General Equilibrium Theory - Unresolved Problems in General Equilibrium

Unresolved Problems in General Equilibrium

Research building on the Arrow–Debreu–McKenzie model has revealed some problems with the model. The Sonnenschein-Mantel-Debreu results show that, essentially, any restrictions on the shape of excess demand functions are stringent. Some think this implies that the Arrow-Debreu model lacks empirical content. At any rate, Arrow-Debreu-McKenzie equilibria cannot be expected to be unique, or stable.

A model organized around the tatonnement process has been said to be a model of a centrally planned economy, not a decentralized market economy. Some research has tried to develop general equilibrium models with other processes. In particular, some economists have developed models in which agents can trade at out-of-equilibrium prices and such trades can affect the equilibria to which the economy tends. Particularly noteworthy are the Hahn process, the Edgeworth process and the Fisher process.

The data determining Arrow-Debreu equilibria include initial endowments of capital goods. If production and trade occur out of equilibrium, these endowments will be changed further complicating the picture.

In a real economy, however, trading, as well as production and consumption, goes on out of equilibrium. It follows that, in the course of convergence to equilibrium (assuming that occurs), endowments change. In turn this changes the set of equilibria. Put more succinctly, the set of equilibria is path dependent... makes the calculation of equilibria corresponding to the initial state of the system essentially irrelevant. What matters is the equilibrium that the economy will reach from given initial endowments, not the equilibrium that it would have been in, given initial endowments, had prices happened to be just right (Franklin Fisher, as quoted by Petri (2004)).

The Arrow–Debreu model in which all trade occurs in futures contracts at time zero requires a very large number of markets to exist. It is equivalent under complete markets to a sequential equilibrium concept in which spot markets for goods and assets open at each date-state event (they are not equivalent under incomplete markets); market clearing then requires that the entire sequence of prices clears all markets at all times. A generalization of the sequential market arrangement is the temporary equilibrium structure, where market clearing at a point in time is conditional on expectations of future prices which need not be market clearing ones.

Although the Arrow–Debreu–McKenzie model is set out in terms of some arbitrary numéraire, the model does not encompass money. Frank Hahn, for example, has investigated whether general equilibrium models can be developed in which money enters in some essential way. One of the essential questions he introduces, often referred to as the Hahn's Problem is : "Can one construct an equilibrium where money has value?" The goal is to find models in which existence of money can alter the equilibrium solutions, perhaps because the initial position of agents depends on monetary prices.

Some critics of general equilibrium modeling contend that much research in these models constitutes exercises in pure mathematics with no connection to actual economies. "There are endeavors that now pass for the most desirable kind of economic contributions although they are just plain mathematical exercises, not only without any economic substance but also without any mathematical value" (Nicholas Georgescu-Roegen 1979). Georgescu-Roegen cites as an example a paper that assumes more traders in existence than there are points in the set of real numbers.

Although modern models in general equilibrium theory demonstrate that under certain circumstances prices will indeed converge to equilibria, critics hold that the assumptions necessary for these results are extremely strong. As well as stringent restrictions on excess demand functions, the necessary assumptions include perfect rationality of individual complete information about all prices both now and in the future; and the conditions necessary for perfect competition. However some results from experimental economics suggest that even in circumstances where there are few, imperfectly informed agents, the resulting prices and allocations may wind up resembling those of a perfectly competitive market(although certainly not a stable general equilibrium in all markets).

Frank Hahn defends general equilibrium modeling on the grounds that it provides a negative function. General equilibrium models show what the economy would have to be like for an unregulated economy to be Pareto efficient.