# Mathematics for Economists

## Lecturer

The lecturer is Claire Naiditch from USTL Lille in Lille (FR).

### Contact

Claire Naiditch University of Lille, Faculty of Economics and Social Sciences
Bât. SH2, Cité scientifique
59655 Villeneuve d’Ascq Cedex
France
claire.naiditch@univ-lille.fr

## Pre-requisites

Students are supposed to have knowledge on standard Mathematics:

• Analysis of functions
• Derivation and integration
• Matrix calculus

## Course contents

Concavity and Convexity

• Concave and convex functions of a single variable
2. Quadratic forms: conditions for definiteness
3. Quadratic forms: conditions for semidefiniteness
• Concave and convex functions of many variables
• Quasiconcavity and quasiconvexity

Static optimization

• Optimization: introduction
• Optimization: definitions
• Existence of an optimum

Interior optima

• Necessary conditions for an interior optimum
• Local optima
• Conditions under which a stationary point is a global optimum

Optimization: Equality constraints

• Two variables, one constraint
• Optimization with an equality constraint: Necessary conditions for an optimum for a function of two variables
• Optimization with an equality constraint: Interpretation of Lagrange multipliers
• Optimization with an equality constraint: Sufficient conditions for a local optimum for a function of two variables
• Optimization with an equality constraint: Conditions under which a stationary point is a global optimum
• Optimization with equality constraints: n variables, m constraints
• The envelope theorem

Optimization: The Kuhn-Tucker conditions for problems with inequality constraints

• Optimization with inequality constraints: The Kuhn-Tucker conditions
• Optimization with inequality constraints: The necessity of the Kuhn-Tucker conditions
• Optimization with inequality constraints: The sufficiency of the Kuhn-Tucker conditions
• Optimization with inequality constraints: Nonnegativity conditions
• Optimization: Summary of conditions under which first-order conditions are necessary and sufficient

Dynamic optimization

• Calculus of variations
• Optimal control and the Maximum Principle

• Osborne, Matin J. 2016. Mathematical methods for economic theory (https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/toc/c)
• Hoy, M., Livernois, J., McKenna, C., Rees, R. and Stengos, T. 2011; Mathematics for Economics. 3rd edition. The MIT Press, Cambridge, Massachussets, USA
• M.D. Intriligator (2002), Mathematical Optimisation and Economic Theory, SIAM, ISBN 978-0-898715-11-8

## Learning outcomes

Upon successful completion of the course, students should be able:

• To formulate and solve a standard static optimization problem in micro- or macroeconomics
• To formulate and solve a standard dynamic optimizing problem in micro- or macroeconomics
• To understand the main properties of matrices that are used in economic analysis

## Organisation

The course consists of six lectures of two hours each. All lectures are concentrated at the beginning of the term. The lectures mainly concentrate on exercises and problems directly related to economics.

## Assessment

Students are evaluated through a written examination.