1. Motivation and Context

Since the Stern Review, a large share of climate-economics debate has been dominated by 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 (a bias against catastrophe-prevention). 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’s contribution is to 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), where the catastrophic state dominates regardless of how small its probability becomes, or the Negligence of Catastrophic Risks (NCR), where the catastrophic state becomes irrelevant.

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 (extinction / subsistence collapse). 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: lim_c→0 u(c) = −∞ (e.g., log utility, CRRA with η ≥ 1)
• Bounded utility: lim_c→0 u(c) > −∞ (e.g., CRRA with 0 < η < 1)

To formalize “vanishing” catastrophe risk, consider a sequence of projects where both the probability of catastrophe p_cat⁽ⁿ⁾ → 0 and the catastrophic payoff c_cat⁽ⁿ⁾ → 0. The question is what happens to the certainty equivalent m_u(P⁽ⁿ⁾) defined implicitly by u(m) = EU(P).

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 then pick p_cat⁽ⁿ⁾ that still goes to zero but slowly enough that p_cat⁽ⁿ⁾·u(c_cat⁽ⁿ⁾) → −∞. Expected utility collapses to −∞, hence the certainty equivalent m → 0.

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 from expected utility.

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 the expected utility sum is dominated by tail realizations—leading to a kind of reverse St. Petersburg paradox.

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. In that sense, Weitzman and Chichilnisky diagnose different sides of the same structural deficiency.

5. Conclusion

Expected utility remains a powerful, elegant framework—but in the presence of catastrophic risks it can become normatively unstable. The core problem is not merely “fat tails” or “parameter uncertainty”, but the interaction between tail behavior and the axiomatic structure of utility near catastrophe.

If policy evaluation is forced to choose between paralysis (tyranny) and indifference (negligence), then the model is not mapping ethical intuition onto decision rules. Any practical solution—whether via bounded utility, explicit truncations, or alternative welfare criteria—should be judged by whether it delivers a finite, non-zero, and robust valuation of catastrophe prevention.