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markovchain

markovchain - ECON 714 MACROECONOMIC THEORY II TA TIM LEE...

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Unformatted text preview: ECON 714: MACROECONOMIC THEORY II TA: TIM LEE FEBRUARY 12, 2010 B & S = Envelope Condition Just a side note in case you’re wondering why we say the B&S theorem is just an envelope condition...remember the FE can be written as V ( k ) = max k u f ( k )- k + β V ( k ) with f.o.c. u f ( k )- k 0* = β V ( k 0* ) at the optimum (if interior). If we write the optimal policy as k 0* = g ( k ) , under the numerous assumptions we made about V it’s differentiable and g is also a continuously differentiable function by the Theorem of the Max. The equations collapse to V ( k ) = u [ f ( k )- g ( k )] + β V [ g ( k )] = u [ f ( k )- g ( k )]- β V [ g ( k )] and if we take the derivative of V w.r.t. k , V ( k ) = u [ f ( k )- g ( k )] f ( k )- g ( k ) + β V [ g ( k )] g ( k ) = u [ f ( k )- g ( k )] f ( k )- u [ f ( k )- g ( k )]- β V [ g ( k )] | {z } = g ( k ) = u [ f ( k )- g ( k )] f ( k ) . Whoopee! 1 Markov Chains I’m taking a lot of stuff from Ljungqvist and Sargent ( 2004 ) and Resnick ( 1992 ). Basic elements:- state space S = { s 1 , s 2 , . . . }- stochastic process { x t } , t = 0, 1, . . .- initial distribution: the row vector μ = ( μ 1 , μ 2 , . . . ) such that μ k ≥ 0, ∞ ∑ k = μ k = 1, μ k = P [ x = s k ]- transition matrix Π = ( π ij , i...
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markovchain - ECON 714 MACROECONOMIC THEORY II TA TIM LEE...

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