Transcript Thermodynamics of Heterotrophic Organisms in the DEB Theory

The theory of Green Accounting
Rui Mota
[email protected]
Tel. 21 841 9442. Ext. 3442
Tiago Domingos
May 2009
What is Sustainable Development?
• Brundtland report (1987) – “Development that meets the needs of the present
without compromising the ability of future generations to meet their own need.”
– Intra- and inter-generational equity
– Anthropocentric
• Sustainability of what?
–
–
–
–
non-declining aggregate output or consumption,
non-declining utility,
non-declining aggregate resources (productive base),
non-increasing pollution, …
• Weak vs. Strong Sustainability
• We choose non-declining utility as the criterion for sustainable development
– some call this Weak Sustainability, but we don’t agree 
– this still misses the intra-generational component
• What is green net national income (GNNI) and what does it measure?
• What is genuine saving and what does it measure?
Welfare in the Ramsey Model
• Assumptions:
–
–
–
–
–
Discounted utilitarianism
Closed economy with no government
No technological progress
No population growth
Competitive economy

max  U (C (t ))e  t dt s.t.
c
0
dK (t )
 F ( K (t ))  C(t ), K (0)  K0  0
dt
• What is the SNA’s conventional net product (or income) measuring?
– NNI = consumption + investment in capital stocks, i.e., Y  C  I .
– Supply of goods, Y, equals demand of goods, C + I. Or resources equal uses.
Welfare in the Ramsey Model

max  U (C )e  t dt
c
0
• Current-value Hamiltonian
s.t.
H (C , K , )  U (C )  
dK
dt
HC  0
• First Order Conditions
• Euler Equation
dK
 F(K )  C
dt
d
   H K
dt
H  H (C , K , ) 
dH
dC
dK
d
 HC
 HK
 H
dt
dt
dt
dt

dH
dK
 
dt
dt

dH
  H U 
dt
National Product and Welfare, Weitzman (1976) QJE
• Define Welfare at time t:


W (t )   U (C ( s ))e
t
   s t 
ds
dH
   H  U   H (t )  W (t ) Hamilton-Jacobi-Bellman equation
dt
Useful Expressions:
- Leibniz Integral Rule:
b ( z ) f
 b( z )
b
a
f
(
x
,
z
)
dx

dx

f
(
b
(
z
),
z
)

f
(
a
(
z
),
z
)
a( z ) z
z a ( z )
z
z
s
-

  ( z ) dz
dx(t )
  (t ) x(t )  a(t )  x(t )   a( s )e t
ds
t
dt
- Integration by parts:
• Noting that


t
df (t ) g (t )
 f ' g  fg ' 
dt
e   ( s  t ) ds 
1

 f ' g  fg   fg '
the HJB equation rewrites as
National Product and Welfare, Weitzman (1976) QJE
 H (t )  W (t ) 


t
H ( t )e
   s t 

ds   U (C ( s ))e
t
   s t 
ds
• The Hamiltonian is the stationary equivalent of the optimal path of utility
• Consider that U (C )  C , hence, the Hamiltonian is exactly the conventional net
product measured in a closed economy with consumption as numeraire, i.e.,
H (t )  C   I , where net investment I 
dK
.
dt
• Welfare significance of net product or income:
– The maximum welfare attainable is what would be obtained if one could
consume all the net product in each period.
• From the HJB equation changes in net product measure changes in welfare.
Genuine Saving in the Ramsey Model
• What is the welfare significance of net saving/investment?
• Net saving/investment – Value of the portion of net product not used in
consumption, i.e.,  dK dt .
• From the definition of welfare and using the HJB equation
dW
dW
dK
 W  U (C ) 

dt
dt
dt
• Moreover, from the HJB equation and the above expression,
dH
dK
 
dt
dt
• With consumption as numeraire,
dW
 Net Investments
dt
• Net investment indicates changes in welfare (Negative net investment
indicates decreasing welfare).
– Measuring negative net saving at a point in time implies that future utility will be less
than current utility over some period of time (Hamilton, 2000 WB).
Nominal and Real Economy, Asheim and Weitzman (2001) EL
• Problems:
– Utility is not observable.
– Hamiltonian is in Utils as numeraire. The national accountant has to convert observable
market prices in money units to utils.
• Define nominal net product y  pC  qI , with nominal prices being
p
MU

, q

where  (t )  0 is the marginal utility of money.

