Chapter 5 Internalizing The Environmental Costs of Pollution

5.1 Business as Usual Paths

We select the following market structures and identify the following emission paths and enviromental damages as our reference point for our coalition analysis

5.2 Benefits from abatement

We address is the issue of the abatement potential associated with smaller industry level climate coalitions. Given how norms have evolved to date, most individual businesses tend to ignore environmental damages caused by other competing firms in a given market. So as a market level externality issue, without some environmental regulation ocean carriers necessarily supply less than the socially optimum pollution abatement level. Subsequently, welfare analysis of any green shipping coalition hinges on the damages attributable to its pollutants and the gains from market level abatement for society. Estimates of environmental damage costs at the market level, generated through firms polluting in the absence of some abatement strategies, represent the societal costs incorporated into STAICO throughout the simulated planning horizon.

5.2.1 Enviromental Damages

we consider a relatively small pollutant damage parameter \(\gamma_D = 1.5\) and approximate the global damages generated by the industry from the accumulation of pollution stock \(D_t (S_t)\) as: \[\begin{equation} \begin{cases} {D_{t}^{CO_2}} = \frac{\gamma_D}{2} \bigg(S_t^{CO_2}\bigg)^2 \\ {D_{t}^{SO_x}} = \frac{\gamma_D}{2} \bigg(S_t^{SO_x}\bigg)^2 \\ \end{cases} \end{equation}\]

The advantages arising from the formation of a pollution coalition among individual firms in a market depends upon the overall level of abatement in our simulation analysis. However, the public good nature of pollution and climate issues allow singletons to leverage the coalition’s abatement endeavours as well as formal signatories, which strengthens free-riding incentives in our framework. STAICO aims to maximize the coalition’s payoff from abatement, with global abatement benefits interpreted as avoided environmental damages due to coalition formation under some form of abatement strategy.

The following benefit function is used to internalise the positive externalities derived from the formation of the coalition:

\[\begin{equation} \begin{cases} \beta_t^{CO_2} = D_t(\overline{S_t^{CO_2}}) - D_t(S_t^{CO_2}) \\ \beta_t^{SO_x} = D_t(\overline{S_t^{CO_2}}) - D_t(S_t^{CO_2}) \\ \end{cases} \end{equation}\]\[\begin{equation} \Rightarrow \begin{cases} \beta_t^{CO_2} = \frac{\gamma_D}{2} \times \bigg[ (\overline{S_t^{CO_2}})^2 - (S_t^{CO_2})^2 \bigg] \\ \beta_t^{SO_x} = \frac{\gamma_D}{2} \times \bigg[ (\overline{S_t^{CO_2}})^2 - (S_t^{CO_2})^2 \bigg] \\ \end{cases} \end{equation}\]

5.2.2 Enviromental Benefits

The global benefits are then allocated to each firm by their market share \(s_i\), with firm-level benefits defined as the market share of the firm times global benefits: \[\begin{equation} \begin{cases} b_{i,t}^{CO_2} = s_{i,t} \times \beta_t^{CO_2} = s_{i,t} \times \frac{\gamma_D}{2} \times \bigg[ (\overline{S_t^{CO_2}})^2 - (S_t^{CO_2})^2 \bigg] \\ b_{i,t}^{SO_x} = s_{i,t} \times \beta_t^{SO_x} = s_{i,t} \times \frac{\gamma_D}{2} \times \bigg[ (\overline{S_t^{CO_2}})^2 - (S_t^{CO_2})^2 \bigg] \\ \end{cases} \end{equation}\]

Note that the benefits from abatement for each firm hinges on the aggregate global emission level, further supporting any inherent free-riding incentives. Each relevant abatement level \(q_{i,t}^{CO_2}\) , \(q_{i,t}^{SO_x}\) can be expressed as :

\[\begin{equation} \begin{cases} q_{i,t}^{CO_2}= \overline{e^{CO_2}_{i,t}} \bigg(V_{i,t}^{BAU} \bigg)- e_{i,t}^{CO_2} \bigg(V_{i,t} \bigg)\\ q_{i,t}^{SO_x } =\overline{e^{SO_x}_{i,t}} \bigg(V_{i,t}^{BAU} \bigg) - e_{i,t}^{SO_x} \bigg(V_{i,t} \bigg)\\ \end{cases} \end{equation}\] such that: \[\begin{equation} \begin{cases} e_{i,t}^{CO_2} \leq \theta_{i,t}^{CO_2} ~\ \forall i \in C\\ e_{i,t}^{SO_x } \leq \theta_{i,t}^{SO_x} ~\ \forall i \in C\\ \end{cases} \end{equation}\]

