General Equilibrium Theory

surplus of consumers and producers at the point of balance for supply and demand curves.

The general equilibrium theory is a branch of microeconomics. It tries to give a global explanation of the behavior of production, consumption and price formation in an economy with one or more markets.

General equilibrium attempts to give a bottom-up explanation, starting with markets and individual agents, while macroeconomics, as expressed by Keynesian economists, It uses a vision from the general to the particular (top-down), where the analysis begins with the most outstanding components. However, many macroeconomic models have a 'goods market' and study, for example, its interaction with the financial market. General equilibrium models often include various commodity markets. Modern general equilibrium models are complex and require computers to help find numerical solutions.

In a market system, the prices and production of all goods, including the price of money and interest, are related. A change in the price of one good, for example bread, can affect another price (for example, the wages of bakers). If the taste of bread depends on who is the baker, the demand for bread can be affected by a change in the wages of bakers and, consequently, in the price of bread. In theory, calculating the equilibrium price of a single good requires an analysis that considers all the millions of different goods that are available.

History of general equilibrium models

The first attempt in neoclassical economics to model the prices of an entire economy was made by Léon Walras. His work The Elements of Pure Economics provides several models, each of which takes into account a greater number of aspects of a real economy (two types of goods, many types of goods, production, growth, money). Some authors (eg Eatwell, 1989, and also Jaffe, 1953) think that Walras was unsuccessful and that the latest models he developed are inconsistent. In particular, the Walras model was a long-period model in which the prices of capital goods are the same regardless of whether they appear as inputs or outputs, and the same profit margin occurs across all lines of industry. In this model, the cost price of each capital good must be equal, in equilibrium, to the demand price. This is inconsistent with what is obtained when the quantities of capital goods are taken as data. However, when Walras introduced capital goods into later models of him, he took the quantities of him as a given, in arbitrary ratios. Kenneth Arrow and Gerard Debreu also took the initial quantities of capital goods as a given, but they adopted a simple model in which the prices of capital goods vary over time and also the interest rate varies from one capital good to other.

Walras was the first to organize a research program that has been followed by many economists of the 20th century. In particular, he raised the need to investigate the conditions necessary for equilibria to be unique and stable.

Walras also proposed a dynamic system by which a general equilibrium can be reached, called tâtonnement or groping process.

The tâtonnement process is a tool to investigate the stability of equilibria. The prices are announced by an auctioneer, and the agents indicate how much they want to offer (supply) or buy (demand) of each of the goods. No transaction or production takes place while prices are unbalanced. On the other hand, the prices of those goods with positive prices and excess supply are reduced, while the prices of the goods with excess demand increase. The question for the mathematician is under what conditions such a process will reach an equilibrium in which demand balances with supply to provide for positively priced commodities and demand does not exceed the supply of zero-priced commodities. Walras could not find a definitive answer to this question (see Unsolved Problems in General Equilibrium, below).

In partial equilibrium analysis, determining the price of a good is simplified by consulting the price of one good, and assuming that the prices of the rest of the goods remain constant. The Marshallian theory of supply and demand is an example of partial equilibrium analysis. Partial equilibrium analysis is appropriate when the first-order effects of a change (for example, the demand curve) do not shift the supply curve. Anglo-American economists became interested in general equilibrium in the late 1920s and early 1930s, after Piero Sraffa showed that Marshallian economists cannot explain the upward slope of the supply curve for a consumer good.

If an industry uses a small amount of a factor of production, a small increase in the volume of output of that industry will not increase the price of that factor. To a first-order approximation, firms in that industry will not experience a decrease in their costs, and the industry's supply curves will not experience an increase. On the other hand, if an industry uses an appreciable amount of the factor of production, an increase in the volume of production will produce a reduction in production costs. But such a factor is likely to be used in substitutes for the industry's product, and an increase in the price of this factor will have effects on the supply of the substitutes. Thus, as Sraffa argued, in these cases the first-order effects of a shift in the original industry's demand curve include a shift in the supply curve of substitutes for the product and corresponding shifts in the supply curve. original industry offering. General equilibrium is suitable for investigating these types of interactions between markets.

Continental European economists made important strides in the 1930s. Walras's proofs of the existence of general equilibrium often relied on accounting for the number of variables and equations. But such strategies are unsuitable for nonlinear systems of equations, and they do not imply that equilibrium prices and quantities cannot be negative, which is a pointless solution. The replacement of certain equations by inequalities and the use of more rigorous mathematics allowed us to improve the modeling of general equilibrium.

