The Role of Capital, Mineral Resources and Environmental Resources in Sustainable Development





Dina Franceschi

Department of Economics

Fairfield University

Fairfield, CT 06430‑5195



James R. Kahn

Environmental Studies Program

and Department of Economics

Washington and Lee University

Lexington, VA 24450









                                                   August 6, 2000












Since the Brundtland Commission’s delineation of the term sustainable development in 1987, virtually every country has incorporated the terms sustainability and sustainable development into their planning vocabulary and criterion for decision-making. However, many issues remain unresolved.  Broad and sweeping references to sustainability and sustainable development do not necessarily translate into implementable policies to achieve these goals. In particular, unresolved issues include developing an understanding of how one sector of the economy can contribute to the sustainable development of the economy as a whole and the role of ecological resources in sustainable development. Our paper provides an initial conceptual examination of these questions by examining the interaction between mining, ecological quality and sustainable development, with specific application to the sustainable development of  the Brazilian Amazon.



















1.0 Introduction                                 

The term and concept of sustainable development has received much attention since the late 1980's, when it became the global theme for a long-term perspective on economic growth.  The definition of the terms has been molded to fit different situations since then[1], but has mostly retained the original thrust behind which sustainable development is driven.  Generally, the interpretation relates “improving the prospects of the current generation without reducing the prospects of future generations” (Brundtland Commission 1987).

The general thrust of the definition of sustainability in the neoclassical economics literature (which incidently, predates the term sustainability and dates back to the early 19th century with the work of Ricardo, Malthus, and John Stuart Mill) is that certain indicators of economic welfare (or development or standard of living) are non-declining over time. If economic welfare (or an alternative indicator of development or standard of living) is a monotonic function of consumption, then sustainability can be defined as a non-declining intertemporal consumption path. Sustainable development can then be viewed as a process of change within the economy which does not violate one of these sustainability criteria (Stern 1997), or more narrowly, can be regarded as an increase in the consumption levels of the current generation which does not result in future declines in consumption levels.  Alternative definitions of sustainability focus on the pursuit of economic efficiency, with some sort of sustainability constraint. For example, Pezzey suggests an operational definition of sustainability which calls for maximizing net present value (of the whole intertemporal stream of welfare (or consumption)), subject to a constraint on non-declining average welfare (or consumption).

The neoclassical economics perspective on sustainability has essentially become a treatise on the role of capital stocks in economic growth (Stern 1997). The models generally define two types of capital, conventional (also called artificial, physical, manufactured or human-made capital) and natural capital. Natural capital is generally viewed as an extractive resource, usually non-renewable such as coal, iron or oil. Natural capital and human-made capital are most often regarded as perfect substitutes (the elasticity of substitution is equal to one). Under these conditions, sustainability is possible if sufficient investment takes place in human-made capital to offset the loss of natural capital. Since artificial capital and natural capital are completely substitutable, one needs only be concerned with the accumulation of the aggregate stock of capital.[2] This has become known as the weak sustainability criterion.  In contrast, the strong sustainability criterion suggests that there are limited opportunities for sustainability, and one must maintain non-declining stocks of both artificial and natural capital in order to ensure a non-declining intertemporal welfare path. 

Stern notes that there has been little empirical work on the substitutability of artificial capital at a scale which is relevant to the sustainability issue. A set of authors[3] focus on the scale issue and provide conceptual arguments as to why artificial capital is not likely to be completely substitutable with natural capital. These arguments are best understood by dividing natural capital into two categories, extractive resources and environmental resources.

This definition implies a particular path for society’s stock of capital assets, where assets can be broadly defined to include the stocks of human-made capital, human capital, extractable resources and environmental resources. In order to allow for an increasing standard of living, the aggregate stock of capital assets must be increasing as well, especially in the face of increasing population. Specific treatments of sustainability  generally have not looked at all four types of capital stocks mentioned above, and have made restrictive assumptions about substitutability among different types of capital.

