Hurwicz, stability, and mechanism design
Ingrao and Israel (1990 Ingrao, B., & Israel, G. (1990). Invisible hand.: Economic equilibrium in the history of science. Cambridge, MA: The MIT Press.
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) claim that historically the same group of ‘utopically’ oriented economists also considered the questions of stability to be redundant, which is only partly true. The most important economist who, one the one hand, was strongly influenced by the ‘utopian-normative’ approach (both relating to the issue of stability and with respect to economic theory in general) and, on the other, made a significant contribution to the theory of stability, was Leonid Hurwicz.
Born in Moscow in the year of the Russian revolution, Hurwicz was always intimately connected to the culture of socialism, be it in half-authoritarian Poland of the 1930s where he studied law, in London and Geneva where he got acquainted to the major critics of market socialism Hayek and Mises, or in the USA, where he was assistant to Lange. A member of the Cowles commission, Hurwicz, was fully initiated into the problems discussed there.
From the standpoint of our discussion Hurwicz is quite a remarkable figure. He received 2007 Nobel prize ‘for having laid the foundations of mechanism design theory,’ in particular for developing a concept of incentive compatibility in the 1970s. Mechanism design theory, as Hurwicz (1973 Hurwicz, L. (1973). The design of mechanisms for resource allocation. American Economic Review, 63, 1–30.
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, p. 1) himself put it, ‘refuses to accept the institutional status quo of a particular time and place as the only legitimate object of interest and yet recognizes constraints that disqualify naïve utopias’. The program formulated by Hurwicz in the 1970s thus included both the constructivist idea of going beyond the existing institutional framework by devising new institutions and an attempt to overcome an abstract utopian (socialist!) ideal of creating a perfect society.
Hurwicz gained reputation in general equilibrium theory after publishing his results on equilibrium stability together with Arrow and with the collaboration of a mathematician H. D. Block (Arrow, Block, & Hurwicz, 1959 Arrow, K. J., Block, H. D., & Hurwicz, L. (1959). On the stability of the competitive equilibrium, II. Econometrica, 27, 82–109.
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; Arrow & Hurwicz, 1958 Arrow, K. J., & Hurwicz, L. (1958). On the stability of the competitive equilibrium, I. Econometrica, 26, 522–552.
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).1717. Arrow, another major progenitor of contemporary general equilibrium theory, was also an adherent of planning. In one of the latest autobiographical texts (Arrow, 2009 Arrow, K. (2009). Some developments in economic theory since 1940: An eyewitness account. Annual Review of Economics, 1, 1–16. 10.1146/annurev.economics.050708.143405.
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), he concedes that the idea of planning was important for him at that time, albeit making certain reservations and claiming that, in fact, despite an enormous influence of Lange and Lerner (and – we would add – Arrow's advisor Harold Hotelling) along with equally enormous amount of intellectual energy spent, socialist ideas and models of planning had little influence on real economic situation. In an unpublished interview taken by our research group in April 2012, Arrow added to the skepticism toward his own socialist past the idea that in the general equilibrium theory he was just developing Hicks who had nothing to do with socialist concerns. However, he also recalled Frederick Taylor's (1929 Taylor, F. M. (1929). The guidance of production in a socialist state. American Economic Review, 19, 1–8.
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) presidential address to the American economic association dealing with the optimal running of socialist economy. Moreover, Hicks was the main reference in stability theory. The ambiguity thus remains, and we could still argue that in 1940–1960s Arrow not only knew about socialist and constructivist interpretations of the general equilibrium model, but also actively participated in its development along these lines. More on that in Mirowski (2002 Mirowski, Ph. (2002). Machine dreams: Economics becomes cyborg science. New York, NY: Cambridge University Press.
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, p. 298f.) and Klein (2013 Klein, D. B. (2013). Kenneth J. Arrow [Ideological profiles of the economics laureates]. Econ Journal Watch, 10, 268–281.
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).
