3.4.1. Inner Minimization

The optimal conditional likelihood ratio that minimizes (5) has the familiar exponentially twisting form17

$$m\_{t+1}^{c} = \frac{e^{\sigma^{\varepsilon} l \ell\_{t+1}}}{\mathbb{E}\_{t} e^{\sigma^{\varepsilon} l \ell\_{t+1}}},$$

or, equivalently,

$$
\mathcal{M}\_{t+1}^{\mathfrak{c}} = \frac{\varepsilon^{\sigma^{\mathfrak{c}} \mathcal{U}\_{t+1}}}{\mathbb{E}\_{t} \mathfrak{e}^{\sigma^{\mathfrak{c}} \mathcal{U}\_{t+1}}} \mathcal{M}\_{t}^{\mathfrak{c}}.\tag{6}
$$

The worst-case martingale distortion in (6) is *pessimistic* in that it attaches higher probabilities to histories with low continuation utilities and lower probabilities to histories with high continuation utilities. Notably, because of its dependence on continuation values, it is also a function of future distortions *m<sup>c</sup> <sup>t</sup>*+*<sup>j</sup>* and future decisions by both the consumer and the government that, as shown later, determine equilibrium prices in the economy.

#### 3.4.2. Outer Maximization with Implied Risk-Sensitive Recursion

Substituting *m<sup>c</sup> <sup>t</sup>*+<sup>1</sup> from (6) into (5) produces the *risk-sensitive* recursion18

$$\mathcal{U}(k\_t, b\_t, Q\_t) = \max\_{\mathbf{c}\_t, \mathbf{E}\_t, k\_{t+1}, b\_{t+1}, Q\_{t+1}} u(\mathbf{c}\_t) + \frac{\beta}{\sigma^c} \mathbb{E}\_t \log \epsilon^{\sigma^c l \ell(k\_{t+1}, b\_{t+1}, Q\_{t+1})} + \lambda\_t^c \mathcal{L}\_t. \tag{7}$$

The first-order conditions (FONCs) for the flow variables {*ct*, *Et*} are

$$
\lambda\_t^\varepsilon = \|u\|\_{\mathcal{U}} \tag{8}
$$

$$(p\_t^\epsilon - \tau\_t^\epsilon)\lambda\_t^\epsilon = \quad \varphi\_t^\epsilon. \tag{9}$$

The envelope conditions are,

$$\mathcal{U}\_{k\_t} \quad = \; \lambda\_t^c \mathbb{R}\_t^k = \mathfrak{u}\_{\mathfrak{c}\_t} \mathbb{R}\_{t'}^k \tag{10}$$

$$\mathcal{U}\_{\mathbb{h}\_{\mathbb{h}}} = \quad \lambda\_{\mathbb{h}}^{\mathbb{c}} = u\_{\mathbb{c}\_{\mathbb{H}'}} \tag{11}$$

$$\mathcal{U}\_{\mathcal{Q}\_{\mathcal{I}}} = \quad \mathcal{q}\_{\mathcal{I}}^{\mathcal{E}}.\tag{12}$$

In conjunction with these envelope conditions, the first-order conditions for the stock variables *bt*+1, *kt*+1, and *Qt*+<sup>1</sup> are, respectively,

$$\beta p\_{t+1} = \beta m\_{t+1}^{\varepsilon} \frac{u\_{\varepsilon\_{t+1}}}{u\_{\varepsilon\_t}},\tag{13}$$

$$\mathbb{E}\_{t} = \beta \mathbb{E}\_{t} m\_{t+1}^{c} \frac{u\_{c\_{t+1}}}{u\_{c\_{t}}} R\_{t+1}^{k} = \mathbb{E}\_{t} \mathfrak{p}\_{t+1} R\_{t+1'}^{k} \tag{14}$$

$$
\sigma\_t^{\mathfrak{c}} \quad = \quad \beta \mathbb{E} \iota^{\mathfrak{c}} m\_{t+1}^{\mathfrak{c}} \varphi\_{t+1}^{\mathfrak{c}}.\tag{15}
$$

Note that *p*ˆ*t*+<sup>1</sup> is the one-period worst-case equilibrium price of a state-contingent claim as shown later in (18). According to the preceding conditions, the price of such a claim is determined by continuation utilities and the random climate cost shock *xt*+1. This is information that the government can exploit to determine optimal fiscal policy.

