Significance Risk and uncertainty are important in pricing climate damages. Despite a burgeoning literature, attempts to marry insights from asset pricing with climate economics have largely failed to supplement—let alone supplant—decades-old climate–economy models, largely due to their analytic and computational complexity. Here, we introduce a simple, modular framework that identifies core trade-offs, highlights the sensitivity of results to key inputs, and helps pinpoint areas for further work.

Abstract Pricing greenhouse-gas (GHG) emissions involves making trade-offs between consumption today and unknown damages in the (distant) future. While decision making under risk and uncertainty is the forte of financial economics, important insights from pricing financial assets do not typically inform standard climate–economy models. Here, we introduce EZ-Climate, a simple recursive dynamic asset pricing model that allows for a calibration of the carbon dioxide ( C O 2 ) price path based on probabilistic assumptions around climate damages. Atmospheric C O 2 is the “asset” with a negative expected return. The economic model focuses on society’s willingness to substitute consumption across time and across uncertain states of nature, enabled by an Epstein–Zin (EZ) specification that delinks preferences over risk from intertemporal substitution. In contrast to most modeled C O 2 price paths, EZ-Climate suggests a high price today that is expected to decline over time as the “insurance” value of mitigation declines and technological change makes emissions cuts cheaper. Second, higher risk aversion increases both the C O 2 price and the risk premium relative to expected damages. Lastly, our model suggests large costs associated with delays in pricing C O 2 emissions. In our base case, delaying implementation by 1 y leads to annual consumption losses of over 2%, a cost that roughly increases with the square of time per additional year of delay. The model also makes clear how sensitive results are to key inputs.

For over 25 y, the dynamic integrated climate–economy (DICE) model (1⇓–3) has been the standard tool for analyzing CO 2 emissions-reductions pathways, and for good reason. One attraction is its simplicity, turning a “market failure on the greatest scale the world has seen” (4) and “the mother of all externalities” (5) into a model involving fewer than 20 main equations, 3 representing the climate system (6). DICE has spawned many variants (7). It has also helped set the tone for what many consider “optimal” CO 2 price paths. The core trade-off between economic consumption and climate damages leads to relatively low CO 2 prices today rising over time.

DICE and models like it have well-known limitations, including how they represent climate risk and uncertainty (7⇓⇓⇓⇓⇓⇓⇓–15). DICE, for example, is not an optimal-control model, as commonly understood by economists employing modern dynamic economic analysis, even though it lends itself to those extensions (9⇓⇓–12). The underlying structure all but prescribes a rising CO 2 price path over time.

One important limitation is the form of the utility function. Constant relative risk aversion (CRRA) preferences, standard in most climate–economy models (1, 7, 16), assume that economic agents have an equal aversion to variation in consumption across states of nature and over time. Evidence from financial markets suggests that this is not the case (17). The risk premium (RP) of equities over bonds points to a fundamental difference in how much society is willing to pay to substitute consumption risk across states of nature compared to over time (18, 19). Some have explained the discrepancy by allowing for extreme events (20⇓–22), and others have looked to more flexible preferences (23⇓⇓–26) or both (27). Our own preference specification follows Epstein and Zin (EZ) (24, 25).

