3. In a signal contribution, Paul A. Samuelson (1938)
presented a theory of choice based, not on the com-
parison between two baskets, but on observable budget
data. His point of departure is that John, by choosing
the budget distribution M, reveals that he prefers M
to any other distribution (such as M′) compatible with
his budget. To this transparent definition, Samuelson
added only an equally transparent axiom:
If a budget
reveals that the basket A is preferred to B, no budget
can reveal that B is preferred to A. Samuelson claimed
that this axiom alone suffices for deriving by integration
the indifference varieties and hence for constructing
an ophelimity function. In fact, the axiom expresses
only a condition equivalent to the Principle of De-
creasing Marginal Rate of Substitution. And as shown
first by Jean Ville (1946) and later, but independently,
by H. S. Houthakker (1950), Samuelson's idea calls for
a stronger axiom (analogous to the transitivity of binary
choice). But soon thereafter it was proved that even
this stronger axiom does not entail the existence of an
ophelimity function. It still leaves large domains for
which there is no comparability among the commodity
baskets (Georgescu-Roegen, 1954b). The problem of
what set of economically meaningful postulates would
make the Antonelli-Pareto idea work still awaits its
solution. If it is ever solved, it will very probably cause
a greater stir in mathematics than in economics.
