## The Arrow-Debreu obsession

from **Lars Syll**

I’ve never yet been able to understand why the economics profession was/is so impressed by the Arrow-Debreu results. They establish that in an extremely abstract model of an economy, there exists a unique equilibrium with certain properties. The assumptions required to obtain the result make this economy utterly unlike anything in the real world. In effect, it tells us nothing at all. So why pay any attention to it? The attention, I suspect, must come from some prior fascination with the idea of competitive equilibrium, and a desire to see the world through that lens, a desire that is more powerful than the desire to understand the real world itself. This fascination really does hold a kind of deranging power over economic theorists, so powerful that they lose the ability to think in even minimally logical terms; they fail to distinguish necessary from sufficient conditions, and manage to overlook the issue of the stability of equilibria.

Almost a century and a half after Léon Walras founded neoclassical general equilibrium theory, economists still have not been able to show that markets move economies to equilibria.

We do know that — under very restrictive assumptions — equilibria do exist, are unique and are Pareto-efficient. After reading Buchanan’s article** **one however has to ask oneself — what good does that do?

As long as we cannot show, except under exceedingly special assumptions, that there are convincing reasons to suppose there are forces which lead economies to equilibria — the value of general equilibrium theory is negligible. As long as we cannot really demonstrate that there are forces operating — under reasonable, relevant and at least mildly realistic conditions — at moving markets to equilibria, there cannot really be any sustainable reason for anyone to pay any interest or attention to this theory.

A stability that can only be proved by assuming “Santa Claus” conditions is of no avail. Most people do not believe in Santa Claus anymore. And for good reasons. Santa Claus is for kids, and general equilibrium economists ought to grow up.

Continuing to model a world full of agents behaving as economists — “often wrong, but never uncertain” — and still not being able to show that the system under reasonable assumptions converges to equilibrium (or simply assume the problem away) is a gross misallocation of intellectual resources and time.

**And then, of course, there is Sonnenschein-Mantel-Debreu!**

So what? Why should we care about Sonnenschein-Mantel-Debreu?

Because Sonnenschein-Mantel-Debreu ultimately explains why New Classical, Real Business Cycles, Dynamic Stochastic General Equilibrium (DSGE) and “New Keynesian” microfounded macromodels are such bad substitutes for real macroeconomic analysis!

These models try to describe and analyze complex and heterogeneous real economies with a single rational-expectations-robot-imitation-representative-agent. That is, with something that has absolutely nothing to do with reality. And — worse still — something that is not even amenable to the kind of general equilibrium analysis that they are thought to give a foundation for, since Hugo Sonnenschein (1972) , Rolf Mantel (1976) and Gerard Debreu (1974) *unequivocally *showed that there did not exist any condition by which assumptions on individuals would guarantee neither stability nor uniqueness of the equlibrium solution.

Opting for cloned representative agents that are all identical is of course not a *real solution* to the fallacy of composition that the Sonnenschein-Mantel-Debreu theorem points to. Representative agent models are — as I have argued at length here — rather an *evasion* whereby issues of distribution, coordination, heterogeneity — everything that really defines macroeconomics — are swept under the rug.

Instead of real maturity, we see that general equilibrium theory possesses only pseudo-maturity. For the description of the economic system, mathematical economics has succeeded in constructing a formalized theoretical structure,

thus giving an impression of maturity, but one of the main criteria of maturity, namely, verification, has hardly been satisfied. In comparison to the amount of work devoted to the construction of the abstract theory, the amount of effort which has been applied, up to now, in checking the assumptions and statements seems inconsequential.

I think: have you considered the idea that an idea about heaven when carefully used can help us understand about the world. We just need to understand the cave story and how to be careful. That is why, we should not underestimate neoclassical economics …. we just need people who understand the cave in addition to the sun. I hope you understand what I am saying. :) You can not work in one of them. You are working in the cave and they are working in the haven. Let’s mix them :)

Plato’s republic can give you idea you need

Arrow-Debreu DOES NOT REPRESENT any kind of “market”, but a CENTRALISED economy. It is NOT RELEVANT (different from “not realist”). Even if its equilibria were stable, the irrelevancy will be the same.

