1
$\begingroup$

$\succsim$ is a countinuous and convex weak order.

$x,y,a$ are vectors in $\mathbb R^n$

We say $a\geq0$ if all directions of the vector $a$ is greater or equal to zero.

We want to prove (or disprove by counterexample) that:

Suppose $x\sim y$ implies $x+a\sim y+a$ for any $a\geq0$ and $x,y\in\mathbb R^n$,

Then the preference is linear.


One definition of linear preference is that $x\sim y$ implies $x+a\sim y+a$ for any $x,y,a$.

$\endgroup$
2
  • $\begingroup$ Should be $x\sim y$ in the title I guess. $\endgroup$
    – VARulle
    Mar 21 '20 at 14:48
  • $\begingroup$ @VARulle Thank you for pointing that out! $\endgroup$
    – High GPA
    Mar 22 '20 at 0:08
2
$\begingroup$

This is not true.

Let $n=1$ and define $u(x)=\min{\{x,0\}}$. Let $\succsim$ be the preference relation represented by $u$. This preference relation is continuous and convex. We also have $x\sim y$ implies $x+a\sim y+a$ for any $a\geq0$ and $x,y\in\mathbb R$. But let $x=0$, $y=1$, and $a=-1$. Then $x\sim y$, but $y+a=0\succ -1=x+a$, thus $x+a\nsim y+a$ and $\succsim$ is not linear.

$\endgroup$
5
  • $\begingroup$ This is a good answer! However, in economics, most if not utility functions are strictly increasing. It would be very helpful if you could give me a hint on how to obtain a counterexample that is also strictly increasing in both elements. $\endgroup$
    – High GPA
    Mar 26 '20 at 1:43
  • 1
    $\begingroup$ @HighGPA: Well, that's a whole new thing, and I guess linearity would then indeed follow. Try posting this as a new question. $\endgroup$
    – VARulle
    Mar 26 '20 at 8:49
  • $\begingroup$ I believe so. I posted that altered question on the Math forum. Do you believe that I could post it on the Econ forum again? math.stackexchange.com/questions/3591086/… $\endgroup$
    – High GPA
    Mar 27 '20 at 21:54
  • $\begingroup$ @HighGPA: If you frame it as a question on monotone, continuous, convex preferences on $\mathbb{R}^n$, then it belongs to economic theory, I'd say. $\endgroup$
    – VARulle
    Mar 27 '20 at 22:49
  • $\begingroup$ Hi, I post a new question and it seems like that I find a proof. Do you mind to check it?economics.stackexchange.com/questions/35670/… $\endgroup$
    – High GPA
    Mar 29 '20 at 2:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.