Renewable Resource Use and Nonseparable Amenity Benefits

Abstract

We incorporate amenity benefits into an overlapping generations model with a renewable resource as a factor of production, source of amenity benefits and store of value. Unlike the conventional renewable resource problems studied under the assumption of additive consumption and amenity benefits, we let amenity benefits affect the utility of consumers in a nonseparable fashion. We examine the role that weights given to consumption and amenities have for harvesting and the resource stock. We characterize dynamics and stability of steady state equilibria with a logistic resource growth function. We demonstrate in parametric and numerical models that the weights given to consumption and amenities in the utility function matter substantially for the steady state equilibrium stock and its stability and dynamics. Both conventional saddle point equilibria and indeterminacy with infinite number of equilibria and saddle-node bifurcation is possible depending on the weights given to consumption and amenities. In addition, we show that for each inefficient equilibrium stock, there is a unique subsidy rate that can move the economy from an inefficient equilibrium to an efficient one. The presence of indeterminacy provides a challenge to resource policies, because the system becomes unpredictable. Therefore, expectations and market psychology may play an important role in resource utilization and provision of amenities.

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Notes

  1. 1.

    There are only a few exceptions where environmental goods and consumption are examined using nonseparable preferences. Bockstael and McConnel (1993) study the implications of nonseparablility for environmental valuation but resource use is not studied. Carbone and Smith (2008) examine the choice of environmental taxes when amenities in the form of air quality enter utility in a non-separable way. Carbone and Smith (2013) examine the demand for environmental quality when ecosystem services contribute to utility in a non-separable way. Both of these papers have a static framework. Kim (2002) demonstrates that environmental taxes typically exacerbate pre-existing tax distortions with nonseparable preferences. In this literature, the role of natural resource capital is not made explicit, however. An example of non-separability in dynamic, non-environmental setting is, for instance, the one sector growth model by Shi (1994).

  2. 2.

    The majority of the Finnish forest landowners produces timber and other forest amenities by choosing ecologically suitable management regimes to promote recreation berry picking, hunting, nature protection and aesthetic values, as shown for instance in Kuuluvainen et al. (1996), and Karppinen (2000). Also, almost all the landowners have adopted voluntary certification of their forests with a special emphasis on forest landscape and forest-related biodiversity conservation. Two recent surveys of empirical analyses of nonindustrial forest landowner behavior by Amacher et al. (2003) and Beach et al. (2005), also show that landowner preferences for nontimber forest benefits have historically been a critical factor in harvesting decisions.

  3. 3.

    Koskela et al. (2002) study an OG model with a renewable resource, but amenity benefits to consumers from the stored resource are not assumed to be present.

  4. 4.

    It would be of interest in our model to analyze the role of the forests as bequests, say, along the lines presented by Weil (1987). See also Amacher et al. (2002), who examine bequeathing in the form of both timber and money.

  5. 5.

    The marginal return of the resource, \({g}'\), depends on the level of the stock. This reflects the fact that a renewable resource, like fish stocks, differs markedly from a conventional asset, whose return in a competitive environment is independent of the amount invested.

  6. 6.

    The cash flow going through the firm is \(p_t x_{t+1} +F(H_t ,L_t )-p_t h_t -w_t L_t -p_t \left[ x_t +g(x_t)\right] \).

    Given the transition equation (6b), maximization of profit leads to conditions presented in (6d).

  7. 7.

    The logistic growth curve is a standard specification of growth in resource economics literature; see e.g. a survey by Brown (2000).

  8. 8.

    The derivation of (12) is rather tedious. We produce the necessary steps in “Appendix”.

  9. 9.

    It can be seen from (9) above that the first period consumption in a steady state is nonnegative, if \((1-\alpha )h-\alpha x\ge 0\). With logistic growth this means that \(a-(1/2)bx-\alpha (1+a-(1/2)bx)\ge 0\).

  10. 10.

    Galor (2007) is a source for a two-dimensional dynamics in economics. In particular, for our analysis we have relied on his Figure 3.19.

  11. 11.

