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Nonlinear Programming and Kuhn-Tucker Conditions in Economic Optimization

This lecture covers advanced concepts in nonlinear programming, focusing on Kuhn-Tucker conditions and their application to economic problems. We explore the significance of nonnegative variables and delve into an example involving peak load pricing for electricity. The discussion emphasizes optimization over time, including the use of the Maximum Principle and Lagrange conditions. Key topics include production functions, control and state variables, as well as the Envelope Theorem. This lecture is essential for understanding the mathematical foundations of economic optimization.

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Nonlinear Programming and Kuhn-Tucker Conditions in Economic Optimization

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  1. Mathe III Lecture 10

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

  3. Non Linear Programming Kuhn – Tucker conditions

  4. Non Linear Programming Kuhn – Tucker conditions

  5. Nonnegative variables

  6. Nonnegative variables

  7. Nonnegative variables For a general problem:

  8. Nonnegative variables

  9. Nonnegative variables Example: Peak Load Pricing The price of a good (electricity) for time period iis given as pi The producer chooses how much to produce in each period (xi), and the maximal capacity of his plant (k). The total cost of producing(x1,…,xn) is C(x1,…,xn). The cost of capacity k is D(k).

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

  11. Nonnegative variables Example: Peak Load Pricing

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

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

  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. The Maximum Principle

  16. The Maximum Principle

  17. derivative w.r.t. zt: derivative w.r.t. yt:

  18. Define the Hamiltonian:

  19. The two Lagrange conditions: The Hamiltonian:

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

  21. Envelope Theorem

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

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

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

  25. The Maximum Principle:

  26. The Maximum Principle:

  27. The Maximum Principle:

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