Essays in Sustainable Development, Summaries of Sustainable Development

Download Essays in Sustainable Development and more Summaries Sustainable Development in PDF only on Docsity! Essays in Sustainable Development Inaugural-Dissertation zur Erlangung des Grades Doctor oeconomiae publicae (Dr.oec.publ.) an der Ludwig-Maximilians-Universität München 2006 vorgelegt von Laura Gariup Referent: Professor Dr. Peter Egger Korreferent: Professor Sven Rady, Ph.D. Promotionsabschlussberatung: 07 Februar 2007 Acknowledgment To my supervisor Professor Peter Egger I express my deepest gratitude for all the wonderful support, professional as well as human, he gave me and because he left me the freedom in continuing to do research on a topic which I am extremely interested in. If these four difficult years can be crowned with the present dissertation, that is also because Professor Sven Rady, from the beginning to the end, has always encouraged and helped me and he has given me precious teachings which I will never forget. For all of that I thank him. I want to thank Professor Marco Runkel because he was one of the first persons who stimulated me to persevere in my work and found always time to listen to my research ideas. It is also especially thanks to him that I had such an enrichment and wonderful experience as that of being a teaching assistant at the chair of Professor Bernd Huber. For another unforgettable experience I thank Professor Sjak Smulders who made my three months stay at the Economic Department of Tilburg University possible. I extremely benefited from his comments and the stimulating environment. For all his engagement I am very grateful to Professor Gerhard Illing. The mathematical part of chapter 4 is based upon derivations in work by Pro- fessor Georg Schlüchtermann. I gratefully acknowledge his assistance with the ex- position and his kind help in preparing the draft. For the precious help with all the administrative questions along these years I am thankful to Ingeborg Buchmayr and to Dirk Roesing with all the technical ones. I 3.6 Conclusion: what the new environmental specifications tell us . . . . 28 3.7 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.7.1 Optimal control problem . . . . . . . . . . . . . . . . . . . . . 30 3.7.2 Derivation of growth rates in a balanced steady state . . . . . 31 3.8 Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.8.1 Optimal control problem . . . . . . . . . . . . . . . . . . . . . 34 3.8.2 Derivation of growth rates in a balanced steady state . . . . . 35 4 Biodiversity loss and stochastic technological processes: a sustain- able growth analysis 39 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Structure of the model . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 The new Hotelling Rule . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 The optimal paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 The optimal paths analysis . . . . . . . . . . . . . . . . . . . . . . . . 54 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.7 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.8 Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.9 Appendix 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 Nature as a knowledge reservoir: a non-scale endogenous growth model with relaxation of knife edge assumptions 72 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 The scale effects and knife edge debate . . . . . . . . . . . . . . . . . 75 5.3 Nature as a knowledge reservoir . . . . . . . . . . . . . . . . . . . . . 77 5.4 Natural knowledge as a prerequisite for sustained growth . . . . . . . 79 5.4.1 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.4.2 Dynamics of technology and natural knowledge . . . . . . . . 82 5.5 The threat from what gets lost: pollution damages on nature as a knowledge reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 IV 5.6 How technological progress influences a knife edge assumption . . . . 87 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Bibliography 91 V Part I Concepts 1 Introduction 4 scientific research. In an economic model this implies inserting an environmental indicator variable, which will be called natural knowledge, not only into the produc- tion function of the final good but also into the production function of the standard technological sector. This model specification, in addition to giving a new explana- tory variable for the growth process, eliminates the presence of scale effects and the recourse to knife edge assumptions about the returns to scale in the produced factors of production. Here, as well as in the previous two chapters, the final goal of the analysis is to conjecture whether the model predicts sustainable growth, and under which assumptions. Each chapter of the second part is therefore a self-contained paper which can be read independently of the others, although the chosen sequence is not casual. It represents an evolution not only in the results of the models (from ones without sustainable development to ones with sustainable development) but also in the focus of the environmental analysis (from the particular role of the regenerative capacity of nature in the regeneration function, to the general one of nature as basis for scientific advancement). 1.2 The sustainability issue It was during the 1970s that the new concept of economic sustainability entered the international political agenda. At that time politicians and researchers recog- nized that the environment plays an important role for the maintenance of economic growth. Nevertheless note that this consciousness was already present in classical economics two centuries before. For all classical economists the central question of research was what determined national wealth and its growth (Perman et al. (2003)), and natural resources were important explanatory variables, as well for Thomas Malthus in his ”Essay on the Principle of Population” (1798), as for David Ricardo in his ”Principles of Political Economy and Taxation” (1817). In the 1970s the connection between natural environment and economic growth returned to the center of attention for many reasons, most important the energy Introduction 5 crisis, environmental catastrophes and discouraging scientific publications, which produced a lot of debate.1 That new sensibility for the relationship between nature and the economic world flowed into the United Nations Conference on the Human Environment in 1972 which was the first of a long series of international confer- ences about the role of the natural environment in the economic development. The successive decades, in fact, are characterized by an increasing awareness of the role of the natural environment and therefore for its preservation, which is testified by all international conventions, programmes, conferences, publications that followed, some of them listed in the appendix to this introduction.2 But it was only in 1987 that the concept of sustainable development was for- malized. In that year the final report of the World Commission on Environment and Development ”Our Common Future” (WCED (1987)) was published.3 It states that ”environment and development are not separate challenges: they are inex- orably linked” and ”attempts to maintain social and ecological stability through old approaches to development and environmental protection will increase instabil- ity.” Therefore the new concept of sustainable development was presented, which is development that ”seeks to meet the needs and aspirations of the present without compromising the ability to meet those of the future.” In the report this new concept is absolutely not associated with reduction of the economic activities, instead: ”Far from requiring the cessation of economic growth, it recognizes that the problems of poverty and underdevelopment cannot be solved unless we have a new era of growth in which developing countries play a large role and reap large benefits.” And further ”The medium term prospects for industrial countries are growth of 3-4 per cent, the minimum that international financial institutions consider necessary if these coun- tries are going to play a part in expanding the world economy. Such growth rates could be environmentally sustainable if industrialized nations can continue the re- 1The Costs of Economic Growth, Mishan (1967); The Limits to Growth, Meadows et al. (1972). 2For a detailed historical reconstruction see UNEP (2002). 