1. Motivation and Context

Since the Stern Review, a large share of climate-economics debate has centered on the “appropriate” discounting of future welfare. That debate matters. But it can also obscure a second structural fault line: how welfare economics handles catastrophic risks—outcomes that are extremely unlikely yet potentially civilization-ending.

Two influential critiques appear to point in opposite directions. Chichilnisky argues that expected utility can be structurally insensitive to rare events. Weitzman’s “dismal” logic shows the reverse: with fat tails and unbounded utility, willingness to pay to reduce catastrophe can blow up, swamping all other considerations.

Buchholz and Schymura show that both intuitions can be correct: the knife-edge sits in the utility specification. Under plausible curvature, EU can force an ethical observer into a corner—either the Tyranny of Catastrophic Risks (TCR) or theNegligence of Catastrophic Risks (NCR).

Ethical intuition tends to demand something in between: take existential risk seriously, but within limits. Individuals do not reduce all activity to eliminate a minute chance of death; analogously, a society should not accept a certainty equivalent near zero just to shave an infinitesimal probability off a remote catastrophe.

2. Theoretical Framework

Consider an ethical observer evaluating uncertain consumption outcomes under the von Neumann–Morgenstern axioms. Let c denote (aggregate) consumption, and normalize the absolute catastrophe to the lower bound c̲ = 0. The observer’s preferences are represented by a strictly increasing, concave utility function u(c).

The pivotal distinction is whether u(c) is bounded from below as c → 0. Isoelastic (CRRA) specifications fall into two qualitatively different classes: unbounded utility (η ≥ 1, including log utility) versus bounded utility (0 < η < 1).

3. Analytical Findings: The Mechanics of Tyranny

The analysis yields two mutually exclusive outcomes—and a deeper generalization that links directly to fat-tailed uncertainty.

3.1 Formal Proofs of the Dichotomy

Define a sequence of potentially catastrophic projects P⁽ⁿ⁾ with a single catastrophic state. Let p_cat⁽ⁿ⁾ → 0 and c_cat⁽ⁿ⁾ → 0.

Proof sketch: choose c_cat⁽ⁿ⁾ ↓ 0 and define probabilities p_cat⁽ⁿ⁾ as a function of −u(c_cat⁽ⁿ⁾). Because u is unbounded below, −u(c) can grow arbitrarily large as c → 0. One can pick p_cat⁽ⁿ⁾ that still goes to zero but slowly enough that p_cat⁽ⁿ⁾·u(c_cat⁽ⁿ⁾) → −∞, collapsing expected utility.

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: if u is bounded below, u(c_cat⁽ⁿ⁾) stays finite. Then p_cat⁽ⁿ⁾·u(c_cat⁽ⁿ⁾) → 0 as p_cat⁽ⁿ⁾ → 0, and the catastrophe term vanishes.

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".

Interpretation: this generalizes the instability beyond a single rare catastrophe. Under fat-tailed uncertainty, it is possible for “average” outcomes to be well-behaved while expected utility is dominated by tail realizations.

4. Theoretical Implications

The parameter η is not a cosmetic choice. In intertemporal welfare economics it governs both risk aversion and inequality aversion, so “plausible” calibration ranges are often motivated by distributional ethics. But those same calibrations can push the model into the unbounded regime.

This yields a trap: to escape tyranny (TCR), one can bound utility (η < 1), but then the model becomes insensitive to vanishing catastrophe probabilities (NCR). To escape negligence, one can make utility unbounded (η ≥ 1), but then catastrophic tails can dominate the evaluation.

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.