G. Define and . =-{(?-*-,-e-*-)-?-?-×-?,-(?-*-,-e-*-)-?-f-?1, ) | rankD ? F (? * , e * ) < dim?}. Then we can write R = ?\?(G) where ? is defined in Lemma 19. An element (?, e) of G is characterised by the fact that the determinant of all the square submatrices of D ? F (?, e) of dimension dim? is equal to 0. Thus, the set G is closed since the determinant is a continuous function

R. Balasko and Y. , Some results on uniqueness and on stability of equilibrium in general equilibrium theory, Journal of Mathematical Economics, vol.22, pp.95-118, 1975.

Y. Balasko, Foundation of the Theory of General Equilibrium, 1988.

Y. Balasko, Economies with price-dependent preferences, Journal of Economic Theory, vol.109, pp.333-359, 2003.

Y. Balasko, General Equilibrium Theory of Value, 2011.

Y. Balasko, Wealth concerns and equilibrium, Journal of Mathematical Economics, vol.59, pp.92-101, 2015.

J. Bonnisseau, Regular economies with non-ordered preferences, Journal of Mathematical Economics, vol.39, pp.153-174, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00187218

J. Bonnisseau and E. L. Mercato, Externalities, consumption constraints and regular economies, Economic Theory, vol.44, pp.123-147, 2010.
URL : https://hal.archives-ouvertes.fr/halshs-00381637

R. Bourlès, Y. Bramoullé, and E. Perez-richet, Altruism in Networks. Econometrica, vol.85, pp.675-689, 2017.

G. Debreu, Economies with a finite set of equilibria, Econometrica, vol.38, pp.387-392, 1970.

G. Debreu, Four aspects of the mathematical theory of economic equilibrium, Proceedings of International Congress of Mathematicians, pp.65-77, 1974.

,. E. Del-mercato, Existence of competitive equilibria with externalities: a differential viewpoint, Journal of Mathematical Economics, vol.42, pp.525-543, 2006.
URL : https://hal.archives-ouvertes.fr/halshs-00355353

M. Dufwenberg, P. Heidhues, G. Kirchsteiger, F. Riedel, and J. Sobel, Other-regarding preferences in general equilibrium, Review of Economic Studies, vol.78, issue.2, pp.613-639, 2011.

E. Fehr and S. Gächter, Fairness and Retaliation: The Economics of Reciprocity, The Journal of Economic Perspectives, vol.14, pp.159-181, 2000.

J. D. Geanakoplos and H. M. Polemarchakis, Pareto improving taxes, Journal of Mathematical Economics, vol.44, pp.682-696, 2008.

V. Guillemin and A. Pollack, Differential topology, 1974.

P. Hammond, The Efficiency Theorems and Market Failure, Elements of General Equilibrium Analysis, ch, vol.6, pp.211-260, 1998.

S. Heap, A. Ramalingam, and B. Stoddard, Endowment inequality in public goods games: A re-examination, Economics Letters, vol.146, pp.4-7, 2016.

P. Heidhues and F. Riedel, Do social preferences matter in competitive markets? Working paper 392, 2007.

E. Hopkins and T. Kornienko, Which inequality? the inequality of endowments versus the inequality of rewards, American Economic Journal: Microeconomics, vol.2, issue.3, pp.106-137, 2010.

E. Hopkins, Inequality and risk-taking behaviour, Game and Economic Behavior, vol.107, pp.316-328, 2018.

J. Laffont, Decentralization with externalities, European Economic Review, vol.7, issue.4, pp.359-375, 1976.

R. Luttmer, Neighbors as negatives: relative earnings and well-being, Quarterly Journal of Economics, vol.120, pp.963-1002, 2005.

A. Maltz, Exogenous Endowment-Endogenous Reference Point, The Economic Journal, vol.130, pp.160-182, 2020.

Y. Masatlioglu and E. A. Ok, A Canonical Model of Choice with Initial Endowments, The Review of Economic Studies, vol.81, issue.2, pp.851-883, 2014.

A. Mas-colell, The theory of general economic equilibrium, A Differentiable Approach, 1985.

N. Sato, Existence of Competitive Equilibrium in Unbounded Exchange Economies with Satiation: A Note. The B.E, Journal of Theoretical Economics, vol.10, issue.1, pp.1-22, 2010.

S. Schecter, On the structure of the equilibrium manifold, Journal of Mathematical Economics, vol.6, pp.1-7, 1979.

S. Smale, Global analysis and economics. IIA: Extension of a theorem of Debreu, Journal of Mathematical Economics, vol.1, pp.1-14, 1974.

J. Sobel, Interdependent Preferences and Reciprocity, Journal of Economic Literature, vol.43, issue.2, pp.392-436, 2005.

R. A. Velez, Fairness and externalities, Theoretical Economics, vol.11, pp.381-410, 2016.

A. Villanacci, L. Carosi, P. Benevieri, and A. Battinelli, Differential Topology and General Equilibrium with Complete and Incomplete Markets, 2002.

A. Villanacci and E. Zenginobuz, Existence and regularity of equilibria in a general equilibrium model with private provision of a public good, Journal of Mathematical Economics, vol.41, issue.4-5, pp.617-636, 2005.

D. J. Zizzo and A. Oswald, Are People Willing to Pay to Reduce Others' Incomes?, Annales d'economie et de statistique, pp.63-64, 2001.