Marginal rate of substitution notation:

Question

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.

Answer 1

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.

Answer 2

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.