The Arrow/Debreu findings are based on a theoretical framework that connects the theory of rational choice with goods modelled as vectors. That theoretical framework is fundamentally flawed. It does not at all represent the choices that real humans make in the real economy. Therefore it can be considered a wrong or false theoretical framework. The mathematical properties of such a framework as researched by Arrow/Debreu are irrelevant.

Let’s simply clean the slate. Where does on find equilibrium for cosmic powered biology surfing Big Bang at life speed? The idea of equilibrium is hog wash borrowed from observation of precipitating salts when that was considered cutting edge science.

Lars, I never miss your essays and like your approach to the critique of economic theory. But when it comes to Arrow-Debreu, your assessment is 100% incorrect.

First problem: Arrow-Debreu is not about stability, you should know that. Arrow-Debreu is not about “markets moving economies to equilibrium”. The Arrow-Debreu model is about proving the existence of equilibrium through the use of a fixed point theorem (Kakutani’s fixed point theorem for upper semi-continuous correspondences). Thus, there is no time and no dynamics (as in stability) in the Arrow-Debreu model.

Second problem: why do you say “we know that under special assumptions equilibria exist”? This seems to imply that the proof of existence of a general competitive equilibria using an Arrow-Debreu model is correct, although it does have “special assumptions”.

According to this, the only defect is that there are special assumptions or, in other words, the model is too restrictive.

Well, that is not accurate. The truth is that even if you accept all of the “special assumptions” (things like convexity, compactness of individual possibility sets, plus all the properties of the mappings involved in the proof, etc.), I was saying that even if one accepts these assumptions you still do not have a good result at the end. The reason is that the procedure used to normalize the resulting price vectors when using the correspondences introduced by Arrow, Debreu, Hahn and Nikaido, destroys the economic interpretation of these mappings. Please take a serious look at the article by Benneti, C., A.Nadal and C. Salas, on “The law of supply and demand in the proof of existence of the general competitive equilibrium” in Ackerman, F. and A. Nadal, The Flawed Foundations of General Equilibrium (Routledge, 2004).

Stop disseminating a simplistic and erroneous view on the Arrow-Debreu model, a view that fits perfectly well with the objectives of mainstream theorists.

Arrow-Debreu and “Kakutani’s fixed point theorem for upper semi-continuous correspondences” are beside the point. Can economists even distinguish between a proportion and a quantity? If they can’t, the relevance of the higher abstract mathematics is moot.

Alejandro:

Sorry that you’re displeased with the article — but it certainly DOES NOT disseminate a “simplistic and erroneous view” on Arrow-Debreu! The main drift of the critique put forward by e.g. yours truly and Alan Kirman is that existence proofs may be fine, but as long as you can’ show that equilibria are stable and that there is some reasonable out of equilibrium dynamics that reverts the economy to them, there is from the perspective of realism and relevance no real resons for us to be immensely interested in those proofs.

As Franklin M. Fisher writes in “The stability of general equilibrium: results and problems”:

“An extremely prominent economist [Milton Friedman – LPS] long ago remarked to me in passing that the study of stability is unimportant because it is obvious that the economy is stable and, if it isn’t, we are all wasting our time. I pass over the question of whether it really is obvious that the economy is stable and observe that the issue of time-wasting by economists is not one of whether the economy is stable but rather of whether the theory is. A principal reason for studying general equilibrium in the first place is to examine the consistency of partial equilibrium analyses. Having powerful theories of the firm, the household, and the market, may not be very useful if all those theories cannot be true at the same time.

