# Preference: Convexity and Monotonicity

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.

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