A Review of Tipping Points and Precaution using HJB equations

Because the welfare indicator is quadratic and the state transition (1) is linear, solutions to the HJB-Eqs. (4) and (8) can be found by postulating quadratic current-value functions \(V_{a}\) and \(V_{b}\).

If \(V_{a} (s) = p_{a} s^{2} + q_{a} s + c_{a} ,p_{a} < 0\), the maximization in Eq. (4) yields the optimal production \(y_{a} = \beta + 2p_{a} s + q_{a}\), as a function of the stock \(s\). After substitution, Eq. (4) must hold for all \(s\), so that the coefficients of \(s^{2}\) and \(s\) must be zero. The solutions for the parameters of \(V_{a}\), and thus \(y_{a}\), become

By substituting optimal production \(y_{a} = \beta + 2p_{a} s + q_{a}\) in Eq. (1), accumulation of the stock of pollution after tipping becomeswhere \(s_{T}\) is the stock of pollution at the tipping point \(T\). The differential Eq. (10) is stable, and the state \(s\) converges to the steady state \(s_{a} = (\beta + q_{a} )/(\delta - 2p_{a} )\), because \(2p_{a} - \delta < 0\). Note that \(p_{a}\) is the negative root of \(2p_{a}^{2} - (r + 2\delta )p_{a} - 0.5\gamma_{a} = 0\). Using (9), the steady state \(s_{a}\) becomes

Figure 1 depicts the optimal solution. The line for the optimal production after tipping \(y_{a} = \beta + 2p_{a} s + q_{a}\) intersects the steady-state line \(y = \delta s\) in \(s_{a}\). For \(s < s_{a}\), the optimal production is larger than \(\delta s\), so that the stock of pollution \(s\) increases. For \(s > s_{a}\), the optimal production is smaller than \(\delta s\), so that the stock of pollution \(s\) decreases. In steady state \(s_{a}\), the optimal production is equal to the natural assimilation \(\delta s\).

Note that the line for the optimal production after tipping \(y_{a} = \beta + 2p_{a} s + q_{a}\) intersects the \(y\)-axis in \(\beta + q_{a}\), which is equal to \(\beta (r + \delta )/(r + \delta - 2p_{a} )\). It follows that if the parameter \(p_{a}\) increases, the line starts higher and declines slower, and the steady state is larger. This results if the risk of tipping is ignored. That problem is the same as the problem after tipping, but it has a lower damage parameter \(\gamma_{b} < \gamma_{a}\), and it starts at time 0 in the initial pollution stock \(s_{0}\). Denoting the parameters of the current-value function of this problem by \(p*\) and \(q*\), it follows that \(p* > p_{a}\) and \(\beta + q* > \beta + q_{a}\), using (9). The steady state is \(s* = \beta (r + \delta )/(\gamma_{b} + \delta (r + \delta )) > s_{a}\), using (11). Figure 1 also depicts the line for the optimal production \(y* = \beta + 2p*s + q*\), in the case the risk of tipping is ignored. It provides a benchmark for the sequel.

The analysis before tipping follows the same route. If \(V_{b} (s) = p_{b} s^{2} + q_{b} s + c_{b} ,p_{b} < 0\), the maximization in Eq. (8) yields the optimal production \(y_{b} = \beta + 2p_{b} s + q_{b}\), as a function of the stock \(s\). After substitution, Eq. (8) must hold for all \(s\), so that the coefficients of \(s^{2}\) and \(s\) must be zero. Equation (8) is different from Eq. (4) in three respects. First, it has parameter \(\gamma_{b}\) instead of \(\gamma_{a}\). Second, the additional term \(- hV_{b} (s)\) implies that \(r\) is augmented with \(h\), which reflects the discounting effect of the hazard rate \(h\). Third, the additional term \(hV_{a} (s)\) implies that the coefficients of \(s^{2}\) and \(s\) on the right-hand side of Eq. (8) have the additional terms \(hp_{a}\) and \(hq_{a}\), respectively. Note that if the value after tipping \(V_{a} (s)\) is zero, the hazard rate \(h\) simply adds to the discount rate \(r\), which reflects the possible interpretation of a discount rate as the probability that life comes to an end. The solutions for the parameters \(p_{b}\) and \(q_{b}\) of \(V_{b}\) (parameter \(c_{b}\) is not needed in the sequel), and thus of \(y_{b}\), become

