Monday, August 11, 2014

Arrow-Debreu Derangement Syndrome

Imagine you had never read any textbook economics, or studied any academic research papers. You didn't know the THEORIES of economics, especially in their mathematical form. But suppose you did know some mathematics, and were also generally well informed about the realities and complexities of real world economies. Now, suppose a demon sat you down and made you read and study the famous theorems concerning the existence of a competitive economic equilibrium as developed in the 1950s by Ken Arrow and Gerard Debreu. What would you think?

My belief is that you would quickly conclude that these theorems probably held little or no importance for understanding any real world economic system. If the demon tried to tell you that these theorems were at the very core of today's theoretical approach to (much of) economics, you'd think he or she was joking. If the demon insisted, you'd suspect you were dealing with an insane demon; and if you discovered the demon was right, you suspect the economics profession of being deranged. At least I would...

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.

I just came across this old post from Yves Smith which makes the point rather nicely:

The scientific pretenses of economics got a considerable boost in 1953, with the publication of what is arguably the most influential work in the economics literature, a paper by Kenneth Arrow and Gérard Debreu (both later Nobel Prize winners), the so-called Arrow-Debreu theorem. Many see this proof as confirmation of Adam Smith’s invisible hand. It demonstrates what Walras sought through his successive auction process of tâtonnement, that there is a set of prices at which all goods can be bought and sold at a particular point in time.42 Recall that the shorthand for this outcome is that “markets clear,” or that there is a “market clearing price,” leaving no buyers with unfilled orders or vendors with unsold goods.

However, the conditions of the Arrow-Debreu theorem are highly restrictive. For instance, Arrow and Debreu assume perfectly competitive markets (allbuyers and sellers have perfect information, no buyer or seller is big enough to influence prices), and separate markets for different locations (butter in Chicago is a different market than butter in Sydney). So far, this isn’t all that unusual a set of requirements in econ-land.

But then we get to the doozies. The authors further assume forward markets (meaning you can not only buy butter now, but contract to buy or sell butter in Singapore for two and a half years from now) for every commodity and every contingent market for every time period in all places, meaning till the end of time! In other words, you could hedge anything, such as the odds you will be ten minutes late to your 4:00 P.M.meeting three weeks from Tuesday. And everyone has perfect foreknowledge of all future periods. In other words, you know everything your unborn descendants six generations from now will be up to.

In other words, the model bears perilous little resemblance to any world of commerce we will ever see. What follows from Arrow-Debreu is absolutely nothing: Arrow-Debreu leaves you just as in the dark about whether markets clear in real life as you were before reading Arrow-Debreu.

And remember, this paper is celebrated as one of the crowning achievements of economics.


  1. Brilliant analysis of the chronic failure of economic theory. I more or less worked this out in the 1960s - confirmed by reading Paul Ormerod's 'Death of Economics' and more recently Mark Buchanan's 'Forecast'. There is an explanation. Economic theory is the ruling 'cult' of the capitalist system. This results in multiple pressures on every institution, including academia. Given the centrality of economics, career interests and corporate sponsorship reduce academic rigour within economics compared with other social sciences. See my (short) review of 'Forecast' at

  2. Economis:

  3. "What follows from Arrow-Debreu is absolutely nothing: Arrow-Debreu leaves you just as in the dark about whether markets clear in real life as you were before reading Arrow-Debreu."

    Strictly true, but if it is the best that economics can come up with then it serves as a (non-strictly-speaking) reductio ad absurdum for the idea of a single market-clearing equilibrium.

    (Puts me in mind of Wittgenstein's Tractatus -- supposed to be a complete working-out of the reference theory of language but better read as RAA).

