Mathematics for Economists

Cake eating problem

We can solve the problem as a dynamic programming problem. With

utility functionrt(ct)$βt^1 u(ct)and transition function

f(w,c)$wc.The Bellman equation is

Vt(w) = max

c 2 [ 0 ,w]

βt^1 u(c)+Vt+ 1 (wc)

.

VT(w) = βT^1 u(w).

Lett=T 1 .By the Inada condition we always have an interior solution

so derivative of the function behind the maximum is zero:

(cT =wcT 1 )

βT^2 u^0 (c)VT^0 (wc)= 0.

βT^2 u^0 (c)βT^1 u^0 (wc)= 0

βT^2 u^0 (cT 1 )=βT^1 u^0 (wcT 1 )=βT^1 u^0 (cT)

u^0 (cT 1 )=βu^0 (cT)