Clearly, the heart of this important consistency question lies in the existence of general equilibrium, and existence theory, fortunately, is a subject which is in pretty satisfactory shape. Nevertheless, there is a sense in which the consistency question cannot be regarded as settled with- out a satisfactory analysis of stability. It is no use knowing that there exist points at which all partial equilibrium propositions can be jointly true, if such points are not attainable. Hence the question of the stability of general competitive equilibrium is a vital one for economic theorists, particularly if the economy is stable, but not only then. If general equilibrium turns out to be stable only under a very restrictive set of assumptions, then, indeed, we will all have been wasting our time, for there will be something wrong with the partial theory that we think we understand.”

Lars, honored to be having this conversation. You are right that stability is the most important problem (this is exactly what Walras stated explicitly). But you (and Kirman) are wrong when you affirm that the proofs of existence are ‘right’. And there are several important reasons why you need to stop and look more closely at the proof of existence. Let me explain.

The fact is that everyone in the general equilibrium camp knows that stability analysis has produced negative results. This is well known since Arrow-Block-Hurwicz (1958-59), Scarf (1960), Hahn-Arrow (1971) and, of course, Sonnenschein-Mantel-Debreu (SMD, 1974). Franklin Fisher’s book on disequilibrium foundations of equilibrium economics is the best and most coherent analysis and summary of the reasons for the failure of stability analysis (especially important is the question of introducing money and trying to get rid of the nefarious auctioneer). As Frank Fisher told me a couple of years ago: nobody has worked on stability any more since twenty years ago because everyone knows (after SMD) this is a dead end street. Still, as ideology, the notion that there are good results in this field blossoms in the ocean of ignorance promoted by mainstream courses.

In very strong contrast, everyone in the camp of the champions of general equilibrium theory will tell you that the proof of existence is a solid, bullet-proof result. Precisely because stability is a dead end street, existence has become the only thing that general equilibrium theory can confidently put on the table as a solid positive result (together with the so-called fundamental theorems of welfare theory). Existence proofs, where there is no time and no dynamics, have a peculiar feature: the mappings are interpreted as being a solid representation of the law of supply and demand and this is enough to conclude that the existence proof is something like a good approximation to the results that stability analysis failed to deliver. So, from this perspective, a critique of the existence proof is important. I mean a serious internal consistency critique, not just a dismissive statement about lack of realism.

The notion that these proofs are fine is amazing in the penmanship of scholars who have written about SMD. Have you checked the mappings used by Nikaido, Arrow-Debreu, Arrow-Hahn and Debreu in their proofs? If the authors who produced these proofs claim that the mappings represent the law of supply and demand, then there is room for SMD here and the mappings will have no economic sense whatsoever. So why continue saying that the proofs are fine but irrelevant? Also read our critique to those mappings: unless you believe that the mathematical sense is enough to use those mappings, you cannot say the proof is right.

By stating that this erroneous view disseminates an erroneous view of the A-D model I don’t mean to be disrespectful. But I mean what I say: talking about the Arrow-Debreu model as if it was essentially a model for the analysis of stability is indeed inaccurate. In fact, the classic Arrow-Debreu model (published in Econometrica 1954) was built in order to use Kakutani’s fixed point theorem in the existence proof, not for stability analysis. A similar model can be constructed (see for example Arrow-Hahn) for dynamic analysis using a different mathematical approach.

I agree that arrow-debreu says nothing about dynamics or stability. I’ve actually seen and have a copy of nadal’s paper though i didn’t quite get the point in my cursory reading (i did look to see if anyone else had comments on it but didnt find any; there are a couple of other papers i have which seem to make some related points). Personally i think arrow-debreu is almost a tautology so i’m fairly convinced (its really just a 2 person 2 goods barter economy, generalized as 2-> infinity, which means a la Boltzmann’s derivation of the second law of thermodynamics, the ‘mechanical’ equililibrium is to an extent the same as the statistical one (at 0 temperature).

I dont remember the name of the operator now but i just remembered—perron-frobenius– you can use it it for both GET and the second law. (interesting stuff on this in physics, possibly ruelle and c beck etc.) (and you can be sure i’m not exactly clear on it—but it seems essentially the same as the idea behind ‘entropic gravity’ (verlinde)—how to get something from nothing, time from timelessness, and show a rock falling from the sky is mathematically the same as your hot coffee cooling off. (This is why s=k ln (omega) is on boltzmann’s tomb, since he was going to cantor’s paradise (omega).

