The Comparative Impact of Integrated Assessment Models’ Structures on Optimal Mitigation Policies

Abstract

This paper aims at providing a consistent framework to appraise alternative modeling choices that have driven the so-called “when flexibility” controversy since the early 1990s, dealing with the optimal timing of mitigation efforts and the social cost of carbon (SCC). The literature has emphasized the critical impact of modeling structures on the optimal climate policy. We estimate within a unified framework the comparative impact of modeling structures and investigate the structural modeling drivers of differences in climate policy recommendations. We use the integrated assessment model (IAM) RESPONSE to capture a wide array of modeling choices. Specifically, we analyse four emblematic modeling choices, namely the forms of the damage function (quadratic vs. sigmoid) and the abatement cost (with or without inertia), the treatment of uncertainty, and the decision framework, deterministic or sequential, with different dates of information arrival. We define an original methodology based on an equivalence criterion to compare modeling structures, and we estimate their comparative impact on two outputs: the optimal SCC and abatement trajectories. We exhibit three key findings: (1) IAMs with a quadratic damage function are insensitive to changes of other features of the modeling structure, (2) IAMs involving a non-convex damage function entail contrasting climate strategies, (3) Precautionary behaviors can only come up in IAMs with non-convexities in damage.

Appendices

Appendix A: First-Order Conditions Resolution

In this part, our calculations follow the two-step resolution method already described in part 2.3. We distinguish before and after uncertainty is resolved.

1.1 A.1 After uncertainty is resolved

After uncertainty is resolved, we know the state of nature ω. The Lagrangian writes:

$$\begin{array}{@{}rcl@{}} &&\mathbf{L}^{\omega}=\sum\limits_{t=t_{i}+1}^{\infty}N_{t}\frac{1}{(1+\rho)^{t}} u\left(\frac{C^{\omega}_{t}}{N_{t}}\right)\\ &&+\sum\limits_{t=t_{i}+1}^{\infty}(\lambda_{A,t}^{\omega}, \lambda_{B,t}^{\omega},\lambda_{O,t}^{\omega})\\ &&\times\left( \begin{array}{l} A_{t+1}^{\omega}-\left(c_{AA}A_{t}^{\omega}+c_{AB}B_{t}^{\omega}+(1-a_{t}^{\omega})\sigma_{t}Y_{t}^{\omega}\right) \\ B_{t+1}^{\omega}-(c_{BA}A_{t}^{\omega}+c_{BB}B_{t}^{\omega}+c_{BO}O_{t}^{\omega}) \\ O_{t+1}^{\omega}-(c_{OB}B_{t}^{\omega}+c_{OO}O_{t}^{\omega}) \end{array} \right) \\ &&+\sum\limits_{t=t_{i}+1}^{\infty}(\nu_{A,t}^{\omega},\nu_{O,t}^{\omega})\\ &&\times\left(\! \begin{array}{l} T_{A,t+1}^{\omega} - ((1-\sigma_{1}\left(\frac{F_{2x}}{\vartheta_{2x}^{\omega}}+\sigma_{2}\right))T_{A,t}^{\omega} +\sigma_{1}\sigma_{2}T_{O,t}^{\omega}\,+\,\sigma_{1} F(A_{t}^{\omega})\!) \\ T_{O,t+1}^{\omega} - (\sigma_{3}T_{A,t}^{\omega} +(1-\sigma_{3})T_{O,t}^{\omega}) \end{array} \right) \\ &&+\sum\limits_{t=t_{i}+1}^{\infty} \mu_{t}^{\omega}\big(-K_{t+1}^{\omega} \,+\, (1\,-\,\delta)K_{t}^{\omega} \\ &&{\kern4pc}+ Y_{t}^{\omega}\left[1\,-\,G(a_{t}^{\omega},a_{t-1}^{\omega})\,-\,D^{\omega}(T_{A,t}^{\omega})\right]\,-\,C_{t}^{\omega}\big) \\ &&+\sum\limits_{t=t_{i}+1}^{\infty} \overline{\tau}^{\omega}_{t}.(1-a_{t})+\underline{\tau}^{\omega}_{t}.a_{t} \end{array} $$

(16)

