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I got that in an exchange economy, conditions as preferences being continuous, strictly convex and strongly monotone and $\sum_i \omega_i\gg 0$ are sufficient conditions for the existence of a Walrasian (or competitive) equlibirum.

Are there necessary conditions for the existence of an equilibrium?

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If the competitive equilibrium exists, it must lie on the contract curve where two consumers' marginal rate of substitution are the same. Also, this equilibrium lies inside the zone of mutual advantages (area where the indifference curves intersect with the endowment) because trade is voluntary. If consumers’ marginal rates of substitution vary continuously with their consumption levels then a competitive equilibrium exists

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    $\begingroup$ Could you please provide a reference for your strong claim "If consumers’ marginal rates of substitution vary continuously with their consumption levels then a competitive equilibrium exists"? $\endgroup$ – Giskard Dec 17 '18 at 18:16

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