3Both the Commission and the report are also called Brundtland after the chairperson of the commission, Gro Harlem Brundtland. She was both Minister of the Environment and Prime Minister of Norway. Introduction 6 cent shifts in the content of their growth towards less material- and energy-intensive activities and the improvement of their efficiency in using materials and energy.” Then after this cornerstone report, the United Nation Conference on Environ- ment and Development in 1992 followed. Among other goals it produced the docu- ment ”Agenda 21” (UN (1993)), which is an action programme for the realization of sustainable development and the creation of a permanent UN agency called Com- mission on Sustainable Development. Ten years later in 2002 the World Summit on Sustainable Development followed. It states again in the ”Report of the World Summit on Sustainable Development” (UN (2002)), the necessity to implement the programs for sustainable development asserting that:”Thirty years ago, in Stock- holm, we agreed on the urgent need to respond to the problem of environmental deterioration. Ten years ago, at the United Nations Conference on Environment and Development, held in Rio de Janeiro, we agreed that the protection of the environment and social and economic development are fundamental to sustainable development, based on the Rio Principles. To achieve such development, we adopted the global programme entitled Agenda 21 and the Rio Declaration on Environment and Development, to which we reaffirm our commitment. The Rio Conference was a significant milestone that set a new agenda for sustainable development.” But, if the Brundtland report is commonly recognized to have put the concept of sustainable development on the international scene, in the literature there are many different definitions of sustainability as recalled by Pezzey (1997). In the standard view the term sustained growth is used to indicate increases in consumption, while sustained development refers to increases in utility. In this dissertation the terms growth and development are used interchangeably, as long as the utility function considers only consumption. To the extent that also the second goal of an increase in the environmental indicator is met, we speak of sustainable growth/development. Introduction 9 1989 Basel Convention on the Transboundary Movements of Hazardous Wastes and their Disposals (Basel), MEA; Inter-governmental Panel on Climate Change established 1990 Second World Climate Conference, Geneva, Switzerland; Global Climate Observing System (GCOS) created 1991 Global Environment Facility established to finance conventions 1992 UN Conference on Environment and Development (the Earth Summit), Rio de Janeiro, Brazil; Convention on Biological Diversity (CBD), MEA; UN Framework Convention on Climate Change 1993 Chemical Weapons Convention 1994 UN Convention to Combat Desertification (UNCCD), MEA; International Conference on Population and Development, Cairo, Egypt; Global Conference on the Sustainable Development of Small Island Developing States, Bridgetown, Barbados 1995 World Summit for Social Development, Copenhagen, Denmark; World Business Council for Sustainable Development created 1996 ISO 14 000 created for environmental management systems in industry; Comprehensive Nuclear Test Ban Treaty 1997 Kyoto Protocol adopted (Kyoto), MEA; Rio + 5 Summit reviews implementation of ”Agenda 21” 1998 Rotterdam Convention on the Prior Informed Consent Procedure for Certain Hazardous Chemicals and Pesticides in International Trade (PIC), MEA 1999 Launch of ”Global Compact” on labour standards, human rights and envi- ronmental protection Introduction 10 2000 Millenium Summit, New York, United States; World Water Forum, The Hague 2001 Stockholm Convention on Persistent Organic Pollutants (POPs), MEA 2002 World Summit on Sustainable Development, Johannesburg, South Africa Chapter 2 Economy and environment joined together 2.1 What an ideal model should encompass As indicated in the previous chapter, to study the sustainability issue the natural environment has to be incorporated into the functional specifications of an optimal growth model. An almost complete and standard model could be constructed with • a production function of the final good (Y ) which is affected by the labor force (L), the human-made capital stock, natural capital and pollution. Re- spectively human-made capital embraces physical capital (K), human capital (H) and technological capital (A); natural capital describes the flows of re- newable (RR) and non-renewable (RNR) resources, and the stocks of renewable (SR) and non-renewable (SNR) resources; pollution in form of stock (P ) and flow (F ), • a growth function respectively for K, H and A, • a growth function for L, • a growth function for SR and SNR, 11 Economy and environment joined together 14 and the necessary condition with respect to the control variables Ct and RNR,t and with respect to the state variables Kt and SNR,t are UC,t = λ1,t (2.1) λ1,tYRNR,t = λ2,t (2.2) ˙λ2,t = ρλ2,t (2.3) ˙λ1,t = ρλ1,t − YKtλ1,t. (2.4) Equations (2.1) and (2.2) describe the static efficiency conditions. The former states that the marginal net benefit of one unit of output either used for consumption or for increases in the capital stock must be equal. The latter condition implies that the marginal value of the resource stock must be equal to the value of the marginal product of the resource. The other two equations instead represent the dynamic efficiency conditions. Equation (2.3) is known as the Hotelling rule and assures that the growth rate of the shadow price of the resources is equal to the utility discount rate. The same happens for the other asset of this economy in equation (2.4). It guarantees in fact that capital appreciation (the growth rate of the shadow price of capital) plus marginal productivity of capital is equal to the discount rate. Another way to see that would be to differentiate equation (2.2) with respect to time and substituting the value with equation (2.3) and (2.4). This operation leads to YKt = ẎRNR,t YRNR,t which states a no-arbitrage condition of equality among rates of return. Then differentiating equation (2.1) with respect to time and inserting equation (2.4), the growth rate of consumption is found: gCt = 1 η (YRNR,t − ρ) where η is the elasticity of marginal utility, −UCC,tCt UC,t . With η being positive, whether consumption is growing, decreasing or stays constant, depends on the difference be- tween the marginal productivity of capital and the rate of time preference. But Economy and environment joined together 15 with capital accumulation the marginal productivity of capital decreases, so Das- gupta and Heal (1974) demonstrated that permanent growth is possible only if the elasticity of substitution σ between the exhaustible resource and capital is greater than 1 and the asymptotic marginal productivity of capital is greater than the rate of time preference. Whether and to what extent substitution between natural and human-made capital is possible, has long been debated between economists (Solow (1986)) and is difficult to imagine for the amenity base and life support services. In the context of the present model it means that we are able to bequeath to future generations substitutes for exhaustible resources, so e.g. it should not be important how much of one specific resource we leave to future generations, but whether we leave them the ability to satisfy the need that we satisfy today with that resource (Perman et al. (2003)). 2.3 Hotelling rule for renewable resources Renewable resources, and nature as an environmental indicator in general, have the capacity to regenerate. Biological populations such as animals or forests reproduce themselves, and natural resources such as water, air, soil are reproduced by bio- chemical and biophysical processes. This means that the functional specification for the change in the resource stock should be substituted with ṠR,t = −RR,t + G(SR,t) where the basic functional form Gt = G(SR,t) states that the resources’ regeneration positively depends on the stock. The current-value Hamiltonian is now slightly different, H = U(Ct) + λ1,t[Y (Kt, RR,t)− Ct]− λ2,t[RR,t −G(SR,t)], Economy and environment joined together 16 and the necessary conditions are UC,t = λ1,t (2.5) λ1,tYRR,t = λ2,t (2.6) λ̇2,t = ρλ2,t −GSR,t λ2,t (2.7) λ̇1,t = ρλ1,t − YKtλ1,t. (2.8) Equation (2.7) is different from the previous model only because the Hotelling rule now is λ̇2,t λ2,t = ρ−GSR,t . Differently from before, the growth rate of the shadow value of resources is smaller because the ability of resources to regenerate themselves decreases their physical scarcity. Chapter 4 presents a complex functional form for the change of the resource stock in order to investigate the problem of biodiversity loss. Even though the structure of that model is different from the present deterministic one, focusing the attention only on the environmental part, we can still appreciate how the Hotelling rule will be modified. The first consideration, following chapter 3, is that GSR,t is not a constant but a function, GSR,t = GS(Pt, SR,t), which is negatively affected by the stock of pollution Pt, so that λ̇2,t λ2,t = ρ−GS(Pt, SR,t). The second step is to treat jointly the negative effects on the resources coming from the harvesting (RR,t) and the environmental degradation, let’s call it Dt, which is a function of the flow of pollution Ft, Pt and SR,t. Therefore the growth rate of the shadow value is increased again: λ̇2,t λ2,t = ρ−GS(Pt, SR,t) + DSR,t . Finally, the positive role of different types of environmentally friendly technolo- gies (T ) is considered. It makes the reduction of the physical scarcity of the resources Chapter 3 Role of the regenerative capacity of nature in the sustainability debate: a Schumpeterian endogenous growth model 19 Role of the regenerative capacity of nature in the sustainabililty debate 20 3.1 Introduction Since the publication of ”The Brundtland Report” in 1987, the growth literature has been using a new concept: sustainable development.1 About since then, a new framework for studying growth has been developed. If the neoclassical growth literature of the ’70s focused predominantly on the optimal use of non-renewable re- sources2 in response to the ”oil crises”, new growth models of the ’90s devote increas- ingly attention to environmental quality problems as a consequence of pollution- induced global changes (greenhouse effect, biodiversity loss). The United Nations Conference on Environment and Development held in Rio de Janeiro in 1992 recognized the pressing environment and development problems of the world and, through adoption of Agenda 21, produced a strategy for sustainable development in the 21st century. After a decade known as the rhetoric decade, the World Summit for Sustainable Development, held in Johannesburg in August 2002, made clear that urgent action is necessary. The main result of the summit was that ”one of the three pillars of sustainable development - the environment - is seriously damaged because of the distortions placed on it by the actions of human population. The collapse of the environmental pillar is a serious possibility if action is not taken as a matter of urgency to address human impacts, which have left: increased pollutants in the atmosphere, vast areas of land resources degraded, depleted and degraded forests, biodiversity under threat, reduction of the fresh water resources, depleted marine resources” (UNEP (2002)). One of the best-established endogenous growth models is the Schumpeterian approach of Aghion and Howitt (1998), henceforth referred to as A&H. They over- come the shortcoming of the Stokey (1998) setup in reaching the combination of two goals which count for the most pragmatic definition of sustainable development (see Brock and Taylor (2004b)): ”... a balanced growth path with the joint result of increasing environmental quality and ongoing growth in income per capita”, the 1The broad concept of sustainable development was discussed also before but that report put it on the international agenda. For a comprehensive survey of definitions see Pezzey (1992). 