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Subsistence, Substitutability and Sustainability in Consumption

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Abstract

We propose a representation of individual preferences with a subsistence requirement in consumption, and examine its implications for substitutability and sustainability. Specifically, we generalize the standard constant-elasticity-of-substitution (CES) utility specification for manufactured goods and environmental services, by adding a subsistence requirement for environmental services. We find that the Hicksian elasticity of substitution strictly monotonically increases with the consumption of environmental services above the subsistence requirement, and approaches the standard CES value as consumption becomes very large. Whether the two goods are market substitutes depends on the level of income. We further show that the subsistence requirement may jeopardize the existence of an intertemporally optimal and sustainable consumption path. Our results have important implications for growth, development and environmental policy.

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Notes

  1. Applying critical-level utilitarianism (Blackorby et al. 1995) for social decision making with endogenous population size, the utility level \(\inf \limits _{S> \bar{S},X\ge 0}U_h(S,X)\) is a natural candidate for the critical utility level—assuming that utility is cardinally measurable and unit-comparable across individuals, and with adequately defined level of consumption for the manufactured good, X.

  2. Although the CES function has originally been proposed as a production function, it is widely used as a utility function since Armington (1969). Note also that specification (3) is itself a special case of the ‘affinely homothetic’ S-branch utility tree (Brown and Heien 1972; Blackorby et al. 1978).

  3. For \(\theta =0\), we include into the definition of \(U_h\) the continuous extension of \(\left[ \alpha \left( S-\bar{S}\right) ^{\theta }+ (1-\alpha ) X^{\theta }\right] ^{1/\theta }\) for \(\theta \rightarrow 0\) which is \(\left( S-\bar{S}\right) ^{\alpha }X^{(1-\alpha )}\).

  4. Hicks (1932[1963]) and Robinson (1933) independently introduced the elasticity of substitution between two inputs to production, which has then been adapted to consumption goods by Hicks and Allen (1934a, b). For a generalization to more than two goods, see Blackorby and Russel (1989).

  5. This implies that the distinction between weak and strong sustainability may not only depend on exogenous parameters, but will become endogenous to the level of consumption, in particular of the environmental service.

  6. Except for \(\theta \rightarrow 1\), where the two are perfect market-substitutes irrespective of any other parameter values.

  7. Optimization problem (28) does have a solution for \(R_0\le \underline{R}\) and \(T>\bar{T}\), but this solution includes \(S_t<\bar{S}\) for some t.

  8. For \(R_0\le R_{\delta }\) this solution is such that \(S_t>0\) and \(\dot{S}_t>0\) for all t, that is, the optimal steady-state stock size \(R_{\delta }\) is approached from below through harvesting a positive and ever increasing (yet at a decreasing rate) amount \(S_t\) that asymptotically approaches the steady-state harvest rate. As a consequence, utility is increasing over time forever (“sustainable consumption”).

  9. Commonly, “weak sustainability” means non-declining utility over time, whereas “strong sustainability” means non-declining stocks over time (Neumayer 2010).

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Acknowledgments

We are grateful to Wolfgang Buchholz, Eli Fenichel, Reyer Gerlagh, Thomas Sterner, Christian Traeger, Rintaro Yamaguchi, two anonymous reviewers, as well as participants at the 2014 BIOECON and the 2015 EAERE conferences for helpful comments. Financial support from the German Federal Ministry of Education and Research under grant 01LA1104C is gratefully acknowledged. MD further thanks the German National Academic Foundation and the German Academic Exchange Service (DAAD) for funding.

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Correspondence to Stefan Baumgärtner.

Appendices

Appendix 1: Proof of Proposition 1

With utility function (3), the marginal rate of substitution for \(S>\bar{S}\) (Eq. 5) is

$$\begin{aligned} {\textit{MRS}} = \frac{\displaystyle \frac{\partial \,U_h(S,X)}{\partial \,S}}{\displaystyle \frac{\partial \,U_h(S,X)}{\partial \,X}} = \frac{\alpha }{1-\alpha } \left[ \frac{ S- \bar{S}}{ X} \right] ^{ \theta -1} = \frac{\alpha }{1-\alpha } \left[ \frac{S}{X}\left( 1-\frac{\bar{S}}{S}\right) \right] ^{\theta -1}\ , \end{aligned}$$
(35)

so that

$$\begin{aligned} \frac{{\textit{MRS}}}{(X/S)}= \frac{\alpha }{1-\alpha } \left( \frac{S}{X}\right) ^{\theta }\left( 1-\frac{\bar{S}}{S}\right) ^{\theta -1} \end{aligned}$$
(36)

and

$$\begin{aligned} \frac{d\,{\textit{MRS}}}{d\,(X/S)}= & {} \frac{\alpha (\theta -1)}{1-\alpha } \left[ \frac{S}{X}\left( 1-\frac{\bar{S}}{S}\right) \right] ^{\theta -2} \left\{ \frac{d(S/X)}{d(X/S)}\left( 1-\frac{\bar{S}}{S}\right) + \frac{S}{X}\frac{d\left( 1-\frac{\bar{S}}{S}\right) }{d(X/S)}\right\} \end{aligned}$$
(37)
$$\begin{aligned}= & {} \frac{\alpha (\theta -1)}{1-\alpha } \left[ \frac{S}{X}\left( 1-\frac{\bar{S}}{S}\right) \right] ^{\theta -2} \left\{ -\left( \frac{X}{S}\right) ^{-2}\left( 1-\frac{\bar{S}}{S}\right) + \frac{S}{X}\frac{d\left( 1-\frac{\bar{S}}{S}\right) }{d(X/S)}\right\} .\quad \end{aligned}$$
(38)

With (36) and (38), the elasticity of substitution (4) becomes

$$\begin{aligned} \sigma (S,X)= & {} \frac{{\textit{MRS}}}{X/S}\ \frac{d\,(X/S)}{d\,{\textit{MRS}}} = \frac{{\textit{MRS}}}{X/S}\ \left( \frac{d\,{\textit{MRS}}}{d\,(X/S)}\right) ^{-1} \end{aligned}$$
(39)
$$\begin{aligned}= & {} \left( \frac{S}{X}\right) ^{\theta }\left( 1-\frac{\bar{S}}{S}\right) ^{\theta -1} \frac{1}{\theta -1}\ \left[ \frac{S}{X}\left( 1-\frac{\bar{S}}{S}\right) \right] ^{2-\theta }\nonumber \\&\times \left\{ -\left( \frac{X}{S}\right) ^{-2}\left( 1-\frac{\bar{S}}{S}\right) + \frac{d\left( 1-\frac{\bar{S}}{S}\right) }{d(X/S)}\frac{S}{X}\right\} ^{-1}\end{aligned}$$
(40)
$$\begin{aligned}= & {} \frac{1}{1-\theta }\left\{ 1-\frac{\frac{X}{S}}{\left( 1-\frac{\bar{S}}{S}\right) } \frac{d\left( 1-\frac{\bar{S}}{S}\right) }{d(X/S)}\right\} ^{-1}\ . \end{aligned}$$
(41)

To calculate the remaining derivative, we transform the problem from the standard variables (SX) into the following variables (wv):

$$\begin{aligned} w&:=\frac{X}{S}\end{aligned}$$
(42)
$$\begin{aligned} v&:= \alpha \left( S-\bar{S}\right) ^\theta + (1-\alpha ) X^\theta , \end{aligned}$$
(43)

where w is the ratio of the two consumption goods and v is a monotonic transformation of the utility function, so that \(v={\textit{constant}}\) is equivalent to \(U(S,X)={\textit{constant}}\). Derivatives under the constraint \(U(S,X)={\textit{const}}.\), i.e. along an indifference curve, are now taken along \(v={\textit{const}}.\), or \(dv=0\).

From (43), using (42), we have

$$\begin{aligned} \left( 1-\frac{\bar{S}}{S}\right)&=\left[ \frac{v}{\alpha S^\theta }- \frac{1-\alpha }{\alpha } w^\theta \right] ^{\frac{1}{\theta }} \ , \end{aligned}$$
(44)

such that

$$\begin{aligned}&\displaystyle \frac{d\left( 1-\frac{\bar{S}}{S}\right) }{d(X/S)} =\frac{d\left[ \frac{v}{\alpha S^\theta }- \frac{1-\alpha }{\alpha } w^\theta \right] ^{\frac{1}{\theta }}}{dw}\end{aligned}$$
(45)
$$\begin{aligned}&\displaystyle =-\left[ \frac{v}{\alpha S^\theta }- \frac{1-\alpha }{\alpha } w^\theta \right] ^{\frac{1}{\theta }-1}\,\left( \frac{v}{\alpha S^{\theta +1}}\,\frac{dS}{dw}+ \frac{1-\alpha }{\alpha }w^{\theta -1}\right) \end{aligned}$$
(46)

