**9. A Comment about Methodology**

In the extant literature, the typical approach to finding analytically tractable solutions to the kind of maximin dynamic programming problems posed in Section 7 is to form their Isaacs-Bellman-Flemming equations that involve guessing and verifying functional forms as well as specifying detailed assumptions regarding preferences and probability distributions. See for example Hennlock (2009) and Li et al. (2016). As a bit of an exception, Anderson et al. (2013) do provide some analytic insights based on their problem's firstorder conditions, especially regarding the importance of the role of deep uncertainty in measuring the TCRE parameter defined in Section 4. However, they leave more detailed conclusions to an evaluation of numerical solutions of their stochastic finite-horizon robust optimization problems.

An innovation of this paper is to derive specific and detailed formulas for the social cost of carbon and both carbon and capital taxes through evaluations of the Euler conditions derived from optimization. As will become apparent in the next three sections, each case will reveal important roles for certain second-order moments in the economy, specifically for a number of covariances between measures of the martingale belief distortions and a variety of economic variables, including net returns to fossil energy and capital that arise as a consequence of evaluating the expectations of products of random variables.

Given some key assumptions and a number of results stated as lemmas in Section 13.1, it becomes possible to sign these covariances, allowing us to determine with fair accuracy the likely signs of ambiguity premiums that must be added to the social cost of carbon, the carbon tax, and any capital subsidy in the various belief regimes analyzed in this paper.

#### **10. The Social Cost of Carbon**

The generic form of the first-order condition for carbon stores *Qt*, based (alternatively) on (56), (61), (66) and (70), is

$$\varphi\_{t} = \beta \mathbb{E}\_{t} n\_{t+1}^{\*\*} (\varphi\_{t+1} + \lambda\_{t+1} \mathbb{x}\_{t+1} D\_{t+1} F\_{t+1}).\tag{82}$$

Defining the marginal-utility scaled shadow price *<sup>t</sup>* = *ϕt*/*λt*, this becomes

$$\begin{split} \boldsymbol{\omega}\_{t} &= \mathbb{E}\_{t} \boldsymbol{n}\_{t+1}^{\ast \ast} \beta \frac{\lambda\_{t+1}}{\lambda\_{t}} (\boldsymbol{\omega}\_{t+1} + \boldsymbol{\chi}\_{t+1} \boldsymbol{D}\_{t+1} \boldsymbol{F}\_{t+1}) \\ &\equiv \quad \mathbb{E}\_{t} \boldsymbol{q}\_{t+1}^{\ast \ast} [\boldsymbol{\omega}\_{t+1} + \boldsymbol{\chi}\_{t+1} \boldsymbol{D}\_{t+1} \boldsymbol{F}\_{t+1}]. \end{split} \tag{83}$$

Formula (83) gives a recursion for worst-case climate-caused damages from the point of view of a planner who may or may not be facing ambiguity, depending on *n*∗∗. The *social cost of carbon* is its forward solution,

$$\omega\_{t} = \mathbb{E}\_{t} \lim\_{T \to \infty} \sum\_{j=0}^{T} (\prod\_{i=0}^{t} p\_{t+i,t}^{\*\*}) x\_{t+j} D\_{t+j} F\_{t+j}. \tag{84}$$

Formula (84) reveals that damages are priced at *p*∗∗ *<sup>t</sup>*+*j*,*<sup>t</sup>* = ∗∗ *t*+*j*,*t πt*+*j*, i.e., the current *t*period robust Arrow–Debreu price of capital in Formula (81), demonstrating an equivalence

between capital and climate damages. In essence, accumulated carbon emissions constitute a negative asset that, in a competitive economy, is optimally priced like any other asset.31 It is convenient to define the instantaneous undistorted social cost of carbon,

Λ*t*+<sup>1</sup> ≡ *t*+1[*t*+<sup>1</sup> + *xt*+1*Dt*+1*Ft*+1], (85)

and also

$$f(\Phi\_{l}, \varepsilon\_{t}) \equiv \frac{1}{1 + \varepsilon\_{t} + \Phi\_{l}[1 - \gamma + \gamma \vartheta\_{t}]} \le f(\Phi\_{l}, 0) < f(0, 0) = 1,\tag{86}$$

where *<sup>ϑ</sup><sup>t</sup>* = *Ht*+*gt ct* ∀*t*. Then use (78) and (79) to write the worst-case social cost of carbon (83) as a distortion of Λ*t*+<sup>1</sup>

$$\begin{split} \boldsymbol{\omega}\_{t} &= \mathbb{E}\_{t} \boldsymbol{n}\_{t+1}^{\*\ast} \mathbb{P}(\boldsymbol{\varepsilon}\_{t+1}, \boldsymbol{n}\_{t+1}^{\*}) \boldsymbol{\Lambda}\_{t+1} \\ &= \boldsymbol{f}(\boldsymbol{\Phi}\_{t}, \boldsymbol{\varepsilon}\_{t}) \mathbb{E}\_{t} \Big[ \boldsymbol{n}\_{t+1}^{\*\ast} \boldsymbol{\Lambda}\_{t+1} \Big( 1 + \boldsymbol{\varepsilon}\_{t+1} + [1 - \gamma + \gamma \boldsymbol{\theta}\_{t+1}] \boldsymbol{n}\_{t+1}^{\*} \boldsymbol{\Phi}\_{t} \Big) \Big]. \end{split} \tag{87}$$

Notably, the SCC is the sum of two parts.

