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Given the marginal rate of substitution of $x$ for $y$ : $\frac{u'(x)}{u'(y)} $

I know one can interpret this as the amount of $y$ one is willing to give up for an additional unit of $x$, or the amount of $y$ that has to be given to compensate the consumer for a loss of $x$.

But mathematically, is it saying, how much the consumer values the next unit of $x$, in terms of the next unit of $y$? or the marginal valuation of $x$ in terms of additional units of $y$?

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    $\begingroup$ The short answer is no. The marginal utility of $y$ is not in units of $y$, but in marginal utils. As such, you can't interpret it in the way you want to. You could say it is how much the consumer values the next unit of $x$, in terms of how much they value the next unit of $y$. $\endgroup$ – BKay Sep 24 at 18:45
  • $\begingroup$ just a follow-up clarification question- how is your last sentence substantive different than what my interpretation? i.e. the marginal valuation of the next unity of x in terms of the next unit of y? $\endgroup$ – Steve Sep 25 at 17:05
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    $\begingroup$ The way I parse your sentence, you are asking about $\frac{u'(x)}{\partial y} $ not $\frac{u'(x)}{u'(y)} $. Contrast the following: 1) (what you said) the marginal valuation of the next unity of x in terms of the next unit of y 2) the marginal valuation of the next unity of x in terms of the marginal value of the next unit of y $\endgroup$ – BKay Sep 25 at 20:11
  • $\begingroup$ Oh ok, that makes perfect sense. Thank you for clarifying! $\endgroup$ – Steve Sep 28 at 16:24
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Writing out the expression with Leibniz notation, we get $\frac{\frac{du}{dx}}{\frac{du}{dy}}$, or the marginal utility of $x$ divided by the marginal utility $y$.

For example, let's imagine I get 10 utils from another apple and 5 utils from another cabbage. In that case, the MRS is therefore $\frac{\frac{10 utils}{1 apple}}{\frac{5 utils}{1 cabbage}}$, or, $\frac{2 cabbage}{1 apple}$, matching the marginal rate of substitution of 2 cabbages per apple.

A bit of simplification will easily get you $\frac{dy}{dx}$, which is the rate of change of $y$ with respect to $x$, the interpretation we're familiar with.

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