**2. Multiplier Preferences**

The representative consumer and the government share a reference probability model, given a joint density *π*(*x<sup>t</sup>* ) of the history of shocks *<sup>x</sup><sup>t</sup>* = *<sup>x</sup>*0, ··· , *xt*, where, as detailed later in Section 4 (see Equation (21)), *xt* is a climate-induced damage shock to the production economy at time *t*. The consumer and the government do not necessarily agree that *π* is the true probability. The government may doubt the model fearing the worst, and the private sector may either have similar doubts or be skeptical in some arbitrary way. Either may then choose an alternative model via some distortion of *π* in a manner described by Hansen and Sargent (2001, 2005, 2008), who invoke the Radon-Nikodym theorem to express any alternative model as as a non-negative measurable mapping *π*ˆ(*x<sup>t</sup>* ) = *Mt*(*x<sup>t</sup>* )*π*(*x<sup>t</sup>* ), with E*tMt* = 1, where, since uncertainty is realized in *<sup>t</sup>* = 0, *<sup>M</sup>*<sup>0</sup> = 1.<sup>9</sup> Being the unconditional likelihood ratio *Mt*(*x<sup>t</sup>* ) = *<sup>π</sup>*ˆ*t*(*x<sup>t</sup>* ) *<sup>π</sup>t*(*xt*) of an alternative density to *<sup>π</sup>*ˆ*t*(*x<sup>t</sup>* ), *Mt*, is a martingale with respect to the reference model *<sup>π</sup>*, E*tMt*+<sup>1</sup> = *Mt*, with the interpretation of a change in measure. The *distorted expectation* of *xt*+1, given history *x<sup>t</sup>* , is<sup>10</sup>

$$\mathbb{E}\_t\left[\mathbf{x}\_{t+1}|\mathbf{x}^t\right] = \mathbb{E}\_t\left[\frac{\mathcal{M}\_{t+1}}{\mathcal{M}\_t}\mathbf{x}\_{t+1}\right].$$

As described in more detail below, disbelief may take two forms, either as skepticism or as pessimism, the latter being a manifestation of worst-case fears. Importantly, throughout, any alternative model *π*ˆ(*x<sup>t</sup>* ) is assumed to be absolutely continuous with respect to the reference model *π*(*x<sup>t</sup>* ). 11

Later, it will be convenient to decompose *Mt* by defining the conditional likelihood ratio *mt*(*xt*+1) <sup>≡</sup> *<sup>M</sup>*(*xt*+1) *<sup>M</sup>*(*xt*) , such that <sup>E</sup>*tmt*+<sup>1</sup> <sup>=</sup> 1.

### *2.1. Skepticism (Random Belief Distortion)*

As indicated in the introduction, for the purpose of this paper, skepticism refers to an arbitrary rejection of the extant approximating model *π*(*x<sup>t</sup>* ) in favor of some other model *Mtπ*(*x<sup>t</sup>* ), where *Mt* is a random variable with properties previously set out, including the assumption of absolute continuity with respect to the true distribution *πt*(*x<sup>t</sup>* ), which means that households can be skeptics but not outright climate change deniers.

#### *2.2. Pessimism (Ambiguity Aversion)*

Pessimism or model doubt refers to a worst-case belief distortion *π*ˆ*t*(*x<sup>t</sup>* ) = *Mtπt*(*x<sup>t</sup>* ) derived from the consumer's having solved a min-max problem shown later.

Following Hansen and Sargent (2008), define discounted *relative entropy* conditional on date zero information as the distance *υ*0(*π*ˆ*t*, *πt*) between *π*ˆ*<sup>t</sup>* and *π<sup>t</sup>* associated with *Mt* over time-*t* information and over an infinite horizon as

$$\begin{aligned} w\_0(\boldsymbol{\pi}\_t, \boldsymbol{\pi}\_t) &= \quad (1 - \beta) \mathbb{E}\_0 \sum\_{t=0}^{\infty} \beta^t M\_t \log M\_t \\ &= \quad \beta \mathbb{E}\_0 [\sum\_{t=0}^{\infty} \beta^t M\_t \mathbb{E}\_t \frac{M\_{t+1}}{M\_t} (\log M\_{t+1} - \log M\_t)], \\ &= \quad \beta \mathbb{E}\_0 [\sum\_{t=0}^{\infty} \beta^t M\_t \mathbb{E}\_t m\_{t+1} \log m\_{t+1}], \end{aligned}$$

where *β* is the subjective discount factor. With this definition, an agent's ambiguity about *π* is represented by a set of joint densities {*Mt*}<sup>∞</sup> *<sup>t</sup>*=<sup>0</sup> satisfying the constraint,

$$\beta \mathbb{E}\_0 M\_t \mathbb{E}\_t [m\_{t+1} \log m\_{t+1}] \le \eta\_t \tag{1}$$

where *η* > 0. In the following, a pessimistic consumer will choose a consumption plan subject to the constraint in (1).12