  4. Mark,
    Several points, starting with that indeed relying on GE in DSGE models to model macro is ridiculous. But you are off on a few points regarding GE.
    One is that the original theorem does not assume or prove either uniqueness or stability. This became a subject of study shortly thereafter by several of the major players, including Arrow and Debreu along with Scarf, Sonnenschein, Mantel, Gale, and Uzawa. Based on several papers and results I shall not detail, the results were very much that conditions for having either are very strict and unlikely to hold in reality, which rather damaged GE as an ongoing micro project, with in many ways game theory moving in to dominate much discussion in more recent years (although it has even worse problems with uniqueness arguably). But the macro DSGEers tend to simply ignore all this and dismiss such concerns, blithely assuming uniqueness and stability and prattling on about how rigorous they are for using GE.
    I note that most of those who showed the problems have kept quiet about the problems with using GE in macro, although one notable exception is Alan Kirman. Arrow himself, probably the most respected living economist by a country mile (who is still professionally active at age 93, which he just reached), has been fully aware of all this, and a sign of his awareness of this and the relevance of complexity approaches is that he was the main economist behind the establishment of the Santa Fe Institute, mostly started by physicists like Gell-Mann, and is still the ultimate godfather lurking behind the economics sectinon of it (I was visiting there earlier there this summer).
    A final note is that you should know that there is a new book out by Roy Weintraub about general equilibrium theory, documenting in more detail something he had shown earlier and that has long been known by cognoscenti, that the late Lionel MacKenzie got to that existence proof ahead of Arrow and Debreu ahead of them and even published his in the previous issue of Econometrica before they did, but has basically gotten no credit for it. Those in the know call it Arrow-Debreu-MacKenzie GE theory, although given his priority it should really be called MAD theory. There is some real dirt in Weintraub's account, which shows the late Debreu playing some dirty academic pool in all of it. The saintly Arrow manages to come out smelling like a rose, more or less. Oh, and MacKenzie was ultimately philosophical about not getting a Nobel like they did and had a pretty good professional life, with most of those he dealt with closely knowing what he did.

  5. A bit more. One reason why economsts are taken with Arrow-Debreu-MacKenzie is that in whichever version one looks at, the conditions for existence actually do make real world sense, such as full information, complete markets, rationality, competition, and so on. You sneered at these assumptions, but in fact we know that when these assumptions do not hold, and indeed they do not fully, there will be problems with the functioning of markets, and indeed some of the problems we have seen such as the financial crashes and crises in recent years can be traced back at least partly to some of these failures. But, of course, the hard core macro DSGE gang tends to simply ignore al this.

    Barkley Rosser

  6. Speaking from the perspective on having had to understand the A-D results while in grad school, let me make one comment. My take-away, and that of most of the people I was in grad school with, was that the A-D results were important, simply *because* they made clear how unlikely reaching a competitive general equilibrium was, how restrictive the assumptions required, of necessity, were. As a result, most of us realized that, in understanding the way in which economic systems actually operated, we had to bring in the frictions of the real world, the limits to human cognition, and the other ways in what the world as it is differs from the world of A-D. (In this, I agree completely with Barkley Rosser, who was one of the people I began grad school with.)

  7. The main reason why Arrow-Debreu-Lucas is so popular is that it proves (by using gross mathematical and logical mistakes here and there as documented by Steve Keen and the Cambridges Capital Controversy) the central truthiness of Economics: that (absent government distortion) people's incomes are uniquely determined by people's productivity.

    In other words that Mozilo, Cayne, Fuld, ONeal, etc. all fully deserve the hundreds of millions of income they awarded themselves, because Arrow-Debreu-Lucas says that is the just compensation for their superior productivity.

    That is superior incomes are exactly deserved by superior people.

    The central truthiness of Economics is the basis for the centrality of Arrow-Debreu-Lucas models; it is an anti-communist model :-).

    Actually as Phil Gramm put it, CEOs can be the most exploited workers in the country:

    «Most of his former colleagues probably can't fathom why Wall Street bankers make tens of millions of dollars in salaries and bonuses each year. How would he justify these fat pay days? "It's simple," he lectures, sounding very much like the Texas A&M economics professor that he was in the 1970s: "In economics, we define labor exploitation as paying people less than their marginal value product.

    I recently told Ed Whitacre [former CEO of AT&T, who retired with a $158 million pay package] he was probably the most exploited worker in American history because he took Southwestern Bell, which was the smallest of the former Bell companies, and he turned it into the dominant phone company on earth. His severance package should have been billions."»

    That exploitation is usually claimed to be the result of government intervention motivated by the envy of inferior people.

  8. To add to this, consider one of the aspects of the Cambridges Capital Controversy, the simple model created by Sraffa.

    In that model the distribution of income is (also) determined by the exogenous rate of interest.

    The very possibility that central banks might with their interest rate policies help rich people become richer is monstrously heretical for those Economists who love Arrow-Debreu-Lucas; for them the rich become richer only because of their harder work and higher productivity; and Greg Mankiw seemed to argue that harder work and higher productivity is the manifestation of genetic superiority of the rich over the poor.

