<p><a href="https://en.wikipedia.org/wiki/Turnpike_theory#:%7E:text=Turnpike%20theory%20refers%20to%20a,the%20initial%20and%20final%20levels." rel="nofollow noreferrer">Turnpike theory</a> uses fairly sophisticated mathematics to obtain results (which may appear surprising to many) about optimal economic growth over the long term.  The theory appears to have its origin in J V Neumann's 1945 paper <em>A Model of General Economic Equilibrium</em>. As an introduction to the topic I suggest pages 1-7 of <a href="https://arxiv.org/pdf/2202.04097.pdf" rel="nofollow noreferrer">Geshkovski B &amp; Zuazua E (2022) <em>Turnpike in Optimal Control of PDEs, Resnets and Beyond</em></a>. The paper also includes a detailed list of references.</p>

CC BY-SA 4.0
2023-05-18T18:30:21.097Z
I rarely refer on this site to my own blog, but readers may find my post [_The Maximum Duration of Constant Consumption_](https://economicdroplets.com/2021/06/28/the-maximum-duration-of-constant-consumption/) an interesting application of the turnpike property.
Adam Bailey
CC BY-SA 4.0
2023-05-18T18:16:09.370Z
2023-05-18T18:16:09.370Z
Adam Bailey
<p>If you're interested in a book, then this is a good one: Microeconomic Theory by Mascollel, Whinston and Green</p>
<p>But if you just want to show mathematicians the joy and potential of applying their skills to economic problems, you can ask them to solve optimization problems such as utility maximization, profit maximization, finding Pareto efficient allocations, and finding competitive equilibrium in different environments. You can find many examples on economics Stack Exchange. If the mathematician is unfamiliar with these concepts, you may need to define the concepts.</p>

CC BY-SA 4.0
2023-05-19T02:27:03.337Z
As a Mathematician starting a Masters in Economics, I can confirm :) +1
Nicolas Torres
CC BY-SA 4.0
2023-05-19T04:14:44.903Z
depending on your viewpoint, either john muth or robert lucas, revolutionized macro-economics through their idea of rational expectations. It was the idea that one can model expectations of an agent by assuming that the agent knows about the econometric model that he-she is directly involved in.
mark leeds
CC BY-SA 4.0
2023-05-18T23:00:15.790Z
2023-05-18T23:00:15.790Z
Amit
<p>In my opinion, <strong>General Economic Equilibrium</strong> is one the most important and beautiful theoretical constructions and achievements of applied mathematics of the past century and beyond.</p>
<p>Beginning with the pioneering work by Léon Walras, his successor in the chair at the University of Losanne Vilfredo Pareto, through Hicks, Samuelson, Arrow, Hahn, Debreu and many others, until present days. Hicks, Samuelson, Arrow, Debreu, are all Nobel laureates in economics.</p>
<p>The <strong>mathematical importance</strong>  of General Economic Equilibrium was acknowledged by <strong>Stephen Smale,</strong> a famous mathematician, Field  medalist in 1966, who was interested in economics.</p>
<p>He included a question of General Equilibrium, the problem of price formation, in his <strong>list of unresolved mathematical problems for the XXI century</strong>:</p>
<p><a href="https://link.springer.com/article/10.1007/BF03025291" rel="nofollow noreferrer">Stephen Smale, <em>Mathematical problems for the next century</em></a></p>
<p>His 8th problem was: <em><strong>Extend the mathematical model of general equilibrium theory to include price adjustments.</strong></em></p>
<p><a href="https://en.wikipedia.org/wiki/Smale%27s_problems" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Smale%27s_problems</a></p>
<p>Standard references of modern General Ecomomic Equilibrium are:</p>
<p>Debreu, G., <em>Theory of Value,</em> 1959, Yale University Press.</p>
<p>Arrow, K. J., Hahn, F.H, <em>General Competitive Analysis</em>, 1971 first edition, North Holland.</p>
<p><span class="math-container">$\;$</span></p>
<p>Debreu was influenced by Bourbaki,  and introduced Bourbakism into Economic Theory, with axiomatic method, topology, convexity, instead of analytical tools.
This approach  led also to controversies, see for instance:</p>
<p><a href="http://tankona.free.fr/weintraub1994.pdf" rel="nofollow noreferrer">Science in Context 7, 2 (1994), pp. 245-272
E. ROY WEINTRAUB AND PHILIP MIROWSKI
The Pure and the Applied:
Bourbakism Comes to
Mathematical Economics</a></p>
<p><span class="math-container">$\;$</span></p>
<p>On the opposite side of the abstract and theoretical field of General Equilibrium,  if we speak of more applied mathematicians, there is nowadays the vast field of <strong>Computational Economics</strong>, see for example the journal of the <em>Society for Computational Economics</em>:</p>
<p><a href="https://www.springer.com/journal/10614" rel="nofollow noreferrer">https://www.springer.com/journal/10614</a></p>

