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,
Pareto's idea was this: if the optimal distribution of
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.)
