Mathematics for Economists

InÖnite cake eating problem

Again we will use the Envelope Theorem. We use the second formulation

V^0 (w) = u^0 (ws)=u^0 (c(w)).

V^0 (wc(w)) = u^0 (c(wc(w))),

wherec(w)is the optimal consumption atw.Hence the Euler equation

u^0 (c(w))=βu^0 (c(wc(w))).

Ifπis the stationary policy function which gives the consumption at cake

sizewthen

u^0 (π(w))=βu^0 (π(wπ(w))).