# How does one calculate an optimal consumption bundle if the MRS only contains one variable?

For example, say the MRS has simplified to the form $MRS(x,y) = -y$, given a utility function \begin{equation*} U(x, y) = e^{(x + ln(y))^{1/3}} \end{equation*}

With a budget line of, say, $2x + y = 10$ which would yield a budget line slope of -2, equating the MRS to the BLS would simply yield $y = 2$, but there is no x to solve for in the equating of the two slopes.

Is the last step, in cases like this, to plug back into the budget constraint? In which case, one would get $x = 4$, yielding an optimal consumption bundle of $(4, 2)$?

• Once you have $y=2$ and $2x+y=10$, don't you think you can infer $x=4$? – Oliv Feb 1 '16 at 7:03
• Indeed. The more interesting question (which leads you to better understand the limits of MRS) is what to do when the consumer's total income is 1. – Giskard Feb 1 '16 at 15:05