## Economic Evolution and Equilibrium: Bridging the Gaptheory,whichformalizesa dynamiceconomicsystemasa systemofdi?erence, or di?erential, equations. There equilibria mean ‘equilibrium trajectories’ of the whole evolution that, in a certain sense, are optimal. A particularly un- tisfactory feature of this conceptualization of an equilibrium, however, is the factthattheintertemporaloptimizingapproachcompletelypredeterminesthe whole future of the economic system. This “closed loop” approach gives rise to the common reproach that economic theory is predominantly concerned with the question of ‘how the economic system ought to behave’ rather than with the question of ‘how does it behave actually’. This is the point at which the new branch of evolutionary economics has made its entrance. In contrast to growth or business cycle theory, evolutionary economics perceives the evolution of the economic system as essentially “open” to true novelties that are unforeseeable by their very nature. This view clearly makes obsolete any conception of equilibrium that resorts to the idea of a ?nal state of rest, or to the idea of an intertemporally optimizing trajectory which is prespeci?ed ab initio by a system of di?erential equations and initial con- tions. To be sure, there have been attempts to reconceptualize the notion of equilibrium from the evolutionary viewpoint. However, these proposals also appear, in one way or another, to hinge on the ideas of rest. This parti- larly applies to the branch of nonlinear dynamics and deterministic chaotic motion. |

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### Contents

1 | |

Notations and Mathematical Preliminaries | 12 |

Conceptualization and Deﬁnition of Evolutions of Economies in Four General Equilibrium Frameworks | 25 |

Introduction to Part I | 27 |

Evolutions in the Traditional Walrasian Exchange Equilibrium Framework | 29 |

41 Evolutions Based on the Model of an Exchange Economy by Arrow and Hahn | 30 |

42 Evolutions Based on Dierkers Version of the Model of an Exchange Economy | 34 |

43 Evolutions Based on a Model of a Large Exchange Economy | 37 |

121 Course Evolutions and Connection Evolutions | 142 |

122 New and Old Commodities | 149 |

The Structure of the Equilibrium Price Set of an Evolution of Exchange Economies | 154 |

Comparison With Related Results in the Literature | 161 |

142 The Regular Approach | 164 |

Conclusions | 175 |

Economic Analysis | 177 |

Introduction to Part III | 178 |

Evolutions in an Exchange Equilibrium Framework Without Walras Law and Homogeneity | 41 |

51 Evolutions Based on a Model of an Exchange Economy Without Walras Law and Homogeneity | 42 |

52 Evolutions Based on a Model With Weakened Boundary Assumptions | 54 |

Evolutions in a General Equilibrium Framework With Production Taxes and Subsidies | 60 |

61 Evolutions Based on a General Equilibrium Model With Production and Taxes | 62 |

62 Evolutions Based on a General Equilibrium Model With Production Taxes and Subsidies | 68 |

Evolutions in the Temporary Fixprice Equilibrium Framework | 71 |

71 Evolutions Based on a Quantity Constrained Micromodel With Effective Demand a la Benassy | 72 |

72 Evolutions Based on a Quantity Constrained MultiSectoral Model | 77 |

Conclusions | 87 |

Formal Analysis | 88 |

Introduction to Part II | 89 |

NearEquilibrium Paths | 95 |

101 Existence of Joining Equilibrium Components and of NearEquilibrium Paths for Each Type of Evolution From Part I | 96 |

102 A Criterion for Identifying Points on Joining Equilibrium Components | 103 |

Equilibrium Paths | 105 |

111 Approximating Evolutions of Exchange Economies With Nice Equilibrium Paths Based on Dierkers Model from Section 42 | 106 |

1111 Polynomial Approximating Exchange Evolutions | 107 |

1112 Piecewise Linear Approximating Exchange Evolutions | 115 |

112 Approximating Evolutions With Equilibrium Paths for the Other Basic Models From Part I | 120 |

113 A Strong Connectedness Result for the Graphs of the Equilibrium Correspondences of the Basic Models From Part I | 126 |

Economic Reﬁnements of the Notion of an Evolution of Economies | 141 |

Applications of the Analytical Results From Part II in the Economists Laboratory | 183 |

171 Extending the PathFollowing Method to the Computation of Equilibria of NonRegular Economies | 184 |

172 Generalizing Comparative Statics to Economies With Multiple Equilibria | 187 |

The Method of Kinetic Analysis of Evolving Economies in Historical Time | 188 |

Evolving Economies in Historical Time | 199 |

191 Evolving Economies in Discrete Historical Time | 203 |

192 Alternative Models of Evolving Economies in Continuous Historical Time | 204 |

1922 The Frequency Model of a Piecewise Continuously Balanced Evolving Economy | 210 |

193 Time Consuming Equilibrium Adjustment Processes | 214 |

194 Frictionless Tuning of Coordination Signals in Evolving Economies in Continuous Historical Time | 217 |

1942 Frictionless Tuning in the Quantity Constrained MultiSectoral Model From Chapter 4 | 221 |

1943 Frictionless Tuning in the General Equilibrium Framework With Production Taxes and Subsidies From Chapter 3 | 222 |

Conclusions | 223 |

General Conclusions and Outlook | 225 |

General Conclusions and Outlook | 226 |

Appendix A1 | 231 |

Appendix B | 241 |

Appendix C | 258 |

References | 263 |

270 | |

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Economic Evolution and Equilibrium: Bridging the Gap Marco Lehmann-Waffenschmidt Limited preview - 2007 |

### Common terms and phrases

achieve Actually agents analysis approximating evolutions Balasko basic models basic set-ups Chapter choose Clearly commodity compact comparative statics connected component connection economy connection evolution construction continuous one-parametrization continuous path converges convex connection convex transition deﬁne Dierker economic system equi equilibrium equivalent self-mapping equilibrium framework equilibrium path equilibrium price equilibrium set evolution of economies Evolutions Based evolving economy excess demand function exchange framework Figure ﬁnitely ﬁrst formal Furthermore geometrical given evolution graph homeomorphism homotopy space intuitive joining equilibrium component kinetic large exchange economy law and homogeneity librium mapping market excess demand Mas-Colell Mas-Colell’s Mathematical means micromodel monotone preference notion path connected piecewise analytical path price vector proof properties Proposition quantity constrained reaction functions reader real number regular economies result s-state Section sector set–up simplex space of economies subset subsidies subspace Theorem theory tion topology tuple Walras correspondence Walrasian exchange zero