Optimum forest program when the carbon sequestration service of a forest has value

Abstract

This paper develops an optimum forest program for even-aged and multi-age forest management when the carbon sequestration service of a forest has value. It is shown that for even-aged forest management, the optimum rotation might be longer or shorter than when the carbon sequestration service is not considered. For multi-age forest management, it is shown that the optimum steady state is characterized by the optimum rotation for an even-aged forest. The optimum rotation is affected by how to evaluate the carbon release after harvesting, which is a central concern for the issue of “harvested wood products” in the United Nations Framework Convention on Climate Change.

Appendix: Proof of Proposition 1

To prove the proposition, we prepare a series of lemmas. Define:

$$ \Upphi(T;\beta)=r^{-1}( 1-{\text{e}}^{-rT}) (1-\beta)g(T)+\beta G(T)-\int\limits_{0}^{T}g(t)\,{\text{e}}^{-rt}\,{\text{d}}t. $$

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Lemma 1

The sign of dV ^C(T)/dT is the same as the sign of $\Upphi(T;\beta)$.

Proof

This follows from:

$$ \frac{{\text{d}}V^{{\text{C}}}(T)}{{\text{d}}T}=\frac{r\,{\text{e}}^{-rT}}{( 1-{\text{e}}^{-rT}) ^{2}} \Upphi(T;\beta). $$

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$\square$

Let:

$$ k=-\lim_{t\rightarrow\infty}g^{\prime}(t)/g(t)\in(0,\infty]. $$

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Lemma 2

The graph $\left( T,\Upphi(T;\beta)\right)$ is either strictly increasing or unimodal. If β ≥ k/(r + k), then $\partial\Upphi(T;\beta)/\partial T>0$ for all T > 0. If β < k/(r + k), then there exists T ^o > 0 such that T ^o > T _I and $\partial\Upphi (T;\beta)/\partial T\,\gtrless\, 0$ if $T\,\lessgtr\, T^{o}$.

Proof

The statement follows from:

$$ \frac{\partial\Upphi(T;\beta)}{\partial T}=\frac{1-{\text{e}}^{-rT}}{r}(1-\beta )g(T)\left[ \frac{g^{\prime}(T)}{g(T)}+\frac{r\beta}{1-\beta}\right]. $$

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Note that $\Upphi(0;\beta)=0$ and $\partial\Upphi(T;\beta)/\partial T>0$ for all $T\in\lbrack0,T_{I}]$ because $g^{\prime}(T)\geq0$ in the interval. Furthermore, note that $g^{\prime}(T)/g(T)$ is strictly decreasing and converges to −k as $T\rightarrow\infty$ by assumption A2(d). Finally, $k\leq r\beta/\left( 1-\beta\right)$ is equivalent to β ≥ k/(r + k). $\square$

Recall $\tilde{\Upphi}$ defined in Eq. (5) in the main text.

Lemma 3

(a) If the carbon release factor β satisfies $\beta\geq\tilde{\Upphi}$ , then dV ^C(T)/aT > 0 for all T > 0 and $T^{C}=\infty$ . (b) If $\beta\in\lbrack0,\tilde{\Upphi})$ , then there is a unique carbon forest rotation $T^{\text{C}}\in(T_{I},\infty)$ such that ${\text{d}}V^{{\text{C}}}(T)/{\text{d}}T\,\gtrless\, 0$ if $T\,\lessgtr\, T^{{\text{C}}}.$

Proof

By Lemmas 1 and 2 and since $\Upphi(0;\beta)=0,$ the existence of T ^C depends on the sign of:

$$ \lim_{T\rightarrow\infty}\Upphi=\beta\lim_{T\rightarrow\infty}G(T)-\int\limits_{0}^{\infty}g(t)\,{\text{e}}^{-rt}\,{\text{d}}t=\lim_{T\rightarrow\infty}G(T)( \beta -\tilde{\Upphi}). $$

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That is, if $\lim_{T\rightarrow\infty}\Upphi\geq0,$ then dV ^C(T)/sT > 0 and $T^{\text{C}}=\infty.$ If $\lim_{T\rightarrow\infty}\Upphi<0,$ then there is $T^{\text{C}}\in(T_{\text{I}},\infty)$ such that $\Upphi(T;\beta)\,\gtrless\, 0$ if $T\,\lessgtr\, T^{\text{C}}.$ Note that $\lim_{T\rightarrow\infty}G(T)>0$ by Assumption A2.$\square$

Define function $\Uppsi(T)$ by:

$$ \Uppsi(T)=r\frac{\partial\Upphi(T;\beta)}{\partial\beta}=-g(T)(1-{\text{e}}^{-rT})+rG(T). $$

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Lemma 4

(a) The carbon forest rotation T ^C as a function of β is continuously differentiable on $\beta\in\lbrack0,\tilde{\Upphi})$ . (b) The sign of ∂T ^C/∂β is the same as the sign of $\Uppsi(T^{{\text{C}}})$ for all $\beta\in\lbrack0,\tilde{\Upphi})$.

