# MRS not invariant under monotonic transformation

Consider U(x, y) = ln(x)ln(y). Is it quasi-linear?

My response was no, since applying an exponential would yield exp(ln(x)ln(y))= exp(ln(x+y)) = x+y.

However, the original MRS was (y/x)(ln(y)/ln(x)). Now it's 1. What did I do wrong?

• Are you sure that $\exp(\ln(x)\ln(y))= \exp(\ln(x+y))$? Nov 14 '21 at 19:10
Note that $$\exp(\ln(x)\ln(y))\ne \exp(\ln(x+y))$$. Instead, $$$$\exp(\ln(x)\ln(y))=x^{\ln y}=y^{\ln x}.$$$$ The MRS of the monotonically transformed utility is still $$\frac{y\ln y}{x\ln x}$$.