Question about an interpretation of the MRS 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 $y$?
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.