**Appendix B. Derivation of Implementability Constraint (37)**

From (36), the expectation of W*t*+<sup>1</sup> with respect to the distorted measure *mt*+1*πt*<sup>+</sup>1, valued at the price of an Arrow–Debreu security *b*, is

$$\begin{split} &\sum\_{\mathbf{x}\_{t+1}}\hat{p}\_{t+1}(\mathbf{x}\_{t+1}|\mathbf{x}^{t})\mathcal{W}\_{t+1}(\mathbf{x}^{t+1}) = \\ &\sum\_{\mathbf{x}\_{t+1}}\hat{p}\_{t+1}(\mathbf{x}\_{t+1}|\mathbf{x}^{t})[b\_{t+1}(\mathbf{x}^{t+1}) + \mathcal{R}\_{t+1}^{k}(\mathbf{x}^{t+1})k\_{t+1}(\mathbf{x}^{t})] \\ &+\sum\_{\mathbf{x}\_{t+1}}\hat{p}\_{t+1}(\mathbf{x}\_{t+1}|\mathbf{x}^{t})(p\_{t+1}^{\varepsilon}(\mathbf{x}^{t+1}) - \tau\_{t+1}^{\varepsilon}(\mathbf{x}^{t+1}))Q\_{t+1}(\mathbf{x}^{t}) \\ &= \sum\_{\mathbf{x}\_{t+1}}\hat{p}\_{t+1}(\mathbf{x}\_{t+1}|\mathbf{x}^{t})[b\_{t+1}(\mathbf{x}^{t+1}) + \mathcal{R}\_{t+1}^{k}(\mathbf{x}^{t+1})k\_{t+1}(\mathbf{x}^{t})] \\ &+\left[p\_{t}^{\varepsilon}(\mathbf{x}^{t}) - \tau\_{t}^{\varepsilon}(\mathbf{x}^{t})\right]\sum\_{\mathbf{x}\_{t+1}}\hat{p}\_{t+1}(\mathbf{x}\_{t+1}|\mathbf{x}^{t})R\_{t+1}^{k}(\mathbf{x}^{t+1})Q\_{t+1}(\mathbf{x}^{t}), \end{split}$$

where (25) and Hotelling's rule (31) are used to get the last line. Application of no-arbitrage condition (14) further simplifies the expression for household wealth:

$$\begin{aligned} \sum\_{\mathbf{x}\_{t+1}} \boldsymbol{\hat{p}}\_{t+1}(\mathbf{x}\_{t+1}|\mathbf{x}^{t}) \mathcal{W}\_{t+1}(\mathbf{x}^{t+1}) &= \sum\_{\mathbf{x}\_{t+1}} \left[ \boldsymbol{\hat{p}}\_{t+1}(\mathbf{x}\_{t+1}|\mathbf{x}^{t}) \boldsymbol{b}\_{t+1}(\mathbf{x}^{t+1}) + \boldsymbol{k}\_{t+1}(\mathbf{x}^{t}) \right] \\ &+ \quad [p\_t^{\varepsilon}(\mathbf{x}^{t}) - \boldsymbol{\tau}\_t^{\varepsilon}(\mathbf{x}^{t})] \boldsymbol{Q}\_{t+1}(\mathbf{x}^{t}), \end{aligned} \tag{A4}$$

indicating that the consumer's expected wealth in the following period includes the unused store of fossil fuel valued at its current after-tax price in addition to the capital stock and bonds carried over into the next period. Using the household's period budget constraint (4) and (A4), household wealth defined in (36) is, equivalently,

<sup>W</sup>*t*(*x<sup>t</sup>* ) = *bt*(*x<sup>t</sup>* ) + *R<sup>k</sup> <sup>t</sup> kt*(*xt*−1)+(*p<sup>e</sup> t*(*x<sup>t</sup>* ) <sup>−</sup> *<sup>τ</sup><sup>e</sup> <sup>t</sup>* (*x<sup>t</sup>* ))*Qt*(*xt*−1) = *ct*(*x<sup>t</sup>* ) <sup>−</sup> *Ht*(*x<sup>t</sup>* ) <sup>−</sup> *gt*(*x<sup>t</sup>* ) + *kt*+1(*x<sup>t</sup>* ) + ∑ *xt*+<sup>1</sup> *<sup>p</sup>*ˆ*t*+1(*xt*+1|*x<sup>t</sup>* )*bt*+1(*xt*+1)+(*p<sup>e</sup> t*(*x<sup>t</sup>* ) <sup>−</sup> *<sup>τ</sup><sup>e</sup> <sup>t</sup>* (*x<sup>t</sup>* ))*Qt*+1(*x<sup>t</sup>* ) <sup>≥</sup> *ct*(*x<sup>t</sup>* ) <sup>−</sup> *Ht*(*x<sup>t</sup>* ) <sup>−</sup> *gt*(*x<sup>t</sup>* ) + <sup>∑</sup>*xt*+<sup>1</sup> *<sup>p</sup>*ˆ*t*+1(*xt*+1|*x<sup>t</sup>* )W*t*+1(*xt*+1) = *ct*(*x<sup>t</sup>* ) <sup>−</sup> *Ht*(*x<sup>t</sup>* ) <sup>−</sup> *gt*(*x<sup>t</sup>* ) + <sup>∑</sup>*xt*+<sup>1</sup> ˆ*t*+1*πt*+1W*t*+1(*xt*+1) = *ct*(*x<sup>t</sup>* ) <sup>−</sup> *Ht*(*x<sup>t</sup>* ) <sup>−</sup> *gt*(*x<sup>t</sup>* ) + *<sup>β</sup>* <sup>∑</sup>*xt*+<sup>1</sup> *mt*+1*πt*+<sup>1</sup> *uct*<sup>+</sup><sup>1</sup> *uct* <sup>W</sup>*t*+1(*xt*+1). (A5)

