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- Welfare economics I: General equilibrium
- File:HS50002 General Equilibrium and Welfare Economics ES 2013.pdf
- General equilibrium theory
- Welfare economics I: General equilibrium

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In economics , general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general equilibrium.

General equilibrium theory contrasts to the theory of partial equilibrium , which only analyzes single markets. In general equilibrium, constant influences are considered to be noneconomic, therefore, resulting beyond the natural scope of economic analysis.

General equilibrium theory both studies economies using the model of equilibrium pricing and seeks to determine in which circumstances the assumptions of general equilibrium will hold. Broadly speaking, general equilibrium tries to give an understanding of the whole economy using a "bottom-up" approach, starting with individual markets and agents. Therefore, general equilibrium theory has traditionally been classified as part of microeconomics.

The difference is not as clear as it used to be, since much of modern macroeconomics has emphasized microeconomic foundations , and has constructed general equilibrium models of macroeconomic fluctuations. General equilibrium macroeconomic models usually have a simplified structure that only incorporates a few markets, like a "goods market" and a "financial market". In contrast, general equilibrium models in the microeconomic tradition typically involve a multitude of different goods markets.

They are usually complex and require computers to calculate numerical solutions. In a market system the prices and production of all goods, including the price of money and interest , are interrelated.

A change in the price of one good, say bread, may affect another price, such as bakers' wages. If bakers don't differ in tastes from others, the demand for bread might be affected by a change in bakers' wages, with a consequent effect on the price of bread. Calculating the equilibrium price of just one good, in theory, requires an analysis that accounts for all of the millions of different goods that are available.

It is often assumed that agents are price takers , and under that assumption two common notions of equilibrium exist: Walrasian, or competitive equilibrium , and its generalization: a price equilibrium with transfers. Walras' Elements of Pure Economics provides a succession of models, each taking into account more aspects of a real economy two commodities, many commodities, production, growth, money.

Some think Walras was unsuccessful and that the later models in this series are inconsistent. In particular, Walras's model was a long-run model in which prices of capital goods are the same whether they appear as inputs or outputs and in which the same rate of profits is earned in all lines of industry. This is inconsistent with the quantities of capital goods being taken as data.

But when Walras introduced capital goods in his later models, he took their quantities as given, in arbitrary ratios. Walras was the first to lay down a research program much followed by 20th-century economists.

In particular, the Walrasian agenda included the investigation of when equilibria are unique and stable— Walras' Lesson 7 shows neither uniqueness, nor stability, nor even existence of an equilibrium is guaranteed. Prices are announced perhaps by an "auctioneer" , and agents state how much of each good they would like to offer supply or purchase demand. No transactions and no production take place at disequilibrium prices. Instead, prices are lowered for goods with positive prices and excess supply.

Prices are raised for goods with excess demand. The question for the mathematician is under what conditions such a process will terminate in equilibrium where demand equates to supply for goods with positive prices and demand does not exceed supply for goods with a price of zero.

Walras was not able to provide a definitive answer to this question see Unresolved Problems in General Equilibrium below. In partial equilibrium analysis, the determination of the price of a good is simplified by just looking at the price of one good, and assuming that the prices of all other goods remain constant. The Marshallian theory of supply and demand is an example of partial equilibrium analysis. Partial equilibrium analysis is adequate when the first-order effects of a shift in the demand curve do not shift the supply curve.

Anglo-American economists became more interested in general equilibrium in the late s and s after Piero Sraffa 's demonstration that Marshallian economists cannot account for the forces thought to account for the upward-slope of the supply curve for a consumer good.

If an industry uses little of a factor of production, a small increase in the output of that industry will not bid the price of that factor up. To a first-order approximation, firms in the industry will experience constant costs, and the industry supply curves will not slope up.

If an industry uses an appreciable amount of that factor of production, an increase in the output of that industry will exhibit increasing costs. But such a factor is likely to be used in substitutes for the industry's product, and an increased price of that factor will have effects on the supply of those substitutes. Consequently, Sraffa argued, the first-order effects of a shift in the demand curve of the original industry under these assumptions includes a shift in the supply curve of substitutes for that industry's product, and consequent shifts in the original industry's supply curve.

