## 4.1 Discretization and Nonlinear Programming Methods

## Source details

**Canonical URL:** [4.1 Discretization and Nonlinear Programming Methods](https://www.imf.org/-/media/files/publications/wp/2018/wp18270.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2018/wp18270.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2018/wp18270.pdf.json)

---

### Objectives and model scope
- Purpose:
  - Examine strategies to (i) keep CO2 emission bounded by a predefined upper bound, (ii) scale up climate policies and allocate resources between mitigation and adaptation when climate risk rises, and (iii) determine dynamic funding and sequencing between traditional infrastructure investment, mitigation and adaptation.
- Modeling approach:
  - Dynamic global model with feedback control (optimal control) extending Integrated Assessment Models (IAM) to explicitly include:
    - Two energy sources: non-renewable (brown) energy from an extractive resource sector and renewable (green) energy produced with private green capital.
    - Government levies lump-sum taxes (eP) to (a) provide direct utility, (b) invest in public capital, and (c) cover administrative expenses.
    - Public capital allocable across three competing uses: traditional infrastructure, mitigation, adaptation.
- Core research questions:
  - Best strategies to keep or steer down CO2 emissions relative to bounds.
  - Optimal split of public investment between infrastructure, mitigation, adaptation across differing initial capital and CO2 conditions.
  - Allowed extraction of fossil energy given emission and temperature constraints.

### Optimal control model specification and dynamics
- Production function:
  - Y = (ν1 G)^β A (A_g K_g + A_u u)^α (K_p)^ζ, with A, A_g, A_u > 0, α, β, ζ > 0, and α + β + ζ < 1; ν1 ∈ (0,1]; u measured by carbon (CO2) content.
- Felicity depends on:
  - Per-capita consumption C, per-capita tax revenue α2 eP used for direct welfare, atmospheric concentration of CO2 (M) above long-run sustainable level, and per-capita public capital expenditure ν2 G allocated to adaptation.
- State vector:
  - X = (K_p, K_g, G, M, R) where K_p = private physical capital per capita, K_g = private green capital per capita, G = public capital per capita, M = CO2 concentration, R = non-renewable resource (fossil energy).
- Control vector:
  - U = (i_p, i_g, eP, u, C) where i_p = investment in physical capital, i_g = investment in green capital, eP = government net tax revenue, u = extraction rate, C = per-capita consumption.
- Public capital allocations:
  - ν1(t), ν2(t), ν3(t) with νk(t) ≥ 0 and ν1(t) + ν2(t) + ν3(t) = 1.
- Dynamics (selected):
  - K̇_p = i_p − (δ_p + n) K_p
  - K̇_g = i_g − (δ_g + n) K_g
  - Ġ = α1 eP − (δ_G + n) G
  - Ṁ = γ u − μ (M − κ̃_M) − θ (ν3 · G)^φ
  - Ṙ = −u
- Control and resource constraints:
  - 0 ≤ u(t) ≤ u_max for all t ∈ [0,T]
  - Resource constraint (mixed control-state equality):
    - s(X,U,ν) := Y − C − i_p − i_g − eP − u ψ R^−τ − χ_p/2 (i_p/K_p − δ_p − n)^2 K_p − χ_g/2 (i_g/K_g − δ_g − n)^2 K_g = 0
- Welfare functional:
  - Welfare = ∫_0^T e^{−(ρ−n)t} [ ... ] dt with integrand f0 using parameters ρ, n, σ, η, ε, ξ, ω, κ, κ̃_M, ̄M capturing direct utility from consumption, transfers, disutility from M, and utility from adaptation spending ν2 G.