#### **Appendix C. The Social Cost of Carbon**

This appendix evaluates the expectation (83) for the four belief regimes:

$$\begin{aligned} \omega\_t &= \left[ f(\Phi\_t, \varepsilon\_t) \mathbb{E}\_t \big[ n\_{t+1}^{\*\*} \Lambda\_{t+1} \big( 1 + \varepsilon\_{t+1} + [1 - \gamma + \gamma \theta\_{t+1}] n\_{t+1}^{\*} \Phi\_t \big) \big] \right], \end{aligned}$$

where <sup>Λ</sup>*t*+<sup>1</sup> <sup>=</sup> *t*+1[(<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*)*t*+<sup>1</sup> <sup>+</sup> *xt*+1*Yt*+1], <sup>E</sup>*t*Λ*t*+<sup>1</sup> <sup>=</sup> <sup>Λ</sup>*<sup>s</sup> <sup>t</sup>*, *<sup>f</sup>*(Φ*t*,*εt*) = <sup>1</sup> 1+*εt*+[1−*γ*+*γϑt*]Φ*<sup>t</sup>* <sup>≤</sup> *<sup>f</sup>*(Φ*t*, 0) <sup>&</sup>lt; *<sup>f</sup>*(0, 0) = 1, *<sup>ϑ</sup><sup>t</sup>* <sup>=</sup> *Ht*+*gt ct* <sup>∀</sup>*t*, and *<sup>ε</sup>t*+<sup>1</sup> <sup>&</sup>gt; 0 only if consumers are pessimistic.40 **A. Homogeneous beliefs**

**RE solution** (*n*<sup>∗</sup> = 1; *n*∗∗ = 1, *ε* = 0, Φ*<sup>t</sup>* = Φ¯ )

$$\begin{split} \boldsymbol{\sigma}\_{t}^{\rm RE} &= \quad f(\boldsymbol{\Phi},0)\mathbb{E}\_{t}[\Lambda\_{t+1}(1+(1-\gamma+\gamma\boldsymbol{\theta}\_{t+1})\boldsymbol{\Phi})] \\ &= \quad f(\boldsymbol{\Phi},0)\mathbb{E}\_{t}[1+(1-\gamma)\boldsymbol{\Phi}]\mathbb{E}\_{t}\Lambda\_{t+1}+\gamma\boldsymbol{\Phi}f(\boldsymbol{\Phi},0)\mathbb{E}\_{t}\Lambda\_{t+1}\boldsymbol{\theta}\_{t+1} \\ &= \quad f(\bar{\boldsymbol{\Phi}},0)[(1+\bar{\boldsymbol{\Phi}}[1-\gamma+\gamma\mathbb{E}\_{t}\boldsymbol{\theta}\_{t+1}])\Lambda\_{t}^{s}+\bar{\boldsymbol{\Phi}}\gamma\boldsymbol{cov}(\Lambda\_{\boldsymbol{\tau}}\boldsymbol{\theta})], \\ &= \quad \Lambda\_{t}^{s}+\gamma\boldsymbol{\Phi}f(\bar{\boldsymbol{\Phi}},0)[(\mathbb{E}\_{t}\boldsymbol{\theta}\_{t+1}-\boldsymbol{\theta}\_{t})\Lambda\_{t}^{s}+\boldsymbol{cov}(\Lambda\_{\boldsymbol{\tau}}\boldsymbol{\theta})], \quad \bar{\boldsymbol{\Phi}}>0, \ \forall t, \\ &= \quad \Lambda\_{t}^{s}, \ \boldsymbol{\Phi}=0 \ \forall t. \end{split} \tag{A6}$$
