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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

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3 Non Linear Programming Kuhn – Tucker conditions

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4 Kuhn – Tucker conditions

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5 Nonnegative variables

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7 For a general problem:

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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).

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

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11 Nonnegative variables Example: Peak Load Pricing

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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

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

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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

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15 The Maximum Principle

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16 The Maximum Principle

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

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18 Define the Hamiltonian:

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19 The two Lagrange conditions : The Hamiltonian:

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

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21 Envelope Theorem

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

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

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

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25 The Maximum Principle:

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26 The Maximum Principle:

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27 The Maximum Principle:

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© Charles van Marrewijk, 2003 1 An Introduction to Geographical Economics Brakman, Garretsen, and Van Marrewijk.

© Charles van Marrewijk, 2003 1 An Introduction to Geographical Economics Brakman, Garretsen, and Van Marrewijk.

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