Thresholds in concave renewable resource models

Ecological Economics 48 (2004) 259 – 267 www.elsevier.com/locate/ecolecon

ANALYSIS

Thresholds in concave renewable resource models Franz Wirl * Industry, Energy and Environment, University of Vienna, Bru¨nnerstr. 72 A-1210 Vienna, Austria Received 4 June 2002; received in revised form 4 September 2003; accepted 15 September 2003

Abstract A number of renewable resource models (some of them just recently published, e.g. about shallow lakes) refer to ‘convexities’ (with respect to the resource stock) as the source for multiple steady states and associated thresholds. This short paper draws attention to the fact that multiple steady states and thresholds are possible in well-behaved (i.e. concave) models too. The ecological consequence is that renewable resource management is sensitive to initial conditions in more cases than so far suggested in the literature. D 2004 Elsevier B.V. All rights reserved. JEL classification: D90; C62; D62 Keywords: Renewable resources; Thresholds; Critical natural capital; Concavity

1. Introduction A number of papers, e.g. starting probably with Lewis and Schmalensee (1982), ranging over Tahvonen and Salo (1996), Tahvonen and Withagen (1996) to recent papers, Brock and Starrett (1999), Dechert and Brock (1999), Ma¨ler (2000), and Ma¨ler et al. (2000), and Rondeau (2001), emphasize the possibility of multiple equilibriums and associated thresholds in models of optimal renewable resource extraction. The common feature of all these papers is that they link the existence of thresholds and of multiple equilibriums to ‘non-concavities’ (i.e. the Hamiltonian of the associated optimal control problem is not jointly concave with respect to state and control so

* Tel.: +43-1-4277-38101; fax: +43-1-4277-38104. E-mail address: [email protected] (F. Wirl). 0921-8009/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2003.09.008

that the sufficient optimality conditions are violated).1 This approach is based on the seminal, theoretical analysis of convex – concave dynamic optimization problems in the works of Skiba (1978) and Dechert and Nishimura (1983); see also the recent and novel treatment by bifurcation techniques in Wagener (2003). Of the recent literature, Brock and Starrett (1999), Dechert and Brock (1999), Ma¨ler (2000), and Ma¨ler et al. (2000), argue that shallow lakes are characterized by a convex –concave dynamic relation and this property (Clark (1990) labels it depensation) can lead to a critical level beyond which the lake ‘collapses’. Rondeau (2001) introduces stock benefits and thus a model very similar to the one below, but assumes a convex –concave utility (due benefits and

1 Clark (1990) mentions multiple equilibriums without this linkage, but the corresponding objective lacks joint concavity.

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damages from wildlife population) to obtain thresholds (‘brinks’) and multiple equilibriums. The common thread of this kind of literature suggests that a history-dependent evolution requires a local convexity, either of utility or of biomass growth. This impression is furthered by the observation that recently published ‘concave’ models, e.g. Li and Lo¨fgren (2000), Heal (1998, 2001) and Ayong Le Kama (2001) do not investigate multiple equilibriums and thresholds although these outcomes are part of the general solution of their model according to the results of this paper. The purpose of this paper is to show that unstable steady states and the ecologically important threshold property can occur in concave renewable resource models too.2 As a consequence, the sensitivity of ecological management with respect to initial conditions extends beyond the familiar models characterized by convex – concave relations, which has obviously important consequences on the design of sustainable eco-systems. The emphasis on thresholds, which characterize critical levels of natural capital, in particular if one of the equilibriums involves a collapse of the eco-system (as e.g. in the works of Brock and Starrett (1999), Dechert and Brock (1999), Ma¨ler (2000), and Ma¨ler et al. (2000)), relates this investigation to those on sustainability. This is one of the most topical issues of ecological economics at least since its ‘definition’ in the famous Bruntland report, WCED (1987); see e.g. Kaufmann and Cleveland (1995), Binswanger (1998), and recent investigations under the heading of ‘critical natural capital’ to which a special issue of this journal is devoted (in particular see the opening by Ekins et al. (2003a,b) and on its applicability). The concept of sustainability has been considered as vague from its very beginning. While Phillis and Andriantiatsaholiniaina (2001) even propose a fuzzy measure, Heal (2001) suggests that sustainability can be linked to optimality. This paper extends the approach in the work of Heal (2001) for stability concerns (see, e.g. Limburg et al. (2002) on nonlinear dynamics but of uncontrolled ecological systems). This approach of optimality and stability analysis allows to apply rigorous and well-established analytical tools, highlights 2

A formally similar point has been made in the context of economic growth in the work of Kurz (1968), which is ignored in the literature on thresholds.

that optimality is not sufficient for sustainability, and can provide some policy guidance, in particular if one of the multiple equilibriums implies extinction.

