Hyperbolic Discounting and the Time-Consistent Solution of Three Canonical Environmental Problems

In this paper I propose a time-consistent method of discounting hyperbolically that contains the discount rate implied by Gamma discounting as a special case. I apply the discounting method to three canonical environmental problems: (i) optimal renewable resource use, (ii) the tragedy of the commons, (iii) economic growth and pollution. I then compare results with those for conventional exponential discounting using the normalization that both methods provide the same present value of an infinite constant flow. I show that, irrespective of potentially high initial discount rates, time-consistent hyperbolic discounting leads always to a steady state of maximum yield, or, if the environment enters the utility function, a steady state where the Green Golden Rule applies. While (asymptotic) extinction is a real threat under exponential discounting it is impossible under time-consistent hyperbolic discounting. This result is also confirmed for open access resources. In a model of economic growth and pollution, hyperbolic discounting establishes the Golden Rule of capital accumulation and the Modified Green Golden Rule.


Introduction
Economic scholars agree that the appropriate choice of the discount rate is of preeminent importance in the evaluation of policies that matter for the distant future. Environmental policies are perhaps the best example for decision problems in which the costs accrue from today onwards and most of the benefits occur in the distant future. For example, any potential disagreement in the assessment of the material effects of global warming is dwarfed by the impact of the discounting method on the implied net present value in monetary terms (Weitzman, 2007;Nordhaus, 2007;Dasgupta, 2008). Conventional economic analysis is based on exponential discounting, i.e. it applies a constant discount rate. This approach, however, is potentially problematic since research in psychology and behavioral economics suggests that time preference rates are declining over time, presumably in a hyperbolic fashion (see Frederick et al., 2002, andDellaVigna, 2009, for surveys).
In environmental economics the discounting debate focuses mostly on the social discount rate (of future income or consumption). Stern (2008), for example, argues that that the discounting of future payoffs for reasons of pure time preference is unethical. The fact that, simultaneously, it appears to be fully rational for the citizens alive today to discount payoffs in the near future leads immediately to the conclusion of declining discount rates. Recent empirical evidence from the housing market suggests that hyperbolic discounting is indeed applied to payoffs in the distant future (Giglio et al., 2015). Moreover, if experts disagree about the "correct" constant social discount rate, the discount rate implied by the average discount factor declines over time (Weitzman, 2001). For these reasons (and others) a panel of leading economists in environmental economics recently came to the conclusion that the discounting of costs and benefits of longhorizon projects should best be done at a hyperbolic rate (Arrow et al., 2014).
The social discount rate (SDR) is conventionally derived from the Euler equation of intertemporal utility maximization such that SDR = ρ+ηg c , in which ρ is the pure rate of time preference (i.e. the utility discount rate), η is the inverse of the elasticity of intertemporal substitution, and g c is the consumption growth rate. In principle, the SDR may decline because the intertemporal elasticity of substitution increases over time or because consumption growth declines over time.
The perhaps most natural reason for a declining social discount rate is that the rate of pure time preference declines over time. In a setting where the elasticity of intertemporal substitution is assumed to be constant and the economy is at its steady state or there is no long-run growth (as in the context of the present paper), a declining rate of time preference is the only way to generate a declining SDR. Drupp et al. (2015) extend the Weitzman (2001) approach by collecting expert assessments of the individual components of the SDR and report disagreement on the rate of pure rate of time preference. Applying Weitzman's (2001) aggregation, this finding implies in the aggregate a declining rate of pure time preference.
An important concern, however, is that hyperbolic discounting leads to suboptimal and inconsistent decision making. In fact, the introduction of hyperbolic discounting in the economics of psychology is motivated by the need to explain preference reversals and time-inconsistent behavior (Frederick et al., 2002). Consequently, scholars tend to believe that hyperbolic discounting inevitably implies an inconsistency problem. 1 Applying hyperbolic discounting to environmental decision making would then entail a continuous reversal of previous plans and suboptimal outcomes (Winkler, 2006). For example, Hepburn et al. (2010) show that naive social planners who apply hyperbolic discounting without being aware of the time-inconsistency problem can manage renewable resources into extinction. Alternatively, it has been suggested that sophisticated social planners, who are aware of the time-inconsistency problem, could achieve consistent decisions by using Markovian strategies in a dynamic game with (not yet existent) future planners (Karp, 2005, Gerlagh andLiski, 2017). This solution, however, is not entirely satisfying either.
The achieved Nash equilibrium is not pareto-optimal and, compared to a simple discounting rule, a Markovian strategy seems to be difficult to implement.
