Concerns for Long-Run Risks and Natural Resource Policy

Abstract

The legislature in many countries requires that short-run risk and long-run risk be considered in making natural resource policy. In this paper, we explore this issue by analyzing how natural resource conservation policy should optimally respond to long-run risks in a resource management framework where the social evaluator has (Duffie and Epstein in Econometrica 60:353–394, 1992; Schroder and Skiadas in J Econ Theory 89:68–126, 1999) continuous-time stochastic recursive preferences. The response of resource conservation policy to long-run risks is reflected into a matrix whose coefficients measure precaution toward short-run risk, long-run risk and covariance risk. Attitudes toward the temporal resolution of risk underly concerns for long-run risks as well as the response of resource conservation policy to future uncertainty. We formally compare the responses of natural resource policy to long-run risks under recursive utility and under time-additive expected utility. A stronger preference for early resolution of uncertainty can prompt a more conservative resource policy as a response to long-run risks. In the very particular case where the social evaluator preferences are represented by a standard time-additive expected utility, long-run risks are not factored in resource conservation policy decisions. Our work also contributes to the so-called Hotelling Puzzle by formally showing that the fundamental Hotelling’s homogeneous resource depletion problem (one without extraction costs, without new discoveries, and without technical progress) can lead to a decreasing shadow price when attitudes toward the temporal resolution of risk are accounted for.

Appendices

Appendix

1.1 Deriving Eq. (10)

To derive Eq. (10), differentiate the maximized Hamilton–Jacobi–Bellman equation (7) with respect to S to obtain

$$\begin{aligned} f_x(x(t),V(t))\frac{\partial x}{\partial S}+f_{V}(x(t),V(t))V_{S}(t)=-\frac{\partial }{\partial S}\Big (\frac{1}{dt}E_{t}dV(S(t))\Big ). \end{aligned}$$

(58)

Assuming that the optimal extraction policy is represented by a smooth function $x(t)=h(S(t))$ of the natural resource stock, the right-hand side of Eq. (58) can be computed from (8) as follows:

$$\begin{aligned} -\frac{\partial }{\partial S}\Big (\frac{1}{dt}E_{t}dV(S(t))\Big )= & {} -\frac{\partial }{\partial S}\left( V_{S}\Big [N(S(t)-x(t)\Big ]+\frac{1}{2}\Gamma ^{2}V_{SS}\right) \nonumber \\= & {} -\underbrace{\left( V_{SS}\Big [N(S(t))-x(t)\Big ]+\frac{1}{2}\Gamma ^{2}V_{SSS}\right) }_{ \frac{1}{dt}E_{t}dV_{S}(S(t))}\nonumber \\{} & {} -V_{S}(t)N'(S(t)+V_{S}(t)\frac{\partial h}{\partial S} \nonumber \\= & {} -\frac{1}{dt}E_{t}dV_{S}(S(t))-V_{S}(t)N'(S(t)+V_{S}(t)\frac{\partial h}{\partial S} \end{aligned}$$

(59)

Plugging Eq. (59) back into Eq. (58) leads to

$$\begin{aligned} f_x(x(t),V(t))\frac{\partial h}{\partial S}+f_{V}V_{S}(t)=-\frac{1}{dt}E_{t}dV_{S}(S(t))-V_{S}(t)N'(S(t))+V_{S}(t)\frac{\partial h}{\partial S}, \end{aligned}$$

(60)

which can be rewritten as

$$\begin{aligned} f_{V}(x(t),V(t))V_{S}= & {} -\frac{1}{dt}E_{t}dV_{S}(S(t))-V_{S}(t)N'(S(t))\nonumber \\{} & {} +\underbrace{ \Big [V_{S}(t)-f_x(x(t),V(t))\Big ]}_{=0}\frac{\partial h}{\partial S}, \end{aligned}$$

(61)

where the last term on the right hand side vanishes since it contains the first order condition (9).

