204A-slides03d

# 204A-slides03d - (3d-P.1 Digression Discrete-Time...

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Unformatted text preview: (3d)-P.1 Digression: Discrete-Time Optimization [For now: As motivation for continuous time. For later: Preview of discrete-time macro.] • Consider optimal consumption and capital accumulation problem over T periods: - Preferences: U = β t − 1 t = 1 T ∑ u ( c t ) = u ( c 1 ) + β u ( c 2 ) + ... + β T − 1 u ( c T )- Budget equations: y t = f ( k t ) = c t + [ k t + 1 − (1 − δ ) k t ] [Simplify the example: A=L=1]- Initial condition: Take k 1 > as given. - Finite horizon T => Capital is useless after period T => Terminal condition k T + 1 = .- Choice variables: k 2 ,..., k T and c 1 ,..., c T . Finite number. • Optimization: Maximize utility subject to the budget equations. - Define T Lagrange multipliers λ t , t=1,…,T, for the T budget equations. - Standard Lagrangian expression: L = β t − 1 t = 1 T ∑ u ( c t ) + λ t t = 1 T ∑ { f ( k t ) − c t + (1 − δ ) ⋅ k t − k t + 1 } where k t + 1 = for t=T; and k 1 > is given for t=1....
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204A-slides03d - (3d-P.1 Digression Discrete-Time...

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