Gorman polar form: Difference between revisions

Revision as of 13:02, 19 November 2015

Revision as of 15:26, 19 November 2015

Line 2:

== 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?

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 should be parallel straight lines. Gorman's first published paper in 1953 developed these ideas in order to answer the question of generalizing a society to a single individual.

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''.

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. Briefly, an individual's (<math>i</math>) resulting expenditure function (<math> e ^ i \left ( p , u ^ i \right ) </math>) must be [[affine transformation|affine]] with respect to utility (<math>u</math>):

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. Briefly,Suppose an individual's (<math>i</math>) resultinghas a utility function <math> u^i</math>. Then, his expenditure function as a function of price (<math> e ^ i \left ( p , u ^ i \right ) </math>) must be an [[affine transformation|affine]] of the utility function, with respectcoefficients tothat utilitydepend (<math>u</math>)on the price vector:

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

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

where both <math>f^i \left (p \right )</math> and <math>g \left (p \right )</math> are [[Homogeneous function|homogeneous]] of degree one in prices (<math>p</math>, a vector). This homogeneity condition is trivial, as otherwise <math>e^i \left (p,u^i \right )</math> would not give linear Engel curves.

where both <math>f^i \left (p \right )</math> and <math>g \left (p \right )</math> are [[Homogeneous function|homogeneous]] of degree one in prices (<math>p</math>, a vector). This homogeneity condition ensures that <math>e^i \left (p\right )</math> gives linear Engel curves.

<math>f^i \left (p \right )</math> and <math>g \left (p \right )</math> have nice interpretations: <math>f^i \left (p \right )</math> is the expenditure needed to reach a reference utility level of zero for each individual (<math>i</math>), while <math>g \left (p \right )</math> is the price index which deflates the excess money income <math>e^i \left (p,u^i \right ) - f^i (p)</math> needed to attain a level of utility <math>\bar{u}</math>. It is important to note that <math>g \left (p \right )</math> is the same for every individual in a society, so the Engel curves for all consumers are parallel.

== Definition ==