## Section 3 documents large within-industry and country heterogeneity in emission intensity

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### Data and sample
- Firm-level data on emissions, balance sheets, and income statements for more than 3,000 listed firms, headquartered in 65 countries, over 2010-2022.
- Emissions: annual firm-level, self-reported following the Greenhouse Gas Protocol from ICE Data Services; focus on CO2 equivalent scope 1 and scope 2 emissions.
- Financials: S&P Compustat Global; energy consumption from DataStream.
- Sample restrictions: exclude observations imputed from industry averages (restrict to firm disclosures); exclude finance, insurance, real state, public administration, utilities, railroad transportation, and local and interurban passenger transit sectors.
- Industry classifications used: SIC (main specification, 4-digit), NAICS, GICS, and Hoberg-Phillips text-based classification.
- Firm-level constructs and special data:
  - Age of capital estimated analogously to a perpetual inventory method.
  - TFP estimated from revenue data as in Asker et al. (2014).
  - For a subset of 23 US manufacturing firms, management practices from the World Management Survey; R&D cost differentials from Lucking (2019).
- Coverage note: 80% of the firms are headquartered in advanced economies.

### Heterogeneity in emission intensity — key findings
- Emission intensity measure: log of CO2 emissions (scope 1 + scope 2) over revenues (megatons per million of USD revenue) in 2019, residualized on country×industry (4-digit SIC) fixed effects.
- Within-industry and within-country dispersion:
  - Difference between the 90th and the 10th percentile within the same industry-country group is 1.85 log points, which translates into 6.37 times larger emissions per unit of revenue.
  - Comparisons: 90-10 percentile difference is 1.56 log points for labor productivity (revenues over wage bill) and 0.51 for revenue-based TFP.
- Aggregate counterfactual (holding output constant):
  - Bringing firms above a given percentile down to that percentile: bringing firms to the 25th percentile would reduce aggregate emissions by 33%.
  - Exercise is exogenous to feasibility, costs, and policy channels; feasibility and costs analyzed in the model.

### Emission intensity and firm characteristics (associations and magnitudes)
- Empirical specification: log(EI_i,t) = φ_{country(ind), industry(l), t} + X_{i,t}β + W_{i,t}γ + ε_{i,t}, with EI measured as Scope 1+2 emissions scaled by revenues (standardized variables); results focus on 2019.
- Raw patterns:
  - Higher emission intensity associated with older physical capital, lower share of intangible assets, and lower TFP.
- Fixed-effects regression magnitudes:
  - A one standard deviation increase in the age of capital is associated with 0.03–0.06 standard deviation higher emissions per unit of revenue.
  - Increases in share of intangible capital associated with lower emissions per unit of revenue.
  - Firms with higher TFP have lower emission intensities.
- Back-of-the-envelope economy-wide magnitudes:
  - If all firms had physical capital vintages in the newest quartile, total emissions would fall by 7%.
  - If all firms were in the top quartile of intangible asset shares, total emissions would fall by 9%.
- Mechanisms (decomposition):
  - Newer physical capital vintages associated with lower emissions per unit of energy (capital-embedded technologies are green-biased).
  - Higher intangible-asset shares lower emissions via reduced energy consumption per unit of revenue (productivity-driven input efficiency), not lower emissions per unit of energy.

### Robustness checks and additional evidence
- Results robust across industry classifications: SIC (4/3/2-digit), GICS (4/2-digit), NAICS (4/2-digit), and Hoberg-Phillips.
- Alternative proxies produce similar results: firm age for capital vintage, R&D expenditures for intangible investment, profitability for production efficiency.
- Sample and control robustness:
  - Holds when restricting to large firms.
  - Robust to financial controls (lagged leverage, liquidity, capitalization, market share).
  - Robust to excluding 2020+ (not driven by COVID) and to focusing on scope 1 only.
  - Robust when emission intensity computed as emissions over total assets or over value added.
  - Firm-level clustering of standard errors yields consistent results.
- Management practices:
  - Within-industry, higher World Management Survey scores associated with lower emission intensities, supporting production-process–efficiency interpretation.

### Instrumental-variable evidence (causal-suggestive)
- IV for R&D (US manufacturing):
  - Instrument: variation in R&D cost across US states due to differences in state R&D tax credits (Lucking, 2019), using headquarter state.
  - First-stage F-statistic = 3.396 (Montiel-Pflueger) — instrument weak.
  - IV (2SLS) and weak-instrument–robust confidence intervals (Wald and Anderson-Rubin) exclude zero, supporting negative causal effect of R&D on emission intensity.
- IV for age of capital:
  - Instrument: firms’ recent growth rates (firms that grew faster likely have newer capital).
  - Tables A11 and A12 suggest causal relationship between newer capital vintages and lower emission intensity; IV coefficients larger than OLS (consistent with attenuation bias).

### Model overview (purpose and structure)
- Purpose: rationalize empirical findings and quantify mitigation-policy implications via a multi-sector heterogeneous-firm general equilibrium model with entry and exit, capital vintages, and intangible knowledge accumulation.
- Key model elements:
  - Finite number of countries j∈{1,...,J}; no trade in goods/assets across countries; discrete time.
  - Households: representative, Cobb-Douglas across sectors, CES over varieties; natural resources (energy) sold at exogenous price m, supplied elastically.
  - Firms: monopolistic competition; production y_si = ω_si (v_si k_vsi)^{κ_s} n_si^{η_s} ℓ_si^{λ_s}, with ω_si intangible knowledge, v_si vintage quality, k_vsi physical capital of vintage v, n_si energy, ℓ_si labor.
  - Emissions: e_si = φ_s n_si v_si^{−ε_v} (dependence on energy, green bias of newer vintages via v^{−ε_v}, sector-country factor φ_s).
  - Investment frictions: cannot operate multiple vintages simultaneously; retiring capital recovers fraction χ; capital depreciation rate δ.
  - Intangible knowledge: ω_si = A_s^{θ_s} a_si; firm knowledge accumulation a' = (1−δ_a)a + μ (ℓ_a γ)^{α_s}, with firm-specific γ drawn from G_s(γ).
  - Government instruments: carbon tax τ_e, sales rebate τ_y (feebate), vintage-specific subsidy τ_v (τ_V for newest vintage), subsidy to intangible investments τ_a; government budget constraint ties taxes/subsidies.
  - Firm dynamic problem: choose vintage, capital investments, intangible investments, labor for production and knowledge, energy, prices; entry cost κ_e and fixed operating cost κ_f determine entry/exit.
  - Capital-goods sector: competitive producers of vintages; capital price q_v = w z_v^{-1}; capital-goods production emits with same intensity as rest of economy.
- Equilibrium margins:
  - Firms adjust via vintage upgrades (reduce emissions per energy), intangible investments (raise productivity, reduce energy per output), capital deepening, and variable input adjustments.
  - Policy effects: subsidies to newest vintage lower q_V and incentivize upgrades; carbon pricing incentivizes energy reductions and affects vintages ambiguously; subsidies to intangible investments increase knowledge accumulation and can reduce physical capital.

