Over-exploitation of natural resources is followed by inevitable declines in economic growth and discount rate

A major challenge in environmental policymaking is determining whether and how fast our society should adopt sustainable management methods. These decisions may have long-lasting effects on the environment, and therefore, they depend critically on the discount factor, which determines the relative values given to future environmental goods compared to present ones. The discount factor has been a major focus of debate in recent decades, and nevertheless, the potential effect of the environment and its management on the discount factor has been largely ignored. Here we show that to maximize social welfare, policymakers need to consider discount factors that depend on changes in natural resource harvest at the global scale. Particularly, the more our society over-harvests today, the more policymakers should discount the near future, but the less they should discount the far future. This results in a novel discount formula that implies significantly higher values for future environmental goods.


Supplementary Note 1. Discounting with two goods
In this Supplementary Note, we consider a social welfare that is a function of the provision of two goods over time, f (t) and c(t) (e.g., a natural resource and a manufactured good), and we derive the formulas for the discount rate and for the prices of the two goods. We consider the general case in which social welfare, U T , is given by the standard form given in Eq. 4 (main text), where u(c, f ) is twice differentiable with respect to both c and f . We consider a small, marginal perturbation that may vary over time and, at time t, it adds to society B(t)/ units to some currency, each of which is used to consume µ units of the natural resource, and (1 − µ) units of the manufactured good at time t. Consequently, the consumption of the two goods over time becomes We assume that the perturbation is marginal, namely, (1 − µ)B(t) c(t) and µB(t) f (t) at all t. Following this perturbation, social welfare becomes We denote w(t) = (1 − µ) ∂u ∂c + µ ∂u ∂f , which implies where U 0 is the utility without the perturbation. It follows that the cumulative discount is given by In turn, dw dt = (1 − µ)u cc dc dt where subscripts of u denote partial derivatives, and the discount rate is given by (1 − µ)u c + µu f + ρ. (A7)

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Note that several authors suggested one discount rate for the manufactured good and a second discount rate for the natural good 37 . These two discounts are equivalent to Eq. (A7) in the two special cases where µ = 0 and where µ = 1, respectively.
Next, we derive formulas for the changes in the prices of the two goods over time. Note that the prices are well-defined in a perfectly competitive market in which the goods are being traded as long as it increases social welfare, and therefore, at any time t, the market price of c, P c (t), and the market price of f , P f (t), are proportional to the respective derivatives of u at time t: Another constraint that is satisfied by the prices is that the same currency is being used at all times (no inflation), which implies that, for all t, In turn, Eqs. A8 and A9 imply Also, using the same currency units (in which the weight of the natural resource is given by µ), the total product is proportional to the value of all products at time t, i.e., To calculate P c , consider a perturbation that affects only c (μ = 0). Substituting into Eq. A6 implies In turn, this must be equal to the rate at which c is discounted 37 , which must equal the market discount rate times the rate of change in the price of c: where or

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Similarly, the change in the price of the natural resource, can be calculated by considering a perturbation withμ = 1, which implies Also, deriving both sides of Eq. A9 with respect to t and substituting Eq. A8 implies Finally, note that the changes in the present values of the goods at a given time, which are given by δ + ν f for the natural resource and δ + ν c for the manufactured good, do not depend on µ (Eqs. A13, A17).

Supplementary Note 2. Sustainable discount rates and relative prices
In this Supplementary Note, we calculate the discount rate that emerges when harvest is sustainable, δ sus , as well as the rates of changes in the prices, ν c and ν f , (Eqs. A14, A16) in two special cases. Specifically, we focus on the the asymptotic rates in which the entire system is under sustainable harvest, which implies that c and f increase exponentially (as in other related studies 32,37 ): In what follows, we calculate the discount rate in the two special cases: where the goods are non-substitutable, and where the goods are partially-substitutable. Note that similar derivations of δ sus for other forms of the utility function can be found in the literature 32,36,37 .

Non-substitutable goods (separable utility function)
For non-substitutable goods, we use the standard, separable utility function given by 12

