Section 'The New Conceptual Perspective on Sustainability: Implications to Environmental Management' focuses on implications of the analysis for environmental management and the final section gives conclusive remarks shedding light on directions for future work. As far as the essay's main goal is concerned, the methodological procedure used integrates two different apparatuses. First, a qualitative approach was undertaken in order to obtain, in the environmental literature, a suitable definition of natural capital. The objective is to clearly define natural capital and connect it to sustainability.

This latter concept follows the premises of the Brundtland Commission A set of important contributions was selected to that end, such as, Lima ; Daly , , , ; Lawn ; Turner, Brouwer, Georgiou and Bateman ; Sahu and Choudhury ; England ; Costantini and Monni ; and Irwin and Ranganathan Second, an analytical approach was used in order to conceive two different models regarding optimal output production growth - one considering output production constrained by the use of a nonrenewable natural resource input, and the other contemplating pollution control over a production process that damages air quality pollution as output paces its path.

To that end, two pioneering models of optimal output growth were intentionally selected due to their innovative approach on optimal environmentally based production growth away back in the seventies. To provide updated support for the two pioneering models used, a set of important recent contributions was used, including Geldrop and Withagen ; Palmada ; Islan ; Charles ; Comolli ; Auty ; Bretschger and Smulders ; and Voinov and Farley ; all using analytical frames jointly treating output production and environmental variables under a single approach - optimal environmentally based output growth.

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The main objective of applying this methodology was to setup a way leading to a new conceptual qualitative perspective allowing for sustainability being appraised even with constrained environmental damage, e. Thus, the analysis to be undertaken in what follows has to be understood, under the methodological procedure here delineated, in the context of a qualitative frame even using two analytical theoretical models in order to reach a new conceptual construct to better understand and analyze sustainability. A general definition of capital is very important to clearly understand natural capital.

Capital here is to be considered as a stock that yields a flow of valuable goods and services into the future, as suggested by England , no matter whether the stock is manufactured or natural. If it is natural, e. Natural capital may also provide services such as recycling waste materials or pollution or even erosion control, which are also considered as sustainable income.

From this definition we can see that the structure and diversity of the system is an important component of natural capital, according to Daly , since the flow of services from ecosystems requires that they function as whole systems. Irwin and Ranganathan propose an interesting action agenda showing ways to sustain ecosystem services.

Another qualification has to do with the distinctive character of natural capital, income and natural resources. All three concepts are distinct, in the sense that natural capital and natural income are just the stock and flow components of natural resources. Renewable natural capital is analogous to machines and is subject to depreciation; nonrenewable natural capital is analogous to inventories and is subject to liquidation. Having defined natural capital, a definition of sustainability is needed in order to establish a logical connection between them.

First of all, it is important to note that, as affirmed by Daly , the stock of total natural capital equals renewable natural capital plus nonrenewable natural capital. The concept of sustainability is related to the maintenance of the constancy of the stock of total natural capital. According to Lawn and Costantini and Monni , a minimum necessary condition for sustainability is the maintenance of the total natural capital stock at or above the current level.

Hence, the constancy of the stock of total natural capital is the key idea behind the sustainability concept. Since the stock of nonrenewable natural capital can be depleted with use, a logical way to maintain constant total natural capital is to reinvest part of the prospects coming from the use of nonrenewable natural capital into renewable natural capital. It is important for operational purposes to define sustainability in terms of constant or nondeclining stock of total natural capital. This is a very significant point, since sustainability implicitly incorporates the notion of intergenerational equity.

According to the Brundtland Commission , the primary implication of sustainability is that future generations should inherit an undiminished stock of 'quality of life' assets. According to England , this broad stock of assets can be measured or interpreted in the following three ways: i as comprising human-made and environmental assets; ii as comprising only environmental assets; or iii as comprising human-made, environmental, and human capital assets. The notion of intergenerational equity, thus, lies at the core of the definition of sustainability.

