*13.1. Preliminary Results*

Having derived formulas for the expected social cost of carbon, and carbon as well as capital taxation, we are now prepared to consider the policy implications for the various belief regimes, with particular focus on any premia or surcharges that may arise as a consequence of belief distortions and ambiguities. The results are summarized in six propositions. As will become apparent, the sign and size of surcharges on the carbon tax and those of the *ex ante* tax on capital will depend on the signs of several key second-order moments, namely covariances involving measures related to the state of the economy, *ς<sup>k</sup>* and *ς<sup>e</sup>* , the average propensity to consume out of disposable income *ϑ*, and measures of the intensity of robustness concerns reflected in the endogenous worst-case belief distortions *mc*, *mp*, *np*, *nPOL*, *nPAT* that arise from evaluating Formulas (87), (100), and (112) containing the expectations of products and ratios of random variables. The resulting expressions involve sums of expectations of their individual components and their covariances, known to be non-trivial because, as shown later, the assumption of endogeneity means that all involved variables are functions of the fundamental climate shock process *xt* and are therefore related to each other. The manner of deconstructing optimal tax policy applied in this paper is novel in the literature, which has mostly had to rely on simulations to quantify policy effects, and should be considered a unique contribution of this paper.

The signs of the covariances needed to evaluate expectations and to prove later propositions are not always clear from intuition and need to be formally derived and stated in the form of several lemmas.

**Assumption 2.** *dct*+<sup>1</sup> *dxt*+<sup>1</sup> ≤ 0*.*

It is generally assumed that consumption declines with global warming (see Frankhouser and Tol 2005; Weitzman 2009).

**Lemma 5.**

$$\frac{d\phi\_{t+j}}{dx\_{t+j}} > 0.$$

**Proof.**

$$\begin{split} \frac{d\boldsymbol{\varrho}\_{t+j}}{d\boldsymbol{x}\_{t+j}} &= \ \beta \mathcal{M}\_{t+j} \frac{d\frac{\boldsymbol{u}\_{\boldsymbol{c}\_{t+j}}}{\boldsymbol{u}\_{\boldsymbol{c}\_{t}}}}{d\boldsymbol{x}\_{t+j}} + \beta \frac{\boldsymbol{u}\_{\boldsymbol{c}\_{t+j}}}{\boldsymbol{u}\_{\boldsymbol{c}\_{t}}} \frac{d\boldsymbol{M}\_{t+j}}{d\boldsymbol{x}\_{t+j}} \\ &= \ \ -\beta \mathcal{M}\_{t+j} \left(\frac{\boldsymbol{c}\_{t}}{\boldsymbol{c}\_{t+j}}\right)^{\gamma} \left(\gamma \boldsymbol{c}\_{t+j}^{-1} \frac{d\boldsymbol{c}\_{t+j}}{d\boldsymbol{x}\_{t+j}}\right) \\ &\quad + \ \beta \left(\frac{\boldsymbol{c}\_{t}}{\boldsymbol{c}\_{t+j}}\right)^{\gamma} \frac{d\boldsymbol{M}\_{t+j}}{d\boldsymbol{x}\_{t+j}} > 0, \end{split}$$

where first terms in each line are positive by Assumption 2, and the second lines are also positive since *dMt*+*<sup>j</sup> dxt*+*<sup>j</sup>* > 0 if *M* = *Mc*, and zero otherwise.

**Assumption 3.** *dkt*+<sup>1</sup> *dxt*+<sup>1</sup> < <sup>0</sup>*.*

Li et al. (2016) assume that capital utilization declines upon a climate cost shock. Frankhouser and Tol (2005, p. 5) observe that "the overall effect of climate change on the accumulation of capital is in principle ambiguous", but that "it seems safe to speculate that the capital accumulation effect will probably be negative". More recent evidence that a significant portion (50%) of total GDP losses can be attributed to disincentives to invest capital is less ambiguous (see Willner et al. 2021).

