Roy's Identity

Relates the Marshallian demand function to the indirect utility function:

$$ x_{i}(p,m)=-\frac{\partial V}{\partial p_{i}}/\frac{\partial V}{\partial m}$$

Derivation
1) Start from the identity: $$u^{*} \equiv V(p, e(p,u^{*}))$$

2) Differentiate with respect to $$p_{i}$$ using the chain rule: $$0 = \partial V(p^{*},m^{*})/\partial p_{i} + \partial V(p^{*},m^{*})/\partial m \cdot \partial e(p^{*},u^{*})/\partial p_{i}$$

3) Re-arrange and combine with following identity: $$ x(p^{*},m^{*})=h^{*}(p^{*},u^{*}) \,$$

4) Arrive at: $$x_{i}(p^{*},m^{*}) \equiv h_{i}(p^{*},u^{*}) \equiv - \frac{ \partial V(p^{*},m^{*})/\partial p_{i}}{\partial V(p^{*}, m^{*})/ \partial m} \equiv \partial e(p^{*},u^{*}) / \partial p_{i} $$.

The last derivation of the equality is shown here.

Source
Roy (1947), "La Distribution du Revenu Entre Les Divers Biens", Econometrica 15, 205-225.