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General Equilibrium Theory of Value

General Equilibrium Theory of Value

Yves Balasko
Copyright Date: 2011
Pages: 192
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  • Book Info
    General Equilibrium Theory of Value
    Book Description:

    The concept of general equilibrium, one of the central components of economic theory, explains the behavior of supply, demand, and prices by showing that supply and demand exist in balance through pricing mechanisms. The mathematical tools and properties for this theory have developed over time to accommodate and incorporate developments in economic theory, from multiple markets and economic agents to theories of production.

    Yves Balasko offers an extensive, up-to-date look at the standard theory of general equilibrium, to which he has been a major contributor. This book explains how the equilibrium manifold approach can be usefully applied to the general equilibrium model, from basic consumer theory and exchange economies to models with private ownership of production. Balasko examines properties of the standard general equilibrium model that are beyond traditional existence and optimality. He applies the theory of smooth manifolds and mappings to the multiplicity of equilibrium solutions and related discontinuities of market prices. The economic concepts and differential topology methods presented in this book are accessible, clear, and relevant, and no prior knowledge of economic theory is necessary.

    General Equilibrium Theory of Valueoffers a comprehensive foundation for the most current models of economic theory and is ideally suited for graduate economics students, advanced undergraduates in mathematics, and researchers in the field.

    eISBN: 978-1-4008-3891-2
    Subjects: Economics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-x)
  3. Preface
    (pp. xi-xii)
  4. CHAPTER 1 Goods and Prices
    (pp. 1-4)

    The aim of this chapter is to develop the main aspects of the economic environment in which economic agents operate. There are two categories of economic agents, consumers and firms. Consumers buy and sell goods with the ultimate goal of consuming those goods. Firms buy goods that they transform into other goods that they later sell. An economy is made up of these consumers and firms. After having developed models of the consumers and firms, we will combine them into a model of an economy with private ownership of production. Before developing these models, it is necessary to be somewhat...

  5. CHAPTER 2 Preferences and Utility
    (pp. 5-18)

    Classical consumer theory is essentially the theory of utility maximization under a budget constraint. This theory starts with the definition of consumers’ preferences. In classical consumer theory, preferences are assumed to be transitive, complete, monotone and convex. These preferences can then be represented by utility functions. The latter are mathematically easier to handle than preferences. Another reason for being interested in utility functions goes back to the early phases of economic theory. Then, it was thought that utility functions could be used as a measure of consumer’s satisfaction or utility. Pareto suggested the term of “ophelimité” instead of utility to...

  6. CHAPTER 3 Demand Functions
    (pp. 19-36)

    Rational consumers are assumed to maximize their preferences subject to the constraints they perceive. With preferences that are representable by utility functions, the consumer’s problem is modeled as one of maximizing a standard utility function subject to a budget constraint. This maximization problem determines the consumer’s demand, a demand that is a function of the consumer’s wealth and market prices.

    These demand functions feature several remarkable properties that make up the bulk of classical consumer theory. The most important ones are Walras law, the weak axiom of revealed preferences (WARP), and the negative definiteness of the Slutsky matrices (ND). An...

  7. CHAPTER 4 The Exchange Model
    (pp. 37-46)

    The exchange model is the simplest of all general equilibrium models. The study of its properties is already interesting for its own sake because it is the typical model of a market. It will also show us the directions to follow when studying more complex models like those that include production or take explicitly into account time and uncertainty. Even more remarkably, many properties of the more complex models often exploit those of the exchange model.

    Consumers are modeled by way of their demand functions. These functions satisfy some or all of the properties identified in the previous chapters but...

  8. CHAPTER 5 The Equilibrium Manifold
    (pp. 47-55)

    We now address the global structure of the equilibrium manifoldE. We first start by motivating these global properties for their economic interest. These global properties can be of a topological nature like pathconnectedness, simple connectedness, and contractibility. They can also take a more practical form like the existence of global coordinate systems for the points of the equilibrium manifold in the same way the points at the surface of the earth can be located through their longitude and latitude. We continue by identifying an important subset of the equilibrium manifold, the set of no-trade equilibria. This enables us to...

  9. CHAPTER 6 Applications of the Global Coordinate System
    (pp. 56-61)

    A global coordinate system for the equilibrium manifold follows from: 1) The determination of the unique fiberF(b) through the equilibrium (p,ω) where$b = \phi (p,\omega ) = (p,p \cdot {\omega _1},\ldots,p \cdot {\omega _m})$; 2) The determination of the location of the equilibrium (p,ω) within the fiberF(b) viewed as a linear space of dimension (ℓ – 1)(m– 1) and, therefore, parameterized by (ℓ – 1)(m– 1) coordinates. If there is little leeway in determining the fiberF(b) through the equilibrium (p,ω), there are different ways of representing the equilibrium (p,ω) within its fiberF(b). This leads us to define coordinate systems (A) and (B) for the...

