Slutsky Equation - Derivation

Derivation

While there are several ways to derive the Slutsky equation, the following method is likely the simplest. Begin by noting the identity where is the expenditure function, and u is the utility obtained by maximizing utility given p and w. Totally differentiating with respect to p_j yields the following:

.

Making use of the fact that by Shephard's lemma and that at optimum,

where is the indirect utility function,

one can substitute and rewrite the derivation above as the Slutsky equation.