Are homothetic preferences monotonic?

Question

I'm trying to understand intuitively what a homothetic preference is, and I am still not quite there. I understand the definition, that a homothetic preference implies that the slope of the indifference curve remains constant along any ray from origin.

But intuitively, like in real life, how would we know that a preference is not homothetic? If its not strictly monotonically increasing (i.e. you don't prefer more of a product to less of it always), does it mean that the preference is not homothetic? Can someone give an example of what exactly is not a homothetic preference so I understand what is?

Answer 1

Are homothetic preferences strictly monotonically increasing?

Homotheticity requires that

$$ \alpha^\gamma U(x,y) = U(\alpha x, \alpha y) $$

This is not defined over the "increasing" part of strictly monotonously increasing. Indeed, you can have decreasing preferences $U(x,y) = -x -y$ for whom homotheticity holds.

Are nomothetic preferences weakly monotonic?

However, the "strict" part (in both jointly) is required, i.e. $U(\alpha x, \alpha y) \neq U(x,y)$ for $\alpha \neq 0$.. You cannot have homothetic and weakly increasing or weakly decreasing preferences.

Proof outline To prove that, we assume otherwise. There is some homothetic $U$ with a strictly increasing part and a flat part. You will see that the $\gamma$ corresponding to the increasing part can potentially be any number (it depends on the functional form). However, the $\gamma$ corresponding to the flat part needs to be 0.[1] Hence, there cannot be a $\gamma$ that holds for the utility function as a whole, for any general increasing part.

[1]: Unless of course, $U(x,y) = 0$ at the flat part. However, as utility functions are immune to monotone transformations, we can ignore such cases.