## wpiea2021173-print-pdf ## Source details **Canonical URL:** [wpiea2021173-print-pdf](https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021173-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2021/english/wpiea2021173-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2021/english/wpiea2021173-print-pdf.pdf.json) --- ### Aggregate production - Final good production per region m: - Y_{m,t} = z_{m,t} ((1−ν_{m}) Y^*_{m,t}^{σ} + ν_{m} E_{m,t}^{σ})^{1/σ} - Y^*_{m,t} = (A_{m,t} L_{m,t})^{1−α_{m}} K_{m,t}^{α_{m}} - Inputs and parameters: - E_{m,t} intermediate energy input; L_{m,t} labor; K_{m,t} capital; A_{m,t} labor-augmenting technical change; z_{m,t} region-specific productivity loss from climate. - ν_{m} and α_{m} region-specific shares; F = 1/(1−σ) is elasticity of substitution between capital-labor bundle and energy. - Factor prices (first-order conditions): - i_{m,t} = α_{m} (1−ν_{m}) z_{m,t}^{σ} (Y^*_{m,t}/K_{m,t}) (Y_{m,t}/Y^*_{m,t})^{1−σ} - w_{m,t} = (1−α_{m}) (1−ν_{m}) z_{m,t}^{σ} (Y^*_{m,t}/L_{m,t}) (Y_{m,t}/Y^*_{m,t})^{1−σ} - P_{m,t} = ν_{m} z_{m,t}^{σ} (Y_{m,t}/E_{m,t})^{1−σ} ### Energy production and research technologies - Fuels: - N fuels with emissions intensity g_i and production costs. - Fuel 1: conventional oil, fixed stock owned by region M (exhaustible). - Fuels i = 2,...,N: linear production e_{m,i,t} = x_{m,i,t} l_{m,i,t}; emissions g_i e_{m,i,t}. - One fuel j is zero-carbon: g_j = 0. - Production cost and research: - Cost of producing fuel i: p_{i,t} = 1/x_{i,t}. - Research cost (i = 2,...,N): - r_{m,i,t} = ϵ_{m,i} (1−χ_{m,i,t})^{η−1} (x_{m,i,t}/\bar{x}_{m,i,t})^{η−1} - Equivalently r_{m,i,t} = ϵ_{m,i} (1−χ_{m,i,t})^{η−1} (p_{m,i,t}/\bar{p}_{m,i,t})^{1−η} - η > 1; ϵ_{m,i} fuel-specific; χ_{m,i,t} government R&D subsidy (χ_{i,t} = 0 for i ≠ j except green). - Return to scale in energy research = 1/(η−1); specification generalizes log research as η → 1. - Energy bundling (CES, elasticity E = 1/(1−ρ)): - E_{m,t} = (∑_{i=1}^n λ_{m,i} e_{m,i,t}^{ρ})^{1/ρ} ### Energy supply and market structure - Input cost function: - c(...) = (τ_{m,t} g_{1,m} + p_{1,m,t}) e_{1,m,t} + ∑_{i=2}^n ((τ_{m,t} g_{i,m} + (1−δ_{i,m,t}) p_{i,m,t}) e_{i,m,t} + ϵ_{i,m} (1−χ_{i,m,t})^{η−1} (p_{i,m,t}/\bar{p}_{i,m,t})^{1−η}) - Define ˆp_{i,m,t} = τ_{m,t} g_i + (1−δ_{i,m,t}) p_{i,m,t} - Purchasing subsidy δ_{i,m,t} and R&D subsidy χ_{i,m,t} assumed zero except for i = j. - First-order conditions (interior if E < η): - e_{i,m,t}/e_{j,m,t} = (λ_j ˆp_{i,m,t} / (λ_i ˆp_{j,m,t}))^{−E} ∀ i - (1−δ_{i,m,t}) e_{i,m,t} = ϵ_i (1−χ_{i,m,t}) \bar{p}_{i,m,t} (p_{i,m,t}/\bar{p}_{m,i,t})^{−η} ∀ i > 1 - r_{i,m,t} ∝ e_{i,m,t} p_{i,m,t}: research spending proportional to expenditure on input. - Energy market: - Output cost C(E) is downward-sloping; energy market is a natural monopoly. - Regulated monopoly price set to average cost: P_{m,t} = C(E_{m,t}) / E_{m,t} - Composition (relative input shares) determined by first-order conditions, independent of monopoly vs. competition; monopoly regulation affects total quantity only. ### Governments: budget and instruments - Region-specific policies: carbon tax τ_{m,t}, production subsidy δ_{j,m,t}, R&D subsidy χ_{j,m,t}. - Governments rebating excess carbon tax revenue as lump-sum transfer Ω_{m,t} (Ω_{m,t} < 0 is lump-sum tax). - Government budget constraint: - Ω_{m,t} + δ_{j,m,y} p_{j,m,t} e_{j,m,t} + (χ_{j,m,t}/(1−χ_{j,m,t})) r_{j,m,t} = ∑_{i=1}^N τ_{m,t} g_{i,m} e_{i,m,t} ### Households: preferences and savings - Households sell labor (inelastic L_{m,t}) and rent capital; capital fully depreciates each period (baseline). - Utility: max ∑_{t=0}^∞ β^t log C_{m,t} subject to C_{m,t} + K_{m,t+1} = w_{m,t} L_{m,t} + i_{m,t} K_{m,t} + Ω_{m,t} - Euler equation: - C_{m,t+1} / C_{m,t} = β i_{m,t+1} - Define factor income ˆY_{m,t} = w_{m,t} L_{m,t} + i_{m,t} K_{m,t}; Γ_{m,t} = Ω_{m,t} / ˆY_{m,t}; savings rate s_{m,t} = K_{m,t+1} / ((1 + Γ_{m,t}) ˆY_{m,t}) - Relationship: (1−s_{m,t+1})/(1−s_{m,t}) = α β s_{m,t} (1 + Γ_{m,t+1}) - If Γ_{m,t} = 0 then s_{m,t} = α β (constant). Government taxes/subsidies can induce non-constant savings because payments to energy-producing firms are not factor payments unless rebated. ### Trade in conventional oil - Region M holds global conventional oil stock R_t. - Oil stock dynamics (Hassler et al. (2020) setup with logarithmic preferences in oil-producing region): - R_{t+1} = β R_t - C_{M,t} = p_{1,t} (1−β) R_t - Global oil market clearing: - ∑_{m=1}^{M−1} e_{1,t} = (1−β) R_t ### Technological diffusion - Domestic diffusion: - p^d_{m,i,t+1}/\bar{p}_{m,i,t} = (∏_k (p_{m,k,t}/\bar{p}_{m,k,t})^{ρ_{ik}})^{ω} - Log form: log \bar{p}^d_{m,t+1} = (I_N − ω ρ) log \bar{p}_{m,t} + ω ρ log p_{m,t} - ρ matrix of intra-fuel spillovers (rows sum to unity); ω dampening parameter (ω < 1 if part of innovation cannot be passed on). - International diffusion (catch-up): - Frontier: \bar{p}^*_{i,t} = min_{m=1,...,M−1} \bar{p}^d_{m,i,t} - Reference cost update: log \bar{p}_{m,t+1} = ζ_{m} log \bar{p}^*_{t} + (1−ζ_{m}) log p^d_{m,t} - ζ_{m} measures speed of diffusion (ζ_{m} = 1 immediate learning; ζ_{m} = 0 no catch-up). ### Climate externality and damages - Global emissions M_t = ∑_{m=1}^M M_{m,t}; atmospheric carbon stock S_t is function of lagged global emissions: - S_t = ∑_{u=0}^∞ (φ_L + (1−φ_L) φ_0 (1−φ)^u) M_{t−u} - Equivalent decomposition: - S^{perm}_t = S^{perm}_{t−1} + φ_L M_t - S^{temp}_t = (1−φ) S^{temp}_{t−1} + (1−φ_L) φ_0 M_t - Temperature dynamics (linear energy budget model): - [T_t; T^{ocean}_t] = [T_{t−1}; T^{ocean}_{t−1}] + [[−σ_1/(η_c λ_c + σ_2), σ_1/(η_c λ_c + σ_2)]; [σ_2/σ_3, −σ_3/σ_3]] [T_{t−1}; T^{ocean}_{t−1}] + [σ_1 η_c log 2 / log(S_{t−1}/\bar{S}); 0] - λ_c is climate sensitivity (doubling of stock relative to pre-industrial raises temperatures by λ_c). - Productivity damages: - log z_{m,t} = −(φ^m_0 + φ^m_1 T_{t−1} + φ^m_2 (T_{t−1})^2) (quadratic in atmospheric temperature) ### Equilibrium definition and aggregate resource constraint - Resource constraint: - C_{m,t} + K_{m,t} + P_{m,t} E_{m,t} = Y_{m,t} - Given policies {τ_m,t, δ_j,m,t, χ_j,m,t}, initial {K_m,0, ̄p_m,0}, global initial {S_0, T_0, R_0}, and exogenous {A_m,t, L_m,t}, an equilibrium is sequences of global and region-specific variables satisfying model equations (1)-(6), (8)-(12), (14)-(24). ### Calibration overview (selected configuration and targets) - Model configuration: - M = 12 regions; region M = oil-producing region (OPEC) - N = 7 fuel types: conventional oil, coal, natural gas, unconventional oil, hydroelectric, nuclear, renewable fuels - Time period = five years; period 0 corresponds to 2015-2019 (flows averaged; stocks use 2015) - Data sources: IEA World Energy Balances, ILO labor data, OPEC proven reserves, and literature for parameter choices/validation. - Productivity growth assumptions: - OECD countries grow at 0.75% per year - Non-OECD converge to same growth path but 60 percent above current levels; convergence rate: 25 percent per decade - Productivity law specified with five-year increments using log A updates (textual formula preserved) - Energy intensity trend: - ν_m,t = (1 + g_ν) ν_m,t−1 with g_ν chosen to match steady annual decline in global energy intensity ≈ 1.5 percent per year since 1970 - Climate physics: - Parameters from Hassler et al. (2020) adjusted to five-year periods - Long-run climate sensitivity λ_c = 3 - Conventional oil and discounting: - Initial R_0 taken from proven reserves; β equivalent to subjective annual discount rate of 0.9 percent ### Common global calibrated parameters (selected) - Economic: - E = 2.00 - F = 0.35 - η = 28.88 - ̄ρ = 0.94 - ω = 1.00 - ζ = 0.45 - β = 0.96 - g_ν = -0.02 (percent annual) - Climate: - σ1 = 0.22; σ2 = 0.15; σ3 = 0.02 - φL = 0.20; φ0 = 0.40; φ = 0.01 - λ_c = 3.00; η_c = 1.85 - ̄S = 581.00 (Gigatonnes) - Initial values: - S_0 = 800.00 (Gigatonnes) - S_perm0 / S_0 = 0.18 - R_0 = 203.67 (Gigatonnes) - T_0 = 0.85 - T_ocean0 = 0.01 ### Region-specific calibration (targets and methods) - ν_m and α_m pinned to IEA energy share and IMF WEO capital share; where unavailable use global average. - Climate externality costs from Nordhaus (2010) analogues. - Energy sector/research calibration: - λ_{m,i} and research cost shifters ϵ_{m,i} chosen to match IEA input prices and fuel usages. - Price units: $ per tonne oil equivalent. - ̄p_{m,i,0} chosen to match EIA-MER efficiency gains (US data) over 2015-2019. - Carbon intensities g_{m,i} computed from IEA emissions/consumption ratios. - Initial capital K_{m,0} set so capital stock consistent with past savings rate; A_{m,0} set so regional output matches data. ### Validation and unmatched moments - β corresponds to around 1 percent per year; period 0 equilibrium oil price: $528 per ton, or $72 per barrel (period-0 model price). - Model implies research share ≈ 1/(η−1) ≈ 3.5 percent of energy expenditures → ≈ 0.5 percent of GDP in energy research given 15 percent energy share of GDP. - IEA World Energy Report suggests about 0.17 percent of world GDP in 2018 (public + publicly-reported private R&D) — possibly lower bound. - UN estimate around 2.1 percent of GDP for research of all kinds. ### Laissez-faire baseline and endogenous technical change (headlines) - Exogenous technical change baseline aligns with integrated assessment models projecting high long-run warming (around 7 degrees Celsius). - Endogenous technical change: - Reduces emissions slightly over model horizon; lowers final temperature by a fraction of one degree. - Endogenous technical change is climate-neutral in long run. - Until ~2100 baseline endogenous technological change is increasingly dirty: global average emissions per unit of energy increase by almost one third (largely driven by coal efficiency gains, especially China). - Second half of simulation: composition shifts toward renewables, offsetting earlier increases. - Geography: population growth and climate externality interact heterogeneously (example: Sub-Saharan Africa vs. Russia). ### Policy simulation roadmap - Section 4: simulate partially successful global climate policies focusing on domestic impacts in emissions-reducing regions (Europe, China), plausible emission reduction targets, and three policy instruments (carbon taxes, production subsidies, R&D subsidies) and their spillover patterns. - Section 5: map policies to global temperatures and global economic activity. --- ### Domestic emissions targets (China and OECD Europe experiment) - Targets and timing: - China: emissions decline to 10 percent of current levels by 2060 and remain thereafter; Chinese emissions must peak in 2030. - European OECD: emissions decline to 10 percent of current levels by 2050 and remain thereafter. - 10 percent floor interpretable as small overshoot or residual removal via CCS; cuts reduce emissions by over 96 percent in 2050 relative to laissez-faire. - Direct temperature effect: - China + OECD Europe ≈ 40 percent of global emissions; holding others fixed yields direct reduction in global temperatures of around 1.5 degrees Celsius in 2200. - Indirect effect via spillovers: - Policy-induced technical change spills over internationally; magnitude depends on instrument mix. - Research subsidies induce more technical change and international spillovers than production subsidies for the same resource cost. - Mapping regional cuts to global outcomes: - Global-temperature