### Calibration overview and parameter identification
- Externally calibrated parameters:
  - ρ Discount rate = 0.04
  - δ_k Depreciation rate of capital = 0.05
  - δ_a Depreciation rate of knowledge = 0.15
  - σ Elasticity of substitution = 8
  - χ Liquidation value = 0.35
  - g_v − g_p Vintage productivity/price growth = 3.2%
- Internal targets and estimation:
  - Calibrate β_s to match sector sales shares per country.
  - Identify factor elasticities using sales, COGS, and factor shares.
  - Assume θ_s = α_s (social returns to intangible knowledge).
  - Estimate g_v via regression of firm TFP on capital age instrumented with 5-year recent growth rate; estimated g_v = 11.9%.
  - Estimate ε_v from log(Emission/Energy) = log(φ_e_s) − ε_v log v via IV.
  - Match distributions of v, γ, ξ to intangibles/tangibles, age of capital, and firm size.

### Policy counterfactuals and calibration
- Policy instruments per country: carbon tax τ_e, feebate (τ_e, τ_y), subsidy to newest vintage τ_V, intangible investment subsidy τ_a.
- Simulation approach:
  - Simulate one instrument at a time; set each to generate the same 15% decline in each country’s total corporate emissions.
  - Note: target is less ambitious than Paris Agreement 45% by 2030; model abstracts from other mitigation margins.

### Quantitative effects of mitigation policies (key reported magnitudes)
- Main comparison (NPVC and fiscal costs; all policies calibrated to achieve a 15% reduction in emissions; NPV computed with a 4% time discount factor):
  - NPV of consumption (percentage change relative to actual economy):
    - Carbon Tax: -.32
    - Carbon Feebate: -.33
    - New Vintage Subsidy: -4.54
    - Intangible Subsidy: -4.07
  - NPV of fiscal transfers (annualized, share of GDP):
    - Carbon Tax: -.40
    - Carbon Feebate: .05
    - New Vintage Subsidy: 5.70
    - Intangible Subsidy: 12.94
  - Long-run consumption (percentage change):
    - Carbon Tax: -.36
    - Carbon Feebate: -.36
    - New Vintage Subsidy: 2.82
    - Intangible Subsidy: -7.10
  - Long-run TFP (percentage change):
    - Carbon Tax: .03
    - Carbon Feebate: .05
    - New Vintage Subsidy: 2.54
    - Intangible Subsidy: 3.90
  - Share of firms updating (percent of total firms):
    - Carbon Tax: 2.32
    - Carbon Feebate: 2.33
    - New Vintage Subsidy: 62.69
    - Intangible Subsidy: .00
- Interpretation:
  - Carbon taxes are an order of magnitude less costly economically (NPVC decline -.32) than subsidies that achieve the same emissions reduction.
  - Subsidies narrow scope and distort allocation, generating higher fiscal costs and transition distortions (e.g., labor reallocation, capital-goods sector expansion generating emissions).

### Vintage upgrade margin — effects on carbon-tax costs and policy levels
- Simulation: carbon tax calibrated to achieve 15% emission reduction under two regimes (firms can upgrade vintages vs. forced to keep initial vintages).
- NPVC effects:
  - When firms cannot upgrade vintages, the carbon tax leads to NPVC drop of 0.42%, a 31% larger cost than when firms can upgrade (0.32%).
- Required average carbon tax (country averages reported in Table 7):
  - With upgrade: 29.09 (dollars)
  - No upgrade: 36.10 (dollars)
  - Allowing upgrades reduces the required average carbon tax by $36.1−$29.1 = $7 (or 20%).
- Upgrade incidence:
  - Only 2.3% of firms choose to upgrade when allowed, yet this yields a 31% difference in the economic cost of the carbon tax.
  - Firms that upgrade are those furthest from the capital vintage frontier and tend to be large, amplifying aggregate effects.
- Table 7 selected entries (NPVC and policy levels):
  - Change in NPVC (Average):
    - Carbon Tax, Upgrade: -.38
    - Carbon Tax, No Upgrade: -.49
    - New Vintage Subsidy, Upgrade: -4.87
    - New Vintage Subsidy, No Upgrade: -23.35
  - Level of Tax/Subsidy (Average):
    - Carbon Tax, Upgrade: 29.09
    - Carbon Tax, No Upgrade: 36.10
    - New Vintage Subsidy, Upgrade: 19.91
    - New Vintage Subsidy, No Upgrade: 69.53
  - IQRs reported for NPVC and policy levels in Table 7.

### Subsidies to newest vintage — intertemporal trade-offs
- Subsidies for newest vintage generate long-run gains but large transition costs:
  - Long-run consumption for New Vintage Subsidy: 2.82 percent.
  - TFP increases by 2.5 percent in long run (reported in text).
- Mechanism: newest vintages are green biased (raise productivity and lower emissions conditional on energy), but widespread upgrading entails large short-term resource diversion from consumption.
- Social discounting:
  - Under market rate 4%, the social discount rate required to make newest-vintage subsidies as appealing as carbon taxes is around 1.5%, with interquartile range across countries of 1.3-4.0.

