A no-arbitrage rule of consumption and a golden rule of capital accumulation are derived under the assumptions that the satisfaction from consumption is spoiled by environmental degradation caused by industrialisation but moderated by cleaning up and greening operations.consumption rule, capital accumulation, environmental degradation, greening

Levy, Amnon

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
Greening the Neoclassical Optimal  
Economic Growth Model 
 
 
 
 
 
 
Amnon Levy 
 
 
 
 
 
 
 
 
 
 
WP 02-05 
 
University of Wollongong 
Department of Economics 
Working Paper Series 2002  
 
GREENING THE NEOCLASSICAL OPTIMAL ECONOMIC GROWTH MODEL 
 
 
 
 
by 
 
Amnon Levy 
University of Wollongong 
 
 
Abstract 
A no-arbitrage rule of consumption and a golden rule of capital accumulation are derived 
under the assumptions that the satisfaction from consumption is spoiled by 
environmental degradation caused by industrialisation but moderated by cleaning up and 
greening operations.  
 
 
 
 
 
 
 
 
Corresponding Author: Amnon Levy, Department of Economics, School of Economics 
and Information Systems, Faculty of Commerce, University of Wollongong, 
Wollongong, NSW 2522, Australia. Tel: 61-2-42213658. E -mail: 
amnon_levy@uow.edu.au 
   1
1. Introduction 
 
The neoclassical optimal growth model constructed by Ramsey (1928) and refined by 
Cass (1965), Koopman (1965) and Phelps (1966) is extended to include environmental 
aspects. The extension is based on two premises. The first is that, unable to migrate, the 
instantaneous satisfaction of the representative agent from consumption is spoiled by the 
rate of degradation of her home physical environment. The second is that the rate of 
environmental degradation rises with the level of industrialisation of the economy, but in 
a degree moderated by cleaning up and greening operations. These premises are 
formalised and integrated into the neoclassical optimal growth model in section 2. The 
properties of the resultant environment-Augmented no-arbitrage rule of consumption and 
golden rule of capital accumulation are discussed in section 3. The model is further 
extended in section 4 to include a natural rehabilitation process. 
 
2. The environment-augmented growth model 
It is assumed that the instantaneous satisfaction (u) of the representative agent from 
consumption (c) is depreciated by the rate of degradation (0 1 £ £ x ) of her physical 
environment. That is, 
u t x t u c t
~
( ) [ ( )] ( ( )) = - 1                 (1) 
where,  u is increasing but concave in c, u u
~
=  when  x = 0 (i.e., pristine environment) 
and u
~
= 0 when  x = 1 (i.e., completely degraded environment that cannot sustain human 
life). It is assumed that, although some of her members of society can migrate (e.g., the 
more daring, the more talented and the younger), the representative agent is confined to 
her initial location. 
   It is also assumed that the rate of environmental degradation rises with the level of 
industrialisation of the economy, represented by the capital-labor ratio (k ), but in a 
degree moderated by the per capita (i.e., the representative agent’s) spending on cleaning 
up and greening operations (g):   2
x t g t k t
.
( ) [ ( )] ( ) = - g g 0 1                   (2) 
where g0 is a positive scalar denoting the adverse marginal effect of industrialization on 
the environment when cleaning up and greening operations are not implemented, and g1 
is a positive scalar representing the marginal moderating effect of cleaning up and 
greening operations on the rate of degradation of the environment. In this context, 
[ ( )] g g 0 1 - g t  represents the environment degradation factor. When the representative 
agent’s spending on cleaning up and greening operations rises with her income and when 
the cleaning up and greening operations have a considerable moderating marginal effect 
on the environment degradation factor, so that for a sufficiently large clean-up spending 
the environment degradation factor is negative, this motion equation is consistent with the 
environmental Kuznets curve hypothesis of an inverted U-shaped relationship between 
the quality of the environment and per capita income (Shafik and Bandyopadhyay, 1992; 
Seldon and Song, 1994; Grossman and Krueger, 1995). 
   The representative agent’s spending on cleaning up and greening operations modifies 
the conventional motion equation of her capital stock as follows: 
k t f k t c t g t n k t
.
( ) ( ( )) ( ) ( ) ( ) ( ) = - - - + d .          (3) 
where  f  is a concave function yielding the per capita income,  d  is the capital 
depreciation rate, and n is the population growth rate. 
   By incorporating these premises into the standard neoclassical optimal growth model, 
the problem of the representative agent having a fixed time-preference rate  r  can be 
presented as choosing the joint trajectories of consumption and cleaning-up and greening 
spending so as to maximize her sum of discounted instantaneous utilities 
 
V e x t u c t dt = -
-
¥
￿
rt
0
1 [ ( )] ( ( ))                (4) 
 
subject to the change in the quality of her environment (equations (2)) and the change in 
her capital stock (equation (3)). Following Kamien and Schwartz (1991, Section 9), if   3
[ ( )] 1- x t  is interpreted as the probability of living beyond t , V  is the expected lifetime 
utility of the representative agent whose life expectancy is uncertain and endangered by 
environmental degradation. 
 