In this paper, I suggest a different solution to the problem and propose the use of timeconsistent hyperbolic discounting. 2 Many economists know the result from Strotz' (1956) seminal paper that only exponential discounting leads to time-consistent decisions if the discount factor is a function of the algebraic distance (t 0 − t) between planning time t 0 and payoff time t.
The "if"-clause, however, has sometimes been forgotten in the following literature such that the conventional wisdom evolved that non-exponential discounting necessarily entails time inconsistency. Here, I suggest a form of hyperbolic discounting to which the theorem of multiplicative 1 For example, Angeletos et al. (2001, p. 54) write that "When a household has a hyperbolic discount function, the household will have dynamically inconsistent preferences, so the problem of allocating consumption over time cannot be treated as a straightforward optimization problem." 2 Here, I use the term time-consistent hyperbolic discounting to define a discounting method that leads in ordinary (i.e. "open loop") decision making to time-consistent solutions. I do not consider discounting problems that imply time-inconsistency in ordinary (or naive) decision making in order to obtain a time-consistent solution by a particular method of sophistication (like Markovian strategies). separability in decision time and payoff time applies (see Burness, 1976;Drouhin, 2015). As a result, decisions are time-consistent.
Time-consistent hyperbolic discounting got so far relatively little attention in environmental economics. A notable exception is Heal (2005) who discusses a form of multiplicatively separable discounting (logarithmic discounting) and praises its attractive properties (p. 1125, p. 1138).
Notwithstanding this prominent "advertisement" there are, to the best of my knowledge, no applications of time-consistent hyperbolic discounting in environmental economics. Here, I fill this lacuna and solve three canonical environmental problems for the case of time-consistent hyperbolic discounting: the optimal use of a renewable resource, the non-cooperative use of an open access resource, and the problem of optimal growth and pollution.
Time-consistent hyperbolic discounting, in contrast to time-inconsistent behavior, leads to quite benign environmental outcomes. At the steady state, it establishes the social optimum for closed access resource management (maximum sustainable yield, Green Golden Rule). I also compare with the solution obtained for exponential discounting. For that purpose I use the normalization that both discounting methods provide the same present value of a constant flow (Myerson et al., 2001). This allows me to disentangle the effects due to the discounting method from the effects due to impatience as such. As a rule, I find that time-consistent hyperbolic discounting leads to a more conservative resource use at the steady state and off the steady state. In particular, there exist situations in which exponential discounting leads to extinction while hyperbolic discounting leads always to sustainable and socially optimal outcomes.
It is perhaps less of a surprise that present-biased hyperbolic discounting implies a conservative resource use at or close to the steady state since, by design, the impact of discounting vanishes at the steady state. However, the result that present-bias turns out to be good for the environment when the economy starts far off the steady state is remarkable. After all, for the near future, hyperbolic discount rates exceed (by far) the present-value equivalent constant discount rate. Seemingly, they should thus lead to overexploitation of the resource. The intuition for the non-obvious result of sustainability lies in the inherent dynamics of the resource stock and the discount rate. The first order conditions exclude extinction in finite time as an optimal outcome (intuitively, utility would jump to minus infinity at the point of extinction of the resource). Thus, only asymptotic extinction could be an optimal outcome. When the discount rate declines hyperbolically, however, there exists always a finite point in time at which the discount rate falls short of any positive natural growth rate of the resource, a condition that ensures sustainability. With further declining discount rate, any resource-management strategy converges towards the long-run social optimum. For conventional exponential discounting, however, asymptotic extinction is a real threat, in particular for common access resources. I explain these features with the help of phase diagrams and numerical implementations of the models.
The conclusion that hyperbolic discounting can be time-consistent and that its application leads to the long-run social optimum has policy relevance since a discounting rule can be easily implemented as an institution. In fact, declining social discount rates are already applied in the evaluation of public projects in the United Kingdom and France (HM Treasury, 2003;Lebegue, 2005). Concrete economic and environmental policies, in contrast, are harder to govern by institutions and may change from day to day, the specific problem at hand, and the responsible policymakers (see Acemoglu et al., 2005 on the importance of institutions vs. policy). This conclusion seems to be of particular importance for the management of open access resources where non-cooperative strategies are likely. It is thus reassuring that the implementation of a hyperbolic discounting rule, given its multiplicative separability, will lead to time-consistent decisions. This means that it will not cause future policy reversals and that there is no need for commitment or sophistication of strategies by the current policymakers.