### Cross-country heterogeneity and policy implications
- Required carbon tax to achieve 15% cut varies across countries:
  - Range: less than $10 to $60 a ton.
  - Interquartile range (IQR): $23.65 to $38.25.
- Dispersion in NPVC costs:
  - NPVC IQR: -.6% to -.1%.
- Vintage-upgrade margin importance varies:
  - More impactful in countries with energy-intensive industries and older technology mixes.
  - Countries with older capital stocks see fewer firms upgrading and smaller reductions in subsidy costs from upgrade margin.
- Intangible-investment subsidies:
  - Lead to large long-run consumption losses: -7.1 percent.
  - Lack green bias; reduce emissions primarily via output reductions given elastic energy supply.

### Model consistency with empirical stylized facts
- Analytical expressions link vintage v and knowledge efficiency 1/γ to emissions per output and TFP:
  - ln(e/y) and lnTFP expressions (equations (18) and (19)) predict that newer vintages and higher knowledge efficiency lower emission intensity and raise TFP.
- Model mechanisms match observed associations between age of capital, intangible shares, TFP, and emission intensities.

*Source: wpiea2023242-print-pdf - Section 3 (https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023242-print-pdf.pdf)*

### Section 3 documents large within-industry and country heterogeneity in emission

### Section 3 documents large within-industry and country heterogeneity in emission intensity

### Data and sample
- Firm-level data on emissions, balance sheets, and income statements for more than 3,000 listed firms, headquartered in 65 countries, over 2010-2022.
- Emissions data: annual firm-level, self-reported following the Greenhouse Gas Protocol from ICE Data Services; focus on CO2 equivalent scope 1 and scope 2 emissions.
- Financials: S&P Compustat Global; energy consumption from DataStream.
- Sample restrictions: exclude observations imputed from industry averages (restrict to firm disclosures); exclude finance, insurance, real state, public administration, utilities, railroad transportation, and local and interurban passenger transit sectors.
- Industry classifications used: SIC (main specification, 4-digit), NAICS, GICS, and Hoberg-Phillips text-based classification.
- Firm-level constructs:
  - Age of capital estimated analogously to a perpetual inventory method (weigh past investments by years since investment and divide by sum of undepreciated investments).
  - TFP estimated from revenue data as in Asker et al. (2014).
  - For a subset of 23 US manufacturing firms, management practices from the World Management Survey; R&D cost differentials from Lucking (2019).
- Coverage note: 80% of the firms are headquartered in advanced economies.

### Heterogeneity in emission intensity (key findings)
- Emission intensity measure: log of CO2 emissions (scope 1 + scope 2) over revenues (megatons per million of USD revenue) in 2019, residualized on country×industry (4-digit SIC) fixed effects.
- Large within-industry and within-country heterogeneity:
  - Difference between the 90th and the 10th percentile within the same industry-country group is 1.85 log points, which translates into 6.37 times larger emissions per unit of revenue.
  - Comparisons to productivity heterogeneity: 90-10 percentile difference is 1.56 log points for labor productivity (revenues over wage bill) and 0.51 for revenue-based TFP.
- Counterfactual (back-of-the-envelope) illustrating aggregate implications:
  - If every firm above a given percentile in its industry-country emission-intensity distribution reduced its emission intensity to that percentile (holding output constant): bringing firms to the 25th percentile would reduce aggregate emissions by 33%.
  - The exercise is exogenous to feasibility, costs, and policy channels; feasibility and policy costs are analyzed later in the model.

### Emission intensity and firm characteristics (associations and magnitudes)
- Empirical specification: log(EI_i,t) = φ_{country(ind), industry(l), t} + X_{i,t}β + W_{i,t}γ + ε_{i,t}, with EI measured as Scope 1+2 emissions scaled by revenues (standardized variables), emissions normalized by lagged revenues, and log assets included as a control. Results focus on 2019 to abstract from COVID effects.
- Raw patterns (binned scatterplots): firms with higher emission intensity operate older physical capital, have lower share of intangible assets, and lower TFP.
- Fixed-effects regressions (summary):
  - A one standard deviation increase in the age of capital is associated with 0.03–0.06 standard deviation higher emissions per unit of revenue (interpreted as legacy machines/processes → higher emissions).
  - Increases in the share of intangible capital to total capital are associated with lower emissions per unit of revenue.
  - Firms with higher TFP have lower emission intensities.
  - Coefficients are robust when all independent variables are included jointly.
- Economic magnitudes (back-of-the-envelope):
  - If all firms had physical capital vintages in the newest quartile, total emissions would fall by 7%.
  - If all firms were in the top quartile of intangible asset shares, total emissions would fall by 9%.
- Decomposition of emission-intensity reductions:
  - Newer physical capital vintages are associated with lower emissions per unit of energy consumed — i.e., capital-embedded technologies are green-biased (may permit cleaner energy use or greater reliance on electricity).
  - Higher intangible-asset shares are not associated with lower emissions per unit of energy; improvements stem from lower energy consumption per unit of revenue (productivity-driven input efficiency).

### Robustness checks and additional evidence
- Industry classification robustness: results stable across SIC granularity (4-digit, 3-digit, 2-digit), GICS 2/4-digit, NAICS 2/4-digit, and Hoberg-Phillips (HP) classification.
- Alternative proxies: using firm age as proxy for capital vintages, R&D expenditures as proxy for intangible investment, and profitability as proxy for production efficiency produce similar results.
- Sample and control robustness:
  - Results hold when restricting to large firms.
  - Robust to financial controls (lagged leverage, liquidity, capitalization, market share).
  - Robust to excluding 2020+ (not driven by COVID) and to focusing on scope 1 only.
  - Robust when computing emission intensities as emissions over total assets or over value added.
  - Firm-level clustering of standard errors yields consistent results.
- Management practices: within-industry, firms with higher World Management Survey scores have lower emission intensities, supporting the production-process–efficiency interpretation rather than pure market-power explanations.

