consumer in the economy. The existence of such a representative agents has been investigated by Muellbauer (1975, 1976). In many ways, the conditions to make this story work are less restrictive than those for the original representative consumer, and indeed Muellbauer’s representative consumer has a representative income that depends on the distribution of income as well as on its mean. However, it is unclear whether the additional complexity of these formulations would commend them to those seeking straightforward interpretations of the COLI concept of a price index.
Unless we place restrictions on the distribution of income, the existence of a representative agent requires that individual h has preferences that can be represented by cost functions of the Gorman “polar form” (Gorman, 1959):
ch(uh, p)=ah(p)+ uhb(p), (1)
where uh is utility, p is a vector of prices, and ah(p) and b(p) are nonnegative linearly homogeneous and quasi-concave functions of p. Taste variation is permitted in the function ah(p) but not in b(p). The representative agent has a cost function that is the average of (1), which is
c(u, p) =a¯(p) + ub(p). (2)
Denote by xh the total expenditure of h.
Suppose that the two price vectors to be compared in the COLI are p1 and p0. The base-period COLI for h, is written
while the representative consumer’s COLI is
where is the population mean of xh0. Straightforward computation then confirms that
so that the representative consumer’s COLI is the plutocratic average of the individual COLIs,