outcome = direct emissions reduction effect + indirect spillover effect from induced technical change and diffusion. ### Why carbon tax rises sharply (Appendix B.2) - Empirical observation: Peak-to-trough emissions fall by ≈ factor ten while taxes increase by ≈ factor thirty. - Three reinforcing mechanisms: - Tax shrinks base; higher taxes needed for same volume reductions. - Taxes raise post-tax input costs so percent increases require larger taxes. - At high taxes input costs ≈ tax across dirty fuels, removing discrimination and composition channel. - Illustration: - Marginal effectiveness falls sharply once tax > $100 per ton. - Tax around $300 required to deliver ~90 percent reductions (rough order referenced). ### Why carbon taxes lower in China - China’s coal intensity (≈ two thirds of all energy) creates high tax base; same emissions reductions can be achieved with smaller tax increases than in OECD Europe. ### Why research subsidies more effective than production subsidies (Appendix B.2) - Contemporary effects similar; dynamic divergence because: - Production subsidies wedge productivity and firms’ input costs, reducing research incentives. - Research subsidies lower future green costs more, compounding over time. - Partial-equilibrium relationships: - dlog E_t / dδ_{j,t} = (η−1) ((1−χ_{j,t})/(1−δ_{j,t})) dlog E_t / dχ_{j,t} - dlog M_t / dδ_{j,t} = (η−1) ((1−χ_{j,t})/(1−δ_{j,t})) dlog M_t / dχ_{j,t} - Fiscal-impact marginal term (equation (29)) typically very small and zero when χ_{j,t} = δ_{j,t}. - Dynamic advantage of research subsidy: - Research subsidy yields larger long-run reductions in relative green input costs by a factor leading to sustained compounding; formula shows negative term −1/(1−δ_{j,t}) indicating stronger long-run relative price effect. ### Domestic decomposition (China and OECD Europe) - Static vs. dynamic channels: - Carbon tax: large contemporaneous composition effect; quantity effect near zero. - Research subsidy: small contemporaneous effect but induces similar dynamic technological change as tax. - Spending carbon tax revenue on research subsidies roughly doubles induced emissions effect versus tax alone. - Welfare/consumption: - Carbon tax raises energy prices, reducing energy consumption. - Research subsidy lowers future energy prices, cushioning welfare costs; explains most long-run consumption improvement in medium run. ### International technological spillovers (section 5) - Spillovers very large in long run: - Even tax-only reductions in Europe + China reduce temperatures in 2200 by nearly additional 2 degrees beyond direct regional cuts. - Mechanism: - China’s coal market drives global coal innovation; reducing Chinese coal demand eliminates negative technological externality. - Chinese coal input costs ≈ 10 percent below next cheapest region pre-policy. - By 2050 carbon tax reduces Chinese coal demand; China stops driving global coal productivity improvements. - Policy package variation: - Funneling tax revenue into research subsidies reduces temperature further by ≈ 1 degree in 2200 (additional to earlier effects) via reduced green costs spilling over internationally. - Production subsidies produce positive long-run international spillovers too (reduce temperature by a little less than 0.5 degree in 2200), because other countries benefit from induced research without domestic offsetting. ### Fuel-by-fuel decomposition of international spillovers (experiment (2), section 5.2) - Method: stepwise monotone substitution of fuel-region-time input price paths (100 steps, 600 model solves for six improvable fuels). - Approximation quality: cumulative partial-equilibrium responses closely match full endogenous technical change for non-active regions (approximation error almost invisible). - Key quantitative findings: - Majority of emissions reduction in non-active regions due to changes in coal and renewables input costs. - Deterrence of coal-technology improvements accounts for ≈ two thirds of decline in emissions in non-active regions. - Contribution to long-run temperature reduction (~4.5 degrees total): - ≈ 3 degrees due to international technological spillovers (mostly deterring dirty-fuel improvement, especially coal). - ≈ 1 degree due to incentives for new green technology. ### Alternative parameterizations of diffusion (section 5.3) - No international diffusion (ζ_m = 0): temperatures in 2200 decline by < 0.5 degrees relative to laissez-faire. - Eaton & Kortum-style lower spillovers (ζ_m = 0.24): long-run temperatures decline by almost 4 degrees. - Trade-link weighted diffusion (import shares): long-run temperatures still decline by almost 4 degrees (China + OECD Europe ≈ 50 percent of imports into other regions). - Faster diffusion (full catch-up with one-period lag): small additional effects (declining marginal returns). - Distributional effect: - Faster diffusion increases global benefits but raises mitigation burden on implementing regions (higher short-term carbon taxes, delayed long-run tax reductions). ### Robustness checks and alternate specifications (section 6) - Main checks: alternate research production functions, non-constant savings, mapping to other models (Fried (2018), Acemoglu et al. (2016)). - Comparison to Fried (2018): - Replication with single-region two-fuel setup; calibrated parameters (example: E = 1.5, F = 0.05, η = 52.4). - Headline aggregate responses near-identical; induced technical change amplifies tax effectiveness (consistent qualitative findings). - Comparison to Acemoglu et al. (2016): - Recalibration with E = 3, F = 1, η = 33.0, shows similar temperature responses to optimal policies across models. - Non-constant savings rate: - Replace log preferences and full depreciation with CRRA and δ_K = 0.5 (five-year period equivalent to 12% annual depreciation). - Consider γ = 2 and γ = 0.5; recompute K_{m,0} and A_{m,0}. - Findings: negligible effect on main conclusions; capital accumulation affects quantity not composition; composition dominates emissions. - Conclusion: main result robust — regional emissions targets can materially mitigate global climate via international diffusion of induced technical change. ### Appendix highlights: energy market structure and partial-equilibrium approximation - Average cost function properties: - With τ_t = 0, C(E_t) = (η/(η−1)) ∑ e_{i,t} p_{i,t}; dlog(C(E_t)/E_t) = −1/η; dlog(C(E_t)) = 1 − 1/η. - Equilibrium conditions require F < η for interior solution; when F < 1 and F < η price regulation needed. - Partial-equilibrium decomposition useful to compute static composition effect, static quantity effect, and dynamic induced technical change. - Static marginal impacts (China example, τ = 0): - Composition dominates emissions responses: e.g., coal d m_{i,t} / d τ_t contribution −3.2 pp; total −3.1 pp. - Memo: research amplification factors: energy composition η/(η−E) = 1.07; total energy η/(η−F) = 1.01. ### Computation and numerical methods - Solution algorithm: - Outer loop on international oil price p_{1,t}; inner loop per region to invert CES demand and compute supply price P^s_{m,t} and adjust E_{m,t}. - Inverting model to meet emissions targets (Appendix D.2): - Double-bisection algorithm on carbon tax and subsidy control θ with policy parameter κ ∈ [0,1], χ_{m,j,t} = κ θ_{m,j,t}, δ_{m,j,t} = (1−κ) θ_{m,j,t}. - Inner loop ensures government budget balance; outer loop hits emissions target. - Converges within a dozen iterations typically; computationally intensive but tractable for two active regions overnight on a modern laptop. - Numerical precision note: 12 bisection steps → error of 0.5^{12} ≈ 0.025% of initial interval; subsidy error ≤ 0.025 percentage points; tax error example ≈ $0.25 if initial upper bound $1000. - Solving with non-constant savings (Appendix D.3): - Euler equation (CRRA, depreciation) and fixed-point iteration to find consistent savings path using backward recursion and terminal balanced-growth formula. *Source: wpiea2021173-print-pdf* ### 2.1 Aggregate production ### 2.1 Aggregate production ### Aggregate production setup - Time is discrete and indexed by t. - In each of m = 1,...,M−1 regions, a continuum of competitive firms produce a final good using a production technology which aggregates energy and a Cobb-Douglas capital-labor bundle with a constant elasticity of substitution. - Gross output: - Y_{m,t} = z_{m,t} ((1−ν_{m}) Y^*_{m,t}^{σ} + ν_{m} E_{m,t}^{σ})^{1/σ} (Equation 1) - Y^*_{m,t} = (A_{m,t} L_{m,t})^{1−α_{m}} K_{m,t}^{α_{m}} is the Cobb-Douglas labor-capital input mixture. - E_{m,t} is the intermediate energy input; L_{m,t} is labor; K_{m,t} is capital; A_{m,t} is labor-augmenting technical change; z_{m,t} is a region-specific aggregate productivity loss due to climate change. - Parameters: - ν_{m} and α_{m} are region-specific shares (identified from data). - F = 1/(1−σ) denotes the elasticity of substitution between the capital-labor bundle and energy. - Factor prices from first-order conditions: - i_{m,t} = α_{m} (1−ν_{m}) z_{m,t}^{σ} (Y^*_{m,t}/K_{m,t}) (Y_{m,t}/Y^*_{m,t})^{1−σ} (Equation 2) - w_{m,t} = (1−α_{m}) (1−ν_{m}) z_{m,t}^{σ} (Y^*_{m,t}/L_{m,t}) (Y_{m,t}/Y^*_{m,t})^{1−σ} (Equation 3) - P_{m,t} = ν_{m} z_{m,t}^{σ} (Y_{m,t}/E_{m,t})^{1−σ} (Equation 4) ### 2.2 Energy production and research technologies ### Fuels and emission intensities - There are N fuels; fuels differ by emissions intensity g_i and production costs. - Fuel 1 is conventional oil, with a fixed stock owned by region M. - Fuels i = 2,...,N are produced with a linear technology: e_{m,i,t} = x_{m,i,t} l_{m,i,t}, where l_{m,i,t} is units of final output used. - Emissions for fuel i: g_i e_{m,i,t}. - One fuel j is zero-carbon ("green"): g_j = 0. - Cost of producing fuel i for given technology: p_{i,t} = 1/x_{i,t}. - Assumption: for a given technology level, each fuel except oil is available at constant input cost (infinite supply assumption for those fuels); oil is exhaustible and modeled explicitly. ### Research in fuel technologies - For fuels i = 2,...,N, firms can conduct research to lower production costs. - Reference level of technology: \bar{x}_{m,i,t}; cost of new learning r_{m,i,t} expressed relative to this level. - Research cost specification: - r_{m,i,t} = ϵ_{m,i} (1−χ_{m,i,t})^{η−1} (x_{m,i,t}/\bar{x}_{m,i,t})^{η−1} - Equivalently: r_{m,i,t} = ϵ_{m,i} (1−χ_{m,i,t})^{η−1} (p_{m,i,t}/\bar{p}_{m,i,t})^{1−η} (Equation 5) - η > 1, ϵ_{m,i} is a fuel-specific cost parameter, χ_{m,i,t} is a government subsidy to research in fuel i. - Simplifying assumption: research subsidies do not vary except for the green technology; set χ_{i,t} = 0 for i ≠ j. - Return to scale in energy research is 1/(η−1). The specification generalizes the log research function recovered as η → 1. ### Energy bundling - Energy-producing firms bundle fuels into energy with a CES production function with elasticity E = 1/(1−ρ): - E_{m,t} = (∑_{i=1}^n λ_{m,i} e_{m,i,t}^{ρ})^{1/ρ} (Equation 6) ### 2.3 Energy supply ### Input cost function and minimization - Climate mitigation policies (carbon taxes, production subsidies, and R&D subsidies) are region-specific and collected from energy-producing firms. - Input cost function: - c(e_{1,m,t},...,e_{N,m,t},p_{1,t},p_{2,m,t},...,p_{N,m,t}) = (τ_{m,t} g_{1,m} + p_{1,m,t}) e_{1,m,t} + ∑_{i=2}^n ((τ_{m,t} g_{i,m} + (1−δ_{i,m,t}) p_{i,m,t}) e_{i,m,t} + ϵ_{i,m} (1−χ_{i,m,t})^{η−1} (p_{i,m,t}/\bar{p}_{i,m,t})^{1−η}) (Equation 