Gorman polar form

Gorman polar form is a functional form for indirect utility functions in economics.

Motivation

Standard consumer theory is developed for a single consumer. The consumer has a utility function, from which his demand curves can be calculated. Then, it is possible to predict the behavior of the consumer in certain conditions, price or income changes. But in reality, there are many different consumers, each with his own utility function and demand curve. How can we use consumer theory to predict the behavior of an entire society? One option is to represent an entire society as a single "mega consumer", which has an aggregate utility function and aggregate demand curve. But in what cases is it indeed possible to represent an entire society as a single consumer?

Formally: consider an economy with

n

{\displaystyle n}

consumers, each of whom has a demand function that depends on his income

m

i

{\displaystyle m^{i}}

and the price system:

x

i

(

p

,

m

i

)

{\displaystyle x^{i}(p,m^{i})}

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

X

(

p

,

m

1

,

…

,

m

n

)

=

∑

i

=

1

n

x

i

(

p

,

m

i

)

{\displaystyle X(p,m^{1},\dots ,m^{n})=\sum _{i=1}^{n}x^{i}(p,m^{i})}

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:

X

(

p

,

m

1

,

…

,

m

n

)

=

X

(

p

,

∑

i

=

1

n

m

i

)

{\displaystyle X(p,m^{1},\dots ,m^{n})=X\left(p,\sum _{i=1}^{n}m^{i}\right)}

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

Early results by Antonelli (1886) and Nataf (1953) had shown that, assuming all individuals face the same prices in a market, their income consumption curves and their Engel curves (expenditure as a function of income) should be parallel straight lines.