#### *3.5. Skeptical Consumer*

Let *M<sup>s</sup>* denote a skeptical consumer's belief distortion. As stated earlier, skepticism represents an arbitrary random distortion *<sup>π</sup>*ˆ*<sup>t</sup>* ≡ *<sup>M</sup><sup>s</sup> <sup>t</sup>π<sup>t</sup>* of the true distribution *πt*. From the point of view of society (or the government), *M<sup>s</sup> <sup>t</sup>*+<sup>1</sup> is an unknowable exogenous variable. However, this household is sure of its beliefs and evaluates (5) by disabling the penalty on belief distortions and letting *<sup>σ</sup><sup>c</sup>* → 0.

The Euler equations are the same as before, except that *m* = *m<sup>s</sup>* is random and unknown to the authorities, and the equilibrium price *p*ˆ of a state-contingent claim based on skepticism as defined here is unrelated to the consumer's continuation utility. Additionally, since *m<sup>s</sup> <sup>t</sup>*+<sup>1</sup> is random, it is unrelated to *xt*+1.

#### *3.6. Arrow–Debreu Prices under Belief Distortions*

The belief distortions of probabilities, ranging from skepticism to deep uncertainty, treated in this paper, add an important dimension to stochastic discounting. This section shows how to construct their corresponding asset-price measures.

For preferences *u*(*ct*) distorted by a martingale process *Mt*+*<sup>j</sup>* (either *M<sup>s</sup> <sup>t</sup>*+*<sup>j</sup>* or *<sup>M</sup><sup>c</sup> t*+*j* ), the *j*-period-ahead stochastic discount factor (MSDF) is,

$$\mathfrak{g}\_{t+j,t} \quad = \ \beta^j \frac{M\_{t+j}}{M\_t} \frac{u\_{\mathfrak{e}\_{t+j}}}{u\_{\mathfrak{e}\_t}} \equiv \frac{M\_{t+j}}{M\_t} \mathfrak{e}\_{t+j,t} \tag{16}$$

where *t*+*j*,*<sup>t</sup>* is the pricing kernel under rational expectations. When *j* = 1, this becomes the familiar one-period-ahead stochastic discount factor SDF.

Let *q*ˆ*t*+*j*(*xt*+*<sup>j</sup>* ) be the *j*-period numeraire

$$\mathfrak{q}\_{t+j}(\mathbf{x}^{t+j}) \quad \equiv \quad \mathfrak{P}^j M\_{t+j} \pi\_{t+j}(\mathbf{x}^{t+j}) \frac{\mathfrak{u}\_{\mathfrak{q}\_{t+j}}(\mathbf{x}^{t+j})}{\mathfrak{u}\_{\mathfrak{q}\_0}(\mathbf{x}^0)}, \ \mathfrak{q}\_0 = 1, M\_0 = 1,\tag{17}$$

and define

$$\begin{aligned} \mathfrak{p}\_{t+j,t}(\mathbf{x}\_{t+j}|\mathbf{x}^{t}) & \equiv \frac{\mathfrak{q}\_{t+j}(\mathbf{x}^{t+j})}{\mathfrak{q}\_{t}(\mathbf{x}^{t})} = \beta^{j} \left( \prod\_{i=1}^{j} m\_{t+i} \frac{\mathfrak{u}\_{\mathfrak{c}}(\mathbf{x}^{t+i})}{\mathfrak{u}\_{\mathfrak{c}}(\mathbf{x}^{t})} \right) \mathfrak{m}\_{t+j}(\mathbf{x}^{t+j}) \\ & = \frac{\mathcal{M}\_{t+j}}{\mathcal{M}\_{t}} \mathfrak{q}\_{t+j,t} \mathfrak{m}\_{t+j}(\mathbf{x}^{t+j}) \end{aligned}$$

as the market's distorted *t* + *j* equilibrium price in (13) of an Arrow–Debreu security in terms of consumption at history *x<sup>t</sup>* , or equivalently,

$$\left. \not p\_{t+j,t} (\mathbf{x}\_{t+j} | \mathbf{x}^t) \right| \quad = \left. \not p\_{t+j,t} \pi\_{t+j} (\mathbf{x}^t) \right| \tag{18}$$

which, for *j* = 1, also corresponds to the first-order condition for capital in (14). For future reference, denote the undistorted rational expectations price by

$$p\_{t+j,t}(\mathbf{x}\_{t+j}|\mathbf{x}^t) \quad = \quad \varrho\_{t+j,t}\pi\_{t+j}(\mathbf{x}^t). \tag{19}$$

