# 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. Feb 1 '16 at 15:05

I wanted to add this as a reply to denesp's comment, but I do not have enough reps.

MRS and a binding BC gives a system of two equations from which we can solve the optimum bundle. In case of income = 10, these two equations have positive solutions, in case of income = 1, these two equations do not have positive solutions. See this:

Income = 1 just makes the choice set small enough that there is no MRS = price ratio, hence the corner solution.

I completely agree that income =1 is really the more fun question.

• This is a nice response, but I believe you have a typo because you say "in case of income = 10" twice.
– BKay
Aug 10 '18 at 10:08
• That was a typo. Fixed it. Thanks for pointing out BKay.
– erik
Aug 10 '18 at 10:32