### Parameter values used (as reported)
- ρ = 0.03
- n = 0.015
- η = 0.1
-  = 1.1
- ω = 0.05
- σ = 1.1
- A ∈ [1,10]
- A_g ∈ [1,5]
- A_u ∈ [100,400]
- α = 0.1
- β = 0.5
- 1 = 1
- τ = 2
- δ_p = 0.1
- δ_g = 0.05
- δ_G = 0.05
- χ_p = 1/(δ_p + n) Ω_p, χ_g = 1/(δ_g + n) Ω_g with Ω_p ∈ [5,15], Ω_g ∈ [5,15]
- α1 = 0.2
- α2 = 0.5
- ̄r = 0.07
- ̃M = 2.5
- γ = 0.9
- μ = 0.01
- κ = 1
- θ = 0.01
- φ ∈ [0.2,1]

### Necessary conditions, Hamiltonian and first-order system
- Current-value Hamiltonian:
  - H(X, λ, U, ν) = f0(X,U,ν) + λ · f(X,U,ν)
- Augmented Hamiltonian with multiplier η for mixed control-state constraint:
  - H(X, λ, η, U, ν) = H(X, λ, U, ν) + η s(X, U, ν)
- Adjoint and optimality:
  - λ̇(t) = (ρ − n) λ(t) − ∂H/∂X − η ∂s/∂X
  - Maximum principle: H(X(t), λ(t), U(t), ν(t)) = max_{(U,ν) ∈ Ω(t)} H(...)
  - Local first-order conditions for interior controls v ∈ {i_p, i_g, eP, C}: ∂H/∂v = 0
- Control admissible set enforces nonnegativity of i_p, i_g, eP; C > 0; 0 ≤ u ≤ u_max; ν_k ≥ 0 and ν1 + ν2 + ν3 = 1; and satisfaction of s(X,U,ν) = 0.

### Stationary (steady) solution and turnpike property
- Stationary system reduces to 13 equations for (K_p, K_g, G, M), (λ_{Kp}, λ_{Kg}, λ_G, λ_M), (i_p, i_g, eP, C), η, with u = 0 at steady state.
- Partial derivatives used include ∂f0/∂C, ∂f0/∂eP, ∂f0/∂G (∂f0/∂G = 0 in steady state because ν2 = 0), and ∂Y/∂K_p, ∂Y/∂K_g, ∂Y/∂G.
- Computed stationary solution (using AMPL and Ipopt with Table 1 parameters):
  - K_p = 2.2164
  - K_g = 0.53731
  - G = 0.66746
  - M = 3.0
  - i_p = 0.25489
  - i_g = 0.034925
  - eP = 0.14462
  - C = 0.74766
  - u = 0
  - λ_{Kp} = 2.24939
  - λ_{Kg} = 2.24939
  - λ_G = 3.6216
  - λ_M = −12.121
  - μ_s = −2.24939
- Turnpike property:
  - Numerical solutions demonstrate turnpike behavior: state and control trajectories stay close to the stationary values above over a large intermediate time interval after initial-condition transients.
  - For free terminal states, K_p, K_g, G trajectories sharply decrease near the terminal time; imposing stationary terminal-state constraints helps approximate infinite-horizon solutions.

### Numerical approach and solver settings
- Approach: “First Discretize then Optimize” to solve the optimal control problem OC(p).
- Tools:
  - Applied Modeling Programming Language (AMPL).
  - Interior-Point optimization solver IPOPT.
- Discretization and integration:
  - Terminal time used in computations: T= 200.
  - Grid sizes used: N= 1000 to N= 5000 grid points.
  - Integration method: trapezoidal rule.
  - Example computation used trapezoidal rule with N= 2000 grid points.
- Solver tolerance and expected accuracy:
  - Error tolerance in IPOPT: tol= 10^−8.
  - Expectation: state variables correct up to 6 or 7 decimal digits.
- Adjoint information:
  - Lagrange multipliers and adjoint variables computed a posteriori in IPOPT enabling verification of necessary optimality conditions.

