04 The Diamond model.pdf

The Diamond model

z Overlapping generations model

1. Assumptions

z Discrete time
z Every individual lives for only two periods
z Population growth rate: n Lt = (1+ n)Lt−1
young old
t t+1 t+2 …
young old

z The young supply the labor, earn the labor e, and divide e into
consumption and saving. The old own the capital, earn their capital e and
consume all wealth.

z The lifetime utility function:
C1−θ 1 C1−θ
U = 1t + 2t+1 , θ> 0, ρ> −1
t 1−θ 1+ ρ 1−θ

z The budget constraint:
1
C1t + C2t+1 = At wt
1+ rt+1



1
2. Household behavior

The Lagrangian:
1−θ 1−θ
C1t 1 C2t+1 ⎡ 1 ⎤
L = + + λ⎢At wt − C1t − C2t+1 ⎥
1−θ 1+ ρ 1−θ⎣ 1+ rt+1 ⎦
FOC:
⎧C −θ= λ
⎪ 1t
⎨ 1 −θ 1
⎪ C2t+1 = λ
⎩1+ ρ 1+ rt+1
1 −θ 1 −θ
⇒ C2t+1 = C1t
1+ ρ 1+ rt+1
1
θ
C2t+1 ⎛1+ rt+1 ⎞
⇒= ⎜⎟
C1t ⎝ 1+ ρ⎠

z The intuitive derivation of the Euler equation:

A small change ∆C in period t: Æ −(1+ rt+1)∆C in period t+1
−θ
The marginal utility of ∆C : C1t
1
The marginal utility of −(1+ r )∆C : C −θ
t+1 1+ ρ 2t+1
The optimization means that the cost of the change must be equal to the benefit:
1
C −θ∆C = C −θ(1+ r )∆C
1t 1+ ρ 2t+1 t+1
θ
C2t+1 1+ rt+1
⇔θ=
C1t 1+ ρ

z The optimal consumption path:
1
θ
1 ⎛1+ rt+1 ⎞
C1t + C1t ⎜⎟= At wt
1+ rt+1 ⎝ 1+ ρ⎠
2
1
A w (1+ ρ) θ