Calculate Her Optimal Level of Consumption in Each Period

1.28 An infinitely lived agent owns 1 unit of a commodity that she consumes over her lifetime. The commodity is perfectly storable and she will receive no more than she has now. Consumption of the commodity in period t is denoted xt , and her lifetime utility function is given by

+ t U(x0,x1,x2,……)=  ln (xt), where 01. t=0

Calculate her optimal level of consumption in each period.

1. Forming the Lagrangian: + t +   ln (xt) + (1- ln xt ) t=0 t=0

t  L =  -  = 0

xt xt

t+1  L =  -  = 0

xt+1 xt+1

2. Set the two ’s equal to each other.

xt+1=  xt where 01

x1=  x0 2 x2=  x1=  x0 t xt= x0

3. Substitute into budget constraint + t 1=  xt = x0 (1+  +….+ +….) t=0 = x0 1-

x0 = (1-)

x1 =  (1-) 2 x2 =  (1-) t xt =  (1-) Thus it converges. THE END