## _wp16164 - 3.1 Model Setup

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### Context, purpose, and methodological approach
- Context:
  - IPCC COP21 (December 2015) targeted “well below” 2°C; Paris Agreement also lowered the targeted average increase to 1.5°C.
  - Even with global emissions halted by 2014, global mean temperature would continue to rise over the next four decades.
  - Next 20 years likely to see average temperatures reach and surpass 1°C above pre-industrial average.
- Purpose:
  - Propose a macroeconomic framework to delineate effects of policy decisions at the environmental–economic nexus and determine countries’ optimal expenditure allocations to mitigation and adaptation.
- Approach:
  - Integrated assessment model (IAM) within a welfare-theoretic framework, embedding emissions driven by extraction of non-renewable resources (Hotelling-style) rather than by carbon intensity of production.
  - Combine social cost of carbon (SCC) literature with resource extraction modeling.
- Key modeling choice:
  - Emissions modeled as CO2-equivalents (CO2-eq); non-CO2 greenhouse gases (CH4, N2O, CO2 (SO2 reference in source), SF6) treated collectively as CO2-eq.
- Solution method overview:
  - Nonlinear Model Predictive Control (NMPC) algorithm used to solve iterated finite-horizon welfare-maximization problems, approximating short-horizon decision-making and reducing reliance on perfect foresight.
  - AMPL used for higher-dimensional joint optimization when necessary.

### Production, resource, and fiscal structure (model primitives)
- Production technology:
  - Output per unit of labor Yt:
    - Yt = A(Ak kt + Au ut)^α · (ν1 · gt)^β
    - Parameters: A, Ak, Au > 0; α + β ≤ 1.
    - ν1 ∈ [0,1] fraction of public capital gt allocated toward production; 1 − ν1 allocated to climate policies.
- Non-renewable resource:
  - Reserve dynamics: ̇Rt = −ut with R0 > 0 and Rt ≥ 0.
  - Marginal extraction cost Ct := C[Rt] with C′R < 0 and C[Rt] = ψ R_t^(−τ).
  - Parameters: ψ, τ > 0; cost approaches infinity as Rt → 0.
- Private capital accumulation (extended Ramsey-Cass-Koopmans law):
  - ̇kt = A(Ak kt + Au ut)^α · (ν1 · gt)^β − eP t − ct − (δk + n) kt − ut · (ψ R_t^(−τ))
  - Variables/parameters: ct (private per capita consumption), eP t (government tax revenue), δk (physical depreciation), n (population growth per capita depreciation).
- Government finances and public capital:
  - Government income: Tt = eP t + iF with iF constant.
  - Government spending Gt ≥ Tt; deficit financed by bonds if strict inequality holds.
  - Tax revenue allocation fractions α1, α2, α3, α4 with α4 = min{1 − α1 − α2 − α3, 0}; model restricts α1 + α2 + α3 ≤ 1.
  - Debt dynamics (per capita): ̇bt = (̄r − n) bt − (1 − α1 − α2 − α3) · eP t with initial stock non-negative.
  - Public capital accumulation: ̇gt = α1 eP t + iF − (δg + n) · g; equilibrium g* = (α1 eP + iF) / (δg + n) > 0.

### Allocation of public capital and emissions dynamics
- Public capital shares:
  - gt allocated to ν1 (production), ν2 (adaptation), ν3 (mitigation) with ν1 + ν2 + ν3 = 1.
  - Mitigation and adaptation projects require only public capital in model.
- Atmospheric CO2-equivalent concentration Mt dynamics:
  - ̇Mt = γ ut − μ (Mt − κ M̃) − θ (ν3 · gt)
  - Parameters and interpretations:
    - γ ∈ (0,1): fraction of emissions added to atmosphere (1 − γ absorbed by oceans).
    - M̃ pre-industrial GHG concentration (approximately 280 ppm) but in model M̃ = 1 in calibration.
    - κ > 1: stabilization ratio imposing natural stabilization above pre-industrial level.
    - μ > 0: decay rate of atmospheric GHG above stabilization.
    - θ: scaling factor linking mitigation public capital ν3 gt to reduction in rise of Mt.

### Social welfare, control problem, and first-order conditions
- Welfare maximization:
  - Max W = ∫_0^∞ e^(−ρ t) · ( [ ct (α2 eP t)^η (Mt − M̃) − ϕ (ν2 gt)^ω ]^(1−σ) − 1 ) / (1 − σ) dt
  - Subject to the state equations and non-negativity gt, kt, ut ≥ 0.
  - ρ ≡ (ρ̄ − n); parameters η, ϕ, ω > 0; 0 < σ ≤ 1; ρ̄ > n > 0 so ρ > 0.
- First-order conditions (current-value Hamiltonian) and costate dynamics:
  - Control FOCs:
    - 0 = ( c (α2 eP)^η (M − M̃) − ϕ (ν2 g)^ω )^(1−σ) / c − λ1
    - 0 = η ( c (α2 eP)^η (M − M̃) − ϕ (ν2 g)^ω )^(1−σ) / eP − λ1 + λ2 α1 − λ3 (1 − α1 − α2 − α3)
    - 0 = −λ1 (ψ R^(−τ)) − λ4 + λ5 γ
  - Costate differential equations:
    - ̇λ1 = λ1 (ρ + δk + n − A α Ak (Ak k + Au u)^(α−1) (ν1 g)^β)
    - ̇λ2 = λ2 (ρ + δk + n) − ω ( c (α2 eP)^η (M − M̃) − ϕ (ν2 g)^ω )^(1−σ) g^(−1) − λ1 A β ν1 (Ak k + Au u)^α (ν1 g)^(β−1) + λ5 θ ν3
    - ̇λ3 = λ3 (ρ + n − ̄r)
    - ̇λ4 = λ4 ρ + λ1 u ψ τ R^(−τ−1)
    - ̇λ5 = λ5 ρ + ϕ ( c (α2 eP)^η (M − M̃) − ϕ (ν2 g)^ω )^(1−σ) (M − M̃)^(−1) + λ5 μ
  - Analytical closed-form solution not generally available; numerical methods applied.

### Numerical solution method and baseline calibration
- Numerical method:
  - NMPC: iterative rolling finite-horizon optimization with horizon N often as short as N ≈ 5; as N → ∞ open-loop pathway converges to infinite-horizon result.
  - NMPC suitable for high-dimensional nonlinear problems and short-horizon decision interpretation.
- Baseline initial stocks and selected parameters (from Table 1) used in simulations:
  - Initial stocks: k0 = 1.3; g0 = 0.5; R0 = 1.5; b0 = 0.8; M0 = 1.3.
  - Selected parameters:
    - ρ = 0.03
    - n = 0.015
    - η = 0.1
    - ϕ = 1.1
    - ω = 0.05
    - σ = 1.1
    - A = 1
    - Ak = 1
    - Au = 40
    - α = 0.5
    - β = 0.5
    - ψ = 1
    - τ = 2
    - δk = 0.075
    - δg = 0.05
    - iF = 0.05
    - α1 = 0.1
    - α2 = 0.7
    - α3 = 0.1
    - ̄r = 0.07
    - M̃ = 1
    - γ = 0.9
    - μ = 0.01
    - κ = 2
    - θ = 0.01

### Scenario analyses — setups and key outcomes
- Scenario 1: No Active Climate Change Policy (BAU)
  - Policy: ν1 = 1, ν2 = 0, ν3 = 0 (all public capital to production).
  - Welfare function adjustment when ν2 = 0:
    - Wt = ( c_t (α2 eP t)^η (M − M̃) − ϕ )^(1−σ) − 1 ) / (1 − σ).
  - Numerical implementation for Figure 1:
    - Horizon N = 15, step length dt = 1/2, repeated through T = 50 iterations.
  - Dynamics summary:
    - Initial private capital declines (̇k < 0) due to consumption, taxation, and limited output.
    - Around t = 4 private capital and consumption rise as extraction boosts output.
    - After t > 12 extraction costs rise; private capital and consumption continue upward thereafter.
    - Extraction rate ut given by −̇R; ut ≤ 3 ≈ 0 initially.
    - Tax revenue increases over time (eP0 value truncated in source).
  - Scenario 1 outcomes (Scenario 1 — illustrative baseline outcomes section):
    - Public capital allocation example: P25 = 0.18.
    - Share directed to investment never enough to cover depreciation; net positive investment possible only because iF > 0.
    - Public debt soars rapidly over time; primary surplus too small to slow per capita debt increase.
    - Carbon emissions and atmospheric CO2 show explosive trajectories: M0 = 1.3 → M25 = 1.73.