    Efficiency outside the steady states is a more subtle issue. Cass (1972) developed a general criterion to check the inefficiency; for an exposition of his criterion in OG models, see e.g. p. 272–273 in Azariadis (1993).

  12. 12.

    Note that if \({\upalpha } = 0.2\), the steady-state stock reduces dramatically and locates between 190 and 320 cubic meters depending on the value of \({\upgamma }\). Thus, all equilibria locate to the left of the MSY point.

  13. 13.

    For details see Galor (2007) p. 88–90, and especially his Fig. 3.19.

  14. 14.

    Note that when \({\upalpha }= 0.05\) the equilibrium is always a sink (proof available from the authors).

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Appendix

Appendix

The derivation of Eq. (12) with the general growth function:

We first reproduce Eq. (8) here

$$\begin{aligned}&f^{\prime }(h_t )u_1 \left[ {f(h_t )-h_t f^{\prime }(h_t )-f^{\prime }(h_t )x_{t+1} ,x_t } \right] \nonumber \\&\quad = f^{\prime }(h_{t+1} )(1+g^{\prime }(x_{t+1} ))\beta u_1 \left[ {f^{\prime }(h_{t+1})(x_{t+1} +g(x_{t+1} )),x_{t+1} } \right] \nonumber \\&\quad \quad +\,\beta u_2 \left[ {f^{\prime }(h_{t+1} )(x_{t+1} +g(x_{t+1} ),x_{t+1}}\right] \end{aligned}$$
(35)

With the utility function \(u(c,x)=c^{\gamma }x^{1-\gamma }\) with \(0<\gamma <1\), (35) can be written as

$$\begin{aligned} \frac{\gamma f^{\prime }(h_t )(c_1^t )^{\gamma }x_t ^{1-\gamma }}{c_1^t }=\gamma f^{\prime }(h_{t+1} )(1+g^{\prime }(x_{t+1} ))\frac{\beta (c_2^t )^{\gamma }x_{t+1} ^{1-\gamma }}{c_2^t }+\frac{\beta (1-\gamma )(c_2^t )^{\gamma }x_{t+1} ^{1-\gamma }}{x_{t+1}}.\nonumber \\ \end{aligned}$$
(36)

We now proceed step by step substituting first for consumptions in the denominators to get

$$\begin{aligned} \frac{\gamma f^{\prime }(h_t )(c_1^t )^{\gamma }x_t ^{1-\gamma }}{f(h_t )-h_t f^{\prime }(h_t )-f^{\prime }(h_t )x_{t+1}}= & {} \gamma f^{\prime }(h_{t+1} )(1+g^{\prime }(x_{t+1} ))\frac{\beta (c_2^t )^{\gamma }x_{t+1}^{1-\gamma }}{f^{\prime }(h_{t+1} )(x_{t+1} +g(x_{t+1} ))}\nonumber \\&+\frac{\beta (1-\gamma )(c_2^t )^{\gamma }x_{t+1} ^{1-\gamma }}{x_{t+1}} \end{aligned}$$
(37)

Cancelling terms out and rearranging we get

$$\begin{aligned} \frac{\gamma (c_1^t )^{\gamma }x_t ^{1-\gamma }}{\frac{f(h_t )}{f^{\prime }(h_t )}-h_t -x_{t+1} }=\frac{\gamma (1+g^{\prime }(x_{t+1} ))\beta (c_2^t )^{\gamma }x_{t+1} ^{1-\gamma }}{(x_{t+1} +g(x_{t+1} ))}+\frac{\beta (1-\gamma )(c_2^t )^{\gamma }x_{t+1} ^{1-\gamma }}{x_{t+1} }. \end{aligned}$$
(38)

Now we assume the following production function \(f(h)=h^{\alpha }\) with \(0<\alpha <1\). Then we get

$$\begin{aligned} \frac{\alpha \gamma }{(1-\alpha )h_t -\alpha x_{t+1} }(c_1^t )^{\gamma }x_t ^{1-\gamma }=\beta \left[ {\gamma \frac{1+g^{\prime }(x_{t+1} )}{x_{t+1} +g(x_{t+1} )}+\frac{1-\gamma }{x_{t+1} }} \right] (c_2^t )^{\gamma }x_{t+1} ^{1-\gamma }. \end{aligned}$$
(39)