By substituting optimal production \(y_{b} = \beta + 2p_{b} s + q_{b}\) in Eq. (1), accumulation of the stock of pollution before tipping becomes

The differential Eq. (13) is stable, because \(2p_{b} - \delta < 0\), and the stock \(s\) aims for the steady state \(s_{b} = (\beta + q_{b} )/(\delta - 2p_{b} )\). However, the stock \(s\) does not converge to this steady state because with probability one, the ecosystem will tip at some point. The optimal production will switch, and the stock will converge to the steady state \(s_{a}\) after tipping. Note that \(p_{b}\) is the negative root of \(2p_{b}^{2} - (r + h + 2\delta )p_{b} - 0.5\gamma_{b} + hp_{a} = 0\). Using (9), the targeted steady state \(s_{b}\) becomes

It is easy to prove the following two lemmas.

It follows that \(s_{b} > s_{a}\) if and only if \(s* > s_{a}\). Moreover, \(s* > s_{b}\) if and only ifand this also holds if and only if \(s* > s_{a}\). Q.E.D.

Respectively. It follows that \(p_{a} < p*\), because \(\gamma_{b} < \gamma_{a}\). Furthermore, the left-hand side of the third equation in (17) is positive for \(p_{b} = p_{a}\) and negative for \(p_{b} = p*\), so that the negative root \(p_{b}\) must lie between \(p_{a}\) and \(p*\), i.e., \(p_{a} < p_{b} < p*\). Q.E.D.

Figure 1 depicts the line for the optimal production before tipping \(y_{b} = \beta + 2p_{b} s + q_{b}\). It is situated between the two other lines. If the initial stock of pollution \(s_{0}\) is small, the production \(y\) starts on this line. When tipping occurs at time \(T\), the stock of pollution \(s\) has accumulated up to \(s(T)\), and the production \(y\) jumps down to the line for the optimal production after tipping \(y_{a} = \beta + 2p_{a} s + q_{a}\). After \(T\), the stock of pollution \(s\) converges to the steady state \(s_{a}\), and the optimal production \(y\) converges to the steady-state level, which is equal to \(\delta s_{a}\). Note that if tipping occurs early, at \(T = T_{1}\), production \(y\) jumps down but it remains higher than the natural assimilation, so that the stock of pollution \(s\) continues to accumulate. However, if tipping occurs late, at \(T = T_{2}\), so that the stock of pollution \(s\) has accumulated beyond \(s_{a}\), production \(y\) jumps down below the natural assimilation, so that the stock of pollution \(s\) decreases and converges to the steady-state level after tipping \(s_{a}\).

The general picture is clear. Anticipating the possibility of tipping and an upward jump in the damage of stock pollution, it is optimal to behave precautionary by lowering the production. This slows down the accumulation of the stock of pollution and reduces the jump to the lower production after tipping.

Figure 2 depicts the dynamics of the stock of pollution \(s\). Differential Eq. (13), for the period before tipping, and (10), for the period after tipping, are linear, and thus easy to solve. Using (14) and (11), the solutions are

Figure 2 depicts the dynamics for the two tipping points, \(T_{1}\) and \(T_{2}\). At \(T_{1}\), the stock of pollution \(s\) is still below \(s_{a}\). Production \(y\) jumps down (see Fig. 1), so that the path has a kink at \(T_{1}\). The stock of pollution \(s\) continues to accumulate and converges to \(s_{a}\) from below. At \(T_{2}\), however, the stock of pollution \(s\) has already accumulated beyond the level \(s_{a}\). Production \(y\) jumps down, below the natural assimilation (see Fig. 1). The stock of pollution \(s\) decumulates and converges to \(s_{a}\) from above. In this model, the possibility of tipping always leads to precaution.

