Limits to Substitution Between Ecosystem Services and Manufactured Goods and Implications for Social Discounting

Abstract

This paper examines implications of limits to substitution for estimating substitutability between ecosystem services and manufactured goods and for social discounting. Based on a model that accounts for a subsistence requirement in the consumption of ecosystem services, we provide empirical evidence on substitution elasticities. We find an initial mean elasticity of substitution of two, which declines over time towards complementarity. We subsequently extend the theory of dual discounting by introducing a subsistence requirement. The relative price of ecosystem services is non-constant and grows without bound as the consumption of ecosystem services declines towards the subsistence level. An application suggests that the initial discount rate for ecosystem services is more than a percentage-point lower as compared to manufactured goods. This difference increases by a further half percentage-point over a 300-year time horizon. The results underscore the importance of considering limited substitutability in long-term public project appraisal.

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Notes

  1. 1.

    That is: even though many parts of natural capital and ecosystem services may be replaceable by technology, Fitter (2013) argues that a number of supporting services (soil formation, water cycling etc.), selected final services (e.g. climate regulation) and goods (e.g. water supply, a safe and enjoyable environment) may be very hard if not impossible to fully substitute (cf. Ayres 2007).

  2. 2.

    Alternatively, Dupoux and Martinet (2014) propose to examine Edgeworth–Pareto substitutability by introducing a specific ‘context-dependent substitutability function’.

  3. 3.

    The extension of \(U_h(E, C)\) for \(\theta \rightarrow \) 0 is a special Stone–Geory case: \(U_h(E, C)= \left( E-\overline{E}\right) ^{\alpha }C^{(1-\alpha )}\).

  4. 4.

    While choice experiments may constitute a suitable approach for estimating the elasticity of substitution, studies have so far not been designed for such purposes. The same is the case for revealed preference studies, with the exception of Martini and Tiezzi (2014), which we address below. Further indicative evidence might be derived from the WTA/WTP disparity (Hanemann 1991).

  5. 5.

    This implies that both goods are ‘normal’, which may not be the case for every single ecosystem service (Horowitz and McConnell 2003).

  6. 6.

    Note that income elasticities are generally not constant but may vary across individuals and also across aggregate measures, as e.g. found in Barbier et al. (2015) and Ready et al. (2002). Broberg (2010) finds that a model with a constant income elasticity does not produce a worse overall fit than those where the income elasticity of WTP is a (non-)linear function.

  7. 7.

    We thereby follow procedure adopted elsewhere to assume that the single ecosystem service components are part of a homogenous ecosystem service good (Baumgärtner et al. 2015a) and to use an unweighted arithmetic mean (see, additionally, Hœokby and Sœderqvist 2003).

  8. 8.

    While these growth rates do not stem from an optimising behaviour of the representative agent and thus only facilitate valuation along a non-optimal trajectory, this approach represents standard practice in the literature (cf. Hoel and Sterner 2007; Traeger 2011).

  9. 9.

    A sensitivity analysis with respect to the subsistence requirement reveals that elasticity of substitution would fall below the threshold of unity after 201 (333) years for a value of \(\overline{E}\) of 0.2 (0.1).

  10. 10.

    The clear result of income elasticities smaller than unity obtained throughout the contingent valuation literature has been challenged by Schläpfer (2006, 2008, 2009). Schläpfer argues that the small income elasticities may be an artefact of the current design of contingent valuation studies, which may lower the income effect. He compares contingent valuation with voting-based studies (Schläpfer and Hanley 2003, 2006) and finds support for an income elasticity of WTP equal to or greater than unity.

  11. 11.

    While the omission of more difficult to measure WTPs for ecosystem services might lead to an overestimate of the overall degree of substitutability, the effect of an over-proportionate availability of studies from developed countries might bias the estimate in the opposite direction. The available evidence so far suggests that the income elasticity of WTP might be higher in high-income populations (Barbier et al. 2015; Ready et al. 2002), suggesting that the current sample mean would underestimate the average degree of substitutability. We cannot [can] confirm this finding when comparing developed and developing countries for estimates derived from all [contingent] valuation studies in Table 1.

  12. 12.

