1. 1

2. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N

3. 3 Non Linear Programming Kuhn – Tucker conditions

4. 4 Kuhn – Tucker conditions

5. 5 Nonnegative variables

6. 6

7. 7 For a general problem:

8. 8

9. 9 Example: Peak Load Pricing The price of a good (electricity) for time period i is given as p i The producer chooses how much to produce in each period ( x i ), and the maximal capacity of his plant ( k). The total cost of producing (x 1,…,x n ) is C(x 1,…,x n ). The cost of capacity k is D(k).

10. 10 Nonnegative variables Example: Peak Load Pricing The producer maximizes:

11. 11 Nonnegative variables Example: Peak Load Pricing

12. 12 The Maximum Principle Optimization over time Stock – state variables Flow – control variables A.K. Dixit: Optimization in Economic Theory, Oxford University Press, 1989. Chapter 10 * * stocks of capital goods consumption, labor supply flow variable production function

13. 13 The Maximum Principle Optimization over time Stock – state variables Flow – control variables

14. 14 The Maximum Principle Optimization over time additively separable utility function The marginal rate of substitution between periods 1,2 is independent of the quantitiy consumed in period 0

15. 15 The Maximum Principle

16. 16 The Maximum Principle

17. 17 derivative w.r.t. z t : derivative w.r.t. y t :

18. 18 Define the Hamiltonian:

19. 19 The two Lagrange conditions : The Hamiltonian:

20. 20 The two Lagrange conditions : The Hamiltonian: From the envelope theorem:

21. 21 Envelope Theorem

22. 22 The two Lagrange conditions : The Hamiltonian: From the envelope theorem:

23. 23 The two Lagrange conditions : The Hamiltonian: From the envelope theorem: Similarly from the envelope theorem:

24. 24 The two Lagrange conditions : The Hamiltonian: Similarly from the envelope theorem:

25. 25 The Maximum Principle:

26. 26 The Maximum Principle:

27. 27 The Maximum Principle: