MATHEMATICS FOR ECONOMICS

The course is part of the mathematical and quantitative activities of the degree programme in Economics and Business. It provides tools for static and dynamic optimization that are useful for formulating, analysing, and solving economic and social problems through rigorous mathematical language. In this way, it contributes to the development of the quantitative skills needed to deal with economic models, decision-making problems, and subsequent applications within the degree programme.

At the end of the course, students will have developed the following competences.

a) Knowledge and understanding

a.1) knowledge of the basic mathematical tools needed to solve optimization problems, such as the method of Lagrange multipliers and Bellman’s dynamic programming method;

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

a.3) interpretation of the tools mentioned in (a.1) and (a.2) in terms of geometric properties, supported by a range of key economic examples.

b) Ability to apply knowledge and understanding

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

b.2) ability to compute, for functions of several variables, maxima and minima on sets defined by systems of equalities/inequalities or positivity constraints;

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

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

b.5) ability to interpret all the properties described above in examples with an economic or managerial focus.

c) Acquired skills — long-term

c.1) increased ability to handle formal language and draw correct logical deductions; consolidation of 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 original problem.

A solid knowledge of the contents of a first-year Mathematics course in a degree programme in economics-related disciplines is a prerequisite for the course. In particular, knowledge of the following topics is required:

* 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 on unconstrained domains and on constrained domains, namely simple compact sets;
* Basic knowledge of sequences and series and of their limits;
* Linear algebra: matrices, operations between matrices and their properties, determinants, inverse matrices.

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 examination consists of a written test and an oral examination. The written test specifically assesses the ability to apply the mathematical tools covered in the course to the solution of exercises and problems; the oral examination assesses the theoretical understanding of the topics, the ability to discuss the written test, and the correct use of mathematical language.

The written test lasts approximately 2 hours and 30 minutes and consists of four exercises, modelled on those solved during the course or assigned in previous exams, all of which are available online on Moodle together with their solutions and point allocation. The test may also include theoretical questions.


During the written test, students may use only the formula sheet provided by the instructor. The use of electronic devices of any kind, including calculators, is not allowed.

In order to be admitted to the oral examination, students must obtain at least 16 points in the written test. The oral examination takes place in the days immediately following the written test: it begins with a discussion of the written test and may then extend, if necessary, to the other topics covered in the course.

written and oral

The instructor is responsible for ensuring the authenticity and originality of all examinations and coursework. In cases of suspected academic misconduct, an additional on-site assessment may be required during the exams, which may differ from the standard format.

The score for the written examination ranges from 0 to 30 points and represents the grade out of thirty. Additional, more complex questions, for a maximum total of 6 points, may be added to allow for the possible award of honours.

The ordinary 30 points are distributed as follows:

* 22-24 points for basic questions;
* 6-8 points for questions of moderate difficulty.

The additional points, up to a maximum of 6, are reserved for more complex questions.

Answers that are not adequately justified will receive no credit. It is therefore important to explain clearly what is being done and why.

The oral examination mainly has the function of confirming the grade obtained in the written examination and, on average, may change that grade by an interval ranging from -3 to +3 points. This indication should be understood as an approximate description of what may happen, not as a formal rule for assigning the final grade.

In particular, should serious discrepancies emerge between the assessment of the written examination and that of the oral examination, the assessment of the oral examination will prevail, without any point constraints.

Teaching activities include lectures and in-class problem-solving sessions.

Tools and materials for reviewing the prerequisites are also made available.

During the course, homework worksheets are assigned, containing exercises on the topics covered. The assigned exercises are discussed and checked weekly during in-class office hours.

Study materials are available on the course Moodle page. In particular, the texts and solutions of exams from previous years are available.

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.

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