### Instrumental variable evidence (causal suggestive results)
- IV strategy for R&D:
  - Instrument: variation in R&D cost across US states due to differences in state R&D tax credits (Lucking, 2019), applied using headquarter state.
  - OLS (US manufacturing): higher R&D (normalized by revenues) associated with lower emission intensity.
  - First-stage: firms in lower-R&D-cost states invest more in R&D (instrument weak; first-stage F-statistic = 3.396 (Montiel-Pflueger)).
  - IV (2SLS) and weak-instrument–robust confidence intervals (Wald and Anderson-Rubin) exclude zero, supporting a negative causal effect of R&D on emission intensity.
- IV for age of capital:
  - Instrument: firms’ recent growth rates used to instrument age of capital (firms that grew faster likely have newer capital).
  - Tables A11 and A12 suggest a causal relationship between newer capital vintages and lower emission intensity, with IV coefficients larger than OLS (consistent with attenuation bias in measured age of capital).

### Model overview (purpose and structure)
- Purpose: rationalize empirical findings and quantify implications of mitigation policies via a multi-sector heterogeneous-firm general equilibrium model with entry and exit, capital vintages, and intangible knowledge accumulation.
- Key elements:
  - Finite number of countries j∈{1,...,J}; no trade in goods/assets across countries; discrete time.
  - Households: representative, Cobb-Douglas across sectors, CES over varieties; natural resources (energy) sold at exogenous price m, supplied elastically.
  - Firms: monopolistic competition producing differentiated varieties; production y_si = ω_si (v_si k_vsi)^{κ_s} n_si^{η_s} ℓ_si^{λ_s}, where ω_si is intangible knowledge, v_si is vintage quality (efficiency unit), k_vsi is physical capital of vintage v, n_si is energy, ℓ_si is labor.
  - Emissions: e_si = φ_s n_si v_si^{−ε_v}, capturing (i) dependence on energy use, (ii) green bias of newer vintages (higher v reduces emissions per energy), and (iii) sector-country common factors φ_s (e.g., grid mix).
  - Investment/frictions: firms cannot operate multiple vintages simultaneously (must replace entire stock to upgrade); retiring capital recovers only fraction χ of value (lumpy, irreversibility); capital depreciates at rate δ.
  - Intangible knowledge: ω_si = A_s^{θ_s} a_si, with a_si produced by firm through labor devoted to knowledge accumulation; aggregate A_s = Σ_i a_si; knowledge accumulation a' = (1−δ_a)a + μ (ℓ_a γ)^{α_s}, with firm-specific γ drawn from G_s(γ).
  - Government instruments: carbon tax τ_e on emissions, sales rebate τ_y (feebate design), vintage-specific subsidy τ_v (e.g., τ_V>0 for newest vintage), subsidy to intangible investments τ_a; government budget constraint balances taxes/subsidies across sectors and firms.
  - Firm dynamic problem: firms choose vintage, capital investments, intangible investments, labor for production and knowledge, energy, and prices; entry cost κ_e and fixed operating cost κ_f determine entry/exit.
  - Capital-goods sector: competitive producers of vintages with linear labor technology; capital price q_v = w z_v^{-1}; capital-goods production generates emissions with same emission intensity as rest of economy.
- Equilibrium properties and policy margins:
  - Firms adjust via (i) vintage upgrades (reducing emissions per energy), (ii) intangible investments (raising productivity and reducing energy per output), (iii) capital deepening and variable input adjustments.
  - Policy effects on capital stock and vintage choice:
    - Subsidies to newest vintage lower q_V and incentivize upgrading/growth of newest-vintage firms.
    - Carbon pricing (tax or feebate) ambiguously affects capital investment across firms: incentivizes energy reductions and capital intensification but raises relative cost of older vintages, inducing downsizing/upgrading incentives.
    - Subsidies to intangible investments increase intangible accumulation and can reduce physical capital in equilibrium.
- Consistency with empirical stylized facts: model setup (vintage green-bias; intangibles lowering emissions per output) is constructed to match observed associations between age of capital, intangible shares, TFP, and emission intensities; analytical characterization and proofs in Appendix A1.

*Source: wpiea2023242-print-pdf - Section 3 (https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023242-print-pdf.pdf)*

### 1.  If a firm updates its vintage, it upgrades to the best vintage V

### 1.  If a firm updates its vintage, it upgrades to the best vintage V

### Vintage upgrade decision
- If a firm updates its vintage, it upgrades to the best vintage V.
- There exists a unique v_s such that a firm with v upgrades to V_s if and only if v < v_s.
- The condition implies that v_s / q_s is increasing in v_s, which implies the newest vintage is always the optimal choice when a firm updates.
- Policies affecting the price of the newest vintage q_V:
  - A subsidy to capital investment τ_V lowers q_V and gives direct incentives for all firms with older capital vintages to upgrade since the value of upgrading (equation (54)) is increasing in τ_V.
  - Carbon taxes and feebates also incentivize upgrades indirectly via the price of energy, because firms with older vintages emit more and face larger cost increases under a carbon tax.

### Optimal choice of energy
- Optimal energy consumption for a firm with capital stock k_v of vintage v (equation (16)):
  - n = Λ_s(ξ) Â_{ρ_s} μ_{λ_s} ŵ_{λ_s} μ_{η_s} (m + τ_e_v^{−ε_v})̂_{η_s+1} (v k_v)̂_{κ_s}
  - Λ_s(ξ) = [ (σ−1)/σ (1+τ_y) (η_s + λ_s) P_s(ξ Y_s)^{1/σ} ]^{σ/(σ+(1−σ)(η_s+λ_s))}
  - Definition: x̂ = x^{σ/(σ−1) − (η_s + λ_s)} for x = ρ, λ_s, η_s
  - Assumption: σ^{σ/(σ−1)} > (η_s + λ_s) ensures well-defined optimal inputs.
- Comparative statics and policy channels:
  - Energy consumption increases in capital stock k_v and in output elasticity to energy η_s.
  - Climate policies affect energy consumption primarily through the price of energy m + τ_e_v^{−ε_v}, which includes the carbon tax.
  - An increase in the price of energy leads firms to substitute away from energy, rely relatively more on capital and labor, and scale down production.
  - Firms operating older, less efficient vintages (higher emissions per unit of energy) are more impacted by a carbon tax and reduce energy consumption by a larger extent.
  - Sectors with higher η_s (more energy-dependent) are more affected, resulting in more pronounced energy-use reductions.

### Optimal investment in intangibles
- Optimal labor invested in intangibles ℓ_a and steady-state accumulated knowledge a (equation (17)):
  - ℓ_a = [ Ω_s(v, γ, ξ, τ_e)̂_{α_s} k̂_{κ_s}_v / ( w(1−τ_a)(r+δ_a) γ̂_{α̂} δ_a ) ]^{1/(1−α̂_s)}
  - a = ( μ ℓ_a / γ )^{α (1/δ_a)}
  - Ω_s(v, γ, ξ, τ_e) is revenue-productivity of capital, defined by p y = Ω_s(v, γ, ξ, τ_e) k̂_{κ_s}_v a^{1/σ σ−1 − (η_s + λ_s)} (Appendix A1.3).
- Properties of Ω_s:
  - Increasing in v and ξ.
  - Decreasing in γ and in the carbon tax τ_e.
  - Depends positively on sectoral price index P_s and production Y_s.
- Comparative statics:
  - Investment in intangibles increases with firm scale k_v.
  - Intangibles accumulation is negatively impacted by higher wage w.
  - Firms with high γ (inefficient at accumulating intangibles) accumulate less and use other inputs relatively more.
- Policy effects:
  - Subsidies to intangible investment τ_a and emissions-targeting policies τ_e incentivize scaling up intangible investments to improve productivity and economize on energy.
  - Declines in energy consumption are especially strong for firms with lower-quality vintages and in sectors with high η_s.

### Optimal pricing decision
- Firms compete monopolistically with local market power; the optimal price is a constant markup over marginal cost.
- Further details on pricing are in Appendix A1.3.

### General equilibrium channels
- Firm-level policy effects propagate through general equilibrium channels:
  - Market shares reallocate across heterogeneous firms and production shifts across sectors.
  - Energy-intensive sectors and firms shrink after a carbon tax; sectors relying on intangible capital expand with intangible investment subsidies.
  - Within industries, firms using newer vintages benefit from subsidies for the newest vintage and grow at the expense of other firms.
  - Entry and exit respond endogenously:
    - A subsidy to the newest vintage increases profitability for new firms.
    - A feebate stimulates entry.
    - A carbon tax may induce exit.

### Consistency with stylized facts (Proposition 1)
- Model implications for cross-firm patterns in steady state (holding capital k and tastes ξ constant):
  - Firms with newer vintages v and/or higher knowledge accumulation efficiency 1/γ emit less per unit of output:
    - ln(e/y) = ln c_{s t} − (ζ_{e v1} + ε_v) ln v + ζ_{e v2} ln(m + τ_e_v^{−ε_v}) + ζ_{e γ} ln γ − ζ_{e k} ln k + ζ_{e ξ} ln ξ  (equation (18))
    - Coefficients: ζ_{e v1} = (κ + α_s ε_{q v}) b_{e s}, ζ_{e v2} = η_s b_{e s}^{−1}, ζ_{e γ} = α_s b_{e s}, ζ_{e ξ} = (1−η_s − λ_s) b_{e s}, ζ_{e k} = (α_s + κ_s) b_{e s}, b_{e s} = [ σ + (σ−1)(η_s + λ_s) ].
  - Firms with newer vintages and higher knowledge efficiency have higher TFP:
    - lnTFP = ln z_{s t} + ζ_{e v1} ln v_i − ζ_{e v2} ln(m + τ_e_v^{−ε_v}) + ζ_{e γ} ln(1/γ) + ζ_{e γ} ln k + 1/σ log ξ  (equation (19))
    - Coefficients: ζ_{e v1} = κ b_e, ζ_{e v2} = η b_{e s}, ζ_{e γ} = α b_{e s}, b_e = 1 − 1/σ.
  - c_{s t} and z_{s t} are common to all firms within a time period, country, and sector (Appendix A1).
- Interpretation:
  - Emission intensity decomposes into emissions per energy and energy per output; vintage quality v and knowledge efficiency 1/γ improve energy efficiency and reduce emissions per unit of energy.
  - Positive empirical association between revenue-based TFP and emission intensity is consistent with both being positively shaped by vintage quality v and knowledge accumulation efficiency 1/γ.
  - Size’s relationship with environmental performance is complex because size depends on vintage quality, knowledge efficiency, and consumers’ tastes.

### Calibration overview
- Calibration goals:
  - Match empirical distribution of firms, importance of new margins (vintage upgrades and intangibles), and green bias of capital-embedded technologies.
  - Match country and sector-level moments; a subset of parameters calibrated externally, others estimated from data.
- Externally calibrated parameters (Table 4):
  - ρ Discount rate = 0.04
  - δ_k Depreciation rate of capital = 0.05
  - δ_a Depreciation rate of knowledge = 0.15
  - σ Elasticity of substitution = 8
  - χ Liquidation value = 0.35 (implies 65% of capital stock is firm-specific)
  - g_v − g_p Vintage productivity/price growth = 3.2%
- Internal calibration targets and granularity (Table 5):
  - Parameters include g_v, ε_v, κ_f, φ_e, κ_e, β_s, α, ρ, κ, η, λ, and distributions G(γ, ξ), with granularity ranging from world-wide to firm-specific.
- Estimation strategy highlights:
  - Calibrate β_s to match sector sales shares per country (re-weight listed firms to match industry shares).
  - Identify elasticities of output to factors using constant returns, average sales, cost of goods sold, and shares to labor, energy, research.
  - Assume social returns to intangible knowledge θ_s = α_s.
  - Estimate g_v via regression of firm TFP on capital age instrumented with 5-year recent growth rate.
  - Estimate ε_v from log(Emission/Energy) = log(φ_e_s) − ε_v log v using same IV approach.
  - Estimate distribution of firm-level variables v, γ, ξ by matching ratios: intangibles/tangible assets, age of capital (v = (1+g_v)^{−age}), and firm size.

### Calibration of counterfactual policies
- Policy instruments calibrated per country: carbon tax τ_e, feebate (τ_e, τ_y), subsidy to newest vintage τ_V, intangible investment subsidy τ_a.
- Simulation approach:
  - Simulate one instrument at a time to isolate properties.
  - Set each policy independently to generate the same 15% decline in each country’s total corporate emissions.
  - Note: target is less ambitious than Paris Agreement 45% by 2030; chosen to retain country sample and because model abstracts from other mitigation margins (e.g., greening electricity grid, housing stock).
- Appendix Figure A7 contains resulting cross-country distribution of carbon taxes and subsidies (not reproduced here).

### Quantitative effects of mitigation policies
- Main quantitative comparison (NPVC and fiscal costs):
  - Carbon tax leads to −0.3% decrease in the net present value of consumption (NPVC).
  - Subsidies for the newest capital vintage and for intangible investments both lead to declines of more than 4% in NPVC (Table 6).
  - Net present values of fiscal costs:
    - Subsidies to the newest vintage: 5.7% of GDP.
    - Intangible investments subsidies: 12.9% of GDP.
- Interpretation:
  - Carbon taxes are an order of magnitude less costly economically than subsidies calibrated to achieve the same emissions reduction; consistent with carbon taxes being first-best mitigation tool.
  - Carbon taxes directly incentivize emissions economization across all margins cost-efficiently.
  - Subsidies are narrower in scope and distort allocation:
    - Newest-vintage subsidies incentivize capital over-accumulation and excessive adoption of the newest vintage.
    - Intangible investment subsidies lead to over-investment in intangible capital.
  - General equilibrium distortions:
    - Subsidies increase demand for labor to produce capital goods or to invest in knowledge, raising real wages and crowding out labor from final goods sectors.
    - Subsidy to the newest vintage expands capital goods sector, generating additional emissions that partially offset final goods sector gains.
  - Higher fiscal costs for intangible investment subsidies stem from needing much larger subsidies to achieve a 15% emissions reduction (see Figure A7).

*Source: wpiea2023242-print-pdf - 1.  If a firm updates its vintage, it upgrades to the best vintage V (IMF Working Paper PDF).*

### section 4—lowers the costs of carbon taxes by an economically meaningful amount. To

### wpiea2023242-print-pdf - section 4—lowers the costs of carbon taxes by an economically meaningful amount. To

### Vintage upgrade margin and impact on carbon taxes
- Simulating a carbon tax calibrated to achieve a 15% reduction in emissions under two regimes: firms allowed to upgrade vintages versus firms forced to keep initial vintages.
- When firms cannot upgrade vintages, the carbon tax leads to a drop in the NPVC by 0.42%, a 31% larger cost than when they can upgrade (0.32%).
- Allowing vintage upgrades reduces the required average carbon tax by $36.1−$29.1 = $7 (or 20%) once the vintage-upgrade margin is accounted for.
- Only a small share of firms upgrade when allowed: 2.3% of firms choose to upgrade, yet this leads to a 31% difference in the economic cost of the carbon tax.
- Firms furthest from the capital vintage frontier are the ones that choose to upgrade; these laggard firms tend to be large, amplifying aggregate effects.

### Key quantitative results from Table 6 (Effects of policies on aggregates)
- NPVs and long-run effects (values denote percentage changes relative to the actual economy, except fiscal transfers in share of GDP and share of firms updating in percent of total firms):
  - NPV of consumption:
    - Carbon Tax: -.32
    - Carbon Feebate: -.33
    - New Vintage Subsidy: -4.54
    - Intangible Subsidy: -4.07
  - NPV of fiscal transfers:
    - Carbon Tax: -.40
    - Carbon Feebate: .05
    - New Vintage Subsidy: 5.70
    - Intangible Subsidy: 12.94
  - Long-run consumption:
    - Carbon Tax: -.36
    - Carbon Feebate: -.36
    - New Vintage Subsidy: 2.82
    - Intangible Subsidy: -7.10
  - Long-run profits:
    - Carbon Tax: -.50
    - Carbon Feebate: -.03
    - New Vintage Subsidy: 2.77
    - Intangible Subsidy: -7.18
  - Long-run labor productivity:
    - Carbon Tax: -.14
    - Carbon Feebate: -.14
    - New Vintage Subsidy: 2.24
    - Intangible Subsidy: 3.12
  - Long-run TFP:
    - Carbon Tax: .03
    - Carbon Feebate: .05
    - New Vintage Subsidy: 2.54
    - Intangible Subsidy: 3.90
  - Share of firms updating:
    - Carbon Tax: 2.32
    - Carbon Feebate: 2.33
    - New Vintage Subsidy: 62.69
    - Intangible Subsidy: .00
- Notes:
  - NPV computed with a 4% time discount factor.
  - Output, consumption, profits, labor productivity, TFP are steady-state weighted averages across sectors and countries using country-specific sector shares and countries GDP.
  - Fiscal transfers are annualized sum of steady-state and transition net subsidies, in percent of steady-state GDP.
  - All policies are calibrated to achieve a 15% reduction in emissions.

### Counterfactuals with and without vintage upgrade margins (Table 7)
- Change in NPVC (Average):
  - Carbon Tax, Upgrade: -.38
  - Carbon Tax, No Upgrade: -.49
  - New Vintage Subsidy, Upgrade: -4.87
  - New Vintage Subsidy, No Upgrade: -23.35
- Change in NPVC (IQR):
  - Carbon Tax, Upgrade: [-.58, -.13]
  - Carbon Tax, No Upgrade: [-.94, -.25]
  - New Vintage Subsidy, Upgrade: [-4.13, -.05]
  - New Vintage Subsidy, No Upgrade: [-20.75, -8.06]
- Level of Tax/Subsidy (Average):
  - Carbon Tax, Upgrade: 29.09
  - Carbon Tax, No Upgrade: 36.10
  - New Vintage Subsidy, Upgrade: 19.91
  - New Vintage Subsidy, No Upgrade: 69.53
- Level of Tax/Subsidy (IQR):
  - Carbon Tax, Upgrade: [23.65,38.25]
  - Carbon Tax, No Upgrade: [36.87,50.31]
  - New Vintage Subsidy, Upgrade: [8.01,13.66]
  - New Vintage Subsidy, No Upgrade: [54.52,75.33]
- Notes:
  - NPVC denotes net present value of consumption; change reported as percentage change of the actual economy.
  - Carbon tax expressed in dollars; subsidy expressed in percentages.
  - A 4% time discount factor is used; averages weighted by country-specific sector shares and by countries GDP.
  - All policies calibrated to achieve a 15% reduction in emissions.