**Assumption 4.** *dHt*+<sup>1</sup> *dxt*+<sup>1</sup> ≤ 0*.*

Labor input likely declines as a result of a climate cost shock since, as documented in Dasgupta et al. (2021); Kjellstrom (2014); Kjellstrom et al. (2009); Somanathan et al. (2018), productive labor is lost as a consequence of warming, justifying

**Assumption 5.** *dEt*+<sup>1</sup> *dxt*+<sup>1</sup> 0*.*

Wilbanks et al. (2008) find that climate warming reduces energy use and production, as verified by the BEA (BEA 2019), although, as an Environmental Protection Agency web page, deleted by the Trump Administration in 2017 but saved and still available on (EPA 2017), reports, climate warming may lead to partially offsetting heating and cooling demands.

**Lemma 6.**

$$\frac{dY\_{z\_{i,t}}}{dx\_t} = -(Q\_0 - Q\_t)D\_tF\_{z\_{i,t}} + \sum\_{j=1}^n \frac{\partial Y\_{z\_{i,t}}}{\partial z\_{j,t}} \frac{dz\_{j,t}}{dx\_t},$$

$$\approx \quad -(Q\_0 - Q\_t)D\_tF\_{z\_{i,t}} < 0,\tag{113}$$

*for all factors of production, k, E, and H.*

**Proof.** The signs of *dzj*,*<sup>t</sup> dxt* are given in Assumptions 3–5, and *∂Yzi*,*<sup>t</sup> <sup>∂</sup>zj*,*<sup>t</sup>* > 0. Recent literature (Dasgupta et al. 2021; Njuki et al. 2020; Zhang et al. 2017) suggests that the indicated sum is likely negative but small, leaving as main driver of climate cost shocks their effects on total factor productivity, here basically represented by the damage function.

**Assumption 6.** *dbt*+<sup>1</sup> *dxt*+<sup>1</sup> > <sup>0</sup>*.*

The effect of a climate shock on debt *bt* can plausibly be said to be positive. In their study of Columbia and Peru, Maldonado and Gallagher (2022) provide some evidence that climate shocks significantly affect public debt trajectories towards significantly higher levels and, in some cases, raise probabilities of increasing debt during climate stress. Nor do developed economies in Europe seem to be immune from this effect. For example, Zenios (2022) combined projections from the IMF World Economic Outlook with simulations of versions of an IAM model obtained from Emmerling et al. (2016) and Gazzotti et al. (2021) to show that climate shocks will raise the sovereign debt-to-GDP ratio in Italy and Cyprus over time.

**Lemma 7.**

$$\frac{dm\_{t+1}^{\epsilon}}{dx\_{t+1}} \ge 0.$$

**Proof.** From (6),

$$\frac{d m^{\varepsilon}\_{t+1}}{d \boldsymbol{x}\_{t+1}} = \frac{\partial m^{\varepsilon}\_{t+1}}{\partial \mathcal{U}\_{t+1}} \frac{d \mathcal{U}\_{t+1}}{d \boldsymbol{x}\_{t+1}} \quad = \quad \sigma^{\varepsilon} m^{\varepsilon}\_{t+1} (1 - m^{\varepsilon}\_{t+1}) \frac{d \mathcal{U}\_{t+1}}{d \boldsymbol{x}\_{t+1}}.$$

The term multiplying *<sup>d</sup>*U*t*+<sup>1</sup> *dxt*+<sup>1</sup> is negative, indicating that decreases in <sup>U</sup>*t*+<sup>1</sup> raise *<sup>m</sup><sup>p</sup> t*+1 toward 1. The envelope conditions (10)–(12) imply

$$\begin{aligned} \frac{d\mathcal{U}\_{t+1}}{d\mathbf{x}\_{t+1}} &= \mathcal{U}\_{k\_{t+1}} \frac{d\mathbf{k}\_{t+1}}{d\mathbf{x}\_{t+1}} + \mathcal{U}\_{b\_{t+1}} \frac{d\mathbf{b}\_{t+1}}{d\mathbf{x}\_{t+1}} + \mathcal{U}\_{Q\_{t+1}} \frac{dQ\_{t+1}}{d\mathbf{x}\_{t+1}} \\ &= \quad \mu\_{c\_{t+1}} [\mathcal{R}\_{t+1}^{k} \frac{d\mathbf{x}\_{t+1}}{d\mathbf{x}\_{t+1}} + \frac{d\mathbf{b}\_{t+1}}{d\mathbf{x}\_{t+1}}] \le 0, \end{aligned}$$

