The optimal rotation for a fully regulated forest is the same as, or shorter than, the rotation for a single even-aged forest stand: comments on Helmedag’s (2018) paper

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This paper discusses Helmedag’s article concerning Faustmann’s formula (Helmedag in Eur J For Res, 2018. He computed the present value of a fully regulated forest, including standing timber and theoretical land values. He showed that the optimal rotation in a fully regulated forest would always be the one that maximized the sustainable forest net revenues. Helmedag concluded that the discount rate would have no significance for the optimal rotation in a fully regulated forest, while his solution would still fulfil the optimality condition implied by Faustmann’s formula. Here, we refute this assertion. In fact, the assumption of a fully regulated forest has no impact on the optimal rotation period, or may even reduce it. We illustrate this by appropriately considering the actual costs of achieving a fully regulated forest with altered rotation. One must not ignore the alterations of the financial flows when changing the underlying rotation age in an established fully regulated forest. When the opportunity costs of the transition period are included, the optimal rotation becomes the same as that of a single even-aged forest, or shorter, depending on the transition regime. Under the optimal transition regime, the diminishing marginal rate of return for extending the rotation period in a fully regulated forest matches the discount rate, when we achieve the Faustmann rotation. We conclude that the optimal rotation period is independent of the status as fully regulated forest, provided efficient harvest operations during the transition period.

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  1. 1.

    We call a single age class a “forest stand” in our paper.

  2. 2.

    An age class usually covers a range wider than 1 year (possibly 20 years). However, in this example we still use the term age class, even if they comprise only one even-aged stand. In another example, we reduce the age classes to a size of 0.2 years.


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We would like to thank Volker Bergen (University of Göttingen) for valuable discussions, the “Deutsche Forschungsgemeinschaft” (DFG) for financial support of the study (KN 586/11-1) and Karen Grosskreutz for the language editing.

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Correspondence to Thomas Knoke.

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Communicated by Martin Moog.


Appendix A

Here, we show how we simplified the following integral:

$$ \mathop \int \limits_{0}^{T} \left( {f\left( T \right) + \frac{{f\left( T \right)e^{ - iT} - L}}{{1 - e^{ - iT} }}} \right)e^{{ - i\left( {T - t} \right)}} {\text{d}}t $$

In a more abstract form, we have:

$$ \mathop \int \limits_{0}^{T} \left( {f\left( T \right) + S\left( T \right)} \right)D\left( t \right){\text{d}}t $$

\( f\left( T \right) \) is the net value of the timber after harvesting at T and \( S\left( T \right) \) is an estimation of the land value (i.e. the theoretical land expectation value according to Faustmann) immediately after harvesting of \( f\left( T \right) \). \( D\left( t \right) \) is the function for the discount factor. To simplify the integral, we rewrite the formula using the compound interest factor \( e^{it} \). Consequently, we first use the original Faustmann version for \( S\left( T \right) \). Faustmann created a periodical rent occurring all T years by using \( e^{iT} \), which he subsequently capitalized appropriately:

$$ S\left( T \right) = \frac{{f\left( T \right) - Le^{iT} }}{{e^{iT} - 1}} $$

We then multiply \( f\left( T \right) \) and \( S\left( T \right) \) by the discount factor \( e^{{ - i\left( {T - t} \right)}} = e^{ - iT} e^{it} \).

$$ f\left( T \right)e^{{ - i\left( {T - t} \right)}} = f\left( T \right)e^{ - iT} e^{it} $$
$$ S\left( T \right)e^{{ - i\left( {T - t} \right)}} = \frac{{\left( {f\left( T \right) - Le^{iT} } \right)e^{ - iT} e^{it} }}{{e^{iT} - 1}} = \frac{{f\left( T \right)e^{ - iT} e^{it} - \left( {Le^{iT} e^{ - iT} e^{it} } \right)}}{{e^{iT} - 1}} = \frac{{f\left( T \right)e^{ - iT} e^{it} - Le^{it} }}{{e^{iT} - 1}} $$
$$ \begin{aligned} \left( {f\left( T \right) + S\left( T \right)} \right)D\left( t \right) & = f\left( T \right)e^{ - iT} e^{it} + \frac{{f\left( T \right)e^{ - iT} e^{it} - Le^{it} }}{{e^{iT} - 1}} \\ & = \frac{{f\left( T \right)e^{ - iT} e^{it} \left( {e^{iT} - 1} \right) + f\left( T \right)e^{ - iT} e^{it} - Le^{it} }}{{e^{iT} - 1}} = \frac{{f\left( T \right)e^{ - iT} e^{it} e^{iT} - f\left( T \right)e^{ - iT} e^{it} + f\left( T \right)e^{ - iT} e^{it} - Le^{it} }}{{e^{iT} - 1}} \\ \end{aligned} $$
$$ = \frac{{e^{it} \left( {f\left( T \right) - L} \right)}}{{e^{iT} - 1}} $$

We can now use this formula as the integral’s function as a simplified form of the original version (see Eq. 2 in the main text):

$$ \mathop \int \limits_{0}^{T} \frac{{e^{it} \left( {f\left( T \right) - L} \right)}}{{e^{iT} - 1}}{\text{d}}t $$

Appendix B

Here, we explain how we computed the optimality condition to transition immediately to the desired age class structure. For this case, we shall maximize:

$$ {\text{FNPV}}^{i} = \left( {\frac{f\left( T \right) - L}{i} - W\left( T \right) - S\left( T \right)T} \right)\frac{A}{T} = \frac{{A\left( {f\left( T \right) - L} \right) - AW\left( T \right)i - AS\left( T \right)Ti}}{Ti} $$