The first part is

*<sup>I</sup>* : *<sup>f</sup>*(Φ*t*,*εt*)(<sup>1</sup> + *<sup>ε</sup>t*+1)E*<sup>t</sup> n*∗∗ *<sup>t</sup>*+1Λ*t*+<sup>1</sup> ,

which discounts future damages valued at the planner's *n*∗∗-distorted price of damages. Based on Table 1, *n*∗∗ is *nPO* for the political planner, 1 for the paternalistic planner, and *n<sup>p</sup>* for the pessimistic planner facing either climate-skeptic or climate-pessimistic consumers.

The second part is

$$III: \quad f(\Phi\_t, \varepsilon\_t) \mathbb{E}\_t \left[ n\_{t+1}^{\ast \ast} n\_{t+1}^{\ast} \Lambda\_{t+1} (1 - \gamma + \gamma \theta\_{t+1}) \Phi\_t \right]\_{\tau}$$

which augments damages in part I with the value of the planner's implementability constraint (if binding) at the *n*∗-distorted price of the net benefits of fossil energy. Based on Table 1, the combined ambiguity distortion *<sup>n</sup>*<sup>∗</sup> × *<sup>n</sup>*∗∗ is *<sup>n</sup>PO* for the political planner, *<sup>n</sup>PA* for the paternalistic planner, *m<sup>p</sup>* for the pessimistic planner facing climate-skeptical consumers, and *m<sup>c</sup>* for the pessimistic planner facing climate-pessimistic consumers.

In the special instance when beliefs are rational and homogeneous, and the Ramsey planner's implementability constraint (39) is not binding (Φ*<sup>t</sup>* = 0), the social cost of carbon is the expected value of Λ*t*+<sup>1</sup> (see for example Golosov et al. (2014)),

$$
\boldsymbol{\omega}\_{t}^{SP} = \mathbb{E}\_{t} \boldsymbol{\varrho}\_{t+1} [\boldsymbol{\omega}\_{t+1} + \mathbf{x}\_{t+1} \boldsymbol{D}\_{t+1} \boldsymbol{F}\_{t+1}] = \mathbb{E}\_{t} \boldsymbol{\Lambda}\_{t+1} \equiv \boldsymbol{\Lambda}\_{t}^{s}.\tag{88}
$$

#### **11. The Carbon Tax**

The conventional formula for the optimal carbon tax, as derived in Golosov et al. (2014), is

$$\text{Cocial} \quad \text{Planner's} \quad \text{Carbon} \\ \text{Tax} \quad : \qquad \pi\_t^{\epsilon - SP} = \varpi\_t^{SP} = \Lambda\_t^s. \tag{89}$$

This section shows that when beliefs are heterogeneous, the preceding formula for the carbon tax is inadequate.

The Ramsey planner's first-order condition (51) with respect to energy *Et* is

$$\left.\gamma\_{E\_t}\lambda\_t - \left.\varphi\_t\right|\_{t} = \left.\nu\_{\tau}\right|\_{t}$$

where *Y* is defined in (20). Advancing one period and taking discounted expectations, the preceding expression implies,

$$\mathbb{E}\_t \beta \left( \mathbb{Y}\_{E\_{t+1}} \lambda\_{t+1} - \varrho\_{t+1} \right) \quad = \quad v.$$

Together, these two expressions imply

$$\mathbb{E}\_{l} \frac{\beta \frac{\lambda\_{l+1}}{\lambda\_{l}} (Y\_{\mathbb{E}\_{l+1}} - \varphi\_{l+1}/\lambda\_{l+1})}{Y\_{\mathbb{E}\_{l}} - \varphi\_{l}/\lambda\_{l}} \quad = \quad \mathbb{E}\_{l} \frac{\varrho\_{l+1}^{\*} (Y\_{\mathbb{E}\_{l+1}} - \varphi\_{l+1})}{Y\_{\mathbb{E}\_{l}} - \varphi\_{l}} = 1. \tag{90}$$

In Formula (90), ∗ *<sup>t</sup>*+1,*<sup>t</sup>* is the discount factor a Ramsey planner applies to future expected excess returns to energy over the social cost of carbon—which I shall call the *Net Social Benefit of Carbon*. The corresponding *t* + *j* equilibrium price of fossil energy is

$$
\hat{p}^\*\_{t+j,t} = \varrho^\*\_{t+j,t} \pi\_{t+j}(\mathbf{x}\_{t+j}|\mathbf{x}^t). \tag{91}
$$

Market equilibrium requires that returns to all assets and activities be equal:

$$\mathbb{E}\_t \left[ \frac{\varrho\_{t+1}^\* \left( Y\_{\mathbb{E}\_{t+1}} - \varpi\_{t+1} \right)}{Y\_{\mathbb{E}\_t} - \varpi\_t} \right] \\ \quad = \quad \mathbb{E}\_t \varrho\_{t+1}^{\*\*} \left( 1 - \delta + Y\_{\mathbb{E}\_{t+1}} \right) = 1, \tag{92}$$

where the right-hand side comes from Euler Equation (55) (or (60), (65), (69)).

