## Abstract

The Chichilnisky criterion is an explicit social welfare function that satisfies compelling conditions of intergenerational equity. However, it is time inconsistent and has no optimal solution in the Ramsey model. By investigating stationary Markov equilibria in the game that generations with Chichilnisky preferences play, this paper shows how, nevertheless, this criterion can be practically implemented in the Ramsey model, leading to attractive consequences. The time-discounted utilitarian optimum is the unique equilibrium path with a high-productive initial stock, implying that the weight on the infinite future in the Chichilnisky criterion plays no role. However, this part of the Chichilnisky criterion may lead to more stock conservation than the time-discounted utilitarian optimum with a low-productive initial stock. Based on the notion of von Neumann–Morgenstern abstract stability, we obtain uniqueness by assuming that each generation coordinates on an almost best equilibrium and takes into account that future generations will do as well.

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## Notes

Jackson and Yariv (2015) provide another perspective on the time inconsistency of the Chichilnisky criterion. They show that any Pareto-efficient and non-dictatorial aggregation of heterogeneous time preferences leads to time inconsistency. A sustainable preference is essentially a Pareto-efficient and non-dictatorial aggregation of positive and zero time preference.

It would be desirable to apply Chichilnisky’s criterion to more general models, e.g., with resources and risk. Given the complexity of doing so even in the Ramsey model, this is a task for future research. An application like Botzen et al. (2014) represents the future by the end period of the DICE model—not the infinite future—and does not address the problem of time inconsistency.

The first term,

*pf*(*k*), is the value of net production, while the negative of the second term, \(- \dot{p}k = (-\dot{p}/p)pk\), is the cost of holding capital, with \(-\dot{p}/p\) being the consumption interest rate. Hence, \(pf(k) + \dot{p}k\) can be interpreted as profit. Note that (3) cannot be defined at time at which*c*and thus*p*is not differentiable.The case where \(\underline{k} = \inf I \notin I\) or \(\underline{k} = \sup I \notin I\) does not correspond to an equilibrium since the

*I*-optimal stock path converges to \(\underline{k}\) which is not in*I*(contradicting Definition 1). The case where \(\underline{k} < \inf I \notin I\) or \(\underline{k} > \sup I \notin I\) does not correspond to an equilibrium since it must generate an*I*-optimal pair (by Proposition 4) and there is no*I*-optimal pair in this case (by Propositions 2).To see this, consider, e.g., the case where there is some \(i \in \{1, \ldots , n\}\) such that \(\underline{k} = \max I_i\). Since \(\{I_1, \ldots , I_n\}\) is a partition of \(\mathbb {R}_{++}\), there is \(j \in \{1, \ldots , n\}\) such that \(\underline{k} = \inf I_j \notin I_j\). However, as pointed out in footnote 4, this does not correspond to a one-attractor equilibrium strategy.

To see this, suppose that there is an extreme point \(k' \le \underline{k}\) of some interval \(I_i\). Footnote 5 implies that \(k' < \underline{k}\). By case (c) of Sect. 4, this implies that \(k' = \max I_i\). Since \(\{I_1, \ldots , I_n\}\) is a partition of the set \(\mathbb {R}_{++}\), there is \(j \in \{1, \ldots , n\}\) such that \(k' = \inf I_j \notin I_j\), where, for every \(k_0 \in I_j\), the solution \(\kappa (\cdot ,k_0) : [0, \infty ) \rightarrow I_j\) to \(k(0) = k_0\) and \(\dot{k} = f(k) - \sigma |_{I_j}(k)\) for \(t \in [0, \infty )\) has the property that \(k_{\infty } = \lim _{t \rightarrow \infty }\kappa (t,k_0)\) satisfies \(k_{\infty } = \max I_j < \underline{k}\) or \(k_{\infty } = \underline{k}\). Furthermore, if \(k_0 \in (k', k_{\infty })\), then \(\dot{\kappa }(t,k_0) > 0\) for all \(t \in (0, \infty )\), so that \(\sigma |_{I_j}(k) < f(k)\) for all \(k \in (k', k_{\infty })\). Therefore, Corollary 1 implies

$$\begin{aligned} V'_{\sigma }(k) = (1 - \alpha )\delta u'(\sigma |_{I_j}(k)) < (1 - \alpha ) f'(k) u'(f(k)) < u'(f(k)) f'(k) = \tfrac{\hbox {d}}{\hbox {d}k}[u\left( f(k)\right) ] \end{aligned}$$by the strict concavity of

*u*since \(f'(k) > \delta \) for \(k \in (k', k_{\infty })\) and \(\alpha > 0\). Combining this observation with steps 1 and 2 of the proof of Lemma 6 contradicts that \(V_{\sigma }\) is continuous at \(k'\).Since, as shown in Sect. 6, \([\underline{k},k] \subseteq K\) for some \(k > \bar{k}\), there are equilibrium strategies for which the stock converges to some \(k_{\infty } > \bar{k}\) for initial stocks \(k_0\) satisfying \(k_0 \in (\bar{k}, \infty )\).

This upper bound is constructed by maximizing the two parts of the C-criterion separately, taking into account that the stock cannot be accumulated beyond \(\hat{k}\) for any \(\hat{k} \ge \underline{k}\) and \(\sigma \in \varSigma \).

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## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

The paper builds on Ekeland et al. (2013) and has been circulated under the title “Generations playing a Chichilnisky game.” We are grateful for helpful discussions with Graciela Chichilnisky and Bård Harstad and useful comments received at SURED 2014, the 9th Tinbergen Institute Conference, the 11th ESWC, the 2nd FAERE conference and a seminar at Gothenburg University. Asheim’s research is part of the activities at the Centre for the Study of Equality, Social Organization, and Performance (ESOP) at the Department of Economics at the University. ESOP is supported by the Research Council of Norway through its Centres of Excellence funding scheme, project number 179552. Ekeland’s research has been supported by the Chaire de Développement Durable and the Laboratoire de Finance des Marchés de l’Énergie at the Université de Paris-Dauphine.

## Appendix: Proofs

### Appendix: Proofs

###
*Proof*

(Proof of Lemma 1) It is easily checked that the set of all *I*-feasible consumption paths is convex. Since *u* is strictly concave, the TDU criterion is strictly concave, and the maximum, if it exists, is unique. \(\square \)

###
*Proof*

(Proof of Lemma 2) Assume that the *I*-feasible pair \((k^{*}(t), c^{*}(t))\) is *I*-competitive and satisfies the CVT condition. For any other *I*-feasible pair (*k*(*t*), *c*(*t*)) we have, using the definition of \(p^{*}(t)\) and the concavity of *u*:

Hence, by Eq. (1) and the property that (3) holds for almost all *t*:

Integrating by parts the right-hand side, and using the fact that \(k\left( 0\right) =k_{0}=k^{*}\left( 0\right) \):

Letting \(T\rightarrow \infty \) (keeping in mind that limits exist) and using the CVT condition:

Hence, \((k^{*}(t), c^{*}(t))\) is *I*-optimal. \(\square \)

###
*Proof*

(Proof of Lemma 4) Assume that the pair \((I, \sigma )\) is a one-attractor stationary Markov equilibrium and fix an arbitrary \(k_0 \in I\), leading to a unique and absolutely continuous capital path, \(\kappa (t;k_0)\) converging to \(k_\infty \).

*Step 1*: \(\sigma \)
*is continuous.* Let \(\tau _\infty (k_0)\) be the finite or infinite time at which \(k_\infty \) is reached. Since \(\sigma \) is Markovian, the capital path from \(k_0 \in I\) is constant if \(k_0 = k_\infty \), increasing on \([k_0, k_\infty )\) if \(k_0 < k_\infty \) and decreasing on \((k_\infty , k_0]\) if \(k_0 > k_\infty \).