    Similar to measuring biodiversity (Bertram and Quaas 2016) one might consider an overall index of ecosystem service abundance with imperfect substitutability among the single ecosystem services.

  13. 13.

    The error range for the sample mean is computed as the standard deviation of the single initial relative price effect estimates.

  14. 14.

    With the given specifications, we approach the subsistence requirement after 363 years.

References

  1. Ayres RU (2007) On the practical limits to substitution. Ecol Econ 61(1):115–128

    Article  Google Scholar 

  2. Barbier EB, Czajkowski M, Hanley N (2015), Is the income elasticity of the willingness to pay for pollution control constant? University of St. Andrews, Department of Geography and Sustainable Development Working Paper No 2015-04

  3. Barton DN (2002) The transferability of benefit transfer: contingent valuation of water quality improvements in Costa Rica. Ecol Econ 42(1):147–164

    Article  Google Scholar 

  4. Baumgärtner S, Klein A-M, Thiel D, Winkler K (2015a) Ramsey discounting of ecosystem services. Environ Resour Econ 61:273–296

    Article  Google Scholar 

  5. Baumgärtner S, Drupp MA, Quaas MF (2015b) Subsistence, substitutability and sustainability in consumer preferences. Environ Resour Econ (forthcoming) http://link.springer.com/article/10.1007%2Fs10640-015-9976-z

  6. Baumgärtner S, Drupp MA, Meya J, Munz M, Quaas MF (2016) Income inequality and willingness to pay for public environmental goods. University of Kiel Economics Working Paper 2016-04

  7. Bertram C, Quaas MF (2016) Biodiversity and Optimal Multispecies Ecosystem Management. Environ Res Econ. doi:10.1007/s10640-015-9988-8

  8. Broberg T (2010) Income treatment effects in contingent valuation: the case of the Swedish predator policy. Environ Resour Econ 46(1):1–17

    Article  Google Scholar 

  9. Carlsson F, Johansson-Stenman O (2000) Willingness to pay for improved air quality in Sweden. Appl Econ 32:661–669

    Article  Google Scholar 

  10. Chiabai A, Travisi C, Markandya A, Ding H, Nunes P (2011) Economic assessment of forest ecosystem services losses: cost of policy inaction. Environ Resour Econ 50(3):405–445

    Article  Google Scholar 

  11. Dasgupta P (2001) Human well-being and the natural environment. Oxford University Press, Oxford

    Google Scholar 

  12. Dasgupta P (2010) Nature’s role in sustaining economic development. Philos Trans R Soc B Biol Sci 365(1537):5–11

    Article  Google Scholar 

  13. Drupp MA, Freeman MC, Groom B, Nesje F (2015) Discounting disentangled. Grantham Research Institute on Climate Change and the Environment Working Paper No. 172

  14. Dupoux M, Martinet V (2014) Context-dependent substitutability: impacts on environmental preferences and discounting. Paper presented at the 2014 WCERE conference

  15. Ebert U (2003) Environmental goods and the distribution of income. Environ Resour Econ 25(4):435–459

    Article  Google Scholar 

  16. Ehrlich PR (1989) The limits to substitution: meta-resource depletion and a new economic-ecological paradigm. Ecol Econ 1(1):9–16

    Article  Google Scholar 

  17. Fenichel EP, Zhao J (2015) Sustainability and substitutability. Bull Math Biol 77(2):348–367

    Article  Google Scholar 

  18. Fitter AH (2013) Are ecosystem services replaceable by technology? Environ Resour Econ 55(4):513–524

    Article  Google Scholar 

  19. Geary RE (1949–1950) A note on ‘A constant utility index of the cost of living’. Rev Econ Stud 18(1):65–66

  20. Gerlagh R, van der Zwaan B (2002) Long-term substitutability between environmental and man-made goods. J Environ Econ Manag 44:329–345

