Sustainable Exploitation of a Natural Resource: A Satisfying Use of Chichilnisky’s Criterion

Abstract

Chichilnisky’s criterion for sustainability has the merit to be, so far, the unique explicit, complete and continuous social welfare criterion that combines successfully the requirement of efficiency with an instrumental notion of intergenerational equity (no dictatorship of the present and no dictatorhsip of the future). But it has one drawback: when applied in the context of renewable resources, and with a constant discount factor, there exists no exploitation path that maximizes this criterion. The present article suggests a way to cope with this problem. The idea is to restrict attention to the set of convex combinations between the optimal discounted utilitarian program and the stationary program leading to the green golden rule. It is shown that an optimal path in this set exists under rather weak sufficient conditions on the fundamentals of the problem. Some ethical properties of this approach are also discussed. In some cases, it turns out that the restricted solution implies no loss of efficiency and benefits intermediate and infinitely distant generations.

The authors would like to thank Hubert Stahn, Antoine Soubeyran, their collegues at LAMETA (in particular those of the PERENE team), the participants at the ISDG worshop in Segovia, September 2006, Gilles Rotillon, Katheline Schubert, Michel De Lara, Vincent Martinet and Luc Doyen, and Graciela Chichilnisky for interesting and helpful comments. Many thanks also to an anonymous referee for his/her constructive suggestions.

Originally published in Economic Theory, Volume 49, Number 2, February 2012 DOI 10.1007/ s00199-010-0573-7.

Appendix

Appendix A: Existence of a Stationary Program Leading the Green Golden Rule

The green golden rule problem reads as

$$\begin{aligned} \max _{\left( c,x\right) }\ U\left( c,x\right) \ , \end{aligned}$$

(30)

subject to

$$\begin{aligned} x =G(x-c)\ , 0 \le c\le x\ . \end{aligned}$$

(31)

It is worth starting the analysis of this problem with an investigation of the possible steady states, noted generically $(c^{s},x^{s})$, before looking for those among them giving the highest utility. Fix $c=0$. G(.) is concave, $G^{\prime }(0)>1$ and lim$_{\ x\rightarrow +\infty }\ G(x)<x,$ therefore there are only two steady states, $x^{s}=0$ which is unstable and another one, $x^{\sup },$ which is globally asymptotically stable: indeed, starting from a positive initial stock lower (resp. larger) than the steady state, the resource monotonically increases (resp. decreases) while consumption is constant (zero actually), therefore the utility function is non-decreasing (resp. non-increasing) along the trajectory and it can be used to construct a Lyapunov function to prove the global asymptotic stability of $x^{\sup }$. Now for any non-zero stationary consumption, $0<c<x_{t},\forall t,$ applying the implicit function theorem to relation ( 31), one can check that the larger the stationary consumption, the lower the stable stationary stock:

$$\begin{aligned} \left. \frac{dx}{dc}\right| _{(c,x^{\sup })}=\frac{G^{\prime }}{ G^{\prime }-1}<0\ , \end{aligned}$$

(32)

where the inequality obtains since at any positive steady state, function G(.) crosses the 45$^\circ$ line from above, hence $0<G^{\prime }<1$. To summarize, as the stationary consumption increases away from zero, the locus of steady states $(c,x^{\sup })$ is characterized by lower stationary stocks. Then, in order to find the best steady state, note that the relevant space of stationary consumptions and stocks is compact:

$$\begin{aligned} x\in \left[ 0,x^{\sup }\right] ,\ c\in \left[ 0,x^{\sup }\right] \end{aligned}$$

(33)

Since the transition function is bijective, we can write:

$$\begin{aligned} c=x-G^{-1}(x)\le x\ \ \text {for}\ \ x\ge 0. \end{aligned}$$

(34)

Then rewrite the problem as follows:

$$\begin{aligned} \max _{x\in \left[ 0,x^{\sup }\right] }\ U\left[ x-G^{-1}(x),x\right] \ . \end{aligned}$$

(35)

There necessarily exists a solution since the function to be optimized is continuous and defined over a compact set. The solution cannot be zero since, by Assumption 4, the above function is strictly increasing at $x=0$. Let us note $0<c^{*}<x^{*}$ this solution ($c^{*}=x^{*}$ is not possible because $G\left( 0\right) =0$ would imply $c^{*}=x^{*}=0$ for the second steady state as well).

