A Model of Eco-Efficiency and Recycling

Cogoy, Mario Working Paper A Model of Eco-Efficiency and Recycling Economics Discussion Papers, No. 2009-9 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Cogoy, Mario (2009) : A Model of Eco-Efficiency and Recycling, Economics Discussion Papers, No. 2009-9, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/27490 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en Discussion Paper Nr. 2009-9 | January 13, 2009 | http://www.economics-ejournal.org/economics/discussionpapers/2009-9 A Model of Eco-Efficiency and Recycling Mario Cogoy University of Trieste, Italy Abstract This paper presents the model of an economy subject to the mass conservation principle. The economic system is related to the environment by a flow of virgin materials into the economy, and by the diffusion of waste into the environment. Ecoefficiency contributes to reducing material waste in all processes. Recycling can reduce the diffusion of waste by feeding it back into the economy. Human capital enhances productivity, eco-efficiency and the quality of all kinds of outputs. Recycling and human capital formation use productive factors and are rooted therefore, as all other activities, in the material basis of the economy. The paper studies an optimal material state of society. JEL: D90, O30, O41, Q00 Keywords: Eco-Efficiency, Recycling, Materials Balances, Material Flows, Human Capital Correspondence: Mario Cogoy, Department of Economics and Statistics - University of Trieste, Piazzale Europa 1 - 34127 - Trieste – Italy, email: [email protected].it © Author(s) 2009. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany 2 1. INTRODUCTION The current anthropogenic pressure on the natural environment calls for effective control of the material dimensions of human activities in advanced industrial societies (van den Bergh 1996; Adriaanse et al. 1997; Fischer-Kowalski 1998; Fischer-Kowalski and Hüttler 1998; Ayres 1999b; Bouman et al. 2000). Different possible technological responses to the task of controlling anthropogenic material flows have been discussed in the literature as e.g.: dematerialisation, eco-efficiency, recycling, industrial ecology and industrial metabolism. Although the analysis of physical constraints on economic activities (Ayres 1998, 1999a; Cleveland and Ruth 1997; Ruth 1993, 1999) is a much debated issue, the development of models combining physical insights with specific tools of economic analysis still remains a very fragmentary field of research, since existing economicphysical models make very different assumptions and focus on very different aspects of the complex interaction between economic and physical analysis. Smith (1972) applies the law of mass conservation to recycling in a model without physical capital. He develops economic conditions for complete or zero recycling. Van den Bergh (1996) presents an experimental model, based on materials balances. Waste can be emitted, recycled or stored. Environmental quality depends on the stocks of renewables and on the stock of pollutants. Simulations are performed with exogenous scenarios. Di Vita (1997) studies the effects of recycling of imported raw materials on the balance of payments and on employment in an open economy. Huhtala (1999) investigates recycling as an alternative to resource extraction. Recycling reduces pollution, but labour is the only input and physical capital does not come into the picture. Nakamura (1999) applies an input-output approach to the study of waste recycling in a static setting without technical progress. Di Vita (2001) focuses on the effects of recycling on the rate of growth. His model includes technical progress, which is endogenously generated by research. Welfare is negatively affected by the dimension of the waste stock, but not by the scale of economic activities. Hosoda (2001) applies the corn-guano model of the post-Sraffian school (Bidard and Erreygers 2001) to an analysis of recycling. The residuals of a first process are used as inputs to a second 3 process. In this way, waste is completely absorbed and unlimited growth is possible, even after exhaustion of landfills. Eichner and Pethig (2001) present a labour-only model based on mass conservation with recycling activities and waste treatment before disposal. They investigate waste markets and market failures in the waste sector. Highfill and Mc Asey (2001) study recycling as an alternative to landfilling in the framework of a growing economy. Economic growth is exogenous and no interrelationship between physical capital, recycling and technical progress is therefore addressed. Eichner (2005) applies a labour-only partial equilibrium model to the analysis of imperfectly competitive markets in the recycling sector. André and Cerdá (2006) analyse the intertemporal shift in the proportion between recyclable and nonrecyclable production inputs in a dynamic economy, not constrained by the mass conservation principle and without capital accumulation and technical progress. An overview of applied materials flows models is given in Bouman et al. (2000). The present