Where, in economics, are partial differential equations used?
I'm particularly interested in micro theory applications, but would also be keen to know of other applications.
Here are a couple of suggestions. There are some PDEs in some recent continuous-time models, for instance in:
I would not be surprised if Yuliy Sannikov used PDEs in some of his other papers (although I have nothing precise in mind).
You can refer to the review by Achdou et al. (Partial differential equation models in macroeconomics, Philosophical Transactions of the Royal Society, 2014) for other references.
In optimal control theory there is the Hamilton–Jacobi–Bellman (HJB) equation which per se is a PDE. (As usual when it comes to advanced Math in conjecutre with Econ) It's more like a cooking recipy to solve PDE's. The HJB equation is both necessary and sufficient for an optimum.
Optimal control is virtually used in all fields of micro (and thus HJB as well). In macro it's usually applied when it comes to micro-foundations. The ramsey(-cass-koopmans) model for instance (I've seen that adressed with HJB equations as well).
Another sub-field where PDEs are typically solved is Differential Game Theory (which goes somewhat in a pure math direction but for instance has applications in Industrial Organization). See here for instance for an intro. That's absolutely not my field but one famous guy of whom I know that uses that in IO is Luca Lambertini.