Are there non monotonic preferences that are strictly convex and continues but the second welfare theorem does not hold for them?

  • $\begingroup$ Try $u_i(x_i, y_i) =-(x_i-1)^2-(y_i-1)^2 $ for $i\in\{1,2\}$ and total endowment of X and Y is $(4,4)$. $((2,2), (2,2))$ is Pareto efficient but not a competitive equilibrium. $\endgroup$ – Amit Feb 12 '17 at 15:28

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