paper attempts to study material flows in an economic model, constrained by the mass conservation principle, and in which technology is determined by human capital. The flows considered are the flows between the natural environment and the economy and the material flows within the economy. The stocks are: the stock of accumulated emissions and the stock of materials temporarily frozen within the economy in the shape of physical capital. Physical capital occupies natural spaces and determines the material scale of the economy. If a recycling sector exists, materials are re-directed from the flow of waste back into the economy as a second kind of material inputs, which is added to the flow of virgin materials, directly extracted from the environment. There is some ambiguity in the literature on the nature of this second kind of material inputs (Converse 1997; Ayres 1999a). Can waste be recycled after diffusion into the environment, or is a previous sequestration of waste before diffusion necessary for recycling activities? If diffused waste is the source of recycling, the difference between virgin materials and recycled materials disappears, since both would have to be extracted from the environment. I shall adopt the view of a combined sequestrationrecycling activity, because I assume that capturing waste materials before dissipation is 4 economically more convenient than retrieving materials after dissipation. In this case, virgin materials and recycled materials are different inputs, since the first originate in nature, while the second originate from sequestration activities. This implies, that recycling is a two-stage process: sequestration comes first and recycling in the stricter sense follows, when materials are extracted, or ‘mined’ out of sequestered materials and transformed into inputs to final output. The material structure of the economy is determined by knowledge and technology, and the level of society’s technological capabilities is measured in this paper by human capital. I shall consider three channels through which human capital influences the material basis of human activities. First of all, human capital contributes to an increase in productivity. This role of human capital has been extensively investigated in economic theory (Lucas 1988; Romer 1989) and will also be considered here. Productivity gains imply however a rising throughput of materials, which has to be accounted for, both backwards, as an increase in material requirements, and also forwards, as a growing production of waste. A second effect of knowledge is to contribute to a more efficient use of materials in the economic process. Human capital can contribute to reduce material losses in the process of transformation of material inputs into useful outputs (Reijnders 1998; Schmidt-Bleek 1993, 1997; Weizsäcker et al. 1997). In this way, human capital enhances ecoefficiency and reduces the environmental impacts of economic activities. A third effect is on consumption. Consumption is not only a material, but also a cultural, aesthetic and social process and is therefore fundamentally influenced by the general level of accumulated social knowledge (Cogoy 1999). In this paper I shall study the optimal material state of the economy in stock-flow equilibrium. This means, that the social planner chooses the optimal level of stocks, human capital included. The optimal level of knowledge is not infinite, since human capital requires a material infrastructure supporting its operations, and is therefore constrained by environmental considerations, as are all physical stocks. For this reason, the optimal level of knowledge is endogenous in this paper. The description of a 5 transitional dynamic path, leading to the optimal material state of the economy is outside the scope of the present paper. Sections 2 and 3 describe the model and derive first-order conditions for optimality. Sections 4 and 5 investigate a simplified reference case and offers an analytical discussion of the solution. The final section concludes and formulates some warnings. 2. THE MODEL 2.1. Material stocks and flows The relationship between the environment and the economy is described in this paper by focussing on two material stocks, and on the flows between them. K is the physical capital stock, and D are accumulated emissions. S is the amount of materials extracted by human activities from the natural environment. By definition: DKS   (1) Although S is nothing more than the negative image of D K  , it is given a positive sign in (1). Equation (1) states that human activities displace materials from the natural environment into two types of sinks: physical capital and accumulated anthropogenic emissions. Now consider flows. (Upper case notation denotes stocks, lower case denotes flows.) v is the flow of virgin materials currently extracted from the natural environment in order to be processed by the economic system, e is the flow of emissions, and a is the flow of materials reabsorbed by natural regeneration from the stock of discharged materials back into natural processes. In stock equilibrium (constant