### Subsidies to newest vintage: intertemporal trade-offs and mechanisms
- Subsidies for the newest vintage generate long-run gains:
  - TFP increases by 2.5 percent (reported in text) and consumption increases by 2.8 percent (Table 6: Long-run consumption for New Vintage Subsidy: 2.82).
- Mechanism:
  - Newer capital vintages are green biased: they raise productivity (lower energy use per unit of output) and lower emissions conditional on energy usage.
  - Capital vintages enter both the emission (Equation 5) and the output (Equation 4) production functions in the model.
  - Short-term costs arise because many firms upgrade and invest in newer vintages, taking resources away from consumption.
- Intertemporal trade-off and social discounting:
  - A social planner more patient than private agents may prefer newest-vintage subsidies due to future gains.
  - Under a fixed market rate of 4%, the social discount rate required to make subsidies to the newest vintage as appealing as carbon taxes is around 1.5%, with an interquartile range across countries of 1.3-4.0%.

### Heterogeneity across countries and policy implications
- Required carbon tax to achieve a 15% emission cut varies widely across 23 countries:
  - Range: less than $10 to $60 a ton.
  - Interquartile range (IQR) for required tax: $23.65 to $38.25.
- Dispersion in NPVC costs is large:
  - NPVC IQR: -.6% to -.1% (reported in text as -.6% to -.1%).
- Importance of vintage-upgrade margin varies by country:
  - Vintage upgrades are more impactful in countries with more energy-intensive industries and technology mixes.
  - Countries with older capital stocks see fewer firms upgrading and smaller reductions in subsidy costs from the upgrade margin.
- Subsidies vs. intangible-investment subsidies:
  - Intangible-investment subsidies (boost productivity without green bias) lead to large long-run consumption losses: -7.1 percent (Table 6: Intangible Subsidy, Long-run consumption: -7.10).
  - Intangible subsidies can reduce emissions only through reducing output (e.g., by increasing share of workers becoming scientists) given elastic energy supply; they lack the green bias of vintage upgrades.
- Policy design must account for country-specific industry and firm-level heterogeneity to accurately calibrate mitigation instruments and assess trade-offs.

### Model and equilibrium features (selected)
- The paper uses a multi-sector heterogeneous-firm general equilibrium model featuring:
  - Two key firm margins: upgrades to newer vintages of capital and intangible investments.
  - Firm-level decisions interact with entry/exit and aggregate markets.
- Market clearing and equilibrium definition highlights:
  - Final goods markets clear for all sectors and countries.
  - Capital goods markets clear across vintages: Σ_{i∈S_s} x_{i v_s} = z_{v_s} ℓ_{v_s} for each vintage v_s.
  - Bonds market clears B_t = 0 and labor market clearing condition S Σ_{s=1} Σ_{i∈S_s} (ℓ^a_{si} + ℓ_{si}) + S Σ_{s=1} Σ_{v_s∈V_s} ℓ_{v_s} = L.
  - An equilibrium consists of value functions U,V^s, policy rules p^s, k', a', ℓ^s, n^s, a mass function M^s, capital good prices {q^v_s}, and a wage w satisfying household optimization, firm problems, capital goods firms’ profit maximization, and consistency of M^s with entry and exit.

### Core conclusions
- Technological factors—especially adoption of newer vintages of physical capital—play a crucial role in explaining persistent firm-level gaps in environmental performance.
- Accounting for endogenous firm technological choices, particularly the vintage-upgrade margin, significantly lowers estimated macroeconomic costs of carbon taxes and changes relative policy rankings.
- Subsidies for the newest capital vintage are costly in transition but can yield substantial long-run gains due to the green bias of capital-embedded technologies; their attractiveness depends on social discounting and country-specific initial conditions.
- Policymaking should incorporate empirically grounded firm heterogeneity, the vintage-upgrade margin, and country-specific technological distributions to evaluate carbon pricing and subsidy instruments effectively.

*Source: wpiea2023242-print-pdf - section 4—lowers the costs of carbon taxes by an economically meaningful amount. To; https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023242-print-pdf.pdf*

### 5.  The final goods (20), bonds, capital goods (21), and labor markets clear (22).

### 5.  The final goods (20), bonds, capital goods (21), and labor markets clear (22).

### A1.3 Model Solution
- Presents the main equations of interest, including the optimal variable input choices, pricing, and dynamic decisions (capital, vintage, and R&D).

