*13.2. Homogeneous Beliefs*

**Proposition 1** (**Benchmark rational expectations**)**.** *If (i) beliefs are homogeneous and rational, and (ii) labor income plus tax rebates as a proportion of consumption do not decrease over time in all periods, then*

*1. The social cost of carbon exceeds the standard formulation* Λ*<sup>s</sup> <sup>t</sup> derived in (88) for a social planner, unless* Φ¯ = 0*,*

$$\begin{split} \boldsymbol{\varpi}\_{t}^{RE} &= \, \, \Lambda\_{t}^{s} + \gamma \Phi f(\Phi, 0) [(\mathbb{E}\_{t} \theta\_{t+1} - \theta\_{t}) \Lambda\_{t}^{s} + \text{cov}(\Lambda\_{\prime} \theta)] \geq \, \Lambda\_{t\prime}^{s} \, \, \Phi > 0, \, \forall t, \\ &= \, \, \, \Lambda\_{t\prime}^{s} \, \, \Phi\_{t} = 0 \, \forall t. \end{split}$$

*2. The carbon tax premium is positive, unless* Φ = 0*,*

$$\begin{aligned} \mathbb{E}\_t \Sigma\_{t+j}^{c-RE} &= \gamma \Phi f(\Phi, 0) \left[ \mathbb{E}\_t \theta\_{t+j} - \theta\_t + \text{cov}(\varrho^c, \theta) \right] > 0, \ \Phi > 0, \ \forall t \\ &= \ 0, \ \Phi = 0 \ \forall t. \end{aligned}$$

*3. The ex ante rate on capital is negative, unless* Φ = 0*,*

$$\begin{aligned} \mathbb{E}\_t \Sigma\_{t+1}^{k-RE} &= -\gamma \Phi f(\Phi, 0) \Big[ \mathbb{E}\_t \theta\_{t+1} - \theta\_t + \text{cov}(\boldsymbol{\xi}^k, \boldsymbol{\theta}) \Big] < 0, \ \Phi > 0, \forall t \\ &= \quad 0, \ \Phi = 0 \,\forall t. \end{aligned}$$

This proposition sets up a background and baseline for comparison with the conclusions for the remaining belief regimes. It also establishes an important distinction between social and Ramsey planning. For a social planner, defined as one for whom the implementability constraint (39) is not binding (Φ = 0)—nor invoked in most of the literature—the policy settings coincide with the formulas derived in Golosov et al. (2014) for the optimal carbon tax and the social cost of carbon, and with the optimal tax on capital derived by Zhu (1992), Chari et al. (1994), and Atkeson et al. (1999). By contrast, a true Ramsey planner, being mindful of preferences and household budgets, adjusts all calculations by factors involving the inter-temporal rate of substitution and the average propensity to consume.

### *13.3. Heterogeneous Beliefs: Skeptical Consumers*

The next proposition establishes that even without ambiguity, skepticism alters optimal policy.

**Proposition 2** (**No ambiguity**)**.** *By producing incentives to spend more on fossil energy and less on capital than is socially optimal, the mere presence of skepticism is an inducement to the Ramsey planner but not a social planner to raise the carbon tax and to subsidize capital, where*

*1. Social cost of carbon is higher than under RE, unless* Φ = 0*:*

$$\begin{aligned} \label{eq:H} \mathcal{O}\_t &=& \mathcal{O}\_t^{RE} > \Lambda\_t^s \; \Phi\_t > 0, \; \forall t, \\ &=& \Lambda\_t^s \; \Phi\_t = 0; \end{aligned}$$

*2. The carbon tax is higher than under RE, unless* Φ = 0*:*

$$\begin{array}{rclcrcl} \mathbb{E}\_t \Xi^c\_{t+j} & \geq & \mathbb{E}\_t \Xi^{c-RE}\_{t+j} \geq 0, \ \Phi\_t > 0 \ \forall t\\ &=& 0, \ \Phi\_t = 0; \end{array}$$

*3. Ex ante capital tax rate is less than under RE, unless* Φ = 0*:*

$$\begin{array}{rclcrcl} \mathbb{E}\_t \Xi\_{t+1}^k & \leq & \mathbb{E}\_t \Xi\_{t+1}^{k-RE} \leq 0, & \Phi\_t > 0 \ \forall t\\ &=& 0, \ \Phi\_t = 0. \end{array}$$

This proposition highlights an intriguing point: having little faith in climate science, climate skeptics might naturally want to pay a lower carbon tax. Yet, with manifestly poetic justice, the very consequence of skepticism by itself is an increase in both the social cost of carbon and the carbon tax, and a decrease in the tax on capital.

The next two propositions provide results for two regimes in which the Ramsey planner faces a climate-skeptical public whose beliefs are not known and are, indeed unknowable. The first regime is political in that the planner believes the unknown beliefs of the private sector to be true. In the second regime, the planner is paternalistic in that it believes the science model to be true.

