Cake eating problem
VT 1 (w) = max
c 2 [ 0 ,w]
βT ^2 u(c)+VT(w c)
=
= max
c 2 [ 0 ,w]
βT ^2 u(c)+βT ^1 u(w c)
=
= βT ^2 u(c(w))+βT ^1 u(w c(w)).
By the envelope theorem if
f(p)$maxx g(p,x)=g(p,x(p))
then
df(p)
dp =
∂g
∂p(p,x(p))+
∂g
∂x(p,x(p))
dx(p)
dp =
∂g
∂p(p,x(p)),