Prescriptive AND
Theory + descriptive
3 comments on declining or lower
discount rates
Thomas Sterner
Country
Agency or sector
rate
Long-term rate
UK
France
Italy
Germany
HM Treasury
Commiss gén. du Plan
Central recommend
Bundesmin. Finanzen
Transportation
Water
3.5%
4%
5%
3%
6%
4%
4%
4%
4%
3.5%
7%
Declining > 30 yrs
Declining > 30 yrs
Spain
Netherlands
Sweden
SIKA* - transport
EPA
Norway
Office of Man & Budget
USA
Canada
Australia
EPA
Treasury Board
Office of Best Practice
2%–3%
8%
7%
Sens. check, >0%
Sens check, 0.5%–
3%
Theoretical
approach
SRTP
SRTP
SRTP
Federal refinancing
SRTP
SRTP
SRTP
SRTP
Gov borrowing
SOC
SRTP
SOC
SOC
Declining rates in France and UK
4.5%
4.0%
4.0%
3.5%
3.5%
3.0%
3.0%
2.5%
2.5%
2.0%
2.0%
1.5%
1.5%
1.0%
1.0%
0.5%
0.5%
0.0%
0
0.0%
0
50
100
150
200
250
300
100
200
300
400
500
Many Issues; Pick the important: I will
focus on 2
• Discounting depends on Growth. There will be no
growth in some sectors. We will not have ”more”
nature nor more time for our children.
• Some of the attraction of growth is that we
become richer than the neighbour. This is a
private motive but does not make sense socially
as the whole society gets richer.
• Disaggregation into Rich and Poor has effects
Two sectors with diff growth rates
C grows; E does not

W   e U (C , E )dt
 t
0
The appropriate discount rate r is then
d
 U C (C , E )
dt
r
U C (C , E )
Relative price effect >>> Typically
lowers discount in slow growth sector
DISCOUNTING and relative income
Ut  u  ct , Rt   u  ct , r (ct , zt )   v  ct , zt 
du/dc captures individual
partial benefit of more c. dv/dc
captures total effect of more c
3 Welfare Functions
Max : w   u  c , r (c , z )  e
p
T

0
d   v  c , z  e  d
T
0
{c0 ,...,cT }
Max : w   u  c , r (c , c )  e
s
T
0
{c0 ,...,cT }
Max : w   u  c  e  d
R
T
0
{c0 ,...,cT }

d   v  c , c  e
T
0

d
Intuition Arrow Dasgupta
•
•
•
•
Rat Race: Work/consume more to beat Jones.
But people will be positional in future too
Beat Jones’s now -->Lose in future
Same optimal growth part IFF
v2t (ct )   v1t (ct )
Defining degree of positionality
Ut  u  ct , Rt   u  ct , r (ct , zt )   v  ct , zt 
u2t r1t
t 
u1t  u2t r1t
We find same results and more..

v1t
1  v1t
1  v1t
 (t )    ln  
 t   0      ln 
t  v10 v10  v20
t  v10

s
1   t  

 
1   0  
We find same results and more..

v1t
1  v1t
1  v1t
 (t )    ln  
 t   0      ln 
t  v10 v10  v20
t  v10

s
• Assume increasing positionality
• Then
ρs > ρp
1   t  

 
1   0  
Assuming Constant Positionality
• Ramsey Discount rate > Optimal Rate
•
ρR = ρS + v12/v1 (cg)
• Generally ρR > ρS > ρp
THREE relevant Discount rates
1. The Privately optimal (assuming z unchanged)
2. The Socially optimal (assuming R unchanged)
3. Ramsey Rule which decision makers use
Comparing 3 discount rates
p

w
ct
1
1 v1t
p
   ln p
   ln
t w c0
t v10
p
v11
v12
v12

(

w
c) t
p
 
=  cg 
cg     g 
cg
p
w c
v1
v1
v1
v11  2v12  v22
v12
d  / dt
  
cg     g 
cg 
v1  v2
v1
1  t
s
 R    cv11 / v1 g     g
Private < Social < Ramsey
p

w
ct
1
1 v1t
p
   ln p
   ln
t w c0
t v10
p
v11
v12
v12

(

w
c) t
p
 
=  cg 
cg     g 
cg
p
w c
v1
v1
v1
v11  2v12  v22
v12
d  / dt
  
cg     g 
cg 
v1  v2
v1
1  t
s
 R    cv11 / v1 g     g