The Management of Fragile Resources: A Long Term Perspective

Abstract

Excessive exploitation diminishes the capacity of natural resources to withstand environmental stress, increasing their vulnerability to extreme conditions that may trigger abrupt changes. The onset of such events depends on the coincidence of random environmental conditions and the resource state (determining its resilience). Examples include species extinction, ecosystem collapse, disease outburst and climate change induced calamities. The policy response to the catastrophic threat is measured in terms of its effect on the long-term behavior of the resource state. To that end, the L-methodology, developed originally to study autonomous systems, is extended to non-autonomous problems involving catastrophic threats.

Appendices

Appendix: Proofs

The proofs in this Appendix extend the arguments of Tsur and Zemel (2014b) to non-autonomous problems involving catastrophic threats.

1.1 A Proof of Property 3

Proof

For any feasible state X we compare the value W(X) obtained by the policy $c=M(X)$ with the value obtained from a small feasible variation of this policy. If the value under the variation policy exceeds W(X), then X does not qualify as an optimal steady state.

For arbitrarily small constants $\epsilon >0$ and $\delta$, consider the variation policy

$$\begin{aligned} c^{\epsilon \delta }(t)={\left\{ \begin{array}{ll}M(X)+\delta /g_c(X,M(X))&{}\quad \text { if }\;t<\epsilon \\ M(X(\epsilon ))&{}\quad \text { if }\;t\ge \epsilon \end{array}\right. }\,. \end{aligned}$$

For the short period $t<\epsilon$, this policy deviates slightly from the constant state policy, then it enters a steady state at $X(\epsilon )$. During the initial period $t<\epsilon$, $\dot{X}=g\left( X(t),M(X)+\delta /g_c(X,M(X))\right) =\delta +o(\delta )$,⁶ hence

$$\begin{aligned} X(\epsilon )-X=\epsilon \delta +o(\epsilon \delta ). \end{aligned}$$

Let $\Gamma (t)\equiv \int _0^t[\rho +h(X(s))]ds$ and $g_c=g_c(X,M(X))$. The contribution of $c^{\epsilon \delta }$ to the objective during $t\in [0,\epsilon )$ is evaluated as

$$\begin{aligned}&\int _0^\epsilon U\left( X(t),M(X)+\delta /g_c\right) e^{-\Gamma (t)}dt = \int _0^\epsilon U\left( X(t),M(X)+\delta /g_c\right) e^{-[\rho +h(X)]t}dt \\&\quad +\,\int _0^\epsilon U\left( X(t),M(X)+\delta /g_2\right) \left[ e^{-\Gamma (t)}-e^{-(\rho +h(X))t}\right] dt, \end{aligned}$$

The first integral in the right can be expressed, recalling (3.3), as

$$\begin{aligned}&\int _0^\epsilon U(X,M(X))e^{-[\rho +h(X)]t}dt+\frac{U_c(X,M(X))}{g_c(X,M(X))}\,\epsilon \delta +o(\epsilon \delta ) \\&\quad =W(X)\left[ 1-e^{-[\rho +h(X)]\epsilon }\right] +\frac{U_c(X,M(X))}{g_c(X,M(X))}\,\epsilon \delta +o(\epsilon \delta ), \end{aligned}$$

and the second integral is $o(\epsilon \delta )$.

The contribution of $c^{\epsilon \delta }$ during the infinite period $t\ge \epsilon$ is

$$\begin{aligned}&\int _\epsilon ^{\infty }U(X(\epsilon ),M(X(\epsilon )))e^{-[\rho +h(X(\epsilon ))]t}dt\\&\quad = \int _\epsilon ^{\infty }[\rho +h(X(\epsilon ))]W(X(\epsilon ))e^{-[\rho +h(X(\epsilon ))] t}dt \\&\quad =\int _\epsilon ^{\infty }[\rho +h(X(\epsilon ))]W(X)e^{-[\rho +h(X(\epsilon ))] t}dt\\&\quad \quad + \int _\epsilon ^{\infty }[\rho +h(X(\epsilon ))]W'(X)\epsilon \delta e^{-[\rho +h(X(\epsilon ))] t}dt+o(\epsilon \delta ). \end{aligned}$$

The first integral on the second line can be expressed as

$$\begin{aligned} W(X)\int _\epsilon ^{\infty }[\rho +h(X(\epsilon ))]e^{-[\rho +h(X(\epsilon ))] t}dt \!=\! W(X)e^{-[\rho +h(X(\epsilon ))]\epsilon }\!=\!W(X)e^{-[\rho +h(X)]\epsilon }+o(\epsilon \delta ) \end{aligned}$$

and the second integral is approximated by $W'(X)\epsilon \delta +o(\epsilon \delta ).$

Summing the contributions of both periods gives

$$\begin{aligned} v^{\epsilon \delta }(X) = W(X)+\left( \frac{U_c(X,M(X))}{g_c(X,M(X))}+W'(X)\right) \epsilon \delta +o(\epsilon \delta ), \end{aligned}$$

or

$$\begin{aligned} v^{\epsilon \delta }(X) - W(X) = L(X)\epsilon \delta /[\rho +h(X)]+o(\epsilon \delta ) \end{aligned}$$

(A.1)

where L(X) is defined in (3.4a).