Assume \(k_0 \ne k_\infty \). Let \(\tau (\cdot ;k_0)\) denote the inverse function of \(\kappa (\cdot ;k_0)\), defined on \([k_0, k_\infty )\) if \(k_0 < k_\infty \) and on \((k_\infty , k_0]\) if \(k_0 > k_\infty \). For fixed \((I, \sigma )\), write \(J(k_0, I, \sigma ) = (1-\alpha ) V(k_0) + \alpha \lim _{\rho \rightarrow 0^+} \left( \rho \int _0^{\infty } e^{-\rho t} u(\sigma (\kappa (t;k_0))) \hbox {d}t \right) \), so that \(V(k_0)\) is the value of the TDU part of the C-criterion when \(\sigma \) is followed from \(k_0\).

if \(k_0 < k_\infty \) and

if \(k_0 > k_\infty \). By means of this change of variable, we will show that \(\sigma \) must be continuous for all values of \(k_0\) if \(\sigma \) is an equilibrium strategy.

For values of \(k_0\) for which \(\sigma \) is continuous, we have that

Therefore, for values of \(k_0\) for which \(\sigma \) is continuous, it follows that

independently of whether \(k_0 < k_\infty \) or \(k_0 > k_\infty \).

Since \(\sigma \) is an equilibrium strategy, it follows that for all values of \(k_0\) for which \(\sigma \) is continuous,

by considering a deviation from \(\sigma \) in a sufficiently short time interval \((0, \varDelta )\). Since \(\sigma \) is an equilibrium strategy, it follows also that for all values of \(k_0\) for which \(\sigma \) is continuous,

since otherwise staying put at \(k_0\) for \(t \in (0, \varDelta )\) by choosing \(c(t) = f(k_\infty )\) would be a profitable deviation. Furthermore, Eqs. (11) and (12) imply that \(V(k_0) - u(\sigma (k_0)) > 0\) if \(k_0 < k_\infty \) and \(V(k_0) - u(\sigma (k_0)) < 0\) if \(k_0 > k_\infty \).

Suppose that \(\tilde{k}\) is a point of discontinuity of \(\sigma \). Write \(c^- = \lim _{k_0 \uparrow \tilde{k}}\sigma (k_0)\) and \(c^+ = \lim _{k_0 \downarrow \tilde{k}}\sigma (k_0)\). It follows from Eqs. (11) and (12) that

Since \(\tilde{k}\) is a point of discontinuity of \(\sigma \), we have that \(c^- \ne c^+\). However, if Eq. (13) is satisfied with strict inequality,^{Footnote 8} then the strict concavity of *u* implies that the equation

is solved by a unique \(\tilde{c}' < f(\tilde{k})\), corresponding to the case where \(\tilde{k} < k_\infty \) and a unique \(\tilde{c}'' > f(\tilde{k})\), corresponding to the case where \(\tilde{k} > k_\infty \). This contradicts that both (14) and (15) can be satisfied and proves that \(\sigma \) must be continuous for \(k_0 \ne k_\infty \).

It remains to be shown that \(\sigma \) is continuous at \(k_\infty \). With \(k_0 = k_\infty \), Eq. (13) is satisfied with equality. Furthermore, *V* is continuous at \(k_\infty \) as (i) \(\sigma (\kappa (t;k_0)) \rightarrow f(k_\infty )\) for all *t* as \(k_0 \rightarrow k_\infty \) in the case where \(\tau _\infty (k_0) = \infty \) for \(k_0 \ne k_\infty \), and (ii) \(\tau _\infty (k_0) \rightarrow 0^+\) as \(k_0 \rightarrow k_\infty \) otherwise. It follows from the strict concavity of *u* and the property that \(\sigma (k_0)\) solves (16) for \(\tilde{k} = k_0\) that \(\sigma (k_0)\) approaches \(f(k_\infty )\) continuously also in case (ii).

