The Arrow-Debreu obsession

29 August, 2014 at 17:14 | Posted in Economics | 6 Comments

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.what if 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.

Mark Buchanan

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.kornai 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.

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6 Comments

  1. “In effect, it tells us nothing at all.”
    .
    I strongly disagree with this conclusion. Arrow-Debreu tells us what a market economy capable of reaching an allocationally efficient general equilibrium in price under uncertainty would look like. A conventional take-away from Arrow-Debreu — one that should not be gainsaid in the breadth of its implications — is that an economy capable of such a general equilibrium under uncertainty would have to have great abundance of insurance. (Taken at face value, Arrow-Debreu seems to suggest that the socialists are right; why all economists after Arrow-Debreu are not socialists is a mystery.)
    .
    Your demand that an analytic theory should be descriptive is misplaced. Analysis, properly, aims at logically determining what are the necessary and sufficient elements of a causal system. It is a priori and never descriptive. Arrow-Debreu does that. It tells us that efficient decision-making under uncertainty requires insurance, and rather a lot of it. It tells us that understanding the system of incentive structures created by institutions requires that we look beneath the distribution of income to the distribution of risk. It tells us that we ought to be vitally interested in what prevents the revelation of information and complete insurance.
    .
    There would be no point in trying to “verify” Arrow-Debreu as a description of the actual economy. It should be obvious that the actual economy is not much like general equilibrium system imagined by Arrow and Debreu. Attention to factual detail in constructing an operational model (to use Popper’s term) capable of describing and measuring the actual economy — a different sort of intellectual and scholarly project — would probably lead to the conclusion that the economy of the industrial revolution isn’t tending in any important respect to a general equilibrium in price. It might lead us to note that the economy is not even a market economy in many respects, but predominately an economy of hierarchies.
    .
    To have a descriptive model and to verify by measurement, economists would have to build operational (not just analytical) models of the actual, institutional economy.

  2. Apropos Buchanan, I wonder whether any physicists or other actual-scientists might take the trouble of writing a parody of what their disciplines would look like if it had followed the model of neoclassical economics.
    You know, one of the supreme theoretical achievements of meteorologists, for example, being a demonstration that, under certain highly restrictive conditions, weather over a given point can be derived from initial conditions.
    (Adding perhaps a Swedish-Central-Bank-Nobel-Prize-winning blogger who dismisses alternative approaches by saying things like, “people like Lorenz are always insisting that we have to add more nuance to our models about how weather agents interact, whereas those of us who were content to use the kinda-sorta-true optimum-equilibrium Humidity Seasonal-Mediterreanean Sealevel (HS-MS) models were broadly right about what to do after Hurricane Sandy”, etc.)

  3. @Bruce Wilder: Arrow-Debreu shows that it suffices to find one global price for each good in order to reach an efficient outcome. This is not socialist at all. In fact, it supports Hayek’s liberal views insofar as it helps explaining why central planning does not lead to a better outcome than a free market.

    • @kronrod You’ve been misinformed. Walras showed that it was sufficient for a process of tatonnement to find a global price for each good; as long as the last market cleared, every market cleared. Arrow-Debreu were tackling the problem of a general equilibrium under uncertainty, where each decision-making agent was making a risky choice, and they showed that, if there was sufficient insurance that each agent could choose and realize the expected value, an unique, efficient equilibrium was possible. Single global prices are gone in Arrow-Debreu replaced by a proliferation of futures contracts. Arrow drew attention to the problem of moral hazard / adverse selection as a source of market failure that helped to explain why there was so little insurance in the actual economy; public provision can overcome that source of market failure, improving efficiency.
      .
      My little joke about socialism is based on social democracy favoring as a defining desideratum, ample public provision of social insurance. It was not a reference to public planning, Hayek’s bête noire. Sorry for the misunderstanding.

      • @Bruce: I’m afraid it’s you who is misinformed. In their seminal paper “Existence of an Equilibrium for a Competitive Market”, they do not mention uncertainty even once. Instead, they show that – under the given circumstances – a free market leads to the optimal outcome and that no centralized Walrasian auctioneer is necessary to do so.

        However, I must admit that I am not familiar with their less famous work. So it could well be that you are right when taking into account those other papers. In that case, it would be nice if you could point me to one.

  4. The Arrow Debreu papers, I
    1]
    ARROW, K.J. [1953] Le role des valeurs boursières pour la repartition la meilleure des risques. In Économetrie, Colloques Internationaux du Centre National de la Recherche Scientifique, Vol. XI, pp. 41-47.

    [2]
    ARROW, K.J. [1963] Uncertainty and the welfare economics of medical care. American Economic Review53: 941-973.

    [3]
    ARROW, K.J. [1963–4] The role of securities in the optimal allocation of risk bearing. Review of Economic Studies31: 91-96.(English translation of Arrow [1953]).

    [4]
    ARROW, K.J., and G. DEBREU [1954] Existence of equilibrium for a competitive economy. Econometrica22: 265-290.

    [5]
    DEBREU, G. [1959] Theory of Value. New York: Wiley.


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