- Scenario 2: Mitigation efforts, no adaptation
  - Public capital split: ν3 = 0.4; ν2 = 0; ν1 = 0.6.
  - Key dynamics and comparisons with Scenario 1:
    - Private capital: deeper initial decline and longer duration; at T = 25 capital stock < two-thirds of value in ν1 = 1 case.
    - Output and consumption: production declines and consumption lower for all t relative to no-policy scenario; consumption stabilizes in Scenario 2 while rising in Scenario 1.
    - Tax revenue: starts at eP0 = 0.09 and remains virtually unchanged (contrasts with Scenario 1’s rise).
    - Debt and public capital: debt soars faster; public capital lower and largely driven by iF.
    - Extraction: private sector increases extraction; extraction rate U-shaped, peaking at t = 13.5.
    - Welfare: Scenario 1 Pareto-preferred to Scenario 2 within considered ranges because welfare functional monotonic in consumption and atmospheric CO2.
  - Interpretation: High ν3 (40%) can induce increased private extraction and reduce welfare relative to focusing on traditional development.

- Scenario 3: Dynamic allocation of ν1 (with ν2 = ν3 = (1 − ν1)/2), fixed tax revenue
  - Setup:
    - ν1 endogenous control; ν2 = ν3 = (1 − ν1)/2.
    - eP t = eP = 0.1 fixed.
  - Optimal path:
    - ν1 falls from 0.95 to below 90% and stabilizes at ν1 = 0.93 → ~7% of public capital for climate purposes (split equally).
  - Robustness:
    - Reducing β from 0.5 to 0.2: ν1 falls from 93% to ~ν1 = 0.83.
    - Increasing disutility of climate change ϵ from 1.1 to 2 with β = 0.2: stabilizing ν1 ~82% with different transitional dynamics.
  - Policy implication:
    - Across parameterizations, 80 to 95% of public investment should go to growth-enhancing infrastructure; remaining 5 to 20% to mitigation and adaptation combined.

- Scenario 4: Fixed ν1 = ̄ν1 = 0.4, endogenous ν2 (adaptation) with ν3 = 1 − ̄ν1 − ν2
  - Setup:
    - ν1 = 0.4 fixed; control ν2; eP t = eP = 0.11; high aversion to climate change ϵ = 2.
  - Optimal evolution:
    - Initially ν2 ≈ 0.4; from t = 4 to t = 12 adaptation ramps up and mitigation scaled back.
    - For t > 12 adaptation reaches maximum ν2 = 0.6 → ν3 = 0 (no further mitigation).
    - Atmospheric concentration increases continuously, reaching M(25) = 2.18 at end of observation period.
  - Robustness:
    - Doubling θ to θ = 0.02: mitigation effort reduced in favor of adaptation but reallocation slower; at end ~20% of public funds to mitigation.
    - Reducing ω from 0.05 to ω = 0.038 with θ = 0.02: reallocation away from mitigation slower.
  - Policy implication:
    - As M increases, adaptation becomes relatively more cost-competitive; funding growth projects that also enhance resilience is important.

- Scenario 5: Joint optimal allocation of ν1, ν2, ν3 using AMPL with nonlinear mitigation effect
  - Motivation:
    - NMPC Matlab implementation faced convergence limits with >3 control variables; AMPL used for fuller joint optimization including tax rate eP.
  - Modeling adjustment:
    - Emission equation modified: ̇Mt = γut − μ(Mt − κ ̃M) − θ(ν3·gt)^φ with 0.20 ≤ φ ≤ 1 and ν1 + ν2 + ν3 = 1 to avoid corner solutions.
  - AMPL results summary:
    - ν1 always above 92%.
    - Residual public capital (1 − ν1) mainly allocated to adaptation; smaller share to mitigation.
    - Welfare level varies with φ; tax rate optimally selected and approximates NMPC results.

### Aggregate findings and policy recommendations
- Core model findings:
  - Majority of public funds should continue to flow to growth-enhancing infrastructure in the model calibrations and scenarios considered.
  - The balance between mitigation and adaptation tends toward adaptation as climate change intensifies because adaptation becomes relatively more cost-competitive as M increases.
  - Diverting too large a share of public capital from productivity-enhancing infrastructure to mitigation can reduce consumption, increase private extraction of non-renewable resources, raise public debt, and lower welfare.
- Practical recommendations for developing economies:
  - Prioritize infrastructure projects that both enhance livelihoods and build resilience (dual-purpose investments).
  - Allocate a modest but non-negligible share of public capital to climate measures: model results suggest approximately 5 to 20% combined for mitigation and adaptation under many parameterizations, with adaptation often larger as atmospheric concentrations rise.
  - Treat optimal allocation as time-varying; controls ν1, ν2, ν3 should not be fixed ex ante.
- Methodological notes:
  - NMPC interpretable as short-horizon decision-making with horizon N (often N ≈ 5) and is computationally suited to the nonlinear high-dimensional problem.
  - AMPL used to handle higher-dimensional controls and to avoid corner (“bang-bang”) solutions via nonlinear mitigation curvature parameter φ.

*Source: _wp16164 - 3.1 Model Setup (excerpt).*

### 3.1 Model Setup ........................................................................................................

### _wp16164 - 3.1 Model Setup

### Context and motivation
- IPCC COP21 Paris meeting in December 2015 produced a universal agreement to limit global temperature increases to ‘well below’ 2°C above the pre-industrial average (UNFCCC, 2015).
- Current action plans appear to remain insufficient to achieve the global 2°C goal, and the Paris Agreement provides a policy implementation framework for mitigating further carbon emissions and adapting to the impacts of climate change (IMF, 2016).
- Even if all global emissions had been halted by 2014, the global mean temperature would still continue to increase over the next four decades (Oppenheimer, 2013).
- The next 20 years will likely see average temperatures reach and surpass 1°C above the pre-industrial average temperature (IPCC Working Group I, 2013, Technical Summary, TFE.8).
- The Paris Agreement lowered the targeted average increase to 1.5°C.

### Paper purpose and high-level approach
- Purpose: to propose a macroeconomic framework capable of delineating the effects of policy decisions at the environmental–economic nexus, determining countries’ optimal expenditure allocations to mitigation efforts and adaptation projects.
- Approach: develop an integrated assessment model (IAM) within a standard welfare-theoretic framework (Fankhauser et al., 1997; Nordhaus and Boyer, 2000).
- Key methodological choice: emissions are driven by the extraction of non-renewable resources (Hotelling, 1931; Pindyck, 1978) rather than by the carbon intensity of production in general, allowing combination of the social cost of carbon (SCC) literature with resource extraction insights (Greiner et al., 2010b).

### Model extensions and distinguishing features
- Adds three elements to a Semmler et al. (2011)-style public-capital growth model:
  - (i) non-renewable energy extraction that emits CO2;
  - (ii) an economic–environment feedback loop generating welfare losses (or benefits) from climate change, with climate change impacting social welfare directly rather than only via private-sector productivity reductions;
  - (iii) utilization of public capital for carbon emission mitigation and adaptation measures to reduce negative impacts.
- Emissions are modeled as CO2-equivalents (CO2-eq), with non-CO2 greenhouse gases noted (methane CH4, nitrous oxide N2O, sulfur dioxide CO2, sulfur hexafluoride SF6) and collectively referred to as CO2-eq.

### Policy dimensions represented
- Mitigation policies: examples include carbon taxes on fossil fuel energy and regulatory measures (e.g., vehicle emissions standards).
- Adaptation policies: examples include government-sponsored R&D, insurance programs in agriculture, urban infrastructure defenses, and public health campaigns.
- The model emphasizes the interplay and potential complementarity/rivalry between mitigation, adaptation, and broader economic development, especially relevant for developing and emerging economies where traditional development investments can have adaptive benefits.

### Tractability and solution method
- High dimensionality of IAMs is addressed by solving jointly for optimal environmental/economic trajectories rather than prescribing exogenous pathways.
- The nonlinear model predictive control (NMPC) algorithm (see Grüne et al., 2015) is used:
  - Agents solve approximately optimal dynamic pathways over short time horizons and update these iteratively (receding horizon approach).
  - NMPC imbues agents with myopia, allowing integration of differing degrees of climatological uncertainty as agents update finite-horizon decisions.
  - The NMPC framework reduces reliance on perfect foresight and approximates a social planner by iteratively updating policy paths.

### Theoretical positioning and welfare interpretation
- The model constitutes a reduced-form IAM similar to Nordhaus and Sztorc (2013) but differs by:
  - Deriving carbon emissions from an optimal resource extraction model à la Hotelling.
  - Directly affecting social welfare through climate impacts (health, ecological loss, living conditions, heightened uncertainty) rather than solely via production losses.
- Public capital is the key driver for climate change adaptation and mitigation, representing investments that can include institutional and social capital needed to serve communities.

