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How does convex preference imply quasi concave utility function (Intuitively and Mathematically)?

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Let $X$ be the convex set of alternatives, let $\succeq$ be a preference relation and let $u(.)$ be a utility function that reflects these preferences, which means that $u(x) \ge u(y)$ if and only if $x \succeq y$.

The preference relation $\succeq$ is convex if

  • For all $y$, $x$ and $z$ in $X$, if $x \succeq y$ and $z \succeq y$ then for all $\alpha \in [0,1]$, $\alpha x + (1-\alpha) y \succeq y$.

  • Equivalently, for all $y$ in $X$, the set of all bundles that are at least as good as $y$, is a convex set.

  • Equivalently, for all $y$ in $X$, the set $U_y = \{x \in X| x \succeq y\}$ is convex.

The utility function $u$ is Quasi-concave if

  • For all $x$, $y$ and $z$ in $X$, if $u(x) \ge u(y)$ and $u(z) \ge u(y)$, then for all $\alpha \in [0,1]$, $u(\alpha x + (1-\alpha) z)\ge u(y)$.

  • Equivalently, for all $y$ in $X$, the set of bundels that give at least as much utility as $y$ is a convex set.

  • Equivalently, for all $y \in X$ the set $V_y = \{x \in X| u(x) \ge u(y)\}$ is convex.

However the set $U_y$ and $V_y$ are the same. $$ V_y = \{x \in X| u(x) \ge u(y)\} = \{x \in X| x \succeq y\} = U_y. $$ As such, convexity of preferences is identical to quasi-concavity of the utility function that reflects these preferences.

Intuitively, convexity of preferences means that all upper contour sets are convex sets. This is by definition equal to convexity of all sets that give at least as much utility level than some given fixed bundle. This is identicial to the notion of quasi-concavity.

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  • $\begingroup$ Hi @tdm. Could you expand a little on the intuition at the end there, if possible? $\endgroup$
    – EB3112
    Jan 8, 2022 at 16:32
  • $\begingroup$ Aren't the second and third points about convex preference relations the exact same thing? As in, they are not equivalent definitions, but the exact same statement with a different notation? Same goes for the second and third points about quasi-concave functions. $\endgroup$
    – Giskard
    Jan 9, 2022 at 14:07
  • $\begingroup$ @Giskard of course, that is why I said "Equivalently". The second is in words, the third in more formal notation. I was just trying to be more pedagogical. $\endgroup$
    – tdm
    Jan 9, 2022 at 17:43
  • $\begingroup$ I applaud your pedagogical intentions. You also wrote "Equivalently" between the first and second points, which are equivalent definitions. But the second and third points are not merely equivalent, but the exact same definition with different notation. Instead of "equivalently" I would prefer to use "or with more formal notation" or "i.e.". But I acknowledge that this is mostly a matter of taste. $\endgroup$
    – Giskard
    Jan 9, 2022 at 18:40

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