The Lagrange multiplier attached to the capital constraint (4) is $\mu _{t}^{\omega }$; the Lagrange multipliers attached to the carbon cycle dynamics constraints (11) are $\lambda _{A,t}^{\omega }$, $\lambda _{B,t}^{\omega }$, and $\lambda _{O,t}^{\omega }$. The Lagrange multipliers attached to the temperature constraints (13) are $\nu _{A,t}^{\omega }$ and $\nu _{O,t}^{\omega }$. The Lagrange multipliers attached to inequality constraints (7) are $\overline {\tau }^{\omega }_{t}$ and $\underline {\tau }^{\omega }_{t}$.

At the beginning of the program, stock variables are inherited from the past, i.e., from the maximization program under uncertainty. Some do not depend on the state of nature:

$$\begin{array}{@{}rcl@{}} A_{t_{i}+1}^{\omega}=\overline{A}_{t_{i}+1},\, B_{t_{i}+1}^{\omega}=\overline{B}_{t_{i}+1},\, O_{t_{i}+1}^{\omega}=\overline{O}_{t_{i}+1},\, K_{t_{i}+1}^{\omega}=\overline{K}_{t_{i}+1} \end{array} $$

(17)

By convention, $a^{\omega }_{t_{i}}=\overline {a}_{t_{i}}$.

We calculate the first-order conditions with respect to the two flow variables: $C_{t}^{\omega }$ and $a_{t}^{\omega }$, and to the six stock variables: $K_{t}^{\omega }$, $a_{t}^{\omega }$, $B_{t}^{\omega }$, $O_{t}^{\omega }$, $T_{A,t}^{\omega }$, and $T_{O,t}^{\omega }$. Recall also that stock variables at t=t _i +1 are initial conditions, so we cannot derive first-order conditions for them at this stage. We get:

• For consumption, ∀t≥t _i +1:

  $$\begin{array}{@{}rcl@{}} &\frac{\partial \mathbf{L}^{\omega}}{\partial C_{t}^{\omega}} = 0 \quad \Leftrightarrow \\ &\mu_{t}^{\omega} = u'\left(\frac{C_{t}^{\omega}}{N_{t}}\right) \frac{1}{(1+\rho)^{t}} \end{array} $$

  (18)

  Then, $\mu _{t}^{\omega }$ is the discounted marginal utility.

• For the abatement capacity, ∀t≥t _i +1:

  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}^{\omega}}{\partial a_{t}^{\omega}} = 0 \quad \Leftrightarrow \\ && \lambda_{A,t}^{\omega}\sigma_{t}= \mu_{t}^{\omega}\partial_{1} G (a_{t}^{\omega},a_{t-1}^{\omega}) + \mu_{t+1}^{\omega}\partial_{2} G (a_{t+1}^{\omega},a_{t}^{\omega})\frac{Y^{\omega}_{t+1}}{Y^{\omega}_{t}}\\ &&{\kern3pc} +\frac{\overline{\tau}^{\omega}_{t}-\underline{\tau}^{\omega}_{t}}{Y^{\omega}_{t}} \end{array} $$

  (19)

  For t=t _i +1, recall the conventional notation that $a_{t_{i}}^{\omega }=\overline {a}_{t_{i}}$. Recall that $\overline {\tau }^{\omega }_{t}>0$ only when $a^{\omega }_{t}=1$, and $\underline {\tau }^{\omega }_{t}>0$ only when $a^{\omega }_{t}=0$.

• For capital, ∀t≥t _i +2:

  $$\begin{array}{@{}rcl@{}} &&\frac{\partial \mathbf{L}^{\omega}}{\partial K_{t}^{\omega}} =0 \quad \Leftrightarrow \\ && \partial_{K}Y_{t}^{\omega}\left(1-\frac{\lambda_{A,t}^{\omega} (1-a_{t}^{\omega})\sigma_{t}}{\mu_{t}^{\omega}}\-G(a_{t}^{\omega},a_{t-1}^{\omega})-D^{\omega}(T_{A,t}^{\omega})\right)\\ &&= (1+\rho)\frac{u'\left(\frac{C_{t-1}^{\omega}}{N_{t-1}}\right)}{u'\left(\frac{C_{t}^{\omega}}{N_{t}}\right)}- (1-\delta) \end{array} $$

