28895 - Mathematical Economics

Learning outcomes

At the end of the course the student has acquired knowledge and skills essential to the study of dynamic economic systems.

In particular, he/she is able to:

- calculate explicitly the solution to systems of linear differential and difference equations;

- study systems of nonlinear differential and difference equations using the phase diagram and through linearization around the steady state;

- solve deterministic dynamic optimization problems in discrete time (dynamic programming) and continuous time (optimal control).

Course contents

1 Ordinary differential equations (ODEs), general concepts

2 Well-posedness of the Cauchy problem

3 Some solvable ODEs

4 Linear ODEs of first- and second-order

5 Eigenvalues and eigenvectors

6 Diagonalization of matrices

7 Stability of ODEs and of systems of ODEs. Linearization at equilibrium points

8 Finite difference approximation of ODEs (stability and convergence)

9 Dynamic optimization in continuous time (Pontryagin maximum principle)

10 Dynamic optimization in discrete time (dynamic programming)

 

Readings/Bibliography

Below is a list of textbooks that cover all the course contents. Nevertheless, buying these textbooks is not strictly necessary. In fact, students can also learn on the slides of the course that will be posted on the course website (to which students may access using their password). Furthermore, the references given below can be substituted with any other introductory material or textbook that explains the basic facts about ordinary differential equations, linear algebra and optimal control.

 

G. Strang, Differential Equations and Linear Algebra, Wellesley-Cambridge Press, 2015

A. C. Chiang, Elements of Dynamic Optimization, McGraw-Hill, 1992

D. Acemoglu, Introduction to Modern Economic Growth, PrincetonUniversity Press, 2008

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2015

 

 

 

Teaching methods

This is the first part of the integrated course in QUANTITATIVE METHODS FOR ECONOMIC ANALYSIS. In particular, the Mathematical Economics course is a 5 CFU course corresponding to 30 hour lessons.

The main theoretical concepts will be explained by projecting slides. Some examples and problems of practical interest will also be presented using the blackboard.

Assessment methods

The learning outcomes are verified through a written exam, which  takes two hours.

Students must demonstrate both theoretical and practical skills. Therefore, the exam consists of three exercises (which may eventually become two if some of them is divided in two parts) and one free-response question. In particular, the exercises will test the students' skills in solving and analyzing ordinary differential equations and systems of ordinary differential equations, as well as the students' abilities in computing eigenvalues and eigenvectors and in diagonalizing matrices. Clearly, as this is a course in mathematics for economics, students are first of all required to think. Therefore, the exam esercises will be only similar, but not exactly identical to those that will be explained in the classroom. By contrast, the free-response question will focus on one (randomly chosen) of the theoretical subjects that will be taught during the course. In answering to it, students must demonstrate the same level of analysis that has been provided by the teacher during the lessons (i.e. all the mathematical arguments and links among formulas must be explained with the due depth and clarity).

Students must be aware that, in order to solve the exam exercises, first of all a solid background of basic calculus and matrix algebra (i.e. the prerequisites of the course) is mandatory. Thus, students are highly recommended to attend the crash-course and to become familiar with all the contents that will be taught during it.

Finally, in line with the fact that students are strongly encouraged to reflect, the use of textbooks and calculators is not permitted.

Teaching tools

The lessons will alternate theoretical concepts with explaining examples and exercises. In addition, in order to make students perceive the practical usefulness of the mathematical models and methods taught, some relevant application problems in economics will be presented and analyzed. Students are recommanded to download the slides of the course before going to the classroom and to attend the lessons with the slides at hand.  

Finally, the course will be preceded by a crash-course, such to provide students with the necessary basic mathematical knowledge (at the undergraduate level). All the students are strongly encouraged to attend it. 

Office hours

See the website of Luca Vincenzo Ballestra