In case of global pollution (like climate change) and independent countries, the game of international pollution control arises which changes the optimal control problem into a differential game. Cooperation is still optimal control but ignoring the transboundary pollution externalities leads to a non-cooperative Nash equilibrium. Furthermore, the Hamilton-Jacobi-Bellman framework implies stock-dependent strategies and yields the Markov-perfect or feedback Nash equilibrium. Solving for the linear symmetric Nash equilibrium in the HJB framework follows the same route as above and leads to higher levels for production and the stock of pollution than under cooperation [7]. Adding the possibility of tipping is somewhat tedious but gives a similar precautionary picture as above. However, Dockner and Long [3] show that for this game, a nonlinear Markov-perfect Nash equilibrium exists with the steady state converging to the steady state of the cooperative outcome if the discount rate converges to zero. Adding the possibility of tipping to this analysis is not straightforward and is left for further research.

The Hamilton-Jacobi-Bellman Eqs. (21) and (22) have the same structure, so that the solution of (21) follows immediately from the solution of (22) by setting \(h = 0\).

Before tipping, the optimality condition in (22) becomes

By substituting the optimal \(y(s)\) from (23) in (22) and differentiating (22) with respect to \(s\), the terms \(U^{\prime}(y)y^{\prime}\) and \(- V_{b}^{\prime } (s)y^{\prime}\) cancel out (envelope theorem), and this yields

For the CRRA utility function \(U(y) = y^{1 - \sigma } /(1 - \sigma )\), \(U^{\prime}(y) = y^{ - \sigma }\), \(U^{\prime\prime}(y) = - \sigma y^{ - \sigma - 1}\), using Eq. (23) again, Eq. (24) becomes the differential equation in \(y\)

Using Eq. (19), the differential equation in the time-domain becomes

The stable manifold for the solution of the dynamical system (19) and (26) toward the steady state of this system yields the same curve as the stable solution of (25).

After tipping, the optimality condition in (21) becomesand in the same way as above (with \(h = 0\)) the differential equations in \(y\) become

After tipping, the solution of the differential Eq. (28) yields the optimal harvesting policy \(y_{a} (s)\). The optimal path of the fish stock converges to the steady state \(s_{a}\) given by \(G_{a}^{\prime } (s_{a} ) = r\). In the steady state \((s_{a} ,G_{a} (s_{a} ))\), using Eq. (28), l’Hôpital’s rule yields a quadratic equation for the slope \(y_{a}^{\prime } (s_{a} )\) of \(y_{a} (s)\):

The phase diagram in the time-domain, with the steady-state conditions \(y = G_{a} (s)\) and \(s = s_{a}\), implies that the slope of \(y_{a}^{\prime } (s_{a} )\) is the positive root of Eq. (30), i.e.,

Note that the logistic growth function \(G(s) = gs(1 - s/k)\) has \(G^{\prime}(s) = g(1 - 2s/k)\) and \(G^{\prime\prime}(s) = - 2g/k\), so that \(2G^{\prime\prime}(s)G(s) = (G^{\prime}(s))^{2} - g^{2}\).

Because \(G_{a}^{\prime } (s_{a} ) = r\), it follows that Eq. (31) becomes

If the elasticity of intertemporal substitution \(\sigma^{ - 1}\) is equal to 0.5, the slope \(y_{a}^{\prime } (s_{a} )\) is the constant \(0.5(r + g)\). In fact, the linear optimal harvesting policy \(y_{a} (s) = 0.5(r + g)s\) is the solution of the differential Eq. (28). Because the policy does not depend on the carrying capacity \(k_{a}\), \(y(s) = 0.5(r + g)s\) is also the optimal harvesting policy in the absence of a tipping point. It follows that for \(\sigma^{ - 1} = 0.5\), basically the same story results as in Polasky et al. [10]. Before tipping, the fishery behaves as if there is no risk of tipping, and the optimal path follows the stable manifold \(y(s) = 0.5(r + g)s\) toward the steady state \(s_{b}\) given by \(G_{b}^{\prime } (s_{b} ) = r\). At the tipping point, the fishery adjusts to the new circumstances. If the fish stock has passed \(s_{a}\), the optimal path follows this stable manifold back toward the steady state \(s_{a}\) given by \(G_{a}^{\prime } (s_{a} ) = r\). Figure 3 shows the stable manifold \(y(s)\). The only difference with a linear fishery is that the optimal path toward \(s_{b}\) is not a moratorium, and the optimal path back to \(s_{a}\) is not instantaneous. Note that this result fits with the analysis of the fishery before tipping in Sect. 3.1. The reason is that according to Eq. (27) the term \(V_{a}^{\prime } (s)\) in Eqs. (25) and (26) is equal to \(U^{\prime}(y_{a} )\), and thus equal to \(y_{a}^{ - \sigma }\). It follows that the last term between brackets in Eqs. (25) and (26) becomes \(h(1 - y_{a}^{ - \sigma } /y_{b}^{ - \sigma } )\), which cancels out because \(y_{a}\) and \(y_{b}\) have the same value for each \(s\), although \(s\) changes direction when tipping occurs. The golden rule before tipping becomes \(G_{b}^{\prime } (s_{b} ) = r\), which is the same as the golden rule \(G_{b}^{\prime } (s*) = r\) for the fishery in the absence of a tipping point.