2Even though there are exceptions see Forster (1980). Role of the regenerative capacity of nature in the sustainabililty debate 21 so-called sustained growth. But their model ignores the lasting effects of cumulative pollutants by modeling the regenerative capacity of the environment as a constant. I will demonstrate that in the Schumpeterian set-up sustainable development, as de- fined above, cannot be reached if, in line with insights from ecological and biological sciences, the regenerative capacity of the environment depends on the lasting effects that pollutants have on the environment. It is argued that a more sophisticated theoretical framework, incorporating different types of innovation, is needed. I will proceed as follows: in section 2 the theoretical background for a regenera- tion function of the environment with a non-constant regenerative capacity will be presented; in section 3 the economic part of the model will be briefly introduced. In section 4 the new concept about the regenerative capacity of the environment will be introduced in the model. In section 5 the main idea will be explicitly modeled and section 6 will comment on the results and conclude. 3.2 The ecological part of a growth model: the regeneration function Two of the four services3 that the environment provides are the waste sink and the life support. The former means the capacity to disperse pollutants. The latter subsumes services like regulation of the hydrological cycle (material cycles of water and phosphorus), regulation of the gaseous composition of the atmosphere (material cycle of carbon), generation and conservation of soils (material cycle of nitrogen). In the growth literature, these two services can be associated with the regenerative capacity of the environment in the regeneration function of the environment. The environment or nature is captured by an environmental quality indicator reflecting all biosystems and their interactions. So, the regenerative capacity of the envi- ronment is different from just the regenerative capacity of one biological population because it includes also the ability of particular types of renewable resources, namely 3The other two are the resource base and the amenity base (see Perman et al. (2003)). Role of the regenerative capacity of nature in the sustainabililty debate 24 3.4 The Schumpeterian approach to the environ- mental quality with endogenous regenerative capacity of the environment As Aghion and Howitt did, I use the specifications proposed by Stokey (1998) for 1) the final good production function with environmental awareness, 2) the flow of pollution production function, and 3) the environmental disutility in the utility function. In contrast, a new regeneration function of the environment is introduced. For- mally, this implies that the final good production function with environmental aware- ness is Y (t) = K (t)α (B (t) (1− n (t)))1−α z (t) where 0 < α < 1 and 0 6 z 6 1 is a measure of the dirtiness of the existing technologies or of the emission standard of the existing techniques. ”The cost of using a cleaner technique is that less output can be obtained per unit of input” (Aghion and Howitt (1998)). The production function for the flow of pollution is P (t) = Y (t) (z (t))γ where γ > 0. The loss of environmental quality E is inserted into the utility function of the representative agent. This captures the amenity base services of the environment. The instantaneous utility function has an additive isoelastic form: u(c, E) = (c)(1−ε) − 1 1− ε − (−E)(1+ω) − 1 1 + ω , with ε, ω > 0. c denotes the consumption and E is defined as the difference between the actual environmental quality and the maximal environmental quality, which could only be reached if all production would cease indefinitely. Role of the regenerative capacity of nature in the sustainabililty debate 25 E is also subject to an ecological threshold of the form Emin ≤ E (t) ≤ 0 because the authors want to take into account that a lower limit exists below which environ- mental quality cannot fall without starting an irreversible deterioration process.4 The new regeneration function of the environment is Ė (t) = −P (t)− θ(t)E (t) where θ(t) = 1 + aE(t). Here, the regenerative capacity of the environment, θ, is not constant any more.5 Since θ depends on E, the establishment of an ecological threshold Emin ≤ E(t) ≤ 0 ∀t implies that the regenerative capacity of the environ- ment, i.e. θ(t), faces a threshold as well.6 So, E is increased by the flow of pollution P and reduced by the regenerative function which now is not any more linear, θE(t), but quadratic, (1 + aE(t))E(t). The social planner’s problem is described (where for concision we drop the time index with all the variables) by: max c,z,n ∫ ∞ 0 e−ρtu (c, E) dt subject to K̇ = Y − c = Kα(B(1− n))1−αz − c Ḃ = ησnB Ė = −P − θE = −Kα(B(1− n))1−αz1+γ − (1 + aE)E and the initial conditions for K, B, E, the ecological threshold Emin ≤ E0 ≤ 0, the requirement that K(t) and B(t) ≥ 0, and the transversality conditions for K, B, E. 4By assumption, see page 165 of Aghion and Howitt (1998); in any case, this threshold is not relevant for the optimal balanced steady state. 5In A&H, θ represents the maximal potential rate of regeneration. This interpretation is no longer suitable here. 6Note that E(t) is a measure of cumulative pollutants as well as of the current state of resources (actual environmental quality). Hence, θ(t)E(t) controls for the loss of environmental quality associated with cumulative and non-cumulative pollutants. Role of the regenerative capacity of nature in the sustainabililty debate 26 The new current-value Hamiltonian is H = u(c, E) + λ [ Kα (B (1− n))1−n z − c ] + µσηnB −ζ [ Kα (B (1− n))1−α zγ+1 + (1 + aE)E ] and the new first order conditions for E and ζ are7 ∂H ∂E = − · ζ + ρζ ⇒ · ζ = ρζ − (−E)ω + ζ (1 + 2aE) , ∂H ∂ζ = · E ⇒ · E = −KαB1−α(1− n)1−αzγ+1 − (1 + aE) E. This leads to a completely different result than A&H. In the original model of A&H a balanced steady state8 with sustainable development is possible even though under four9 really special assumptions, because sustained development (gK , gc, gy > 0)10 is combined with environmental improvement (gE < 0 ). Proposition 1 In this model along the balanced steady state, there is no sustainable development defined as joint achievement of sustained development and environmen- tal improvement. Hence, an improvement of the environmental quality (gE < 0) and, at the same time, sustained development (gK , gc, gy > 0) cannot be achieved. But rather, there is a constant environmental quality (gE = 0) and non-sustained development (gK , gc, gy = 0). Proof: See Appendix 1 to this chapter. 7We only present the most relevant first order conditions, here. The other ones are presented in Appendix 1 to this chapter. 8That is where all variables growth at a constant rate. 9First assumption: ε−1 > 0, second assumption: ησ−ρ > 0, third assumption: (ε−1)(ησ−ρ) < θ [ ε(1 + ω) + ε+ω (1−α)γ ] and fourth assumption: Emin ≤ E0 ≤ 0. 10g means growth rates. Role of the regenerative capacity of nature in the sustainabililty debate 29 that are suited to manage the stock of damage should be targeted. Role of the regenerative capacity of nature in the sustainabililty debate 30 3.7 Appendix 1 3.7.1 Optimal control problem The problem exhibits three state variables, K(t), B(t) and E(t), and three control variables, c(t), n(t) and z(t). Formally, it can be written as max c,n,z ∞∫ 0 u(c, E)e−ρtdt subject to · K = Y − c = Kα(B(1− n))1−αz − c · B = σηnB · E = −Kα(B(1− n))1−αzγ+1 − (1 + aE)E K0, B0 ≥ 0 E0 ∈ (Emin, 0) lim t→∞ e−ρtλK = 0 lim t→∞ e−ρtµB = 0 lim t→∞ e−ρtζE = 0 E(t) ∈ [Emin, 0] ∀t. The current-value Hamiltonian is H = u(c, E)+λ [ Kα (B (1− n))1−α z − c ] +µσηnB−ζ [ Kα (B (1− n))1−α zγ+1 + (1 + aE)E ] . The necessary conditions are: Role of the regenerative capacity of nature in the sustainabililty debate 31 Static part: ∂H ∂c = 0 ⇒ ∂u(c, E) ∂c − λ = 0 ⇒ λ = c−ε (3.1) ∂H ∂z = 0 ⇒ λ− ζ (γ + 1) zγ = 0 ⇒ ζ = λ (1 + γ) zγ (3.2) ∂H ∂n = 0 ⇒ µησB = (1− α) γ 1 + γ λY 1− n (3.3) Dynamic part: ∂H ∂K = − · λ + ρλ ⇒ · λ λ = ρ− α Y K ( 1− ζ λ zγ ) (3.4) ∂H ∂B = − · µ + ρµ ⇒ · µ = ρµ− µηnσ − (1− α) λ Y B γ 1 + γ (3.5) ∂H ∂E = − · ζ + ρζ ⇒ · ζ = ρζ − (−E)ω + ζ(1 + 2aE). (3.6) A&H instead have ∂H ∂E = − · ζ + ρζ ⇒ · ζ = ρζ − (−E)ω + ζθ. The three resource constraints are ∂H ∂λ = · K ⇒ · K = KαB1−α(1− n)1−αz − c (3.7) ∂H ∂µ = · B ⇒ · B = σηnB (3.8) ∂H ∂ζ = · E ⇒ · E = −Kα(B(1− n))1−αzγ+1 − (1 + aE)E. (3.9) A&H instead have ∂H ∂ζ = · E ⇒ · E = −Kα(B(1− n))1−αzγ+1 − θE. 3.7.2 Derivation of growth rates in a balanced steady state From gK ≡ · K K = Y K − c K Role of the regenerative capacity of nature in the sustainabililty debate 34 3.8 Appendix 2 3.8.1 Optimal control problem The problem exhibits four state variables, K(t), B(t), E(t) and S(t), and three control variables c(t), n(t) and z(t). Formally, it can be written as max c,n,z ∞∫ 0 u(c, E)e−ρtdt subject to · K = Y − c = Kα(B(1− n))1−αz − c · B = σηnB · E = −P + bS−δE · S = φPS K0, B0, S0 ≥ 0 E0 ∈ (Emin, 0) lim t→∞ e−ρtλK = 0 lim t→∞ e−ρtµB = 0 lim t→∞ e−ρtζE = 0 