CC BY-SA 4.0
2023-05-29T20:49:25.437Z
As a Mathematician starting a Masters in Economics, General Equilibrium is the best I have encountered so far :) +1
Nicolas Torres
CC BY-SA 4.0
2023-05-18T20:25:44.270Z
2023-05-19T14:21:53.767Z
2023-05-19T14:21:53.767Z
BakerStreet
<p>Lucas and Rational Expectations. So elegant that it rebuilt macroeconomics, laying the foundations for &quot;modern macro&quot;.
<a href="https://julia.quantecon.org/multi_agent_models/rational_expectations.html" rel="nofollow noreferrer">https://julia.quantecon.org/multi_agent_models/rational_expectations.html</a></p>
<p>Robert E Lucas, Jr. and Edward C Prescott. Investment under uncertainty. Econometrica: Journal of the Econometric Society, pages 659–681, 1971.
<a href="http://pages.stern.nyu.edu/%7Edbackus/GE_asset_pricing/adjustment%20costs/LucasPrescott%20Econometrica%2071.pdf" rel="nofollow noreferrer">http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/adjustment%20costs/LucasPrescott%20Econometrica%2071.pdf</a></p>
<p>Two more:</p>
<ul>
<li><p>Search-theoretical models of money are a narrow field of application of the above, probably looking for mathematical minds
<a href="http://homepage.ntu.edu.tw/%7Eyitingli/file/Money/introduction_2020_new_r1.pdf" rel="nofollow noreferrer">http://homepage.ntu.edu.tw/~yitingli/file/Money/introduction_2020_new_r1.pdf</a></p>
</li>
<li><p>An algebraic topology formulation: not an &quot;application within economics&quot; but rather a linking from discrete math proof in economic theory (Arrow's impossibility theorem; social choice in MWG) to another area of math. So it's &quot;pure&quot; math!
<a href="https://www.sciencedirect.com/science/article/abs/pii/S0096300305002936" rel="nofollow noreferrer">https://www.sciencedirect.com/science/article/abs/pii/S0096300305002936</a></p>
</li>
</ul>
<p>meta: mathematical elegance (like Lucas) has done more damage to the reputation of the economics profession amongst social sciences and society, than helped move the discipline forward<br />
(belongs in a comment, but i'm not allowed)</p>

CC BY-SA 4.0
2023-06-20T22:54:14.653Z
2023-06-20T22:54:14.653Z
dcl04
<p>Could you recommend economic papers or books that exhibit exceptionally creative applications of mathematics? In other words, if you were to demonstrate to a mathematician the joy and potential of applying their skills to economic problems, what sources would you suggest?</p>

CC BY-SA 4.0
2023-05-19T11:11:15.653Z
How to best bring that mathematician to the dark side will depend very much on the individual.
Michael Greinecker
CC BY-SA 4.0
2023-05-19T11:48:04.583Z
Only when you come to the dark side can you find opportunities to lighten things up.
Amit
CC BY-SA 4.0
2023-05-19T22:21:44.660Z
Some of the waterpipe experiments on economics relied heavily on mathematical models
Richard
<microeconomics><mathematical-economics><reference-request><soft-question>
What are the most creative uses of Mathematics in Economics
Pavel Kocourek
# What are the most creative uses of Mathematics in Economics

(Asked by: username_0 on 2023-05-18)

Could you recommend economic papers or books that exhibit exceptionally creative applications of mathematics? In other words, if you were to demonstrate to a mathematician the joy and potential of applying their skills to economic problems, what sources would you suggest?

**Tags:** microeconomics, mathematical-economics, reference-request, soft-question

> **Comments:**
> **username_1**: How to best bring that mathematician to the dark side will depend very much on the individual.  
> **username_2**: Only when you come to the dark side can you find opportunities to lighten things up.  
> **username_3**: Some of the waterpipe experiments on economics relied heavily on mathematical models  

---

## 4 answers

### Answer by username_4 on 2023-05-18. Score: 4

[Turnpike theory](https://en.wikipedia.org/wiki/Turnpike_theory#:%7E:text=Turnpike%20theory%20refers%20to%20a,the%20initial%20and%20final%20levels.) uses fairly sophisticated mathematics to obtain results (which may appear surprising to many) about optimal economic growth over the long term. The theory appears to have its origin in J V Neumann's 1945 paper *A Model of General Economic Equilibrium*. As an introduction to the topic I suggest pages 1-7 of [Geshkovski B & Zuazua E (2022) *Turnpike in Optimal Control of PDEs, Resnets and Beyond*](https://arxiv.org/pdf/2202.04097.pdf). The paper also includes a detailed list of references.

