# Marginal rate of substitution notation:

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.

• 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. Mar 17, 2020 at 6:33

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.