5.3 Abatement vector

Regarding the coalition structure \(C\), we formalise the optimal level of slow steaming \(V_{i,t}\) and the equilibrium \(SO_x\) and \(CO_2\) abatement strategy vector \(q^{∗}\) for the players as:

\[\begin{equation} \begin{cases} q_{i,t}^{CO_2}= \overline{e^{CO_2}_{i,t}} \bigg(V_{i,t}^{BAU} \bigg)- e_{i,t}^{CO_2} \bigg(V_{i,t} \bigg)\\ q_{i,t}^{SO_x } =\overline{e^{SO_x}_{i,t}} \bigg(V_{i,t}^{BAU} \bigg) - e_{i,t}^{SO_x} \bigg(V_{i,t} \bigg)\\ \end{cases} \end{equation}\]

where \(\overline{e^{CO_2}_{i,t}} , \overline{e^{SO_x}_{i,t}}\) depict the \(i^{th}\) signatory firm’s sulfur and carbon business as usual (BAU) emission levels in period \(t\) , \(q_{i,t}^{CO_2}, q_{i,t}^{SO_x }\) their abatement level, and \(e_{i,t}^{CO_2}, e_{i,t}^{SO_x}\) their emission levels owing to their membership decision.

Following the accumulation of pollution stocks and the industry’s incurred environmental damages, firms engage in a game of Partial Agreement Nash Equilibrium (PANE). The PANE framework means that in equilibrium, no player has an incentive to deviate from their chosen abatement strategy. Deriving the optimal abatement strategy vector from a global standpoint will enable us to distinguish between winners and losers for particular abatement schemes. The solution will also allow us to distribute abatement responsibilities between heterogeneous firms to achieve emissions targets at the least cost.

5.4 Marginal Abatement Cost Curves (MACC)

The marginal abatement cost curve shows the net costs associated with a unit tonne of each of carbon and sulfur reduced over a single time period (year). For maritime shipping, it also reflects slow steaming’s maximum abatement potential as a policy to implement within pollution coalitions. To derive our shipping sector’s customised MACCs, we follow , and ’s methodology. First, we determine “business as usual” (BAU) emissions based on projections of market demand, meaning the emissions inventory created in the absence of slow steaming and coalition formation. Then, we compute the carbon and sulfur abatement potential and the associated costs of an emission threshold ( ; ).

Thus, our \(MAC\) curves in \(\$/ton ~\ CO_2\) and \(\$/ton ~\ SO_x\) are :

\[\begin{equation} \begin{cases} MAC_{i,t}^{CO_2 } = \frac{ \Pi^{BAU}_{i,t} \big(V^{BAU}_{i,t},~\ N^{BAU}_{i,t}\big) - \Pi_{i,t} \big(V_{i,t},~\ N_{i,t}\big)} {\overline{e^{CO_2}_{i,t}} \big(V_{i,t}^{BAU} \big)- e_{i,t}^{CO_2} \big(V_{i,t} \big)}\\ \\ MAC_{i,t}^{SO_x} = \frac{ \Pi^{BAU}_{i,t} \big(V^{BAU}_{i,t},~\ N^{BAU}_{i,t}\big) - \Pi_{i,t} \big(V_{i,t},~\ N_{i,t}\big)} {\overline{e^{SO_x}_{i,t}} \big(V_{i,t}^{BAU} \big) - e_{i,t}^{SO_x} \big(V_{i,t} \big)}\\ \end{cases} \end{equation}\]

