The Slutsky Equation decomposes the a price change into an income effect and a substitution effect:
Derivation
Basic idea: Express Hicksian demand as Marshallian demand using expenditure function and then differentiate.
1) Start with following identity:
2) Differentiate with respect to
3) Note that by Shephard's Lemma, we can rewrite this as:
4) Since , we get:
5) After rearranging: