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Econ-167 · economics · Economics

Persistent climate damages

LLM status Open unverified

Statement

Original subject group: Climate, environment and energy

Audit classification: Background; proposed extension. Required a complete climate intervention and formalized finite persistence tests.

Definitions and precise research target

Editorial formalization. Consider the countries with annual national accounts and temperature records from 1960 through 2025. For country $i$, let $Y_{it}(\delta)$ be real output per resident in year $t$ under a climate intervention that permanently raises its expected annual temperature by $\delta$ degrees Celsius from a common initial year. The intervention preserves the baseline distribution of nonclimate shocks and permits households, firms, and governments to adapt; migration and capital accumulation are therefore outcomes. Let $Y_{it}(0)$ denote the corresponding baseline. For every horizon $h\in\{1,\ldots,50\}$, define \[\theta_h=\left.\frac{\partial}{\partial\delta}E[\log Y_{i,t+h}(\delta)-\log Y_{i,t+h}(0)]\right|_{\delta=0}.\] Assume output is positive and the displayed mean derivative exists. The expectation uses fixed baseline population weights across countries. The question is whether a credible identification strategy and economically disciplined extrapolation can distinguish an approximately constant response $\theta_h$ after a prespecified lag from a response with persistently negative first differences over the stated horizon. Predeclare a tolerance for approximate constancy; every finite sequence is bounded, so boundedness cannot distinguish these cases. Weather fluctuations alone need not identify this intervention. Specify restrictions connecting temporary temperature shocks to permanent climate changes, account for cross-country spillovers, and report uncertainty when those restrictions are relaxed. The target is this finite-horizon response, not a universal claim that every warming episode permanently reduces growth.

Fix a baseline date $t_0$, country set $I$, and weights $\omega_i\geq0$, $\sum_i\omega_i=1$. The climate intervention must specify the joint temperature, precipitation, and extremes process, not only its mean: for example change one declared global physical parameter and map it to every country. Define $g_h(\delta)=\sum_i\omega_iE[\log Y_{i,t_0+h}(\delta)]$ and $\theta_h=g_h'(0)$. Assume positive output and integrability on a neighborhood of zero; differentiability of the expectation is an assumption, not an automatic exchange of derivative and expectation. Fix lag $h_0<50$ and $\epsilon>0$: level persistence means $\max_{h\geq h_0}|\theta_h-\theta_{h_0}|\leq\epsilon$; continuing growth loss means $\theta_{h+1}-\theta_h\leq-2\epsilon$ for all $h=h_0,\ldots,49$. These classes need not exhaust possible response profiles. 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 temporary-shock variant. Replace permanent warming by a one-year temperature perturbation at $t_0$; define $\theta_h^{\rm pulse}=\partial_\delta E[\log Y_{t_0+h}(\delta)-\log Y_{t_0+h}(0)]|_0$. Compare permanent and pulse responses only under a declared linear time-invariant response law.
  2. Editorial adaptation variant. Introduce constrained and unconstrained adaptation regimes $a\in\{0,1\}$, with common climate paths. Identify $\theta_h^1-\theta_h^0$; holding migration/capital fixed changes the target.

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-167-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-167

From weather impulses to persistent climate losses: finite-horizon identification and sensitivity

Problem ID: Q54-163. Status: unresolved; rigorous restricted research attempt.

Target and source scope

The target distinguishes an approximately constant fifty-year response of log output from continuing negative first differences under a complete permanent climate intervention. These are finite-horizon statements; merely calling a response bounded cannot distinguish them. Climate macroeconomics reviews emphasize damage and adaptation [BS]. We establish what can be learned in a finite linear response model, then give a constructive failure of the weather-to-climate extrapolation assumption. This is neither a country-level causal estimate nor a proof of a universal growth effect.