7) - \bar{p}_{i,w,t} is the technologically-determined reference cost; δ_{i,m,t} and χ_{i,m,t} are purchasing and R&D subsidies for fuel i in region m (assumed zero except for i = j). - Energy-producing firms minimize c subject to the CES bundling constraint. Define ˆp_{i,m,t} = τ_{m,t} g_i + (1−δ_{i,m,t}) p_{i,m,t}. ### First-order conditions and implications - If E < η, interior solution with first-order conditions: - e_{i,m,t}/e_{j,m,t} = (λ_j ˆp_{i,m,t} / (λ_i ˆp_{j,m,t}))^{−E} ∀ i (Equation 8) - (1−δ_{i,m,t}) e_{i,m,t} = ϵ_i (1−χ_{i,m,t}) \bar{p}_{i,m,t} (p_{i,m,t}/\bar{p}_{m,i,t})^{−η} ∀ i > 1 (Equation 9) - Rearrangement shows r_{i,m,t} ∝ e_{i,m,t} p_{i,m,t}: research spending is proportionate to expenditure on a given input. - The output cost function C(E) is downward-sloping; energy market is a natural monopoly. - Monopolist priced capped by regulation yielding price equals average cost when monopoly profits are set to zero: - P_{m,t} = C(E_{m,t}) / E_{m,t} (Equation 10) - Composition of energy (relative input shares) is determined by the first-order conditions (8) and (9) and is independent of monopoly/comparative assumptions; only total quantity depends on monopoly regulation. ### 2.4 Governments ### Budget and policy instruments - Governments in regions m = 1,...,M−1 raise revenues from the carbon tax and spend on subsidies for production and R&D. - Governments run balanced budgets, rebating excess carbon tax revenue over subsidies as lump-sum transfer Ω_{m,t} (Ω_{m,t} < 0 denotes a lump-sum tax). - Government period budget constraint: - Ω_{m,t} + δ_{j,m,y} p_{j,m,t} e_{j,m,t} + (χ_{j,m,t}/(1−χ_{j,m,t})) r_{j,m,t} = ∑_{i=1}^N τ_{m,t} g_{i,m} e_{i,m,t} (Equation 11) ### 2.5 Households ### Preferences and budget - Households in regions m = 1,...,M−1 sell labor and rent capital; receive lump-sum transfers Ω_{m,t}. - Labor supply is inelastic of size L_{m,t}. Capital depreciates fully each period. - Preferences are logarithmic over consumption. Household problem: - max ∑_{t=0}^∞ β^t log C_{m,t} subject to C_{m,t} + K_{m,t+1} = w_{m,t} L_{m,t} + i_{m,t} K_{m,t} + Ω_{m,t} - Euler equation: - C_{m,t+1} / C_{m,t} = β i_{m,t+1} (Equation 12) ### Savings, transfers, and implication for growth - Define factor income ˆY_{m,t} = w_{m,t} L_{m,t} + i_{m,t} K_{m,t}; Γ_{m,t} = Ω_{m,t} / ˆY_{m,t}; savings rate s_{m,t} = K_{m,t+1} / ((1 + Γ_{m,t}) ˆY_{m,t}). - Relationship: - (1−s_{m,t+1})/(1−s_{m,t}) = α β s_{m,t} (1 + Γ_{m,t+1}) (Equation 13) - If Γ_{m,t} = 0 then s_{m,t} = α β (constant). Government taxes/subsidies effectively alter household income composition and can produce a non-constant savings rate because payments to energy-producing firms are not factor payments unless rebated. ### 2.6 Trade in conventional oil ### Oil stock dynamics and market clearing - Region M contains the global stock of conventional oil R_t; oil production follows Hassler et al. (2020). - With logarithmic preferences in oil-producing region, region M sells fraction (1−β) of the stock each period and consumes the proceeds: - R_{t+1} = β R_t (Equation 14) - C_{M,t} = p_{1,t} (1−β) R_t (Equation 15) - Global oil market clearing: - ∑_{m=1}^{M−1} e_{1,t} = (1−β) R_t (Equation 16) ### 2.7 Technological diffusion ### Domestic diffusion - Two domestic diffusion channels: 1. Within-fuel inter-temporal knowledge spillovers (progress in a given fuel enters public knowledge with a lag). 2. Intra-fuel spillovers (developments in one technology have applications in others). - Input cost excluding imported technologies p^d_{m,i,t+1}/\bar{p}_{m,i,t} follows: - p^d_{m,i,t+1}/\bar{p}_{m,i,t} = (∏_k (p_{m,k,t}/\bar{p}_{m,k,t})^{ρ_{ik}})^{ω} (Equation 17) - In logs: log \bar{p}^d_{m,t+1} = (I_N − ω ρ) log \bar{p}_{m,t} + ω ρ log p_{m,t} (Equation 18) - ρ is the matrix of intra-fuel spillovers (rows sum to unity); ω is a dampening parameter (ω < 1 if part of innovation cannot be passed on). ### International diffusion (catch-up) - Global fuel-specific frontier technologies \bar{p}^*_{t} defined by: - \bar{p}^*_{i,t} = min_{m=1,...,M−1} \bar{p}^d_{m,i,t} ∀ i (Equation 19) - Reference cost in each region is weighted average of domestic progress and catch-up: - log \bar{p}_{m,t+1} = ζ_{m} log \bar{p}^*_{t} + (1−ζ_{m}) log p^d_{m,t} (Equation 20) - ζ_{m} measures region-specific speed of diffusion from the international frontier (ζ_{m} = 1 implies immediate learning; ζ_{m} = 0 implies no catch-up). ### 2.8 The climate externality ### Carbon stock and temperature dynamics - Global emissions M_t = ∑_{m=1}^M M_{m,t}; atmospheric carbon stock S_t is a function of lagged global emissions. - Following Golosov et al. (2014): - A fraction φ_L of emissions remains permanently; of the remainder, fraction 1−φ_0 leaves immediately and the rest decays at rate φ: - S_t = ∑_{u=0}^∞ (φ_L + (1−φ_L) φ_0 (1−φ)^u) M_{t−u} (Equation 21) - Equivalently S_t = S^{perm}_t + S^{temp}_t where: - S^{perm}_t = S^{perm}_{t−1} + φ_L M_t - S^{temp}_t = (1−φ) S^{temp}_{t−1} + (1−φ_L) φ_0 M_t - Temperature and ocean temperature dynamics (linear energy budget model): - [T_t; T^{ocean}_t] = [T_{t−1}; T^{ocean}_{t−1}] + [[−σ_1/(η_c λ_c + σ_2), σ_1/(η_c λ_c + σ_2)]; [σ_2/σ_3, −σ_3/σ_3]] [T_{t−1}; T^{ocean}_{t−1}] + [σ_1 η_c log 2 / log(S_{t−1}/\bar{S}); 0] (Equation 22) - \bar{S} is pre-industrial atmospheric carbon; λ_c is climate sensitivity (doubling of stock relative to pre-industrial raises temperatures by λ_c). ### Productivity damages from temperature - Region-specific productivity declines are quadratic in atmospheric temperature: - log z_{m,t} = −(φ^m_0 + φ^m_1 T_{t−1} + φ^m_2 (T_{t−1})^2) (Equation 23) *Source: wpiea2021173-print-pdf - 2.1 Aggregate production* ### 2.9 Equilibrium ### 2.9 Equilibrium ### Aggregate resource constraint and equilibrium definition - Aggregate resource constraint (equation (24)): - C_m,t + K_m,t + P_m,t E_m,t = Y_m,t - Given region-specific policies {τ_m,t, δ_j,m,t, χ_j,m,t} and initial values {K_m,0, ̄p_m,0}, global initial values {S_0, T_0, R_0}, and exogenous variables {A_m,t, L_m,t}, an equilibrium is a sequence of: - Global variables: {S_t, T_t, R_t, p_1,t, {p*_i,t}_{N i=2}} - Region-specific variables: {Y_m,t, E_m,t, C_m,t, K_m,t+1, Ω_m,t, s_m,t, Γ_m,t+1, i_m,t, w_m,t, P_m,t, {e_i,m,t, p_i,m,t, p_d i,m,t, ̄p_i,m,t, r_i,m,t}_{N i=2}} - These variables must satisfy equations (1)-(6), (8)-(12), and (14)-(24). ### Calibration overview (section 3) - Model configuration: - M = 12 regions - Region M is the oil-producing region (OPEC countries) - N = 7 fuel types: conventional oil (imported from OPEC), coal, natural gas, unconventional oil, hydroelectric power, nuclear power, renewable fuels - Time period = five years; period 0 corresponds to 2015-2019 (flow variables use the average over this period; stock levels use 2015) - Main data sources and calibration targets: - IEA World Energy Balances for fuel-specific energy inputs, carbon dioxide emissions, aggregate energy consumption, and input prices (spotty coverage) - ILO labor force data and projections for labor forces and growth rates - OPEC annual statistical bulletin for proven reserves of conventional oil - Hassler et al. (2020), Golosov et al. (2014), Meinshausen et al. (2009), Papageorgiou et al. (2017), Van der Werf (2008), Acemoglu et al. (2016), Fried (2018), Eaton and Kortum (1999), Comin and Hobijn (2010), Dechezleprêtre et al. (2013), Nordhaus (2010) inform parameter choices and validation (as referenced in text) - Exogenous variables and productivity growth: - Labor force: period 0 set using ILO; first-period growth from ILO projections; thereafter growth converges smoothly to long-run growth rate of zero - Global population growth in 2070 set to 0.2% - Labor productivity assumptions: - OECD countries grow at 0.75% per year - Non-OECD countries converge to a path that grows at the same rate but is 60 percent above current levels - Convergence rate: 25 percent per decade - Productivity law specified: - log A_m,t = log A_m,t−1 + log(1 + 5×0.0075) if m is in OECD - log A_m,t = log A_m,t−1 + log(1 + 5×0.0075) + 0.25(log A_m,t−1 − log ̄A_m,t−1) if m not in OECD - Where log ̄A_m,t = log 1.6 + log A_m,0 + log(1 + 5×0.0075) t - Energy intensity trend: - Energy share parameter ν evolves exogenously as ν_m,t = (1 + g_ν) ν_m,t−1 - g_ν chosen so that ν decreases at a rate matching steady annual decline in global energy intensity of around 1.5 percent per year since 1970 - Climate physics: - Climate model parameters taken from Hassler et al. (2020) adjusted to five-year periods - Long-run climate sensitivity λ_c set to 3 (matching median from Meinshausen et al. (2009)) - Initial and pre-industrial carbon stocks and initial temperatures set to standard literature values - Conventional oil stock and discounting: - Initial stock of conventional oil R_0 taken from proven reserves of conventional oil (OPEC) - Resultant β is equivalent to a subjective annual discount rate of 0.9 percent ### Common global parameters (table of calibrated values) - Economic parameters: - Intra-fuel elasticity of substitution, E = 2.00 - Elasticity of energy demand, F = 0.35 - Research function parameter, η = 28.88 - Cross-fuel spillovers, ̄ρ = 0.94 - Persistence of innovation, ω = 1.00 - Rate of international technological diffusion, ζ = 0.45 - Discount rate, β = 0.96 - Change in energy share (percent annual), g_ν = -0.02 - Climate parameters: - Persistence of atmospheric temp., σ1 = 0.22 - Convergence of air to ocean temp., σ2 = 0.15 - Permanent emissions share, σ3 = 0.02 - Temporary emissions share, φL = 0.20 - Emissions decay rate, φ0 = 0.40 - Convergence of ocean to air temp., φ = 0.01 - Long run climate sensitivity, λ_c = 3.00 - Short run climate sensitivity, η_c = 1.85 - Pre-industrial carbon stock (Gigatonnes), ̄S = 581.00 - Initial values: - Carbon stock (Gigatonnes), S_0 = 800.00 - Permanent share of carbon stock, S_perm0 / S_0 = 0.18 - Conventional oil stock (Gigatonnes), R_0 = 203.67 - Atmospheric temp. vs. pre-industrial avg., T_0 = 0.85 - Oceanic temp. vs. pre-industrial avg., T_ocean0 = 0.01 ### Region-specific calibration (targets and methods) - Region-specific parameters match regional aggregates and fuel-specific moments: - Energy share ν_m and capital share α_m pinned down by IEA energy share and IMF World Economic Outlook capital share - Where data unavailable, global average used - Climate externality costs taken from closest analogous regions in Nordhaus (2010) - Energy sector and research calibration: - Production weights λ_m,i and research cost shifters _m,i chosen to match IEA input prices and fuel usages - Price units: dollars per tonne oil equivalent (cost to produce the same energy as burning one ton of oil) - Initial reference cost of production ̄p_m,i,0 chosen so period-0 fuel-specific input cost reductions equal efficiency gains from EIA-MER over 2015-2019 (US data) - Carbon intensities computed as ratio of carbon emissions to consumption for each fuel from IEA data - Initial capital and productivity: - Initial capital K_m,0 chosen so capital stock is consistent with model’s past savings rate - Initial labor productivity A_m,0 set so regional output equals the data - Targets for region-specific calibration (Table 2): - Capital share, α_m: Matches labor share of GDP - Energy share parameter, ν_m: Matches energy expenditure share in GDP - Climate externality, φ_i: From Nordhaus (2010) - Fuel-specific productivities, λ_m,i: Matches IEA fuel consumption - Carbon intensities, g_m,i: Matches IEA aggregate emissions - Research cost parameter, _m,i: Matches IEA input costs - Capital stock, K_m,0: Savings rate × Y_t−1 - Labor productivity, A_0,m: Matches output - Reference cost of production, ̄p_m,i,0: Matches EIA-MER productivity growth ### Unmatched moments and validation - Discount factor and oil price: - Value of β corresponds to around 1 percent per year (consistent with literature); chosen to match global supply and demand of conventional oil - Period 0 equilibrium oil price in model: $528 per ton, or $72 per barrel - Average oil price during 2015-2019 was a little less than $60 (not targeted) - Research expenditure share: - Model implies fraction 1/(η−1) of energy expenditures go to research: around 3.5 percent - Given energy shares in GDP around 15 percent, this implies approximately 0.5 percent of GDP spent on energy research - IEA World Energy Report suggests around 0.17 percent of world GDP in 2018 (public and publicly-reported private R&D) — possibly a lower bound - UN data estimate around 2.1 percent of GDP spent on research of all kinds - Model value remains somewhat higher than IEA public measures but may be consistent if energy is relatively research-intensive ### Laissez-faire baseline and role of endogenous technical change (section 3.2) - Baseline scenarios solved: - Laissez-faire: no carbon taxes or green energy subsidies - Version with fuel-neutral exogenous technical change: input prices of all fuels decline exogenously at the rate of labor productivity growth (except conventional oil, price determined by global markets) - Key global outcome comparisons (Figure 2 summary): - Exogenous technical change model aligns closely with other integrated assessment models: - Golosov et al. (2014) and Hassler et al. (2020) project a temperature increase of around 7 degrees Celsius - Golosov et al. (2014) project annual emissions in 2200 of around 600 Gt CO2 - Adding endogenous technical change: - Reduces emissions slightly over the model horizon, lowering final temperature by a fraction of one degree - Endogenous technical change per se is climate-neutral in the long run - Until around 2100, baseline endogenous technological change is increasingly dirty: global average emissions per unit of energy increase by almost one third - Driven largely by increasing efficiency in coal, particularly in China where coal is the largest fuel market and innovation incentives are high - In the second half of the simulation, energy composition shifts more towards renewables, eventually offsetting the increase in emissions intensity up to 2100 - Driven by innovation in renewables, especially in Europe; diffusion increases global renewables market over time - Geographic heterogeneity: - Important drivers: population growth and climate externality (often act in opposite directions) - Marginal cost of higher temperature is increasing; ongoing climate change acts as a larger drag on growth - Example: Sub-Saharan Africa has higher population growth than Russia but experiences a more severe productivity loss from climate change ### Next analysis steps outlined - Section 4 will simulate outcomes under partially successful global climate policies, focusing on: - Domestic impact in emissions-reducing regions (e.g., Europe and China) - Plausible emission reduction targets and the economics behind different policy combinations (model allows for three different climate change mitigation policies) - How different ways of achieving emissions targets imply different technological spillovers to the rest of the world - Section 5 will analyze the mapping from these policies to global temperatures and global economic activity *Source: wpiea2021173-print-pdf - 2.9 Equilibrium* ### 4.1 Domestic emissions targets ### 4.1 Domestic emissions targets ### Emissions-target scenario and timing - Emissions in China decline to 10 percent of their current levels by 2060 and remain at those levels thereafter. - Emissions in the European OECD model regions decline to 10 percent of their current levels by 2050 and remain at those levels thereafter. - Chinese emissions are required to peak in 2030. - These targets imply similar per capita emissions for China and OECD Europe because China’s lower per capita energy usage is almost exactly offset by a higher emissions intensity. - The assumption that emissions fall to 10 percent rather than exactly zero is for technical convenience and can be interpreted as either a small overshoot of the target or residual removal via carbon capture and storage; relative to laissez-faire, the cuts reduce emissions by over 96 percent in 2050. ### Direct effect on global temperatures - China and OECD Europe account for around 40 percent of global emissions, so their reductions have a sizable direct impact on global temperatures. - Recomputing global temperatures with lower emissions in China and OECD Europe but holding emissions fixed elsewhere yields a small direct effect on global temperatures. - The direct effect reduces temperatures by only around 1.5 degrees Celsius in 2200. ### Indirect effect via spillovers and technical change - Technical change induced by policies in China and OECD Europe spills over to other regions, altering production possibility frontiers worldwide. - The magnitude of spillovers depends on the policy mix used to achieve emissions reductions: taxes, production subsidies, and research subsidies differ in their effects on the direction and international diffusion of technical change. - Research subsidies induce more technical change and hence more international spillovers than production subsidies for the same resource cost. ### Mapping regional cuts to global outcomes - The mapping from regional emissions cuts to global temperatures combines the direct effect (emissions reductions in China and OECD Europe) and the indirect effect (policy-induced technical change that spills over internationally). - The overall global-temperature outcome therefore depends on both the magnitude of regional emissions reductions and the policy instruments used to achieve them. *Source: wpiea2021173-print-pdf - 4.1 Domestic emissions targets.* ### Appendix B.2). ### Appendix B.2) ### Why does the carbon tax rise so sharply in the medium term? - Peak-to-trough, emissions in both regions fall by around a factor of ten while taxes increase by a factor of thirty. - The static intuition: the marginal effectiveness of the carbon tax decreases as it gets higher, so ever higher taxes are needed to meet ambitious emission reduction targets. - Three mutually reinforcing mechanisms driving the sharply rising tax: - The carbon tax deters pollution, shrinking the tax base; reducing emissions by a given volume therefore requires ever higher percentage increases in post-tax input costs. - Taxes themselves raise post-tax input costs; a given percentage increase in input costs requires a larger increase in carbon taxes. - At high carbon taxes, input costs become almost proportional to the tax for all dirty fuels, so the tax no longer discriminates between dirty fuel types and suppresses the composition effect within dirty fuels. - Empirical/static illustration: - The marginal effectiveness of the first dollar of a carbon tax is highly effective but declines sharply, particularly once the tax is over $100 per ton. - A carbon tax of around $300 is required to deliver reductions in the order of 90 percent (roughly the peak of the policy path referenced). ### Static responses and fuel composition (Figure 6 discussion) - Low carbon taxes discriminate sharply between fuels, producing notable drops in coal usage even for small taxes. - Two key factors in the composition channel (from equation (25) discussion): - The emissions weight of the fuel, w_{M i,t}, since impact on emissions of a given percentage change in fuel usage scales with the share of emissions. - The implicit subsidy to the fuel under laissez-faire, g_i / \hat p_{i,t}, interpreted as the number of tonnes of CO2 that a single dollar can put into the atmosphere (i.e., an implicit subsidy on pollution). - Coal stands out: at around four times the next-worst fuel, coal has by far the highest implicit subsidyg_i / \hat p_{i,t} because coal is dirty (g_i is high) and cheap (\hat p_{i,t} is low), so small taxes have large marginal effects via coal. - At higher taxes, post-tax price is mostly tax; a one dollar increase produces the same percent increase in the price of all fuels, causing all dirty fuels to decline in constant proportion—hence declining marginal effectiveness and the need for sharply rising taxes to eliminate residual fossil-fuel use. - Model uncertainty implication: near-zero elasticity of emissions with respect to the tax at high tax rates implies required tax to hit a given emissions target varies considerably with model misspecification; there is less uncertainty for less ambitious targets. The low and slowly growing level in the first decade is more robust than the peak tax. ### Why are carbon taxes lower in China? - China’s intensive use of coal (accounting for around two thirds of all energy) generates a correlation between emissions share w_{i,t} and implicit subsidyg_i / \hat p_{i,t}, strengthening the composition channel. - Because China has a much larger carbon tax base, the same emissions reductions can be accomplished with smaller tax increases than in OECD Europe. ### Why are research subsidies more effective than production subsidies? - Main empirical finding: spending carbon tax revenue on research subsidies permits a larger and sooner reduction in carbon taxes than spending on production (purchasing) subsidies. - Contemporaneous effects of research and production subsidies are near-identical; dynamic difference arises because production subsidies introduce a wedge between productivity and firms’ input costs and therefore do not incentivize research to the same extent—future research gains build on past research, amplifying the effect of research subsidies over time. - Partial-equilibrium relationships (as presented): - dlog E_t / dδ_{j,t} = (η−1) ( (1−χ_{j,t})/(1−δ_{j,t}) ) dlog E_t / dχ_{j,t} (equation (27)) - dlog M_t / dδ_{j,t} = (η−1) ( (1−χ_{j,t})/(1−δ_{j,t}) ) dlog M_t / dχ_{j,t} (equation (28)) - These imply the two subsidy policies are substitutes at rate (η−1)(1−χ_{j,t})/(1−δ_{j,t}) for contemporaneous aggregate emissions and energy. - Fiscal impact marginal effect (equation presented): - (η−1) ( (1−χ_{j,t})/(1−δ_{j,t}) ) dlog s_t / dχ_{j,t} − dlog s_t / dδ_{j,t} = χ_{j,t}/η χ_{j,t} (1−δ_{j,t} η) + δ_{j,t} (η−1) − 1 / (1−δ_{j,t}) (equation (29)) - This term is typically very small and exactly zero when χ_{j,t} = δ_{j,t}; thus an extra dollar on either policy will have very similar effects and identical ones when research and production subsidies are equal. - Intuition for offsetting effects: - A dollar of research subsidy goes further than a dollar of production subsidy because research applies to only the research component of energy costs; the research share of energy costs equals 1/(η−1), so a dollar can fund a reduction in research costs that is η−1 times as large as if spent on production subsidies. - However, a one percent improvement in research efficiency reduces production costs by 1/(η−1), so