We recall the transition equation \(x_{t+1} =x_t +g(x_t )-h_t\). We solve for the harvest, and substitute it to (39). We do that first on the left-hand side of (39) to get

$$\begin{aligned} \frac{\alpha \gamma }{(1-\alpha )(x_t +g(x_t )-x_{t+1} )-\alpha x_{t+1} }(c_1^t )^{\gamma }x_t ^{1-\gamma }=\beta \left[ {\gamma \frac{1+g^{\prime }(x_{t+1} )}{x_{t+1} +g(x_{t+1} )}+\frac{1-\gamma }{x_{t+1}}}\right] (c_2^t )^{\gamma }x_{t+1} ^{1-\gamma },\nonumber \\ \end{aligned}$$
(40)
Table 3 Land area under FSC certification in millions of hectares

which by cancelling terms yields

$$\begin{aligned} \frac{\alpha \gamma }{(1-\alpha )(x_t +g(x_t ))-x_{t+1} }(c_1^t )^{\gamma }x_t^{1-\gamma }=\beta \left[ {\gamma \frac{1+g^{\prime }(x_{t+1})}{x_{t+1}+g(x_{t+1} )}+\frac{1-\gamma }{x_{t+1}}}\right] (c_2^t)^{\gamma }x_{t+1}^{1-\gamma }.\nonumber \\ \end{aligned}$$
(41)

We now use the budget constraints (11a) and (11b) from the text, and plug them into (41) to get

$$\begin{aligned}&\frac{\alpha \gamma }{(1-\alpha )(x_t +g(x_t ))-x_{t+1} }\left( \left[ {(1-\alpha )(x_t +g(x_t ))-x_{t+1} } \right] (x_t +g(x_t )-x_{t+1} )^{\alpha -1}\right) ^{\gamma }x_t ^{1-\gamma }\nonumber \\&\quad =\alpha ^{\gamma }\beta \left[ {\gamma \frac{1+g^{\prime }(x_{t+1})}{x_{t+1}+g(x_{t+1})}+\frac{1-\gamma }{x_{t+1}}}\right] \left[ {x_{t+1}+g(x_{t+1})-x_{t+2}} \right] ^{(\alpha -1)\gamma }\left[ {x_{t+1} +g(x_{t+1})} \right] ^{\gamma }x_{t+1} ^{1-\gamma }.\nonumber \\ \end{aligned}$$
(42)

We collect terms and rewrite

$$\begin{aligned}&\frac{\alpha \gamma x_t ^{1-\gamma }}{\left[ {(1-\alpha )(x_t +g(x_t ))-x_{t+1}}\right] ^{1-\gamma }}(x_t +g(x_t )-x_{t+1} )^{(\alpha -1)\gamma }\nonumber \\&\quad =\alpha ^{\gamma }\beta \left[ {\gamma \frac{1+g^{\prime }(x_{t+1} )}{x_{t+1}+g(x_{t+1})}+\frac{1-\gamma }{x_{t+1} }} \right] \left[ {x_{t+1} +g(x_{t+1})-x_{t+2}} \right] ^{(\alpha -1)\gamma }\left[ {x_{t+1}+g(x_{t+1})} \right] ^{\gamma }x_{t+1} ^{1-\gamma }.\nonumber \\ \end{aligned}$$
(43)

This is a second-order nonlinear difference equation, which governs the behavior of the stock over time. Rewriting (43) slightly we obtain Eq. (12) in the text (Table 3).

