Slutsky Equation

The Slutsky Equation decomposes the a price change into an income effect and a substitution effect:

$${\partial x_i(\mathbf{p}, m) \over \partial p_j} = {\partial h_i(\mathbf{p}, u) \over \partial p_j} - {\partial x_i(\mathbf{p}, m) \over \partial m } x_j(\mathbf{p}, m),\,$$

Derivation
Basic idea: Express Hicksian demand as Marshallian demand using expenditure function and then differentiate.

1) Start with following identity:

$$ h_{i}(p,u)= h_{i}(p,V(p,m)) = x_{i}(p,m) = x_{i}(p, e(p,u)) \,$$

2) Differentiate with respect to $$p_{j} \,$$

$$ \partial h_{i}(p_{i},u)/\partial p_{j} = \partial x_{i}(p, m)/\partial p_{j}+\partial x_{i}(p, m)/ \partial m \cdot \partial e(p,u)/ \partial p_{j} $$

3) Note that by Shephard's Lemma, we can rewrite this as:

$$ \partial h_{i}(p_{i},u)/\partial p_{j} = \partial x_{i}(p,m)/\partial p_{j}+\partial x_{i}/ \partial m \cdot h_{j}(p,u) $$

4) Since $$h_{j}(p,V(p,m)) \equiv x_{j}(p,m) \,$$, we get:

$$ \partial h_{i}(p_{i},u)/\partial p_{j} = \partial x_{i}(p,m)/\partial p_{j}+\partial x_{i}(p,m)/ \partial m \cdot x_{j}(p,m) $$

5) After rearranging:

$$ \partial x_{i}(p,m)/\partial p_{j} = \partial h_{i}(p_{i},u)/\partial p_{j} -\partial x_{i}/ \partial m \cdot x_{j}(p,m) $$