2. The model We consider the following renewable resource extraction model: Z l max expðrtÞuðhðtÞ; RðtÞÞdt; ð1Þ

fhðtÞz0g 0

˙ ¼ gðRðtÞÞ  hðtÞ; Rð0Þ ¼ R0 ; RðtÞz0: RðtÞ

ð2Þ

That is, a society (or a monopolistic firm) harvests (h) a renewable resource stock (R). The resource grows according to the biological growth function g(.) that is positive over [0, R¯], and where R¯ corresponds to the carrying capacity=unharvested steady state, lim RðtÞ ¼ R¯ for h(t)=0 and R0>0. Instantat!l neous utility (u) consists of two parts: 1. consumptive utility (or profits) due to harvesting; 2. non-consumptive uses (bird watching, forests for recreation and as a protection against landslides, etc.) provided by the resource stock. Variants of this renewable resource extraction model have been studied in literally hundreds of papers and Bach (2001) is a special application to tropical forests. Berck (1981) is the first that allows explicitly for nonconsumptive benefits (but of a separable nature, u(h, R)=u1(h)+u2(R), and overlooks the multiple equilibriums); Clark et al. (1979) include the resource stock in Eq. (1) but to account for its impact on the catch rate and in a way that the resulting objective is not jointly concave. These two papers are extended in the work of Wirl (1995, 1999) to study limit cycles in a higher dimensional framework. The other recent examples, which account for stock effects, investigate the following: Heal (1998) concentrates on different intertemporal welfare objectives, Li and Lo¨fgren (2000) focuses on the consequences of declining discount rates (hyperbolic discounting) on socially optimal, intertemporal environmental policy making, Rondeau (2001) considers utility functions u that are nonconcave with respect to R, and Ayong Le Kama and

F. Wirl / Ecological Economics 48 (2004) 259–267

Alain (2001) combines Eqs. (1) and (2) with a Ramsey model of optimal saving leading to two states (capital and resource). The already mentioned recent applications to (shallow) lakes emphasize the convex– concave shape of g but neglect non-consumptive benefits from the resource stock. Since the emphasis of this paper is on ‘concave’ renewable resource models, the following assumptions are made: Assumption 1 . The benefit function u(h, R) is C2, increasing in both arguments and concave in (h, R) and strongly concave in the harvest, uhh<0. Assumption 2 . The growth function g(.) is C2, positive and concave over the open interval (0, R¯), has two roots, g(0)=g(R¯)=0, and satisfies g V(0)>r. These assumptions are in line with the textbook model of renewable resource extraction. In particular, g(.) is of the familiar inverted-U shape, and g V(0)>r implies the existence of a positive steady state of the resource stock. An important consequence of Assumptions 1 and 2 is that the (current value) Hamiltonian of the optimal control problem (Eqs. (1) Eqs. (2)), which is defined as Hðh; R; kÞuuðh; RÞ þ kf ðh; RÞ;

ð3Þ

is concave in (h, R), since the resource shadow price k is positive for non-satiating utilities (see below). As a consequence and in contrast to the convex – concave models, the first order conditions are sufficient if the limiting transversality condition is satisfied (see Feichtinger and Hartl, 1986). The numerical calculations and figures plus some analytical derivations are based on linear-quadratic and separable benefits u and the familiar logistic growth g: uðh; RÞ ¼ h  1=2ah2 þ wR;

ð4Þ

gðRÞ ¼ Rð1  RÞ:

ð5Þ

The simple and standard specifications in Eqs. (4) and (5) stress that it does not require complicated functional forms to derive the following results. Whether u refers to utility or to profits, specification (4) implies saturation at hsat=1/a. Assumption 1 requires uh>0Zh<1/a, which is satisfied for a<4, because the maximum sustainable yield equals 1/4=

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max{R(1R), Ra[0, 1]} for the logistic growth (Eq. (5)); we will relax the assumption about non-satiating utility for particular purposes. The resource stock provides direct and according to Eq. (4) linear benefits (wR); this linearity is not crucial and similar results can be obtained for a concave relation, e.g. for w(R – 1/2R2) that induces satiation of non-consumptive uses at the carrying capacity R¯=1.

