# Content

Economists (and other scientists) often study the changes over time in variables like national income, interest rate, oil production... The laws of motion governing these variables are usually expressed in terms of one or more differential or difference equation(s).

• In the first part of this course we will give a review of linear algebra, functions and Taylor's formula. We have a brief look at numerical methods for solving systems of linear and nonlinear equations and study Brouwer's fixed point theorem and study optimization problems (unconstrained and constrained).
• In the second part of this course we will study (ordinary) differential equations and special systems of differential equations and discuss the connections between differential equations, the classical calculus of variations and optimal control theory.
• In the third part of this course we will study difference equations and dynamic programming.

# Organsational Matters

(Weekly) lessons
Individual discussions and questions via Skype/Zoom

Performance review
Exercises must be solved and you should hand in the solutions. The solutions will be marked. All these marks together (arithmetic mean) give the preliminary mark. At the end of the term there will be an examination. Your overall mark will be the arithmetic mean of the preliminary mark and the examination mark.

# Examination

General Information
There is an examination (duration 90 minutes). The date of the examination will be fixed with the participants of the course.

Allowed electronic means:
simple pocket calculator (einfacher Taschenrechner, according to Merkblatt Hilfsmittel)

Allowed non-electronic means:
open-book

# Complementary literature

### General

Sydsaeter K., Hammond P., Seierstad A., Strom A.: Further Mathematics for Economic Analysis, Prentice Hall.Chiang,

Alpha C.: Fundamental Methods of Mathematical Economics, McGraw-Hill International Editions.

# Course Arc

### 1. Part: Fundamentals

Please solve the following problems and submit your solutions of the marked questions.

#### Week 11: Linear algebra and vector spaces

Keywords
vector, matrix, eigenvalue, eigenvector, diagonalization, linear transformation, vector space, metric, norm, inner product, complete space, continuous function, uniformly continuous function, convergent function, uniformly convergent function
Preparation

Please read the following Handout 1 and study (at least) the pages 1-8.

#### Week 13: Functions and Taylor's formula

Keywords
partial derivative, gradient, Hesse matrix, differential, directional derivative, chain rule, implicit function and derivative, Taylor formula, unconstrained optimization
Preparation
Meeting (Skype or Zoom)
Make an appointment

#### Week 14: Numerical methods

Keywords
Zero, fixed point, self mapping, Lipschitz continuous, Lipschitz constant, contraction, Banach's fixed point theorem, Jacobi matrix, Newton method
Preparation
Meeting (Skype or Zoom)
Make an appointment

#### Week 16: Brouwer's fixed point theorem

Keywords
continuous self mapping, compact and convex set, fixed point
Preparation
Please read the following Handout 4 and study (at least) the pages 1-6.
Meeting (Skype or Zoom)
Make an appointment

#### Week 17: Static optimization

Keywords
local and global extremal points, unconstrained optimization, constrained optimization, Lagrange method, Kuhn-Tucker method
Preparation
Meeting (Skype or Zoom)
Make an appointment

## 2. Part: Differentiable processes, differential equations and optimal control

Please solve the following problems and submit your solutions of the marked questions.

#### Week 18: Differentiable processes and differential equations

Keywords
differentiable processes, differential equations, systems of differential equations, autonomous systems, phase plane analysis, equilibrium point, linear systems with constant coefficients
Preparation
Please read and study the following Handout 6 (at least pages 1-11)
Meeting (Skype or Zoom)
Make an appointment

#### Week 19: Optimal control problems

Keywords
functional, increment, calculus of variations, fundamental theorem of calculus of variations, Euler equation, optimal control problems, performance measure, Pontryagin's Minimimum Principle
Preparation
(at least the first part ,,Functionals and calculus of variations)
Meeting (Skype or Zoom)
Make an appointment

## 3. Part: Discrete processes, difference equations and discrete time optimization

Please solve the following problems and submit your solutions of the marked questions.

#### Week 20: Discrete processes and difference equations

Keywords
differential and difference equations, linear difference equations, stability, systems of difference equations, autonomous systems, linear systems with constant coefficients
Preparation