Cake eating problem

As the constraints are linear Slaterís condition holds. The Lagrange

function is

L=

T

∑

t= 1

βt^1 u(ct)+λ

w 1 

T

∑

t= 1

ct+wT+ 1

!

T

∑

t= 1

μtctφwT+ 1.

Di§erentiating

∂L

∂ct

= λβt^1 u^0 (ct)μt= 0

∂L

∂wT+ 1 = λφ=^0.

Ifφ=0 thenλ= 0 ,but in this caseβt^1 u^0 (ct)=0 which is impossible

asu^0 > 0 .Henceφ>0 which implies thatwT+ 1 = 0 .By the Inada

conditionct>0 hence μt= 0 .So

βt^1 ^1 u^0 (ct 1 )=λ=βt^2 ^1 u^0 (ct 2 ).