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As a math student interested in economics I was recently reading some of the papers by Graciela Chichilnisky on social choice theory and some of her papers applying topology to other areas of economics. Does anyone know of other economists engaged in similar projects? That is, would anyone be able to point me towards areas in econ which use pure math in substantial ways or researchers who regularly use pure math in their work? I know Graciela Chichilnisky has a PhD in pure math. Are there other working economists with a similar background? I am aware of the large amount of applied math / statistics at use, but I am more interested in the theory side of things (topology, analysis, algebra, etc).

For context, I am asking because I am thinking of applying to econ PhD programs, and I'm curious if pure math has a place in modern economics research.

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    $\begingroup$ I am discouraged from replying to a Q by 'Throwawayaccount'. $\endgroup$
    – Giskard
    Apr 23 at 6:27
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    $\begingroup$ I would check out Ariel Rubinstein's work. Pretty influential (especially his stuff before the popularization of discounting). He has some freely available textbooks on his website: arielrubinstein.tau.ac.il $\endgroup$
    – Brennan
    Apr 23 at 17:26
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    $\begingroup$ Isn't using math to address problems in another domain applied math by definition, regardless of the mathematical subjects involved or the academic pedigree of the practitioner? $\endgroup$ Apr 23 at 22:09
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    $\begingroup$ @John Bollinger This is one possible meaning of applied mathematics. But often mathematicians by 'applied mathematics' mean mathematics involved with data and numerical applications. $\endgroup$ Apr 23 at 22:19
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    $\begingroup$ @John Bollinger . That's a valid observation. What happens though is sometimes the math exists because of the application to the field. For example, take the results in optimal control theory having to do with Bellman optimality and Pontyagrin's Maximum principle. A lot of that material came about because of the attempt ( in the 50's and 60's ) to apply optimal control theory to economics. Otherwise, who knows ? So, my point is that the application of mathematics to another field can sometimes result in theoretical mathematical breakthroughs. $\endgroup$
    – mark leeds
    Apr 25 at 16:57

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To get an idea of the research in the field of mathematical economics you can look at some journals devoted to the subject, as the Journal of Mathematical Economics and Economic Theory.

You can realize that, in this field, there is a heavy use of advanced mathematical theory, not only applications of well-known mathematical tools, even if not trivial, but an original mathematical research.

The Fields medalist Stephen Smale, is perhaps the most famous example of a mathematician who worked in the field of economics.

In his famous Mathematical Problems for the Next Century. Mathematical Intelligencer. 20 (2): 7–15 (1998), Smale included an economic problem among the open mathematical problems of the XXI century: the 8th problem of the Smale’s problems is an economic question:

Extend the mathematical models of General Equilibrium Theory to include price adjustment.

(general equilibrium theory is one of the fields of economics where there is a heavy use of mathematics).

Another well-known mathematician who subsequently devoted himself to economics was Charambolos Aliprantis.

Aliprantis is the founder of the above mentioned journal Economic Theory, and he is known for his use of functional analysis and Riezs spaces in economic theory, see for example his books Locally Solid Riesz Spaces With Applications to Economics or Positive Operators, Riesz Spaces, and Economics