**Proposition 3** (**Political planner).** *Ignorance of private beliefs believed to be true leads to the following policy alterations:*

*1. The social cost of carbon contains an ambiguity premium for both Ramsey and social plans:*

$$\begin{aligned} \label{eq:SDAC-} \phi\_t &=& \phi\_t^{RE} + \Phi f(\Phi, 0) \Big[ (1 - \gamma) \text{cov}(n^{PO}, \Lambda) + \gamma \text{cov}(n^{PO}, \Lambda \theta) \Big] \\ &+& f(\Phi, 0) \text{cov}(n^{PO}, \Lambda) \ge \phi\_t^{RE}, \,\forall t, \\ &=& \Lambda\_t^s + \text{cov}(n^{PO}, \Lambda) \ge \Lambda\_{t, \prime}^s \Phi = 0; \end{aligned}$$

*2. The carbon tax is higher than under RE, unless* Φ = 0*:*

$$\begin{array}{rcl} \mathbb{E}\_t \Xi^{\epsilon - P0}\_{t+j} & \geq & \mathbb{E}\_t \Xi^{\epsilon - RE}\_{t+j} \geq 0, \ \Phi\_t > 0 \ \forall t, \\\ & = & 0, \ \Phi\_t = 0; \end{array}$$

*3. Ex ante capital tax is lower than under RE, unless* Φ = 0*:*

$$\begin{array}{rcl} \mathbb{E}\_t \Xi\_{t+1}^{k-PO} & \leq & \mathbb{E}\_t \Xi\_{t+1}^{k-RE} \leq 0, \ \Phi > 0 \ \forall t\\ & = & 0, \ \Phi = 0. \end{array}$$

The preceding proposition establishes that a political planner's ignorance about private beliefs, even if held to be correct, justifies a positive ambiguity premium for the social cost of carbon, activated by correlations between the planner's worst-case belief multiplier *nPO* and the social cost of carbon that would apply if rational expectations prevailed. In addition, unless the government is a social planner, ambiguity raises the carbon tax and lowers the capital tax.

**Proposition 4** (**Paternalistic planner).** *Ignorance of private beliefs that the planner also believes to be false leads to the following policy alterations:*

*1. The social cost of carbon contains an ambiguity premium, unless* Φ = 0*:*

$$\begin{aligned} \mathcal{O}\_t &=& \mathcal{O}\_t^{RE} + \Phi\_t f(\Phi\_t, 0) \left[ (1 - \gamma) \text{cov}(n^{PA}, \Lambda) + \gamma \text{cov}(n^{PA}, \Lambda \theta) \right] \\ &\ge& \mathcal{O}\_t^{RE} \, \Phi\_t > 0, \forall t\_\prime \\ &=& \Lambda\_t^s \, \Phi\_t = 0; \end{aligned}$$

*2. The carbon tax contains an ambiguity premium, unless* Φ = 0*:*

$$\begin{split} \mathbb{E}\_t \Xi\_{t+j}^{c-PA} &\geq \quad \mathbb{E}\_t \Xi\_{t+j}^{c-RE} + \gamma \Phi\_t f(\Phi\_t, 0) \text{cov}(n^{PA}, \zeta^c \theta) \geq \mathbb{E}\_t \Xi\_{t+1}^{c-RE} \\ &= \quad \geq 0, \ \Phi\_t > 0 \ \forall t, \\ &= \quad 0, \ \Phi\_t = 0; \end{split}$$

*3. The ex ante capital tax contains an ambiguity subsidy, unless* Φ = 0*:*

$$\begin{aligned} \mathbb{E}\_t \Xi\_{t+1}^{k-PA} &\leq \quad \mathbb{E}\_t \Xi\_{t+1}^{k-RE} - \Phi\_t f(\Phi\_t, 0) \left[ 1 + \gamma \text{cov}(n^{PA}, \xi^k) + \gamma \text{cov}(n^{PA}, \xi^k \theta) \right] \\ &\leq \quad \mathbb{E}\_t \Xi\_{t+1}^{k-RE} \leq 0, \quad \Phi\_t > 0 \; \forall t \; \\ &= \quad 0, \; \Phi\_t = 0. \end{aligned}$$

The main distinction between a political and a paternalistic planner, as defined in this paper, is that for the latter, an ambiguity premium for the social cost of carbon applies only if the government is a Ramsey planner and not a social planner. For both types of Ramsey planner, a positive ambiguity premium on the social cost of carbon is optimal because the planner's worst-case martingale belief distortion correlates with the certainty-equivalent version of the social cost of carbon.

The next proposition introduces the possibility of the government itself having doubts about the model.