From Eq. (61), it follows that :

$$\begin{aligned} \frac{1}{dt}E_{t}dV_{S}(S(t))=\Big [-f_{V}(x(t),V(t))-N'(S(t))\Big ]V_{S}(t) \end{aligned}$$

(62)

Using the first-order Eq. (9), we may replace $V_{S}$ by $f_{x}(x,V)$, to obtain:

$$\begin{aligned} \frac{1}{f_{x}(x(t),V(t))}\frac{1}{dt}E_{t}df_{x}(x(t),V(t))=-f_{V}(x(t),V(t))-N'(S(t)). \end{aligned}$$

(63)

Using the Multivariate Itô Lemma, the left-hand side of Eq. (63) can be computed as:

$$\begin{aligned} \frac{1}{f_{x}(x(t),V(t))}\frac{1}{dt}E_{t}df_{x}(x(t),V(t))= & {} \Big (\frac{x(t)f_{xx}(t)}{f_{x}(t)}\Big )\left[ \frac{1}{x(t)}\frac{1}{dt}E_{t}dx(t)\right] \nonumber \\{} & {} +V(t)\frac{f_{xV}(t)}{f_{x}(t)}\left[ \frac{1}{V(t)}\frac{1}{dt}E_{t}dV(t)\right] \nonumber \\{} & {} + \frac{1}{2}x^{2}(t)\frac{f_{xxx}(t)}{f_{x}(t)}\sigma _{x}^{2}(t)+\frac{1}{2}V^{2}(t)\frac{f_{xVV}(t)}{f_{x}(t)}\sigma _{V}^{2}(t)\nonumber \\{} & {} + x(t)V(t)\frac{f_{xxV}(t)}{f_{x}(t)}\sigma _{x}(t)\sigma _{V}(t). \end{aligned}$$

(64)

Plugging (64) back into Eq. (63) leads to

$$\begin{aligned}{} & {} \Big (\frac{x(t)f_{xx}(t)}{f_{x}(t)}\Big )\left[ \frac{1}{x(t)}\frac{1}{dt}E_{t}dx(t)\right] \nonumber \\{} & {} \quad +V(t)\frac{f_{xV}(t)}{f_{x}(t)}\left[ \frac{1}{V(t)}\frac{1}{dt}E_{t}dV(t)\right] \nonumber \\{} & {} \quad +\frac{1}{2}x^{2}(t)\frac{f_{xxx}(t)}{f_{x}(t)}\sigma _{x}^{2}(t)+\frac{1}{2}V^{2}(t)\frac{f_{xVV}(t)}{f_{x}(t)}\sigma _{V}^{2}(t)+x(t)V(t)\frac{f_{xxV}(t)}{f_{x}(t)}\sigma _{x}(t)\sigma _{V}(t)\nonumber \\{} & {} \quad = f_{V}(x(t),V(t))-N'(S(t)). \end{aligned}$$

(65)

It follows that

$$\begin{aligned} \underbrace{\frac{1}{x(t)}\frac{1}{dt}E_{t}dx(t)}_{Expected\,pace\,of\,depletion\,of\,natural\,capital}= & {} \underbrace{\mu _{f}(t)}_{Certainty-equivalent\, pace\,at\,time\,t}\nonumber \\{} & {} +\underbrace{{\mathcal {R}}_{f}(t)}_{Risk-\,driven\,pace\, gap\,at\,time\,t}, \end{aligned}$$

(66)

where

$$\begin{aligned} \mu _{f}(t)= & {} \left[ \frac{-x(t)f_{xx}(t)}{f_{x}(t)}\right] ^{-1}\left[ f_{V}(x(t),V(t))-N'(S(t))\right. \nonumber \\{} & {} \left. +\left[ \frac{V(t)f_{xV}(t)}{f_{x}(t)}\right] \left( \frac{1}{V(t)}\frac{1}{dt}E_{t}dV(t)\right) \right] , \end{aligned}$$

(67)

and

$$\begin{aligned} {\mathcal {R}}_{f}(t)= & {} \frac{1}{2}\left[ \underbrace{-\frac{x(t)^{2}(t)f_{xxx}(t)}{x(t)f_{xx}(t)}}_{weight\,on\,short{-}run\,risk}\underbrace{\sigma _{x}^{2}(t)}_{Short{-}run\,risk}\right. \nonumber \\{} & {} \left. \underbrace{- \frac{V^{2}(t)f_{xVV}(t)}{x(t)f_{xx}(t)}}_{weight\,on\,long{-}run\,risk} \underbrace{\sigma _{V}^{2}(t)}_{Long{-}run\,risk}\underbrace{- \frac{2x(t)V(t)f_{xxV}(t)}{x(t)f_{xx}(t)}}_{weight\,on\,covariance\,risk}\underbrace{\sigma _{xV}(t)}_{Covariance\,risk} \right] . \end{aligned}$$

(68)