### Numerical strategy and roadmap
- Section 3.1 (this section) describes the detailed model and economic interpretations for formalizations.
- Section 3.2 establishes the analytical solution to the degree possible without linearizing dynamics.
- Section 4 reports numerical results obtained via the NMPC algorithm and iterative finite-horizon solution of the nonlinear system.

*Source: _wp16164 - 3.1 Model Setup (excerpt).*

### 3.1    Model Setup

### _wp16164 - 3.1    Model Setup

### Production technology and inputs
- Output per unit of labor at time t≥0: Yt.
- Production function:
  - Yt = A(Ak kt + Au ut)^α · (ν1 · gt)^β
  - Parameters: A, Ak, Au > 0; α and β satisfy α + β ≤ 1.
  - Ak and Au specify the efficiency of private capital and the non-renewable resource in production.
  - ν1 ∈ [0,1] is the fraction of public capital gt allocated toward production; 1−ν1 is allocated to climate change policies.

### Non-renewable resource stock and extraction costs
- Reserve dynamics:
  - ̇Rt = −ut with R0 > 0 and Rt ≥ 0.
- Marginal cost of extraction (Hotelling-style cost rising as resources deplete):
  - Ct := C[Rt], with C′R < 0 and C[Rt] = ψ R_t^(−τ)
  - Parameters: ψ, τ > 0.
  - Assumption: cost of extraction approaches infinity as Rt → 0.

### Private capital accumulation
- Extended Ramsey-Cass-Koopmans-type law of motion for private capital kt:
  - ̇kt = A(Ak kt + Au ut)^α · (ν1 · gt)^β − eP t − ct − (δk + n) kt − ut · (ψ R_t^(−τ))
- Variables and parameters appearing:
  - ct: private per capita consumption.
  - eP t: government tax revenue.
  - δk: physical depreciation rate of private capital.
  - n: per capita depreciation due to population growth.
  - Last term is total extraction cost at time t.

### Government income, spending, debt dynamics, and public capital
- Government income:
  - Tt = eP t + iF
  - iF represents financial support from abroad (treated as constant).
- Government spending Gt ≥ Tt; strict inequality implies bond-financed deficit spending.
- Allocation of tax revenue fractions (α1, α2, α3, α4):
  - α1: fraction of tax revenue on infrastructure (public capital).
  - α2: fraction for direct social transfers and social services (increase welfare).
  - α3: fraction for administrative costs (no direct utility impact).
  - α4 = min{1 − α1 − α2 − α3, 0} used to service debt.
  - Model restricts to “well-disciplined” case avoiding primary deficits:
    - α1 + α2 + α3 ≤ 1
- Debt dynamics (world interest rate ̄r, per capita terms):
  - ̇bt = (̄r − n) bt − (1 − α1 − α2 − α3) · eP t
  - Initial stock of debt is non-negative.
- Public capital accumulation:
  - ̇gt = α1 eP t + iF − (δg + n) · g
  - Equilibrium level: g* = (α1 eP + iF) / (δg + n) > 0

### Allocation of public capital across uses
- Public capital gt allocated among:
  - ν1: infrastructure supporting production.
  - ν2: climate change adaptation.
  - ν3: climate change mitigation.
- Shares are exogenously determined and satisfy:
  - ν1 + ν2 + ν3 = 1
- Mitigation and adaptation modeled as projects requiring only public capital.

### Emissions dynamics and mitigation
- CO2-equivalent atmospheric concentration Mt dynamics:
  - ̇Mt = γ ut − μ (Mt − κ M̃) − θ (ν3 · gt)
  - γ ∈ (0,1): fraction of emissions added to atmosphere (1 − γ absorbed by oceans).
  - M̃ denotes pre-industrial atmospheric GHG concentration (approximately 280 ppm).
  - κ > 1: stabilization ratio imposing natural stabilization above pre-industrial level.
  - μ > 0: decay rate of atmospheric GHG above stabilization.
  - θ: scaling factor linking mitigation public capital ν3 gt to reduction in the rise of Mt.

