The Marginal Rate of Substitution


Recall that the MRS(x) corresponds geometrically to the slope of the indifference curve at x. For a given utility function u(x1,x2) the MRS is given by

MRS(x1,x2) = - u1(x1,x2) / u2(x1,x2) ,

where u1(x1,x2) and u2(x1,x2) denote the partial derivatives of the utility function with respect to the first and the second argument, respectively.

Also recall that u1(x1,x2) is referred to as the marginal utility of good 1, denoted by MU1(x1,x2) and that and that u2(x1,x2) is referred to as the marginal utility of good 2, denoted by MU2(x1,x2).

The MRS can therefore be computed by dividing the partial derivative of the utility function with respect to x1 by the partial derivative of the utility function with respect to x2 and by multiplying this expression by -1.

For example, the partial derivatives for the Cobb-Douglas utility function u(x1,x2)=x1x2 are given by u1(x1,x2)=x2 and u2(x1,x2)=x1. Thus, MRS(x1,x2)=-x2/x1 for the particular Cobb-Douglas utility function u(x1,x2)=x1x2.

What is therfore the MRS at (1,4) for this utility function? This can be determine by inserting 1 for x1 and 4 for x2 in the above expression. Thus, MRS(1,4)=-4. Similarly, MRS(2,2)=-1 and MRS(4,2)=-1/2.