# MRS and IC : The language sounds contradictory

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?

• Can you specify the reference? I have the 10th edition of NS here and it contains no such verbatim phrase. Jul 12, 2021 at 7:46
• @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. Jul 12, 2021 at 7:52
• 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"... Jul 12, 2021 at 8:24
• I guess there is a typo in "we need to exchange x at the MRS for every unit of x". Jul 12, 2021 at 11:35
• 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
Jul 12, 2021 at 12:43

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 $$-\frac{dy}{dx}=MRS_{xy}$$.
• 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? Jul 12, 2021 at 8:40
• @Giskard, no, 2 units of $y$ for a unit of $x$, as I wrote. Jul 12, 2021 at 8:47
• 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? Jul 12, 2021 at 9:00
• 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. Jul 12, 2021 at 11:03