The proof of Corollary 1

We reproduce the condition, \(\textit{RHS}(0)>\textit{LHS}(0)\), here with a slight abuse of notation as \(\textit{LHS}(0;\gamma )\equiv \alpha ^{1-\gamma }\gamma \left( {\frac{1+a}{a-\alpha (1+a)}} \right) ^{1-\gamma }<\alpha ^{\gamma }\beta (1+a)\equiv \textit{RHS}(0;\gamma )\). We first note that \(\textit{RHS}(0;0)=\beta (1+a)>0=\textit{LHS}(0;0), \frac{\lim }{\gamma \rightarrow 1}{} \textit{LHS}(0;\gamma )=1\) and \(\frac{\lim }{\gamma \rightarrow 1}{} \textit{RHS}(0;\gamma )=\alpha \beta (1+a)<1\). Thus we have \(\frac{\lim }{\gamma \rightarrow 1}{} \textit{RHS}(0;\gamma )<\frac{\lim }{\gamma \rightarrow 1}{} \textit{LHS}(0;\gamma )\). Since both \(\textit{RHS}(0;\gamma )\) and \(\textit{LHS}(0;\gamma )\) are continuous functions of \(\gamma \), there must exist at least one value of \(\hat{{\gamma }}\in (0,1)\) such that \(\textit{RHS}(0;\hat{{\gamma }})=\textit{LHS}(0;\hat{{\gamma }})\). We define \(\gamma ^{*}=\inf (\hat{{\gamma }})\). From above it follows that \(\gamma ^{*}>0\), and furthermore that \(\textit{LHS}(0;\gamma )>\textit{LHS}(0;\gamma )\) for all \(\gamma \in \left[ {0,\gamma ^{*}} \right) \). We differentiate \(\textit{RHS}_\gamma (0;\gamma )=\ln \alpha \left[ {\alpha ^{\gamma }\beta (1+a)} \right] <0\). Differentiating the left-hand side we get \(\textit{LHS}_\gamma (0;\gamma )=\left\{ {-\ln \alpha +\frac{1}{\gamma }-\ln \left[ {\frac{\alpha (1+a)}{a-\alpha (1+a)}} \right] } \right\} \alpha ^{1-\gamma }\gamma \left( {\frac{1+a}{a-\alpha (1+a)}} \right) ^{1-\gamma }\). \(\textit{LHS}_\gamma (0;\gamma )>0\), if \(\frac{1}{\gamma }-\ln \left[ {\frac{\alpha (1+a)}{a-\alpha (1+a)}} \right] >0\), and the fact that \(0<\alpha <1\). Since \(\textit{RHS}(0;0)>\textit{LHS}(0;0)\), there is thus a unique \(0<\gamma ^{**}<1\) such that \(\textit{RHS}(0;\gamma ^{**})=\textit{LHS}(0;\gamma ^{**})\) and for \(\gamma ^{**}<\gamma<~1 \quad \textit{RHS}(0;\gamma )<\textit{LHS}(0;\gamma )\). Q.E.D.

The characterization of the function \(\gamma =z(\alpha )\):

We first note the following facts:

$$\begin{aligned}&\frac{\lim }{\alpha \rightarrow a/(1+a)}z(\alpha )=0, \frac{\lim }{\alpha \rightarrow a/(2(1+a))}z(\alpha )=\infty \\&z^{\prime }(\alpha )=-\frac{1}{\left\{ {\ln \left[ {1\bigg /{\left( \frac{a}{\alpha (1+a)}-1\right) }}\right] }\right\} ^{2}}\left( {\frac{1}{\alpha }+\frac{1+a}{a-\alpha (1+a}} \right) <0 \end{aligned}$$

and

$$\begin{aligned} z''(\alpha )=+\frac{2}{\left\{ {\ln \left[ {1\bigg /{\left( \frac{a}{\alpha (1+a)}-1\right) }} \right] } \right\} ^{3}}\left( {\frac{1}{\alpha }+\frac{1+a}{a-\alpha (1+a}} \right) \left( {-\frac{1}{\alpha ^{2}}+\frac{(1+a)^{2}}{(a-\alpha (1+a))^{2}}} \right) . \end{aligned}$$

We rewrite the second derivative as

$$\begin{aligned} z''(\alpha )=+\frac{2}{\left\{ {\ln 1\bigg /{\left( \frac{a}{\alpha (1+a)}-1\right) }} \right\} ^{3}}\left( {\frac{1}{\alpha }+\frac{1+a}{a-\alpha (1+a}} \right) \left( {\frac{a(-a+2\alpha (1+a)}{\alpha ^{2}(a-\alpha (1+a))^{2}}} \right) . \end{aligned}$$

Since \(\alpha \) must lie on the open interval \(\left( {a/(2(1+a),a/(1+a)} \right) , z''(\alpha )>0\), the function \(z(\alpha )\) is decreasing and strictly convex as depicted in Fig. 2.

