34484 - Advanced Mathematics for Economics

Learning outcomes

At the end of the course, the student will have the knowledge of the mathematical concepts and techniques that are of central importance in modern economic analysis. The student will be able, in particular, to apply fruitfully tools and concepts from static optimization, dynamical systems, and dynamic optimization to a wide range of models which will be analyzed in the Economics courses.

Course contents

Preliminaries

Short revision of linear algebra: vector spaces, linear transformations, eigenvalues and eigenvectors.

Complex numbers and circular functions.

Short revision of multivariate calculus: partial differentiation, total differential; quadratic forms and their sign, second-order Taylor expansion.

Short revision of integral calculus.

 

Static models and static optimization

Solution of linear equilibrium models. Analysis of the solution of nonlinear models and dependence on parameters: Comparative Statics and the Implicit-Function Theorem. Implicit function differentiation.

Convex sets and their topological properties. Concave and quasiconcave functions.

Optimization in several variables. Unconstrained extrema: stationary points, Hessian matrix. Constrained optimization. Equality constraints and Lagrange multipliers. Inequality constraints and Kuhn-Tucker conditions.  Comparative statics for parametrized optimization problems: value function and Envelope Theorems

Some applications to Microeconomics.

Dynamic models

Discrete-time and continuous-time dynamical systems. First-order and higher-order difference equations. Systems of difference equations. First-order and higher-order differential equations. Systems of differential equations. Structure of the solution in the linear case. Nonlinear autonomous systems. Asymptotic behaviour, equilibria and stability. Local analysis by linearization. Saddle equilibria, stable and unstable spaces. Phase diagrams and qualitative analysis. Cycles and chaos.

Economic applications. Neoclassical growth models. An introduction to dynamic Perfect-Foresight models.

Intertemporal optimization

Dynamic optimization in discrete time. Some important examples of intertemporal optimization problems in Economics. Tools and solution methods: Optimal Control and the Maximum Principle, Dynamic Programming. Extensions to the stochastic case.

Elements of dynamic optimization in continuous time.

Economic applications. Optimal growth models. Optimal intertemporal consumption.

 

Readings/Bibliography

C. P. SIMON, L. E. BLUME, Mathematics for Economists, Norton, 1994.

K. SYDSAETER, P. HAMMOND, A. SEIERSTAD, A. STROM, Further Mathematics for Economic Analysis, Financial Times/Prentice Hall, 2nd Edition, 2008.

Other references. For a more introductory treatment of dynamic analysis and dynamic optimization techniques, students may also consider:

M. HOY, J. LIVERNOIS, C. McKENNA, R. REES, A. STENGOS, Mathematics for Economics, MIT Press, 2nd Edition, 2001 (Chapters 17 to 25);

A. K. DIXIT, Optimization in Economic Theory, Oxford University Press, 2nd Edition, 1990 (Chapters 10-11);

M.W. KLEIN, Mathematical Methods for Economics, Addison-Wesley, 2nd Edition 2002 (Chapters 13 to 15).

Teaching methods

Classroom lessons

The exercises and problems presented and discussed in the classroom are important to properly understand all the parts of the program. In the written exam, the student will be required to solve specific exercises using the tools and techniques learnt in the classroom.

Assessment methods

The examination consists of a written part and an oral part.

Office hours

See the website of Roberto Dieci