# University of Virginia Library

#### V

1. Before the emergence of the modern school of
utility, thoughts on value, demand, and exchange ordi-
narily reflected the economic conditions prevailing
during each period. A turning point in this respect took
place following the process of mathematization fos-
tered by that school. Utility theory has ever since been
its own source of new ideas, suggested primarily (and,
at times, exclusively) by its mathematical framework.
An excellent illustration is the observation made by
Irving Fisher in his doctoral dissertation (1892). The
pure geometry of Edgeworth's diagram led Fisher to
note that in order to determine the optimal budget
distribution we do not need to know how many utils
each isoline represents: the knowledge of the isolines
as such suffices. This simple geometrical truth caused
the first serious dent in the idea that a cardinally
measurable utility is indispensable for explaining value.

It was, however, Vilfredo Pareto (Manuale di econo-
mia politica,
1905) who first constructed a consumer
theory which does not require the notion of utility at
all. His point of departure is that an individual con-
fronted with two baskets of commodities will always
either prefer one basket or be indifferent as to which
one he gets. Given this faculty of binary choice, Pareto
reasoned that, by asking the individual to choose be-
tween M and every other possible basket, we can
determine an indifference curve, i.e., a curve that rep-
resents the loci of all baskets “indifferent” in relation
to M. The procedure does not refer in any way to
utility. And once the indifference curves are deter-
mined, they help determine the optimal distribution
of any budget in exactly the same manner as the utility
isolines. Furthermore, we can construct a function
V(x1,x2,..., xn) such that its value is constant on each
indifference curve, just as the utility function
U(x1,x2,..., xn) is constant on each isoline. The only
difference is that V is not uniquely determined—any
increasing function of V, say V2, would do.

It is for the function V that Pareto coined the term
“ophelimity.” But, as was argued in subsequent devel-
opments, we may still speak of utility and of V as its
ordinal, instead of cardinal, measure. This means that
the value of V simply orders all baskets according to
the individual's preferences. Today the notion of an
ordinal utility dominates consumer theory, the central
problem of which is how to derive an ophelimity
function from directly observable budget data.

2. In fact, this problem is relatively old. It was first
formulated in a neglected memoir of an Italian engi-
neer, G. Antonelli (1886). And, as happens quite often,
the glory went to the more famous rediscoverer of the
idea, in this case to Pareto (1905). In simple terms,
every possible budget has been determined by obser-
vation, every indifference curve can be determined by
the tangential artifice shown in Figure 2 for C3. But
he ignored the fact that this artifice (which in mathe-
matics is called “integration”) is not always available
for more than two commodities. An obvious para-
dox—known as the integrability problem—thus arose
to intrigue many a mathematical economist. Some light
was cast on Pareto's theory of choice and the integra-
bility problem when it was shown (Georgescu-Roegen,
1936) that Pareto's argument failed to include two
axioms (1) that any commodity may be substituted for
another so that the first and the second basket be
completely indifferent, and (2) that the binary choice
is transitive. (Choice is transitive if A being chosen over
B and B over C, A is chosen over C.)

3. In a signal contribution, Paul A. Samuelson (1938)

457

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.