**Proposition 5** (**Pessimistic planner—skeptical consumer).** *Ignorance of private beliefs that the government believes to be false, in combination with a planner's doubts about the model, leads to the following policy alterations:*

*1. The social cost of carbon contains an ambiguity premium in both Ramsey and social plans:*

$$\begin{array}{rl} \mathcal{O}\_{t} & \geq & \mathcal{O}\_{t}^{RE} + f(\Phi\_{t}, 0) cov(n^{p}, \Lambda) \\ & + & f(\Phi\_{t}, 0) \Phi\_{t}[(1 - \gamma) cov(m^{p}, \Lambda) + \gamma cov(m^{p}, \vartheta \Lambda)] \geq \mathcal{O}\_{t}^{RE}, \ \Phi\_{t} > 0 \\ & \geq & \mathcal{O}\_{t}^{RE} + cov(n^{p}, \Lambda) \geq \mathcal{O}\_{t}^{RE}, \ \Phi\_{t} = 0 \ \forall t. \end{array}$$

*2. The carbon tax contains a positive or negative ambiguity premium, unless* Φ = 0*:*

$$\begin{split} \mathbb{E}\_{t} \Xi\_{t+j}^{\varepsilon} &\geq \quad \mathbb{E}\_{t} \Xi\_{t+j}^{\varepsilon-RE} + (1-\gamma) \Phi\_{t} f(\Phi\_{t}, 0) \left[ \operatorname{cov}(m^{p}, \frac{1}{n^{p}}) + \operatorname{cov}(\boldsymbol{\xi}^{\varepsilon}, \frac{m^{p}}{n^{p}}) \right] \\ &+ \quad \gamma \Phi\_{t} f(\Phi\_{t}, 0) \left[ \operatorname{cov}(\boldsymbol{\theta}, \boldsymbol{\xi}^{\varepsilon}) + \operatorname{cov}(m^{p}, \frac{1}{n^{p}}) + \operatorname{cov}(\boldsymbol{\xi}^{\varepsilon} \boldsymbol{\theta}, \frac{m^{p}}{n^{p}}) \right] \\ &\overset{\geq}{\leq} \quad \Xi\_{t+j}^{\varepsilon-RE}, \quad \boldsymbol{\Phi}\_{t} > 0 \; \forall t \\ &= \quad 0, \; \Phi\_{t} = 0. \end{split}$$

*3. The ex ante capital tax is a subsidy in both Ramsey and social plans:*

$$\begin{split} \mathbb{E}\_{t} \Sigma\_{t+1}^{k-R-s} &\leq \quad \mathbb{E}\_{t} \Sigma\_{t+1}^{k-RE} - f(\Phi\_{t}, 0)[cov(\boldsymbol{\xi}^{k}, \boldsymbol{n}^{p}) + (1-\gamma)cov(\boldsymbol{\xi}^{k}, \boldsymbol{m}^{p})] \\ &- \quad \gamma \Phi\_{t} f(\Phi\_{t}, 0)cov(\boldsymbol{\theta}, \boldsymbol{\zeta}^{k} \boldsymbol{m}^{p})] \leq \mathbb{E}\_{t} \Sigma\_{t+1}^{k-RE} \leq 0, \ \Phi\_{t} \geq 0, \\ &= \quad -cov(\boldsymbol{\zeta}^{k}, \boldsymbol{n}^{p}) - (1-\gamma)cov(\boldsymbol{\zeta}^{k}, \boldsymbol{m}^{p}) \leq 0, \ \Phi\_{t} = 0. \end{split}$$

Consistent with Propositions 3 and 4, the marginal contribution of an increased correlation between the government's ignorance of skeptical private beliefs *mpπ* with the normalized discounted excess return to fossil energy over the social cost of carbon *ς<sup>e</sup>* or the discounted gross return to capital *ςk*, is to raise both the social cost of carbon and the carbon tax, and to lower the *ex ante tax* on capital. The marginal contribution of correlations involving the planner's own model doubts is to raise the social cost of carbon and to lower both the carbon tax and the *ex ante* tax on capital.

In the next and final proposition, the government remains pessimistic, but consumers are pessimistic rather than skeptical.

#### *13.4. Heterogeneous Beliefs: Pessimistic Consumers*

**Proposition 6** (**Pessimistic planner—pessimistic consumer).** *The combination of pessimism in the private sector and the government's pessimism has the following implications:*

