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Stiglitz and the full force of Sonnenschein-Mantel-Debreu

from Lars Syll

In his recent article on Where Modern Macroeconomics Went Wrong, Joseph Stiglitz acknowledges that his approach “and that of DSGE models begins with the same starting point: the competitive equilibrium model of Arrow and Debreu.”

This is probably also the reason why Stiglitz’ critique doesn’t go far enough.

It’s strange that mainstream macroeconomists still stick to a general equilibrium paradigm more than forty years after the Sonnenschein-Mantel-Debreu theorem — SMD — devastatingly showed that it  is an absolute non-starter for building realist and relevant macroeconomics:

SMD theory means that assumptions guaranteeing good behavior at the microeconomic level do not carry over to the aggregate level or to qualitative features of the equilibrium …

24958274Given how sweeping the changes wrought by SMD theory seem to be, it is understandable that some very broad statements about the character of general equilibrium theory were made. Fifteen years after General Competitive Analysis, Arrow (1986) stated that the hypothesis of rationality had few implications at the aggregate level. Kirman (1989) held that general equilibrium theory could not generate falsifiable propositions, given that almost any set of data seemed consistent with the theory …

S. Abu Turab Rizvi

New Classical-Real Business Cycles-DSGE-New Keynesian microfounded macromodels 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 SMD unequivocally showed that there did not exist any condition by which assumptions on individuals would guarantee either stability or uniqueness of the equilibrium solution.

Opting for cloned representative agents that are all identical is of course not a real solutionto the fallacy of composition that SMD 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.

Of course, most macroeconomists know that to use a representative agent is a flagrantly illegitimate method of ignoring real aggregation issues. They keep on with their business, nevertheless, just because it significantly simplifies what they are doing. It reminds — not so little — of the drunkard who has lost his keys in some dark place and deliberately chooses to look for them under a neighbouring street light just because it is easier to see there.

Frank-Ackerman_0General equilibrium is fundamental to economics on a more normative level as well. A story about Adam Smith, the invisible hand, and the merits of markets pervades introductory textbooks, classroom teaching, and contemporary political discourse. The intellectual foundation of this story rests on general equilibrium, not on the latest mathematical excursions. If the foundation of everyone’s favourite economics story is now known to be unsound — and according to some, uninteresting as well — then the profession owes the world a bit of an explanation.

Frank Ackerman

Almost a century and a half after Léon Walras founded general equilibrium theory, economists still have not been able to show that markets lead economies to equilibria. We do know that — under very restrictive assumptions — equilibria do exist, are unique and are Pareto-efficient. But — what good does that do? As long as we cannot show that there are convincing reasons to suppose there are forces which lead economies to equilibria — the value of general equilibrium theory is nil. 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.

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.

kirmanThe full force of the Sonnenschein, Mantel, and Debreu (SMD) result is often not appreciated. Without stability or uniqueness, the intrinsic interest of economic analysis based on the general equilibrium model is extremely limited …

The usual way out of this problem is to assume a “representative agent,” and this obviously generates a unique equilibrium. However, the assumption of such an individual is open to familiar criticisms (Kirman 1992; Stoker 1995), and recourse to this creature raises one of the basic problems encountered on the route to the place where general equilibrium has found itself: the problem of aggregation. In fact, we know that, in general, there is no simple relation between individual and aggregate behavior, and to assume that behavior at one level can be assimilated to that at the other is simply erroneous …

The very fact that we observe, in reality, increasing amounts of resources being devoted to informational acquisition and processing implies that the standard general equilibrium model and the standard models of financial markets are failing to capture important aspects of reality.

Alan Kirman

  1. September 19, 2017 at 4:22 pm

    Again and again Lars Syll explains how neoclassical economics is wrong and completely mistaken. It’s all right. Readers of this blog are, as long as I guess, critical to the present state of economics and by consequence to general equilibrium theory and to consensus macroeconomics. The next step that Syll should take is to show how to reconstruct economics. If not, he simply ends by multiplying people who do not believe in economics of any brand. There will be a total cynicism or apathy. This is very dangerous, because this kind of social psychology easily goes to chauvinism of all kinds.