Najam, Papa and Taiyab and Najam, Runnalls and Halle developed important contributions related to sustainability definitions and their relations to governance and globalization. Holmberg and Samdbrook emphasize that the Brundtland Commission , -The World Commission on Environment and Development -, was the first entity to give geopolitical significance to the use of the sustainable development concept, and thus is an important benchmark on environmental issues.

It is clear and desirable that item iii above is the most relevant one to consider under the given definition of sustainability. According to Daly , human-made capital, renewable and nonrenewable natural capital and diverse ecosystem services all interact with human capital and productive processes to determine the production level of market goods and services of a country. The specific form of this interaction is very important to sustainability.

As suggested by Sahu and Choudhury , linking those more general arguments with the definition of total natural capital given above and owing to the intergenerational issue, the frame developed up to this point is crucial for an appropriate definition of sustainability.

We see the interconnections between natural capital and sustainability. It is necessary to have the definition of the former in order to achieve the latter, and to reach the minimum necessary condition for sustainability the maintenance of the stocks of total natural capital is a requirement. A tangent issue is related to the traditional way to conceive and measure standard production growth.

It is well known that the measure of welfare via gross national product [GNP] misconceives the relevance of natural capital, despite its significance in terms of the production of real goods and services in the ecological-economic system. To deal with this shortcoming, there has been recent interest in improving national income and welfare measures to account for natural capital depletion and other corrections of mismeasured variables of economic welfare. As a consequence, a new index Index of Sustainable Economic Welfare [ISEW] has been used to allow for those corrections related to the depletion of nonrenewable resources and long-run environmental damage.

When depletion of natural capital, pollution costs, and income distribution effects are accounted for, the USA is seen as making no improvements at all. Therefore, it is possible that if we continue to ignore natural capital, we may well push welfare down while we think we are building it up. England shows the importance of the ISEW-index to recent research on environmental economics. The ISEW-index is presented in Daly and Coob and, according to Harris , such a measure has not yet been used in developing countries. Boyd also shows what is needed to take into account when green gross domestic product GGDP is under focus.

Another relevant issue concerns the constraints posed by measurement problems on quantifying environmental assets. As posted by Turner et al. Issues regarding environmental measurability will be discussed under the emergence of the so-called contingent valuation approach in 'Section Integrating the Qualitative-Analytical Approaches towards a New Conceptual Perspective on Sustainability'. Having given the relevant definitions of natural capital and sustainability, Section 'Environmentally Based Output Growth Models: an Analytical Apparatus' presents two environmentally balanced output growth models considering, in one perspective, a finite and depletable natural resource, and in another, pollution control as a way of augmenting the stock of a renewable natural resource fresh air.

The choice of both models was intentional, due to their pioneering contribution applying optimal constrained output growth to environmental issues and also the fact that they fit perfectly to the essay's main contribution of jointly considering separate theoretical pieces and contemplating an integrative perspective. The first model of production growth by Anderson will be examined, and in the second model, output growth with pollution controls by Forster will be analyzed.

Both models make use of a mathematical method called optimal control theory to address issues on environmental-production growth. The main goal is to show how standard production growth has to be slowed down when constraints on natural resource uses and pollution generation are imposed.

## An Almost Practical Step Toward Sustainability

Furthermore, this result is a key factor for the analysis of sustainability conceived here. To meet the sustainability criterion, at the same time that we know that rapid production growth leads to depletion of the stocks of natural resources and pollutes the environment, production processes accumulation of physical capital have to face constraints. The possibility of using productive factors e.

Two classes of environmentally based output growth models will be analyzed in this section: i production growth using finite and depletable natural resources and ii output growth with pollution as waste generation. The first pioneering model comes from Anderson , who explores the implications for production growth of accounting explicitly for the depletion of a nonreproducible natural resource, such as a fossil fuel reserve. Stiglitz uses a similar construction to model production growth in the presence of exhaustible natural resources. More recently, Palmada makes extensive use of the quantitative tools used in optimal growth models and applies them to formalize optimal allocations of different natural resources, such as air, water and forests during production growth phases.