We attempt to broaden the discussion of sustainable development by simultaneously examining the role of all types of capital assets. We then discuss the implications of this reconsideration for sustainable development of the Brazilian Amazon, a region rich in environmental and extractable resources, but with a relatively small stock of human-made capital.

A particularly important aspect of our approach will be to differentiate between extractable resources (such as minerals and fossil fuels) and environmental resources (such as ecosystems). Environmental resources provide a broad range of ecological services such as nutrient cycling, habitat protection, carbon sequestration and watershed protection. The importance of this distinction between extractable resources and environmental resources, as discussed later in the paper, is that artificial (human-made) and  human capital are likely to be good substitutes for extractable resources, but none of these types of capital are likely to be close substitutes for environmental resources.





2.0 Traditional Approaches

2.1 Neoclassical Growth and Development Models

Neoclassical growth and development models tend to focus solely on problems of artificial (human-made) capital accumulation. These models developed rules for allocating current output between consumption and investment in artificial capital.  For example, the classic Harrod-Domar model of development sets up a system of economic growth equations based on the full employment of capital and labor where the steady-state rates of growth of capital, labor and output are equal.  The conclusions of the model, given exogenous factors, are that the warranted rate of growth is equal to the ratio of the propensity to save to the capital-output ratio.  Variations on the paradigm include, Solow’s factor substitution model which resolves the same question but with added flexibility in inputs (K and L), and Kaldor’s saving rate adjustment model which allows flexibility in the saving mechanism to compensate for disproportional labor force growth.  Recall however, that the majority of the classic growth models examined savings and investment from purely the capital-profit or labor-wage standpoint.

The policy implications of these models are that development can be viewed as a process of artificial capital accumulation and of removal of obstacles to the artificial capital accumulation process.  Poverty of nations was viewed as a self-perpetuating process, a vicious cycle of poverty which is depicted in Figure 1. In this view of economic development processes, low income and high current consumption needs lead to low savings, low investment in artificial capital and a lack of artificial capital formation. The low stock of artificial capital implies a continually low marginal product of labor, with a correspondingly low level of income. In fact, since human-made capital depreciates, capital stocks may fall over time due to insufficient investment to counter depreciation, and as a result, income may be falling over time. In this perspective on development the solution to the problem is simple: inject capital and break the cycle of poverty. This was essentially the basis of the international development programs of the 1960s and 1970s. Unfortunately, in many places such as Sub-Saharan Africa, this development program was largely unsuccessful, with the current standard of living generally lower than it was three of four decades in the past. While rapid population growth, tribal conflict, civil war, AIDS and government corruption may have been important factors in the lack of economic development in some countries, another important factor is that development policy may have focused too much on the accumulation of human-made capital, to the exclusion of consideration of other types of capital, particularly environmental capital.

Figure 2 illustrates the shortcomings of these traditional models by showing how environmental degradation can effect the cycle of poverty. Low income and high current consumption needs lead to environmental degradation from over-exploitation of renewable resource systems, causing their (often irreversible) collapse. Also, inefficient production technologies lead to high emissions of waste products, causing urban environmental problems and contamination of aquifers.

This environmental degradation lowers the marginal product of labor. This is particularly important in the rural areas of developing countries, where desertification, deforestation, fishery collapses and soil erosion/fertility loss lead to losses in agricultural productivity. In addition, the human health consequences of air pollution and drinking water contamination make labor less

productive and generate costs such as those associated with the treatment of health problems. Finally, the environmental resources provide ecological services such as maintenance of climate, biodiversity, nutrient cycling, waste assimilation, primary productivity, maintenance of atmospheric chemistry, soil formation, protection of water resources and flood protection, all of which are important to basic life processes, quality of life and economic productivity.