*Step 2:*
\((\kappa (t;k_0), \sigma (\kappa (t;k_0)))\)
*satisfies* Eq. (3) *for almost all*
\(t \in [0, \infty )\). Since \((I, \sigma )\) is a one-attractor stationary Markov equilibrium, then there exists \(\varDelta > 0\) such that

for every *I*-admissible choice \(c_{k_0,\varDelta }\) where the solution \(k : [0, \infty ) \rightarrow I\) to \(k(0) = k_0\) and \(\dot{k} = f(k) - c_{k_0, \varDelta }(t)\) for \(t \in [0, \infty )\) satisfies that \(k(\varDelta ) = k_1 := \tau (\varDelta ,k_0)\), since then \(\sigma (\kappa (t,k_0)) = c_{k_0, \varDelta }(t)\) for \(t \in (\varDelta , \infty )\).

There exists a pair \(\hat{k} : [0, \varDelta ] \rightarrow I\) and \(\hat{c} : [0, \varDelta ] \rightarrow \mathbb {R}_{+}\) such that (i) \(\hat{c}\) is absolutely continuous, so that the associated present-value price path \(\hat{p}(t)\) defined \(\hat{p}(t) = e^{-\delta t}u'(\hat{c}(t))\) is differentiable almost everywhere, (ii) Eq. (3) is satisfied for almost all \(t \in [0, \varDelta ]\), and (iii) \(\hat{k}(0)=k_0\) and \(\hat{k}(\varDelta )=k_1\). By (10) and the strict concavity of the TDU criterion, we have that

if \((\kappa (t,k_0), \sigma (\kappa (t,k_0)))\) does not coincide with \((\hat{k}(t),\hat{c}(t))\) for \(t \in [0,\varDelta ]\), contradicting (17). Hence, the pair \((\kappa (t,k_0), \sigma (\kappa (t,k_0)))\) satisfies Eq. (3) for almost all \(t \in [0, \varDelta ]\). Since \(k_0 \in I\) is arbitrary, it follows that \((\kappa (t;k_0), \sigma (\kappa (t;k_0)))\) satisfies Eq. (3) for almost all \(t \in [0, \infty )\).

*Step 3:*
\((\kappa (t;k_0), \sigma (\kappa (t;k_0))\)
*is*
*I*-*competitive and satisfies the CVT condition.* By Step 1, \(\sigma : I \rightarrow \mathbb {R}_{++}\) is continuous, so that, for every \(k_0 \in I\), the solution \(\kappa (t;k_0)\) has the properties that \(\sigma (\kappa (t;k_0))\) is an absolutely continuous function of *t* and, by Step 2, \((\kappa (t;k_0), \sigma (\kappa (t;k_0))\) satisfies Eq. (3) for almost all \(t \in [0, \infty )\). Moreover, by Definition 1, \((\kappa (t;k_0), \sigma (\kappa (t;k_0))\) is *I*-feasible, so that \((\kappa (t;k_0), \sigma (\kappa (t;k_0))\) is *I*-competitive. Finally, by Definition 1, \(\kappa (t;k_0)\) converges, implying that \((\kappa (t;k_0), \sigma (\kappa (t;k_0))\) satisfies the CVT condition. \(\square \)

###
*Proof*

(Proof of Corollary 1) Assume that the pair \((I, \sigma )\) is an equilibrium, and let \(k_0 \in \hbox {int}I\). Let \(V(k_0)\) be defined as in the proof of Lemma 4. We need to show that \(V'(k_0)\) exists and equals \(\delta u'(\sigma (k_0))\).

*Case 1:*
\(k_0 \ne k_\infty \). The result follows from (12) of the proof of Lemma 4.

*Case 2:*
\(k_0 = k_\infty \) Since \(k_0 \in \hbox {int}I\), we are in case (a) of the cases considered in the text preceding the corollary, so that \(k_0 = \underline{k}\). The result follows since

is differentiable as a function of \(k_0\) at \(k_0 = \underline{k}\) and \(V(k_0) \ge u(f(k_0))\) for all \(k_0 \in I\).

\(\square \)

###
*Proof*

(Proof of Lemma 5) Included in the main text. \(\square \)

###
*Proof*

(Proof of Lemma 6) Let \(\sigma = \{(I_1, \sigma _1), \ldots , (I_n, \sigma _n)\}\) be a multiple-attractor equilibrium strategy.

*Step 1: At a point of discontinuity*
\(k'\)
*of*
\(V_{\sigma }, V_{\sigma }(k') = u(f(k'))\). By Propositions 2 and 4 and the observation that \(V_{\sigma }\) cannot be discontinuous at \(\underline{k}\), if \(k'\) is a point of discontinuity, then there is \(i \in \{1, \ldots , n\}\) such that *either*
\(I_i \subset (0, \underline{k})\) and \(k' = \max I_i\)
*or*
\(I_i \subset (\underline{k}, \infty )\) and \(k' = \min I_i\). In both cases,

since \(\sigma |_{I_i}(k') = f(k')\) and \(\delta \int _0^{\infty } e^{- \delta t} \hbox {d}t = 1\).

*Step 2: For all*
\(k' \in (0, \underline{k}), V_{\sigma }(k') \ge u(f(k'))\). By Propositions 2 and 4, if \(k' \in (0, \underline{k})\), then there is \(i \in \{1, \ldots , n\}\) such that \(k' \in I_i\), where the solution \(\kappa (\cdot , k') : [0, \infty ) \rightarrow I_i\) to \(k(0) = k'\) and \(\dot{k} = f(k) - \sigma |_{I_i}(k)\) for \(t \in [0, \infty )\) has the property that \(k_{\infty } = \lim _{t \rightarrow \infty }\kappa (t,k')\) satisfies \(k_{\infty } = \max I_i < \underline{k}\) or \(k_{\infty } = \underline{k}\). In either case,

since \(k' \le k_{\infty }\), and both *f* and *u* are strictly increasing. Moreover, since the pair (*k*(*t*), *c*(*t*)) with \(k(t) = k'\) and \(c(t) = f(k')\) for all \(t \in [0, \infty )\) is \(I_i\)-feasible, it follows from the \(I_i\)-optimality of \((\kappa (t,k'), \sigma _i(\kappa (t,k')))\) that

Hence, by (18) and (19), \(V_{\sigma }(k') \ge (1 - \alpha )u(f(k')) + \alpha u(f(k')) = u(f(k'))\).

The two steps imply that discontinuity of the value function \(V_{\sigma }\) for \(k \in (0, \underline{k})\) is inconsistent with \(V_{\sigma }\) being upper semi-continuous. Combined with Lemma 5, this establishes the result since, as argued in footnote 5, \(V_{\sigma }\) cannot be discontinuous at \(\underline{k}\). \(\square \)

###
*Proof*

(Proof of Lemma 7) It follows from Corollary 1 and the definition of \(v_{k_\infty }(k_0)\) that

for \((k_\infty , k_0) \in [\underline{k}, \infty ) \times (k_\infty , \infty )\), by the strict concavity of *u* and the properties of *I*-competitive paths in the phase diagram in (*k*, *c*)-space. \(\square \)

###
*Proof*

(Proof of Lemma 9) Consider the \(\varepsilon \)-system \((D, \succ _{\varepsilon })\) for the Ramsey–Chichilnisky game, where \(\varepsilon \) is some positive number. Suppose that \(\succ _{\varepsilon }\) is *not* strictly acyclic, that is, there exists an infinite sequence \(\{(k_1,\sigma _1), (k_2,\sigma _2), \ldots , (k_j,\sigma _j) , \ldots \}\) of elements in *D* such that, for all \(j \in \mathbb {N}, (k_{j+1},\sigma _{j+1}) \succ _{\varepsilon } (k_j,\sigma _j)\). In each of two exhaustive cases, this leads to a contradiction.