Formula (92) is a version of Hotelling's welfare-optimal rule and posits that, in equilibrium, the expected socially optimal pre-tax gross return on the extant stock of fossil fuel remaining in the ground, which society discounts with stochastic discount factor ∗, is equal to the expected pre-tax gross return to capital in place, discounted using the government's SDF ∗∗. By comparison, Hotelling's rule, when derived for laissez-faire in (31), posits an equivalence between the private after-tax return to fossil fuel in the ground and the private after-tax gross return to the extant stock of capital, discounted using the private sector's distorted SDF ˆ*t*+1. Since both <sup>∗</sup> *<sup>t</sup>*+<sup>1</sup> and ˆ*t*+<sup>1</sup> may reflect ambiguous beliefs, formula (92) is potentially a robust version of Hotelling's rule. Note that the expected sign of the numerator is positive (or negative) in period *t* + 1 if its sign is positive (or negative) in period *t*.

Market equilibrium for the price of fossil energy *p<sup>e</sup>* = *YE* was previously derived as obeying the difference Equation (28),

$$\mathcal{Y}\_{\mathbb{E}\_t} - \tau\_t^{\varepsilon} = \mathbb{E}\_t \hat{\varrho}\_{t+1} (\mathcal{Y}\_{\mathbb{E}\_{t+1}} - \tau\_{t+1}^{\varepsilon}),\tag{93}$$

while, from (90), a socially optimal energy price must obey the rule

$$\|\boldsymbol{\chi}\_{\boldsymbol{E}\_{t}} - \boldsymbol{\mathcal{O}}\_{t}\| \quad = \quad \mathbb{E}\_{t} \boldsymbol{\varrho}\_{t+1}^{\*} \left(\boldsymbol{\chi}\_{\boldsymbol{E}\_{t+1}} - \boldsymbol{\mathcal{O}}\_{t+1}\right).$$

Since, in equilibrium, both expressions must hold, subtract the first from the second equation to eliminate *YEt* :

$$\begin{split} \pi\_t^\varepsilon - \mathcal{O}\_t &= \mathbb{E}\_t \boldsymbol{\varrho}\_{t+1}^\* [\boldsymbol{Y}\_{\boldsymbol{E}\_{t+1}} - \mathcal{O}\_{t+1}] - \mathbb{E}\_t [\boldsymbol{\varrho}\_{t+1} (\boldsymbol{Y}\_{\boldsymbol{E}\_{t+1}} - \boldsymbol{\pi}\_{t+1}^\varepsilon)] \\ &+ \quad E\_t \boldsymbol{\varrho}\_{t+1} \boldsymbol{\varrho}\_{t+1} - \boldsymbol{E}\_t \boldsymbol{\varrho}\_{t+1} \boldsymbol{\varrho}\_{t+1} \\ &= \quad \mathbb{E}\_t \boldsymbol{\varrho}\_{t+1} [\boldsymbol{\pi}\_{t+1}^\varepsilon - \boldsymbol{\varpi}\_{t+1}] + \mathbb{E}\_t \boldsymbol{\varrho}\_{t+1}^\* [\boldsymbol{Y}\_{\boldsymbol{E}\_{t+1}} - \boldsymbol{\varpi}\_{t+1}] - \mathbb{E}\_t \boldsymbol{\varrho}\_{t+1} (\boldsymbol{Y}\_{\boldsymbol{E}\_{t+1}} - \boldsymbol{\varpi}\_{t+1}). \end{split}$$

Adding and subtracting *Et*ˆ*t*+1*t*+<sup>1</sup> yields

$$\mathbb{E}\left[\mathbf{r}\_{t}^{\varepsilon}-\boldsymbol{\varpi}\_{t}\right] = \mathbb{E}\_{t}\boldsymbol{\upbeta}\_{t+1}\left[\mathbf{r}\_{t+1}^{\varepsilon}-\boldsymbol{\varpi}\_{t+1}\right] + \mathbb{E}\_{t}Z\_{t+1\prime} \tag{94}$$

where

$$Z\_{t+1} = \left(\varrho\_{t+1}^\* - \varrho\_{t+1}\right) (Y\_{E\_{t+1}} - \varpi\_{t+1})\_{\prime t}$$

is the difference between the government's discounted *Net Social Benefit* of fossil fuel and the private sector's discounted *Net Social Benefit*, where, from (90), *YEt*<sup>+</sup>*<sup>j</sup>* ≥ *t*+*j*. Intuitively, the carbon tax exceeds the social cost of carbon in every period if ∀*j* > 0, the social discount factor is higher than the private discount factor, ∗ *<sup>t</sup>*+*<sup>j</sup>* ≥ ˆ*t*+*j*. So if, in every period, the private sector is more myopic, i.e., less patient with respect to returns to fossil fuels than the planner, the government will add a premium to the carbon tax above the expected social cost of carbon . Exactly how and by how much, is determined as follows.

