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 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]] as(the amoney functionrequired ofto pricereach (<math>a ecertain ^utility ilevel \leftin (a pcertain \rightprice ) </math>system) must be anlinear [[affine transformation]] ofin the utility function, with coefficients that depend on the price vector:

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