Mathematics for Economists

Cake eating problem

In the general case using the Fermat principle for the max

βTt^1 u^0 (cTt)=VT^0 t+ 1 (wTt+ 1 )

By the Bellman equation

VTt+ 1 (w) = max

c 2 [ 0 ,w]



βTtu(c)+VTt+ 2 (wc)



=

= max

u 2 [ 0 ,w]



βTtu(wu)+VTt+ 2 (u)



.

By the Envelope Theorem

VT^0 t+ 1 (w)=βTtu^0 (wu(w))=βTtu^0 (cTt+ 1 )

hence

βTt^1 u^0 (cTt)=βTtu^0 (cTt+ 1 )