<p>No, it is not. The verbatime citation is <em>The slope of this curve represents the rate at which the individual is willing to trade <span class="math-container">$x$</span> for <span class="math-container">$y$</span> while remaining equally well off.</em></p>
<p><em>To trade <span class="math-container">$x$</span> for <span class="math-container">$y$</span></em> here means to give up some <span class="math-container">$\Delta x$</span> to receive <span class="math-container">$\Delta y$</span> per unit of <span class="math-container">$\Delta x$</span>. Letting <span class="math-container">$\Delta x\rightarrow 0$</span> the rate is <span class="math-container">$-\frac{dy}{dx}=MRS_{xy}$</span>.</p>

CC BY-SA 4.0
2021-07-12T08:40:08.677Z
Let the utility function of a person be $U(x,y) = 2x + y$. We can probably agree that $|MRS_{xy}| = 2/1$. Would you say the person is willing to trade 2 units of $x$ for a unit of $y$? Or even that the person is willing to trade $x$ for $y$ at a 2:1 ratio?
Giskard
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2021-07-12T08:47:09.817Z
@Giskard, no, 2 units of $y$ for a unit of $x$, as I wrote.
VARulle
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2021-07-12T09:00:01.383Z
Where did you write this? Seems to me like you wrote "*To trade $x$ for $y$ means to give up some $\Delta x$ to receive $\Delta y$*." and then you implied the rate of trading was equal to the marginal rate of substitution? And "*some* $\Delta x$" implies this is the quantity of $x$ traded?
Giskard
CC BY-SA 4.0
2021-07-12T09:31:12.197Z
@Giskard, exactly. But I don't understand your problem with this. I'm not a native English speaker, so maybe I'm unclear or wrong, but where exactly?
VARulle
CC BY-SA 4.0
2021-07-12T11:03:39.733Z
Neither am I :) But I believe "to trade specified_quantity of $x$ for $y$" implies the same slope as "to trade $\frac{1}{\text{specified_quantity}}$ of $y$ for $x$". The difference in trading direction is, in my opinion, the basis of the question.
Giskard
CC BY-SA 4.0
2021-07-12T11:30:48.330Z
@Giskard, well, I trusted the Wikipedia entry https://en.wikipedia.org/wiki/Marginal_rate_of_substitution#As_the_slope_of_indifference_curve stating that *MRS of X for Y is the amount of Y which a consumer can exchange for one unit of X locally* (substituting "trade" for "exchange").
VARulle
CC BY-SA 4.0
2021-07-12T12:26:02.173Z
But this is the other direction, trading an amount of $Y$ for $X$, not the direction of trading an amount of $X$ for $Y$.
Giskard
CC BY-SA 4.0
2021-07-12T13:09:02.767Z
@Giskard: But the *direction* is not important. I can always receive or give up some amount of either good. The question is, of which good do I receive or give up *one unit*. Then the compensating amount of the other good (given up or received) is the MRS. Granted, this was not very clear in my answer, so I edited it accordingly now.
VARulle
CC BY-SA 4.0
2021-07-12T14:12:01.193Z
"*The question is, of which good do I receive or give up one unit.*" This was also the point I was making.
Giskard
CC BY-SA 4.0
2021-07-12T14:20:04.113Z
As we are considering the "derivative" of $y$ with respect to $x$, you look at how much $y$ changes at the margin when $x$ increases at the margin. So, the natural interpretation would be indeed: At the margin, how much $Y$ (in terms of $X$) would I trade (give up) for  $X$.  But, indeed, the direction of trade ultimately doesn't matter, because the relationship is symmetric via the inverse.
BrsG
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2021-07-13T02:50:14.480Z
@BrsG Well the direction is under control if I have to say. This is precisely why I used the $\pm, \mp$ in my post and included both *give and get* in my interpretation.
not tdm's twin
CC BY-SA 4.0
2021-07-13T08:02:42.297Z
@BrsG, I don't want to be nitpicking, but this is not the "natural" way of reading a derivative. The limit in the definition is a two-sided limit, so the $\Delta x$ can have either sign.
VARulle
CC BY-SA 4.0
2021-07-12T08:33:17.130Z
2021-07-12T13:07:33.490Z
2021-07-12T13:07:33.490Z
VARulle
<p>The MRS<span class="math-container">$_{xy}$</span> is defined as <span class="math-container">$\left(-\frac{dy}{dx}\right)$</span> and Nicholson/Snyder (NS) writes it as <em>the amount of <span class="math-container">$x$</span> we can trade for <span class="math-container">$y$</span> while remaining equally well off</em>.</p>
<p>However, the analytical definition however tells something different, that in a sufficiently close neighbourhood of <span class="math-container">$(x,y)$</span>, <span class="math-container">$y\mp \left(-\frac{dy}{dx}\right) = m(x \pm 1) + c$</span>, which can be interpreted as: the amount of <span class="math-container">$y$</span> we need to give or get (trade off) for a unit of <span class="math-container">$x$</span> (while remaining equally well off).</p>
<p>The two statements sound contradictory, the first one says we need to exchange <span class="math-container">$x$</span> at the MRS for every unit of <span class="math-container">$x$</span> and the second one says quite the opposite, that we substitute <span class="math-container">$x$</span> at the MRS for every unit of <span class="math-container">$y$</span>.</p>
<p>Is it a mistake in the book by NS?</p>

