The Role of Industrial and Market Symbiosis in Stimulating CO₂ Emission Reductions

Abstract

An increasing concern for climate change puts pressure on industrial firms to achieve carbon emission reductions. These could be realized through cooperation among firms in industrial chains, which leads to industrial symbiosis. By taking a real options approach, we make the timing component of the investment decisions explicit. This is important in assessing the impact of carbon-reducing investment over a specific time-span. We show that a joint venture between a CO₂ emitting firm and a firm that can use the CO₂ will result in a higher probability that an investment in CO₂ capture will take place within a specific time period, which reduces the amount of CO₂ emitted substantially. We also show that, in addition to industrial symbiosis, cooperation between firms can benefit from “market symbiosis” as well, in the sense that investments are more likely to take place in markets that are positively correlated. This is an important result, given that the EU has set binding targets to its Member States for reducing their emissions.

Appendices

Appendix 1: Proof of Proposition 2

To keep notation simple we identify $X=P_U$ and $Y=P_D$. We restrict attention to the set E of points in ${\mathbb{R}}^2_+$ where all the first and second-order partial derivatives of $\varphi$ are well-defined. Note that for every $t\ge 0$ it holds that $(X_t,Y_t)\in E$, ${\mathbb{P}}$-a.s.

For any $\varphi \ge F_J$ that is $C^2$-a.e., the HJB equation (13) can be re-written as a pair of variational inequalities on ${\mathbb{R}}^2_+$:

$$ {\left\{ \begin{array}{ll} {{\mathscr {L}}}\varphi (x,y)-r\varphi (x,y)\le 0 &{\quad}\text {when }\varphi (x,y)=F_J(x,y),\text { and}\\ {{\mathscr {L}}}\varphi (x,y)-r\varphi (x,y)=0 &{\quad}\text {when } \varphi (x,y)>F_J(x,y). \end{array}\right. } $$

Let

$$ {{\mathcal {C}}}= \{(x,y)\in {\mathbb{R}}^2_+|\varphi (x,y)>F_J(x,y)\}, $$

and define the stopping time

$$ \tau _{{\mathcal {C}}}= \inf \{t\ge 0|(X_t,Y_t)\not \in {{\mathcal {C}}}\}. $$

Noting that

$$ {\mathbb{E}}\left[ e^{-r\tau _{{\mathcal {C}}}}\varphi (X_{\tau _{{\mathcal {C}}}},Y_{\tau _{{\mathcal {C}}}})\right] =0, $$

on $\{\tau _{{\mathcal {C}}}=\infty \}$, it follows from Dynkin’s formula (see, e.g., Øksendal 2000) and the fact that $\varphi =F_J$ on $\partial {{\mathcal {C}}}$ (by a.s.-continuity of sample paths) that

$$\begin{aligned} {\mathbb{E}}\left[ e^{-r\tau _{{\mathcal {C}}}}J_V(X_{\tau _{{\mathcal {C}}}},Y_{\tau _{{\mathcal {C}}}})\right]&= {\mathbb{E}}\left[ e^{-r\tau _{{\mathcal {C}}}}\varphi (X_{\tau _{{\mathcal {C}}}},Y_{\tau _{{\mathcal {C}}}})\right] \\ &= \varphi (x,y)+{\mathbb{E}}\left[ \int _0^{\tau _{{\mathcal {C}}}}e^{-rt}\left( {{\mathscr {L}}}\varphi (X_t,Y_t)-r\varphi (X_t,Y_t)\right) {\rm d}t\right] \\ &= \varphi (x,y). \end{aligned}$$

Now consider any other stopping time $\tau \in {{\mathcal {M}}}$. Then another application of Dynkin’s formula shows that

$$\begin{aligned} {\mathbb{E}}\left[ e^{-r\tau }J_V(X_{\tau },Y_{\tau })\right]& \le {\mathbb{E}}\left[ e^{-r\tau }\varphi (X_{\tau },Y_{\tau })\right] \\ &= \varphi (x,y)+{\mathbb{E}}\left[ \int _0^{\tau }e^{-rt}\left( {{\mathscr {L}}}\varphi (X_t,Y_t)-r\varphi (X_t,Y_t)\right) {\rm d}t\right] \\ &\le \varphi (x,y). \end{aligned}$$

Therefore, it follows that $V_J=\varphi$ and that the optimal stopping time in (14) is $\tau _{{\mathcal {C}}}$, as claimed. $\square$

Appendix 2: Numerical Implementation

Our numerical implementation of the optimal stopping problem (12) uses the Markov chain approximation method as described in, e.g., Kushner (1997) and Kushner and Dupuis (2001). Our optimal stopping problem is of the form

$$ V(x,y)= \sup _{\tau \in {{\mathcal {M}}}}{\mathbb{E}}\left[ -\int _0^{\tau }e^{-rt}c(X_t){\rm d}t+e^{-r\tau }F(X_\tau ,Y_\tau )\right] , $$