5.5 The new optimisation problem

5.5.1 The coaltiion

The coalition considers the environmental impacts of its members attempts to maximize profits. Collectively, the members face a multi-objective optimization model based on individual profit maximization, as well as \(CO_2\) and \(SO_x\) emissions reduction such that outcomes are driven by the fleets’ sailing speed. Under the coalition and leveraging slow-steaming as the abatement policy, signatories choose the emission path that maximises the sum of their payoffs:

\[\begin{equation} \max_{{ N_{j,t} ,V_{j,t} }} \Pi^{C} = \sum_{t=1}^{T} \sum_{j=1}^{n'} {(1+r)^{-t}} \bigg[ ~\ \rho_t ~\ X^{AB}_{j,t} + \bigg( s_{j,t} ~\ \beta_t^{CO_2} + s_{j,t} ~\ \beta_t^{CO_2} \bigg)\\ - \bigg( \eta_t^{fuel} ~\ \frac{X^{AB}_{j,t}} {k_j} ~\ d ~\ \phi_j ~\ (V_{j,t})^2 \bigg) - \bigg( \eta_t^{MGO} ~\ \frac{X^{AB}_{j,t}} {k_j} ~\ d ~\ F_A ~\ \frac{1}{V_{j,t}} \bigg) - \bigg( CO_{j} ~\ N_{j,t} \bigg) \bigg] \forall j \in C \end{equation}\] such that :

{Coalition Demand :} \[\begin{equation} X_{C,t}^{A,B} = \sum_{j=1}^{n'} X_{j,t}^{A,B}= \sum_{j=1}^{n'} s_{j} Y_t^{A,B} = \sum_{j=1}^{n'} s_{j} \times (\frac{GDP_t}{GDP_0})^v \times (\frac{\rho_t}{\rho_0})^η \times Y_0^{AB} ~\ \forall j \in C \end{equation}\]
{Coalition Supply and Operational Constraint :} \[\begin{equation} \begin{cases} V_j^{min} \leq V_{j,t} \leq V_j^{max} ~\ \forall j \in C \\ \\ N_{j,t} \in Z^+ ~\ \forall j \in C \\ N_{j,t} \geq N_{{min}_{j,t}}^{vessel} = \frac{X^{AB}_{j,t} \times \bigg( \frac{d }{V_{j,t}} +t_{port} \bigg)} {k_j \times \tau } ~\ \forall j \in C \\ \end{cases} \end{equation}\]

{Abatement vector and emissions } \[\begin{equation} \begin{cases} q_{j,t}^{CO_2}= \overline{e^{CO_2}_{j,t}} \bigg(V_{j,t}^{BAU} \bigg)- e_{j,t}^{CO_2} \bigg(V_{j,t} \bigg)\\ q_{j,t}^{SO_x } =\overline{e^{SO_x}_{i,t}} \bigg(V_{j,t}^{BAU} \bigg) - e_{j,t}^{SO_x} \bigg(V_{j,t} \bigg)\\ \end{cases} \end{equation}\] such that: \[\begin{equation} \begin{cases} e_{j,t}^{CO_2} = \frac{X^{AB}_{j,t}} {k_j} \times d \bigg( \epsilon^{CO_2 }_{fuel} \times \phi_j \times (V_{j,t})^2 + \epsilon^{CO_2 }_{MGO} \times \frac{F_A}{V_{j,t}} \bigg) \leq \theta_{j,t}^{CO_2} ~\ \forall j \in C\\ e_{j,t}^{SO_x } = \frac{X^{AB}_{j,t}} {k_j} \times d \bigg( \epsilon^{SO_x }_{fuel} \times \phi_j \times (V_{j,t})^2 + \epsilon^{SO_x }_{MGO} \times \frac{F_A}{V_{j,t}} \bigg) \leq \theta_{j,t}^{SO_x} ~\ \forall j \in C\\ \end{cases} \end{equation}\]

5.5.2 Singeltons

Singletons, on the other hand don’t have to abate according to the coalition’s emission threshold. Instead, they just choose a vessel speed that maximises their net present value in the market. Separately, they set up their emission paths as the following:

\[\begin{equation} \begin{aligned} \max_{{ N_{j',t} ,V_{j',t} }} \pi^{j'} = {} & \sum_{t=1}^{T} {(1+r)^{-t}} \bigg[ ~\ \rho_t ~\ X^{AB}_{j',t} \\ & + \bigg( s_{j',t} ~\ \beta_t^{CO_2} + s_{j',t} ~\ \beta_t^{CO_2} \bigg)\\ & - \bigg( \eta_t^{fuel} ~\ \frac{X^{AB}_{j',t}} {k_j'} ~\ d ~\ \phi_j' ~\ (V_{j',t})^2 \bigg) \\ & - \bigg( \eta_t^{MGO} ~\ \frac{X^{AB}_{j',t}} {k_j'} ~\ d ~\ F_A ~\ \frac{1}{V_{j',t}} \bigg) \\ & - \bigg( CO_{j'} ~\ N_{j',t} \bigg) \bigg] \forall j' \notin C \end{aligned} \end{equation}\] such that :