The papers is organized as follows. In the next Section I introduce time-consistent hyperbolic discounting. In Section 3 I apply the method to the optimal management of a renewable resource.
In Section 4 I discuss the exploitation of an open access resource under hyperbolic discounting and in Section 5 I discuss the problem of optimal economic growth and pollution. For all cases I illustrate off-steady-state behavior with numerical examples and establish the superiority of time-consistent hyperbolic discounting vis a vis conventional exponential discounting. Section 6 concludes the paper.

Welfare and Discounting
Consider a social planner who maximizes welfare understood as utility from consumption u(c) experienced over an infinite time horizon and discounted to the present.
with the discount factor D(t 0 , t) depending on calender time t and decision time t 0 . As a reference case we consider conventional exponential discounting at a constant rate such that D(t, t 0 ) = e −ρ(t−t 0 ) . The main focus, however, is on hyperbolic discounting where the discount factor is given by the following multiplicatively separable function of decision time and calender time: Notice that any sequence of social planners starts with a discount factor of one at their individual decision time t 0 and that the discount factor declines hyperbolically with calender time. The parameters α and β are useful to calibrate discounting. The discount rate is defined as ρ(t) = −(dD/dt)/D = and it is obtained as for exponential discounting. (3) Notice that the hyperbolic discount rate does not depend on decision time t 0 . It declines in calender time t and a larger α or β implies a higher discount rate at any time. Since the discount rate is independent from decision time, decisions are time-consistent. The crucial feature that provides time consistency is that the discount factor is multiplicatively separable in decision time and calender time (Drouhin, 2015;Burness, 1976).
In order to make inferences, welfare has to be finite, which in turn requires β > 1. To see this, assume that a steady state of constant utility exists. Then, for The feature that the utility integral is finite although the discount rate vanishes in the long-run simplifies the following analysis. It allows to apply standard optimization techniques and avoids the use of less-well known methods like the catching-up criterion (Seierstad and Sydsaeter, 1987).
For a fair comparison between exponential and hyperbolic discounting we apply the equivalentpresent-value argument made by Myerson et al. (2001); see also Caliendo and Findlay (2014) and Strulik (2015). This means that the value of the discounting parameters is determined such that a constant infinite stream (of, for example, utility) provides the same present value. The The condition of a finite present value of an infinite utility stream is thus equivalent to the condition that the associated equivalent constant time preference rate is positive (β > 1). The normalization of parameters is needed in order to disentangle the effect of the discounting method (constant versus hyperbolic) from pure impatience. Here we are interested in how the hyperbolic decline of the discount rate affects the management of a renewable resource.
As discussed in the Introduction, Weitzman (2001) has shown that individual disagreement (or uncertainty) about the correct constant discount rate leads in the aggregate to a hyperbolically declining discount rate. The Weitzman case is represented by setting α = σ 2 /µ and β = µ 2 /σ 2 which leads to the discount rate where µ is the mean and σ the standard deviation of the individual discount rates. 3

Optimal Harvesting of a Renewable Resource
We first consider renewable resource growth according to the Verhulst (1938) model (see e.g. Wilen, 1985). This means that absent any harvesting (or pollution) the resource stock (or environmental quality), denoted by x, grows logistically until it reaches its carrying capacity κ.
The change of the resource stock is given by g(x) = rx(1 − x/κ) where r denotes the maximum natural growth rate. Notice that the maximum sustainable yield from the resource is attained where g(x) reaches a maximum, i.e. where g x = 0. 4 Consumption c is equal to the harvesting rate such that the change of stock is obtained aṡ The social planner maximizes (1) subject to (2) and (6), given the initial stock x 0 and the non-negativity conditions, x ≥ 0, c ≥ 0. The associated Hamiltonian function is given by in which λ is the shadow price of the resource. The first order conditions are: ( Differentiating the first condition with respect to time and eliminating λ andλ leads to the following consumption rule:ċ in which η ≡ −(∂ 2 u/∂c 2 )c/(∂u/∂c) is the elasticity of marginal utility, i.e. the inverse of the elasticity of intertemporal substitution. Without loss of generality we assume that η is constant.
Notice that by log-differentiating of (7) the consumption rule is independent from decision time t 0 , implying that consumption plans are time-consistent. This is a natural outcome of multiplicative separability. Optimal resource management is determined by resource growth (6) and consumption growth (9), together with the transversality condition lim t→∞ λ(t)x(t) = 0.
Consumption growth is high when the return on investment in the resource stock, i.e. the return on foregone consumption, reflected by the first term in parenthesis in (9), is high or when the discount rate, reflected by the second term, is low.
