Land conversion pace under uncertainty and irreversibility: too fast or too slow?

Abstract

In this paper stochastic dynamic programming is used to investigate land conversion decisions taken by a multitude of landholders under uncertainty about the value of environmental services and irreversible development. We study land conversion under competition on the market for agricultural products when voluntary and mandatory measures are combined by the Government to induce habitat conservation. We show that land conversion can be delayed by paying landholders for the provision of environmental services and by limiting the individual extent of developable land. It is found, instead, that the presence of ceilings on aggregate conversion may lead to runs which rapidly exhaust the targeted amount of land. We study the impact of uncertainty on the optimal conversion policy and discuss conversion dynamics under different policy scenarios on the basis of the relative long-run expected rate of deforestation. Interestingly, we show that uncertainty, even if it induces conversion postponement in the short-run, increases the average rate of deforestation and reduces expected time for total conversion in the long run. Finally, we illustrate our findings through some numerical simulations.

Appendix

1.1 A.1 Proof of Proposition 1

Let $V(A,B;\bar{A})$ be twice-differentiable in $B$ and consider a short interval $dt$ where no conversion takes place.⁵⁰ So, by applying a standard dynamic programming approach, the farmer’s value function in (5) can be rewritten as follows:⁵¹

$$\begin{aligned} rV(A,B;\bar{A})~dt=\Delta \pi (A,B;\bar{A})~dt+E_{0}[dV(A,B;\bar{A})] \end{aligned}$$

(14)

Expanding $dV(A,B;\bar{A})$ using Ito’s Lemma, the solution to (14) must solve the following differential equation:

$$\begin{aligned}&\frac{1}{2}\sigma ^{2}B^{2}V_{BB}(A,B;\bar{A})+\alpha BV_{B}(A,B;\bar{A})-rV(A,B;\bar{A}) \nonumber \\&\quad +\left[ (1-\lambda )\delta A^{-\gamma }+(\lambda \eta _{2}-\eta _{1})B\right] =0 \end{aligned}$$

(15)

Using standard arguments the solution of (15) is (see Dixit and Pindyck 1994):⁵²

$$\begin{aligned} V(A,B;\bar{A})=Z(A)B^{\beta }+(1-\lambda )\frac{\delta A^{-\gamma }}{r} +(\lambda \eta _{2}-\eta _{1})\frac{B}{r-\alpha } \end{aligned}$$

(16)

where $\beta$ is the negative root of the characteristic equation $Q(\beta )=\frac{1}{2}\sigma ^{2}\beta (\beta -1)+\alpha \beta -r=0$ and $Z(A)$ is a constant to be determined.

To determine $Z(A)$ and $B^{*}(A)$ some suitable boundary conditions on (16) are required. First, further land development, by increasing the number of competing farmers in the market, keeps the value of being an active farmer below $(1-\lambda )c$ (value-matching condition). Formally, this is equivalent to impose:

$$\begin{aligned} Z(A)B^{*}(A)^{^{\beta }}+\frac{(1-\lambda )\delta }{r}A^{-\gamma }+(\lambda \eta _{2}-\eta _{1})\frac{B^{*}(A)}{r-\alpha }=(1-\lambda )c \end{aligned}$$

(17a)

Second, by differentiating the value-matching condition (17a) totally with respect to $A$, we obtain

$$\begin{aligned} \frac{\partial V(A,B^{*}(A);\bar{A})}{\partial A}=V_{A}(A,B^{*}(A); \bar{A})+V_{B}(A,B^{*}(A);\bar{A})\frac{{\small dB}^{*}{\small (A)}}{ dA}=0\qquad \quad \end{aligned}$$

(17b)

Now, let’s recall that marginal rents for an active farmer must be null at $B^{*}(A)$ (see e.g. Proposition 1 in Bartolini (1993) and Grenadier (2002, p. 699) i.e.:

$$\begin{aligned} V_{A}(A,B^{*}(A);\bar{A})=Z^{\prime }(A)B^{*}(A)^{\beta }-(1-\lambda )\frac{\delta \gamma A^{-(\gamma +1)}}{r}=0 \end{aligned}$$

(17c)

Substituting (17c) into (17b) yields the following condition (extended smooth-pasting condition):

$$\begin{aligned} \frac{\partial V(A,B^{*}(A);\bar{A})}{\partial A}=V_{B}(A,B^{*}(A); \bar{A})\frac{{\small dB}^{*}{\small (A)}}{dA}=0 \end{aligned}$$

(17d)

where $V_{B}(A,B^{*}(A);\bar{A})=\beta Z\left( A\right) B^{*}(A)^{\beta -1}+\frac{\lambda \eta _{2}-\eta _{1}}{r-\alpha }$.

Third, considering the limit on conversion, $\bar{A}$, imposed by the Government, the last boundary condition to be considered is:

$$\begin{aligned} Z(\bar{A})=0 \end{aligned}$$

(18)

Note that condition (17d) plays the same role of a complementary-slackness condition in standard programming. That is, it must hold whenever control takes place.

Assume for the moment that (17d) holds over the interval $[A_{0},\bar{A}]$ with $\frac{{\small dB}^{*}{\small (A)}}{dA}\ne 0$. This implies that $V_{B}(A,B^{*}(A);\bar{A})=\beta Z(A)B^{*}(A)^{^{\beta -1}}+\frac{ \lambda \eta _{2}-\eta _{1}}{r-\alpha }=0$. Note that since $\frac{\lambda \eta _{2}-\eta _{1}}{r-\alpha }<0$, then, as conjectured, $Z(A)<0$. Hence, each landholder exercises the option to convert at the level of $B^{*}(A)$ where the value, $V(A,B^{*}(A);\bar{A})$ is tangent to the conversion cost, $\left( 1-\lambda \right) c$. This critical threshold is given by the solution of the system including equations (17a) and $V_{B}(A,B^{*}(A);\bar{A})=0$. Solving it, we obtain

$$\begin{aligned} B^{*}(A)=\frac{\beta }{\beta -1}\left( r-\alpha \right) \frac{1-\lambda }{\eta _{1}-\lambda \eta _{2}}\left[ \left( \frac{\hat{A}}{A}\right) ^{\gamma }-1\right] c, \end{aligned}$$

(19)

which, as one can easily verify, is a continuous decreasing function of $A$.

Now, consider a landholder following the smooth-pasting policy in (19) and suppose he is the last able to convert his plot. By using (16) and (18) it follows that

$$\begin{aligned} V(\bar{A},B^{*}(\bar{A});\bar{A})-\lim _{B\rightarrow B^{*}(\bar{A} )}V(A,B;\bar{A})>0 \end{aligned}$$

(20)

where $V(\bar{A},B^{*}(\bar{A});\bar{A})=(1-\lambda )\frac{\delta \bar{A} ^{-\gamma }}{r}+(\lambda \eta _{2}-\eta _{1})\frac{B^{*}(\bar{A})}{ r-\alpha }$

Note that the inequality (20) holds due the presence of the term, $Z(A)B^{\beta }$, accounting for future market entries. The upward jump in value in (20) contradicts the optimality of $B^{*}(A)$ by violating the smooth-pasting condition $V_{B}(A,B^{*}(A);\bar{A})=0$. Note in fact that in this case $V_{B}(\bar{A},B^{*}(\bar{A});\bar{A})=\frac{\lambda \eta _{2}-\eta _{1}}{r-\alpha }<0$. Hence, by using (17a), at $A=\bar{A}$ the optimal conversion threshold is given by

$$\begin{aligned} B^{*}(\bar{A})=\left( r-\alpha \right) \frac{1-\lambda }{\eta _{1}-\lambda \eta _{2}}\left[ \left( \frac{\hat{A}}{\bar{A}}\right) ^{\gamma }-1\right] c \end{aligned}$$

(21)

Furthermore, as all landholders are equal, all expect such upward jump in value occurring at $B=B^{*}(\bar{A})$. Therefore, as $B^{*}(A)$ is decreasing in $A,$ by starting at $\bar{A}$ each landholder would maximizes the value of his conversion option by pre-empting the entry of others. By playing backward, pre-emption is then a (perfect) Nash equilibrium strategy up to a certain $A=A^{+}$ where $V(A^{+},B^{*}(A^{+});\bar{A})-\lim _{B\rightarrow B^{*}(A^{+})}V(A,B;\bar{A})=0$. Note that this implies that: (a) the candidate policy in the interval $[A^{+}, \bar{A}]$ is to impose $\frac{dB^{*}(A)}{dA}=0$; (b) the optimal conversion threshold for all the discrete amount of conversion in the interval $[A^{+},\bar{A}]$ is the value-maximizer $B^{*}(\bar{A})$; (c) at $A^{+}$ the necessary condition for optimality, $V_{B}(A,B^{*}(A); \bar{A})=0,$ holds again.