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where γ is the relative significance of f , and η is the elasticity of utility with respect to consumption, characterizing how fast an increase in utility diminishes. In turn, note that the second partial derivatives of u are given by u cf = u f c = 0, u cc = −ηu c /c, and u f f = −ηu f /f . Substitution of these partial derivatives into Eq. A7 implies Specifically, if both goods grow exponentially at the same rate, g f = g c , then which retrieves Ramsey's discount formula 14,16 . Otherwise, if 0 ≤ g f < g c , then substitution of Finally, to calculate the changes in the prices, note that Eq. A13 implies Specifically, if δ = δ sus and g f = g c , then ν c = ν f = 0, whereas if g f < g c and t → ∞, then Partially-substitutable goods (non-separable utility function) For partially-substitutable goods, we consider a non-separable utility function 12 , where, as with the separable utility function, 0 < γ < 1 is the relative significance of f , and η is the elasticity of utility with respect to consumption. Note that the partial derivatives of u are given by u c = (1 − γ)(1 − η)u/c and u f = γ(1 − η)u/f . In turn, the second partial derivatives of u are given by In turn, substitution of u c and u f into Eq. B11 and multiplying both the numerator and the denominator by cf /u implies Some algebra (collecting terms that are identical in both square brackets) implies Note that, in the special case where g f = g c , we obtain Ramsey's discount formula, δ today = ηg c + ρ. If g f < g c and t → ∞, then f c, and it follows that Finally, to calculate the rate of change in prices, note that Similarly, Specifically, note that, if δ = δ sus and g f = g f , then ν c = ν f = 0, whereas if g f < g c and t → ∞, then

Supplementary Note 3. Proof of the theorem
In this Supplementary Note, we prove the theorem (Theorem in the main text). We begin with proving six Lemmas and three Corollaries.
Lemma 1. Assume that u(c, f ) is monotonically increasing, twice differentiable with respect to both c and f , and satisfies u f f < 0 and u cf ≤ 0. Consider a perturbation such that f (t 0 ) increases at a given time t 0 , while c(t 0 ) does not change. Then, the cumulative discount at time t 0 , ∆(t 0 ), which is given by Eqs. A5, A3 where 0 < µ ≤ 1, increases due to the perturbation.
Proof of Lemma 1. According to Eq. A5, the discount increases as w decreases. Therefore, we need to show that w (given by Eq. A3) decreases as f increases. Specifically, In turn, the assumptions that u f f < 0 and u cf < 0 imply that dw/df ≤ 0, which complete the proof of Lemma 1.
Lemma 2A. Assume that u(c, f ) is monotonically increasing and twice differentiable with respect to both c and f , where c is given by Eq. 9 and f is given by Eq. 6. Also assume that, as c → ∞ while f remain fixed, u cc /u f f → 0 and u f c /u f f → 0. Consider a perturbation that occurs at a given time t 0 , increases f (t 0 ) by H 0 , and decreases c(t 0 ) by K(t 0 ), where 0 ≤ K(t) ≤ K max at all t. Then, for sufficiently large t 0 and sufficiently small H 0 , ∆(t 0 ) (Eqs. A5, A3) increases due to the perturbation. Specifically, if g f = 0 and C 1 and C 2 are bounded from above, then, for sufficiently large t, ∆(t) increases as the total harvest,Ĥ = αH s + H n , increases (regardless of whether the increase inĤ is due to an increase in H n or H s ).
Proof of Lemma 2A. To show that ∆ increases withĤ ifĤ is sufficiently small and t is sufficiently large, we need to show that dw(t 0 )/dĤ < 0. In turn, it follows from Eq. A3 that Note that, since K is bounded from above, dc/dĤ is negative and is bounded from below, regardless of the values of H n and H s that determineĤ. Specifically, there exists K max > 0 such that dc/dĤ ≥ −K max . Therefore, it remains to show that, when t → ∞, for any K max > 0, or, equivalently,

