The Neo-Walrasian General Equilibrium School

The Gale-Nikaido Lemma

Only the briefest of outlines is possible here.  For more details, consult our history of general equilibrium theory.

"Neo-Walrasian" economics refers to the strain of general equilibrium theory (often referred to by its acronyms, G.E. or G.E.T.) that emerged in the post-war period.  Its roots stretch back to the Lausanne School of Léon Walras and Vilfredo Pareto around the turn of the century.  After a period of stagnation, it re-emerged in two forms in 1930s, one more "Walrasian", advanced by the Vienna Colloquium, and another more "Paretian" that was championed particularly at the L.S.E., Chicago and Harvard

The "Neo-Walrasian" school which emerged in the 1940s and 1950s in the United States, notably under the auspices of the Cowles Commission, merged these two traditions and endowed it with a new mathematical apparatus of axomatic reasoning and convex structures (notably the "separating hyperplane theorem").   

During these early years, the Neo-Walrasians, notably Tjalling Koopmans (1951), Kenneth Arrow (1951), Gérard Debreu (1951, 1954), recast the old Paretian theories of the consumer, production and the welfare theorems, in this new clothing.  The Vienna question, the existence of equilibrium, was proved by Arrow and Debreu (1954), Lionel McKenzie (1954), David Gale (1955) and Hukukane Nikaido (1956), using fixed-point theorems created around this time.  In later years, Herbert Scarf (1967, 1973) would used fixed point-theorems as the basis of  his methods for computing equilibrium via simplical subdivisions, thus initiating the field of applied general equilibrium.

Impressed with their initial successes, in the late 1950s and 1960s, the Neo-Walrasians pressed on.  Their subsequent efforts until the 1970s can be divided  to what we call the "Hicksian" programme and the "Edgeworthian" programme.

The Hicksian programme refers to the incorporation of "Grand Themes" like stability, uncertainty, money, capital, macroeconomics, growth, etc. into general equilibrium theory, the project effectively initiated by John Hicks in Value and Capital (1939) which, in turn, harked back to the grand vision set out in Léon Walras's Elements of Pure Economics (1874).  The mathematics employed in this time period were of a different hue and in some ways simpler, not going much beyond differential equations and linear algebra.   In this sense, there was a temporary return to the "Paretian" type of general equilibrium theory.

Local and global stability of equilibrium was pursued in the late 1950s and early 1960s, notably by Kenneth Arrow and Leonid Hurwicz (1958, 1959) and Lionel McKenzie (1960).  This research project nonetheless floundered under the weight of powerful counterexamples provided by Herbert E. Scarf (1960) and David Gale (1963).  As a result, alternative forms of stability (non-tatonnement) were developed by Frank Hahn and  Takashi Negishi (1962) and Hirofumi Uzawa (1963).

The incorporation of capital theory, in the form of an intertemporal equilibrium, was effectively accomplished by Edmond Malinvaud (1953). The incorporation of  monetary theory was pursued by Don Patinkin (1956) but, after some problems were raised by Frank H. Hahn (1965),  the theory of money in equilibrium turned to the analysis of transactions costs and sequence economies (e.g. Hahn, 1971, 1973; Grandmont, 1977).  Travelling along a different path, Jacques Drèze (1975) and Jean-Pascal Benassy (1975) extended general equilibrium theory to the Keynesian macroeconomic realm via the use of "rationing" equilibrium, what has been called "Non-Walrasian" general equilibrium theory.

Sequence economies were also a consideration that arose from uncertainty and information theory.  The incoporation general equlibrium under  uncertainty was initiated via the "state-preference" approach was introduced by Kenneth Arrow (1953) and  Gérard Debreu (1959).  In a series of famous articles, Roy Radner (1967, 1968, 1972) incorporated uncertainty, information and asset markets into sequential general equilibrium theory.   Not only did he introduce the concept of what has become known as spot-asset market "Radner" equilibrium, he also introduced information sets and "rational expectations" equilibrium.   Oliver D. Hart (1975) noted some problems with spot-financial market equilibrium, while Sanford J. Grossman and Joseph E. Stiglitz (1980) identified other problems with rational expectations equilibrium.

The tombstone of the Hicksian programme arrived in the early 1970s, via a series of articles by Hugo Sonnenschein (1972, 1973), followed up by the work of Rudolf Mantel (1974) and Gerard Debreu (1974) on aggregation.  They came up with the dreadfully dispiriting "Debreu-Sonnenschein-Mantel" theorem which effectively posited that reasonable underlying economies can generate all kinds of the weirdly-shaped market excess demand functions.