3. The environment-augmented no-arbitrage and golden rules 
The solution to this optimal control problem leads to the following no-arbitrage rule: 
 
c t
f k n
u c t u c t
k t
k t
x t
g t
u c t u c t
.
( )
'( ) ( )
"( ( )) / '( ( ))
( )
( )
( )
[ ( )]
"( ( )) / '( ( ))
=
- + +
-
-
+
-
Ø
º
Œ
ø
ß
œ -
-
r d g
g g
1
1 1
0 1
.      (5) 
 
(See the Appendix for a detailed derivation.) 
 
   As it is appealing to analyze the rate of change of consumption, the following 
convenient specification of the instantaneous utility function is considered 
 
u c = b                    (6) 
 
where  0 1 < < b . In this case, the environment-augmented no-arbitrage rule (5) of 
consumption, can be rendered as 
 
c t
c t
f k n
g t
k t
k t
x t
A B
.
( )
( )
[ '( ) ( )]
( )
( )
( )
( )
=
-
￿
Ł
￿
￿
ł
￿ - + + -
-
-
￿
Ł
￿
￿
ł
￿ +
-
Ø
º
Œ
ø
ß
œ
1
1 1
1
1
0 1
1 b
r d
g g
b g
1 2 4444 4 3 44444 1 2 44444 4 3 444444
  (7) 
 
The first term (A) on the right-hand side of equation (7) is the conventional no-arbitrage 
rule of consumption obtained by the aforementioned optimal growth literature. It states 
that the instantaneous change in consumption corresponds to the difference between the 
marginal product and the user cost of capital, discounted by the degree of concavity of   4
the instantaneous utility function. The second term (B) modifies this conventional no-
arbitrage rule for the case where environmental aspects are taken into account.  
   Recalling that  Sign B Sign g t ( ) [ ( )] = - g g 0 1 , the environment-augmented no-arbitrage 
rule (7) recommends a smaller, the same, or a larger rate of consumption change than that 
advocated by the conventional no-arbitrage rule as the optimal spending on cleaning up 
and greening operations  is smaller than, equal to or greater than the stationary level 
g g 0 1 / , respectively. However, this environment-augmented no-arbitrage rule does not 
enable an analytical derivation of the optimal relationship between the quality of the 
representative agent’s environment (1- x) and her income ( f k ( ) ) which might, or might 
not coincide, with the ad hoc environmental Kuznets curve hypothesis.  
   The no-arbitrage rule (7) indicates further that the change in the quality of the 
environment induced by the accumulation of capital makes the difference between the 
capital-labour ratios in the environment-augmented optimal growth model and the 
conventional optimal growth model, except in steady state.  By setting c
.
 to be equal to 
zero, the stationary capital-labor ratio satisfies  
 
f k n g
k
k
x
ss ss
ss
ss
ss
'( ) ( ) [ ] = + + + - +
-
Ø
º
Œ
ø
ß
œ r d g g
g
0 1
1
1
1
        (8) 
 
where the subscript  ss denotes steady state levels. Recall that  gss = g g 0 1 , the ratio of 
the marginal effect of industrialization on the environment when cleaning up and greening 
operations are not implemented and the marginal moderating effect of cleaning up and 
greening operations on the rate of degradation of the environment. Consequently, the 
second term on the right-hand-side is equal to zero and the stationary capital-labor in the 
proposed environment-augmented growth model satisfies the familiar golden rule of 
equality between the marginal product of per capita capital and its user cost – the sum of 
the rates of time preference, capital depreciation and population growth. That is, as in the 
standard neoclassical optimal growth model  
   5
k f n ss = + + - ' ( ) 1 r d .                (9) 
 
4. Natural rehabilitation 
If  there exists a natural environmental rehabilitation process the environment-augmented 
stationary capital-labour ratio  may  differ from the stationary capital-labour ratio 
advocated by the  conventional golden rule. For simplicity, let the natural rehabilitation 
rate be constant: namely, a positive scalar m . By subtracting mx t ( )  from the right-hand-
side of the environment degradation motion equation (2), the no-arbitrage rule is now 
rendered as 
 
c t
c t
f k n
g t
k t
k t x t
x t
A B
.
'
( )
( )
[ '( ) ( )]
( )
( )
( ) ( )
( )
=
-
￿
Ł
￿
￿
ł
￿ - + + -
-
-
￿
Ł
￿
￿
ł
￿ +
-
-
Ø
º
Œ
ø
ß
œ
1
1 1
1
1
0 1
1 b
r d
g g
b g
m
1 2 4444 4 3 44444 1 2 444444 4 3 4444444
  (10) 
 
and the spending on cleaning up and greening operation in steady state declines with the 
ratio of the stationary degradation rate to the stationary per capita capital: 
 
g
x
k ss
ss
ss
= -
￿
Ł
￿
￿
ł
￿
g
g
m
g
0
1 1
 .                (11) 
 