lim t→∞ e−ρtξS = 0 E(t) ∈ [Emin, 0] ∀t. The current-value Hamiltonian is H = u(c, E) + λ [ Kα (B (1− n))1−α z − c ] + µσηnB − ζ [ P − bS−δE ] + ξφPS. We recall that P = Y zγ = Kα (B (1− n))1−α z1+γ. The necessary conditions are: Role of the regenerative capacity of nature in the sustainabililty debate 35 Static part: ∂H ∂c = 0 ⇒ ∂u(c, E) ∂c − λ = 0 ⇒ λ = c−ε (3.23) ∂H ∂z = 0 ⇒ λ Y z − ζ(1 + γ) P z + ξφ(1 + γ) P z S = 0 ⇒ ζ = λY (1 + γ)P + ξφS (3.24) ∂H ∂n = 0 ⇒ −λ(1− α) Y (1− n) + µσηB + ζ(1− α) P (1− n) − ξφ(1− α) P (1− n) S = 0 ⇒ µησB = λ(1− α) Y (1− n) − ζ(1− α) P (1− n) + ξφ(1− α) P (1− n) S = 0 ⇒ µησB = (1− α) γ 1 + γ λY 1− n (3.25) Dynamic part: ∂H ∂K = − · λ + ρλ ⇒ · λ λ = ρ− α Y K ( 1− ζ λ zγ + ξ λ zγφS ) (3.26) ∂H ∂B = − · µ + ρµ ⇒ · µ = ρµ− (1− α)λ Y B − µσηn + ζ P B (1− α)− ξφ P B (1− α) ⇒ · µ = ρµ− µηnσ − (1− α) λ Y B γ 1 + γ (3.27) ∂H ∂E = − · ζ + ρζ ⇒ · ζ = ρζ − (−E)ω − ζbS−δ (3.28) ∂H ∂S = − · ξ + ρξ ⇒ · ξ = ρξ − ξφP + ζbδS−δ−1E. (3.29) The three resource constraints are ∂H ∂λ = · K ⇒ · K = KαB1−α(1− n)1−αz − c (3.30) ∂H ∂µ = · B ⇒ · B = σηnB (3.31) ∂H ∂ζ = · E ⇒ · E = −P + bS−δE (3.32) ∂H ∂ξ = · S ⇒ · S = φPS. (3.33) 3.8.2 Derivation of growth rates in a balanced steady state From gK ≡ · K K = Y K − c K Role of the regenerative capacity of nature in the sustainabililty debate 36 a balanced steady state · gK = 0 is found either when gK = gY = gc, (3.34) as in the endogenous growth literature is assumed, or when gK = 0, gY = gc. (3.35) From (3.23), we obtain gλ = −εgc. (3.36) From (3.28) the growth rate of ζ is gζ = ρ− (−E)ω ζ − bS−δ and therefore in balanced steady state the growth rate of gζ must satisfy −ωgE + gζ = 0 (3.37) and gS = 0. (3.38) From gB ≡ · B B = ησn and · gB = 0, we get gn = 0. (3.39) From (3.27) and · gµ = 0, we obtain gY − gB + gλ − gµ = 0. With (3.36) and knowing that gY = gc, we obtain gc (1− ε)− gB = gµ. (3.40) From (3.32) and · gE = 0, we find −Y zγ E (gY + γgz − gE)− δbS−δgS = 0. Chapter 4 Biodiversity loss and stochastic technological processes: a sustainable growth analysis 39 Biodiversity loss and stochastic technological processes 40 4.1 Introduction The two most critical environmental challenges that our society faces nowadays are human-induced global ones: the greenhouse effect and the biodiversity loss. We develop a stochastic endogenous growth model to investigate the biodiversity loss challenge for the purpose of sustainability. In fact we investigate the conditions for an optimal growth path to be sustainable. There are four motivations for developing such a model. First, the problem of biodiversity loss has not received attention from the growth literature yet (in con- trast to the greenhouse effect). Second, the analysis of the biodiversity loss requires an approach that is fundamentally different from that one used for assessing the greenhouse effect. Third, an optimal growth model is the best way to study sustain- ability, since we can use such a model to derive conditions under which sustained growth can go hand in hand with environmental improvement. Finally, the chosen stochastic approach is very effective to incorporate different types of technological progress and to find an analytical solution. In a deterministic version we should add a state variable for each different type of technological progress. This would increase the complexity of the model and, as already discussed in Chapter 2, the chance to find a general mathematical solution, see Pezzey (1992). The importance of biodiversity loss as an indicator of environmental sustainabil- ity has only recently come to the limelight of research. For instance, the current observed rate of extinction per century just for birds and mammals is 100-1000 times the ”natural” background rate, based on fossil records, see Townsend et al. (2003). Furthermore, the tropical moist forest clearance and burning due to land conversion (one of the major causes of biodiversity loss) will increase over the next 50 years by 30 percent and 20 percent, respectively, see Tilman et al. (2001). Why do we need a different approach to account for biodiversity loss? This can be seen immediately from recalling that the two major reasons for biodiversity loss (at the level of species, communities, and ecosystems) are overexploitation of renewable resources (which in the model corresponds to the variable R) and habitat disruption. Biodiversity loss and stochastic technological processes 41 The latter can be directly caused by the flow of pollution (Γ), or indirectly by the stock of pollution (P). The consideration of biodiversity loss requires viewing renewable resources from a broader perspective (including biological populations and water, soil and atmosphere). Hence, it is insufficient to analyze either the optimal use of renewable resources or the lasting effects of pollution problems resulting from that use. In fact both aspects are simultaneously relevant for biodiversity loss. This is inherently different from the greenhouse effect whose cause is the cumulation of pollutants rather than the exhaustion of non-renewable resources per se. There is a notable disconnection between the ”70s growth models” being inter- ested in the optimal use of non-renewable resources (Dasgupta and Heal (1974), Stiglitz (1974), Solow (1974a)) and the present pollution-induced awareness intrin- sic in the ”90s growth models” on environmental quality (Bovenberg and Smulders (1995), Stokey (1998), Aghion and Howitt (1998), Brock and Taylor (2004a)). Brock and Taylor (2004b) provide a critical discussion of this ‘unbundling’ of interests in environmental issues. Obviously, the case of renewable resources calls for a simulta- neous treatment of ”optimal use” and ”quality degradation” issues. This chapter combines the ideas of the standard environmental quality litera- ture of economic growth (which investigates pollution awareness) with that of the ”corn-eating” framework (used in the analysis of optimal use of renewable resources, see Pezzey (1992)). Thereby, three standard shortcomings will be overcome: (i) we investigate the joint effect of harvesting and induced pollution degradation on re- newable resources; (ii) in contrast to previous research, the recruitment curve will not treat the environment as invariant (Townsend et al. (2003)) or, put differently, the regenerative capacity (Clark (1990)) will not be exogenous and fixed but ”con- ditional on the particular environment circumstances that happen to prevail, and [it will] change if any of those circumstances change” (Perman et al. (2003));1 (iii) the regeneration function will be affected by the scientific and technological advance- ments, unlike in previous research. We introduce three possible types of environ- 1In the previous chapter we have demonstrated the importantce of the non-constancy of the regenerative capacity. Biodiversity loss and stochastic technological processes 44 of environmental friendly innovations: σ1µP−ζ t Sκ t dq1,t, σ2Rtdq2,t, and σ3ΓP ξ t Sρ t dq3,t. 4) The variations in the random cumulated number of innovations (dq1,t,dq2,t, dq3,t) follow a non stationary Poisson process2 with arrival rates λi(Nt)(i = 1, 2, 3), where Nt denotes the fraction of labor devoted to R&D as in Lafforgue (2004): P(qi,t − qi,s = k) = ( ∫ t s λi(Nu)du)k k! e− R t s λi(Nu)du (4.3) for 0 ≤ s ≤ t, and q1,t − ∫ t 0 λ1(Ns)ds, q2,t − ∫ t 0 λ2(Ns)ds, q3,t − ∫ t 0 λ3(Ns)ds are in- dependent martingales (E[qi,T |qi,t] = qi,t for T ≥ t). The intensity function λi(.) is assumed to be increasing and concave in Nt. In the time intervals between the innovation jumps of the technological sector, the resource evolves deterministically. Labor can be devoted to either production or R&D activities. Therefore, the following constraints must hold: Nt ≥ 0, Lt ≥ 0, 1− Lt −Nt ≥ 0. We consider pollutants as an inevitable consequence of human activity. Following the argument of the law of thermodynamics and the considerations of Common (1995) about the environmental impact, in the very long run, we model the evolution of the stock of pollution as: · P = ΓP. Note that this specification is consistent with an inverted U-shape of the environ- mental Kuznets curve. However, it also covers the case where no such inverted U-shaped environmental Kuznets curve prevails.3 What is important here is that 2A Poisson process (qt) is a time dependent family of identically and independent distributed (iid) random variables with integer values q0 = 0. The increments qt − qs and qv − qu are stochas- tically independent and stationary. 