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In turn, Eq. C4 follows from the assumption that Specifically, this applies to increasing harvest as this is a special case where K(t) = max{C 1 (x 1 (t)), C 2 (x 2 (t))}, which completes the proof of Lemma 2A.
Lemma 2B. Assume that u(c, f ) is monotonically increasing and twice differentiable with respect to both c and f , where c is given by Eq. 9 and f is given by Eq. 6. Also assume that, Consider a perturbation that occurs at a given time t 0 and increases f (t 0 ) by H 0 e g f and decreases c(t 0 ) by K(t), where 0 ≤ K(t) ≤ K max at all t. Then, for sufficiently large t and sufficiently small H 0 , ∆(t) (Eqs. A5, A3) increases with H 0 . Specifically, this implies that if C 1 and C 2 are bounded from above, then, for sufficiently large t, ∆(t) increases as total harvest,Ĥ = αH s + H n , increases (regardless of whether the increase inĤ is due to an increase in H n or H s ).
Proof of Lemma 2B. As in Lemma 2A, we need to show that, for sufficiently large t 0 , dw(t 0 )/dĤ < 0. In turn, Since u c > 0 (u is monotonically increasing) and decreasing with both c and f (u f c < 0 and u cc < 0), and since dc/dĤ is bounded, it follows that Therefore, Eq. C6 implies that, for sufficiently large t, dw/dĤ > 0. Specifically, this applies to increasing harvest as this is a special case where K(t) = max{C 1 (x 1 (t)), C 2 (x 2 (t))}, which completes the proof.
Lemma 3. Assume that u(c, f ) is monotonically increasing and twice differentiable with respect to both c and f , where c is given by Eq. 9 where C 1 and C 2 are bounded from above and f is given by Eq. 6. Also assume that, for all f 0 , if c(t) = c 0 exp(g c t) and Note that, as a special case where g f = 0, Eq. C9 becomes Then, for sufficiently large t 0 , a sufficiently small increase in harvest at t 0 , either sustainable or non-sustainable, results in an increase in u(t 0 ).
Proof of Lemma 3. We need to show that, for any level of the total harvest,Ĥ = αH s + H n , there exists t , such that increasingĤ would increase u for any t > t . Increasing harvest increases u if and only if du/dĤ > 0, and therefore, we need to show that Since C 1 and C 2 are bounded from above, dc/dĤ is bounded from below, namely, there exists M > 0 such that dc/dĤ > −M . Therefore, to complete the proof, we need to show that, for sufficiently large t, or, equivalently, for any M > 0. However, this follows directly from Eq. C9, which completes the proof of Lemma 3.
Corollary 1. Assume that social welfare, U T , is given by Eq. 4, where ρ is a constant and u follows the assumptions of Lemma 3. Then, optimal harvest dictates that, for sufficiently large t, the entire area is under harvest, namely, H n (t) + H s (t) = x 1 (t) + x 2 (t).
Proof of Corollary 1. This Corollary follows directly from Lemma 3. Specifically, increasing sustainable harvest at a sufficiently large t 0 increases u(t 0 ). Moreover, this increase in sustainable harvest necessarily increases U T as it does not affect future values of u (sustainable harvest does not affect the dynamics of x 1 and x 2 ). Therefore, optimal harvest dictates that harvest increases until it hits the constraint where H n + H s = x 1 + x 2 . This completes the proof of Corollary 1.
Corollary 2. Assume that social welfare, U T , is given by Eq. 4, where ρ is a constant and u follows the assumptions of Lemma 3. Then, following the market dynamics, there exists a time t after which all shared resources are exhausted (x 2 (t) = 0 for all t > t ).
Proof of Corollary 2. Lemma 3 implies that, for sufficiently large t, an increase in harvest increases u. Specifically, since α < 1, increasing non-sustainable harvest of the shared resource at time t 0 increases u(t 0 ), even if it comes on the account of sustainable harvest (and even if harvest non-sustainably is more expensive, λ < 1). (Note that the non-sustainable harvest may decrease the future values of u, and therefore, may not be preferable using optimal harvest; however, the managers in the market solution over-harvest despite the future reduction in u due to the non-sustainable harvest of the shared resource.] Therefore, for sufficiently large t, each manager is better off harvesting the shared resource non-sustainably. Thus, there exists t such that the entire shared resource is being exhausted. This completes the proof of Corollary 2. Lemma 4. Assume that social welfare, U T , is given by Eq. 4, f is given by Eq. 6, c is given by Eq. 9 where C 1 and C 2 are constants, and H n and H s are non-negative and satisfy the constraints given by Eqs. 7, 8. Assume that, for sufficiently large t, both cu f c /u f and f u f f /u f are monotone with t. Also assume that u(c, f ) is non-decreasing in both c and f and that u f c ≤ 0 and u f f ≤ 0. Furthermore, assume that H n ≥ 0 and H s ≥ 0 are subject to the constraints given by Eqs. 7, 8 and follow optimal harvest. Then, there exists t such that if x(t ) = x 1 (t ) + x 2 (t ) increases, then total harvest,Ĥ = H n + αH s increases at all t > t . Namely, for sufficiently large t , if H opt small (t) denotes the total optimal harvest in a system with a given x(t ) = x small , and H opt large (t) denotes the optimal harvest in a system that is identical except that Proof of Lemma 4. The idea behind the proof is to show that, for sufficiently large t,Ĥ = H n + αH s is non-increasing with time. Since U T is time invariant when T → ∞, the optimal harvest,Ĥ opt , depends on time only implicitly via the state variable x (Ĥ opt =Ĥ opt (x(t))).
