I would like to know which mathematical skills are most important to understanding different types of theoretical economics literature. An ideal answer would list disciplines used for different 'genres' of literature; for example: "Mathematical finance requires elementary linear algebra and stochastic calculus" (is this accurate?).
For microeconomic theory, a starting point would be
- Basic multivariate calculus,
- topology/set theory
- real analysis,
- probability theory,
- a basic knowledge of linear algebra.
For 'applied' micro theory (theoretical industrial organisation, applications in information economics, etc.), or for reading of a graduate-level microeconomics text (such as Mas-Colell, Whinston, and Green's "Microeconomic Thoery"), a good understanding of the basics of these areas will suffice. Most material will be set in something like a Euclidean space and will draw on a fairly restricted set of methods and results from the above domains. You can already get a long way with a good understanding of the basic continuity properties of the reals ($\epsilon$-$\delta$ proofs, open and closed intervals, intermediate value theorem, basic fixed-point theorems, etc.), the ability to do partial differentiation, integration, and to calculate conditional probabilities and expectations.
As one turns to the journal literature and looks more towards pure (rather than applied) economic theory, the level of mathematical sophistication increases somewhat and it becomes more likely that you will encounter more exotic (from the point of view of economics) mathematical methods and results.
For macroeconomics, it depends a bit on the part you're working in
basic requirements are always
- Basic multivariate calculus
- Real analysis
- Probability theory and stochastic calculus
- Beginnings of linear algebra
If you go down the Macro/Econometrics route, your linear algebra knowledge should be quite profound.
Then, you should have (because discrete time is almost everywhere a standard tool) working knowledge in
- difference equations
- Bellman Equations
If you go theoretical, that also requires
- Functional analysis
If instead you're going to continuous time (this will be the case for Macro-Finance, also likely Growth. For mainstream macro, it seems to be the trend, but is yet unclear), you will need knowledge of differentiable equations, in particular
- Solution methods to First and second order ODE
- Derivation and Solutions to Kolmogorov Forward Equation (Fokker Plank)
- Working knowledge with Wiener processes
Where again, if you want to do purely theoretical work, your knowledge (in particular with probability theory) needs to be much more profound.