### Illustrative solution structure and general policy dynamics
- Typical dynamic behavior:
  - State and control variables move toward a reasonable steady state that keeps CO2 concentration under control.
  - Once non-renewable fossil fuel extraction becomes economically unproductive (with u(t)≈0) the model converges to the turnpike corresponding to the long-run steady state until finite terminal time dynamics appear.
  - To eliminate unwanted terminal dynamics, subsequent analysis prescribes terminal states equal to the steady state solution.
- Ordering and timing of public policies:
  - Infrastructure policy initially gives way to expanding mitigation efforts which are large in initial periods; adaptation policy typically kicks in later.
  - Ratios of macroeconomic variables remain in a reasonable range across simulations.
- Typical allocation and evolution of public investment:
  - In several scenarios mitigation and adaptation together start by accounting for 35% of the public investment in infrastructure, gradually falling to 15% (in one scenario) and to 10% in another scenario, before converging asymptotically to zero over time.
  - Mitigation is the focus of public investment policy in the beginning and accounts for more than twice the investment of adaptation in many scenarios.
  - Optimal solutions in multiple scenarios leave some of the fossil fuels (R) in ground.

### Scenario summaries with prescribed terminal states (Section 4.3)
- Common terminal stationary targets used:
  - Kp(tf) = 2.2164, Kg(tf) = 0.53731, G(tf) = 0.66746 (stationary values from computed steady state).
- 4.3.1 Small Initial Capital Stocks and High Emission Stock
  - Initial and terminal conditions:
    - Kp(0) = 1, Kg(0) = 0.02, G(0) = 0.2, M(0) = 3.25, R(0) = 1
    - Kp(tf) = 2.2164, Kg(tf) = 0.53731, G(tf) = 0.66746
  - Key outcomes:
    - Policy steers CO2 concentration down toward the steady state after some initial increase.
    - Output, consumption, and investments remain fairly stable after brief initial transitions associated with low initial capital.
    - Infrastructure spending lower at beginning; large initial efforts at mitigation and rising adaptation efforts.
    - Mitigation initially accounts for more than twice the investment of adaptation.
    - Combined mitigation and adaptation start at 35% of public infrastructure investment, gradually falling to 15%, then asymptotically to zero.
    - Some fossil fuels remain unextracted.
- 4.3.2 Large Initial Capital Stocks and High Emission Stock
  - Initial and terminal conditions:
    - Kp(0) = 3, Kg(0) = 0.5, G(0) = 1.0, M(0) = 3.25, R(0) = 1
    - Kp(tf) = 2.2164, Kg(tf) = 0.53731, G(tf) = 0.66746
  - Key outcomes:
    - CO2 concentration steered down toward steady state without initial increase.
    - Excess initial capital induces slow reduction of capital during initial periods, imparting a downward slope to consumption, output, and capital stock during initial periods.
    - Infrastructure spending lower at beginning; large mitigation efforts and rising adaptation efforts.
    - Mitigation initially accounts for more than twice investment of adaptation.
    - Combined mitigation and adaptation start at 35% of public infrastructure investment, gradually falling to 10%, then asymptotically to zero.
    - Some fossil fuels remain unextracted.
- 4.3.3 Small Initial Capital Stocks and Low Emission Stock
  - Initial and terminal conditions:
    - Kp(0) = 1, Kg(0) = 0.02, G(0) = 0.2, M(0) = 2.6, R(0) = 1
    - Kp(tf) = 2.2164, Kg(tf) = 0.53731, G(tf) = 0.66746
  - Key outcomes:
    - Policy prevents CO2 concentration from rising unboundedly.
    - Output, consumption, and investments remain fairly stable after brief initial transitions.
    - Infrastructure spending lower at beginning with large initial mitigation efforts and no adaptation initially.
    - Mitigation starts accounting for over 35% of public infrastructure investment, gradually falling to zero over the initial periods.
    - Some fossil fuels remain unextracted.

### Conclusions — policy implications and main findings (Section 5)
- Controllability of CO2:
  - With proper policy actions the CO2 can be steered down when capital stocks are large and actual CO2 concentration is above a target level.
  - CO2 can be controlled not to exceed an upper limit so that emissions stay bounded by a predefined upper bound.
- Role of mitigation and adaptation timing:
  - Early enacting of mitigation effort is vital for controlling atmospheric CO2 content.
  - Adaptation policy typically increases in importance at a later time.
  - Infrastructure investment efforts are typically delayed.
- Financing and renewable transition:
  - Provides dynamic estimates of how scaling up mitigation and adaptation can be funded and allocated between traditional and climate-related infrastructure, mitigation, and adaptation.
  - Successful mitigation policy implies phasing in renewable energy; the analysis explored how much traditional fossil energy should be left in situ to satisfy CO2 emission and temperature constraints.
- Methodological contribution:
  - Enlarged IAM and used solution methods for higher-dimensional nonlinear control problems.
  - Showed numerical solutions for finite horizon decision model exhibit turnpike properties similar to infinite horizon models.
- Caveats:
  - Control actions may fail if tipping points and thresholds produce regime shifts.
- Reference note:
  - IPCC (2018) finding that climate policies face great challenges in not surpassing upper limits of atmospheric CO2 concentration is noted.

