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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?

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    $\begingroup$ How to best bring that mathematician to the dark side will depend very much on the individual. $\endgroup$ May 19, 2023 at 11:11
  • $\begingroup$ Only when you come to the dark side can you find opportunities to lighten things up. $\endgroup$
    – Amit
    May 19, 2023 at 11:48
  • $\begingroup$ Some of the waterpipe experiments on economics relied heavily on mathematical models $\endgroup$
    – Richard
    May 19, 2023 at 22:21

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Turnpike theory 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. The paper also includes a detailed list of references.

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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.

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    $\begingroup$ As a Mathematician starting a Masters in Economics, I can confirm :) +1 $\endgroup$ May 19, 2023 at 2:27
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    $\begingroup$ 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. $\endgroup$
    – mark leeds
    May 19, 2023 at 4:14
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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

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.

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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

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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

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    $\begingroup$ As a Mathematician starting a Masters in Economics, General Equilibrium is the best I have encountered so far :) +1 $\endgroup$ May 29, 2023 at 20:49
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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

Two more:

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)

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