2.2 Sustainability and Exhaustible Resources

The conventional treatment of exhaustible resources in terms of sustainability dates back to the early works of Thomas Malthus and David Ricardo, where resource scarcity and diminishing returns dominated the analysis.  These models focused on finite resource supply and whether continued economic growth was possible as resources became more scarce.  Work by Barnett and Morse (1963) built on these ideas by emphasizing the inevitable development of, and switching to good substitutes for the resources that become characterized by increasing scarcity.  Utilizing a model based on the free substitution of inputs (labor, artificial capital and exhaustible resources), Barnett and Morse found that technological innovation in discovery, extraction and production techniques lowered the costs of extractable resources, greatly outweighing the cost increasing effects of depletion.  These influential findings largely refuted the previously held notion that Malthusian limits to growth existed and would constrain or collapse current consumption and development rates.  Barnett and Morse’s conclusions have served as the point of embarkation for further exploration of the relationship between sustainability and exhaustible resource use, including works by John Hartwick (1977,1993, 1994), Raymond Mikesell (1994), John Pezzey (1989), John Tilton (1996) and Vincent, Panayotou and Hartwick (1997).

The assumption of substitutability between inputs is not an inconsequential one.  The fundamental premise underlying the work of Barnett and Morse is the interchangeability between labor (human capital), artificial capital, and extractable natural resources as inputs to the production process.  Two issues regarding substitution to achieve sustainability in these models are of concern. 

The most important issue is that the models do not incorporate the environmental or ecological resources as inputs to the production process. As discussed earlier, ecological resources provide an array of important services to a variety of different production processes, day to day living and other ecological systems.  However, the inclusion of environmental resources or ecological services into the production function is not in itself sufficient to rectify the shortcomings of these economic models of sustainability. In addition, one must consider the notion of complete substitutability among the different types of capital.[4]

Kahn and O’Neill (1999) argue that at the large scales defined by the current level of economic activity, artificial capital can not provide an adequate substitute for environmental resources, as it  is completely infeasible for human activity to provide the ecological services at the level of natural activity. For example, although artificial capital can be used to process point source wastes in a sewage treatment plant, it would be completely infeasible for all the non-point sources of nutrients (both natural and anthropogenic) to be processed by treatments plants. Perhaps the best example of the inability of human-made capital to substitute for natural capital can be found in flood protection in the Mississippi River flood plain. Flood protection was provided by environmental resources in terms of wetlands which acted as giant sponges, absorbed sudden large influxes of rainwater, storing them, and gradually releasing them to be carried downstream over a prolonged period of time. As wetlands were converted to farmland and other economic purposes, systems of dikes and levees were developed to try to control the flooding problem, however they were not capable of doing so. In fact the “one hundred year flood” which occurred in 1993 was generated by a “20 year” rainfall, indicating a vast reduction in flood control protection, despite the provision of the artificial capital devoted to the production of flood control. It should be noted that scale is not the only source of concern. Ecological complexity argues against complete substitutability between environmental capital and other types of capital (see Kahn and O’Neill 1999).

Further, the traditional neoclassical approach (which suggests the mitigation of scarcity associated with exhaustible resources) is based on the premise that the market can allocate supply and demand and create appropriate incentives through the pricing system.  In the classic literature of resource scarcity, including the work by Barnett and Morse, as an exhaustible resource or mineral resource base becomes depleted, prices rise.  The rise in price is primarily the response of increased costs of extraction of the relatively less abundant source (relative in the sense of the deposit being less rich than deposits of the past).  As price of one product rises, either firms find better extraction technology to control costs or buyers switch to another product that can satisfy their needs at a lower price.  In the case of environmental or ecological resources, no comparable market mechanism exists to send appropriate signals or create incentives to mitigate scarcity.  Most often ecological systems do not have prices associated with their services, because they are external to markets.  The public goods (and often open-access) nature of ecosystems often precludes sustainable long term usage.  Again, the fact that these systems are exceptions to the classic market mechanisms heightens the need for their explicit inclusion in the formula for sustainability.                                                                                                                           


3.0 A Theoretical Treatment of Sustainability

Much time and effort in the recent past has been devoted to finding the optimal rate of exhaustible resource extraction for intergenerational sustainability.  Solow, Hartwick, Stiglitz, Pezzey, and others have contributed to this literature.  Hartwick in particular has derived a saving and investment rule for economies utilizing exhaustible resources.  The Hartwick model and its results focus on the generation of sustainability through a replacement mechanism.  That is, the rule defines the level of investment and savings which must be generated by the current generation to replace the natural capital (exhaustible resource stock) they have consumed, in order to maintain a constant level of consumption for future generations (Hartwick 1977 and 1994).

