we-solve (beta)

Econ-105 · economics · Economics

Clean-technology policy sequences

LLM status Open unverified

Statement

Original subject group: Climate, environment and energy

Audit classification: Background; proposed extension. Specified nonanticipating policies and expectation constraints for the stochastic optimization.

Definitions and precise research target

Editorial formalization. Consider an economy with fossil and clean production sectors over annual periods $t=0,\ldots,30$. The state includes installed capital $K_t$, clean-sector knowledge $Z_t$, household assets, and political support $S_t$. A policy path $\pi$ specifies carbon taxes, clean-research subsidies, infrastructure investment, and transfers each year. Knowledge spillovers and irreversible investment are explicitly modeled. Let $W(\pi)$ be discounted household welfare with stated distributional weights, and $G_t(\pi)$ emissions. For cumulative emissions budget $B$, fiscal limits $F_t$, and implementability threshold $s_0$, the proposed problem is \[\pi^*\in\arg\max_{\pi\in\mathcal P}W(\pi)\quad\text{subject to}\quad E[\sum_{t=0}^{30}G_t(\pi)]\leq B,\quad E[\operatorname{Cost}_t(\pi)]\leq F_t,\quad E[S_t(\pi)]\geq s_0.\] Here $\mathcal P$ is the stated set of administratively feasible sequences, and $\operatorname{Cost}_t$ is net public expenditure. Establish feasibility and whether the welfare supremum is attained; if it is not, report the supremum and approximating paths rather than assert an optimizer exists. Determine whether early innovation and infrastructure spending changes the welfare ranking of later carbon-tax paths after equilibrium entry and distributional effects. Estimate or bound learning, spillover, and support parameters instead of choosing them to justify a preferred sequence.

The state $x_t=(K_t,Z_t,\text{assets}_t,S_t)$ follows a declared law $x_{t+1}=f(x_t,u_t,\xi_{t+1})$, where controls $u_t$ specify each tax, subsidy, public investment, and transfer. A policy is a nonanticipating measurable rule $u_t=\pi_t(x_0,\xi_1,\ldots,\xi_t)$, rather than a sequence allowed to observe future shocks. With shocks present, use the displayed expected constraints, $E\sum_tG_t\leq B$, $E\operatorname{Cost}_t\leq F_t$, and $E S_t\geq s_0$, with almost-sure constraints as a stricter alternative. Welfare $W=E\sum_i\omega_i\sum_t\beta^tu_i(c_{it})$ uses fixed positive weights and integrable utilities. Political support is a defined model state or measured probability, with an explicit transition law. Optimization in a completely known finite menu is routine; parameter recovery and implementability motivate the agenda. 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 commitment variant. Compare a committed thirty-year path with annually revised nonanticipating rules under a stated political transition model. Investor expectations must change consistently.
  2. Editorial risk-constraint variant. Replace expectation constraints by $P(\sum_tG_t\leq B)\geq1-\alpha$ and $P(S_t\geq s_0\ \forall t)\geq1-\gamma$. The stricter feasible set may be empty. Fix $\alpha,\gamma\in(0,1)$ before comparison.

References and source audit

1. Broad transformation and policy discussion; precise sequence optimization is editorial. Locator: Economic Journal 132(644), 1259–1289; author-institution publication summary. Publisher retrieval failed; author-institution metadata and summary inspected. URL: https://academic.oup.com/ej/article/132/644/1259/6519262.

2. 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.

  1. Nicholas Stern. 2022. A Time for Action on Climate Change and a Time for Change in Economics. Economic Journal 132(644), 1259–1289; author-institution publication summary. https://academic.oup.com/ej/article/132/644/1259/6519262. DOI: https://doi.org/10.1093/ej/ueac005.
  2. 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.

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-105-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-105

An explicit threshold for innovation before carbon pricing

Problem ID: Q58-103. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

Question and a two-date restriction

The catalogue asks how innovation, infrastructure, carbon taxes, transfers, and political support should be sequenced over a long transition with uncertain parameters. Here we solve a deterministic two-date restriction with a knowledge spillover, irreversible early research, an emissions requirement, and a fixed activation cost. We then characterize fiscal and political constraints through a convex program. This is a model result; it supplies no empirical learning coefficient or general thirty-year optimum.

Dates are $0,1$. Abatement is $a_0,a_1\ge0$, and cumulative required abatement is $D>0$. Early public research is $r\ge0$, installed irreversibly at date zero and giving knowledge $Z_1=r$. Research is usable only at date one. Abatement and research costs, in date-zero resource units, are \[ C(a_0,a_1,r)=\frac{c_0a_0^2}{2}+\frac{k r^2}{2} +\beta\left(\frac{c_1a_1^2}{2}-\vartheta r a_1\right), \tag{1} \] where $c_0,c_1,k>0$, $\beta\in(0,1]$, and $\vartheta\ge0$. The emissions constraint is $a_0+a_1\ge D$. Baseline emissions minus $D$ is the cumulative budget. Cost (1) represents a local quadratic technology; its domain and convexity conditions are part of this restriction, not empirical assertions about global technology.

If research is used, an additional fixed resource cost $f\ge0$ is paid. The government can choose regime 0, requiring $r=0$ and no activation cost, or regime 1, allowing $r\ge0$ and adding $f$. Assume \[ kc_1>\beta\vartheta^2,\qquad d_0=\beta c_1,\qquad d_1=\beta c_1-\frac{\beta^2\vartheta^2}{k}>0. \tag{2} \] The quadratic form is then strictly positive definite. A representative household has linear utility in remaining resources. Taxes and rebates are internal transfers, so minimizing real cost is equivalent to maximizing this restricted household welfare.

Solving the sequencing choice

Theorem (Exact innovation threshold): In regime $j\in\{0,1\}$, without additional fiscal or political constraints, the unique cost-minimizing abatement levels are \[ a_{0j}=\frac{d_jD}{c_0+d_j},\qquad a_{1j}=\frac{c_0D}{c_0+d_j}. \] Research is $r_0=0$ in regime 0 and $r_1=\beta\vartheta a_{11}/k$ in regime 1. The corresponding minimized variable costs are \[ V_j(D)=\frac{D^2}{2}\frac{c_0d_j}{c_0+d_j}. \] Therefore activation is strictly optimal exactly when \[ f<\Delta(D):=\frac{D^2}{2} \frac{c_0^2(d_0-d_1)}{(c_0+d_0)(c_0+d_1)}. \tag{3} \] If $\vartheta>0$, activation lowers date-zero abatement and raises date-one abatement, even though innovation itself occurs early.
Proof: For fixed $a_1$, complete the square: \[ \frac{k r^2}{2}-\beta\vartheta r a_1 =\frac{k}{2}\left(r-\frac{\beta\vartheta a_1}{k}\right)^2 -\frac{\beta^2\vartheta^2a_1^2}{2k}. \] The minimizing $r$ is nonnegative, so regime 1 reduces to $c_0a_0^2/2+d_1a_1^2/2$. Regime 0 has the same form with $d_0$. Both coefficients are positive. At a minimum $a_0+a_1=D$: if the sum were larger, proportional contraction would reduce the positive quadratic objective. Substitute $a_1=D-a_0$ and differentiate the strictly convex quadratic to obtain the displayed levels and values. Subtract $V_1$ from $V_0$ to obtain (3). Strict decrease of $d_1$ relative to $d_0$ lowers $dD/(c_0+d)$ and raises its complementary share, proving the sequencing statement. ∎

The comparison conditions on the same emissions requirement. Relaxing the budget in one regime would change the economic question. Nor does early research necessarily imply early abatement: the knowledge spillover changes the intertemporal marginal-cost ratio.

Proposition (Implementation by taxes and a research payment): Suppose firms take public knowledge $r$ as given and independently minimize \[ c_0a_0^2/2-\tau_0a_0,\qquad c_1a_1^2/2-\vartheta r a_1-\tau_1a_1 \] over nonnegative abatement. Every interior optimum in Theorem 2.1 is implemented by \[ \tau_0=c_0a_0,\qquad \tau_1=c_1a_1-\vartheta r. \] In the research regime $\tau_0=\beta\tau_1$ and $\tau_1>0$. A research contractor with cost $kr^2/2$ supplies the optimal $r$ when paid marginal subsidy $\beta\vartheta a_1$ per unit of research, taking its announced value as given.
Proof: The two private objectives are strictly convex, so their first-order conditions uniquely give the stated abatement choices. At the optimum $c_0a_0=d_1a_1=\beta(c_1a_1-\vartheta r)$; positivity follows from (2). The contractor maximizes a linear payment times $r$ minus $kr^2/2$, whose unique maximizer is $\beta\vartheta a_1/k$. ∎

This is a commitment implementation, with taxes on baseline emissions equivalent to the displayed abatement incentives up to a fixed term. The research payment generally costs more fiscally than the contractor's real resource cost. That distinction matters once public spending is constrained.