*Case 1. There exists*
\(j \in \mathbb {N}\)
*such that*
\(k_{j} \in (0, \underline{k}]\). If \(k \in (0, \underline{k}\,]\), then \((k, \sigma ) \in D\) only if \(\sigma = \{(I_1, \sigma |_{I_1}), \dots , (I_n, \sigma |_{I_n})\}\) satisfies that there is \(\kappa \in (\underline{k}, \infty ) \cup \{\infty \}\) such that \(I_1 = (0,\kappa )\) and \(\sigma |_{I_1} = \sigma _{\underline{k}}|_{(0,\kappa )}\). Furthermore, by Lemma 8(i), \(k'\) is reachable from \(k \in (0, \underline{k}]\) if and only if \(k' \in (0, \underline{k}]\). Therefore, for any \(\varepsilon > 0\), if \(k_j \in (0, \underline{k}\,]\), then there is no \((k_{j+1}, \sigma _{j+1}) \in D\) such that \((k_{j+1}, \sigma _{j+1}) \succ _{\varepsilon } (k_j, \sigma _j)\). This establishes the contraction in this case.

*Case 2. There does not exist*
\(j \in \mathbb {N}\)
*such that*
\(k_{j} \in (0, \underline{k}]\). By Lemma 8(ii), \(\{k_j\}_{j \in \mathbb {N}}\) is a non-increasing sequence; hence, there is \(\hat{k} \in [\underline{k}, \infty )\) such that \(\hat{k} = \lim _{j \rightarrow \infty }k_j\). The set \(\{V_\sigma (\hat{k}) : \sigma \in \varSigma \}\) is bounded, with—as argued in Sect. 6—\(\upsilon _{\underline{k}}(\hat{k})\) being the greatest lower bound, and \((1-\alpha )\sup _{A\left( \hat{k}\right) } \delta \int _{0}^{\infty }e^{-\delta t}u(c(t))\hbox {d}t + \alpha u(f(\hat{k}))\) being an upper bound.^{Footnote 9} For any \(\zeta > 0\) and any \(\sigma \in \varSigma , V_\sigma (k)\) is Lipschitz continuous on \([\hat{k}, \hat{k} + \zeta ]\). Hence, for any \(\varepsilon > 0\), the existence of an infinite sequence \(\{(k_j,\sigma _j), (k_{j+1},\sigma _{j+1}), \ldots , (k_{j+\ell }, \sigma _{j+\ell }), \dots ) \}\) with \(\hat{k} \le \cdots \le k_{j+\ell } \le \cdots \le k_{j+1} \le k_j \le \hat{k} + \zeta \) such that, for all \(\ell \in \mathbb {N}, (k_{j+\ell +1},\sigma _{j+\ell +1}) \succ _{\varepsilon } (k_{j+\ell },\sigma _{j+\ell })\) contradicts that the set \(\{V_\sigma (\hat{k}) : \sigma \in \varSigma \}\) is bounded, for \(\zeta > 0\) chosen sufficiently small. \(\square \)

###
*Proof*

(Proof of Lemma 10) If \(k \in (0, \underline{k}\,]\), then \((k, \sigma ) \in D\) if \(\sigma = \{(\mathbb {R_{++}}, \sigma _{\underline{k}})\}\) and \((k, \sigma ) \in D\) only if \(\sigma = \{(I_1, \sigma |_{I_1}), \ldots , (I_n, \sigma |_{I_n})\}\) satisfies that there is \(\kappa \in (\underline{k}, \infty ) \cup \{\infty \}\) such that \(I_1 = (0,\kappa )\) and \(\sigma |_{I_1} = \sigma _{\underline{k}}|_{(0,\kappa )}\). Furthermore, \(k'\) is reachable from \(k \in (0, \underline{k}]\) if and only if \(k' \in (0, \underline{k}]\). Therefore, for any \(\varepsilon > 0\), if \(k \in (0, \underline{k}\,]\) and \((k, \sigma ) \in D\), then there is no \((k', \sigma ') \in D\) such that \((k', \sigma ') \succ _{\varepsilon } (k, \sigma )\). This establishes the lemma. \(\square \)

