Gorman polar form: Difference between revisions

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== 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:<ref name=Alp>{{cite web | url=http://ocw.mit.edu/courses/economics/14-452-economic-growth-fall-2009/recitations/MIT14_452F09_rec2.pdf | title=Gorman's Aggregation Theorem | date=2009 | accessdate=2 December 2015 | author=Simsek, Alp}}</ref> consider an economy with <math>n</math> consumers, each of whom has a [[demand function]] that depends on his income <math>m^i</math> and the price system:

:<math>x^i(p,m^i)</math>

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

:<math>X(p,m^1,\dots,m^n) = \sum_{i=1}^n {x^i(p,m^i)}</math>

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:

:<math>X(p,m^1,\dots,m^n) = X(p, \sum_{i=1}^n {m^i}</math>

Under what conditions is it possible to represent the aggregate demand as a function of the price and the aggregate income?

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 curve]]s (expenditure as a function of income) should be parallel straight lines. This means that we can calculate an income-consumption curve of an entire society just by summing the curves of the consumers. In other words, suppose the entire society is given a certain income. This income is somehow distributed between the members of society, then each member selects his consumption according to his income-consumption curve. If the curves are all parallel straight lines, the aggregate demand of society will be ''independent of the distribution of income among the agents''.

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*{{cite journal |last=Honohan |first=Patrick |authorlink=J. Peter Neary |last2=Neary |first2=J. Peter |title=W. M. Gorman (1923–2003) |journal=The Economic and Social Review |volume=34 |issue=2 |year=2003 |pages=195–209 |url=http://www.esr.ie/Vol34_2Neary.pdf }}

*{{cite journal |last=Nataf |first=A. |year=1953 |title=Sur des questions d’agrégation en économétrie |journal=Publications de l’Institut de Statistique de l’Université de Paris |volume=2, Fasc. Vol. 4 |pages=5–61 }}

* {{cite web | url=http://ocw.mit.edu/courses/economics/14-452-economic-growth-fall-2009/recitations/MIT14_452F09_rec2.pdf | title=Gorman's Aggregation Theorem | date=2009 | accessdate=2 December 2015 | author=Simsek, Alp}}

[[Category:Utility function types]]