MATHEMATICAL METHODS FOR ECONOMICS 2022-2023

The course has changed its structure. Hence we invite all students, in particular those who are not of the first year, to check carefully the new rules.

The web page of the same course of the previous year is here:

http://docenti.luiss.it/gozzi/teaching-english/mathematical-methods-for-economics-2021-22/

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Content of this page (apart from the up-to-date news about the course which are given above in inverse chronological order under the title “Hot News”).

(A) Teachers and time table

(B) Precourse, self-evaluation test and tutoring

(C) Office hours and forums

(D) Textbooks

(E) Preliminary program and effective one

(F) Exam rules

(G) The material for the first part of the course

(H) The material for the second part of the course

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(A) Teachers and time table.

Teacher: prof. Fausto Gozzi;

Assistants: dr. Leonardo De Carlo and dr. Antonio Veredice;

The course starts Monday, September 19 and ends Friday, December 2 with the following schedule:

Monday 17.45-19.15 (Room A403)

Tuesday 8.30-10.00 (Room A209)

Thursday 15.00-16.15 (Room A101)

Friday 15.15-16.15 (Online room 22)

Moreover the lecture lost on November 1, 2022 will be recovered:

Wednesday, November 2, h. 16.30-18.00 (Room A401)

(B) Precourse, self-evaluation test and tutoring.

The course starts giving for granted the basic arguments taught at the bachelor degree in Economics and Business at Luiss. These are similar to the ones taught at the standard mathematics courses of the italian “laurea triennale” in economics and similar subjects.

More precisely we assume that all students know :

– Calculus for functions of one variable (sequences and series, functions and their properties, derivatives and integrals in one variable, drawing graphs, searching extremals and zeros);
– Basic calculus for several variable functions (functions and their properties, basic topology, directional derivatives, gradient, differentials in several variables);
– Basic linear algebra concepts (vector spaces and their bases, linear dependence and independence of vectors, matrices, rank, determinant, linear systems, Rouché-Capelli Theorem, inner product).

It is our experience that some students may not remember well such topics, as in many cases they studied mathematics only the first year, while mathematics is a skill that needs to be trained frequently.

Till last year the University has provided a precourse (with a teacher) summarizing part of the basic arguments needed to be able to understand the lectures of the course. This year the precourse has not been organized (hence there is no teacher for it) but the material of the precourse of the previous year is still online (even if not updated) and the students are invited to look at the precourse material online at this site:

https://learn.luiss.it/course/view.php?id=1814

It is compulsory for all students to study the above material, unless they perfectly remember all the topics treated there. The textbook contains all such basic prerequisites in Chapters 1 to 14 and Appendices A1 to A6.

The students are also suggested to take the final self-evaluation test of the precourse. The students who get a grade strictly less than 18 in the self-evaluation test must contact TAs to agree about a recovery plan that can help them to understand the course.

(C) Office hours and forums

The office hours below are valid only during the course (i.e. from September 19, 2022 till December 2 2022).

After the course the office hours will be only by appointment (to agree personally or by email). During the course it is always possible to ask questions after the lectures and/or request supplementary office hours by email. In some cases office hours will be advertised in the section “Hot News” at the beginning of this web page.

Professor: Fausto Gozzi. Tuesday h.14.00-15.00 (face-to-face in the office 517 viale Romania). Email: fgozzi at luiss dot it.

Assistants.
Leonardo De Carlo: Wednesday 17-18, room T1 at the building in via Lisbona n.7. Email: leonardo_d3_carlo@protonmail.com

Antonio Veredice: Thursday after the lectures. Email: antonio.veredice@uniroma1.it

To interact with the professor and the assistants and/or among them, the students can also use the forums which are activated on the page of this course on the luiss learn website. Every student can post and/or ask questions on the general forum. This should be the standard way to ask questions about the course. Of course, for private questions, the students should use the email.

(D) Textbooks

Main textbook
Authors: Simon, Blume,
Title: Mathematics for economists
Editor: Norton

Other textbooks

Authors: Lawrence E. Spence, Arnold J. Insel, Stephen H. Friedberg Title: Elementary Linear Algebra
Editor: Pearson Education

Author: James Stewart
Title: Calculus (International Metric Edition)
Editor: Brooks/Cole; International Edition

Author: Michael C. Delfour
Title: Introduction to Optimization and Differential Calculus
Editor: SIAM

(E) Preliminary program and effective one

The Syllabus with the preliminary program is here:

https://www.luiss.it/cattedreonline/corso/M432/0/19FLME1/2022

Final program of the same course of last year

FinalContentofLecturesENGECFINFall2021

The program for this year will be a bit different.

Here you find the content of lectures, together with references, updated every 1-2 weeks

Week1ContentofLecturesENGECFINFall2022

Week2ContentofLecturesENGECFINFall2022

Week3ContentofLecturesENGECFINFall2022

Week4ContentofLecturesENGECFINFall2022

Week5ContentofLecturesENGECFINFall2022

Week8ContentofLecturesENGECFINFall2022

Week9ContentofLecturesENGECFINFall2022

Week10ContentofLecturesENGECFINFall2022

Final program of the course

FINALContentofLecturesENGECFINFall2022

(F) Exams rules

There will be no mid term exam.

The exam consists of a written test and an oral test. Both tests must be carried out face-to-face.

It is strongly suggested to take the exam in the winter session (December ot January).

For this reason there is a two-point bonus for those who, enrolled at the academic year 2022-23, take the exam within the February session.

On the Written Test
The exam program consists of two parts: the first part (P1) is about:

– Recap on the calculus of several variables;

– Implicit Function Theorem;

– Static Optimization.