If the elasticity of intertemporal substitution \(\sigma^{ - 1}\) is smaller than 0.5, the slope \(y_{a}^{\prime } (s_{a} )\) in Eq. (31) is smaller than \(0.5(r + g)\), and the stable manifold \(y_{a} (s)\) bends away below the line \(0.5(r + g)s\) (Fig. 4). It passes below the steady state \((s*,G_{b} (s*))\), so that harvesting \(y\) jumps down if tipping occurs in that steady state. The steady-state stock of fish before tipping \(s_{b}\) cannot be equal to \(s*\), because it does not fit the golden rule of the fishery before tipping which is given byaccording to Eqs. (27) and (25), where \(y_{b} (s_{b} ) = G_{b} (s_{b} )\). The second term on the right-hand side of Eq. (33) is not zero in this case. If harvesting \(y\) jumps down at the tipping point from \(G_{b} (s_{b} )\) to the stable manifold \(y_{a} (s)\), this term in Eq. (33) becomes negative, so that the steady-state stock of fish before tipping \(s_{b}\) is larger than \(s*\), because \(G_{b}^{\prime } (s_{b} ) < r\). Figure 4 pictures this situation and denotes the steady state before tipping by \(s_{b2}\). The steady state \(s_{b2}\) lies between \(s*\) and the intersection of the stable manifold \(y_{a} (s)\) and the growth function \(G_{b} (s)\). It means that harvesting becomes precautionary, because it aims for a higher steady-state stock of fish than in the absence of a tipping point. If the hazard rate \(h\) increases, the steady state \(s_{b2}\) moves closer to the intersection point, so that precaution increases.

If the elasticity of intertemporal substitution \(\sigma^{ - 1}\) is larger than 0.5, the slope \(y_{a}^{\prime } (s_{a} )\) in Eq. (31) is larger than \(0.5(r + g)\). The stable manifold \(y_{a} (s)\) bends away above the line \(0.5(r + g)s\). It passes above the steady state \((s*,G_{b} (s*))\), so that harvesting \(y\) would jump up if tipping occurs in that steady state. The golden rule in Eq. (33) does not fit. If harvesting \(y\) jumps up from \(G_{b} (s_{b} )\) to the stable manifold \(y_{a} (s)\) at the tipping point, the second term on the right-hand side of Eq. (33) is positive, so that the steady-state stock of fish before tipping \(s_{b}\) is smaller than \(s*\), because \(G_{b}^{\prime } (s_{b} ) > r\) (see Fig. 4). This steady state \(s_{b1}\) lies again between \(s*\) and the intersection of the stable manifold \(y_{a} (s)\) and the growth function \(G_{b} (s)\), but to the left of \(s*\). It follows that the harvesting does not become precautionary but increases exploitation and aims for a lower steady-state stock of fish than in the absence of a tipping point. The analysis yields the following proposition [14]:

Note that the elasticity of intertemporal substitution \(\sigma^{ - 1}\) can be interpreted as the price elasticity. For this interpretation, sufficient flexibility (\(\sigma^{ - 1} > 0.5\)) allows for an increase in exploitation because the later necessary reduction is not that harmful.

