# Advanced Mathematics for Economics (HS24)

### Content

- In the first part of this course we will give a review of
**linear algebra**. - In the second part we look at
**functions and Taylor's formula**. - In the third part we study
**optimization problems**(unconstrained and constrained).

### Complementary literature

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.

### Organsational Matters

- Exercises can (
**not have to**) be solved.**If**you hand in the solutions of all exercises then 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, if (arithmetic mean of the preliminary mark and the examination mark) > (examination mark)
- the examination mark, otherwise

- If you do not hand in the solutions of the exercises, then your examination mark is your overall mark.

There is **no compulsory attendance** at all.

### Examination

When? **October 18, 2024 at 4:00 p.m.**

Where? **Auditorium (WWZ)**

Duration? **90 minutes**

**Allowed electronic means:**

simple pocket calculator (einfacher Taschenrechner, according to Merkblatt Hilfsmittel)

**Allowed non-electronic means:**

open-book

You can download a **Mock examination**.

## Course Arc

#### 1. Linear algebra

**Keywords**

vector, matrix, eigenvalue, eigenvector, diagonalization, linear transformation, spectral theorem for symmetric matrices, quadratic forms and definitness**Not relevant for examination**

generalized eigenvalues**Preparation**

Please study the following **Handout 1**.**Exercises**

Please solve the **following problems** and hand in your solutions of the marked questions by 20.09.2024 at the latest.**Timetable**__Mo, 02.09.24, 09:15-12:00 (Seminarraum S14, WWZ)__**1. Part** Theoretical foundations (presentation of the handout): **vector, matrix, linear maps and matrices, eigenvalue, eigenvector****2. Part** Self-study (of the handout), questions and work on the exercises

__Tue, 03.09.24, 09:15-12:00 (Seminarraum S14, WWZ)__**1. Part** Theoretical foundations (presentation of the handout): **diagonalization, spectral theorem for symmetric matrices****2. Part** Self-study (of the handout), questions and work on the exercises

__We, 04.09.24, 09:15-12:00 (Seminarraum S14, WWZ)__**1. Part** Theoretical foundations (presentation of the handout): **quadratic forms****2. Part** Self-study (of the handout), questions and work on the exercises

#### 2. Part: Functions and Taylor's formula

**Keywords**

gradient, Hesse matrix, differential, directional derivative, chain rule, implicit function theorem and derivative, Taylor formula, concave and convex functions, unconstrained optimization**Not relevant for examination**

Quasi-concave and quasi-convex functions**Preparation**

Please study the following **Handout 2.****Exercises**

Please solve the **following problems** and hand in your solutions of the marked questions by 20.09.2024 at the latest.**Timetable**__Thu, 05.09.24, 09:15-12:00 (Seminarraum S14, WWZ)__**1. Part** Theoretical foundations (presentation of the handout): **partial derivative, gradient, Hesse matrix, differential, chain rule**,** implicit function theorem**, **directional derivative****2. Part** Self-study (of the handout), questions and work on the exercises__Fri, 06.09.24, 09:15-12:00 (Seminarraum S14, WWZ)__**1. Part** Theoretical foundations (presentation of the handout): **Taylor formula, concave and convex functions ****2. Part** Self-study (of the handout), questions and work on the exercises

__Mo, 09.09.24, 09:15-12:00 (Seminarraum S14, WWZ)__**1. Part** Theoretical foundations (presentation of the handout): **unconstrained optimization****2. Part** Self-study (of the handout), questions and work on the exercises

#### 3. Part: Static Optimization

**Keywords**

local and global extremal points, unconstrained optimization, constrained optimization, Lagrange method, Karush-Kuhn-Tucker method**Preparation**

Please study the following **Handout 3.****Exercises**

Please solve the **following problems** and hand in your solutions of the marked questions by 20.09.2024 at the latest.**Timetable**__Tue, 10.09.24, 09:15-12:00 (Seminarraum S14, WWZ)__**1. Part** Theoretical foundations (presentation of the handout):** local and global extremal points, unconstrained optimization, constrained optimization, Lagrange method****2. Part** Self-study (of the handout), questions and work on the exercises__Wed, 11.09.24, 09:15-12:00 (Seminarraum S14, WWZ)__**1. Part** Theoretical foundations (presentation of the handout): **Karush-Kuhn-Tucker method****2. Part** Self-study (of the handout), questions and work on the exercises__Thu, 12.09.24, 09:15-12:00 (Seminarraum S14, WWZ)__**Only** Self-study (of the handout), questions and work on the exercises