Transcript interest rate

Doctoral Program and Advanced Degree in Sustainable Energy Systems
Doctoral Program in Mechanical Engineering
Doctoral Program in Environmental Engineering
Ecological Economics
Lecture 08
Tiago Domingos
Assistant Professor
Environment and Energy Section
Department of Mechanical Engineering
Present Value of Money
• Present Value
– Conversion of money flows to their present value
T
1
VFt
t


1

i
t 0
NPV  
•
•
•
•
•
NPV: Net Present Value
i: interest rate
VFt(t): value at instant t
If VF is in current prices then i is the nominal interest rate
If VF is in constant prices then i is the real interest rate
Nominal vs. real interest rates
The Fisher relation
1  nominal   (1  real)(1  inflation
For low rates, this is approximately equal to
nominal  real  inflation
)
Present Value of Utility
• Welfare
– Discrete time
1
U t 
t
t 0 1   
T
W
– Continuous time
T
W   e-tU t dt
t 0
Discount Rate
Justification and Components
•
The same monetary flow at different instants does not have the same
value (time preference)
– CONSUMPTION: Uncertainty
• Being alive in the future (individual vs. society)
• Preferences in the future
• Value of the benefit or the cost
– CONSUMPTION: Impatience
– PRODUCTION: Capital productivity (opportunity cost of capital)
•
Under certain conditions, the discount rate is equal to the real market
interest rate
consumption
discount rate
pure time
preference rate
C
r   
C
utility
discount rate
.
  pL
variation in survival
probability
per capita consumption
growth rate
elasticity of the marginal
utility of consumption
Turner et al. (1994), pp. 102-106.
Discount Rate
Calculation Methods
•
Pure rate of time preference
– “ A universal point of view must be impartial about time, and impartiality
about time means that no time can count differently from any other. In
overall good, judged from a universal point of view, good at one time cannot
differ from good at another. Hence...the [pure time] discount rate...must be
nought”
•
Broom (1992, p. 92) cit. in Pearce e Ulph (1998, p. 273)
Variation in survival probability
– Increase in mortality risk with age or
– Average population mortality rate
•
Elasticity of the marginal utility of consumption
– Individual saving behaviour
– Social judgement on income transfers between people
•
Per capita consumption growth rate
– Long run average of growth rate in real per capita consumption
Discount Rate
Elasticity of the Marginal Utility of Consumption
U(C)
U(C)
 0
0
C
C
Utility of
consumption
U(C)>0
U C  
a
C 1 
1 
Marginal utility of
consumption
Variation in marginal
utility of consumption
U’(C)>0
U’’(C)<0
U ' C   aC  
U ' ' C   aC   1
Elasticity of the marginal
utility of consumption

C
U ' ' C 
U ' C 

Discount Rate
Ethical Interpretation of the Elasticity of Marginal Utility
• Consider two families, with consumption resp. equal
to C1 and C2, with C1 = 2C2. The ratio of the marginal
utilities
of consumption is:

aC1


2
aC2 
• For different values of μ , this ratio is equal to:
μ
0,5
0,7
1,0
1,5
2,0
5,0
10,0
2- μ
0,7
0,6
0,5
0,35
0,25
0,03
negl.
Discount Rate
Estimate for the United Kingdom
.
r  p  L  c
p
.
L

c
r
Pure rate of
time
preference
Variation in
survival
probability
Elasticity of
marginal
utility
Per capita
consumption
growth rate
Consump.
discount
rate
Lower
bound
0,0
0,0
0,7
1,3
0,9
Best
estimate
0,3
-1,1
0,8
1,3
2,4
Upper
bound
0,5
-1,2
1,5
2,2
5,0
Estimate
Pearce, D., D. Ulph (1998). A social discount rate for the United Kingdom. In D.
Pearce, Economics and Environment, Edward Elgar, Cheltenham, UK, pp. 268-285.
Discount Rate
Criticisms
• The future is given a very low value
• Limits to growth
– It may not make sense to assume that economic growth will
continue indefinitely (i.e., at a constant rate, i.e.,
exponentially)
• Negative discount rates
– Countries with economic de-growth
– The future has more value than the present
Multisector Optimal Growth
• m-dimensional consumption bundle, including everything that
influences well-being.
– Includes all non-market commodities, e.g, produced at home, environmental
services, …
• n-dimensional capital vector:
– Includes man-made capital, natural resources, human capital
(education and knowledge) and foreign capital. Time is included as a
capital, to depict technological progress in production.
• Attainable production possibilities
C(t ),I(t )  S (K(t ), t )
• The model