**Lemma 1** (**The carbon tax).** *The carbon tax is the sum of two terms: (1) the expected social cost of carbon <sup>t</sup> and (2) a premium χt:*

$$
\hat{\pi}\_{l}^{\varepsilon} = \ \ \ \ \hat{\sigma}\_{l} + \chi\_{l\prime} \tag{95}
$$

*where, utilizing (17),*

$$\begin{split} \chi\_{t} & \equiv \quad \mathbb{E}\_{t} \sum\_{j=0}^{\infty} \prod\_{i=0}^{j} \varrho\_{t+i} Z\_{t+j} = \mathbb{E}\_{t} \sum\_{j=0}^{\infty} \beta^{j} \left( \prod\_{i=1}^{j} m\_{t+i} \frac{u\_{c}(\mathbf{x}^{t+i})}{u\_{c}(\mathbf{x}^{t})} \right) Z\_{t+j} \\ & = \quad \mathbb{E}\_{t} \sum\_{j=0}^{\infty} \frac{M\_{t+j}}{M\_{t}} \varrho\_{t+j,t} Z\_{t+j} = \mathbb{E}\_{t} \sum\_{j=0}^{\infty} \hat{\rho}\_{t+j,t} (\varrho\_{t+j}^{\*} - \varrho\_{t+j}) (Y\_{E\_{t+j}} - \varpi\_{t+j}) \\ & = \quad \mathbb{E}\_{t} \sum\_{j=0}^{\infty} \hat{\rho}\_{t+j,t} Z\_{t+j}. \end{split} \tag{96}$$

**Proof.** Solve the difference Equation (94).

The first component of *τ*ˆ*<sup>e</sup>* is the Social Cost of Carbon previously derived that may or may not already contain an ambiguity-related premium. The second component adds a further ambiguity premium depending on belief regime. In essence, a government uses this formula to impose a premium on (or grant a discount toward) the carbon tax if cumulative *expected* differentially discounted *net private benefits* of fossil fuel are positive (or negative), where the sign and size of the premium (or discount) *χ<sup>t</sup>* depends on the signs and sizes of all future *Zt*+*<sup>j</sup>* priced at *p*ˆ*t*+*j*,*<sup>t</sup>* that need to be determined. Note that in Formula (96), *p*ˆ*t*+*j*,*t*(*YEt*<sup>+</sup>*<sup>j</sup>* − *t*+*j*) is the (possibly belief distorted) market value of the excess of private returns of fossil fuels over their social cost in period *t* + *j*. The premium *χ<sup>t</sup>* is the expected sum of all such terms, each multiplied by the difference between the government's and the private sector's discount factor. In general, the premium is positive if the public is myopic compared with the planner: ∗ *<sup>t</sup>*+*<sup>j</sup>* − ˆ*t*+*<sup>j</sup>* > 0 in all periods. Intuitively, a skeptical consumer who disbelieves the seriousness of climate change, will tend to use more energy than warranted from society's point of view because it myopically undervalues future climate costs. By contrast, a pessimistic private sector would opt to use less. As will become apparent later, this calculus is modified, if the authority itself has ambiguity about private beliefs.

An asset-pricing interpretation of (95) is that the optimal carbon tax is the sum of two possibly robust asset prices: (1) expected cumulative fossil fuel damages per unit of carbon valued at prices *p*∗∗ *<sup>t</sup>*+*<sup>j</sup>* in (84), and (2) cumulative net private benefits per unit of carbon over the social cost of carbon, valued at *p*ˆ*t*+*<sup>j</sup>* and weighted by the difference in discount factors ∗ *<sup>t</sup>*+*<sup>j</sup>* − ˆ*t*+*j*. The interpretation of *<sup>τ</sup>*ˆ*<sup>e</sup> <sup>t</sup>* as the potentially robust price of an underlying asset—the government-imposed cap on emissions—extends a result by Belfiori (2017), who also derived an equivalence between the optimal carbon tax and the optimal price of traded carbon permits in an economy with a cap and trade.

The case of a social planner under rational expectations, in which Φ = 0 and beliefs are undistorted and homogeneous, is a suitable benchmark for comparison. However, it is exceptional in that in all other policy/belief regimes, the government may either impose a premium or give a concession. The premium (or concession) in Formula (96) is the expected value of the sum of products of random variables over all *t* + *j*, so the sign is not obvious from inspection, although it can be deciphered via decomposition into covariance components derived in Appendix D, which distinguishes between two cases for each of the possible policy regimes when either (1) the wealth constraint in (39) is binding (Φ*<sup>t</sup>* > 0), implying Ramsey planning, or (2) it is not (Φ*<sup>t</sup>* = 0), implying social planning.