CC BY-SA 4.0
2021-07-12T07:46:18.950Z
Can you specify the reference? I have the 10th edition of NS here and it contains no such verbatim phrase.
VARulle
CC BY-SA 4.0
2021-07-12T07:52:57.143Z
@VARulle Fig. 3.2, Page 92, 11th Ed. NS: *The curve $U_1$ represents those combinations of $x$ and $y$ from which the individual derives the same utility. **The slope of this curve represents the rate at which the individual is willing to trade $x$ for $y$ while remaining equally well off.** This slope (or, more properly, the negative of the slope) is termed the marginal rate of substitution. In the figure, the indifference curve is drawn on the assumption of a diminishing marginal rate of substitution.*
not tdm's twin
CC BY-SA 4.0
2021-07-12T08:24:26.310Z
Well, "*the rate at which the individual is willing to trade x for y*" is not the same as "*the amount of x we can trade for y*"...
VARulle
CC BY-SA 4.0
2021-07-12T11:35:06.033Z
I guess there is a typo in "*we need to exchange x at the MRS for every unit of x*".
VARulle
CC BY-SA 4.0
2021-07-12T12:43:58.117Z
I used to use NS for a micro class. The book has quite some (minor) mistakes & typo's. I guess this is one of them. $dy/dx$ measures how much of $y$ we need to give up to obtain an additional unit of $x$, keeping utility fixed.
tdm
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2021-07-12T13:25:21.693Z
@tdm, but is this inconsistent with "the rate at which the individual is willing to trade $x$ for $y$"? After all, $\frac{dy}{dx}$ also measures how much of $y$ we need to receive to give up a unit of $x$. If I am willing to trade my car for 3 bicycles, isn't then my $MRS_{cars,bicycles}=3$?
VARulle
CC BY-SA 4.0
2021-07-12T13:45:14.673Z
@VARulle I don't think there can be a consensus as this seems to be more semantics related than logic. Personally, however, I would read it as "the rate at which the individual is willing to trade x for *one unit* of $y$". In other words, I would interpret it as: how much do I need to give away of $x$ to get 1 more unit of $y$ (and keeping utility fixed). This would then be the inverse of $dy/dx$. (I'm not a native English speaker though).
tdm
CC BY-SA 4.0
2021-07-13T02:48:34.567Z
@VARulle Well I think it does satisfy the case when you get (receive) $y$ in exchange for $x$. This is precisely why I used the $\pm, \mp$ in my post instead of just $+$ or $-$. (If you have to give up one unit of $y$ for every MRS units of $x$, then $m \to \frac{1}{m} = \frac{dx}{dy} = \frac{1}{\text{MRS}}$.
not tdm's twin
CC BY-SA 4.0
2021-07-13T03:00:14.243Z
@tdm Either interpretation takes care of both cases: give and get. This is essentially what I tried to describe with the use of $\pm, \mp$ in my post. (In NS's interpretation, we would do something like I pointed in my previous comment.) So I think there can be a consensus.
not tdm's twin
CC BY-SA 4.0
2021-07-13T08:05:22.977Z
I think we should continue this discussion at the English language SE...
VARulle
<microeconomics><utility>
MRS and IC : The language sounds contradictory
not tdm's twin
# MRS and IC : The language sounds contradictory

(Asked by: username_0 on 2021-07-11)

The MRS$\_{xy}$ is defined as $\left(-\frac{dy}{dx}\right)$ and Nicholson/Snyder (NS) writes it as *the amount of $x$ we can trade for $y$ while remaining equally well off*.

However, the analytical definition however tells something different, that in a sufficiently close neighbourhood of $(x,y)$, $y\mp \left(-\frac{dy}{dx}\right) = m(x \pm 1) + c$, which can be interpreted as: the amount of $y$ we need to give or get (trade off) for a unit of $x$ (while remaining equally well off).

The two statements sound contradictory, the first one says we need to exchange $x$ at the MRS for every unit of $x$ and the second one says quite the opposite, that we substitute $x$ at the MRS for every unit of $y$.

Is it a mistake in the book by NS?

