Gorman polar form: Difference between revisions

Revision as of 15:31, 19 November 2015

Revision as of 20:24, 1 December 2015

Line 18:

:<math> v^i \left (p,m^i \right ) = \frac {m^i-f^i(p)}{g(p)} </math>,

where <math>m</math> is the amount of income available to the individual and is equivalent to the expenditure (<math>e^i \left (p,u^i \right )</math>) in the previous equation. This is what Gorman called “the polar form of the underlying utility function.” Gorman's use of the term ''polar'' was in reference to the idea that the indirect utility function can be seen as using polar rather than Cartesian (as in direct utility functions) coordinates to describe the indifference curve. Here, income (<math>m^i</math>) is analogous to the radius and prices (<math>p</math>) to an angle.

== Examples ==

Two types of preferences that have the Gorman polar form are:<ref name=Varian>{{Cite Varian Microeconomic Analysis 3}}</ref>{{rp|154}}

1. [[Homothetic preferences]] (particularly: linear, Leontief and Cobb-Douglas). The indirect utility function has the form:

::<math>v(p, m) = v(p)\cdot m</math>

2. [[Quasilinear utilities]]. The indirect utility function has the form:

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

Both are clearly special cases of the Gorman form.

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