Consequently, the  stationary capital-labor ratio in this environment-augmented optimal 
growth model satisfies 
 
f k n
x
k k
k x
x
ss
ss
ss ss
ss ss
ss
'( ) ( ) = + + + +
-
-
Ø
º
Œ
ø
ß
œ r d m
g
m 1
1 1
.          (12) 
 
The golden (i.e., stationary) capital-labor ratio in  this version of the environment-
augmented optimal growth model is equal to the golden capital-labor ratio in the standard 
neoclassical optimal growth model if and only if   6
 
k
x
x
x
ss
ss
ss
kss ss
= -
-
m
g
1
1
.                  (13) 
 
Recalling that  f  is concave, the golden, environment-augmented, capital-labor ratio is 
smaller (larger) than the golden capital-labor ratio in the standard neoclassical optimal 
growth model if 
k
x
ss
ss
 is larger (smaller) than m
g
-
- 1
1
x
x
ss
kss ss
.   7
References 
Cass, D. 1965. Optimum growth in an aggregative model of capital accumulation. Review 
of Economic Studies 32, 233-240. 
Kamien, M. I., Schwartz, N. L., 1991. Dynamic Optimization, Amsterdam, North-
Holland, Amsterdam. 
Koopman, T. 1965. On the concept of optimal economic growth. In The Econometric 
Approach to Development Planning, Amsterdam: North Holland. 
 
Phelps, E.S. 1966. Golden Rules of Economic Growth, New York: W. W. Norton. 
Ramsey, F. 1928. A mathematical theory of saving. Economic Journal 38, 543-559.  
Seldon, T.M., Song, D. 1994. Environmental quality and development: is there a Kuznets 
curve for air pollution emissions? Journal of Environmental Economics and 
Management 27:2, 147-152. 
Shafik, N., Bandyopadhyay, S. 1992. Economic growth and environmental quality: time 
series and cross-country evidence. Background Paper prepared for World Bank, 
World Development Report 1992: Development and the Environment, New York: 
Oxford University Press.  
Grossman, G.M., Krueger, A.B. 1995. Economic growth and the environment. Quarterly 
Journal of Economics CX:2, 353-377. 
   8
Appendix 
 
The Hamiltonian associated with the optimal control problem is  
 
H e x u c f k c g n k g k t = - + - - - + + - -r l d l g g ( ) ( ) [ ( ) ( ) ] ( ) 1 1 2 0 1     (A1) 
 
where  l1 and l2  are the shadow values of the representative agent’s capital and rate of 
environmental degradation, respectively, and where the time index  t  is omitted for 
convenience. 
 
The adjoint equations are: 
 
l
¶
¶
l d l g g
.
[ '( ) ( )] ( ) 1 1 2 0 1 = - = - - + - -
H
k
f k n g            (A2) 
and 
 
l
¶
¶
r
.
( ) 2 = - = - H
x
e u c t .                (A3) 
The optimality conditions are: 
 
¶
¶
l r H
c
e x u c t = - - = - ( ) '( ) 1 0 1               (A4) 
and  
 
¶
¶
l l g
H
g
k = - - = 1 2 1 0.                (A5) 
 
By differentiating the optimality condition (A4) with respect to  t the following singular 
control equation is obtained:   9
- - - + - - = - - - e u c x e x u c e x u c c t t t r r r r l '( ) ( ) '( ) ( ) "( )
. . .
1 1 0 1 .      (A6) 
 
By substituting equations (A2), (A4) and (A5) into equation (A6) for l
.
1, l1 and for l 2, 
respectively, and rearranging terms, the no-arbitrage rule (5) is obtained. 
 

A mathematical theory of saving.

Dynamic Optimization,

Economic growth and environmental quality: time series and cross-country evidence. Background Paper prepared for World Bank, World Development Report 1992: Development and the Environment,

Economic growth and the environment.

Environmental quality and development: is there a Kuznets curve for air pollution emissions?

Golden Rules of Economic Growth,

On the concept of optimal economic growth.

Optimum growth in an aggregative model of capital accumulation.

Greening the Neoclassical Optimal Economic Growth Model

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Greening the Neoclassical Optimal Economic Growth Model

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