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### 204A-slides03d

Course: ECON 240A, Fall 2009
School: UCSB
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Word Count: 489

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Discrete-Time Digression: 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: T U = t 1u(ct ) = u(c1 ) + u(c2 ) + ... + T 1u(cT ) - Preferences: t =1 yt = f ( kt ) = ct + [ kt+1 (1 )kt ] - Initial condition: Take k1 &gt; 0 as given. - Budget equations: [Simplify the...

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Discrete-Time Digression: 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: T U = t 1u(ct ) = u(c1 ) + u(c2 ) + ... + T 1u(cT ) - Preferences: t =1 yt = f ( kt ) = ct + [ kt+1 (1 )kt ] - Initial condition: Take k1 > 0 as given. - Budget equations: [Simplify the example: A=L=1] - Finite horizon T => Capital is useless after period T => Terminal condition kT +1 = 0 . - Choice variables: k2 ,..., kT and c1 ,..., cT . 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: T t 1 t =1 t =1 for t=T; and k1 L = where kt+1 =0 T u(ct ) + t { f ( kt ) ct + (1 ) kt kt+1 } > 0 is given for t=1. - Optimality conditions: (i) L / ct = 0 for t=1,,T; (ii) L /kt = 0 for t=2,,T; (iii) satisfy the budget equations. - Claim: These conditions are analogous to the conditions of the Maximum principle. (3d)-P.1 Discrete-time Optimality Conditions T Repeat: L = t =1 t 1 (3d)-P.2 T u(ct ) + t { f ( kt ) ct + (1 ) kt kt+1 } t =1 i. Differentiate with respect to ct (for t=1,,T): L c t = t 1u ' (ct ) t = 0 => t = t 1u ' (ct ) ii. Differentiate with respect to kt (for t=2,,T): - Note that kt appears twice: in the period-t constraint and in the period-(t-1) constraint. L = [ { f ( k ) c + (1 ) k k }] + [ { f (k ) c t t t t+1 t 1 t 1 + (1 ) k t 1 kt }] k t k t t k t t 1 = { t f ' ( kt ) + (1 )} t 1 = 0 t t 1 = t [ f ' ( kt ) ] - Write as: iii. Differentiate with respect to the multipliers t (for t=1,,T): L t = f ( kt ) ct + (1 ) kt kt+1 = 0 => kt+1 k t = f ( k t ) k t ct Three parts each analogous to the Maximum Principle: (i) Optimal consumption => Marginal utility = Shadow value. (ii) Optimal capital => Change in multiplier proportional to return on capital. => Recovers the budget equations (iii) Optimal multiplier Important insight: Discrete-time Euler Equations [Result worth remembering for later.] Derivation: combine and t = t 1u ' (ct ) => u ' (ct ) = t / t 1 t 1 = t { f ' ( kt ) + (1 )} = t {1 + f ' ( kt ) } = t (1 + rt ) - Apply to any periods t and t+1: u ' (ct ) = t / t 1 = (1 + rt +1 ) t +1 / t 1 = (1 + rt +1 ) t +1 / t = (1 + rt +1 ) u ' (ct +1 ) Result: or => u' (ct ) = (1 + rt+1 ) u' (ct+1 ) u ' (ct ) = [1 + f ' (kt +1 ) ] u ' (ct +1 ) Marginal utility now = Discounted marginal utility next period * (1 + interest rate) where (1 + interest rate) = gross return on capital = 1+ MPK depreciation. Equivalent: => u ' ( ct ) u ' ( ct +1 ) = 1 + rt +1 = 1 + f ' (kt +1 ) Marginal rate of substitution now vs. next period = Gross return on capital. Intuition: Slope of indifference curve = Slope of budget line. Equivalent: u ' ( ct ) u ' ( ct +1 ) = (1 + rt +1 ) Marginal utilities declining over time iff (1 + rt +1 ) > 1 Consumption increasing. => Consumption growth is positive if and only if (1 + rt+1 ) > 1 . (3d)-P.3
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