InÖnite cake eating problem

One can generalize the problem like we want to calculate

T

∑

t= 0

βtF(t,xt,xt+ 1 )!max

wherex 0 is given andT∞.If it is maximal and the optimum is at an

interior point then using that the same variablext+ 1 appears in two terms

βt

∂F

∂xt+ 1 (t,xt,xt+^1 )+β

t+ 1 ∂F

∂xt(t,xt+^1 ,xt+^2 )=^0.

This is a second order di§erential equation called Euler equation. If

T<∞then we also have that

FxT+ 1 (t,xT,xT+ 1 )= 0.

In this case one can solve this equation with backward induction.