Traditionally, sustainability was approached using the steady state notion of constant consumption over time, where utility is defined as a function of consumption.


 The Hartwick model seeks to examine the possibilities of maintaining at least a constant level of consumption over time, which can be expressed as                    


Hartwick’s capital theory of investment model begins with a stock of exhaustible resources, S(t), where the continuous extraction of the stock is the flow of the resource, R(t).


Production is defined as needing two inputs, R and K, exhaustible resource flow and artificial capital stocks, respectively.


If Equation (4) is differentiated with respect to time, this gives the time derivative of production, as in equation (5)


Hartwick then defines the reinvestment rule as a level of investment set equal to a chosen proportion (α) of the rents associated with the production of exhaustible resources. With extraction costs assumed to equal zero, rents would be equal to the product of resource extraction (R) and the marginal product of resource flows (FR). The reinvestment rule would then be given by equation (6).


Consumption can be measured as a residual or the difference between production and investment:



The model then employs Hotelling’s efficiency rule for exhaustible resources and some clever mathematical manipulation, to find [5]





This equation merits further discussion. If the marginal product of resources is positive, and since 0 £ α £ 1, the sign of  will depend on the sign of FR. FR, the marginal product of resource flows, will be positive. Therefore the sign of  will depend on the sign of . If the stock of resources is finite (and technology is assumed constant as in the Hartwick model) then  must eventually become negative. Therefore, the greatest possible value of  is zero, which will occur when  α = 1, or when all the rents are reinvested in artificial capital. Intuitively, this simply means that as long as the current generation is reinvesting exactly the rents derived from the extraction of the exhaustible resource, consumption can be maintained at a constant level over the time horizon chosen.

In Hartwick’s model, can not be increasing. However, if one allowed the stock of resources to increase (through technological innovation) then  could be increasing and the reinvestment rule would change, as  α = 0 would maximize . Of course, if technological innovation required investment in artificial capital, then the optimal proportion of reinvestment or rents would be somewhere between zero and one, exclusive of the extreme values.

Obviously, these models do not incorporate environmental resources or ecological services, which, as discussed earlier, have important implications for sustainability. Environmental resources can be added to the Hartwickian approach, and this will add considerable insight to the understanding of the necessary and sufficient conditions for sustainability.

Environmental resources can be incorporated by taking the same initial mathematical construct with the addition of a term that relates the extraction of the exhaustible resource to the depletion of ecological services.  We will follow Hartwick’s model,  rather than developing a new model, in order to facilitate a comparison of results. We start with the definition of the ecological resource stock as being proportional to the exhaustible resource stock.  As the mineral resource stock (S) is extracted, it gets smaller and more land is used and greater degradation is generated, thus causing the ecological resource stock (E) to get smaller as well.   Also, ecological services are some function of the ecological resource stock.  If ecological services are modeled as enhancing utility or consumption directly, consumption can now be defined as production plus ecological services minus the change in capital.







Similar mathematical analysis using the state equation for exhaustible resource extraction (1), the same reinvestment rule (2) and Hotelling’s efficiency rule results in a potentially non-sustained rate of consumption over time. 


If 0 £ α < 1 and both the time derivatives of R and E are negative, then the time derivative of consumption must be negative. Utilizing the Hartwick rule of total reinvestment of resource rents, where α=1, the change in consumption moves with a change in the ecological stock.