General equilibrium is designed to investigate such interactions between markets. Continental European economists made important advances in the s. Walras' proofs of the existence of general equilibrium often were based on the counting of equations and variables. Such arguments are inadequate for non-linear systems of equations and do not imply that equilibrium prices and quantities cannot be negative, a meaningless solution for his models. The replacement of certain equations by inequalities and the use of more rigorous mathematics improved general equilibrium modeling.

McKenzie in the s. In such an approach, the interpretation of the terms in the theory e. Three important interpretations of the terms of the theory have been often cited.

First, suppose 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 equilibrate 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 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 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, however, its proponents argue that it is still useful as a simplified guide as to how real economies function.

Some of the recent work in general equilibrium has in fact explored the implications of incomplete markets , which is to say an intertemporal economy with uncertainty, where there do not exist sufficiently detailed contracts that would allow agents to fully allocate their consumption and resources through time.

While it has been shown that such economies will generally still have an equilibrium, the outcome may no longer be Pareto optimal. The basic intuition for this result is that if consumers lack adequate means to transfer their wealth from one time period to another and the future is risky, there is nothing to necessarily tie any price ratio down to the relevant marginal rate of substitution , which is the standard requirement for Pareto optimality.

Under some conditions the economy may still be constrained Pareto optimal , meaning that a central authority limited to the same type and number of contracts as the individual agents may not be able to improve upon the outcome, what is needed is the introduction of a full set of possible contracts. Hence, one implication of the theory of incomplete markets is that inefficiency may be a result of underdeveloped financial institutions or credit constraints faced by some members of the public.

Research still continues in this area. Basic questions in general equilibrium analysis are concerned with the conditions under which an equilibrium will be efficient, which efficient equilibria can be achieved, when an equilibrium is guaranteed to exist and when the equilibrium will be unique and stable. In a pure exchange economy, a sufficient condition for the first welfare theorem to hold is that preferences be locally nonsatiated.

The first welfare theorem also holds for economies with production regardless of the properties of the production function. Implicitly, the theorem assumes complete markets and perfect information. In an economy with externalities , for example, it is possible for equilibria to arise that are not efficient. The first welfare theorem is informative in the sense that it points to the sources of inefficiency in markets. Under the assumptions above, any market equilibrium is tautologically efficient.

Therefore, when equilibria arise that are not efficient, the market system itself is not to blame, but rather some sort of market failure. Even if every equilibrium is efficient, it may not be that every efficient allocation of resources can be part of an equilibrium. However, the second theorem states that every Pareto efficient allocation can be supported as an equilibrium by some set of prices.

In other words, all that is required to reach a particular Pareto efficient outcome is a redistribution of initial endowments of the agents after which the market can be left alone to do its work. This suggests that the issues of efficiency and equity can be separated and need not involve a trade-off.

The conditions for the second theorem are stronger than those for the first, as consumers' preferences and production sets now need to be convex convexity roughly corresponds to the idea of diminishing marginal rates of substitution i. Even though every equilibrium is efficient, neither of the above two theorems say anything about the equilibrium existing in the first place.

To guarantee that an equilibrium exists, it suffices that consumer preferences be strictly convex. With enough consumers, the convexity assumption can be relaxed both for existence and the second welfare theorem.

Similarly, but less plausibly, convex feasible production sets suffice for existence; convexity excludes economies of scale. Proofs of the existence of equilibrium traditionally rely on fixed-point theorems such as Brouwer fixed-point theorem for functions or, more generally, the Kakutani fixed-point theorem for set-valued functions.

See Competitive equilibrium Existence of a competitive equilibrium. Starr applied the Shapley—Folkman—Starr theorem to prove that even without convex preferences there exists an approximate equilibrium. The Shapley—Folkman—Starr results bound the distance from an "approximate" economic equilibrium to an equilibrium of a "convexified" economy, when the number of agents exceeds the dimension of the goods.