Modern concept of general equilibrium in economics

Balance price on a free market.

The modern concept of general equilibrium is provided by a model developed jointly by Kenneth Arrow, Gerard Debreu, and Lionel W. McKenzie in the 1950s. Gerard Debreu presents this model in his work The Theory of Value (1959) as an axiomatic model, following the mathematical style promoted by Bourbaki. In this approach, the interpretation of the terms in the theory (commodities, prices) is not fixed by the axioms.

Three important interpretations of the terms of the theory have often been cited. First, suppose that the raw materials are distinguishable by the region where they are delivered. From this it follows that the Arrow-Debreu model is a spatial model of, for example, international trade.

Second, suppose raw materials are distinguished according to the point in time at which they are delivered. That is, imagine that all markets clear at a certain initial instant in time. In this model, agents buy and sell contracts. For example, a contract specifies a good to be delivered and the date by which it must be delivered. The Arrow-Debreu model of intertemporal equilibrium contains forward markets for all commodities at all dates. There is no market at any future date.

Third, suppose contracts specify the states of nature that affect whether a commodity must be delivered: "a contract for the transfer of a now-specific commodity, as well as its physical characteristics, its location, and its date, event whose occurrence conditions the completion of the transfer. This new definition of a raw material allows to obtain a theory free from the risk of any probabilistic concept... " (Debreu, 1959).

These interpretations can be combined. Therefore, it can be said that the complete Arrow-Debreu model is applicable when the goods are identified according to when they must be delivered, where they must be delivered, and in what circumstances they must be delivered, as well as their intrinsic nature. Thus, there will be a complete pricing system for contracts such as "1 ton of red winter wheat, delivered January 3 to Minneapolis, if there is a hurricane in Florida during December". A general equilibrium model with complete markets of this kind still seems to be far from being an adequate way of describing the workings of real economies. However, its authors argue that it is still useful as a simplified guide to how real economies work.

Some of the recent work on general equilibrium has explored the implications of incomplete markets, that is, an intertemporal economy with uncertainty, where there are no sufficiently detailed contracts to allow agents to correctly allocate their demands and resources over time. Although it has been shown that such economies will generally continue to be in equilibrium, the outcome may no longer be Pareto optimal. The basic explanation for this result is that if consumers lack adequate means to transfer their abundance from one point in time to another and the future is risky, there is nothing to tie any price to the relevant marginal substitute index, which is the standard requirement for Pareto optimality. However, under some conditions the economy may still be Pareto optimal, meaning that a central authority limited to the same type and number of contracts as individual agents may not improve the outcome. What is needed, rather, is the introduction of a complete system of possible contracts. Thus, one of the implications of incomplete markets theory is that inefficiency may be the result of underdeveloped financial institutions or credit crunch suffered by some members of the public. Research in this area is still ongoing.

Properties and characteristics of general equilibrium

The basic questions in general equilibrium analysis concern the conditions under which an equilibrium will be efficient, what efficient equilibria can be reached, when the existence of an equilibrium is guaranteed, and when the equilibrium will be unique and stable.

First Fundamental Theorem of Welfare Economics

The first fundamental theorem of welfare states that market equilibriums are efficient according to the Pareto criterion. In a pure exchange economy, a sufficient condition for the first welfare theorem to hold is that consumer preferences are not locally satisfied. The first welfare theorem is also valid for economies with production, regardless of the properties of the production function. Additional implicit assumptions are that consumers are rational, markets are complete, there are no externalities, and information is perfect. For example, in an economy with externalities it is possible to find equilibrium points that are not efficient.

While it is true that these assumptions are unrealistic, what the theorem basically asserts is that the sources of inefficiency found in the real world are not due to the very nature of the market system, but to some kind of market failure.

Second Fundamental Theorem of Welfare Economics

Although every equilibrium is efficient, it is not true that every efficient allocation of resources will be an equilibrium. The second theorem indicates that every efficient allocation can be sustained by a certain set of prices. In other words, all that is required to achieve a particular result is a redistribution of agents' initial endowments after which the market will adjust without intervention. This suggests that efficiency and equity can be addressed separately without favoring one over the other. However, the conditions for the second theorem are stronger than the necessary conditions for the first theorem, since consumer preferences now need to be convex (convexity roughly corresponding to the idea of decreasing marginal utility, or to prefer "averages over extremes").