*1. The social cost of carbon contains an ambiguity premium in both Ramsey and social plans:*

$$\begin{array}{rl} \boldsymbol{\omega}\_{t} & \geq & [1 - f(\boldsymbol{\Phi}\_{t\prime}\boldsymbol{\varepsilon}\_{t})\boldsymbol{\varepsilon}\_{t}] \boldsymbol{\sigma}\_{t}^{RE} + f(\boldsymbol{\Phi}\_{t\prime}\boldsymbol{\varepsilon}\_{t})[\boldsymbol{\varepsilon}\_{t} + \boldsymbol{\varepsilon}\boldsymbol{\upsilon}(\boldsymbol{n}^{p}, \boldsymbol{\Lambda})] \\ & + & f(\boldsymbol{\Phi}\_{t\prime}\boldsymbol{\varepsilon}\_{t})\boldsymbol{\Phi}\_{t}[(1 - \boldsymbol{\gamma})\boldsymbol{\varepsilon}\boldsymbol{\upsilon}(\boldsymbol{m}^{c}, \boldsymbol{\Lambda}) + \boldsymbol{\gamma}\boldsymbol{\upsilon}\boldsymbol{\upsilon}(\boldsymbol{m}^{c}, \boldsymbol{\theta}\boldsymbol{\Lambda})] \\ & \geq & \boldsymbol{\alpha}\_{t}^{RE}, \ \boldsymbol{\Phi}\_{t} > 0, \\ & \geq & \frac{1}{1 + \boldsymbol{\varepsilon}\_{t}}[\boldsymbol{\alpha}\_{t}^{RE} + \boldsymbol{\varepsilon}\_{t} + \boldsymbol{\upsilon}\boldsymbol{\upsilon}(\boldsymbol{n}^{p}, \boldsymbol{\Lambda})] > \boldsymbol{\Lambda}\_{t}^{s}, \ \boldsymbol{\Phi}\_{t} = 0, \ \forall t. \end{array}$$

*2. If* Φ > 0*, the premium on the carbon tax may be positive or negative, and is negative otherwise:*

$$\begin{split} \mathbb{E}\_{t} \Sigma\_{t+j}^{\varepsilon} &\geq \quad \mathbb{E}\_{t} \Sigma\_{t+j}^{\varepsilon-RE} \\ &\quad - \, f(\Phi\_{t}, \varepsilon\_{t}) f(\Phi\_{t}, 0) [1 + \Phi\_{t}(1 - \gamma + \gamma \mathbb{E}\_{t} \theta\_{t+1} + \gamma \text{cov}(\boldsymbol{\xi}^{\varepsilon}, \boldsymbol{\theta}))] \varepsilon\_{t} \\ &\quad + \, f(\Phi\_{t}, \varepsilon\_{t}) \Big[ \varepsilon\_{t} + \operatorname{cov}(\boldsymbol{\xi}^{\varepsilon}, \frac{1}{n\varepsilon^{\varepsilon}}) + \Phi\_{t} [(1 - \gamma) \operatorname{cov}(\boldsymbol{\xi}^{\varepsilon}, \frac{1}{n^{\mathbb{P}}}) + \gamma \operatorname{cov}(\boldsymbol{\theta}\_{\mathbb{S}}^{\varepsilon}, \frac{1}{n^{\mathbb{P}}})] \Big] \\ &\overset{\leq}{\geq} \, \, \mathbb{E}\_{t} \Sigma\_{t+j}^{\varepsilon-RE}, \, \Phi\_{t} > 0, \, \forall t \\ &\quad = \, \, \frac{1}{1 + \varepsilon\_{t}} \operatorname{cov}(\boldsymbol{\xi}^{\varepsilon}, \frac{1}{n^{\mathbb{P}}}) < 0, \, \: \Phi\_{t} = 0. \,\, \forall t \end{split}$$

*3. The ex ante capital tax rate may be positive or negative:*

$$\begin{split} \mathbb{E}\_{t} \mathbb{E}\_{t+1}^{k-R-p} &\leq \quad \mathbb{E}\_{t} \mathbb{E}\_{t+1}^{k-RE} \\ &\quad - f(\Phi\_{t}, \varepsilon\_{t}) f(\Phi\_{t}, 0) \Big[1 + \Phi\_{t} \Big(1 - \gamma + \gamma [\mathbb{E}\_{t} \Phi\_{t+1} + \operatorname{cov}(\boldsymbol{\varsigma}^{k}\_{\boldsymbol{\varsigma}}, \boldsymbol{\theta})] \Big) \Big] \mathbb{1} \\ &\quad - f(\Phi\_{t}, \varepsilon\_{t}) \Big[\operatorname{cov}(\boldsymbol{\varsigma}^{k}, \frac{n^{p}}{m^{\boldsymbol{\varsigma}}}) + \operatorname{cov}(\frac{1}{m^{\boldsymbol{\varsigma}}}, n^{p}) + (1 + \varepsilon\_{t}) \operatorname{cov}(\boldsymbol{\varsigma}^{k}\_{\boldsymbol{\varsigma}}, n^{p}) \Big] \\ \overset{\leq}{\geq} &\quad \mathbb{E}\_{t} \mathbb{E}\_{t+1}^{k-RE}, \quad \Phi\_{t} > 0 \\ &= \quad - \frac{1}{1 + \varepsilon\_{t}} \Big[\operatorname{cov}(\boldsymbol{\varsigma}^{k}, \frac{n^{p}}{m^{\boldsymbol{\varsigma}}}) + \operatorname{cov}(\frac{1}{m^{\boldsymbol{\varsigma}}}, n^{p}) \Big] \\ &\quad - \operatorname{cov}(\boldsymbol{\varsigma}^{k}, n^{p}) \overset{\leq}{\geq} 0, \quad \Phi\_{t} = 0. \end{split}$$

