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2 Uncertain climate dynamics

2.1 approximation to climate dynamics

We use exponentially weighted average of each of response functions as coefficients \(\{\theta_\ell\}_{\ell=1}^L\) in our computations. The discount rate \(\delta=0.01\) and the number of climate models \(L = 144\).

The histogram of those coefficients are represented below:

from src.plots import plot2
plot2()

2.2 Stochastic climate pulses

To explore uncertainty, we introduce explicit stochasticity as a precursor to the study of uncertainty. We capture this randomness in part by an exogenous forcing processes that evolves as:

\[dZ_t = \mu_z(Z_t) dt + \sigma_z(Z_t) dW_t\]

where \(\{ W_t : t \ge 0\}\) a multivariate standard Brownian motion. We partition the vector Brownian motion into two subvectors as follows:

\[dW_t = \begin{bmatrix} dW_t^y \cr dW_t^n \cr dW_t^k \end{bmatrix}\]

where the first component consists of the climate change shocks and the second component contains the technology shocks. Consider an emissions “pulse” of the form

\[\left(\iota_y \cdot Z_t \right) {\mathcal E}_t \left( \theta dt + \varsigma \cdot dW_t^y\right)\]

where \({\mathcal E}_t\) is fossil fuel emissions and \(\iota_y \cdot Z = \{ \iota_y \cdot Z_t : t\ge 0\}\) is a positive process which we normalize to have mean one. The \(\iota_y\cdot Z\)-process captures “left out” components of the climate system’s reaction to an emission of \({\mathcal E}_t\) gigatons into the atmosphere while the \(\varsigma \cdot dW\) process captures short time scale fluctuations.

We will use a positive Feller square root process for the \(\iota_y\cdot Z\) process in our analysis.

Within this framework, we impose the “Matthews’ approximation” by making the consequence of the pulse permanent:

\[dY_t = \mu_y(Z_t, {\mathcal E}_t) dt + \sigma_y(Z_t, {\mathcal E}_t) dW_t^y\]

where

\[\begin{align*} \mu_y(z, e) & = e \left(\iota_y \cdot z \right) \theta \cr \sigma_y(z, e) & = e \left(\iota_y \cdot z \right) \varsigma' \end{align*}\]

Throughout, we will use uppercase letters to denote random vector or stochastic processes and lower case letters to denote possible realizations. Armed with this “Matthews’ approximation”, we collapse the climate change uncertainty into the cross-model empirical distribution reported in the figure above. We will eventually introduce uncertainty about \(\theta\).

Remark 1

For a more general starting point, let \(Y_t\) be a vector used to represent temperature dynamics where the temperature impact on damages is the first component of \(Y_t\). This state vector evolves according to:

\[dY_t = \Lambda Y_t dt + {\mathcal E}_t \left(\iota_y \cdot Z_t \right) \left(\Theta dt + \Phi dW_t^y \right)\]

where \(\Lambda\) is a square matrix and \(\Theta\) is a column vector. Given an initial condition \(Y_0\), the solution for \(Y_t\) satisfies:

\[Y_t = \exp \left( t \Lambda \right) Y_0 + \int_0^t \exp\left[ (t-u) \Lambda \right] \left(\iota_y \cdot Z_u \right) {\mathcal E}_u \left(\Theta du + \Phi dW_u^y \right)\]

Thus under this specification, the expected future response of \(Y\) to a pulse at date zero is:

\[\exp \left( u \Lambda \right) \Theta\]

It is the first component of this function that determines the response dynamics. This generalization allows for multiple exponentials to approximate the pulse responses. Our introduction of a multiple exponential approximation adapts for example, Joos et al. [6] and Pierrehumbert [7]. 1

As an example, we capture the initial rise in the emission responses by the following two-dimensional specification

\[\begin{align*} dY_t^1& = Y_t^2 dt \cr dY_t^2 & = - \lambda Y_t^2 dt + \lambda \theta {\mathcal E}_t dt \end{align*}\]

which implies the response to a pulse is:

\[\theta \left[ 1 - \exp( - \lambda t) \right] {\mathcal E}_0\]

A high value of \(\lambda\) implies more rapid convergence to the limiting response \(\theta {\mathcal E}_0\). This approximation is intended as a simple representation of the dynamics where the second state variable can be thought of as an exponentially weighted average of current and past emissions. 2 (For more detailed computations, see Appendix C)

Remark 2

The approximation in Geoffroy et al. [8] includes the logarithm of carbon in the atmosphere as argued for by Arrhenius [9] which is not directly reflected in the linear approximation to the temperature dynamics that we use. The pulse experiments from Joos et al. [6] show a more than proportional change in atmospheric carbon when the pulse size is changed. It turns out that this is enough to approximately offset the logarithmic Arrhenius adjustment so that the long-term temperature response remains approximately proportional for small pulse sizes. See also Pierrehumbert [7] who discusses the approximate offsetting impacts of nonlinearity in temperature and climate dynamics.

1

See equation (5) of Joos et al. [6] and equations (1)-(3) of Pierrehumbert [7]. Pierrehumbert puts the change in radiative forcing equal to a constant times the logarithm of the ratio of atmospheric \(CO_2\) at date \(t\) to atmospheric \(CO_2\) at baseline date zero. His Figures 1 and 2 illustrate how an approximation of the Earth System dynamics by three exponentials plus a constant tracks a radiative forcing induced by a pulse into the atmosphere at a baseline date from the atmosphere works quite well with half lives of approximately six, sixty five, and four hundred and fifty years.

2

In independent work, Dietz and Venmans [10] and Barnett et al. [11] have used such simplified approximations within an explicit economic optimization framework. The former contribution includes the initial rapid upswing in the impulse response functions. The latter contribution abstracts from this. Barnett et al. instead explore ways to confront uncertainty, broadly-conceived, while using the Matthews approximation.