EZ Preferences Here, we use EZ preferences and focus on climate uncertainties. We approach climate change as an asset pricing problem with atmospheric CO 2 as the “asset.” The value of an investment in reducing CO 2 emissions depends on the state of nature, represented by its fragility θ t . That, in turn, helps determine the discount rate applied to the damages that would have occurred without the investment. Our representative agent maximizes a recursive utility U t based on consumption c t and expectations E t over future utility for times t ∈ 0,1,2 , … , T − 1 : U t = 1 − β c t ρ + β E t U t + 1 α ρ α 1 ρ . [1]Parameters α and ρ measure the agent’s willingness to substitute consumption across states of nature and across time, respectively. (See Methods for the final-period utility U T and further derivations.) CRRA preferences are a special case, with α = ρ . Unlike with CRRA, Eq. 1 implies that CO 2 prices no longer collapse to zero with increasing risk aversion (RA) and equity risk premia (Fig. 1A). The same goes for the portion of CO 2 prices explained by RA (Fig. 1B). Fig. 1. Risk calibration. A shows how using EZ preferences, unlike CRRA, results in increasing 2015 CO 2 prices, in 2015 US$, with increasing RA, translated into the implied equity RP using Weil’s conversion (19), while holding implied market interest rates stable at 3.11%. B shows how the percentage of the 2015 CO 2 price explained by RA, as opposed to expected damages (EDs), increases with equity RP for EZ utility, while decreasing for CRRA (Risk Decomposition). EZ preferences have since found their way into the climate–economic literature (9⇓⇓–12, 28⇓⇓⇓⇓⇓⇓–35). Some have embedded EZ into DICE (28, 35), and others employ supercomputers to solve (9⇓⇓–12). The complexity typically does not allow for analytic solutions (34). We here follow a simple binomial-tree model with a long history in financial modeling application (36). It is precisely this modeling choice—standard in financial economics but novel to climate–economic applications—that leads to our fundamentally differing CO 2 price paths. Mitigating climate risk provides a hedge, leading to high CO 2 prices early on. As uncertainties decline over time, so do CO 2 prices.

Model The setting for EZ-Climate is a standard endowment economy (37). In each period, the agent is endowed with a certain amount of the consumption good c ¯ t . However, she is not able to consume the full c ¯ t for 2 reasons: climate change and climate policy. In periods t ∈ 1,2 , … , T , a portion of c ¯ t may be lost due to climate-change damages, which are, in turn, a function of cumulative radiative forcing ( C R F t ) up to time t and of fragility θ t : D t CRF t , θ t . Up to period T − 1 , the agent may elect to spend some of c ¯ t to reduce her impact on the future climate, κ t , which, in turn, depends on mitigation x t . The resulting consumption c t , after D t and κ t are taken into account, is given by: c 0 = c ¯ 0 1 − κ 0 x 0 , [2] c t = c ¯ t 1 − κ t x t 1 − D t C R F t , θ t , for t ∈ 1,2 , … , T − 1 , and [3] c T = c ¯ T 1 − D T ( C R F T , θ T ) . [4] C R F t is a function of mitigation, x s , in each period from 0 to t, calibrated to a combination of Representative Concentration Pathway (RCP) scenarios (Climate Damages). The agent maximizes utility given by Eq. 1 in each of T periods by selecting, at each time and in each state, a level of mitigation x t , creating in essence a “ 2 T + 1 − 1 ”-dimensional optimization problem. Fig. 2A shows our base case, which uses a 7-period tree with decision nodes from 2015 through 2300. At each node, more information about θ t and the resulting climate damages is revealed, before uncertainty is resolved at the beginning of the penultimate period in 2300. In the “easy” spirit of EZ-Climate, the limited number of decision points makes the solution both tractable and quickly solvable. Fig. 2. Model tree structure. A shows a diagram of the binomial tree structure (with probability p = 1 2 ) used in solving the model for each state of nature θ t across time t ∈ 1,2 , … , T , corresponding to years 2015, 2030, 2060, 2100, 2200, 2300, and final period 2400. Note the “recombining” tree structure, highlighted in the first 2 periods: Damage functions in any particular state are independent of the path taken, but x t and the resulting U t are path-dependent. B shows the CO 2 price in the base case ( p e a k T = 6 , d i s a s t e r t a i l = 18 , and E I S = 0.9 ). EZ-Climate provides an accessible, modular framework (7) that is dependent on key economic inputs—chiefly, RA and the elasticity of intertemporal substitution (EIS)—and 2 main climate-related ones: mitigation costs and climate damages. Costs depend primarily on assumptions around backstop technologies (38) and technological change. Damages depend on the full climate–economic chain from economic output to CO 2 emissions, from emissions to concentrations, from concentrations to C R F , and from C R F to climate damages lowering economic consumption. While all of these calibrations are important, and uncertainties abound, a key addition is allowing for potentially catastrophic risk in form of climatic tipping points (TPs) (39, 40) (Climate Damages).