#### **4. Firms**

The economy's output is produced by a continuum of atomistic firms with a Cobb-Douglas production technology:

$$Y\_t \quad = (1 - D\_l(T\_l - T\_0))F(k\_l, H\_l, E\_l, Q\_l) = (1 - D\_l(Q\_l))k\_l^{\
u}E\_l^{\nu}H\_l^{1-\mu-\nu},\tag{20}$$

where *kt* is the stock of capital, *Ht* is hours of labor, *Et* is the flow of fossil energy, and *Qt* is the remaining stock of carbon energy in the ground. As explained presently, *Dt*(*Qt*) is a damage function measuring the proportion of GDP lost due to the change in average global temperatures *Tt* − *T*<sup>0</sup> since the beginning of the Industrial Revolution at *t* = 0.

Dietz and Venmans (2019) observe that "climate has delivered two important and related insights. First, global warming appears to be approximately linearly proportional to cumulative emissions of carbon dioxide. Second, the temperature response to an emission of CO2 appears to be approximately instantaneous and then constant as a function of time". This conclusion accords with Matthews et al. (2012), and Collins et al. (2013), who earlier defined the *Transient Climate Response to Cumulative Carbon Emissions* (TCRE)19

$$
\lambda\_t^{TC} = \frac{T\_t - T\_0}{Q\_0 - Q\_t}.
$$

,

where 0 <sup>≤</sup> *<sup>Q</sup>*<sup>0</sup> <sup>−</sup> *Qt* <sup>=</sup> <sup>∑</sup>*t*−<sup>1</sup> *<sup>i</sup>*=<sup>0</sup> *Ei* is accumulated carbon emissions since the beginning of the Industrial Revolution.20 The TCRE parameter is generally assumed to be a stochastic variable, due to uncertainties surrounding climate modeling.

The preceding motivates a damage function having the following exponential form

$$D\_l(Q\_l) = \mathbf{e}^{\mathsf{f}\_l(T\_l - T\_0)} = \mathbf{e}^{\mathsf{f}\_l \lambda\_l^{\mathrm{TC}}(Q\_0 - Q\_l)} \equiv \mathbf{e}^{\mathbf{x}\_l(Q\_0 - Q\_l)}, \ \frac{\partial D\_l}{\partial Q\_l} = -\mathbf{x}\_l \mathbf{e}^{\mathbf{x}\_l(Q\_0 - Q\_l)} < 0,\tag{21}$$

where *ζ<sup>t</sup>* is a stochastic parameter that translates the damaging effects of temperature changes into units of GDP, and *xt* = *ζtλTC <sup>t</sup>* combines the damage parameter with the TCRE parameter *λTC <sup>t</sup>* . <sup>21</sup> As posited in Section 2, I shall consider *xt* to be a random variable with either known distribution *πt*(*xt*) or unknown distribution giving rise to ambiguities described earlier. Confining the source of economic damage to a single catch-all variable follows the practice of a number of authors, including Li et al. (2016), who create a single source of model uncertainty with a stochastic variable *γ* that reduces their end-of-period capital stock. Others who have used similar exponential formulations include Golosov et al.'s (2014), and Anderson et al. (2013).<sup>22</sup> To the extent that there exist possibilities of catastrophic tipping points that would likely upend all calculations and planning, the underlying distribution of *xt* may be taken as Knightian and unknowable by either the private sector or the government or both. Tipping points, analyzed by Lemoine and Traeger (2016), and Cai et al. (2013), are abrupt nonlinear climate disruptions that pose a potentially existential threat to humanity in ways that may override concerns with belief and skepticism. A further source of extreme uncertainty is *polar amplification* analyzed by Brock and Xepapadeas (2017).

