Renewable resource management and extinction

JOURNAL OF ENVIHONMENTAL ECONOMICS AND MANAGEMENT 5, 198-205 ( 1978) zyxwvutsrqponmlkjihgfedcbaZ NOTE Renewable Resource and Extinction zyxwvutsrqponmlkjihgfedc Management zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ COLIN W. CLARK Department of Mathematics, The University of British Columbia, Vancouver, Canada V6T 1 W5 AND GORDON R. MUNRO’ Department of Economics, The University of British Columbia, Vancouver, Canada V6T 1W5 R.eceived October 31, 1977; revised February 23, 1978 Several authors have noted that extinction of a biological resource could be consistent with a policy of maximizing the discounted present value of economic rent. However, the arguments put forward in support of this assertion have hitherto been based on autonomous models. In this note we discuss the nonautonomous case, which turns out to be considerably more difficult to analyze. In studies of optimization models in renewable resource management, several authors have recently observed that a profit maximizing policy could in theory lead to the extinction of the resource stock.2 According to these authors, a necessary and sufficient condition for the optimality of an extinction policy is simply that the own rate of interest of the resource stock be less than the discount rate, at all stock levels greater than zero. Examples of such “ economically exhaustible,” but biologically renewable, resources may include whales and other cndangercd species of wildlife (Clark [S]). Upon reexamining the argument’s in support of this proposition, wc find that they have all been based on autonomous models, that is to say, on models in which the parameters are assumed to remain constant over time. The real world is, of course, far from autonomous. In this note we wish to explore briefly the consequences of relaxing the restrictive autonomy assumption. (We shall, however, retain two other questionable assumptions : (a) that future parameter values, while not necessarily constant, are known with certainty, and (b) that questions of intergenerational equity are adequately roprcsented by the usual presentvalue criterion.3) 1All correspondence should be addressed to Prof. Munro at the above address. * Clark [4, 51; Beddington et al. [3]; Smith [9] ; Neher [7]. 3 See below. 198 0095-0696/ 78/ 0052-0198$02.00/ O Copyright 0 1978 by Academic Press, Inc. All rights of reproduction in +ny form reserved. RENEWABLE RESOURCES AND 199 EXTINCTION Since biological extinction is by definition irreversible, t’hr above necessary and sufficient conditions, resting as t,hey do upon current’ information alone, prove in fact to be quite inadequate, unless one can assume that the parameters of t’he model are rigidly time invariant. Unfortunately, as we shall observe, the own-rate-of-interest decision rule apparently has no simple operational extension to the general situation-each case must be investigated on its own. Our renewable resource model is formulated as follows [S] : * maximize / a(t){?,(t) - c(x, Olh(Odt (I) zyxwvutsr 0 subject to dx,‘ dt = F(x) - h(t), 0 L h(t) x(0) = x0 (2) (3) 5 hnm, 0 < r(t), where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA a(t) = exp[ -~iS(s)ris] (4) and 6(t) = discount rate at time t, x(t) = biomass at time t (the state variable), P(t) = price of a unit of harvested resource at t.ime t, c(x, t> = unit harvest cost at time t, h(t) = rate of harvest at time t (the control variable), F(x) = natural growth rate of biomass, h LllaX= maximum feasible harvest rate. t) are assumed to In this model, the functions F(x), 6(t), p(t), c(x, t), and h,,,(x, be given, whereas the state variable z(t) and control variable h(t) are to be determined, for all t > 0, by means of the maximization requirement. It will be observed that the model is linear in the control variable h(t) ; this linearity hypothesis, while somewhat restrictive, does allow for definitive analytical results in the nonautonomous case. It does not seem possible to derive equally definitive results for a nonlinear nonautonomous model. WC shall assume that the biological production function F(x) satisfies F(0) = F(K) = 0, F(x) > 0 (0 < x < K), F” (x) < 0 (5) This implies that a ‘(minimum viable population” does not exist, or in other zyxwvutsrqponmlkji words, that the biomass can be reduced to an arbitrarily small posit,ive level without leading to the extinction of the resource. Taken literally, this assumption is obviously unrealistic. Nevertheless the present model seems worthy of study, first as an approximation to the case where the minimum viable population is small, and second because the previous studies mentioned above have asserted the possible optimality of extinction even under this extreme assumption. For the autonomous zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC cast [S], the optimal harvest policy is characterized in terms of an optimal equilibrium biomass level CC”,which must sahisfy the “modi- CLARK AND MUNRO zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR 200 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA fied golden-rule” equation4 c’(x*)F(x*) F’ (x*) - __--- P - = (6) 6. ccx*> In certain cases this equation may fail to possess a solution x* > 0, and in these cases the optimal stock level is x * = 0, i.e., extinction is optimal (except in the uninteresting case where p < c(x) for all x). As shown by Clark [5], the conditions P > c(0) and 6 > F’(0) (7) are necessary in order to have x * = 0, while the slightly stronger conditions P > c(0) and 6 > 2F’(O) (3) are sufficient. In all cases, the optimal approach to equilibrium is the most rapid approach (or “bang-bang” approach), utilizing h(t) = 0 or h,,, to adjust the initial biomass ~(0) as rapidly as possible to the optimal level x*.