3. Optimality conditions and stability analysis The first order optimality conditions for the control problem (Eqs. (1) Eqs. (2)) are (for an interior policy, see, e.g. Seierstad and Sydsaeter, 1986): Hh ¼ uh  k ¼ 0;

ð6Þ

k˙ ¼ ðr  gVÞk  uR :

ð7Þ

The Hamiltonian maximizing condition (6) defines implicitly the optimal control, h ¼ FðR; kÞ; with the partial derivatives FR ¼ uhR =uhh and Fk ¼ 1=uhh < 0;

ð8Þ

and ensures k=uh>0 since u is increasing in h. Substituting Eq. (8) into the state (6) and costate (7) differential equations yields the canonical equations (listing the arguments), R˙ ¼ gðRÞ  FðR; kÞ; k˙ ¼ ðr  g VðRÞÞk  uR ðFðR; kÞ; RÞ; with the associated Jacobean: 0 1 uhR 1 gV þ  B C uhh uhh C: J ¼B @ u2 uhR A hR  uRR  kg W r  g V uhh uhh

ð9Þ ð10Þ

ð11Þ

Since trðJ Þ ¼ r > 0; at least one of the two eigenvalues of J,

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 e12 ¼ trðJ ÞF trðJ Þ2  4detðJ Þ ; 2

ð12Þ

ð13Þ

is positive (or has positive real parts) so that all steady

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states of the dynamic system (Eqs. (9) and (10)) are unstable (strictly speaking). The determinant (using the shorthand HRR=URR+kg W) is

detðJ Þ¼

½gV2 uhh þ 2gVuhR þ HRR  rgVuhh  ruhR : uhh ð14Þ

A negative determinant, det(J ) <0, implies that one of the eigenvalues in Eq. (13) is negative. This property characterizes a saddle point, which although unstable has a stable manifold of dimension one that determines the optimal policy; with some abuse of language and notation ‘stability’ refers in the following always to the conditional stability of a saddle point, since this is the maximum of stability that the canonical Eqs. (9) and (10) allow for. A positive determinant, det( J )> 0, is then necessary and sufficient to exclude even this conditional stability: both eigenvalues are positive (for any concave framework, see Feichtinger and Wirl, 2000) so that the associated steady state is an unstable node. The square bracket in Eq. (14) is a negative definite quadratic form, because H is concave given the assumptions about u and g (and since k >0 for positive interior harvests due to Eq. (6) and H hh =u hh and H hR =u hR ). Therefore, det( J )>0 is only possible if either (i) gV> 0, or (ii) uhR < 0. Condition (ii) states that a larger resource stock reduces the marginal utility from exploiting and consuming this resource; i.e. more forests and fish or a better environment in general reduce the marginal benefit from cutting an additional tree, catching an additional fish or releasing a unit of waste into the ecosphere. Since such preferences seem unlikely (only interdependencies of uhRz0 are sensible), an instability due to a negative mixed derivative of u can be ruled out in the context of renewable resource exploitation (but not in other instances, see Feichtinger and Wirl, 2000). Therefore, the following analysis is simplified by assuming separable benefits (as in the works of Berck, 1981 and Wirl, 1999): Assumption 3. Benefits u are separable, i.e. uhR=0.

Assumption 3 leaves condition (i) as the only route to instabilities, multiple equilibriums and thresholds. Furthermore, this condition (i) seems easy to satisfy, after all, it is satisfied in the standard renewable resource model, uR=0, that yields the characterization r =g V>0 for the optimal steady state resource stock; this familiar solution is denoted R* with the associated stationary harvest h*=g(R*). Yet this common and traditional steady state of an economically optimal harvesting policy is not capable to generate an instability, because substituting this condition r =g V into the Jacobean using in addition Assumption 3 yields: detðJ Þ ¼ 