Let $H=50$. A scalar physical intervention index $x_t$ moves a declared joint climate process (temperature, precipitation and extremes) along one fixed direction. It does not merely relabel mean temperature. Let $y_h$ be baseline-country-weighted mean log output at horizon $h$, using fixed weights and positive underlying output. The restricted response law is \[ y_h(x)-y_h(0)=\sum_{j=0}^{h-1}\psi_j x_{h-j},\qquad1\leq h\leq H. \tag{1} \] The coefficients incorporate all adaptation, migration and capital responses in this specified direction. Equation (1) assumes causal linearity, time invariance, no anticipation and invariance across the experimental and permanent regimes. Cross-country spillovers are already included in $y_h$; it would be incorrect to estimate isolated-country coefficients and sum them without this restriction. All sums are finite. One realization of (1) sets country log outputs to an integrable baseline plus this finite linear response, then exponentiates, ensuring strictly positive output.

A unit pulse has $x_1=\delta,x_{t>1}=0$ and derivative per unit $\delta$ equal to $\psi_{h-1}$. A permanent intervention has $x_t=\delta$ for every $t\geq1$ and derivative $\theta_h$. The derivative exists directly from (1), without exchanging an infinite sum or an expectation and derivative.

The exact pulse-to-step reduction

Theorem: Under (1), for every $1\leq h\leq H$, \[ \theta_h=\sum_{j=0}^{h-1}\psi_j,\qquad \theta_{h+1}-\theta_h=\psi_h\quad(1\leq h<H). \tag{2} \] Fix a lag $h_0<H$ and tolerance $\epsilon>0$. The level-persistence condition in the target is equivalent to \[ \max_{h_0\leq h\leq H} \left|\sum_{j=h_0}^{h-1}\psi_j\right|\leq\epsilon. \tag{3} \] Continuing growth loss is equivalent to $\psi_h\leq-2\epsilon$ for every $h=h_0,\ldots,H-1$. If causal pulse responses are known through horizon $m+1$, where $0\leq m<H-1$, and there are no restrictions on the remaining coefficients, then every vector $(\theta_{m+2},\ldots,\theta_H)\in{\mathbb{R}}^{H-m-1}$ is compatible with the known pulse responses.
Proof: Substitute $x_t=\delta$ in (1) and differentiate the resulting finite linear expression. Subtract successive partial sums to get (2), and subtract the $h_0$ partial sum to obtain (3). This also proves the first-difference characterization. For the last assertion, take any proposed later response vector. Set $\psi_{m+1}=\theta_{m+2}-\sum_{j=0}^{m}\psi_j$, and thereafter $\psi_{h-1}=\theta_h-\theta_{h-1}$ for $h=m+3,\ldots,H$. These choices preserve every observed pulse coefficient and give exactly the prescribed step responses. They define an admissible finite response law and positive output by the construction above. Thus compatibility is constructive rather than an assertion based only on counting parameters. ∎

