Figure 1. Welfare of the current self (self 0) 1 α L LOP: self 0 1

s
α
sE : Market
sOP =s*: planner=self 0
𝛽𝑐
0
𝛽𝑙
1
𝛽𝑐
L
LE
1
Market
l*: planner
LOP: self 0
0
𝛽𝑐
Figure 1. Welfare of the current self (self 0)
1
𝛽𝑙
s
α
sOP the self in the far future
sE
Market
sOP =s*: planner=self 0
𝛽𝑐
0
𝛽𝑐
𝛽̂
𝛽𝑙
1
L
LE
1
Market
l*: planner
𝐿∞ : the self in the far future
0
𝛽𝑐
𝛽̂
Figure 2: Welfare of the self in the far future
1
𝛽𝑙