### Social welfare and control problem
- Aggregate welfare maximization over controls (ct, ut, eP t):
  - Max W = ∫_0^∞ e^(−ρ t) · ( [ ct (α2 eP t)^η (Mt − M̃) − ϕ (ν2 gt)^ω ]^(1−σ) − 1 ) / (1 − σ) dt
  - Subject to state equations: (2), (4), (7), (9), (6) and non-negativity gt, kt, ut ≥ 0.
  - ρ ≡ (ρ̄ − n) is intertemporal discount rate net of population growth.
  - Parameters/assumptions for welfare function:
    - η, ϕ, ω > 0.
    - 0 < σ ≤ 1 (diminishing marginal returns to utility).
    - ρ̄ > n > 0 so that ρ > 0.

### First-order conditions and costate dynamics
- Current-value Hamiltonian H with costate variables λ1 (k), λ2 (g), λ3 (b), λ4 (R), λ5 (M).
- First-order conditions for controls:
  - 0 = ( c (α2 eP)^η (M − M̃) − ϕ (ν2 g)^ω )^(1−σ) / c − λ1
  - 0 = η ( c (α2 eP)^η (M − M̃) − ϕ (ν2 g)^ω )^(1−σ) / eP − λ1 + λ2 α1 − λ3 (1 − α1 − α2 − α3)
  - 0 = −λ1 (ψ R^(−τ)) − λ4 + λ5 γ
- Costate differential equations:
  - ̇λ1 = λ1 (ρ + δk + n − A α Ak (Ak k + Au u)^(α−1) (ν1 g)^β)
  - ̇λ2 = λ2 (ρ + δk + n) − ω ( c (α2 eP)^η (M − M̃) − ϕ (ν2 g)^ω )^(1−σ) g^(−1) − λ1 A β ν1 (Ak k + Au u)^α (ν1 g)^(β−1) + λ5 θ ν3
  - ̇λ3 = λ3 (ρ + n − ̄r)
  - ̇λ4 = λ4 ρ + λ1 u ψ τ R^(−τ−1)
  - ̇λ5 = λ5 ρ + ϕ ( c (α2 eP)^η (M − M̃) − ϕ (ν2 g)^ω )^(1−σ) (M − M̃)^(−1) + λ5 μ
- Analytical closed-form solution not generally available; authors proceed to numerical methods.

### Numerical solution method and baseline simulation setup
- Numerical method: Nonlinear Model Predictive Control (NMPC).
  - Iterative rolling finite-horizon optimization (horizon N often as short as N ≈ 5).
  - As N → ∞, open-loop pathway converges to infinite-horizon result.
  - NMPC suited to high-dimensional nonlinear problems and interpreted as short-horizon decision-making.
- Baseline initial stocks and parameters used in simulations:
  - k0 = 1.3
  - g0 = 0.5
  - R0 = 1.5
  - b0 = 0.8
  - M0 = 1.3
- Selected simulation parameter values (as listed in Table 1):
  - ρ = 0.03
  - n = 0.015
  - η = 0.1
  - ϕ = 1.1
  - ω = 0.05
  - σ = 1.1
  - A = 1
  - Ak = 1
  - Au = 40
  - α = 0.5
  - β = 0.5
  - ψ = 1
  - τ = 2
  - δk = 0.075
  - δg = 0.05
  - iF = 0.05
  - α1 = 0.1
  - α2 = 0.7
  - α3 = 0.1
  - ̄r = 0.07
  - M̃ = 1
  - γ = 0.9
  - μ = 0.01
  - κ = 2
  - θ = 0.01

### Scenario: No Active Climate Change Policy (BAU) — setup and selected dynamics
- Policy specification:
  - ν1 = 1, ν2 = 0, ν3 = 0 (entire public capital devoted to production; no mitigation or adaptation).
- Welfare functional adjustment to prevent zero welfare when ν2 = 0:
  - Wt = ( c_t (α2 eP t)^η (M − M̃) − ϕ )^(1−σ) − 1 ) / (1 − σ) in the case ν2 = 0.
- Numerical implementation details for Figure 1:
  - Horizon N = 15, step length dt = 1/2, repeated through T = 50 iterations.
- Description of optimal time-path behavior under BAU:
  - Initial private capital stock declines (̇k < 0) due to consumption and taxation and limited output from low non-renewable resource availability.
  - Around t = 4 both private capital and consumption rise in tandem as increased extraction allows output to grow faster than consumption.
  - After t > 12 higher extraction costs and diminishing marginal products reduce use of non-renewable resource; private capital and consumption continue upward in a virtuous cycle thereafter.
  - Extraction rate ut is given by the absolute slope of Rt (−̇R); ut ≤ 3 ≈ 0 initially.
  - Tax revenue slightly augmented over time, increasing from eP0 = (value truncated in source text).