For example, in economies with a large consumption side, nonconvexities in preferences do not destroy the standard results of, say Debreu's theory of value. In the same way, if indivisibilities in the production sector are small with respect to the size of the economy, [. The derivation of these results in general form has been one of the major achievements of postwar economic theory. In particular, the Shapley-Folkman-Starr results were incorporated in the theory of general economic equilibria [13] [14] [15] and in the theory of market failures [16] and of public economics.

Although generally assuming convexity an equilibrium will exist and will be efficient, the conditions under which it will be unique are much stronger. The Sonnenschein—Mantel—Debreu theorem , proven in the s, states that the aggregate excess demand function inherits only certain properties of individual's demand functions, and that these Continuity , Homogeneity of degree zero , Walras' law and boundary behavior when prices are near zero are the only real restriction one can expect from an aggregate excess demand function.

Any such function can represent the excess demand of an economy populated with rational utility-maximizing individuals. There has been much research on conditions when the equilibrium will be unique, or which at least will limit the number of equilibria. One result states that under mild assumptions the number of equilibria will be finite see regular economy and odd see index theorem. Furthermore, if an economy as a whole, as characterized by an aggregate excess demand function, has the revealed preference property which is a much stronger condition than revealed preferences for a single individual or the gross substitute property then likewise the equilibrium will be unique.

All methods of establishing uniqueness can be thought of as establishing that each equilibrium has the same positive local index, in which case by the index theorem there can be but one such equilibrium. Given that equilibria may not be unique, it is of some interest to ask whether any particular equilibrium is at least locally unique. If so, then comparative statics can be applied as long as the shocks to the system are not too large.

As stated above, in a regular economy equilibria will be finite, hence locally unique.

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General equilibrium theory is, of course, at the very heart of our fledgling science of economics, and welfare economics provides the normative basis for all.

A good basic understanding of general equilibrium theory is a fundamental and indispensable background for advanced work in virtually any sub-field of economics; and a thorough understanding of the methods of welfare economics, particularly in a general equilibrium context, is indispensable for investigators undertaking applied policy analysis. This book addresses these needs and requirements by emphasizing the basic underpinnings of general equilibrium and welfare economics. In particular, the theory of choice, which is fundamental to both areas, is developed in a very comprehensive and rigorous fashion. Chichilnisky, Mathematical Reviews, Issue j. All the chapters begin with introducing the readers into the topic.

In economics , general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general equilibrium. General equilibrium theory contrasts to the theory of partial equilibrium , which only analyzes single markets. In general equilibrium, constant influences are considered to be noneconomic, therefore, resulting beyond the natural scope of economic analysis. General equilibrium theory both studies economies using the model of equilibrium pricing and seeks to determine in which circumstances the assumptions of general equilibrium will hold. Broadly speaking, general equilibrium tries to give an understanding of the whole economy using a "bottom-up" approach, starting with individual markets and agents. Therefore, general equilibrium theory has traditionally been classified as part of microeconomics. The difference is not as clear as it used to be, since much of modern macroeconomics has emphasized microeconomic foundations , and has constructed general equilibrium models of macroeconomic fluctuations.

A good basic understanding of general equilibrium theory is a fundamental and indispensable background for advanced work in virtually any sub-field of economics; and a thorough understanding of the methods of welfare economics, particularly in a general equilibrium context, is indispensable for investigators undertaking applied policy analysis. This book addresses these needs and requirements by emphasizing the basic underpinnings of general equilibrium and welfare economics. In particular, the theory of choice, which is fundamental to both areas, is developed in a very comprehensive and rigorous fashion. Chichilnisky, Mathematical Reviews, Issue j.

Part I. General Equilibrium in a Pure Exchange Economy 1. Positive Analysis 1.

Pham, Ngoc-Sang : Credit limits and heterogeneity in general equilibrium models with a finite number of agents. We introduce two-period general equilibrium models with heterogeneous producers and financial frictions. Any agent can borrow to realize their productive project but the debt repayment does not exceed a fraction so-called credit limit of the project's value. Our framework allows us to investigate the aggregate and distributional effects of credit limits and heterogeneity of agents. The connection between credit limits, welfare, and efficiency is also explored.

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