stocks): e a v   (2) Equation (2) states that in stock equilibrium virgin materials inputs into the economy must be equal to dispersed waste and to natural absorption. I assume that absorption is a linear function of the stock of pollutants: Da   10    (3) 6 where  is the rate of absorption. From (2) and (3): De   (4) Dv   (5) A constant capital stock requires: Kcy K   (6) where y are materials embodied in final output, c is aggregate physical consumption, and K K  is linear capital depreciation. This is the familiar equation, stating that net investment is zero if gross investment is equal to capital depreciation. It has to be interpreted here as a materials balance equation, since all quantities are expressed in mass units. From the point of view of materials balances the role of capital on consumption is a negative one, since a larger capital stock will require a larger flow of materials replacing worn-out capital. Capital is not only a container of materials however, but also a productive factor, and this role will be studied in the next section. Virgin materials are extracted from the environment and processed up to the point that they may serve as inputs to the production of final output. From the materials balance point of view, the output of processed and refined virgin materials is a fraction of the amount of materials extracted from the environment: vmV   10    (7) where V m are useful materials, transformed and refined out of the flow of virgin materials v , and  is an eco-efficiency coefficient, depending on human capital. If  is equal to one, the process is perfectly efficient and no materials are lost in transformation. If  is equal to zero, the process is perfectly inefficient, no output comes out of the process, and all inputs are transformed into waste. Waste from extraction is obviously:   v  1.  is a complex measure of eco-efficiency, since it evaluates more than one aspect of materials efficiency with one variable only. If fewer materials are lost in transformation,  will rise. But  will also rise, if process waste is directly channelled as input to other 7 processes. In other words,  measures materials efficiency both at plant level and also at the level of interconnected plants (industrial ecology and industrial metabolism, cf. Ayres 1989; Ayres and Simonis 1994; Ayres and Ayres 1996; Erkman 1997). If materials are discarded from one production plant and directly channelled to another, they cannot be considered as waste in a strict sense. The definition of waste is confined in this paper to those materials which are either emitted into the environment, or sequestered for future recycling. If industrial ecology were perfectly successful, process waste would be reduced to zero and  would be equal to one. Depreciating capital and consumption would then be the only sources of waste. If a recycling activity exists, one fraction of total waste is sequestered and directed to the recycling sector, in a similar way, as virgin material inputs are extracted from the environment and submitted to the process of refining and transformation. Useful recycled materials are a fraction of the materials “mined” from sequestered waste: rmR   (8) where R m are recycled materials, suitable for serving as inputs to the final output sector and r is the flow of materials sequestered from waste. I assume that the eco-efficiency coefficient  is the same in all processes. Waste from the recycling process is therefore:   r  1. I assume that society chooses how to divide waste between dispersion and sequestration. Such a choice is a choice under technological constraint however, since waste materials can only be sequestered, if capital and labour are allocated to this purpose. Recycling substitutes virgin material inputs. At the same time the scale of the economy is affected, since recycling capital will be required. If society chooses to recycle, waste of different types will enter the sequestration process. It is certainly not meaningful to mix all sorts of waste and have a uniform mixture of materials out of which recycled materials can be “mined”. It will be probably more reasonable to have separate storage facilities for different types of materials (Craig 2001). Bent nails will have to be straightened, and not ground and mixed with sand before extracting iron out of the mixture. The recycling sector, as it is modelled in the present paper, uses capital 8 and labour in order to reprocess waste of different types, collected in different sequestration facilities. I assume that these operations deliver processed materials of the same quality as the output of the virgin materials sector, so that processed virgin materials and processed waste are perfect substitutes. Complete recycling implies: 0    eav and therefore 0  D. It will be shown in section 5, that this option is unlikely to be economically optimal. A consequence of this is that the issue of complete recycling (Bianciardi et al. 1993, 1996; Converse 1996, 1997; Washida 1998) turns out to be of little economic relevance. Even if complete recycling were technically possible, it can hardly be