MATHEMATICS FOR ECONOMICS

1. Knowledge and understanding:

The course presents from a theoretical point of view the fundamental elements of static optimization, dynamic systems and dynamic optimization by studying some of their most relevant microeconomic and macroeconomic applications. Examples and ways of working will be presented in the classroom during lectures and exercises.

2 Applying knowledge and understanding:

At the end of the course, the student has acquired knowledge and skills essential to studying advanced economic systems. In particular, he/she knows how to correctly apply the formulation studied in the representation and modeling of real systems. In detail, the student can: -solve constrained optimization problems with nonlinear objectives; -explicitly compute the solution of systems of differential equations; -investigate the stability of steady-state solutions for linear and nonlinear systems; -formulate and study dynamic optimization problems in economics.

3. Making judgements:

The course aims to provide students with the tools of optimization and dynamic modeling, to stimulate the ability to read and interpret a theoretical economic model, and to develop autonomous critical judgment in the context of the topics covered.

4. Communication skills:

By the end of the course, the student should have acquired the technical vocabulary related to the topics covered and a good ability to communicate his or her statements and considerations related to the program carried out in lectures and explored in depth in the recommended texts.

5. Learning skills

By the end of the course, the student should have acquired a good capacity for autonomy in studying the discipline, in reading, interpreting and analyzing mathematical models in economics.

Elements of linear algebra and Multi-variable Functions; free and constrained optimization; continuous-time and discrete-time dynamical systems; introduction to dynamic optimization in economics.

Should the teaching be given in a blended or distance learning mode, necessary variations from what has been stated above may be introduced to comply with the planned syllabus.

Part I:

(1) Simon, Blume, Mathematics for economists, Norton & Company 2010

Parts II and III:

(3) Leonard, Van Long, Optimal Control Theory and Static Optimization in Economics, Cambridge 2012

(4) Bischi, Lamantia, Radi, Lecture notes on Dynamical Systems in Economics and Finance (provided on request by the lecturer).

The final assessment is in two stages both of which are mandatory. In the first stage, the student must take a written examination with open-ended answers on the main topics of the program. The second stage of the exam, which is accessed after passing the written exam, consists of an interview with the teachers related to the exposition of theoretical topics and the execution of exercises.

The written exam aims to assess the student's analytical ability and problem solving skills in mathematics. The oral examination aims to assess the student's achievement of specific course skills, knowledge of the topics covered and methodological rigor achieved.

The written assignment is graded in thirtieths based on the correctness of the development, clarity and completeness of the discussion. The oral test, also graded in thirtieths, aims to assess the mastery, clarity of exposition and level of accuracy of preparation achieved by the student. The exam is considered passed if the student achieves a score of at least 18/30 on both tests. The final grade, in thirtieths, is an average of the two tests (written test and oral test) taken by the student.

An on-going test is scheduled for those attending.

1. What does it mean for a set of vectors to be linearly independent? How can this be verified?
2. Define the span of a set of vectors and the basis of a vector space.
3. What is the relationship between rank, the dimension of the kernel, and the dimension of the domain?
4. Explain the concept of a vector subspace.
5. Define a linear transformation between vector spaces. What properties must it satisfy?
6. What is the relationship between linear transformations and matrices? How is a linear transformation represented in a basis?
7. Explain the concept of image and kernel of a linear transformation. What is their geometric meaning?
8. When is a linear transformation invertible? What conditions must be satisfied?
9. What is meant by the matrix associated with a linear transformation?
10. Define eigenvalue and eigenvector of a matrix. What is their geometric interpretation?
11. What is the difference between simple eigenvalues and multiple eigenvalues?
12. Define what is meant by an eigenspace. What are the algebraic and geometric multiplicities of an eigenvalue?
13. Define the gradient of a function. What is its geometric interpretation?
14. What does the differential of a multivariable function represent?
15. Explain the meaning of the Hessian matrix and its role in convexity analysis.
16. What are the first-order necessary conditions for a stationary point in an unconstrained problem?
17. What are the second-order conditions for classifying a stationary point (minimum, maximum, saddle)?
18. State Sylvester’s criterion for defining the positivity of a matrix.
19. What is the difference between convexity and concavity of a function?
20. State the implicit function theorem.
21. What are the necessary conditions to apply the implicit function theorem? (existence, differentiability, nonzero Jacobian)
22. Explain the geometric meaning of the implicit function theorem.
23. How is the implicit function theorem used to compute partial derivatives of implicitly defined variables?
24. What is the link between the implicit function theorem and level curves?
25. What are the first-order necessary conditions for an optimization problem with equality constraints (Lagrange theorem)?
26. What are the second-order conditions and what do they guarantee?
27. Explain the economic meaning of Lagrange multipliers.
28. What are the first-order necessary conditions for an optimization problem with inequality constraints (Kuhn-Tucker theorem)?
29. What is the difference between Kuhn-Tucker conditions and Lagrange conditions?
30. What is meant by constraint qualification and why is it important?
31. State the envelope theorem and explain its economic meaning.
32. What is the difference between differentiating the value function with respect to a parameter and differentiating the objective function?
33. What is the interpretation of Lagrange multipliers in the context of the envelope theorem?
34. Define an ordinary differential equation (ODE) and explain the difference between order and degree.
35. What is the difference between linear and nonlinear differential equations?
36. State the existence and uniqueness theorem for ODEs (Picard-Lindelöf).
37. Explain the concept of general solution and particular solution of an ODE.
38. What is the difference between homogeneous and nonhomogeneous equations? Give an example.
39. What are the main methods for solving first-order linear ODEs?
40. Define a linear system of differential equations. What is its matrix form?
41. What is the role of eigenvalues and eigenvectors in solving a linear system of ODEs?
42. Explain the concept of stability of an equilibrium point in a linear system.
43. What is the difference between autonomous and non-autonomous systems?
44. How are the critical points of a two-dimensional linear system classified based on eigenvalues?
45. Define the stability of an equilibrium point in a dynamic system.
46. What is the difference between Lyapunov stability and asymptotic stability?
47. Explain the concept of bifurcation and give an example.
48. What is an attractor? What is the difference between a global and a local attractor?
49. Why is the Hartman-Grobman theorem important in the study of nonlinear systems?
50. What is the meaning of the Jacobian’s eigenvalues in a dynamic system?
51. State Pontryagin’s Maximum Principle.
52. What is the role of the costate variable (or shadow price) in an optimal control problem?
53. Explain the difference between present-value and current-value formulations.
54. What does transversality mean in optimality conditions?
55. What is the difference between the Hamilton-Jacobi-Bellman approach and Pontryagin’s approach?