Proof of Proposition 4.1

Proof

$(1)\,\rightrightarrows \, (2):$ Assume that at  time  t,  $\frac{1}{x(t)}\frac{1}{dt}E_{t}dx(t)-\mu (t)\le (\ge ) 0$  for any  positive  semi-definite  matrix  $\Sigma (t).$ Since $\frac{1}{x(t)}\frac{1}{dt}E_{t}dx(t)-\mu (t)\ge 0 =-\frac{1}{2}trace\Big ({\mathcal {W}}_{f}(t) \Sigma (t)\Big ),$ it follows that

$$\begin{aligned} -\frac{1}{2}trace\Big ({\mathcal {W}}_{f}(t) \Sigma _f(t)\Big ) \le (\ge )\,\, 0 \,for\,any\, \Sigma (t). \end{aligned}$$

(69)

For any given column vector $\Psi (t)$ of size $2\times 1$,⁴² assume that the positive semi-definite matrix is of the form $\Sigma (t)=\Psi (t)\Psi '(t).$ Since ${\mathcal {W}}_{f}(t) \Psi (t)\Psi '(t)=\Psi '(t){\mathcal {W}}_{f}(t)\Psi (t),$ then it follows from (69) that

$$\begin{aligned} -\frac{1}{2}\Psi '(t){\mathcal {W}}_{f}(t)\Psi (t)\,\le (\ge )\,\,0\,\,\,\,for\,\,any\,\,\Psi (t), \end{aligned}$$

(70)

where the row vector $\Psi '(t)$, of size $1\times 2,$ is the transpose of the vector $\Psi (t).$ The inequalities shown in (70) are satisfied for any $\Psi (t)$ if and only if the time-t matrix ${\mathcal {W}}_{f}(t)$ is positive semi-definite (negative semi-definite).

$(2)\,\rightrightarrows \,(1)$: The converse is immediate. $\square$

Proof of Proposition 4.2

Proof

$(1)\,\rightrightarrows \, (2):$ Assume that $\left[ \frac{1}{x(t)}\frac{1}{dt}E_{t}dx(t)-\mu _{f}(t)\right] - \left[ \frac{1}{{\tilde{x}}(t)}\frac{1}{dt}E_{t}d{\tilde{x}}(t)-\mu _{g}(t)\right] \le 0$ for any positive semi-definite matrix $\Sigma (t).$

Since $\left[ \frac{1}{x(t)}\frac{1}{dt}E_{t}dx(t)-\mu _{f}(t)\right] - \left[ \frac{1}{{\tilde{x}}(t)}\frac{1}{dt}E_{t}d{\tilde{x}}(t)-\mu _{g}(t)\right] =-\frac{1}{2}trace\Big \{\left[ {\mathcal {W}}_{f}(t)-{\mathcal {W}}_{g}(t)\right] \Sigma (t)\Big \},$ it follows that at time t, 

$$\begin{aligned} -\frac{1}{2}trace\Big \{\left[ {\mathcal {W}}_{f}(t)-{\mathcal {W}}_{g}(t)\right] \Sigma (t)\Big \}\le \,0\,\,for\,any\,\,\Sigma (t). \end{aligned}$$

(71)

For any column vector $\Psi (t)$ of size $2\times 1$, choosing $\Sigma (t)$ of the form $\Psi (t)\Psi '(t)$ and pursuing to (71) leads to

$$\begin{aligned} -\frac{1}{2}\Psi '(t)\left[ {\mathcal {W}}_{f}(t)-{\mathcal {W}}_{g}(t)\right] \Psi (t)\le \,0\,\,for\,\,any\,\,\Psi (t), \end{aligned}$$

(72)

where the row vector $\Psi '(t)$, of size $1\times 2,$ is the transpose of the vector $\Psi (t).$

The inequality (72)   holds for any vector $\Psi (t)$ if and only if the time-t symmetric matrix ${\mathcal {W}}_{f}(t)-{\mathcal {W}}_{g}(t)$ is positive semi-definite. From proposition 4.1, it is readily seen that ${\mathcal {W}}_{f}(t)$ and ${\mathcal {W}}_{g}(t)$ are positive semi-definite.