economically meaningful. The same reasoning as before also applies to final output, where refined materials (virgin and recycled) are transformed to useful goods:   RV mmy   (9) Waste in the final output sector is:     RV mm   1 . From (7) to (9) we get:   rvy  2  (10) Equation (10) defines output in terms of eco-efficiency and the material flows into the economy. The quadratic exponent of  for final output obviously follows from the assumption, that two stages are required: materials refinement (virgin and/or recycled) and final production. At each of these stages some materials are lost. The relationship between material stocks and flows between the economy and the natural environment and within the economy is represented in Figure 1: Figure 1 It can be seen from Figure 1 that there are in this model three sources of waste. A first source of waste is in the transformation process, where materials are moulded into the desired shape: if some materials are ‘lost’, while others are given a useful economic shape, these losses represent a source of waste. y r v   is waste from transformation, since r v  are materials (virgin and recycled) entering the transformation process, and 15 materials processing sectors, and less will be left for final output. For this reason, ecoefficiency plays an important role at aggregate level, since, given aggregate capital and labour, the system will be more productive if a greater portion of aggregate factors is employed in final output production. Equations (5), (17) and (28), together with technology functions (22) to (24) describe a feasible stock-flow equilibrium, constrained by the mass conservation principle. 2.6. Preferences I assume that welfare is determined for each individual by the stream of qualified percapita consumption z , and by the state of the environment. The state of the environment is a public good and affects all of the identical N individuals in the same way. Physical capital affects welfare in two ways: a direct, and an indirect one. The indirect effect has been extensively analysed in economic theory: capital enhances labour productivity and contributes in this way to output and consumption. On the other hand however, physical capital encroaches upon natural spaces, spoils landscapes, destroys biotopes, and for this reason also has direct negative effects on welfare. Physical capital is, in other words, a necessity, not a pleasure3. This suggests the idea of an optimal level of physical capital, which will have to be determined by a compromise between its positive contribution to production and its negative effects on the material scale of the economy. The stock of accumulated emissions D is another argument of the welfare function. With these premises in mind, preferences can be modelled as:   DKzUU ,, 0,0,0,0,0,0  DDDKKKzzz UUUUUU (29) At this point, the difference between physical and human capital may be summarized as follows. Both types of capital enhance productivity in all sectors, in which they are employed. Physical capital directly encroaches upon the environment by occupying natural spaces, whereas human capital only indirectly affects the environment, insofar as physical capital is needed for the maintenance of knowledge. 16 3. THE SOCIAL PLANNER'S PROBLEM 3.1. The problem In order to determine the optimal stationary material state of society, the social planner maximizes (29), subject to (5), (17), (28) and technology functions (22) to (24). It is intuitive, that optimal human capital is finite in the model presented in this paper. Maintenance and renewal of depreciated human capital require capital and labour. Human capital is limited by the same environmental constraints limiting the quantity of physical capital and is endogenously determined therefore along with the other stocks. In order to give an intuitive interpretation of the first-order conditions, I shall use a more general form for (17) and (28):   Krvzz ,,,,   (17’)   0,,,,, HKrvF  (28’) 3.2. First order conditions First-order conditions for this problem are:    D KzK K v vz U zUU F F zU  (30)   KzK K r rz zUU F F zU  (31)     KzK K H zzUU F FHFF zzU                   (32) First-order condition (30) equates marginal benefits to consumption from an increase in v to marginal environmental degradation deriving from induced increases in stocks K and D . Kz zU is the direct negative marginal effect on consumption, deriving from the burden of capital depreciation. Similarly, (31) compares benefits and costs of a marginal increase in recycling. The lack of a second member on the right hand side is due to the fact that recycling does not increase the pollution stock. Equation (32) compares the 17 consumptive benefits deriving from a marginal increase in eco-efficiency to the negative effects of an induced marginal increase in physical capital. Of course, this effect is mitigated by a raise in productivity  . First order conditions, together with the problem’s constraints, determine the optimal values for variables: K , D , H , v , r , z ,  ,  and  . 