<p>It is implicit in the interpretation: </p>

<p>Mas-Collel: the amount that must be given (+)  to compensate for a reduction (-). </p>

<p>Reny: The rate at which good j  can be exchanged (+ &amp; -) for good i. </p>

<p>The derivation from total differentiation only requires the utility to be constant, so the derivative must be negative to express that if the quantity of good <span class="math-container">$i$</span> increases, the quantity of good <span class="math-container">$j$</span> should decrease.</p>

<p>Another reason why some authors omit the negative sign is that in equilibrium  <span class="math-container">$|MRS_{ij}|=|\frac{P_i}{P_j}|$</span> So either you use the derivation from total differentiation and also the negative ratio of prices, or leave both sides as positive and focus on interpretation. Most people prefer the latter. </p>

CC BY-SA 4.0
2020-03-16T16:05:08.480Z
2020-03-16T16:05:08.480Z
Regio
<p>The above answer by Regio covers most of the answer</p>

<p>Just to emphasize further :
Consider a 2 goods case. We define MRS as the opportunity cost of consuming one more unit of good 1. Opportunity cost means "What Am I Giving Up ?". Since you are consuming only two goods, you are giving up some amount of good 2 in order to consume this one more unit of good 1.  </p>

<p>This is why the (-) sign is omitted while stating MRS. MRS is conventionally defined as the absolute value of the slope of the Indifference Curve. The (-) sign in the slope implies a negative relationship between good 1 and good 2 along the same Indifference Curve. This negative relationship or the " giving up of one to consume more of another" is implicit in the definition of opportunity cost. Hence only the absolute value is reported. </p>

CC BY-SA 4.0
2020-03-16T21:45:22.813Z
2020-03-16T21:45:22.813Z
Poorvi
<p>I am having a dumb doubt in writing some slides for an undergraduate class. 
I want to be consistent with the use in microeconomics but this easy thing is really bugging me:</p>

<ol>
<li>Mas-colell pag. 54</li>
</ol>

<p><span class="math-container">$ MRS_{lk} = \frac{\frac{\partial u}{\partial x_l}}{\frac{\partial u}{\partial x_k}} $</span> </p>

<p>"tells us the amount of good k that the consumer must be given to compensate her for a one-unit marginal reduction in her consumption of good l"</p>

<ol start="2">
<li>Reny advanced micro pag. 18</li>
</ol>

<p><span class="math-container">$MRS_{ij}(x) ≡ \frac{\frac{\partial u}{\partial x_i}}{\frac{\partial u}{\partial x_j}}$</span></p>

<p>" ..MRS_{ij}(x) is again a positive number, and it tells us the rate at which good j can be exchanged per unit of good i with no change in the consumer’s utility"</p>

<p>I know also how to derive this result with the total differentiation but one thing really confuses me: </p>

<p><span class="math-container">$du = \frac{\partial u}{\partial x_i}*dx_i+\frac{\partial u}{\partial x_j}*dx_j$</span></p>

<p><span class="math-container">$ 0 = \frac{\partial u}{\partial x_i}*dx_i+\frac{\partial u}{\partial x_j}*dx_j $</span></p>

<p><span class="math-container">$ \frac{\partial u}{\partial x_i}*dx_i = -\frac{\partial u}{\partial x_j}*dx_j$</span></p>

<p><span class="math-container">$ \frac{dx_j}{dx_i} = - \frac{\frac{\partial u}{\partial x_i}}{\frac{\partial u}{\partial x_j}}$</span></p>

<p>The last equation expresses the differential increase in <span class="math-container">$x_j$</span> correspondent to a marginal increase in <span class="math-container">$x_i$</span> just as the two formulas above. Where does the minus go in the two books? Am I neglecting something?
Thanks. </p>

