Economics

MASSIMILIANO FREZZA

The course provides students with advanced mathematical knowledge essential for the quantitative analysis of economic and financial phenomena and for the construction of interpretative and predictive models.

Specific Objectives (Dublin Descriptors):

Knowledge and understanding. Building upon the mathematical foundations acquired during the undergraduate program, students will be able to understand more advanced concepts such as the theory of linear transformations, the analysis of eigenvalues and eigenvectors, the study of multivariable functions, and both constrained and unconstrained optimization techniques. They will also develop solid knowledge and understanding of quadratic forms and ordinary differential equations. These mathematical tools will be applied to effectively address economic and financial problems.
Applying knowledge and understanding. Students will be able to apply the mathematical methods learned to solve complex problems in economics and finance. They will use mathematical tools such as eigenvalues and eigenvectors or differential equations to model and analyze economic and financial phenomena. They will be able to compute limits, differentiate multivariable functions, and apply optimization techniques to solve problems related, for instance, to resource allocation and portfolio management. In addition, they will be able to apply financial mathematics models to evaluate investments and manage cash flows over time, rigorously addressing economic and financial dynamics in practical and decision-making contexts.
Making judgments. Students will develop the ability to critically analyze mathematical models applied to economics and finance. They will be able to assess the robustness of proposed solutions, interpret the results of financial analyses, and make informed judgments about the validity and reliability of the mathematical methods used in various economic contexts. Moreover, they will acquire the competence to examine and evaluate the applicability of theoretical and practical models, considering the economic and financial implications of the proposed solutions and identifying the strengths and limitations of the methods employed.
Communication skills. Students will be able to express the results of mathematical analyses applied to economics and finance clearly and rigorously. They will be capable of presenting financial models, explaining the methodologies used, and accurately illustrating the economic implications of their calculations. They will adapt their communication appropriately to both specialist and non-specialist audiences, highlighting the practical implications of mathematical models for financial resource management and economic decision-making.
Learning skills. Students will be able to autonomously deepen their understanding of the mathematical techniques and quantitative models learned, keeping up to date with new methodologies applied to economics and finance. They will develop a critical and independent study method, useful for addressing academic and professional challenges in quantitative fields, and for rapidly adapting to the evolution of mathematical models and practical applications in the economic and financial domains.

Students enrolling in this course are expected to have:
1. Basic knowledge of high school mathematics, including algebra, elementary functions (linear, quadratic, exponential, logarithmic), and basic geometry.
2. Familiarity with elementary calculus, such as limits, derivatives, and simple integrals (even at an introductory level).
3. An understanding of basic economic concepts, including supply and demand, utility, and marginal analysis (recommended but not strictly required).
4. Logical reasoning skills and the ability to follow formal arguments and problem-solving procedures.

Linear Algebra
- Vector spaces and subspaces
- Concepts of dimension and basis of a vector space
- Change of basis in a vector space and its effect on the components of a vector
- Linear transformations between two vector spaces
- Association of a matrix to a linear transformation
- Kernel and image of a linear application and their properties
- Basis of the image and kernel
Eigenvalue and eigenvector
- Definitions and properties
- Eigenvalues and eigenvectors of symmetric matrices
- Diagonalization
Quadratic forms
- Positive and negative (semi) definiteness
- Criteria for positive and negative (semi) definiteness
- Canonical form of a quadratic form
- Definiteness with equality constraints
Functions and Calculus of several variables
- Functions between Euclidean spaces
- Basics from domain, limits and continuity
- Directional derivative: partial derivative as particular case
- Gradient vector and Hessian matrix
- Tangent plane
- Homogeneous functions and Euler´s theorem
- Multivariable Taylor Polynomial

Basics of optimization
- Unconstrained optimization: first and second order conditions
- Constrained optimization: Lagrange’s multipliers

Ordinary differential equations
- Linear differential equations of order n: homogeneous and non-homogeneous case
- First order linear differential equations: constant and variable coefficients
- Bernoulli differential equation
- Linear differential equations of order greater than 1 with constant coefficients

Reference books:

- J. Bergin, Mathematics for Economists with Applications, Routledge, 2015, ISBN: 9780415638289 (Chapters 2-10-11-16-17)
- S. Lang, Linear Algebra, Springer-Verlag, 3rd Ed., 1987,ISBN 0-387 -96412-6 , (Chapters 1-2-3-4-6)
- C. P. Simon, L. Blume, Mathematics for Economists, Norton, 1994, ISBN-10 ‏ : ‎ 0393117529 (Chapters 14-17-18)
- Lecture support notes made available on the course platform

The supplementary course materials (lecture slides, any recordings, recommended readings, etc.) are available on the following page: https://classroom.google.com/c/ODA1NTY1Njg3MjQz

