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The modern conception of general equilibrium is provided by a model developed jointly by Kenneth Arrow and Gerard Debreu in the 1950s. Gerard Debreu presents this model in Theory of Value (1959) as an axiomatic model, following the style of mathematics promoted by Bourbaki. In such an approach, the interpretation of the terms in the theory (e.g., goods, prices) are not fixed by the axioms.
Three important theorems have been proved in this framework. First, existence theorems show that equilibria exist under certain abstract conditions. The first fundamental theorem of welfare states that every market equilibrium is Pareto optimal under certain conditions. The second fundamental theorem of welfare states that every Pareto optimum is supported by a price system, again under certain conditions. These conditions were stated in the language of mathematical topology. The proofs used such concepts as separating hyperplanes and fixed point theorems.
Three important interpretations of the terms of the theory have been often cited. First, supposed commodities are distinguished by the location where they are delivered. Then the Arrow-Debreu model is a spatial model of, for example, international trade.
Second, suppose commodities are distinguished by when they are delivered. That is, suppose all markets equilibriate at some initial instant of time. Agents in the model purchase and sell contracts, where a contract specifies, for example, a good to be delivered and the date at which it is to be delivered. The Arrow-Debreu model of intertemporal equilibrium contains forward markets for all goods at all dates. No markets exist at any future dates.
Third, suppose contracts specify states of nature which affect whether or not a commodity is to be delivered: "A contract for the transfer of a commodity now specifies, in addition to its physical properties, its location and its date, an event on the occurrence of which the transfer is conditional. This new definition of a commodity allows one to obtain a theory of [risk] free from any probability concept..." (Debreu 1959)
These interpretations can be combined. So the complete Arrow-Debreu model can be said to apply when goods are identified by when they are to be delivered, where they are to be delivered, and under what circumstances they are to be delivered, as well as their intrinsic nature. So there would be a complete set of prices for contracts such as "1 ton of Winter red wheat, delivered on 3rd of January in Minneapolis, if there is a hurricane in Florida during December". A general equilibrium model with complete markets of this sort seems to be a long way from describing the workings of real economies.
Research building on the Arrow-Debreu 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 essentially arbitrary. Some think this implies that the Arrow-Debreu model lacks empirical content. At any rate, Arrow-Debreu equilibria cannot be expected to be unique, stable, or determinate.
A model organized around the tatonnnement process has been said to be a model of a centrally planned economy, not a decentralized market economy. Some research has tried, not very successfully, to develop general equibrium 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 Arrow-Debreu model of intertemporal equilibrium, in which forward markets exist at the initial instant for goods to be delivered at each future point in time, can be transformed into a model of sequences of temporary equilibrium. Sequences of temporary equilibrium contain spot markets at each point in time. Roy Radner found that in order for equilibria to exist in such models, agents (e.g., firms and consumers) must have unlimited computational capabilities.
Although the Arrow-Debreu model is set out in terms of some arbitrary numeraire, 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. The (unsatisfied) goal is to find models in which whether or not money exists alters equilibrium solutions, perhaps because the initial position of agents depends on monetary prices, for example, when they have debts.
Some critics of general equilibrium modeling contend that much research in these models constitutes exercises in pure matheamtics 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). Georescu-Roegen cites as an example a paper that assumed more traders than there are points on a real line.
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 completely unrealistic. The necessary assumptions include perfect rationality of individuals; complete information about all prices both now and in the future; and the conditions necessary for perfect competition.
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. Note that Hahn's defense drops any claim that general equilibrium models describe actual capitalist economies.
Some economists reject equilibrium theory outright in favour of more pragmatic models based more closely on observation of the economy.
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