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 Edit

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 Edit

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

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