- S. Lang, Introduction to linear algebra,, Springer-Verlag , 2rd Ed., 1986 (Printed on acid-free paper)
(Chapters 1-2-3-4-5-6-7-8-9)

- C.L. Byrne, A First Course in Optimization, 2015, CRC Press Taylor & Francis Group, ISBN 978-1-4822-2658-4 (Chapters 5-8)

The course will be delivered through a combination of the following teaching methods:
- Lectures
Formal classes where the instructor presents key mathematical concepts, theories, and techniques relevant to economics and finance. Lectures are designed to build foundational understanding and provide structured explanations of the course material.
- Interactive Exercises and In-Class Activities
During selected lectures or tutorials, students may engage in guided exercises or collaborative problem-solving tasks to reinforce comprehension and encourage active participation.
- Assignments and Homework
Regular problem sets will be assigned to help students consolidate their understanding and apply the techniques learned in class to practical problems.

Attendance to the course is strongly recommended.

The exam is conducted through a written test at the end of the lectures, lasting two hours, consisting of four questions, each of which may be subdivided into multiple points. The questions can be theoretical or practical and may include open-ended questions. In the case of open-ended questions, the student must provide comprehensive answers, explaining in detail the techniques employed and the methodologies used. For practical exercises, students are required to present all the mathematical steps necessary to reach the final solution, including any explanatory comments. The evaluation is on a scale from 0 to 30 with honors, with a score of at least 18 considered passing. Each test is evaluated based on the achievement of specific learning objectives. The student must demonstrate a solid methodological approach in tackling and solving the proposed problems, as well as the ability to critically select and apply the most suitable methodologies and models for solving specific situations. Additionally, the student must be able to correctly apply the most appropriate tools and procedures. The evaluation also takes into account the coherence of the answers, their consistency with the knowledge the student is expected to have acquired by the end of the course, and the use of precise and context-appropriate technical language. The exam format is the same for all students, whether attending or non-attending

QUESTION 1

1a) Describe the process of diagonalization of a linear transformation.

1b) Let T be a linear transformation represented by the matrix A=[1,1,-1; 1,-3,1; 1,1,-3] with respect to the canonical basis:

- Calculate the eigenvalues.
- Calculate the eigenvectors.

1c) Determine whether the transformation T is diagonalizable and, in this case, write its diagonal form by performing all calculations.

In detail:
In point 1.a, level 1 of Bloom's taxonomy is achieved ("Remember," which indicates the minimum required level for the question). In point 1.b, levels 2/3 of Bloom's taxonomy are reached ("Understand/Apply," indicating an intermediate evaluation). In point 1.c, levels 4/5 of Bloom's taxonomy are achieved ("Analyze/Evaluate," representing the highest evaluation for the question)

• Topic 1: Review of Linear Algebra and Basic Calculus.   Linear  Transformations  Association of a matrix to a linear transformation . Kernel and image of a linear transformation and their properties Basis of the image and the kernel  (Weeks 1-2-3, September 21th -October 2th)
  • Books: 1) Course materials uploaded on Classroom (p. 1-131). 2) Serge Lang (Chapters 1-2-3-4)3) Bergin (Chapter 2)


• Topic 2 - Eigenvalues and eigenvectors. (October 13th).  Fundamental theorem on independence of eigenvalues. Diagonalization of a linear transformation. Quadratic forms (Weeks 4-5, October 5th -16th).
  • Books: 1) Course materials uploaded on Classroom (pp. 132-229). 2) Serge Lang (Chapter 5)3) Bergin (Chapter 10)


• Topic 3  Functions of several variables. Domain, brief introduction to limits and continuity , directional derivatives. Gradient and tangent plane, Hessian matrix, Gradient formula, Marginal product , Jacobian matrix homogeneous functions, Euler theorem, Taylor polynomial for functions of multiple variables  (Weeks 6-7-8, October 19th-November 6th).
  • Books: 1)Course materials uploaded on Classroom (pp. 230-308).2) Simon and Blume(Chapter 14)


• Topic 4: Introduction to Optimization,Unconstrained optimization: first and second-order conditions.
  Constrained optimization: Lagrange multipliers (Weeks 9-10, November 9th- November 20th).
  • Books: Course materials uploaded on Classroom (pp. 309-348)2) Bergin (Chapters 11-12)3)Simon and Blume (Chapters 17-18.4)


• Topic 5: Introduction to Ordinary Differential Equations  .First-order linear differential equations: constant and variable coefficients.  Bernoulli differential equation .
  Second-order linear differential equations (Weeks 11-12, November 23 th- December 4th).
  • Books: 1) Course materials uploaded on Classroom (pp. 349-409)2) Bergin (Chapter 16-17.4)