As in Proposition 5, which concerned skeptical beliefs, in this belief regime, the marginal effect of private-sector pessimism *m<sup>c</sup>* is to raise the social cost of carbon. However, the effect on carbon and capital taxation is the opposite, producing a reduction in the carbon tax and an increase in the tax on capital. The intuition is that increased consumer doubts about the climate model tend to motivate carbon consumption below the socially optimal level and to increase capital spending above its socially optimal level. The net effect of an increase in *<sup>t</sup>* and a decrease in E*t*Ξ*<sup>e</sup> <sup>t</sup>*+*<sup>j</sup>* may or may not in the end produce a lower carbon tax itself, because from (95) is the sum of two terms: *<sup>t</sup>* + *χt*, where *χ* is the sum of terms that contain expected future values of Ξ*<sup>e</sup> t*+*j* .

Proposition 6 echoes Proposition 5 in that the marginal contribution of any correlation between the planner's own model doubts *n<sup>p</sup>* and asset returns represented by *ς<sup>k</sup>* and *ς<sup>e</sup>* is to likewise raise the social cost of carbon and to lower both the carbon tax and the *ex ante* tax on capital. This becomes apparent if the role of private beliefs is de-activated by setting *<sup>m</sup><sup>p</sup>* ≡ 1 or *<sup>m</sup><sup>c</sup>* ≡ 1, respectively, leading to

$$\begin{array}{rcl}\Box\_{\mathbb{H}} &=& \Box\_{t}^{RE} + f(\Phi\_{t}, 0)cov(n^{p}, \Lambda), \\ \mathbb{E}\_{t}\Xi\_{t+j}^{\varepsilon} &\geq& \mathbb{E}\_{t}\Xi\_{t+j}^{\varepsilon-RE} \end{array} \tag{115}$$

$$+\quad f(\Phi\_t, 0)\Phi\_t \left[ (1 - \gamma) \text{cov}(\boldsymbol{\xi}^{\varepsilon}, \frac{1}{n^p}) + \gamma \text{cov}(\boldsymbol{\theta} \boldsymbol{\xi}^{\varepsilon}, \frac{1}{n^p}) \right] ,\tag{116}$$

$$\mathbb{E}\_t \Xi\_{t+1}^{k-R-s} \quad \le \quad \mathbb{E}\_t \Xi\_{t+1}^{k-RE} - f(\Phi\_t, 0) \text{cov}(\mathfrak{g}^k, n^p). \tag{117}$$

The belief regimes in the three preceding formulas, representing policies stripped of any effects due to belief distortions in the private sector—skeptical or pessimistic—are most closely related to the extant literature on robust climate policy and so serve best for comparisons, to which I now turn.

In a policy regime most similar to that treated in Proposition 5, with *m<sup>p</sup>* set equal to 1 but differing in some details, Hennlock (2009) attributes doubts about the model not to a policy authority *per se* but to a utility-maximizing representative consumer representing society who computes robust feedback rules that, as in (115), generate an ambiguity

premium on the expected social cost of carbon, so that with nonlinear damage, policy becomes more responsive to changes in climate. Li et al. (2016) study a dynamic optimization problem that is nearly identical to the planning models in Section 7, but under the assumption that the government is a social planner (Φ = 0 in the present paper) and not a Ramsey planner. They find that even a relatively small increase in the concern about model uncertainty can cause a significant drop in optimal energy extraction and a rise in the socially optimal carbon tax, which in this paper would also correspond to the result in (115). Likewise, Cai and Lontzek (2019) find that with empirically plausible parameterizations of Epstein-Zin preferences to represent attitudes towards risk, the uncertainty associated with anthropogenic climate change implies carbon taxes much higher than associated with deterministic models, while Lemoine and Traeger (2016) conclude that a government's aversion to Knightian uncertainty in the face of an ambiguous tipping point increases the optimal tax on carbon dioxide emissions, but only by a small amount. As with Li et al. (2016), since the derived tax effects come via changes in the SCC, those two conclusions also best correspond to formula (115). Heal and Millner (2013) do not consider taxation *per se*, but find that the value of abatement (that would presumably include carbon taxation) rises as ambiguity aversion increases. Rezai and van der Ploeg (2017) take a somewhat different approach to modeling ambiguity and consider a so-called agnostic policy authority—essentially a government having ambiguity about the approximating climate model—as facing potential models ranging from denialist to scientific. If such a government pursues max-max policies, it imposes higher carbon taxes as a precaution. Later, with the help of Nordhaus's (1993) DICE model to simulate carbon taxation, Rezai and van der Ploeg (2019) broaden their earlier results and consider an agnostic planner who adopts Pascal's Wager (Pascal's (1670)) 37 with the question: what would such an agnostic but rational planner—one who does not know or care which model is correct but who wishes to avoid the worst—do when faced with some probability that the approximating model, adhered to by so-called deniers, is false? Their conclusion is that the *hedge-your-bet* optimal carbon tax is quite close to the optimal tax derived in a non-denialist scientific setting, even if the probability of the model being false is a mere ten percent. Further, when ambiguity about whether scientists or deniers are correct rises, as represented by a parameter of constant relative ambiguity aversion, the optimal carbon pricing policy moves ever closer to the science-based policy.