    Article  Google Scholar 

  21. Gollier C (2010) Ecological discounting. J Econ Theory 145:812–829

    Article  Google Scholar 

  22. Guesnerie R (2004) Calcul economique et d’eveloppement durable. Revue E’con 55:363–382

    Google Scholar 

  23. Hammitt J, Liu J-T, Liu J-L (2001) Contingent valuation of a Taiwanese wetland. Environ Dev Econ 6:259–268

    Article  Google Scholar 

  24. Hanemann WM (1991) Willingness to pay and willingness to accept: how much can they differ? Am Econ Rev 81(3):635–647

    Google Scholar 

  25. Heal G (2009a) The economics of climate change: a post-Stern perspective. Clim Change 96(3):275–297

    Article  Google Scholar 

  26. Heal G (2009b) Climate economics: a meta-review and some suggestions for future research. Rev Environ Econ Policy 3(1):4–21

    Article  Google Scholar 

  27. Hoel M, Sterner T (2007) Discounting and relative prices. Clim Change 84:265–280

    Article  Google Scholar 

  28. Hœokby S, Sœderqvist T (2003) Elasticities of demand and willingness to pay for ecosystem services in Sweden. Environ Resour Econ 26(3):361–383

    Article  Google Scholar 

  29. Horowitz JK, McConnell KE (2003) Willingness to accept, willingness to pay and the income effect. J Econ Behav Organ 51(4):537–545

    Article  Google Scholar 

  30. Jacobsen J, Hanley N (2009) Are there income effects on global willingness to pay for biodiversity conservation? Environ Resour Econ 43(2):137–160

    Article  Google Scholar 

  31. Kopp RE, Golub A, Keohane NO, Onda C (2012) The influence of the specification of climate change damages on the social cost of carbon. Economics: The Open-Access, Open-Assessment E-Journal 6

  32. Kovenock D, Sadka E (1981) Progression under the benefit approach to the theory of taxation. Econ Lett 8:95–99

    Article  Google Scholar 

  33. Kristrœm B, Riera P (1996) Is the income elasticity of environmental improvements less than one? Environ Resour Econ 7:45–55

    Article  Google Scholar 

  34. Lindhjem H, Tuan TH (2012) Valuation of species and nature conservation in Asia and Oceania: a meta-analysis. Environ Econ Policy Stud 14(1):1–22

    Article  Google Scholar 

  35. Liu S, Stern DI (2008) A meta-analysis of contingent valuation studies in coastal and bear-shore marine ecosystems, CSIRO Working Paper Series 2008–2015

  36. Mäler K-G (2008) Sustainable development and resilience in ecosystems. Environ Resour Econ 39(1):17–24

    Article  Google Scholar 

  37. Martini C, Tiezzi S (2014) Is the environment a luxury? An empirical investigation using revealed preferences and household production. Resour Energy Econ 37:147–167

    Article  Google Scholar 

  38. MEA (2005) Ecosystems and human well-being: synthesis. Island Press, Washington

  39. Millner A (2013) On welfare frameworks and catastrophic climate risks. J Environ Econ Manag 65(2):310–325

    Article  Google Scholar 

  40. Neumayer E (2007) A missed opportunity: the Stern review on climate change fails to tackle the issue of non-substitutable loss of natural capital. Glob Environ Change 17:297–301

    Article  Google Scholar 

  41. Neumayer E (2010) Weak versus strong sustainability: exploring the limits of two opposing paradigms, 3rd Edn. Edward Elgar, Cheltenham

  42. Pezzey JCV, Anderies JM (2003) The effect of subsistence on collapse and institutional adaptation in population-resource societies. J Dev Econ 72:299–320

    Article  Google Scholar 

  43. Ready RC, Malzubris J, Senkane S (2002) The relationship between environmental values and income in a transition economy: surface water quality in Latvia. Environ Dev Econ 7(1):147–156

    Article  Google Scholar 

  44. Revankar NS (1971) A class of variable elasticity of substitution production functions. Econometrica 39(1):61–71

    Article  Google Scholar 

  45. Rockström J et al (2009) A safe operating space for humanity. Nature 461:472–475

    Article  Google Scholar 

  46. Schläpfer F, Hanley N (2003) Do local landscape patterns affect the demand for landscape amenities protection? J Agric Econ 54(1):21–34

    Article  Google Scholar 

  47. Schläpfer F, Hanley N (2006) Contingent valuation and collective choice. KYKLOS 59(1):115–135

    Article  Google Scholar 

  48. Schläpfer F (2006) Survey protocol and income effects in the contingent valuation of public goods: a meta-analysis. Ecol Econ 57(3):415–429

