Gorman polar form: Difference between revisions

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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. Suppose an individual <math>i</math> has a utility function <math> u^i</math>. Then, his [[expenditure function]] (the money required to reach a certain utility level in a certain price system) must be linear in the utility:

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

== Gorman's form of the indirect utility function ==

== Definition ==

Inverting this formula gives the [[indirect utility function]] (utility as a function of price and income):

:<math> v^i \left (p,m^i \right ) = \frac {m^i-f^i(p)}{g(p)} </math>,