Results and Discussion Fig. 2B shows CO 2 prices for each node of the tree in our base case. The 2015 CO 2 price comes from a single node in the tree. In each subsequent period, the price is set in expectation over all possible states of nature θ t in that given period. All grouped nodes at a given time have the same θ t and, thus, the same damage for a given amount of C R F . The price itself is path-dependent. Fig. 2B also shows the costs associated with bad θ t draws in latter periods. Bad news is costly. Bad news late, when it is more difficult to counteract with more active policy, is worse. It is precisely the inability to know upfront when good or bad news arrives that accounts for the insurance value of early mitigation and, thus, the role that the resolution of risk over time plays in the declining CO 2 price. Declining CO 2 Price. Unlike most modeled CO 2 price paths, ours typically rise briefly before declining over time. One partial explanation is the move from CRRA to EZ preferences. CRRA preferences duplicate the decline only with a RA = 10 9 ∼ 1.1 , when EZ collapses to CRRA. For higher RA, consistent with those estimated from models calibrated to financial-market data, early CRRA prices collapse to near zero (Fig. 3A). But going from CRRA to EZ preferences is not the only explanation, implied by the fact that CRRA price paths stay flat over time. Fig. 3. Declining CO 2 price paths. A shows how EZ utility here leads to CO 2 prices that start high and decline over time, regardless of assumed RA, a feature mimicked only by unrealistically low RA = 10 9 ∼ 1.1 , when EZ and CRRA utilities coincide (Economic Parameters). B and C show the importance of EIS and the rate of pure time preference (δ), respectively. B varies real interest rates from 2.74% (EIS = 1.2) to 3.77% (EIS = 0.6) to keep c ¯ = 1.5 % . C fixes EIS at 0.9, while δ varies from 0.25% to 0.75% (SI Appendix, Figs. S5 and S6). Others have pointed to reasons for declining CO 2 price paths including producer behavior (41), the need for directed technological change from “dirty” to “clean” sectors (42), or inertia (43). We here find 2 factors driving the declining CO 2 price paths: the resolution of uncertainty, combined with technological progress that makes mitigation significantly cheaper over time. Our base case assumes exogenous technological progress φ 0 = 1.5 % and endogenous progress φ 1 = 0.015 per year, linked to average mitigation efforts to date (Eq. 19). The combination makes mitigation costs diminishingly small hundreds of years out, helping to drive the declining price paths (Fig. 4A and SI Appendix, Fig. S3). Fig. 4. CO 2 price sensitivities. A shows the implications of technological change and TP assumptions. Setting φ 0 = 0 % φ 1 = 0 increases early-year and final-period prices, flattening the price path. Multiple TPs act akin to fattening the tail of the damage function, steepening the price path. They also interact with the no-technological-change assumption, increasing final-period prices. B shows that 2015 CO 2 prices depend crucially on “catastrophic” climate risk assumptions, set to p e a k T = 6 ○ C and d i s a s t e r t a i l = 18 in the base case (Climate Damages). C, by contrast, shows the minimal implications of extending the final period from 2300 to 2400 for t = 6 and 7, respectively, in the base case to 2400 and 2700. Another reason for declining price paths is the assumed nature of TPs in the base case. Each node has a certain probability of hitting a TP, given by Eq. 22. Once hit, there is no reversing the resulting damages. That structure increases prices in early years, decreasing them later, as it introduces a nonconcavity into the damage function (37). Allowing for multiple TPs exacerbates that result in the base case, as it fattens the tail of the damage function (Fig. 4A). Assuming no technological change, meanwhile, increases final-period prices, more so with multiple TPs. While the declining CO 2 price path is a persistent feature across model specifications (SI Appendix, Figs. S3, S5, and S6), the absolute CO 2 price in early years depends crucially on a number of calibration choices. Fig. 3 shows the importance of economic parameters, chiefly, EIS and the pure rate of time preference (δ). Fig. 4B shows the sensitivity of the initial CO 2 price to assumptions around “catastrophic” climate risk. Our base case assumes 6 °C for the “peak temperature” ( p e a k T ) and 18 for the d i s a s t e r t a i l calibrations. While there is seeming convergence around 6 °C as an upper bound for what could conceivably be quantified (see, for example, https://helixclimate.eu/), declaring it equivalent to a “global TP” is at best unduly conservative (11, 15, 40), at worst arbitrary. Much more work is needed to justify any one particular parameter value and, thus, any one CO 2 price (7). Our goal with EZ-Climate is to provide a simple, modular framework to think about climate risks, uncertainties, TPs, and their implications for CO 2 prices. Social Cost of Delay. The optimal-control nature of EZ-Climate also allows for a calculation of the social cost of delay in implementing CO 2 prices. Unlike prior efforts (2, 7), we do not look to the CO 2 price for estimating that cost. In fact, doing so can be positively misleading. After constraining the price to $0 in the first period, the price in the second period is lower than in the unconstrained case. The price reflects the marginal benefits of additional emissions reductions, which are now lower. We here instead quantify the cost of delay by constraining mitigation to zero in the first period and asking how much additional consumption would be required during that period in order to bring the utility of the representative agent to the level of the unconstrained solution. Table 1 shows the annual consumption loss during the constrained first period. For a 10-y delay, the equivalent annual consumption loss over the first constrained period is ∼23%: Each year of delay increases the annual consumption loss over the entire constrained period by ∼2.3%. It also increases the time interval of the loss, thus leading to a slightly more than quadratic rate of increase in the deadweight loss of utility over time. In rough monetary terms, delaying implementation by only 1 y costs society approximately $1 trillion. A 5-y delay creates the equivalent loss of approximately $24 trillion, comparable to a severe global depression. A 10-y delay causes an equivalent loss in the order of $10 trillion per year, approximately $100 trillion in total. Table 1. Social cost of delay by first-period length