if the negative effect on capital outweighs the presumably positive effect on debt. Otherwise, the result follows from *<sup>σ</sup><sup>c</sup>* < 0 and *<sup>m</sup><sup>c</sup>* ≤ 1.33

**Lemma 8.**

$$\frac{dn\_{t+1}^{PO}}{d\mathfrak{x}\_{t+1}} \;>\; \quad 0.$$

**Proof.** Evaluate

*dnPO t*+1 *dxt*+<sup>1</sup> <sup>=</sup> *<sup>∂</sup>nPO t*+1 *∂*V*t*+<sup>1</sup> *d*V*t*+<sup>1</sup> *dxt*+<sup>1</sup> + *∂nPO t*+1 *∂*V*t*+<sup>1</sup> *d*Υ*t*+<sup>1</sup> *dxt*+<sup>1</sup> = *σnPO <sup>t</sup>*+1(<sup>1</sup> <sup>−</sup> *<sup>n</sup>PO <sup>t</sup>*+1) V*kt*+<sup>1</sup> *dkt*+<sup>1</sup> *dxt*+<sup>1</sup> + VΥ*t*+<sup>1</sup> *d*Υ*t*+<sup>1</sup> *dxt*+<sup>1</sup> <sup>+</sup> <sup>Φ</sup>¯ *<sup>d</sup>*Υ*t*+<sup>1</sup> *dxt*+<sup>1</sup> = *σnPO <sup>t</sup>*+1(<sup>1</sup> <sup>−</sup> *<sup>n</sup>PO <sup>t</sup>*+1) (<sup>1</sup> <sup>−</sup> *<sup>δ</sup>* <sup>+</sup> *Ykt*<sup>+</sup><sup>1</sup> )(*uct*<sup>+</sup><sup>1</sup> <sup>+</sup> <sup>Ω</sup>*ct*+1Φ¯ ) *dkt*+<sup>1</sup> *dxt*+<sup>1</sup> + *σnPO <sup>t</sup>*+1(<sup>1</sup> <sup>−</sup> *<sup>n</sup>PO <sup>t</sup>*+1) Φ*t d*Υ*t*+<sup>1</sup> *dxt*+<sup>1</sup> − Φ*t*+<sup>1</sup> *d*Υ*t*+<sup>1</sup> *dxt*+<sup>1</sup> = *σnPO <sup>t</sup>*+1(<sup>1</sup> <sup>−</sup> *<sup>n</sup>PO <sup>t</sup>*+1) (<sup>1</sup> <sup>−</sup> *<sup>δ</sup>* <sup>+</sup> *Ykt*<sup>+</sup><sup>1</sup> )(*uct*<sup>+</sup><sup>1</sup> <sup>+</sup> <sup>Ω</sup>*ct*+1Φ¯ ) *dkt*+<sup>1</sup> *dxt*+<sup>1</sup> > 0,

which follows from *<sup>σ</sup>* < 0, *<sup>n</sup>PO* ≤ 1, envelope conditions (52) and (54), Euler condition (50) with *ξ* = 0, from Assumption 3. 34, and from the previous result that Ω*ct*+<sup>1</sup> > 0.<sup>35</sup>

The preceding result accords with the properties 0 < *nPO <sup>t</sup>*+<sup>1</sup> <sup>≤</sup> 1, lim*σ*↑<sup>0</sup> *<sup>n</sup>PO <sup>t</sup>*+<sup>1</sup> → 1, lim*σ*↓−<sup>∞</sup> *<sup>n</sup>PO <sup>t</sup>*+<sup>1</sup> → 0, and the partial derivative

$$\frac{\partial n\_{t+1}^{PO}}{\partial \sigma} = \left(\mathcal{V}\_{t+1} + \Phi\_t \mathcal{Y}\_{t+1}\right) (1 - n\_{t+1}^{PO}) n\_{t+1}^{PO} \ge 0. \tag{114}$$

As one might expect intuitively, increased ambiguity aversion (*<sup>σ</sup>* ↓) reduces *<sup>n</sup>PO* and drives it towards zero, a result that echoes those in Gilboa and Schmeidler (1989) and Millner et al. (2012).

