# Check if a utility function represents a monotone preference

Given a function $$u(x_1, x_2) = x_1 +x_2 + \min(2x_1, x_2)$$, how do we mathematically prove that it monotonic or not?

Is there is a general algebraic technique to show monotonicity of suchlike functions?

• For problems like this, you always start with a definition. What's the definition of a monotonic transformation?
– Art
Oct 19 '19 at 4:19
• @Art I don't think monotonic transformation is what you meant. There is no need to transform anything. Oct 19 '19 at 9:05
• @Giskard Oops my bad. This is the definition I meant. The question implies that the function $u(x_1, x_2)$ could be said to be monotonic or not. My understanding is that a "monotonic function" applies to $f: \mathbb{R} \to \mathbb{R}$ only. So I guess what I'm suggesting is to change title to "Check if a utility function represents a monotone preference."
– Art
Oct 19 '19 at 9:52

Using this definition of monotone preferences (Thanks to User Art). Let $$X\subseteq \mathbb{R}^n$$ be the set of possible consumption bundles, and $$x,y\in X$$. We have weak monotone preferences if $$y>>x \implies y\succ x$$, and strict preferences if $$y\geq x \implies y\succ x$$. If we have a utility function $$u: X\to \mathbb{R}$$, then $$y\succ x$$ is the same as $$u(y)> u(x)$$.
Regarding notation for two bundles $$x,y\in X, x=(x_1,x_2,...,x_n), y=(y_1,y_2,...,y_n)$$, we say $$y>>x$$ if $$y_i>x_i$$ forall $$i=1,2,...,n$$ and we say $$y\geq x$$ if $$y_i\geq x_i$$ forall $$i=1,2,...,n$$ and there is a $$j$$, such that $$y_j>x_j$$. (This notation is not universal)