Download presentation

Presentation is loading. Please wait.

Published byErin Whitler Modified over 2 years ago

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:

Similar presentations

Presentation is loading. Please wait....

OK

Week 1.

Week 1.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google