The theory of Green Accounting

Rui Motarmota@ist.utl.pt

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

• What is the SNA’s conventional net product (or income) measuring?– NNI = consumption + investment in capital stocks, i.e., .– Supply of goods, Y, equals demand of goods, C + I. Or resources equal uses.

0max ( ( )) t

cU C t e dt 00 0

( ) ( ( )) ( ), ( )dK tF K t C t K K

dt s.t.

Y C I

Welfare in the Ramsey Model

• Current-value Hamiltonian

• First Order Conditions

• Euler Equation

0max ( ) t

cU C e dt ( )dK

F K Cdt s.t.

( , , ) ( ) dKH C K U C

dt

0CH

Kd

Hdt

C KdH dC dK dH H C K H H Hdt dt dt dt

( , , )

dH dKdt dt

dHH U

dt

National Product and Welfare, Weitzman (1976) QJE

• Define Welfare at time t:

• Noting that the HJB equation rewrites as

dHH U

dt

s t

tW t U C s e ds ( ) ( ( ))

( ) ( )H t W t

Useful Expressions:

- Leibniz Integral Rule:

-

- Integration by parts:

( ) ( )

( ) ( )( , ) ( ( ), ) ( ( ), )

b z b z

a z a z

f b af x z dx dx f b z z f a z zz z z z

( )( ) ( ) ( ) ( ) ( ) ( )s

tz dz

t

dx tt x t a t x t a s e ds

dt

( ) ( ) ' ' ' 'df t g tf g fg f g fg fg

dt

Hamilton-Jacobi-Bellman equation

1( )s t

te ds

National Product and Welfare, Weitzman (1976) QJE

• The Hamiltonian is the stationary equivalent of the optimal path of utility

• Consider that , hence, the Hamiltonian is exactly the conventional net product measured in a closed economy with consumption as numeraire, i.e.,

, where net investment .

• 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.

( ) ( )H t W t s t s t

t tH t e ds U C s e ds ( ) ( ( ))

( )U C C

( )H t C I dKI

dt

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., .

• From the definition of welfare and using the HJB equation

• Moreover, from the HJB equation and the above expression,

• With consumption as numeraire,

• 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).

dK dt

( )dW dW dKW U Cdt dt dt

dW Net Investmentsdt

dH dKdt dt

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 , with nominal prices being

where is the marginal utility of money.

• Do changes in nominal net product indicate welfare changes?

• 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.

y pC qI

MUp q

, 0( )t

ry pC qI W

Nominal interest rate

Nominal and Real Economy• Define real net product , with real prices being

where is the price index that deflates nominal into real prices.

• Do changes in real net product indicate welfare changes?

• This is the definition of a Divisia price index. Hence,

• Growth in real net product means that welfare is increasing, if .

• 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

Y PC QI

,p qP Q

0( )t

RY PC QI W

Real interest rate

2 0d p p C pC pCPCdt pC

R

Y RQI W

0R

Chained CPI, Asheim (2007) EDE

• Real and nominal prices

• A Divisia price index can be approximated in discrete time by a chain index, which is re-based every year.

This implies that

• The usual CPI has a fixed base year, e.g., year 0.

• Chained CPI may be approximated by

Which implies that

1111 1t t tt

tt t t

p p Cp C CPIp C p C

0

0 0

tt

p C CPIp C

1 1 0 11

0

t t t tt

t t t t

CPI p C p CCPI

CPI p C p C

11

t tt

t t

p CCPI

p C

pP

11

1

tt

t tt

CPI CPI

0

tt

CPICPI

The usual way to deflate 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

• The model

0max ( ( )) t

cU C t e dt

C( ),I( ) (K( ), )t t S t t

s.t.K Iddt

C( ),I( ) (K( ), )t t S t t

00K( ) K

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:

un-sustainable development.

