# Solve for the bundle component that makes one as 'well off' as earlier

Suppose we have a (presumably time independent?) utility function $U(x_1,x_2)$ for consumer Rita.

1.

What is Rita's MRS of $x_2$ for $x_1$?

$$MRS_{x_1, x_2} = \frac{MU_{x_1}}{MU_{x_2}} = \frac{\frac{\partial U}{\partial x_1}}{\frac{\partial U}{\partial x_2}}$$

(or in some strange books $-\frac{MU_{x_1}}{MU_{x_2}}$)

Is that right?

2.

If Rita consumed $a$ of $x_1$ and $b$ of $x_2$, what was her MRS?

$$MRS_{x_1, x_2}(a,b) = \frac{MU_{x_1}(a,b)}{MU_{x_2}(a,b)}$$

Is that right?

3.

3.1 If Rita is currently consuming $c$ units of $x_1$, how many units of $x_2$ must she consume in order to leave her just as well off as she was in #2?

3.2 What is the MRS at $(c,x_2)$?

What's the equation to solve here?

$$U(a,b) = U(c,x_2')$$

?

Or

$$\frac{MU_{x_1}(a,b)}{MU_{x_2}(a,b)} = \frac{MU_{x_1}(c,x_2')}{MU_{x_2}(c,x_2')}$$

?

I'm thinking the former because 'well off' seems to call for 'utility' and not 'MRS', and the latter answers 3.2 easily...

Anyway, if the former then for 3.2 I just compute

$$\frac{MU_{x_1}(c,x_2')}{MU_{x_2}(c,x_2')}$$

?

• I think your answers to 1 and 2 are correct. For 3.1, your intuition that "as well off as" should be interpreted "indifferent" is correct, and indifferent is about comparing utility levels, not MRS. – Herr K. Feb 23 '16 at 23:17
• Thanks @HerrK. And 3.2? – BCLC Feb 24 '16 at 16:49
• Your answer to 3.2 looks correct. – Herr K. Feb 24 '16 at 17:25
• @HerrK. Post as answer? – BCLC Feb 24 '16 at 17:28