Finally, Anderson et al. (2013), who study a model similar in spirit to the models in this paper, except for an additional robustness channel capable of affecting growth, find that a planner's increased deep uncertainty about the model can result in either a decrease or an increase in the optimal carbon tax, depending on other factors, such as market features and social preferences. In the present paper, the conclusions are driven by similar forces involving preferences and market features, but in the form of second-order moments represented by the covariances between worst-case martingale belief multipliers *m* and relevant market features, such as net asset-returns to fossil energy and capital.

Table 2 summarizes the preceding propositions. A principal leitmotif of this paper is belief heterogeneity, a main driver of ambiguities in all regimes studied here. Even if (as a mental, albeit unrealistic exercise in Proposition 2) one were to assume away ambiguity, the very presence of heterogeneity in beliefs between the private sector and the government, leading to a positive spread between their respective discount factors, is sufficient to alter the policies of a Ramsey planner, though not those of a social planner. As a consequence, the government increases its estimate of the social cost of carbon and adds a premium to the carbon tax, while raising the capital subsidy rate.

A government with its own doubts about the model is compelled to raise the social cost of carbon and acquires further motives to either raise or lower taxes, depending on the case. However, in all instances, the planner will raise the social cost of carbon. The political and paternalistic planners of Propositions 3 and 4 face ambiguity because of their ignorance of arbitrary private beliefs held by consumers who regard the scientific model with skepticism and discount the future at relatively higher rates. They therefore use relatively more fossil energy and invest in relatively less capital than is socially optimal. To encourage socially optimal choices, the government taxes carbon and subsidizes capital.

The pessimistic planners in Propositions 5 and 6 are identical except with respect to the kind of beliefs they face. The former operates under ambiguities that result from doubts about the model itself and from its ignorance of private beliefs. The latter government confronts a single ambiguity caused by its own doubts about the model, but being constrained by consumers who have known pessimistic beliefs, this government effectively manages two kinds of ambiguities, its own and those of the public. In both cases, the government's doubts about the model, indexed by *np*, provide motives to raise the social cost of carbon and the capital subsidy but to lower the carbon tax. A pessimistic government's ignorance of the beliefs of skeptical consumers in Proposition 5 motivates policies that mimic those of the political and paternalistic planners in Propositions 3 and 4.

Pessimistic consumers discount the future less than do skeptical or rational consumers and therefore use less carbon energy and invest more in capital relative to socially optimal rates and quantities. So the presence of private-sector pessimism is an inducement for the government to lower carbon taxes and to raise the capital tax. Given that the pessimistic government's own doubts produce motives in the opposite direction, the net effect of private and government doubts can be ambiguous.

**Table 2.** Propositions 1–6.


#### **14. A Feedback from Taxes to Consumers' Pessimistic Beliefs**

When faced with pessimistic consumers, the policies of a Ramsey planner (Φ > 0) present an instance of a possible two-way feedback between carbon tax policy and privatesector pessimism via the debt channel.38 For example, consider a positive surprise in debt *bt* <sup>&</sup>gt; <sup>E</sup>*t*−1*m<sup>c</sup> <sup>t</sup> bt*. From (77), the shadow value to the taxing authority of the consumer's utility *ε<sup>t</sup>* must drop, implying a drop in the consumer's worst-case utility U*t*, hence an increased pessimism via (6).

The tax implications follow from formulas (79), (86), (87), (95), (100) and (112), which show that the social cost of carbon and the carbon tax are decreasing functions of *εt*, and that the *ex ante tax* on capital is an increasing function of *εt*, so that a *t*-period surprise increase in debt that causes a drop in *ε<sup>t</sup>* and an increase in consumer pessimism should be associated with an increase in the SCC and the carbon tax, and a drop in the tax on capital.

The taxing authority is motivated by two considerations. On the one hand, for purely fiscal reasons, it wants to raise taxes in those states against which it is cheaper to issue debt. It therefore raises tax rates in high-debt states caused by climate shocks, and conversely lowers taxes in low-debt states when climate is calmer. The government also has a goal of setting the prices of carbon and capital in ways that are optimal for society. It turns out that these twin goals coincide: by manipulating debt and taxes to raise (or reduce) the household's utility and turning it more (or less) pessimistic and less (or more) willing to use fossil energy but more (or less) willing to spend on capital, the government manages both, an optimal allocation and a need to smooth debt and taxes over time.

### **15. Conclusions**

This paper is about the climate policy implications of belief heterogeneity and ambiguity in a dynamic market economy governed by a benign welfare maximizing authority, here referred to as a Ramsey planner. To keep this paper focused, many important features relevant for practical policy, such as non-linearities in the mechanisms from the burning of fossil fuels to climate change, are set aside, as are some macroeconomic issues, such as the implications of either exogenous or directed endogenous technological change on growth.39 Nor do I account for substitution among available types of fossil energy inputs, such as oil, gas, and coal, and green energy from sun, water, and even nuclear power. Finally, following many examples in the literature, to keep the presentation manageable, all sources of uncertainty have been combined into a single "climate-cost shock" variable representing all *CO*2-related economic climate damages, including those related to the dynamics of *CO*2, productivity shocks other than from climate, and cost shocks from alternative sources of energy, including renewable energy.

This paper, has, in the main, sought to adhere to the spirit of its antecedents, including Stern (2007), Nordhaus (2008), Acemoglu et al. (2012), von Below (2012), van der Ploeg and Withagen (2014), Golosov et al. (2014), Belfiori (2017), Rezai and van der Ploeg (2017), Barrage (2018), and Cai and Lontzek (2019). Golosov et al. (2014), whose model I consider a benchmark for comparisons, proved that the optimal carbon tax (expressed as a proportion of GDP) depends solely on the social cost of carbon, the carbon persistence parameter, and a discount factor. While this is true here as well when beliefs are homogeneous and rational, this paper's value added is an accounting of how belief distortions about the underlying model will alter that discount factor, depending on type of ambiguity.

The principal insight in this paper is that dissonance in beliefs, expressed as skepticism or doubt about forces governing economic outcomes arising from climate cost shocks, produce ambiguity and create wedges between the private sector's discount factor and the government's discount factor and therefore between the way in which consumers and society differentially price two assets with uncertain pay-offs: one from physical capital, whose returns are positive, and the other from atmospheric carbon accumulation, whose returns are negative.

Ambiguity that is due to the government's ignorance of optimistically distorted private beliefs raises the expected social cost of carbon and the carbon tax, which is offset by a lowering of capital taxes. This conclusion carries a certain political irony: a public that confidently denies or minimizes the fact of climate change may actually see its carbon taxes increased.

Ambiguity that arises as a consequence of neither government nor private sector trusting the scientific model has mixed effects depending on the relative degree of pessimism in either sector. So, while any doubts held by either the planner or the private sector produce an ambiguity premium to the social cost of carbon, their differential tax effects are mixed. A combination of government doubt about the model and consumer skepticism motivates lowering the tax on capital, but has ambiguous effects on any carbon tax premium. The combination of fear of model misspecification on the part of both the government and the private sector justifies an ambiguity premium for the social cost of carbon, but opposing effects of some key correlations in the economy leave the net effect on carbon tax premiums ambiguous. The source of this disparity of outcomes derives from two channels: (i) an indirect positive effect through the social cost of carbon and (ii) a direct positive *or* negative effect on the tax itself, depending on the source of ambiguity.

In all cases discussed in this paper, the perspective of Arrow–Debreu asset pricing theory illuminates an equivalence between Pigouvian carbon taxation and optimal pricing of carbon permits in a cap-and-trade economy. This correspondence extends to conditions of Knightian uncertainty, when derived asset prices must meet the test of robustness.

Being theoretic, this paper begs the question of just how applicable the findings herein might be to the real world. In the absence of counter-factual history, the best approach is stochastic simulations of a properly calibrated DSGIE climate-economy model governed by alternative Ramsey regimes discussed here. Alternatively, solving Isaacs-Bellman-Flemming equations associated with the belief regimes in Section 7 as Hennlock (2009) and Li et al. (2016) have done, also holds promise.

Finally, a word of caution. This paper is normative in that it posits not what is but what should be, based on the ideal of welfare maximization implemented by a benign authority heedful of consumers' preferences and budgets. In the reality, such a Ramsey planner may be mere fiction, even in nominal democracies, such as the United States, where an overriding authority, the Supreme Court, has recently ruled that the Executive has no authority to implement carbon policies without explicit and detailed instructions by a legislature that has shown little inclination to address the approaching climate catastrophe. In the real world, policy regimes may even turn rogue: in 2017, the Envronmental Protection Agency under the previous US Administration scrubbed its website of all references to climate. The EPA's website EPA (2017), cited earlier, is available only because it was copied and preserved on another website.

