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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 2016/2017

- Course ID
- SEM0058A
- Teacher
- Paolo Ghirardato (Titolare del corso)
- Modular course
- Year
- 1° anno
- Teaching period
- Secondo semestre
- Type
- Caratterizzante
- Credits/Recognition
- 6
- Course disciplinary sector (SSD)
- SECS-S/06 - metodi matematici dell'economia e delle scienze att. e finanz.
- Delivery
- Tradizionale
- Language
- Inglese
- Attendance
- Facoltativa
- Type of examination
- Scritto
- 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 general equilibrium theory and in dynamic macroeconomics, that is, some of the more fundamental results in the discipline of Economics.

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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 42 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 first semester (while the course is being taught), and it is articulated in a midterm administered in early November and a final exam administered in early December. The remaining four sessions (from January until September) comprise a single comprehensive examination. The details for each type of examination are provided below.

Midterm+Final (November+December): 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 January and September): 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]

Part I: Topology in Metric Spaces

-Metrics and norms, metric spaces and normed spaces

-Topological notions in metric spaces, open and closed sets, sequences and convergence

-Continuity and its characterization

-Connectedness, completeness and compactness, and their consequences for continuous functions

Part II: Some Mathematics for Economics-A brief look at constrained optimization, with equality and inequaliy constraints

-Convexity of sets and functions, and the separating hyperplane theorems

-Fixed point theorems and the contraction mapping theorem

Part III: Applications

-Basic results on General Equilibrium analysis

-A dynamic choice problem

## 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 required readings for the course:

-N.L. Carothers, Real Analysis, Cambridge University Press,Cambridge 1999, Chapters 1-8 and 10

-Massimo Marinacci and Luigi Montrucchio, Notes on Calculus on Vector Spaces, unpublished manuscript, chapters 9 and 10 and section 6.6

-C.D. Aliprantis and K.C. Border, Infinite-Dimensional Analysis, third edition, Springer-Verlag, 2007, part of Chapter 5- Oggetto:
## Class schedule

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## Note

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