  (20)

• For the carbon stocks, ∀t≥t _i +2:

  $$\begin{array}{@{}rcl@{}} && \frac{\partial\mathbf{L}^{\omega}}{\partial A_{t}^{\omega}} = 0 \quad \Leftrightarrow \\ && \lambda_{A,t-1}^{\omega} = \lambda_{A,t}^{\omega}c_{AA} +\lambda_{B,{t}}^{\omega}c_{BA} + \nu_{A,t}^{\omega}\sigma_{1}F'(A_{t}^{\omega}) \end{array} $$

  (21)
  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}^{\omega}}{\partial B_{t}^{\omega}} = 0 \quad \Leftrightarrow \\ &&\lambda_{B,t-1}^{\omega} = \lambda_{A,t}^{\omega} c_{AB}+\lambda_{B,{t}}^{\omega}c_{BB} + \lambda_{O,{t}}^{\omega} c_{OB} \end{array} $$

  (22)
  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}^{\omega}}{\partial O_{t}^{\omega}} = 0 \quad \Leftrightarrow \\ &&\lambda_{O,t-1}^{\omega} = \lambda_{B,{t}}^{\omega}c_{BO} + \lambda_{O,{t}}^{\omega} c_{OO} \end{array} $$

  (23)

• For the temperatures, ∀t≥t _i +2:

  $$\begin{array}{@{}rcl@{}} && \frac{\partial\mathbf{L}^{\omega}}{\partial T_{A,t}^{\omega}} = 0 \quad \Leftrightarrow \\ && \nu_{A,t-1}^{\omega} = \nu_{A,t}^{\omega} \left(1-\sigma_{1}\left(\frac{F_{2x}}{\vartheta_{2x}^{\omega}}+\sigma_{2}\right)\right)\\ &&{\kern3pc} + \nu_{O,t}^{\omega} \sigma_{3} + \mu_{t}^{\omega} \partial_{T} D^{\omega}(T_{A,t}^{\omega})Y_{t}^{\omega} \end{array} $$

  (24)
  $$\begin{array}{@{}rcl@{}} && \frac{\partial\mathbf{L}^{\omega}}{\partial T_{O,t}^{\omega}} = 0 \quad \Leftrightarrow \\ && \nu_{O,t-1}^{\omega} =\nu_{A,t}^{\omega} \sigma_{1}\sigma_{2} + \nu_{O,t}^{\omega} (1-\sigma_{3}) \end{array} $$

  (25)

1.2 A.2 Before Uncertainty is Resolved

The Lagrangian of the maximization program then equals the expectation over the possible states of nature of the sum of the objective function and a cluster of dynamic equations.

The Lagrangian writes:

$$\begin{array}{@{}rcl@{}} \mathbf{L}_{u} &=& \sum\limits_{t=t_{0}}^{t=t_{i}}\mathbb{E}\left[\frac{1}{(1+\rho)^{t}} u(C_{t}^{\omega})+V(\omega)\right] \end{array} $$

(26)