To pursue this, define a new variable Ξ*<sup>e</sup> <sup>t</sup>*+*<sup>j</sup>* as the weighted proportional difference between the two stochastic discount factors ∗ *<sup>t</sup>*+*<sup>j</sup>* and ˆ*t*+*j*:

$$\Xi\_{t+j}^{\varepsilon} \equiv \varsigma\_{t+j}^{\varepsilon} \frac{\varrho\_{t+j}^{\*} - \hat{\varrho}\_{t+j}}{\varrho\_{t+j}} = \varsigma\_{t+j}^{\varepsilon} \left( \frac{\varrho\_{t+j}^{\*}}{\varrho\_{t+j}} - 1 \right),\tag{97}$$

where

$$\mathfrak{g}\_{t+j}^{\varepsilon} \equiv \frac{\mathfrak{g}\_{t+j}(\mathbf{Y}\_{E\_{t+j}} - \mathfrak{o}\_{t+j})\_{\prime}}{\mathbb{E}\_{t}\mathfrak{g}\_{t+j}(\mathbf{Y}\_{E\_{t+j}} - \mathfrak{o}\_{t+j})\_{\prime}} \Longrightarrow \mathbb{E}\_{t}\mathfrak{g}\_{t+j}^{\varepsilon} = 1,\tag{98}$$

is the *j*-th period's normalized discounted excess return to fossil energy over the social cost of carbon. Utilizing (78), Ξ*<sup>e</sup> <sup>t</sup>*+*<sup>j</sup>* is

$$\Xi\_{t+j}^{\varepsilon} = \varrho\_{t+j}^{\varepsilon} (\frac{\Psi(\varepsilon\_{t+j'} n\_{t+j}^\*)}{m\_{t+j}} - 1),\tag{99}$$

where Ψ(*εt*+*j*, *n*<sup>∗</sup> *t*+*j* ) is defined in (79). Notice that Ξ*<sup>e</sup>* depends on the government's belief multiplier *n*∗ which varies according to policy regime, as shown in Table 1, and on the private sector's belief distortion *mt*+<sup>1</sup> = *m<sup>c</sup> <sup>t</sup>*+<sup>1</sup> when consumers are pessimistic and *mt*+<sup>1</sup> = *m<sup>s</sup> <sup>t</sup>*+<sup>1</sup> when they are skeptical. Importantly, whereas *<sup>m</sup><sup>s</sup>* is an exogenous martingale process, *m<sup>c</sup>* is the pessimistic consumer's worst-case multiplier that depends on continuation utility as derived in (6). From the preceding, using (78) and (79), the expected value of Ξ*e <sup>t</sup>*+*<sup>j</sup>* is

$$\begin{split} \mathbb{E}\_{t} \mathbb{E}\_{t+j}^{\varepsilon} &= \mathbb{E}\_{t} \frac{\varrho\_{t+j}^{\*} - \varrho\_{t+j}}{\varrho\_{t+j}} \xi\_{t+j}^{\varepsilon} \\ &= \mathbb{E}\_{t} \left[ \left( \frac{1}{m\_{t+j}} \frac{1 + \varepsilon\_{t+j} + [1 - \gamma + \gamma \theta\_{t+j}] n\_{t+j}^{\*} \Phi\_{t+j-1}}{1 + \varepsilon\_{t} + [1 - \gamma + \gamma \theta\_{t}] \Phi\_{t}} - 1 \right) \xi\_{t+j}^{\varepsilon} \right]. \end{split} \tag{100}$$

Note that (100) gives the difference between two non-centered covariances with the normalized discounted excess return to fossil energy, one involving the government's and the other the private sector's stochastic discount factor.

Lemma 2 is key to determining the sign of the carbon tax premium. Based on (95),

#### **Lemma 2.**

$$\text{sign } \chi\_t = \text{sign}[\mathfrak{t}\_t^\varepsilon - \mathfrak{o}\_t] \ = \text{sign } \mathbb{E}\_t \Xi\_{t+j}^\varepsilon \,\forall j. \tag{101}$$

**Proof.** First,

$$\mathbb{E}\_t Z\_{t+j} \overset{\geqslant}{\underset{\sim}{\rightleftharpoons}} 0, \quad \text{if} \quad \mathbb{E}\_t \mathbb{E}\_{t+j} \overset{\geqslant}{\underset{\sim}{\rightleftharpoons}} 0, \ j = 1, \cdots \infty.$$

To show this, multiply and divide *Zt*+*<sup>j</sup>* by ˆ*t*+*j*E*t*ˆ*t*+*j*(*YEt*<sup>+</sup>*<sup>j</sup>* <sup>−</sup> *t*+*j*):

$$\begin{split} Z\_{t+j} &= \quad Z\_{t+j}\frac{\dot{\varrho}\_{t+j}}{\dot{\varrho}\_{t+j}}\frac{\mathbb{E}\_{\dot{\varrho}}\dot{\varrho}\_{t+j}(Y\_{E\_{t+j}} - \varpi\_{t+j})}{\mathbb{E}\_{\dot{\varrho}}\dot{\varrho}\_{t+j}(Y\_{E\_{t+j}} - \varpi\_{t+j})} \\ &= \quad \left[ \frac{\dot{\varrho}\_{t+j}(Y\_{E\_{t+j}} - \varpi\_{t+j})}{\mathbb{E}\_{\dot{\varrho}}\dot{\varrho}\_{t+j}(Y\_{E\_{t+j}} - \varpi\_{t+j})} \frac{\varrho\_{t+j}^{\star} - \dot{\varrho}\_{t+j}}{\dot{\varrho}\_{t+j}} \right] \mathbb{E}\_{\dot{\varrho}}\dot{\varrho}\_{t+j}(Y\_{E\_{t+j}} - \varpi\_{t+j}) \\ &= \quad \left[ \begin{matrix} \varrho\_{t+j}^{\star}\frac{\varrho\_{t+j}^{\star} - \dot{\varrho}\_{t+j}}{\dot{\varrho}\_{t+j}} \end{matrix} \right] \mathbb{E}\_{\dot{\varrho}}\dot{\varrho}\_{t+j}(Y\_{E\_{t+j}} - \varpi\_{t+j}) \\ &= \quad \quad \quad \quad \quad \quad \dot{\Xi}\_{t+j}^{\star}\dot{\Xi}\_{t}\dot{\varrho}\_{t+j}(Y\_{E\_{t+j}} - \varpi\_{t+j}), \\ \implies \quad \quad \quad \quad \quad \quad \dot{\Xi}\_{t}Z\_{t+j} &= \quad \quad \quad \quad \dot{\Xi}\_{t}\dot{\Xi}\_{t+j}^{\star}\dot{\Xi}\_{t}\dot{\varrho}\_{t+j}(Y\_{E\_{t+j}} - \varpi\_{t+j}), \end{matrix} \tag{102}$$

