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?

  • 1
    $\begingroup$ For problems like this, you always start with a definition. What's the definition of a monotonic transformation? $\endgroup$
    – Art
    Oct 19 '19 at 4:19
  • $\begingroup$ @Art I don't think monotonic transformation is what you meant. There is no need to transform anything. $\endgroup$
    – Giskard
    Oct 19 '19 at 9:05
  • $\begingroup$ @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." $\endgroup$
    – 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)

Now, see if you can use this to answer your question


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