# Inferior goods, monotonic utility and strict concavity [duplicate]

Is it possible to have an inferior good under the assumption of a utility function which is strictly monotonically increasing and strictly quasiconvex?

I do not think so, but I could not find a formal proof.

*My underlying idea is that whit an inferior good, there exist a point where the utility consuming less of one good is better and then its derivative negative at that point, moreove, I think that this behaviour would break the concavity shape.

• I was reading those examples and they do not fulfill the convexity assumption. Dec 12, 2016 at 21:53
• Perhaps you have not spent enough time on the first example $$U(x,y) = \alpha_1 \ln(x-\gamma_x)- \alpha_2 \ln(\gamma_y - y)$$ which does fulfill all your conditions for $0 < \alpha_1 < \alpha_2$. Dec 12, 2016 at 21:59
• But the second derivative of U(x,y) over y is: alpha2/(gamma-y)^2 which is positive in its domain. Dec 12, 2016 at 22:42
• That is not the definition of convexity for multivariate functions. Please read the paper linked in the answer, it actually explains why convexity holds. Dec 13, 2016 at 6:28