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Econ-180 · economics · Economics

Economic consequences of climate tipping points

LLM status Open unverified

Statement

Original subject group: Climate, environment and energy

Audit classification: Background; proposed extension. Specified irreversible hazard timing, population weights, equilibrium selection and model-set bounds.

Definitions and precise research target

Editorial formalization. Consider a calibrated world economy with a finite set of regions $r$, annual periods $t=0,\ldots,100$, and one specified physical tipping event, such as a major ice-sheet transition. Let $T_t$ be global mean temperature, $S_t\in\{0,1\}$ the irreversible event indicator, and $q(T_t)$ its conditional transition probability when $S_t=0$. Consumption $C_{rt}>0$, adaptation investment $A_{rt}\geq0$, capital, and migration are equilibrium outcomes. A stated damage map links the event to sea-level rise and regional production. For discount factor $\beta\in(0,1)$, positive regional welfare weights $w_r$, and increasing concave utility $u_r$, define the incremental expected welfare loss \[L=E\!\left[\sum_{t=0}^{100}\beta^t\sum_r w_r\{u_r(C_{rt}^{0})-u_r(C_{rt}^{1})\}\right].\] Superscripts denote otherwise identical economies respectively excluding and including the event risk; the expectation integrates physical and economic shocks. Determine an identified or transparently bounded range for $L$ using defensible transition probabilities and endogenous adaptation. Separate uncertainty about physical hazards from uncertainty about economic propagation, and test whether allowing migration, insurance, and correlated regional damages reverses policy rankings. This is a proposed quantitative model comparison for one event, rather than a claim that tipping points lack existing economic analyses.

Use a filtered probability space and a physical state $X_t$. Define $P(S_{t+1}=1\mid\mathcal F_t,S_t=0)=q(X_t)\in[0,1]$ and $S_{t+1}=1$ whenever $S_t=1$; temperature-only hazards are a restriction. Declare carbon dynamics, sea-level mapping, production, household budgets, adaptation technology, and a measurable equilibrium-selection rule. Couple the risk/no-risk experiments by common exogenous shocks, setting the hazard to zero in regime 0. Regional consumption is per capita, with fixed person weights or explicitly modeled population weights; utility sums must use the same population accounting convention. Require $E\sum_t\beta^t\sum_r w_r|u_r(C_{rt}^a)|<\infty$. For admissible calibrated models $\mathcal M$, the target interval is $[\inf_{M\in\mathcal M}L(M),\sup_{M\in\mathcal M}L(M)]$; neither sign nor narrowness is assumed. All expectations use a stated baseline population and probability law. Identification means every admissible data-generating model with the same observed-data law yields the same displayed target; otherwise report the identified set. Exact equations, horizons and assumptions are editorial, rather than source-given canonical conjectures.

Variants and assumption changes

  1. Editorial interacting-events variant. Replace $S$ by $S\in\{0,1\}^J$ and specify joint transition probabilities, allowing one event to change another's hazard. Compare total $L$ with summed single-event losses; equality is not assumed.
  2. Editorial ambiguity variant. Replace $P$ by a declared nonempty set $\mathcal P$ and report $\inf_{P\in\mathcal P}L_P,\sup_{P\in\mathcal P}L_P$, separating hazard and equilibrium-selection ambiguity.

References and source audit

1. Climate macroeconomics survey; no exact fifty-year or tipping-event conjecture verified. Locator: NBER working-paper record and abstract; AEA forthcoming record, DOI 10.1257/jel.20261831. Primary indexed record verified; direct NBER/PDF retrieval failed. AEA forthcoming abstract inspected; no final journal publication year asserted. URL: https://www.nber.org/papers/w33567.

2. SCC synthesis discusses damage persistence, tipping, and nonmarket substitution; proposed estimands are editorial. Locator: Introduction; Section 1.1; Section 2. Full text or primary publication page inspected. URL: https://pmc.ncbi.nlm.nih.gov/articles/PMC11670083/.

  1. Adrien Bilal; James H. Stock. 2025. A Guide to Macroeconomics and Climate Change. NBER working-paper record and abstract; AEA forthcoming record, DOI 10.1257/jel.20261831. https://www.nber.org/papers/w33567. DOI: https://doi.org/10.3386/w33567.
  2. Frances C. Moore; Moritz A. Drupp; James Rising; Simon Dietz; Ivan Rudik; Gernot Wagner. 2024. Synthesis of evidence yields high social cost of carbon due to structural model variation and uncertainties. Introduction; Section 1.1; Section 2. https://pmc.ncbi.nlm.nih.gov/articles/PMC11670083/. DOI: https://doi.org/10.1073/pnas.2410733121.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-180-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-180

Irreversible tipping risk with endogenous protection: sharp hazard bounds and the value of adaptation information

Problem ID: Q54-176. Status: unresolved; rigorous restricted research attempt.

Target and tractable economic restriction

The target measures the incremental welfare cost of one physical tipping event while allowing adaptation, migration and uncertain propagation. Climate macroeconomics treats these as important economic adjustments [BS]. We solve a finite protective-investment problem with endogenous adaptation and hazard ambiguity. Its simple consumption account permits complete proofs. Endogenous migration, capital prices and general-equilibrium feedbacks are absent; the result is a restricted reduction, not a numerical estimate of any real tipping event.

Fix $T=100$ and $\beta\in(0,1)$. Regional weights $w_r>0$ are fixed person weights. Consumption utility is linear, $u_r(c)=c$, an increasing concave special case. Regions have deterministic positive endowments sufficiently large to cover all losses and protection expenditures specified below, so consumption is strictly positive in every state. An irreversible event starts from $S_0=0$ and has known-date conditional hazards $q_s\in[0,1]$, $s=0,\ldots,T-1$. Independent uniform variables $U_s$ common to the compared models implement \[ S_{s+1}=S_s+(1-S_s){\mathbf 1}\{U_s\leq q_s\}. \tag{1} \] The hazards can represent a fixed temperature/physical-state path. Their independence from protection and economic choices is a substantive restriction. No-risk comparison sets every $q_s=0$ with the same endowments.

A social decision maker chooses one protection plan $a$ from a finite nonempty set $\mathcal A$ before the uniforms are revealed. It prescribes real adaptation expenditure with discounted weighted cost $K(a)\geq0$ and residual event damage $D_t(a)\geq0$ in date-$t$ weighted consumption units. The menu includes $a^0$ with $K(a^0)=0$. Plans have no benefits when the event never occurs. All costs and damages are finite. Consumption losses are exactly the real cost and event damage; no financial transfers are counted a second time. Linear utility and fixed weights make these sums an exact utility difference in this restricted economy. The finite-menu optimal plan is a behavioral selection; ties can be resolved by a fixed index rule without affecting the value.

Optimal loss and sharp rectangular hazard bounds

Write \[ F_t(q)=1-\prod_{s=0}^{t-1}(1-q_s),\quad F_0=0, \quad J(a,q)=K(a)+\sum_{t=1}^T\beta^tD_t(a)F_t(q). \tag{2} \]
Theorem: The optimal incremental welfare loss relative to the no-risk optimum is \[ L(q)=\min_{a\in\mathcal A}J(a,q)\geq0. \tag{3} \] It is continuous and nondecreasing in every hazard coordinate. Suppose the admissible physical models are exactly the nonempty box $\mathcal Q=\prod_s[\underline q_s,\overline q_s]\subset[0,1]^T$, with the protection technologies held fixed. Its sharp loss set is \[ [L(\underline q),L(\overline q)]. \tag{4} \] In particular the endpoint protection plans are reoptimized, not held at the plan optimal for an arbitrary central hazard vector.
Proof: By (1), remaining untipped through date $t$ requires $U_s>q_s$ for $s<t$. Independence gives probability $\prod_{s<t}(1-q_s)$, proving (2). Under no risk, the least attainable cost is zero because costs are nonnegative and $a^0$ is available. For any plan, expected utility loss under risk is therefore $J(a,q)$. Optimality and finiteness of the menu give (3), including attainment and nonnegativity. Each $F_t$ is a continuous polynomial and is nondecreasing in each $q_s$. Since $D_t(a)\geq0$, every $J(a,q)$ has these properties. If $q\leq q'$ coordinatewise then $J(a,q)\leq J(a,q')$ for each $a$; taking minima preserves the inequality. A finite minimum of continuous functions is continuous, proving the stated properties of $L$.

Every admissible loss lies between the two corner losses by monotonicity, and both corners are admissible. The line segment $q(v)=(1-v)\underline q+v\overline q$, $v\in[0,1]$, stays in the box. Continuity and the intermediate value theorem give every loss between the endpoints. Thus the range is exactly (4), not only a pair of outer bounds. ∎

The same corner reasoning applies to independent interval restrictions on $K(a)$ and $D_t(a)$ if their endpoints remain feasible positive-consumption technologies. Correlations among hazards, damage parameters and adaptation costs invalidate a Cartesian-box sharpness claim. One must retain those joint restrictions rather than taking individually convenient endpoints.

Proposition (Marginal hazard cost at a stable adaptation choice): Suppose a particular $a^*$ uniquely minimizes $J(a,q)$ at some $q\in(0,1)^T$. Then $L$ is differentiable in a neighborhood of $q$ and \[ \frac{\partial L}{\partial q_s}(q) =\sum_{t=s+1}^{T}\beta^tD_t(a^*) \prod_{\substack{0\leq v<t\\v\ne s}}(1-q_v) \geq0. \tag{5} \] At an adaptation tie differentiability is not asserted.
Proof: There is a strictly positive value gap between the unique minimizer and every other plan, unless the menu contains only that plan. Continuity and finiteness imply that all gaps stay positive on a sufficiently small neighborhood. On that neighborhood $L(q')=J(a^*,q')$. Differentiate its finite polynomial expression (2). Terms with $t\leq s$ do not depend on $q_s$; each remaining derivative is the product in (5). Every factor and damage is nonnegative, proving the sign. The singleton case is immediate. ∎
Endogenous protection enters (5) through the chosen plan and its residual damage. There is no separate derivative of a discrete optimal plan while that plan is locally unchanged. At a tie the minimum may have a kink.

Information about event timing and contingent protection

Let $\tau\in\{1,\ldots,T,\infty\}$ be the first tipped date, and let $p_\tau$ be its probabilities under (1). For any plan define its realized loss \[ \ell(a,\tau)=K(a)+\sum_{t=\tau}^T\beta^tD_t(a),\qquad \ell(a,\infty)=K(a). \tag{6} \] Consider an intentionally strong information benchmark: the decision maker learns $\tau$ before choosing a plan, at no information cost and with the same feasible menu. This is not a feasible real-time adaptation policy; it bounds the benefit of any weaker signal that only informs the choice of an ex-ante plan.
Proposition (Exact perfect-information value): Define \[ L_{\rm PI}=\sum_\tau p_\tau\min_{a\in\mathcal A}\ell(a,\tau). \] Then $0\leq L_{\rm PI}\leq L(q)$, and equality $L_{\rm PI}=L(q)$ holds if and only if at least one plan minimizes $\ell(a,\tau)$ for every $\tau$ with $p_\tau>0$. If a signal $Z$ is observed before choosing from the same menu, its optimal expected loss $L_Z$ satisfies $L_{\rm PI}\leq L_Z\leq L(q)$ whenever ignoring $Z$ is feasible.
Proof: For every fixed $a$, the pointwise minimum in (6) is no greater than $\ell(a,\tau)$. Taking expectations and minimizing over constant $a$ gives $L_{\rm PI}\leq L(q)$, and all losses are nonnegative. Let $a^*$ attain the finite minimum defining $L(q)$. The gap is \[ L(q)-L_{\rm PI}=\sum_\tau p_\tau [\ell(a^*,\tau)-\min_a\ell(a,\tau)]. \] All terms are nonnegative, and there are finitely many. The sum is zero if and only if every positive-probability term is zero. This proves necessity of a common optimizer; its sufficiency follows by using it. For a signal-dependent decision $a(Z)$, each realized loss is at least $\min_a\ell(a,\tau)$, so its expectation is at least $L_{\rm PI}$. Ignoring the signal attains $L(q)$; optimizing can only lower that loss. A minimizing rule exists by choosing the lowest conditional expected loss among finitely many plans, with a fixed measurable tie rule. ∎
True after-event adaptation has different timing and often different feasible actions. The proposition is valid for it only after encoding complete feasible contingent plans in $\mathcal A$ and keeping that menu fixed across the information comparison. It must not be read as permission to choose early investments using future shocks.

A transparent sign restriction and remaining gaps

Nonnegative loss in (3) depends on nonnegative real costs and damages, unchanged baseline resources, and no protection co-benefits. It is not a sign theorem for the catalogue's unrestricted equilibrium models. Adding innovation, migration gains, insurance distortions or altered policy responses can change the comparison. Likewise the deterministic hazard path restriction excludes feedback from economic emissions to tipping probabilities. The numerical sizes of $q_s$ and $D_t(a)$ are not identified by the algebra. Historical nonoccurrence of an event does not determine its conditional hazard or economic propagation.

A serious application would separately bound the physical transition law, validate regional residual-damage maps, model migration with consistent population weights and solve the selected risk and no-risk equilibria. The finite theory shows exactly how an independently justified rectangular hazard set propagates through endogenous protection, and why information and adaptation flexibility have distinct values. It leaves the full identified range and calibrated policy rankings unresolved.

The indexed NBER record and primary AEA forthcoming record/abstract of [BS] were inspected on 9 October 2026; the direct NBER fetch failed. Only the broad loss, mitigation and adaptation agenda is attributed to that source. The finite model and all proofs above are supplied here and make no empirical or priority claim.

References

Completion Estimate

20%

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