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## QUANTITATIVE METHODS FOR ECONOMICS - MODULO DI MATHEMATICAL ECONOMICS

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## Quantitative Methods for Economics - Mathematical Economics

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### Academic year 2017/2018

- Course ID
- SEM0058A
- Teacher
- Paolo Ghirardato (Lecturer)
- Modular course
- Year
- 1st year
- Type
- Distinctive
- Credits/Recognition
- 6
- Course disciplinary sector (SSD)
- SECS-S/06 - metodi matematici dell'economia e delle scienze att. e finanz.
- Delivery
- Formal authority
- Language
- English
- Attendance
- Optional
- Type of examination
- Written
- Prerequisites
- A good knowledge of basic calculus (Matematica Generale), of the foundations of probability calculus and statistical inference (Statistica), and possibly of the basics of optimization and linear algebra (Matematica per leconomia)
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### Sommario del corso

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## Course objectives

This course is a 6-credit module of a 2-module (12 credit) course aimed at introducing and developing many of the analytical tools which are used throughout theoretical and applied Economics. In this module, particular stress will be posed on the development of the analytical tools necessary for understanding and proving some of the basic results in Mathematical Economics and their application to dynamic Macroeconomics.

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## Results of learning outcomes

At the end of the course, the student is expected to be capable of:

-using the basic tools and results to pose, formalize and solve an economic model

-knowing the extent to which the results obtained in the previous step are dependent on the assumption that s/he has made about the behavior of the economic agents

-knowing therefore the extent to which the results are suggestive of real-world economic phenomena

-being able to think about possible and useful generalizations of the posited model

-being able to communicate such findings using appropriate and clear mathematical notation and language

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## Course delivery

The course-work is articulated in 48 hours of formal in-class lecture time, and in at least as many hours of at-home work solving practical exercises.

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## Learning assessment methods

Generalities:

The course grade is determined solely on the basis of written examinations. The objective of the examination is to test the student's ability to do the following:

1) Present briefly the main ideas, concepts and results developed in the course, also explaining intuitively the meaning and scope of the definitions and the arguments behind the validity of the results

2) Use effectively the concepts and the result to answer questions in basic general equilibrium analysis --e.g., proving the existence and structure of an equilbrium of an economy-- or similar issues --e.g., solving a simple dynamic consumption-saving problem

Practicalities:

There are 5 possible exam sessions in each academic year. The first session takes place during the second semester (while the course is being taught), and it is articulated in a midterm administered in late March and a final exam administered in late May. The remaining four sessions (from June until February) comprise a single comprehensive examination. The details for each type of examination are provided below.

Midterm+Final (March+May): Each of the two exams lasts 90 minutes, and it is articulated in 3 questions. Each question has an essay part, and some of the questions also have a more practical ("exercise") part. Each question is scored 0-10, so that the maximum score for each exam is 30. There is no minimum score in the midterm for admisssion to the final exam. Once both exams are graded, the final score in 60ths is computed, and it is transformed into 30ths, taking also into account the general class performance in the two exams (i.e, giving some weight to relative, as well as absolute performance).

Comprehensive examinations (4 sessions between June and February): Each exam lasts 165 minutes, and it is articulated in 6 questions. Each question has an essay part, and some of the questions also have a more practical ("exercise") part. Each question is scored 0-10, so that the maximum score for the exam is 60. The final score in 60ths is computed, and it is transformed into 30ths, taking also into account the general class performance in the two exams (i.e, giving some weight to relative, as well as absolute performance).

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## Support activities

Weekly homework sets will be assigned, and their solution will be posted and (if time allows) discussed in class.

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## Program

[Please refer to the course's web page, indicated in the "Altre Informazioni" for a more up-to-date syllabus and reference list]

The following is the planned syllabus for the course:

- Preliminaries [Carothers, Chapter 1; Sundaram, Chapter 1 and Appendix B]
- Basic notions of Topology in metric spaces [Sundaram, Chapter 1 and Appendix C; Carothers, Chapters 3-8]
- The Weierstrass and Contraction Mapping Theorems [Sundaram, Chapter 3, Section 12.4; Marinacci-Montrucchio, Section 6.6]
- Convexity and separation [Sundaram, Chapters 1 and 7; Aliprantis-Border, Chapter 5]
- The Maximum Theorem [Sundaram, Chapter 9]
- Dynamic Programming I: Finite-horizon problems. The consumption-savings problem [Sundaram, Chapter 11]
- Dynamic Programming II: The contraction mapping theorem, and infinite-horizon problems. The optimal growth model [Carothers, pp.97-99 for the Contr. Mapping Thm.; Sundaram, Chapter 12]

## Suggested readings and bibliography

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[Please refer to the course's web page, indicated in the "Altre Informazioni" for a more up-to-date syllabus and reference list]

The following are the suggested readings for the course (see the syllabus above for the specific chapters/sections used). The perusal of Stokey and Lucas's

*Recursive Methods in Economic Dynamics*(Harvard Univeristy Press, 1989), Chapters 1-5 is also suggested as it provides additional examples and motivation.- R. Sundaram,
*A First Cours on Optimization Theory*, Cambridge University Press, Cambridge 1996 - N.L. Carothers,
*Real Analysis*, Cambridge University Press, Cambridge 1999 - Massimo Marinacci and Luigi Montrucchio,
*Notes on Calculus on Vector Spaces*, unpublished manuscript - C.D. Aliprantis and K.C. Border,
*Infinite-Dimensional Analysis*, third edition, Springer-Verlag, 2007 (only part of Chapter 5)

- R. Sundaram,
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## Note

For more information, see:

http://sites.carloalberto.org/ghirardato/didattica/qme/qme.html- Oggetto: