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Why Hypothesis 3 (convexity of preferences) is important to find the general competitive equilibrium?

Why Hypothesis 3 (convexity of preferences) is important to find the general competitive equilibrium?

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Convexity of preference is important to find the general competitive equilibrium because in order to guarantee that equilibrium exists the consumer preferences should be convex. The convexity preference can only be relaxed when there are larger consumers.

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