    • Rob
      September 19, 2017 at 11:34 pm

      The first step in envisioning the solution to any problem is accurately define the origin, nature, and gravity of the problem. If this is not done correctly I find it hard to see how a solution that will be useful can be created-envisioned. I believe Lars Syll is providing this service — admirably and clearly — so a common person such as myself can understand the issues.
      I think it is unfair to lay the entire burden (or a substantial degree) of reconstructing economics upon him and holding him responsible for widespread cynicism and apathy in society (I am mostly speaking of what I see here in the US). That existed already after decades of observing predatory capitalism ravage the lives of our family members and fellows on an almost daily basis while politicians took handouts in the form of campaign donations and did the bidding of their corporate overlords, all justified by the “greed is good” mantra taught in MS courses in managerial economics. I think you misplace the blame for apathy and cynicism.

      As for leading common men and women such as myself to not “believe in economics” I think you overestimate what the common person in the street (here in the US or in Japan, where I lived and my wife came from) understands economics is period. Once again misplace the blame for loss of faith, if there really ever was any. I have a BA in business and accounting, worked as a software engineer, started my own corporation, and all having only studied one intro course on macroeconomics and one intro course on microeconomics. It wasn’t until the Great Recession that I began, a little over a year ago, a deep dive into the history and field of economics with the goal of knowing it deeply to understand just what went wrong. And I have Lars Syll invaluable in this endeavor.

      As for what it will take to reconstruct economics I think it will take nothing short of a cultural renaissance that takes the symmetry of truth, beauty, and goodness seriously enough to launch a deep and sustained philosophical inquiry into the history, meaning, and future of capitalism as we know it today. And part of that — and I would argue a foundational part — is the philosophical critique Lars Syll and other brilliant minds are offering today. In my limited and partially informed view I think it is going to take the best of humanities thinkers in science, philosophy, and religion to reconstruct any form of lasting understanding of economics, if that is, it is not already too late.

      For after watching the demagogic President of the United States — Donald Drumpf — address the UN today, I fear we may have woke up too late and may be about to repeat the mistakes of Germany circa 1934.

  2. September 19, 2017 at 4:49 pm

    I think we have to accept some degree of division of labour here. I’m an economic methodologist and science theoretician. “Methodology is about the way in which we do economics, what we think is a good or bad argument and how we choose between theories. It’s not just about whether to use this technique or that technique. Among other things, the methodology we use assumes a particular way of understanding how the economy works and what kind of knowledge is possible. I think it is always important for economists to understand the basis of what they do, and to be able to justify their own choice of methodology.” (Sheila Dow)

    • September 20, 2017 at 4:41 pm

      I admit that division of labor is necessary. I am even thinking that a very bad tradition has bee generated in economics. When somebody presents his or her new model, he or she put all relevant variables in the model. As he or she is inevitably specialized in a field or two, all other parts are patchworks of rudimentary ideas. If these rudimentary parts are well established theories, it is a normal routine in which a science proceeds. In the case of economics, those parts are not the results of serious scientific examination or selection (of competing theories). It is often products of whim of ideas but accepted as plausible only because it is a long tradition.

      A good division of labor in economics is not only necessary but probably the only way that may lead to a renovation of economics toward more realistic one. I am not blaming that you are not almighty. I am asking if there is more constructive orientation than always repeating to point out neoclassical flaws.

      As a methodologist, you know that equilibrium framework has now no big margin of development. Then why do you not search other more suitable frameworks of analysis. It is not necessary that you yourself find some. You may find someone who is trying such an attempt. It may be only in a starting phase. But if you think that it may have a good possibility, you can let people know on this new trend. It may help young economists to challenge to construct a new theory with new method. I believe this is also a function of a methodologist in economics.

  3. September 19, 2017 at 7:00 pm

    What is the meaning of “market” in this discussion:

    1. An organised system of bidding to buy or sell a commodity, where supply and demand regulate the price and the quantity transferred.

    2. A network of many varied agents and commodities, where agents buy, sell, and transform commodities as a means to satisfy their diverse needs.