The Edgeworthian programme refers to the efforts to examine the relationship between a Walrasian competitive equilibrium and the solutions obtained via alternative exchange process (notably those from  game theory). The mathematical tools of choicethat were introduced in this effort in the 1960s and 1970s -- i.e.  measure theory and non-standard analysis -- were substantially more demanding than anything most economists had been used to.

The central effort was the attempt to prove "Edgeworth's conjecture", i.e. that the core would shrink to the set of Walrasian Equilibrium if we increased the degree of competition (here formalized as the number of agents) to infinity.   In 1963, Gerard Debreu and Herbert E. Scarf set the ball rolling with their proof of core convergence within the context of a "replicated" economy (i.e. arbitrarily large numbers of agents of each type). In 1964, Robert Aumann proved the equivalence of the Edgeworthian core and the Walrasian equilibria when we have a continuum (uncountably infinite number) of agents.  This "new" definition of "perfect competition" required the introduction of measure theory -- notably Lyapunov's Theorem -- into economics. Edgeworth's conjecture in more general forms has been pursued by other economists since (esp. Truman Bewley (1973), Werner Hildenbrand (1974), Donald J. Brown and Abraham Robinson (1972, 1974), Robert M. Anderson (1978)).

Taking a leaf from this experience, the Hicksian themes were rejuvenated in the 1970s.   By upping the mathematical ante, general equilibrium theorists hoping to solve the nottier problems and loosen some of the assumptions which had been made previously.

The first major step was proving existence of general equilibrium with an infinite number of commodities.  Capital and uncertainty theory had raised several questions about infinite-dimensional commodity spaces.   The problem of infinite commodities was posed by  Gérard Debreu (1954) and pursued by Truman Bewley (1969, 1972), Bezalel Peleg and Menachem Yaari (1970), Andreu Mas-Colell (1986), William R. Zame (1987) and many others.  This required the pursuit of infinite-dimensional vector space theory.

A second effort was also initated by  Gérard Debreu (1970).  Namely, from Sard's theorem of differential topology, Debreu proved that although equilibria are not generally unique, they are locally unique, i.e. there is (usually) a finite number of equilibria. The Debreu paper launched the thousand ships of global analysis, which effectively resurrected the differential calculus in economics.   Egbert Dierker (1973) laid down the conditions on (absolute) uniqueness of equilibrium while others, notably Yves Balasko and the Steve Smale, pursued other themes on the continuity properties of equilibrium sets, stability, etc.

Attempts to resurrect old Paretian welfare theory were also pursued during this time. Duncan Foley (1970) extended general equilibrium theory to the theory of Lindahl pricing of public goods.   However, as David Starrett (1972) noted, many externality cases imply significant non-convexities in the production set.  Removal of the convexity axiom has turned out to be rather tricky.  However, following a conjecture by M.J. Farrell, Ross M. Starr (1969) proved the "convexifying" effects of a large number of firms, each with (mildly) non-convex production sets. (of course, Lyapunov's theorem guarantees convexity in the continuum case).   However, at the individual level non-convexity remains hard to model. Two approaches were developed in the 1980s, one relying on "non-smooth" differentials (e.g. Cornet, 1982) and another involving "integral" activity analysis (e.g. Scarf, 1986).  One of the results of this research programme has been the resurrection of the old Paretian "marginal cost pricing" theorem of the 1930s. 

  Following the dictums of "Ockam's Razor", attempts were made to derive standard results with ever more general assumptions on preferences and technology.   Besides the convexity case mentioned earlier, removal of the completeness and transitivity axiom from preferences was accomplished in one blow by Andreu Mas-Colell (1974).

The innovations continued.  New characterizations of competitive equilibrium were also achieved during this time.  We already know, from Aumann (1964) that competitive equilibrium can be characterized as a core with an infinite number of agents.    Later economists characterized equilibrium as limiting cases of other game-theoretic solution concepts, e.g. with the set of fair net trades by David Schmeidler and Karl Vind (1972), with the Shapley value  by Robert J. Aumann and Lloyd S. Shapley (1974), with the bargaining set by Andreu Mas-Colell (1989), for instance .   Unfortunately, these solution concepts are for "cooperative" games and, furthermore, require infinite number of agents.  Efforts have been made throughout , while we usually like to think that Walrasian equilibrium is non-cooperative and can be achieved with less than infinite number of agents. 

To this end, alternative characterizations have been introduced.  Duncan Foley (1967) characterized competitive equilibrium as a "no-envy" allocation.  Joseph M. Ostroy (1980) characterized quite interestingly as a "no-surpus" allocation.  A latter-day effort has been to linking it with sequential bargaining theory (e.g. Douglas Gale, 1986; Martin J. Osborne and Ariel Rubinstein, 1994), but much remains to be done.

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