Consumption can actually be enhanced by a growth in ecological services (e.g., allowing the Everglades Agricultural Area to revert to wetlands and increasing water flows to heavily populated South Florida) or conversely diminished by a negligence in ecological protection (e.g. continued drainage and wetland conversion which reduces water flows to South Florida).  The results become ambiguous however, when the reinvestment of rents rule is relaxed.  Depending on the relative magnitudes of the two terms, the change in consumption could be negative or positive.

The interpretation of equation (12) places the sustainability debate in context. At one extreme would be the “Technological Optimists” who would argue that is positive, that both  and FR are large, and that this would compensate for a negative  At the other extreme would be the “Environmental Pessimists” who would argue that is negative, that  can’t be positive due to the finiteness of crustal mineral deposits, and that  must be negative.

A less extreme approach would recognize the sign of  as a policy choice.  can be positively influenced by investment in research and development, artificial capital and human capital. However, care must be taken so that the process of accumulating these stocks of capital do not negatively impact environmental capital stocks, and their ability to produce ecological services such as as maintenance of climate, biodiversity, nutrient cycling, waste assimilation, primary productivity, maintenance of human capital, and the protection of environmental capital. The development of policies to influence technology towards a smaller ecological footprint is an important part of this process. In addition, policy can invest directly in environmental capital by remediating existing problems, reforestation, creation of wetlands, carbon sequestration, and so forth. If the above conditions are met, then it is theoretically possible for to be positive and sustainable development can be an achievable goal. However, if the sign of  is a policy choice, then sustainability must be pursued with an active policy agenda and not simply left to market forces, as the market forces which alleviate scarcity of mineral resources are not applicable to environment resources.                                                                                                 

A similar result with similar implications occurs when the representation of ecological services enters the model through the production function as opposed to entering into consumption directly.   Production in this case is a function of the capital flow, exhaustible resource flow and exhaustible resource stock.


Relying on the same relationship between exhaustible resource stock and ecological resource stock, the change in consumption relationship after mathematical manipulation again shows reduced consumption over time with the addition of ecological services depletion.


The possibility of declining consumption again occurs, the result being determined by the magnitudes of the terms.  In any case, however, the change in consumption over time is either improved or diminished by the marginal contribution to production of the change in ecological services over time.  Assuming that the marginal product of the environmental asset is positive, when α=1, the change in consumption again moves with the change in the environmental stock.




One can gather considerable insights to the areas in which policy can contribute to sustainability by looking at Equations (12) and (16).  Policy must strive to make the sum of both

terms of this equation positive. This may not seem possible given the initial assumptions of the model, but the assumptions can be relaxed to allow for technological change. First, technological innovation can change the sign of the time derivative of R. Although the crustal abundance of mineral resources can not be increased, technological innovation can increase the availability of the resource stock S over time, implying that R need not be declining over time. R can also be increasing over time due to the judicious use of renewable resources, including the sun’s energy. Secondly, the time derivative of the environmental resource is actually equal to -B*R, where B measures how much extractive resource depletion/usage depletes the stock of environmental resources. The more that technological innovation can reduce the environmental impact of extractive resource use (B), the smaller the second term of the equations (12) and (16) and the less it would offset a potentially positive first term. It is important not to lose sight of the substitutability arguments that were made above. If artificial capital and extractive resources are poor substitutes for ecological services, then as environmental resources decline, FE will become very large, and consumption will inevitably have to decline. 

The role of the magnitude of FE merits further discussion to discover the full implications of FE for determining the sign of . Figure 3 shows the total product function for ecological services, where the slope of the function is equal to FE. As with other inputs, diminishing marginal productivity is expected, and is reflected in the slope of the function in Figure 3. Note that if ecological services are a monotonic function of environmental resources, E could represent either ecological services or environmental capital in this graph.













