Cake eating problem

We have a cake of sizew 1 .We want to eat it inT periods. Our utility for

the consumption plan(ct)Tt= 1 is

T

∑

t= 1

βt^1 u(ct).

The constraints are

T

∑

t= 1

ct+wT+ 1 =w 1

wT+ 1  0 ,ct 0.