I need an example of a

  • Convex, non-monotonic preference

  • Non-convex, monotonic preference

I figured that an example of non-convex, monotonic utility preference could be $U(x,y)=x^2+y^2$. For convex, non-monotonic preference, I cannot think of a standard example. As far as my understanding goes, lexicographic preferences are convex, but I don't think these are non monotonic. Please help me figure out the examples so that I can conceptually understand monotonicity and convexity.

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    $\begingroup$ $-(x-1)^2 -(y-1)^2$ is an example of convex, non- monotonic preference $\endgroup$ – Amit Jun 16 '19 at 9:01
  • $\begingroup$ @Amit Thank You! $\endgroup$ – S.Rana Jun 16 '19 at 12:46
  • $\begingroup$ @Amit Also post answers you think are correct as answers. $\endgroup$ – Giskard Jun 16 '19 at 13:51
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    $\begingroup$ It's an example of an utility function that represents convex, non-monotonic preferences, actually. $\endgroup$ – Pedro Cavalcante Jun 16 '19 at 20:31

$u_1(x, y) = -(x-1)^2 - (y-1)^2$

$u_2(x, y) = -|x-1| - |y-1|$

are examples of utility functions that are concave, and therefore represent the preferences that are convex. Also, these preferences are not monotonic.

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