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QUESTION: What are the major or systematic applications of post-1960s mathematics to microeconomics?

For example, in the late 19th century, Fisher first used the mathematical ideas of Gibbs to construct modern utility theory. In the 20th century, Mas-Colell incorporated topological ideas to study general equilibrium. What about the late 20th, early 21st century?

For example, consider directed graph theory, measure theory, topology, the category theory and modern homology or cohomology, topos methods, functional integration, etc.

Note 1: econometrics/statistics, without modelling, is excluded. The only modern mathematics used there is random walk theory, and the ergodic problem, solved via complex analysis. RW and EP are not specific to economics.

Any appropriate economics publication is an answer. This included also those published in non-strictly economics journals, e.g. the Journal of Mathematical Psychology.

Note 2: Yes, I know, this type of work is rarer (not to be confused with obscurity: some of it is well known). That is what makes it easy to miss such a reference when it is published. Hence the question.

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  • $\begingroup$ I think that most of this kind of research has been relegated to Journal of Mathematical Economics in the case of Micro. In econometrics a lot of functional analysis is being used, in top journals but pure topology a little less. Long gone days of Hildenbran, Kannai, MasColell, Debreu, Chichilnisky, Anderson and Arrow. $\endgroup$
    – user157623
    Commented Nov 23, 2014 at 16:21
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    $\begingroup$ I'm voting to close as too broad. It is not very clear to me exactly what you would like to include or exclude and what motivates those criteria. $\endgroup$
    – Jyotirmoy Bhattacharya
    Commented Nov 23, 2014 at 16:31
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    $\begingroup$ The title is more concise than the body of the question, where the focus broadens dramatically. Perhaps you should consider re-working the body of the question. $\endgroup$
    – Alecos Papadopoulos
    Commented Nov 23, 2014 at 18:15
  • $\begingroup$ @GuidoJorg What about simply "What are the major applications of post-1960s mathematics to microeconomics?" For me the references to Mas-Colell and Fisher and the many exclusions in the questions make it harder to decide what would qualify as an answer. $\endgroup$
    – Jyotirmoy Bhattacharya
    Commented Nov 24, 2014 at 2:57
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    $\begingroup$ Done. Is the question better structured now? $\endgroup$
    – user218
    Commented Nov 24, 2014 at 8:41

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I strongly suspect that an emerging important area for applications of measure theory will be in approximate dynamic programming techniques. Approximate dynamic programming (aka "reinforcement learning" in the computer science literature) has been the direction of research work in the last ~10-20 years of the dynamic programming literature. Economics is only just now starting to adopt some of these advances. For example of the direction of the DP literature, see Bertsekas' most recent 4th edition expansion of his dynamic programming series, or Powell's Approximate DP: Solving the Curse of Dimensionality. Economists are just starting to pick up some of these tools, both directly and indirectly, and I suspect that they will have a growing impact on the literature over the next few years. Some of the analytical background for convergence of these methods is topology and dynamical systems.

A good example of theoretical contribution to this type of literature from economists is Pál and Stachurski (2013), Fitted Value Function Iteration With Probability One Contractions (ungated version here). Peruse that paper and you can see the importance of a good grasp of measure theory. Stachurski's book Economic Dynamics is actually a very nice exposition of dynamic programming from this perspective, building at a pace which works for multiple levels of graduate student/professional (measure theory comes in formally at the end I believe -- I'm still working towards those insights).

Hopefully this answers your question to some degree. I'm afraid that the phrase "post-1960s mathematics" is somewhat ambiguous to me (due to my own lack of knowledge of history of maths literature), so if I've completely missed the mark, my apologies!

