The Hartwick Rule and Sustainability. An Ongoing Development

Term Paper (Advanced seminar), 2014

16 Pages, Grade: 1,3



1 Introduction

2 Sustainability

3 The Foundations of the Hartwick Rule

4 The Hartwick Rule
4.1 The basic Model
4.2 Extensions to the basic Model by Hartwick
4.3 Extensions to the basic Model by other Economists

5 Conclusion

6 References

1 Introduction

”Contemplation of the world’s disappearing supplies of minerals, forests, and other exhaustible assets has led to demands for regulation of their exploitation. The feeling that these products are now too cheap for the good of future generations, that they are being selfishly exploited at too rapid a rate, and that in consequence of their excessive cheapness they are being produced and consumed wastefully has given rise to the conservation movement” (Hotelling 1931, 139).

Already in 1931 Hotelling described aptly the problem concerning sustainable management of resources in order to ensure intergenerational equity. More than 80 years later the problem still has not been solved but a lot of work has been done in order to find a remedy. One major contributor is John Hartwick with his rule concerning how to keep consumption constant if the economy runs partially on exhaustible resources. This paper starts with a short introduction to three different concepts of sustainability. It continues to give an overview of relevant research prior to Hartwick. The main part consists of the original Hartwick rule which is discussed in some depth. In the following priority is given to adaptations to the rule done by Hartwick himself over those by other economists. Obviously, it is not possible to cover every work in this regard. Rather, the work chosen tackles those aspects that are viewed most critically and were fairly close in time to release to the original Hartwick rule.

2 Sustainability

A commonly accepted basic definition of the concept of sustainability dates back to the Brundtland Report from 1987 in which the United Nations stated that ”sustainable development is development that meets the need of the present without comprising the ability of the future generations to meet their own needs” (WCED 1987, 43). Although it sounds straight forward there are very diverse concepts to be found in the current discourse. Generally accepted is the notion of intergenerational equity. Each generation should be able to achieve a well-being that is at least as high as the one of the predecessors. Further it is mostly accepted that the capital stock is heterogeneous in the way that it can be divided in natural and physical capital, where natural capital captures environmental resources. The underlying assumption is that these two kinds of capital can be substituted, the how and how far being subject of discussion, and thus allowing reducing one kind of capital and still be able to achieve sustainability as long as the other kind of capital is being increased accordingly. The sum of the two matters. Thus, sustainability is achieved when the sum of natural and physical capital is maintained. When dealing with environmental economics there are three important kinds of sustainability which will be presented.

Weak Sustainability The main difference between the concepts of weak, strong, and environmental sustainability lies in the elasticity of substitution between natural and physical capital. Weak sustainability assumes that the elasticity of substitution is 1 or above. Thus, it is theoretically possible to decrease natural capital close to zero and still be sustainable if physical capital can be increased. The maintenance of the constant aggregated level is usually achieved through investment (Tietenberg 2000, 96).

Strong Sustainability In contrast to weak sustainability the substitution of natural and physical capital is limited when discussing strong sustainability. Here, the focus lies on the preserving of the value of the natural capital stock (Tietenberg 2000, 96). This is plausible since natural capital does provide some functions which physical capital cannot provide. Hence, the natural capital is often claimed to be ’critical natural capital’ which is not to be reduced in order to ensure intergenerational equity (Gutés 1996, 147).

Environmental Sustainability Whilst strong and weak sustainability focus on sustaining aggregate values of capital stocks environmental sustainability aims at maintaining ”physical flows of individual resources” (Tietenberg 2000, 96). This is closer to an ecological notion than weak and strong sustainability as it is necessary that every kind of natural resource is managed in a way that allows replenishment (e.g. fishing quotas that do not exceed fish stock growth). Obviously, the substitution between different capital stocks is strongly limited in this case.

The discussion of the Hartwick rule is placed in the discussion around weak sustainability. The validity of the concept of weak sustainability is not debated in this paper. It is left to the reader to decide whether or not he agrees with it.1

3 The Foundations of the Hartwick Rule

When discussing the Hartwick Rule and sustainability it is a discussion of weak sustainability. In order to fully grasp this it is necessary to examine the background of Hartwick Rule. This rule is based on neoclassical models which assume substitutability between man-made and natural capital (Faucheux et al. 1997, 529). The following part provides an overview over the development of models which ultimately lead to the Hartwick Rule.

Hotelling The work of Hotelling in 1931 is often called seminal in the field of resource economics. As such it does not surprise1 that the Hartwick Rule also draws on it. Hotelling saw that the problem of exhaustible resources could not be tackled by a ”static equilibrium type of economic theory” since the ”indefinite maintenance of a steady state of production is a physical impossibility, and which is therefore bound to decline” (Hotelling 1931, 139). One of the many questions he posed was how the perfect rate of extraction should look like considering the resource would be finite (e.g. a mine). In order to provide the answer he constructed the following model. The underlying assumption is that the resource owner wants to maximise his present value of his future profits. The force of interest2 is set to γ, therefore [Abbildung in dieser Leseprobe nicht enthalten] describes the present value of a unit of profit to be obtained after a period t as long as interest rates are constant over time. Thus, the resource owner is indifferent whether he receives p0 now or [Abbildung in dieser Leseprobe nicht enthalten] in period t. Hence, p can be formulated as

