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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"

  1. 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.

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    $\begingroup$ 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. $\endgroup$ – Giskard Mar 17 at 6:33
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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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The above answer by Regio 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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