30449 - MATHEMATICS - MODULE 2 (APPLIED MATHEMATICS)

A review of one variable differential calculus, integral calculus, linear algebra and complex numbers is suggested.

Modeling in applied science always requires mathematical and statistical tools. In particular, Economic Theory makes a large use of some classical mathematical topics such as Mathematical Programming and Dynamical Systems. The development of technology has increased computing power making manageable and convenient the use of many mathematical techniques. Notwithstanding, the implementation and the comprehension of any model needs the knowledge of the available tools. The course goal is providing the main mathematical and analytical tools which are used in models applied to Economics and Decision Theory.

• Differential calculus for functions of n real variables: partial derivatives, first order and second order differential.
• Implicit functions.
• Unconstrained optima. Constrained optima: classical programming and differentiable non linear programming.
• Dynamical systems: ordinary differential equations, finite difference equations. Glossary and properties.
• Solving separable and linear autonomous equations.
• Stability: the linear autonomous case, linearization method in the non linear autonomous case. One dimensional autonomous systems: phase diagram, stair step and cobweb diagrams.

• Recognize the mathematical model and its main properties.
• Identify a model and the assumptions that must hold in order that the model may be correctly applied.
• Reproduce the correct procedures  for solving a static optimization problem, for assessing the asymptotic behavior of a dynamical system or for finding its trajectories.

• Apply the learned calculus methods to  solve an optimization problem, to assess the asymptotic behavior of a dynamical system, etc;.
• Demonstrate the main properties of a model.
• Formulate in a formal way the assumptions that are needed for properly applying  the model.

• Face-to-face lectures
• Exercises (exercises, database, software etc.)

Teaching and learning activities for this course are divided into (1) face-to-face-lectures, (2) in class exercises.  

1. During the lectures convenient examples and applications allow students to identify the quantitative patterns and their main logical-mathematical properties.
2. The in class exercises allow students to properly apply the analytical tools in practice.

The students’ assessment is based on a written exam (100% of the final grade), consisting of exercises and open questions aimed to assess students’ ability to:

• Apply the analytical  tools in order  to solve optimization problems and differential/difference equations.
• Describe the notions and the methods learned.
• Justify in a proper manner the achieved conclusions.

Students can take a partial written exam and complete the written exam at the end of the course. In this case the weight is: 50% for the partial exam and 50% for the end of term exam.

• E. CASTAGNOLI, M. MARINACCI, E. VIGNA, Principles of Mathematics and Economics, Milano, dispense Egea, 2013, (ISBN 978-88-6407-192-3).
• Teaching notes authored by instructors are available on line.