The partial derivatives for the equation (22):

We compute the partial derivatives from (22)

$$\begin{aligned} F_1 (x_{t+1} ,x_t )= & {} 1+a-bx_{t+1} -A^{\prime }(x_{t+1} )B(x_{t+1} ,x_t )-A(x_{t+1} )B_1 (x_{t+1} ,x_t ) \end{aligned}$$
(44)
$$\begin{aligned} F_2 (x_{t+1} ,x_t )= & {} -A(x_{t+1} )B_2 (x_{t+1} ,x_t ). \end{aligned}$$
(45)

What is then left to compute is the derivative, \(A^{\prime }(x_{t+1})\), and the partial derivatives \(B_1 (x_{t+1} ,x_t)\) and \(B_2 (x_{t+1}, x_t)\) from above.

We start the computations with \(A^{\prime }(x_{t+1})\), and after some manipulations end up with

$$\begin{aligned} A^{\prime }(x_{t+1})= & {} \beta ^{\frac{1}{(1-\alpha )\gamma }}\frac{1}{(1-\alpha )\gamma }\left\{ {\left[ {\frac{1+a-(1+\gamma )(1/2)bx_{t+1}}{\left[ {1+a-(1/2)bx_{t+1}}\right] ^{1-\gamma }}} \right] } \right\} ^{\frac{1-\gamma (1-\alpha )}{(1-\alpha )\gamma }}\nonumber \\&\times \left\{ {\frac{-2(1+a)+(1+\gamma )(1/2)bx_{t+1} }{\left[ {1+a-(1/2)bx_{t+1}}\right] ^{2-\gamma }}}\right\} (1/2)\gamma b. \end{aligned}$$
(46)

Given the carrying capacity, the resource can exhibit either positive or zero growth permanently over time, so that we can safely restrict the stock to provide nonnegative growth: \(g(x)=ax-(1/2)bx^{2}\ge 0\). It then follows that \(A^{\prime }(x_{t+1})\) is unambiguously negative. The following steps prove this assertion: \(g(x)=ax-(1/2)bx^{2}\ge 0\Rightarrow x\left[ a-(1/2)bx\right] \ge 0\Rightarrow -a+(1/2)bx\le 0\). We rewrite \(-2(1+a)+(1+\gamma )(1/2)bx= -2-a-a+(1/2)bx+\gamma (1/2)bx=-a+(1/2)bx-a+\gamma (1/2)bx-2.\) Since \(0<\gamma <1\), it is clear that \(-2(1+a)+(1+\gamma )(1/2)bx<0\). This shows that \(A^{\prime }(x_{t+1} )<0\), since \(a-(1/2)bx_{t+1}>0\) for all \(x<\tilde{x}\).

Next we compute \(B_1 (x_{t+1} ,x_t)\)

$$\begin{aligned} B_1 (x_{t+1} ,x_t )= & {} \left\{ {\frac{\alpha ^{1-\gamma }\gamma x_t ^{1-\gamma }}{\left[ {(1-\alpha )((1+a)x_t -(1/2)bx_t ^{2})-x_{t+1} }\right] ^{1-\gamma }}}\right\} ^{\frac{1}{(\alpha -1)\gamma }}\nonumber \\&\times \left\{ {\frac{1-\gamma }{(\alpha -1)\gamma }\frac{(1+a)x_t -(1/2)bx_t ^{2}-x_{t+1}}{(1-\alpha )((1+a)x_t -(1/2)bx_t ^{2})-x_{t+1} }-1} \right\} . \end{aligned}$$
(47)

We check out that \(B_1 (x_{t+1} ,x_t )<0\). The term, \((1-\alpha )((1+a)x_t -(1/2)bx_t ^{2})-x_{t+1} \), is the first period consumption, which must be positive at the optimum because of the Inada conditions in the utility function. Then it clearly follows that also the term, \((1+a)x_t -(1/2)bx_t ^{2}-x_{t+1} \), must be positive. Since \(\alpha <1, B_1 (x_{t+1} ,x_t )\) must be negative.