Totally differentiating (43), and using \(dv=0\), yields

$$\begin{aligned}&0&=\theta \,\left[ \alpha \left( S-\bar{S}\right) ^{\theta -1}+ (1-\alpha )w^\theta \, S^{\theta -1}\right] \,dS+\theta (1-\alpha )\,S^\theta \,w^{\theta -1}\,dw \end{aligned}$$
(47)
$$\begin{aligned} \Leftrightarrow&\frac{dS}{dw}&=-\frac{(1-\alpha )S^\theta \,w^{\theta -1}}{\alpha \left( S-\bar{S}\right) ^{\theta -1}+(1-\alpha )w^\theta \, S^{\theta -1}} \end{aligned}$$
(48)

Using (46) and (48), we have:

$$\begin{aligned}&-\frac{\frac{X}{S}}{\left( 1-\frac{\bar{S}}{S}\right) } \frac{d\left( 1-\frac{\bar{S}}{S}\right) }{d(X/S)}\nonumber \\&\quad = \frac{\frac{X}{S}}{\left( 1-\frac{\bar{S}}{S}\right) }\,\left[ \frac{v}{\alpha S^\theta }- \frac{1-\alpha }{\alpha } w^\theta \right] ^{\frac{1}{\theta } -1}\,\left( \frac{v}{\alpha S^{\theta +1}}\,\frac{dS}{dw}+ \frac{1-\alpha }{\alpha }w^{\theta -1}\right) \end{aligned}$$
(49)
$$\begin{aligned}&\quad =-\frac{X}{S}\,\left( 1-\frac{\bar{S}}{S}\right) ^{-\theta }\,\left( \frac{v}{\alpha S^{\theta +1}}\,\frac{(1-\alpha )S^\theta \,w^{\theta -1}}{\alpha \left( S-\bar{S}\right) ^{\theta -1}+(1-\alpha ) S^{\theta -1}\,w^\theta }-\frac{1-\alpha }{\alpha }w^{\theta -1}\right) \end{aligned}$$
(50)
$$\begin{aligned}&\quad =-\left( \frac{X}{S}\right) ^\theta \,\left( 1-\frac{\bar{S}}{S}\right) ^{-\theta }\, \left[ \frac{v(1-\alpha )}{\alpha ^2 S\,\left( S-\bar{S}\right) ^{\theta -1}+\alpha (1-\alpha ) X^{\theta }}- \frac{1-\alpha }{\alpha } \right] \end{aligned}$$
(51)
$$\begin{aligned}&\quad =-X^\theta \,(S-\bar{S})^{-\theta }\, \left[ \frac{v(1-\alpha ) - \alpha (1-\alpha ) S\,\left( S-\bar{S}\right) ^{\theta -1} - (1-\alpha )^2 X^{\theta }}{\alpha ^2 S\,\left( S-\bar{S}\right) ^{\theta -1}+\alpha (1-\alpha ) X^{\theta }} \right] \end{aligned}$$
(52)
$$\begin{aligned}&\quad =-X^\theta \,\left( S-\bar{S}\right) ^{-\theta }\, \frac{(1-\alpha )\left( S-\bar{S}\right) ^{\theta }- (1-\alpha ) S \left( S-\bar{S}\right) ^{\theta -1}}{\alpha S\,\left( S-\bar{S}\right) ^{\theta -1}+ (1-\alpha ) X^{\theta }}\end{aligned}$$
(53)
$$\begin{aligned}&\quad =(1-\alpha )X^\theta \,\frac{\frac{S}{S-\bar{S}}-1}{\alpha S \left( S-\bar{S}\right) ^{\theta -1}+ (1-\alpha )X^\theta } . \end{aligned}$$
(54)

