Heterogeneous Beliefs and Climate Catastrophes

Abstract

We study how heterogeneous beliefs about the causes and extent of global warming affect local mitigation and adaptation strategies and therefore global climate dynamics. Local policies are determined by expectations of policy makers about future climate. There are three types of expectations: strong skeptic, weak skeptic and ‘science-based’. Strong skeptics deny human-induced climate change and a possibility of a climate catastrophe. Weak skeptics believe that industrial emissions cause global warming, but deny catastrophic climate change. Science-based policy makers, considering the warning of the scientific community, account for both: human influence on climate and possible catastrophic shifts. Aggregate behavior of policy makers determines the total emission level which influences global climate dynamics. The paper argues that even if there are only skeptical policy makers the climate catastrophe can still be avoided.

Appendices

Appendix 1: Border Collision Bifurcations

The map (18) is non-smooth as its right hand side is non-differentiable due to restrictions on the variables $a^{[i]}$ and $e^{[i]}$. Such a map may undergo bifurcations as the system parameters vary and a variable ($a^{[i]}$ or $e^{[i]}$) hits the boundary of its domain. Such bifurcations are called border collision bifurcations, i.e. bifurcations of periodic orbits colliding with boundaries of the system domain or with boundaries of basins of attraction of other periodic orbits. Figure 19 shows bifurcation diagrams with two border collision bifurcations marked as ‘BCB1’ and ‘BCB2’. The first border collision bifurcation corresponds to the emission level $e^{[2]}$ hitting its upper boundary ${\bar{e}}$. This leads to a discontinuous jump in the steady state levels of the variables. The second border collision bifurcation ‘BCB2’ corresponds to the adaptation level $a^{[1]}$ hitting its lower boundary 0. This leads to occurrence of a stable limit cycle.

Fig. 19

Border collision bifurcations with respect to $c_a$

The bifurcation curves represent the long run behavior of the system (18) for a range of $c_a$ other parameters being fixed. These curves has been computed by running (18) using various initial conditions. The terminal time has been set large enough to let the system settle down in a stable attractor.

This article does not aim at providing the full bifurcation analysis of the system (18), but rather an overview of the ‘most interesting’ types of system behavior. A reader interested in bifurcations of non-smooth systems is referred to Di Bernardo et al. (2008) and Leine and Nijmeijer (2004).

Appendix 2: Neimark-Sacker Bifurcations

It can be shown that the bifurcation corresponding to a disappearance of a stable limit cycle for $c_a\approx 30$ in Fig. 13 is a Neimark-Sacker bifurcation. Note that it occurs in the parameter region where $a^{[1]}=0$, $a^{[2]}=0$ and $a^{[3]}>0$ (see Fig. 20). By taking these conditions into account the system (18) can be simplified to

$$\begin{aligned} \left( \begin{array}{c} p_t\\ e^{[3]}_t\\ W^{[1]}_t\\ W^{[2]}_t\\ W^{[3]}_t\\ n^{[1]}_t\\ n^{[2]}_t \end{array} \right) \mapsto \left( \begin{array}{c} (1-b)p_t+\frac{p_t^2}{1+p_t^2}+n^{[1]}_t\bar{e}+n^{[2]}_t\frac{1}{c_p}+n^{[3]}_te^{[3]}_t\\ \frac{1+a_t^{[3]}}{c_p}\\ \log \left( \bar{e}\right) -c_pp_t+\delta W^{[1]}_t\\ \log \left( \frac{1}{c_p}\right) -c_pp_t+\delta W^{[2]}_t\\ \log \left( e^{[3]}_t\right) -\frac{c_pp_t}{1+a^{[3]}_t}-c_aa^{[3]}_t+\delta W^{[1]}_t\\ \gamma \frac{e^{\beta W^{[1]}_{t}}}{\sum _{i=1}^3 e^{\beta W^{[i]}_{t}}}+(1-\gamma )n^{[1]}_{t}\\ \gamma \frac{e^{\beta W^{[2]}_{t}}}{\sum _{i=1}^3 e^{\beta W^{[i]}_{t}}}+(1-\gamma )n^{[2]}_{t} \end{array} \right) \end{aligned}$$

(23)

with

$$\begin{aligned} a^{[3]}_t=\frac{1+\sqrt{1+4c_ac_p\left[ (1-b)p_t+ \frac{p^2_t}{1+p_t^2}+(n^{[1]}_t\bar{e}+n^{[2]}_t \frac{1}{c_p}+n^{[3]}_te^{[3]}_t)\right] )}}{2c_a}-1. \end{aligned}$$

(24)

The system (23) is a smooth system and can be studied using standard methods of the bifurcation theory (see Kuznetsov 1995). Namely, a Neimark-Sacker bifurcation curve can be computed. Figure 20 shows curves marked $a^{[i]}=0$. Each curve separates two regions: for any pair $(c_a,c_p)$ that lies above the curve the following holds $a^{[i]}=0$, otherwise $a^{[i]}>0$ is true. The curve ‘NS’ contains pairs of critical parameter values $(c^{NS}_a,c^{NS}_p)$ such that the system (23) exhibit a Neimark-Sacker bifurcation for $c_a=c^{NS}_a$ and $c_p=c^{NS}_p$ which results in (dis)appearance of a stable limit cycle (see Fig. 13 for $c_a\approx 30$). The ‘NS’-curve has been computed using MATCONT continuation software in MATLAB (see Dhooge et al. 2003).

Fig. 20

Bifurcations diagram in the $(c_a,c_p)$-plane, same as in Fig. 14 but with the Neimark-Sacker bifurcation curve added