###
*Proof*

(Proof of Lemma 11) Using Proposition 6, Lemma 7 and expressions (8) and (9), it follows that, for all \(\kappa \in (\underline{k}, \bar{k} \, ], u(f(\kappa )) = \max _{\sigma \in \varSigma }V_\sigma (\kappa )\). Fix \(k \in (\underline{k}, \bar{k}\,]\). If \(\sigma = \{(I_1, \sigma |_{I_1}), \ldots , (I_n, \sigma |_{I_n})\} \in \varSigma \), then \((k, \sigma ) \in G_{\epsilon }\) if \(u(f(k')) - V_\sigma (k') \le \varepsilon \) for all \(k' \in (\underline{k}, k\,]\), since then there is no \((k', \sigma ') \in D\) such that \((k', \sigma ') \succ _{\varepsilon } (k, \sigma )\), using the observation in the proof of Lemma 10 that no domination is possible for \(k' \in (0, \underline{k})\).

It follows that, for all \(k' \in (\underline{k}, k\,], (k', \sigma ') \in G_\varepsilon \), provided that \(\sigma ' = \{(I'_1, \sigma '|_{I'_1}), \ldots , (I'_n, \sigma '|_{I'_n})\} \in \varSigma \) satisfies that \(u(f(k'')) - V_{\sigma '}(k'') \le \varepsilon \) for all \(k'' \in (\underline{k}, k'\,]\). This is consistent with \(I'_n = [k' , \infty )\) and \(\sigma '|_{I'_n} = \sigma _{k'}|_{I'_n}\) so that \(V_{\sigma '}(k') = u(f(k'))\). This shows that \((k, \sigma ) \in G_{\epsilon }\) only if \(u(f(k')) - V_\sigma (k') \le \varepsilon \) for all \(k' \in (\underline{k}, k\,]\), since otherwise there is \((k', \sigma ') \in G_\varepsilon \) such that \((k', \sigma ') \succ _{\varepsilon } (k, \sigma )\).

###
*Proof*

(Proof of Lemma 12) Suppose \(k \in (\bar{k}, \infty ), (k, \sigma ) \in G_\varepsilon \) and \(v_{\bar{k}}(k') - V_\sigma (k') > \varepsilon \) for some \(k' \in (\underline{k}, k]\). By internal \(\varepsilon \)-stability, there does not exist \((k', \sigma ') \in G_\varepsilon \) where \(\sigma ' = \{(I_1, \sigma '|_{I_1}), \ldots , (I_n, \sigma '|_{I_n})\}\) satisfies that \(I_n = [\bar{k}, \infty )\). It follows from Lemmas 10 and 11 that, by choosing sufficiently many points of discontinuity in \((\underline{k}, \bar{k}\,]\), there exists \(\sigma ' = \{(I_1, \sigma '|_{I_1}), \ldots , (I_n, \sigma '|_{I_n})\}\) such that, for all \(k'' \in [0, \bar{k}\,], (k'', \sigma ') \in G_\varepsilon \). Hence, by external \(\varepsilon \)-stability, there exists \(k'' \in (\bar{k}, k'\,]\) and \((k'', \sigma '') \in G_\varepsilon \) such that \(V_{\sigma ''}(k'') - v_{\bar{k}}(k'') > \varepsilon \). However, by Lemmas 5 and 7, \(v_{\bar{k}}(k'') > V_\sigma (k'')\) for \(k'' \in (\bar{k}, k'\,]\) so that \(V_{\sigma ''}(k'') - V_{\sigma }(k'') > \varepsilon \) and \((k'', \sigma '') \succ _{\varepsilon } (k, \sigma )\). Since \((k, \sigma ), (k'', \sigma '') \in G_\varepsilon \), this contradicts internal \(\varepsilon \)-stability and shows that \(v_{\bar{k}}(k') - V_\sigma (k') \le \varepsilon \) for all \(k' \in (\underline{k}, k]\) if \(k \in (\bar{k}, \infty )\) and \((k, \sigma ) \in G_\varepsilon \). \(\square \)

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Asheim, G.B., Ekeland, I. Resource conservation across generations in a Ramsey–Chichilnisky model.
*Econ Theory* **61**, 611–639 (2016). https://doi.org/10.1007/s00199-016-0965-4

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DOI: https://doi.org/10.1007/s00199-016-0965-4