  10. CHAPTER 7 The Broad Picture
    (pp. 62-73)

    In this chapter, them-tuple$({f_i})$of demand functions defining the exchange model belongs to$\mathcal{E}{_r}$, i.e., the demand function${f_i}$is bounded from below (B) for every consumer and satisfies desirability (A) for at least one consumer. These additional properties will give to the natural projection the very important property of properness. The combination of smoothness and properness will suffice to yield what is now known as the theory of regular economies following [24].

    By definition, the continuous map π :E→ Ω is proper if the preimage of every compact set is compact. A set is compact...

  11. CHAPTER 8 The Fine Picture
    (pp. 74-81)

    In the previous chapter, we have seen that, form-tuples of demand functions$({f_i}) \in \mathcal{E}{_r}$,the natural projection π :E→ Ω is a smooth proper map and the properties that we have developed so far are essentially properties of any smooth proper map.

    In this chapter, them-tuple$({f_i})$of demand functions defining the exchange model is restricted to belong to$\mathcal{E}{_c}$. In addition to the assumptions made in the previous chapters (recall that$\mathcal{E}{_c}$is a subset of$\mathcal{E}{_r}$), the demand function${f_i}$satisfies the weak axiom of revealed preferences (WARP) for every consumer, and the slightly...

  12. CHAPTER 9 Production with Decreasing Returns
    (pp. 82-95)

    This chapter is devoted to the theory of the firm. Production consists in the transformation of goods known as inputs into other goods, the outputs. The firm is the center of productive activity. Its activity is represented by a vector in the commodity space. This chapter starts with the definition of the firm’s production set. This set consists of the activities that are technologically feasible for the firm. A few very general properties are satisfied by all or almost all production sets. Despite their generality, these properties simplify the analysis of production. If the firm’s production set describes all activities...

  13. CHAPTER 10 Equilibrium with Decreasing Returns
    (pp. 96-107)

    The approach to the study of the general equilibrium model with private ownership of smooth production with decreasing returns is to adjust consumers’ individual demand functions for production. The “exchange model” defined by these production adjusted demand functions is then shown to be equivalent to the original general equilibrium model with private ownership of production.

    The production adjusted demand functions are very close to satisfy the properties considered in the first part of this book, properties that guarantee that the main properties of the exchange model are satisfied. As a result, the general equilibrium model with private ownership of smooth...

  14. CHAPTER 11 Production with Constant Returns
    (pp. 108-123)

    The assumption of decreasing returns to scale is often justified by the fact that the quantities of some factors used in the production process are fixed and cannot be increased in order to produce more outputs. The latter are then produced with less efficient use of resources. This is typically the case in short-run models where, by definition, capacity and, more generally, capital is not variable. The situation is totally different in long-run models where all production factors, including capital, can be varied. In growth models for example, the output is a homogenous function of degree one of the capital...

  15. CHAPTER 12 Equilibrium with Constant Returns
    (pp. 124-144)

    This chapter is devoted to the study of an equilibrium model where privately owned firms feature either smooth decreasing or constant returns to scale. Profit of the constant returns to scale firms being equal to zero at equilibrium, the equilibrium of the model does not depend on the ownership structure of these firms. In addition, the convex conical production sets of these firms sum up into a convex cone. It is as if the production sector operating under constant returns consists of a unique firm. The general equilibrium model with decreasing and constant returns to scale firms is essentially the...

  16. Postscript
    (pp. 145-148)

    The general equilibrium model studied in this book, a model that we can call the standard model, offers us a description of modern economies with their competitive markets and firms that is sufficiently general to be economically relevant without being too complex to make its study intractable. Many simpler models exist, but their relevance is doubtful at best because those models are too simplistic. There also exist models that are more complex than the standard model, but these models are so difficult to study that not many of their properties have been identified.

    The standard model has been around for...

  17. APPENDIX A Notation
    (pp. 149-150)
  18. APPENDIX B Point-set Topology
    (pp. 151-151)
  19. APPENDIX C Smooth Manifolds
    (pp. 152-154)
  20. APPENDIX D Singularities of Smooth Maps
    (pp. 155-158)
  21. APPENDIX E Convexity
    (pp. 159-165)
  22. APPENDIX F Miscellany
    (pp. 166-166)
  23. References
    (pp. 167-170)
  24. Index
    (pp. 171-175)