**1. Introduction**

*"The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt." Bertrand Russell*

That global warming poses an existential threat to the future of humanity and the planet is now a universally accepted scientific fact. Yet significant segments of society still harbor doubt and skepticism about what scientists tell us, with some, including a previous US Administration, either minimizing the threat or denying it altogether, and others fearing the worst.1

In the minds of many, doubt or skepticism is justified by uncertainties surrounding climate science, including the extent of terrestrial carbon uptake, mankind's role in this, the relationship between carbon and temperature, and, ultimately any claimed economic damages.2 The nature and causes of resistance to accepting bad news and environmental threats have been much discussed in the literature (see Meyer and Kunreuther (2017), Kunreuther et al. (1978), Kaufmann et al. (2017)). At the simplest level, disbelief may be motivated by myopic economic self interest, such as a refusal to contemplate the possibly enormous costs of carbon abatement, and an unwillingness to accept significant taxation on fossil energy (see Lifton (2017)). Alternatively, the much publicized threats of pending climate-related disasters have likely contributed to a fear in some sectors of society that the worst of climate catastrophes will ensue. For the analyst, some humility may be in order,

**Citation:** von zur Muehlen, Peter. 2022. Prices and Taxes in a Ramsey Climate Policy Model under Heterogeneous Beliefs and Ambiguity. *Economies* 10: 257. https://doi.org/10.3390/ economies10100257

Academic Editor: Ralf Fendel

Received: 2 July 2022 Accepted: 8 October 2022 Published: 17 October 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

because, as Millner et al. (2010) have noted, "our knowledge of the climate system is not of sufficiently high quality to be described with unique probability distributions, and that formal frameworks that account for aversion to ambiguity are normatively legitimate". In a recent paper, Pindyck (2017) warned that excessive reliance on Integrated Assessment Models (IAMs), first introduced by Nordhaus (1993), "create a perception of knowledge and precision that is illusory, possibly fooling policy-makers into thinking that the forecasts that models generate have some kind of scientific legitimacy. The same goes for any illusion that the probability distributions underling uncertainty can even be known". It is this last thought that mostly motivates the present paper.

For purposes of exposition, I shall give specific meaning to the terms *skepticism* and *doubt*. Skeptics are said to be individuals who distort their probability assessments in favor of different outcomes than predicted by science. Because such beliefs are arbitrary, they are also, in principle, unknown to the government, thereby creating an ambiguity for the policy maker (see Hansen and Sargent (2012)). By doubt I shall mean the kind of pessimism regarding the trustworthiness of a given model that Hansen and Sargent (2012) have described as emanating from a belief that nature is likely to produce worst-case outcomes. Although the literature has focused solely on ambiguities inhabiting the policy authority, both, government and private agents, may be pessimistic regarding the climate model. Pessimists in this paper make decisions by playing a game against an imagined malevolent opponent. A powerful and rational motive for pessimism is the prospect of tipping points that Dietz et al. (2021) have described as the most important source of uncertainty, one capable of throwing off all modeling descriptions and justifying preparedness for the worst.3 In this paper, all pessimistic players use Hansen and Sargent's (2008) min-max strategies to compute their policies, meaning they are revealed to the authority, assumed to be a Ramsey planner, in the form of additional implementability constraints under which such a planner must operate.

The preceding sources of ambiguity give rise to four potential belief regimes:

	- (a) Political planner: The government strategically assumes private beliefs to be true. Its ignorance of private beliefs produces ambiguity.
	- (b) Paternalistic planner: The government trusts and adheres to the true model. However, the presence of unknown private beliefs creates ambiguity for policy.
	- (c) Pessimistic planner: Having doubts about the model and facing unknown private beliefs, the planner confronts two sources of ambiguity.
	- (a) Pessimistic planner: The planner's own doubts doubts about the model and the private sector's pessimistic doubts that constrain the Ramsey planner's policy produce two sources of ambiguity.

The framework for studying policy with ambiguity is based on work by Hansen and Sargent (2005, 2007, 2008), a general rationale for pursuing robust climate policy under deep uncertainty having been provided by Workman et al. (2021). In their theory, deviations of private-sector beliefs from some approximating or reference distribution—the true scientific distribution—are represented as martingale multiplier distortions *M*, later defined as likelihood ratios having well defined properties. A justification for this approach to modeling ambiguity is based on a theorem by Strzalecki (2011), who axiomatized the robust control criterion of multiplier preferences introduced by Hansen and Sargent (2001), relating them to other classes of preferences studied in decision theory, in particular, the variational preferences introduced by Maccheroni et al. (2006), and proving them to be equivalent to multiplier distortions of probabilities.

