Gorman polar form: Difference between revisions

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'''Gorman polar form''' is a [[functional form]] for [[indirect utility function]]s in [[economics]]. Imposing this form on [[utility]] allows the researcher to treat a society of utility-maximizers as if it consisted of a single [[Representative agent|'representative' individual]]. [[W. M. Gorman|Gorman]] showed that having the [[function (mathematics)|function]] take Gorman polar form is both [[necessary and sufficient]] for this condition to hold.

== 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 | accessdateaccess-date=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>vx^i(p, m^i) = v(p) + m</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\left(p, \sum_{i=1}^n m^i \right)</math>

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 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''.

== Gorman's form of the expenditure function ==

Gorman's first published paper in 1953 developed these ideas in order to answer the question of representing a society by a single individual. In 1961, Gorman published a short, four-page paper in ''Metroeconomica'' which derived an explicit expression for the functional form of preferences which give rise to linear Engel curves. The [[expenditure function]] of each consumer <math>i</math> (the amount of money required to reach a certain utility level in a certain price system) must be linear in utility:

:<math> e^i \left (p, u^i \right ) = f^i(p) + g(p) \cdot u^i </math>,

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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>

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 case of the Gorman form.

Indeed, the Marshallian demand function for the nonlinear good of consumers with quasilinear utilities does 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) = -\frac{dv(p)/dm}{v(p)/dp_i} = -\frac{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:

::<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>

In the special case in which all agents have the same utility function <math>u(x,m)=u(x)+m</math>, the aggregate utility function is:

::<math>U(x,M) = n \cdot u\left(\frac{x}{n}\right) + M</math>

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

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

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

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

which is also a special case of the Gorman form.

BothParticularly: linear, Leontief and Cobb-Douglas utilities are clearlyhomothetic specialand casesthus ofhave the Gorman form.

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

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== Application ==

Many applications of Gorman polar form are summarized in various texts and in Honohan and Neary's article.<ref>{{cite citedjournal at|last=Honohan the|first=Patrick end|author-link=J. ofPeter thisNeary article|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 |url-status=dead |archive-url=https://web.archive.org/web/20050110001924/http://www.esr.ie/Vol34_2Neary.pdf |archive-date=2005-01-10 }}</ref> These applications include the ease of estimation of <math>f^i(p)</math> and <math>g(p)</math> in certain cases. But the most important application is for the theorist of economics, in that it allows a researcher to treat a society of utility-maximizing individuals as a single individual. In other words, under these conditions a community [[indifference curve|indifference mapping]] is guaranteed to exist.

== See also ==

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== References ==

{{Reflist}}

*{{cite book |last=Antonelli |first=G. B. |year=1886 |title=Sulla Teoria Matematica dell’Economiadell'Economia Politica |___location=Pisa }} English translation in {{cite book |editor1-first=J. S. |editor1-last=Chipman |editor2-first=L. |editor2-last=Hurwicz |editor3-first=M. K. |editor3-last=Richter |editor4-first=H. F. |display-editors = 3 |editor4-last=Sonnenschein |title=Preferences, Utility and Demand: A Minnesota Symposium |___location=New York |publisher=Harcourt Brace Jovanovich |year=1971 |pages=333–360 |isbn= }}

*{{cite journal |first=W. M. |last=Gorman |title=On a class of preference fields |journal=Metroeconomica |volume=13 |year=1961 |pages=53–56 |doi= }}

*{{cite journal |last=Honohan |first=PatrickW. |authorlink=JM. Peter Neary |last2last=NearyGorman |first2title=J.On Petera |title=W.class M.of Gormanpreference (1923–2003)fields |journal=The Economic and Social ReviewMetroeconomica |volume=3413 |issue=2 |year=20031961 |pages=195–20953–56 |urldoi=http: 10.1111//wwwj.esr1467-999X.ie/Vol34_2Neary1961.pdf tb00819.x}}

*{{cite journal |last=Nataf |first=A. |year=1953 |title=Sur des questions d’agrégationd'agrégation en économétrie |journal=Publications de l’Institutl'Institut de Statistique de l’Universitél'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]]

[[Category:Consumer theory]]