(14)

where

$$ \left[\begin{array}{l}{\rm d}X/X\\ {\rm d}Y/Y\end{array}\right]= \left[\begin{array}{l}\alpha _1\\ \alpha _2\end{array}\right]{\rm d}t+ \left[\begin{array}{ll}\sigma _1 &{\quad} \rho \sigma _1\sigma _2\\ \rho \sigma _1\sigma _2 &{\quad} \sigma _y\end{array}\right] \left[\begin{array}{l}{\rm d}W_1\\ {\rm d}W_2\end{array}\right], $$

and $x\mapsto c(x)$ represents the running costs, which are here given by the costs of CO₂ emissions.

For numerical stability is desirable to use the following transformation of variables:

$$ U:=\log (X),\quad \text {and}\quad V:=\log (Y). $$

Letting the functions ${\hat{F}}:{\mathbb{R}}_2\rightarrow {\mathbb{R}}$ and ${\hat{c}}:{\mathbb{R}}\rightarrow {\mathbb{R}}$ be such that

$$ {\hat{F}}(u,v) := F\left( e^u,e^v\right) ,\quad \text {and}\quad {\hat{c}}(u):=c\left( e^u\right) ,\quad \text {all } u,v\in {\mathbb{R}}, $$

we can then rewrite (14) as

$$ {\hat{V}}(u,v)= \sup _{\tau \in {{\mathcal {M}}}}{\mathbb{E}}\left[ -\int _0^{\tau }e^{-rt}{\hat{c}}(U_t){\rm d}t+e^{-r\tau }{\hat{F}}(U_\tau ,U_\tau )\right] , $$

where

$$ \left[\begin{array}{l}{\rm d}U\\ {\rm d}V\end{array}\right]= \left[\begin{array}{ll}\alpha _1-\sigma _1^2/2\\ \alpha _2-\sigma _2^2/2\end{array}\right]{\rm d}t+ \left[\begin{array}{ll}\sigma _1 &{} \rho \sigma _1\sigma _2\\ \rho \sigma _1\sigma _2 &{} \sigma _2\end{array}\right] \left[\begin{array}{l}{\rm d}W_1\\ {\rm d}W_2\end{array}\right], $$

which has the property that

$$ V\left( x,y\right) ={\hat{V}}(\log (x),\log (y)),\quad \text {all } x,y\in {\mathbb{R}}_{++}. $$

Note that a straightforward application of Ito’s lemma gives that

$$ {\rm d}U = (\mu _1-\sigma _1^2/2){\rm d}t+\sigma _1{\rm d}W_1,\quad \text {and}\quad {\rm d}V = (\mu _2-\sigma _2^2/2){\rm d}t+\sigma _2{\rm d}W_2. $$

The basic idea is to replace the the continuous-time stochastic process (U, V) by a discrete-time Markov chain $(U^h,V^h)$, where $h>0$ is the step-size on a grid over

$$ G := [{\underline{u}},{\overline{u}}]\times [{\underline{v}},{\overline{v}}] $$

for some ${\underline{u}}<{\overline{u}}$ and ${\underline{v}}<{\overline{v}}$ that ensure that G is large enough to produce an accurate approximation to ${\hat{V}}$. In our case we choose

$$ {\underline{u}}={\underline{v}}=-10,\quad {\overline{u}}=\log (P_U^*),\quad \text {and}\quad {\overline{v}}=\log (P_D^*). $$

Any point (u,v) on the grid is such that

$$ u\in {{\mathcal {N}}}^h_u:=\{{\underline{u}},{\underline{u}}+h,\ldots ,{\overline{u}}-h,{\overline{u}}\},\quad \text {and} \quad v\in {{\mathcal {N}}}^h_v:=\{{\underline{v}},{\underline{v}}+h,\ldots ,{\overline{v}}-h,{\overline{v}}\}. $$

Introducing, for each $a\in {\mathbb{R}}$, the notation $a^+:=\max \{0,a\}$ and $a^-:=\max \{-a,0\}$, the approximating Markov chain on the grid has the following transition probabilities on ${{\mathcal {N}}}^h_u\times {{\mathcal {N}}}^h_v$:

$$\begin{aligned} p^h(u\pm h,v|u,v)&= \frac{(\sigma _1^2-|\rho |\sigma _1\sigma _2)/2 +h\left( \alpha _1-\sigma _1^2/2\right) ^+}{Q^h(u,v)},\\ p^h(u,v\pm h|u,v)&= \frac{(\sigma _2^2-|\rho |\sigma _1\sigma _2)/2 +h\left( \alpha _2-\sigma _2^2/2\right) ^+}{Q^h(u,v)},\\ p^h(u+h,v+h|u,v)&= p^h(u-h,v-h|u,v) = \frac{\rho ^+\sigma _1\sigma _2}{2Q^h(u,v)},\\ p^h(u+h,v-h|u,v)&= p^h(u-h,v+h|u,v) = \frac{\rho ^-\sigma _1\sigma _2}{2Q^h(u,v)}, \end{aligned}$$

where

$$ Q^h(u,v)= \sigma _1^2+\sigma _2^2-|\rho |\sigma _1\sigma _2+ \left| \alpha _1-\tfrac{1}{2}\sigma _1^2\right| h +\left| \alpha _2-\tfrac{1}{2}\sigma _2^2\right| h. $$