**Tags:** microeconomics, utility

> **Comments:**
> **username_1**: Can you specify the reference? I have the 10th edition of NS here and it contains no such verbatim phrase.  
> **username_0**: @username_1 Fig. 3.2, Page 92, 11th Ed. NS: *The curve $U_1$ represents those combinations of $x$ and $y$ from which the individual derives the same utility. **The slope of this curve represents the rate at which the individual is willing to trade $x$ for $y$ while remaining equally well off.** This slope (or, more properly, the negative of the slope) is termed the marginal rate of substitution. In the figure, the indifference curve is drawn on the assumption of a diminishing marginal rate of substitution.*  
> **username_1**: Well, "*the rate at which the individual is willing to trade x for y*" is not the same as "*the amount of x we can trade for y*"...  
> **username_1**: I guess there is a typo in "*we need to exchange x at the MRS for every unit of x*".  
> **username_2**: I used to use NS for a micro class. The book has quite some (minor) mistakes & typo's. I guess this is one of them. $dy/dx$ measures how much of $y$ we need to give up to obtain an additional unit of $x$, keeping utility fixed.  
> **username_1**: @username_2, but is this inconsistent with "the rate at which the individual is willing to trade $x$ for $y$"? After all, $\frac{dy}{dx}$ also measures how much of $y$ we need to receive to give up a unit of $x$. If I am willing to trade my car for 3 bicycles, isn't then my $MRS_{cars,bicycles}=3$?  
> **username_2**: @username_1 I don't think there can be a consensus as this seems to be more semantics related than logic. Personally, however, I would read it as "the rate at which the individual is willing to trade x for *one unit* of $y$". In other words, I would interpret it as: how much do I need to give away of $x$ to get 1 more unit of $y$ (and keeping utility fixed). This would then be the inverse of $dy/dx$. (I'm not a native English speaker though).  
> **username_0**: @username_1 Well I think it does satisfy the case when you get (receive) $y$ in exchange for $x$. This is precisely why I used the $\pm, \mp$ in my post instead of just $+$ or $-$. (If you have to give up one unit of $y$ for every MRS units of $x$, then $m \to \frac{1}{m} = \frac{dx}{dy} = \frac{1}{\text{MRS}}$.  
> **username_0**: @username_2 Either interpretation takes care of both cases: give and get. This is essentially what I tried to describe with the use of $\pm, \mp$ in my post. (In NS's interpretation, we would do something like I pointed in my previous comment.) So I think there can be a consensus.  
> **username_1**: I think we should continue this discussion at the English language SE...  

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## 1 answers

### Answer by username_1 on 2021-07-12. Score: 1

No, it is not. The verbatime citation is *The slope of this curve represents the rate at which the individual is willing to trade $x$ for $y$ while remaining equally well off.*

*To trade $x$ for $y$* here means to give up some $\Delta x$ to receive $\Delta y$ per unit of $\Delta x$. Letting $\Delta x\rightarrow 0$ the rate is $-\frac{dy}{dx}=MRS\_{xy}$.

> **Comments:**
> **username_3**: Let the utility function of a person be $U(x,y) = 2x + y$. We can probably agree that $|MRS_{xy}| = 2/1$. Would you say the person is willing to trade 2 units of $x$ for a unit of $y$? Or even that the person is willing to trade $x$ for $y$ at a 2:1 ratio?  
> **username_1**: @username_3, no, 2 units of $y$ for a unit of $x$, as I wrote.  
> **username_3**: Where did you write this? Seems to me like you wrote "*To trade $x$ for $y$ means to give up some $\Delta x$ to receive $\Delta y$*." and then you implied the rate of trading was equal to the marginal rate of substitution? And "*some* $\Delta x$" implies this is the quantity of $x$ traded?  
> **username_1**: @username_3, exactly. But I don't understand your problem with this. I'm not a native English speaker, so maybe I'm unclear or wrong, but where exactly?  
> **username_3**: Neither am I :) But I believe "to trade specified_quantity of $x$ for $y$" implies the same slope as "to trade $\frac{1}{\text{specified_quantity}}$ of $y$ for $x$". The difference in trading direction is, in my opinion, the basis of the question.  
> **username_1**: @username_3, well, I trusted the Wikipedia entry https://en.wikipedia.org/wiki/Marginal_rate_of_substitution#As_the_slope_of_indifference_curve stating that *MRS of X for Y is the amount of Y which a consumer can exchange for one unit of X locally* (substituting "trade" for "exchange").  
> **username_3**: But this is the other direction, trading an amount of $Y$ for $X$, not the direction of trading an amount of $X$ for $Y$.  
> **username_1**: @username_3: But the *direction* is not important. I can always receive or give up some amount of either good. The question is, of which good do I receive or give up *one unit*. Then the compensating amount of the other good (given up or received) is the MRS. Granted, this was not very clear in my answer, so I edited it accordingly now.  
> **username_3**: "*The question is, of which good do I receive or give up one unit.*" This was also the point I was making.  
> **username_4**: As we are considering the "derivative" of $y$ with respect to $x$, you look at how much $y$ changes at the margin when $x$ increases at the margin. So, the natural interpretation would be indeed: At the margin, how much $Y$ (in terms of $X$) would I trade (give up) for  $X$.  But, indeed, the direction of trade ultimately doesn't matter, because the relationship is symmetric via the inverse.  
> **username_0**: @username_4 Well the direction is under control if I have to say. This is precisely why I used the $\pm, \mp$ in my post and included both *give and get* in my interpretation.  
> **username_1**: @username_4, I don't want to be nitpicking, but this is not the "natural" way of reading a derivative. The limit in the definition is a two-sided limit, so the $\Delta x$ can have either sign.  

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