$$\begin{array}{@{}rcl@{}} &&+\sum\limits_{t=t_{0}}^{t_{i}}(\lambda_{A,t},\lambda_{B,t},\lambda_{O,t})\left( \begin{array}{l} \overline{A}_{t+1}-\left(c_{AA}\overline{A}_{t}+c_{AB}\overline{B}_{t}+(1-\overline{a}_{t})\sigma_{t}\overline{Y}_{t}\right) \\ \overline{B}_{t+1}-(c_{BA}\overline{A}_{t}+c_{BB}\overline{B}_{t}+c_{BO}\overline{O}_{t})\\ \overline{O}_{t+1}-(c_{OB}\overline{B}_{t}+c_{OO}\overline{O}_{t}) \end{array} \right)\\ &&+ \sum\limits_{t=t_{0}}^{t=t_{i}}\mathbb{E}\left[(\nu_{A,t}^{\omega},\nu_{O,t}^{\omega})\left( \begin{array}{lllll} T_{A,t+1}^{\omega} - ((1-\sigma_{1}(\frac{F_{2x}}{\vartheta_{2x}^{\omega}}+\sigma_{2}))T_{A,t}^{\omega} +\sigma_{1} \sigma_{2} T_{O,t}^{\omega}+\sigma_{1} F(\overline{A}_{t})) \\ T_{O,t+1}^{\omega} - (\sigma_{3}T_{A,t}^{\omega} +(1-\sigma_{3})T_{O,t}^{\omega}) \end{array} \right) \right] \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} &&+\sum\limits_{t=t_{0}}^{t=t_{i}}\mathbb{E}\left[\mu_{t}^{\omega}(-\overline{K}_{t+1} + (1-\delta)\overline{K}_{t} + \overline{Y}_{t}\left[1-G(a_{t}^{\omega},a_{t-1}^{\omega})-D^{\omega}(T_{A,t}^{\omega})\right]-C_{t}^{\omega})\right]\\ &&+\sum\limits_{t=t_{0}}^{t_{i}} \overline{\tau}_{t}.(1-\overline{a}_{t})+\underline{\tau}_{t}.\overline{a}_{t} \end{array} $$

(28)

We calculate the first-order conditions with respect to all endogenous variables: flow variables $C_{t}^{\omega }$ and $\overline {a}_{t}$, and stock variables $\overline {K}_{t}$, $\overline {a}_{t}$, $\overline {B}_{t}$, $\overline {O}_{t}$, $T_{A,t}^{\omega }$, and $T_{O,t}^{\omega }$. The derivation will be specific for stock variables at t _i +1, and for the flux variable $\overline {a}_{t_{i}}$ due to the inertia in abatement cost, for we have to take into account their impact on V(ω). We get:

• For consumption, ∀t≤t _i :

  $$\begin{array}{@{}rcl@{}} && \frac{\partial \mathbf{L}^{\omega}}{\partial C_{t}^{\omega}} = 0 \quad \Leftrightarrow \\ && \mu_{t}^{\omega} = u'\left(\frac{C_{t}^{\omega}}{N_{t}}\right) \frac{1}{(1+\rho)^{t}} \end{array} $$

  (29)

• For the abatement capacity, ∀t<t _i :

  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}_{u}}{\partial \overline{a}_{t}} = 0 \quad \Leftrightarrow \\ \lambda_{A,t}\sigma_{t} &= & \mathbb{E}\left[\mu^{\omega}_{t}\right]\partial_{1} G(\overline{a}_{t},\overline{a}_{t-1}) \\ &&+ \mathbb{E}\left[\mu^{\omega}_{t+1}\right]\partial_{2} G(\overline{a}_{t+1},\overline{a}_{t})\frac{\overline{Y}_{t+1}}{\overline{Y}_{t}}+\frac{\overline{\tau}_{t}-\underline{\tau}_{t}}{\overline{Y}_{t}} \end{array} $$

  (30)

  For t=t _i , actions, which are decided at the beginning of the period, cannot take into account the information that arrives in this period:

  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}_{u}}{\partial \overline{a}_{t_{i}}} = 0 \quad \Leftrightarrow \\ &&\lambda_{A,t_{i}}\sigma_{t_{i}} = \mathbb{E}\left[\mu^{\omega}_{t_{i}}\right] \partial_{1} G(\overline{a}_{t_{i}},\overline{a}_{t_{i}-1})\\ &&{\kern2.6pc} +\mathbb{E}\left[\mu^{\omega}_{t_{i}+1}\partial_{2} G(a^{\omega}_{t_{i}+1},\overline{a}_{t_{i}})\right]\frac{\overline{Y}_{t_{i}+1}}{\overline{Y}_{t_{i}}}+\frac{\overline{\tau}_{t_{i}}-\underline{\tau}_{t_{i}}}{\overline{Y}_{t_{i}}} \end{array} $$

  (31)

  because $\frac {\partial V(\omega )}{\partial \overline {a}_{t_{i}}}=\frac {\partial L^{\omega }}{\partial \overline {a}_{t_{i}}}=\mu ^{\omega }_{t_{i}+1}\partial _{2} G(a^{\omega }_{t_{i}+1},\overline {a}_{t_{i}})\overline {Y}_{t_{i}+1}$.

