Proof that two utility functions $u(x_1,x_2) \text{ and } v(x_1,x_2)$ with the same MRS represent the same preference I am having some difficulty proving that two utility functions with the same marginal rate of substitution represent the same preference. I think I need two use some argument related to strictly increasing monotonic transformation, but I am really lost. If someone could help me, it would be great.
Thank you very much!
I think that you need to also assume that the utilities are non-decreasing in the goods. That is, if $x'_1&gt;x_1$, then $u(x'_1,x_2)&gt;u(x_1,x_2)$.
The utility functions having the &quot;same preferences&quot; means
$\forall (x_1,x_2), (x'_1,x'_2): u(x_1,x_2) &gt; u (x'_1,x'_2) \text{ iff } v(x_1,x_2) &gt; v(x'_1,x'_2) $
So let's assume $u(x_1,x_2) &gt; u (x'_1,x'_2) $. Take a curve going from $(x_1,x_2)$ to a point $(x'_1, x''_2)$ by changing $x_1$ to $x'_1$ and having the second coordinate change according the MRS so that this curve has constant value of $u$ [1]. Then since $u$ is constant along this curve, $u(x'_1,x''_2) = u(x_1,x_2)$. Since $u(x_1,x_2) &gt; u(x'_1,x'_2)$, $u(x'_1,x''_2) &gt; u(x'_1,x'_2)$, and from the non-decreasing utility property, we have $x''_2 &gt;x'_2$.
Since this curve was defined by the MRS of $u$, and the MRS of $u$ is equal to that of $v$, we also have that $v(x'_1,x''_2) = v(x_1,x_2)$, and so  $v(x_1,x_2) &gt; v(x'_1,x'_2)$
[1] This can be done, for example, by taking $C = (x_1+ t, x_2+\int_0^tMRS_{x_1x_2}dt)$. Then taking $t=x'_1-x_1$, we get the point $\left(x'_1, \int_0^{x'_1-x_1}MRS_{x_1x_2}dt \right)$