A numerical example may help to get the picture. Using \(G^{\prime}(s) = g(1 - 2s/k)\), it follows that \(s* = 30\) and \(s_{a} = 24\), if \(g = 0.05\), \(r = 0.02\), \(k_{b} = 100\) and \(k_{a} = 80\). For \(h = 0.05\), it follows from Eq. (33), by calculating the stable manifolds \(y_{a} (s)\) numerically, that \(s_{b1} = 28.65\), if \(\sigma^{ - 1} = 1\), and \(s_{b2} = 31.53\), if \(\sigma^{ - 1} = 0.33\).

Polasky et al. [10] show in the fishery model with fixed-price revenues as utility that an endogenous or stock-dependent hazard rate \(h(s)\), with \(h^{\prime}(s) < 0\), leads to precaution before tipping. The sections above show that a fishery model with a more general utility can lead to precaution or the opposite for a constant hazard rate \(h\). If the opposite results here, it is interesting to see what the net effect is in case the hazard rate \(h\) depends on the stock of fish \(s\). It is easy to implement a stock-dependent hazard rate \(h(s)\) in the Hamilton-Jacobi-Bellman Eq. (22), and to derive the extension of the golden rule for the fishery which yields, for the new current-value functions,

Because \(h^{\prime}(s_{b} ) < 0\) and \(V_{b} (s_{b} ) > V_{a} (s_{b} )\) the third term on the right-hand side of Eq. (34) is negative which pushes \(s_{b}\) upwards. This is the precautionary effect in Polasky et al. [10], because the second term is zero in the linear fishery. However, Sect. 3.2 shows that the sign of the second term of Eq. (34) can go both ways in case of a more general utility. It follows that the precautionary effect is either stronger or reduced or even turned around for a stock-dependent hazard rate \(h(s)\). The net result depends on the parameters of the fishery (see [14].

The problem with extending the analysis to a differential game with \(n\) fishing actors is that the HJB Eqs. (21) and (22) in individual harvest \(y = y_{i} ,i = 1,...,n,\) have the term \(G(s) - ny\) instead of \(G(s) - y\), so that differentiation of these HJB equations with respect to \(s\) yields differential Eqs. (25) and (27) in the state-domain that do not have the equivalent differential Eqs. (26) and (28) in the time-domain. It follows that (25) and (27) have multiple stable solutions. This is also the result of Dockner and Long [3] in the game of international pollution control. As mentioned in Sect. 2.3, Dockner and Long [3] show that in this way, the tragedy of the commons can be solved to some extent. However, Wirl [14] shows that this not only depends on the availability of nonlinear strategies but that it is crucial to have an increasing elasticity of marginal utility, which holds in the game of international pollution control. Since the fishery here has a constant elasticity of intertemporal substitution, which is the inverse of the elasticity of marginal utility, this result may not hold.

The analysis with a possible tipping point becomes much more complicated and is left for further research. It is easier, however, to perform this analysis for an open-loop Nash equilibrium. In that case, the strategies of the other actors are exogenous inputs which do not affect the analysis in the state-domain, and the HJB framework is only used to implement the hazard rate. This approach was applied to the Ramsey growth model and will be shortly discussed at the end of the next section.

For the constant hazard rate \(h\), the Hamilton-Jacobi-Bellman equations after and before tipping becomewhere \(V_{a}\) and \(V_{b}\) denote the current-value functions after and before tipping.

The golden rules of capital accumulation are \(F_{a}^{\prime } (k_{a} ) = r\) after tipping and \(F_{b}^{\prime } (k*) = r\) in the absence of tipping, while before tipping the golden rule becomeswhere \(k_{a}\), \(k*\) and \(k_{b}\) denote the respective steady-state capital stocks.

In the steady state \((k_{a} ,F_{a} (k_{a} ))\) after tipping, the slope of the stable manifold is

The production function \(F(k) = A\sqrt k\) has \(F^{\prime}(k) = 0.5Ak^{ - 0.5}\) and \(F^{\prime\prime}(k) = - 0.25Ak^{ - 1.5}\) so that \(F^{\prime\prime}(k)F(k) = - (F^{\prime}(k))^{2}\).