The second part (P2) is about:

– Linear Operators and Spectral Theory

– Linear Dynamical Systems

– Nonlinear Dynamical Systems

The written test will be also divided in such two parts and will contain also some exercises on applied models.

It is possible to take separately the two parts in the same academic year.

The written test may include theory questions and is completely “closed-book”: it is not allowed to consult books and/or notes and the use of electronic devices of any kind is not allowed. People using any of such devices or violating rules will be carefully prosecuted.

The student who has achieved a sufficient mark in the written test can choose to:

(a) register the grade (without taking the oral exam);
(b) take the oral exam.

In case (a) the maximum grade that can be registered is 25 (therefore a student with a grade higher than 25 in the written test who decides not to take the oral exam could register a maximum of 25).

Important note.

In the case (a) the choice to register the grade without taking the oral exam must be validated by the professor. For some students the professor may decide that the oral exam is needed in any case.

On the oral exam

If the student decides to take the oral exam (case (b) above), the final mark, expressed out of thirty, will depend on the combined result of the two tests, and therefore may be higher or lower than the mark of the written test. The student may also be rejected (regardless of the grade of the written exam) if the oral exam is seriously insufficient. In this case, the student will have to repeat the written test in a subsequent exam.
The questions proposed in the oral exam concern both theory topics and practical exercises.
A sufficient grade obtained in the written test is maintained until the last useful exam session of the current academic year and no later than that date (without any exception). Therefore, it is possible to plan any oral exam in a subsequent session with respect to the one in which the written exam was taken, as long as it relates to the same academic year.
The student has the right to refuse the grade of the written test and/or the oral test. The refusal of the grade obtained in the oral test entails the need to take the written test again in any of the subsequent sessions foreseen in the exam calendar.

Part of the course, as emphasized in the content of lectures, is called “optional” in the following sense:
1) Students that wants to get a degree less or equal than 28 can avoid studying this part and will not receive questions about it. The grade will be at most 28. If the grade plus the bonus is greater than 28, the student can register only 28.
2) Students that wants to get a degree possibly more than 28 must study this part and will receive one or more questions about it. Then the degree may be more than 28.
If a student does not answer correctly to at least one question of the optional part, the grade cannot be more than 28, including the bonus.

It is possible, to take the oral exam with a reduced program, which we call the “basic program” which excludes the material of weeks 5 and 11, as indicated in the Content of Lectures. Students that wants to take the exam on the basic program can get a maximum grade of 25.

Hence the students have the following options for the oral exam

1) Standard program: the one given below, except the optional parts which are clearly indicated in bold and italic. With this program the maximum grade is 28.

2) Full program: the one given below, including the optional parts which are clearly indicated in bold and italic. With this program the maximum grade is 30 cum laude.

3) Basic program: the standard one minus the arguments of weeks 5 and week 11. With this program the maximum grade is 25.

The option can be communicated by the students directly at the beginning of the oral exam.

Further Rules

The two point bonus for those who, enrolled at the academic year 2022-23, take the exam within the February session does not change the above threesholds, i.e.:

– if a student only takes the written test the sum of the grade plus the bonus cannot exceed 25. The same if she/he takes the oral exam with the basic program

– if a student takes the oral test with the standard program the sum of the grade plus the bonus cannot exceed 28.

The written test can be repeated at each available date but every delivery cancels the previous grades.

The oral exam (or the registration) can be done at every available date (but in the same academic year), not necessarily the same of the written exam.

The oral exam cancels all the previous written/oral tests. Hence if a student does not take the vote of the oral exam she/he has to repeat also the written exam.

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(G) The material for the first part of the course

(G1) Slides

Weeks 1-2-3 (Sept 19-23 and 26-30 and Oct 3-7)

Week1-2EC2022SlidesFGMethodsFirstPart

Slides of Friday, September 30 and of week 3, with notes of the online lectures:

AnnotatedWeek3Part1EC2022SlidesFGMethodsFirstPart

AnnotatedWeek3Part2EC2022SlidesFGMethodsFirstPart

Whiteboards of the lectures of 26-27 September:

ECLavagne 26 Set ore 19

ECLavagne 27 Sett 22 ore 10

Weeks 4-5 (October 10-14 and 17-21)

Weeks 4 and 5 2022ECSlidesFGMethodsFirstPart

Annotated Friday 14 Oct 22 Week 4ECSlidesFGMethodsFirstPart

Applications of Week 5

Rapaoport model on cooperation/competition: page 1-26 of the slides below:

ECBeamerCooperazione2022

Pollution and taxes: see the slides below:

ECexamplepollution22

Videos shown at the lectures of Week 5.

1)Giorgio Vallortigara (Neuroscientist) minutes from 22.10 to 25.50 (but we suggest to listen to the whole video which is very interesting)

https://www.youtube.com/watch?v=0OU4cKGQ85U

2)Telmo Pievani (Biology and Phylosophy of Science) minutes from 0 to 3 and from 24.45 to 26.10 (but we suggest to listen to the swhole vidoo which is very interesting)

https://www.youtube.com/watch?v=leifNqRcBZw

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(G2) Tutorials

Tutorial1part1MMEF2021FG

SolutionsTutorial1part1MMEF2021FG

Tutorial2MMEFfall2021FGJacobianIFT

SolutionTutorial2MMEFfall2021FGJacobianIFT

Tutorial3MMEFfall2021UnconstrainedOptimization

SolutionsTutorial3MMEFfall2021UnconstrainedOptimization

Tutorial4MMEFConstrainedOptimization1EqualityFall2021

SolutionsTutorial4MMEFConstrainedOptimization1EqualityFall2021

Tutorial10MMEFConstrainedOptimization2EnvelopeFall2021