Abstract
The Green Golden Rule treats the total resource stock as a homogenous stock.
However, the total stock consists of various stocks which have different values and different rates of transformation. Hence, the incorporation of resource-stock diversity is necessary for the determination of correct optimal conditions. The GGR is extended to incorporate the diversity of renewable resource stock: the economically optimal conditions are developed for a three-stock and an n-stock forest resource, and economics implications of these conditions are discussed. The EGGR prescribes higher level of conservation compared to the GGR.
Keywords: Sustainability, forest, Green Golden Rule, and optimum stock levels
JEL classification: D90
1.0
Introduction
Sustainability issues were discussed by Ciriacy-Wantrup and Krutilla even prior to the publication of The Limits to Growth that generated a new wave of economists’ interest in sustainability. Papers included in a special issue of the Review of Economic
Studies (Volume 41 (128), 1974), including Dasgupta and Heal (1974) and Stiglitz
(1974), laid foundations for the future debate on sustainability. The next contribution in this sequence was Hartwick’s rule (Hartwick, 1977), and all this literature has been reviewed in Pezzey and Toman (2003) and Kant and Berry (2005). The first wave of economic literature on sustainability considered utility to be a function of natural resource consumption only and used discounted utilitarian framework to address intergenerational equity.
In the next round, economists such as Krautkraemer (1985) and Beltratti et al.
(1993) included “resource stock”, in addition to consumption, as a source of utility.
Chichilnisky (1997) proposed an axiomatic approach of sustainable development and suggested Chichilnisky’s criterion to address intergenerational equity. Chichilnisky et al.
(1995) extended the Golden Rule (GR) of economic growth to include environmental
1

stock, and termed the new rule as the Green Golden Rule (GGR). The GR refers to a growth path of the highest maintainable level of consumption per capita while the GGR refers to the highest indefinitely maintainable level of instantaneous utility – which includes utility from the consumption and the stock of environmental/natural resources.
The GGR is a valuable contribution to the economic literature on sustainability issues but it is insensitive to the diversity of natural resource stocks and different utilities derived from them - either by consumption or by stock. Heal (1998, p. 47) recognised this limitation by acknowledging that most ecosystems are considerably more complex than assumed in the GGR formulation, and suggested that this is a topic for further research.
We make an attempt to capture some complexity of ecosystems by incorporating the diversity of natural resource stocks and utilities from them, and propose the Enhanced
Green Golden Rule (EGGR). For the sake of tractability and interpretation of results, first we develop the EGGR for an age-class-based three stocks of a forest, and later we generalize these results for a forest of n stocks. In the next section, we introduce the concepts related to structure of an age-class-based multiple stocks forest, and it is followed by the EGGR for a three stocks and n stocks forest, respectively.
2.0 The structure of an age-class-based three stocks forest
The magnitude of various products and services, provided by forests, depends on the total forest stock (generally means timber stock) as well as on the structure of the stock. Age-class distribution is the most common practice of analyzing forest stock structure, and the young, mature, and old forest stocks are the common age-classes.
Young stock is generally consumed as fuel wood and pole crop, whereas mature and old stocks are used for construction material and furniture. Similarly, mature and old stocks provide recreation and habitat for big mammals whereas young stock provides habitat for small mammals. Hence, we consider a forest resource consisting of these three stocks and assume that their consumptions are not additive.
Suppose the young, mature, and old stocks are S
1
, S
2
, and S
3
, respectively, and the growth function of each stock is logistic with
*
S ,
1
S
2
*
, and S
3
* as maximum stocks. As the young stock grows, some trees in the young (or mature) stock will grow but will remain
2

in the young (or mature) stock and the remaining trees will cross over to the mature (or old) stock. We assume that no tree will move from the young stock to old stock directly.
Hence, the total growth of young stock is only that growth which remains in young stock, the total growth of mature stock is the sum of its own partial growth which remains in the mature stock and partial growth of young stock that crosses over to the mature stock; and the total growth of the old stock is the sum of its own growth and the partial growth of the mature stock that crosses over to the old stock. Suppose

