# 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

Economics Asked by Jackaba on May 9, 2021

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 $$x'_1>x_1$$, then $$u(x'_1,x_2)>u(x_1,x_2)$$.

The utility functions having the "same preferences" means

$$forall (x_1,x_2), (x'_1,x'_2): u(x_1,x_2) > u (x'_1,x'_2) text{ iff } v(x_1,x_2) > v(x'_1,x'_2)$$

So let's assume $$u(x_1,x_2) > u (x'_1,x'_2)$$. 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 $$u(x_1,x_2) > u(x'_1,x'_2)$$, $$u(x'_1,x''_2) > u(x'_1,x'_2)$$, and from the non-decreasing utility property, we have $$x''_2 >x'_2$$.

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 $$v(x_1,x_2) > v(x'_1,x'_2)$$

[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 $$left(x'_1, int_0^{x'_1-x_1}MRS_{x_1x_2}dt right)$$

Answered by Acccumulation on May 9, 2021