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This book is a substantially revised and enlarged version of the monograph General Equilibrium with Increasing Returns, published by Springer-Verlag as a Lecture Notes volume in 1996. It incorporates new topics and the most recent developments in the field. It also provides a more systematic analysis of the differences between production economies with and without convex production sets. Five out of twelve chapters are new, and most of the remaining ones have been reformulated. An outline of contents appears in chapter 1. As its predecessor, this book contains a formal and systematic exposition of the main results on the existence and efficiency of equilibrium, in production economies where production sets need not be convex. There is an explicit attempt at making of it a suitable reference both for graduate students and researchers interested in theory (not necessarily specialists in mathematical economics). With this twofold purpose in mind, the work has been written according to three key principles: (i) To provide a uhified approach to the problems involved. For that we construct a basic model that is rich enough to encompass the different models appearing throughout, and to derive all the results as coroilaries of a reduced number of general theorems. (ii) To maintain a relatively low mathematical complexity. Thus, when the estimated cost of generality exceeds the benefit of simplicity, we shall state and prove the theorems under assumptions that need not be the most general ones.

Inhaltsverzeichnis

Frontmatter

1. Preliminaries

Abstract
Most societies organize their economic activity through the functioning of markets. Myriads of individual economic agents make decisions according to their private interests, whose interaction results in an allocation of resources. The production and exchange of commodities is at the centre of the picture: consumers demand commodities and supply labour services, firms produce commodities according to their technological knowledge, and commodities flow among agents by means of an exchange process which is realized through markets and prices.
Antonio Villar

2. Consumers

Abstract
A consumer is an individual agent (a single household or a family) who takes consumption decisions, that is, decisions referring to the demand for goods and services and the supply of different types of labour. It will be assumed that there is a fixed number m of consumers, indexed by i = 1,2,..., m.
Antonio Villar

3. Production and Supply

Abstract
Production refers to a process by which certain commodities (inputs) are transformed into different ones (outputs). Input commodities may include raw materials, elements of fixed capital (land, machinery, buildings), energy, other produced commodities, and different types of labour. Outputs are produced commodities that can be consumed, stored or used as new inputs. Note that “transformation”, here, has to be interpreted according to the notion of commodities adopted in chapter 1. In particular, transporting a commodity to a different place, or keeping it until another period, constitute transformations. Also, an element of fixed capital that is not totally used up in the production process appears as an additional commodity among the outputs (hence joint production is the rule).
Antonio Villar

4. Competitive Equilibrium

Abstract
This chapter is devoted to the analysis of competitive markets, that is, markets in which consumers and firms behave as payoff maximizers at given prices. The next sections provide a positive answer to the old question concerning the capability of prices and markets to coordinate economic activity in a decentralized framework. It will be shown that, under a set of well specified assumptions, markets are in themselves adequate institutions for the efficient allocation of resources. This may be called the Invisible Hand Theorem,, which summarizes the most relevant features of competitive markets: competitive equilibria constitute a non-empty subset of the set of efficient allocations.
Antonio Villar

5. Equilibrium with Non-Convex Firms

Abstract
General equilibrium models face serious difficulties in the presence of nonconvex technologies, when there are finitely many firms and non-convexities are not negligible. Such difficulties are both analytical and theoretical and are mainly concerned with the fact that the supply correspondence may not be convex-valued or even defined, so that the existence of competitive equilibrium will typically fail. This implies that, if we want to analyze a general equilibrium model allowing for non-convex technologies, we must permit the firms to follow more general rules of behaviour, and suitably re-define the equilibrium notion. This will imply, however, that the identification between equilibrium and optimum will no longer hold (the Invisible Hand Theorem now splits into two halves). Thus, the existence of equilibria under nonconvex technologies, and the analysis of their properties of optimality now become two very different questions.
Antonio Villar

6. Marginal Pricing

Abstract
When production sets are convex, profit maximization at given prices implies the choice of production plans for which marginal rates of transformation coincide with market prices. This is precisely a necessary condition for the efficiency of market allocations (if this were not the case, it would be possible to reallocate commodities more productively). When the behaviour of competitive firms is modelled in terms of inverse supply mappings, profit maximization amounts to selecting those prices that support efficient production plans. These supporting vectors are “marginal prices”, because they correspond to the marginal rates of transformation.
Antonio Villar

7. Increasing Returns and Monopolies

Abstract
The presence of increasing returns in market economies may lead to the creation of natural monopolies. When productivity increases with size, efficiency calls for a single firm to serve the market. This is a fact already pointed out by John Stuart Mill: “It is obvious, for example, how great economy of labour would be obtained if London were supplied by a single gas or water company instead of the existing plurality. While there are even as many as two, this implies double establishments of all sorts, when one only, with a small increase, could probably perform the whole operation equally well. Were there only one establishment, it could make lower charges, consistent with making the rate of profit now realized. But would it do so? Even if it did not, the community in the aggregate would still be a gainer.” [Mill (1848), quoted in Quinzii (1992)].
Antonio Villar

8. Loss-Free Pricing Rules

Abstract
This chapter provides an application of the pricing rule approach to the analysis of unregulated market economies with non-convex production sets. Loss-free pricing rules provide a natural framework for this analysis, because the equilibrium of firms implies non-negative profits. When inaction is possible and firms are privately owned, production equilibria can be associated with non-negative profits.
Antonio Villar

9. Competition and Increasing Returns

Abstract
This chapter provides an application of the loss-free pricing rules presented in chapter 8, to the analysis of competitive markets when there are increasing returns to scale. In order to do so, it will be assumed that non-convex firms exhibit a particular type of increasing returns to scale for which constrained profit maximization and average cost-pricing are compatible (distributive production sets).
Antonio Villar

10. Non-Convexities as Public Goods

Abstract
The inefficiency of marginal pricing equilibrium allocations, in economies with non-convex firms, conveys the message that production decisions affect the allocation of resources in a way that is not fully captured by the price system. Therefore, the presence of increasing returns to scale, fixed costs, or other forms of non-convexities, can be interpreted as an instance of externalities or as a public good feature.
Antonio Villar

11. Input-Output Analysis

Abstract
The input-output analysis, initiated by Wassily Leontief in 1936, can be regarded as a first application of the principles of general equilibrium theory to the empirical analysis. Its goal is the study and quantification of the structural relations between the economic sectors that constitute the economy of a country. The methodological approach consists of representing the structural relations between industries by means of linear equations, whose coefficients are obtained empirically. When these coefficients are stable (eg when constant returns to scale prevail), equilibrium relations appear as the solutions to linear equation systems. The key tool for the computation of these coefficients is the input-output table, that is briefly presented below.1
Antonio Villar

12. The Limits of the Economy

Abstract
The existence of equilibrium in production economies has been shown assuming, among other things, that the set of attainable allocations is compact. Even though this is a very reasonable property, it has been introduced as an axiom rather than as a consequence of some primitive assumptions on production and consumption sets. We deal with this point here, analyzing in detail conditions under which the aggregate consumption and production sets are closed, and the set of feasible allocations is compact.
Antonio Villar

Backmatter

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