½uRR þ kgW

< 0: uhh

ð15Þ

Therefore, any steady state at which gV=r is stable, moreover, this conventional stationary outcome R* is also unique given the concavity of the growth function and its existence is guaranteed for gV(0)>r (according to Assumption 2). Furthermore, g V>r is impossible at a stable steady state for a (positively valued) resource, because the corresponding stationary resource stock would then be below the level associated with a policy interested in the catch only. Hence, only a steady state satisfying r >g V>0 is a candidate for an unstable equilibrium. Rewriting the determinant (Eq. (14)) in the following way: HRR uhR detðJ Þ ¼ g Vðr  g VÞ  þ ðr  2g VÞ ; ð16Þ uhh uhh and accounting for the simplification of a separable utility function in Assumption 3 yields the following condition for instability: gVðr  gVÞ > HRR =uhh :

ð17Þ

This instability criterion (Eq. (17)) can be easily met. It will be demonstrated in detail for Eqs. (4) and (5), which imply the optimal harvest h=F(R, k)= (1k)/a (due to Eq. (6)) and the following canonical equations system: R˙ ¼ Rð1  RÞ  ð1  kÞ=a;

ð18Þ

k˙ ¼ kðr  gVÞ  w; gV¼ 1  2R:

ð19Þ

This system has (up to) three steady states, because substituting the stationary solution of Eq. (19), El=w/

F. Wirl / Ecological Economics 48 (2004) 259–267

(r– gV), into Eq. (18) results in a cubic equation (in R); the corresponding roots are cumbersome and thus suppressed. The Jacobean as well as its determinant simplify to: 0 1 1 B gV a C J ¼@ ð20Þ A; 2k r  gV detðJ Þ ¼ gVðr  gVÞ  2w=½aðr  gVÞ :

ð21Þ

The following phase diagram and bifurcation analysis is similar to that of Clark (1990, pp. 185– 187). However, Clark’s model lacks, as already mentioned, joint concavity so that it is difficult to figure out which of the mechanisms, either the (implicit) increasing returns in fishing, or the condition established in this paper, is the driving force behind multiple equilibriums and thresholds. The R˙=0 isocline, k=1aR(1R) in the case of Eqs. (4) Eqs. (5), is U-shaped and hits the ordinate at that level of the costate, which corresponds to no exploitation. This implies k!l for a utility function satisfying the Inada condition, uh!l for h! 0, and k=1 for the linear-quadratic specification used in Fig. 1. This U-shaped isocline has its bottom at the maximum sustainable yield, Rmsy, and is thus decreasing below this level and increasing beyond (for any growth of bio-mass). Moreover, the minimum of the R˙=0 isocline is in the positive

263

quadrant due to Assumption 1 (in general, if consumption of the maximum sustainable yield is below the saturation level). The signs of the elements of the Jacobean in Eq. (20) imply that the resource stock is increasing above and decreasing below the R˙=0 isocline (these properties hold generally and not only for the particular example). The k˙ =0 isocline, k=w/ [rg(R)], is declining over the relevant domain r> gV and has a pole at R*. Again the laws of motions follow from the Jacobean such that the costate increases above this isocline and decreases below it. Summarizing, the dynamic orientations shown by the arrows in Fig. 1 result and the optimal strategies are given by the bold saddle point paths (accounting for the non-negativity of exploitation, h=(1k)/az0Z kV1). The results of this phase diagram analysis can be summarized as follows: Proposition 1. Concave renewable resource models (i.e. models (1) and (2) satisfying Assumptions 1 –3) allow for multiple equilibriums and thresholds. The u threshold is determined by an unstable steady state Rl (a node) in the domain r>g V>0, i.e. between the intertemporally optimal stationary resource stock R* (satisfying g V(R*)=r) in the absence of direct (‘nonconsumptive’) benefits and the maximum sustainable yield, denoted Rmsy (characterized by g V(Rmsy)=0). A u threshold Rl leads to history-dependent evolutions with different steady state outcomes: R!Rl(the u ‘high’ steady state resource stock) for R0>Rl , and

Fig. 1. Phase diagram in the (R, k) plane with multiple steady states for Eqs. (4) and (5).

