Mathe III Lecture 1 Mathe III Lecture 1

2 WS 2005/6 Avner Shaked Mathe III Math III

3 Tutorien für Mathematik III Im WS 05/06 Tutor: ChongDae KIM Mo. 11:00 Uhr Uhr HS N. Mo Uhr Uhr HS N. Di Uhr Uhr HS N. Di Uhr Uhr HS N.

4 / Homepage address with PowerPoint Presentations: /

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7 Bibliography K. Sydsaeter, P.J. Hammond: Mathematicsfor Economic Analysis Excellent, Comprehensive R. Sundaram: A First Course in Optimization Theory A. de la Fuente: Mathematical Methods and Model for Economists A. K. Dixit: Optimization in Economic Theory Mathematical,covers less than Sydsaeter & Hammond, more of dynamic programming New, theoretical, good in dynamics Short, concentrates on Lagrange, Uncertainty & Dynamic Prog. A. C. Chiang: Elements of Dynamic Optimization

8 Bibliography K. Sydsaeter, P.J. Hammond: Mathematics for Economic Analysis R. Sundaram: A First Course in Optimization Theory A. de la Fuente: Mathematical Methods and Model for Economists A. K. Dixit: Optimization in Economic Theory A. C. Chiang: Elements of Dynamic Optimization

9 Difference Equations (Sydsaeter.& Hammond, Chapter 20, Old Edition) Differential Equations (Sydsaeter.& Hammond, Chapter 21 Old Edition) Constrained Optimization (Sydsaeter.& Hammond, Chapter 18) Uncertainty (Dixit, Chapter 9) The Maximum Principle, Dynamic Programming (Dixit, Chapters 10,11) Calculus of Variations (Chiang, Part 2)

10 Difference Equations The state today is a function of the state yesterday The state at time t is a function of the state at t-1 Or: The state at time t is a function of the states of the previous k periods: t-1, t-2, t-3…,t-k, and possibly of the date t

11 The solution to the equation: is an infinite vector satisfying the above equation for

12 Example: Interest rate saving For a given x 0 :

13 Example (cntd.):

14 ? ?  t……?

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16 Mathematical Induction

17 Mathematical Induction Etc. Etc. Etc. Modus Ponens (Abtrennregel)

18  

19 ? 

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21 €1 for 1 period €1 for 2 periods €1 for t-1 periods € x 0 for t periods The solution to the difference equation: is: Example (cntd.):

22 First Order Difference Equations etc. etc. etc.

23 The difference equation x t =f(t, x t-1 ) has a unique solution with a given value x 0. Theorem: i.e. For each value x 0 there exists a unique vector, x 1, x 2, x 3, ……. satisfying the difference equation. Existence & Uniqueness

24 First Order Difference Equations Linear Equations with Constant Coefficients

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27 Example: A Model of Growth

28 Proportional Growth Rate

29 Equilibrium & Stability ??? An Equilibrium A Stationary State

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