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Are there any (specific) math open problems that mathematical (micro)economics / finance / econometrics researchers wish mathematicians could solve for their discoveries to flourish? If positive, which math open problems are related to which applied topics and how they are related?
When I ask that, it really comes to my mind the case of stochastic analysis and option asset pricing problems in mathematical finance.
People have argued that if $P \neq NP$ then efficient markets are impossible and certain equalibria may not exist. However, they may hold approximately, so I'm not sure if this qualifies. Additionally, if it turns out that $P=NP$ then certain economic optimization problems (e.g. in logistics) become easily solvable. On the other hand, if $P \neq NP$ then it might imply that encryption is very safe and therefore certain forms of economic arrangements will still be possible in the face of technological advance.
Here's a quotation from a paper on Partial Differential Equations in Macroeconomics. One of the authors is a Fields Medalist.
Macroeconomic models with heterogeneous agents share a common mathematical structure which, in continuous time, can be summarized by a system of coupled nonlinear partial differential equations (PDEs): (i) a Hamilton–Jacobi–Bellman (HJB) equation describing the optimal control problem of a single atomistic individual and (ii) an equation describing the evolution of the distribution of a vector of individual state variables in the population (such as a Fokker–Planck equation, Fisher–KPP equation or Boltzmann equation). While plenty is known about the properties of each type of equation individually, our understanding of the coupled system is much more limited.
The paper obviously isn't about microeconomics (though it does briefly mention some topics in micro as well), but it does talk about genuine open problems in mathematics whose answers have implications in economics.
A lot of mathematicians and, here, computer scientists study open problems in game theory. Here is an article that talks about the potential solution of an open problem in mechanism design: http://news.mit.edu/2016/faculty-profile-constantinos-daskalakis-0204
Other problems in game theory, however, have proven more susceptible to analysis from a computational perspective. In 2012, after coming to MIT, Daskalakis and his students solved a 30-year-old problem in economics, a generalization of work that helped earn the University of Chicago’s Roger Myerson the Nobel Prize in economics. That problem was how to structure auctions for multiple items so that, even if all the bidders adopt strategies that maximize their own returns, the auctioneer can still extract the greatest profit.
Maybe it would be needed first of all to get ride of this:
http://www.paecon.net/PAEReview/issue63/Barzilai63.pdf
He claims to have demonstrated that any calculus on a utility space, wether they cardinal, ordinal or expected, is logically inconsistent, mathematically impossible:
Abstract
By formally defining the relevant mathematical spaces and models we show that the operations of addition and multiplication, and the concepts that depend on these operations, are not applicable on ordinal, cardinal, and expected utility. Furthermore, expected utility’s scale construction rule is self-contradictory.