Existence

Although every equilibrium is efficient, neither of the two previous theorems says anything about what the existing equilibrium is. To ensure equilibrium exists we need consumer preferences to be continuous, increasing, and convex (although with a large number of consumers this condition can be relaxed both for existence and for the second theorem of welfare economics) and with endowments positive. Similarly, though less plausibly, feasible production systems must be convex, precluding the possibility of economies of scale.

Proofs for the existence of equilibrium generally rely on fixed point theorems such as Brouwer's fixed point theorem, or its generalization (the Kakutani fixed point theorem). Indeed, one can quickly go from a general theorem on the existence of equilibrium to Brouwer's fixed point theorem. For this reason, many mathematical economists consider proving existence to be a stronger result than proving the two fundamental theorems.

Uniqueness

Although (assuming convexity) there will be an equilibrium that will generally be efficient, the conditions under which it will be unique are much stronger. Although the topic is highly technical, a simple analysis shows us that the presence of wealth/abundance effects (which is the characteristic that most clearly distinguishes general equilibrium analysis from partial equilibrium) raises the possibility of the existence of multiple equilibria.. When the price of a given good changes, two effects occur. First, the relative attraction between the different commodities is modified and, second, the distribution of wealth/abundance of individual agents is altered. These two effects can offset or reinforce each other in such a way that more than one set of prices constitutes an equilibrium.

A result known as the Sonnenschein-Mantel-Debreu Theorem indicates that the aggregate demand function inherits only certain characteristics from the individual demand function, and that these (continuity, degree zero homogeneity, Walras' law, and behavior of the limit when prices are close to zero) are not enough to guarantee the uniqueness of the equilibrium.

Much research has been done on the conditions under which the equilibrium will be unique, or at least when the number of possible equilibria is limited. One result indicates that, under smooth conditions, the number of equilibria will be finite and odd (see the Index Theorem). Furthermore, if an economy as a whole, characterized by an aggregate excess demand function, possesses either the revealed preference characteristic (which is a much stronger condition than individual revealed preferences) or the gross surrogate characteristic, then equilibrium it will be unique. All methods of establishing uniqueness can be considered to establish that each equilibrium has the same positive local index, in which case there can be, by the index theorem, only one equilibrium point.

Determination

Since equilibria may not be unique, it is interesting to determine whether a specific equilibrium is at least unique for a specific location. If this is so, comparative statics can be applied as long as the perturbations to the system are not too large. As previously indicated, in a regular economy the equilibria will be finite and therefore locally unique. Debreu determined that "most" of the economies are regular. However, recent work by Michael Mandler (1999) has challenged this claim. The Arrow-Debreu-McKenzie model is neutral among the production function models, it is continuously differentiable, and it is built from linear combinations of fixed coefficient processes. Mandler accepts that in both production models, the initial endowments will not be consistent with a continuous series of equilibria, except for a set with zero Lebesgue measure. However, the endowments change in the model over time, and this evolution of endowments is determined by the decisions of the agents (eg, firms) of the model.

In this model, agents have an interest in equilibria that are indeterminate:

"Indetermination is not only a technical nuisance, but rather undermines the assumption of price taking of competitive models. Since arbitrarily small manipulations of supply factors can significantly increase the price of a factor, the owners of a factor will not consider prices as parametric. " (Mandler, 1999, p. 17)

When technology is modeled by linear combinations of processes with fixed coefficients, optimal agents will drive endowments such that a continuum of equilibria exists:

"The dotations where an indetermination occurs are systematically presented over time and therefore cannot be neglected; the model of Arrow-Debreu-McKenzie will be influenced by the dilemmas of price factor theory." (Mandler, 1999, p. 19)

Critics of general equilibrium analysis question its practical applicability, based on the possibility of non-unique equilibria. Supporters have pointed out that this appearance is in fact a reflection of the complexity of the real world and is therefore an attractive realistic feature of the model.

Stability

In a typical general equilibrium model, the prices that prevail "when general conditions in the economy stabilize" are those prices that make the demands of different consumers for different merchandise compatible.

From this, the question arises as to what has been the process by which the economy has reached that equilibrium status, that is, the process by which prices and allocations have reached that level in which the markets are empty. Which is related to knowing what would be the behavior in the face of transitory events that modify the economy. Would prices return to the levels they were before the events that disrupted the economy? This is the question of equilibrium stability, and it can easily be seen that it is related to uniqueness.

