# What are some important mathematics results that were first developed in Economics?

Economics is known to import mathematical methods that were proven to be useful in other areas.

There is any important result in mathematics that was first developed in the context of Economics research and then explored in other applied fields such as Physics or Engineering?

I know a lot of examples of mathematical results that have been first developed in economics, mostly result in set-valued analysis and convex analysis. My ignorance of engineering and physics keeps me from listing many examples there. Many results in optimal transport have certainly been used in many other areas.

The Kakutani fixed-point theorem should also make the list in my humble opinion. Now Shizuo Kakutani was a pure mathematician who published this purely mathematical result in a pure math journal:

Kakutani, Shizuo. "A generalization of Brouwer’s fixed point theorem." Duke mathematical journal 8.3 (1941): 457-459.

So why should it count? Well, the Kakutani fixed point theorem is largely a variant (with a simplified proof) of a Lemma John von Neumann introduced in Karl Menger's mathematical colloquium in order to solve an economic growth model.

Von Neumann, John. "Uber ein ökonomisches Gleichungssystem." Ergebn. Math. Kolloq. Wien. Vol. 8. 1937.

One can actually show the existence of solutions to the model by simpler methods, as David Gale eventually did, but here is the result of von Neumann in modern language:

Theorem: Let $C$ and $K$ be nonempty convex and compact subsets of Euclidean spaces (of not necessarily the same dimension). Let $E$ and $F$ be closed subsets of $C\times K$ such that for each $x\in C$, the set $E_x=\{y\in K\mid (x,y)\in E\}$ is nonempty, convex, and compact and such that for each $y\in K$, the set $F^y=\{x\in C:(x,y)\in F\}$ is nonempty, convex, and compact. Then $E\cap F\neq\emptyset$.

That Kakutani's fixed point theorem implies the result of von Neumann is shown in Kakutani's paper. But Kakutani's fixed point theorem is also an easy consequence of von Neumann's theorem. If $C=K$, then the assumptions on $E$ say that $E$ is the graph of an upper hemicontinuous correspondence with nonempty, convex, and compact values from $C$ to itself. That this correspondence has a fixed-point is equivalent to a point of the form $(x,x)$ being in the graph. Now if we let $F$ being the diagonal of $C$, that is $F=\{(x,x)\mid x\in C\}$, then $E$ and $F$ satisfy the conditions of von Neumann's result, so the intersection must be empty and the correspondence with graph $E$ must have a fixed-point.

A reprint of von Neumann's paper can be found in this book. An English translation can be found here.

The Envelope theorem is arguably one, https://en.wikipedia.org/wiki/Envelope_theorem . I would say Blackwell's papers on comparisons of experiments are a significant mathematical contribution--extending Hardy,Littlewood and Polya's earlier results, and paving the way for further investigation.

Econometricians have made statistical contributions as well; see e.g. https://en.wikipedia.org/wiki/Heckman_correction .

More recently, I saw a paper presented last year on how a certain type of stochastic PDE could be solved by reinterpreting it as a (stochastic) differential game. I would expect to see more related to this (the relationship between differential games and stochastic pdes) in the coming years, especially given the recent surge of interest in mean-field games.

• The comparison of experiments result is a result of statistics proven by a statistician. – Michael Greinecker Mar 13 '18 at 8:37
• @MichaelGreinecker This is not strictly true. Blackwell extended the results of Hardy, Littlewood and Polya to finite dimensional spaces. Around the same time, Stein and Sherman proved related results, and, among other things, answered questions posed by Blackwell. Blackwell certainly made a significant contribution himself though. – user11305 Mar 13 '18 at 11:19
• Later on these results were extended to various infinite dimensional settings. – user11305 Mar 13 '18 at 11:20
• Actually, Blackwell referred to a Rand report by Bohnenblust, Shapley, Sherman in game theory (but not economics), not to Hardy, Littlewood, and Polya. And Blackwell was a statistician, not an economist. The result has actually been rediscovered in a variety of different contexts. – Michael Greinecker Mar 13 '18 at 11:46
• @MichaelGreinecker I am simply attributing credit in the same manner that the literature does. Moreover, in both Blackwell (1951) and (1953), he refers to both HLP and BSS (as well as Stein and Sherman in the first of his papers). – user11305 Mar 13 '18 at 12:06