Please say after me – Sonnenschein-Mantel-Debreu

21 Jul, 2012 at 10:53 | Posted in Economics | 23 Comments

Can you say Sonnenschein-Mantel-Debreu?


Because that probably also means that you can understand 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. After all – as Nobel laureate Robert Solow noted in “The State of Macroeconomics” (Journal of Economic Perspectives 2008:243-249) – “a modern economy is populated by consumers, workers, pensioners, owners, managers, investors, entrepreneurs, bankers, and others, with different and sometimes conflicting desires, information, expectations, capacities, beliefs, and rules of behavior.” So, representative agent models are rather an evasion whereby issues of distribution, coordination, heterogeneity – everything that really defines macroeconomics – are swept under the rug.

Conclusion – don’t believe a single thing of what these microfounders tell you until they have told you how they have coped with – not evadedSonnenschein-Mantel-Debreu!

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!


  1. Gross substitution axiom is a cop-out. It is specifically tailored to address a macro outcome, as opposed to arising from pure micro considerations–it has no intuitive meaning in the micro context. In that sense, it violates the idea of microfoundations. basically it boils down to this–I have a micro-based model, but it does not generate all the stuff that i really believe it should. let us see, what assumptions we can make at the micro level to get the macro outcome that we desire!!!

    BTW, even the gross substitution axiom is insufficient to ensure that aggregate demand functions are well-behaved.

  2. “unequivocally showed that there did not exist any condition by which assumptions on individuals would guarantee neither stability nor uniqueness of the equlibrium solution.”

    What does this mean in english? If it says that by ignoring the amount of money people have in their pockets for consumption, you can tell something about real world, it is of course true.

  3. “unequivocally showed that there did not exist any condition by which assumptions on individuals would guarantee neither stability nor uniqueness of the equlibrium solution.” Do you really mean neither and not either? And although I basically agree with you, I think Pontus is right on the limited point that if you say “there did not exist any condition” and he then cites a condition, he has shown that statement is wrong. Even if his example has no practical importance.

  4. See also Matias Vernengo’s new post – Stock-Flow with Consistent Accounting (SFCA) models


    SFCA proved to be considerably more successful than conventional, in particular Dynamic Stochastic General Equilibrium (DSGE) models, in predicting the Great Recession (see here paper by Dirk Bezemer).

    As noted by Gennaro, the fundamental principle of SFCA models is that:

    “in the economy – and therefore in models representing the economy – everything comes from somewhere and goes somewhere else: ‘there are no black holes.’ This obvious principle has relevant implications: one is that the debt of somebody is a credit for somebody else.”

    Note that this fundamental principle has more to do with the fully consistent accounting part of the model, than with the relation of stocks and flows. But stock-flow relations are also essential, since flow decisions of spending are tied to stocks. Private agents can spend if they have access to stocks of credit, of accumulated assets, that is, some stock of wealth. The State often has the power to spend and accumulate a stock of debt, since it can decide (Functional Finance and Chartalist approaches, which are implicit in Godley’s work, become important here) the token in which debts are denominated.

    • It’s a modelling technique and it demonstrates a *different viewpoint* on the economy. By taking a different viewpoint things that weren’t easy to see suddenly become clearer.

      In this case the central problem of our current system. There are lots of unrepayable debts, therefore there are an equivalent amount of unrecoverable assets.

      Yet nobody is talking about who has these unrecoverable assets and how we deal with them.

      We know debtor’s prisons don’t work from the Victorian experience. Yet we are currently imposing that solution on entire nations.

      We invented limited liability entities, bankruptcy procedures, administration and ‘chapter 11’ processes to allow unrepayable debts and unrecoverable assets to meet their end in a controlled fashion.

      But we clearly don’t have that in place for currency pegged countries in the Euro, or for big banking institutions.

      We should be talking about how banks and nation states in a currency union are liquidated to get rid of the unrecoverable assets.

      Or we should be talking about how those unrecoverable assets can be made good by swapping the bad liabilities for good ones (which requires the currency issuer to step in and use its ‘deus ex machina’ capabilities in some fashion).

  5. So how come you keep on insisting that “representative agent models are rather an evasion whereby issues of […] coordination […] are swept under the rug.”

    As I just showed, using a first-year undergraduate textbook example, this is factually wrong. For a professor of economics to misunderstand such a trivial insight is baffling!

    It is truly amazing to see how someone can build a career (!) on trivial misunderstandings. But I guess it is as Upton Sinclair said: “It is difficult to get a man to understand something, when his salary depends upon his not understanding it!”

    Within certain schools of heterodox think it appears that ignorance really is bliss.

    • Coordination between actors when one ussumes that “they are all the same”? Woah! That’s a really tough research agenda! Are you sure you’re not overstretching your aspirations? And how relevant and realistic such a model must be for understanding modern complex economies. One has to be impressed by the state of presentday Cambridge economics. Marshall, Keynes, Robinson, Sraffa? What are they compared to the bright young guys of present day’s Cantabrigian brigades of microfounded macroeconomists!

    • I’m new to this blog, and I’m not sure to buy everything Lars Syll says. He possibly may overstate the implications of the Sonneschein-Mantel-Debreu argument.
      But his critique is relevant, while the Pontus defense of NC, NK, RBC, MF and DSGE models misses the point.