1
and

2
are the proportions of growth of S
1
and S
2 that remain in S
1 and S
2
, respectively, ( 1
 
1
) is the proportion of growth of S
1
that adds to the growth of S
2
; and ( 1
 
2
) is the proportion of growth of
S that adds to the growth of
2
S
3
. The growth functions of the three stocks could be represented as:
R
1  
1

1
S
1
( 1

S
S
1
*
1 ) where, 0

S

1
S
1
*
(1)
R
2
R
3

( 1
 
1
)

1
S
1
( 1

S
1
S
1
*
)
 
2

2
S
2
( 1

S
2
S
2
*
) where, 0

S
2

S
2
* (2)

( 1
 
2
)

2
S
2
( 1

S
2
S
2
*
)
 
3
S
3
( 1

S
3
S
3
*
) where, 0

S
3

S
3
*
(3) where

1
,

2 and
 are characteristic growth coefficients of young, mature and old
3 stocks, respectively.
3.0 The Enhanced Green Golden Rule (EGGR) for a three-stocks forest
We modify the economic model of Chichilnisky et al. (1995) to incorporate three stocks, and assume that the consumption and levels of stocks contribute to utility.
Suppose the utility function U ( C
1 t
, C
2 t
, C
3 t
, S
1 t
, S
2 t
, S
3 t
) is a strictly concave. For succinctness, we use the notation U ( C t
, S t
) . We also assume that the utility function is additively separable in consumptions and stocks. Suppose the production of man-made capital K occurs according to the linear homogeneous production function t
F ( K t
, S
1 t
, S
2 t
, S
3 t
) , and capital accumulation is expressed as
K t

F ( K t
, S t
)

C t
(4)
3

The rates of change of the three stocks of forest are expressed as:
S

1 t

R
1 
C
1 t
S

2 t

R
2 
C
2 t
S

3 t

R
3 
C
3 t
(5)
(6)
(7)
Similar to the GGR, in which society is only concerned with the long-run values of the consumption and levels of forest stocks, we seek a path to maximize the long run utility, t lim
 
U ( C t
, S t
) . The solution is specified by the following proposition:
Proposition 1: There exist values of *
1
*
2
*
3
*
1
*
2
*
3 characterized by
U s
1
U c
1
U s
2
 
R
1 s
1
 
R 2 s
2

U c
2

U
U
U c
2
U s
3
 
R
3 s
3
,
U c
3 such that c
1 c
3
U c
2
R s
1
2 ,
R 3 s
2
, and t lim
 
U ( K t
, C t
, S t
) = U *
1
*
2
*
3
*
1
*
2
*
3 is a necessary and sufficient condition for a feasible path (K t
, C t
, S t
) V t to be a solution of the problem maximize t lim
 
U ( K t
, C t
, S t
) over all feasible paths.
Proof: As indefinitely maintainable value of C
1
, C
2
, C
3 satisfy R
1 
C
1
, R
2 
C
2
, and R
3 
C
3
and S
1
, S
2
, S
3
, this means that the problem Max t lim
 
U ( K t
, C t
, S t
) , over feasible paths,
Reduces to Max U (C, S)
Subject to the constraints given by equations 9, 10, and 11.
R
1 
C
1
R
2 
C
2
(9)
(10)
4

R
3 
C
3
(11)
Similar to the GGR, the stock of capital is not a concern because any stock of capital can be accumulated over a sufficiently long period. The set of (S, C) satisfying the constraint in (9), (10, and (11) is compact, so this problem is well-defined. Hence, the maximum is characterized by the first order conditions:
U s
1
U c
1
U s
2
U c
2
 
