Whatever happened to stability analysis?

Alejandro Nadal

Once upon a time, stability of the general equilibrium was considered an important element in the education of students in economics. Today it seldom receives the attention it deserves and this is regrettable. Stability is one of the most important aspects of neoclassical theory because it addresses the question of just how the mechanism of free competition in the marketplace actually leads to the formation of equilibrium prices.

This crucial aspect of microeconomics is seldom covered adequately (if at all) in recent textbooks and university programs, whether at the undergraduate or post-graduate levels. Most students spend years learning how individual agents maximize, or exploring cases of oligopoly, or playing around with game theory, but when it comes to stability, their teachers skirt around the main issues.

As a result, a cloud of confusion persists. Students come to believe that somewhere in the sacred scriptures of the discipline there exists a theory that accurately reproduces just how the market forces of competition guide an economy through a price adjustment process that leads to the formation of equilibrium prices. In fact, if stability analysis received the attention it deserves, students would be able to see that it is the most important failure of general equilibrium theory.

Stability was typically introduced to students as a property of general equilibrium. The equilibrium was stable if the economic forces activated after it was disturbed returned the economy to the original (equilibrium) position. Local stability responded more to this definition, while global stability implied that the equilibrium position would be reached regardless of the starting point.

As Léon Walras (1969) explained (at the end of Lesson 11), demonstrating how the mechanism of free competition led to equilibrium prices was essential. But this was easier said than done. Hicks in the 1930s and Samuelson in 1948 were able to make some progress. But Hicks’ contribution in static stability was not associated with any adjustment process. Samuelson showed that stability analysis required an analysis of the evolution of excess demands over time and introduced the typical price adjustment equations used in modern formulations. However, he did not provide the conditions under which such a system of equations would converge to a general equilibrium.

In 1958-9 two papers, by Arrow and Hurwicz and Arrow, Block and Hurwicz, showed how under certain conditions an economy could converge to equilibrium. But these were extreme conditions: gross substitution (GS) for all goods or the validity of the weak axiom of revealed preferences (WARP) at the market level. In the key passage summarizing their results, Arrow and Hurwicz wrote: “none of the results so far obtained contradicts the proposition that under perfect competition, with the customary assumptions as to convexity, etc., the system is always stable”.

A year later, Scarf (1960) published his counterexample showing how unjustified this conjecture was. The extreme conditions of GS and WARP turned out to be indispensable, at least with the market processes described by Arrow and his colleagues. The ordinary structural conditions of the general equilibrium model were not enough to ensure convergence.

Other aspects of the model leave much to be desired. Perfect competition implies that no firm is able to modify prices, so in models in this tradition (called tâtonnement models) price adjustment is the responsibility of a fictitious character called the auctioneer, an agent that is incompatible with the notion of a private and decentralized economy. Tâtonnement models exclude transactions out of equilibrium, so that agents are stupid and believe prices announced by the auctioneer are equilibrium prices (also, initial allocations of individual agents remain unchanged until equilibrium is attained).

In the sixties a different tack was followed. Trading models were developed by Hahn and Negishi, Fisher (1983) and others in which agents were allowed to engage in transactions during the price formation process (i.e. out of equilibrium). The conditions for stability are less stringent (no GS, no WARP), but an “orderly market hypothesis” is introduced and the fictitious auctioneer is still required. Because the process changes initial holdings, the arrival point of equilibrium is path-dependent. More important, trading out of equilibrium requires the introduction of money, a serious problem in general equilibrium theory. Typically, when confronted with this revelation, students are perplexed: What? Money was always absent in my microeconomics courses?

The stability debate reached its climax with the papers published by Sonnenschein,  Mantel and Debreu in 1973-4. These results show that the usual assumptions of GET allow the dynamics of the classic tâtonnement process to be essentially arbitrary. To avoid this, additional restrictions must be imposed on excess demand functions.

The failure of stability theory is of relevance to macroeconomics. The notion that in the presence of rigidities markets fail to operate properly is the reciprocal of the belief that stability is a property of markets. The ‘rigidity’ view is pervasive in macroeconomics, from conventional Keynesianism to believers in the micro-foundations of macroeconomics and the new synthesis with its DSGE models (where transversality conditions impose stability).

This is what underlies Milton Friedman’s view that the natural rate of unemployment is “the level that would be ground out by the Walrasian system of general equilibrium equations, provided there is embedded in them the actual structural characteristics of the labor and commodity markets”. Maintaining ignorance about the limitations of stability theory comes in handy when perpetuating the mythology of market theory.

As Mundell once remarked, stability analysis is the most successful failure of general economic theory. It is also the best example of how an academic community pushes the most serious problems of mainstream theory under the rug and gets away with it. Students should learn to look under the rug. The ability to improve our understanding of economic processes depends on efforts to uncover the failures of mainstream theoretical constructs.

