Module 24-M-Opt Optimization for Quantitative Economics

Non-official translation of the module descriptions. Only the German version is legally binding.

This module introduces the student to the mathematical foundations of (i) convergence : continuity of functions and the underlying topological structure in metric spaces and (ii) convexity and optimization and their use in economic models. The students learn about the foundations of calculus in metric spaces and convex analysis. This should provide a deep understanding as well as techniques, which allow the students to deal with optimization problems with and without constraints.

The module consists in one lecture with the following content:

A. Convergence in metric spaces

Metric spaces, distance, norm on a vector space, open and closed sets, sequences in a metric space, continuity, uniform continuity. Compact sets in a metric space. Complete spaces, Contractions. Finite dimensional vector space. Complement to calculus : Frechet differentiability and the Implicit function theorem

B. Convexity and optimization

B.1. Convexity of sets and functions. Convex sets. Examples : budget sets, balls, production sets. Convex and concave functions, graph, epigraph and hypograph. Quasiconvex and quasiconcave functions. Strictly convex and quasi convex functions. Characterization of a convex funtion with its first order derivative. Characterization of a convex funtion with its second order derivative. Topological properties of convex sets. Projection on a closed convex set. Separation theorems. Orthogonality and polarity. The bipolar theorem. Farkas lemma.

B.2. Optimization under constraints

B.2.1. Unconstrained optimization. Global and local maximum (minimum). First order necessary conditions. Second order necessary condition and second order sufficient condition. Global maxima for concave (convex) functions. Examples.

B.2.2. Constrained optimization. Convexity conditions and Slater condition. The Kuhn-Tucker problem in convex programming (statement without proof). Applications of Kuhn-Tucker Theorem in consumer theory and producer theory. More examples of Applications of Kuhn-Tucker Theorem. Linear programming. Quadratic programming

Books:
Simon, C., Blume, L., Mathematics for Economists, (1994) Norton.De La Fuente, A., Mathematical Methods and Models for Economists, 2nd Ed. (2005) Cambridge University Press.

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