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


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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  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.


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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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Correspondence to Moritz A. Drupp.

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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:

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

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:

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

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).

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

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

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

and the partial derivates:

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

Employing (21), the marginal WTP is given by

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

Plugging this into Eq. (20) yields

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

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:

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

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)

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

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)

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

and SOCs

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

are used to derive the respective elasticities of marginal utility

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

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

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


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

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):

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

Equation (14):

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

Equations (17) and (16):

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

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):

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

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

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

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):

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

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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).

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

JEL Classification

  • Q01
  • Q57
  • H43
  • D61
  • D90