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$\succsim$ 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 $x,y,a$.

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  • $\begingroup$ Should be $x\sim y$ in the title I guess. $\endgroup$
    – VARulle
    Mar 21, 2020 at 14:48
  • $\begingroup$ @VARulle Thank you for pointing that out! $\endgroup$
    – High GPA
    Mar 22, 2020 at 0:08

1 Answer 1

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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.

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  • $\begingroup$ 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, 2020 at 1:43
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    $\begingroup$ @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, 2020 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, 2020 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, 2020 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, 2020 at 2:46

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