Settingċ =k = 0 in (6) and (9) and applying the non-negativity constraints, we obtain the trivial steady state, (c = 0, k = 0), and a unique positive steady state at: Here we have made use of the fact that the discount rate declines to zero in the long run, lim t→∞ ρ(t) = 0. Performing the same analysis for the case of exponential discounting, we arrive at the consumption ruleċ and the steady state Comparing the solutions (10) and (12) and recalling that g x = 0 where x = κ/2 verifies the following result.
Proposition 1. Time-consistent hyperbolic discounting leads to steady-state consumption of the maximum sustainable yield. Exponential discounting leads to steady-state consumption of less than the maximum yield. Ifρ > r there exists no sustainable steady state for exponential discounting. There exists always a sustainable steady state for hyperbolic discounting.
Notice that this result is not due to impatience as such since we normalized the discount rates.
The result reflect just the impact of the discounting methods. Intuitively, hyperbolic discounting gives more weight to the far future, which is when the steady state is (asymptotically) reached.
Thus, if a steady state exists, harvesting less than maximum yield is not an expression of impatience but an expression of applying a constant discount rate.
We next consider off-steady consumption and the possibility of extinction. Inspection of (12) shows that there exists no positive steady-state under exponential discounting if the discount rate exceeds the maximum natural growth rate (i.e. forρ > r). It is thus tempting to believe that extinction is more frequent under hyperbolic discounting since initially the (very) high discount rate exceedsρ. However, the opposite is true. To see this, consider the phase diagram in Figure   1.A. Theẋ = 0-isocline is given where c = g(x) and displayed as the hump-shaped curve. The resource declines above the isocline and grows below it. Theċ = 0-isocline is a horizontal line through x = κ(r − ρ(t))/(2r). Consumption grows to the left of the isocline and declines to the right of it. The resulting arrows of motion are depicted in Figure 1.A. They show that the intersection of the isoclines is a saddlepoint.
Strictly speaking, however, theċ = 0-isocline cannot be depicted in a phase diagram because it is a non-autonomous equation. As time proceeds and the discount rate declines, x = κ(r − ρ(t))/(2r) grows and the isocline moves to the right. Figure 1 shows an initial isocline where ρ(0) exceeds r, indicating an unsustainable situation. As time proceeds, the isocline shifts to the right and, once it crosses the ordinate, sustainability is ensured. With further proceeding time, the isocline moves to x = κ/2 and consumption converges towards the maximum sustainable yield. Figure 1.B: Exponential Discounting and Asymptotic Extinction Figure 1.B illustrates the case of asymptotic extinction for exponential discounting. In this case,ρ > r and theċ = 0-isocline lies to the left of the c-axis. Aside from the trajectory that goes through the origin, all other trajectories hit either the c-axis or the x-axis in finite time. Any trajectory that hits the c-axis is suboptimal because it leads to a jump of c when the resource gets extinct and c ≥ 0 becomes binding with equality (the trajectory violates the Weierstrass-Erdmann condition). Any trajectory that hits the x-axis is suboptimal because it violates the transversality condition. To see this note that at the intersection c = 0, x > 0, and g x < 0. From (8) the shadow price λ grows at rate −g x . Thus with x constant and λ growing at a positive rate, the transversality condition lim t→∞ λ(t)x(t) is violated. The outcome that extinction in finite time is suboptimal is very intuitive. Given the arrows of motion in Figure   1.B., the only solution left is asymptotic extinction. As the economy moves along a path to the origin (along path {c * } in Figure 1.B), both x and c vanish in sync such that the resource becomes extinct asymptotically. Along the trajectory g x > 0 such that λ declines continuously.
With both x and λ converging to zero, the transversality condition is fulfilled. Figure 1.B. can also be employed to understand why (asymptotic) extinction is impossible under hyperbolic discounting. As long as theċ = 0-isocline lies to the left of the c-axis, the only optimal solution is given by a trajectory like {c * } along which x and c decline at declining rates.
However, the origin is reached only in infinite time while the rightward movingċ = 0-isocline crosses the c-axis in finite time at the moment when ρ(t) falls short of r. This means that there exists always a finite moment of time at which an initially unsustainable harvesting policy becomes sustainable. Then, the rightward shift of theċ = 0-isocline continues until it comes to a rest at the long-run optimum x * . The following proposition summarizes these observations.
