Gorman polar form: Difference between revisions

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

=== [[Quasilinear utilities]] ===

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

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

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

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

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

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

Bothwhich areis clearlya special cases of the Gorman form.

Indeed, the demand functions of consumers with quasilinear utilities do not depend on the income at all:

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

::<math>vx_i(p, m^i) = v(v_i')^{-1}(p) + m</math>

Hence, the aggregate demand function 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:

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

where the function <math>U</math> satisfies the equality:

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

Both are clearly special cases of the Gorman form.

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

The indirect utility function has the form:

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

which is also a special cases of the Gorman form.

Particularly: linear, Leontief and Cobb-Douglas utilities are homothetic and thus have the Gorman form.

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