the larger funded change has a proportionally smaller effect, offsetting the first effect. - Dynamic difference: even if contemporaneous aggregate emissions and energy are unchanged by switching subsidies at the appropriate rate, the research subsidy lowers future cost of green fuels relative to dirty ones more than a purchasing subsidy: - (η−1) ( (1−χ_{j,t})/(1−δ_{j,t}) ) dlog(p_{j,t}/p_{i,t}) / dχ_{j,t} − dlog(p_{j,t}/p_{i,t}) / dδ_{j,t} = −1 / (1−δ_{j,t}) < 0 - Thus research subsidies produce larger long-run reductions in relative green input costs and compound over time, making research subsidies much more effective dynamically. ### Decomposing the dynamic effect of policies: domestic impact - Method: partial-equilibrium impulse responses computed period-by-period via finite differences; summing period-policy specific impulse responses gives total partial-equilibrium response (approximation error is small relative to general equilibrium outcomes). - Channels distinguished: static composition, static quantity, and dynamic induced technical change. Key parameter sensitivities: - Elasticity of substitution E principally drives the composition effect. - Elasticity of demand F impacts the quantity effect. - Convexity of the research cost function η determines induced technical change. - Domestic findings (China and OECD Europe): - The contemporaneous impact on emissions of the carbon tax far outweighs that of the research subsidy in all periods—carbon tax has large static composition effect; quantity effect is almost zero. - Dynamic effects of tax and research subsidy are much more similar in size: research subsidy directly targets the technological component of input costs and induces technological response similar to the carbon tax; induced technological-change emissions impacts from research subsidies and carbon taxes are approximately the same. - Spending carbon tax revenues on research subsidies roughly doubles the induced emissions effect versus the carbon tax alone. - Impact on energy consumption (proxy for household consumption): - Research subsidies and carbon taxes operate in different directions; research subsidy explains almost all long-run improvement in household consumption in the medium run. - Carbon tax raises the price of energy overall, reducing total energy consumption contemporaneously and in future. Research subsidy permanently lowers the price of energy overall, counteracting carbon-tax-induced increases in energy prices and cushioning welfare costs. ### The impact of international technological spillovers - Global temperature response to regional policies: - International technological spillovers are very large, especially in the long run: even when emissions are reduced only in Europe and China via a carbon tax alone, temperatures in 2200 are reduced by nearly another 2 degrees beyond that due to the regional emission reductions alone. - Mechanism: Chinese coal market drives global innovation in coal technologies; in laissez-faire, coal innovation in China is a negative externality because it diffuses and reduces the cost of pollution worldwide. - Coal input costs in China are on average around 10 percent below the next cheapest region (Figure 8b discussion). - By 2050, the carbon tax reduces Chinese coal demand and China is no longer driving global productivity improvements in coal; the negative technological externality is eliminated. - Coal-specific concern: the model cannot predict specific technologies, but plausible coal cost-reducing developments include automation, improved communication/transport, materials science improvements. The Chinese coal market’s expenditure size (equivalent to approximately 2 percent of GDP) means even small efficiency improvements can generate large incentives for innovation. - Variation across policy packages: - Funneling carbon tax revenues into research subsidies reduces temperatures further—around a further 1 degree in 2200—because increased green research reduces the cost of green energy which spills over internationally; this effect dominates a countervailing impact where research subsidies reduce the long-run price of coal by expanding energy markets. - Production subsidies, while less useful domestically for emissions targets, can produce positive long-run international spillovers: they lower temperature by a little less than one half of one degree in 2200 because other countries benefit from the extra research induced by production subsidies without experiencing the domestic offsetting channel (lower prices and higher energy consumption). *Appendix B.2, wpiea2021173-print-pdf* ### 5.2 Decomposing the dynamic effect of policies: international spillovers ### 5.2 Decomposing the dynamic effect of policies: international spillovers ### Fuel-by-fuel decomposition of international spillovers (experiment (2)) - Method: - Trace impact of changing productivity of each fuel on each region by repeatedly re-solving the model with exogenous input prices. - Construct a monotone sequence of fuel-region-time input price paths from laissez-faire to experiment (2) and solve the model for each step, changing fuels one-by-one and recording incremental impacts on regional emissions. - Sequence uses 100 steps; with six improvable fuels this requires solving the model 600 times. - Approximation quality: - The resulting cumulative change in emissions is a very good approximation to the full endogenous technical change model for non-active regions (not China or OECD Europe). The Approximation Error bar in Figure 9 is almost invisible. - The decomposition captures some second-round effects (e.g., increased incentives for renewables innovation in non-active regions due to larger domestic markets) but the primitive shock is foreign policy changes; thus the exercise isolates international technological spillovers. - Key quantitative findings (Figure 9; India and the United States shown as representative non-active regions): - The bulk of the reduction in emissions in non-active regions arises from changes in input cost of two fuels: coal and renewables. - Input price changes go in opposite directions (up for coal, down for renewables) but contributions to emissions have the same sign because coal is polluting and renewables are not. - Deterrence of improvements in coal technologies typically accounts for around two thirds of the decline in emissions. - The split in other non-active regions is very similar. ### Contribution of international spillovers to long-run temperature outcomes - Headline comparison: - International technological spillovers can be an effective substitute for international policy cooperation. - Best-case outcome: long-run temperatures below 3 degrees above pre-industrial norms, versus over 7 degrees in the baseline. - Decomposition of the roughly 4.5 degree reduction in long-run temperatures: - Around 3 degrees is due to international technological spillovers. - Incentives for new green technology play a secondary role: accounting for only around 1 degree Celsius in 2200. - The most important channel is dissuading productivity improvements in dirty fuels, especially coal. --- ### 5.3 Alternative parameterizations of international technological diffusion ### Five alternative calibrations considered and their impacts - No international diffusion (ζ_m = 0 for all regions): - Benchmarks general equilibrium channels excluding technological spillovers. - Temperatures in 2200 with zero spillovers decline by less than 0.5 degrees Celsius relative to laissez-faire. - The remaining difference is principally climate damages via endogenous feedback of temperature to output. - Lower spillovers calibrated to Eaton and Kortum (1999) unconditional adoption rate of 21 years (ζ_m = 0.24 for all regions): - Long-run temperatures decline by almost 4 degrees Celsius. - ζ_m = 0.24 is about halfway between headline and no-spillovers cases, yet the long-run temperature impact is around two-thirds of the way to the headline result. - Implies relationship between spillovers and long-run temperature is likely concave; small spillovers have larger marginal effects. - Trade-link (imports)-weighted diffusion (equation change to use import shares w^I_m,k; import shares computed using UN COMTRADE data for 2015-2019): - Long-run temperatures still decline by almost 4 degrees Celsius. - Rationale: China and OECD Europe together account for around 50 percent of imports into other regions, so they remain major exporters and policy in these regions substantially affects global technology paths. - Faster diffusion so all regions fully catch up to the frontier with a one period lag: - Two experiments: diffusion fast for all fuel types; diffusion fast for green fuels only. - Both produce relatively small additional effects, reflecting declining marginal effectiveness of ever-faster diffusion. - These experiments represent outer bounds on what knowledge-spread programs might achieve. - Distributional effect on burden of mitigation: - Faster diffusion amplifies global benefits from regional mitigation but increases burden on regions that implement policies. - Because emissions-reducing regions hinder dirty-technology development, they fall behind the global frontier in dirty energy production; knowledge about dirty-fuel innovations flows back into them while green-technology innovations flow out. - Faster diffusion reduces cost of fossil-fuel inputs in Europe and China, necessitating higher carbon taxes. - Example quantitative impacts (Figure 10b, OECD Europe subset): - High spillovers for all technologies would necessitate an increase of around $50/ton CO2 in the short term. - Long-run cuts in carbon taxes are delayed by around two decades under high spillovers. - In contrast, in absence of international spillovers, European countries could meet emissions targets while cutting carbon taxes around a decade earlier than in the headline case — but at considerable global environmental cost. --- ### 6 Robustness (overview and targeted checks) ### Modeling assumptions and robustness strategy - Strong modeling assumptions noted: production function for research, full capital depreciation, log preferences of households. - Robustness checks show main results are not substantively impacted by these assumptions. ### 6.1 Alternate forms of endogenous technical change - Research sector assumptions in baseline: - Research treated as an unremunerated factor of production (a pure cost) with a simple constant-returns cost structure. - Guarantees households derive no income from payments to the energy sector, aiding a constant savings rate; simplicity facilitates computation. - Recalibration exercises: - Recalibrate model to match deep parameters of two global integrated assessment models: Fried (2018) and Acemoglu et al. (2016). - These alternative specifications use households’ labor as an input to innovation and more sophisticated innovation production. - Replicate baseline and policy experiments from those papers; results correspond closely to the baseline model. - Conclusion: simplifications do not quantitatively affect policy responses; main conclusions are robust to alternate endogenous technical change specifications. ### 6.1.1 Comparison to Fried (2018) - Fried (2018) key setup and finding: - Global one-region general equilibrium model with two fuel types: fossil fuel and green energy. - Endogenous