  9. blissex. You may have just coined "Arrow-Debreu-Lucas," but it is not a coherent formulation. For one thing, Lucas does not invoke GE, even if he does insist on rational expectations, which is implied by A-D-M. But his are ultimately toy models.

    Where you are off-base is on this matter of justifying income (and wealth) inequality. You are right that there are some who make the interpretation you do, but it is not consistent with the original formulation of the theory. The theory assumes that there is a given initial allocation, but it does not at all say that this allocation is somehow justified or virtuous. The resulting equlibirum may be efficient, but it is not necessarily just, and the obvious reason is that it may be based on an unjust initial allocation.

    I would point out in this regard, blissex, in case you are unaware of the fact, that Kenneth Arrow has pretty much always described himself as a "socialist" and has always supported substantial programs to increase equality of income and proper social safety nets. Lumping him in with the laiissez-faire attitudes is not only inaccurate but completely unfair.

  10. «matter of justifying income (and wealth) inequality. You are right that there are some who make the interpretation you do, but it is not consistent with the original formulation of the theory.»

    But the design or purpose of the original formulation of the theory is mostly if not entirely irrelevant in this discussion, being off topic.

    Our blogger was asking why «the economics profession was/is so impressed by the Arrow-Debreu results», and I was trying to give «The main reason why Arrow-Debreu-Lucas is so popular».

    It may have been designed for whichever other reason, but I reckon that its nearly mandatory popularity is due simply to its supporting, by abstruse contortions, the central truthiness of Economics.

    Surely no right-thinking Economist would stand being accused of being objectively a Communist by supporting theories in which the distribution of income is not solely determined by factor productivity (absent envious government intervention).

    Now to "prove" that factor income is or should be determined solely by productivity requires a phenomenal array of extremely narrow special case assumption (which cannot even be logically or mathemtically consistent) of extremely improbable applicability to their real world.

    The splendid coincidence is that Arrow-Debreu-Lucas theory is founded upon that vast list of peculiar special cases assumptions, as our blogger noted at the beginning.

    Whether the coincidence is now or was then deliberate, or entirely fortunate, is beyond the point for the present discussion.

    The singular value of Arrow-Debreu-Lucas style theories for a large number of Economists is that it "proves" the central truthiness of Economics, something that they know in their heart is true, and this justifies Arrow-Debreu-Lucas, whatever its orginal formulation was for or whoever participated to its formulation.

  11. Making Arrow-Debreu-Lucas the core of Economics also has an inestimable advantage precisely because it is founded on a vast depth of extremely peculiar and (logically or mathematically) inconsistent assumption, and it takes a lot of "courage" to adhere to it.

    It is difficult to progress in an academic career in Economics without publishing papers based in some way or another on Arrow-Debreu-Lucas, and it much easier to get tenure and endowed chairs and consultancy and speaking gigs from corporates by exalting it and the central truthiness of Economics.

    Now suppose some aspiring Economist starts thinking that Arrow-Debreu-Lucas is just a faith based crude propaganda tool, but one that she has to use to publish and get tenure; and then once she gets tenure she announces that all her previous work is based on a faith based crude propaganda tool which she used only to push forward her career. What happens next is that her "aligned" head of department brings her before the Ethics Committee of the University and she is then instantly deprived of tenure for having confessed to gross fraud in her tenure submission's publications.

    Once you have "aligned" yourself with the central truthiness of Economics and built a career where Arrow-Debreu-Lucas is the "intellectual foundation" of your success it is very difficult to change opinion even slightly without paying a huge price, and your research assistants and phd students paying also a huge price by being associated with you.

    The practical courses to avoid being trapped by being drawn into using Arrow-Debreu-Lucas and the central truthiness of Economics are only two.

    The first is to ignore Arrow-Debreu-Lucas and the central truthiness of Economics entirely, never compromise with it, and to start from the beginning doing work and research at non-"aligned" institutions, thus deciding early to give up any chance to become very rich and famous thanks to Ken Lay endowed chairs, Citigroup directorships, Countrywide consulting contracts, AIG speaking fees, ...

    The other is the middle way of "flanking" the core of the Economics profession specializing in so-far unpopular subjects and giving them fancy new names and never using Arrow-Debreu-Lucas, so both "aligned" Economists don't feel the central verity of Economics threatened and one's publication record is not related to the central truthiness of Economics. This does not happen only in Economics; I read a beautiful summary of this way of doing things somewhere, I'll try to find it again.