While $\epsilon >0$, the sign of $\delta$ can be freely chosen. Thus, if $L(X)\ne 0$ we can set $\text {sign}(\delta )=\text {sign}(L(X))$ to ensure that $v^{\epsilon \delta }(X)>W(X)$ hence X is not an optimal steady state. It follows that only the roots of $L(\cdot )$ qualify as candidates for optimal steady states. The only exceptions are the bounds $\underline{X}$ and $\bar{X}$. Choosing $\delta >0$ is not feasible at $\bar{X}$ because this policy would lead the process outside the feasible domain. Therefore, $\bar{X}$ cannot be excluded as an optimal steady state if $L(\bar{X})>0$. A similar argument holds for the lower bound $\underline{X}$ if $L(\underline{X})<0$.

B Proof of Property 4

Proof

Consider $S(t)=\exp \left( -\int _0^t h(X(s))ds\right)$ as a second state variable and let $\lambda$ and $\mu$ denote the current-value co-states corresponding to $X(\cdot )$ and $S(\cdot )$, respectively. The current-value Hamiltonian corresponding to the problem of maximizing the objective (2.4) subject to the dynamic constraint (2.1) is

$$\begin{aligned} \mathcal {H}=U(X,c)S+\lambda g(X,c)-\mu h(X)S. \end{aligned}$$

(B.1)

The necessary conditions for (an interior) optimum include:

$$\begin{aligned}&\displaystyle U_c(X,c)S+\lambda g_c(X,c)=0, \end{aligned}$$

(B.2)

$$\begin{aligned}&\displaystyle \dot{\lambda }-\rho \lambda =-[U_X(X,c)S+\lambda g_X(X,c)]+\mu h'(X)S. \end{aligned}$$

(B.3)

and

$$\begin{aligned} \dot{\mu }-\rho \mu =-U(X,c)+\mu h(X). \end{aligned}$$

(B.4)

The last equation is integrated from t to $\infty$, yielding

$$\begin{aligned} \mu (t)=v(X(t)), \end{aligned}$$

where v(X) is the value obtained for the maximal objective when the initial stock is X. Denoting the normalized shadow price by

$$\begin{aligned} \Lambda \equiv \lambda /S, \end{aligned}$$

the necessary conditions take the form

$$\begin{aligned}&U_c(X,c)+\Lambda g_c(X,c)=0,\end{aligned}$$

(B.5)

$$\begin{aligned}&\dot{\Lambda }=[g_X(X,c)-(\rho +h(X))]\frac{U_c(X,c)}{g_c(X,c)}-U_X(X,c)+h'(X)v(X)\equiv \zeta (X,c).\nonumber \\ \end{aligned}$$

(B.6)

At an optimal interior steady state $\hat{X}$, where $c=M(\hat{X})$ and $v(\hat{X})=W(\hat{X})$, we find

$$\begin{aligned} \zeta (\hat{X},M(\hat{X}))=-L(\hat{X})=0, \end{aligned}$$

(B.7)

which agrees with $\Lambda (\cdot )$ being stationary at the steady state.

Next, we express the optimal control c as a function of the state variable X, say $c(t)=C(X(t))$ ⁷ where

$$\begin{aligned} C(\hat{X})=M(\hat{X}). \end{aligned}$$

(B.8)

Define the functions

$$\begin{aligned} A(X)= & {} g_c(X,C(X))U_{cc}(X,C(X))-U_c(X,C(X))g_{cc}(X,C(X)), \end{aligned}$$

(B.9)

$$\begin{aligned} B(X)= & {} g_c(X,C(X))U_{cX}(X,C(X))-U_c(X,C(X))g_{cX}(X,C(X)). \end{aligned}$$

(B.10)

According to assumption (2.2), the expression $A(X)/g_c(X,C(X))$ is strictly negative, which ensures that $\mathcal {H}$ is concave in c. Taking the time derivative of (B.5) and using (B.6) to eliminate $\dot{\Lambda }$, we find

$$\begin{aligned} C'(X)\frac{A(X)}{g_c^2(X,C(X))}+\frac{B(X)}{g_c^2(X,C(X))}+\frac{\zeta (X,C(X))}{g(X,C(X))}=0. \end{aligned}$$

(B.11)

