Gorman polar form: Difference between revisions

Revision as of 10:34, 12 October 2022

Revision as of 20:26, 29 August 2023

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Indeed, the Marshallian 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]] ===