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Mathe III. Lecture 10. 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. Non Linear Programming. Kuhn – Tucker conditions. Non Linear Programming. Kuhn – Tucker conditions. Nonnegative variables.
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Mathe III Lecture 10
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
Non Linear Programming Kuhn – Tucker conditions
Non Linear Programming Kuhn – Tucker conditions
Nonnegative variables For a general problem:
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).
Nonnegative variables Example: Peak Load Pricing The producer maximizes:
Nonnegative variables Example: Peak Load Pricing
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
The Maximum Principle Optimization over time Stock – state variables Flow – control variables
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
derivative w.r.t. zt: derivative w.r.t. yt:
The two Lagrange conditions: The Hamiltonian:
The two Lagrange conditions: The Hamiltonian: From the envelope theorem:
The two Lagrange conditions: The Hamiltonian: From the envelope theorem:
The two Lagrange conditions: The Hamiltonian: From the envelope theorem: Similarly from the envelope theorem:
The two Lagrange conditions: The Hamiltonian: Similarly from the envelope theorem: