## Abstract

Global CO_{2} emissions are understood to be the largest contributor to anthropogenic climate change, and have, to date, been highly correlated with economic output. However, there is likely to be a negative feedback between climate change and human wealth: economic growth is typically associated with an increase in CO_{2} emissions and global warming, but the resulting climate change may lead to damages that suppress economic growth. This climate–economy feedback is assumed to be weak in standard climate change assessments. When the feedback is incorporated in a transparently simple model it reveals possible emergent behaviour in the coupled climate–economy system. Formulae are derived for the critical rates of growth of global CO_{2} emissions that cause damped or long-term boom–bust oscillations in human wealth, thereby preventing a soft landing of the climate–economy system. On the basis of this model, historical rates of economic growth and decarbonization appear to put the climate–economy system in a potentially damaging oscillatory regime.

## 1. Introduction

It is widely accepted that climate change will have major impacts on humankind. Depending on the magnitude of twenty-first century climate change, negative impacts are expected on water, food, human health [1,2] and ultimately economic growth [3,4]. Global CO_{2} emissions, which are the largest contributor to anthropogenic climate change [5], have, to date, been highly correlated with economic output [6]. As a result there is a negative feedback between climate change and economic growth that is mediated by CO_{2} emissions: an increase in human wealth causes an increase in emissions and global warming, but the warming damages human wealth, slowing its rise or even making it fall.

This climate–economy feedback is typically neglected in a standard climate change assessment [7], which is largely a serial process going from socioeconomic scenarios to emissions to climate change to impacts [8]. Some integrated assessment models do include the feedback but typically only weakly [3]. A feasible sensitivity of the economy to the climate results in important emergent properties, which are the subject of this paper. Dangerous rates of change [9] can be defined as those rates that cause long-term oscillations, thereby preventing a ‘soft landing’ of the climate–economy system at a new equilibrium state [10].

Using a simple model of the coupled climate–economy system, this paper derives formulae for the critical rates of growth of global CO_{2} emissions that define the edges of stable and unstable regimes. On the basis of this model and estimates of the historical rates of economic growth and decarbonization, which together have led to the historical rates of growth of CO_{2} emissions, the climate–economy system appears to be in a potentially damaging oscillatory regime.

The model is defined in §2 and the stability of its equilibria is analysed in §3. The model is calibrated to the twentieth century data in §4 and projections are presented in §5. Appendix A relates the results to those from the widely used and more detailed dynamic integrated model of climate and the economy (DICE) model [3].

## 2. Model definition

The model presented here describes the global human–environment system with just three state variables: atmospheric CO_{2} concentration (), global warming () and global wealth (), interdependent as in figure 1*a*. Schematically (figure 1*b*) the model has a similar form to a predator–prey model [11]. Global wealth has the role of the prey. It supplies the ‘predator’ of pollution and is reduced by the pollution’s impacts.

### (a) Model equations

Global warming is assumed to increase with the atmospheric CO_{2} concentration according to the standard logarithmic dependence on CO_{2} [12,13], moderated by the extra outgoing radiation from the higher temperature on Earth. Equilibrium climate sensitivity [14] is **Δ***T*_{2*CO2} (for a doubling of CO_{2} from an assumed pre-industrial level of *C*_{PI}), approached on a characteristic climate time scale of *τ*_{T}, which is set by the thermal capacity of the oceans.
2.1

Atmospheric carbon dioxide () increases in proportion to global CO_{2} emissions (), but the excess above the pre-industrial level is reduced by the combined effect of land and ocean carbon sinks with an assumed characteristic time scale of *τ*_{C} years. For the sake of simplicity, nonlinear effects of climate change on land carbon sinks are neglected [15]. In reality, CO_{2} is removed from the atmosphere on a large range of time scales [16]. A single effective time scale of *τ*_{C}=50 years is assumed as this is broadly consistent with the observation that about half of historical anthropogenic CO_{2} emissions have remained airborne [17].
2.2

CO_{2} emissions, , increase with global wealth, , which is human and material capital. Initially, , where *η* is a constant carbon intensity, which is the amount of CO_{2} required to service each unit of wealth. Consistent with historical records of emissions [18–20], the carbon intensity is assumed to fall exponentially over time by a constant, positive decarbonization rate of *μ* per year, so that after *t* years, one unit of wealth can be serviced by CO_{2} emissions of *e*^{−μt} times the amount currently required. If grows at a faster rate than *μ* per year, then emissions will increase as
2.3

Global wealth () grows through net investment in social capital, technology and productivity [21,22], as shown by the positive feedback loop on the right-hand side of figure 1*a*. An increase in occurs when world production exceeds world consumption and depreciation. Within the model is theoretically infinite, only constrained by the condition of natural resources, i.e. by .