Proposition 2. For a sufficiently high discount rate,ρ > r, exponential discounting implies (asymptotic) extinction. Extinction is impossible under time-consistent hyperbolic discounting irrespective of (potentially large) initial discount rates. Instead, any optimal harvesting policy converges to the maximum sustainable yield.
We next illustrate adjustment dynamics with an example. 5 For that purpose I set r = 0.04, κ = 10, and η = 2 as well as µ = 0.04 and σ = 0.03 (i.e. α = 0.0225 and β = 1.777). Using (4), the equivalent constant discount rate is obtained asρ = 0.0175. Blue (solid) lines in Figure   2 show the optimal trajectories for the benchmark hyperbolic case. The discount rate declines continuously from 4 percent to zero (lower left panel). Consumption c is initially high due to the high present bias and then declining towards its steady state level (upper left panel). The resource stock first declines due to the high initial consumption and then recovers gradually to the level of κ/2 = 5 (upper right panel) where consumption equals the maximum sustainable yield of 0.1 (upper left panel). Now consider an increase of α to 0.06. Applying (5), the equivalent constant discount rate is obtained asρ = 0.0467. This value exceeds the maximum natural growth rate (r = 0.04) and thus extinction cannot be avoided when a constant discount rate is applied. The resulting trajectories for hyperbolic discounting are shown by red (dashed) lines in Figure 2. Although the social planner initially uses a very high discount rate of almost 11 percent, extinction is avoided. Before the resource is completely exploited, the discount rate has fallen below the natural growth rate (lower left panel) and the resource starts growing again (upper right panel).  Parameters: r = 0.04; κ = 10; η = 2. Blue (solid) lines: µ = 0.04, σ = 0.03 (i.e. α = 0.0225, β = 1.777). Red (dashed) lines: higher initial discount rate α = 0.06 (implying unsustainability for exponential discounting atρ = α(β − 1)). The lower right panel shows the phase diagram.
Finally, we extend the model by considering utility experienced from the resource stock (capturing, for example an interest in species diversity or a clean environment). The welfare functional (1) is replaced by W (t 0 ) = ∫ ∞ t 0 D(t 0 , t)u(c, x)dt and maximization with respect to (2) and (6) provides the costate equation, which replaces (8). The first order condition (7) remains valid. Eliminating λ we obtain from (7) and (13): Proposition 3. Time-consistent hyperbolic discounting establishes the Green Golden Rule as the steady state solution of optimal renewable resource management.
For the proof we exploit that consumption and the resource stock stay constant at the steady state such that the left hand side of (14) is zero. Then, at the steady state, g x = −u x /u c = dc/dx.
The slope of the resource growth function is tangent to the planners indifference curve at the highest achievable utilityū * , see Figure 3. This outcome is known as the Green Golden Rule (Heal, 2005).

Figure 3: Hyperbolic Discounting and the Green Golden Rule
To proceed further, assume that utility is separable in c and x and isoelastic, u = c 1−η /(1−η)+ x 1−ψ /(1 − ψ). Then, (14) implies theċ = 0-isocline is given by c = (2rx/κ + ρ(t) − r) 1/η x ψ/η with consumption increasing above and falling below the isocline. Theẋ = 0-isocline remains as for the simple model. We obtain the phase diagram shown in Figure 3. Since x is non-negative, extinction is no threat. As time proceeds and ρ(t) = αβ/(1 + αt) declines, theċ-isocline shifts to the right towards an intersection with the resource curve g(x) where g(x) is tangent to the highest attainable indifference curveū * where the Green Golden Rule holds. Thus, hyperbolic discounting generalizes the Green Golden Rule. For exponential discounting the Green Golden Rule applies only for the knife-edge case ofρ = 0. For time-consistent hyperbolic discounting the Green Golden Rule applies for any discounting function ρ(t).

Hyperbolic Discounting and the Tragedy of the Commons
In this section we explore the use of a renewable resource when property rights are not defined (or cannot be enforced). Suppose that the resource x is harvested by i = 1, . . . , n agents. The agents are symmetric, i.e. they share all parameters but follow their own resource management calculus. Here we are interesting in the non-cooperative solution (since the cooperative solution coincides with the one for the simple model from the previous section). To simplify the problem, we return to the assumption that the resource does not enter the utility function. In a stylized way, the scenario could be interpreted as a game between social planners who agreed on a discounting rule but who cannot commit to any particular policy. As discussed in the Introduction, a discounting rule can be imagined as implementation of a hard-to-change institution while the actual policy (the "daily" use of the resource) cannot be contracted. We are interested in time-consistent policies, such that the planners play Markovian (feedback) strategies c i (x, t). 6 Summarizing, planner i maximizes This leads to the following first order condition and costate equation: The feedback terms ∂c j /∂x indicate that planners take the response of the other players into account. Eliminating the costate variable and applying symmetry, (17) and (18) can be sum- in which c = c i for all i = 1, . . . , n.
