Objectives

The course provides an introduction to the mathematical theories and methods used in modern economics and finance. The objective of the course is to equip students with the mathematical toolkit required for later coursework.

Contents

Part I. Real Analysis

a. Mathematical Foundations

Set Theory, Mathematical Logic, Proof Theory

b. Linear Algebra

Vector Spaces, Matrix Algebra, Systems of Linear Equations, Quadratic Forms and Definiteness, Eigenvalues and Eigenvectors, Projections, Matrix Decompositions

c. Calculus

Topology, Differential Calculus, Integral Calculus

Part II. Stochastic Analysis

a. Probability Theory

Probability Foundations, Probability Measure, Random Variable and Distribution, Multivariate Random Variables, (Higher Order) Moments and Integration, Conditioning and Information

b. Statistics

Statistics and Random Sampling, Properties of Estimators, Stochastic Processes, Convergence Concepts for Stochastic Processes, Laws of Large Numbers and Central Limit Theorems, Large Sample Properties of Estimators, Classes of Estimators

Part III. Optimization

a. Static Optimization

Unconstrained Optimization, Constrained Optimization, Lagrange Method, Kuhn-Tucker Method, Saddle Point Method, Regularity and Sensitivity Analysis

b. Dynamic Optimization (optional)

Fixed-Point Theorems, Contractions, Bellman's Principle of Optimality, Bellman Equation, Dynamic Programming, Solution Methods, Regularity

References

The main course reference are the lecture notes available on the course website. They cover all relevant material for the course. In addition, mathematical concepts can be acquired in books, online courses, and other references. There are numerous textbooks on the mathematics for economics and finance. No single book is comprehensive.

The recommended textbooks on optimization are:

• (Good review, basic level) Simon, C. P. and L. Blume, 1994, Mathematics for Economists, Norton, New York.

• (Recommended level; more material than we can cover) Sydsaeter, K., P. Hammond, A. Seierstad, and A. Strom, 2005, Further Mathematics for Economic Analysis, Prentice Hall.

• (Good short book; focused on optimization) Dixit, A. K., 1990, Optimization in Economic Theory, 2nd Edition, Oxford University Press.

• (Good book; focused on optimization) Sundaram, R. K., 1999, A First Course in Optimization Theory, Cambridge University Press.

• (Advanced level) Fuente, A. de la, 2000, Mathematical Methods and Models for Economists, Cambridge: Cambridge University Press.

The recommended textbooks on probability and statistics are:

• (Recommended; more material than we can cover) Casella, G. and R. L. Berger, 2001, Statistical Inference, 2nd edition, Brooks Cole.

• (Good short book; focused on introduction to econometrics) Gallant, A. R., 1997, An Introduction to Econometric Theory, Princeton University Press.

• (Good math and statistics appendix) Green, W., 2010, Econometric Analysis, Princeton University Press.

Further references are available upon request.

Evaluation

First attempt

Exam:

Written 3h00 hours

Documentation:

Allowed

Calculator:

Allowed with restrictions

Evaluation:

Your grade will be determined based on the criteria laid out in the syllabus available on the Moodle course website. There will be optional tasks during the semester and a compulsory final exam. The optional tasks yield a bonus on the final grade, that is, the final grade will take into account the optional tasks only if they provide the student a higher grade than the final exam grade.

Retake

Exam:

Written 3h00 hours

Documentation:

Allowed

Calculator:

Allowed with restrictions

Evaluation:

Your grade will be determined based on the criteria laid out in the syllabus available on the course website and distributed in class.