The climate–economy feedback loop is closed by assuming that global warming suppresses economic growth. The key model assumption is that the net rate of growth in wealth, , depends on the level of global warming and falls as global warming increases.

The actual ranges of temperature allowing economic growth or forcing economic contraction are unknown, so is only specified as far as assuming that at one (positive) level of global warming the rate of economic growth will fall to the decarbonization rate *μ*, at which point, by equation (2.3), the growth of CO_{2} emissions will be zero.
2.4
2.5
2.6

### (b) Non-dimensional form of model

In non-dimensional variables, the model is a closed or ‘autonomous’ system: 2.7 2.8and 2.9 where 2.10 2.11 2.12 2.13 2.14 2.15and 2.16

## 3. Model equilibria

Without making any approximations, the model has two equilibrium points, i.e. combinations of *C*, *T* and *E* which are in balance, and so can (in theory) be permanent. These points are obtained by setting (2.7)–(2.9) to zero and solving for *C*, *T* and *E*. The equilibrium points are (0,0,0) and, using the assumptions in equations (2.5–2.6), the positive equilibrium
3.1

At the equilibrium level of emissions, CO_{2} concentration and global warming are constant. From equation (2.14) so that a constant value of *E* means increases exponentially at rate *μ*.

### (a) Zero equilibrium is unstable

The zero equilibrium (no emissions, a pre-industrial level of CO_{2} and no warming) is unstable. A small level of emissions grows exponentially at a rate *χ*(0) without (initially) any significant impact on , because the accumulated emissions are initially small, so the radiative forcing is small, and the increase in temperature only emerges over the time scale *τ*_{T}.

### (b) Stability of positive equilibrium

The positive equilibrium may be stable or unstable. If it is stable, differences from the equilibrium get smaller over time, so that configurations of the variables *C*, *T* and *E*, which are only slightly different from the equilibrium configuration, tend over time towards the equilibrium. If it is unstable, differences from the equilibrium get larger over time, so that the equilibrium is practically unattainable.

Whether the equilibrium is stable or unstable depends on the relative time scales for economic growth, global warming and the carbon cycle. In dimensionless variables, the stability depends on the relationship between *χ* and *ϕ*. For,
3.2
3.3
by equation (3.1). By equations (2.5), (2.6) and (2.16)
3.4
Then (proven in §3*c*) the equilibrium is stable if and only if
3.5
Equations (3.3) and (3.5) show that the stability of the equilibrium depends entirely on the damage function *χ*, and the relative time scales of the warming and carbon cycles, *ϕ*. The equilibrium gets less stable the higher the level of warming at which emissions can continue to grow, and the more severe the change in the damage function near the equilibrium. In other words, the slacker the control, but the more suddenly it is applied, the less stable is the equilibrium. This confirms the idea of instability being a function of delays in responses to an oncoming limit [10]. There are many physical analogies to this (e.g. when braking in a car smoothly and early, or suddenly and at the last moment).

Even if the equilibrium is stable, the system oscillates on its way to achieving the equilibrium unless (proven in §3*c*)
3.6
where
3.7
It follows that the system does not oscillate near the equilibrium if
3.8
and does oscillate near the equilibrium if
3.9

### (c) Proof of stability conditions

#### (i) Jacobian of linearized system

Standard linear stability theory [23] proves equations (3.5)–(3.6). By the Hartman–Grobman theorem [24], the qualitative behaviour of small displacements from the equilibrium is the same as the behaviour of small displacements in the linearized system. Let ** s**(

*t*) be the state of the system at time

*t*and

*s*

_{e}be the equilibrium point.

Let *s*_{d}(*t*)=**s**(*t*)−**s**_{e}. Let and . Then let **J** be the Jacobian matrix evaluated at **s**_{e}:
3.10
3.11
3.12
by equation (3.1).

If, as , so that the linearized system tends to the equilibrium point then for the nonlinear system. If , so that the fixed point is unstable for the linearized system, then it is also unstable for the nonlinear system.

or according to whether the eigenvalues of **J** have negative or positive real part. **s**_{d} spirals towards 0 or away from it if any of the eigenvalues of **J** have non-zero imaginary parts.