It remains to show the existence of a stationary plan leading to the GGR. Clearly a possible, though not unique, such plan is the linear one:

$$\begin{aligned} c_{t}=\frac{c^{*}}{x^{*}}x_{t}\ . \end{aligned}$$

(36)

Plugging back this plan into the dynamics, one has:

$$\begin{aligned} x_{t}=G\left[ (1-\frac{c^{*}}{x^{*}})x_{t}\right] =g\left( x_{t}\right) . \end{aligned}$$

(37)

Function g(.) is similar to function G(.) upon a positive linear transformation of its arguments (since $c^{*}<x^{*}$): its dynamic properties are the same, the economy converges towards the golden rule $\left( c^{*},x^{*}\right) .$ $\square$

Appendix B: Admissibility of Convex Combinations

To state that $\left\{ c_{t}^{\gamma }\right\} _{t=0}^{\infty }$ is admissible it must be proven that: (i) $c_{t}^{\gamma }\le x_{t}\ ,\forall t$ and (ii) $\lim _{t\rightarrow \infty }c_{t}^{\gamma }=c^{\gamma }<\infty .$ Part (i) is true since

$$\begin{aligned} c_{t}^{\gamma }\le \max \left\{ \phi _{DU}\left( x_{t}\right) ,\phi _{GGR}\left( x_{t}\right) \right\} \le x_{t}. \end{aligned}$$

(38)

As for part (ii), note that

$$\begin{aligned} \lim _{t\rightarrow \infty }c_{t}^{\gamma }= & {} \gamma \lim _{t\rightarrow \infty }\phi _{DU}\left( x_{t}\right) +\left( 1-\gamma \right) \lim _{t\rightarrow \infty }\phi _{GGR}\left( x_{t}\right) \\= & {} \gamma x^{DU}+\left( 1-\gamma \right) x^{GR}<\infty . \end{aligned}$$

$\square$

Appendix C: Continuity of Convex Combinations

This property can be established recursively. Note first that

$$\begin{aligned} \gamma \phi _{DU}\left( x_{0}\right) +\left( 1-\gamma \right) \phi _{GGR}\left( x_{0}\right) \end{aligned}$$

(39)

varies continuously with $\gamma .$ Therefore, $x_{1}^{\gamma }$ is also a continuous function of $\gamma$ because G(.) is a continuous function. Now it is easy to see that if this property holds for $x_{s}^{\gamma }\ ,\forall s\le t$ for some t,  it must hold for $x_{t+1}^{\gamma }$ as well by Assumption 2, which completes the proof. $\square$

Appendix D: Existence of a Restricted Optimal Program for CSWC

Under Assumptions 1, 2 and 3, each term in the series

$$\begin{aligned} J_{T}=\sum _{t=0}^{T}\beta ^{t}U(c_{t}^{\gamma },x_{t}^{\gamma }) \end{aligned}$$

(40)

is continuous in $\gamma .$ Let $U^{\sup }=\max \left\{ \left| \underline{U}\right| ,\left| \overline{U}\right| \right\} .$ Under the boundedness condition in Assumption 1

$$\begin{aligned} J_{\infty }^{s}= & {} \theta \sum _{t=0}^{\infty }\left\| \beta ^{t}U(c_{t}^{\gamma },x_{t}^{\gamma })\right\| _{\infty }+\left( 1-\theta \right) \lim _{t\rightarrow \infty }U(c_{t}^{\gamma },x_{t}^{\gamma }) \\\le & {} \theta U^{\sup }\sum _{t=0}^{\infty }\beta ^{t}+\left( 1-\theta \right) U^{\sup }=\frac{\theta U^{\sup }}{1-\beta }+\left( 1-\theta \right) U^{\sup }<\infty \ , \end{aligned}$$

where $\left\| .\right\| _{\infty }$ denotes the sup norm. By definition:

$$\begin{aligned} J^{s}= & {} \theta \sum _{t=0}^{\infty }\beta ^{t}U(c_{t}^{\gamma },x_{t}^{\gamma })+\left( 1-\theta \right) \lim _{t\rightarrow \infty }U(c_{t}^{\gamma },x_{t}^{\gamma }) \\\le & {} J_{\infty }^{s}=\theta \sum _{t=0}^{\infty }\left\| \beta ^{t}U(c_{t}^{\gamma },x_{t}^{\gamma })\right\| _{\infty }+\left( 1-\theta \right) \lim _{t\rightarrow \infty }U(c_{t}^{\gamma },x_{t}^{\gamma })<\infty . \end{aligned}$$

