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.