max  U (C (t ))e  t dt s.t.
c
0
dK
I
dt
C(t ),I(t )  S (K(t ), t )
Criteria for Sustainability, Pezzey (2004) EDE
• An economy is sustainable at time t if and only if the representative
agent’s current utility does not exceed the maximum level of utility which
can be sustained forever from t onwards.
• One-sided sustainability test:
dY
QI  0 or
0 
dt
un-sustainable development.
• Multisector results in real terms.
– Real Net Income,
– Genuine Saving,
Y  PC  QI
QI
–
Consumption
dY
R dW
 RQ  I 
dt
 dt
Variation in
Real Net Income
Variation in
Welfare
Investment
Welfare Relationships
• Both Genuine Savings and Green NNI are related to future consumption.
• These relationships can be used to empirically check the theory.
• If genuine saving is negative (or green NNI deacreases) then current
consumption will decrease in the future.
Small Open Economy
• Include
– stocks of commercial forests,
– welfare costs of air emissions,
• The capital stocks are
K : ( K , K f ,S)
:
– Domestic man-made capital,
dK
 I  CFC
– Net foreign capital dt
held privately or by the government,
dK f
 rK fresources
 X  M  QR  (R X  R M )
– Stock of commercialdtnatural
r – interest rate
• Production
dS
 G(S)  R d  R X
dt
I  F ( K , R d  R M )  M  X  C  a  f (R d  R X ,S)
Small Open Economy
U (C) : U (C ,E)
• Households’ utility function
depends on material
consumption rate and (negatively) on the flow of emissions
• The vector of emissions
depends on production and
E( F ( ),a)
abatement expenditure.
• Maximize welfare subject to the above relations and having as
controls consumption,
, all forms of extraction,
C (t )
Rd (t ), R X (t ), RM (t )
, abatement
expenditure
andMtrade
balance
.
a(t )
(t )  X (t )
• Conventional (SNA) NNI:
NNI : C  K  K f
• Green Net National Income:

R
t
Y

NNI

(Q

f
)

S

e

E

Q
R
• Genuine Saving (Adjusted Net Saving):


Q K  Q  NNI  C  (Q  f R )  S  Q t
t
R
Small Open Economy – Table of symbols
C (t )
K
Consumption rate at time t
Man-made capital,
Kf
U ()
Utility
E()
Rate of emissions of air pollutants
e
F ()
a
Ri , i  d , X , M
MX
Marginal cost of abatement = Marginal damage cost
Production function
Abatement expenditure
Extraction of natural resources for domestic use, exports and
from imports.
Imports - Exports
r
Constant nominal interest rate
S
Stock of resources
R
Constant real interest rate
QR
Resource price
f (R d  R X ,S)
fR
Net foreign capital
Cost of extraction of resource
Marginal cost of abatement
Small Open Economy
• Starting from conventional SNA aggregates:
– Deduct the damage from flow pollution emissions, e E
– Deduct (add) the value of rents from resource depletion (or

R
not),
(Q  f R )  S
140000
120000
100000
GNI
CFC
Million €
80000
Air emissions
Forest Depl.
60000
Tech. Progress
GNNI
40000
Pot GNNI
GNNI, T=100
20000
0
1990
-20000
1995
2000
2005
Social costs vs Marginal abatement
Constant MDC
• Models point to measure emissions at the
– Marginal cost of abatement (MCA), or
– Marginal social cost (MSC) = Marginal benefit of abatement (MBA), a.k.a. Marginal
Damage Costs (MDC)
• Measurement away from the optimum
– c, over-polluting (assumed current state) => a is upper bound
– d, under-polluting => b is lower bound
GNNI and GS in Portugal – Air Emissions
• How to value a unit of emissions?
– Marginal benefit of avoided emission,
– Marginal cost of emission (MDC), or
– Marginal abatement costs?
• Marginal cost of emission per emitted pollutant [€2000/ton]:
[€2000/t]
SO2
NH3
NOx
VOC
PM2,5
Best
6872
7399
2040
1150
44000
Low
High
3472
9972
3699
10999
1140
3040
450
1550
22000
64000
GNNI and GS in Portugal – Air Emissions
70
60
SO2
% of total cost
50
NH3
40
NOx
VOC
30
PM 2,5
20
10
0
1990
1995
2000
2005
GNNI and GS in Portugal – Forests
• National Forest Inventory 2005/06
1400
1200
Conifers
Eucalyptus
10^3 ha
1000
800
600
400
200
• Average Volumes:
0
1990 1991 1992 1993
1994 1995 1996 1997
1998 1999 2000 2001
2002 2003 2004 2005
[m3/ha]
95/98
05/06
Conifers
88.5
82.5
Eucalyptus
55
55
GNNI and GS in Portugal – Forests
100
Coniferous
Eucalyptus
50
million €
0
1990
1995
2000
2005
-50
-100
-150
-200
The depreciation of commercial forests in Portugal is on average
10% of the contribution of forestry to national product (around
4%).
GS in Portugal
40000
35000
30000
GS, no Qt
25000
Million €
GS
20000
GS, T=100
15000
Potential GS
10000
5000
0
-5000
1990
1995
2000
2005
• Without the
value of time – Decreasing tendency throughout the period
-10000
and negative GS after 2002.
• With the value of time – Decreasing tendency until 2001, but GS are
always positive.
Sustainability Message
• In 1993, SO2 costs of emissions, which represent around 30% of total
emission costs, decreased substantially. Increases welfare but does not
counteract the loss of production.
What’s Missing?
•
•
•
•
•
•
•
The depletion of water resources.
The depletion of biodiversity.
Depletion of stocks of fish.
Inclusion of the value of ecosystem services.
Soil quality.
Distributional issues (intragenerational concerns).
...