Coupled Climate-Economy-Ecology-Biosphere Modeling: A Dynamic and Stochastic Approach

Abstract

Much of the work on climate change and its economic impacts so far has been done on the basis of equilibrium theories, in the climate as well as the economic realm. Increasingly, though, the climate sciences community has come to realize that natural climate variability is an important issue in assessing climate evolution on time scales of years-to-decades. Somewhat less broad and consensual is the incipient use of nonequilibrium, stochastic-dynamic models in the macroeconomic literature. The purpose of this chapter is to cover some of the advances in the recent climate and economic literature on the use of such models to address climate change mitigation and adaptation. The chapter stresses the importance of taking into account the nonlinearities in both the climate and economic system, as well as in the coupling. Some of these issues are illustrated by reviewing work on a coupled climate–economy–biosphere (CoCEB) model with random shocks, designated as CoCEB-S. This review emphasizes the latter model’s results on the comparative efficacy of approaches to abatement – such as low-carbon technology, deforestation reduction, or carbon capture and storage – and it uses the evolution of the inclusive wealth index as its key policy evaluation tool.

Random Differential Equations

The most common way of introducing randomness into the governing equations of physical, as well as biological and socioeconomic processes, is that of stochastic differential equations (SDEs). An SDE takes the form

$$ \mathrm{d}x=f\left(x;\mu \right)\mathrm{d}t+\sigma (x)\mathrm{d}W, $$

where μ is a parameter and dW is Brownian motion, with W(t) a Wiener process. If σ is constant, the noise is additive (e.g., Hasselmann 1976), if it depends on x it is multiplicative (Ghil et al. 2008; Chekroun et al. 2011).

Instead, what we used herein is a particular form of a random differential equation (RDE),

$$ \dot{x}\left(t,\omega \right)=f\left(t,x\left(t,\omega \right);\mu \left(\omega \right)\right),\quad x\left(0,\omega \right)={x}_0\left(\omega \right), $$

(16)

where ω denotes a fixed realization of a stochastic process (e.g., Strand 1970). The path wise solutions of the deterministic ODE (16) are then likewise realizations of a stochastic process. Unlike SDEs, which require using the stochastic calculus for their solution (e.g., Särkkä and Solin 2019), RDEs can be formulated and analyzed path wise in terms of the standard deterministic calculus. Still, a strong connection between RDEs and SDEs does exist; see Imkeller and Schmalfuss (2001) for the conjugacy between the two and the existence of global attractors.

It follows from the considerations above that, in principle, an RDE could be solved numerically with well-known deterministic numerical schemes, such as Euler or Runge-Kutta methods. To achieve a low discretization error, such schemes, however, require sufficient smoothness of f, which is normally not the case in the presence of the jumps induced by the stochastic process whose realization is identified by ω. Hence, more sophisticated methods are usually required for achieving good convergence of the numerical solution in discrete time nΔt to the correct one of (16) in continuous time t (Grüne and Kloeden 2001; Han and Kloeden 2017). For simplicity – and given all the approximations involved in our CoCEB-S model Eqs. (14a)–(14h) – we consider here only a one-step numerical Euler–Maruyama (EM) scheme (Bayram et al. 2018), and we use a time step Δt = 1 y.