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.