CC BY-SA 4.0
2020-03-17T06:33:40.223Z
Both current answers make it seem like this is a trivial thing, so I would like to say I also find the quoted notation annoying. When I describe MRS to my students I treat it as a negative number, though I explain that in informal explanations I will be talking about the absolute value. In lecture notes for mathematically mature students I would use your notation, not Mas-collel's or Reny's.
Giskard
<microeconomics><utility><consumption>
Marginal rate of substitution notation:
KArrow'sBest
# Marginal rate of substitution notation:

(Asked by: username_0 on 2020-03-16)

I am having a dumb doubt in writing some slides for an undergraduate class.
I want to be consistent with the use in microeconomics but this easy thing is really bugging me:

1. Mas-colell pag. 54

$ MRS\_{lk} = \frac{\frac{\partial u}{\partial x\_l}}{\frac{\partial u}{\partial x\_k}} $

"tells us the amount of good k that the consumer must be given to compensate her for a one-unit marginal reduction in her consumption of good l"

2. Reny advanced micro pag. 18

$MRS\_{ij}(x) ≡ \frac{\frac{\partial u}{\partial x\_i}}{\frac{\partial u}{\partial x\_j}}$

" ..MRS\_{ij}(x) is again a positive number, and it tells us the rate at which good j can be exchanged per unit of good i with no change in the consumer’s utility"

I know also how to derive this result with the total differentiation but one thing really confuses me:

$du = \frac{\partial u}{\partial x\_i}\*dx\_i+\frac{\partial u}{\partial x\_j}\*dx\_j$

$ 0 = \frac{\partial u}{\partial x\_i}\*dx\_i+\frac{\partial u}{\partial x\_j}\*dx\_j $

$ \frac{\partial u}{\partial x\_i}\*dx\_i = -\frac{\partial u}{\partial x\_j}\*dx\_j$

$ \frac{dx\_j}{dx\_i} = - \frac{\frac{\partial u}{\partial x\_i}}{\frac{\partial u}{\partial x\_j}}$

The last equation expresses the differential increase in $x\_j$ correspondent to a marginal increase in $x\_i$ just as the two formulas above. Where does the minus go in the two books? Am I neglecting something?
Thanks.

**Tags:** microeconomics, utility, consumption

> **Comments:**
> **username_1**: Both current answers make it seem like this is a trivial thing, so I would like to say I also find the quoted notation annoying. When I describe MRS to my students I treat it as a negative number, though I explain that in informal explanations I will be talking about the absolute value. In lecture notes for mathematically mature students I would use your notation, not Mas-collel's or Reny's.  

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## 2 answers

### Answer by username_2 on 2020-03-16. Score: 2 | Accepted answer

It is implicit in the interpretation:

Mas-Collel: the amount that must be given (+) to compensate for a reduction (-).

Reny: The rate at which good j can be exchanged (+ & -) for good i.

The derivation from total differentiation only requires the utility to be constant, so the derivative must be negative to express that if the quantity of good $i$ increases, the quantity of good $j$ should decrease.

Another reason why some authors omit the negative sign is that in equilibrium $|MRS\_{ij}|=|\frac{P\_i}{P\_j}|$ So either you use the derivation from total differentiation and also the negative ratio of prices, or leave both sides as positive and focus on interpretation. Most people prefer the latter.


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### Answer by username_3 on 2020-03-16. Score: 2

The above answer by username_2 covers most of the answer

Just to emphasize further :
Consider a 2 goods case. We define MRS as the opportunity cost of consuming one more unit of good 1. Opportunity cost means "What Am I Giving Up ?". Since you are consuming only two goods, you are giving up some amount of good 2 in order to consume this one more unit of good 1.

This is why the (-) sign is omitted while stating MRS. MRS is conventionally defined as the absolute value of the slope of the Indifference Curve. The (-) sign in the slope implies a negative relationship between good 1 and good 2 along the same Indifference Curve. This negative relationship or the " giving up of one to consume more of another" is implicit in the definition of opportunity cost. Hence only the absolute value is reported.


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