The analysis to be conducted below follows the standard procedure of considering a one-sector economy, such as in the Bretschger and Smulders analysis of optimal uses of nonrenewable resources, as well as in Farzin and Akao and Voinov and Farley , both treating explicitly environmentally based output production models using optimal control in a one-sector economy.

## Models of Economic Growth with Environmental Assets (Economics, Energy and Environment)

The main objective of these models is to find an optimal capital accumulation trajectory that maximizes the present value of per capita consumption over a finite-planning horizon, subject to some specific terminal conditions on the stocks of traditional capital and natural resources. It is worth noting that when a depletable natural resource is considered, the infinitely time-period horizon used in optimal growth models, as suggested in Chiang , is no longer applicable. For an accurate analysis of the mathematical modeling of growth and sustainability, Islan is an important reference.

Other models of optimal output production growth with finite and depletable natural resources are due to Le Van, Schubert and Nguyen , whose focus relies on developing countries and poverty, and Auty , who analyzes the inverse relation between low income countries and natural resource wealth.

The problem of the optimal model by Anderson is formulated by assuming a Leontief production function:. Sa, Reis and Palma show how technology could optimally control for exhaustion of a nonrenewable natural resource in a competitive sector, in the same way technological progress enters in Anderson's model here analyzed.

From equation 1 , if , we will have:. Equation 2 tells us that the rate of output Y t is a function of physical capital and labor over time and equation 2' states that the rate of resource depletion is proportional to the rate of output production. The saving-investment identity, i.

Now, the optimal growth problem is to find the optimal path for s t the control variable that maximizes the following present value of consumption over the planning horizon [0, T]:. We can rewrite 4 in its intensive form. Thus, the optimal growth problem is:. Thus, i is the equation of physical capital accumulation in its intensive form and ii is the new version of 2'. The set of transversality conditions involves a complex mathematical procedure that it is not feasible to deal with here.

Its detailed analysis, which involves an optimal control problem with several constraints and end-point transversality conditions, is presented in Chiang The next step is to setup the current Hamiltonian. In optimal dynamic output growth models, the practice of using Hamiltonians is analogous to the use of Lagrangians in static optimization setups. Applications of the optimal dynamic versions in the context of environmental economics are done by Geldrop and Withagen and Islan in analyzing mathematical models of natural capital and sustainability using Hamiltonians with renewable and nonrenewable natural resources constraints.

The two costates are the shadow-price of physical capital stock and depletable natural resource, respectively. The current Hamiltonian is:. Clearly, this current Hamiltonian brings the depletable resource constraint in the very last part of the equation and the new end-point restrictions. Because of the necessity of considering the transversality conditions, to maximize H c at each point in time with respect to s t , we need the following decision rules:. Taking partial derivatives of H c with respect to the two state variables and using 8 :.

Using the decision rules stated in equation 7 , and taking into account the conditions in equation 9 [s t can be eliminated from the first equation in 9 and i in equation 5 ], we derive the two relevant loci of motion:. When the nonreproducible stock of natural resources is considered, the result shows a tendency to postpone capital accumulation and spend time on production growth paths where capital is used less intensively than in models of unconstrained natural resource uses.

Therefore, the basic result, coming from this production growth model accounting for depletable natural resource uses, points to a general slowdown trend of the production growth pace. This is so because the constraint poses a limiting restriction on the use of the considered depletable resources, which leads to a reduced rate of physical capital accumulation and increased rate of savings less consumption , while acting as the control variable, drives per capita consumption downwards.

It should be emphasized that this behavior is the optimal one, in terms of maximizing the present value of the consumption stream over time and at the same time satisfying the relevant constraints. It is optimal to slow down the country's capital accumulation decreasing production when depletable natural resources are considered. More recent contributions have shown this same result in different contexts, such as Comolli in investigating the relations between natural and physical capital during specific economic growth phases, and also Farzin and Akao as far as optimal exhaustion of a nonrenewable is concerned within a finite time horizon plan.

