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Question

Is there an example of consumer preferences over consumption bundles $(x,y)\in \Bbb R^2$ that would be concave when $x$ is abundant relative to $y$ and convex otherwise?

Are there known situations when this happens? I have never heard of that.


Context

I've been recently thinking a lot about analyzing functions $f:\Bbb R \to \Bbb R$ with positive third-order derivative instead of functions with positive second-order derivative (convex functions). Such functions are either entirely convex / concave, or there is a unique point (inflection point) at which concavity switches into convexity. It turns out that such functions have many nice properties, for example they have at most three roots and most one local maximum and one local minimum. Therefore I expect that if a utility function $u(x,y)$ had positive/negative third-order derivative along lines in the space of bundles, then it would still be easy to analyze the consumer choice problem as it is in the case of convex preferences.

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  • $\begingroup$ What are concave preferences? Convexity of preferences and convexity of functions are very different things. $\endgroup$ 2 days ago
  • $\begingroup$ @MichaelGreinecker I would regard preferences to be concave iff any lower contour set is concave. If the preference is represented by a utility function $u$, then $u$ is quasi-concave. I understand that concavity of $u$ is sufficient condition for concavity of the preferences, but not necessary. Note that function property of having positive third-order derivative can be also generalized to its "quasi" (i.e. ordinal) form. $\endgroup$ 2 days ago
  • $\begingroup$ Quasi-concavity of a representing utility function is equivalent to the convexity (!) of the represented preferences. I don’t know what concave sets are either. $\endgroup$ 2 days ago
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    $\begingroup$ @MichaelGreinecker I remember Varian made a hypothetical example of sardines and ice cream. Each one is delicious however combining them makes people worse off.. $\endgroup$
    – T123
    2 days ago
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    $\begingroup$ @T123 There are many reasons why non-convex preferences matter. It is not clear that this specific form is particularly useful. $\endgroup$ 2 days ago

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