Characteristics

Abstract

If Eve had not insisted that ‘an apple is an apple is an apple’, Adam would probably have brought down the wrath of Jehovah himself by characterizing things, unsurprisingly, by their characteristics, thus bringing man-made order into chaos. There remains the problem: what characteristics? Here a little mathematics is useful. Suppose goods Y are sold only in bundles X, and that there are a_ij units of Y_j in X_i. The total quantity of it will be

$$ {y}_j\sum_i{a}_{ij}{x}_i, $$

and the total value of a bundle X_i

$$ {p}_i\sum_j{a}_{ij}{q}_j, $$

in an obvious notation, so that expenditure

$$ \sum_i{p}_i{x}_i=\sum_{ij}{a}_{ij}{q}_i{x}_i=\sum_j{q}_j{y}_j, $$

for all q, x, and expenditure, m, is invariant, a fact brought to the attention of young economists at large by Samuelson when he published his Foundations just after the war. That immediately suggested that we might instead consider the X as goods, thought of as bundles of characteristics Y. Were we to try ${y}_j={f}^j(x),{p}_i={g}^i(q)$, instead of Eqs. 1 and 2, the notion that total expenditure should be invariant would yield $m=\sum {g}^i(q){x}_i\equiv \sum {q}_j{f}^j(x)$, so that ${\partial}^2m/\partial {x}_i\partial {q}_j={f}_i^j(x)={g}_j^i(q)={a}_{ij}$ say, the subscripts denoting differentiation, and hence back to the linear characteristics model (Eqs. 1 and 2), which had already been used extensively, if implicitly, by Rowntree when studying working-class budgets in York before and after World War I, by Miss Schulz in her monthly ‘human needs’ budgets during World War II, by numerous nutritionists, and finally by Stigler, in a paper cited in Koopmans’ Cowles Commission monograph on activity analysis in 1951 as a precursor of linear programming, a topic very fashionable among young economists at the time.