    Textbook markets like crude oil or electricity on the grid are the first kind. They’re artificial, and manage to generate some sort of continuity if not equilibrium. They’re amenable to calculus so they’re well studied.

    Natural markets in villages or whole societies are of the second kind. The point of these markets is to discover and satisfy diverse needs in an emergent way, and to the extent we have both bread and smartphones they work quite well. Natural markets require graph theory to study.

    I’m increasingly of the belief that Adam Smith’s “invisible hand” refers to the market’s magical ability to mine iron and produce steel, for a rivet, of an engine, in a ship, which gets an Irish miner to America where he opens a pub, without him or anyone else grasping or planning the whole value cycle.

    That’s a much more magnificent concept of the market than as a price-quantity regulator, which any bureaucrat could do.

  4. Paul Davidson
    September 19, 2017 at 9:38 pm

    Any form of general equibrium theory cannot be a correct microfoundation for a Macroeconomic analysis of the system we live in. Why? Because the future of our economic system is UNCERTAIN –as Keynes insisted!. While all forms of general equilibrium theory requires decision makers to know the future,. either with absolute certainty or at least actuarial [probabilistic] CERTAINTY. . Thus in a simple Walrasian system, each decision maker pends all his income on entering into spot and forward real contracts for any specified date from today to eternity.[A real contract is on that would exist in a barter economy.] Since the decision maker knows the future in any general equilibrium system, he/she knows the conditions in any market on any date in the future

    In our world all market transactions are organized via the use of money denominated spot and forward contracts!

    In their book “General Competitive Analysis” Arrow and Hahn insist that a “serious monetary theory” can be made only if the analyst recognizes that all contracts are made in money denominated terms.

    Keynes’s LIQUIDITY theory i a serious money theory meeting Arrow and Hahn’s criteria. Liquidity is the ability to meet all one’s spot and forward money contract obligations as they come due!

    The institution of money denominated contracts helps decision makers deal with an uncertain future. By entering into spot and forward money denominated contracts, the decision maker provides himself/herself with legal certainty of future cash inflows and outflows!

    Just think of how the reader of this tries to maintain apositive balance in his/hr checking account — so as to have liquidity to meet contractual commitments

  5. September 20, 2017 at 1:07 am

    Kirman: “…the problem of aggregation. In fact, we know that, in general, there is no simple relation between individual and aggregate behavior, and to assume that behavior at one level can be assimilated to that at the other is simply erroneous…

    It is impossible to overstate the importance of this point. Perhaps the most serious error in logic that an economist can be guilty of is relying on a fallacy-of-composition when constructing a theoretical projection.

    Example: A substantial income tax increase—if it were imposed on only a single rich person—would indeed impose a very significant penalty on that individual in the form of lost purchasing power. But if all rich people are required to pay the same percentage ‘penalty’, then actually none of them would experience any loss of purchasing power in the marketplace.

    This, because the marketplace always auctions the scarcest luxuries/assets to the highest bidders. If all of the 1% were required to pay income taxes at steeply progressive marginal rates, they would still end up with the highest disposable incomes in the land.

    In a market economy, that’s all the money you need to claim the scarcest goods & services the economy has to offer. They’d simply maintain their lifestyles at lower prices. They would actually experience no loss of purchasing power in the marketplace.

    How might a practiced general equilibrium modeler try to incorporate this particular observation re: the non-aggregation of “consumption effects” when income tax rates on rich people change?

    That’s a good question…

  6. September 20, 2017 at 1:38 am

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

    Forgive me for getting a little philosophical here, but why can’t reality economists just leave general equilibrium models by the wayside? Showing how various macro economic variables respond to each other after one or more of them changes is not un-interesting, to be sure…

    But please explain to me why it wouldn’t make a hell of a lot more sense for economists to simply focus their attention on trying to explain how particular—or various—desirable economic outcomes can be brought about?

    Or maybe advise policy makers on how they might be able to act to optimize the productive output of the economy (or the economic welfare of the economy’s participants) and maintain it at those optimal levels indefinitely? In a continuously changing economy, who really cares whether or not moments of equilibrium are briefly experienced?