3Although the empirical finding of an inverted U-shaped environmental Kuznets curve is uncon- troversial for some particular forms of pollution (see Grossman and Krueger (1995)), recent papers cast doubt on an application of the concept to pollution in general (see Bertinelli and Strobl (2005), Stern (2004)). Biodiversity loss and stochastic technological processes 45 the emission of pollutants does not converge to zero but to a maybe even small level Γ > 0 as income grows. Equation (4.3) tells us that the current probability of a new successful innovation increases with the effort devoted to R&D activities. As long as no such effort is undertaken, the probability of success is zero (λi(0) = 0). In each point of time either a new innovation is developed (dqi,t = 1) with probability λi(Nt)dt, or is not (dqi,t = 0) with probability [1−λi(Nt)dt]. In this last case equation (4.2) is reduced to its deterministic components and becomes dSt = µP−ζ t Sκ t dt−Rtdt− ΓP ξ t Sρ t dt where the technological progress does not mitigate the negative effects on the re- generative function. Assume that the consequences of successful innovations are instantaneous. Then, each time a new innovation occurs, qi is instantaneously in- creased by one unit and dt = 0. Thus, the availability of resources instantaneously grows but in a discontinuous manner since the stock trajectory jumps upward at each new success of the technological process. The size of such a jump is given by ∆1St = σ1µP−ζ t Sκ t , ∆2St = σ2Rt, ∆3St = σ3ΓP ξ t Sρ t . The discrete changes in the availability of resources are assumed to be proportional to the size of the lasting effects, to maintain the notion that the R&D activity is proportional to the severity of the lasting effects. Each of the three possible types of innovations happens independently of each other. On average, the positive effects of innovations may balance the lasting pressures of harvesting and pollutants. The instantaneous utility function of the infinitely lived representative agent is characterized by u(Ct) = C1−γ t 1− γ ; γ > 0; γ 6= 1 where Ct is the consumption quantity of the final good at date t and 1 γ is the elasticity of intertemporal substitution of consumption. Biodiversity loss and stochastic technological processes 46 4.3 The new Hotelling Rule The program (Ω) of the social planner is to maximize the expected present value of the utility V (S) = max Rt,Nt≥0 E ∞∫ 0 u(Ct)e −δtdt subject to Ct = F (Rt, 1−Nt) dSt = (µP−ζ t Sκ t dt+σ1µP−ζ t Sκ t dq1,t)− (Rtdt−σ2Rtdq2,t)− (ΓP ξ t Sρ t dt−σ3ΓP ξ t Sρ t dq3,t) Rt, Nt, 1−Nt, St ≥ 0 ∀t ≥ 0 where future utility flows are discounted at rate δ > 0 and one control variable is redundant through Lt = 1−Nt. Using the dynamic programming technique (Merton (1990)) and the results of Sennewald and Waelde (2005) and Sennewald (2005) we find the Hamilton-Jacobi-Bellman equation associated with the value function of (Ω), V (St). δV (St) = max Rt,Nt≥0 {u(Ct) + 1 dt EdV (St)}. (4.4) If we expand the stochastic differential dV (St), equation (4.4) becomes δV (S) = max Rt,Nt≥0 {u(Ct) + V ′(S)[µP−ζ t Sκ −Rt − ΓP ξ t Sρ] + +λ1(Nt)∆1V (Ŝ) + λ2(Nt)∆2V (Ŝ) + λ3(Nt)∆3V (Ŝ)} (4.5) where ∆1V (Ŝ), ∆2V (Ŝ), and ∆3V (Ŝ) are the respective instantaneous increases in social welfare due to the development of a new environmental friendly innovation, reducing the permanently negative effects of pollutants on the regenerative capacity µ, the harvesting pressure, and the temporary emission intensity: ∆1V (Ŝ) = V (S + σ1µP−ζ t Sκ t )︸ ︷︷ ︸ V1(Ŝ) − V (S) ∆2V (Ŝ) = V (S + σ2Rt)︸ ︷︷ ︸ V2(Ŝ) − V (S) ∆3V (Ŝ) = V (S + σ3ΓP ξ t Sρ)︸ ︷︷ ︸ V3(Ŝ) − V (S). Biodiversity loss and stochastic technological processes 49 Hence, the transversality condition holds if the expression in the parenthesis on the right hand side is positive. Note that µκP−ζ t Sκ−1 − ρ ΓP ξ t Sρ−1 t > 0 is a sufficient condition for this. How- ever, even if the effect on the regeneration rate and the direct effect of pollutants on S is negative, µκP−ζ t Sκ−1 − ρΓP ξ t Sρ−1 t < 0, the transversality condition holds as long as it is smaller in absolute value than the positive technological effect, λ1(Nt)σ1µP−ζ t κSκ−1 t V ′ 1(Ŝt) V ′(St) + λ3(Nt)σ3ΓP ξ t ρSρ−1 t V ′ 3(Ŝt) V ′(St) . 4.4 The optimal paths For finding an analytical solution of the optimal policy functions of harvesting and R&D effort, we set λi(Nt) = λiNt, with λi ∈ [0, 1]. In fact only a linear functional form for the intensity allows us to solve (Ω) analytically. So, an increase in the R&D effort leads to a higher probability of technological innovation, but leaving the marginal probability unchanged. For analytical simplicity we also restrict ourself to the case where σ2 = σ3 = 0 because the main result of the positive effect of newly developed innovations will not be affected and no additional qualitative insight will be gained. Proposition 4 The optimal paths of harvesting, R&D effort, and consumption are unique and regular. They are 1−N∗ t = L∗ t = (1− θ)(1− γ)mt ληt (4.10) R∗ t = mtSt (4.11) C∗(St) = mt[ (1− γ)(1− θ) ληt ]1−θSθ t (4.12) where ηt = (1 + σ1µP−ζ t Sκ−1 t )θ(1−γ) − 1 mt = 1 γ [δ − ληt − θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t )]. Biodiversity loss and stochastic technological processes 50 Proof: See Appendix 2 to this chapter. Since ηt and µt are time dependent, the optimal allocation of labor in the pro- duction sector and in the R&D sector are not constant. But since we have to meet the constraints 0 ≤ N∗, L∗ ≤ 1, we have to characterize a feasible set of parameters that guarantees the existence of an interior optimal solution. Proposition 5 An interior optimal solution exists if and only if δt ∈ {δt, δt} where, for µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t > 0,the lower bound is δt =  θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + ληt ifγ < 1 0 ifγ > 1 and the upper bound is for γ < 1 δt = θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + λ(ηt + ηtγ (1− γ)(1− θ) ) and for γ > 1 δt =  0 ifλ ≤ Ω θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + λ(ηt + ηtγ (1−γ)(1−θ) ) ifλ > Ω where Ωt = −θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) [1−θ(1−γ)] (1−γ)(1−θ) ηt . See figures 4.1a) and 4.1b). For µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t = 0 δt =  ληt ifγ < 1 0 ifγ > 1 and both for γ < 1 and γ > 1 δt = [1− θ(1− γ)] (1− γ)(1− θ) ηtλ. See figures 4.1c) and 4.1d). Biodiversity loss and stochastic technological processes 51 For µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t < 0 and γ < 1 δt =  0 ifλ ≤ Υ θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + ληt ifλ > Υ δt =  0 ifλ ≤ Φ θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + λ(ηt + ηtγ (1−γ)(1−θ) ) ifλ > Φ where Υt = −θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) ηt and Φt = −θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) [1−θ(1−γ)] (1−γ)(1−θ) ηt . See figure 4.1e). For γ > 1 δt =  θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + ληt ifλ ≤ Ψ 0 ifλ > Ψ and δt = θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + λ(ηt + ηtγ (1− γ)(1− θ) ) where Ψt = −θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) ηt . See figure 4.1f). Proof: See Appendix 3 to this chapter. The lower and upper constraints delimitate our feasible set and are time de- pendent. But since we are interested in balanced steady state solutions, we use now the definition given to the evolution of pollutants gP = Γ = · P/P and as- sume that also gS = · S/S is a constant. If and only if −ζgP = (1 − κ)gS and ξgP = (1 − ρ)gS hold, we have that P−ζ t Sκ−1 t = constant and P ξ t Sρ−1 t = constant, and therefore we find a balanced steady state solution for the optimal paths where both the feasible set for the parameters and the optimal path are no more time Biodiversity loss and stochastic technological processes 54 4.5 The optimal paths analysis We first characterize the exact optimal paths and then find their asymptotic be- havior. In a balanced steady state, according to solutions (4.10) and (4.11), our stochastic differential equation (4.2), if we recall from Appendix 2 to this chapter that P−ζSκ−1 = A and P ξSρ−1 = B, is dSt = µAStdt−mStdt− ΓBStdt + σ1µAStdqt (4.13) where we recall m = 1 γ (δ − λη − (1− γ)θ(µA− ΓB)). As long as there is no innovation (no jump) dqt = 0, then equation (4.13) has the solution S(t, 0) = S0e (µA−m−ΓB)t. When an innovation is developed dqt = 1 and the availability of resources is instan- taneously increased by σ1 percent. Then the resources follow the optimal trajectory S∗(t, qt) = (1 + σ1µA)qtS(t, 0) = (1 + σ1µA)qtS0e (µA−m−ΓB)t, (4.14) the optimal harvesting trajectory is R∗(t, qt) = mS∗(t, qt) = m(1 + σ1µA)qtS0e (µA−m−ΓB)t, (4.15) and the one for consumption reads C∗(t, qt) = mθ(1 + σ1µA)θqtSθ 0e θ(µA−m−ΓB)t(1−N∗)1−θ. (4.16) Since the exact trajectories (4.14),(4.15) and (4.16) are piecewise discontinuous (they jump upwards at the instant an innovation is developed) and their asymptotic behav- ior is undetermined (qt tends to infinity in probability over an infinite time horizon and S0e (µA−m−ΓB)t could decline to zero if µA < m+ΓB), we compute the smoothed trajectories. Hence, we consider the paths of the expected value of S∗t , R ∗ t and C∗ t . Integrating (4.13) and computing the expected value, we get St = S0 exp((µA−m− ΓB + λN∗σ1µA)t) Biodiversity loss and stochastic technological processes 55 and therefore the solution for (4.15) (the expected optimal extraction rate path) is Rt = mSt (4.17) and in a similar way the solution for (4.16) (the expected optimal consumption path) is Ct = (µS0) θ(1−N∗)1−θ exp({θ(µA−m− ΓB) + λN∗[(1 + σ1µA)θ − 1]}t). (4.18) The corresponding growth rates are constant over time. For R and S they are gS = gR = (4.19) = µA−m− ΓB + λN∗σ1µA = = µA− ΓB − 1 γ (δ − λη − (1− γ)θ(µA− ΓB))(1 + (1− γ)(1− θ) η σ1µA) + λσ1µA and for C the growth rate is gC = (4.20) = θµA− θΓB − θm + λN∗(1 + σ1µA)θ − λN∗ = = θ(µA− ΓB) + λ[(1 + σ1µA)θ − 1] + −1 γ (δ − λη − (1− γ)θ(µA− ΓB))(θ + (1− γ)(1− θ) η [(1 + σ1µA)θ − 1]. We are interested in the signs of those rates. In particular, the aim of this chapter is to determine the necessary conditions for sustainable growth, i.e., positive consumption growth and increasing resources over time. Therefore we find the two functions that guarantee gS > 0 and gC > 0. Using (4.19) (δ − λη − (1− γ)θ(µA− ΓB))[1 + (1− γ)(1− θ) η σ1µA] < γ(µA− ΓB) + γλσ1µA if and only if δ < δS = (µA− ΓB)((1− γ)θ + γ Q ) + λ(η + γσ1µA Q ) (4.21) where Q = 1 + (1−γ)(1−θ) η σ1µA. Using (4.20) (δ−λη−(1−γ)θ(µA−ΓB))[1+ (1− γ)(1− θ) η [(1+σ1µA)θ−1]] < θγ(µA−ΓB)+γλ[(1+σ1µA)θ−1] Biodiversity loss and stochastic technological processes 56 if and only if δ < δC = θ(µA− ΓB)((1− γ) + γ Z ) + λ(η + γ[(1 + σ1µA)θ − 1] Z ) (4.22) where Z = θ + (1−γ)(1−θ) η [(1 + σ1µA)θ − 1]. Now, recall that the original constraints δ(λ) and δ(λ) assure a feasible set of parameters. Together with the new non-negativity constraints δS(λ) and δC(λ) given in (4.21) and (4.22), we can study the sustainability issue in the δ and λ space. For the sake of better readability, let us give this system of four equations again: δ(λ) = (µA− ΓB)θ(1− γ) + λη δ(λ) = (µA− ΓB)θ(1− γ) + λ(η + ηγ (1− γ)(1− θ) ) δS(λ) = (µA− ΓB)(θ(1− γ) + γ Q ) + λ(η + γσ1µA Q ) δC(λ) = θ(µA− ΓB)((1− γ) + γ Z ) + λ(η + γ[(1 + σ1µA)θ − 1] Z ). When γ < 1 and µA − ΓB > 1 (see figure 