$(2)\,\rightrightarrows \, (1):$ The converse is immediate. $\square$

Derivation of the Closed Form Solution for a Hotelling’s Homogeneous Resource Depletion Problem with Schroder and Skiadas (1999) Recursive Utility

To find a closed-form solution to the Bellman equation, we conjecture that the value function is of the form

$$\begin{aligned} V(t)=(\Phi S(t))^{\gamma (1+\alpha )}>0, \end{aligned}$$

(73)

for some positive parameter $\Phi$ to be determined. Substituting the educated guess (73) into (49), it turns out that the optimal extraction policy is given by

$$\begin{aligned} x(t)=\left( \gamma ^{\frac{1}{\gamma -1}}\Phi ^{\frac{\gamma }{\gamma -1}}\right) S(t). \end{aligned}$$

(74)

Substituting the educated guess (73) into the maximized Bellman equation (50) leads to

$$\begin{aligned}{} & {} (1+\alpha ) S^{\gamma \alpha +1}\Big [\frac{ \left( \left( \gamma ^{\frac{1}{\gamma -1}}\Phi ^{\frac{\gamma }{\gamma -1}}\right) S\right) ^{\gamma } }{\gamma }(\Phi S)^{\gamma \alpha }-\beta \Phi ^{\gamma (1+ \alpha )}\Big ] \nonumber \\{} & {} \quad - \left[ \gamma ^{\gamma -1}\Phi ^{\frac{\gamma }{\gamma -1}}\right] S \gamma (1+\alpha )\Phi \Phi ^{\gamma (1+\alpha )-1}S^{\gamma (1+\alpha )-1}\nonumber \\{} & {} \quad + \frac{1}{2}\sigma ^{2}\gamma (1+\alpha ) (\gamma (1+\alpha )-1)\Phi ^2\Phi ^{\gamma (1+\alpha )}S^{\gamma (1+ \alpha )-2}S^{2} \nonumber \\{} & {} \quad =0. \end{aligned}$$

(75)

It follows that

$$\begin{aligned} (1+\alpha )(\Phi S)^{\gamma (1+ \alpha )} \Big [\gamma ^{\gamma -1}\Phi ^{\frac{\gamma }{\gamma -1}}\left( \frac{1}{\gamma }-\gamma \right) -\beta +\frac{1}{2}\sigma ^{2}\gamma (\gamma (\alpha +1)-1) \Big ]=0, \end{aligned}$$

(76)

which gives

$$\begin{aligned} \Phi =\left[ \frac{ \beta + \frac{1}{2}\sigma ^{2}\gamma ( 1- \gamma (\alpha +1)) }{\left( \frac{ 1}{\gamma } -\gamma \right) \gamma ^{\gamma -1}}\right] ^{\frac{\gamma -1}{\gamma } }. \end{aligned}$$

(77)

Substituting Eqs. (73) and (74) into Eq. (48), the expected rate of change of the shadow price of the resource stock, assuming $\gamma >0,$ is obtained as:

$$\begin{aligned} \underbrace{\frac{1}{V_{S}(t)}\frac{1}{dt}E_tdV_{S}(t)}_{Expected\,rate\,of\,change\,of\,the\,resource\,shadow\,price}= & {} \underbrace{ \underbrace{\beta }_{Discount\, rate}}_{>0}+ \underbrace{\underbrace{(1-\gamma )\sigma ^{2}}_{Premium\,for\,Short{-}Run\,\, risk}}_{>0}\nonumber \\{} & {} - \underbrace{ \underbrace{ \alpha \left[ \gamma ^{\frac{1}{\gamma -1}}\Phi ^{\frac{\gamma }{\gamma -1}}+\gamma \sigma ^{2}+\beta \right] }_{Premium\,\, for\,\,temporal\,\, resolution \,\, of\,uncertainty}}_{\lesseqqgtr \,0}, \end{aligned}$$

(78)

where the expression of $\Phi$ is given by Eq. (77).

1.1 The Transversality Condition

Pertaining to (53) and (41), along the optimal path, the stochastic dynamic of the resource stock takes the form

$$\begin{aligned} dS(t)=-\gamma ^{\frac{1}{\gamma -1}}\Phi ^{\frac{\gamma }{\gamma -1}}S(t)dt+\sigma S(t)dB(t),\,\,S_{0}>0\,\,given. \end{aligned}$$

(79)

Therefore

$$\begin{aligned} S(t)=S_{0}\,e^{\left( - |\gamma |^{\frac{1}{\gamma -1}}\Phi ^{\frac{\gamma }{\gamma -1}}-\frac{1}{2}\sigma ^{2}\right) t+\sigma B(t)}\,>0. \end{aligned}$$