Because \(F_{a}^{\prime } (k_{a} ) = r\), the slope of the stable manifold \(c_{a}^{\prime } (k)\) in the steady state is \(2r\), if the elasticity of intertemporal substitution \(\sigma^{ - 1}\) is equal to 2. The same story follows as in Sect. 3 for the fishery, but the threshold is 2 now instead of 0.5 because of the different functional forms. The linear stable manifold \(c_{a} (k) = 2rk\) coincides with the stable manifold before tipping. The economy before tipping targets for the same capital stock as in the absence of a tipping point. If \(\sigma^{ - 1} < 2\), the targeted capital stock \(k_{b}\) is larger than \(k*\), which implies precautionary saving in this case. However, if \(\sigma^{ - 1} > 2\), the targeted capital stock \(k_{b}\) is smaller than \(k*\), and the opposite occurs.

This result depends of course on the specific functional form of the production function \(F(k) = A\sqrt k\). However, for a general increasing concave production function \(F\), the slope of the stable manifold \(c_{a} (k)\) increases for an increasing elasticity of intertemporal substitution \(\sigma^{ - 1}\). In general, a value of \(\sigma^{ - 1}\) exists for which the stable manifold \(c_{a} (k)\) intersects the production function \(F_{b}\) in the steady state \((k*,F_{b} (k*))\). For a lower \(\sigma^{ - 1}\) the stable manifold \(c_{a} (k)\) passes below this steady state, so that consumption \(c\) jumps down and the targeted capital stock \(k_{b}\) is larger than \(k*\). However, for a higher \(\sigma^{ - 1}\) the stable manifold \(c_{a} (k)\) passes above this steady state, so that consumption \(c\) jumps up and the targeted capital stock \(k_{b}\) is smaller than \(k*\).

Because the elasticity of intertemporal substitution can be interpreted as the inverse of intergenerational inequality aversion, optimal saving is precautionary if this inequality aversion is high. This implies, for example, that the prospect of possible climate tipping requires to save more and consume less, if the intergenerational inequality aversion is sufficiently high.

The hazard rate \(h\) for climate tipping depends on the stock of accumulated greenhouse gases \(s\), with \(h^{\prime}(s) > 0\). Greenhouse gases \(e\) result from the use of fossil fuels (also \(e\) for a proper choice of dimensions) as input factor in the production function \(F(k,e)\). The accumulation of the stock of pollution \(s\) becomeswhere \(\delta\) denotes the natural assimilation rate, and \(s_{0}\) the initial stock of pollution.

The economy produces a stream of greenhouse gas emissions by using fossil fuels but for a constant hazard rate \(h\) and in the absence of direct damage of pollution, this does not affect the analysis. For each level of the capital stock \(k\), and a given price of fossil fuels \(d\), maximization of \(F(k,e) - de\) over \(e\) yields the optimal production function \(\tilde{F}(k)\), net of costs and depreciation. The problem is same as in Sect. 4.1. Because of the possible shock to total factor productivity at the climate tipping point, the optimal production function can shift from \(\tilde{F}_{b}\) before tipping to \(\tilde{F}_{a}\) after tipping.

For a stock-dependent hazard rate \(h(s)\), an incentive arises to lower the risk of climate tipping and reduce the stream of greenhouse gas emissions. The analysis becomes more complicated, because it has a stock-dependent hazard rate \(h(s)\) and two state Eqs. (36) (with \(F(k,e)\)) and (41). After tipping, the problem is the same again as in Sect. 4.1, because pollution does not play a role anymore, but before tipping, the Hamilton–Jacobi-Bellman equation becomes significantly different:

The current-value function before tipping \(V_{b}\) has two state variables, \(k\) and \(s\), and \(V_{a}\) denotes the current-value function after tipping.

