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

3. 3 The Maximum Principle: A Reminder

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

5. 5 Example Solve for c Two differential equations in k,π

6. 6 Example Another way: differentiate

7. 7 Example Another way: X X

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

9. 9 Example Another way: k c k’

10. 10 Example Another way: k c k’ k*

11. 11 Example Another way: k c k*

12. 12 Example Another way: k c k*

13. 13 Example Another way: k c k*

14. 14 Example Another way: k c k*

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

16. 16 Example Another way: k c k*

17. 17 Example Another way: k c k*

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

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

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

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

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

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

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

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

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

27. 27 Another approach to dynamic programming

28. 28 Another approach to dynamic programming Backwards Induction