<p>Writing out the expression with Leibniz notation, we get <span class="math-container">$\frac{\frac{du}{dx}}{\frac{du}{dy}}$</span>, or the marginal utility of <span class="math-container">$x$</span> divided by the marginal utility <span class="math-container">$y$</span>.</p>
<p>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 <span class="math-container">$\frac{\frac{10 utils}{1 apple}}{\frac{5 utils}{1 cabbage}}$</span>, or, <span class="math-container">$\frac{2 cabbage}{1 apple}$</span>, matching the marginal rate of substitution of 2 cabbages per apple.</p>
<p>A bit of simplification will easily get you <span class="math-container">$\frac{dy}{dx}$</span>, which is the rate of change of <span class="math-container">$y$</span> with respect to <span class="math-container">$x$</span>, the interpretation we're familiar with.</p>

CC BY-SA 4.0
2020-09-24T01:41:25.490Z
2020-09-24T18:39:27.500Z
2020-09-24T18:39:27.500Z
H Huang
<p>Given the marginal rate of substitution of <span class="math-container">$x$</span> for <span class="math-container">$y$</span> : <span class="math-container">$\frac{u'(x)}{u'(y)} $</span></p>
<p>I know one can interpret this as the amount of <span class="math-container">$y$</span> one is willing to give up for an additional unit of <span class="math-container">$x$</span>, or the amount of <span class="math-container">$y$</span> that has to be given to compensate the consumer for a loss of <span class="math-container">$x$</span>.</p>
<p>But mathematically, is it saying, how much the consumer values the next unit of <span class="math-container">$x$</span>, in terms of the next unit of <span class="math-container">$y$</span>? or the marginal valuation of <span class="math-container">$x$</span> in terms of additional units of <span class="math-container">$y$</span>?</p>

CC BY-SA 4.0
2020-09-24T18:45:54.763Z
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
CC BY-SA 4.0
2020-09-25T17:05:29.110Z
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?
Steve
CC BY-SA 4.0
2020-09-25T20:11:18.453Z
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
CC BY-SA 4.0
2020-09-28T16:24:38.910Z
Oh ok, that makes perfect sense. Thank you for clarifying!
Steve
<microeconomics><mathematical-economics><consumer-theory><marginal>
Question about an interpretation of the MRS
Steve
# Question about an interpretation of the MRS

(Asked by: username_0 on 2020-09-24)

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 $y$?

**Tags:** microeconomics, mathematical-economics, consumer-theory, marginal
> **Comments:**
> **username_1**: 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 $y$.  
> **username_0**: 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?  
> **username_1**: 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**  
> **username_0**: Oh ok, that makes perfect sense. Thank you for clarifying!  

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## 1 answers

### Answer by username_2 on 2020-09-24. Score: 2 | Accepted answer

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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