technical change in both fuel types and non-energy intermediates; partial monopoly sellers of fuel-specific technologies; specialized labor (“scientists”) hired for research. - Discoveries are private for one period (five years) then available free. - Key finding: induced technical change amplifies carbon tax effectiveness by almost 20 percent. - When technological change is exogenous, a constant $30 per ton CO2 carbon tax can reduce emissions by 30 percent in 20 years. - With endogenous technical change, the same outcome is achieved with a tax of only $24.5. - Replication approach in current model: - Modify model to a single region and two fuels (clean and dirty) to match Fried (2018). - Match parameters to analogues in data where possible (elasticities of substitution and demand). - For parameters without direct analogues, approximate using values at average growth rates. - For parameters not explicitly reported, match to IEA data for the US on fuel-specific consumption, prices, and emissions. - Green fuel aggregates all zero-carbon energy sources, including nuclear. - Key parameter matches (table excerpt as described in text): - Intra-fuel elasticity of substitution, E: 1.5 (Identical) - Elasticity of energy demand, F: 0.05 (Energy elasticity of substitution) - Research function parameter, η: 52.4 (Matches returns to scale at avg. growth) - Cross-fuel research spillovers, ρ12, ρ21: 0.025 (Matches spillovers at avg. growth) - Research dampening parameter, ω: 1 (Identical) - Production function weights, λ_i: Fossil 0.29, Green 0.71 (Match US energy production shares) - Initial reference prices, ̄p_i,0: Fossil 287, Green 1000 (Approx fuel prices, USD/ton oil equiv.) - (Text on carbon intensities is present but truncated in the source.) *Source: wpiea2021173-print-pdf - 5.2 Decomposing the dynamic effect of policies: international spillovers* ### 3.40 Match current US emissions ### 3.40 Match current US emissions ### Replication of Fried (2018) - The model is calibrated to replicate Fried (2018) and is subjected to the same carbon tax policies. - Table 4: Percentage deviation from baseline in response to a permanent carbon tax, simplified model versus Fried (2018). Carbon taxes for exogenous and endogenous models are respectively $30 and $24.5 per ton CO2. - Headline aggregate responses are near-identical; this holds both when the endogenous growth channel is shut down and when endogenous innovation is active. - Key percentage deviations (Model versus Fried 2018): - Aggregates: - Emissions: Model exogenous -31.8, Fried 2018 -30.0; Model endogenous (in 20 yrs.) -29.8, Fried 2018 -30.0; Model endogenous (long run) -37.3, Fried 2018 -36.9 - Energy usage: Model exogenous -0.5, Fried 2018 -0.7; Model endogenous (in 20 yrs.) -0.4, Fried 2018 -0.6; Model endogenous (long run) -0.4, Fried 2018 -0.7 - Prices: - Energy price: Model exogenous 9.5, Fried 2018 14.9; Model endogenous (in 20 yrs.) 7.7, Fried 2018 13.0; Model endogenous (long run) 7.3, Fried 2018 14.6 - Green energy price: Model exogenous -0.0, Fried 2018 0.6; Model endogenous (in 20 yrs.) -1.1, Fried 2018 -1.0; Model endogenous (long run) -3.6, Fried 2018 -2.6 - Green versus fossil: - Relative price: Model exogenous -0.0, Fried 2018 0.2; Model endogenous (in 20 yrs.) -3.7, Fried 2018 -7.0; Model endogenous (long run) -12.1, Fried 2018 -17.0 - Relative usage: Model exogenous 45.5, Fried 2018 78.0; Model endogenous (in 20 yrs.) 42.8, Fried 2018 79.2; Model endogenous (long run) 53.5, Fried 2018 112.9 - Research input: - Fossil technology: Model exogenous 0.0, Fried 2018 0.0; Model endogenous (in 20 yrs.) -27.2, Fried 2018 -60.5; Model endogenous (long run) -28.8, Fried 2018 -29.9 - Green technology: Model exogenous 0.0, Fried 2018 0.0; Model endogenous (in 20 yrs.) 11.9, Fried 2018 53.3; Model endogenous (long run) 12.6, Fried 2018 23.8 - Observations on differences: - Energy price responses are typically around half those in Fried due largely to the inelasticity of energy demand in Fried (F = 0.05). - Emissions decline substantially despite almost unchanged energy usage, driven mainly by composition effects. - Small differences in initial fuel shares and fuel-specific parameter calibration can generate large differences in percentage changes of impulse responses; these differences tend to dissipate over time. ### Comparison to Acemoglu et al. (2016) - The model is re-calibrated to compare with Acemoglu et al. (2016), a continuous time general equilibrium model with product-specific clean/dirty technologies and a climate externality (temperature affects aggregate productivity). - Main exercise: feed the optimal carbon tax and research subsidy policies from Acemoglu et al. (2016) into the author’s model and compare temperature outcomes. - Temperature outcomes: - Under laissez-faire (taxes and subsidies of zero) temperatures rise by almost 10 degrees Celsius over 200 years. - With Acemoglu et al. (2016) optimal policies, temperatures across the two models vary by barely more than one tenth of one degree Celsius across the next 200 years, peaking at a little under 2 degrees Celsius above pre-industrial levels mid-way through the next century. - Key parameter mappings (Table 5 equivalents): - Intra-fuel elasticity of substitution, E = 3 (Central estimate in Papageorgiou et al. (2017)) - Elasticity of energy demand, F = 1 (Products Cobb-Douglas in inputs) - Research function parameter, η = 33.0 (Matches returns to scale at avg. growth) - Cross-fuel research spillovers, ρ12, ρ21 = 0 (Identical) - Research dampening parameter, ω = 1 (Identical) - Fuel-specific parameters (Production function weights, λi): Fossil 0.39, Green 0.61 (Match global energy production shares) - Initial reference prices, p̄i,0: Fossil 433, Green 1392 (Global fuel prices, USD/ton oil equiv.) - Carbon intensities, gi: 2.7 (Match current global emissions) - Conclusion of comparison: despite structural differences, the main aggregate responses to policy changes are much the same between models. ### Non-constant savings rate (capital depreciation and CRRA preferences) - Motivation: relax strong assumptions that yield a constant savings rate (log preferences, full depreciation). - Household problem replaced with: - max ∑_{t=0}^∞ β^t C_{m,t}^{1−γ}/(1−γ) s.t. C_{m,t} + K_{m,t+1} − (1−δ_K)K_{m,t} = w_{m,t} L_{m,t} + i_{m,t} K_{m,t} + Ω_{m,t} - Investment I_{m,t} = K_{m,t+1} − (1−δ_K)K_{m,t} - Euler equation becomes: - (C_{m,t+1}/C_{m,t})^γ = β(i_{m,t+1} + 1 − δ_K) (equation (31)) - Resource constraint replaced by: C_{m,t} + I_{m,t} = Y_{m,t} - Calibration choices and scenarios: - Set δ_K = 0.5 (equivalent to a 12 percent annual depreciation rate with five-year periods). - Consider γ = 2 and γ = 0.5. - Recompute initial K_{m,0} and A_{m,0} so capital-output ratio and initial output levels match targets. - Compute equilibrium for laissez-faire and policy settings previously found to hit regional emissions targets (using carbon taxes to fund research subsidies). - Key findings: - These extensions have a negligible effect on the main findings of the paper; policies computed under the constant-savings model still nearly meet emissions targets under the more general assumptions. - Global temperature and household consumption impacts of technological diffusion from emissions-reducing regions are almost identical across model versions (see Figure 12). - Capital accumulation interacts primarily with total energy usage (quantity) and not energy composition; because composition is the quantitatively dominant determinant of emissions, capital matters little for emissions. - Climate mitigation policies raise the marginal productivity of capital in the long run; the Euler equation implies the savings rate falls slightly in the long run — by around one percentage point — without reducing welfare because higher TFP allows higher consumption now and later. ### Conclusions and caveats - Central result: - Regional pursuit of emissions targets can materially mitigate global climate change via international diffusion of induced technological change. - Successful emissions-reduction policies in China and OECD Europe could, in the calibrated model, avoid worst-case spiraling catastrophic climate change. - Result is robust to assumptions about speed/patterns of technological diffusion, determinants of the savings rate, and the nature of endogenous technical change. - Implications: - Positive international policy spillovers make individual climate mitigation programs more appealing by amplifying their global effect. - Technological spillovers increase the marginal gain of staying in a climate agreement as participation declines, making agreements more stable against partial non-participation. - Important caveats and extensions for future work: - Trade is excluded (other than oil) and might influence outcomes via export-driven research incentives or overseas demand altering domestic research effort; net effect is ambiguous. - Fuel-specific capital (stranded assets) could slow reallocation from high- to low-carbon fuels; a lower intra-fuel elasticity of substitution E would require more aggressive policies to meet emissions targets, making mitigation costlier but leaving the technological effect conditional on meeting targets largely unchanged. - Further exploration of trade effects and fuel-specific capital is left for future work. *Source: wpiea2021173-print-pdf - 3.40 Match current US emissions* ### References ### wpiea2021173-print-pdf - References ### Literature themes - The references compile literature on directed technical change, endogenous technological change, climate policy, the social cost of carbon, technology diffusion, and empirical studies of temperature effects on production. Key recurring topics include optimal carbon taxes, induced technical change, and energy–environment modeling. - Notable methodological focuses in the cited literature include general equilibrium models of optimal taxes, CES production specifications, diffusion of ideas, patent-based evidence on technology transfer, and empirical estimations of climate damages. ### Appendix A — The structure of the energy market - Main results: - The average cost function is downward-sloping; the per unit price of fuel inputs and research expenditure satisfies: - C(E_t)/E_t = V_t + R_t/E_t = ∑_{i=1}^N \hat p_{i,t} e_{i,t}/E_t + ∑_{i=1}^N p_{i,t} e_{i,t}/((η−1)E_t) - V_t = (∑_{i=1}^N λ_i^{1/(ρ−1)} \hat p_{i,t}^{ρ/(ρ−1)})^{(ρ−1)/ρ} = ∑_{i=1}^N \hat p_{i,t} e_{i,t}/E_t - When τ_t = 0 the upper bound on the (absolute) slope of C(E_t) occurs; with τ_t = 0, ˆp_{i,t} = p_{i,t} and: - C(E_t) = (η/(η−1)) ∑_{i=1}^N e_{i,t} p_{i,t} - dlog(C(E_t)/E_t) = −1/η - dlog(C(E_t)) = 1 − 1/η - Market structure and pricing: - Firms maximize Π_t = P_t(E_t)E_t − C(E_t) subject to Π ≥ 0. - Aggregate demand assumed CES with elasticity F: E_t = A_{D,t} (P_t)^{−F}. - With cost function C(E) = A_S E^{1−1/η}, monopolist profit Π_t(E) = A_{D,t} E^{1−1/F} − A_S E^{1−1/η}. - Equilibrium requires F < η. If 1 < F < η, profits are concave and an interior solution exists. If typical parameterization F < 1 and F < η, profits are maximized as E_t → 0, so price regulation (cap) is required to guarantee an interior solution. - For τ_t > 0 define cost elasticity 1/η′; because elasticity of cost is maximized at τ_t = 0 we have η′ > η. Thus condition F < η implies F < η′ and equilibrium conditions