It fails because people are not identical, goods are not all substitutible, infinity does not exist in the real world, etc. (unless, of course, you are one of the 4000 who are get refugee status in heaven with the jehovahs witnesses).

(of course, its approximately true—you can assume all people are essentially greedy angels with the best intentions and perfect dis/information, or that they are ‘dumb as dirt’ and operate in a zero information market, and you get the same result—so what does that tell you? It shows they have free will, but (a la Yakovenko’s interpretation of cobb-douglas utlity from a stat mech perspective) free will is distributed according to the law of large numbers, so you can believe in half a chance of being an exceptional ruler (‘law of series’, or make discoveries, like columbus, via data mining (slutsky-yule)). )

i I also learned GET from reading Kirman (who i also a read a paper by —i think in J theor biology which was more my area) with a physicist (weisbuch) using a sort of ising model (from physics) describing some sort of market (fishes or something). So i learened GET from Kirman who now more or less rejects it. Also note that the D in arrow debreu is the same as the one in SMD. Maybe we could have a prediction market betting on the battle between forces run by General Equilibrium versus those run by the General Strike.

Alejandro: Thanks for taking your time and further clarifying your position re Arrow & Debreu.

I actually don’t think we substantially disagree on the issues concerning the history and aspirations of A & D. Indeed, e.g., the A & D kind of market clearing equilibrium is, as you state, nothing else than a (Kakutani) fixed point solution of an axiomatic equation system and there is no explanation whatsoever (especially not in Debreu) of how one reaches that fixed point equilibrium. The silence is total — as on so many other real world factors (market institutions, money, path dependendencies, non-additivity relations between agents, etc., etc).

The difference in our evaluations seems to hinge on how to evaluate the existence proofs. You may be right, but I have chosen to give the benefit of the doubt, because to me the basic problem with the A & D model/theory is not if the existence proof is right or not, but that there, in any case, is no explanation of HOW the equilibrium prices are established and that this renders the model/theory empty and uninteresting from a real world perspective.

As Michio Morishima wrote already fifty years ago:

“If economics successfully devise a correct general equilibrium model, even if it can be proved to possess an equilibrium solution, should it lack the institutional backing to realise an equilibrium solution, then the equilibrium solution will amount to no more than a utopian state of affairs which bear no relation whatsoever to the real economy.”

If we want economics to be a study of scalar aggregates (GDP, unemployment, deficit, etc.) then economists might be excused to try and build models of the scalar macro based on simulating a large population of individuals that have more or less corresponding micro versions of the same scalar variables (individual incomes, loans, household budgets, etc.)

But economics of scalar aggregates aren’t very interesting. At best your macro model will have some tolerable accuracy, and so what? An interesting economy isn’t about scalar aggregates, it’s about flow graphs: Who is connected to whom? Who is isolated (unemployed) or in a clique (self-sufficient economy). How does a market create and destroy edges between producers and consumers of a good, and how do some markets do this better than others? How long are the paths from producers to consumers and what does that tell us? What does the graph of potential demand look like, and what does it take for real investment to materialize and make the value flow real?

Sorry to say this as a relative outsider, but arguing about the deficiencies of macro-from-micro models is yesterday’s battle. We don’t need more accurate macro. We need to pay less attention to macro as scalar aggregates, its dogma and its predictions, and start learning about real economies which are graphs.