Plugging this into (41) yields

$$\begin{aligned} \sigma (S,X)&=\frac{1}{1-\theta }\left\{ 1-\frac{\frac{X}{S}}{\left( 1-\frac{\bar{S}}{S}\right) } \frac{d\left( 1-\frac{\bar{S}}{S}\right) }{d(X/S)}\right\} ^{-1}\end{aligned}$$
(55)
$$\begin{aligned}&=\frac{1}{1-\theta }\left\{ 1+(1-\alpha )X^\theta \,\frac{\frac{S}{S-\bar{S}}-1}{\alpha S \left( S-\bar{S}\right) ^{\theta -1}+ (1-\alpha )X^\theta } \right\} ^{-1}\end{aligned}$$
(56)
$$\begin{aligned}&=\frac{1}{1-\theta }\,\left[ \frac{\alpha S\left( S-\bar{S}\right) ^{\theta -1}+(1-\alpha )X^\theta }{\alpha S\left( S-\bar{S}\right) ^{\theta -1}+(1-\alpha )X^\theta \,\frac{S}{S-\bar{S}}}\right] \end{aligned}$$
(57)
$$\begin{aligned}&=\frac{1}{1-\theta }\,\left[ \frac{\alpha \left( \frac{S-\bar{S}}{X}\right) ^{\theta -1}+(1-\alpha )\frac{X}{S}}{\alpha \left( \frac{S-\bar{S}}{X}\right) ^{\theta -1}+(1-\alpha )\frac{X}{S-\bar{S}}}\right] \end{aligned}$$
(58)
$$\begin{aligned}&= \frac{1}{1-\theta }\,\left[ 1-\frac{ (1-\alpha )\displaystyle \frac{\bar{S}}{S}}{\alpha \displaystyle \left[ \frac{S-\bar{S}}{X}\right] ^{\theta }+(1-\alpha )}\right] . \end{aligned}$$
(59)

Appendix 2: Proof of Proposition 2

Results (8), (9), (11), (12), (14) and (15) can easily be verified.

Proof of Result (10):

$$\begin{aligned} \frac{d\,\sigma }{d\, S}= \frac{1}{1-\theta }\, \frac{1-\alpha \bar{S}}{S^2}\,\frac{\left[ \displaystyle \alpha \left( \frac{\theta S}{X} \left[ \frac{S-\bar{S}}{X}\right] ^{\theta -1}+ \displaystyle \left[ \frac{S-\bar{S}}{X}\right] ^{\theta }-1 \right) +1 \right] }{\left( 1-\alpha + \alpha \displaystyle \left[ \frac{S-\bar{S}}{X}\right] ^{\theta }\right) ^2} >0\quad \text { for }\theta \ge 0. \end{aligned}$$
(60)

Proof of Result (13):

$$\begin{aligned} \frac{d\,\sigma }{d\,\bar{S}}= \frac{1}{\theta -1}\, \frac{1-\alpha }{S}\,\frac{\left[ \displaystyle \frac{\theta \alpha \bar{S}}{X} \left[ \frac{S-\bar{S}}{X}\right] ^{\theta -1}+ (1-\alpha )+ \alpha \displaystyle \left[ \frac{S-\bar{S}}{X}\right] ^{\theta } \right] }{\left( 1-\alpha + \alpha \displaystyle \left[ \frac{S-\bar{S}}{X}\right] ^{\theta }\right) ^2} <0\quad \text { for }\theta \ge 0. \end{aligned}$$
(61)

Appendix 3: Proof of Proposition 3

The consumer requires \(m= p_S \bar{S}\) to meet her subsistence needs \(\bar{S}\). This means that up to this level of income she is not willing to substitute S for X, i.e. \(\sigma ^{\star }=0\). For \(m> p_S \bar{S}\), she faces the utility maximization problem

$$\begin{aligned} {\max \limits _{S, X} \, U_h(S,X)} \qquad \text {s.t.} \qquad p_S S + p_X X \le m\ . \end{aligned}$$
(62)

The Lagrangian and first-order conditions are:

$$\begin{aligned} {\mathcal {L}}(S,X, \mu )= & {} \left[ \alpha \left( S-\bar{S}\right) ^{\theta }+ (1-\alpha ) X^{\theta }\right] ^{1/\theta } + \, \mu (m - p_S S - p_X X ) \end{aligned}$$
(63)
$$\begin{aligned} \frac{\partial \, {\mathcal {L}}}{\partial \, S}= & {} 0 \ \Leftrightarrow \ \alpha (S- \bar{S})^{(\theta -1)} \left[ \alpha (S- \bar{S})^{\theta } + (1- \alpha ) X^{\theta } \right] ^{(1/\theta -1)} = \mu p_S\end{aligned}$$
(64)
$$\begin{aligned} \frac{\partial \, {\mathcal {L}}}{\partial \, X}= & {} 0 \ \Leftrightarrow \ (1-\alpha ) X^{(\theta -1)} \left[ \alpha (S- \bar{S})^{\theta } + (1- \alpha ) X^{\theta } \right] ^{(1/\theta -1)} = \mu p_X \end{aligned}$$
(65)
$$\begin{aligned} \frac{\partial \, \mathcal { L}}{\partial \,\mu }= & {} 0 \ \Leftrightarrow \ p_S S + p_X X = m \end{aligned}$$
(66)

