Mathematical Economics I

Course Type	Course Code	No. Of Credits
Discipline Elective	SLS2EC239	4

Course coordinator and team: Jyotirmoy Bhattacharya

Does the course connect to, build on or overlap with any other courses offered in AUD?

The course allows students to appreciate at a deeper level methods which would have been introduced in other core and elective courses such as Microeconomics I & II, Macroeconomics I & II and Econometrics and Data Analysis. It provides a foundation for other electives on mathematical methods and on areas of economics which use these methods intensively.

Specific requirements on the part of students who can be admitted to this course: (Pre-requisites; prior knowledge level; any others – please specify)

Students must have had mathematics at the 10+2 level followed by an undergraduate course on mathematical methods of economics or must have equivalent mathematical background.

No. of students to be admitted (with justification if lower than usual cohort size is proposed): As per SLS norms.

Course scheduling (semester; semester-long/half-semester course; workshop mode; seminar mode; any other – please specify):

As per course scheduling norms for MA economics program.

How does the course link with the vision of AUD?

It contributes to training in the rational study of society by enhancing students skills in mapping economic phenomena to consistent formal systems with clear rules of deduction. It also contributes to interdisciplinarity by showing how the same mathematical language can be used to describe very different situations, thus helping students appreciate the surprising structural similarities that can be often found underlying the apparent phenomenological diversity of the world.

How does the course link with the specific programme(s) where it is being offered?

The construction and criticism of models is central to the practice of economics and economic models are presented in the language of mathematics. The empirical testing of economic ideas too is based on mathematics in the form of mathematical statistics. The core courses in the M.A. programme make students aware of the basic minimum of mathematical methods used in economics.

This course along with other electives in the mathematical methods sequence is meant for students who wish to specialize in the use of mathematical techniques. They cover mathematical methods at a higher level of abstraction and formalism compared to core courses and aim to help students attain a higher level of mathematical maturity, specially with respect to the ability to discover and communicate rigorous proofs.

Course Details:

Summary: The pedagogical strategy of this course is to focus in depth on one area of mathematics instead of a whistle-stop tour through many different topics. It is hoped that sustained study of one major area will be more effective in improving the mathematical maturity of students by giving them time to master details and engage with more challenging material.

Linear algebra has been chosen as the core topic of the first course in this elective sequence because of its broad applicability in economics, with applications ranging from dynamical systems in macroeconomics to regression models in econometrics. It is also a prerequisite for multivariate calculus and optimization which are likely to be the topics of later electives on mathematical methods.

The core of this course will be a presentation of linear algebra from the viewpoint of abstract vector spaces and linear transformations. The emphasis would be on building structural and geometric intuition and not on rote calculations. It is expected that students would already have some prior knowledge of linear algebra. The course will reinforce this prior knowledge and use the axiomatic development of linear algebra as an introduction to the modern approach to the study of mathematical structures. The core theoretical material on linear algebra would be followed by applications and extensions based on student and instructor interests.

Aims

To improve students’ ability to learn and communicate rigorous mathematical arguments.
To make students aware of the major results of linear algebra and their interconnections.
To make students aware of the applications of linear algebra to economics.

Learning Outcomes

After having successfully completed this course students would be able to:

State the axioms defining a vector space and give examples of vector spaces drawn from different areas of mathematics and applications.
State the definition of linear transformations and give examples of linear transformation from different areas of mathematics and applications.
Demonstrate the ability to analyse problems in linear algebra using standard techniques such as induction on dimension, decomposition of spaces into direct sums and changes of basis to make manifest the structure of transformations.
Discover proofs for problems similar to those covered in course and to be able to communicate them with sufficient rigour and clarity.
Model problems in terms of vector spaces and transformations and apply standard results from linear algebra for their analysis.

Overall structure (course organisation, rationale of organisation; outline of each module):

The course follow the generally accepted structure of courses in intermediate linear algebra, with the major modules being:

Introduction to abstract vector spaces and linear maps.
Eigenvalues: diagonalization and triangularization.
Inner-product spaces.
Structure of operators on finite-dimensional inner-product spaces.

Applications and extensions

In modules (1)-(4) topics related to computation and applications to economics will be interleaved with the purely theoretical material. Module (5) will discuss applications and extensions that pull together material from the entire course.

Assessment Plan:

Assessment	Objective	Weight
Class Tests	To test understanding of proofs discussed in lectures and the text and to be able to apply similar proof methods to new problems.	35 % each for best two of three tests
Term Paper	To test ability to independently study mathematical literature and to be able to effectively communicate an overview of an area through a judicious choice of results, examples and counterexamples.	30%

Contents (week wise plan with readings):

Readings

[A], Axler, S. (2015) Linear Algebra Done Right, 3^rd ed., Springer Verlag
[S], Strang, G. (2007) Linear Algebra and Its Applications, 4^th ed., Cengage India
[TB], Trefethen, L.N an Bau, D.(1997)Numerical Linear Algebra, SIAM

Lecture Plan

Week	Topic	Reading
1.	Vector spaces: abstract definition and concrete examples. Subspaces and direct sums. Quotients.	[A], Ch. 1,
2.	Linear independence, spanning sets and bases. Dimension. Definition of linear maps	[A], Ch. 2 and 3
3	Linear maps: null space and range, the rank-nullity theorem. Invertibility and isomorphic spaces.	[A], Ch. 3
4.	Matrices. System of linear equations. LU decomposition.	[S], Ch. 2 [TB], Lectures 1,20
5.	Review of complex numbers. Polynomials: fundamental theorem of calculus (statement only), division algorithm and GCD.	[A], Ch. 4
6.	Eigenvalues. Triangularization and diagonalization.	[A], Ch. 5
7.	Inner products: inner products, norms, orthogonality.	[A], Ch. 6,
8.	The Gram-Schmidt algorithm. QR decomposition. The least-squares problem.	[TB], Lectures 6-8,11 [S], Ch. 3
9.	Operators on inner-product spaces: Spectral Theorem for self-adjoint and normal operators.	[A], Ch. 7, [TB], Lecture 4 [S], Ch. 5
10.	Positive operators, singular value decomposition	[A], Ch. 7, [TB], Lecture 4 [S], Ch. 6

The remaining weeks will be dedicated to extensions and applications depending on the interests of students and instructors. Some possibilities are:

Linear inequalities: Existence theorems including Farkas’s Lemma. Applications to optimization, linear models of production and finance. Reference: Vohra, R.V. (2005) Advanced Mathematical Economics, Routledge, Ch. 2
Convexity. Definition of convex sets and functions and their basic properties, the separating hyperplane theorem and its applications, duality and applications to optimization. Reference: Corbae, D., Stinchcombe, M. and Zeeman, J. (2009) An Introduction to Mathematical Analysis for Economic Theory and Econometrics, Princeton University Press, Chapter 5.
Dynamics in Discrete Time. Solutions and stability analysis of difference equations and linear rational expectations models. Reference: Miao, J. (2014) Economic Dynamics in Discrete Time, MIT Press, Chapters 1–2.
Dynamics in Continuous Time. Solutions and stability analysis of systems of differential equations. Reference: Shone, R. (2002) Economic Dynamics, 2^nd ed., Cambridge University Press, Chapter 4.
Pedagogy:
1. Instructional strategies: Lectures and problem sets
2. Special needs (facilities, requirements in terms of software, studio, lab, clinic, library, classroom/others instructional space; any other – please specify):

Classroom with a projector

1. Expertise in AUD faculty or outside : AUD Faculty
2. Linkages with external agencies (e.g., with field-based organizations, hospital; any others) NA