Duality
Over and above his substantive contributions, a recurring theme in Gorman's writings was the need to select the appropriate technical tools for the problem at hand. Typically, this meant using “dual” tools, functions defined over prices rather than quantities.
As households and firms typically take prices as given, it is much easier to understand their behaviour in terms of expenditure, cost and profit functions than in terms of primal utility and production functions. The latter only take account of tastes and technology, the former add optimising behaviour. Gorman was not alone in advocating this approach, but he was one of its most ardent proponents. The great virtue of duality is that it avoids matrix inversion, which he called ‘the only technically difficult operation in general equilibrium theory' (Gorman 1986a). Even a cursory comparison between modern textbooks and Hicks's Value and Capital or Samuelson's Foundations shows how much more powerful are dual methods.A nice example of the value of the dual approach was Gorman's contribution to the issue of household equivalence scales. Such scales, which attempt to correct consumption patterns for differences in household composition, had been used for years in applied budget studies, though without any theoretical foundation. Barten (1964) pioneered the exploration of such scales in the context of utility theory. But Barten used the primal approach, expressing utility as a function of consumption per “equivalent adult”, where the scale which determines equivalence varies between commodities. Gorman (1976) argued that the insight of an ‘otherwise obtuse' schoolmaster he once had put it better: ‘When you have a wife and baby, a penny bun costs threepence' (ibid.: 215). Leaving aside the banality (and, to a modern ear, the sexism) of the aphorism, Gorman noted that it gets to the heart of the issue: differences in household composition are better thought of as altering the effective prices which must be paid, rather than the effective number of consumers. This approach, implemented using the expenditure function, led to a substantial simplification and extension of Barten's results.[170]
Expenditure and profit functions are usually the appropriate tools.
However, in some problems, quantities may be the exogenous variables. In such cases, the appropriate technical tool is the distance function, defined implicitly as the scalar by which an arbitrary consumption bundle must be deflated to yield a target level of utility: u[ql'd(q,u0)] = u0. This can be viewed as the natural inverse of the direct utility function. But it also turns out to bear a dual relationship to the expenditure function. Just as (by Shephard’s Lemma) the price derivatives of the expenditure function equal the optimal quantities, so the quantity derivatives of the distance function equal the optimal shadow prices. Gorman developed this concept in full, independently of others. In Gorman (1965a), he gave what appears to be the first statement of the duality between cost and distance functions, while in Gorman (1970a) he examined the properties of the distance function in detail. These papers however remained unpublished, so modern treatments typically give precedence to Debreu (1951) and Malmquist (1953) and pass over Gorman’s pioneering explorations.Gorman’s emphasis in all this was on the need for careful thought about which theoretical tools were appropriate for a particular problem. As he wrote in unpublished notes for a 1986 seminar at University College Dublin, doing economic theory ‘is like eating an apple pie. If you know there is one in the fridge, and where the light switches are, there is nothing to it. Look around when you next visit a strange house, in case you should feel hungry in the night’ (Gorman 1986b).
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