Two aspects of this total product function merit further discussion. First, at high levels of ecological services, the losses associated with a small diminution of ecological services are also

likely to be small. In fact, if increases in ecological services become redundant at high levels of ecological services, the slope of the total product function may equal zero. Second, there may be

a threshold below which production falls dramatically with further reduction in ecological services. This threshold is likely to exist because of the complexity and nonlinearity of ecological behavioral functions.[6]

Both characteristics have important implications for equation (12). If is negative, the product of FE and will be small in the vicinity of E2 and large in the vicinity of E1. This illustrates the underlying cause of the assertion that a non-negative is more likely when the level of ecological services is high than when the level of ecological services is low.


4.0 Implications for Sustainable Development of Amazonia Policy, the Case of Amazonia

The theory in the preceding sections can be used as a guide to develop sustainable development policies. Even without an empirical analysis, it is possible to construct guidelines for development based on an assessment of capital stocks in comparison to capital needs. We choose the Amazon region of Brazil as our case study, due to the threats to ecological resources associated with the development process. In this region, there is a strong  need for a sustainable development process to protect the global environmental resources associated with the rainforest.

Given the central role of all types of capital in the process of sustainable development, the first step in analyzing this question is to assess capital stocks in the region. Although the Amazon region is large and heterogeneous, characteristics of the capital stocks are shared throughout the region. In general, the region is characterized by low levels of artificial capital, low levels of human capital, abundant extractive resources and abundant (but threatened) environmental capital. In the following analysis, we will show how capital augmentation plans should be developed, and how they may need to differ throughout the region.  

The first question is how to create the surplus to increase artificial capital and human capital. Since this paper has focused on extractive resources, these will be discussed first. Currently, mineral production in the Brazilian Amazon is a significant source of regional and national GDP. The worlds largest tin mine is in Pitinga, Amazonas, and the world’s largest iron mine is in Carajas, Pará.  Primary mineral production currently accounts for 3% of GDP, total secondary production accounts for 33% of GDP, and there is significant potential to increase the amount of production. Furthermore, 25% of all exports come from the mining industry.  In light of the previous discussion of the Hartwick Rule and our modification of the Hartwick Rule, an expansion of mining can lead to sustainable development if two conditions are fulfilled.  First, the rents from the mining must be reinvested in other forms of capital. Second, the mining activity must not result in significant damage to the rainforest or river systems.  In other words, the expansion of mining could result in a movement towards sustainable development as long as these two conditions of reinvestment and environmental protection are met.

For other types of extractive activities, it is more difficult to obtain a surplus, and environmental impacts can be even more devastating (Franceschi 1999). For example, the clearing of rainforest for agricultural field crops leads to permanent loss of rainforest, but is also unsustainable in another dimension. Poor soils, and the breakage of the nutrient cycle associated with removing the forest cover implies unproductive agriculture, with crop production often limited to 2-4 years. Under these conditions, it is not possible for large scale agriculture to generate a surplus with which to generate funds for investment, and in fact leads to much more loss of forest per dollar of current GDP or per job generated in comparison to mining.                       Of course, sustainable development does not require that the surplus used for capital investment be generated from within the region. These funds for investment can come from other parts of Brazil or from abroad. In fact, the policies which have been developed to create and sustain the Manaus Free Trade Zone[7] represent a transfer from southern Brazil to the Amazon region that results in an increase in the stock of artificial capital in Manaus. Although environmental protection was not the original motivation for the policies, they have resulted in a slowing of deforestation in the area (Rivas 1998), and Amazonas remains at 97% of its original forest cover, in comparison to other Amazonian states which have lost over 20% of their original forest cover.

Our discussion of sustainable development in the Amazon region would not be complete without discussion of subsistence level activities. For example, small farmers in Rondonia are primarily responsible for deforestation in that state (Caviglia, 1999). These farmers are migrants from south Brazil and utilize traditional field crop methods instead of the sustainable agroforestry techniques employed by Indians and Caboclos[8].  Caviglia shows that the income that can be generated from sustainable agroforestry exceeds that from traditional field crop methods, but these migrant farmers continue to use the unsustainable techniques. Here, an investment in human capital would contribute to sustainable development through the transmittal of knowledge of the Indian/caboclo sustainable techniques to the migrant farmers who only know the unsustainable temperate agricultural techniques of southern Brazil.


