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class notes - ECO475 notes Steven Jens December 8, 2009...

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Unformatted text preview: ECO475 notes Steven Jens December 8, 2009 Contents 1 Intro 3 2 Solow Model 3 3 Representative Consumers Decision 5 3.1 Static Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Dynamic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2.1 Method 1: Value Function Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.2 Method 2: Guess and Verify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Dynamic Programming 8 4.1 Finite Horizon Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Theorem of the Maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3 Infinite Horizon Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Metric Spaces 10 5.1 Contraction Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Contraction Mapping Theorem proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3 SLP theorem 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.4 Blackwell proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 Sequential Problem 12 6.1 Properties of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.2 Euler, Transversality, and the Sequential Problem . . . . . . . . . . . . . . . . . . . . . . . 15 7 Continuous Time Dynamic Programming 16 7.1 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7.1.1 Example: continuous nominal growth model . . . . . . . . . . . . . . . . . . . . . . 17 7.2 Application: Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1 8 Competitive Equilibrium 20 8.1 Arrow-Debreu Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 8.2 Arrow-Debreu Equilibria are Pareto Efficient . . . . . . . . . . . . . . . . . . . . . . . . . 21 8.3 Social Planners Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8.4 Negishis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 8.5 Sequential Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 8.6 Proof that ADE SME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 9 Equilibrium 24 9.1 Neoclassical Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 9.2 Competitive Equilibrium in Neoclassical Growth Model . . . . . . . . . . . . . . . . . . . 26 9.3 First Welfare Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 9.3.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 9.4 Second Welfare Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9....
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class notes - ECO475 notes Steven Jens December 8, 2009...

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