1. Motivation and Context

Expected Utility (EU) Theory remains the standard theoretical tool for cost-benefit analysis under conditions of risk, particularly within the realm of environmental economics. However, this framework has long been the subject of critical discourse. Beyond the well-known paradoxes highlighted by behavioral economics, such as the Allais or Ellsberg paradoxes, EU theory faces a profound challenge when applied to ethical decisions concerning "catastrophic risks".

The debate has generated more heat than light regarding how society should evaluate risks that entail extremely high losses with negligibly small probabilities—a scenario characteristic of extreme climate change. The "dismal theorem" posits that, within the EU framework, society would be willing to pay an infinite amount to avoid such fat-tailed risks, potentially paralyzing economic progress. This paper investigates whether EU theory can handle such catastrophic risks in an ethically appealing way or if it inevitably leads to implausible outcomes.

2. Theoretical Framework

To disentangle this problem, we analyze the interaction between the probability distribution of catastrophic events and the underlying von Neumann-Morgenstern (vNM) utility function. The analysis assumes a "social planner" or ethical observer evaluating projects where the payoff in a "bad" state (c₂) approaches a lower bound of zero (representing a complete catastrophe, such as extinction), while the probability of this state (p₂) also approaches zero.

Ethical intuition suggests that willingness to pay to avoid such risks should be non-zero but limited—a "sensible" middle ground. However, the mathematical properties of EU theory create a sharp dichotomy based on the boundedness of the utility function.

3. Analytical Findings: The Mechanics of Tyranny

Our formal analysis reveals that EU theory fails to provide a robust framework for balancing catastrophic risks against standard economic trade-offs. We identify two mutually exclusive, yet equally unacceptable, outcomes.

3.1 Formal Proofs of the Dichotomy

To rigorously demonstrate these outcomes, we define a sequence of potentially catastrophic projects P_c⁽ⁿ⁾ where the probability of the catastrophe p_k+1⁽ⁿ⁾ and the payoff in the catastrophic state c_k+1⁽ⁿ⁾ both converge to zero.

Proof Sketch: We choose a sequence of payoffs converging to zero. We define a corresponding sequence of probabilities linked to the utility of these payoffs. As n approaches infinity, the expected utility term for the catastrophic state approaches negative infinity. Consequently, the limit of the entire expected utility is negative infinity, which directly implies that the certainty equivalent goes to zero.

Implication: If the utility function is unbounded below (implying sufficiently high risk aversion, typically η ≥ 1), the evaluation is dominated by the catastrophic event regardless of how small its probability becomes.

Proof Sketch: Since the utility of the catastrophic outcome is bounded, the product of the vanishing probability and the bounded utility must go to zero. Therefore, the expected utility converges solely to the utility of the non-catastrophic states.

Implication: With bounded utility functions, the catastrophic risk eventually has no impact at all on the evaluation, effectively "throwing the baby out with the bathwater".

4. Theoretical Implications

These findings suggest a fundamental deficiency in the standard welfare-economics model when applied to extreme risks. There is an "archimedean point" in the choice of the utility function that determines the outcome. To avoid NCR, one must assume a utility function unbounded below (risk aversion ≥ 1). This level of risk aversion is typically required to ensure intergenerational equity and avoid ethically indefensible savings rates.

However, assuming this plausible level of risk aversion makes TCR unavoidable. Consequently, EU theory forces the ethical observer into a corner: one must either accept an implausibly low degree of risk aversion (leading to negligence) or accept that rare catastrophes will completely dominate economic analysis.

5. Conclusion

We conclude that there are severe limits to expected-utility analysis in the context of climate change and other catastrophic risks. The approach cannot reconcile the belief that "catastrophic risks are important" with the belief that "the price to reduce catastrophic risks is finite". While alternatives to EU theory exist, many tend to weigh catastrophic risks even more heavily, potentially aggravating the tyranny rather than resolving it. Thus, a straightforward and unambiguous way out of this dilemma does not yet seem to exist within the standard economic paradigm.