#### **3. Households**

#### *3.1. CRRA Preferences*

Consumers derive utility from consumption *ct*, given a constant-elasticity preference function *u*(*ct*), 13

$$\begin{aligned} \mathfrak{u}(c\_t) &= \begin{array}{c} c\_t^{1-\gamma} \\ \hline 1-\gamma' \end{array} 0 < \gamma < 1, \\ &= \ \log c\_t; \ \gamma \to 1, \end{aligned}$$

with elasticities *cc* = −*uccc*/*uc* = *γ* (implying constant relative risk aversion).

If *γ* = 1 (logarithmic preferences), *cc* = 1. For the record, the constant intertemporal elasticity of substitution for consumption is 1/*γ*. I will assume that *γ* ≤ 1, which accords with much of the literature on long-run risk—the kind this paper is most concerned with.14

#### *3.2. The Household's Budget Constraint*

The household owns three assets: (1) the stock of depreciating capital with a net yield [(<sup>1</sup> − *<sup>τ</sup><sup>k</sup> <sup>t</sup>* (*x<sup>t</sup>* ))(*rt*(*x<sup>t</sup>* ) − *<sup>δ</sup>*)]*kt*(*xt*−1), where *rt* is the real rental rate on capital *kt*(*xt*−1) left over from last period, and *τ<sup>k</sup> <sup>t</sup>* is the tax on capital; (2) a government bond *b*(*x<sup>t</sup>* ), defined as an Arrow–Debreu security promising one unit of consumption in period *t* + 1, if the state is *xt*+<sup>1</sup> and zero otherwise, and (3) the resource *Qt* of fossil fuels from which it draws *Et* units every period, according to the law of motion<sup>15</sup>

$$Q\_{t+1} = Q\_t - E\_{t\prime} \tag{2}$$

which, by the assumption of exhaustibility, implies

$$\sum\_{t=0}^{\infty} E\_t \le Q\_0. \tag{3}$$

The household sells *Et* to the firm at a price *p<sup>e</sup> <sup>t</sup>* with after-tax revenue (*p<sup>e</sup> t*(*x<sup>t</sup>* ) − *τe <sup>t</sup>* (*x<sup>t</sup>* ))*Et*(*x<sup>t</sup>* ), where *τ<sup>e</sup> <sup>t</sup>* is an excise (carbon) tax per unit of energy.

The household receives income from (1) inelastically supplied labor *Ht* at the competitive wage *wt* = 1, (2) rent from capital, (3) revenues from the sale of fossil energy, and (4) a lump-sum transfer from the government *gt*. It spends its resources on consumption *ct*, new capital *kt*+1, and the purchase of a new Arrow security *bt*+1, trading at the state-contingent

price *<sup>p</sup>*ˆ*t*+1(*xt*+1|*x<sup>t</sup>* ) to be defined later. Summarizing, the household's one-period budget constraint is

$$\begin{aligned} 0 &\le \quad H\_t(\mathbf{x}^t) + b\_t(\mathbf{x}^t) + \mathbf{g}\_t(\mathbf{x}^t) + \mathbf{R}\_t^k(\mathbf{x}^t)k\_t(\mathbf{x}^{t-1}) \\ &\quad + \quad (p\_t^\varepsilon - \mathbf{r}\_t^\varepsilon)[Q\_l(\mathbf{x}^t) - Q\_{l+1}(\mathbf{x}^t)] - c\_l(\mathbf{x}^t) - k\_{l+1}(\mathbf{x}^t) - \mathbb{E}\_l \boldsymbol{\mathcal{p}}\_{l+1} b\_{l+1}(\mathbf{x}^{t+1}) \\ &\equiv \quad \mathcal{L}\_t(\mathbf{x}^t), \end{aligned} \tag{4}$$

where *R<sup>k</sup> t*(*x<sup>t</sup>* ) = <sup>1</sup> + (<sup>1</sup> − *<sup>τ</sup><sup>k</sup> <sup>t</sup>* )(*rt*(*x<sup>t</sup>* ) − *δ*) is the after-tax gross return to capital.

#### *3.3. The Consumer's Maximization Problem*

This section studies the two types of consumer introduced earlier: those who are skeptical of the model and form some arbitrary belief distortion *M* to the true distribution *π* according to Section 2.1, and those who are pessimistic and play a game against a malevolent force to determine a worst-case value for *M* following Section 2.2. It is convenient to set this problem up for the latter and then show the former to be a special case.
