Cake eating problem
IfT=3 then by the Euler equations
1
c 1 =β
1
c 2 =β
21
c 3
and the resource equation is
c 1 +c 2 +c 3 =w 1
This implies that
c 1 =
w 1
1 +β+β^2
,c 2
βw 1
1 +β+β^2
,c 3 =
β^2 w 1
1 +β+β^2
The value function is
V 1 =lnc 1 +βlnc 2 +β^2 lnc 3 =A 3 +B 3 lnw 1.