From conditions (64) and (65), we obtain

$$\begin{aligned} \frac{\alpha }{1-\alpha } \left[ \frac{ (S- \bar{S})}{ X} \right] ^{\theta -1} = \, \frac{p_S}{p_X} . \end{aligned}$$
(67)

Rearranging gives

$$\begin{aligned} X = (S- \bar{S}) \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} }\ . \end{aligned}$$
(68)

Inserting (68) into (66) and solving for S yields the Marshallian demand function

$$\begin{aligned} S^{\star }(m, p_S, p_X)= \frac{m+ p_X \bar{S} \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } }{ p_S + p_X \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } }\ . \end{aligned}$$
(69)

Inserting (69) into (68) yields the Marshallian demand function

$$\begin{aligned} X^{\star }(m, p_S, p_X) = \frac{m- p_S \bar{S} }{ p_X + p_S \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ -\frac{1}{ \theta -1} } }\ . \end{aligned}$$
(70)

Inserting (69) and (70) into Eq. (7) yields the Hicksian elasticity of substitution in the utility-optimal allocation (\(X^{\star }\), \(S^{\star }\)):

$$\begin{aligned} \displaystyle \displaystyle \sigma ^{\star }= & {} \displaystyle \frac{1}{1-\theta }\,\left[ 1-\frac{ \displaystyle (1-\alpha )\,\bar{S} \, \, \displaystyle \frac{ p_S + p_X \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } }{m + p_X \bar{S} \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } } }{(1-\alpha )+\alpha \displaystyle \left[ \frac{\displaystyle \frac{m+ p_X \bar{S} \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } }{ p_S + p_X \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } } -\bar{S}}{\displaystyle \frac{m- p_S \bar{S} }{ p_X + p_S \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{-1}{ \theta -1} } } }\right] ^{\theta }}\right] \end{aligned}$$
(71)
$$\begin{aligned} \displaystyle \displaystyle= & {} \displaystyle \frac{1}{1-\theta }\,\left[ 1-\frac{ (1-\alpha )\,\bar{S} \, \, \left( p_S + p_X \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } \right) }{ \left( m + p_X \bar{S} \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } \right) \left( \alpha \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{-\theta }{ \theta -1} } +(1-\alpha ) \right) } \right] \end{aligned}$$
(72)
$$\begin{aligned}= & {} \frac{1}{1-\theta }\,\left[ \frac{m \left( (1-\alpha ) + \alpha \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{-\theta }{ \theta -1} } \right) }{ (1-\alpha ) \bar{S} \left( p_S + p_X \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } \right) + m \left( (1-\alpha ) + \alpha \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{-\theta }{ \theta -1} } \right) } \right] \ .\nonumber \\ \end{aligned}$$
(73)

Solving the equation \(\sigma ^{\star }=1\) for \(m=m^P\) we obtain (22).

From Eq. (73) it can easily be seen that as income m goes to infinity, the elasticity of substitution \(\sigma ^{\star }\) approaches standard CES result in absence of a subsistence requirement:

$$\begin{aligned} \displaystyle \sigma ^{\star } \rightarrow \frac{1}{1-\theta } \quad \text {for}\ m\rightarrow \infty \ . \end{aligned}$$
(74)

Furthermore, the elasticity of substitution monotonically increases with income,

$$\begin{aligned} \frac{d\,\sigma ^{\star }}{d\, m}= \, \frac{1}{1-\theta }\, \left[ \frac{ (1-\alpha ) \bar{S} \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{2\theta +1}{ \theta -1} } \left( p_X + p_S \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } \right) \left( (1-\alpha ) + \alpha \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{\theta }{ \theta -1} } \right) }{ \left( \alpha m + (1-\alpha ) \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{\theta }{ \theta -1} } \left( m + p_S \bar{S} + p_X \bar{S} \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1}} \right) \right) ^2 } \right] >0. \end{aligned}$$
(75)

and decreases with an increasing subsistence requirement

$$\begin{aligned} \frac{d\,\sigma ^{\star }}{d\, \bar{S}}= \, \frac{1}{\theta -1}\, \left[ \frac{ (1-\alpha ) m \left( p_S + p_X \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } \right) \left( (1-\alpha ) + \alpha \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{\theta }{ \theta -1} } \right) }{ \left( \alpha m \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{\theta }{ \theta -1} } + (1-\alpha ) \left( m + p_S \bar{S} + p_X \bar{S} \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1}} \right) \right) ^2} \right] <0. \end{aligned}$$
(76)

