## Differential Topology and General Equilibrium with Complete and Incomplete MarketsGeneral equilibrium In this book we try to cope with the challenging task of reviewing the so called general equilibrium model and of discussing one specific aspect of the approach underlying it, namely, market completeness. With the denomination "general equilibrium" (from now on in short GE) we shall mainly refer to two different things. On one hand, in particular when using the expression "GE approach", we shall refer to a long established methodolog ical tradition in building and developing economic models, which includes, as of today, an enormous amount of contributions, ranging in number by several 1 thousands • On the other hand, in particular when using the expression "stan dard differentiable GE model", we refer to a very specific version of economic model of exchange and production, to be presented in Chapters 8 and 9, and to be modified in Chapters 10 to 15. Such a version is certainly formulated within the GE approach, but it is generated by making several quite restrictive 2 assumptions • Even to list and review very shortly all the collective work which can be ascribed to the GE approach would be a formidable task for several coauthors in a lifetime perspective. The book instead intends to address just a single issue. Before providing an illustration of its main topic, we feel the obligation to say a word on the controversial character of GE. First of all, we should say that we identify the GE approach as being based 3 on three principles . |

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### Contents

PREREQUISITES | 3 |

MANIFOLDS IN EUCLIDEAN SPACES | 29 |

DIFFERENTIALS | 49 |

REGULAR VALUES | 79 |

MANIFOLDS WITH BOUNDARY | 107 |

SARDS THEOREMAND TRANSVERSALITY | 133 |

HOMOTOPY AND DEGREE THEORY | 159 |

Part II | 207 |

NUMERAIRE ASSETS | 307 |

Existence of equilibria | 314 |

Regular economies | 322 |

Relevance and robustness of the indeterminacy result | 339 |

REAL ASSETS | 355 |

RESTRICTED PARTICIPATION | 395 |

PLANNER INTERVENTION | 417 |

461 | |

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### Common terms and phrases

40 in Chapter Assumption u2 commodity Consider constraint contained Corollary defined Definition deg(f deg(g degree denoted described diffeomorphism differential dimension element endowment equations equilibrium equilibrium allocations euclidean spaces Example exchange economy existence of equilibria exists an open fact finite full measure subset full rank full row rank function f function theorem given hence homotopy household h incomplete markets inverse inverse function theorem jacobian matrix Lemma Let f maximization problem measure zero Moreover nominal assets notation numeraire numeraire assets Observe open and full open neighborhood open set open subset parametrization Pareto optimal Pareto optimal allocation perturb Proof properties Proposition prove quasiconcave real indeterminacy regular economies regular value Remark restriction result Section solution set submanifold subspace surjective tangent space Theorem 40 topological space topology transversality uh th utility functions value for f variables vector