    Article  Google Scholar 

  49. Schläpfer F (2008) Contingent valuation: a new perspective. Ecol Econ 64(4):729–740

    Article  Google Scholar 

  50. Schläpfer F (2009) Contingent valuation: confusions, problems, and solutions. Ecol Econ 68(6):1569–1571

    Article  Google Scholar 

  51. Sharif M (1986) The concept and measurement of subsistence: a survey of the literature. World Dev 14(5):555–577

    Article  Google Scholar 

  52. Sœoderqvist T, Scharin H (2000) The regional willingness to pay for a reduced eutrophication in the Stockholm archipelago, Beijer Discussion Paper No. 128

  53. Steger TM (2000) Economic growth with subsistence consumption. J Dev Econ 62(2):343–361

  54. Stern DI (1997) Limits to substitution and irreversibility in production and consumption: a neoclassical interpretation of ecological economics. Ecol Econ 21(3):197–215

    Article  Google Scholar 

  55. Stern N (2007) The economics of climate change: the stern review. Cambridge University Press, Cambridge

    Google Scholar 

  56. Sterner T, Persson M (2008) An even sterner review: introducing relative prices into the discounting debate. Rev Environ Econ Policy 2(1):61–76

    Article  Google Scholar 

  57. Stone JRN (1954) A not on economics growth with subsistence consumption. Econ J 64:511–527

    Article  Google Scholar 

  58. ten Brink P (ed) (2011) The economics of ecosystems and biodiversity in national and international policy making. Earthscan, London

    Google Scholar 

  59. Traeger CP (2011) Sustainability, limited substitutability, and non-constant social discount rates. J Environ Econ Manag 62(2):215–228

    Article  Google Scholar 

  60. Wang H, Whittington D (2000) Willingness to Pay for Air Quality Improvement in Sofia, Bulgaria. World Bank Policy Research Working Paper 2280, Washington DC: The World Bank

  61. Wang H, Shi Y, Kim Y, Kamata T (2013) Valuing water quality improvement in China. A case study of Lake Puzhehei in Yunnan Province. Ecol Econ 94:56–65

    Article  Google Scholar 

  62. World Commission on Environment and Development (WCED) (1987) Our common future. Oxford University Press, Oxford. http://www.un-documents.net/ocf-02.htm

  63. Weikard H-P, Zhu X (2005) Discounting and environmental quality: when should dual rates be used? Econ Model 22:868–878

    Article  Google Scholar 

  64. Weitzman ML (2009) On modeling and interpreting the economics of catastrophic climate change. Rev Econ Stat 91(1):1–19

    Article  Google Scholar 

  65. Weitzman ML (2010) What is the damages function for global warming and what difference might it make? Clim Change Econ 1(1):57–69

    Article  Google Scholar 

  66. Whitehead JC, Haab TC, Huang J-C (2000) Measuring recreation benefits of quality improvements with revealed and stated behavior data. Resour Energy Econ 22:339–354

    Article  Google Scholar 

  67. Yu X, Abler D (2010) Incorporating zero and missing responses into CVM with openended bidding: willingness to pay for blue skies in Beijing. Environ Dev Econ 15:535–556

    Article  Google Scholar 

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Acknowledgments

I am very grateful to Stefan Baumgärtner, Ben Groom and Martin Quaas for their support. Furthermore I thank Mikolaj Czajkowski, Simon Dietz, Reyer Gerlagh, Christian Gollier, David Löw-Beer, Frikk Nesje, Eric Neumayer, Martin Persson, Paolo Piacquadio, Till Requate, Felix Schläpfer, Gregor Schwerhoff, Thomas Sterner and participants at the 2014 SURED, the 2014 WCERE and the IfW Centenary Conference for helpful comments. Financial support from the German National Academic Foundation, the DAAD and the BMBF under grant 01LA1104C is gratefully acknowledged.

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Appendix 1: Relationship Between the Income Elasticity, the CES Substitutability Parameter and the Elasticity of Substitution

Appendix 1: Relationship Between the Income Elasticity, the CES Substitutability Parameter and the Elasticity of Substitution

This Appendix clarifies the relationship between the income elasticity of WTP, the CES substitutability parameter and the elasticity of substitution.