where, from (90), *YEt*<sup>+</sup>*<sup>j</sup>* <sup>≥</sup> *t*+*j*. It follows that <sup>E</sup>*tZt*+*<sup>j</sup>* and <sup>E</sup>*t*Ξ*<sup>e</sup> <sup>t</sup>*+*<sup>j</sup>* have the same sign. Finally, with *<sup>t</sup>* defined in (84), the lemma follows from Formulas (96) and (102).

In the absence of any known time-dependent anomalies, there is no reason to believe that the sign of E*t*Ξ*<sup>e</sup> <sup>t</sup>*+*<sup>j</sup>* is different at different times, justifying

**Assumption 1.** *sign* E*<sup>t</sup>* ∑<sup>∞</sup> *<sup>j</sup>*=<sup>0</sup> Ξ*<sup>e</sup> <sup>t</sup>*+*<sup>j</sup>* <sup>=</sup> *sign* <sup>E</sup>*t*Ξ*<sup>e</sup> <sup>t</sup>*+*<sup>j</sup>* ∀*j*

Since in (96), the sign of *χ<sup>t</sup>* depends on the sign of all its future expected elements, E*<sup>t</sup> <sup>p</sup>*ˆ*t*,*t*+*jZt*+*j*, we require

$$\textbf{Lemma 3.}\text{ }\text{If }\mathbb{E}\_t\Sigma\_{t+j}^{\varepsilon}\overset{\ge}{\le}0\text{ for any }j\ge 0\text{, then }\chi\_t\overset{\ge}{\le}0\text{ for any }j\ge 0.$$

**Proof.** First, in (96),

$$\mathbb{E}\_t \mathfrak{p}\_{t+j,t} Z\_{t+j} \overset{\simeq}{\underset{\prec}{\rightleftharpoons}} 0, \quad \mathrm{if} \quad \forall j \ge 1, \quad \mathbb{E}\_t \mathfrak{p}\_{t+j,t} \mathbb{E}\_{t+j}^{\varepsilon} \overset{\simeq}{\underset{\prec}{\rightleftharpoons}} 0.$$

The lemma follows by applying Assumption 1, and (96) and (102).

A suitable formula for the carbon tax in terms of the Ξ*<sup>e</sup> <sup>t</sup>*+*<sup>j</sup>* is found by substituting (96) and (102) into (95):

$$\mathfrak{A}\_t^\varepsilon = \varpi\_t + \mathbb{E}\_t \sum\_{j=0}^\infty \mathfrak{p}\_{t+j,t} \Xi\_{t+j}^\varepsilon \mathfrak{d}\_{t+j} (Y\_{E\_{t+j}} - \varpi\_{t+j}).\tag{103}$$

The essence of the preceding is that if we know the sign of Ξ*<sup>e</sup> <sup>t</sup>*+*<sup>j</sup>* for any *j*, then we shall know the sign of any premium *χ<sup>t</sup>* over the standard carbon tax. The implications of this for the different belief regimes are stated in several propositions in Section 13.

#### **12. An** *Ex Ante* **Tax on Capital**

While *p<sup>e</sup> t*(*x<sup>t</sup>* ) and *τ<sup>e</sup> <sup>t</sup>* (*x<sup>t</sup>* ) are uniquely determined in (24) and (95), respectively, it is easy to demonstrate, following Chamley (1986), Zhu (1992), and Chari et al. (1994), that a state-by-state capital income tax is not uniquely determined because an implementable allocation {*b*, *k*, *Q*} that uniquely determines household wealth W in (36) can be obtained by a multiplicity of capital tax and bond policies {*τk*, *<sup>b</sup>*} at prices {*p*ˆ,*r*}. <sup>32</sup> However, across states of nature, the history-dependent value of tax payments is fully determined. So with this in mind, define the *effective* or *ex ante* tax rate on capital as the ratio of the prices of two "assets", one yielding a stream of tax revenues and the other yielding a stream of gross before-tax capital returns, conditional on history *x<sup>t</sup>* , where the latter is defined by

$$
\bar{\mathbf{r}}\_{t+1}^k(\mathbf{x}^t) \equiv \frac{\mathbb{E}\_t \mathfrak{p}\_{t+1,t}(\mathbf{x}\_{t+1}, \mathbf{x}^t) \mathbf{r}\_{t+1}^k(\mathbf{x}^{t+1}) (\mathbf{Y}\_{k\_{t+1}}(\mathbf{x}^{t+1}) - \delta)}{\mathbb{E}\_t \mathfrak{p}\_{t+1,t}(\mathbf{x}\_{t+1}, \mathbf{x}^t) (\mathbf{Y}\_{k\_{t+1}}(\mathbf{x}^{t+1}) - \delta))}. \tag{104}
$$