Equation (19) is hard to interpret since it involves the solution of a partial differential equation.
For a special case, however, we can obtain a closed-form solution for the optimal harvesting strategy. Suppose the utility function is iso-elastic, u = c 1−η /(1 − η), and consider the case that η = 2. Fortunately, this assumption has empirical relevance. A recent meta-analysis of 2735 published estimates of the intertemporal elasticity of substitution found the world average of η at 2.0 (Havranek et al., 2015). As shown in the Appendix, for η = 2 we obtain the optimal consumption strategy with ρ(t) = αβ/(1 + αt) in the case of hyperbolic discounting and ρ(t) =ρ in the case of exponential discounting. 7 As shown in (20), the consumed share of the resource is large if the resource regenerates quickly (for high r) or if individuals discount at a high rate. Comparing discounting methods we see that exponentially discounting individuals consume a constant share of the resource at all times whereas hyperbolically discounting individuals consume a declining share of the resource.
The consumed share of the resource declines in the number of competing players but the totally consumed resource, nc, increases in the number of players. Two players consume the share 2/3(r +ρ), three players consume the share 3/4(r +ρ) etc. The feature that the total use of the resource increases in the number of players is an expression of non-cooperative behavior. For any player, the opportunity cost of harvesting a unit less is that the other players are induced to consume more when there are more resources. Formally this is reflected by the ∂c j /∂x term in (18). This feedback effect drives down the value of the resource λ for every player and increases consumption. The increased consumption due to non-cooperative behavior leads to overuse of the resource and potentially to extinction, i.e. the tragedy of the commons (Hardin, 1968).
Extinction, however, depends on the method of discounting.
Proposition 4. When a renewable resource is harvested competitively without property rights, there exists always a sustainable steady state x * * > 0 in the case of hyperbolic discounting, for any number of competing individuals. For exponential discounting there exists no sustainable steady state if the discount rate is sufficiently high or if the number of competing individuals is sufficiently high, i.e. for r < nρ.
For the proof, insert (20) into (16) and solve forẋ = 0. This provides the steady state solution: with ρ = 0 in case of hyperbolic discounting and ρ =ρ in case of exponential discounting.
Hyperbolic discounting eliminates extinction as a steady-state phenomenon because ρ(t) vanishes in the long-run. Hyperbolic discounting, however, does not solve the overuse of the commons.
7 The solution is a non-degenerate Markov strategy. Markovian strategies are decisions rules c(x(t), t) in which the choice of player's current action is conditioned on current time and the state vector (Dockner et al., 2000 p. 30 and pp. 59-60). The presence of the state variable eliminates the commitment problem. Notice that the presence of a time-dependent discount rate does not affect commitment because discounting according to (2) implies a time-consistent solution.
For n > 1, the steady state resource stock x * * falls short of the x * , the level that provides the maximum sustainable yield. Figure 4 illustrates these results. The steady state is obtained where the linear consumption strategy c(x) intersects the natural growth rate g(x). As time proceeds and the discount rate declines, the consumption profile gets flatter and converges towards the intersection at x * * in the long run. Under exponential discounting, (asymptotic) extinction occurs for nρ > r, a relatively mild condition when there are many players. In this case, consumption policy c(x)

Figure 4: Hyperbolic Discounting and the Tragedy of the Commons
lies everywhere above the g(x) curve and x and c decline at declining rates. With the intuition developed in the context of Proposition 2, it is clear that extinction is impossible for timeconsistent hyperbolic discounting. While x can be only exhausted asymptotically there exists always a finite time at which nρ(t) < r, which guarantees long-run sustainability.
I next illustrate adjustment dynamics with an example. As for the simple model, I set r = 0.04; κ = 10; η = 2 and consider first the case of two players (n = 2). Blue (solid) lines in Figure   5 show adjustment dynamics for µ = 0.04 and σ = 0.03 (implying α = 0.0225 and β = 1.777).
The resource approaches quickly the steady state (below the maximum yield of κ/2 = 5) as the discount rate declines. The lower right panel shows the linear consumption strategy c(x).