The agent’s income is exogenously given and denoted by Y. The consumption good is traded on a market at given price \(p>0\), while the consumption of the ecosystem service is fixed at an exogenously given level \(E>0\). The agent maximizes its utility subject to the budget constraint and fixed level of the ecosystem service:

$$\begin{aligned} \max _{E,C}\ U_h (E,C)\quad {\text {s.t.}}\quad pC=Y\ {\text {and}}\ E\ {\text {fixed}}. \end{aligned}$$
(19)

Following Ebert (2003) the income-equivalent total WTP for the ecosystem service at level E is defined as the WTP w per unit times the total number E of units:

$$\begin{aligned} {\text {WTP}} = w \, E. \end{aligned}$$
(20)

The marginal willingness to pay w is implicitly defined as the virtual price that yields the ecosystem service level E as the ordinary (unconditional) Marshallian demand in the hypothetical choice problem where the ecosystem service is considered a private market good. It can be derived from the agent’s indirect utility function V(pEY) by an extension of Roy’s identity (Ebert 2003: 440).

$$\begin{aligned} w = \dfrac{\partial V(p, E, Y) / \partial E}{\partial V(p, E, Y) / \partial Y}. \end{aligned}$$
(21)

With utility function (Eq. 2) the indirect utility function is

$$\begin{aligned} V(p, E, C)=\left[ \alpha \left( E-\overline{E}\right) ^{\theta }+ (1-\alpha ) \, \left( \frac{Y}{p}\right) ^{\theta }\right] ^{1/\theta } \end{aligned}$$
(22)

and the partial derivates:

$$\begin{aligned} \dfrac{\partial V(p, E, Y) }{\partial E}= & {} \alpha \left( E-\overline{E}\right) ^{\theta -1} \left[ \alpha \left( E-\overline{E}\right) ^{\theta }+ (1-\alpha ) \, \left( \frac{Y}{p}\right) ^{\theta }\right] ^{1/\theta -1} \end{aligned}$$
(23)
$$\begin{aligned} \dfrac{\partial V(p, E, Y)}{\partial Y}= & {} (1-\alpha ) \, p^{-\theta } Y^{\theta -1} \left[ \alpha \left( E-\overline{E}\right) ^{\theta }+ (1-\alpha ) \, \left( \frac{Y}{p}\right) ^{\theta }\right] ^{1/\theta -1} \end{aligned}$$
(24)

Employing (21), the marginal WTP is given by

$$\begin{aligned} w = \dfrac{\alpha }{1 - \alpha } \,p^{\theta } \left( E-\overline{E}\right) ^{\theta -1} Y^{1-\theta } \end{aligned}$$
(25)

Plugging this into Eq. (20) yields

$$\begin{aligned} {\text {WTP}}(Y) = wE = \dfrac{\alpha }{1 - \alpha } \,p^{\theta } E \left( E-\overline{E}\right) ^{\theta -1} Y^{1-\theta }. \end{aligned}$$
(26)

With utility function 2, the agents total WTP for the ecosystem service at level E then depends on income Y and the other model parameters as follows:

$$\begin{aligned} {\text {WTP}}(Y) = \nu \,Y^{1- \theta } \quad {\text {with}}\quad \nu =\dfrac{\alpha }{1 - \alpha } \,p^{\theta } E \left( E-\overline{E}\right) ^{\theta -1}. \end{aligned}$$
(27)

The (constant) income elasticty of WTP, denoted \(\xi \), is thus given by \(1- \theta \). We have therefore established that, as in the CES case, \({\text {if}} \quad \xi \ \lesseqqgtr 1 \ \ \ \ {\text {then}} \ \ \ \ \theta \gtreqqless 0\). Yet, as the inverse of the elasticity of substitution not only depends on \(\theta \) but on all other parameters and variables of the model as follows (cf. Baumgärtner et al. 2015b: 6)

$$\begin{aligned} \displaystyle \displaystyle \frac{1}{\sigma (E,C)} \ = \ 1-\theta \,\left[ 1-\frac{(1-\alpha )\displaystyle \frac{\overline{E}}{E}}{\alpha \displaystyle \left[ \frac{E-\overline{E}}{C}\right] ^{\theta }+(1-\alpha )}\right] ^{-1} {\text {for E }> \overline{E}}, \end{aligned}$$
(28)

there is no straightforward relationship between the income elasticty of WTP and the elasticity of substitution. We can only unambiguously state that \(\sigma (E,C)<1\) iff \(\theta \le 0\).