extend to τ_t > 0. ### Appendix B — Further details and calibration - Region-specific calibration parameters and initial conditions (Table A.1) include α_m, log ν_m, log K_{m,0}, log A_{m,0}. Selected entries: - Asia Pacific: α_m 0.41, logν_m −18.66, logK_m,0 14.18, logA_m,0 10.13 - China: α_m 0.41, logν_m −18.68, logK_m,0 15.27, logA_m,0 10.52 - European OECD: α_m 0.39, logν_m −20.62, logK_m,0 15.61, logA_m,0 11.95 - India: α_m 0.41, logν_m −17.48, logK_m,0 13.66, logA_m,0 9.68 - USA: α_m 0.38, logν_m −20.39, logK_m,0 15.63, logA_m,0 12.41 - Fuel-specific productivities λ_{m,i} (Table A.2) and pollution intensities g_{m,i} (Table A.3) are provided by region and fuel. Selected λ entries (Conv., Oil, Coal, Dom. Oil, Gas, Hydro, Nuclear, Renewables): - Asia Pacific: 0.33 0.09 0.24 0.23 0.23 0.17 0.88 - China: 0.41 0.28 0.29 0.20 0.58 0.36 0.75 - USA: 0.37 0.13 0.34 0.15 0.25 0.69 0.63 - Reference input costs, 2012 USD per tonne oil equivalent, ̄g_{m,i} (Table A.4) present numerical costs by region and fuel. Selected pollution-intensity/cost figures: - Coal (Asia Pacific) 143.78; Conv. Oil (Asia Pacific) 465.47; Renewables (Asia Pacific) 1469.39 - Coal (China) 156.85; Conv. Oil (China) 475.82; Renewables (China) 1466.01 - USA: Coal 175.97; Conv. Oil 408.07; Dom. Oil 154.59; Renewables 1454.99 - Additional figures (A.1–A.6) illustrate regional outcomes under laissez-faire, fuel consumption and emissions under laissez-faire, policy decomposition of emissions changes, household consumption relative to laissez-faire, optimal policies in Acemoglu et al. (2016), and emissions versus target under alternate saving-rate assumptions. ### Appendix C — A partial equilibrium approximation - Purpose: Study equilibrium in the energy market alone to compute impacts of policies on emissions and energy usage and to decompose impacts into three channels: static composition effect, static quantity effect, and dynamic induced technical change. - Model equations (single-region, suppressing m): - Cost-minimization first-order conditions (reproduced): - e_{i,t}/E_t = (ˆp_{i,t} V_t/λ_i)^{−E} (Equation 35) - (1−δ_{i,t}) e_{i,t} = φ_i (1−χ_{i,t}) \bar p_{i,t} (p_{i,t}/\bar p_i)^{−η} (Equation 36) - r_{i,t} = ((1−δ_{i,t})/(η−1) p_{i,t}) e_{i,t} (Equation 37) - ˆp_{i,m,t} = τ_{m,t} g_i + (1−δ_{i,m,t}) p_{i,t} (Equation 38) - Energy demand with constant elasticity F: E_t = A_t (P_t)^{−F} (Equation 39) - Price equals average cost under regulation (zero profit): P_t = V_t + R_t/E_t (Equation 40) - Technological diffusion: log \bar p_{t+1} = (I_N − ω_ρ) log \bar p_t + ω_ρ log p_t (Equation 41) - Static derivatives (responses to contemporaneous policy changes): - Aggregate demand derivative: dlogE = −F dlogP (Equation 42) - From pricing: dlogP = w_P dlogV + (1−w_P) dlogR − (1−w_P) dlogE, where w_P = V/P (Equation 43) - dlogV = ∑_{i=1}^N w_i dlog ˆp_i (Equation 44) - dlogR = ∑_{i=1}^N v_i (dloge_i + dlogp_i) (Equation 45) - Key constraint combining these: ∑_{i=1}^N w_i dloge_i = −F w_P ∑_{i=1}^N w_i dlog ˆp_i − F(1−w_P) ∑_{i=1}^N v_i dlogp_i + F(1−w_P) ∑_{i=1}^N (w_i − v_i) dloge_i (Equation 46) - Carbon tax (derivative w.r.t. a one dollar increase in τ): - Relative logs: ̃x_i = log x_i − log x_j for clean fuel j. - Changes in composition: d ̃e_i = −E d ̃ˆp_i (Equation 47) and dloge_i = −η dlogp_i ∀i (Equation 48), so d ̃e_i = −η d ̃p_i ∀ i ≠ j (Equation 49). - Post-tax price differential: d ̃ˆp_i = d ̃p_i + g_i ˆp_i − g_i τ ˆp_i dlogp_i (Equation 50). - Define H_i = (E/(η−E))(g_i ˆp_i). Then d ̃p_i = H_i (1 − τ dlogp_i) / (1 + H_i τ) ≡ Q_i(τ)(1 − τ dlogp_j) (Equations 51–52). - Quantities and post-tax price equivalents: d ̃e_i = −η Q_i(τ)(1 − τ dlogp_j) and d ̃ˆp_i = (η/E) Q_i(τ)(1 − τ dlogp_j) (Equations 53–54). - Carbon-tax compositional effect initially scales with g_i ˆp_i and is mitigated as τ increases because the tax becomes a larger share of post-tax price; the discrimination between polluting fuels vanishes as τ grows. - Aggregate price response for clean fuel p_j: - dlogp_j = Z(w_P,τ)/(F − η + τ Z(w_P,τ)) (Equation 56), where Z(w_P,τ) = (η − F + F w_P (E − η)/E) \bar Q(τ) − η F (1 − w_P) (\bar Q(τ) − \bar Q_0(τ)), with \bar Q(τ) = ∑ w_i Q_i(τ), \bar Q_0(τ) = ∑ v_i Q_i(τ). - As τ increases, dlogp_j → 1/τ: the carbon tax increasingly resembles a value tax and discrimination across polluting fuels diminishes. - Procedure to compute marginal aggregate responses at any τ: trace static allocation impact in period t; calculate how reference prices evolve in t+1, t+2,...; evaluate how those price changes affect allocations in subsequent periods. *Source: wpiea2021173-print-pdf - References* ### 1. Using definitions ofd ̃x ### 1. Using definitions of d̃x ### Static derivatives and marginal impacts - Key rearrangement: - dlog p_i = Q_i(τ) + (1−τ Q_i(τ)) dlog p_j - Substitution for dlog p_j using equation (56) yields dlog p_i. - Derivatives for E_t, M_t follow from: - dlog M_t = ∑_{i=1}^N w^M_{i,t} dlog e_{i,t} - dlog E_t = ∑_{i=1}^N w_{i,t} dlog e_{i,t} - where w_{i,t}, w^M_{i,t} are the shares of fuel i in expenditures and emissions respectively (so w^M_{i,t} = g_i e_{i,t} / M_t). - Table A.5 (static marginal impact of the carbon tax at τ_t = 0, worked example using averages 2015-2019, China) — selected data and shares: - Energy, e_{i,t} (MtOE): Coal 1956, Nat. Gas 1655, Oil 393, Zero-Carbon 1529, Total 76 - Price, p_{i,t} ($ per TOE): 1493, 675, 1313, 64 - Emissions, m_{i,t} (Mt CO2): 7409, 3641, 2800, 0, 9053 - Emissions intensity, g_i (tonnes CO2 per TOE): 3.79, 2.21, 2.37, 0.00 - Expenditure share, w_{i,t}: 0.28, 0.06, 0.26, 0.41 - Emissions share, w^m_{i,t}: 0.82, 0.04, 0.14, 0.00 - Social cost, g_{i,t} / \hat p_{i,t} (tonnes CO2 per $): 0.0250, 0.006, 0.005, 0.000 - Avg. social cost, ∑_i w_{i,t} g_i / \hat p_{i,t}: 0.009, 0.009, 0.009, 0.009 - Marginal responses (Energy consumption, d e_{i,t} / d τ_t, (pp, contrib.)): - Coal: -1.1; Nat. Gas: 0.0; Oil: 0.1; Zero-Carbon: 0.6; Total: -0.3 - Composition effect: Coal -1.0; Nat. Gas 0.0; Oil 0.2; Zero-Carbon 0.7; Total -0.0 - Quantity effect: Coal -0.1; Nat. Gas -0.0; Oil -0.1; Zero-Carbon -0.1; Total -0.3 - Change in emissions, d m_{i,t} / d τ_t, (pp, contrib.): - Coal: -3.2; Nat. Gas: 0.0; Oil: 0.1; Zero-Carbon: 0.0; Total -3.1 - Composition effect: Coal -3.0; Nat. Gas 0.0; Oil 0.1; Zero-Carbon 0.0; Total -2.8 - Quantity effect: Coal -0.2; Nat. Gas -0.0; Oil -0.0; Zero-Carbon -0.0; Total -0.3 - Memo: Research amplification factors: - Energy composition, η/(η−E): 1.07 - Total energy consumption, η/(η−F): 1.01 - Special case τ = 0 (simplified derivatives): - dlog p_{i,t} / d τ_{i,t} = E (1/(η−E)) (g_i / \hat p_{i,t} − ∑_{i=1}^N w_{i,t} g_i / \hat p_{i,t}) + F ((η−1)/η) (1/(η−F)) ∑_{i=1}^N w_{i,t} g_i / \hat p_{i,t} dlog e_{i,t} / d τ_{i,t} = −E (η/(η−E)) (g_i / \hat p_{i,t} − ∑_{i=1}^N w_{i,t} g_i / \hat p_{i,t}) − F ((η−1)/η) (η/(η−F)) ∑_{i=1}^N w_{i,t} g_i / \hat p_{i,t} - First line: Change in energy composition (labelled) - Second line: Change in total energy consumption (labelled) - Responses of aggregates at τ = 0: - dlog M_t / d τ_t = −E (η/(η−E)) (∑_{i=1}^N (w^M_{i,t} − w_{i,t}) g_i / \hat p_{i,t}) − F ((η−1)/η) (η/(η−F)) ∑_{i=1}^N w_{i,t} g_i / \hat p_{i,t} - First term: Effect of change in energy composition - Second term: Effect of change in total energy consumption - dlog E_t / d τ_t = −F ((η−1)/η) (η/(η−F)) ∑_{i=1}^N w_{i,t} g_i / \hat p_{i,t} - Effect of change in total energy consumption - The first of these is equation (25) in the main text. Table A.5 computes the elements for China. ### Research subsidy (changes in χ_j) - Units conversion used: - d/d χ_j = −1/(1−χ_j) d/d log(1−χ_j) - Differentiated relations (notation: d = derivative w.r.t. log(1−χ_j)): - d ̃e_i = −E d ̃\hat p_i ∀ i - d ̃\hat p_i = d ̃p_i − g_i τ / \hat p_i dlog p_i ∀ i - d ̃e_i = −η d ̃p_i − 1 ∀ i ≠ j - d log e_j = 1 − η d log p_j - Solve system for d ̃p_i in terms of d log p_j: - d ̃p_i = −(1 + J_i τ dlog p_j) / (η − E + J_i τ) - Define J_i = E g_i / \hat p_i - Rewritten for i ≠ j: - d ̃p_i = C_i(τ) + D_i(τ) dlog p_j - d ̃e_i = −η C_i(τ) − η D_i(τ) dlog p_j − 1 - d ̃\hat p_i = (η/E) C_i(τ) + (η/E) D_i(τ) dlog p_j + 1/E - Where: - C_i(τ) = −1/(η − E + J_i τ) - D_i(τ) = J_i τ C_i(τ) - C_i(0) = −1/(η − E) - D_i(0) = 0 - lim_{τ→∞} C_i(τ) = −1/η - lim_{τ→∞} D_i(τ) = −E/η - Pre-tax price change solution: - dlog p_j = B/A + ∑_{i≠j} M_i / A - Definitions: - A = F − η − η ∑_{i≠j} w_i D_i(τ) + F ∑_{i≠j} (w_P(η/E) w_i + (1−w_P) v_i) D_i(τ) + F(1−w_P) η ∑_{i≠j} (w_i − v_i) D_i(τ) - B = −( w_j + (1−w_j) F w_P (1/E) + F(1−w_P) (v_j − w_j) ) - M_i = ((1 − F w_P / E − F(1−w_P)) η w_i + F(1−w_P)(η − 1) v_i) C_i(τ) - τ = 0 simplifications (composition and quantity channels) — green fuel j: - dlog p_{j,t} / d χ_{j,t} = ( (1−w_{j,t})/(1−χ_{j,t}) )(−1/(η−E)) + ( w_{j,t}/(1−χ_{j,t}) )(−1/(η−F)) - dlog e_{j,t} / d χ_{j,t} = ( (1−w_{j,t})/(1−χ_{j,t}) )(E/(η−E)) [Change in energy mix] + ( (w_{j,t})/(1−χ_{j,t}) )(F/(η−F)) [Change in total energy] - Other fuels i ≠ j at τ = 0: - dlog e_{i,t} / d χ_{j,t} = −( (w_{j,t})/(1−χ_{j,t}) )(E/(η−E)) + ( (w_{j,t})/(1−χ_{j,t}) )(F/(η−F)) - Aggregates for research subsidy (τ = 0): - dlog E_t / d χ_{j,t} = ( w_{j,t} / (1−χ_{j,t}) ) (F/(η−F)) - dlog M_t / d χ_{j,t} = ( w_{j,t} / (1−χ_{j,t}) ) (F/(η−F) − E/(η−E)) ### Production subsidy (changes in δ_j) - Units conversion: - d/d δ_j = −1/(1−δ_j) d/d log(1−δ_j) - Differentiated relations: - d ̃e_i = −E d ̃\hat p_i ∀ i - d ̃\hat p_i = d ̃p_i − 1 − g_i τ / \hat p_i dlog p_i ∀ i - d ̃e_i = −η d ̃p_i + 1 ∀ i ≠ j - d log e_j = −η d log p_j − 1 - Solve for d ̃p_i: - d ̃p_i = κ_i(τ) + L_i(τ) dlog p_j - d ̃e_i = 1 − η κ_i(τ) − η L_i(τ) dlog p_j - d ̃\hat p_i = −1/E + (η/E) κ_i(τ) + (η/E) L_i(τ) dlog p_j - Where: - κ_i(τ) = (1 − E)/(η − E + J_i τ) - L_i(τ) = −J_i τ / (η − E + J_i τ) - Pre-tax price change: - dlog p_j = B' / A' + ∑_{i≠j} M'_i / A' - Definitions structurally similar to research subsidy with replacements of C_i by κ_i and D_i by L_i. - τ = 0 simplifications for green fuel j: - dlog p_{j,t} / d δ_{j,t} = −( (1−w_{j,t})/(1−δ_{j,t}) )(E − 1)/(η − E) + ( (w_{j,t})/(1−δ_{j,t}) )(1 − F)/(η − F) - dlog e_{j,t} / d δ_{j,t} = ( (1−w_{j,t})/(1−δ_{j,t}) )(η − 1)(E/(η − E)) [Change in energy mix] + ( (w_{j,t})/(1−δ_{j,t}) )(η − 1)(F/(η − F)) [Change in total energy] - Other fuels i ≠ j at τ = 0: - dlog e_{i,t} / d δ_{j,t} = −( (w_{j,t})/(1−δ_{j,t}) )(η − 1)(E/(η − E)) + ( (w_{j,t})/(1−δ_{j,t}) )(η − 1)(F/(η − F)) - Aggregates: - dlog E_t / d δ_{j,t} = w_{j,t} (η − 1)/(1−δ_{j,t}) (F/(η − F)) - dlog M_t / d δ_{j,t} = w_{j,t} (η − 1)/(1−δ_{j,t}) (F/(η − F) − E/(η − E)) - Relationship between contemporaneous responses to the two subsidies: - dlog E_t / d δ_{j,t} = (η − 1) ( (1−χ_{j,t})/(1−δ_{j,t}) ) dlog E_t / d χ_{j,t} - dlog M_t / d δ_{j,t} = (η − 1) ( (1−χ_{j,t})/(1−δ_{j,t}) ) dlog M_t / d χ_{j,t} - This is equation (28) in the main text. ### Dynamic derivatives and impulse responses - Marginal impulse responses for period t+s with respect to policy x_t: - dy_{t+s} / d x_t = ( dy_{t+s} / d log \bar p'_{t+s} )_{N×1} × ∏_{k=1}^{s−1} ( d log \bar p_{t+k+1} / d log \bar p_{t+k} )_{N×N} × ( d log \bar p_{t+1} / d x_t )_{1×N} (equation (58)) - All but the last term are independent of x_t. - Long-run effects differ only to the extent contemporaneous production cost effects differ across policies. - Response of reference prices (differentiating (41)): - d log \bar