When α rises to 0.06, implying an equivalent constant discount rate ofρ = 0.0467, extinction becomes inevitable with exponential discounting. Hyperbolic discounting however leads to sustainable consumption. Results are shown by red (dash-dotted) lines in Figure 5. The initial discount rate is now above 10 percent and far above the constant equivalent rate. The consumption strategy c(x) gets steeper (lower right panel) and individuals consume more initially (upper left panel). However, as time proceeds, consumption declines at a faster rate than the resource stock due to the declining interest rate. In the long-run, the system approaches the same steady state as in the benchmark case. Parameters: r = 0.04; κ = 10; η = 2 and n = 2 players. Blue (solid) lines: µ = 0.04, σ = 0.03 (i.e. α = 0.0225, β = 1.777). Red (dashed) lines: higher initial discount rate α = 0.06 (implying unsustainability for exponential discounting atρ = α(β − 1)). Green (dash-dotted) lines: high discount rate (α = 0.06) and n = 10 players. The lower right panel shows the consumption strategy c(x).
Finally, we increase the number of competing individuals to n = 10 and keep the high initial discount rate (α = 0.06). Results are shown by green (dash-dotted) lines in Figure 5. Initial consumption is now much higher because of the increased opportunity cost of foregone consumption (upper left panel and lower right panel) and the resource declines steeply. However, since the discount rate declines as well and faster, sustainability is ensured and the system converges toward the steady state at x * * = κ/11 = 0.9. Summarizing, missing property rights for nonrenewable resources (the environment) are less of a tragedy when non-cooperative individuals apply time-consistent hyperbolic discounting.

Economic Growth, Pollution, and the Environmental Kuznets Curve
In this section we investigate the role of time-consistent hyperbolic discounting in the neoclassical growth model augmented by pollution and environmental quality based on Brock (1977) and Becker (1982). In order to minimize changes with regard to the previous section we treat environmental quality as a renewable resource (as in Xepapadeas, 2005): in which g(x) is an environmental regeneration function and p is pollution. For simplicity, we model pollution as an input in production such that aggregate output is given by y = f (k, p), in which k is the aggregate capital stock.
The economy is populated by a large number of individuals (normalized to one) who derive utility from consumption and a clean environment. Their lifetime utility at decision time t 0 is given by The social planner maximizes lifetime utility of the representative individual taking into account (22) and the accumulation of new capital by net in which δ is the depreciation rate. For exponential discounting it has been shown that this type of problem can lead to a steady state where a higher discount rate improves environmental quality (Smulders, 2007;Asheim, 2008). The reason is that higher impatience increases consumption and reduces investment such that the economy converges towards a lower steady-state capital stock, implying less pollution and higher environmental quality.
The Hamiltonian associated with the maximization problem is given by To further simplify the exposition, I follow Smulders (2007) and impose iso-elastic functions for production and utility, f (k, p) = k γ p ϵ , u(c, x) = (cx ϕ ) 1−η /(1 − η). Then, as shown in the Appendix, the solution leads to an augmented Ramsey rule for consumption (24) and an equation of motion for optimal pollution (25): Proposition 5. Hyperbolic discounting establishes the Golden Rule of capital accumulation and the modified Green Golden Rule as a steady-state outcomes. Exponential discounting is neither consistent with the Golden Rule nor with the modified Green Golden Rule.
For the proof first note that, at the steady state,ċ =ẋ = 0 and ρ(t) = 0. Then, (24) reduces to γy/k − δ = 0. Let s denote the savings rate such that c = (1 − s)y and (23), evaluated at the steady state, becomesk = y − (1 − s)y − δk = 0, implying sy/k − δ = 0. Hence, s = γ. The result that the savings rate equals the capital share in production is known as the Golden Rule of capital accumulation. That it cannot be fulfilled given exponential discounting is textbook knowledge (see, e.g., Barro and Sala-i-Martin, 2004).
Using the information that γk γ−1 p ϵ − δ = 0 and noting that, at the steady state,ṗ =k = 0, equation (25) reduces to where the last equality follows from c = (1 − s)y and s = γ. As shown in the Appendix, this condition, in conjunction with the Golden Rule, maximizes utility at the steady state. It has therefore been dubbed the modified Green Golden Rule (Smulders, 2007). That exponential discounting cannot meet the Green Golden Rule is evident from the fact that, at the steady state supported by exponential discounting, we have (from (24) with ρ(t) =ρ) that γk γ−1 p ϵ − δ =ρ, implying that the right hand side of (26) equalsρ ̸ = 0.