Appendix 2: Derivation of the Good-Specific Discount Rates

To derive the good-specific discount rates \(\rho _E (t)\) and \(\rho _C (t)\), we gather the necessary inputs:

The FOCs of u(ECt)

$$\begin{aligned} u_E= & {} \alpha (E_t - \overline{E})^{\theta -1} \left[ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } \right] ^{\frac{1-\eta -\theta }{\theta }} \end{aligned}$$
(29)
$$\begin{aligned} u_C= & {} (1-\alpha ) C_{t}^{\theta -1} \left[ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } \right] ^{\frac{1-\eta -\theta }{\theta }} \end{aligned}$$
(30)

and SOCs

$$\begin{aligned} u_{EE}= & {} -\alpha (E_t - \overline{E})^{\theta -2} \, \, (\alpha \eta (E_t - \overline{E})^{\theta } \nonumber \\&+\,(1-\theta )(1- \alpha ) C_t^{\theta } ) \left[ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } \right] ^{\frac{1-\eta -2\theta }{\theta }} \end{aligned}$$
(31)
$$\begin{aligned} u_{CC}= & {} -(1-\alpha ) C_{t}^{\theta -2} \, \, (\alpha (1-\theta ) (E_t - \overline{E})^{\theta } \nonumber \\&+\, \eta (1- \alpha ) C_t^{\theta } ) \left[ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } \right] ^{\frac{1-\eta -2\theta }{\theta }} \end{aligned}$$
(32)
$$\begin{aligned} u_{EC} = u_{CE}= & {} (1-\alpha ) \alpha C_{t}^{\theta -1} (E_t - \overline{E})^{\theta -1} (1-\eta - \theta ) \left[ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } \right] ^{\frac{1-\eta -2\theta }{\theta }}\nonumber \\ \end{aligned}$$
(33)

are used to derive the respective elasticities of marginal utility

$$\begin{aligned}&\psi _{EE} := -\frac{u_{EE}(\cdot )E_t}{u_E(\cdot )}= \frac{E_t}{E_t -\overline{E}} \left[ \frac{\alpha \eta (E-\overline{E})^{\theta }+(1-\alpha )(1-\theta )C_t^{\theta }}{ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } } \right] \end{aligned}$$
(34)
$$\begin{aligned}&\psi _{CC} := -\frac{u_{CC}(\cdot )C_t}{u_C(\cdot )}= \frac{\alpha (1-\theta ) (E-\overline{E})^{\theta }+(1-\alpha )\eta C_t^{\theta }}{{ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } } } \end{aligned}$$
(35)
$$\begin{aligned}&\psi _{EC} := -\frac{u_{EC}(\cdot )C_t}{u_E(\cdot )}= \frac{ (1-\alpha ) C_t^{\theta } (\eta + \theta -1)}{{ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } } } \end{aligned}$$
(36)
$$\begin{aligned}&\psi _{CE} := -\frac{u_{CE}(\cdot )E_t}{u_C(\cdot )}= \frac{\alpha E (E-\overline{E})^{\theta -1} (\eta + \theta -1)}{{ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } } }\,. \end{aligned}$$
(37)

Using these, the good-specific discount rates are given by (cf. Eqs. (8) and (9)):

$$\begin{aligned} \rho _E (t)= & {} \delta + \frac{E_t}{E_t -\overline{E}} \frac{\alpha \eta (E-\overline{E})^{\theta }+(1-\alpha )(1-\theta )C_t^{\theta }}{ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } } g_E (t) \nonumber \\&+ \frac{ (1-\alpha ) C_t^{\theta } (\eta - (1-\theta ))}{{ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } } } g_C (t) \end{aligned}$$
(38)

and

$$\begin{aligned} \rho _C (t) = \delta \,+\, \frac{\alpha (1-\theta ) (E-\overline{E})^{\theta }+(1-\alpha )\eta C_t^{\theta }}{{ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } } } g_C (t)\, \,+\,\, \frac{\alpha E (E-\overline{E})^{\theta -1} (\eta - (1-\theta ))}{{ \alpha (E_t - \overline{E})^{\theta } +(1- \alpha ) C_t^{\theta } } } g_E (t). \end{aligned}$$
(39)