• Multisector results in real terms.– Real Net Product, – Genuine Saving,

–

0 0QI or Y

P C Q IY

Q I RY R W

Q I

Small Open Economy

• Include – stocks of commercial forests,– stocks of minerals,– welfare costs of air emissions,– value of technological progress.

• The capital stocks are :– Domestic man-made capital, – Net foreign capital held privately or by the government,

– Stock of commercial natural resources

• Production

K : ( , ,S)fK K

K I CFC

Q (R R )f f R X MK rK X M

S G(S) R Rd X

( ,R R , ) (R R ,S)d M d XK F K t M X C a f CFC

Time as a capital in the production function represents technological progress

Small Open Economy

• Households’ utility function depends on material consumption rate and (negatively) on the flow of emissions

• The vector of emissions depends on production and abatement expenditure.

• Maximize welfare subject to the above relations, and having as controls, consumption, , all forms of extraction, , abatement expenditure and .

• Conventional (SNA) NNI:

• Green Net National Income:

• Genuine Saving (Adjusted Net Saving):

• The value of time

(C) : ( ,E)U U C

E( ( ),a)F

( )C t R ( ), R ( ), R ( )d X Mt t ta( )t ( ) ( )M t X t

: fNNI C K K

R(Q f ) S e ER tY NNI Q

R(Q f ) St R tQ K Q NNI C Q

( )( )t R s tst

Q t F e ds

Small Open Economy – Human Capital

• The model of the small open economy can be extended to include human capital as a form of capital, , i.e. assuming .

• Following the world bank model, human capital changes according to

• 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.

K : ( , , ,S)H fK K KHK

( )HdK q m

dt

Education expenditures m are transformed in human capital by q(.)

Small Open Economy – Table of symbols( )C t Consumption rate at time t

( )U Utility

E( ) Rate of emissions of air pollutants

( )F Production function

a Abatement expenditure

, , ,iR i d X M Extraction of natural resources for domestic use, exports and from imports.

M X Imports - Exports

r Constant nominal interest rate

S Stock of resources

K Man-made capital,

RQ Resource price

R Constant real interest rate

(R R ,S)d Xf Cost of extraction of resource

e Marginal cost of abatement = Marginal damage cost

Rf Marginal cost of abatement

fK Net foreign capital

For more information check

Small Open Economy

• Starting from conventional SNA aggregates:– Deduct the damage from flow pollution emissions, – Deduct (add) the value of rents from resource depletion (or

not), – Add the value of technological progress .

e E

R(Q f ) SR

tQ

-20000

0

20000

40000

60000

80000

100000

120000

140000

1990 1995 2000 2005

Mill

ion

€

GNI

CFC

Air emissions

Forest Depl.

Tech. Progress

GNNI

Pot GNNI

GNNI, T=100

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] Best Low High

SO2 6872 3472 9972

NH3 7399 3699 10999

NOx 2040 1140 3040

VOC 1150 450 1550

PM2,5 44000 22000 64000

GNNI and GS in Portugal – Air Emissions

0

10

20

30

40

50

60

70

1990 1995 2000 2005

% o

f tot

al c

ost

SO2

NH3

NOx

VOC

PM2,5

GNNI and GS in Portugal – Forests

• National Forest Inventory 2005/06

• Average Volumes:

Coniferous

Eucalyptus

95/98 05/06[m3/ha]88.555 55

82.5

0

200

400

600

800

1000

1200

1400

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

10^3

ha

Conifers

Eucalyptus

GNNI and GS in Portugal – Forests

-200

-150

-100

-50

0

50

100

1990 1995 2000 2005

mill

ion

€ConiferousEucalyptus

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,

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

1990 1995 2000 2005

Mill

ion

€

T=17

T=20

T=40

T=60T=80

T=100

T=120

Potential Qt, T=17

GS in Portugal

• 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.

-10000

-5000

0

5000

10000

15000

20000

25000

30000

35000

40000

1990 1995 2000 2005

Mill

ion

€GS, no Qt

GS

GS, T=100

Potential GS