MATHEMATICS FOR ECONOMICS

The course proposes some topics of static (in several variables) and dynamic optimization, and their applications in the economic and social fields.

At the end of the course, the student will have developed the following skills.

a) Knowledge and understanding

a.1) knowledge of the basic mathematical tools necessary to solve optimization problems, for example, the Lagrange multiplier method or the Bellman dynamic programming method;

a.2) knowledge of the preliminary tools for the study of (a.1), such as eigenvalues ​​and eigenvectors of a matrix, implicit function theorem;

a.3) Interpretation of said tools in (a.1) and (a.2) in terms of geometric properties, and with the support of a range of crucial economic examples;


b) Ability to apply knowledge and understanding:

b.1) ability to compute eigenvalues ​​and eigenvectors of matrices;

b.2) ability to calculate, for functions of several variables: maximums and minimums in sets defined by systems of equalities / inequalities, or positivity constraints;

b.3) ability to calculate, for discrete dynamical systems, the equilibrium points and the trajectories of the system;

b.4) ability to calculate optimal strategies for controlled discrete dynamical systems;

b.5) ​​ability to interpret all the properties described above in examples of economic or managerial vocation.


c) (Lifelong) learning skills

c.1) increased ability to handle a formal language and draw correct logical deductions; consolidate rigorous rational reasoning.

c.2) increased ability to translate an economic problem into formal terms, solve it, and interpret the solution in terms of the initial question.

A solid knowledge of the contents of a first year Mathematics course is a prerequisite for the course. In particular, knowledge of:
- Differential calculus for functions of (one and) several variables;
- Optimization for functions of one variable (including first and second order conditions);
- Optimization for functions of (at least) two variables in unconstrained domains, and in constrained domains (simple compact sets).
- Basic knowledge of sequences and series and their limits;
- Linear algebra: matrices, operations between matrices and their properties, determinants, inverses;
- (possibly) integration of functions in one variable, integration by parts and by substitution, integration of simple rational functions.

1. Optimization for functions of several variables.

1.1 Chain rule for functions of several variables; implicit function theorem.
1.2 The Lagrange multipliers Method, for one or more equality / inequality constraints, or for positivity constraints.
1.3 Economic examples

2. Discrete Dynamical Systems.
2.1 Eigenvalues ​​and eigenvectors of matrices; linear approximations of functions in several variables; difference equations.
2.2 Dynamic systems (in particular, linear systems), equilibrium points and trajectories.
2.3 Economic examples.

3. Dynamic optimization.
3.1 Controlled Dynamical systems.
3.2 Bellman's method of dynamic programming.
3.3 Economic examples.

Sydsaeter, Hammond, Seierstad, e Strom. "Essential Mathematics for Economic Analysis". Pearson Education. (2012). Fourth Edition. Chapters 12, 14, 17.

Sydsaeter, Hammond, Seierstad, e Strom. "Further Mathematics for Economic Analysis". Pearson Education. (2008). Second Edition. Chapters 1, 11, 12.

Ronald Shone, "Economic Dynamics Phase Diagrams and their Economic Application", Second Edition, (2002) Cambridge University Press. Chapters 3, 5, 6.

Lecture notes.

The written exam is mandatory (approximately 2.5 hours in duration) and consists of 4 exercises similar to those solved during the course (or in previous exams, available online in Moodle with associated solutions) and theoretical questions.

During the exam, students may use a formula sheet provided by the instructor.

Generally, there are extra points available (from 3 to 6) for receiving honors, totaling 33-36 points.

The available points are distributed as follows:

18-20 points for basic questions;
6-8 points for moderately difficult questions;
6-8 points for more complex questions.
If answers are not properly justified, they are worth zero. Therefore, it's important to explain what is being done and why.

The oral exam is optional for both the student and the instructor. In case of assessment doubts, I can ask the student to take it. If the student wants to improve the grade, or if they have a grade that is not fully satisfactory but still higher than 16, they can request to take it.

Frontal lesson, practise sessions.

Review tools and materials for pre-requirements.

Assignment of weekly homework, whise solutions is checked at the earliest office hours.
Office hours are held in a classroom and are public.

Study materials are available on the moodle page of the course.

Italian

Students are required to register on the e-learning page of the course on the University platform (moodle.unive.it) on which educational and organizational updates will be published.

written and oral

This subject deals with topics related to the macro-area "Natural capital and environmental quality" and contributes to the achievement of one or more goals of U. N. Agenda for Sustainable Development