4. REFERENCE CASE Given the number of variables and the non-linearity of the system, finding a solution may become quite an intricate business. In the next section I shall present a graphic discussion of a simplified reference case, which allows to study the basic structure of the model. For the simplified reference case I shall make following assumptions: a) Non depreciating physical capital Non-depreciating physical capital implies: 0 K  (33) b) Technology functions I shall assume that the upper bound of  is 1, and model eco-efficiency as: H H     (34) which implies:     1 H   2 1 1      H (35) Productivity and consumption quality are assumed to be bounded and fixed multiples of eco-efficiency:         (36)      (37) 18  and  are arbitrarily chosen upper bounds of productivity and consumption quality. c) Preferences I shall model utility as a logarithmic function of qualified consumption, net of quadratic environmental damage from stocks:   22 2 1 log DqKqzqU DKz  (29’) where i q are weights of the arguments in the utility function. With these assumptions (17), (28) and (30) to (32) become:   rv N z 3  (38)                1 2 1rvrvNK (39)             222 11KrvqrvvqqNK KDz     (40)         2 1KrvqNKq Kz     (41)               2 1 2 1 221 1 3 KNKrvq NKq K z                   (42) For a given size of the population, (38) to (42) yield optimal values for z , v , r ,  and K . For a graphic analysis of the solution, (39) to (42) can be more conveniently manipulated into:       01 22   zDD qrvqvq (43)                       1131 1211 2 2r v (44) 19                                 2 22122 2 212 2 2 11111 1 vqvqqq vNq DDzK D (45)           01111 222  zKDD qKqvqvq  (46) Physical and qualified per-capita consumption are:     vNq q b D z     1 22 (47)     vNq q z D z     1 23 (48) The logic of the solution is as follows. Equations (43) and (44) yield an efficient locus of points in v /  space. Each point on the locus is associated with a different level of recycling. Equation (45) identifies the optimal point on the efficient locus (and therefore optimal recycling) for an exogenously given size of the population. Equation (46) can then be solved for optimal physical capital. Equations (47) and (48) determine the values of physical and qualified per-capita consumption. 5. A STUDY OF EQUATIONS (43) TO (47) 5.1. Efficient locus of non-negative recycling in v /  space Consider first (43) and (44). Setting 0  r yields:           D z q q v1 (49)                  1131 1 2 v (50) Since the graph of (49) is monotonically declining, whereas the graph of (50) is monotonically rising, the two equations will only have a 0  v, 0   solution, if              13 11 D z q q (51) 20 It can be also shown, that if (51) is not satisfied, then a solution 0   , 0  v, 0  r exists. Setting 0   into (43) and (44) yields:      D z q q rr v1 22 2 2         (52)                13 21 r v (53) The value of v in (52) tends to zero, as r tends to infinity, whereas in (53) v declines to zero for a finite value of r . Therefore (53) cuts (54) from above if the inequality in (51) is reversed. From this I conclude that (43) and (44) have either a 0  r,0  v, 0   or a 0   , 0  v, 0  r solution, depending on the direction of the inequality in (51). I shall call this point the origin of the solution locus. For increasing values of r the crossing points of (43) and (44) generate an efficient locus in v /  space (Figure 2). Figure 2 Point Q represents the origin of the locus in the case where (51) holds. If (51) does not hold, point Q is on the vertical axis. Pairs of v and  on this locus satisfy:                  1131 1 2 v (54) and also:           D z q q v1 (55) Inequalities (54) and (55) define an area of non-negative recycling in v /  space. Inserting r from (43) into (44) yields the equation of this locus:                      01211 1111 2 2 2 2     z DD q vqvq (56) Derivation of the implicit function (56) yields: 21        , , 2 1 vg vg d dv  (57) where:                              312211 211312111, 2 2 1 z DD q vqvqvg (58)                 11112, 2 2vqvg D (59) It can be seen, that:   0, 1  vg , and that:   0, 2  vg if :       2 1112 1        v (60) A quick check shows, that if (54) is satisfied, (60) is satisfied as well. This implies, that the locus is monotonically falling. It can be seen that complete recycling ( 0   vD ) can only be efficient at 1   . The reason for this is simple: at 0  D marginal damage from pollution is zero and there is no incentive therefore to prevent some materials to diffuse into the environment. 