September 01 2014,

My sympathies go out to the ever-convincing and probably very frustrated Lars Syll. What a thankless task to have to explain over & over again to critics, who are “neither fish nor foul”, that, whatever they think they’re talking about, it’s not economics. The ever relevant Gunnar Myrdal put a stake through the heart of general equilibrium pseudo- economics in 1957. In the clearest possible language. I’ve referenced him many times. He was not the first giant of economic thought to do so. Nor the last. Why waste any more time on that mind-game ?Pavlos, whoever you are: You are almost hopelessly confused about the difference between explanation and description. Almost a throwback to the admirable, but pleasantly addled Gustave Schmoeller of the “German Historical School”. And remember the great and clear-minded Immanuel Kant: “Empirical perceptions [inclusive of graphical illustration & time series] in themselves, are directionless without [falsifiable] theory. Theories, without empirical perceptions are hollow”. Please excuse my translation & inserts. This great aphorism is the immediate ancestor of the PARADIGM as an understanding of the development of human thought through real time, about particular subjects. Only a new Paradigm, can take us out from under the dead weight of General Equilibrium mummery.

Norman L. Roth, Toronto, Canada.

Please GOOGLE: [1] Economics of Technology, Norman Roth {2} Economics of Technos, Norman Roth (3) Origins of Markets, Norman Roth [4] telos & technos, roth

Mark Buchanan asks why pay any attention to the A-D results. “The attention, [he suspects], must come from some prior fascination with the idea of competitive equilibrium, and a desire to see the world through that lens, a desire that is more powerful than the desire to understand the real world itself”.

I can imagine different people having different motivations. The simplest equilibrium model is rain falling into a windless pool, each drop having negligible and short-term effect but the level of the pool of wealth gradually rising. This is such a comforting explanation of commerce that it stills thought: most people pay no more attention to the problems caused by it. For chronic worriers the motivation is likely to different: wishful thinking. Clever young men, feeling empowered and distinguished, may be seeking to prove their manhood by translating the comforting image into comforting words and mathematics. Relatively few will be sceptical enough to compare any of these models with the reality: to see, as Pavlos has, that monetary economics involves extensive flows of money rather than local oscillations such as raindrops produce. Perhaps only sceptics disturbed by experiencing at least vicariously the hardships caused by flows going to the wrong places are likely to twig that circulation creates circuital paths which localises flows, and that the misconception that money is valuable has created a strange attraction (in the mathematics of chaos a strange attractor) which is creating the dynamic positional equilibrium of black holes like Rupert Murdoch’s pockets. [If you didn’t laugh, you’d have to cry].

PS. That was for fun, but try reading James Gleick’s ‘Chaos: The Making of a New Science”, p.134 (in the Cardinal edition, 1988).

Lars, yes we do not substantially disagree. But hey, never give the benefit of the doubt when you can see for yourself… And the comprehensive critique of the proof of existence is not a minor point considering the absurd popularity of the original Arrow-Debreu model. Why it overflows into the discussions in macroeconomics with the (still undead) microfoundations project, etc. Perhaps we can talk about this one day here in Mexico City where you have an open invitation. Keep up the very good work

Alejandro, thanks for your comments and your invitation :)

p.s. i googled that paper and came across the ‘triplecrisis’ blog where the paper ‘existence of supply and demand in general equilibrium’ is discussed a bit in 2011 (i have a comment there too, similar in part to the one above). I only followed one link posted there to a 2008 paper, which is claimed to support the view of that paper. However my reading is that they actually support the arrow-debreu result (in a fairly standard way, in just a few pages) but then say this does not hold for a constructible (as in math—eg Bishop) system. This is pretty well known (and is related to the issues of discreteness and infinity i mentioned). The assumptions of course are very restrictive (like perfect information). They are obviously almost irrelevant to real economies (where things like ‘money illusion’ and related SMD type issues exist).