(80)

Since $e^{-\nu t} E\left( |V(t)|\right) = e^{-\nu t} E\Big [ (\Phi S(t))^{\gamma (1+\alpha )} \Big ] =e^{-\nu t}\Phi ^{\gamma (1+\alpha )} E\Big [ S(t)^{\gamma (1+\alpha )} \Big ]$, it follows, using Eq. (80), that

$$\begin{aligned} e^{-\nu t} E\left( |V(t)|\right)= & {} \Phi ^{\gamma (1+\alpha )} S_{0}^{\gamma (1+\alpha )}\,e^{\left( -\nu +\gamma (1+\alpha )\left[ - |\gamma |^{\frac{1}{\gamma -1}}\Phi ^{\frac{\gamma }{\gamma -1}} -\frac{1}{2}\sigma ^{2}\right] \right) t}\,\,E\left( e^{ \gamma (1+\alpha )\sigma B(t)}\right) . \end{aligned}$$

(81)

Noting that the term $E\left( e^{ \gamma (1+\alpha )\sigma B(t)}\right)$ is the expected value of a Geometric Brownian Motion, it follows that

$$\begin{aligned} e^{-\nu t}E\left( |V(S(t)|\right)= & {} \Phi ^{\gamma (1+\alpha )} S_{0}^{\gamma (1+\alpha )}\,e^{\left( -\nu +\gamma (1+\alpha )\left[ - |\gamma |^{\frac{1}{\gamma -1}}\Phi ^{\frac{\gamma }{\gamma -1}} -\frac{1}{2}\sigma ^{2}\right] + \frac{1}{2}(\gamma (1+\alpha )\sigma )^{2}\right) t}. \end{aligned}$$

(82)

Then the transversality condition (see footnote 32) is verified if the exponent in (82) is negative; that is

$$\begin{aligned} \nu >\gamma (1+\alpha )\left[ - |\gamma |^{\frac{1}{\gamma -1}}\Phi ^{\frac{\gamma }{\gamma -1}} -\frac{1}{2}\sigma ^{2}\right] + \frac{1}{2}(\gamma (1+\alpha )\sigma )^{2}, \end{aligned}$$

(83)

or

$$\begin{aligned} \nu >\gamma (1+\alpha )\left[ - \left[ \frac{ \beta + \frac{1}{2}\sigma ^{2}\gamma (\gamma (\alpha +1)-1) }{\gamma - \frac{ 1}{\gamma } }\right] -\frac{1}{2}\sigma ^{2}\right] + \frac{1}{2}(\gamma (1+\alpha )\sigma )^{2}, \end{aligned}$$

(84)

by replacing the term $|\gamma |^{\frac{1}{\gamma -1}}\Phi ^{\frac{\gamma }{\gamma -1}}$ by its expression obtained from Eq. (77).

Computational Simulation for the Stationary Optimal Policy with a Schroder and Skiadas (1999) Recursive Utility and a Renewable Resource Growing Nonlinearly

For the computational simulation of the stationary optimal policy with Schroder and Skiadas (1999) recursive utility and a renewable resource growing nonlinearly, we have developed an algorithm that combines the Crank and Nicolson (1974) scheme and the numerical techniques by Achdou et al. (2021). The numerical algorithm computes the value function at each step n iteratively as the solution to a system of linear equations, $L*V^{n+1}=U*V^{n}$, where L and U are large tridiagonal matrices.⁴³

Fig. 3

a Stationary Value Function, b Stationary Extraction Policy. We consider the recursive utility $f(x,V)=(1+\alpha )\Big [\frac{x^{\gamma }}{\gamma }V^{\frac{\alpha }{1+\alpha }}-\beta V\Big ]$, with preference parameters $\beta =3\%$, $\alpha =-0.037$, $\gamma =0.5$, variance of volatility shocks $\Gamma =2$, and nonlinear natural resource growth function $N(S)=r*S^p$, with $r=0.2$ and $p=0.5$

The numerical simulations, shown in Fig. 3, suggest that the value function V(S) of the stationary resource problem is concave, and the extraction policy x(S) depends nonlinearly upon the stock.

We use MATLAB to perform the necessary computations. Our numerical algorithm converges after 3768 iterations, as illustrated in Fig. 4.⁴⁴

Fig. 4

A threshold of $10^{-6}$ verifies the convergence to the stationary value function