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.

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(p, E, Y) 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 $\theta \le 0$.

1.1 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(E, C, t)

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

(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)

1.2 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)