~ In the nonautonomous case [S] the optimal harvest policy is characterized in : terms of the silzgular solution X* = z*(t), which satisfies zyxwvutsrqponmlkjihgfedcbaZY aT/ax F’ (x*) + __ = s(t) _ ;;;;, an/ah a h = F(z*), (9) L where ~(2, h, 0 = (~(0 - c(x, t)lh (IO) which represents the current flow of economic rent. The singular path z*(t) is “myopic” in the sense of Arrow [l, 21, and is the optimal stock path (following an appropriate initial adjustment) provided that it is feasible relative to the control constraints (3). Consider, for example, a singular path as depicted in Fig. 1. At t = 0 the myopic rule provides the optimal stock level x*(O) = 0. At time T, however, the optimal stock x*(t) becomes positive, and is assumed to remain positive thereafter. What is the optimal policy under these assumptions? Consider, for a given initial stock level x = x (0), the conservation policy which allows the stock to recover (using h = 0) until t = to (Fig. 1) and subsequently harvests the stock so as to follow the myopic path r*(t). Let J(x) denote the marginal value of this conservation policy. Clearly extinction is not an optimal policy unless J(O ), P(O) - ~(0, 0) > zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR (11) i.e., unless the marginal net return from harvesting the last unit of the resource exceeds the marginal net benefit to be enjoyed from a policy of conservation. This is the additional condition which must be added to both the necessary and sufficient conditions (7) and (8) of the autonomous model. Although Eq. (11) appears straightforward, it is not always easy to determine, for a specific case, whether the condition is in fact satisfied. Let us give an example by way of illustration. We shall assume that all parameters other than price, p(t), are constant through time. The price is initially stable, but at some 4The left side of Eq. 6 See also Spence (6) is t,he own rate of interest, of the resource and Starrett [lo]. biomass 5. RENEWABLE RESOURCES AND EXTINCTION 201 zyxwvutsr FIGTJRE1 point in the future, t = T, the price is expected to start increasing. Specifically let us suppose that, while 6 > 2r, where r = F’(O), we have: $(0/p(t) = 0 for = k for all O<t<T (12) t > T, where k is a positive constant such that 6-l?<<. (13) zyxwvutsrqp Then in the limit as t -+ 30, Eq. (9) reduces to da/ ax F’(z*) + lim-= F’(z*) t-a. &r/ah = 6 - ]c, so that by (5) and (13) we have lim zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 2*(t) = x*~ > 0. (14) t-m Under these assumptions, extinction is never an optimal policy, no matter how large 6 or T may be (although the amount of biomass conserved at t = 0 may be very small).6 A formal proof of this proposition is given in the Appendix, but we can attempt an intuitive explanation. Because of the ultimate increase in price, for small x the own rate of interest of the resource stock will eventually exceed the discount rate. Since this condition will prevail forever, after a finite t.ime lapse, it follows that conservation of some (possibly very small) “breeding stock” is worthwhile. If the price increase does not continue indefinitely, so that ultimately condition (12) ceases to be valid, then extinction can eventually become optimal (and indeed could be optimal at t = 0) unless other parameter shifts occur. Thus in a sense one is “riding the back of a tiger”-if 6 > 2r, the threat of extinction at some time will be ever present. As an alternative example, we can suppose that all parameters remain constant, except for the discount rate, 6 = 6(t). If 6(O) > 2r and 6(t) < r for t > T, a similar argument shows that extinction is not optimal, even though z*(O) = 0. From the discussion so far it might appear that we can formulate a general rule to the effect that if the singular path x*(t) ultimately becomes strictly positive 6It will be shown in the Appendix that the optimal level of conservation, e, becomes exponentially small as 5” becomes large. Consequently, the above nonextinction theorem becomes progressively more unrealistic as 1’ ---) w if in fact there is a significantly large minimum viable population. CLARK AND MUNRO 202 te T Time, t zyxwvutsrqponmlkjihgfedcbaZYXWVU FIGURE 2 (z*(t) 2 3 > 0 for all t > !!“I) for any reason, then conservation will necessarily be warranted. Unfortunately this is not true, as the following simple example shows. Again suppose all parameters arc constant over time, except for price p(t), which satisfies for zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK O<t<T r,(L) = Pl for all t > T. = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF p2 (19 Here pl and p2 are constant’s such that’ p2 < pl, and the corresponding levels are for x*(t) = 0 Olt<T = x*1 > 0 myopic t > T. for all Then the myopic path has the same appearance as in Fig. 2. For this example, extinction at’ t = 0 is the optimal policy, even though servation would ultimately become optimal. To see this, let us define: PVl = present value of the extinction policy, PV2 = present value of any chosen conservation PVs = present value of the same conservation that p(t) = pl for all t 2. 