Financing Climate Change Policies: A Multi-phase Integrated Assessment Model for Mitigation and Adaptation

Abstract

We present an extended integrated assessment model (IAM) that optimizes climate financing policies over multiple phases of discrete policy action. We build on Semmler et al. (Control systems and mathematical methods in economics. Springer, 2018) which develops a single-phase model of the optimal allocation of infrastructure expenditure to carbon-neutral physical capital, climate change adaptation, and emissions mitigation. That model is solved by discretizing the optimal control problem and applying large-scale optimization techniques. A new algorithm, the arc parameterization method (APM), allows us to extend the model to multiple regimes, operationalized through a 3-phase policy environment. The first policy regime is defined by a limited set of policy tools. In the second regime, green bonds (i.e. climate-focused financing) are introduced, and in the final regime, green bonds are repaid with a new tax. We demonstrate that this multi-phase model incorporating climate, fiscal and financial policies is superior to single-phase models.

This paper was developed while Willi Semmler was a visiting scholar at the IIASA in July and August 2016. He would like to thank the IIASA for its hospitality. A related version of this paper was worked out while Willi Semmler was a visiting scholar at the IMF Research Department in April 2016. The authors would like to thank Prakash Loungani and his colleagues for many helpful insights.

Appendix: Multi-phase Optimal Control Problems and Their Numerical Solution

Multi-process (multi-phase) optimal control problems have been studied by Clarke and Vinter [4, 5] and later by Augustin and Maurer [1]. Suppose that a dynamic economic process on a given time interval [0, T] consists of $(s+1)$ phases (regimes) that switch at the transition times $t_{k}\in (0,T),\,k=1,...,s$. The switching times are ordered according to

$$\begin{aligned} 0=t_{0}<t_{1}<t_{2}<\cdots<t_{s}<t_{s+1}=T. \end{aligned}$$

(6.11)

In each interval $[t_{k},t_{k+1}],\,k=0,1,\ldots s,$ the dynamics and objectives may be different.

Let $x\in \mathbb {R}^{n}$ be the state variable and $u\in \mathbb {R}^{m}$ the control variable. The dimensions of the state vector and control vector may be different in different phases. For simplicity, we refrain here from discussing this general case and assume the same dimensions in each subinterval. Hence, the dynamics of the economic process in the interval $[t_k,t_{t+1}]$ is given by the ordinary differential equation,

$$\begin{aligned} \dot{x}(t)=f_{k} \big ( x(t),u(t) \big ) ,\quad t_{k}\le t\le t_{k+1}\quad (k=0,1,\ldots ,s), \end{aligned}$$

(6.12)

where the right-hand side of the ODE is a $C^{1}$ function $f_{k}:\mathbb {R}^{n}\times \mathbb {R}^{m}\rightarrow \mathbb {R}^{n}.$ The time $t_{k}$ in (6.12) is understood from the right, while the time $t_{k+1}$ is taken from the left. The initial condition and terminal constraints are given as

$$\begin{aligned} x(0)=x_{0},\quad \psi (x(T))=0. \end{aligned}$$

(6.13)

We further impose control constraints in each interval,

$$\begin{aligned} u_{k,\min }\le u(t)\le u_{k,\max }\,,\quad t_{k}\le t\le t_{k+1}\quad (k=0,1,\ldots ,s), \end{aligned}$$

(6.14)

with $-\infty \le u_{k,\min }<u_{k,\max }\le +\infty$.

A continuous state trajectory x(t) on the whole interval [0, T] is obtained by imposing the continuity condition

$$\begin{aligned} x(t_{k})=x(t_{k}-),\qquad k=1,\ldots ,s. \end{aligned}$$

(6.15)

Note that the continuity of the state variables in (6.15) does not automatically ensure the continuity of the control variables; in fact, these can jump, as we demonstrate below, when the system transitions between policy phases are studied. Moreover, we can prescribe interior (transition) conditions for the state variables by

$$\begin{aligned} \varphi _k(x(t_k-) = 0 ,\quad k= 1,\ldots ,s , \end{aligned}$$

(6.16)

with $C^{1}$ functions $\varphi _k : \mathbb {R}^n \rightarrow \mathbb {R}.$

In each interval one may also have different objectives which are defined by functions $L_{k}:\mathbb {R}^{n}\times \mathbb {R}^{m}\rightarrow \mathbb {R}^{n},\;k=0,1,\ldots ,s$. Then the optimal multi-phase control problem is defined by the following objective:

$$\begin{aligned} J(x,u) = \max _{u} \left\{ \sum _{k=0}^{k=s}\;\int \limits _{t_{k}}^{t_{k+1}}\,e^{-r_k\cdot t}L_{k}(x(t),u(t))\,dt \right\} , \end{aligned}$$

(6.17)

subject to the constraints (6.12)–(6.15), and $r_k > 0$ for $k = 0, 1, \ldots , s$.

To solve the optimal multi-phase control problem, we implement the arc-parametrization in Maurer et al. [12] in conjunction with discretization and nonlinear programming methods. Although they apply the arc-parametrization method (APM) only to bang-bang control problems, the APM can easily be extended to continuous, multi-phase control problems as follows. Let

$$\begin{aligned} \xi _{k}=t_{k+1}-t_{k},\quad k=0,1,\ldots ,s, \end{aligned}$$

(6.18)

denote the arc lengths (or, arc durations) of the multi-process. The time interval $[t_{k},t_{k+1}]$ is mapped onto the fixed interval $[k/(s+1),(k+1)/(s+1)]$ by the linear transformation

$$\begin{aligned} t&= a_{k}+b_{k}\tau , \quad \tau \in \left[ \frac{k}{s+1},\frac{k+1}{s+1}\right] , \end{aligned}$$

(6.19)

where $a_k = t_k - k \xi _k$ and $b_{k}=(s+1)\xi _{k}$. Taken together, the complete time interval [0, T] is thereby mapped onto the unit interval [0, 1]. Identifying $x(\tau )=x(a_{k}+b_{k}\tau )=x(t)$ in the relevant intervals, we obtain the scaled ODE system

$$\begin{aligned} \frac{dx}{d\tau } = \xi _{k}(s+1) \cdot f_{k} \big (t_{k}+\tau \cdot \xi _{k}, x(\tau ), u(\tau ) \big ) \end{aligned}$$

(6.20)

for $\tau \in \left[ \frac{k}{s+1},\frac{k+1}{s+1}\right]$. Note that $\xi _{k}$ are treated as optimization variables if the transition times $t_{k}$ are free.

The time transformation leads us to rescale the objective function (6.17) as follows:

$$\begin{aligned} J(x,u)= \max _{u} \left\{ \sum _{k=0}^{k=s} \int \limits _{\frac{k}{s+1}}^{\frac{k+1}{s+1}} \xi _{k}(s+1)e^{-r_k \cdot (a_{k}+b_{k}\tau )}L_{k}(x(\tau ),u(\tau ))\,d\tau \right\} , \end{aligned}$$

(6.21)

and subject to (6.12)–(6.15) and rescaled according to (6.19). For the purposes of exposition, we fix the transition points $t_k$ to reflect the exogenous (and often sub-optimally protracted) nature of introducing and implementing new policies.