Title: Mathematics for Economics
Instructors: Matteo Brachetta, Alessia Cafferata, Mauro Rosestolato, Maria-Laura Torrente
Credits (CFU): 4 CFU
Lectures: 32 hours (8 hours for each module)
Homework: 64 hours
Period Taught: November 2024 – January 2025
Course Description and objectives
This course is the part of the Mathematics for Economics sequence for the PhD program in Economics and Quantitative Methods.
The course is divided in four modules.
The first module (Maria-Laura Torrente) aims to provide the student with the knowledge of advanced mathematical methods to successfully deal with economic models from a quantitative viewpoint. More specifically, the course will introduce the student to both the unconstrained and the constrained optimization of functions of several variables.
The second module (Matteo Brachetta) aims to study some applications of optimization techniques to problems in Finance, with a special focus to portfolio theory. After a brief review of random variables and utility functions, static maximization of expected utility is investigated. In particular, this part covers asset allocation problems with concave utility functions. Depending on the background of the students, some numerical experiments might be explored.
The third module (Alessia Cafferata) aims to provide the basic knowledge to model different economic problems under the formalism of dynamical systems in one and two dimensions in discrete time. After a brief review of linear and nonlinear difference equations and the introduction of deterministic chaos, different kinds of maps and local bifurcations will be investigated. Students are expected to learn the basic discrete-time formulation of dynamic models for the study of economic and social systems and to learn an autonomous approach to the problems related to the analysis of systems that evolve over time. Some economic applications of dynamic models will be explored.
The fourth module (Mauro Rosestolato) aims to give the student a taste of optimal control theory in continuous time via dynamic programming. After some motivating examples, the general optimal control problem is introduced. Bellman's principle of optimality, the Hamilton-Jacobi-Bellman equation, and their combined use to find an optimal control strategy will be presented and applied to simple cases.
Prerequisites
It is expected that students have prior knowledge of calculus and optimization in one variable and basic notions of linear algebra.
Course Materials
Lecture slides will generally contain all the material students are expected to learn.
In turn, the lectures will often refer to and closely follow the relevant chapters in:
- Simon, C., Blume, L.: Mathematics for Economists (1994), Norton and Company
- Peccati, L., D'Amico, M., Cigola, M. : Maths for Social Sciences (2018), Springer
- Knut Sydsaeter, Peter Hammond, Atle Seierstad, Arne Strom, "Further mathematics for Economic Analysis" (2nd edition), Pearson
- Bischi, G.I., Lamantia, F., Radi, D.: Lectures Notes on Dynamical Systems in Economics and Finance (2014), available online.
Assessment
Students will be assessed in the final exam, which will consist of questions on the the theoretical and modeling features treated in the course as well as exercises.