Linking the concept of sustainability derived in Section 'Natural Capital and Sustainability: a Qualitative Conceptual Approach' with the result of this environmentally sounded growth model by Anderson , slowing down the pace of output growth is feasible and desirable, for the stock of nonrenewable natural resources cannot be totally depleted and production activity is in its course, albeit at a slower pace. It is also possible to rule the rate of depletion of the nonrenewable natural resource in such a way that the rate of regeneration of renewable natural capital is always higher, and thus augmentation of total natural capital is obtained.

This arrangement would at least preserve the constancy of the total stock of natural capital, a pre-requisite to sustainability as shown in Section 'Natural Capital and Sustainability: a Qualitative Conceptual Approach'. The second model deals with an important feature not considered in standard production growth models. Following Forster , we present an optimal physical capital accumulation model taking into account the possibility of waste generation pollution. As Forster states, "It is naive to think that no wastes are produced and fairly obvious that the free disposal assumption of the neoclassical growth model is not satisfied in the real world" p.

Again, the choice of this optimal output model was intentional, due to its pioneering role in optimal environmental economics. Other recent models of pollution generation under optimal environmentally based output growth can be cited, such as Lyon and Lee ; and Chakravorty, Moreaux and Tidball Making use of the usual procedure, we begin, following Foster , assuming a standard production function:. Once again, it is assumed that this production function is well behaved, in the sense that all standard characteristics apply.

It is also assumed that the labor force is a constant proportion of a constant population. The produced output can be either consumed C t , invested in physical capital stock I t or in pollution control E t. Therefore, an additional restriction must be imposed in the following way:. At this stage we have the equations to setup the optimal control problem, but it is reasonable to suppose that physical capital also produces pollution in addition to output.

It is also worth noting that by devoting output to pollution control, the community can lower the amount of pollution generated, refreshing air quality. Note that there is no stock accumulation of pollutant in this model, which is a recognizable shortcoming. But, as in Forster , it can be easily introduced without substantial changes. Therefore, following Foster , we can formulate an equation for pollution determination as:. Finally, the last equation to consider in order to setup the optimal control problem is the linearly separable utility function, assumed to be a function of consumption C t and pollution P t ,:.

Now we are ready to state the optimal control problem. The objective is to maximize the discounted flow of utility over an infinite time horizon. The problem is to find an optimal path for the variables in order to:. To analyze the solution for this problem, we need to formulate the current Hamiltonian, which in this case is as follows:.

We have a similar problem as the one we derived in the last model of optimal capital accumulation in the presence of a depletable resource. The only difference is that the very last two terms in 17 and the fact that transversality conditions do not have a role to play, as stated in Chiang , given the infinite-horizon feature of this problem. The derivation of the optimal conditions leads to the following equations of motion for the two loci in consumption and capital accumulation:.

Laura Castellucci. Water Allocation in Rivers under Pressure. Dustin Evan Garrick. The Economics of Sustainability. John C. Beyond Smoke and Mirrors. Burton Richter. Joseph Ofungwu. Microeconomic Policy. Solomon Cohen. Sustainability and Energy Politics. Giorel Curran. Nicholas Stern. Schumpeter's Price Theory. Harry Bloch. Peter Pogany. Income Distribution in Macroeconomic Models. Giuseppe Bertola. Social Fairness and Economics. Lance Taylor. Analytical Political Economy. Roberto Veneziani. Groups and Markets.

Hans Gersbach. Meta-Analysis in Environmental Economics. Peter Nijkamp. Finance and the Macroeconomics of Environmental Policies. Understanding Environmental Policy Convergence. Modernization Science. Chuanqi He. Environmental Kuznets Curves. Michael P. Routledge Handbook of International Education and Development.

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