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  • $\begingroup$ None of those names rang a bell ( My only real experience in economics is from time series econometrics ) but still an interesting and useful answer. Thanks. $\endgroup$
    – mark leeds
    Apr 23 at 16:54
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    $\begingroup$ @Giskard Good observation. I don't know who wrote that Wikipedia article, they maybe cosiders the works mentioned not satisfactory. But maybe the problem is that for other mathematical problems what is required are proofs, and we can say if a conjecture has ben proved or not, or it has been partially proved, that I think means under some restrictive assumption. In a economic problem as the 8th, when can we say 'It has been solved'? It is not a proof, it implies to build satisfactory models. $\endgroup$ Apr 23 at 20:46
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    $\begingroup$ @Bakerstreet: I agree with you wholeheartedly about applied math often meaning what you described. I was just pointing out that there are sometimes cases where an attempt to apply something new to a field, leads to a breakthrough that was not necessarily expected. An example is from RE. In 1960, John Muth applied an interesting expectations concept to an economic problem. It took 10 years but that paper eventually led to what is often referred to as the "rational expectations" revolution. en.wikipedia.org/wiki/John_Muth $\endgroup$
    – mark leeds
    Apr 26 at 3:33
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    $\begingroup$ What I find maybe more interesting than anything is the evolution of the fields themselves. For example, one could have studentEE with a Ph.D in EE, studentST with a Ph.D in statistics and studentEC with a Ph.D in econometrics, Yet, in each case, the theses could be pretty close as far as the topic, whatever it is. But this is a topic for another day !!!!! $\endgroup$
    – mark leeds
    Apr 27 at 7:14
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    $\begingroup$ @BakerStreet Thank you for this lovely answer. Of course I know Stephen Smale for his math, but I have somehow completely missed his contributions to economics. It has been an interesting read. The journals and books that you linked look promising and I will spend some time going through them. Thank you again! $\endgroup$ May 9 at 15:25
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Game theory, as suggested by the name, is not applied. Fixed point theorems, measure theory and other abstract areas abound. A lot of very succesful game theorists have math PhDs.

A lot of people will claim that game theory has applications, and this is true for a few topics like auction-theory, but a lot of it is purely abstract, the alleged aplications being illustrative stories whose highest math level is around 'averages' and 'which of these two numbers is bigger', so that a business student can grasp it without a lot of additional study.

There are also 'applied game theory' fields, but I would argue these seldom use high level math, there is a strong delineation.

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    $\begingroup$ As a matter of personal taste, I really like the storytelling aspect of game theory, and also the math puzzles; what wierds me out is the strong insistance on selling it as if the famous lessons learned required high level math and were particularly applicable. $\endgroup$
    – Giskard
    Apr 23 at 6:38
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    $\begingroup$ I agree. The first thing my students learn in my game theory course is that game theory has many applications, but almost all of them are intra-scientific applications. $\endgroup$
    – VARulle
    Apr 23 at 9:26
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    $\begingroup$ @VARulle I would even argue with this viewpoint of many intra-scientific applications; I worked in a competition authority for a while and accordingly read IO papers. Nash-equilibrium is a horrible applied solution concept, not at all robust. In some cases welfare effects depend on second order beliefs of the agents. Even most refinements are model specific, as in they work in some class of games but not in other, very similar classes, so it is not hard to get two opposing results in very similar theoretical models. Yet it was everywhere. But maybe you weren't thinking of the IO field. $\endgroup$
    – Giskard
    Apr 23 at 9:37
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Nowdays, great part of economist has a mathematical background. I believe reason comes from the research around general equlibrium theory. I can remember Gérard Debreu Who was a bourbakist, von Neumann, Solow, and so on... It is easy to take a look on recenti economists winning Nobel prize in Economy and to read about their background...

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  • $\begingroup$ The Q is not about whether knowing mathemathics is beneficial for economics research, but whether you can do pure math research in economics. E.g.; a math background is useful for econometrics research, but a lot of econometrics is applied. $\endgroup$
    – Giskard
    Apr 23 at 7:11
  • $\begingroup$ @Giskard My answer Is a sort of bayesan replay! In any case, pure math Is beneficial for economics such as for other fields: physics, chemistry, ... $\endgroup$ Apr 23 at 8:52
  • $\begingroup$ Chicken or the egg !?! $\endgroup$ Apr 23 at 9:06
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    $\begingroup$ As someone said in the comments, some econometrcians are pretty into math but I'm not sure if it would fall under pure math. They looks at things like alpha-mixing conditions and stochastic convergence theory of various econometric models. Halbert White and James Davidson are two people that come to mind but I'm not any type of math person so they might be view as applied possibly by a pure math person. Maybe check out their backgrounds and see what you think. Unfortunately, Halbert White passed a way a few years ago at a relatively young age. A gigantic loss for the econometrics community. $\endgroup$
    – mark leeds
    Apr 23 at 9:47
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    $\begingroup$ I apologize for not giving you credit. I just read the comments and really don't pay attention to who said what. Since credit matters to some, I'll do that in the future. My apologies. $\endgroup$
    – mark leeds
    Apr 23 at 12:03

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