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Unformatted text preview: 1. Consider the standard cake eating problem with a modification of the transition equation for the cake W = RW c (1) if R > 1. the cake yields a positive return, whereas if R (0 , 1) the cake depreciates. (a) If preferences are of the form u ( c ) = c 1 σ 1 σ , calculate the optimal sequence of consumption. What happens when σ = 1? and when σ = 0? To solve the constrained optimization problem, need to find the sequence of { c t } T 1 { W t } T +1 2 that satisifies V t ( W ) = max T X t =1 β ( t 1) u ( c t ) (2) subject to the transition equation (1), which holds for t = 1 , 2 , 3 ,...,T. Also, subject to nonegative constrants on comsuming the cake given by c t ≥ 0 and W t +1 ≥ Adding period 0 consumption, the solution to (2) can be rewitten as V t +1 ( W ) = max { c t } T t { u ( c t ) + βV t ( W ) } (3) By using the Legrange multipliers, combining the first order conditions we obtain the Euler equation that is necessary condition of optimality for any t . u ( c t ) = R β u ( c t +1 ) (4) By letting u ( c ) = c 1 σ 1 σ , ⇒ u ( c ) = c σ , subsitute in (4) obtain the following: ( c t ) σ = R β ( c t +1 ) σ (5) c t +1 = ( R β ) 1 σ c t (6) Let ˜ β = ( R β ) 1 σ , and using properites of geometric sequence then (6) can be expressed as consumpution of period T + 1 in terms of period 0. c t +1 = ( ˜ β ) T c (7) Using the sequence of transition equations from (1), multiply each equation by R T i where i = 0 , 1 ,...,T ....
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This note was uploaded on 12/14/2011 for the course FIN 5515 taught by Professor Staff during the Spring '10 term at FSU.
 Spring '10
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