*Source: wp18270 - 4.1 Discretization and Nonlinear Programming Methods (PDF chapter/section).*

### 4.1 Discretization and Nonlinear Programming Methods ...............................................11

### 4.1 Discretization and Nonlinear Programming Methods

### Introduction: objectives and model scope
- Purpose: examine strategies to (i) keep CO2 emission bounded by a predefined upper bound, (ii) scale up climate policies and allocate resources between mitigation and adaptation when climate risk rises, and (iii) determine dynamic funding and sequencing between traditional infrastructure investment, mitigation and adaptation.
- Modeling approach: dynamic global model with feedback control (optimal control) extending Integrated Assessment Models (IAM) to explicitly include:
  - Two energy sources: non-renewable (brown) energy from an extractive resource sector and renewable (green) energy produced with private green capital.
  - Government levies lump-sum taxes (eP) to (a) provide direct utility, (b) invest in public capital, and (c) cover administrative expenses.
  - Public capital can be allocated across three competing uses: traditional infrastructure, mitigation, adaptation.
- Core research questions addressed by numerical and analytical methods:
  - Best strategies to keep or steer down CO2 emissions relative to bounds.
  - Optimal split of public investment between infrastructure, mitigation, adaptation across differing initial capital and CO2 conditions.
  - Allowed extraction of fossil energy given emission and temperature constraints.

### Optimal control model specification
- Time: continuous on finite horizon T.
- Production function (equation (1)):
  - Y = (ν1 G)^β A (A_g K_g + A_u u)^α (K_p)^ζ, with A, A_g, A_u > 0, α, β, ζ > 0, and α + β + ζ < 1.
  - ν1 ∈ (0,1] is fraction of public capital used for traditional productivity-enhancing infrastructure.
  - u is fossil fuel resource extracted and used, measured by carbon (CO2) content.
- Felicity (utility) depends on:
  - Per-capita consumption C.
  - Per-capita tax revenue α2 eP used for direct welfare.
  - Atmospheric concentration of CO2 (M) above long-run sustainable level.
  - Per-capita public capital expenditure ν2 G allocated to adaptation.
- State vector (5-dimensional):
  - X = (K_p, K_g, G, M, R) where K_p = private physical capital per capita, K_g = private green capital per capita, G = public capital per capita, M = CO2 concentration, R = non-renewable resource (fossil energy).
- Control vector (5 basic controls):
  - U = (i_p, i_g, eP, u, C) where i_p = investment in physical capital, i_g = investment in green capital, eP = government net tax revenue, u = extraction rate, C = per-capita consumption.
- Public capital allocations (controls):
  - ν1(t): standard infrastructure, ν2(t): adaptation, ν3(t): mitigation with νk(t) ≥ 0 and ν1(t) + ν2(t) + ν3(t) = 1 for all t.
- Dynamics (equations (2)–(6)):
  - K̇_p = i_p − (δ_p + n) K_p
  - K̇_g = i_g − (δ_g + n) K_g
  - Ġ = α1 eP − (δ_G + n) G
  - Ṁ = γ u − μ (M − κ̃_M) − θ (ν3 · G)^φ
  - Ṙ = −u
- Control constraint on extraction rate (equation (8)):
  - 0 ≤ u(t) ≤ u_max for all t ∈ [0,T]
- Resource constraint (mixed control-state equality, equation (10)):
  - s(X,U,ν) := Y − C − i_p − i_g − eP − u ψ R^−τ − χ_p/2 (i_p/K_p − δ_p − n)^2 K_p − χ_g/2 (i_g/K_g − δ_g − n)^2 K_g = 0
- Welfare functional (equation (11)) with integrand f0 (equation (13)):
  - Welfare = ∫_0^T e^{−(ρ−n)t} [ ... ] dt where the integrand uses parameters ρ, n, σ, η, ε, ξ, ω, κ, κ̃_M, ̄M and captures direct utility from consumption, transfers, disutility from M, and utility from adaptation spending ν2 G.

