Gorman polar form: Difference between revisions

Revision as of 08:04, 21 December 2015

Revision as of 13:11, 11 February 2016

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When the utility function of agent <math>i</math> has the form:

::<math>u_i(x, m) = u_i(x) + m</math>

the indirect utility function has (assuming an interior solution) the form:

::<math>v_i(p, m) = v_i(p) + m</math>

which is a special cases of the Gorman form.

Indeed, the marshallian demand functionsfunction for the nonlinear good of consumers with quasilinear utilities dodoes 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^i) = (-dv(p)/dm)/(v(p)/dp_i) = -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>

The entire society can be represented by a single representative agent with quasilinear utility function: