Green accounting:
Green NNP and genuine savings
Lectures in resource economics
Spring 2004, Part 4
G.B. Asheim, nat.res. 4, updated 27.03.2004
1
National accounting gives a distorted picture of
„
savings if changes in stocks of natural and environmental resources are not included in national accounts
„
development of productive capacity if changes
in environmental amenities are not included.
Problems:
„
„
How to measure genuine (or comprehensive)
savings and what does it indicate?
How to measure green (or comprehensive) net
national product and what does it indicate?
G.B. Asheim, nat.res. 4, updated 27.03.2004
2
Solow claims in “An almost practical step
towards sustainability” (1992/1993):
„
Claim 1: Properly defined national product
measures the maximum current level of consumer
satisfaction that can be sustained forever …
[and is therefore] … a measure of sustainable
income given the state of the economy.
„
Claim 2: Properly defined and properly calculated,
this year’s net national product can always be
regarded as this year’s interest on society’s total
stock of capital.
G.B. Asheim, nat.res. 4, updated 27.03.2004
3
1
Hence, two questions:
„
Question 1: Does Green NNP measure
sustainable income (the maximum level of
consumption that can be sustained forever)?
Answer: Yes, approximately. Although Green
NNP tends to overestimate sustainable income.
„
Question 2: Can Green NNP be regarded as
interest on society’s genuine wealth?
Answer: No. Change in Green NNP can be
regarded as interest on genuine savings.
4
G.B. Asheim, nat.res. 4, updated 27.03.2004
Two models of optimal growth
Model (a): One sector model
K& t = Q( K t ) − Ct
K& t = Q( K t , Rt ) − Ct
Model (b): Dasgupta-Heal-Solow model
S&t = − Rt
∞
Dasgupta & Heal (1974),
Solow (1974)
S 0 = ∫ Rt dt
0
Assumptions on the technology: Q( K t , Rt )
Consequences of discounted utilitarianism.
5
G.B. Asheim, nat.res. 4, updated 27.03.2004
NNP in the one sector model (model (a))
( K 0 ) = C0 + K& 0
Green NNP at time 0: Y0 = Q
123
1
424
3
Net
product
Expenditure
Response to question 1:
A consumption level equal to Q( K 0 ) is sustainable by keeping K& t = 0 for all times.
Q
Q( K )
Response to question 2:
QK K
Interest rate: i = QK ( K ) ( < Q( K ) / K )
Interest on capital:
iK = QK ( K ) ⋅ K < Q( K )
123
K
Green NNP
G.B. Asheim, nat.res. 4, updated 27.03.2004
6
2
Savings in the one sector model (model (a))
Genuine savings at time 0: K& 0
Even though Green NNP is not interest on
capital (if the production function exhibits
decreasing returns to scale), it holds that:
Y&0 = QK ( K 0 ) ⋅ K& 0 = i0 K& 0
{
{
change
in
green
NNP
Interest
on
genuine
savings
This is a general result, holding in all models
under comprehensive accounting.
7
G.B. Asheim, nat.res. 4, updated 27.03.2004
NNP in the D-H-S model (model (b))
Interest rate:
i = QK ( K , R)
p = QR ( K , R)
p&
Hotelling’s rule: i =
p
Resource price:
Green NNP at time 0:
Y0 = Q( K 0 , R0 ) − p0 R0 = C0 + K& 0 + p0 S&0
1442443 144244
3
Net
product
Expendi ture
Response to question 1: I will return to this.
8
G.B. Asheim, nat.res. 4, updated 27.03.2004
Response to question 2: NNP as interest on
wealth in the D-H-S model (model (b))?
Assume for this page only that the production function
exhibits constant returns to scale (CRS):
Q( K , R) = QK ⋅ K + QR ⋅ R = iK + pR
Green NNP = Y = Q( K , R) − pR = iK < i (1
K + pS )
424
3
Wealth
„
„
Observation: Green NNP need not be equal to
interest on wealth, even under CRS.
Query: If net product = income, is iK national income (so that no income accrues to resource owners)?.
G.B. Asheim, nat.res. 4, updated 27.03.2004
9
3
Savings in the D-H-S model (model (b))
Genuine savings at time 0: K& 0 + p0 S&0 = K& 0 − p0 R0
Y& =
d
dt
(C + K& − pR ) = (Q( K , R) − pR )
d
dt
= QK ⋅ K& + QR ⋅ R& − p& R − pR&
= iK& − ipR (since QK = i, QR = p, and p& = ip )
= i ( K& − pR)
Independently of whether CRS is imposed, it holds that:
C& 0 + dtd (K& 0 − p0 R0 ) = i0 (K& 0 − p0 R0 )
142
4 43
4
144
42444
3
Interest on
genuine savings
change in
green NNP
10
G.B. Asheim, nat.res. 4, updated 27.03.2004
Genuine savings = PV of future changes in
consumption C& t + dtd (K& t − pt Rt ) = it (K& t − pt Rt )
t
− iτ dτ ⎞
⎛ &
⎜⎜ (K t − pt Rt )e ∫0 ⎟⎟ = 0
⎠
⎝
t
∞
− iτ dτ
K& 0 − p0 R0 = ∫ C& t e ∫0 dt
14243
0
1
4243
Genuine
− iτ dτ
⇒ C& t e ∫0
+
By integrating:
t
d
dt
savings
„
Present value of
future changes
in consumption
Observation: If development is sustained (non-decr.
cons.), then genuine savings is non-negative, and
Green NNP is non-decreasing since Y&0 = i0 (K& 0 − p0 R0 ) .
This holds under comprehensive accounting.
11
G.B. Asheim, nat.res. 4, updated 27.03.2004
Response to question 1: What about Green
NNP as a measure of sustainable income?
∞
Assume discounted utilitarianism: max ∫ U (Ct )e − ρt dt
0
t
Then:
− iτ dτ
U C (Ct )e − ρt = U C (C0 )e ∫0
Or equivalently: it = ρ −
Since
∞
d
dt
t
− iτ dτ
K& 0 − p0 R0 = ∫ C& t ⋅ e ∫0 dt ,
it follows that U C (C0 ) ⋅ (K& 0 − p0 R0 ) =
144424443
0
Genuine savings
in utility terms
G.B. Asheim, nat.res. 4, updated 27.03.2004
∞
(U C (Ct ) )
U C (Ct )
∫ dtd (U (Ct ) )e
− ρt
dt
1442443
0
Present value of
future changes in utility
12
4
What about Green NNP as a measure of sustainable income (cont)?
d
dt
(U (C )e ) = − ρU (C )e
− ρt
t
− ρt
t
+
d
dt
(U (Ct ) )e − ρt
By integrating, it follows that:
U (C0 ) +
∞
∫ dtd (U (Ct ) )e
0
− ρt
∞
dt = ρ ∫ U (Ct )e − ρt dt
Since U C (C0 ) ⋅ (K& 0 − p0 R0 ) =
0
∞
− ρt
∫ (U (Ct ) )e dt
d
dt
0
it follows that
U (C0 ) + U C (C0 ) ⋅ (K& 0 − p0 R0 ) = ρ ∫ U (Ct )e − ρt dt
1444442444443
1044244
3
Utility NNP
∞
This is Weitzman’s (1976) main result.
Utility interest on
utility wealth
(M.L. Weitzman, On the welfare significance of national product in a dynamic economy, QJE, 1976)
13
G.B. Asheim, nat.res. 4, updated 27.03.2004
What about Green NNP as a measure of sustainable income (cont)?
∞
Since
∫e
− ρt
dt = 1/ρ
0
it follows that Weitzman’s result can be rewritten as:
∞
∞
0
0
− ρt
− ρt
∫ (U (C0 ) + U C (C0 ) ⋅ (K& 0 − p0 R0 ))e dt = ∫ U (Ct )e dt
Interpretation: Utility NNP is the
stationary equivalent of future utility.
Let CtS denote the max. sustainable consumption at time t.
If actual consumption equals max. sustainable consumption
∞
at time 0; i.e., C0S = C0 , and (Ct )t =0 is the unique optimal
path, then genuine savings are positive: K& 0 − p0 R0 > 0. Why?
G.B. Asheim, nat.res. 4, updated 27.03.2004
14
Conclusions (on Green NNP as a measure of sustainable income)
Genuine savings can be positive, i.e, K& 0 − p0 R0 > 0 , even
though consumption exceeds max. sustainable consumption.
Y0 = C0 + K& 0 − p0 R0 > C0 does not imply that
consumption is sustainable since the resource price
depends on the path that will be followed.
(G.B. Asheim, Net national product as an indicator of sustainability, SJE, 1994)
Under discounted utilitarianism
& − p R >0
„ positive genuine savings K
0
0 0
& − p R )> 0
„ positive NNP growth Y&0 = i0 (K
0
0 0
is a necessary, but not sufficient, condition for sustainable
(J.C.V. Pezzey, One-sided Sustainability Tests with Amenities and
development.
Shifts in Technology, Trade and Population, forthcoming in JEEM)
G.B. Asheim, nat.res. 4, updated 27.03.2004
15
5