### Parameter values used (as reported)
- ρ = 0.03 (Pure discount rate)
- n = 0.015 (Population Growth Rate)
- η = 0.1 (Elasticity of transfers and public spending in utility)
-  = 1.1 (Elasticity of CO2-eq concentration in (dis)utility)
- ω = 0.05 (Elasticity of public capital used for adaptation in utility)
- σ = 1.1 (Intertemporal elasticity of instantaneous utility)
- A ∈ [1,10] (Total factor productivity)
- A_g ∈ [1,5] (Efficiency index of green capital)
- A_u ∈ [100,400] (Efficiency index of the non-renewable resource)
- α = 0.1 (Output elasticity of inputs, A_g K_g + A_u u)
- β = 0.5 (Output elasticity of public infrastructure, ν1 G)
- 1 = 1 (Scaling factor in marginal cost of resource extraction)
- τ = 2 (Exponential factor in marginal cost of resource extraction)
- δ_p = 0.1 (Depreciation rate of physical capital)
- δ_g = 0.05 (Depreciation rate of private capital)
- δ_G = 0.05 (Depreciation rate of public capital)
- χ_p = 1/(δ_p + n) Ω_p, χ_g = 1/(δ_g + n) Ω_g with Ω_p ∈ [5,15], Ω_g ∈ [5,15]
- α1 = 0.2 (Proportion of tax revenue allocated to new public capital)
- α2 = 0.5 (Proportion of tax revenue allocated to transfers and public consumption)
- ̄r = 0.07 (World interest rate (paid on public debt))
- ̃M = 2.5 (equilibrium concentration)
- γ = 0.9 (Fraction of greenhouse gas emissions not absorbed by the ocean)
- μ = 0.01 (Decay rate of greenhouse gases in atmosphere)
- κ = 1 (Atmospheric concentration stabilization ratio (relative to ̃M))
- θ = 0.01 (Effectiveness of mitigation measures)
- φ ∈ [0.2,1] (exponent in mitigation term (ν3 G)^φ)

### Necessary conditions, Hamiltonian and first-order system
- Current-value Hamiltonian (equation (15)):
  - H(X, λ, U, ν) = f0(X,U,ν) + λ · f(X,U,ν)
- Augmented Hamiltonian with multiplier η for mixed control-state constraint (equation (16)):
  - H(X, λ, η, U, ν) = H(X, λ, U, ν) + η s(X, U, ν)
- Adjoint (costate) dynamics and first-order conditions (equations (18)–(20)):
  - λ̇(t) = (ρ − n) λ(t) − ∂H/∂X − η ∂s/∂X
  - Maximum principle: H(X(t), λ(t), U(t), ν(t)) = max_{(U,ν) ∈ Ω(t)} H(...)
  - Local first-order conditions for interior controls v ∈ {i_p, i_g, eP, C}: ∂H/∂v = 0
- Control admissible set (equation (17)) enforces nonnegativity of i_p, i_g, eP; C > 0; 0 ≤ u ≤ u_max; ν_k ≥ 0 and ν1 + ν2 + ν3 = 1; and satisfaction of s(X,U,ν) = 0.