Finally, to determine $A^{+}$ consider that $A^{+}$ splits the interval $[A_{0},\bar{A}]$ into an interval where landholders rush and a run takes place and an interval where landholders follows a “smooth-pasting” conversion policy. This means that $A^{+}$ must satisfy both (19) and (21), i.e., $B^{*}(A^{+})=B^{*}(\bar{A})$.

Substituting and solving for $A^{+}$, we obtain

$$\begin{aligned} A^{+}=\left[ \frac{(\beta -1)\bar{A}^{-\gamma }+\hat{A}^{-\gamma }}{\beta }\right] ^{-\frac{1}{\gamma }} \end{aligned}$$

(22a)

Finally, let’s consider the following two scenarios: $\hat{A}\le \bar{A}$ and $\hat{A}>\bar{A}$. From (22a) it follows that:

$$\begin{aligned} \frac{\beta }{\beta -1}\left[ \left( \frac{\hat{A}}{A^{+}}\right) ^{\gamma }-1\right] =\left( \frac{\hat{A}}{\bar{A}}\right) ^{\gamma }-1 \end{aligned}$$

(22b)

Studying (22b) we can state that since $\frac{\beta }{\beta -1}>0$:

• if $\hat{A}\le \bar{A}$ then it must be $\bar{A}\le A^{+}$. This implies that there is no run taking place. Land will be converted smoothly according to (19) up to $\hat{A}$ since $\frac{\delta }{r} A^{-\gamma }\le c$ for $A\ge \hat{A}$;

• if $\hat{A}>\bar{A}$ then it must be $A^{+}<\bar{A}$. In this case, land is converted smoothly up to $A^{+}$ where landholders start a run to convert land up to $\bar{A}$.

1.2 A.2 Long-run distributions

Let $h$ be a linear Brownian motion with parameters $\mu$ and $\sigma$ that evolves according to $dh=\mu dt+\sigma dw$. Following Harrison (1985, pp. 90–91); see also Dixit (1993, pp. 58–68)] the long-run density function for $h$ fluctuating between a lower reflecting barrier, $a\in (-\infty ,\infty )$, and an upper reflecting barrier, $b\in (-\infty ,\infty )$, is represented by the following truncated exponential distribution:

$$\begin{aligned} f\left( h\right) =\left\{ \begin{array}{l@{\quad }l} \frac{2\mu }{\sigma ^{2}}\frac{e^{\frac{2\mu }{\sigma ^{2}}h}}{e^{\frac{ 2\mu }{\sigma ^{2}}b}-e^{\frac{2\mu }{\sigma ^{2}}a}} &{} \quad \mu \ne 0, \\ \frac{1}{b-a} &{} \quad \mu =0. \end{array} \right. \end{aligned}$$

(23)

We are interested to the limit case where $a\rightarrow -\infty .$ In this case, from (23) a limiting argument gives:

$$\begin{aligned} f\left( h\right) =\left\{ \begin{array}{l@{\quad }l} \frac{2\mu }{\sigma ^{2}}e^{-\frac{2\mu }{\sigma ^{2}}\left( b-h\right) } &{} \mu >0, \\ 0 &{} \mu \le 0. \end{array}\right. \quad \text{ for } -\!\infty <h<b \end{aligned}$$

(24)

Hence, the long-run average of $h$ can be evaluated as $E\left[ h\right] =\int _{\Phi }hf\left( h\right) ~dh$, where $\Phi$ depends on the distribution assumed. In the steady-state this yields:

$$\begin{aligned} E\left[ h\right] \!=\!\int _{-\infty }^{b}hf\left( h\right) ~dh\!=\!\int _{-\infty }^{b}h\frac{2\mu }{\sigma ^{2}}e^{-\frac{2\mu }{\sigma ^{2}}\left( b-h\right) }~dh\!=\!\frac{2\mu }{\sigma ^{2}}e^{-\frac{2\mu }{\sigma ^{2}} b}\int _{-\infty }^{b}he^{\frac{2\mu }{\sigma ^{2}}h}~dh\!=\!b\!-\!\frac{2\mu }{\sigma ^{2}}\nonumber \\ \end{aligned}$$