> **Comments:**
> **username_4**: I rarely refer on this site to my own blog, but readers may find my post [_The Maximum Duration of Constant Consumption_](https://economicdroplets.com/2021/06/28/the-maximum-duration-of-constant-consumption/) an interesting application of the turnpike property.  

---

### Answer by username_2 on 2023-05-18. Score: 3

If you're interested in a book, then this is a good one: Microeconomic Theory by Mascollel, Whinston and Green

But if you just want to show mathematicians the joy and potential of applying their skills to economic problems, you can ask them to solve optimization problems such as utility maximization, profit maximization, finding Pareto efficient allocations, and finding competitive equilibrium in different environments. You can find many examples on economics Stack Exchange. If the mathematician is unfamiliar with these concepts, you may need to define the concepts.

> **Comments:**
> **username_5**: As a Mathematician starting a Masters in Economics, I can confirm :) +1  
> **username_6**: depending on your viewpoint, either john muth or robert lucas, revolutionized macro-economics through their idea of rational expectations. It was the idea that one can model expectations of an agent by assuming that the agent knows about the econometric model that he-she is directly involved in.  

---

### Answer by username_7 on 2023-05-18. Score: 3

In my opinion, **General Economic Equilibrium** is one the most important and beautiful theoretical constructions and achievements of applied mathematics of the past century and beyond.

Beginning with the pioneering work by Léon Walras, his successor in the chair at the University of Losanne Vilfredo Pareto, through Hicks, Samuelson, Arrow, Hahn, Debreu and many others, until present days. Hicks, Samuelson, Arrow, Debreu, are all Nobel laureates in economics.

The **mathematical importance** of General Economic Equilibrium was acknowledged by **Stephen Smale,** a famous mathematician, Field medalist in 1966, who was interested in economics.

He included a question of General Equilibrium, the problem of price formation, in his **list of unresolved mathematical problems for the XXI century**:

[Stephen Smale, *Mathematical problems for the next century*](https://link.springer.com/article/10.1007/BF03025291)

His 8th problem was: ***Extend the mathematical model of general equilibrium theory to include price adjustments.***

<https://en.wikipedia.org/wiki/Smale%27s_problems>

Standard references of modern General Ecomomic Equilibrium are:

Debreu, G., *Theory of Value,* 1959, Yale University Press.

Arrow, K. J., Hahn, F.H, *General Competitive Analysis*, 1971 first edition, North Holland.

$\;$

Debreu was influenced by Bourbaki, and introduced Bourbakism into Economic Theory, with axiomatic method, topology, convexity, instead of analytical tools.
This approach led also to controversies, see for instance:

[Science in Context 7, 2 (1994), pp. 245-272
E. ROY WEINTRAUB AND PHILIP MIROWSKI
The Pure and the Applied:
Bourbakism Comes to
Mathematical Economics](http://tankona.free.fr/weintraub1994.pdf)

$\;$

On the opposite side of the abstract and theoretical field of General Equilibrium, if we speak of more applied mathematicians, there is nowadays the vast field of **Computational Economics**, see for example the journal of the *Society for Computational Economics*:

<https://www.springer.com/journal/10614>

> **Comments:**
> **username_5**: As a Mathematician starting a Masters in Economics, General Equilibrium is the best I have encountered so far :) +1  

---

### Answer by username_8 on 2023-06-20. Score: 1

Lucas and Rational Expectations. So elegant that it rebuilt macroeconomics, laying the foundations for "modern macro".
<https://julia.quantecon.org/multi_agent_models/rational_expectations.html>

Robert E Lucas, Jr. and Edward C Prescott. Investment under uncertainty. Econometrica: Journal of the Econometric Society, pages 659–681, 1971.
[http://pages.stern.nyu.edu/~dbackus/GE\_asset\_pricing/adjustment%20costs/LucasPrescott%20Econometrica%2071.pdf](http://pages.stern.nyu.edu/%7Edbackus/GE_asset_pricing/adjustment%20costs/LucasPrescott%20Econometrica%2071.pdf)

Two more:

* Search-theoretical models of money are a narrow field of application of the above, probably looking for mathematical minds
  [http://homepage.ntu.edu.tw/~yitingli/file/Money/introduction\_2020\_new\_r1.pdf](http://homepage.ntu.edu.tw/%7Eyitingli/file/Money/introduction_2020_new_r1.pdf)
* An algebraic topology formulation: not an "application within economics" but rather a linking from discrete math proof in economic theory (Arrow's impossibility theorem; social choice in MWG) to another area of math. So it's "pure" math!
  <https://www.sciencedirect.com/science/article/abs/pii/S0096300305002936>

meta: mathematical elegance (like Lucas) has done more damage to the reputation of the economics profession amongst social sciences and society, than helped move the discipline forward  
(belongs in a comment, but i'm not allowed)


---

