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Mathe III Lecture 1

Mathe III Lecture 1. Mathe III Lecture 1. Math III. WS 2006/7 Avner Shaked. Mathe III. Mathematik III Chong-Dae Kim Donnerstag 9.15 – 10.45 Uhr – HS A Donnerstag 10.45 – 12.15 Uhr – HS A Donnerstag 12.15 – 13.45 Uhr – HS A Freitag 13.00 – 14.30 Uhr – HS G.

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Mathe III Lecture 1

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  1. Mathe III Lecture 1 Mathe III Lecture 1

  2. Math III WS 2006/7Avner Shaked Mathe III

  3. Mathematik III Chong-Dae Kim Donnerstag 9.15 – 10.45 Uhr – HS A Donnerstag 10.45 – 12.15 Uhr – HS A Donnerstag 12.15 – 13.45 Uhr – HS A Freitag 13.00 – 14.30 Uhr – HS G

  4. Homepage address with PowerPoint Presentations: http://www.wiwi.uni-bonn.de/shaked/ http://www.wiwi.uni-bonn.de/shaked/

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

  6. 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

  7. 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)

  8. 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

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

  10. Interest rate saving Example: For a givenx0:

  11. Example (cntd.):

  12.    t……? ? ?

  14. Mathematical Induction

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

  16.   

  17. ? 

  18. Example (cntd.): €1 for 1 period €1 for 2 periods €1 for t-1 periods € x0 for t periods The solution to the difference equation: is:

  19. First Order Difference Equations etc. etc. etc.

  20. Theorem: Existence & Uniqueness The difference equation xt =f(t , xt-1) has a unique solution with a given value x0 . i.e. For each value x0there exists a unique vector , x1 , x2 , x3 ,…….satisfying the difference equation.

  21. First Order Difference Equations Linear Equations with Constant Coefficients

  22. Example: A Model of Growth

  23. Proportional Growth Rate

  24. Equilibrium & Stability ? ? ? An Equilibrium A Stationary State

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