5.2 The solution point as a function of the size of the population An easy graphic interpretation of (45) can be given after manipulating it into:                                          2 2 12 2 12 2 2 1 1 1 1vqq v q v q Nq DD z K D (61) For any value of  such that 10    , a unique value of v exists, setting the right hand side of the equation equal to the left hand side, since the right hand side is monotonically declining from infinity to zero as v increases from zero to the positive value reducing the expression in square brackets to zero. Without the necessity of calculating derivatives, one can also see, that 0  d dv , and that the graph shifts downwards for rising values of N. 22 For the case where (51) holds, the graph of (61) passing through the origin of the locus of efficient points is represented in Figure 3: Fig 3 If the population increases, the graph shifts downwards. Eco-efficiency increases and virgin material inputs into the economic system are substituted by recycled materials. It must be noted, that the size of the population in the origin of the efficient locus is strictly positive, and recycling is zero. Therefore, if (51) holds, there is for non-negative recycling a lower limit to the size of the population. In the case where (51) does not hold, the origin of the efficient locus is on the vertical axis and the size of the population in the origin is zero. 5.3. Physical capital Solving equation (46) for v yields:                112 114 2 22 2 2 D zKDDD q qKqqqq v (62) For values of K z q q K   this equation yields two branches, one ascending, the other descending. The locus of bifurcation points is given by:       112 v (63) Figure 4 represents the graph of (62) together with the locus of bifurcation points (63) and the efficient locus (56). Figure 4 The graph shows, that physical capital declines as the growth of the population shifts the solution point downwards along the efficient locus. This is because physical capital is substituted by labour and human capital as the population increases. If physical capital and virgin material inputs both decline, this means, that the state oft the environment improves. It may seem strange that in the stationary state the quality of the environment is better with a larger size of the population. This result follows 23 however from the assumption of good substitutability between consumption and environmental quality in the social welfare function. When individual consumption declines, an improvement in environmental conditions helps reduce the loss in welfare. There is an exception however to the decline of physical capital with an increase of the population. If the origin of the efficient locus is above the locus of bifurcation points, physical capital rises first and later declines as the population increases, as is shown in Figure 5. Figure 5 5.4. Physical per capita consumption. Substituting N from (47) into (45) yields:                                                         2 2212 2 2 12 32 2 12 2 2 1111 11 vqvqqbq vqq DDzK zD (64) This equation identifies points in v /  space for given values of physical per capita consumption. Any point on the efficient locus is associated with one level of per capita consumption. The graphs of (56) and (64) must therefore cross. How this looks like, depends however on the value of the Cobb-Douglas parameter  . The first derivative of (64) is:           , , 14 3 vg vg v d dv   (65) where:                      2 22 2 2 3 111212234 1112,     zD D qvq vqvg (66)                 112231114, 22 4zDD qvqvqvg (67) v /  space can be divided in sections, depending on the signs of    , 3vg and    , 4vg . 5.4.1. The   0, 3  vg locus. 24 The first derivative of   0, 3  vg is:                22 3 2 0, 1112 3422 3    zD D vg qvq vq d dv     (68) The graph of   0, 3  vg is downwards sloping, as is shown in Figure 6. Figure 6 5.4.2. The   0, 4  vg locus. If 3 2   ,    , 4vg is always positive. If 3 2   the locus:   0, 4  vg is given by:                     1118 111323232 22 2 2 D DzDD q qqqq v (69) Equation (69) consists of two branches. The bifurcation is located at the value of  which sets the expression under the square root equal to zero, i.e.:                12 32 14 1 2 z D q q (70) The expression under the square root rises monotonically from zero to infinity as  declines from 3 2   to zero. The bifurcation point shifts therefore from 1   for 3 2   to    for 0   . Figure 7 represents the graph of the   0, 4  vg locus with 3 2   . Figure 7 5.4.3. The division of v /  space. The graphs of   0, 3  vg and of   0, 4  vg can be combined and generate different patterns, depending on the value of  . Three possible patterns can be distinguished. a) 31 REFERENCES Adriaanse, A. et al. (1997). Resource Flows: the Material Basis of Industrial Economies. 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Earthscan, London 35 Extraction K Production y=2(v+r) Recycling v r y-c v K (1-)(v+r) c (1-)v (1-)r e r VD a waste The stock-flow model Figure 1 36 0 0.5 1  v Q The locus of efficient points (red) if inequality (51) holds Figure 2 0 0.5 1  v Q Equation (61) (blue) with increasing N Figure 3 37 0 0.5 1 v  Q Bifurcation locus (violet) and different levels of physical capital (gold) Figure 4 0 0.5 1 v  Q Origin of the efficient locus Q above the bifurcation locus Figure 5 38 0 0.5 1  v   0, 3  vg   0, 3  vg The   0, 3  vg locus (pink) Figure 6 0 0.5 1  v  declining   0, 4  vg   0, 4  vg The   0, 4  vg locus (cyan) Figure 7 39 0 0.5 1  v 0  d dv 0  d dv Q Physical per capita consumption with a “large”  Figure 8 0 0.5 1  v 0  d dv 0  d dv 0  d dv Q A Physical per capita consumption with a “small”  Figure 9 40 0 0.5 1  v Q 0  d dv 0  d dv Physical per capita consumption with an “intermediate” value of  Figure 10