As indicated earlier, the study of ambiguity in the context of climate policy is not without precedent, but discussions have generally been limited to cases when only the planner has doubts about the approximating climate-economic model. Such doubts include concerns about potential mis-specification of alternative models and ambiguity over

how much weight to assign to each of these models, while agents themselves are usually assumed to have rational beliefs (see Millner et al. (2012), Brock and Durlauf (2015), Cai et al. (2013), Cai and Lontzek (2019), Anderson et al. (2013), Berger et al. (2016), Li et al. (2016), Lemoine and Traeger (2016), Rezai and van der Ploeg (2017), and Barnett et al. (2020)). Hennlock's (2009) is no exception in that, although he attributes deep uncertainty to the consumer, the government, being a direct extension of the consumer, remains the consumer's sole agent, so that, in effect, it is the planner who is modeled as having doubts about the model. In a bit of a twist to the approaches taken by other researchers, Rezai and van der Ploeg (2017) studied the implications of adopting max-min, max-max, and min-max regret policies when the planner faces alternative models ranging from science-based paradigms to denialist imaginings, concluding that max-min or min-max regret climate policies that rely on a non-skeptic view of global warming lead to a substantial and moderate amount of caution, respectively, while max-max policies produce policies that do not match the beliefs of climate skeptics. Later, Rezai and van der Ploeg (2019) applied a version of *Pascal's wager* and asked: what would an agnostic but rational planner—one who does not know or care which model is correct—do when faced with some probability that the approximating model, adhered to by so-called deniers, is false? Their conclusions are briefly described in Section 13.

The literature on optimal climate policy has generally followed the tradition of welfare analysis based on expected utility maximization within the framework of an integrated climate assessment model, with government defined as a social planner (vid. Golosov et al. 2014) seeking to maximize the expected welfare of society unconstrained by private decisions or market outcomes. An alternative is to assume that the government is a Ramsey planner, likewise seeking to maximize consumer welfare, akin to the authority introduced by Chari et al. (1994) to study optimal dynamic capital taxation, but under constraints imposed by market equilibrium.4 This paper studies both versions of government, in which each type of planner must acknowledge the possibly distorted beliefs held by the private sector.

In its essentials, the description of the economy here follows the recent literature on optimal carbon taxation, foremost among them Nordhaus (2008), Acemoglu et al. (2012), von Below (2012), Golosov et al.'s (2014), van der Ploeg and Withagen (2014), and Belfiori (2017, 2018). The analytical framework is a by now familiar dynamic stochastic integrated general equilibrium (DSIGE) model similar to those in Anderson et al. (2013) and Golosov et al. (2014), which in turn are based on RICE—Regional Dynamic Integrated Model of Climate and the Economy—developed by Nordhaus (1993, 2008, 2007).

The government's fiscal policy tools include bond finance and taxes on carbon and capital. I include a tax on capital because within a Ramsey planning framework, capital and Pigouvian carbon taxation are tightly linked: (1) the government's stochastic discount factor for the return to capital and for the expected social cost of carbon (SCC) is the same, and (2) the SCC and taxes on capital and carbon are influenced by the same shadow price.5

In essence, this paper will show how the planner's ignorance of the model or of skeptical private beliefs creates an endogenous gap between the government's and the household's discount factors, leading to a gap in their respective Arrow–Debreu pricing of carbon and capital that contributes to an *ambiguity premium* over the standard certaintyequivalent formulation of the expected social cost of carbon.6 This finding is related to recent papers by von Below (2012), Barrage (2018), and Belfiori (2017), who proved in different contexts that the optimal tax on capital is negative, and the optimal tax on carbon is higher than the standard Pigou rate, if the government's subjective discount rate is *exogenously* lower than the public sector's.7 The underlying reason is that climate change decreases the returns to capital, so that individuals, who are too impatient from a social point of view, i.e., skeptical, do not save enough without a capital subsidy and, at the same time, burn too much fossil fuel unless the latter is taxed sufficiently. The optimal policy response is therefore to tax capital less and to tax carbon more.<sup>8</sup> In the reverse case, a pessimistic public motivated to over-invest and to under-utilize carbon may justify

lower carbon and higher capital taxation, unless the government is also pessimistic. In this paper, any disparity in discount factors between the government and the private sector is endogenously driven, in this instance by heterogeneity in beliefs, fear of mis-specification, and ambiguity.