We thus form the first derivation of Eq. (30), and later set it equal to zero:

$$ \frac{{{\text{d}}\left( {{\text{FNPV}}^{i} } \right)}}{{{\text{d}}T}} = \frac{{\left[ {Af^{\prime} \left( T \right) - Ai\left( {W^{\prime}\left( T \right) + S^{\prime}\left( T \right)T + S\left( T \right)T^{\prime}} \right)} \right]Ti - \left[ {iT^{\prime}\left( {A\left( {f\left( T \right) - L} \right) - Ai\left( {W\left( T \right) + S\left( T \right)T} \right)} \right)} \right]}}{{\left( {Ti} \right)^{2} }} $$

Considering \( W^{\prime}\left( T \right) = f\left( T \right) \) and \( T^{\prime} = 1 \), we obtain for the optimal rotation, \( T^{i} \):

$$ 0 = \left[ {f^{\prime}\left( {T^{i} } \right) - i\left( {f\left( {T^{i} } \right) + S^{\prime}\left( {T^{i} } \right)T^{i} + S\left( {T^{i} } \right)} \right)} \right]T^{i} - \left[ {\left( {f\left( {T^{i} } \right) - L} \right) + i\left( {W\left( {T^{i} } \right) + B\left( {T^{i} } \right)T^{i} } \right)} \right] $$

Finally, we have:

$$ f^{\prime}\left( {T^{i} } \right) = i\left( {f\left( {T^{i} } \right) + S^{\prime}\left( {T^{i} } \right)T^{i} + S\left( {T^{i} } \right)} \right) + \frac{{\left( {f\left( {T^{i} } \right) - L} \right) - i\left( {W\left( {T^{i} } \right) + S\left( {T^{i} } \right)T^{i} } \right)}}{{T^{i} }} $$

To be complete, for \( S^{\prime}\left( T \right) \) we consider, given that \( S\left( T \right) = \frac{{f\left( T \right) - Le^{iT} }}{{e^{iT} - 1}} \):

$$ S^{\prime}\left( T \right) = \frac{{\left( {f^{\prime}\left( T \right) - iLe^{iT} } \right)\left( {e^{iT} - 1} \right) - \left[ {\left( {f\left( T \right) - Le^{iT} } \right)ie^{iT} } \right]}}{{\left( {e^{iT} - 1} \right)^{2} }} $$
$$ S^{\prime}\left( T \right) = \frac{{f^{\prime}\left( T \right)\left( {e^{iT} - 1} \right) - f\left( T \right)ie^{iT} + Lie^{iT} }}{{\left( {e^{iT} - 1} \right)^{2} }} = \frac{{f^{\prime}\left( T \right)\left( {e^{iT} - 1} \right)}}{{\left( {e^{iT} - 1} \right)^{2} }} - \frac{{f\left( T \right)ie^{iT} - Lie^{iT} }}{{\left( {e^{iT} - 1} \right)^{2} }} $$
$$ S^{\prime}\left( T \right) = \frac{{f^{\prime}\left( T \right)}}{{e^{iT} - 1}} - \frac{{f\left( T \right)ie^{iT} - Lie^{iT} }}{{e^{iT} - 1}}\frac{1}{{e^{iT} - 1}} = \frac{{f^{\prime}\left( T \right)}}{{e^{iT} - 1}} - i\left( {f\left( T \right) + \frac{{f\left( T \right) - Le^{iT} }}{{e^{iT} - 1}}} \right)\frac{1}{{e^{iT} - 1}} $$
$$ S^{\prime}\left( T \right) = \frac{{f^{\prime}\left( T \right)}}{{e^{iT} - 1}} - i\left( {f\left( T \right) + S\left( T \right)} \right)\frac{1}{{e^{iT} - 1}} = \frac{{f^{\prime}\left( T \right) - i\left( {f\left( T \right) + S\left( T \right)} \right)}}{{e^{iT} - 1}} $$

Appendix C

When transitioning from one rotation to another rotation, we can compute the FNPV by considering all the appropriately discounted net revenues (first curly parentheses in Eq. 38) and the appropriately discounted FPV (second curly parentheses), which we obtain after the transition period is complete. The variable \( \Delta T \) controls the size of an age class; it is convenient to consider only sizes leading to natural numbers for the frequency of age classes. A value of \( \Delta T = 0.2 \) informs fine-grained considerations, which means that 1 year can be split into five “age classes”. Equation 38 considers such refined sizes of age classes, which are necessary for approximating the very exact rotation periods computed by Helmedag (i.e. 10.666 or 11.296). However, this is arbitrary and assumes a continuous growth over the year, ignoring any vegetation period. This would apply to the tropics, but not to Central Europe.

$$ {\text{FNPV}} = \left\{ {\mathop \sum \limits_{k = 1}^{T / \Delta T} \left( {f\left( T \right)\left[ {a\left( T \right) - k\Delta a} \right] + f\left( {T + \Delta T} \right)\left( {k - 1} \right)\Delta a} \right)e^{ - ik\Delta T} } \right\} + \left\{ {\frac{{\left( {f\left( {T + \Delta T} \right) - L} \right)a\left( {T + \Delta T} \right)}}{{e^{i \cdot \Delta T} - 1}}e^{ - iT} } \right\} $$

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Knoke, T., Paul, C., Friedrich, S. et al. The optimal rotation for a fully regulated forest is the same as, or shorter than, the rotation for a single even-aged forest stand: comments on Helmedag’s (2018) paper. Eur J Forest Res (2019) doi:10.1007/s10342-019-01242-x

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  • Faustmann
  • Optimal rotation
  • Land rent theory
  • Forest rent theory
  • Opportunity costs