And finally we compute \(B_2 (x_{t+1} ,x_t )\)

$$\begin{aligned}&B_2 (x_{t+1} ,x_t )=\left\{ {\frac{\alpha ^{1-\gamma }\gamma x_t ^{1-\gamma }}{\left[ {(1-\alpha )((1+a)x_t -(1/2)bx_t ^{2})-x_{t+1} } \right] ^{1-\gamma }}} \right\} ^{\frac{1}{(\alpha -1)\gamma }}\left[ {1+a-bx_t } \right] \nonumber \\&\quad +\frac{1}{(\alpha -1)\gamma }\left\{ {\frac{\alpha ^{1-\gamma }\gamma x_t^{1-\gamma }}{\left[ {(1-\alpha )((1+a)x_t -(1/2)bx_t ^{2})-x_{t+1}}\right] ^{1-\gamma }}} \right\} ^{\frac{1+(1-\alpha )\gamma }{(\alpha -1)\gamma }} \left[ {(1+a)x_t -(1/2)bx_t ^{2}-x_{t+1}}\right] \nonumber \\&\quad \times \left\{ {\frac{(1-\gamma )\alpha ^{1-\gamma }\gamma x_t ^{-\gamma }}{\left[ {(1-\alpha )((1+a)x_t -(1/2)bx_t ^{2})-x_{t+1}} \right] ^{1-\gamma }}-\frac{(1-\gamma )((1-\alpha )((1+a)-bx_t )\alpha ^{1-\gamma }\gamma x_t^{1-\gamma }}{\left[ {(1-\alpha )((1+a)x_t -(1/2)bx_t ^{2})-x_{t+1}}\right] ^{2-\gamma }}}\right\} .\nonumber \\ \end{aligned}$$
(48)

It seems that we cannot sign this without further analysis/assumptions.

The evaluation of the functions A and B, and their derivatives in the steady state:

$$\begin{aligned} A(x^{*})=\beta ^{\frac{1}{(1-\alpha )\gamma }}\left\{ {\left[ {\frac{1+a-(1+\gamma )(1/2)bx^{*}}{\left[ {1+a-(1/2)bx^{*}} \right] ^{1-\gamma }}}\right] }\right\} ^{\frac{1}{(1-\alpha )\gamma }} \end{aligned}$$
(49a)
$$\begin{aligned} B(x^{*},x^{*})=\left\{ {\frac{\alpha ^{1-\gamma }\gamma }{\left[ {a-(1/2)bx^{*}-\alpha (1+a-(1/2)bx^{*})} \right] ^{1-\gamma }}\left( {\frac{1}{ax^{*}-(1/2)b(x^{*})^{2}}}\right) ^{(1-\alpha )\gamma }} \right\} ^{\frac{1}{(\alpha -1)\gamma }}.\nonumber \\ \end{aligned}$$
(49b)

