<p>I would like to add two additional points: </p>

<p>The Arrow-Endhoven-Theorem which is clearly based on economic considerations and a stronger version of the Karush-Kuhn-Tucker Theorem (which is not based on economics). </p>

<p>Another contribution which is at least partly based on economics (but not based on a pure economist) is the proof of the Mini-Max-Theorem by von Neumann (but as far as I know this result is widely ignored today - or not?).    </p>

<p>p. s. I don't want to argue about the Envelope Theorem but I think I saw a statement somewhere that the basic theorem has it's origin somewhere else. I will look into this later on.   </p>

CC BY-SA 3.0
2018-03-13T07:57:41.650Z
2018-03-13T07:57:41.650Z
PAS
<p>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 <a href="https://en.wikipedia.org/wiki/Transportation_theory_(mathematics)" rel="noreferrer">optimal transport</a> have certainly been used in many other areas.</p>

<p>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:</p>

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

<p>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.</p>

<blockquote>
  <p>Von Neumann, John. "Uber ein ökonomisches Gleichungssystem." Ergebn.
  Math. Kolloq. Wien. Vol. 8. 1937.</p>
</blockquote>

<p>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:</p>

<p><strong>Theorem:</strong> 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 $E\cap F\neq\emptyset$.</p>

<p>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.</p>

<p>A reprint of von Neumann's paper can be found in <a href="http://www.springer.com/in/book/9783709173305" rel="noreferrer">this book</a>. An English translation can be found <a href="http://piketty.pse.ens.fr/files/VonNeumann1945.pdf" rel="noreferrer">here</a>.</p>

CC BY-SA 3.0
2018-03-14T10:49:36.180Z
2018-03-14T10:49:36.180Z
Michael Greinecker
<p>Economics is known to import mathematical methods that were proven to be useful in other areas.</p>

<p>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? </p>

<reference-request><mathematical-economics>
What are some important mathematics results that were first developed in Economics?
psn
# What are some important mathematics results that were first developed in Economics?

(Asked by: username_0 on 2018-03-13)

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?

**Tags:** reference-request, mathematical-economics

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## 2 answers

### Answer by username_2 on 2018-03-14. Score: 6 | Accepted answer

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](https://en.wikipedia.org/wiki/Transportation_theory_(mathematics)) 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](http://www.springer.com/in/book/9783709173305). An English translation can be found [here](http://piketty.pse.ens.fr/files/VonNeumann1945.pdf).


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### Answer by username_1 on 2018-03-13. Score: 2

I would like to add two additional points:

The Arrow-Endhoven-Theorem which is clearly based on economic considerations and a stronger version of the Karush-Kuhn-Tucker Theorem (which is not based on economics).

Another contribution which is at least partly based on economics (but not based on a pure economist) is the proof of the Mini-Max-Theorem by von Neumann (but as far as I know this result is widely ignored today - or not?).

p. s. I don't want to argue about the Envelope Theorem but I think I saw a statement somewhere that the basic theorem has it's origin somewhere else. I will look into this later on.


---