Appendix 4: Proof of Proposition 4

The cross-price derivatives of the Marshallian demand functions for S and X are obtained from Eqs. (69) and (70) and have the following properties:

$$\begin{aligned} \frac{d\,S^{\star }}{d\, p_X}= & {} \frac{\theta }{\theta -1}\, \left[ \frac{ ( p_S \bar{S} - m) \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1} } }{ \left( p_S + p_X \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{1}{ \theta -1}} \right) ^2 } \right] \quad \gtreqqless \, 0 \quad \text {for} \quad \theta \gtreqqless 0 \end{aligned}$$
(77)
$$\begin{aligned} \frac{d\,X^{\star }}{d\, p_S}= & {} \frac{1}{\theta -1}\, \left[ \frac{ \bar{S} \left( (1-\theta ) p_X + p_S \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{-1}{ \theta -1} } \right) - \theta m \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{-1}{ \theta -1} } }{ \left( p_X + p_S \left[ \frac{\alpha }{1-\alpha } \frac{p_X}{p_S} \right] ^{ \frac{-1}{ \theta -1}} \right) ^2 } \right] , \end{aligned}$$
(78)

with

$$\begin{aligned} \frac{d\,X^{\star }}{d\, p_S}&\left\{ \begin{array}{r} \rightarrow \infty \\ =-(1-\alpha )\frac{\overline{S}}{p_X} < 0 \\ < 0\end{array}\right\} \,&\quad \text {for} \quad \theta \begin{Bmatrix} \rightarrow 1 \\= 0 \\ < 0 \end{Bmatrix}. \end{aligned}$$
(79)

Solving the equation \(dX^{\star }/dp_S=0\) for m, we obtain (26). It is further easy to verify that \(\frac{d^2 X^{\star }}{dp_S dm}>0\) for \(\theta >0\).

Appendix 5: Proof of Proposition 5

Use \(S_t^\star \) to denote the solution of the optimization problem (28) that satisfies \(S_t^\star >\bar{S}\) for all \(t\le T\), and \(R_t^\star \) to denote the corresponding development of the resource stock. Then \(R_t^\star \le \hat{R}_t\) where the dynamics of \(\hat{R}_t\) are described by \(\dot{R}_t=g(R_t)-\bar{S}\) with \(R_0^\star =\hat{R}_0=R_0\) given. Furthermore,

$$\begin{aligned} \frac{\partial \hat{R}_0}{\partial t}=g\left( \hat{R}_0\right) -\bar{S}<0 \end{aligned}$$
(80)

and hence

$$\begin{aligned} \frac{\partial \hat{R}_t}{\partial t}&=g\left( \hat{R}_t\right) -\bar{S}<0 \end{aligned}$$
(81)
$$\begin{aligned} \frac{\partial ^2\hat{R}_t}{\partial t^2}&=g'\left( \hat{R}_t\right) \, \frac{\partial \hat{R}_t}{\partial t}<0. \end{aligned}$$
(82)

Thus there must be some \(\hat{T}<\infty \) where \(\hat{R}_t=0\), and hence \(R_t^\star =0\) for all \(t>\hat{T}\). This contradicts the assumption \(S_t^\star >\bar{S}\) for all \(t>\bar{T}\) with \(\bar{T}\le \hat{T}\).

Appendix 6: Proof of Proposition 6

Because of Proposition 5 (statement 1), an optimal solution where the subsistence requirement is exceeded forever, i.e. with \(T\rightarrow \infty \), exists only for \(R_0>\underline{R}\). In this case, Proposition 5 (statement 2b) implies that if \(g(R_{\delta })<\underline{R}\) the optimal harvest rate is \(S_t=\bar{S}=const.\) and the optimal steady state stock level is \(\underline{R}\), which is monotonically approached (Acemoglu 2009). Thus, the subsistence requirement is not exceeded at any time.

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Baumgärtner, S., Drupp, M.A. & Quaas, M.F. Subsistence, Substitutability and Sustainability in Consumption. Environ Resource Econ 67, 47–66 (2017). https://doi.org/10.1007/s10640-015-9976-z

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