*Source: _wp16164 - 3.1    Model Setup (excerpt).*

### 0.11  toe

### _wp16164 - 0.11  toe

### Scenario 1 (No policy action) — illustrative baseline outcomes
- Public capital allocation example: P25 = 0.18.
- Share directed to investment in public capital is never enough to cover depreciation of existing public capital stock; positive net investment only possible because of free inflows from abroad, iF >0.
- Public debt trajectory: public debt soars rapidly over time, indicating the primary surplus is too small to slow down continuous increase in per capita debt.
- Government adjustment channel constrained: with allocation of public resources predetermined, government can only lower outstanding debt by increasing tax revenue.
- Carbon emissions and atmospheric CO2: explosive trajectories.
  - Total atmospheric greenhouse gas concentration increases from initial level M0 = 1.3 to M25 = 1.73.

### Scenario 2 (Mitigation efforts, no adaptation)
- Public capital split in this scenario:
  - ν3 = 0.4 (40% of public capital usage to mitigation efforts).
  - ν2 = 0 (no adaptation).
  - remainder to public infrastructure: ν = 0.6.
- Macroeconomic and welfare consequences:
  - Private capital: initial decline of private capital stock is much deeper and lasts longer than in no-policy-action scenario; at T = 25 the capital stock is less than two-thirds of the final value observed for ν1 = 1.
  - Output and consumption: production declines because public infrastructure capital for production is reduced; for all t, consumption is lower in presence of mitigation efforts versus no social spending on climate change mitigation; gap widens over time (consumption stabilizes in Scenario 2 but is still rising in Scenario 1).
  - Tax revenue: starts at eP0 = 0.09 and remains virtually unchanged during the simulation, contrasting with Scenario 1’s continuous rise in government income.
  - Debt and public capital: with less tax revenue, debt soars faster and public capital stock is lower than under no action; buildup of public capital mainly driven by free foreign funds so its path is relatively inelastic to domestic investment changes.
  - Non-renewable resource extraction: compensated in private sector by increased extraction; extraction rate follows U-shaped time path, peaking at t = 13.5 (inflection point of R curve).
- Welfare comparison:
  - Scenario 1 is Pareto-preferred to Scenario 2 because social welfare function (equation (20)) is monotonic in consumption and atmospheric CO2 within considered ranges (greater consumption and less CO2 preferred).
  - Interpretation: high allocation of public capital to mitigation (ν3 = 40%) can induce increased private extraction of non-renewable resources and reduce welfare relative to focusing on traditional economic development.

### Dynamic allocation of public capital (Scenario 3 and robustness)
- Scenario 3 setup:
  - ν1 (share to output promotion) introduced as control variable.
  - Remaining (1−ν1)×100% of public capital evenly divided between mitigation and adaptation: ν2 = ν3 = (1−ν1)/2.
  - Tax revenue fixed at eP t = eP = 0.1 for computational tractability.
  - Other parameters per Table 1 (as in source).
- Optimal path results (Figure 3 summary):
  - ν1 falls from initial 0.95 to below 90% and stabilizes at ν1 = 0.93 → approximately 7% of public capital used for climate change-related purposes (equally divided to mitigation and adaptation).
- Robustness checks:
  - Reducing public capital weight in production from β = 0.5 to β = 0.2: optimal ν1 falls from 93% to approximately ν1 = 0.83 (black line in Fig. 3) — a 60% reduction in public capital’s output elasticity translates into an 11% decline in allocation share.
  - Increasing disutility of climate change from ϵ = 1.1 to ϵ = 2 with β = 0.2: stabilizing value of ν1 little changed at 82% (cyan in Fig. 3), dynamics reversed (initial jump in ν1 then decline and recovery).
- Policy implication from Scenario 3:
  - Across parameterizations, between 80 and 95% of public investment should go toward growth-enhancing infrastructure; remaining 5 to 20% to climate mitigation and adaptation.

### Scenario 4 (Fixed ν1, endogenous split of ν2 and ν3) — focus on adaptation vs mitigation mix
- Scenario setup:
  - ν1 fixed at ̄ν1 = 0.4.
  - Government controls ν2 (adaptation); ν3 = 1 − ̄ν1 − ν2.
  - Tax revenue set at eP t = eP = 0.11.
  - High aversion to climate change maintained: ϵ = 2.
- Optimal evolution (Figure 4 summary):
  - Initially ν2 near its initial value of 0.4.
  - From t = 4 to t = 12 adaptation ramps up and mitigation is scaled back.
  - For t > 12, maximum adaptation reached and maintained at ν2 = 0.6 → implies ν3 = 0 (no further mitigation).
  - Outcome driver: marginal disutility of atmospheric GHG is inversely proportional to M (see equation (21)); as M increases, relative utility loss falls making adaptation increasingly cost-competitive with mitigation (marginal mitigation cost fixed at θ in equation (9)).
  - Atmospheric concentration example: continuous and undamped increase reaching M(25) = 2.18 at end of observation period (not shown in figure).
- Robustness checks:
  - Doubling mitigation cost θ to θ = 0.02: relative mitigation effort gradually reduced in favor of adaptation, but reallocation toward adaptation occurs more slowly; at end of simulation approximately 20% of public fund toward mitigation.
  - Reducing adaptation elasticity from ω = 0.05 to ω = 0.038 while keeping θ = 0.02: reallocation away from mitigation occurs more slowly (green path in Fig. 4).
- Policy implication:
  - Focus gradually shifts from mitigation to adaptation as atmospheric concentration increases; funding growth projects that also enhance resilience to climate change is important.

### Scenario 5 (Optimal joint allocation of ν1, ν2, ν3 using AMPL)
- Motivation:
  - Full joint optimization of tax rate eP and fractions ν1, ν2, ν3 requires more control variables than NMPC Matlab implementation can reliably handle; AMPL employed to solve more complex problem.
- Modeling adjustment to avoid corner “bang-bang” solutions:
  - Introduced nonlinear mitigation effect in emission equation:
    - ̇Mt = γut − μ(Mt − κ ̃M) − θ(ν3·gt)φ  with 0.20 ≤ φ ≤ 1 and ν1 + ν2 + ν3 = 1.
  - Limited φ range selected for reasonable numerical results.
- AMPL results (Figure 5 summary):
  - ν1 remains always above 92% (upper left panel of Fig. 5).
  - Residual public capital (1 − ν1) mainly goes to adaptation (ν2) and a smaller share to mitigation (ν3) (top right and lower left panels).
  - Welfare level varies with mitigation cost curvature parameter φ (lower right panel).
  - Tax rate (not shown) is also optimally selected and approximates NMPC results from earlier scenarios.

### Aggregate conclusions and policy recommendations
- Core findings:
  - In model calibrations and scenarios considered, the bulk of public funds should continue to flow to growth-enhancing infrastructure.
  - The balance between mitigation and adaptation tends toward adaptation as climate change becomes more extreme (adaptation becomes relatively more cost-competitive as M increases).
  - Suboptimal allocations that divert too large a share of public capital away from productivity-enhancing public infrastructure to mitigation can reduce consumption, increase private extraction of non-renewable resources, raise public debt, and lower welfare.
- Practical policy recommendations for developing economies:
  - Prioritize infrastructure projects that enhance economic livelihoods and also build resilience to climate change (dual-purpose investments).
  - Allocate a modest but non-negligible share of public capital to climate-related measures — model results suggest approximately 5 to 20% for mitigation and adaptation combined under many parameterizations, with adaptation often receiving the larger share as atmospheric concentrations rise.
  - Recognize that optimal allocation likely changes over time; controls ν1, ν2, ν3 should not be fixed ex ante.
- Methodological and computational notes:
  - NMPC implementation in Matlab faced convergence limits with more than three control variables; AMPL can handle higher-dimensional control optimization and was used to obtain joint optimal allocations.
  - Nonlinear mitigation cost specification (parameter φ) helps avoid corner (“bang-bang”) solutions when ν3 is an endogenous decision variable.

*Italic: Source — content unit from IMF working paper PDF _wp16164 - 0.11  toe (text provided above).*

### References

### _wp16164 - References

### Adaptation, Mitigation, and Integrated Assessment
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### Damage Functions, Welfare, and Economic Impacts
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### Climate Policy, Fiscal and Financial Implications
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### Assessments, Reports, and Legal Instruments
- IPCC (2012), Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation. Cambridge University Press.
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- IPCC Working Group II (2007), “Climate change 2007:  Impacts, adaptation, vulnerability.”
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*Source: _wp16164 - References (IMF).*

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