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The two papers Arrow and Hurwicz produced were overtly inconclusive and positively biased: they admitted taking notice only of some cases, but in all of them equilibrium achieved by the tatonnement process was proved to be globally stable (meaning that the system converges to it from any point, and not just from its neighborhood). Even more important in our context is the idea of the system stability different from the equilibrium stability dealt with before. This generalization is justified on purely conceptual terms: ‘When there are two or more equilibria, it cannot be the case that all equilibrium points are globally stable’ (Arrow & Hurwicz, 1958 Arrow, K. J., & Hurwicz, L. (1958). On the stability of the competitive equilibrium, I. Econometrica, 26, 522–552.
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, p. 524, cf. Feiwel, 1987 Feiwel, G. (Ed.). (1987). Arrow and the ascent of modern economic theory. Houndmills: Macmillan.
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, p. 263). But even if some equilibria are unstable, the cases considered suggested global stability of the whole edifice that gave hope for its relevance and even a certain degree of realisticness.
However, that was not the case. As Hurwicz' colleague Scarf's (1960 Scarf, H. (1960). Some examples of global instability of the competitive equilibrium. International Economic Review, 1, 157–172. 10.2307/2556215.
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) and Gale's (1963 Gale, D. (1963). A note on global instability of competitive equilibrium. Naval Research Logistics Quarterly, 10, 81–87. 10.1002/nav.3800100107.
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) examples showed, no general result on the stability of the tatonnement can be established. In general, it turned out that for the tatonnement to be globally stable market excess demand functions should conform to the weak axiom of revealed preferences – ‘a property that is very special indeed’ (Fisher, 2011 Fisher, F. (2011). The stability of general equilibrium: What do we know and why is it important? In P.Bridel (Ed.), General equilibrium analysis: A century after Walras (pp. 34–45). London: Routledge.
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, p. 37). However, at least it gave an idea of how a stable economy, representational issues put aside, should look like – or, we would add, should be made to look like.
The latter was precisely the response of Hurwicz! His idea was to experiment with various models in order to construct, design an economic system that would be on the whole stable (cf. Feiwel, 1987 Feiwel, G. (Ed.). (1987). Arrow and the ascent of modern economic theory. Houndmills: Macmillan.
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, p. 262).1818. Interestingly, the notorious normative problem of Walrasian tatonnement – the question of who changes the prices – is circumvented here. The issue is, rather, whether the process as such, guided by the social planner or market forces, is stable.
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He concedes that Scarf's and Gale's examples somehow discouraged him:
[O]nce it was noted that there can be instabilities, this opened the question whether one could think of alternative stable mechanisms that one could consider either from a normative, descriptive, or computational point of view. (Feiwel, 1987 Feiwel, G. (Ed.). (1987). Arrow and the ascent of modern economic theory. Houndmills: Macmillan.
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, p. 268)
What is given here is a particularly instructive classification. A descriptive point of view invoked by Hurwicz implies the process of finding a model that would adequately capture the functioning of a real economy. A computational point of view is characteristic for some mathematical economists that were mostly interested in constructing a viable algorithm for computing equilibrium prices. The best example of such research is the work of Scarf (1973 Scarf, H. E. (1973). The computation of economic equilibria(with the collaboration of T. Hansen). Cowles Monograph No. 24. New Haven, CT and London: Yale University Press.
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). It is of particular interest for our story that even Scarf's negative results on stability were a part of this computational programme: as Scarf (1991 Scarf, H. (1991). The origins of fixed point methods. In J. K.Lenstra, A. H. G.Rinnooy Kan, & A.Schrijver (Eds.), History of mathematical programming. A collection of personal reminiscences. Amsterdam: North-Holland.
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) himself noted, his interest in finding counter-examples of tatonnement stability were guided by a conjecture that this Walrasian idea is in fact algorithmically viable and hence, can be computationally effective. If a general equilibrium model together with tatonnement were globally stable one could hope for a constructive solution of an equilibrium problem, i.e., of computing equilibria referring simply to the respective excess demands. But this hope was not realized. |