<p>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</p><p>+ t U(x0,x1,x2,……)=  ln (xt), where 01. t=0</p><p>Calculate her optimal level of consumption in each period.</p><p>1. Forming the Lagrangian: + t +   ln (xt) + (1- ln xt ) t=0 t=0</p><p> t  L =  -  = 0</p><p>xt xt</p><p> t+1  L =  -  = 0</p><p>xt+1 xt+1</p><p>2. Set the two ’s equal to each other.</p><p> xt+1=  xt where 01</p><p> x1=  x0 2 x2=  x1=  x0 t xt= x0</p><p>3. Substitute into budget constraint + t 1=  xt = x0 (1+  +….+ +….) t=0 = x0 1- </p><p> x0 = (1-)</p><p> x1 =  (1-) 2 x2 =  (1-) t xt =  (1-) Thus it converges. THE END</p>