Mathe III

1. Mathe III Lecture 12

2. No lecture on Wed February 8th Thursday 9th Feb 14:15 - 17:00 Thursday 9th Feb 14:15 - 17:00

3. The Maximum Principle: A Reminder

4. Example capital production function depreciation rate consumption F(k) k

5. Example Two differential equations ink,π Solve for c

6. Example Another way: differentiate

7. Example Another way: X X

8. Example c k Another way: No t !!!!!!

9. Example c k Another way: k’

10. Example c k Another way: k* k’

11. Example c k Another way: k*

12. Example c k Another way: k*

13. Example c k Another way: k*

14. Example c k Another way: k*

15. Example c k Another way: Stationary point k*

16. Example c k Another way: k*

17. Example c k Another way: k*

18. Example c k k(0),c(0) ???? k(0), is given c(0), is chosen k → 0 c → 0 k(0) k*

19. Example c k k(0),c(0) ???? k(0), is given c(0), is chosen k → 0 c → 0 k(0) k*

20. Richard E. Bellman 1920-1983 Another approach to dynamic programming

21. Another approach to dynamic programming For a given timeτ < T define the problem:

22. Another approach to dynamic programming Lagrange: (equating the derivative w.r.t. zt to 0) But: ??????

23. Another approach to dynamic programming But: ?????? The Lagrangian of the original problem:

24. Another approach to dynamic programming But: ?????? The Lagrangian of the original problem:

25. Another approach to dynamic programming but this is the (first order) condition for maximizing the Hamiltonian

26. Another approach to dynamic programming Calculating the Bellman value functions is equivalent to the maximum principle (Hamiltonian)

27. Another approach to dynamic programming

28. Another approach to dynamic programming Backwards Induction