5.0 Conclusions

Although the evolution of sustainable development policies must consider a spectrum of micro-economic, macro-economic, ecological, cultural, institutional, and social issues, our paper suggests that a central focus of sustainable development should be the accumulation of human capital, artificial capital, and natural capital, while protecting the stock of environmental capital. Mining activity can make an important contribution to sustainable development by generating a surplus for other investment activities, but care must be taken so that the mining activity does not result in too large a reduction in the stock of environmental capital.  Above all, our results show that the inclusion of the strong sustainability criterion requires greater care be taken in the sustainable development planning process.  In particular, simple reinvestment of rents from productive activities into artificial capital is not enough in most cases to support development for the long term.  Reinvestment or preservation in the environmental capital stock can be as important for productivity as human or artificial capital.















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Caviglia, Jill L., 1999, Sustainable Agriculture in Brazil: Economic Development and Deforestation, New Horizons in Environmental Economics Series, Edward Elgar Publishing, United Kingdom.

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Appendix (to be listed on a website, not included in paper)




The replication of Hartwick’s results begins with his definitions of consumption and saving.


where Q = F (K(t), R(t)) and “K dot” denotes the change in capital over time or investment.  The change in consumption over time then, can be defined as,


This is subject to the same equation of motion of exhaustible resource flow, equation (1) in the text,  


and the same capital reinvestment rule as equation (2) in the text,




Substituting into equation (i), we have


Using Hotelling’s rule for efficiency in the extraction of exhaustible resources,


and simplifying Hartwick finds,





Now define the ecological stock to be proportional to the exhaustible resource stock, as mineral resources are extracted from the ground forest is cut down.


In this case, the ecological resource stock impacts consumption directly.  The consumption equation, a revision of (i), will be,


(The representation of time is suppressed for simplicity.)

Taking the time derivative gives,


After substitution of the reinvestment rule (ii), our equation of motion (iii), Hotelling’s efficiency rule (iv), and simplification the resulting change in consumption equation is equation (3) in the text,


This can be interpreted as the change in consumption over time being supported or reduced by the rate of enhancement or depletion of the ecological stock.




The third case incorporates the condition of the ecological stock being proportional to the exhaustible stock into the production function.  Where production is now defined as,


Using (i) and (iii) we have,



Taking the time derivative gives,


Substituting (iii) and (iv) and simplifying gives a somewhat similar equation for the change in consumption over time as the second case.


Or, by substituting (ii) we have the text equation (4),


This essentially means that the change in consumption over time moves with the change in the flow of the marginal productivity of ecological services over time.  Again, depending on relative magnitudes, the change in consumption over time could potentially be negative, indicating a decrease in utility for society or an unsustainable path.


[1] See Pezzey’s (1989) compilation of 35 definitions of sustainable development.

[2] Of course, the quantity and quality of human capital will be important as well, but human capital is generally not discussed in these neoclassical examinations of necessary and sufficient criteria for sustainability.

[3] Daly, Kahn and O’Neill, Franceschi and Kahn, among others.

[4] See Pearce and Warford for a comprehensive discussion of substitutability among inputs. An important aspect of their discussion is their focus on environmental capital and its lack of substitutability with other forms of capital.

[5] The Hotelling efficiency rule says that in a competitive market environment, the rate of return to capital and the rate of return to mineral resources are equal. An appendix which details all the derivations can be found at http://.......  (Note: the appendix follows this submitted version of the paper for the benefit of the reviewers)

[6] See Kahn and O’Neill (1999) for further discussion

[7] The Federal Government lifts tariffs on imported components and provides other financial incentives to encourage assembly and manufacturing activity in Manaus.

[8] Caboclos are people of mixed Indian and Portuguese descent who have been rainforest inhabitants for many generations, and whose families have been in the rainforest for as long as three or four hundred years.