#### (ii) Characteristic equation of Jacobian

This section shows that the eigenvalues of **J** depend on the size of
3.13
because the characteristic equation of the Jacobian is
3.14
The eigenvalues *λ* of **J**(**s**_{e}) are solutions to its characteristic equation, i.e.
3.15
3.16
3.17

Consider the cubic in equation (3.17). When *h*=0, the roots of the cubic are 0, −*ϕ* and −1. Since *ϕ*>0 (by equation (2.15)), and the coefficient of *λ*^{3} is positive, the graph of the cubic (when *h*=0) is one of the two in figure 2. *h* is a positive constant so it can be considered a height that shifts the graph up the vertical axis, as in figure 3. As *h* increases, the root at zero becomes negative, so equation (3.17) never has a non-negative real root. Also, as *h* increases, the largest negative root gets larger, i.e. tends towards , so that equation (3.17) always has at least one negative root. Since equation (3.17) always has a negative, real root, the cubic (3.17) factorizes into a linear part and a quadratic part. The nature of the other roots depends on the solution of the quadratic part, which depends on *h*. So, the height *h* determines whether the cubic has, in addition to the real negative root: two real negative roots, or one repeated negative root, or two complex conjugate roots.

#### (iii) Condition for stability: no roots with positive real part

This subsection proves equation (3.5). The equilibrium is stable if **J** has no eigenvalues with positive real part. In order to prove (3.5), it is shown that equation (3.17) has no solutions with positive real part, if and only if
3.18
To prove equation (3.18), the cubic in equation (3.17) can be factorized as
3.19
where
3.20
and
3.21
By the quadratic formula, {*λ*^{2}+(*ϕ*+1−*p*)*λ*+*ϕ*−(*ϕ*+1−*p*)*p*} has roots with negative real part if and only if
3.22
3.23
If *h*=(1+*ϕ*)*ϕ* then the roots of equation (3.15) are −*p*=−(1+*ϕ*) and . As discussed above using figure 3, as *h* decreases, *p* decreases, so if *h*<(1+*ϕ*)*ϕ*, this implies that *p*<1+*ϕ*. Together with equation (3.23) this proves (3.18).

#### (iv) Condition for no oscillation: no complex roots

This subsection proves (3.6). Small displacements from the equilibrium **s**_{e} tend smoothly to 0, with no oscillations, so long as *h* is small enough for all the roots of (3.17) to be real and negative. Thus *h* must be smaller than it is in the borderline case, where (3.15) has a negative root −*p* and two repeated negative roots −*q*. By considering how the sketches in figure 2 are shifted upwards by *h*>0, the repeated negative roots have the value of *λ* at the turning point between 0 and point *B*, so that
3.24
In the borderline case, equation (3.15) is of the form (*λ*+*p*)(*λ*+*q*)^{2}. So, *h*<*pq*^{2} and it remains to show that the equation for *pq*^{2} in equation (3.7) is correct. This is done by factorizing (3.15) into a linear and quadratic part, as before, and then comparing coefficients of powers of *λ*. In the borderline case,
3.25
Expanding,
3.26
Equating coefficients of powers of *λ*,
3.27
3.28and
3.29One can obtain *pq*^{2} from equations (3.27) and (3.28). Substituting *p* from equation (3.27) into equation (3.28) gives
3.30
3.31
3.32
The negative root of equation (3.32) must be taken. For, in any case,
3.33
3.34
3.35
3.36
So,
3.37
3.38
by equation (3.27)
3.39

#### (v) Approximate values for the stable equilibrium to have no oscillations

This subsection justifies equations (3.8) and (3.9). Via binomial expansion of (1−*ϕ*+*ϕ*^{2})^{3/2} in equation (3.39), for small *ϕ*
3.40
For large *ϕ*, via binomial expansion of *ϕ*^{3}(1/*ϕ*^{2}−1/*ϕ*+1)^{3/2},
3.41
Furthermore, it appears empirically, as shown in figure 4, for all *ϕ* in the probably relevant range for the model, that
3.42

### (d) Period of oscillations

The period of the oscillations towards the stable equilibrium is 2*π*/*ω* where *ω* is the imaginary part of the complex eigenvalues [23].

By applying the quadratic formula to the quadratic part of equation (3.19), 3.43 where 3.44 Hence 3.45 3.46 3.47 So the period of oscillations towards the stable equilibrium is 3.48

### (e) Critical values in dimensional variables

The above conditions for stability and the period of oscillations may be expressed in the original dimensions by applying equations (2.10)–(2.16) to the non-dimensional results.