The typical firm rents capital *kt* from consumers and buys energy *Et* from households in order to maximize an expected infinite stream of profits. Hours of labor *Ht* are supplied inelastically at the going wage 1—a standard assumption in this literature (see Golosov et al. (2014) and Li et al. (2016)).

$$\begin{aligned} &\max\_{k\_t, H\_t, \mathcal{E}\_t} \mathbb{E}\_0 \sum\_{t=0}^{\infty} \hat{\eta}(\mathbf{x}\_t) [Y\_t - r\_t k\_t - H\_t - p\_t^\varepsilon E\_t], \\ &= \max\_{k\_t, H\_t, \mathcal{E}\_t} \mathbb{E}\_0 \sum\_{t=0}^{\infty} \hat{\eta}\_t(\mathbf{x}\_t) \Big( (1 - e^{-\mathbf{x}\_t (Q\_t - Q\_0)}) k\_t^\varepsilon E\_t^\nu H\_t^{1 - \mathbf{a} - \boldsymbol{\nu}} - r\_t k\_t - H\_t - p\_t^\varepsilon E\_t \Big), \end{aligned}$$

where *q*ˆ(*xt*|*x*0) is the belief-distorted and possibly robust numeraire defined earlier in (17). The first-order conditions with respect to {*kt*, *Ht*, *Et*} are

$$r\_{\rm t} = \quad \mathcal{Y}\_{k\_{\rm t}} = a \frac{\mathcal{Y}\_{\rm t}}{k\_{\rm t}} \, \tag{22}$$

$$1\_{\mathcal{I}} = \,^\prime \mathcal{Y}\_{H\_{\mathcal{I}}} = (1 - \mathfrak{a} - \nu) \frac{\mathcal{Y}\_{\mathcal{I}}}{H\_{\mathcal{I}}} \, , \tag{2.3}$$

$$p\_t^\varepsilon \quad = \quad \Upsilon\_{E\_t} = \nu \frac{\mathcal{Y}\_t}{E\_t}. \tag{24}$$

#### **5. Two Versions of Hotelling's Rule**

Denote by *R<sup>e</sup> <sup>t</sup>*+<sup>1</sup> the rate of return to energy, net of the carbon tax *<sup>τ</sup><sup>c</sup> t* ,

$$R\_{t+1}^{\varepsilon} = \frac{p\_{t+1}^{\varepsilon} - \tau\_{t+1}^{\varepsilon}}{p\_t^{\varepsilon} - \tau\_t^{\varepsilon}}.\tag{25}$$

Additionally, since, for either *m* = *m<sup>s</sup>* or *m* = *mc*, (9) and (15) imply

$$p\_t^\varepsilon - \tau\_t^\varepsilon = \beta \mathbb{E}\_t m\_{t+1} \frac{u\_{\varepsilon\_{t+1}}}{u\_{\varepsilon\_t}} (p\_{t+1}^\varepsilon - \tau\_{t+1}^\varepsilon),\tag{26}$$

it follows that

$$\mathbb{E}\_t \mathfrak{p}\_{t+1} \mathbb{R}\_{t+1}^\varepsilon = 1,\tag{27}$$

where expectations are taken with respect to *π*(*x<sup>t</sup>* ), as before. Substituting (24) for period *t* and *t* + 1 in (26) and using the result *p<sup>e</sup>* = *YE* from (24), yields the firm's dynamic rule for optimal energy use, given taxes,

$$\mathcal{Y}\_{\mathbb{E}\_t} - \tau\_t^{\epsilon} \quad = \quad \mathbb{E}\_t \mathfrak{d}\_{t+1} (\mathcal{Y}\_{\mathbb{E}\_{t+1}} - \tau\_{t+1}^{\epsilon}), \tag{28}$$

which is a version of Hotelling's rule that will become relevant later for determining the social cost of carbon.

If *mt*+<sup>1</sup> = 1, the preceding equation confirms Hotelling's original formula that with zero or constant taxes, energy consumption falls over time at the subjective rate of discount. If consumers are climate-skeptic, a multiplier *m<sup>s</sup> <sup>t</sup>*+<sup>1</sup> < 1 effectively lowers the private-sector's discount factor for future benefits of fossil fuel in favor of current returns, leading to increased current consumption relative to rational expectations. If the consumer is pessimistic with *mt*+<sup>1</sup> = *m<sup>c</sup> <sup>t</sup>*+1, energy expenditures are delayed, following the same argument.