If there are multiple equilibria, then some of them will be unstable. If an equilibrium is unstable and there is a shock or event, the economy will tend towards a different system of allocations and prices once the event is over and the convergent process is over. However, stability depends not only on the number of equilibria, but also on the type of process that drives the price change (for a specific type of price adjustment process). Therefore, some researchers have focused on those plausible adjustment processes that will guarantee the stability of the system, that is, prices and allocations that always converge to a certain equilibrium. However, if there is more than one equilibrium, the point at which the process ends will depend on the initial condition of the system.

Unsolved problems in general equilibrium

Research on the Arrow-Debreu-McKenzie model has revealed some problems with the model. The result known as the Sonnenschein-Mantel-Debreu theorem indicates that essentially any constraint on the form of excess demand functions is constraining. Some think that this implies that the Arrow-Debreu model lacks empirical content. However, Arrow-Debreu-McKenzie equilibria cannot be expected to be unique or stable.

It has been mentioned that a model posed around the tatonnement process is a model of a centrally planned economy, not a decentralized market economy. Some research has tried, without much success, to develop general models of equilibrium with other processes. Specifically, some economists have developed models in which agents can trade at prices that are out of equilibrium, and such trades can affect the equilibria toward which the economy tends. Particularly significant are the Hahn process, the Edgeworth process, and the Fischer process.

The data determining the Arrow-Debreu equilibria include beginning inventories of capital goods. If production and trade occur out of equilibrium, these inventories will change, further complicating the analysis.

In a true economy, however, trade, as well as production and consumption, continue in conditions beyond balance. Therefore, in the course of convergence to balance (assuming this happens), inventories change. This in turn changes the balance set. I mean, the balance set depends on the trajectory... This dependence on the trajectory makes the calculation of the balances that correspond to the initial state of the system essentially irrelevant. What matters is the balance that will reach the economy based on the initial inventories, not the balance in which the initial inventories would have been given, in the event that the initial inventories would have had correct prices (Franklin Fischer, according to Petri's quote, 2004).

The Arrow-Debreu model, in which all trading takes place in futures contracts made at time zero, requires the existence of a large number of markets. It is equivalent in the case of complete markets to a concept of sequential equilibrium, in which spot markets for goods and assets are opened for each event that occurs on a date (which are not equivalent in the case of incomplete markets).; Market clearing then requires the entire price sequence to clear all markets at all times. A generalization of the sequential markets scheme is the transitory equilibrium method, in which market clearing at a point in time is conditional on future price expectations, which need not necessarily be market clearing values.

Although the Arrow-Debreu-McKenzie model is stated in terms of some arbitrary numeral, the model does not encompass money. Frank Hahn, for example, has investigated whether general equilibrium models can be developed in which money participates as a central element. The goal is to find models in which the existence of money can alter the equilibrium solutions, perhaps because the initial position of the agents depends on money prices.

Some critics of the general equilibrium model claim that much of the research on these models is nothing more than mathematical exercises with no connection to real economies. "Today there are efforts that are considered great economic contributions, although they are mere mathematical exercises, without any economic substance and without any mathematical value" (Nicholas Georgescu-Roegen, 1979).

Although modern models in general equilibrium theory show that in certain circumstances prices will converge to equilibrium, critics argue that the assumptions necessary to obtain these results are extremely restrictive. Like the stringent constraints on excess demand functions, the necessary assumptions include perfect rationality of individual complete information about all prices now and in the future, and the necessary conditions for perfect competition. However, some results from experimental economics indicate that even in circumstances where there are few imperfectly informed agents, the resulting prices and allocations often resemble those of a perfectly competitive market.

Hahn Frank defends the general equilibrium that he models, considering that it provides a negative function. General equilibrium models show what would have to happen for an unregulated economy to be Pareto efficient.

Computable general equilibrium

Until the 1970s, general equilibrium analysis was essentially theoretical, notwithstanding the work of Leif Johansen in 1960. The application of the equilibrium scheme in the search for answers to practical questions of public policy became popular only until 1969, when Herbert Scarf presented an efficient numerical algorithm for calculating equilibriums in complex economies. In parallel, the advances in computational power and the development of national statistical systems --which made available to the general public the input-output tables necessary to implement large-scale models and great sectoral detail-- have provided an important incentive to include so-called "applied general equilibrium modelling" as part of the usual tools of public analysts.