    How can it reasonably be stated that equilibrium analysis is necessary in order for economists to give good advice to policy makers? I just don’t see it…

  7. September 20, 2017 at 4:06 pm

    James Kroeger

    It is the question of inertia which works in thinking and scientific pursuits. One of typical reaction to the advice that it would be better to abandon equilibrium framework is this: Without equilibrium and rationality we loose tools of analysis and no economic reasoning is possible. This is not my invention. A famous economist who once occupied the presidency of Japan Economic Association expressed this.

    Of course, this is only conservative reaction from the part of mainstream economists. There are many economists (including me) who reject equilibrium as the framework of analysis.

    What I hope for Lars Syll is to study those heterodox economists and find a hopeful theory or trend and introduce them for wider range of people, including young economists who are wondering to find a possible breakthrough. This must be possible for a methodologist like him.

    • September 21, 2017 at 12:31 am

      Former JEA president: “Without equilibrium and rationality we lose tools of analysis and no economic reasoning is possible.”

      How so?

      Yes, it is useful to view purchase transactions as moments of “equilibrium” within particular markets, but it is difficult to see a necessary connection between this phenomenon and the attempts that are made in general equilibrium models to map “overall inflation” (the Price Level) as a function of changing macro variables.

      I would dare say that virtually all of the general equilibrium models that show up in textbooks fail to recognize the supremely important fact that rates of price inflation vary across different income groups. It is indeed possible for one income group to experience robust, double digit rates of price inflation at the same time that another income group is experiencing disinflation, or even deflation.

      When such a reality is unfolding—as we have seen in recent decades—how can a general equilibrium model really be of any use to us, since nearly all of them assume away this possibility, depicting inflation instead as a mysterious effect which spreads out evenly over the entire economy?

      I would argue that the “tools of analysis” and “economic reasoning” provided by general equilibrium models based on demonstrably wrong assumptions re: prices across across all income groups should be considered more than a little bit suspect…

      • September 21, 2017 at 2:48 am

        James Kroeger,

        I am not supporting such an attitude (to conserve equilibrium framework at all prices). I refuse to use “equilibrium” notion in my research. Even though, economics is possible, For example, please read two of my papers:

        (1) The Revival of Classical Theory of Values

        (2) The New Theory of International Values: An Overview

        (1) is a brief explanation how a new theory of prices is possible. It is totally different from the ordinary supply and demand equality formulation. However, in fact, it is not new at all because it is in essence an old theory at the time of classical political economy.

        I add (2), because it shows that the new theory of value is not constrained in the case of domestic values (theory of prices within a country). The theory can be generalized to hold in the international trade situation.

        Although my tow papers are not yet so ‘refined’ as to argue that “rates of price inflation vary across different income groups”, my theory of international values can explain why there emerges a big difference of wages. (Wages of highest income countries are often 30 to 70 times higher than that of poorest countries).

        In my impression, what you want to say by your proposition that “rates of price inflation vary across different income groups” is the question of consumption patterns. The composition of consumption bundles differ much according to income groups and, in such a case, the inflation rates measured by those different bundles are of course different.

  8. Rob Reno
    September 20, 2017 at 11:18 pm

    Lars, are there any books you highly recommend from the WEA Paperbacks series?

    • September 21, 2017 at 4:24 pm

      I appreciate most of them, but if I have to single out two of them, I would especially recommend reading “A Philosophical Framework for Rethinking Theoretical Economics and Philosophy of Economics” by Gustavo Marques and ‘Narrative fixation in economics’ by Edward Fullbrook :)

      • Rob Reno
        September 21, 2017 at 5:03 pm

        Thank you so much Lars, that helps me focus my reading.

  9. September 21, 2017 at 6:07 pm

    A correction of Syll’s slip in relation to SMD theorem:

    Lars Syll stated in this post “Stiglitz and the full force of Sonnenschein-Mantel-Debreu” as follows:

    “SMD unequivocally showed that there did not exist any condition by which assumptions on individuals would guarantee either stability or uniqueness of the equilibrium solution.”

    This statement is not exact and, in that sense, an error. There exist various conditions under which the aggregate demand function well behaves.