p_{t+1} / d log \bar p_t = I + ω ρ ( d log p_t / d log \bar p_t − I ) (equation (59)) - Derivatives w.r.t. reference prices (differentiating (47) and (49)): - d ̃e_i = −E d ̃\hat p_i ∀ i - d ̃\hat p_i = d ̃p_i − g_i τ / \hat p_i dlog p_i ∀ i - d ̃e_i = −η d ̃p_i + 1 − η ∀ i ≠ j - d log e_j = η − 1 − η d log p_j - Solve for d ̃p_i: - d ̃p_i = −(η − 1 + J_i τ dlog p_j) / (η − E + J_i τ) - For i ≠ j: d ̃p_i = Ĉ_i(τ) + D_i(τ) dlog p_j - d ̃e_i = −η Ĉ_i(τ) − η D_i(τ) dlog p_j − η + 1 - d ̃\hat p_i = (η/E) Ĉ_i(τ) + (η/E) D_i(τ) dlog p_j + (η − 1)/E - Where: - Ĉ_i(τ) = (1 − η)/(η − E + J_i τ) = (η − 1) C_i(τ) - D_i(τ) = J_i τ/(η − E + J_i τ) - Solution for d log p_j with these terms: - d log p_j = ˆB / ˆA + ∑_{i≠j} ˆM_i / ˆA - Definitions differ from R&D subsidy case by multiplicative factors and replacement of C_i by Ĉ_i. - Special case τ_t = 0 ∀ t (dynamic derivatives simplify): - d log p_{i,t} / d log \bar p_{i,t} = (η − 1)( (1 − w_{i,t})/(η − E) + w_{i,t}/(η − F) ) - d log p_{i,t} / d log \bar p_{k,t} = (η − 1)( − w_{k,t}/(η − E) + w_{k,t}/(η − F) ) for i ≠ k - Interpretation: propagation breaks into composition (proportional to 1/(η − E)) and quantity (proportional to η − F) effects. - Computing impulse responses: - Analytical marginal impulse responses use substitution into equation (58). - Policy decomposition in Figures 7 and A.3 computed via finite difference: divide policy package into small increments; add up marginal impulse responses recomputing weights and prices after each step. - Approximation errors between full general equilibrium solution and summed partial equilibrium incremental responses are very small (validation noted). ### Stability of impulse responses - Reference price propagation matrix near τ_t = 0: - d log \bar p_{t+1} / d log \bar p_t = I + ω ρ ( d log p_t / d log \bar p_t − I ) = I + ( (E − 1)/(η − E) ) ω ρ + ω (η − 1) ( (η − F)^{−1} − (η − E)^{−1} ) W ≡ D - W = 1' w is the matrix of post-tax expenditure weights. - Stability condition: equilibrium dynamics stable if maxeig(D) < 1. - Necessary condition via trace(D): - ∑ eig(ρ) < ( (η − 1)/(E − 1) ) ( (E − F)/(η − F) ) ∈ [1, E(η − 1)/(η (E − 1)) ] (equation (60)) - Rows of ρ sum to unity; left-hand side range is [1, N] depending on linear independence of ρ. - No spillovers (ρ = I) makes upper bound N attained, shrinking the parameter space for stability. - Complete spillovers (identical rows of ρ) make sum of eigenvalues 1, favoring stability. - Intuition: with no spillovers a fuel can obtain an insurmountable technological lead and self-reinforce (temporary policies can have permanent effects); large spillovers diffuse advances across fuels, preventing runaway progress in one fuel. ### Dynamics with symmetric technology diffusion - Symmetric ρ structure: - ρ_{ij} = { ̄ρ + ρ_d if i = j; ̄ρ if i ≠ j } - ρ_d = 1 − N ̄ρ - ρ = ρ_d I + ̄ρ 1_{N×N} - Decomposition of D and eigenvalues: - D = (1 + a ρ_d) I + (a ̄ρ 1_{N×N} + b W) - eig(D) = {1 + a ρ_d + λ | λ ∈ eig(a ̄ρ 1_{N×N} + b W)} - a ̄ρ 1_{N×N} + b W has identical rows ⇒ single non-zero eigenvalue a N ̄ρ + b - maxeig(D) = (1 + a ρ_d) + max(a N ̄ρ + b, 0) = 1 + ω max[ ( (E − 1)/(η − E) ) ρ_d, (F − 1)/(η − F) ] - Note: maximum eigenvalue always greater than 1 (there is always growth in long-run impulse response). - Growth rate depends on technological diffusion ρ_d: - If ρ_d < (E − 1)/(F − 1) × (η − E)/(η − F), growth determined by interaction of η and F (aggregate channel). - If ρ_d above threshold, interaction driven by η and E (composition channel). - If F = 1, no change in energy expenditure P_t E_t ⇒ no increase in aggregate energy R&D. - If E = 1, composition channel fails (expenditure shares constant). - Large η (returns to R&D decline) drives dynamic R&D effect toward zero. ### Computation — basic solution algorithm - Two iterative processes per period: outer guess for international oil price p_{1,t}; inner region-level iteration on energy consumption E_{m,t} to clear markets. - Pseudo-code outline: - Outer Loop: 1. Guess the international oil price, p_{1,t}. For the first iteration, set p_{1,t} = p_{1,t−1}. - Inner Loop: in each region m i. Guess energy consumption E_{m,t}. ii. Invert the CES energy demand curve to find the price P^d_{m,t} at which demand equals E_{m,t}. iii. Compute energy firms’ price P^s_{m,t} from cost minimization given total energy consumption E_{m,t} and global oil price p_{1,t}. iv. Increase (decrease) E_{m,t} if P^d_{m,t} is more (less) than P^s_{m,t}. v. If P^d_{m,t} − P^s_{m,t} ≠ 0 return to step ii. *Source: wpiea2021173-print-pdf - 1. Using definitions of d̃x* ### 2. Compute global oil demand by summing regional demands. ### 2. Compute global oil demand by summing regional demands. ### D.2 Inverting the model to meet emissions targets - Purpose: Back out policies (carbon tax and subsidies) that deliver a specific emissions target and a balanced government budget. - Approach: Double-bisection algorithm in each period to compute tax and subsidy levels. - Policy parameterization: - Fix κ ∈ [0,1]. - Solve for subsidy control θ_{m,j,t} ∈ [0,1] with χ_{m,j,t} = κ θ_{m,j,t} and δ_{m,j,t} = (1−κ) θ_{m,j,t}. - Ratio of research to production subsidies given by κ/(1−κ). - Numerical necessity: Government revenue from the carbon tax and expenditure on subsidies depends on equilibrium allocations, so budget-balancing subsidies given a carbon tax must be solved numerically. This requires nesting a government balance loop inside an emissions-target loop (double-bisection). Exception: if there is only a carbon tax, the inner (government balance) loop can be skipped. - Algorithm (Outer loop: iterate on the carbon tax to hit the emissions target) 1. Guess upper and lower bounds on the carbon tax in each active region, τ_{lower,m,t}, τ_{upper,m,t}. On the first iteration, set τ_{lower,m,t} = 0. 2. Let τ^*_{m,t} = 1/2 (τ_{lower,m,t} + τ_{upper,m,t}). - Algorithm (Inner Loop: iterate on the subsidy control to achieve budget balance) i. Guess upper and lower bounds on the subsidy control θ_{lower,m,t}, θ_{upper,m,t} in each active region. On the first iteration, set θ_{lower,m,t} = 0, θ_{upper,m,t} = 1/max(κ,1−κ). ii. Let θ^*_{m,t} = 1/2 (θ_{lower,m,t} + θ_{upper,m,t}) for each region. iii. Solve the period t global general equilibrium model with tax and subsidy policies given by τ^*_{m,t}, θ^*_{m,t}. iv. For each m, update the upper (lower) bound for the subsidy control replacing it with θ^*_{m,t} if the government balance is in deficit (surplus). v. If θ_{lower,m,t} − θ_{upper,m,t} is small enough stop, else return to step ii. - After inner loop convergence: 3. For each m, update the upper (lower) bound for the tax rate control replacing it with τ^*_{m,t} if region m emissions are below (above) target. 4. If τ_{lower,m,t} − τ_{upper,m,t} is small enough stop, else return to step 2. - Computational notes: - The double-bisection nests the algorithm in Appendix D.1 (itself a double loop over multiple regions), producing high computational cost. - Advantages of bisection: robust, reliably converging across cases, guaranteed to find a solution of given precision within fixed time. - Both tax and subsidy policies solve to acceptable tolerances within a dozen iterations, putting a hard cap of 144 solutions in each period. - With almost forty periods to solve, the process is lengthy but manageable because constant savings rates allow period-by-period solution. - For two regions setting emissions targets, this process converges overnight on a modern laptop. - If savings rates were endogenous, the process would need dozens of outer iterations and computing solutions with multiple parameter values would take months. - Limited parallelization scope: each bisection and period relies sequentially on previous iterations. - Footnote numeric detail: Bisection shrinks the error by half with each step, implying a 12-step error of 0.5^{12} '0.025% of the initial interval. - Because subsidies are constrained to the unit interval, this means an error on the subsidy rate of no more than 0.025 percentage points. - If the initial upper bound on taxes is $1000, then this means an error on the tax rate of around 25 cents. ### D.3 Solving the model with non-constant savings rate - Modified Euler equation (equation (31) reproduced): (C_{m,t+1}/C_{m,t})^{γ} = β(i_{m,t+1} + 1 − δ_{K}) (61) - Substituting for savings rate, rental rate for capital, and households’ period budget constraint gives: s_t = 1/(1 + Γ_t) ( α(1 + g_{t+1})/β − 1 ((1 + g_{t+1})(1−s_{t+1})/(1−s_t))^{γ} − (1−δ_{K}) − (1−δ) k_t ) ≡ f(s_t; s_{t+1}, k_t, g_{t+1}, Γ_t) - Where (1 + g_{t+1}) = Ŷ_{t+1} / Ŷ_t is the growth in earned income and k_t = K_t / Ŷ_t is the capital to (earned) income ratio. - Fixed point characterization: - Given current capital ratio k, income growth g′, unearned income ratio Γ, and next period savings s′, the savings rate s which solves the Euler equation is the solution to the fixed point problem: s = f(s; s′, k, g′, Γ). - Long-run balanced growth path formula: - On a long-run balanced growth path with constants ̄g, ̄k, ̄Γ, the long-run savings rate ̄s is: ̄s = α β (1 + ̄g) ( ̄g + δ_{K} ) / ( 1 + ̄g + ̄Γ ( ̄g + δ_{K} ) ( (1 + ̄g)^{γ} − β (1 − δ_{K}) ) ) ≡ h( ̄k, ̄g, ̄Γ ) - Special case: when γ = 1, δ_{K} = 1, Γ = 0, this simplifies to ̄s = α β. - Solution strategy with non-constant savings: - Assume model reaches a balanced growth path in each region by the end of the simulation (simulation period is nearly two centuries). - Simulate the model for a given path of savings rates. - Use terminal period to define long-run values ̄g, ̄k, ̄Γ in each region. - Compute terminal period savings rates s_T using h(·). - Initiate backward recursion using the terminal s_T to compute sequence of savings rates consistent with the Euler equation back to period 1. - If this sequence is sufficiently close to the assumed sequence used in the simulation, accept; otherwise update the savings rate guess and iterate until convergence. - Pseudo-code (Part 1: Check the Euler equation errors on the current guess) 1. Guess the sequence of savings rates for all regions, s_{m,t}. 2. Solve the general equilibrium model fixing the savings rates at these guesses (modify algorithm in Appendix D.1 by replacing the savings rate α β with the appropriate entry from this sequence). 3. Use model values for capital, income growth, and unearned income share to compute Euler equation errors in every region and period, using error_{m,t} = s_{m,t} − f(s_{m,t}, s_{m,t+1}, k_{m,t}, g_{m,t+1}, Γ_{m,t}). 4. If the Euler equation errors are small enough, stop. Otherwise, continue. - Pseudo-code (Part 2: Update the savings rate guess) 5. Compute a new guess for terminal savings in each region using s^*_{m,T} = h(k_{m,T}, g_{m,T}, Γ_{m,T}). 6. Iterate backwards to update the savings rate to s^*_{m,t} using the Euler equation and simulated values for capital, income growth, and unearned income share, solving in each period and region the fixed point problem s^*_{m,t} = f(s^*_{m,t}, s^*_{m,t+1}, k_{m,t}, g_{m,t+1}, Γ_{m,t}). - Note: This fixed point problem is solved via an iterative loop of its own. 7. Return to step 1 using the updated guess for the savings rate, replacing s_{m,t} with s^*_{m,t}. *Source: wpiea2021173-print-pdf - 2. Compute global oil demand by summing regional demands.* --- _Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021173-print-pdf.pdf_