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Unformatted text preview: Stochastic Optimization Prof. Lutz Hendricks August 10, 2009 L. Hendricks () Stochastic Optimization August 10, 2009 1 / 58 Stochastic Optimization We add shocks to the growth model. Recursive methods are needed. The resulting model is used to study business cycles asset pricing L. Hendricks () Stochastic Optimization August 10, 2009 2 / 58 Model The household (or planner) maximizes max E ∞ ∑ t = β t u ( c t ) (1) subject to k t + 1 = f ( k t , θ t ) & c t (2) Productivity shocks take on N discrete values θ t 2 f θ 1 , ..., θ N g (3) and follow a Markov process Pr ( θ t + 1 = θ j j θ t = θ i ) = Ω ij E denotes the expectation given information at date 0. L. Hendricks () Stochastic Optimization August 10, 2009 3 / 58 What are the choice variables? The planner cannot choose sequences f c t , k t g because we don&t know the realizations of θ t . He must choose state contingent plans . For every history s t = ( s , ..., s t ) (4) where s t = ( k t , θ t ) , choose c t = c ( s t ) (5) and k t + 1 = κ ( s t ) (6) L. Hendricks () Stochastic Optimization August 10, 2009 4 / 58 Two period example L. Hendricks () Stochastic Optimization August 10, 2009 5 / 58 Two period example The problem is max E 2 ∑ t = 1 β t & 1 u ( c t ) subject to k 2 = f ( k 1 ) & c 1 c 2 = f ( k 2 , θ 2 ) θ 1 known L. Hendricks () Stochastic Optimization August 10, 2009 6 / 58 Two period example Write out the expectation explicitly: max β u ( c 1 ) + N ∑ j = 1 Pr & θ 2 = θ j ¡ β u & c 2 & θ j ¡¡ + λ 1 [ f ( k 1 , θ 1 ) & c 1 & k 2 ] + N ∑ j = 1 Pr & θ 2 = θ j ¡ λ 2 & θ j ¡ ¢ f & k 2 , θ j ¡ & c 2 & θ j ¡£ The household chooses c 1 , k 2 and c 2 & θ j ¡ . L. Hendricks () Stochastic Optimization August 10, 2009 7 / 58 Budget constraints Note the constraint terms: Pr & θ 2 = θ j ¡ λ 2 & θ j ¡ ¢ f & k 2 , θ j ¡ & c 2 & θ j ¡£ (7) This looks like the household only needs to satisfy the constraint with some probability. Not true  the constraint must hold in every history. But this is just notation. De&ne ˜ λ 2 & θ j ¡ = Pr & θ 2 = θ j ¡ λ 2 & θ j ¡ to see this. L. Hendricks () Stochastic Optimization August 10, 2009 8 / 58 Firstorder conditions c 1 : u ( c 1 ) = λ 1 k 2 : λ 1 = ∑ Pr & θ 2 = θ j ¡ λ 2 & θ j ¡ f k & k 2 , θ j ¡ c 2 & θ j ¡ : β u & c 2 & θ j ¡¡ = λ 2 & θ j ¡ Euler equation: u ( c 1 ) = β ∑ Pr & θ 2 = θ j ¡ u & c 2 ( θ j ) ¡ f k & k 2 , θ j ¡ = β E ¢ u ( c 2 ) f k ( k 2 , θ 2 ) j θ 1 £ L. Hendricks () Stochastic Optimization August 10, 2009 9 / 58 Many periods L. Hendricks () Stochastic Optimization August 10, 2009 10 / 58 Many periods A history of length t is s t . The household chooses c & s t ¡ and k & s t ¡ to maximize ∑ s t p & s t ¡ β t u & c & s t ¡¡ subject to x & s t ¡ + c & s t ¡ = f & k & s t ¡ , θ & s t ¡¡ , 8 s t k & s t + 1 , s t ¡ = x & s t ¡ , 8 s t , s t + 1 The last constraint ensures that k & s t + 1 ¡ is the same for all s t + 1 .....
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This note was uploaded on 10/29/2009 for the course ECON 720 at UNC.
 '09
 LUTZHENDRICKS

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