition of the real problem in economics. Yes, virtually everyone here knows there are problems with neo-classical economics, but economics itself is not the problem. 99% of the present economic problems we face could be rapidly resolved by changing the monetary paradigm. And it’s obvious why:

      1) The monetary paradigm of Debt Only for the sole form and vehicle for the distribution of credit-money hasn’t changed for over 5000 years. This despite repeated historically identified systemic economic collapses due to inability to service debts.

      2) Giving a public monopoly on credit creation is problematic…unless it is firmly and completely logically aligned with an unimpeachable ethic like the natural philosophical concept of grace. A private monopoly on credit creation is an utter contradiction of free enterprise theory and such an obvious ignoring of Lord Acton’s dictum that power corrupts and absolute power corrupts absolutely, so if we want free enterprise and individual and systemic freedom we will proceed with post haste in ending the present idiocy.

      3) Monetary Gifting as the new monetary paradigm strategically implemented with a 50% Discount/Rebate monetary policy at the point of retail sale immediately and terminally ends the two deepest and seemingly unresolvable problems of modern economies, that is, individual and systemic monetary scarcity and chronic price and asset inflation. The practical and epistemological need to bring directness of monetary policy to the individual and to enterprise is perfectly analogous to bringing directness to one’s relationship with god that resolved the monopoly powers and structural indirectness demanded by the catholic church prior to the Reformation.

      As I have posted here numerous times the new insight regarding the terminal ending point of the entire economic process at retail sale, and hence the realizations that it is also the point of the summing of all economic factors and the supreme leveraging, pivoting and so paradigm changing power point for monetary policy perfectly aligns with the fact that a new tool, insight and/or policy has always accompanied paradigm changes.

      Why blather on about the complexities of economics when the new monetary paradigm enables us to resolve its worst problems, and prosperously take a huge deep breath while also being able to begin addressing the ecological and energy/entropy problems because finance is now a public utility that serves us instead of being the civilization long dominating parasite it still is today?

      • Yoshinori Shiozawa
        August 3, 2019 at 4:47 am

        Dear Craig,

        I have no intention to criticize Lars personally. I am always talking about research strategy or science strategy for a new economics. I repeatedly pointed the same thing as comments to Lars’s posts, but he does not change his opinion much and I have to repeat my position. That’s all.

  6. Craig
    August 2, 2019 at 7:16 pm

    “The next target should be to show a new direction for the construction of a new economics. If not, his efforts enhance disbelief on economics in general including heterodox economics.”


    “No science emerges from nothing. A science grows.”

    Again correct, but sciences grow until they become stymied by their own orthodoxies and internal inconsistencies and so need necessarily to be transformed by a new paradigm in that area of human endeavor.

    Isn’t 5000 years of inevitable economic failure due to the monetary paradigm long enough to wait for a new one?

    • Yoshinori Shiozawa
      August 3, 2019 at 4:59 am

      Sciences “need necessarily to be transformed by a new paradigm in that area of human endeavor.”

      Yes, indeed. Who prepares paradigm shift? In case of economics, it is economists (it doesn’t matter if they are professional or not, academic or not). We have to think how to emulate young aspirants to a good direction. Lars’s methodological philosophy does not seem to be a good strategy in this regards. We should also keep in mind this aphorism: It takes a theory to beat a theory.

  7. Ken Zimmerman
    August 3, 2019 at 12:02 pm

    This is all pointless and dreary. Only the people, all the people who create economies and re-create them as they fail, can tell us, show us what they are, how they work, and what they mean. Choose your tools to accomplish these tasks. The results will surprise and amaze those who want to be scientists, and scare the hell out of the others.

  8. Craig
    August 3, 2019 at 6:05 pm

    “Economists need to do the same – abandon current methodology borrowed from science and develop a new methodology suited for the study of human beings and societies.”

    This is a wisdom insight. And the “new methodology” you are seeking is in fact the integrative mental process of wisdom itself which includes, synthesizes, maintains the ethics of and ultimately transcends science. Wisdom is the best and most ethical integration of philosophy and policy, the ideal and the practical and of the mental modes of science and human consciousness itself. In other words of the truths, workabilities, applicabilities and highest ethical considerations in apparent OPPOSITES.

    What economics needs is not a static one moment in time ideology, but rather to cognite on the wisdom state of the dynamic, interactive, integrative free flow of factors aka grace and then find the place in the economic/productive process that the most liquid factor in a monetary economy, namely money, will facilitate both its ending point (retail sale) and its continual re-starting/free flowingness.

    IOW we must have a Wisdomics-Gracenomics that effects a new monetary paradigm.

  9. August 4, 2019 at 9:25 am

    Rather than “Mainstream theoretical economics is still under the spell of the Bourbaki tradition in mathematics.” I would say that ‘Mainstream theoretical economics is still under the spell of a perverted version of Bourbaki, which might be corrected by paying more attention to it and its successors in mathematics.” or maybe “Mainstream theoretical economics is still under the spell of the social scientists’ interpretation of the Bourbaki tradition in mathematics.” (Personal opinion, of course, but this aspect of this blog does bug me, particularly as I haven’t been able to comment!)

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