Gorman polar form: Difference between revisions

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'''Gorman polar form''' is a functional form for [[indirect utility function|indirect utility functions]]s in [[economics]]. Imposing this form on [[utility]] allows the researcher to treat a society of utility-maximizers as if it consisted of a single [[individual]]. [[W. M. Gorman]] showed that having the [[function (mathematics)|function]] take Gorman polar form is both a [[necessary and sufficient]] for this condition to hold.

== Motivation ==

Early results by Antonelli (1886) and Nataf (1953) had shown that assuming all individuals face the same prices in a market, their income consumption curves and their [[Engel curve|Engel curves]]s should be parallel straight lines. Gorman's first published paper in 1953 developed these ideas in order to answer the question of generalizing a society to a single individual.

In 1961, Gorman published a short, four-page paper in Metroeconomica which derived an explicit expression for the functional form of preferences which give rise to linear Engel curves. Briefly, an individual's (<math>i</math>) resulting expenditure function (<math> e ^ i \left ( p , u ^ i \right ) </math>) must be [[affine transformation|affine]] with respect to utility (<math>u</math>):

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== Proof of linearity and equality of slope of Engel curves ==

To prove that the [[Engel curve|Engel curves]]s of a function in Gorman polar form are [[linear]], apply [[Roy's identity]] to the 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 u^i(p,m^i)}{\partial p_n}}{\frac{\partial u^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>