Sharp bounded extrapolation and decisive certificates

Suppose outside evidence restricts the unobserved coefficients to $\psi_j\in[-M\rho^{j-m-1},M\rho^{j-m-1}]$ for $j=m+1,\ldots,H-1$, where $M\geq0$ and $0\leq\rho<1$ are sensitivity inputs. No data-driven choice of these constants is claimed. More generally let every coefficient have an interval $[l_j,u_j]$, including singleton intervals for known ones. Assume these restrictions are independent across coefficients.
Proposition (Sharp joint set and finite-horizon certification): Under (1), the joint identified set is the image of $\prod_{j=0}^{H-1}[l_j,u_j]$ under the invertible cumulative-sum map (2). For every $h$, its marginal interval is $[\sum_{j<h}l_j,\sum_{j<h}u_j]$. In the geometric-tail model, for $h\geq m+2$ this is \[ \left[\theta_{m+1}-M\frac{1-\rho^{h-m-1}}{1-\rho},\quad \theta_{m+1}+M\frac{1-\rho^{h-m-1}}{1-\rho}\right]. \tag{4} \] All compatible models satisfy level persistence if and only if, for every $h=h_0,\ldots,H$, \[ \sum_{j=h_0}^{h-1}l_j\geq-\epsilon, \qquad \sum_{j=h_0}^{h-1}u_j\leq\epsilon. \tag{5} \] All compatible models satisfy continuing growth loss if and only if $u_j\leq-2\epsilon$ for all $j=h_0,\ldots,H-1$.
Proof: Every coefficient vector in the box is admissible by assumption and constructs a response law; the converse is part of the class definition. Cumulative summation is invertible by successive differencing, proving the joint characterization. The minimum and maximum of a sum with coefficient one occur at the lower and upper box corners. Continuous interpolation attains intervening scalar values. Summing the geometric bounds gives (4). For (5), every relevant partial sum ranges exactly between the displayed lower and upper sums. Containment of each such interval in $[-\epsilon,\epsilon]$ is sufficient. If any containment fails, the corner attaining that offending endpoint is an admissible model violating persistence, so containment is necessary. Equation (2) and the same endpoint argument prove the growth-loss statement. ∎
Existence of some compatible persistent model is a different question from (5). It is a finite linear feasibility problem with the box restrictions and the two inequalities in (3) at each horizon. Likewise existence of a compatible growth-loss model adds $\psi_j\leq-2\epsilon$ to the box. These feasibility reductions follow directly from the exact joint set; no asymptotic approximation or independent-coordinate confidence claim is being made. If coefficient intervals form a simultaneous $1-\alpha$ confidence region under a separately justified sampling design, these universal certificates make a false certification with probability at most $\alpha$: on the coverage event the true vector is among those certified. This implication needs no independence across the estimated coefficients.

Why all weather impulses may still be insufficient

Proposition (An observationally invisible climate channel): There exist two causal positive-output models agreeing on every finite weather-path experiment in which only weather is manipulated, but having arbitrarily different responses to the permanent climate intervention, even when both responses exist at zero and satisfy no anticipation.
Proof: Let $w_t$ be a mean-zero weather deviation and $c_t$ a physical persistent climate state, with observed annual temperature $T_t=w_t+c_t$. Fix precipitation and extremes at their baseline processes in this declared intervention. Thus setting $c_t=\delta$ raises expected annual temperature by exactly $\delta$ while preserving the weather law. All available weather experiments set $c_t=0$ but may assign arbitrary finite paths to $w_t$. In model A, set log output at date $h$ to $b_h+\sum_{j<h}\psi_j w_{h-j}$. In model B add $\sum_{j<h}\eta_j c_{h-j}$, where $\eta_j$ are arbitrary real constants. Take deterministic finite $b_h$, or integrable random baselines common to the models. Exponentiating gives positive outputs; expected log outputs are finite. Every available experiment yields exactly the same observable law in the two models, not just the same mean. Define the complete climate intervention to set $c_t=\delta$ permanently while preserving the baseline weather distribution. It changes model A's mean log output by zero and model B's by $\delta\sum_{j<h}\eta_j$. Choosing $\eta_j$ by successive differencing creates any desired finite response sequence. Both models use current and past states only, and their derivatives are immediate. ∎
The extra state may represent persistent ocean conditions or an announced long-run climate distribution to which investment responds. This is a logical counterexample, not evidence that this omitted channel exists. It locates the assumption doing the work: the direction manipulated by weather evidence must span the complete climate intervention, with an invariant response law. Longer weather records alone do not prove that.

Remaining empirical and economic gaps

Equations (2)--(5) solve a finite identification and sensitivity problem conditional on causal pulse coefficients and invariance. Identifying those coefficients from the 1960--2025 observational panel still requires credible exogeneity or instruments, physical intervention definitions, treatment of spatial dependence, and a joint uncertainty region. Endogenous adaptation outside the historical support can break (1). Uncertain $M,\rho$ should be reported as sensitivity ranges, not estimated structural facts without additional evidence. The two persistence classes do not exhaust all profiles. The full target remains unresolved.

The NBER direct page was inaccessible during this attempt. Its indexed primary record and the AEA forthcoming record/abstract were checked on 9 October 2026. The source below supports the broad agenda only; no fifty-year theorem or coefficient bound is attributed to it. No numerical experiment is needed for the finite algebraic results.

References

Completion Estimate

35%

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