Hotelling's (1931) original rule states that the price of an exhaustible resource net of extraction costs should rise at the rate of interest, which, on average, is above the rate of real GDP growth. By Equation (14), the *t*-period Arrow–Debreu price of a claim on a unit of capital is

$$P\_t^{k\_\*} = \mathbb{E}\_t \sum\_{j=0}^{\infty} \hat{\rho}\_{t+j,t} \mathcal{R}\_{t+j\prime}^k \tag{29}$$

while Equation (27) implies that the *t*-period Arrow–Debreu price of a claim on a unit of carbon energy is

$$P\_t^{\varepsilon} \quad := \quad \mathbb{E}\_t \sum\_{j=0}^{\infty} \mathfrak{p}\_{t+j,t} R\_{t+j}^{\varepsilon}. \tag{30}$$

From Equations (14) and (27) follows a version of Hotelling's rule in terms of returns:

$$\begin{array}{ll}\text{V version 1}: & \quad \mathbb{R}^{\varepsilon}\_{t+1} = \mathbb{R}^{k}\_{t+1'} \; \forall t \,, \end{array} \tag{31}$$

while Equations (29) and (30) restate Hotelling's rule in terms of Arrow–Debreu contingent prices:

$$\text{Verson 2}: \qquad P\_{t+1}^{\varepsilon} = P\_{t+1\prime}^{k} \,\,\forall t. \tag{32}$$

Note that in each case, returns and prices are distorted by the consumer's beliefs, be they skeptical *m<sup>s</sup>* or pessimistic *mc*.

These two versions of Hotelling's rule (31) and (32) constitute binding no-arbitrage conditions that require prices and returns to capital and fossil fuel to be equal: in equilibrium, the return to fossil fuel left in the ground for one more period equals the return to the next unit of capital. This rule illuminates two important features of a competitive market for exhaustible energy when there is uncertainty and skepticism regarding the underlying model: (i) the pricing of energy resources continues to obey the laws of asset markets requiring equality of returns to all activities, including capital, bonds, and energy stores, but (ii) the market now uses a martingale-distorted and possibly robust stochastic discount factor to evaluate the expected future returns to all assets.

**Remark 1.** *Hotelling's rule assures efficient allocation but not necessarily socially optimal outcomes if it fails to internalize costs to society created by private economic activity. Later, Section 11 shows how the government can remedy this failure with a public version of Hotelling's rule that includes a social accounting of all costs.*

The next few sections discuss how a Ramsey planner implements competitive equilibrium depending on assumptions about heterogeneity in beliefs, the degree of doubt about the model by either the government or the public, and by how ignorant the planner is about private beliefs.

#### **6. The Ramsey Planner's Constraints**

*6.1. National Income Identity and the Government Budget Constraint*

In all belief regimes considered here, a Ramsey planner commits to policy in period 0 by choosing a competitive equilibrium that maximizes the consumer's expected utility over time. This means the government chooses allocations that satisfy the natural resource constraint (2) and (3) and the national income resource constraint that output exhausts consumption plus investment plus government spending, which, to keep things simple, is assumed to be entirely devoted to a lump-sum transfer *gt*,

$$Y\_t = c\_t + k\_{t+1} - (1 - \delta)k\_t + \mathcal{g}\_t. \tag{33}$$

The government's budget constraint requires that spending and the redemption of bonds from the preceding period be covered by tax receipts, lump-sum transfers, and new issuance of bonds:

$$b\_{t} = -\mathbb{E}\_{t} \mathfrak{p}\_{t+1} b\_{t+1} - \mathfrak{g}\_{t} + \mathfrak{r}\_{t}^{\varepsilon} E\_{t} + \mathfrak{r}\_{t}^{k} (r\_{t} - \delta) k\_{t}.$$

When *bt* < 0, the government is a lender. Solved forward, the preceding equation becomes the government's dynamic budget constraint,

$$b\_{t} \leq \mathbb{E}\_{t} \sum\_{j=0}^{\infty} \boldsymbol{\eta}\_{t+j} [\boldsymbol{\tau}\_{t+j}^{k} (\boldsymbol{r}\_{t+j} - \boldsymbol{\delta}) \boldsymbol{k}\_{t+j} - \boldsymbol{\tau}\_{t+j}^{\varepsilon} (\boldsymbol{Q}\_{t+j+1} - \boldsymbol{Q}\_{t+j}) - \boldsymbol{g}\_{t+j}].\tag{34}$$

Note the added term involving receipts from the carbon tax.