R
1 s
1
 
R
2 s
2

U c
2
U c
1

U c
3
U c
2
R s
1
2
R
3 s
2
(12)
(13)
U s
3
 
R
3 s
3
(14)
U c
3
This completes the proof of Proposition 1.
We term the solution provided by equations (12), (13), and (14) as the Enhanced Green
Golden Rule (EGGR). The left-hand side (LHS) of each equation signifies the marginal rate of substitution (MRS) between consumption and stock-level, and the right hand side
(RHS) corresponds to the marginal rate of transformation (MRT) of the respective stock.
The EGGR gives the same optimality condition for the old (or terminal) stock (equation
14) as of the GGR; the MRS between consumption and stock-level is equal to the MRT of the stock with respect to itself. However, the optimality conditions for young and mature stocks (equations 12 and 13) are different than the GGR conditions. For young stock, the EGGR requires that the MRS between the consumption and stock-level is equal to the MRT of the young stock with respect to itself plus the MRT of mature stock with respect to young stock expressed in the terms of young stock (normalized by the ratio of the marginal utilities of mature stock and young stock). We call the RHS of Equation (12) as the Normalized Composite Marginal Rate of Transformation (NCMRT) (composite of the ROT of the young and mature stocks and normalized to express in the units of young stock). Hence, for young and mature stocks, the EGGR requires the equality of the MRS between consumption and stock-level with the NCMRT of the stock, and therefore, the
EGGR optimal values of young and mature stocks are higher than the GGR optimal values if these two stocks were treated separately.
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4.0 A Generalized EGGR for a forest of n stocks
In many cases, forest managers divide the total forest stock into more than three classes and based on features other than age such as wildlife-habitat, biodiversity, canopy, and density. Intuitively, the results of Section 3 can be generalized for n-stocks, provided the growth of each stock follows the same growth function – logistic - as assumed in Section
2. A generalized EGGR for n-stocks can be written as:
U s
1
U c
1
U s
2
U c
2


 
R
1 s
1
 
R
2 s
2

U c
2
U c
1

U c
3
U c
2
R s
1
2
R
3 s
2
U s n

1
 
R n s n

1 U c n

1
U s n
U c n
 
R n s n

U c n
U c n

1
R n s n

1
The interpretation of these n-equations is the same as of the three equations in Section 3.
In the case of terminal (nth) stock, the EGGR requires the equality of the MRS between consumption and stock-level and the MRT of the stock with respect to itself. In the case of all non-terminal stocks, the EGGR requires the equality of the MRS between consumption and stock-level of n-2 stock to the NCMRT of n-2 stock (composite of the
MRT of the n-2 and n-1 stocks with respect to n-2 stock and expressed in the units of n-2 stock). Hence, for all non-terminal stocks the EGGR gives higher stock-levels than the
GGR stock-levels when these stocks were treated separately
5.0 Conclusions
The paper has extended the GGR to include multiple stocks of a renewal resource, and the EGGR has all the properties of the GGR and other similar concepts discussed by
Chichilnisky et al. (1995). In addition, the EGGR provides many useful results. First, the optimal stock-levels, as per the EGGR, for all non-terminal stocks are bigger than the
6

optimal-levels by the GGR, and therefore the optimal-levels are also bigger than the maximum sustainable yield levels. Hence, in general, the EGGR prescribes higher level of conservation of natural resources as compared to the GGR. Second, the EGGR provides optimal stock-levels of different stocks, and therefore can be used in designing management strategies for ecological sustainability such as biodiversity and habitat conservation. Third, the EGGR incorporates marginal utilities of consumption as well as stock-levels of different stocks, the optimal levels of stocks will depend upon the social valuation of consumption and stocks. The marginal utilities of stock-levels and consumptions of different stocks of a forest resource will vary across communities as well as across different categories of forests, the EGGR can be used to design community-specific and forest-specific resource conservation plans. Fourth, the EGGR is based on a broad concept of diversity of stock, and therefore, these results can be used for different types of diversities of forests, such as old-growth forest versus other forests, diversity of forests based on diversity of flora or fauna, diversity of forests based on categories of industrial wood (lumber and pulp wood), diversity of forest based on traditional and/or Aboriginal values.
In summary, the EGGR is more sensitive to ecological complexities of forest ecosystems, and incorporates ecological dimension of sustainability in economic decisions making.
An important task for future research may be to develop specific optimal conditions for different types of utility functions and resource growth functions.
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