The derivatives in steady state can be expressed as follows

$$\begin{aligned} A^{\prime }(x^{*})= & {} \beta ^{\frac{1}{(1-\alpha )\gamma }}\frac{1}{(1-\alpha )\gamma }\left\{ {\left[ {\frac{1+a-(1+\gamma )(1/2)bx^{*}}{\left[ {1+a-(1/2)bx^{*}} \right] ^{1-\gamma }}} \right] } \right\} ^{\frac{1-\gamma (1-\alpha )}{(1-\alpha )\gamma }}\nonumber \\&\times \left\{ {\frac{-2(1+a)+(1+\gamma )(1/2)bx^{*}}{\left[ {1+a-(1/2)bx^{*}} \right] ^{2-\gamma }}} \right\} (1/2)\gamma b \end{aligned}$$
(50a)
$$\begin{aligned} B_1 (x,x)= & {} \left\{ {\frac{\alpha ^{1-\gamma }\gamma x^{1-\gamma }}{\left[ {(1-\alpha )((1+a)x-(1/2)bx^{2})-x} \right] ^{1-\gamma }}} \right\} ^{\frac{1}{(\alpha -1)\gamma }}\nonumber \\&\quad \times \left\{ {\frac{1-\gamma }{(\alpha -1)\gamma }\frac{(1+a)x-(1/2)bx^{2}-x}{(1-\alpha )((1+a)x-(1/2)bx^{2})-x}-1} \right\} \end{aligned}$$
(50b)
$$\begin{aligned} B_2 (x,x)= & {} \left\{ {\frac{\alpha ^{1-\gamma }\gamma }{\left[ {(1-\alpha )((1+a)-(1/2)bx)-1} \right] ^{1-\gamma }}} \right\} ^{\frac{1}{(\alpha -1)\gamma }}\left[ {1+a-bx} \right] \nonumber \\&+\frac{1}{(\alpha -1)\gamma }\left\{ {\frac{\alpha ^{1-\gamma }\gamma }{\left[ {(1-\alpha )((1+a)-(1/2)bx)-1} \right] ^{1-\gamma }}} \right\} ^{\frac{1+(1-\alpha )\gamma }{(\alpha -1)\gamma }}\left[ {ax-(1/2)bx^{2}} \right] \nonumber \\&\times \left\{ {\frac{(1-\gamma )\alpha ^{1-\gamma }\gamma x}{\left[ {(1-\alpha )((1+a)-(1/2)bx)-1} \right] ^{1-\gamma }}+\frac{(1-\gamma )((1-\alpha )((1+a)-bx)\alpha ^{1-\gamma }\gamma x}{\left[ {(1-\alpha )((1+a)-(1/2)bx)-1} \right] ^{2-\gamma }}} \right\} .\nonumber \\ \end{aligned}$$
(50c)

The proof of Lemma 1

The first period consumption is

$$\begin{aligned} c_1^t =w_t -p_t x_{t+1}. \end{aligned}$$
(51)

Given the pricing relations (the subsidy included) this can be written as

$$\begin{aligned} c_1^t =f(h_t )-h_t f^{\prime }(h_t )-\frac{f^{\prime }(h_{t+1} )}{1-\mu _t }x_{t+1}. \end{aligned}$$
(52)

Manipulating this expression we obtain

$$\begin{aligned} c_1^t =f^{\prime }(h_t )\left[ {\frac{f(h_t )}{f^{\prime }(h_t )}-h_t -\frac{x_{t+1} }{1-\mu _t }} \right] . \end{aligned}$$
(53)

Given the production function, \(f(h)=h^{\alpha }\), and expressing (53) in a steady state we get

$$\begin{aligned} c_1 =\alpha h^{\alpha -1}\left[ {\frac{h^{\alpha }}{\alpha h^{\alpha -1}}-h-\frac{x}{1-\mu }} \right] , \end{aligned}$$
(54)

and finally we end up with

$$\begin{aligned} c_1 =h^{\alpha -1}\left[ {(1-\alpha )h-\frac{\alpha }{1-\mu }x} \right] . \end{aligned}$$
(55)

Since the optimal first period consumption is positive we have proved the claim. Furthermore, since\((1-\alpha )g(x)-\alpha x/(1-\mu )\ge 0\), the following holds for the subsidy rate

$$\begin{aligned} \mu \le 1-\frac{\alpha }{1-\alpha }\frac{x}{g(x)}. \end{aligned}$$
(56)

We define \(\bar{{\mu }}\) such that (56) holds as an equality.

The Proof of Proposition 2

We first note the following properties for the functions in (34):

$$\begin{aligned}&\textit{RHS}'(\mu )<0, \textit{RHS}''(\mu )<0\frac{\lim }{\mu \rightarrow 0}{} \textit{RHS} (\mu )>0, \frac{\lim }{\mu \rightarrow 1} \textit{RHS}(\mu )=0,\\&\quad \textit{LHS}^{\prime }(\mu )=(\gamma -1)\alpha \gamma \left[ {(1-\alpha )g(x)-\frac{\alpha }{1-\mu }x} \right] ^{\gamma -2}\left( {-\frac{\alpha }{(1-\mu )^{2}}x} \right) >0. \end{aligned}$$

