Hunting the Auctioneer

Alejandro Nadal

“For the Snark’s a peculiar creature, that won’t / Be caught in a commonplace way./ Do all that you know, and try all that you don’t: / Not a chance must be wasted today!

Lewis Carroll, The Hunting of the Snark

Of all the assumptions of general equilibrium theory, the Walrasian auctioneer may very well be the most important one. Because macroeconomics has been absorbed by neoclassical microeconomics, today the role of the auctioneer is also embedded in macro models. For example, the work of Lucas and Kydland and Prescott relies heavily on the presence of an auctioneer. It is therefore no exaggeration to say that this assumption runs through the spinal chord of orthodox economic theory, both micro and macro.

Although everyone seems to agree that this is not a good assumption, the auctioneer is also one of orthodoxy’s less well-understood figures. Both friend and foe repeatedly misunderstand its role and its intimate relation to stability theory in general equilibrium models.

To general equilibrium, the auctioneer is both essential and destructive. The possibility of engaging a dynamic price adjustment process depends crucially on the presence of the auctioneer, but this figure acts as a centralizing agent that contradicts the very objective of the theory, that is, to show how market forces in a decentralized market economy lead to equilibrium prices and allocations. Given the destructiveness of this fictitious character, one wonders why orthodox economists have not ridden themselves of this figure. The answer is simple enough and must be found in the difficulties encountered in a world without the auctioneer.

Typically, students in economics are taught that this figure adjusts prices during the dynamics of price formation. Thus, the auctioneer announces prices and collects information about the economic plans of households and firms. He then recalculates prices and adjusts them according to the law of supply and demand until the system reaches equilibrium. Because the model relies on the assumption of perfect competition agents cannot adjust prices. Thus the relevance of the question raised by Koopmans (1957): if agents cannot change prices, who then adjusts prices? So the figure of the auctioneer is presented as an uncomfortable assumption, but one that is needed to proceed with the analysis of price formation.

This is misleading, as the auctioneer also appears in the Arrow-Debreu model that relies on differential topology to demonstrate the existence of a general competitive equilibrium using Kakutani’s fixed point theorem (see for example Debreu, 1982: 134 and Arrow-Debreu, 1954: 275). The footprint of the auctioneer can also be found in the construction of the individual agents in Arrow-Debreu models as shown in Nadal (2004) and in the trading process once equilibrium prices have been determined. Thus, the auctioneer possesses features that make it almost ubiquitous in general equilibrium theory. Its work goes well beyond the price formation processes.

Because of the omnipresence of the auctioneer we begin to suspect that orthodoxy will have a difficult time trying to get rid of this figure. The best effort to do without the auctioneer is Franklin Fisher (1983). But the end result is not satisfactory as new and equally radical hypotheses have to be introduced. The resilience of the auctioneer is related to the deep functions it performs in general equilibrium theory. Understanding those logical functions is crucial for a serious critique of the neoclassical paradigm.

To start, the auctioneer does not simply reveal information about “prices”. It announces systems of relative prices. This entails that there is a single price for every class of commodities. Thus, the auctioneer reveals price vectors that have a distinctive property: the price of two commodities expressed in terms of each other is equal to the ratio of the prices of these two commodities expressed in a third one. In Walras’ (1969) notation we have pi,j = pi,k/pj,k. This means that direct and indirect barter render the same result (changing trading itineraries does not alter the end result).

It is important to recall that the system of prices exists in equilibrium as well as in disequilibrium. Why should goods exhibit the property of having a single price even before the market process begins? Clearly one should not expect this to happen as this is a characteristic that would make sense in equilibrium but not before this position is attained.

If the auctioneer disappears and the modeller allows for multiple prices for each commodity, agents will calculate their economic plans with different prices and it will be impossible to calculate excess demand in each market, bringing down the entire adjustment process.

So, the price adjustment process that is supposed to lead to equilibrium prices depends crucially on the presence of the auctioneer. This is true for so-called tâtonnement models in which the auctioneer cries out prices that are used as parameters by all agents in calculating their plans. Here no trading takes place until equilibrium is attained.

According to Steve Keen (2011:180) this ruse is clearly artificial. Yes, it is, but Keen’s analysis is insufficient. He fails to understand that even with the presence of the auctioneer and the absurd assumption forbidding trade at disequilibrium prices, general equilibrium theory is unable to show that in the general case the tâtonnement process leads to equilibrium prices. Thus, the papers by Arrow and Block (1958) and by Arrow, Block and Hurwicz (1959) could prove global stability in two very special cases: gross substitutes (GS) and the weak axiom of revealed preferences (WARP) at the market level. Both assumptions are well known to be absurd. Failure to complete the critique of stability theory perpetuates the mythology that general equilibrium theory did indeed demonstrate the convergence to equilibrium prices. It did not!