(25)

1.3 A.3 Proof of Proposition 3

Taking the logarithm of (11) we get:

$$\begin{aligned} \ln \xi&= \ln \left[ \frac{\beta }{\beta -1}\left( 1-\lambda \right) \frac{P_{A}\left( A\right) }{r}-\frac{\eta _{1}-\lambda \eta _{2}}{ r-\alpha }B\right] \nonumber \\&= \ln \left[ \frac{\eta _{1}-\lambda \eta _{2}}{r-\alpha }\right] +\ln \left[ J-B\right] \end{aligned}$$

(26)

where $J\mathbf = \frac{\beta }{\beta -1}\left( r-\alpha \right) \Psi \frac{ P_{A}\left( A\right) }{r},\,\Psi =\frac{1-\lambda }{\eta _{1}-\lambda \eta _{2}}$ and $J>B$. Rewriting $\ln \left[ J-B\right]$ as $\ln \left[ e^{\ln J}-e^{\ln B}\right]$ and expanding it by Taylor’s theorem around the point ( $\widetilde{\ln J},\widetilde{\ln B}$) yields:

$$\begin{aligned} \ln \left[ J-B\right] \simeq v_{0}+v_{1}\ln J+v_{2}\ln B \end{aligned}$$

where

$$\begin{aligned} v_{0}&= \ln \left[ e^{\widetilde{\ln J}}-e^{\widetilde{\ln B}}\right] - \left[ \frac{\widetilde{\ln J}}{1-e^{\widetilde{\ln B}-\widetilde{\ln J}}}+ \frac{\widetilde{\ln B}}{1-e^{-(\widetilde{\ln B}-\widetilde{\ln J})}}\right] \\ v_{1}&= \frac{1}{1-e^{\widetilde{\ln B}-\widetilde{\ln J}}}, v_{2}= \frac{1}{1-e^{-(\widetilde{\ln B}-\widetilde{\ln J})}},\,\frac{v_{2}}{v_{1}}= \frac{1-v_{1}}{v_{1}}<0 \end{aligned}$$

By substituting the approximation into (26) it follows that:

$$\begin{aligned} \ln \xi \simeq \ln \frac{\eta _{1}-\lambda \eta _{2}}{r-\alpha } +v_{0}+v_{1}\ln J+v_{2}\ln B \end{aligned}$$

(27)

Now, by Ito’s lemma and the considerations discussed in the paper on the competitive equilibrium, $\ln \xi$ evolves according to $d\ln \xi =v_{2}d\ln B=v_{2}[(\alpha -\frac{1}{2}\sigma ^{2})~dt+\sigma dw]$ with $\ln \hat{\xi }$ as upper reflecting barrier. Setting $h=\ln \xi$, the random variable $\ln \xi$ follows a linear Brownian motion with parameter $\mu =v_{2}(\alpha -\frac{1}{2}\sigma ^{2})>0$ and has a long-run distribution with (24) as density function.

Solving (27) with respect to $\ln A$ we obtain the long-run optimal stock of deforested land, i.e.:

$$\begin{aligned} \ln A\simeq \frac{\ln \left[ \frac{\eta _{1}-\lambda \eta _{2}}{r-\alpha } \right] +v_{0}+v_{1}\ln \left[ \frac{\beta }{\beta -1}\left( r-\alpha \right) \Psi \frac{\delta }{r}\right] +v_{2}\ln B-h}{\gamma v_{1}} \end{aligned}$$

(28)

From (28) by some manipulations we can show that at $\xi (\tilde{ B},\tilde{A})=\hat{\xi }$