0. Then it is easy to see that can be optimal.7 PVI > PV, policy, policy, under > PV2 and hence con- the assumption no conservation policy APPENDIX The nonextinction the following lemma, LEMMA. theorem stated in this note can be proved on the basis in which the notation is the same as before. of Let and assume that d(O+) = limq5(2) z-0 = 1 and +‘(O+) > - 00. Then t, 5 l/r In l/h + constant, (17) 7TO be honest,, we must admit that the singular path for this example is not precisely as shown in Fig. 2. Inasmuch as $(T) = - m, an appropriate vertical segment (extending to z = - m ) should be added at t = 2’ to the graph in Fig. 2 for the present example (cf. Clark and Mnnro [6, Fig. 21). RENEWABLE whew RESOURCES (SW Fig. 2) t, is dfJined Proof. 20.3 zyxwvutsrqp EXTINCTION by thr quatio,? s dx/dt = F(x), z(0) = E, x(t.) = x*1. Here x*1 denotes an arbitrary AND 0 L t 5 f,, given positive number <K (where F(K) = 0). The hypotheses imply that 1 - 4(x) = 4(O) - $(x) < c1.2, where cl = constant. Hcncc 1 1 l-$(X) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM Cl constant 5 cs = < __rM (x) +(x) -_--=-_---- F(x) TX for 0 5 x 5 x*1. Thercforc we have t, =lfdt =[*I& <l” ‘ [ b + c2] dx _< ilnf + ca. Q.E.D. It may bc remarked that hypothesis (16) of the lemma means that the relative growth rate F(x)/x equals r at x = 0 (in the limit,), and does not fall off too rapidly for x > 0. This is an important technical ingredient of our theory, implying that the growth rate of the biomass is suit’ably maintained for positive stock levels. We can now est!ablish our nonextinction theorem for the case of varying price p(t). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA THEOREM 1. Let the growth function Assume that p(t) remain constant satisjies con,ditions F(x) satisfy (12) and over time. Then extinction the hypotheses (IS), of the above lemma. and that the other parameters of the resource is nonoptimal. Proof. Let e > 0 denote the amount’ of the resource conserved at timr t = 0, following an initial adjustment, and define PV (,) 2~s the present valw of the following conservation policy (see Fig. 2) : h(t) = 0 for all o<t<t. = h*l = F(x*l) for all t 2 t, (18) assuming an initial stock lovcl x(0) = 6. The biomass lrvcl x*, is defined as in Eq. (14). Then 51 PV (E) = e-“ “ [p(t) / tc - c(~*~)]F(x~*)dt. (1% If c: is small, the present value of t’he policy of immediate extinction is given approximately by PV,,t ‘v [P(O) - c(0, O)-Jf. (20) We will prove that’ (21) CLARK AND MUNRO zyxwvutsrqponmlkjihgfedcbaZYXWVUTS 204 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA from which it, follows t)hat, for sufficiently PV(t) small E > 0 > PV,,t, i.e., the extinction policy is not optimal.8 Now for t 2 T we have [l - S$] p(t) - c(z*d = P(t) >PO[l-$$I C@(t) = = CQI(T)&-~', where ch is a positive constant. From this we obtain9 IV(E) XI A- (6- k)l& > c5 J It = C6e-(6--k)l’. Using inequality (17) of the lemma, we then obtain IV(E) > CTt(*--k)‘r (22) and Eq. (21) then follows easily from this together with condition (13). Q.E.D. A few comments on the foregoing proof may be in order. First, although t, necessarily recedes toward + cc, as e -+ 0, the rate of increase of 1, must not be “ too fast” ; otherwise Eq. (21) cannot be derived. Now t, is determined by the growth rate of the biological stock, and the lemma shows that tf grows sufficiently slowly-provided that the biological growth rate does not fall off too rapidly as CCincreases. Second, it is clear that the derivation of Eq. (21) requires exponential growth of p(t) at a suitable rate, for all t from T to + CC.The strict requirement of Eq. (12), however, can clearly be weakened, for example to an inequality: for all p(t) 2 lip(t) The constant, CTappearing in inequality t 2 T. (22) is seen to be of the form C~7e-kT--rz(6-k) c7=- 6-k The optimal conservation . level t = E* (at t = 0) is determined dPV(t)/ dt = [p(O) - by c(O)]. From this it follows that exp - ________ 1 rkT + ~(6 - k) E* N zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA as T-++a, [ r - (6 - Ic) as commented upon earlier (see footnote 6). * It is not asserted here that the conservation policy of Eq. (18) is itself optimal-indeed not since the optimal policy must ultimately follow the singular path z*(t). 9 If in fact k > 6, t,his calculation shows that, PV(,) = + 30. it. is RENEWABLE RESOURCES AND EXTINCTION Alt’hough we have confined our at,tention to varying prices, similar proofs Do deal with changes in other paramct,ers, such rate. ACKNOWLEDGMENT 205 IVC could apply as the discount The authors wish to acknowledge the support of the Canada Council. The support was given through the Universit,y of British Columbia’s (Department of Economics) Program in Natural Resource Economics. 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