The second derivative is

$$\begin{aligned} \textit{LHS}''(\mu )= & {} (\gamma -1)(\gamma -2)\alpha \gamma \left[ (1-\alpha ) g(x)-\frac{\alpha x}{1-\mu })\right] ^{\gamma -3}\left( \frac{\alpha x}{(1-\mu )^{2}}\right) ^{2}\\&+\,(\gamma -1)\alpha \gamma \left[ (1-\alpha ) g(x)-\frac{\alpha x}{1-\mu })\right] ^{\gamma -3} \left( -\frac{2\alpha x}{(1-\mu )^{3}}\right) \end{aligned}$$

which in turn yields \(\textit{LHS}''(\mu )>0\). Furthermore, we get, and \(\frac{\lim }{\mu \rightarrow \bar{{\mu }}}{} \textit{LHS}(\mu )=\infty \). It then follows that for any feasible level of the stock there is a unique subsidy rate. Figure 5 describes the behaviour of the equilibrium functions. Q.E.D.

Fig. 5
figure5

The uniqueness of the subsidy rate

The subsidy rate and the level of the stock are inversely related meaning that raising the subsidy will decrease the level of the stock. Differentiating the functions \(\textit{LHS}(\mu )\) and \(\textit{RHS}(\mu )\) with respect to x we get

$$\begin{aligned} \frac{\partial \textit{LHS}(\mu )}{\partial x}= & {} \alpha \gamma (\gamma -1)\left[ {(1-\alpha )g^{\prime }(x)-\frac{\alpha }{1-\mu }} \right] \left[ {(1-\alpha )g(x)-\frac{\alpha }{1-\mu }x} \right] ^{\gamma -2}>0\nonumber \\ \end{aligned}$$
(57)
$$\begin{aligned} \frac{\partial \textit{RHS}(\mu )}{\partial x}= & {} \gamma \alpha ^{\gamma }(1-\mu )^{1-\gamma }\beta \left[ {\gamma \frac{1+g^{\prime }(x)}{x+g(x)}+\frac{1-\gamma }{x}} \right] \left[ {x+g(x)} \right] ^{\gamma -1}\left[ {1+g^{\prime }(x)} \right] \nonumber \\&\quad +\,\,\alpha ^{\gamma }(1-\mu )^{1-\gamma }\beta \left[ {\gamma \frac{g'' (x)}{x+g(x)}-\gamma \frac{\left[ {1+g^{\prime }(x)} \right] ^{2}}{\left[ {x+g(x)} \right] ^{2}}-\frac{1-\gamma }{x^{2}}} \right] \left[ {x+g(x)} \right] ^{\gamma }.\nonumber \\ \end{aligned}$$
(58)

Collecting the joint terms, and cancelling we can rewrite (58) as

$$\begin{aligned} \frac{\partial \textit{RHS}(\mu )}{\partial x}=\gamma \alpha ^{\gamma }(1-\mu )^{1-\gamma }\beta \left[ {x+g(x)} \right] ^{\gamma }\left\{ {\frac{1-\gamma }{x^{2}}\left[ {\gamma \frac{1+g^{\prime }(x)}{1+\frac{g(x)}{x}}-1} \right] +\gamma \frac{g''(x)}{x+g(x)}} \right\} <0.\nonumber \\ \end{aligned}$$
(59)

The last sign follows from the fact that the growth function is a strictly concave function, which means \(g^{\prime }(x)<g(x)/x\). The signs of the derivatives of these functions with respect to the stock mean that the \(\textit{LHS}(\mu )\) function shifts up, when the level of the stock is increased, and the \(\textit{RHS}(\mu )\) function shifts down, when the stock is raised. All in all, this means that there is a negative relationship in equilibrium with the rate of subsidy and the level of the stock. Figure 5 clarifies this result. Q.E.D.

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Amacher, G., Ollikainen, M. & Puhakka, M. Renewable Resource Use and Nonseparable Amenity Benefits. Environ Resource Econ 69, 637–659 (2018). https://doi.org/10.1007/s10640-016-0095-2

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Keywords

  • Overlapping generations
  • Renewable resources
  • Nonseparable preferences
  • Indeterminacy

JEL Classification

  • D90
  • Q20
  • C62