### Stationary (steady) solution and characteristics
- Stationary equations reduce to 13 equations for variables (K_p, K_g, G, M), (λ_{Kp}, λ_{Kg}, λ_G, λ_M), (i_p, i_g, eP, C), η, with u = 0 at steady state (as Ṙ = −u and stationary Ṙ = 0 implies u = 0).
- Partial derivatives used to form first-order conditions include:
  - ∂f0/∂C, ∂f0/∂eP, ∂f0/∂G (∂f0/∂G = 0 in steady state because ν2 = 0)
  - ∂Y/∂K_p, ∂Y/∂K_g, ∂Y/∂G as specified (equation (26)).
- Computed stationary solution (using AMPL and Ipopt with Table 1 parameters) (equation (27)):
  - K_p = 2.2164
  - K_g = 0.53731
  - G = 0.66746
  - M = 3.0
  - i_p = 0.25489
  - i_g = 0.034925
  - eP = 0.14462
  - C = 0.74766
  - u = 0
  - λ_{Kp} = 2.24939
  - λ_{Kg} = 2.24939
  - λ_G = 3.6216
  - λ_M = −12.121
  - μ_s = −2.24939
- Turnpike property observed:
  - Numerical solutions demonstrate turnpike behavior: state and control trajectories stay close to the stationary values in (27) over a large intermediate time interval after initial-condition transients.
  - For free terminal states, K_p, K_g, G trajectories sharply decrease near the terminal time; imposing stationary terminal-state constraints helps approximate infinite-horizon solutions.

*Source: wp18270 - 4.1 Discretization and Nonlinear Programming Methods (PDF chapter/section).*

### 4.1    Discretization and Nonlinear Programming Methods

### 4.1    Discretization and Nonlinear Programming Methods

### Numerical approach and solver
- Adopted approach: “First Discretize then Optimize” to solve the optimal control problem OC(p) defined in (12)–(14).
- Modeling and solver tools:
  - Applied Modeling Programming Language (AMPL).
  - Interior-Point optimization solver IPOPT (Wächter and Biegler [17]).
- Discretization and integration:
  - Terminal time used in computations: T= 200.
  - Grid sizes used: N= 1000 to N= 5000 grid points.
  - Integration method: trapezoidal rule.
- Solver tolerance and expected accuracy:
  - Error tolerance in IPOPT: tol= 10^−8.
  - Expectation: state variables correct up to 6 or 7 decimal digits.
- Adjoint information:
  - Lagrange multipliers and adjoint variables computed a posteriori in IPOPT enabling verification of necessary optimality conditions.

### Illustrative solution structure and turnpike property
- With free terminal states (example computation):
  - Integration method and grid: trapezoidal rule with N= 2000 grid points.
  - Initial conditions used: T= 200 : Kp(0) = 2.5, Kg(0) = 0.3, G(0) = 0.8, M(0) = 3.25, R(0) = 1.
- Observed dynamic behavior:
  - State and control variables move toward a reasonable steady state that keeps CO2 concentration under control.
  - Turnpike property: once non-renewable fossil fuel extraction becomes economically unproductive (with u(t)≈0) the model converges to the turnpike corresponding to the long-run steady state solution described in Section 3, until finite terminal time dynamics appear later.
  - To eliminate unwanted terminal dynamics, subsequent analysis prescribes terminal states equal to the steady state solution.

### Policy dynamics across simulations (general patterns)
- Ordering and timing of public policies:
  - Infrastructure policy initially gives way to expanding mitigation efforts which are large in initial periods; adaptation policy typically kicks in later.
  - Ratios of macroeconomic variables remain in a reasonable range across simulations.
- Typical allocation and evolution of public investment in mitigation and adaptation:
  - In several scenarios mitigation and adaptation together start by accounting for 35% of the public investment in infrastructure, gradually falling to 15% (in one scenario) and to 10% in another scenario, before converging asymptotically to zero over time.
  - Statements about mitigation share:
    - “Mitigation is the focus of the public investment policy in the beginning as it accounts for more than twice of investment than adaptation.”
    - In the low-emission small-capital case: mitigation starts with accounting for over 35% of the public investment in infrastructure, gradually falling to zero over the initial periods.
- Fossil fuel extraction:
  - Optimal solution in multiple scenarios leaves some of the fossil fuels (R) in ground.

