# 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 $$y$$?

• 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 $y$.
– BKay
Sep 24, 2020 at 18:45
• 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? Sep 25, 2020 at 17:05
• 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
– BKay
Sep 25, 2020 at 20:11
• Oh ok, that makes perfect sense. Thank you for clarifying! Sep 28, 2020 at 16:24

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.