InÖnite cake eating problem

In the cake eating problem withu(x)=lnx

T

∑

t= 0

βtln(xt+ 1 xt)!max,x 0 =w.

Di§erentiating with respect toxt+ 1

βt

1

xt+ 1 xt+β

t+ 1 1

xt+ 2 xt+ 1 (^1 )=^0

1

ct

=β^1

ct+ 1

.