Conclusion Our conclusion could mimic that of DICE, introduced over 25 y ago (1), with one crucial difference: Like with DICE, and despite crucial recent advances (7, 35), “it should be emphasized that this analysis has a number of important qualifications,” especially, ironically, “the economic impact of climate change” (1). Unlike DICE, EZ-Climate does not “[abstract] from issues of uncertainty” (1). It embraces them, following a simple binomial-tree framework long used in the finance literature (36). The simple, modular framework also highlights the sensitivity of CO 2 prices to key inputs. There is no single, correct, “optimal” price path. One persistent feature, however, is declining price paths. That puts the focus on near-term action and on the large costs of delay.

Acknowledgments For helpful comments and discussions, we thank Jeffrey Bohn, V. V. Chari, Don Fullerton, Ken Gillingham, Christian Gollier, William Hogan, Christos Karydas, Dana Kiku, Gib Metcalf, Robert Socolow, Adam Storeygard, Christian Traeger, Martin Weitzman, Richard Zeckhauser, Stanley Zin, and seminar participants at American Economic Association meetings; the Environmental Defense Fund; ETH Zürich; Global Risk Institute; Harvard; the Journal of Investment Management conference; New York University; Tufts; University of Illinois Urbana–Champaign; and University of Minnesota. We also thank Oscar Sjogren, Weiyu Wan, and Shu Ye for helping prepare our code for distribution via https://gwagner.com/EZClimate.

Footnotes Author contributions: K.D.D., R.B.L., and G.W. designed research, performed research, contributed new analytic tools, and wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1817444116/-/DCSupplemental.