Appendix 3: Proof of Proposition 1

We now derive the properties of the relative price effect, \(\Delta \rho (t) = (1- \theta ) \left[ g_C (t) - \frac{E_t}{E_t -\overline{E}}\right. \left. g_E (t) \right] \) for \(E_t>\overline{E}>0\) and \(\theta <1\), as presented in Proposition 1:

Equation (13):

$$\begin{aligned} \frac{\partial \Delta \rho (t)}{\partial \overline{E}} \ \, =\ \, - (1-\theta ) \frac{g_E (t) E_t}{(E_t - \overline{E})^2} \ \, \gtreqqless \ \, 0 \quad {\text {for}} \quad g_E (t)\ \, \lesseqqgtr \ \, 0 \end{aligned}$$
(40)

Equation (14):

$$\begin{aligned} \frac{\partial \Delta \rho (t)}{\partial \theta } \ \, =\ \, g_E (t) \frac{E_t}{E_t -\overline{E}} - g_C (t)\ \, \gtreqqless \ \, 0\ \, \, {\text {for}} \ \, g_E (t) \frac{E_t}{E_t -\overline{E}} \ \, \gtreqqless \ \, g_C (t) \end{aligned}$$
(41)

Equations (17) and (16):

$$\begin{aligned} \dot{ \Delta \rho (t)}= & {} (1- \theta ) \left[ \dot{g_C (t)} - \dot{g_E (t)} \frac{E_t}{E_t -\overline{E}} + \frac{g_E (t) \dot{E_t} \overline{E}}{(E_t -\overline{E})^2} \right] \, \nonumber \\= & {} (1- \theta ) \left[ \dot{g_C (t)} - \dot{g_E (t)} \frac{E_t}{E_t -\overline{E}} + \frac{{g_E (t)}^2 E_t \overline{E}}{(E_t -\overline{E})^2} \right] . \end{aligned}$$
(42)

For the special case of \(\dot{g_C (t)} = \dot{g_E (t)} =0\) and \(g_E (t) \ne 0\), we obtain for Eq. (15):

$$\begin{aligned} \dot{ \Delta \rho (t)} \, = \, \frac{g_E (t) \dot{E_t} \overline{E}}{(E_t -\overline{E})^2}\, =\, \frac{{g_E (t)}^2 E_t \overline{E}}{(E_t -\overline{E})^2}\, \, > \, \,0. \end{aligned}$$
(43)

For \(g_E (t)> 0\), \(E_t \rightarrow \infty \) as \(t \rightarrow \infty \). Therefore, we obtain for Eq. (17):

$$\begin{aligned} \lim _{E_t\rightarrow \infty } \dot{ \Delta \rho (t)} \ \, = \ \ (1- \theta ) \left[ \dot{g_C (t)} - \dot{g_E (t)} \right] \,\ \, = \ \, \Delta \dot{ \rho }^{CES} (t). \end{aligned}$$
(44)

For \(g_E (t)< 0\), and bounded (changes in) growth rates, \(E_t \rightarrow \overline{E}\) in finite t. Therefore, we obtain for Eq. (16):

$$\begin{aligned} \lim _{E_t\rightarrow \overline{E}} \dot{ \Delta \rho (t)} \, = \infty . \end{aligned}$$
(45)

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Drupp, M.A. Limits to Substitution Between Ecosystem Services and Manufactured Goods and Implications for Social Discounting. Environ Resource Econ 69, 135–158 (2018). https://doi.org/10.1007/s10640-016-0068-5

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Keywords

  • Dual discounting
  • Ecosystem services
  • Limited substitutability
  • Project evaluation
  • Subsistence
  • Sustainability

JEL Classification

  • Q01
  • Q57
  • H43
  • D61
  • D90