$$\begin{aligned} 1&= \exp \left( \frac{v_{0}}{v_{1}}\right) \left( \frac{\frac{\eta _{1}-\lambda \eta _{2}}{ r-\alpha }}{\hat{\xi }}\right) ^{\frac{1}{v_{1}}}\left[ \frac{\beta }{\beta -1} \left( r-\alpha \right) \Psi \frac{\delta }{r}\right] A^{-\gamma }B^{\frac{ v_{2}}{v_{1}}} \\&= \exp \left( \frac{v_{0}}{v_{1}}\right) \left[ \frac{\beta }{\beta -1}\left( r-\alpha \right) \Psi \right] ^{-\frac{v_{2}}{v_{1}}}\frac{\delta }{r}c^{-\frac{1}{ v_{1}}}A^{-\gamma }B^{\frac{v_{2}}{v_{1}}} \\&= \exp \left( \frac{v_{0}}{v_{1}}\right) \left( \frac{J}{\frac{\delta }{r}A^{-\gamma }}\right) ^{- \frac{v_{2}}{v_{1}}}\frac{\delta }{r}c^{-\frac{1}{v_{1}}}A^{-\gamma }B^{ \frac{v_{2}}{v_{1}}} \\&= \exp \left( \frac{v_{0}}{v_{1}}\right) J^{-\frac{v_{2}}{v_{1}}}\left( \frac{\delta }{rc} A^{-\gamma }\right) ^{\frac{1}{v_{1}}}B^{\frac{v_{2}}{v_{1}}} \\&= \exp (v_{0})J^{-v_{2}}\left( \frac{\hat{A}}{A}\right) ^{\gamma }B^{v_{2}} \\&= \frac{\widetilde{J}-\widetilde{B}}{\widetilde{J}^{v_{1}}\widetilde{B} ^{v_{2}}}J^{-v_{2}}\left( \frac{\hat{A}}{A}\right) ^{\gamma }B^{v_{2}} \\&= \left( \frac{\tilde{A}}{A}\right) \left( \frac{B}{\widetilde{B}}\right) ^{-\frac{1}{\gamma }[1-\left( \frac{\tilde{A}}{\hat{A}}\right) ^{\gamma }]} \end{aligned}$$

and

$$\begin{aligned} \frac{A}{\tilde{A}}=\left( \frac{B}{\widetilde{B}}\right) ^{-\frac{1}{\gamma }[1-\left( \frac{ \tilde{A}}{\hat{A}}\right) ^{\gamma }]} \end{aligned}$$

Note that since $\tilde{A}<\hat{A}$ then $-\frac{1}{\gamma }[ 1-(\frac{ \tilde{A}}{\hat{A}})^{\gamma }] <0$.

Taking the expected value on both sides of (28) leads to:

$$\begin{aligned} E\left[ \ln A\right] \simeq \frac{\ln \left[ \frac{\eta _{1}\!-\!\lambda \eta _{2}}{r\!-\!\alpha }\right] \!+\!v_{0}\!+\!v_{1}\ln \left[ \frac{\beta }{\beta \!-\!1}\left( r\!-\!\alpha \right) \Psi \frac{\delta }{r}\right] \!+\!v_{2}\left[ B_{0}\!+\!\left( \alpha \!-\! \frac{1}{2}\sigma ^{2}\right) t\right] \!-\!E\left[ h\right] }{\gamma v_{1}} \end{aligned}$$

Since by (25) $E(h)$ is independent on $t$, differentiating with respect to $t$, we obtain the expected long-run rate of deforestation:

$$\begin{aligned} \frac{1}{dt}E\left[ d\ln A\right] \simeq \frac{\alpha -\frac{1}{2}\sigma ^{2} }{\gamma }\frac{v_{2}}{v_{1}}=-\frac{\alpha -\frac{1}{2}\sigma ^{2}}{\gamma } e^{\widetilde{\ln B}-\widetilde{\ln J}} \quad \text{ for } \alpha <\frac{1}{2} \sigma ^{2} \end{aligned}$$

By the monotonicity property of the logarithm, $\widetilde{B}$must exists such that $\ln \widetilde{B}=\widetilde{\ln B}$. Furthermore, by plugging $\widetilde{B}$ into (7), we can always find a surface $\tilde{A}$ and $\widetilde{J}=\frac{\beta }{\beta -1}\left( r-\alpha \right) \Psi \frac{ P_{A}( \tilde{A}) }{r}$ such that a linearization along ($\widetilde{\ln B},\widetilde{\ln J}$) is equivalent to a linearization along ($\ln \widetilde{B},\ln \widetilde{J}$), where $\widetilde{\ln J}=\ln \widetilde{J}$. This implies that by setting $(\widetilde{B},\tilde{A})$, the long-run average rate of deforestation can be written as:

$$\begin{aligned} \frac{1}{dt}E\left[ d\ln A\right]&= -\frac{\alpha -\frac{1}{2}\sigma ^{2}}{ \gamma }\frac{\widetilde{B}}{\widetilde{J}}=-\frac{\alpha -\frac{1}{2}\sigma ^{2}}{\gamma }\frac{1}{1+\frac{\beta }{\beta -1}( r-\alpha ) \Psi \frac{c}{\widetilde{B}}} \\&= -\frac{\alpha -\frac{1}{2}\sigma ^{2}}{\gamma }\frac{\frac{P_{A}(\tilde{A}) }{r}-c}{\frac{P_{A}(\tilde{A}) }{r}}=-\frac{ \alpha -\frac{1}{2}\sigma ^{2}}{\gamma }\Bigg (1-\frac{c}{\frac{\delta }{r}\tilde{A }^{-\gamma }}\Bigg ) \end{aligned}$$

where $\frac{P_{A}(\tilde{A}) }{r}=\frac{\widetilde{B} }{\frac{\beta }{\beta -1} ( r-\alpha ) \Psi }+c$ and $\tilde{A}< \hat{A}$.

1.4 A.4 The impact of uncertainty on the distribution of $\xi$

Rearranging (27) yields

$$\begin{aligned} \ln \xi \simeq U_{\xi }+v_{2}\ln B \end{aligned}$$

(29)

where $U_{\xi }=\ln \frac{\eta _{1}-\lambda \eta _{2}}{r-\alpha } +v_{0}+v_{1}\ln J.$

By some manipulations:

$$\begin{aligned} e^{\ln \xi }&= e^{U_{\xi }+v_{2}\ln B} \nonumber \\ \xi&= e^{U_{\xi }}B^{v_{2}} \end{aligned}$$

(30)

Using Ito’s lemma

$$\begin{aligned} d\xi&= e^{U_{\xi }}\left[ v_{2}B^{v_{2}-1}dB+\frac{1}{2} v_{2}(v_{2}-1)B^{v_{2}-2}(dB)^{2}\right] \\&= e^{U_{\xi }}B^{v_{2}}v_{2}\left\{ \left[ \alpha +\frac{1}{2} (v_{2}-1)\sigma ^{2}\right] ~dt+\sigma dw\right\} \\&= \xi v_{2}\left\{ \left[ \alpha +\frac{1}{2}(v_{2}-1)\sigma ^{2}\right] ~dt+\sigma ~dw\right\} \end{aligned}$$

Calculating first, second moment and variance for $\xi$ we obtain:

$$\begin{aligned} E(\xi )&= \xi (0)e^{v_{2}[\alpha +\frac{1}{2}(v_{2}-1)\sigma ^{2}]t} \\ E(\xi ^{2})&= \xi ^{2}(0)e^{2v_{2}[\alpha +(v_{2}-\frac{1}{2})\sigma ^{2}]t} \\ Var(\xi )&= \xi ^{2}(0)e^{2v_{2}\left[ \alpha +\frac{1}{2}(v_{2}-1)\sigma ^{2}\right] t}(e^{v_{2}^{2}\sigma ^{2}t}-1) \end{aligned}$$

Note that since $\alpha +\frac{1}{2}(v_{2}-1)\sigma ^{2}<0$ and $v_{2}<0$ then $E(\xi )$ is increasing in $t$. Finally, by deriving $Var(\xi )$ with respect to $\sigma$ it is easy to check that

$$\begin{aligned} \frac{{\small \partial Var(\xi )}}{{\small \partial \sigma }}{\small =2v}_{2} {\small \sigma te}^{2v_{2}\left[ \alpha +\frac{1}{2}(v_{2}-1)\sigma ^{2} \right] t}{\small \xi }^{2}{\small (0)}\left[ {\small (v}_{2}{\small -1)(e} ^{v_{2}^{2}\sigma ^{2}t}{\small -1)+v}_{2}{\small e}^{v_{2}^{2}\sigma ^{2}t} \right] {\small >0} \end{aligned}$$

That is, as $\sigma$ soars $Var(\xi )$ increases and so does the probability of hitting $\hat{\xi }$ which in turn implies an increase in the long run average deforestation rate.

1.5 A.5 Additional tables

With land conversion run (Table 7).

Table 7 Optimal forest stock and long-run average rate of deforestation under second-best with $B=200$ and $c=500$

Without land conversion run (Tables 8 and 9)

Table 8 Optimal forest stock and long-run average rate of deforestation under second-best with $B=75$ and $c=1500$

Table 9 Optimal forest stock and long-run average rate of deforestation under second-best with $B=200$ and $c=1{,}500$