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R!Rl (the ‘low’ steady state resource stock) for u R0
threshold, while any initial condition to the right will trigger an evolution to a high level equilibrium (a large biomass, a clean environment, etc.)—that is important from an economic and ecological point of view, and that dwarfs the differences, which concern details of the optimal strategies. The phase diagram analysis is complemented by a bifurcation analysis of the steady states and their (local) stability properties. The corresponding numerical examples prove the existence of multiple equilibriums and thresholds. Assuming a=3.9<4 ensures a positive marginal utility at all feasible steady states. Hence, saturation associated with specification (4) is not crucial, because the same result holds if this linear-quadratic utility function is only a local approximation that is extrapolated in a continuously and differentiable way satisfying nonsatiability and the law of diminishing marginal utilities globally. In addition, r = 0.9 < gV(0) =1 so that Assumption 2 is satisfied and a positive steady state, R*=0.05, is guaranteed for the pure harvesting case (w=0) since gV=12R=r=0.9; it will be shown below that small discount rates allow for an instability too. Fig. 2 shows a bifurcation diagram for the parameter w. This parameter weighs ‘green’ preferences (and thus allows for an easy interpretation) and shifts the k˙ =0 isocline upward leaving the R˙=0 isocline unaffected so that different kinds of

Fig. 2. Steady states and their stability properties depending on the parameter w that measures the direct benefits from the resource stock R; r = 0.9, a=3.9.

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intersections can be easily traced. Starting at very low values of w, only a single and stable steady state exists, after all the limiting case w=0 is known to yield a single steady state, here Rl=R*=0.05. This uniqueness is retained for small spillovers due to a single intersection of the isoclines. Yet for w=0.0225, the isocline k˙ =0 is shifted upward to the point that this isocline becomes tangent to R˙=0 (in this example at the maximum sustainable yield3, R=Rmsy=1/2). At w = 0.11297. . ., this isocline is raised such that it is again tangent of R˙=0 (now at level below the sustainable yield and above R˙=0 such that the determinant of the Jacobean is again zero) so that further increases of w allow only for a single steady state. As a consequence, three steady states result for 0.02250, which is uncommon in pure economic resource models but natural for ecological considerations, is responsible for this ‘strange’ result rather than the arithmetic condition, r > g V> 0. Proposition 2 . Assume that u=u(h) thus uR=0 for all R, that uV(hmsy)<0 (satiating over feasible stationary harvest levels), and that the concavity requirements of Assumptions 1 and 2 hold, then the stationary outcome depends on the initial conditions if an unstable steady state (determining the threshold) satisfying r >gV>0 exists. Proposition 2 highlights that it is the domain r >g V> 0 and thus this kind of ‘growth’ that fosters instability. That is, stock spillovers are helpful, however, it is not the stock effect per se but the associated move of the steady state into the critical domain, the open interval (R*, Rmsy). However, multiple equilibriums and thresholds in the absence of resource spillovers require to drop the assumption of non-satiating utility of consumption. In order to prove the existence claimed in Proposition 2, we choose the utility parameter a in the example in Fig. 1 such that consumption equal to the maximum 3

This follows arithmetically from Eqs. (18) and (19).

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sustainable yield exceeds the saturation level, hsat< hmsyug(Rmsy), yet ensuring a positive marginal utility at the traditional, optimal stationary harvest policy h*ug(R*) so that uVðh*Þ > uVðhsat Þ ¼ 0 > uVðhmsy Þ:

ð22Þ

This condition (22) can be always met for the specification (4) by choosing a such that the saturating exploitation hsat=1/a is between the ‘conventional’ steady state optimum and the maximum sustainable harvest, hsata( g(R*), g(Rmsy)). The fact that satiation can induce multiple equilibriums is already noted in the work of Clark (1990, p 184). Nevertheless, the analysis is here performed for three reasons: first, our model ensures concavity of the Hamiltionan, (the shadow price k remains non-negative along all paths, satiating consumption corresponds to k=0, see the phase diagram in Fig. 3); second, to verify that (R*, Rmsy) is the critical domain; and third to prove that small discount rates allow also for multiple equilibriums. The phase diagram in Fig. 3 is drawn in the state-control plane, because this highlights better the consequences of satiation and a stationary harvest satisfying Eq. (22). Neglecting the spillover, uR=0, and eliminating the costate from the canonical Eqs. (9) and (10) using Eq. (6), yields a differential equations system in state and control: R˙ ¼ gðRÞ  h;

ð23Þ

uVðhÞ h˙ ¼ ðr  gVðRÞÞ: uWðhÞ

ð24Þ

The R˙=0 isocline is determined by the biological growth function, h=g(R). There are two candidates for h˙=0, either r = g V(R) and thus R=R*, or uV= 0 so that h=hsat. The second possibility, h=hsat, intersects R˙=0 twice—due to the construction (Eq. (22)) and the property of an inverted U-shape of g—first below at R1 and then at R2 above the maximum sustainable yield, R*
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Fig. 3. Phase diagram of Eqs. (23) and (24) and thus without stock effects with an unstable steady state R1 in the domain r >gV>0.