      The point is that if you base a model on a science-fiction mechanism (the representative agent), you may hope to describe life on Mars, but not on earth.
      It does not mean that those models have no validity at all, or can’t be tuned to describe some specific mechanisms, but that as a global framework describing the general economy, they have very little relevance.

      As for the following comment “For a professor of economics to misunderstand such a trivial insight is baffling!”, I would have hoped that in the current state of the world economy and of economics, economists would be prepared to admit that they all face trivial things they don’t understand.
      As a former economist, the only thing I’m sure of is that I’m sure of nothing.

    • The entire point of the representative agent framework was to treat all macroeconomic actors as no different than a single person. Yes, the prisoner’s dilemma and other games feature rational actors who lower utility due to coordination failure, but we’re literally pretending the economy is one guy.

      • I agree with pontus about SMD though —obviously, to me at least—i dont think NC, DSGE, RE etc models are ‘correct’, though in some cases they are approximations, and possibly even adequate ones.
        The SMD result is just part of general equilibrium theory, just as the fact that the 3 body problem has chaotic solutions doesn’t contradict newtonian determinism for some other cases. Similarily, its fine to note that, in the general case, peano arithmatic exhibits Godelian incompleteness, and while some optimal solutions to the traveling salesman can be found, in general its NP-complete.

        What is interesting about the representative agent idea is that in fact, many or possibly all models with heterogeneous agents can be translated into the representative agent model whcih exhibits the same behavior (just as all higher math can be rewritten in terms of ‘0s’ and ‘1s’.

        upton sinclair was right

  6. “It simply means that they are all the same.” Exactly. You nailed it!

  7. “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.”

    Bold statement, but wrong as usual. If individual excess demand functions satisfy the gross substitution property (which holds for a variety of preferences), there is a unique equilibrium.

    Anyway, I still don’t know why you keep on insisting that representative agents exclude the analysis of cooperation failures. This is again patently wrong. Only a very naive level of abstraction would lead you to that conclusion (e.g. “there’s only one agent, so how can there be coordination problems?”). Just to give you a counter example where this logic goes wrong: In the classic prisoners’ dilemma both agents are identical in all respects. Thus we consider a representative agent. Yet, and as you (hopefully!) know, there are deep coordination problems in this framework.

    Representative agent doesn’t mean that there is only one agent. It doesn’t mean that all agents act in harmony. It simply means that they are all the same. And they all try to do there utmost to achieve there goals, without (much) concern what happens to others (cf. Prisoners’ dilemma). Coordination arise frequently and easily in representative agent models.

    • “Bold statement, but wrong as usual. If individual excess demand functions satisfy the gross substitution property (which holds for a variety of preferences), there is a unique equilibrium.”

      How realistic is the assumption?

      This is a recurring meme: “actually, no you don’t understand, because if you introduce another assumption, everything’s fine.” Real science progresses by relaxing assumptions, not adding them.

      • “Real science progresses by relaxing assumptions, not adding them.”

        Gosh, what an empty slogan.

        So if you have n equation in n+1 unknowns, and I point out that you need another equation to have a determined system, it’s not “real science”? Real science would remove “yet another equation”?

        That sounds like another Chuck Norris joke (“When Chuck Norris solves systems of equations he needs only n equations for n+1 unknowns”). Are you for real?

        • You’re not pointing out we need another equation, you’re adding another assumption – SMD can be negated IF x criteria is satisfied.

          You’ve simply put words in my mouth and evaded the thrust of the matter completely.

          • I’m saying that you have a very simplistic view of what qualifies as “real science”. In all types of deductive reasoning you have a collection of statements, and from these you derive conclusions. Of course, as the set of initial statements is reduced, your conclusions might change, disappear, or even remain. That’s just how deductive reasoning works, and that’s not a litmus test for what qualifies as “science”.

          • By the way, if you make a statement that “there are no conditions such that this and that holds” and I bring forth a condition such that exactly “this and that” holds, I’ve falsified your statement. Karl Popper thought of this a real science.

          • That statement was obviously not a comprehensive overview of my opinion on what constitutes science, but a comment on the nature of how economics progresses wrt assumptions.

            “there are no conditions such that this and that holds”

            I didn’t say that. Anyway the real question is whether or not the assumption is realistic, which you’ve opted out of asking, instead preferring to spuriously extrapolate my statements and put words into my mouth to try and make me look stupid or ignorant.

          • opted out of answering*

          • Listen buddy. Lars made that statement, and I responded to Lars. You claimed that my reply was more of “a recurring meme” than “real science”. I argue that if someone makes a statement claiming it “holds without exception”, it is enough to point out an exception. That’s called falsification. And Popper called that science (and not “memes”).

            Now if you, or Lars, wants to withdraw the initial statement and instead conclude that “the SMD theorem reveals that no unique equilibrium exist except under unreasonable assumption”, that’s fine. But then your statement is weaker and also subjective (“unreasonable”).

            The SMD theorem states that if there are m agents and n commodities, and preferences satisfy very weak restrictions, then there is nothing which guarantees a unique equilibrium. Fine. This is a negative result. But there is no reason extrapolating this result to situations it doesn’t cover. That just appears unserious.

          • The original question *I* asked was how realistic the condition was. You can have the other debate with Lars.

    • Ok, say the equilibrium is unique. Is it stable?

  8. Lets assume that new classical, real business cycles, DSGE, and new-Keynesian micro-founded macromodels are bad substitutes for real macroeconomic analysis. I got that part. Please remind me what real macroeconomic analysis consists of. That will help me with the next part.

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