    On this point, Stiglitz’s following statement is more accurate:

    “That is, a according to the Mantel-Sonnenschein theorem, in the absence of some restriction, such as the “representative agent” assumption (where all individuals are assumed identical), virtually any aggregate function can be consistent with the standard competitive model. See Mantel (1974), Sonnenschein (1972) and Kirman (1992).
    There is also a large literature describing the very restrictive conditions under which such household aggregation can be done.” (Stiglitz “Where Modern Macroeconomics went Wrong” p. )

    Stiglitz is not explicit on the “large literature.” Readers who want to know the point more in deatail are adviced to read

    Alain Kirman (1992) Whom or What a Represtnative Individual Represent? Journal of Economic Perspectives 6(2): 117-136.

    For example, let me cite two examples:

    “Hildenbrand (1983) has shown that if the income distribution is decreasing -that is if each successively higher income class contains a smaller proportion of agents-then the so-called “law of demand” holds.” (Kirman 1992 p.130)

    “[Gramont] gives an example in which if agents have very spread out preferences, then aggregate excess demand will have the well known “gross substitutability” property and equilibria will be unique and stable. … for his result [to hold], it is enough that agents respect their budget constraints.” (Kirman 1992 pp.130-131)

    In sum, there exist conditions that would guarantee either stability and uniqueness of the equilibrium solution. The trouble for macroeconomics is that they are very restrictive.

  10. September 21, 2017 at 6:11 pm


    Stiglitz’s statement appears as footnote 60.

  11. September 22, 2017 at 12:59 am

    Responding to Yoshinori, above…

    I haven’t had a chance to read through your second paper, Yoshinori, but I did want to correct what seems to be a misinterpretation of a claim I have made:

    “In my impression, what you want to say by your proposition that “rates of price inflation vary across different income groups” is the question of consumption patterns.”

    I would point out that price inflation, properly understood, is a phenomenon that is totally derived from disposable incomes inflation. It therefore has nothing to do with consumer preferences, if that is what you were suggesting…

    I emphasize the fact that actual price inflation in any particular market can only occur if all those who have been able to participate in that market have acquired more disposable dollars/yen with which to bid up higher prices charged.

    No variance in “consumption patterns” is necessary to explain this, if I understand correctly your meaning in using those words. I think it is fair to say that all economic participants have the same basic “consumption pattern”; they just have different disposable dollar/yen accumulations.

    Theories of price determination generally acknowledge that both supply and demand come together to determine the ultimate market price. But price inflation is a “special case” within the global topic price determination, because it focuses only on price changes which are solely determined by changes occurring on the demand side of the equation, i.e., changes in disposable money income/accumulations.

    Just ran out of time…

    • September 24, 2017 at 3:17 pm

      @James Kroeger

      In my post on September 21, 2017 at 2:48 am, I have only hinted that your problem may be related to consumption bundles. Normal inflation rates are defined by price rise percentage with weights by a bundle of consumption. If you are thinking other inflation concept you should first define your “inflation.” If not, nobody will understand what you want to say. You are using your term “inflation” in a very different meaning than that used commonly.

  12. augustin
    February 13, 2019 at 7:45 am

    This post is silly. SMD only says that there is no restriction on the kinds of functions that can be excess demand functions, beyond the usual four properties. All this means is that meaningful conclusions cannot come from the assumption of market clearing alone. One needs to make assumptions about the distribution of agents’ characteristics, the nature of technology, and so forth, in order to draw conclusions from GE models. (Similarly for Brown and Matzkin). The fact that you can’t get something for nothing, the SMD claim, says nothing about the fruitfulness of models with assumptions that delimit in some ways the nature of the objects constituting the model.

  13. Michael Handelman
    July 22, 2020 at 1:07 am

    The problem with Stiglitz is that he seems to think it is possible to combine Arrow-Debreu with a Schumpeterian (“dynamic”) perspective. So he recognizes in “Freefall” (page 271), that neoclassical models don’t incorporate dynamic, technological innovation, yet he simultaneously believes (I guess), one can graft a Schumpeterian dynamism onto neoclassical theory. So he is stuck in severe performative contradictions or eclecticism in how he is attempting to combine incompatible accounts of reality.

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