4.2a)), δ(λ) and δ(λ) have the same positive intercept at λ = 0, namely θ(1− γ)(µA − ΓB). Since Q,Z > 0 and γ > 0 the intercept of δS(λ) and δC(λ) is higher than that of δ(λ), δ(λ). The slope of δ(λ) is bigger than that of δ(λ). Also, the slope of both δS(λ) and δC(λ) are bigger than that of δ(λ) since γσ1µA Q > 0 and γ[(1+σ1µA)θ−1] Z > 0. They exhibit a bigger intercept than δ(λ) so that they cannot intersect with δ(λ). Comparing the slopes of δS(λ) and δ(λ), γη (1− γ)(1− θ) ≥ γσ1µA Q ⇔ (4.23) η (1− γ)(1− θ) + σ1µA ≥ σ1µA. The last condition always holds with strict inequality, since η(1 − γ) > 0. Thus, the slope of δ(λ) is strictly bigger than that of δS(λ). Knowing that the intercept of δS(λ) is bigger than that of δ(λ), then the two functions have an intersection point. Comparing δC(λ) and δ(λ), γη (1− γ)(1− θ) ≥ γ[(1 + σ1µA)θ − 1] Z ⇔ (4.24) η (1− γ)(1− θ) + (1 + σ1µA)θ − 1 ≥ (1 + σ1µA)θ − 1. Biodiversity loss and stochastic technological processes 59 )c )d )e )f )b)a   *I       1                    1 C C C C C C S S S S S S *I *1I *1I*2I *2I Figure 4.2: Sustainability areas for γ <,> 1 and µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t >, =, < 0 Biodiversity loss and stochastic technological processes 60 4.6 Conclusion Thanks to all the three new considerations in the regeneration function (connection between harvesting and quality degradation of renewable resources; non-constancy of the regenerative capacity; technological processes that directly affect the availability of resources), we obtain a new Hotelling rule. In fact, in this model, the rate of growth of the resource rent is not simply equal to the social discount rate net of the regeneration rate of the resource. But rather, it has to be corrected for two crucial factors of influence: (i) the effect on the growth rate through technological progress, and (ii) the sum of the indirect effects of pollutants on the regeneration rate and the direct one on the stock of resources. The correction connected with the direct and indirect effects of pollutants increases the growth rate of the resource rent due to the fact that lasting effects of pollutants increase the physical scarcity of the resource. The effect of technology as such reduces the growth rate of the resource rent because at the time a new innovation is developed, the physical scarcity of resources is instantaneously reduced. As a result, the adverse effects of pollutants on the regeneration function may slow down the use of the resource, but uncertainty on the return of innovations may speed up the harvesting. We have also determined the necessary conditions for sustainable growth i.e., positive consumption growth and positive resources growth over time. For that rea- son as in Lafforgue (2004), we have firstly found an analytical solution of the optimal policy functions of harvesting, R&D effort and consumption; secondly characterized the smoothed optimal paths (the exacted ones being only piecewise continuous and asymptotically undetermined); thirdly computed the growth rates. As final result we have that, if the marginal probability of innovations is high enough compared with the degree of impatience of society, the expected positive effect of R&D activi- ties overrides expected negative effects of harvesting and environmental degradation (caused by the direct and indirect impacts of pollutants) so that the smoothed tra- jectory of renewable resources and consumption increases over time. Biodiversity loss and stochastic technological processes 61 4.7 Appendix 1 We start with the description of the eight possible states according to the three independent Poisson process (q1,t), (q2,t), (q3,t). For this we consider a given time t and a later time t + dt. Event in ”dt” State Probability Only jump of (q1,t) I λ1(Nt)dt(1− λ2(Nt))dt(1− λ3(Nt))dt Only jump of (q2,t) II λ2(Nt)dt(1− λ1(Nt))dt(1− λ3(Nt))dt Only jump of (q3,t) III λ3(Nt)dt(1− λ1(Nt))dt(1− λ2(Nt))dt Only jump of (q1,t) and (q2,t) IV λ1(Nt)dtλ2(Nt)dt(1− λ3(Nt))dt Only jump of (q1,t) and (q3,t) V λ1(Nt)dtλ3(Nt)dt(1− λ2(Nt))dt Only jump of (q2,t) and (q3,t) VI λ2(Nt)dtλ3(Nt)dt(1− λ1(Nt))dt Jump of (q1,t), (q2,t), (q3,t) VII λ1(Nt)dtλ2(Nt)dtλ3(Nt)dt No jump at all VIII (1− λ1(Nt))dt(1− λ2(Nt))dt(1− λ3(Nt))dt For the derivative V ′(S) at time t + dt we have V ′(S)|t+dt = V ′(S)|t + V ′′(S)dSt. This gives according to the above table the following representation of V ′(S)|t+dt depending of the state V ′(S)|t+dt =  V ′(S)|t + µP−ζ t Sκ t V ′′(S)dt−RtV ′′(S)dt− ΓP ξ t Sρ t V ′′(S)dt state VIII V ′ 1,2,3 = V ′(St + σ1µP−ζ t Sκ t + σ3ΓP ξ t Sρ t + σ2Rt) state VII V ′ 2,3 = V ′(St + σ3ΓP ξ t Sρ t + σ2Rt) state VI V ′ 1,3 = V ′(St + σ1µP−ζ t Sκ t + σ3ΓP ξ t Sρ t ) state V V ′ 1,2 = V ′(St + σ1µP−ζ t Sκ t + σ2Rt) state IV V ′ 3(Ŝ) = V ′(St + σ3ΓP ξ t Sρ t ) state III V ′ 2(Ŝ) = V ′(St + σ2Rt) state II V ′ 1(Ŝ) = V ′(St + σ1µP−ζ t Sκ t ) state I Biodiversity loss and stochastic technological processes 64 Inserting back into (4.32) gives (1− θ) ( θ 1− θ )θ(1−γ) (λ∆1V (Ŝ))θ(1−γ) V ′(S)θ(1−γ) · Lθ(1−γ) · L−γ−(1−γ)θ = λ∆1V (Ŝ). After rearrangements we get Lγ = ( 1− θ λ∆1V (Ŝ) )1−θ(1−γ)( θ V ′(S) )θ(1−γ) and finally L = ( 1− θ λ∆1V (Ŝ) ) 1−θ(1−γ) γ ( θ V ′(S) ) θ(1−γ) γ . (4.34) We insert (4.34) into (4.33): R = ( 1− θ λ∆1V (Ŝ) ) (1−θ)(1−γ) γ ( θ V ′(S) ) 1−(1−θ)(1−γ) γ . (4.35) We now insert this into the HJB equation (4.5) and, using the abbreviations G1 = µP−ζ t Sκ t and G3 = ΓP ξ t Sρ t , we get δV (S) = Rθ(1−γ)L(1−θ)(1−γ) 1− γ + V ′(S)G1 − V ′(S)R− V ′(S)G3 + λ(1− L)∆1V (Ŝ). With (4.34) and (4.35) we realize after rearrangements that δV (S) = ( γ 1− γ )[ 1− θ λ∆1V (Ŝ) ] (1−θ)(1−γ) γ [ θ V ′(S) ] θ(1−γ) γ (4.36) + λ∆1V (Ŝ) + G1 S SV ′(S)− G3 S SV ′(S). Thus, δV (S) = ( γ 1− γ )[ 1− θ λ∆1V (Ŝ) ] (1−θ)(1−γ) γ [ θ V ′(S) ] θ(1−γ) γ (4.37) + λ∆1V (Ŝ) + µP−ζ t Sκ−1 t [SV ′(S)]− ΓP ξ t Sρ−1 t [SV ′(S)]. Now assume that for all t ≥ 0 P−ζ t Sκ t = AtSt (4.38) P ξ t Sρ t = BtSt, (4.39) with At, Bt continuous. Condition (4.38) guarantees that the HJB equation (4.5) defines a necessary condition for an optimal path. For the sufficient condition no Biodiversity loss and stochastic technological processes 65 additional assumption is required (see Sennewald (2005)). Then equation (4.37) turns into δV (S) = ( γ 1− γ )[ 1− θ λ∆1V (Ŝ) ] (1−θ)(1−γ) γ [ θ V ′(S) ] θ(1−γ) γ (4.40) + λ∆1V (S) + µAt[SV ′(Ŝ)]− ΓBt[SV ′(S)]. Equation (4.40) shows an ordinary differential equation for V . To solve it we use the approach V (S) = ΨSθ(1−γ) (4.41) where Ψ ∈ R is unknown, and needs to be determined below. We compute the derivative and ∆1V (Ŝ): V ′(S) = θ(1− γ)ΨSθ(1−γ)−1 (4.42) ∆1V (Ŝ) = Ψ[(1 + σ1µAt) θ(1−γ) − 1]Sθ(1−γ). (4.43) Insertion of (4.41), (4.42) and (4.43) into (4.40) gives δΨSθ(1−γ) = ( γ 1− γ )( 1− θ λ ) (1−θ)(1−γ) γ [Ψ[(1 + σ1µAt) θ(1−γ) − 1]]− (1−θ)(1−γ) γ · S− (1−θ)(1−γ) γ θ(1−γ)θ θ(1−γ) γ (θ(1− γ)Ψ)− θ(1−γ) γ S θ(1−γ) γ (1−θ(1−γ) + λΨ[(1 + σ1µAt) θ(1−γ) − 1]Sθ(1−γ) + µAtθ(1− γ)Ψ[S · Sθ(1−γ)−1]− ΓBtθ(1− γ)Ψ[S · Sθ(1−γ)−1]. Collecting terms that involve S, we obtain in a first step δΨSθ(1−γ) = ( 1− θ λ ) (1−θ)(1−γ) γ [Ψ[(1 + σ1µAt) θ(1−γ) − 1]]− (1−θ)(1−γ) γ · ( γ 1− γ )θ θ(1−γ) γ (θ(1− γ)Ψ)− θ(1−γ) γ Sθ(1−γ) (4.44) + λΨ[(1 + σ1µAt) θ(1−γ) − 1]Sθ(1−γ) + µAtθ(1− γ)ΨSθ(1−γ) − ΓBtθ(1− γ)ΨSθ(1−γ). Biodiversity loss and stochastic technological processes 66 In (4.44) each term contains Sθ(1−γ) by which we divide to derive δΨ = ( 1− θ λ ) (1−θ)(1−γ) γ [(1 + σ1µAt) θ(1−γ) − 1]− (1−θ)(1−γ) γ · ( γ 1− γ )θ θ(1−γ) γ (θ(1− γ))− θ(1−γ) γ Ψ− θ(1−γ) γ Ψ− (1−θ)(1−γ) γ (4.45) + λ[(1 + σ1µAt) θ(1−γ) − 1]Ψ + µAtθ(1− γ)Ψ− ΓBtθ(1− γ)Ψ. Now note that Ψ− θ(1−γ) γ − (1−θ)(1−γ) γ = Ψ · Ψ− 1 γ . Hence, each term in (4.45) contains Ψ. We divide again and use the abbreviation x = σ1µAt, y = µAt and z = ΓBt to derive δ − λ[(1 + x)θ(1−γ) − 1]− (y − z)θ(1− γ) = (4.46) Ψ− 1 γ ( γ 1− γ )( 1− θ λ ) (1−θ)(1−γ) γ [(1 + x)θ(1−γ) − 1]− (1−θ)(1−γ) γ θ θ(1−γ) γ (θ(1− γ))− θ(1−γ) γ . This implies Ψ = [ γ(1− γ)− γ+θ(1−γ) γ [ 1−θ λ[(1+x)θ(1−γ)−1] ] (1−θ)(1−γ) γ δ − λ[(1 + x)θ(1−γ) − 1]− (y − z)θ(1− γ) ]γ. (4.47) Now we use the expression V (S) = ΨSθ(1−γ) and insert (4.47) into the expressions for L and R given in (4.34) and (4.35), respectively: R = ( 1− θ λΨ[(1 + x)θ(1−γ) − 1]Sθ(1−γ) ) (1−θ)(1−γ) γ ( θ θ(1− γ)ΨSθ(1−γ)−1 ) 1−(1−θ)(1−γ) γ = = ( 1 1− γ ) 1 γ ( (1− θ)(1− γ) λ[(1 + x)θ(1−γ) − 1] ) (1−θ)(1−γ) γ Ψ− 1 γ S. (4.48) Collecting terms, this can be rewritten as R = δ − λ[(1 + x)θ(1−γ) − 1]− (y − z)θ(1− γ) γ S. (4.49) And for L we compute L = ( 1− θ λΨ[(1 + x)θ(1−γ) − 1]Sθ(1−γ) ) 1−θ(1−γ) γ ( θ θ(1− γ)ΨSθ(1−γ)−1 ) θ(1−γ) γ = = ( 1− θ λ[(1 + x)θ(1−γ) − 1] ) 1 γ ( λ[(1 + x)θ(1−γ) − 1] (1− γ)(1− θ) ) θ(1−γ) γ Ψ− 1 γ . (4.50) Collecting terms we derive L = (1− θ)(1− γ) λ[(1 + x)θ(1−γ) − 1]γ (δ − λ[(1 + x)θ(1−γ) − 1]− (y − z)θ(1− γ)). (4.51) Biodiversity loss and stochastic technological processes 69 According to (4.10) again, we have that L∗ t ≤ 1 ⇔ 1− (1− γ)(1− θ)mt ληt ≥ 0. This condition is equivalent to ληt − (1− γ)(1− θ) 1 γ (δ − ληt) ≥ 0 and the upper bound on δ becomes δt = [1− θ(1− γ)] (1− γ)(1− θ) ηtλ. Case 4: µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t = 0 and γ > 1. When γ > 1 then ηt < 0. Hence according to equation (4.10) L∗ t ≥ 0 ⇔ mt ≥ 0. The lower bound on δ becomes δt = 0, as δt = ληt is negative. According to (4.10), we have that L∗ t ≤ 1 ⇔ 1− (1− γ)(1− θ)mt ληt ≥ 0. This condition is equivalent to again ληt − (1− γ)(1− θ) 1 γ (δ − ληt) ≥ 0 and the upper bound on δ becomes δt = [1− θ(1− γ)] (1− γ)(1− θ) ηtλ. In the same way we proceed for the last two cases. Case 5: µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t < 0 and γ < 1. When γ < 1 then ηt > 0. Hence according to equation (4.10) L∗ t ≥ 0 ⇔ mt ≥ 0. The lower bound on δ becomes δt =  0 ifλ ≤ Υ θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + ληt ifλ > Υ Biodiversity loss and stochastic technological processes 70 where Υt = −θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) ηt , which