Gorman polar form: Difference between revisions

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'''Gorman polar form''' is a functional form for [[indirect utility function]]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 [[Representative agent|'representative' individual]]. [[W. M. Gorman|Gorman]] showed that having the [[function (mathematics)|function]] take Gorman polar form is both [[necessary and sufficient]] for this condition to hold.

== Motivation ==

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:<math>x^i(p,m^i)</math>

The aggregate demand of society is, in general, a function of the price system and the entire distribution of incomes:

:<math>X(p,m^1,\dots,m^n) = \sum_{i=1}^n {x^i(p,m^i)}</math>

To represent the entire society as a single consumer, the aggregate demand must be a function of only the prices and the ''total'' income, regardless of its distribution:

:<math>X(p,m^1,\dots,m^n) = X\left(p, \sum_{i=1}^n {m^i} \right)</math>

Under what conditions is it possible to represent the aggregate demand in this way?

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which is a special case of the Gorman form.

Indeed, the marshallianMarshallian demand function for the nonlinear good of consumers with quasilinear utilities does not depend on the income at all (in this quasilinear case, the demand for the linear good is linear in income):

::<math>x_i(p, m) = (-\frac{dv(p)/dm)/(}{v(p)/dp_i)} = -\frac{1/(}{dv(p)/dp_i)} = (v_i')^ {-1}(p)= v_i'(p)^{-1}</math>

Hence, the aggregate demand function for the nonlinear good also does not depend on income:

::<math>X(p, M) = \sum_{i=1}^n{(v_i')^{-1}(p)}</math>

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::<math>(U')^{-1}(p) = \sum_{i=1}^n{(v_i')^{-1}(p)}</math>

In the special case in which all agents have the same utility function <math>u(x,m)=u(x)+m</math>, the aggregate utility function is:

::<math>U(x,M) = n \cdot u\left(\frac{x \over }{n}\right) + M</math>

=== [[Homothetic preferences]] ===