# Uses of convex analysis in Economics

I'm taking kind of a crash-course in convex analysis to complement my mathematical skills and was wondering if anyone knew about nice ways in which this kind of tools were used in Economics. To be more precise, some of the things I've seen so far are not strictly in the area of convex analysis but are very related, like dual spaces, weak topology, subdiferentials and Hahn-Banach theorem.

The only example I know of is the duality between the UMP and EMP in consumer theory (and of course the firm maximization and cost minimization problems). I also think that Hahn-Banach is used in the proof of the first welfare theorem.

Has anyone here used this kind of mathematical concepts in their work or has seen any interesting recent usage of them?

## 2 Answers

A partial answer: convex analysis is extensively used in axiomatic decision theory, at least in its recent developments. Most of these papers focus on individual behavior. You can have a look for instance at the following papers on ambiguity-averse preferences:

• "Maxmin Expected Utility with Non-Unique Prior" (Gilboa & Schmeidler)
• "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences" (Maccheroni, Marinacci & Rustichini)
• "A Smooth Model of Decision-Making under Ambiguity" (Klibanoff, Marinacci & Mukerji)
• "Ambiguity in the Small and in the Large" (Ghirardato and Siniscalchi)

Here is a paper that applies convex analysis to a model of trade under ambiguity aversion: "Subjective Beliefs and Ex-Ante Trade" (Rigotti, Shannon & Strzalecki).

Beyond models of ambiguity aversion, virtually all recent work in axiomatic decision theory makes use of convex analysis, and applies its tools to study various phenomena: regret aversion (Sarver, Ergin), cost of thinking (Ortoleva), random choice (Gul, Pesendorfer)... Please tell me if you want more precise suggestions.

For the mathematical part, a very good reference is Convex Analysis by Rockafellar (1970). It is cited by most of the papers above ;-).

Convex analysis shows up all over the place in economics, and not just in decision theory.

Explicit references to Rockafellar or equivalents show up quite often in theory papers, from the classic Myerson (1981) to, say, Bergemann, Brooks and Morris (2015), or Mathevet, Perego and Taneva (2017).

Daskalakis, Deckelbaum and Tzamos (2016) also use Fenchel-Rockafellar duality to make headway on analysing the problem of a monopolist with multiple goods, which has been a long-standing open problem in economics.