Abbildung in dieser Leseprobe nicht enthalten

The price p can be understood as the net price after paying fixed costs (e.g. extraction cost) which sets

relative prices at different t. Obviously, these depend on p0 which is determined through demand in t0 and total supply of the resource denoted by a (Hotelling 1931, 140 & 141). p has to grow at rate γ in order to ensure that present value of a unit is the same in all periods. If this is violated there are gains to be made by shifting the extraction to other periods (Devarajan & Fisher 1981, 66). Assuming that q (the quantity taken at t) can be described as q = f (p,t) and T being the time of final depletion of the resource, a can be written as T can thus be easily obtained by

Abbildung in dieser Leseprobe nicht enthalten

as q is zero in T (Hotelling 1931, 141). Based on this the Hotelling Rule was formulated which states that ” the price of an exhaustible resource must grow at a rate equal to the rate of interest, both along an efficient extraction path and in a competitive resource industry equilibrium” (Devarajan & Fisher 1981, 66). With this rule Hotelling aimed at insuring that the excessive use of finite resources was curbed.

Solow After Hotelling there were forty years in which resource economics did not receive much attention but after the report of the Club of Rome ”The limits of Growth”, published in 1972, and the oil crisis in 1973 this field of research attracted notice again. One of the economists was Solow who was looking for a way of infinite growth under the constraint of finite natural resources (Gutés 1996, 148). The underlying production function is defined as

Abbildung in dieser Leseprobe nicht enthalten

where Q describes the output, K capital, L labour, and R rate of flow of natural resources taken from an existing pool. The interesting case to be considered is that the average product of R has no upper bound but R = 0 entailing that Q = 0. One of the ways to provide this is the following Cobb-Douglas function3

Abbildung in dieser Leseprobe nicht enthalten

with m being the rate of labour-augmenting technical progress. Here the natural resource R is an ”essential resource”4 (Solow 1974, 34 & 35). Q describes the net output which can also be written as

Abbildung in dieser Leseprobe nicht enthalten

with C specifying the aggregate consumption.5 Reformulating leads to the problem of finding a constant[Abbildung in dieser Leseprobe nicht enthalten]6 constrained by

Abbildung in dieser Leseprobe nicht enthalten

with n being the population growth rate. Thus, Solow aims at finding the largest per head consumption which can be maintained forever with the problem of finite natural resources and without using capital that is not there (Solow 1974, 35). Since there is no straight forward solution to this problem Solow tackles different cases.7 In order to tackle the problem on a general basis he looks at a per capita output. The Cobb-Douglas function thus reads as follows

Abbildung in dieser Leseprobe nicht enthalten

Assuming no technical progress it is obvious that as the natural resource per capita diminishes the output per capita can only be maintained if capital per capita accumulation increases fast enough to balance this decline. Solow assumes a constant population (n = 0) and no technical progress (m = 0). For simplicity the elasticity of output with respect to exhaustible resources h will be notated b and the elasticity of output with respect to reproducible capital [Abbildung in dieser Leseprobe nicht enthalten] receives the notation a. The relationship between the elasticities is defined as [Abbildung in dieser Leseprobe nicht enthalten](Solow 1974, 37). Solow shows with this model that any Q can be maintained as long as the initial capital stock is large enough. Further, he shows that in case that a is considerably larger than b the early generations should use up R fast in order to built up a high capital stock K. To be more precise, since the net output is kept constant a share of generated output [Abbildung in dieser Leseprobe nicht enthalten] is consumed and the rest is net investment (Solow 1974, 39). Assuming population growth but no technical progress no constant consumption per capita is sustainable. Yet, constant population combined with technical progress should be able to maintain a constant consumption and net output (Solow 1974, 40 & 41).

Stiglitz Like Solow, Stiglitz worked with a Cobb-Douglas function

Abbildung in dieser Leseprobe nicht enthalten

where λ describes the rate of technical progress which is assumed to be constant and Q is the aggregate net output that is to be used either for investment or consumption. Thus,

Abbildung in dieser Leseprobe nicht enthalten


1 For further information on the discussion whether weak sustainability is diametrical to sustainability in terms of protecting the environment see Gutés 1996.

2 Also known as the instantaneous rate of interest (Cairns & Davis 2007, 461).

3 An alternative would be the function Q = F(K, L)Rh with 0 < h < 1 and F homogeneous of degree 1 − h. Solow prefers to use the Cobb-Douglas function as it facilitates to accommodate for technical progress (Solow 1974, 34).

4 Dasgupta and Heal coined this term in (Solow 1974, 34)

5 Note that a dot on top of a variable always describes the time derivative of this variable.

6 Note that a minuscule denotes a per capita variable.

7 See page 35 and 36 for more details (Solow 1974).

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The Hartwick Rule and Sustainability. An Ongoing Development
University of Trier
Environmental Economics
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B.A., B.Sc. Esther Schuch (Author), 2014, The Hartwick Rule and Sustainability. An Ongoing Development, Munich, GRIN Verlag,


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