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Chapter 5. The Equilibrium Manifold

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General Equilibrium Theory of Value
This chapter is in the book General Equilibrium Theory of Value
“Chapter5” — 2011/6/1 — 12:48 — page 47 — #1CHAPTER 5The Equilibrium Manifold5.1 INTRODUCTIONWe now address the global structure of the equilibrium manifoldE.We first start by motivating these global properties for their economicinterest. These global properties can be of a topological nature like path-connectedness, simple connectedness, and contractibility. They can alsotake a more practical form like the existence of global coordinate systemsfor the points of the equilibrium manifold in the same way the points atthe surface of the earth can be located through their longitude and lati-tude. We continue by identifying an important subset of the equilibriummanifold, the set of no-trade equilibria. This enables us to uncover theremarkable structure of the equilibrium manifold as a collection of linearspaces parameterized by the no-trade equilibria. We apply this remark-able structure to define a global coordinate system for the equilibriummanifold. As in the previous chapter, them-tuple of demand functions(fi)characterizing the exchange model is assumed all throughout thischapter to belong toE, i.e., the only assumptions are smoothness (S) andWalras law (W).5.2 GLOBALPROPERTIES ANDTHEIRINTEREST5.2.1 PathconnectednessA topological space is pathconnected if it is always possible to link twoarbitrarily chosen points of this space by a continuous path. For exam-ple, every convex set is pathconnected. Indeed, the segment linking twoarbitrary points is, because of the convexity assumption, contained inthe set and therefore defines a continuous path linking these two points.What is the economic meaning for the equilibrium manifold to be path-connected? Let us consider the two equilibria(p,ω)and(p′,ω′).Wecould assume that(p,ω)describes a current equilibrium of the econ-omy while(p′,ω′)is an equilibrium that is aimed for a later date. Buthow is the economy going to move from the first to the second equilib-rium? Of particular interest are continuous trajectories that belong to theequilibrium manifold. The existence or non-existence of such continuous

“Chapter5” — 2011/6/1 — 12:48 — page 47 — #1CHAPTER 5The Equilibrium Manifold5.1 INTRODUCTIONWe now address the global structure of the equilibrium manifoldE.We first start by motivating these global properties for their economicinterest. These global properties can be of a topological nature like path-connectedness, simple connectedness, and contractibility. They can alsotake a more practical form like the existence of global coordinate systemsfor the points of the equilibrium manifold in the same way the points atthe surface of the earth can be located through their longitude and lati-tude. We continue by identifying an important subset of the equilibriummanifold, the set of no-trade equilibria. This enables us to uncover theremarkable structure of the equilibrium manifold as a collection of linearspaces parameterized by the no-trade equilibria. We apply this remark-able structure to define a global coordinate system for the equilibriummanifold. As in the previous chapter, them-tuple of demand functions(fi)characterizing the exchange model is assumed all throughout thischapter to belong toE, i.e., the only assumptions are smoothness (S) andWalras law (W).5.2 GLOBALPROPERTIES ANDTHEIRINTEREST5.2.1 PathconnectednessA topological space is pathconnected if it is always possible to link twoarbitrarily chosen points of this space by a continuous path. For exam-ple, every convex set is pathconnected. Indeed, the segment linking twoarbitrary points is, because of the convexity assumption, contained inthe set and therefore defines a continuous path linking these two points.What is the economic meaning for the equilibrium manifold to be path-connected? Let us consider the two equilibria(p,ω)and(p′,ω′).Wecould assume that(p,ω)describes a current equilibrium of the econ-omy while(p′,ω′)is an equilibrium that is aimed for a later date. Buthow is the economy going to move from the first to the second equilib-rium? Of particular interest are continuous trajectories that belong to theequilibrium manifold. The existence or non-existence of such continuous

Chapters in this book

  1. Frontmatter i
  2. Contents v
  3. Preface xi
  4. Chapter 1. Goods and Prices 1
  5. Chapter 2. Preferences and Utility 5
  6. Chapter 3. Demand Functions 19
  7. Chapter 4. The Exchange Model 37
  8. Chapter 5. The Equilibrium Manifold 47
  9. Chapter 6. Applications of the Global Coordinate System 56
  10. Chapter 7. The Broad Picture 62
  11. Chapter 8. The Fine Picture 74
  12. Chapter 9. Production with Decreasing Returns 82
  13. Chapter 10. Equilibrium with Decreasing Returns 96
  14. Chapter 11. Production with Constant Returns 108
  15. Chapter 12. Equilibrium with Constant Returns 124
  16. Postscript 145
  17. Appendix A. Notation 149
  18. Appendix B. Point-set Topology 151
  19. Appendix C. Smooth Manifolds 152
  20. Appendix D. Singularities of Smooth Maps 155
  21. Appendix E. Convexity 159
  22. Appendix F. Miscellany 166
  23. References 167
  24. Index 171
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https://doi.org/10.1515/9781400838912.47

Chapters in this book

  1. Frontmatter i
  2. Contents v
  3. Preface xi
  4. Chapter 1. Goods and Prices 1
  5. Chapter 2. Preferences and Utility 5
  6. Chapter 3. Demand Functions 19
  7. Chapter 4. The Exchange Model 37
  8. Chapter 5. The Equilibrium Manifold 47
  9. Chapter 6. Applications of the Global Coordinate System 56
  10. Chapter 7. The Broad Picture 62
  11. Chapter 8. The Fine Picture 74
  12. Chapter 9. Production with Decreasing Returns 82
  13. Chapter 10. Equilibrium with Decreasing Returns 96
  14. Chapter 11. Production with Constant Returns 108
  15. Chapter 12. Equilibrium with Constant Returns 124
  16. Postscript 145
  17. Appendix A. Notation 149
  18. Appendix B. Point-set Topology 151
  19. Appendix C. Smooth Manifolds 152
  20. Appendix D. Singularities of Smooth Maps 155
  21. Appendix E. Convexity 159
  22. Appendix F. Miscellany 166
  23. References 167
  24. Index 171
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