The equilibrium point for and is, from equation (3.1) 3.49 From equation (3.5), the equilibrium is stable if and only if 3.50 From equation (3.8), the system does not oscillate near the equilibrium if 3.51 and, from equation (3.9), the system does oscillate near the equilibrium if 3.52 From equation (3.48), the period of oscillations towards the stable equilibrium is 3.53

Let equal the left-hand side of equations (3.50)–(3.52). If the additional simplification is made (as in equation (4.1)) that the damage function *χ* is linear with respect to global warming, with economic growth no greater than a background rate of *ξ* per year, then
3.54
3.55
and
3.56
therefore
3.57
If *δ* is small, so that growth is possible even with a high level of global warming, then
3.58
If *δ* is large, so that growth is choked off at a relatively low level of global warming, then
3.59
So, for example, if the damage function is linear, and the impact of global warming is severe, then
3.60
3.61
3.62

## 4. Model parameters

Clearly, from equation (3.50), the parameters chosen for the model determine whether its equilibrium is stable or not. On the left-hand side of equation (3.50), a higher level of tolerable global warming or a decrease in the decarbonization rate are destabilizing, as they allow longer lags before the system has to adjust. A higher level of climate sensitivity and a steeper damage function are also destabilizing, as they imply a faster pace of change to which the system must adjust. On the right-hand side of equation (3.50), greater thermal or carbon cycle inertia is destabilizing, as it means the system can only adjust slowly. The initial conditions, including the initial carbon intensity, affect the system’s trajectory, but do not affect the stability of the equilibrium.

The numerical simulations in this section use parameters based on the following:

— an initial carbon intensity

*η*of 0.025 ppmv/$ trillion, consistent with current CO_{2}rises of approximately 2 ppmv per year;— initial levels of CO

_{2}concentration, global warming and global wealth of 380 ppmv, 0.7 K, $160 trillion [25];— a central estimate for the equilibrium climate sensitivity of

**Δ***T*_{2*CO2}=3 K, approached on a characteristic climate time scale of*τ*_{T}=50 years;— a pre-industrial level of CO

_{2}of 280 ppmv;— a characteristic carbon time scale of

*τ*_{C}=50 years, consistent with a fixed airborne fraction;— a decarbonization rate

*μ*=1% yr^{−1}, based on records of economic and CO_{2}emissions growth for the late twentieth century [18–20]; and— a linear expansion (or damage) function for wealth of 4.1

Equation (4.1) assumes the rate of economic growth is reduced by a fraction *δ* for each kelvin of global warming. The orthodox climate prediction chain essentially assumes that *δ*≈0, such that there is no feedback from climate change to economic growth. The actual fraction *δ* is unknown, though it is constrained by twentieth century data. Rearranging the linear damage function in equation (4.1),
4.2
4.3
Using the observed values of *μ*, and for the late twentieth century, which are 1 per cent, 0.7 K and 3 per cent, and allowing for a 10 per cent error in each measurement, then
4.4
which constrains (*ξ*−*μ*) and *δ* in figure 7 to the brown region marked as observed.

Low-level climate change impacts imply an exponential growth of at a constant background rate of *ξ* (equation (4.1)). In the late twentieth century, the actual global economic growth rate averaged about 3 per cent per year [26]. However, the *Special Report on Emissions Scenarios* [27] translates into a wide range of growth rates for the twenty-first century.

## 5. Model results

Figure 5 compares the model projections of the twenty-first and twenty-second centuries for the standard no-feedback case (dashed lines) with projections when *δ*=0.5, a value that would produce an equilibrium global warming of . A low background economic growth rate of *ξ*=1% per year is considered (in green) as is a high background economic growth rate of *ξ*=4% per year (in black). In both cases, the closure of the climate–economy feedback loop significantly affects the projections, especially in the twenty-second century. However, the emergent dynamics are very different in the low and high growth cases.

In the low growth rate case, the impact of climate change on economic growth leads to a *soft landing* at the equilibrium in which the negative climate–economy feedback loop counteracts the background economic growth rate, the CO_{2} emission rate stabilizes, and the economy grows at the decarbonization rate of *μ* per year. By contrast, in the high growth case, the negative feedback loop is too slow to balance the background growth rate. This leads to an overshoot of the climate equilibrium that precedes an economic crash (figure 5*c*, black solid line). The high growth rate case projects an economic depression for the whole of the twenty-second century, although rather ironically the CO_{2} concentration and climate recover as a result.

These are very striking emergent behaviours of the climate–economy system, so it would be natural to ask whether they are strongly dependent on the simplifications made here. In order to assess this, the widely used DICE integrated assessment model [3] was applied to similar scenarios of economic growth and decarbonization. Under the simplifying assumptions listed in appendix A, the DICE model produces qualitatively similar emergent behaviour (figure 9).

Figure 6 shows the long-term consequences of these different economic growth rates. At background growth rates of emissions above about 5% per year, the fixed-point equilibrium becomes unstable, undergoing a Hopf bifurcation to a stable periodic orbit of permanent booms and busts. In the low growth rate case, the economy overshoots the fixed-point equilibrium but approaches it with a smooth landing (green lines in figure 6). For intermediate growth rates, the economy undergoes damped oscillations about its equilibrium state.