is the intersection point between the δt = 0 line and the δt = θ(1−γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + ληt line. According to (4.10) again, we have that L∗ t ≤ 1 ⇔ 1− (1− γ)(1− θ)mt ληt ≥ 0. This condition is equivalent to ληt − (1− γ)(1− θ) 1 γ (δ − ληt − θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t )) ≥ 0 and the upper bound on δ becomes δt =  0 ifλ ≤ Φ θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + λ(ηt + ηtγ (1−γ)(1−θ) ) ifλ > Φ where Φt = −θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) [1−θ(1−γ)] (1−γ)(1−θ) ηt , which is the intersection point between the δt = 0 line and the δt = θ(1−γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + λ(ηt + ηtγ (1−γ)(1−θ) ) line. Case 6: µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t < 0 and γ > 1. When γ > 1 then ηt < 0. Hence according to equation (4.10) L∗ t ≥ 0 ⇔ mt ≥ 0. The lower bound on δ becomes δt =  θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + ληt ifλ ≤ Ψ 0 ifλ > Ψ where Ψt = −θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) ηt , which is the intersection point between the δt = θ(1−γ)(µP−ζ t Sκ−1 t −ΓP ξ t Sρ−1 t )+ληt line and the δt = 0 line. To find the new upper bound on δ again, L∗ t ≤ 1 ⇔ 1− (1− γ)(1− θ)mt ληt ≥ 0. Biodiversity loss and stochastic technological processes 71 This condition is equivalent to ληt − (1− γ)(1− θ) 1 γ (δ − ληt − θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t )) ≥ 0 and the upper bound on δ becomes δt = θ(1− γ)(µP−ζ t Sκ−1 t − ΓP ξ t Sρ−1 t ) + λ(ηt + ηtγ (1− γ)(1− θ) ). Nature as a knowledge reservoir 74 (1956) and di Maria and Valente (2006) of Acemoglu (2003). Along this line of research, nature even though being recognized as producing important services,1 is constrained, ultimately, to be a limiting factor2 to growth, either because natural resources are exhaustible or because the environmental qual- ity is strongly negatively affected by pollution.3 Another interesting point is that with this interpretation of nature as a limiting factor no insight can be gained to improve the answers the growth literature gives to its two above mentioned funda- mental questions.4 But if the attention is concentrated on the inestimable role that nature plays for the advancement of the sciences, e.g. as a knowledge reservoir, then this new interpretation can help in climbing up the quality ladder for growth models and thus contributes to explaining the sources of the growth process. Along with attributing a new role to nature in the context of economic growth, we take into consideration that as Romer (2006) writes: ”The principal conclusion of the Solow model is that the accumulation of physical capital cannot account for either the vast growth over time in output per person or the vast geographic differences in output per person.” We therefore exclude physical capital from our analysis and endogenise 1The four big categories of environmental services are: resource base that enters directly into the production function of output, waste sink which enters into the environmental quality function, amenity base service entering the utility function and the life support that can enter either in the production function directly or in the regenerative capacity function. 2Along the direct negative effect of nature on growth, there is also a possible indirect one, through environmental policy. Whether stringent environmental policies have a negative direct input effect or a win-win outcome (the Porter hypothesis) is long debated. Ricci (2004) surveys the related literature. It should be noted that if a positive effect is found, it relies either on knife edge assumptions, that should be avoided, or on an indirect effect through the standard explanatory variables for growth. 3The empirical literature is also debating the existence or not of the so called ”curse of natural resources” when nature is seen as supplier of raw materials; see Gylfason (2001), Gylfason (2004), Sachs and A.M.Warner (1995), Bretschger (2006), Brunnschweiler (2006). Instead, the paper of Bloom and Sachs (1998) stresses the role of a better understanding of how climate and natural ecology work for development policies. 4This conceptually means that the sustainable development literature is a follower of the growth literature. Nature as a knowledge reservoir 75 the Solow effectiveness of labor (A), interpreting it as technology. In addition, the introduction of the positive role of nature in the production function of technologies eliminates the low empirical success of the scale endogenous growth literature and the necessity of knife edge assumptions about the returns to scale to the produced factors in the production function of technologies. In the following section the debate about scale effects and non-robustness (need of knife edge assumptions) will be briefly summarized; in section 3 the role of nature as a knowledge reservoir will be illustrated. Section 4 presents the basic model with natural knowledge and its dynamic implications for economic growth. In section 5 a more detailed version of the model is introduced in order to investigate in section 6 the role that the technological sector can play in the presence of environmental constraints. Section 7 concludes. 5.2 The scale effects and knife edge debate The endogenous growth literature is motivated by the desire to explain what in the Solow model is exogenous and the driving force for sustained growth, namely the technological progress. The standard endogenous growth literature, also referred to as first-generation R&D-based growth models, is based upon the knife edge assump- tion of constant returns to scale in the produced factors of production.5 In addition to that, these models imply scale effects, because the scale of the economy (L), or the fraction of the resources it gives to the R&D sector (LR L ), influences the long-run growth rate. In fact, both the horizontal innovation approach of Romer (1990),6 where the manufacturing sector and the innovative sector are described by Y = Kα(ALY )(1−α) Ȧ = λALR, λ > 0, 5For a detailed survey and discussion see Groth (2004) and Jones (1999). 6The production function of the final good is the result of static efficiency for Y =( A∑ i=1 xα i ) L (1−α) Y , 0 < α < 1, where xi, the input of intermediate good, is equal to x = K A for all i. Nature as a knowledge reservoir 76 and the vertical innovation approach of Grossman and Helpman (1991) and Aghion and Howitt (1992),7 with Y = Kα(AQLY )(1−α) Q̇ = λQLR, λ > 0 lead to the steady state result gy = ẏ y = λsRL, where sR = LR L . Therefore policy can affect the long-run growth rate by influencing sR, which is the fraction of labor devoted to the innovative sector. But Jones’s critique (Jones (1995a) and Jones (1995b)) claims that the assumed scale effects are contradicted by empirical evidence. He proposes an alternative with decreasing returns to scale Ȧ = λAϕLR, ϕ < 1 LR = sRL, L̇ L = n ≥ 0. This produces that in steady state gy = n 1− ϕ and thus the scale effects are cleared out. The response to Jones was the second generation R&D-based models, Dinopoulos and Thompson (1998), Howitt (1999), Peretto (1998), Young (1998). These models, connecting the horizontal and the vertical innovation approach, managed to get rid only of the scale effect arising from the scale of the economy (L) but not of that deriving from the fraction of the resources devoted to the different R&D sectors (sQ). In fact, given the production function Y = Kα(AQLY )1−α 7Also here the production function of the final good is the result of static efficiency for Y =( Q∑ i=1 xα i ) L (1−α) Y , 0 < α < 1, where xi = x = K A Q and A, the number of different intermediate goods, is fixed and Q is the quality attached to the latest version of intermediate good. Nature as a knowledge reservoir 79 many others); in the pharmaceutical industry for the production of antibiotics, vac- cines, active ingredients and therapeutical approaches; in new microbial processes (in the chemical industry for the production of enzymes, amino acids, nucleotide or steroids); in the utilization and conversion of crude oil, natural gas and cellulose; in the growing sectors of gene and biotechnology; in the treatment of wastes (see Dixon (1996), Schlegel (1992)). The relevance of studying nature can also be seen in the growing sector of biomechanics9 which has lead to the development and production of nanostructures and in computer sciences to robotics and cybernetics.10 5.4 Natural knowledge as a prerequisite for sus- tained growth 5.4.1 Model structure Having in mind the economic implications of the environment illustrated in the pre- vious section and the economic evidence about the factor capital (K) mentioned in the introduction, the model structure is straightforward. It is described by four variables, namely output (Y ), labor (L), knowledge (A) in the standard interpreta- tion of technology (see Romer (1990), Grossman and Helpman (1991), Aghion and Howitt (1992)), and knowledge (D) in the interpretation of basic scientific research which arises by the study and the understanding of nature. In the analysis to follow A will be called technology and D natural knowledge. There are, therefore, three sectors: the final good sector where the output is produced; the standard R&D sector which is mostly private and characterized by the strength of developing tech- nologies which have a clear target for their utilization in the production of goods; the 9Well known examples of drawing from nature in engineering are Leonardo da Vinci’s flying machines and ships. 