In this formula, the *ex ante* tax rate *τ*¯ *<sup>k</sup> t*+1(*x<sup>t</sup>* ) is the ratio of expected tax revenues to expected gross returns from capital, conditional on history *x<sup>t</sup>* and valued at possibly distorted Arrow–Debreu prices *p*ˆ.

With the use of (14), (104) is re-written as

$$
\begin{split}
\bar{\pi}\_{t+1}^{k}(\mathbf{x}^{t}) & \equiv \quad \frac{\mathbb{E}\_{t}\boldsymbol{\hat{p}}\_{t+1,t}(\mathbf{x}\_{t+1},\mathbf{x}^{t})[1-\delta+\mathsf{Y}\_{k\_{t+1}}(\mathbf{x}^{t+1})]-1}{\mathbb{E}\_{t}\boldsymbol{\hat{p}}\_{t+1,t}(\mathbf{x}\_{t+1},\mathbf{x}^{t})(\mathbf{Y}\_{k\_{t+1}}(\mathbf{x}^{t+1})-\delta)}, \\
&= \quad \frac{\mathbb{E}\_{t}\boldsymbol{\hat{\varrho}}\_{t+1}[1-\delta+\mathsf{Y}\_{k\_{t+1}}]-1}{\mathbb{E}\_{t}\boldsymbol{\hat{\varrho}}\_{t+1}(\mathbf{Y}\_{k\_{t+1}}-\delta)}.
\end{split} \tag{105}
$$

Based on consumption-Euler Equations (50) or (68),

$$\begin{split} 1 &= \mathbb{E}\_{l} n\_{l+1}^{\ast \ast} \beta \frac{\lambda\_{l+1}}{\lambda\_{l}} (1 + \mathbb{Y}\_{k\_{l+1}} - \delta) \\ &= \mathbb{E}\_{l} n\_{l+1}^{\ast \ast} \beta \frac{[1 + \varepsilon\_{l+1}] u\_{\varepsilon\_{l+1}} + \Omega\_{\varepsilon\_{l+1}} \Phi\_{l+1}}{[1 + \varepsilon\_{l}] u\_{\varepsilon\_{l}} + \Omega\_{\varepsilon\_{l}} \Phi\_{l}} (1 + \mathbb{Y}\_{k\_{l+1}} - \delta) \\ &= \mathbb{E}\_{l} n\_{l+1}^{\ast \ast} \beta \frac{1 + \varepsilon\_{l+1} + [1 - \gamma + \gamma \theta\_{l+1}] n\_{l+1}^{\ast} \Phi\_{l+1} - 1}{1 + \varepsilon\_{l} + [1 - \gamma + \gamma \theta\_{l}] \Phi\_{l}} (1 + \mathbb{Y}\_{k\_{l+1}} - \delta), \end{split} \tag{106}$$

the government discounts *gross* returns to capital in period *t* + 1 using ∗∗, derived in (79), which is also used to discount future damages resulting from fossil use in (84), as shown earlier, where the distorting factor *n*∗∗ *<sup>t</sup>*+1varies according to belief regime as detailed in Table 1.

Substituting (107) into (105), an alternative expression for *τ*¯ *<sup>k</sup> <sup>t</sup>* is

$$\tau\_{t+1}^{k} = \frac{\mathbb{E}\_t[(\varrho\_{t+1} - \varrho\_{t+1}^{\ast \ast})(1 - \delta + \mathcal{Y}\_{k\_{t+1}})}{\mathbb{E}\_t[\varrho\_{t+1}(\mathcal{Y}\_{k\_{t+1}} - \delta)]}. \tag{107}$$

Noteworthy in this expression is that it echoes the expression in (96) determining the sign of the carbon tax premium, likewise depending on the difference between the private sector's discount factor and that of the government, ˆ*t*+<sup>1</sup> − ∗∗ *<sup>t</sup>*+1. Simply put, the numerator in (107) is the difference between two non-centered covariances between the stochastic pre-tax return 1 − *δ* + *Ykt*<sup>+</sup><sup>1</sup> and the private sector's and the government's SDF, respectively. In general terms, the *ex ante* tax on capital is negative if the private sector is more myopic, i.e., less patient with respect to returns to capital than the government, specifically, if the covariance of private returns with the private stochastic discount factor is less than the corresponding covariance with the planner's stochastic discount factor. Conversely, if private returns are better correlated with the government's stochastic discount factor, or are equal to it, then the average tax is positive or zero. Earlier literature by Chamley (1986), Judd (1985), and Atkeson et al. (1999), concludes that in a deterministic economy with time-additive preferences, the optimal *ex ante* tax on capital is zero, except possibly at time 0. Subsequently, Chari et al. (1994) and Zhu (1992) found that in a stochastic economy with homogeneous beliefs, the tax is positive (or negative) if a weighted average of the change in period elasticities of the utility function is positive (or negative). Below, I shall show how a divergence in beliefs between the private sector and government produces a gap between the government's and the private sector's discount factors that contribute additional motives to subsidize or penalize capital, depending on the source of ambiguity.