I’m trying to remember exactly the recipe for the first simplest counterexample to AD produced in the 60’s by Mantel (i think) showing cycles can exist (which is almost analogous to results in population genetics) which i saw described in a book by Ingrao and Israel (the invisible hand). Its possible this shows AD is false (ie there are ‘exceptional cases’ like sets of measure zero or singularities for some particular parameter values–as in chaos theory—some standard systems which do have fixed point solutions may not for differnent values of paramters). But i don’t recall if the Mantel system actually fits all conditions of AD or tweaks them somehow. (I think it actually has a sort of disguised discreteness or finiteness constraint so it doesn’t fit the AD axioms. (In fact I think has to do with the ‘fact’ known from ergodic theory that most of these existence theorems are disguised versions of the law of large numbers (as was pointed out when Birkhoof and von Neumann independently proved the ergodic theorem of Boltzmann (based on translating his ‘molecular chaos’ assumption in mechanics into math). The did prove it, but it was based on assuming it at the outset. Mantel’s system is quite finite. You won’t find any laws of large numbers for things lioke clasical ‘ramsey theory’; though of (Pairs and Harrington) if you have big enough ramsey sets they become undecidable, and one can even consider this ‘dynamically’ so you pass through NP phases on the way to the transfinite. In this case ‘god’ or say ‘omega’ can find the equilibrium distribution, but humans don’t know god’s mind—and if god doesn’t exist, they can just decide what the equilbrium is for themselves (act like Turing’s oracle, and select a large cardinal axiom—maybe different ones for different countries, so in some places saying ‘i am a liar’ is true, while in others its false, the way other truths, truisms or laws vary).

Its also ineresting to me from an ‘information economics’ view how discussions like this operate i academia—what papers get cited, which don’t, etc. In ‘orthodox ‘ fields people typically just asjk the ‘pope’ or something who to pay attention to, and who to ignore, because ‘father knows best, and like santa claus he knows what and who is right and wrong’. Even if (like AD) as a truism its in general logically false or practically innacurate, its more or less an ok rule of thumb algorithm. Religion may not be scientific but its not going away anytime soon because science isn’t everything—Rush Limbaugh i gather makes 25M$/yr.

PS—its not mantel but Herbert Scarf (1960). He doesn’t disprove AD but rather shows in general the system is nonergodic—an equilibrium does exist, but not for some initial conditions (which leads to the whole issue of how you define equilibrium and ergodicity—crtuchfeild of SFI wrote a paper describing like 10 or 20 variants of ‘ergodicity’, alot of it coming from FPU discussions of the 40’s. (J B Rosser listed on RWER more or less said the same thing in a comment on the paper on Economists view. Interestingly i even know this though, like Scarf I’ve never taken a course in economics except one MOOC with Ariely which i basically dropped out of, partly because I didnt want his certification as competent , any more than i want a kiss of death or a free ride on a sinking ship. ).

Nadal’s paper to me is similarily just about what you mean by say’s law. To me its a tautology and hence true by definition, but if you interpret the terms differently then its not true.(I think also shows that Robert Lucas was right, as was Gary Becker and Eugene Slutsky and Robert Sapolsky—its all in your mind, so look on the bright side—-the half empty glass turns out to be half full. There’s no involuntary unemployment or recessions, by the SMD. The only technical and conceptual issue is how you deal with the problem of ‘different minds’ or ‘many worlds’ (hugh everett, lack of a cardinal ordering of utlitity; but Jimi Hendrix solved this by showing that a 6 could actually be a 9, and magnetic poles can reverse, so the north can become the south, the center a periphery (world systems) and ‘the boundary of a boundary is zero (john wheeler), and you can get it for bitcoins ).

also all this can be proven from the generic axiom ‘colorless green ideas sleep furiously’ which most people see as common sense, (a version of the golden rule, or gold’s theorem (linguistics) —- or in the standard dialect—‘blonds have more fun’). As M L (‘rodney’) King put it, its all about the color of your character (where of course ‘character’ refers to the trace of your group matrix, describing the symmetries of the chromatic polynomial for your graph, or geneology (see Ruth Bari (RIP) (on wikipedia, mother of Judi Bari (EF! activist blown up possibly by either corporate executives or maybe someone else) and also Gina Kolata (science journalist for NY Times, and a bit of a corporate and status quo shill like Nicholas Wade —they write the history of good fortune for the fortunate (the exceptional americans, via peer review), and they sell too—-if you cant say anything positive, dont say anything at all, and besides we wont publish it).

Thanks for this post and the comments.

