Editing Ordinal utility (section)

== Marginal rate of substitution ==
Suppose a person has a bundle <math>(x_0,y_0)</math> and claims that he is indifferent between this bundle and the bundle <math>(x_0-\lambda\cdot\delta,y_0+\delta)</math>. This means that he is willing to give <math>\lambda\cdot\delta</math> units of x to get <math>\delta</math> units of y. If this ratio is kept as <math>\delta\to 0</math>, we say that <math>\lambda</math> is the ''[[marginal rate of substitution]] (MRS)'' between ''x'' and ''y'' at the point <math>(x_0,y_0)</math>.<ref name=KeeneyRaiffa1993>{{Cite book  |last1=Keeney |first1=Ralph L. |last2=Raiffa |first2=Howard |title=Decisions with Multiple Objectives |year=1993 |isbn=978-0-521-44185-8}}</ref>{{rp|82}}

This definition of the MRS is based only on the ordinal preference relation – it does not depend on a numeric utility function. If the preference relation is represented by a utility function and the function is differentiable, then the MRS can be calculated from the derivatives of that function:
:<math>MRS = \frac{v'_x}{v'_y}.</math>
For example, if the preference relation is represented by <math>v(x,y)=x^a\cdot y^b</math> then <math>MRS = \frac{a\cdot x^{a-1}\cdot y^b}{b\cdot y^{b-1}\cdot x^a}=\frac{ay}{bx}</math>. The MRS is the same for the function <math>v(x,y)=a\cdot \log{x} + b\cdot \log{y}</math>. This is not a coincidence as these two functions represent the same preference relation – each one is an increasing monotone transformation of the other.

In general, the MRS may be different at different points <math>(x_0,y_0)</math>. For example, it is possible that at <math>(9,1)</math> the MRS is low because the person has a lot of ''x'' and only one ''y'', but at <math>(9,9)</math> or <math>(1,1)</math> the MRS is higher. Some special cases are described below.