10The sector of biomechanics, or also said bionics from the connection of biology and electronics, is nowadays one of the most promising sectors, especially if we consider that the overlap between biology and technology in terms of mechanisms used is only 10% approximately. Nature as a knowledge reservoir 80 scientific research sector which is mostly non-private and fundamentally motivated by the intrinsic human aspiration of enlightenment and improvement of the human condition. Normally in this sector, gained knowledge that afterwards is found to be relevant for the development of a specific market good (either directly in the production function or indirectly through the development of a new invention) is only a side product rather than the target of research,11 such as the discovery of the first antibiotic12 or the invention of new materials/tissue.13 The production function of the final good is Y (t) = D(t)α[A(t)(1− aA − aD)L(t)]1−α, (5.1) that for technologies is Ȧ(t) = D(t)β(aAL(t))γA(t)θ, (5.2) and finally the production function of natural knowledge is Ḋ(t) = (aDL(t))χD(t)ε. (5.3) Population growth is exogenous and follows the standard differential equation L̇(t) = nL(t), with n exhibiting a positive value. Fraction aA of the labor force is used for the invention of technologies; fraction aD for the discovery of natural knowledge; and 1 − aA − aD is used for the production of the final good.14 The three production functions have a standard Cobb-Douglas specification and follow the standard literature, see e.g.Romer (2006). The production function for the final 11Indeed in all natural sciences, as Pasteur said, the relationship between basic and applied research is very close: ”Il n’y a pas des sciences appliquees... Mais il y a des applications de la science”, see Schlegel (1992). 12Penicillin, which is a substance produced by the mould Penicillium notatum, was the first antibiotic discovered by Alexander Fleming in 1928 by chance on a nutrient agar which was thrown away after the study of another bacterium. 13As to the promising expectations arising from the study of the echinoderm sea cucumber, see Thurmond and Trotter (1996) and Jangoux et al. (2002). 14So, in a very elementary interpretation, we could also rename the variables A as inventions and D as discoveries. Nature as a knowledge reservoir 81 good presents constant returns to labor and to natural knowledge for a given tech- nology, with 0 < α < 1. Together with the introduction of the technological process as Harrod-neutral, the model therefore exhibits constant returns to scale in the pro- duction function of the final good for both factors of production, namely technology and natural knowledge. Thus, on net, whether this economy has increasing, de- creasing or constant returns to scale to the produced factors depends on the returns to scale it has in the production function of knowledge, equation (5.2), and so on (β + θ) T 1. Note that in this model, we will see that the type of returns to scale to the produced factors is no more a key determinant for the existence of a balanced growth path, thus there is no need for a knife edge assumption like β + θ = 1 or decreasing returns, β + θ < 1. There is no specific assumption with respect to the type of returns to scale to natural knowledge and labor in the production function of technology, and therefore, β ≥ 0 and γ ≥ 0. The same applies for the production function of natural knowledge where the returns to scale to labor could be decreasing, constant or increasing, χ ≥ 0. There are good reasons for all three possibilities therefore, we do not impose a specific formulation. Finally, the parameters θ and ε, which represent the contribution of existing inventions to the success of the standard R&D sector and the contribution of ex- isting discoveries to the advancement of scientific research, are also not subject to any assumption leaving them to be positive or negative. In fact, the contribution could be positive if we believe that inventions or discoveries in the past make future improvements easier; or it could also be negative if we assume that the bigger the stock of improvements, the more difficult to add new ones. Nature as a knowledge reservoir 84 arrows point east; below the locus, it is falling and therefore the arrows point west. Similarly, equation (5.7) corresponds to the set of points where gD is constant. Above the locus, gD is falling and the arrows point south; below the locus, it is rising and the arrows point north. Thus, the two schedules divide the space into four regions. The arrows point southwest in the first quadrant, southeast in the second, northeast in the third, and northwest in the fourth. The balanced steady state is (g∗A, g∗D) and this point is stable. For any values of gA and gD, the dynamics of the system takes it back to the balanced steady state. The model does not imply scale effects because the long-run growth rates are not permanently influenced by changes in the resources devoted either (as in the early new growth literature) to the R&D sector (aA) or the scientific research sector (aD). At the same time the existence of the equilibrium is independent of the type of returns to scale in the produced factors of production in the production function of technology: increasing if β + θ > 1, constant if β + θ = 1, decreasing if β + θ < 1. This overcomes both the knife edge assumption of the endogenous growth literature where β+θ = 1 and the assumption of Jones’s critique (Jones (1995a)) that β+θ < 1. This is because the driving force of the economy is now the production function of natural knowledge where the only limitation is the human thinking capacity. 5.5 The threat from what gets lost: pollution dam- ages on nature as a knowledge reservoir The production function for the final good is again Y = Dα[A(1− aA − aD)L]1−α and the production function for technology is maintained as before Ȧ = Dβ(aAL)γAθ. The production function of natural knowledge as potential for the maximal scientific improvement, differently from the basic model, additionally captures the realistic Nature as a knowledge reservoir 85 feature that something of the nature is destroyed due to pollution damages as result of the human activity. Therefore precious sources of information get lost, reduc- ing the basis of scientific knowledge. These damages are modeled as an inevitable consequence of the output production. Thus the new function is Ḋ = (aDL)χDε − d{Dα[A(1− aA − aD)L]1−α}. (5.11) For finding an equilibrium the dynamics of the model must be studied.15 The new growth rate for gD is gD = aχ DLχDε−1 − dDα−1A1−α[(1− aA − aD)L]1−α (5.12) and therefore balanced steady state implies that the growth rate of gD must satisfy χgL + (ε− 1)gD = 0 (5.13) and (α− 1)gD + (1− α)gA + (1 + α)gL = 0, (5.14) hence gD = χ 1− ε gL (5.15) gD = gA + gL. (5.16) Equations (5.15) and (5.16) are indicating the constraints that must be satisfied for having ˙gD = 0. This happens only in one point (g′A, g′D) which is the intersection point of the two equations if χ 1−ε > 1 (even though it does not mean any restriction on the type of returns to scale on labor (χ) in equation (5.3), recalling that from the basic model ε < 1). Thus g′D = χ 1− ε gL and g′A = ( χ 1− ε − 1)gL. 15As we ate interested in the existence of equilibrium at this stage, we will not investigate issues of convergence. Nature as a knowledge reservoir 86 The locus of points where gD is constant is, as before, the straight line with intercept − γ β gL and slope 1−θ β : ġA = 0 ⇒ gD = −γ β gL + 1− θ β gA. (5.17) Summing all the information together the second proposition follows. Proposition 7 This economy possesses an equilibrium ( ˙gD = 0 and ġA = 0) with (g∗A, g∗D) = (g′A, g′D) where A and D grow steadily but if and only if β + θ < 1. Proof: If g′A and g′D are substituted into equation (5.17), it follows χ 1− ε gL = −γ β gL + 1− θ β ( χ 1− ε − 1)gL where it must be that 1− θ β = ( χ 1−ε + γ β ) ( χ 1−ε − 1 ) > 1 (5.18) and therefore β + θ < 1. See figure 5.2a). This economy can perform sustainable growth without scale effects, but it needs decreasing returns to scale in the two produced factors of production, namely A and D, in the production function of technologies. This brings the model back to run on the same assumption as Jones (1995b), yet not to the early new growth literature with β +θ = 1. Now the main relevant force of the economy is the constraint arising from the loss of useful information d{Dα[A(1− aA− aD)L]1−α from equation (5.11). But because the production function of the final good has constant returns to scale in A and D, in the end, whether that constraint is too strong or not for having sustaiable growth depends on the returns to scale of A and D in the production function of technologies. Therefore only with decreasing returns the limiting effect is not too severe for the growth process. Nature as a knowledge reservoir 89 drives this economy is d{DαA1−α−λ[(1− aA − aD)L]1−α} from equation (5.19). Dif- ferently from the previous section where no mitigation arising from the technological progress was considered, here, in the environmental constraint, the returns to scale on D and A are always decreasing, independent of whether λ is >, =, < 1 − α. This guarantees that the environmental constraint is not strong enough to affect sustainable growth predictions. 5.7 Conclusion A new model structure is developed where nature is given a positive interpretation as a knowledge reservoir which is a maximal source for scientific improvement. Three different versions of the model, which do not predict scale effects, are presented to investigate how the role of the returns to scale in the produced factors of production changes. Starting with no constraints at all on the production function of natural knowledge, we move to their inclusion, ending with the recognition of the positive role that the technological progress has in the mitigation of the environmental threat. It is demonstrated that only in the case with the environmental constraint and without technological mitigation, a specific assumption about the returns to scale in the produced factor of production is needed in order to guarantee sustainable growth. 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