**Lemma 9.**

$$\frac{d n\_{t+1}^{PA}}{d \mathfrak{x}\_{t+1}} \;>\; \quad 0.$$

**Proof.** From (39) and results (41),

$$\begin{split} \frac{d n\_{t+1}^{PA}}{d \mathfrak{x}\_{t+1}} &= \ \frac{\partial n\_{t+1}^{PA}}{\partial \mathbf{Y}\_{t+1}} \frac{d \mathbf{Y}\_{t+1}}{d \mathfrak{x}\_{t+1}} = \sigma \Phi\_t n\_{t+1}^{PA} (1 - n\_{t+1}^{PA}) \frac{d \mathbf{Y}\_{t+1}}{d \mathfrak{x}\_{t+1}} \\ &= \ \sigma \Phi\_t n\_{t+1}^{PA} (1 - n\_{t+1}^{PA}) \left( \Omega\_{\mathfrak{c}\_{t+1}} \frac{d \mathbf{c}\_{t+1}}{d \mathfrak{x}\_{t+1}} \right) \ge 0. \end{split}$$

Given *σ* < 0 and Assumption 2, this implies a positive correlation between *nPA* and *x*.

By the preceding arguments,

**Lemma 10.**

$$\frac{\partial n\_{t+1}^p}{\partial \mathbf{x}\_{t+1}} \ge 0.$$

**Proof.** The proof follows argument similar to the proof of Lemma 10.

The preceding discussion leads up to the following two lemmas:

**Lemma 11.** *Given definition (109),*

$$\frac{d\zeta\_{t+1}^k}{dx\_{t+1}} \stackrel{>}{\leftharpoons} 0.$$

**Proof.** Based on Lemma 5 and the result in (113),

$$\begin{aligned} \frac{d\varrho\_{t+1}^{\vec{k}}}{d\mathfrak{x}\_{t+1}} &= \frac{d\varrho\_{t+1}(1-\delta+\mathcal{Y}\_{\mathbb{k}\_{t+1}})}{d\mathfrak{x}\_{t+1}}\\ &= \quad (1-\delta+\mathcal{Y}\_{\mathbb{k}\_{t+1}})\frac{d\varrho\_{t+1}}{d\mathfrak{x}\_{t+1}}+\mathfrak{e}\_{t+1}\frac{d\mathcal{Y}\_{\mathbb{k}\_{t+1}}}{d\mathfrak{x}\_{t+1}}>0. \end{aligned}$$

**Lemma 12.**

$$\frac{d\varsigma\_{t+j}^c}{d\mathbf{x}\_{t+j}} > 0 \; \forall j > 0.$$

**Proof.** Use Lemma 5 and the result in (113) and note that, from Formula (98), the sign of *dς<sup>e</sup> t*+*j dxt*+*<sup>j</sup>* is the same as the sign of

$$\frac{d\mathfrak{d}\_{t+j}(Y\_{E\_{t+j}} - \varpi\_{t+j})}{d\mathbf{x}\_{t+j}} = (Y\_{E\_{t+j}} - \varpi\_{t+j})\frac{d\mathfrak{d}\_{t+j}}{d\mathbf{x}\_{t+j}} + \mathfrak{d}\_{t+j}\frac{dY\_{E\_{t+j}}}{d\mathbf{x}\_{t+j}} > 0.$$

**Lemma 13.** *<sup>d</sup>*Λ*t*+*<sup>j</sup> dxt*+*<sup>j</sup>* ≥ 0*.*

**Proof.** This follows from its definition in (85).