Figure 7 shows the location of these three regimes in the parameter space defined by the background growth rate of CO_{2} emissions (*ξ*−*μ*), and the fractional suppression of economic growth per unit of global warming, *δ*, for the case *τ*_{C}=*τ*_{T}=50 years. Conditions (3.50)–(3.52) define how these regimes depend upon these time scales of the carbon cycle and climate response. For higher *τ*_{C} and *τ*_{T} the soft landing region of the parameter space shrinks and the unstable region expands.

The possible parameters are constrained by the historical level of global warming and economic growth. In the absence of any intervention, a soft landing requires the economy to be almost insensitive to global warming, such that *δ* is less than about 0.05 K^{−1}. This is equivalent to requiring that the economy can withstand global warming of more than 20 K without contracting. It therefore seems likely that the climate–economy system is currently in an oscillatory regime, with the possibility of an economic crash if growth is faster in the future or if the damage function for wealth is more steep or nonlinear than we have supposed.

Figure 8 shows the stability regimes in the parameter space defined by the background economic growth rate and the decarbonization rate, for two values of *δ*. It is clear that decarbonization raises the threshold under which a soft landing is possible.

Even damped oscillations are likely to be damaging to the long-term well-being and security of humanity [28], so how can they be avoided? Figure 7 suggests two main ways to ensure a soft landing for the climate–economy system. The first is to reduce the sensitivity of the economy to climate damages through adaptation (such that *δ*<0.05 K^{−1}). The second is to reduce the background growth rate of CO_{2} emissions to rates that can be gradually counteracted by the climate–economy feedback loop. This requires that the rate of decarbonization of the economy approaches the background rate of economic growth (such that *ξ*−*μ*<0.5% per year, for the parameters used in this paper). This in itself requires either large increases in the rate of decarbonization (through conventional mitigation or carbon capture and storage) or reductions in the background rate of global economic growth.

## 6. Conclusion

The inclusion of even a relatively weak feedback loop between climate change and economic growth leads to projections for the twenty-first and twenty-second centuries that differ fundamentally from the standard no-feedback case. The climate–economy feedback permits a climate equilibrium state in which the background economic growth rate is counteracted by climate change impacts on the economy. Economic growth in this climate state is equal to the rate of decarbonization, so mitigation efforts are essential to ensure long-term sustainable growth. However, figure 8 suggests that decarbonization will not be enough to ensure a soft landing on this sustainable trajectory. Instead overshoot oscillations or even instabilities are possible under historical rates of economic growth, and feasible levels of economic damage owing to climate change. We conclude that navigating the climate–economy system to a soft landing will require massive efforts in both mitigation and adaptation, but may also require lower but more sustainable rates of global economic growth.

## Acknowledgements

This work was funded by the Natural Environment Research Council, the Met Office and the University of Exeter. We would like to thank Prof. John Thuburn and the Applied Maths seminar group at the University of Exeter for constructive comments.

## Footnotes

One contribution of 13 to a Theme Issue ‘The Anthropocene: a new epoch of geological time?’.

- This journal is © 2011 The Royal Society

## Appendix A. Relation to a more sophisticated integrated assessment model: DICE

The well-known DICE model [3], if suitably simplified and with a scaled-up damage function, exhibits similar behaviour to the three-variable model presented in this paper. Figure 9 compares a simplified version of the global DICE model (dashed lines) with the our model (continuous lines). The simplifications made to the DICE model are the following.

— The DICE capital share is set to one, removing the sensitivity to the DICE exogeneous population growth. This is defensible if population is treated as a component of global wealth, with the bulk of productivity differences attributed to the accumulation of social infrastructure [21].

— The DICE exogeneous productivity growth rate is set to zero. This is no more arbitrary than setting the exogeneous decarbonization rate to a constant.

— The DICE carbon intensity is set to reduce by 10 per cent per decade, consistent with the historical record.

— The DICE savings rate is fixed at 23 per cent, approximately the level set by the optimized DICE model.

— The DICE damage function is

*multiplied by*10. This is the most striking change, but part of the increase in the damage-to-production function is due to the fact that the DICE-99 depreciation function is fixed, whereas it is reasonable to suppose that the replacement cost of assets will increase along with the production cost.

The parameters for our model are the same as those in §5 except that

— the damage coefficient is set to 0.15 (implying a climate-induced recession at 7 K of global warming); and

— the background economic growth rate is set to 5 per cent.