Incidentally Keen appears to be unaware that there is a class of general equilibrium models that do away with the assumption forbidding trade outside of equilibrium positions. We are referring here to non-tâtonnement models such as those built in the footsteps of Hahn and Negishi (1962). Non-tâtonnement processes were indeed more realistic, and do not require the GS and WARP assumptions. In spite of this, orthodoxy was never terribly fond of these models because they exhibit the property of path dependency due to the redistributive effects of trading under disequilibrium prices. The main point here is that because of its essential functions, even non-tâtonnement models rely on the auctioneer.

Once equilibrium prices have been formed in tâtonnement models the auctioneer can find no solace because there is no guarantee that the equilibrium allocation will be effectively realized. Because the matrix of excess demands may not allow for pairwise trading to occur, there is no assurance that the individual plans of the agents in the model will be fulfilled (see Ostroy and Starr 1974). In the absence of money, the intervention of an external agent that acts as a clearinghouse is required. It seems once again that the auctioneer is truly indispensable in general equilibrium theory.

In non-tâtonnement models there is a different story. Trade takes place in disequilibrium situations until favourable opportunities are exhausted. But because the conditions for bilateral barter may not be present, a means of exchange is required. This, in turn, is problematic and takes us into the delicate question of why general equilibrium models cannot tolerate the introduction of money. That is a different story and theme for another entry in this blog.

The economic crisis is still far from generating a thorough revision of the main paradigms of orthodoxy. A nice way to start is with a thorough revision of assumptions such as the fictitious auctioneer.

Links and References

Arrow, K. and H. D. Block (1958)

“On the Stability of the Competitive Equilibrium, I”, Econometrica, 26 (522 – 552)

Arrow. K., H. D. Block and L. Hurwicz (1959)

“On the Stability of the Competitive Equilibrium II”, Econometrica, 27 [82-109)
Arrow, K. and G. Debreu (1954)

“Existence of an Equilibrium for a Competitive Economy”, Econometrica, 22 (3) [265-90]

Debreu (1982)

“Existence of a Competitive Equilibrium”, in Arrow. K. and M. Intrilligator (eds) Handbook of Mathematical Economics, Amsterdam: North Holland.

Fisher, Franklin (1983)

Disequilibrium Foundations of Equilibrium Economics. Cambridge University Press.

Hahn, F. and T. Negishi (1962)

“A Theorem on Non-Tâtonnement Stability”, Econometrica, 30 (3) [463-469]

Keen, Steve (2011)

Debunking Economics. London: Zed Books.

Koopmans T. C. (1957)

Three Essays on the State of Economic Science. New York: McGraw Hill

Nadal, A. (2004)

“Behind the Building Blocks: Commodities and Individuals in General Equilibrium Theory”, in Ackerman, F. and A. Nadal, The Flawed Foundations of General Equilibrium. London: Routledge.

Ostroy, J. M. and Starr, R. M. (1974)

“Money and the Decentralization of Exchange”, Econometrica, 42 [1093-113]

Walras, Léon (1969)

Elements of Pure Economics (or The Social Theory of Wealth). Trans. W. Jaffé. New York: Augustus Kelley.

7 Responses to “Hunting the Auctioneer”

  1. Herman says:

    Another illuminating post on the problems of Walrasian GE. Thank you Prof Nadal. I really have nothing to add, and am commenting just in case the absence of comments is mistaken for an absence of interest or even worse, perception of the absence of importance. On the last point, one of the silliest claims on the *un*importance of stability in GE –and one still made with some frequency — comes from Milton Friedman:

    Franklin Fisher(2011):
    “Milton Friedman remarked to me long ago that the study of the stability of general equilibrium is unimportant, first, because it is obvious that the economy is stable, and, second, because if it isn’t stable we are all wasting our time. He should have known better. In the first place, it is not at all obvious that the actual economy is stable. Apart from the lessons of the past few years, there is the fact that prices do change all the time.”

    The clearly religious insistence on GET (with the emphasis on equilibrium) in macro means that people who know better need to keep hammering away to expose the fact that equilibrium macro is built on foundations of quicksand.