Finally, we investigate adjustment dynamics. For simplicity we retain the functional form of g(x) from (6), i.e.ẋ = rx The economy is described by the dynamic system (23)- (25) and (27) with y = f (k, p) = k γ p ϵ .
Since four dimensions cannot be analyzed using phase diagram techniques, we start right away with a numerical analysis. I set r = 0.04, κ = 10, η = 2 as for the simple model from Section 3.
I set the capital share γ to 0.6, capturing a broad definition of capital, and I set the pollution externality to 0.3. In order to start the economy at a relatively early phase of industrialization, I set the initial value for capital at 30 percent of steady-state level and the initial value of environmental quality to 70 percent of the pristine level κ. Adjustment dynamics are shown in Figure 6.
It is interesting to inspect first the adjustment dynamics implied by using the equivalent constant discount rate (of 0.0175). They are shown by red (dashed) lines in Figure 6. The upper panels show the concave adjustment paths for consumption and capital stock, a behavior, well known from the standard neoclassical growth model. Pollution is strongest in the early phase of industrialization and declines to almost steady-state level in a century (lower left panel). Consequently, environmental quality x falls initially, overshoots mildly and adjusts from below towards its steady-state level. The lower right panel shows the implied association between income and environmental quality. The economy starts at the North-West corner with low income and high environmental quality and converges with overshooting behavior of environmental quality towards the steady state of high income. This overshooting behavior is known as the environmental Kuznets curve. 8 Solid (blue) lines in Figure 6 show the adjustment dynamics for hyperbolic discounting. Individuals consume initially more due to the high present-bias (upper left panel). This means 8 Brock and Taylor, 2005;Egli and Steger, 2007. The standard environmental Kuznets curve refers to a u-shaped association between income and the pollution stock. Here, this is expressed as an inverted u-shaped association between income and environmental quality. Parameters: r = 0.04; κ = 10; η = 2; γ = 0.6; ϵ = 0.1; ϕ = 0.3; k(0) = 0.3k * , x(0) = 0.7x * . Blue (solid) lines: µ = 0.04, σ = 0.03 (i.e. α = 0.0225, β = 1.777). Red (dashed) lines: same economy with exponential discountingρ = α(β − 1) = 0.0175). that they invest and pollute initially a little less (upper right and lower left panel). The discount rate, however, declines quickly and after less than 20 years, consumption falls short of the level implied by exponential discounting. Investment, in turn, exceeds the level implied by exponential discounting and after about 60 years the capital stock exceeds the level implied by exponential discounting (upper right panel). The economy becomes richer and cleaner at about the same time as pollution falls below the level implied by exponential discounting. After a century, the exponentially discounting economy basically converged to its steady state while the hyperbolically discounting economy keeps growing and becoming cleaner. As a result, the environmental Kuznets curve becomes more pronounced than for exponential discounting (lower right panel).

Conclusion
In this paper I proposed a time-consistent method to discount at a declining rate and applied it to three canonical environmental problems. I compared the solutions to those obtained under conventional exponential discounting, applying the normalization that both methods provide the same present value of a constant flow. I showed that hyperbolic discounting leads to a more conservative use of resources and the environment than exponential discounting and that it establishes the social optimum at the steady state (consumption of maximum yield, implementation of the Golden Rule and the Green Golden Rule). The unconditional convergence towards the social optimum occurs despite initially high present-bias and discount rates that by far exceed the equivalent constant rate. The reason is that in finite time the declining discount rate falls short of any positive natural growth rate of the resource and extinction in finite time can be ruled out by the optimality conditions. This means that any trajectory irrespective of potentially very high initial discount rates converges eventually to the long-run social optimum.
Under exponential discounting, however, asymptotic extinction is a real threat, in particular when property rights are not defined or hard to enforce and many parties exploit an exhaustible resource non-cooperatively. For exponential discounting, the Green Golden Rule (the long-run social optimum) applies only for the knife-edge case ofρ = 0 whereas, for time-consistent hyperbolic discounting, any discounting function ρ(t) converges towards the Green Golden Rule.
These results are policy relevant since they show that the concern of time-inconsistent, suboptimal, and potentially disastrous outcomes of hyperbolical discounting can be resolved by the use of multiplicatively separable discount factors. Multiplicative separability of the discount factor ensures time consistency and socially optimal solutions at the steady state. A discounting rule is relatively easy to implement as an institution and then hard to change. These features are in particular important when long-run decisions are made subject to a slow changing state variable, such as the environment. Finally, using (A.8) andẏ/y = γk/k + ϵṗ/p we obtain (25) in the text.