How is Pure Mathematics Used in Economics?

I saw a Reddit post asking about some applications of abstract mathematical results. I wrote a response and wanted to share it here as well.¹ The reason I wrote this and I’m sharing it here is that people around me, even highly educated ones, often have surprising misconceptions about economics. Some think of it as an extension of statistics or physics, some as some kind of anthropology, among other things. So this is a first attempt at providing a better representation of the toolkit that economists use.

Foundation/Existence (fixed point theorems, topology)

We are very interested in existence theorems. The concepts that we deal with are often some equilibria that can be expressed as fixed point but using correspondences instead of functions “What I do is optimal given what you do and what you do is optimal given what I do” can be expressed as a fixed point of the argmax’s of optimization problems that are indexed/constrained by what other player chooses.

We also want our fixed points to have certain properties (“equilibrium selection”) and we don’t always agree on what those properties should be. So there are many fixed point theorems being proved for these purposes in a small but important subfield of game theory that uses algebraic topology.

Optimization and Structure (functional analysis, abstract algebra)

Optimization problems we deal with are sometimes about optimizing over functions. So functional analysis enters the picture, especially in decision theory, asset pricing, and dynamic programming. For instance, we might be choosing how much of our income we should save and invest every month to maximize our lifetime well-being; we would be looking for an optimal investment policy function. Of course, we might simply have some data and we might be looking for the function that provides the best fit. In either case, we would be choosing some function to optimize another.

Abstract algebra is also used in surprising ways, although its applications are somewhat less common and less predictable (excluding linear algebra; we use that everyday for breakfast, lunch and dinner). In general, decision theory uses orders a lot, results on lattices are often very useful. One famous result is called Topkis’s theorem and it provides a way to do comparative statics in non-differentiable environments by exploiting some lattice structure.²

Information and Knowledge (measure theory, epistemic logic)

We rarely deal with certainty so one could say that measure theory is dense in economic theory, even thought it wouldn’t be entirely accurate; it is certainly used in every subfield of economics.

We are also interested in what kind of knowledge and/or beliefs lead people to play certain types of equilibria (most notably Nash equilibria). So epistemic logic become useful tools here. This paper doesn’t really use any of the epistemic logic stuff but it is fairly easy to read and it can give a sense of the problems I am talking about.³

Geometry (differential manifolds, tropical geometry)

Sometimes, we want there to be a price vector that clears the market: Given the price vector $p$, when everyone chooses quantities of commodity that they would like to buy and sell, the economy is not consuming more of any commodity than is available. (This can also be achieved using a fixed point theorem but there are other approaches.) It can be as simple as using the separating hyperplane theorem but predictably, things can get more complicated beyond the simplest cases. I’ve seen papers and books that use differential manifolds⁴ to deal with these issues but I don’t really know much about these myself.⁵ I’ve seen recent-ish papers that use abstract convexities (Richter and Rubinstein) and tropical geometry (Baldwin and Klemperer) to deal with these kinds of issues.

What to expect

I’m sure I’m forgetting some obvious examples and I obviously tried to exclude more applied fields like optimal transport, graph theory, etc. but judgement may have been uneven across fields above. I may make another post on how applied mathematics is used in economics in the future and I hope I can also dive deeper into how each of these fields are actually utilized in separate posts.