Alejandro Nadal’s recent book, Rethinking Macroeconomics for Sustainability, is available from Zed Books.


Arrow, K. and L. Hurwicz (1959)

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

Debreu, G. (1974), “Excess demand functions”, Journal of Mathematical Economics. 1. (15-21) http://ideas.repec.org/a/eee/mateco/v1y1974i1p15-21.html

Fisher, F. (1983), Disequilibrium Foundations of Equilibrium Economics. Cambridge University Press.

Friedman, Milton (1968)

Hahn and Negishi (1962)

Hicks, John (1939), Value and Capital. Oxford: Clarendon Press.

Mantel, R. (1974), “On the characterization of aggregate excess demand,”  Journal of Economic Theory. 7. (348-353)http://econpapers.repec.org/article/eeejetheo/v_3a7_3ay_3a1974_3ai_3a3_3ap_3a348-353.htm

Samuelson, P. (1947), Foundations of Economic Analysis. Harvard University Press.

Scarf, H. (1960), “Some examples of global instability of competitive equilibria”. International Economic Review, 1 [157 – 172]

Sonnenschein, H. (1973), “Do Walras’ identity and continuity characterize the class of community excess demand functions?” Journal of Economic Theory. 6. (345-354), http://ideas.repec.org/a/eee/jetheo/v6y1973i4p345-354.html

Walras, León (1969), Elements of Pure Economics. New York: Augustus Kelley.

16 Responses to “Whatever happened to stability analysis?”

  1. Walt says:

    This is a great post. I think if you did a survey of economists, you would find over 50% under the impression that tatonnement or something like it always converges, and that’s why GE is a good model of the economy.

  2. Ellen says:

    I agree, fantastic post. This is such an important lesson for students of economics to learn.

  3. Alan says:

    Hard to believe the concept of equilibrium has survived the GFC and all the contradictory evidence. Neoclassical economics must rule in some hypothetical space, but let’s get back to the real, serious condition of our present economies.

  4. Rakesh Bhandari says:

    Very intriguing post, but as a non-economist, may I ask you to explain why the assumptions of gross substitution and revealed preference and orderliness are required for the adjustment process to arrive at equilibrium prices? I have a book in front of me by Bruno Ingrao and Giorgio Israel that covers this ground (“The Invisible Hand: Economic Equilibrium in the History of Science”), but I am not yet convinced that the economists have themselves isolated the most important problems in regards to the assumptions implicit in existence, uniqueness and stability analysis.
    Thank you.

  5. Herman says:

    Given the fact the GET is a purely intellectual construct that bears no relation to any economy that has ever existed anywhere on the planet, academic economists’ obsession with it is indeed very puzzling.

    Perhaps it is because of the severe herding pressures in the profession and the resulting Groupthink that causes otherwise sensible people to forget the distinction between mathematics and science and make nonsensical claims like:

    “A characteristic feature that distinguishes economics from other scientific fields is that, for us, the equations of equilibrium constitute the center of our discipline.”
    —[Andreu Mas-Colell et. al. 1995, Pg 620, ‘Tatonnement and Stability’]

    Frank Hahn, who has been a pioneer of GET and one of its staunch defenders, had some very harsh things to say about how it is misused by modern macro people (‘Lucasians’):

    “Although I never believed it when I was young and held scholars in great respect, it does seem to be the case that ideology plays a large role in economics. How else to explain Chicago’s acceptance of not only general equilibrium but a particularly simplified version of it as ‘true’ or as a good enough approximation to the truth? Or how to explain the belief that the only correct models are linear and that the von Neuman prices are those to which actual prices converge pretty smartly? This belief unites Chicago and the Classicals; both think that the ‘long-run’ is the appropriate period in which to carry out analysis. There is no empirical or theoretical proof of the correctness of this. But both camps want to make an ideological point. To my mind that is a pity since clearly it reduces the credibility of the subject and its practitioners.”

    [from the introduction to
    General Equilibrium: Problems and prospects
    edited by Fabio Petri and Frank Hahn ]

  6. Although I appreciate the post, I think this story needs a denouement chapter with references to work by Smale, Kirman & Koch, Rizvi, and Saari.

    I’m not sure Hahn is fair to Sraffians in the last comment. Some have argued for reasons to be interested in Sraffa’s prices that don’t depend on them being the limit points of some (rapidly) converging dynamic process.

  7. Rakesh Bhandari says:

    Robert, what would those reasons for interest be?