I think we need to say clearly that the very use of the word equilibrium in mainstream economics (market clearing condition or supply-demand ‘balance’) is an abuse of language: remember that, in the usual sense of the term, equilibrium is a property of a dynamical system (a time-invariant solution of a dynamical system); one computes the equilibrium states of a system from its very dynamics, a differential equation (continuous time) or a difference equation (discrete time). So strictly speaking, it’s simply meaningless to talk about ‘equilibrium’ in a timeless framework. Mathematical economists are well aware of this potential confusion, starting from Walras himself: this is why he invented the utterly pseudo-scientific mythology of the ‘tatonnement’ process, to make his notion of ‘equilibrium’ coincide with the standard notion of equilibrium.

Therefore, rather than saying that in general ‘general equilibrium’ has not been proved to be stable, the negative results on stability are more exactly interpreted by saying that Walras’s project of justifying his use of the term ‘equilibrium’ has failed (even if one accepts a mythological dynamical system in fictitious time), and that we should stop referring to the ‘competitive solution’ (Arrow-Debreu, 1954) as ‘equilibrium’ in the usual sense of the term. Walrasian equilibrium is not an equilibrium in the standard sense. Arrow-Debreu (1954), for instance, clearly imply this when they warn the reader that: ‘Neither the uniqueness nor the stability of the competitive solution is investigated in this paper’, because (and this is the important point) ‘The latter study would require specification of the dynamics of a competitive market as well as the definition of equilibrium’.

The need to distinguished between the two notions may justify the modifier ‘competitive’.

Now while waiting for mainstream economists to justify the ‘relevance’ (Guerrien) and use

of this notion of equilibrium (by working out a realistic dynamics of the economy leading the economy towards it, as Syll was arguing) we should stick to the standard notion of equilibrium. For proving the existence of a ‘general equilibrium’ says no more about the economy than proving the possibility of ‘international peace’ says about world affairs.

Unfortunately, in their criticizing neoclassical economics, many critics have come to dismiss the important mathematical notion of equilibrium in general. This is like dismissing French altogether on the grounds that one is living among people who are misusing French terms.

Sorry for being long,

Your devoted Student.

mistake at the end: ‘A devoted Reader’ rather than ‘Your devoted Student’ (to avoid confusion)

Is it not a bit trivial to say that a hyperbolic fixed-point of a continuous dynamic system exists and omit to demonstrate that the Jacobian of the system is Hurwitz stable at that fixed-point?

Athens Journal of Business and Economics April 2017

143

Arrow-Debreu Model versus Kornai-critique

By József Móczár

More than forty-five years have passed since János Kornai published his book entitled „Anti-equilibrium” (Kornai, 1971). This was the first scientific work in the international literature that provided a comprehensive critique on the general equilibrium theory as described in Debreu’s theory of price and the Arrow-Debreu model and opened a debate on the validity of the mainstream neoclassical model. Frank Hahn’s response was the most severe to the critique. Kornai, insisting on his original critique, reflected on Hahn’s response in his own autobiography (Kornai, 2008). In this paper, we review the Arrow-Debreu model and its background, reconstruct the major points of the Kornai vs. Hahn debate, including its historical preliminaries, and examine the validity of criticisms and rebuttals. As we will see, the recent theories have not always verified Hahn’s objections and some Nobel Prize lectures in economics recently showed that both the neoclassical theory and the general equilibrium theory in the sense of Arrow-Debreu model was wrong on either empirical or theoretical grounds (Offer and Söderberg, 2016). We also show Kornai’s newest results towards an alternative model of detailed resource allocation, DRSE contrary to the general equilibrium (Kornai, 2014).

Keywords: general equilibrium theory, Arrow-Debreu model, Anti-Equilibrium, Kornai vs. Hahn debate, Walrasian equilibrium, Kornai’s new equilibrium states, ex post and ex ante models, DRSE model, ergodic dynamic system.

Arrow-Debreu DOESN’T PROOVE the unicity of equilibrium. Please, make an effort, and remember that…