Expected utility and catastrophic risk in a stochastic economy–climate model

Abstract

We analyze a stochastic dynamic finite-horizon economic model with climate change, in which the social planner faces uncertainty about future climate change and its economic damages. Our model (SDICE*) incorporates, possibly heavy-tailed, stochasticity in Nordhaus’ deterministic DICE model. We develop a regression-based numerical method for solving a general class of dynamic finite-horizon economy–climate models with potentially heavy-tailed uncertainty and general utility functions. We then apply this method to SDICE* and examine the effects of light- and heavy-tailed uncertainty. The results indicate that the effects can be substantial, depending on the nature and extent of the uncertainty and the social planner’s preferences.

Introduction

The current economy–climate debate raises many difficult issues. Only one of these is discussed in the current paper, namely the question if and how abatement, consumption, and investment policies are affected by catastrophic risk. Economy–climate policies are typically analyzed using Integrated Assessment Models (IAMs) that describe the complex interplay between climate and the economy. Our paper augments a widely used deterministic IAM by incorporating (potentially heavy-tailed) risk related to future climate change and its associated economic damage, and analyzes its impact on the policy variables.

Our model is based on Nordhaus’ (Nordhaus, 2017a, Nordhaus, 2017b) dynamic integrated model of climate and the economy (DICE), which has become an important benchmark IAM, not only in the theoretical literature but also serving as a tool for economy–climate policy analysis by the US government. The DICE model is our starting point and the deterministic version of our model reduces to DICE. To represent uncertainty and motivated by the developments in Manne and Richels (1992), Nordhaus (1994), Roughgarden and Schneider (1999), Kelly and Kolstad (1999), Keller et al. (2004), Mastrandrea and Schneider (2004), Leach (2007), Weitzman (2009), and in particular Ackerman et al. (2010), we introduce to DICE random shocks featuring potentially heavy-tailed risk. We refer to the resulting stochastic model as SDICE*. We initially focus on uncertainty through the damage-abatement fraction and, later, in an extension of this base stochastic model, we shall also account for uncertainty in the damage parameter, an uncertain emissions-to-output ratio, and uncertainty in technological efficiency.

To solve the stochastic dynamic economy–climate model thus obtained, we embed the associated optimization problem into a general class of stochastic dynamic finite-horizon optimization problems. We next develop a regression-based method for solving such problems. Our solution method is flexible in the sense that it allows for a wide class of utility functions and that it imposes only weak assumptions on the stochasticity, permitting both light- and heavy-tailed risks and stochastic parameters.

In the context of SDICE* we show formally that heavy-tailed risk is only compatible with some utility functions, and in particular that it is not compatible with power utility. To do so, we invoke the general decision-theoretic results of Ikefuji et al. (2015) and apply these to the current setting. We propose to use the Pareto utility function to represent preferences in the presence of heavy-tailed risk. This utility function was introduced by Ikefuji et al. (2013) and advocated by Cerreia-Vioglio et al. (2015). Pareto utility avoids the drawbacks ‘near the edges’ that standard families of utility functions such as power and exponential utility exhibit, and is particularly suited for heavy-tailed risk analysis.

Our four main findings are as follows. First, the introduction of light-tailed uncertainty through the damage-abatement fraction of SDICE* leads to a reduction of abatement, while optimal consumption and investment are relatively less affected. Conditional upon the shocks realizing their expected value, namely zero, the pattern remains the same in later periods, and this applies to both power and Pareto utility. Although lower abatement has a negative propagation effect in our economy–climate model, the social planner sacrifices some abatement to maintain consumption and investment, under uncertainty about the damage-abatement fraction. Compared to a Pareto utility maximizer, the power utility maximizer has a stronger motive to avoid a low damage-abatement reduced output, and therefore consumes less in early periods and invests and abates more in all periods. The changes remain small as long as the shocks take values close to their expectation, that is, in the ‘center’ of the distribution.

Second, when the light-tailed shocks take larger negative values, the optimal policies are more affected: pronounced differences occur in the optimal policy and state variables at the ‘edges’, both within and between models. In particular, a power utility maximizer has a stronger motive to abate as a precaution (Kimball, 1990) than a Pareto utility maximizer, and this amplifies with adverse realizations of the shocks. This effect is the result of a trade-off between maintaining current consumption and taking intensified precautionary action under adverse circumstances. We find that the power utility maximizer tends to favor a relatively larger substitution from current consumption to intensified precautionary action when compared to Pareto utility. This effect is distinct from the higher levels of abatement observed for a power utility maximizer in the center of the distribution (and under the deterministic model) which it further amplifies.

Third, allowing for heavy-tailed uncertainty making catastrophic risk scenarios more pronounced, our first main finding broadly remains valid under Pareto utility, while our second main finding gets reinforced, with power utility becoming incompatible in this case. Indeed, for a power utility maximizer, the expectation of the intertemporal marginal rate of substitution becomes infinite when considering heavy-tailed uncertainty in the SDICE* model.