### 4.3    Solutions with Prescribed Terminal States — Scenario summaries

### 4.3.1 Small Initial Capital Stocks and High Emission Stock
- Prescribed terminal stationary values (27) used as terminal conditions for Kp, Kg, G:
  - Kp(0) = 1, Kg(0) = 0.02, G(0) = 0.2, M(0) = 3.25, R(0) = 1,
  - Kp(tf) = 2.2164, Kg(tf) = 0.53731, G(tf) = 0.66746.
- Key outcomes:
  - Policy steers CO2 concentration down toward the steady state after some initial increase.
  - Output, consumption, and investments remain fairly stable after brief initial transitions associated with low initial capital.
  - Infrastructure spending lower at beginning; large initial efforts at mitigation and rising adaptation efforts.
  - Mitigation initially accounts for more than twice the investment of adaptation.
  - Combined mitigation and adaptation start at 35% of public infrastructure investment, gradually falling to 15%, then asymptotically to zero.
  - Some fossil fuels remain unextracted.

### 4.3.2 Large Initial Capital Stocks and High Emission Stock
- Initial and terminal conditions:
  - Kp(0) = 3, Kg(0) = 0.5, G(0) = 1.0, M(0) = 3.25, R(0) = 1,
  - Kp(tf) = 2.2164, Kg(tf) = 0.53731, G(tf) = 0.66746.
- Key outcomes:
  - CO2 concentration steered down toward steady state without initial increase.
  - Excess initial capital induces slow reduction of capital during initial periods, imparting a downward slope to consumption, output, and capital stock during initial periods.
  - Infrastructure spending lower at beginning; large mitigation efforts and rising adaptation efforts.
  - Mitigation initially accounts for more than twice investment of adaptation.
  - Combined mitigation and adaptation start at 35% of public infrastructure investment, gradually falling to 10%, then asymptotically to zero.
  - Some fossil fuels remain unextracted.

### 4.3.3 Small Initial Capital Stocks and Low Emission Stock
- Prescribed terminal stationary values (27) and initial conditions:
  - Kp(0) = 1, Kg(0) = 0.02, G(0) = 0.2, M(0) = 2.6, R(0) = 1,
  - Kp(tf) = 2.2164, Kg(tf) = 0.53731, G(tf) = 0.66746.
- Key outcomes:
  - Policy prevents CO2 concentration from rising unboundedly.
  - Output, consumption, and investments remain fairly stable after brief initial transitions.
  - Infrastructure spending lower at beginning with large initial mitigation efforts and no adaptation initially.
  - Mitigation starts accounting for over 35% of public infrastructure investment, gradually falling to zero over the initial periods.
  - Some fossil fuels remain unextracted.

### 5    Conclusions — policy implications and main findings
- Controllability of CO2:
  - With proper policy actions the CO2 can be steered down when capital stocks are large and actual CO2 concentration is above a target level.
  - CO2 can be controlled not to exceed an upper limit so that emissions stay bounded by a predefined upper bound.
- Role of mitigation and adaptation timing:
  - Early enacting of mitigation effort is vital for controlling atmospheric CO2 content.
  - Adaptation policy typically increases in importance at a later time.
  - Infrastructure investment efforts are typically delayed.
- Caveats:
  - Control actions may fail if tipping points and thresholds produce regime shifts (as shown in Greiner et al. (2010) and Nordhaus (2008)).
- Financing and renewable transition:
  - Provides dynamic estimates of how scaling up mitigation and adaptation can be funded and allocated between traditional and climate-related infrastructure, mitigation, and adaptation.
  - Successful mitigation policy implies phasing in renewable energy; the analysis explored how much traditional fossil energy should be left in situ to satisfy CO2 emission and temperature constraints.
- Methodological contribution:
  - Enlarged IAM and used solution methods for higher-dimensional nonlinear control problems.
  - Showed numerical solutions for finite horizon decision model exhibit turnpike properties similar to infinite horizon models.
- Reference to external assessment:
  - Notes IPCC (2018) finding that climate policies face great challenges in not surpassing upper limits of atmospheric CO2 concentration.

*Source: IMF working paper chapter "4.1    Discretization and Nonlinear Programming Methods" (wp18270).*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18270.pdf_