  8. I suspect the percentage of the imaginary survey proposed by Walt is more like 70%. One possible explanation is that due to the problems mentioned in this entry, stability became a subject of secondary importance in microeconomics texts after the seventies. Ironically, what Walras considered to be the essential point in general equilibrium became a pariah in its own castle.
    Rakesh, you are right, economists have not been able to isolate the most important problems inherent to the assumptions used to build their models. However, almost every serious theorist agrees that gross substitution and WARP are problematic. GS or WARP are used indistinctly to prove a lemma used in the proof of global stability. Essentially that lemma says that the product of any vector of excess demands and the vector of equilibrium prices will always be positive. This is of course meaningless from the economic standpoint, but the result allows Arrow-Block-Hurwicz to show that prices converge to equilibrium. The cost is too high. As for the orderly markets assumption in Hahn and Negishi, that soounds much more reasonable, but it requires the introduction of money, something that has its own intractable problems.
    As for the dénouement suggested by Robert, yes, perhaps we need something for the finale. I’m not sure Smale is a good candidate, though. In his analysis of global economies, the use of Newton’s method to calculate equilibrium points does not help at all because, as Arrow and Hahn say (General Competitive Analysis) this process does not mimick the invisible hand: the price of a good may be raised even though it is in excess supply. Smale’s objective to bring together existence theorems, algorithms and dynamic questions falls short of a sensible representation of market processes.
    Perhaps Franklin Fisher’s work is much more important and relevant candidate for the proposed dénouement. He also reaches negative results as it shows the elimination of the auctioneer is not an easy task.

  9. Rakesh Bhandari says:

    Thank you for the reply Alejandro.
    Here is something interesting, I thought.

    “….social conformity or emulation–such as wanting a consumer good because other people already have it–can create positive feedback in the market, with the potential for destabilization. In the simplest terms, erratic and fragile market bubbles or ‘cascades’ can occur if individuals consider the behavior of others to be a better source of information than their own knowledge or preferences. ” Frank Ackerman, “Still dead after all these years: interpreting the failure of general equilibrium theory” Journal of Economic Methodology, 9:2, 119-139, 2002, p. 134

    There is another point in which I am interested. Let’s say that there is excess supply.

    The adjustment mechanism is supposed to be reduced disequilibrium prices leading a downward quantity adjustment which in turn leads back to the equilibrium price.

    But wouldn’t the response of some of the producers to such disequilibrium be an attempt to shift their supply curve right, such that they could cover their costs and sell an increased quantity at this lower disequilibrium price?

    In that case, disequilibrium would be overcome not by producing less but by producing more!

    The consequence of that would be the unviability of some firms and bankruptcy.

    Stability analysis seems to efface what is such a crucial part of the adjustments that the market is always making, namely bankruptcy.

  10. Walt says:

    Don’t just talk about it, Rakesh. Prove a theorem that your proposed mechanism works, and then we’ll talk. Walras’ original tatonnement idea sounded great too, until people showed it didn’t actually work.

  11. Rakesh, I strongly recommend Franklin Fisher’s book Disequilibrium Foundations of Equilibrium Economics (Cambridge, 1983). Here you will find how general equilibrium theory fares when we introduce individual agents that can become aware of disequilibrium (and arbitraging) opportunities.

  12. Rakesh Bhandari says:

    Thank you Alejandro. I shall indeed check it out from the library along with your book tomorrow. Look forward to reading both.

  13. Kevin P. Gallagher says:

    I think it was Tobin, after Hicks of course, that made the final tweaks to IS-LM in a Walrasian context in order to pave the way for New Keynesian DSGE models. Tobin is to be applauded for a lot of things, but in this case, in the wake of the crisis, maybe not. Schiller and Ackerlof have a provoking book suggesting that the “animal spirits” of finance were the parts of the General Theory ignored by the Keynesian synthesis types, suggesting that they too may not be off the hook for this crisis. As Paul Davidson showed last week, the New Keynesians feel vindicated now..

    Since Prof. Nadal is too modest to recommend it, I will. Another great synthesis of these arguments is a book he wrote with fellow TCB blogger Frank Ackerman:

    The Flawed Foundations of General Equilibrium: Critical essays on economic theory
    By Frank Ackerman and Alejandro Nadal
    Routledge, June 2004;


    My favorite essay in this collection, aside from the devastating critique of GE, is the article on the internal contradictions of the open economy model. Perhaps we can coax Prof. Nadal to write about that some other time…

  14. Fred says:

    Ok, I am going to ask some very ignorant questions. Perhaps the answers are already identified in prior works, thus any responses could appropriately guide me.

    How can stability exist if it is predicated upon an economic environment aggregating heterogeneous preferences and free will of the constituents of the economy?

    Furthermore, if prices are the ballasting effect towards stability, then does price stability really require offsetting preferences?

    How can a system be stable if a convergence of preferences leads to instability? (The obvious caveat being a convergence to uniform preferences ultimately exhibiting stability…)

    Thanks in advance for your responses.

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