Also, x is non-increasing with time (Eq. 7). Therefore, ifĤ opt decreases with time, this implies thatĤ opt decreases with x.
More formally, note that Corollary 2 implies that, for sufficiently large t, H n + H s = x, and therefore,Ĥ where H n and H s denote optimal non-sustainable harvest and optimal sustainable harvest, respectively. In turn, substituting Eq. C15 into Eq. 6 implies and therefore, where we used dx/dt = −H n (Eq. 7).
Next, note that postponing one unit of harvest by a small (infinitesimal) unit of time, from t 0 to t 0 + dt, implies that one harvests αdt at t 0 plus 1 at t 0 + dt instead of 1 at t 0 . When H n is positive at both t 0 and t 0 + dt, it implies that the increase in utility due to extra 1 at t 0 equals an increase in utility due to extra 1 +δ at t 0 + dt, whereδ is the rate by which the value of a unit of S10 fish increases over time. Therefor, either (i) δ < α and H n = 0 or (ii) H n (t) > 0 andδ(t) = α: In turn,δ it is given by the market discount plus the rate of increase in the price of f ,δ = δ + ν f , which is given by Eq. A7 with µ = 1 (Supplementary Note 1): Equivalently, we can writeδ where we denote In turn, it follows that where Furthermore, substituting Eq. C17 into Eq. C23 implies Next, note that the limit A ∞ = lim t→∞ A(t) exists. Specifically, since u f c ≤ 0, u f f ≤ 0, u c ≥ 0 and u f ≥ 0, it follows that A c (t) and A f (t) are non-negative. Furthermore, since cu f c /u f and f u f f /u f are monotone for sufficiently large t, it follows that the limits of A c (t) and A f (t) exists (might be infinite), and we denote Moreover, we first assume that and afterward, we find the asymptotic expansion of H n and we show that it is non-increasing as x decreases, and then, we show that Eq. C27 is consistent and indeed follows from the asymptotic expansion of H n . These considerations imply that A(t) has a limit given by We distinguish the following three cases. First, in the case where A ∞ < 0, it follows from Eq. C23 that, for sufficiently large t, In turn, f (t) is given by Eq. 6, which implies that dĤ/dt ≤ 0 if and only if Eq. C29 holds. Therefore, if A ∞ < 0 there exists t such that dĤ/dt < 0 for all t > t . Second, in the case where A ∞ > 0, note that a strict equality in Eq. C25 cannot hold forever. Specifically, the equality implies that dH n /dt > 0, which cannot hold forever as the resource is limited (dx/dt = x and H n ≤ x). This implies that after any t there have to be intervals where H n = 0. However, any continuous deviation of H n from zero yields a continuous deviation of the lefthand-side of Eq. C25 from zero, which would still be greater than A(t) for sufficiently large t. On the other hand, any discontinuous deviation would create an effectively infinitely large derivative. Therefore, if A ∞ < 0, then there exists a t such that H n (t) = 0 (and dĤ(t)/dt = 0) for all t > t . Third, in the case where A ∞ = 0, we use the assumption that for sufficiently large t, both cu f c /u f and f u f f /u f are monotone with t. This condition implies that there exists a time t , such that A(t) does not switch signs for all t > t (namely, either (i) A ∞ < 0 for all t > t or (ii) A ∞ ≥ 0 for all t > t ). If (i) holds, then, for sufficiently large t, dH n /dt = 0 follows from the same considerations as in Case I. If (ii) holds, then, for sufficiently large t, dH n /dt < 0 follows from the same consideration as in Case II.
We have seen that, in all three cases,Ĥ(x(t)) decreases with time, and therefore, it decreases with x. It remain to show that the asymptotic expansion of H n where t → ∞ is either H n = 0 or is given by H n ∼ hx where h is a constant, and that this implies that Eq. C27 holds. Specifically, we have already seen that, in Case I, H n (t) = 0 if t is sufficiently large, and therefore, we restrict attention to the case A ∞ > 0. First, we derive H n (x) = h(x)x with respect to t, which implies Therefore, it follows from Eq. C25 with equality that S12 or, after reducing both nominator and denominator by a factor x, Next, we look for an asymptotic solution where h is a constant (dh/dt = 0): Namely, which implies or and since only h 1 ≥ 0, we get h = A ∞ .
Substituting H n = hx implies Note that c 0 exp(g c t)/x → ∞ as t → ∞, which implies that (1/c)(dc/dt) → g c as t → ∞. This completes the proof of Lemma 4.
Corollary 3. Assume that social welfare, U T , is given by Eq. 4, where ρ is a constant, f is given by Eq. 6 and c is given by Eq. 9 where C 1 = C 2 = constant. Consider two identical systems with the only difference being that one is subject to market harvest and the other is subject to optimal harvest. Denote x market the value of x = x 1 + x 2 that follows from market dynamics and x opt the value of x that follows socially optimal harvest (see Methods). Then, for any t, x opt ≥ x market .