Whatever Happened to Stability Analysis?

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 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 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 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 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.

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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 H. D. Block (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.