Fourth, in the center of the distribution the impact of uncertainty in the damage-abatement fraction dominates the impact of uncertainty in the damage parameter and an uncertain emissions-to-output ratio and closely resembles the impact of uncertainty through technological efficiency. At the edges, an uncertain damage parameter impacts consumption, but leaves abatement nearly unaffected, while uncertainty in the emissions-to-output ratio significantly impacts abatement but has relatively little effect on consumption. Furthermore, when adverse scenarios for technological efficiency realize, optimal abatement is suppressed compared to the adverse scenarios in which the damage-abatement fraction is relatively small.

Although there are many papers on climate policy under uncertainty, the literature on the interplay between climate and the economy under uncertainty is much smaller. The existing IAMs which explicitly include uncertainty can be divided into three classes: (i) stochastic dynamic IAMs with learning, but no consideration of catastrophe; (ii) deterministic IAMs considering catastrophe; and (iii) stochastic dynamic IAMs considering tipping points.

In class (i), Kelly and Kolstad (1999) explore Bayesian learning about the relationship between greenhouse gas levels and global mean temperature changes, analyze when uncertainty is resolved, and show that the expected learning time depends on the variance of the temperature realizations and varies directly with the emission policy. Extensions of Kelly and Kolstad (1999) are provided in Keller et al. (2004), Leach (2007), and Traeger (2014). Jensen and Traeger (2014b) study the effects of climate sensitivity uncertainty, learning, and temperature stochasticity separately, and find precautionary savings in the presence of the stochasticity of temperature, while Bayesian learning about climate sensitivity raises the abatement rate and hence the optimal carbon tax.

In class (ii), Mastrandrea and Schneider (2004), Ackerman et al. (2010), Dietz (2011), Hwang et al. (2013), and Gillingham et al. (2015) study the implication of catastrophic risks in IAMs. These papers focus on examining the shape of the damage function and the climate sensitivity parameter. We mention in particular the relevant contribution by Ackerman et al. (2010), who analyze the impact of parameter uncertainty in the specification of the damage function and/or in the temperature equation on the optimal policies. Their approach consists in first simulating the parameter(s) of interest by drawing from a prespecified probability distribution, and then deterministically solving DICE for each realization of the parameter(s), thus obtaining a ‘distribution’ of the optimal policies. This approach provides an assessment of the sensitivity and robustness of the optimal policies to parameter assumptions within the context of a deterministic economy–climate model. Also, Gillingham et al. (2015) conduct an extensive Monte Carlo analysis for six IAMs, to analyze how model output responds to model misspecification due to parameter uncertainty, by estimating surface-response functions. The current paper, in contrast, solves a stochastic optimization problem. Our social planner takes potentially heavy-tailed stochasticity in the damage-abatement fraction (and the damage parameter, the emissions-to-output ratio and technological efficiency, in extensions of the model) already into account when solving for the optimal policies.

In class (iii), Lemoine and Traeger (2014), Lontzek et al. (2015), Cai et al., 2012, Cai et al., 2016, and Berger et al. (2017) explore how the risk of stochastically uncertain environmental tipping points affects climate policy, using a stochastic IAM based on the DICE model. Berger et al. (2017) adopt non-expected utility preferences to accommodate aversion to both risk and ambiguity when analyzing tipping elements in climate change, employing a two-period model in which uncertainty resolves in 2100. The paper by Cai et al. (2012) is particularly relevant for us. They extend conventional economy–climate analysis based on deterministic IAMs to allow for a range of stochastic features. In particular, they conduct an extensive analysis of carbon emission policies in a stochastic environment. A key distinction between their work and ours is that they only allow shocks with a bounded probability distribution, thus ruling out the normal or the Student distribution, in order to avoid catastrophic risk scenarios (‘tail events’). In contrast, risks with unbounded support and potentially featuring heavy tails, as well as the catastrophic risk scenarios they may induce, are at the heart of our analysis.

Our paper also relates to the literature on numerical methods for dynamic programming and stochastic optimal control. The algorithm that we develop for solving SDICE* is inspired by the Least Squares Monte Carlo (LSMC) approach introduced by Longstaff and Schwartz (2001); see also Carriere (1996), Clément et al. (2002), and Powell (2011) for further details, including convergence results. LSMC has been successfully applied to a variety of problems in financial economics and operations research; see e.g., Brandt et al. (2005), who use LSMC to solve a portfolio choice problem with non-standard preferences, Laeven and Stadje (2014), who solve problems of optimal portfolio choice and indifference valuation under general asset price dynamics and in the presence of model uncertainty using LSMC, and Krätschmer et al. (2018), who employ LSMC to analyze model uncertainty in optimal stopping.

The paper is organized as follows. In Section 2 we succinctly summarize DICE. In Section 3 we introduce uncertainty into DICE. In Section 4 we provide a formal description of a general class of stochastic dynamic finite-horizon economy–climate models, allowing for heavy-tailed uncertainty and general utility functions and embedding SDICE* as a special case, and develop a regression-based method for solving such models. In Section 5 we show, in the context of our model, that heavy-tailed uncertainty is not compatible with all utility functions, in particular power utility, and propose an alternative utility function: Pareto utility. In Section 6 we present the results of our SDICE* model and discuss their implications, while some extensions are presented in Section 7. Section 8 concludes.