# Utility functions and positive monotone transformations

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 $$v(x_1,x_2)$$ (I think I got the same MRS if I use the chain rule, but I'm not sure?), and how can I 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?

• Note that ln(x1)+2*ln(x2) = ln(x1)+ln(x2^2) = ln (x1 * x2^2) and then verify that MRS is invariant to any positive monotone transforation. – Bayesian Nov 26 '20 at 16:41

Note that $$ln(x_1)+2*ln(x_2) = ln(x_1)+ln(x_2^2) = ln (x_1 * x_2^2),$$ and note that $$MRS_v = \frac{g'(x_1 * x_2^2) x_1 * 2 x_2}{g'(x_1 * x_2^2) x_2^2}$$ such that the derivative of $$g$$ cancels out.