#### Milestone Year

### 1964

#### Mathematical Economics

WIS scientists made significant contributions to mathematical economics;
three such contributions are described next.

Classical economic theory is based in an essential way on the assumption
that markets are "perfect". In particular, no small number of agents has a
significant effect on the market. Presumably, if the market is "big" then
the market does not differ significantly from a perfect one. In order to
make this notion precise one needs to obtain appropriate limit theorems.
In particular, a topology (notion of proximity) has to be introduced on
economic agents` characteristics, in particular their preference relations.
In 1964, a WIS scientist constructed a natural topology and obtained
sufficiently general limit theorems, leading to a fruitful line of research.

The "Law" of diminishing returns to scale was used by classical authors
as to apply to the amount of utility derived from consumption.
The notion that one could assign numerical values in an invariant manner
to utility was attacked and almost abandoned. Nonetheless,
as a WIS scientists showed in 1977, if only concave utility functions
are considered, then the situation changes. There are three "canonical"
ways of determining whether such a utility exists at all; and all
constructions lead to the same "canonical" (aka "cardinal") utility
function, and many classical concepts may be rehabilitated.

Various attempts had been made trying to extend a preference ordering from
a set (i.e, comparing pairs of alternatives in this set) to the power set
(i.e., comparing pairs of subsets of alternatives). In 1984, it was shown
by a WIS scientist and his visitor that no such extension exists,
under rather mild and natural assumptions.