’Equilibrium prevails if all plans and expectations of all economic subjects are fulfilled so that, at given data, no economic subject feels inclined to revise his plans or expectations.’-Erik Lindahl

Let us talk about Walrasian economic equilibrium and about the dynamic process that governs the evolution of market prices, possibly towards an equilibrium. As the quote above by Erik Lindahl (1891-1960), a Swedish economist, suggests, economic equilibrium is a rather general and abstract concept, which is a state where nobody is willing to move or alter her policy.

The neoclassical theory of economic equilibrium is arguably the intellectual cornerstone of modern economic theory. The modern theory of economic equilibrium, including the Welfare theorems, is the rigorous equivalent of the ’invisible hand’ of Adam Smith. Equilibrium concepts are common in any dynamical systems context, be it a Walrasian equilibrium in neoclassical economic theory, or a Nash equilibrium in game theory. Analytically defined economic equilibrium is originally a concept that originated from the 19th century physical sciences.

Although the father of modern economics is indeed usually cited to be Adam Smith, the first semi-rigorous economic theory on equilibrium was put forward by Leon Walras in the late 19th century. The next major breakthrough was in 1954, when the existence of competitive equilibrium was proven by Arrow and Debreu. In spite of the fame of Debreu and Arrow, actually the mathematical theory of general equilibrium was put forward to a great extent by Abraham Wald in 1936. Moreover, Johnny von Neumann, 1945, contributed to the theory of equilibrium greatly as well.

The existence theorem for a competitive equilibrium is a substantial achievement as such, but nevertheless it does not tell much about the workings of a true market economy. We know very little about the adjustment process of prices as such. Especially little we do know about the dynamics of it. The canonical theory since 1940’s assumes that there is a ”trial and error” or ’tatonnement’ process, where a Walrasian auctioneer adjusts the prices so that the excess demand is driven to zero according to the following dynamics

where p is the price vector, lambda is a constant and Z is the excess demand function for the economy.

Even though the equation above seems rather innocent, Scarf in 1960, among others has shown that it is not generally globally stable. A globally stable equilibrium means (assuming it exists) in this context that if you start with some initial set of prices and excess demands, the dynamics will lead to an equilibrium always.

The lack of a stable equilibrium is a major problem, as it implies that in general, there is no market clearing. It is actually rather peculiar, that economic theory contains such an essential pathology, given how lightly people usually assume that demand and supply will balance each other. Therefore, we would need to advance on the dynamic adjustment process, because actually we know rather little about theoretical economies.

According to the Fields medalist and mathematician Stephen Smale, the problem of lack of knowledge with general equilibrium theory is really severe, and he has included it on his famous list of eighteen unsolved problems in mathematics.

In the 1970’s Stephen Smale published extensively on the problems around general equilibrium theory. In particular, he held the view that the main unsolved problem in mathematical economics was the lack of understanding of dynamics of general equilibrium.

’I feel equilibrium theory is far from satisfactory. For one thing the theory has not

successfully confronted the question, ”How is equilibrium reached?” Dynamic

considerations would seem necessary to resolve this problem.’ -Stephen Smale

There has been a consistent line of research in non-tatonnement through last decades, although thin, regarding Walrasian exchange and adjustment of prices. Even though the price adjustment process is of fundamental importance, the amount of research published on this particular issue is not however so large. It is worrying, as price adjustment towards equilibrium is a key issue, not least because the market clearing does not occur always.

So, my leftist friends, study more mathematics, and you can challenge Mr. Smith!

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