• For capital, ∀t≤t _i :

  $$\begin{array}{@{}rcl@{}} && \frac{\partial\mathbf{L}_{u}}{\partial \overline{K}_{t}} = 0 \quad \Leftrightarrow \\ && \partial_{K} Y_{t} \left(1 - \frac{\lambda_{A,t}(1-\overline{a}_{t})\sigma_{t}}{\mathbb{E}[\mu^{\omega}_{t}]} - G(\overline{a}_{t}, \overline{a}_{t-1})- \frac{\mathbb{E}[\mu^{\omega}_{t} D^{\omega}(T_{A,t})]}{\mathbb{E}\left[\mu^{\omega}_{t}\right]} \right)\\ && = \frac{\mathbb{E}\left[\mu^{\omega}_{t-1}\right]}{\mathbb{E}\left[\mu^{\omega}_{t}\right]} -(1-\delta) \end{array} $$

  (32)

  For t=t _i +1,

  $$\begin{array}{@{}rcl@{}} && \frac{\partial\mathbf{L}_{u}}{\partial \overline{K}_{t_{i}+1}} = 0 \quad \Leftrightarrow \\ &&\partial_{K} Y_{t_{i}+1} \left(1 - \frac{\mathbb{E}[\lambda^{\omega}_{A,t_{i}+1}(1-a^{\omega}_{t_{i}+1})\sigma_{t_{i}+1}]}{\mathbb{E}\left[\mu^{\omega}_{t_{i}+1}\right]}\right. \\ &&{\kern4pc} - \left.\frac{\mathbb{E}[\mu^{\omega}_{t_{i}+1} G(a^{\omega}_{t_{i}+1}, \overline{a}_{t_{i}})]}{\mathbb{E}\left[\mu^{\omega}_{t_{i}+1}\right]} - \frac{\mathbb{E}[\mu^{\omega}_{t_{i}+1} D^{\omega}(T_{A,t_{i}+1})]}{\mathbb{E}\left[\mu^{\omega}_{t_{i}+1}\right]} \right)\\ &&= \frac{\mathbb{E}\left[\mu^{\omega}_{t_{i}}\right]}{\mathbb{E}\left[\mu^{\omega}_{t_{i}+1}\right]} -(1-\delta) \end{array} $$

  (33)

  because $\frac {\partial V(\omega )}{\partial \overline {K}_{t_{i}+1}}=\frac {\partial L^{\omega }}{\partial \overline {K}_{t_{i}+1}}= \mu ^{\omega }_{t_{i}+1}(1-\delta + \partial _{K} Y_{t_{i}+1}(1 - G(a^{\omega }_{t_{i}+1}, \overline {a}_{t_{i}})- D^{\omega }(T_{A,t_{i}+1}) )) - \lambda ^{\omega }_{A,t_{i}+1}(1-a^{\omega }_{t_{i}+1}) \sigma _{t_{i}+1} \partial _{K} Y_{t_{i}+1}$.

• For the atmospheric carbon multiplier, ∀t≤t _i :

  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}_{u}}{\partial \overline{A}_{t}} = 0 \quad \Leftrightarrow \\ &&\lambda_{A,t-1} = \lambda_{A,t}c_{AA}+ \lambda_{B,t}c_{BA}\\&&+\mathbb{E}\left[\nu_{A,t}^{\omega}\sigma_{1}F'(\overline{A}_{t})\right] \end{array} $$

  (34)
  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}_{u}}{\partial \overline{B}_{t}} = 0 \quad \Leftrightarrow \\ && \lambda_{B,t-1} = \lambda_{A,t} c_{AB}+\lambda_{B,{t}}c_{BB} + \lambda_{O,{t}} c_{OB} \end{array} $$

  (35)
  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}_{u}}{\partial \overline{O}_{t}} = 0 \quad \Leftrightarrow \\ &&\lambda_{O,t-1} = \lambda_{B,{t}}c_{BO} + \lambda_{O,{t}} c_{OO} \end{array} $$

  (36)