**Funding:** This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** A previous version of the paper, entitled "Confronting Climate Skepticism or Fear: Ramsey Carbon Taxation and Fiscal Policy under Ambiguity", was presented at the Troxler Symposium, "Economic Analysis, Environmental Uncertainties, and Policy Implications" at East Carolina University, 8 November 2019. I would like to thank all participants for comments and insights.

**Conflicts of Interest:** The author declares no conflict of interest.

### **Appendix A. The Household's Intertemporal Budget Constraint**

Solve (4) forward for *bt*, starting with *t* = 0:

*bt* ≥ *T*−1 ∑ *t*=0 Π*t*−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> ∑ *xj*+<sup>1</sup> *p*ˆ*j*+1(*xj*+1) [*ct*(*x<sup>t</sup>* ) − *Ht* − *gt*] + Π*T*−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> ∑ *xj*+<sup>1</sup> *<sup>p</sup>*ˆ*j*+1*xj*+1,0 <sup>∑</sup> *xT*−<sup>1</sup> *kT* + ∑ *xT p*ˆ*T*+1*xT*+1*bT*+<sup>1</sup> + Π*T*−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> ∑ *xj*+<sup>1</sup> *p*ˆ*j*+1*xj*+<sup>1</sup> ∑ *xT* [*pe <sup>T</sup>* <sup>−</sup> *<sup>τ</sup><sup>e</sup> <sup>T</sup>*]*QT*−1(*xT*) + *T*−1 ∑ *t*=1 Π*t*−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> ∑ *xj*+<sup>1</sup> *p*ˆ*j*+1*xj*+<sup>1</sup> <sup>1</sup> − ∑ *xt p*ˆ*t*+1*xt*+1*R<sup>k</sup> t kt* + *T*−1 ∑ *t*=1 Π*t*−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> ∑ *xj*+<sup>1</sup> *p*ˆ*j*+1*xj*+<sup>1</sup> <sup>1</sup> − ∑ *xt p*ˆ*t*+1*xt*+1*R<sup>k</sup> t* [*pe <sup>t</sup>* <sup>−</sup> *<sup>τ</sup><sup>e</sup> <sup>t</sup>* ]*Qt*, (A1)

where *<sup>p</sup>*<sup>0</sup> <sup>=</sup> 1, and Hotelling's rule (31) ( *<sup>p</sup><sup>e</sup> t*+1−*τ<sup>e</sup> t*+1 *pe t*−*τ<sup>e</sup> t* = *R<sup>k</sup> <sup>t</sup>*+1) is used in the last term. By the no-arbitrage condition (14), the last two lines are zero. The third line vanishes under the assumption of resource exhaustibility, where lim*T*→<sup>∞</sup> *QT*−<sup>1</sup> = 0. The second line disappears as a consequence of the household's no-Ponzi game condition

$$\lim\_{T \to \infty} \sum\_{\mathbf{x}^T} \mathfrak{d}\_T(\mathbf{x}^T) [k\_{T+1}(\mathbf{x}^T) + \sum\_{\mathbf{x}^T} \mathfrak{d}\_{T+1}(\mathbf{x}\_{T+1}|\mathbf{x}^T) b\_{T+1}(\mathbf{x}^T)] = 0. \tag{A2}$$

This leaves the first line to be evaluated. Note, from (17), the t-step ahead pricing kernel,

$$\begin{array}{rcl}\hat{q}\_{t}(\mathbf{x}^{\boldsymbol{t}},\mathbf{x}\_{0}) & \equiv & \Pi\_{j=0}^{t-1}\hat{p}\_{j+1,0}(\mathbf{x}^{\boldsymbol{j}})\\ &=& \Pi\_{j=0}^{t-1}\beta m\_{j+1}(\mathbf{x}^{\boldsymbol{j}+1})\pi\_{j}(\mathbf{x}^{\boldsymbol{j}})\frac{u\_{\boldsymbol{c}\_{j+1}}(\mathbf{x}^{\boldsymbol{j}+1})}{u\_{\boldsymbol{c}\_{j}}(\mathbf{x}^{\boldsymbol{j}})}\\ &=& \beta^{t}M\_{t}(\mathbf{x}^{\boldsymbol{t}})\pi\_{t}(\mathbf{x}^{\boldsymbol{t}})\frac{u\_{\boldsymbol{c}\_{t}}(\mathbf{x}^{\boldsymbol{t}})}{u\_{\boldsymbol{c}\_{0}}(\mathbf{x}^{0})}, \quad q\_{0}=1,M\_{0}=1,\end{array} \tag{A3}$$

is the price of an Arrow–Debreu contract written at *t* = 0. Substituting the last line in (A3) into the term in parentheses in the first line of (A1) then yields an expression for the household's intertemporal budget constraint shown in (35).