The next section provides formal definitions of pessimism and skepticism as understood in this paper, using multiplier preferences introduced by Hansen and Sargent (2001, 2005, 2008). Section 3 derives the Euler conditions for a consumer who may be skeptical (Section 3.5) or pessimistic (Section 3.4), as defined in Section 2. Section 4 presents a three-factor production function subject to damaging climate-related total productivity shocks in a model of a firm renting capital and purchasing energy from the household. Section 5 uses results from Sections 3.3, 3.6, and 4 to derive two versions of Hotelling's rule. Subject to constraints derived in Section 6, Section 7 presents the Ramsey planner's Euler conditions for the three belief regimes under study. Section 8 derives the possibly distorted equilibrium prices of carbon damage and capital. In anticipation of the main results, Section 9 comments on this paper's methodology and approach, which contain some innovations. Section 10 derives the expected social cost of carbon, including an ambiguity premium that governments in all four policy/belief regimes will implement. Section 11 establishes the conditions under which the planner may or may not impose an additional ambiguity-related carbon tax premium over the social cost of carbon. Section 12 presents a number of conditions under which a planner may or may not raise the subsidy rate on capital, where it will become apparent that such conditions mirror those that drive results for the carbon tax premium. Finally, before the paper's conclusion, Section 14 describes a reverse feedback from taxation to beliefs whereby a planner facing an economy with pessimistic agents is able to manipulate debt and taxes to affect pessimistic beliefs.

Throughout, references to state-conditioned distorted (including robust) Arrow–Debreu prices reflect the basic theme in this paper: that a Ramsey planner's Pigouvian tax policy under ambiguity is able to implement allocations via equilibrium pricing of an underlying asset with unknown returns that have an equivalence in a cap and trade economy. In this respect, this paper is most closely related to Barnett et al. (2020), who use asset pricing methods not only to impute market valuations but also to ascertain social valuations, as this paper intends. As in Barnett et al. (2020), the asset prices in this paper can be viewed as equivalent to shadow prices of the expected discounted values of stochastic processes impinging on the economy.

Climate policy that is motivated by a planner's own deep or Knightian uncertainty is an interesting and important topic and has been fairly exhaustively treated in the literature. Furthermore, as has also been shown elsewhere in the literature, the effects of ambiguities originating from the planner turn out not always to be clear-cut, depending on specific features related to preferences and returns in the economy. More importantly, as maintained in this paper, an equally or possibly more urgent issue for policy must be the role of beliefs held in the private sector, because they affect consumption and investment decisions, where heterogeneity between private and government beliefs will surely impact policy. That public acceptance of climate science and its policy prescriptions are not unanimous is uncontroversial and has been well documented. However, little attention has been paid to its implications, particularly its effects on ambiguity in optimal climate policy. By altering inter-temporal rates of substitution or pricing kernels that determine consumption decisions and household wealth, belief distortions in the private sector impact a planner's implementability constraints and become more salient by the addition of ambiguities that arise when those beliefs are unknowable to the government. Whatever deep uncertainty may or may not already inhabit the mind of a planner, a welfare maximizing policy authority would be remiss in ignoring the effects of private-sector belief heterogeneity and associated ambiguities on its own policy decisions.

This paper then is the first systematic attempt to analyze the policy implications of ambiguities arising from belief heterogeneity in the private sector regarding the nature of anthropomorphic climate change. Significantly, this paper distinguishes between mere skepticism producing myopic behavior in the economy and true doubt as manifested by worst-case fears leading to increased foresight. The analytic choice to investigate the implications of belief heterogeneity and associated ambiguities by evaluating the planner's Euler conditions follows the example of Anderson et al. (2013). However, the particular approach leading to a role for second-order moments as factors in optimal policy is an innovation of this paper. The covariances between multiplier distortions and variables in the economy that arise as relevant to policy will be shown to allow more detailed descriptions of the effects of ambiguity on the social cost of carbon and on carbon and capital taxation. As will become clear later, ambiguities resulting from belief heterogeneity and distortions in the private sector produce ambiguity-related premiums on the social cost of carbon and, separately, on the carbon tax and on capital subsidies.