  For t=t _i +1,

  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}_{u}}{\partial \overline{A}_{t_{i}+1}} = 0 \quad \Leftrightarrow\\ && \lambda_{A,t_{i}} = \mathbb{E}\left[ \lambda_{A,t_{i}+1}^{\omega}c_{AA}+ \lambda_{B,t_{i}+1}^{\omega}c_{BA}\right.\\&& \left.\qquad\quad+\nu_{A,t_{i}+1}^{\omega}\sigma_{1}F'(\overline{A}_{t_{i}+1})\right] \end{array} $$

  (37)
  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}_{u}}{\partial \overline{B}_{t_{i}+1}} = 0 \quad \Leftrightarrow \\ && \lambda_{B,t_{i}} = \mathbb{E}\left[\lambda^{\omega}_{A,t_{i}+1} c_{AB}+\lambda^{\omega}_{B,{t_{i}+1}}c_{BB} + \lambda^{\omega}_{O,{t_{i}+1}} c_{OB}\right] \end{array} $$

  (38)
  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}_{u}}{\partial \overline{O}_{t_{i}+1}} = 0 \quad \Leftrightarrow \\ && \lambda_{O,t_{i}} = \mathbb{E}\left[\lambda^{\omega}_{B,{t_{i}+1}}c_{BO} + \lambda^{\omega}_{O,{t_{i}+1}} c_{OO}\right] \end{array} $$

  (39)

• For the temperatures, ∀t≤t _i +1:

  $$\begin{array}{@{}rcl@{}} &\frac{\partial\mathbf{L}_{u}}{\partial T_{A,t}^{\omega}} = 0 \quad \Leftrightarrow \\ &\nu_{A,t-1}^{\omega} =& \nu_{A,t}^{\omega} \left(1-\sigma_{1}\left(\frac{F_{2x}}{\vartheta_{2x}^{\omega}}+\sigma_{2}\right)\right)\\ && +\nu_{O,t}^{\omega} \sigma_{3} + \mu_{t}^{\omega} \partial_{T} D^{\omega}(T_{A,t}^{\omega})Y_{t}^{\omega} \end{array} $$

  (40)
  $$\begin{array}{@{}rcl@{}} &&\frac{\partial\mathbf{L}_{u}}{\partial T_{O,t}^{\omega}} = 0 \quad \Leftrightarrow \\ &&\nu_{O,t-1}^{\omega} =\nu_{A,t}^{\omega} \sigma_{1}\sigma_{2} + \nu_{O,t}^{\omega} (1-\sigma_{3}) \end{array} $$

  (41)

Appendix B: The Social Cost of Carbon

1.1 B.1 Theoretical Definition

The social cost of carbon (SCC) is “the additional damage caused by an additional ton of carbon emissions. In a dynamic framework, it is the discounted value of the change in the utility of consumption denominated in terms of current consumption” [33].

1.2 B.2 Definition in RESPONSE

At time t, if there is an additional ton of carbon in the atmosphere, this will increase A _t+1 by a unit and thus decrease the welfare from t+1. This variation of welfare is captured by ∂ W _t+1/∂ A _t+1=−λ _A,t+1 c _{A A} −λ _B,t+1 c _{B A} −ν _A,t+1 σ ₁ F ^′(A _t+1). Once on an optimal path, this is equal, thanks to (21) or (34), to −λ _A,t . So, on an optimal path, the social cost of carbon is also related to the abatement cost thanks to (19) or (30). The SCC has to be counted in current utility units.

More precisely, the equations for SCC at the different stages of the model are given below.

After uncertainty is resolved (t≥t _i +1), for each state of nature ω, the SCC is:

$$\begin{array}{@{}rcl@{}} SCC^{\omega}_{t} &=&\frac{\lambda_{A,t}^{\omega}}{\mu_{t}^{\omega}}\\ &=&\frac1{\sigma_{t}}\left( \partial_{1}G(a_{t}^{\omega},a_{t-1}^{\omega}) + \frac{\mu_{t+1}^{\omega}}{\mu_{t}^{\omega}}\frac{Y^{\omega}_{t+1}}{Y^{\omega}_{t}} \partial_{2} G(a_{t+1}^{\omega},a_{t}^{\omega})\right.\\ &&\left.\qquad+\frac{\overline{\tau}^{\omega}_{t}-\underline{\tau}^{\omega}_{t}}{\mu_{t}^{\omega}Y^{\omega}_{t}}\right) \end{array} $$