Political support, fiscal limits, and a complete certificate

To make this distinction explicit, use direct public procurement of research at real cost $kr^2/2$; the contractor-subsidy implementation above is a separate unrestricted benchmark. Let $t\ge0$ be a date-zero transfer to households burdened by adjustment. Assume support is the known index \[ S=s_{\rm base}+t-\chi a_0,\qquad \chi\ge0, \] and that a dollar of transfer has marginal public-finance resource cost $\lambda\ge0$ beyond its transfer value. Support must exceed $s_0$, and research plus transfers must obey a fiscal limit $F$. In a fixed activation regime $j$, add constraints \[ t\ge s_0-s_{\rm base}+\chi a_0,\qquad kr^2/2+t+jf\le F, \] and minimize $C+\lambda t+jf$ subject also to the emissions requirement and nonnegativity. In regime 0 impose $r=0$.
Proposition (Feasibility and constrained optimality): In regime 1 this is a convex optimization problem under (2). If its feasible set is nonempty, a minimum is attained. If there is a point satisfying all inequality constraints strictly, a feasible point is optimal if and only if it has nonnegative multipliers $\eta,\zeta,\xi$ and $\rho_0,\rho_1,\rho_r,\rho_t$ satisfying complementarity and \begin{align*} c_0a_0-\eta+\chi\zeta-\rho_0&=0,\\ \beta c_1a_1-\beta\vartheta r-\eta-\rho_1&=0,\\ kr-\beta\vartheta a_1+\xi kr-\rho_r&=0,\\ \lambda-\zeta+\xi-\rho_t&=0. \end{align*} The multipliers $\eta,\zeta,\xi$ correspond respectively to $D-a_0-a_1\le0$, $s_0-s_{\rm base}+\chi a_0-t\le0$, and $kr^2/2+t+f-F\le0$; the $\rho$ multipliers correspond to nonnegativity. Complementarity means each multiplier times its own constraint residual is zero.
Proof: The Hessian of $C$ has positive $c_0$ block and $(r,a_1)$ block determinant $\beta k c_1-\beta^2\vartheta^2>0$, so $C$ is strictly convex in these three variables. Every constraint defines a closed convex set. Along a minimizing sequence with bounded objective, positive definiteness bounds $(a_0,a_1,r)$, while the fiscal constraint bounds $0\le t\le F-f$. A convergent subsequence is feasible and attains the minimum.

Form the Lagrangian with the specified signs. Differentiation yields the equations. If the equations and complementarity hold, convexity gives that the candidate globally minimizes the Lagrangian; feasibility and nonnegative multipliers then give its objective no greater than that of any feasible competitor. Conversely, strict feasibility permits separation of the convex set of achievable strict objective improvement and constraint residuals from the negative orthant. Normalize the separating objective coefficient to one; strict feasibility rules out a zero coefficient. The remaining coefficients are nonnegative multipliers. The supporting condition gives Lagrangian minimization and complementarity, and differentiability gives the equations. ∎

The theorem compares two constrained regime values after feasibility is checked; (3) need not survive binding fiscal or political constraints. The expectation constraints of the full catalogue become deterministic ones here. Uncertain learning, entry, household heterogeneity, and changing political coalitions are not hidden inside these coefficients.

Evidence still needed

The model predicts a threshold in the fixed cost and in the required cumulative reduction, conditional on an identified $\vartheta$. Recovering that coefficient requires variation in early research or knowledge that is separated from anticipated future demand and concurrent infrastructure. Estimating $\chi$ requires a declared measurement model for support. A calibration that selects these coefficients solely to make (3) hold would not answer the empirical agenda. Stochastic nonanticipating policies, distributional welfare weights, and the full 31-date transition remain unresolved.

References

Completion Estimate

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