Renewable Resource Use and Nonseparable Amenity Benefits

Gregory Amacher; Markku Ollikainen; Mikko Puhakka

doi:10.1007/s10640-016-0095-2

Environmental and Resource Economics

April 2018, Volume 69, Issue 4, pp 637–659 | Cite as

Renewable Resource Use and Nonseparable Amenity Benefits

Authors
Authors and affiliations

Gregory Amacher
Markku Ollikainen
Mikko Puhakka

Article

First Online: 30 November 2016

Accepted: 13 November 2016

209 Downloads

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.

Keywords

Overlapping generations Renewable resources Nonseparable preferences Indeterminacy

JEL Classification

D90 Q20 C62

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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

	Africa	Asia	Europe	Latin America	North America
Land area 2016	7.60	8.40	90.70	12.87	68.80
Land area 2012	7.42	5.0	63.97	9.51	60.54
Four year change in land area (%)	2.4	68	41.8	35.3	13.6

Source: Forest Stewardship Council (2012, 2016). Facts and Figures. Available at https://us.fsc.org/en-us/what-we-do/facts-figures

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
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.

References

Amacher G, Koskela E, Ollikainen M (2002) Optimal forest policies in an overlapping generations economy with timber and money bequests. J Environ Econ Manag 44:346–369CrossRefGoogle Scholar
Amacher G, Conway C, Sullivan J (2003) Econometric analyses of nonindustrial forest landowners: is there anything left to study? J For Econ 9:137–164Google Scholar
Amacher G, Ollikainen M, Koskela E (2009) The economics of forest resources. MIT Press, CambridgeGoogle Scholar
Azariadis C (1993) Intertemporal macroeconomics. Blackwell Publishers, OxfordGoogle Scholar
Beach RH, Pattanayak SK, Yang JC, Murray BC, Abt RC (2005) Econometric studies of non-industrial private forest management: a review and synthesis. For Policy Econ 7:261–281CrossRefGoogle Scholar
Bennett RL, Farmer REA (2000) Indeterminacy with non-separable utility. J Econ Theory 93:118–143CrossRefGoogle Scholar
Bockstael N, McConnel K (1993) Public goods as characteristics of non-market commodities. Econ J 103:1244–1257CrossRefGoogle Scholar
Brown GM (2000) Renewable natural resource management and use without markets. J Econ Lit 38:875–914CrossRefGoogle Scholar
Carbone J, Smith VK (2008) Evaluating policy interventions with general equilibrium externalities. J Public Econ 92:1254–1274CrossRefGoogle Scholar
Carbone J, Smith VK (2013) Valuing nature in a general equilibrium. J Environ Econ Manag 66:72–89CrossRefGoogle Scholar
Cass D (1972) On capital overaccumulation in the aggregative, neoclassical model of economic growth: a complete characterization. J Econ Theory 4(1972):200–223CrossRefGoogle Scholar
Forest Stewardship Council (2012) Facts and figures. https://us.fsc.org/en-us/what-we-do/facts-figures
Forest Stewardship Council (2016) Facts and figures. https://us.fsc.org/en-us/what-we-do/facts-figures
Galor O (2007) Discrete dynamical systems. Springer, HeidelbergCrossRefGoogle Scholar
Karppinen H (2000) Forest values and the objectives of forest ownership. The Finnish Forest Research Institute, Research Reports No 757Google Scholar
Kehoe TJ, Levine DK (1990) The economics of indeterminacy in overlapping generations models. J Public Econ 42:219–243CrossRefGoogle Scholar
Kemp M, Van Long N (1979) The under-exploitation of natural resources: a model with overlapping generations. Econ Rec 55:214–221CrossRefGoogle Scholar
Kim S (2002) Optimal environmental regulation in the presence of other taxes: the role of non-separable preferences and technology. Contrib Econ Analy Policy 1:1–25Google Scholar
Koskela E, Ollikainen M, Puhakka M (2002) Renewable resources in an overlapping generations economy without capital. J Environ Econ Manag 43:497–517CrossRefGoogle Scholar
Koskela E, Ollikainen M, Puhakka M (2008) Saddles and bifurcations in an overlapping generations economy with a renewable resource. Finn Econ Pap 21:3–21Google Scholar
Krautkraemer JA (1985) Optimal growth, resource amenities and the preservation of natural environments. Rev Econ Stud 52:153–170CrossRefGoogle Scholar
Kurz M (1968) Optimal economic growth with wealth effects. Int Econ Rev 9:348–357CrossRefGoogle Scholar
Kuuluvainen J, Karppinen H, Ovaskainen V (1996) Landwoner objectives and nonindustrial private timber supply. For Sci 42:300–309Google Scholar
Majumdar M, Mitra T (1994) Periodic and chaotic programs of optimal intertemporal allocation in an aggregative model with wealth effects. Econ Theory 4:649–676CrossRefGoogle Scholar
Mourmouras A (1991) Competitive equilibria and sustainable growth in a life-cycle model with natural resources. Scand J Econ 93:585–591CrossRefGoogle Scholar
Mourmouras A (1993) Conservationist government policies and intergenerational equity in an overlapping generations model with renewable resources. J Public Econ 51:249–268CrossRefGoogle Scholar
Nyarko Y, Olson LJ (1991) Stochastic dynamic models with stock-dependent rewards. J Econ Theory 55:161–168CrossRefGoogle Scholar
Nyarko Y, Olson LJ (1994) Stochastic growth, when utility depends on both consumption and the stock level. Econ Theory 4:791–797CrossRefGoogle Scholar
Olson LJ, Knapp KC (1997) Exhaustible resource allocation in an overlapping generations economy. J Environ Econ Manag 32:277–292CrossRefGoogle Scholar
Shi S (1994) Weakly nonseparable preferences and distortionary taxes in a small open economy. Int Econ Rev 35:411–428CrossRefGoogle Scholar
Weil P (1987) Love thy children. reflections on the Barro debt neutrality theorem. J Monet Econ 19:377–391CrossRefGoogle Scholar
Woodford M (1984) Indeterminacy of equilibrium in the overlapping generations model: a survey. Unpublished, May 1984. http://www.columbia.edu/~mw2230/Woodford84.pdf

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Gregory Amacher
- 1
Markku Ollikainen
- 2
Mikko Puhakka
- 3
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1.College of Natural ResourcesVirginia Polytechnic Institute and State UniversityBlacksburgUSA
2.Department of Economics and ManagementUniversity of HelsinkiHelsinkiFinland
3.Department of EconomicsUniversity of OuluOuluFinland

Renewable Resource Use and Nonseparable Amenity Benefits

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