{Firm Level Demand :} \[\begin{equation} X_{j',t}^{A,B}= s_{j'} Y_t^{A,B} = s_{j'} \times (\frac{GDP_t}{GDP_0})^v \times (\frac{\rho_t}{\rho_0})^η \times Y_0^{AB} ~\ \forall j' \notin C \end{equation}\]
{Firm Level Supply and Operational Constraint :} \[\begin{equation} \begin{cases} V_j'^{min} \leq V_{j',t} \leq V_j'^{max} ~\ \forall j' \notin C \\ \\ N_{j',t} \in Z^+ ~\ \forall j' \notin C \\ N_{j',t} \geq N_{{min}_{j',t}}^{vessel} = \frac{X^{AB}_{j',t} \times \bigg( \frac{d }{V_{j',t}} +t_{port} \bigg)} {k_j' \times \tau } ~\ \forall j' \notin C \\ \end{cases} \end{equation}\]

{Firm level Abatement path and emissions } \[\begin{equation} \begin{cases} q_{j',t}^{CO_2}= \overline{e^{CO_2}_{j',t}} \bigg(V_{j',t}^{BAU} \bigg)- e_{j',t}^{CO_2} \bigg(V_{j',t} \bigg)\\ q_{j',t}^{SO_x } =\overline{e^{SO_x}_{j',t}} \bigg(V_{j',t}^{BAU} \bigg) - e_{j',t}^{SO_x} \bigg(V_{j',t} \bigg)\\ \end{cases} \end{equation}\]such that: \[\begin{equation} \begin{cases} e_{j',t}^{CO_2} = \frac{X^{AB}_{j',t}} {k_j} \times d \bigg( \epsilon^{CO_2 }_{fuel} \times \phi_j' \times (V_{j',t})^2 + \epsilon^{CO_2 }_{MGO} \times \frac{F_A}{V_{j,t}} \bigg) \forall j' \notin C\\ e_{j',t}^{SO_x } = \frac{X^{AB}_{j',t}} {k_j'} \times d \bigg( \epsilon^{SO_x }_{fuel} \times \phi_j' \times (V_{j',t})^2 + \epsilon^{SO_x }_{MGO} \times \frac{F_A}{V_{j',t}} \bigg) ~\ \forall j' \notin C\\ \end{cases} \end{equation}\]

Environmental damages and benefits from abatement

This game formulation allows us to investigate an inter-temporal optimisation problem internalising environmental damages in conjunction with the shipper’s profit maximization problem. The production/externality feedback loop between the environment and the shipping market ensures a non-cooperative game between the signatories and the coalition through the singletons’ abatement endeavours, the latter of which gives rise to free-riding incentives.

{Industry level emission path }

\[\begin{equation} \begin{cases} E_t^{CO_2} = \sum_{i=1}^{n'} e_{j,t}^{CO_2} + \sum_{i=1}^{n''} e_{j',t}^{CO_2} \\ E_t^{SO_x} = \sum_{i=1}^{n' } e_{j,t}^{SO_x} + \sum_{i=1}^{n''} e_{j',t}^{SO_x} \end{cases} \end{equation}\]

{Industry level pollution stock }

\[\begin{equation} \begin{cases} S_{t+1}^{CO_2} = E_t^{CO_2} + (1 - \delta) \times S^{CO_2}_{t}\\ S_{t+1}^{SO_x} = E_t^{SO_x} + (1 - \delta) \times S^{SO_x}_{t}\\ \end{cases} \end{equation}\]

Industry level environmental damages and abatement benefits

\[\begin{equation} \begin{cases} \beta_t^{CO_2} = D_t(\overline{S_t^{CO_2} }) - D_t(S_t^{CO_2}) = \frac{\gamma_D}{2} \times \bigg[ (\overline{S_t^{CO_2}})^2 - (S_t^{CO_2})^2 \bigg] \\ \beta_t^{SO_x} = D_t(\overline{S_t^{SO_x}}) - D_t(S_t^{SO_x}) = \frac{\gamma_D}{2} \times \bigg[ (\overline{S_t^{SO_x}})^2 - (S_t^{SO_x})^2 \bigg] \\ \end{cases} \end{equation}\]