3.1 Aggregation

In his first published paper, Gorman (1953), he provided the definitive answer to a key question in economics: when does a society of utility-maximising individuals behave as if it was a single individual? In other words, when does a community indifference map exist? He showed that a necessary and suffi­cient condition is that, assuming all individuals face the same prices, their income-consumption or Engel curves should be parallel straight lines.

The location of the h superscripts on the right-hand side is crucial. Individuals can differ greatly in their responses to price changes as far as the f^h functions are concerned. However, their differences must be independent of income (or utility): all individuals must have the same g_i function, so that at the margin they have identical responses to changes in u. Hence, aggregate demands have the same form as (21.1):

where X_i, F_i and U are the sums over all individuals of the corresponding micro terms.

Gorman returned to this question in Gorman (1961a), now using the much more powerful tools of duality which he and others had developed in the interim. This short paper is bedtime reading by contrast with the 1953 paper, yet it contains what is probably his best-known contribution. Here Gorman derived an explicit expression for the form of preferences which give rise to linear Engel curves. He showed that individual h’s expenditure func­tion must take the simple form:

where the functions f^h(p) and g(p) are homogeneous of degree one in prices (so ensuring that this property is exhibited by the expenditure function itself), and their derivatives equal the coefficients in (21.1).

which Gorman called ‘the polar form of the underlying utility function' (ibid.: 54). With this unconventional term, Gorman was drawing attention to the fact that using what we would now call the indirect utility function amounts to switching from Cartesian to a form of polar coordinates in describing the indifference surface. Specifically, expenditure I may be taken as analogous to the radius and the vector of prices p to the angle in solid geometry. In any case, the term “Gorman polar form” has come to be universally applied to the functional form in (21.4).^[158]

By construction, the Gorman polar form plays a central role in consumer theory, and it has also been hugely important in empirical work. On the one hand, special cases with particular functional forms for f^h(p) and g(p) proved amenable to estimation, even before the advent of high-speed computers. Gorman himself showed that, if the marginal propensities to consume (which equal pg/g) are constant, then the function g(p) can be written as a geometric mean of prices:

The linear expenditure system, developed by R.C. Geary amongst others, is a further special case, corresponding to the combination of (21.5) with a lin­ear form for fp).^[159] On the other hand, Gorman's results did not prove a barrier to extending the theory to more general demand systems which avoid the implausible restrictions on income effects of (21.3).

the sum of individual demands, was replaced by the weaker requirement that they generate only the same budget shares. This in turn has spawned a huge empirical literature applying members of Muellbauer’s family and its exten­sions, such as the “Almost Ideal Demand System” of Deaton and Muellbauer (1980).

Gorman (1968a) also explored the conditions that must be satisfied for the existence of an aggregate stock of a fixed factor such as capital. The necessary and sufficient condition turns out to be formally very similar to that for aggre­gation of demands over individual consumers. Each firm must have a restricted profit function similar in form to (21.3), where utility is replaced by a func­tion of the amount of capital used by the firm. In his own words, this result ‘certainly does not help justify the practice of fitting aggregate production functions’ (ibid.: 167). This contribution of Gorman to the capital theory controversies of the 1960s lacked the fireworks of those that emanated from the two Cambridges (England and Massachusetts), but it is probably of more lasting importance.

3.1