  12. blissex,

    I do not know why you insist on using "Arrow-Debreu-Lucas" formulation. This may be relevant for DSGE models, but Arrow certainly is well aware of the problems with those, quite aside from the issue of income and wealth distributions.

    Let me repeat: A-D-M (not A-D-L) does imply that factors will be paid their marginal products, but that does not imply what the final distribution of either wealth or income will be. That is because that final distribution depends on the initial allocation of ownership of those factors, and that may be completely unjust and unfairl, a Pharaoh economy where Pharaoh owns all factors, including all the labor as slaves. Maybe Lucas might think that is great, but Ken Arrow most certainly would not.

  13. OK, now I think I see what is "rosserjb"'s point: that Arrow should not be tainted by association with the use that most "aligned" Economists make of an approach that bears his name.

    As to that I can think of different responses:

    * After all the Arrow-Debreu (-Lucas) based approach bear his name and he got a Rijksbank Nobel-memorial prize in large part because of it. It is not totally unfair to taint him with that association :-).

    * Regadless of its long standing use by "aligned" Economists to provide propaganda for their political persuasion, Arrow-Debreu (never mind the Lucas extension) is a terrible, terrible model. Perhaps the (relatively) young Arrow thought it was amusing to speculate along the Walras, Marshall, JBClark line, but if so it was largely wasted effort.

    Altogether I have read quite of Arrow's later work (a long time ago), and overall he seems a reasonable economist, and indeed not very "aligned".

    Also after all he and Debreu (and Mackenzie) only invented the extremely peculiar assumptions that enable most "aligned" Economists to assert with a straight face the central truthiness of Economics; perhaps the approach should be called after the Walras/JBClark/Lucas model, not the Arrow-Debreu conditions.

    But that's not what everybody calls it, and poor Arrow gets dragged in.

    BTW the "-Lucas" I add because of a "technicality": the original Walras/JBClark model with the Arrow-Debreu conditions deals purely with static modeling. Embellishing it with the Lucas approach allows "aligned" Economists to dispense with any attempt to introduce dynamics, and to extend "tatonnement" across eternity (fancy that!). Thus it is central to the dominant approach (the one described as "pure shit" in a linked post, I guess) of most "aligned" Economists.

    Which is invaluable for "aligned" Economists, because I doubt very much whether there is any model in which time matters in which it is possible to "prove" the central truthiness of Economics even with more absurd or unmathematical conditions than the Arrow-Debreu ones.

    The difficulty of course is that if time actually matters then capital matters, and if capital matters, then we have an endowment problem at every time period (instead of "we dont speak about that" for the initial endowments), and then money does matter, and a lot of other issues that infringe on the central truthiness of Economics.

    PS: Mark Thoma's blog has just added a link to a very recent overview of Arrow's work here:

    The author's overview of the Arrow-Debreu conditions and the significance of general equilibrium models is "interesting" and perhaps unfair to Arrow.

  14. I just an interesting "coincidence" about Arrow: he was ethnically a Romanian Jew and politically a Socialist and his work on General Equilibrium happened during the Red Scare...

    An aside: I just read Debreu's Wikipedia page and really Laughed Out Loud at:

    «breakthrough paper, entitled Existence of an Equilibrium for a Competitive Economy, together with Kenneth Arrow, in which they provided a definitive mathematical proof of the existence of a general equilibrium»

    «in general, aggregate excess demand functions vanish at a finite number of points – basically, he showed that economies have a finite number of price equilibria»

    Such is the shameless propaganda that gets passed around.

  15. Apologies for going on and on but this Arrow-Debreu story is far from banal and simple to discuss.

    Let's look again at the period of time in which the Arrow-Debreu conditions were invented and their political significance.

    The Arrow-Debreu conditions essentially "prove" that the "economy" is always in General Equilibrium, that is there is no possibility of aggregate excess supply in any market.

    Translated for the public, that says "there is no involuntary unemployment, the unemployed are moochers and looters", and "two Chicago economists proved that bastard english economist Keynes completely wrong" and "the capitalist system is the best and fairest system ever"; the paper was finally published in 1954, in full anti-socialist witch-hunt times.

    Only a few year before in 1947 the moderate, centrist Keynesian textbook by Tarshis was witch-hunted out of USA universities as "aligned" interested reckoned keynesianism was a stepping stone to socialism.

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