Equation (B.11) is a first order differential equation, which together with (B.8) defines C(X) for all X in the relevant neighborhood. Indeed, for $X\ne \hat{X}$ the coefficient of $C'(X)$ is non vanishing while the other two terms of (B.11) are finite, hence the derivative $C'(X)$ is well defined. A difficulty with its evaluation at $\hat{X}$ arises because the function $g(\cdot ,\cdot )$, appearing at the denominator of the last term, vanishes at $\hat{X}$. However, in an unconstrained steady state, $L(\hat{X})=0$ and the singularity is removed because $\zeta (\hat{X},C(\hat{X}))=\zeta (\hat{X},M(\hat{X}))=-L(\hat{X})=0$ (cf. (B.7)) This term, then, can be evaluated using l’Hôpital’s rule. Using (B.7), we find

$$\begin{aligned} \frac{d\zeta (\hat{X},C(\hat{X}))}{dX}=-L^\prime (\hat{X})+\zeta _c(\hat{X},C(\hat{X}))[C'(\hat{X})-M\,'(\hat{X})], \end{aligned}$$

while (3.2) implies

$$\begin{aligned} \frac{dg(X,C(X))}{dX}=g_X(X,C(X))+g_c(X,C(X))C'(X)=g_c(X,C(X))[C'(X)-M\,'(X)].\end{aligned}$$

It follows that

$$\begin{aligned} \lim _{X\rightarrow \hat{X}}\left\{ \frac{\zeta (X,C(X))}{g(X,C(X))}\right\} =\frac{1}{g_c(\hat{X},C(\hat{X}))} \left( \frac{-L^\prime (\hat{X})}{C'(\hat{X})-M\,'(\hat{X})}+\zeta _c(\hat{X},C(\hat{X}))\right) . \end{aligned}$$

The last term on the right hand side is obtained by taking the derivative of (B.6) with respect to c,

$$\begin{aligned} \zeta _c(X,C(X))=-A(X)\frac{\rho +h(X)-g_X(X,C(X))}{g_c^2(X,C(X))}-\frac{B(X)}{g_c(X,C(X))}, \end{aligned}$$

which, using (3.2), reduces (B.11) in the limit $X\rightarrow \hat{X}$ to

$$\begin{aligned} \frac{A(\hat{X})}{g_c(\hat{X},C(\hat{X}))}\left( C'(\hat{X})-M\,'(\hat{X})-\frac{\rho +h(X)}{g_c(\hat{X},C(\hat{X}))}\right) +\frac{-L^\prime (\hat{X})}{C'(\hat{X})-M\,'(\hat{X})}=0. \end{aligned}$$

Denoting

$$\begin{aligned} \Delta (X)\equiv C'(X)-M\,'(X), \end{aligned}$$

(B.12)

we obtain the quadratic equation

$$\begin{aligned} \Delta ^2(\hat{X})-\frac{\rho +h(X)}{g_c(\hat{X},C(\hat{X}))}\Delta (\hat{X})-\frac{g_c(\hat{X},C(\hat{X}))L^\prime (\hat{X})}{A(\hat{X})}=0. \end{aligned}$$

(B.13)

To determine which of the solutions of (B.13) corresponds to the stable steady-state slope-difference $\Delta (\hat{X})$, observe that the state $\hat{X}$ is attractive only if $g_c(\hat{X},C(\hat{X}))\Delta (\hat{X})\le 0$. To see this, consider a state just below the steady state, say $X_{\varepsilon }=\hat{X}-\varepsilon$. To approach $\hat{X}$ from below requires $\dot{X}=g(X_{\varepsilon },C(X_{\varepsilon }))>0$. Recalling that $g(X_{\varepsilon },M(X_{\varepsilon }))=0$, this implies $g_c[C(X_{\varepsilon })-M(X_{\varepsilon })]>0$, while $g_c[C(\hat{X})-M(\hat{X})]=0$. Recalling that $g_c$ is bounded away from 0, we confirm that $g_c\Delta (\hat{X})\le 0$.

Next, we write the solutions of (B.13) as

$$\begin{aligned} g_c(\hat{X},C(\hat{X}))\Delta (\hat{X})=\frac{\rho +h(X)}{2}\left( 1\pm \sqrt{1+\frac{4L^\prime (\hat{X})g_c^3(\hat{X},C(\hat{X}))}{[\rho +h(X)]^2A(\hat{X})}}\right) . \end{aligned}$$

(B.14)

Since $A(\hat{X})/g_c^3<0$, the argument of the square-root operator above does not fall short of unity only if $L^\prime (\hat{X})\le 0$. In this case, we have one non-positive solution for $g_c\Delta (\hat{X})$ which can provide the boundary value $C'(\hat{X})=M\,'(\hat{X})+\Delta (\hat{X})$ for the differential equation (B.11). In contrast, if $L^\prime (\hat{X})>0$, the argument falls short of unity and the two solutions in (B.14) are either positive or complex, hence (B.11) does not yield a solution that converges to $\hat{X}$. This rules out the possibility that $L^\prime (\hat{X})>0$ at a stable steady state, verifying Property 4.