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    $\begingroup$ I'm following up on Stachurski's book :) Will reply in a day or so. $\endgroup$
    – user218
    Commented Nov 25, 2014 at 3:58
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    $\begingroup$ @GuidoJorg: I flipped through Stachurski to give you some pointers to specific places, and realized that I'd had a brain fart -- was thinking of applications of measure theory, not topology. Have edited answer to reflect this. My apologies! Let me know if Q still fulfills your question (appears to with your edit, but wanted to check!). Also, wanted to note that this is technically, generally, applications in macro (but I think that line will be blurring as some of these methods advance). $\endgroup$
    – CompEcon
    Commented Nov 25, 2014 at 13:52
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    $\begingroup$ Measure theory is fine :) BTW, I got the book. I also found a couple other recent monographs that appear related, and one on topology. Looking through them and will be back with feedback, accept the answer, etc. $\endgroup$
    – user218
    Commented Nov 27, 2014 at 18:06
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    $\begingroup$ I like Stachurski's book. It compares well with other recent mathematical economics literature: I just finished with several other books 1990s-2010 which claimed to be mathematically modern theoretical contributions (dealing with incomplete markets general equilibrium, sensitivity to initial conditions, investment in technology, etc.); but they were almost all quite disappointing variants of standard Keynsian models, with the usual problems of such models, and all of them applied mathematics, if at all, superficially and not very elegantly. $\endgroup$
    – user218
    Commented Dec 2, 2014 at 3:36
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This was too long for comment. "Post 1960" seems an arbitrary and very high bar for an applied field, including micro theory. Most of the topics you name would not be considered contemporary mathematics. For example, measure theory started with Lebesgue's thesis and is over a century old. Topology is even older and started with Poincare, who introduced homology groups. Both are taught to undergrads today, like calculus. (The mathematics used by Mas-Colell et al. in GE is analysis, rather than topology.)

The externality of research programs that drive modern mathematics since the mid-20th century to the applied community is indirect at best. The point of view and techniques motivated by, for example, non-commutative geometry, Langland's program, Poincare conjecture, the Baum-Connes conjecture, the twin prime conjecture (Fields medals have been awarded post 1960 for progress on these problems), etc.---will probably never be seen outside mathematics. Mathematical finance, of course, remains mathematics but that is quite removed from the economic point of view.

Edit It turns out that, addressing your question directly, there has been applications of topology to social choice theory, initiated by Chichilnisky, et. al. Here is a JET paper on the topic by a topologist:

http://math.uchicago.edu/~shmuel/TSC.pdf .

Maybe someone with expertise in topology can comment further.

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Loeb spaces have been used to model situations with a continuum of agents. See http://eml.berkeley.edu/~anderson/Book.pdf and the chapters by Sun on economic applications in the book Nonstandard Analysis for the Working Mathematician.

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Measure theory is widely used in the problem of fair division (aka "cake-cutting"). See the many papers about fairness in economics journals.

For a particular example, see Tatsuro Ichiishi and Adam Idzik, "Equitable allocation of divisible goods", JME 1999.

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Beside the work of Chichilnisky mentioned by Michael, another interesting use of topology in social choice theory appears in the work of Redekop on Arrow's theorem on economic domains.

  • Redekop, J. (1991). Social welfare functions on restricted economic domains. Journal of Economic Theory, 53, 396–427.
  • Redekop, J. (1993a). Arrow-inconsistent economic domains. Social Choice andWelfare,10, 107–126.
  • Redekop, J. (1993b). The questionnaire topology on some spaces of economic preferences. Journal of Mathematical Economics, 22, 479–494.
  • Redekop, J. (1993c). Social welfare functions on parametric domains. Social Choice andWelfare,10, 127–148.
  • Redekop, J. (1995). Arrow theorems in economic environments. In W. A. Barnett, H. Moulin, M. Salles, & N. J. Schofield (Eds.), Social choice,welfare,and ethics (pp. 163–185). Cambridge: Cambridge University Press.
  • Redekop, J. (1996). Arrow theorems in mixed goods, stochastic, and dynamic economic environments. Social Choice and Welfare,13, 95–112.

Arrow's impossibility theorem was originally proven for an abstract set of alternative, allowing every possible preference profile over this set of alternatives. The question that Redekop (and others) asked was : is there an equivalent of Arrow's theorem when the alternatives are bundles of goods, and agent have "classical" preferences over those goods (monotonic, convex, continuous, selfish,...).

More precisely, the question was whether there would exists a social welfare function satisfying the three Arrovian axioms (Independence of Irrelevant Alternative, Weak Pareto and Non-Dictatorship) on these Economic domains (see Le Breton, Michel, and John A. Weymark. "Chapter Seventeen-Arrovian Social Choice Theory on Economic Domains." Handbook of social choice and welfare 2 (2011): 191-299 for a great review, which this answer is based upon).

Roughly, Redekop's work shows that, for some of those economic problems, if a domain of preferences admits an Arrovian social welfare function, the domain must be "small" in some topological sense. For instance, in Redekop (1991), he introduces an ingenious topology on sets of preferences he dubbed the questionnaire topology, and shows that, in a public goods economy, if a domain of preferences admits an Arrovian social welfare function, then the domain must be nowhere dense according to this topology (i.e. the closure of the domain contains no open set).

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