gence to either the traditional steady state solution, R!R*, or R!R2. This verifies the existence of a threshold separating the two attracting and stable steady states. In addition, it is worth noting that starting at R0a(R*, R1) allows for sustaining the harvest h0=R0(1R0)>h*=R*(1R*) forever. Nevertheless, choosing instead the saddlepoint strategy (i.e. starting closer to hsat and thus at a level exceeding h0) and heading to the lower consumption level R* is optimal for R00.

4. Final remarks Recent and past investigations of optimal renewable resource management stress the possibility of thresholds, multiple equilibriums and of history dependence, which are of obvious importance for ecological considerations. They all refer either to a (locally) convex utility or to (locally) convex growth of the bio-mass (‘depensation’). This suggests that a (locally) convex shape (with the consequence that the sufficient optimality conditions do not hold) is the only source for a threshold and the purpose of this paper is to draw attention to another mechanism that is of particular relevance for ecological problems. The mechanism for multiple equilibriums and thresholds in concave renewable resource models is growth, r>gV> 0, that characterizes steady states between the traditional optimal harvest policy and the maximum sustainable yield. It arises in particular if direct (non-consumptive) resource benefits (of a moderate) magnitude is introduced. Summing up, also concave renewable resource models allow for an unstable equilibrium with economic and ecological consequences similar to the convex frameworks found in the literature, i.e. the existence of a threshold such that it is optimal to attain a high level equilibrium (a clean lake, a large biomass,

F. Wirl / Ecological Economics 48 (2004) 259–267

etc.) for starting above this threshold and a low level equilibrium (possibly extinction) for an initial condition below the threshold level. The important policy implication is that the long-run consequences of even optimally managed renewable resources are sensitive to initial conditions in more cases than so far suggested in the literature. Acknowledgements The author acknowledges helpful suggestions and comments from three anonymous referees and the editor of the journal (Prof. Cleveland). References Ayong Le Kama, A.D., 2001. Sustainable growth, renewable resources and pollution. Journal of Economic Dynamics and Control 25, 1911 – 1918. Bach, C.F., 2001. Economic incentives for sustainable management: a small optimal control model for tropical forestry. Ecological Economics 30, 251 – 265. Berck, P., 1981. Optimal management of renewable resources with growing demand and stock externalities. Journal of Environmental Economics and Management 11, 101 – 118. Binswanger, H.C., 1998. Making sustainability work. Ecological Economics 27, 1 – 3. Brock, W.A., and Starrett, D., 1999. Nonconvexities in Ecological Problems. Mimeo, University of Wisconsin. Clark, C.W., 1990. Mathematical Bioeconomics. The Optimal Management of Renewable Resources. Wiley, New York. Clark, C.W., Clarke, F.H., Munro, G.R., 1979. The optimal exploitation of renewable resource stocks: problems of irreversible investment. Econometrica 47, 25 – 47. Dechert, D.W., and Brock, W.A., 1999. Lakegame. Mimeo. Dechert, D.W., Nishimura, K., 1983. Complete characterization of optimal growth paths in an aggregative model with a non-concave production function. Journal of Economic Theory 31, 332 – 354. Ekins, P., Simon, S., Deutsch, L., Folke, C., De Groot, R., 2003. A framework for the practical application of the concepts of critical natural capital and strong sustainability. Ecological Economics 44, 165 – 185. Ekins, P., Folke, C., De Groot, R., 2003. Identifying critical natural capital. Ecological Economics 44, 159 – 163. ¨ konomischer Gustav, F., Hartl, R.F., 1986. Optimale Kontrolle O Prozesse. de Gruyter, Berlin. Feichtinger, G., Wirl, F., 2000. Instabilities in concave, dynamic, economic optimization. Journal of Optimization Theory and Applications 107, 277 – 288. Hartl, R.F., Kort, P., Feichtinger, G., Wirl, F., 2004. Multiple equilibria and thresholds due to relative investment costs: non-concave – concave, focus – node, continuous – discontinuous. Journal of Optimization Theory and Applications (in press).

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