Proof

Fix $\beta\in\lbrack0,\tilde{\Upphi}).$ Then, there is unique T ^C(>T _I) by Lemma 3. By the total derivative of the first-order condition:

$$ \frac{{\text{d}}V^{{\text{C}}}(T^{{\text{C}}})}{{\text{d}}T}=\frac{{\text{e}}^{-rT^{{\text{C}}}}}{\left( 1-{\text{e}}^{-rT^{{\text{C}}}}\right) ^{2}}\Upphi(T^{{\text{C}}};\beta)=0, $$

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we have:

$$ \frac{{\text{d}}^{2}V^{{\text{C}}}(T^{{\text{C}}})}{{\text{d}}T^{2}}{\text{d}}T^{{\text{C}}}+\frac{{\text{e}}^{-rT^{{\text{C}}}}}{\left( 1-{\text{e}}^{-rT^{{\text{C}}}}\right) ^{2}}\Uppsi(T^{{\text{C}}})\,{\text{d}}\beta=0. $$

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Because d² V ^C(T ^C)/dT ² ≠ 0, we have statement (a) by the implicit function theorem. Furthermore, ∂T ^C/∂β satisfies:

$$ \frac{\partial T^{\text{C}}}{\partial\beta}=-\frac{[ {\text{e}}^{-rT^{\text{C}}}/( 1-{\text{e}}^{-rT^{\text{C}}}) ^{2}] \Uppsi(T^{\text{C}})}{\text{d}^{2}V^{\text{C}}(T^{\text{C}})/\text{d}T^{2}}. $$

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Because d² V ^C(T ^C)/dT ² < 0, we have statement (b).$\square$

Recall T _L defined in Eq. (6) in the main text.

Lemma 5

Assume $\beta\in\lbrack0,\tilde{\Upphi}).$ Then, $\Uppsi(T)\,\lessgtr\, 0$ if and only if $T\,\lessgtr\, T_{L}.$

Proof

Note that $\Uppsi(0)=0, \lim_{T\rightarrow\infty}\Uppsi(T)>0,$ and:

$$ \Uppsi^{\prime}(T)=g(T)(1-{\text{e}}^{-rT})( r-g^{\prime}(T)/g(T)). $$

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By assumption, A2(d) $g^{\prime}(T)/g(T)$ is strictly decreasing. Therefore, if $g^{\prime}(0)/g(0)\leq r,$ then $\Uppsi^{\prime}(T)>0$ for all T > 0, which implies T _L = 0 and $\Uppsi(T)>0$ for all T > 0. Similarly, if $g^{\prime }(0)/g(0)>r,$ then $T_{\text{L}}\in(0,\infty],$ and $\Uppsi(T)\ \lessgtr\ 0$ if and only if $T\ \lessgtr\ T_{\text{L}}.$ The proof ends by gathering both cases. $\square$

Lemma 6

Assume $\beta\in\lbrack0,\tilde{\Upphi}).$ Then, T ^C > max{T _I, T _L}.

Proof

By Lemma 3(b), T ^C > T _I. Then, we show T ^C > T _L. The case of T _L = 0 is obvious. For the case of T _L > 0, use the fact that T ^C satisfies:

$$ 0=\Upphi(T^{\text{C}};\beta)=-r^{-1}(1-\beta)\Uppsi(T^{\text{C}})+G(T^{\text{C}})-\int\limits_{0}^{T^{\text{C}} }g(t)\,{\text{e}}^{-rt}\,{\text{d}}t. $$

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Then, we have:

$$ r^{-1}(1-\beta)\Uppsi(T^{\text{C}})=G(T^{\text{C}})-\int\limits_{0}^{T^{C}}g(t)\,{\text{e}}^{-rt}\,{\text{d}}t>0. $$

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Therefore, $\Uppsi(T^{\text{C}})>0.$ This implies T ^C > T _L by Lemma 5. Therefore, T ^C > max{T _I, T _L}. $\square$

Lemma 7

∂T ^C/∂β > 0 for all $\beta \in(0,\tilde{\Upphi})$ and $\lim_{\beta\nearrow\tilde{\Upphi}}T^{C}(\beta)=\infty.$

Proof

Assume $\beta\in(0,\tilde{\Upphi}).$ Then, T ^C > T _L by Lemma 6. Therefore, $\Uppsi(T^{\text{C}})>0$ by Lemma 5, which implies ∂T ^C/∂β > 0 by Lemma 4. Denote by β(T ^C) the inverse function of T ^C(β). Using Eq. (48) and the first-order condition $\Upphi(T^{\text{C}};\beta(T^{\text{C}}))=0,$ we have:

$$ \beta(T^{\text{C}})=\frac{\int\nolimits_{0}^{T^{\text{C}}}g(t)\,{\text{e}}^{-rt}\,{\text{d}}t-r^{-1}( 1-{\text{e}}^{-rT^{\text{C}} }) g(T^{\text{C}})}{G(T^{C})-r^{-1}( 1-{\text{e}}^{-rT^{\text{C}}}) g(T^{\text{C}})}. $$

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Then, $\beta(T^{\text{C}})\rightarrow\tilde{\Upphi}$ as $T^{\text{C}}\rightarrow\infty.$ This implies $\lim_{\beta\rightarrow\tilde{\Upphi}}T^{\text{C}}(\beta)=\infty.$ $\square$

Proof of Proposition 1

Propositions 1(a) and (b) follow from Lemma 3. Proposition 1(c) follows from Lemmas 4(a), 6, and 7.$\square$