The optimality conditions before tipping become

The term \(- V_{bs} (k,s)/V_{bk} (k,s)\) in the second condition of (43) can be interpreted as the social cost of carbon. A carbon tax \(\tau\) equal to this term corrects for this externality. It pushes down the stream of greenhouse gas emissions \(e\), and thus the accumulated stock of greenhouse gases \(s\) and the hazard rate \(h(s)\). Implementing this tax \(\tau\), the optimal (lower) use of fossil fuel \(e\) yields the optimal production function \(\tilde{F}_{b} (k)\), net of costs and depreciation, which drives the capital accumulation (36). Note that \(\tilde{F}_{b} (k)\) depends on the tax \(\tau\) now. By substituting the optimal consumption \(c\) and use of fossil fuels \(e\) from (43) in (42) and then by partially differentiating (42) with respect to \(k\) and with respect to \(s\), some terms cancel out using (43) (envelope theorem), and this yields in the time-domain (omitting the time dependence of \(k\), \(s\) and \(\tau\))

Using (43), the first equation of (44) yields the system of differential equations

This system is familiar but requires the paths of the tax \(\tau (t)\) and the stock of greenhouse gases \(s(t)\). The path of the tax \(\tau (t)\) determines the path of the emissions \(e(t)\) and thus the path of the stock \(s(t)\), by using Eq. (41). Because \(\tau = - V_{bs} (k,s)/V_{bk} (k,s)\), Eqs. (44) yield

Note that for a constant hazard rate \(h\), the second term on the right-hand side of (46) is zero, so that \(\tau = 0\) and the system reduces to the case of a constant hazard rate above.

Using (45), the golden rule of capital accumulation becomeswhere \(\tau_{b}\) and \(s_{b}\) denote the steady-state values of the tax \(\tau\) and the stock of greenhouse gases \(s\). The golden rule (47) for a stock-dependent hazard rate \(h(s)\) is like the golden rule for a constant hazard rate \(h\) but has important differences. The steady-state tax \(\tau_{b}\) lowers the use of fossil fuel. This has two effects. First, it lowers the steady-state stock of greenhouse gases \(s_{b}\) and thus the hazard rate \(h(s_{b} )\) which implies that if saving is precautionary, this result is mitigated. Second, it affects the level of production \(\tilde{F}_{b}\) in the steady state, so that it affects whether precautionary saving results or not, because this level of production also determines whether consumption jumps down or not when tipping occurs. Remember that when consumption jumps down, the second term on the right-hand side of Eq. (49) is negative, so that the marginal production \(\tilde{F}_{bk}\) is lower than \(r\) and the capital stock \(k_{b}\) increases.

The dynamical system (41)–(45)–(46) is very difficult to analyze. Adding the option of renewables as an alternative for fossil fuel in the production function and calibrating the model with data for the world economy in 2010, van der Ploeg and de Zeeuw [9] analyze the system numerically, for different hazard rates and different shocks to total factor productivity. They conclude that the optimal policy response to the possibility of climate tipping is twofold. First, the carbon tax should be much higher than what results in the absence of climate tipping (about 60 instead of 15 dollar per ton CO2). Second, the optimal policy requires precautionary saving (an additional capital accumulation of about 25%). They also show that a higher elasticity of intertemporal substitution (or a lower intergenerational inequality aversion) lowers precautionary saving but for a range of reasonable values, saving remains precautionary in this analysis.

The same problem arises as in the previous sections. However, as mentioned in Sect. 3.4, it is possible to perform the analysis by restricting the differential game to the open-loop Nash equilibrium. In that case, the HJB framework is only used to implement the hazard rate. The strategies of the other actors are exogenous time-dependent inputs. It follows that the value functions are time-dependent, and the problem is not stationary, but otherwise the analysis is the same as above. In case of two actors, this leads to two sets of dynamical equations in consumption \(c\), capital stock \(k\) and tax \(\tau\), and a joint dynamical equation in the pollution stock \(s\) [7]. An interesting application is to compare cooperative and non-cooperative behavior of the North and the South when facing the joint risk of climate tipping. Assuming that the North is in a further stage of development and less vulnerable to climate catastrophes, the result is as follows. The carbon taxes are lower in the absence of cooperation, as is to be expected, so that precautionary saving increases. If the North and South cooperate, they aim for a high common carbon tax but initially allow for a lower carbon tax in the South, so that the South can catch up. If the North and South do not cooperate, the North always has a low carbon tax. The South has initially a low carbon tax, giving priority to growth, but later a high carbon tax, giving priority to prevent climate tipping.