Now mimic expression (100) and define the weighted proportional difference between the private and the government's stochastic discount factors,

$$\Xi\_{t+1}^{k} \equiv \varsigma\_{t+1}^{k} \frac{\varrho\_{t+1} - \varrho\_{t+1}^{\ast \ast}}{\varrho\_{t+1}} = \varsigma\_{t+1}^{k} (1 - \frac{\varrho\_{t+1}^{\ast \ast}}{\varrho\_{t+1}}) = \varsigma\_{t+1}^{k} (1 - \frac{n\_{t+1}^{\ast \ast}}{m\_{t+1}} \Psi(\varepsilon\_{t+1}, n\_{t+1}^{\ast})),\tag{108}$$

where

$$\varepsilon\_{t+1}^{k} \equiv \frac{\hat{\varrho}\_{t+1} \left[1 - \delta + \mathcal{Y}\_{k\_{t+1}}\right]}{\mathbb{E}\_{t} \left[\hat{\varrho}\_{t+1} (1 - \delta + \mathcal{Y}\_{k\_{t+1}})\right]},\tag{109}$$

is the normalized discounted gross return to capital, such that E*tς<sup>k</sup> <sup>t</sup>*+<sup>1</sup> = 1, and where Ψ(*εt*<sup>+</sup>1, *n*<sup>∗</sup> *<sup>t</sup>*+1) is defined in (79). The next lemma proves that the sign of the capital tax equals the sign of *Et*Ξ*<sup>k</sup> <sup>t</sup>*+1:

#### **Lemma 4.**

$$\mathbb{E}\_{t+1}^{k} \stackrel{\prec}{\underset{\succ}{\rightleftarrows}} 0, \quad \text{if} \quad \mathbb{E}\_{t} \mathbb{E}\_{t+1}^{k} \stackrel{\prec}{\underset{\succ}{\rightleftarrows}} 0. \tag{110}$$

**Proof.** Use (109) in (108) and write,

$$\begin{split} \mathbb{E}\_{t} \mathbb{E}\_{t+1}^{k} & \equiv & \mathbb{E}\_{t} [\mathbf{c}\_{t+1}^{k} \frac{\hat{\varrho}\_{t+1} - \varrho\_{t+1}^{\*\ast}}{\hat{\varrho}\_{t+1}}] \\ &= & \mathbb{E}\_{t} [\frac{\hat{\varrho}\_{t+1} [1 - \delta + \mathbf{Y}\_{k\_{t+1}}]}{\mathbb{E}\_{t} [\hat{\varrho}\_{t+1} (1 - \delta + \mathbf{Y}\_{k\_{t+1}})]} \frac{\hat{\varrho}\_{t+1} - \varrho\_{t+1}^{\*\ast}}{\hat{\varrho}\_{t+1}}] \\ &= & \frac{\mathbb{E}\_{t} [\hat{\varrho}\_{t+1} (\mathbf{Y}\_{k\_{t+1}} - \delta)]}{\mathbb{E}\_{t} [\hat{\varrho}\_{t+1} (1 - \delta + \mathbf{Y}\_{k\_{t+1}})]} \mathbb{E}\_{t+1}^{k} \end{split} \tag{111}$$

where the last term is obtained by multiplying and dividing by <sup>E</sup>*t*[ˆ*t*+1(*Ykt*<sup>+</sup><sup>1</sup> <sup>−</sup> *<sup>δ</sup>*)] and using (107). The proof follows by observing that the term multiplying *τ*¯ *<sup>k</sup> <sup>t</sup>*+<sup>1</sup> is positive.

For later use, re-write (111) as

$$\begin{split} \mathbb{E}\_{t} \boldsymbol{\Xi}\_{t+1}^{k} &= \mathbb{E}\_{t} \boldsymbol{\varepsilon}\_{t+1}^{k} \left( 1 - \frac{n\_{t+1}^{\ast \ast}}{m\_{t+1}} \frac{1 + \varepsilon\_{t+1} + [1 - \gamma + \gamma \vartheta\_{t+1}] n\_{t+1}^{\ast} \Phi\_{t}}{1 + \varepsilon\_{t} + [1 - \gamma + \gamma \vartheta\_{t}] \Phi\_{t}} \right) \\ &= \mathbb{E}\_{t} \boldsymbol{\varepsilon}\_{t+1}^{k} \left( 1 - f(\Phi\_{t}, \varepsilon\_{t}) \frac{n\_{t+1}^{\ast \ast}}{m\_{t+1}} (1 + \varepsilon\_{t+1} + [1 - \gamma + \gamma \vartheta\_{t+1}] n\_{t+1}^{\ast} \Phi\_{t}) \right. \end{split} \tag{112}$$

#### **13. Policy Implications of Belief Heterogeneity and Ambiguity**