(42)

For t=t _i +1, this formula is rewritten as:

$$\begin{array}{@{}rcl@{}} SCC^{\omega}_{t_{i}+1} =\frac1{\sigma_{t_{i}+1}}\bigg(\partial_{1}G(a_{t_{i}+1}^{\omega},\overline{a}_{t_{i}}) & +& \frac{\mu_{t_{i}+2}^{\omega}}{\mu_{t_{i}+1}^{\omega}}\frac{Y^{\omega}_{t_{i}+2}}{\overline{Y}_{t_{i}+1}} \partial_{2} G(a_{t_{i}+2}^{\omega},a_{t_{i}+1}^{\omega})\\ &+&\frac{\overline{\tau}^{\omega}_{t_{i}+1}-\underline{\tau}^{\omega}_{t_{i}+1}}{\mu_{t_{i}+1}^{\omega}\overline{Y}_{t_{i}+1}}\bigg) \end{array} $$

(43)

Before uncertainty is resolved, ∀t≤t _i the SCC is:

$$\begin{array}{@{}rcl@{}} SCC_{t} &=& \frac{\lambda_{A,t}}{\mathbb{E}[\mu^{\omega}_{t}]}\\ &=& \frac1{\sigma_{t}}\left(\partial_{1} G(\overline{a}_{t},\overline{a}_{t-1})\right.\\ &&\left. + \frac{\mathbb{E}[\mu_{t+1}^{\omega}\partial_{2}G(a_{t+1}^{\omega},\overline{a}_{t})]}{\mathbb{E}[\mu^{\omega}_{t}]} \frac{\overline{Y}_{t+1}}{\overline{Y}_{t}}+\frac{\overline{\tau}_{t}-\underline{\tau}_{t}}{\mathbb{E}[\mu^{\omega}_{t}]\overline{Y}_{t}}\right) \end{array} $$

(44)

For t≤t _i −1, the formula simplifies to:

$$\begin{array}{@{}rcl@{}} SCC_{t} &=& \frac{1}{\sigma_{t}}\left( \partial_{1}G(\overline{a}_{t},\overline{a}_{t-1})\right. \\ &&\left.+\frac{\mathbb{E}[\mu_{t+1}^{\omega}]}{\mathbb{E}[\mu^{\omega}_{t}]}\frac{\overline{Y}_{t+1}}{\overline{Y}_{t}} \partial_{2}G(\overline{a}_{t+1},\overline{a}_{t}) +\frac{\overline{\tau}_{t}-\underline{\tau}_{t}}{\mathbb{E}[\mu^{\omega}_{t}]\overline{Y}_{t}}\right) \end{array} $$

(45)

Comparing the formula in the uncertain case with the certain case, we can give an interpretation of the uncertain social cost in terms of the social cost if uncertainty had been resolved. The uncertain social cost corresponds to the mean of social cost in the different states of nature, averaged by utility in these states of nature (prices in different states of nature are not comparable, only utility units can be added, this is done by using the multiplier $\mu ^{\omega }_{t}$). So if we defined $SCC^{\omega }_{t}$ as previously (see the formulas after uncertainty is resolved), this interpretation of the social cost is tantamount to the formula: $SCC_{t}=\frac {\mathbb {E}[\mu ^{\omega }_{t}SCC^{\omega }_{t}]}{\mathbb {E}[\mu ^{\omega }_{t}]}$ before uncertainty is resolved.

1.3 B.3 Computation

To get the value of $SCC^{\omega }_{t} =\frac {\lambda _{A,t}^{\omega }}{\mu _{t}^{\omega }}$, we use the shadow prices associated to the concentration dynamics for $\lambda _{A,t}^{\omega }$ and to the capital dynamics for $\mu _{t}^{\omega }$ computed by GAMS.

Appendix C: Results: Distributions of Abatment and SCC According to Modeling Structures for Different Dates

Table 5 Boxplot results for abatement (% of potential emissions) according to modeling structures

Table 6 Boxplot results for SCC (/tCO₂) according to modeling structures