Note that the transition probabilities are non-negative if $\rho =0$ or if

$$ |\rho |<\frac{\sigma _1}{\sigma _2}<|\rho |^{-1},\quad \text {when }\rho \ne 0. $$

Given a grid ${{\mathcal {N}}}_u^h\times {{\mathcal {N}}}_v^h$, our time disretization is chosen such that our discrete-time Markov chain approximation of (14) (to be defined below) converges (weakly) to (14). It turns out (see, e.g., Kushner and Dupuis 2001) that a sequence of time points with (potentially state-dependent) time intervals of length

$$ \Delta t^h(u,v) := h^2/Q^h(u,v), $$

achieves this.

From Proposition 2 we know that the solution to (14) should satisfy the Hamilton–Jacobi–Bellman (HJB) equation

$$ {\hat{V}}(u,v) = \max \left\{ {\hat{F}}(u,v),{{\mathscr {L}}}{\hat{V}}(u,v)+(1-r){\hat{V}}(u,v)-{\hat{c}}(u)\right\} , $$

(15)

here written as a fixed-point equation, where ${{\mathscr {L}}}$ is the characteristic operator of (U,V), i.e.

$$ {{\mathscr {L}}}\varphi := \frac{1}{2}\sigma _1^2\varphi ''_{11}+\frac{1}{2}\sigma _2^2\varphi ''_{22} +\rho \sigma _1\sigma _2\varphi ''_{12}+(\alpha _1-\sigma _1^2/2)\varphi '_1 +(\alpha _2-\sigma _2^2/2)\varphi '_2. $$

Using the transition probabilities of our Markov chain approximation we can discretize the characteristic operator for functions $\varphi ^h$ defined on the grid ${{\mathcal {N}}}_u^h\times {{\mathcal {N}}}_v^h$:

$$\begin{aligned}{\hat{{{\mathscr {L}}}}}^h\varphi ^h(u,v)&:= p(u+h,v|u,v)\varphi ^h(u+h,v)+p(u,v+h|u,v)\varphi ^h(u,v+h)\\ &\quad+\,p(u-h,v|u,v)\varphi ^h(u-h,v) +p(u,v-h|u,v)\varphi ^h(u,v-h)\\ &\quad +\,1_{\rho \ge 0}\left[ p(u+h,v+h|u,v)\varphi ^h(u+h,v+h)+p(u-h,v-h|u,v)\varphi ^h(u-h,v-h)\right] \\ &\quad +\,1_{\rho < 0}\left[ p(u+h,v-h|u,v)\varphi ^h(u+h,v-h)+p(u-h,v+h|u,v)\varphi ^h(u-h,v+h)\right] . \end{aligned}$$

We now replace the HJB equation (15) by the discrete-time approximation

$$ W^h(u,v)= (TW^h)(u,v), $$

(16)

where the operator T is given by

$$ (TW^h)(u,v):=\max \left\{ {\hat{F}}^h(u,v),(1-r\Delta t^h(u,v)){{\mathscr {L}}}^hW^h(u,v)-c(u)\Delta t^h(u,v)\right\} . $$

Using Blackwell’s theorem (Aliprantis and Border 2006, Theorem 3.53), it is fairly easy to show that the operator T is a contraction mapping. From the Banach fixed point theorem (Aliprantis and Border 2006, Theorem 3.48) it follows that the fixed-point problem (16) has a unique fixed point. In addition, repeated application of T leads to convergence to the fixed point $W^h$, which then acts as our approximation to the value function V.

To summarize, we start with an initial guess $W^h_0$ on the grid ${{\mathcal {N}}}^h_u\times {{\mathcal {N}}}^h_v$. From this initial guess we extract an initial guess of the continuation region, $G_0$, by extracting all points $(u,v)\in {{\mathcal {N}}}^h_u\times {{\mathcal {N}}}^h_v$ for which $W_0^h(u,v)>{\hat{F}}(u,v)$. We then compute a new iteration, $W_1^h$ by applying the operator T, i.e.

$$ W^h_1 := TW^h_0. $$

This procedure is repeated until the change between $W^h_n$ and $W^h_{n-1}$ (in the sup norm) falls below 1. That is, our final approximation is to the nearest $1,000.