Mathematics for Economists

InÖnite cake eating problem

Example

Solve the problem withu(c)=c 11 σσ

We guess thatV(x)=αx 1 ^1 σσ and the optimal policy isπ(x)=λx,with

0 <λ< 1.

αw

1 σ

1 σ=c 2 max[ 0 ,w]



c^1 σ

1 σ+

βα

1 σ(wc)

1 σ



.

Di§erentiating

cσ+βα(wc)σ( 1 )= 0 ,(βα)^1 /σ(wc)=c

(βα)^1 /σw=c



1 +(βα)^1 /σ



c= (βα)

 1 /σ

1 +(βα)^1 /σ

w= w

1 +(βα)^1 /σ

=λw