  2. Thanks Herman for your encouragement.
    The comment by Friedman is quite revealing. In 1994 I had a conversation with Partha Dasgupta about general equilibrium theory. (I remember we were in the Marquette Building in Chicago). He asked me: “Why are you so obsessed by stability theory? When I look out the window I don’t see markets falling apart, I see a stable economy”. To this I replied: “How curious. When I look out the window I see the sun revolving around the Planet Earth!” He was puzzled. I don’t think he ever understood the point I was trying to make.
    Franklin Fisher is one of the most serious scholars and he deserves credit for his intellectual honesty and creative work in GET.

  3. Paul S says:

    Great post!

    Also, great comments.

    I’m curious, why does Friedman think that economists (I’m assuming) would all be wasting their time if the economy is unstable? Isn’t the potential for instability a reason in itself for studying the economy?

  4. Thanks Paul S. That’s a very interesting point. So, according to Friedman, Hyman Minsky was wasting his time?
    But to me there is another shocking part of Friedman’s remark: the notion that “it is obvious that the economy is stable”. This is extraordinary. Is this how these great thinkers work?
    There is nothing obvious in science. We need analytical tools to see if the economy is stable or not. And the analytical instruments that existed when Friedman was around were embedded in the works of Arrow, Block, Hurwicz, Hahn and Negishi. In addition, there was the work of Sonneschein, Debreu and Rolf Mantel, who showed that the general equilibrium edifice would never serve to settle the question about stability (with or without the auctioneer). Friedman had zero knowledge of this type of work. And I’m afraid macroeconomists doing DSGE, RCB and NK models display the same degree of ignorance when it comes to general equilibrium theory.

  5. Herman says:

    A major problem with the Friedman quote is that he is conflating two distinct meanings of the word “stability”. One is the common English usage of the word, which could mean “When I look out the window I don’t see markets falling apart,” (Dasgupta). The other is the technical sense of stability in a GE model, say Arrow-Debreu, which essentially means that the economy has a built-in tendency to converge to an equilibrium. As Profs Nadal and Fisher note, identifying stability in this sense is far from obvious. There is also no reason to believe, for example, that a lack of stability in a GE model would necessarily imply that markets should “fall apart”. Of course it is also not at all obvious that markets are stable in the ordinary English sense of the word. And I absolutely agree with you that instability (in either sense) clearly is a reason in itself for studying the economy.

    But why should Friedman have a monopoly on blurring the meaning of “stability”? We can have some fun with it too:

    Suppose most market economies are stable, but GET says that stability is highly unlikely, then it follows that GET is not a good model of the economy and must be rejected. However, somehow I don’t think that Friedman would have accepted this inevitable implication of his assertion that the economy is “obviously stable”.

    Or suppose that Friedman insists that despite the above, GET is, presumably for theological reasons, the only acceptable way of modelling the economy. Then it follows that the observed “stability”, such as it is, is due to reasons external to the model, ie it is probably due to the wider social context within which a market economy is embedded — things like the institutions, laws, norms and so on. And so GET is irrelevant. But again I doubt if Friedman would have accepted this implication of his assertion.

    The post was about the Auctioneer, but the comments are about stability. Stability has been a major theme in Prof Nadal’s work so I hope he forgives the derailment.

  6. hi Herman, that’s a very nice point and I think it is very helpful in discussions on GET.
    Yes, the post was on the auctioneer but then the discussion centered on stability. That’s up to a certain point normal because the role of the auctioneer is especially obvious in stability analysis. In reality the role of the auctioneer is more complex. It begins in the primal stage when the general equilibrium model is being built (intervening in how individual agents are defined), then continues into the price formation process (stability), and culminates in the trading process (in the absence of money the auctioneer morphs into a clearinghouse that makes trading possible). We’ll get back to the first and last functions over blog posts.
    The typical critique of the auctioneer is limited to the notion that this is an artificial figure (see Keen’s comment). That’s not the only problem: even with this fictitious, artificial and abstract figure, GET models fail to prove stability in the general case.
    Imagine somebody saying “Bear with me and be patient, I’m building a model that will prove how a price vector converges to equilibrium. I will need to introduce an artificial figure called the auctioneer”. Personally, I would say, “OK, I’ll be patient. Show me how it works”. Later the same person would say, “There, you see, we can only show stability in two abhorrent cases, GS and WARP at the market level”. Here I would say: “Hey, I waited and was patient, and I accepted the introduction of all sorts of abusive and arbitrary assumptions. I even accepted the presence of the auctioneer. And yet, at the end of the day, even with the help of all these assumptions, we see your model does not work!” My point: the problem is not only the auctioneer, it is the general equilibrium model.

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