Economics-utility function

Question

We let $g(z)$ be a strictly monotonous function so: $$\frac{dg(z)}{dz}>0$$ Consumer 1 has preferences given by the utility function $u(x_1,x_2)=ln(x_1)+2ln(x_2)$, while consumer 2 has preferences given by n $v(x_1,x_2)=g(x_1x_2^2)$. Then I have to show that consumer 2 got same preferences as consumer 1. I think I have to use MSR on $x_1x_2^2$ and on $v(x_1,x_2)$. For MSR on $x_1x_2^2$ I get: $$MRS=-\frac{\frac{\partial }{\partial x_1}}{\frac{\partial }{\partial x_2}}=-\frac{x_2^2}{2x_1x_2}$$ But How can I find MSR on $v(x_1,x_2)$ (maybe with the chain rule?) and use this to conclude that the ranking of the two indifference curves is the same when $g(z)$ is monotonous? I hope that someone can help me?

Answer 1

Observe consumer 2's utility is a positive monotonic transformation of consumer 1's utility:

$$v(x_1,x_2)=g(e^{u(x_1x_2)}),$$

and you are done.