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 = t 1 t = 1 T u ( c t ) = u ( c 1 ) + u ( c 2 ) + ... + T 1 u ( c T )- Budget equations: y t = f ( k t ) = c t + [ k t + 1 (1 ) 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 t , t=1,,T, for the T budget equations. - Standard Lagrangian expression: L = t 1 t = 1 T u ( c t ) + t t = 1 T { f ( k t ) c t + (1 ) 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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