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

<p>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:</p>

<p><a href="https://i.sstatic.net/VlF0q.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/VlF0q.jpg" alt="enter image description here"></a></p>

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

<p>I completely agree that income =1 is really the more fun question. </p>

CC BY-SA 4.0
2018-08-10T10:08:20.773Z
This is a nice response, but I believe you have a typo because you say "in case of income = 10" twice.
BKay
CC BY-SA 4.0
2018-08-10T10:32:32.423Z
That was a typo. Fixed it. Thanks for pointing out BKay.
erik
CC BY-SA 4.0
2018-08-10T08:44:35.410Z
2018-08-10T10:31:37.313Z
2018-08-10T10:31:37.313Z
erik
<p>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*}</p>

<p>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.</p>

<p>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 $(4, 2)$?</p>

CC BY-SA 3.0
2016-02-01T07:03:53.097Z
Once you have $y=2$ and $2x+y=10$, don't you think you can infer $x=4$?
Oliv
CC BY-SA 3.0
2016-02-01T15:05:23.107Z
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
<utility><consumption>
How does one calculate an optimal consumption bundle if the MRS only contains one variable?
bloopton
# How does one calculate an optimal consumption bundle if the MRS only contains one variable?

(Asked by: username_0 on 2016-02-01)

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 $(4, 2)$?

**Tags:** utility, consumption

> **Comments:**
> **username_1**: Once you have $y=2$ and $2x+y=10$, don't you think you can infer $x=4$?  
> **username_2**: 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.  

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## 1 answers

### Answer by username_3 on 2018-08-10. Score: 2

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:

[![enter image description here](https://i.sstatic.net/VlF0q.jpg)](https://i.sstatic.net/VlF0q.jpg)

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.

> **Comments:**
> **username_4**: This is a nice response, but I believe you have a typo because you say "in case of income = 10" twice.  
> **username_3**: That was a typo. Fixed it. Thanks for pointing out username_4.  

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