So the series

$$\begin{aligned} J_{T}^{s}=\theta \sum _{t=0}^{T}\beta ^{t}U(c_{t}^{\gamma },x_{t}^{\gamma })+\left( 1-\theta \right) U(c_{T}^{\gamma },x_{T}^{\gamma }) \end{aligned}$$

(41)

converges normally towards the function $J^{s}$, which implies that it also converges uniformly, that is

$$\begin{aligned} \forall \epsilon>0, \exists N\in \varvec{N}, \forall \gamma \in \left[ 0,1\right] ,\ \ \forall T>N,\ \left| J_{T}^{s}-J^{s}\right| <\epsilon . \end{aligned}$$

(42)

This last property ensures that $J^{s}$ is a continuous function of $\gamma .$ The demonstration is then completed, since the continuous mapping of a compact set is itself a compact set, with a maximal point. $\square$

Appendix E: Proof of Proposition 3

The problem is to show that the coefficient of the optimal linear feedback,

$$\begin{aligned} \mu =\frac{ \left( 1-\alpha \right) \left( \alpha \beta -1\right) \left( \theta \beta -\beta +1\right) }{\left( 1-\theta \right) \left[ \alpha \beta \left( 1-\beta \right) +\beta \right] +\theta \alpha -1+\pi \left[ \alpha \beta \left( 1+\alpha -2\theta \right) -\alpha \left( 1-\theta \right) \left( 1+\alpha \beta ^{2}\right) \right] }\ , \end{aligned}$$

(43)

is actually a convex combination

$$\begin{aligned} \gamma \frac{1-\alpha \beta }{1+\pi \alpha \beta }+\left( 1-\gamma \right) \frac{1-\alpha }{1+\pi \alpha }\ ,\ \ \gamma \in \left[ 0,1\right] \ , \end{aligned}$$

(44)

of the coefficients of the discounted utilitarian optimal feedback and of the green golden rule.

Solving the equation

$$\begin{aligned}&\gamma \frac{1-\alpha \beta }{1+\pi \alpha \beta }+\left( 1-\gamma \right) \frac{1-\alpha }{1+\pi \alpha } \\&\quad =\frac{ \left( 1-\alpha \right) \left( \alpha \beta -1\right) \left( \theta \beta -\beta +1\right) }{\left( 1-\theta \right) \left[ \alpha \beta \left( 1-\beta \right) +\beta \right] +\theta \alpha -1+\pi \left[ \alpha \beta \left( 1+\alpha -2\theta \right) -\alpha \left( 1-\theta \right) \left( 1+\alpha \beta ^{2}\right) \right] } \end{aligned}$$

for $\gamma$, one finds

$$\begin{aligned} \gamma =\frac{\left( \alpha -1\right) \theta \left( \pi \alpha \beta +1\right) }{D}\ , \end{aligned}$$

(45)

where

$$\begin{aligned} D= \begin{array}{c} \beta -\pi \alpha +\alpha \beta \left( 1+\pi -\beta +\pi \alpha -\pi \alpha \beta \right) -1 \\ +\,\theta \left( \alpha -\beta +\pi \alpha -\alpha \beta -2\pi \alpha \beta +\alpha \beta ^{2}+\pi \alpha ^{2}\beta ^{2}\right) \end{array} \ . \end{aligned}$$

(46)

Let us investigate the properties of the above expression, viewed as a function $\gamma \left( \theta \right)$. Note first that

$$\begin{aligned} \gamma \left( 0\right) = 0\ ,\ \ \ \ \gamma \left( 1\right) =1\ . \end{aligned}$$

(47)

Also,

$$\begin{aligned} \gamma ^{\prime }\left( \theta \right) =\frac{N}{D^{2}}>0\ . \end{aligned}$$

(48)

where

$$\begin{aligned} N=\left( \pi \alpha \beta +1\right) \left( \pi \alpha +1\right) \left( 1-\alpha \beta \right) \left( 1-\beta \right) \left( 1-\alpha \right) >0\ . \end{aligned}$$

(49)

This means that $\gamma \left( \theta \right) \in \left[ 0,1\right] ,$ hence the result. $\square$