• Do changes in nominal net product indicate welfare changes?
Nominal interest rate






r 

y  pC   
qI  W





• Positive nominal net product does not measure welfare improvements.
• Net product should be measured in constant consumption prices.
• Only consumption goods should be used as quantity weights in the price
index.
Nominal and Real Economy
• Define real net product Y  PC  QI , with real prices being
P
p

, Q
q
 (t )  0 is the price index that deflates nominal into
 where
real prices.
• Do changes in real net product indicate welfare changes?



Y  PC    





PC 



Real interest rate


 QI  R W





d  p  p  C   pC
 pC


0


dt   
2
 pC

• This is the definition of a Divisia price index. Hence, Y  RQI 
R

W

• Growth in real net product means that welfare is increasing, if R  0 .
• Net saving measures the instantaneous change in welfare. Both nominal
and real net saving have the same welfare significance.
• Change in Net Product is equal to interest on Net Saving. Check theory
Chained CPI, Asheim (2007) EDE
• Real and nominal prices
P
p

• A Divisia price index can be approximated in discrete time by a chain
index, which is re-based every year.



 t 1
 p C  pt 1  pt   Ct
1  

 CPI t 1  1
t
 p C
pt  Ct
This implies that

pt 1  Ct
 CPI t 1
pt  Ct
t


 t 1
 CPIt 1   t   CPI
t
 1
• The usual CPI has a fixed base year, e.g., year 0.
pt  C0
 CPI t
p0  C0
• Chained CPI may be approximated by

CPIt 1
p C
p C
 CPIt 1  t 1 0  t 1 t
CPIt
pt  C0
pt  Ct
CPI t


Which implies that
t
The usual way to deflate
CPI 0
is a good proxy
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 (variables in bold are vectors):
– 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 )
K(0)  K 0
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.
• This is implied by sustainability as forever non-declining utility.
• One-sided sustainability test:

QI  0 or Y  0  un-sustainable development.
• Multisector results in real terms.
– Real Net Product, Y  P  C  Q  I
– Genuine Saving, Q  I

– Y  RQ  I 
R


W
Small Open Economy
• Include
–
–
–
–
stocks of commercial forests,
stocks of minerals,
welfare costs of air emissions,
value of technological progress.
• The capital stocks are K : ( K , K f ,S) :

– Domestic man-made capital, K  I  CFC
– Net foreign capital held privately or by the government,

K f  rK f  X  M  Q R  (R X  R M )
– Stock of commercial natural resources

S  G(S)  R d  R X
• Production

K  F ( K ,R d  R M , t )  M  X  C  a  f (R d  R X ,S)  CFC
Time as a capital in the production function represents
technological progress
Small Open Economy
• Households’ utility function U (C) : U (C ,E) depends on material
consumption rate and (negatively) on the flow of emissions
• The vector of emissions E( F ( ),a) depends on production and abatement
expenditure.
• Maximize welfare subject to the above relations, and having as controls,
consumption, C (t ) , all forms of extraction, Rd (t ), R X (t ), RM (t ), abatement
expenditure a(t ) and M (t )  X (t ) .
• Conventional (SNA) NNI:
NNI : C  K  K f

Y  NNI  (Q  f R )  S  e E Q t
• Green Net National Income:
R


• Genuine Saving (Adjusted Net Saving): Q K  Q  NNI  C  (Q  fR )  S Qt
• The value of time

Q (t )   Fs e R( s t )ds
t
t
t
R
Small Open Economy – Human Capital
• The model of the small open economy can be extended to include human
capital as a form of capital, K H , i.e. assuming K : ( K , K H , K f ,S) .
• Following the world bank model, human capital changes according to
dK H
 q( m )
dt
Education expenditures m are
transformed in human capital
by q(.)
• In our expressions for GNNI and GS,
– The GNNI is not altered because education expenditures are already included in the
conventional NNI, as government spending. In the above model, this expenditures are
considered an investment and not a form of consumption. This implies a reallocation
of education expenditures that does not alter the value of NNI.
– In GS just add the value of education expenditures since they are a form of
investment.
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
For more information
check
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
not), (Q R  fR )  S
– Add the value of technological progress Q t .
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
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
0
1990 1991 1992 1993
1994 1995 1996 1997
1998 1999 2000 2001
2002 2003 2004 2005
• Average Volumes:
[m3/ha]
95/98
05/06
Coniferous
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%).
GNNI and GS in Portugal – Technological progress
• The integral may be approximated by,
45000
40000
35000
Million €
30000
25000
20000
15000
T=17
T=20
T=40
10000
T=60
T=80
T=100
5000
T=120
Potential Qt, T=17
0
1990
1995
2000
2005
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
-10000
• Without the value of time – Decreasing tendency throughout the period
and negative GS after 2002.
• With the value of time – Decreasing tendency until 2001, but GS are
always positive.