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→Proof of linearity and equality of slope of Engel curves: we apply roy's to indirect utility, not simply utility. |
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== Proof of linearity and equality of slope of Engel curves ==
To prove that the [[Engel curve]]s of a function in Gorman polar form are [[linear]], apply [[Roy's identity]] to the [[indirect utility function]] to get a [[Marshallian demand function]] for an individual (<math>i</math>) and a good (<math>n</math>):
:<math>x^i_n(p,m^i) = -\frac{\frac{\partial v^i(p,m^i)}{\partial p_n}}{\frac{\partial v^i(p,m^i)}{\partial m^i}} = \frac{\partial f^i(p)}{\partial p_n} + \frac{\partial g(p)}{\partial p_n}\cdot\frac{m-f^i(p)}{g(p)}</math>
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