### Firm’s choice of variable inputs
- Operating costs function:
  - C(ℓ,n)=wℓ+mn+τ_e n^v^(−ε_v) (23)
- First order conditions (μ is the Lagrange multiplier associated with the production technology):
  - λ_s μ y / ℓ = w (24)
  - η_s μ y / n = m+τ_e e n (25)
- Combination:
  - η_s / λ_s ℓ = (m+τ_e v^(−ε_v) / w) n
- Substituting into the production function yields:
  - y = A^ρ_s a (ℓ)^λ_s + η_s μ^(η_s λ_s) ( (w / (m+τ_e v^(−ε_v))) )^η_s (v k_v)^κ_s
- Solutions for ℓ and n:
  - ℓ = Γ^ℓ_{s v} y^{1/(η_s+λ_s)}
  - n = Γ^n_{s v} y^{1/(η_s+λ_s)}
  - where Γ^ℓ_{s v} and Γ^n_{s v} are functions of parameters and (v k_v)
- Definition for later:
  - ˜Γ^ℓ_{s v} = A^{−ρ/(η_s+λ_s)} s^{a−1/(λ_s+η_s)} μ^{λ_s/(η_s(λ_s+η_s))} ( (m+τ_e v^(−ε_v) / w) )^{η_s/(η_s+λ_s)} v^{−κ_s/(η_s+λ_s)}
  - Γ^ℓ_{s v} = ˜Γ^ℓ_{s v} (a k_v^{κ_v})^{−1/(λ_s+η_s)}
- Operating costs from (23):
  - C(ℓ,n)=wℓ [ μ (1+η_s/λ_s) ] = Γ_{s v} y^{1/(η_s+λ_s)} (26)
  - Γ_{s v} = Γ^ℓ_{s v} [ μ w μ (1+η_s/λ_s) ] (27)
  - ˜Γ_{s v} = ˜Γ^ℓ_{s v} [ μ w μ (1+η_s/λ_s) ] (28)
  - Γ_{s v} = ˜Γ_{s v} (a k_v^{κ_v})^{−1/(λ_s+η_s)}
- Operating profits:
  - p^{1−σ}_{si} ξ_s P^σ_s Y_s (1+τ_y) − Γ_{s v} [ p^{−σ}_{si} ξ_s P^σ_s Y_s ]^{1/(η_s+λ_s)} (47)
- First order condition for optimal price:
  - (σ−1) p^{−σ}_{si} ξ_s P^σ_s Y_s (1+τ_y) = Γ_{s v} σ/(η_s+λ_s) p^{−σ(η_s+λ_s)/(η_s+λ_s)−1}_{si} [ ξ_s P^σ_s Y_s ]^{1/(η_s+λ_s)} (29)
  - Rearranged:
    - p^{1−σ}_si + σ/(η_s+λ_s) si = Γ_{s v} (1+τ_y) σ/(σ−1)(η_s+λ_s) [ ξ_s P^σ_s Y_s ]^{1/(η_s+λ_s)−1} (30)
    - p_{si} = σ/(σ−1) Γ_{s v} (1+τ_y)(η_s+λ_s) y^{1/(η_s+λ_s)−1}_{si} (31)
- Constant markup interpretation:
  - p_{si} = σ/(σ−1) MC_{si}
  - MC_{si} = Γ_{s v} (1+τ_y)(η_s+λ_s) y^{1/(η_s+λ_s)−1}_{si}
- Equilibrium firm size y_{si} (derivation steps preserved in source):
  - y_{si} = [ μ^{σ/(σ−1)} Γ_{s v} (1+τ_y)(η_s+λ_s) 1/P_s ]^{−1} (ξ_{si} Y_s)^{1/σ} ... (full expression preserved in source)
- Firm value in output terms:
  - p_{si} y_{si} = σ/(σ−1) Γ_{s v} (1+τ_y)(η_s+λ_s) y^{1/(η_s+λ_s)}_{si}
- Implied optimal levels of labor and energy:
  - ℓ = Λ_s ξ Aˆρ_s [ μ^{λ_s}/w ]ˆ{λ_s+1} [ μ^{η_s} (m+τ_e v^(−ε_v)) ]ˆ{η_s} (v k_v)ˆκ_s
  - n = Λ_s ξ Aˆρ_s [ μ^{λ_s}/w ]ˆ{λ_s} [ μ^{η_s} (m+τ_e v^(−ε_v)) ]ˆ{η_s+1} (v k_v)ˆκ_s
  - Λ_{s ξ} = [ σ−1 σ (1+τ_y) P_s (ξ_{si} Y_s)^{1/σ} ]^{σ/(σ+(1−σ)(η_s+λ_s))}

### Optimal Capital and Research Intensity
- Operating profits and related expressions:
  - π_{si}+κ_s = p y(1+τ_y) − Γ_{s v} y^{1/(η_s+λ_s)} (32)
  - = p y(1+τ_y) [ 1 − (σ−1)/σ (λ_s+η_s) ] (33)
  - = μ^{σ/(σ−1)} 1/(η_s+λ_s) Γ_{s v} y^{1/(η_s+λ_s)}_{si} (34)
  - = Ω_{s v ξ} (a k_v^{κ_s})^{−1/(η_s+λ_s) + 1/(λ_s+η_s)} σ/(σ+(1−σ)(η_s+λ_s)) (35–37)
  - Ω_{s v ξ} = [ σ−1 σ (1+τ_y)(λ_s+η_s) P_s (ξ_{si} Y_s)^{1/σ} ]^{σ/(σ+(1−σ)(η_s+λ_s))} μ^{σ/(σ−1)} 1/(η_s+λ_s) ˜Γ^{(1−σ)(η_s+λ_s)/(σ+(1−σ)(η_s+λ_s))}_{s v}
- Assumption:
  - σ/(σ−1) > η_s+λ_s
- Firm dynamic problem (profit function and laws of motion):
  - v(a,k,v,γ,ξ)=max_{ℓ_a,x_v',v',a',k'} { Ω_{s v ξ} kˆκ_s_v a^{1/σ σ−1 −(η_s+λ_s)} − X w q_w x_w − w(1−τ_a) ℓ_a + 1/(1+r) v(a',k'_v',v',γ,ξ) } (38)
  - k'_{v'} = (1−δ_k) k_v' + x_v' (39)
  - a' = (1−δ_a) a + μ_{ℓ_a} γ^{α_s} (40)
- FOCs and envelope conditions (μ_k, μ_a denote Lagrange multipliers for capital and knowledge laws of motion):
  - q_v = μ_k (41)
  - w(1−τ_a) = μ_a α ℓ_a^{α−1} γ^{−α} (42)
  - 1/(1+r_v) v_k(a',k',v,γ) = μ_k (43)
  - 1/(1+r_v) v_a(a',k',v,γ) = μ_a (44)
  - v_k(a,k,v,γ) = Ω_{s v ξ} ˆκ_s k^{ˆκ_s −1}_v a^{1/σ σ−1 −(η_s+λ_s)} + μ_k (1−δ_k) (45)
  - v_a(a,k,v,γ) = Ω_{s v ξ} ˆκ_s k^{ˆκ_s}_v a^{1/σ σ−1 −(η_s+λ_s) −1} + μ_a (1−δ_a) (46)
- Resulting condition (steady-state):
  - Ω_{s v ξ} ˆκ_s k^{ˆκ_s −1}_v a^{1/σ σ−1 −(η_s+λ_s)} + q_v (1−δ_k) = (1+r) q_v (47)
- Closed-form for k_v:
  - k_v = [ Ω_{s v ξ} ˆκ_s a^{1/σ σ−1 −(η_s+λ_s)} / (q_v (r+δ_k)) ]^{1/(1−ˆκ_s)} (48)
- Knowledge and labor conditions:
  - Ω_{s v ξ} ˆα_s k^{ˆκ_s}_v ℓ^{ˆα_s −1}_a γ^{ˆα_s} ˆδ_a + w(1−τ_a)(1−δ_a) = (1+r) w(1−τ_a) (49)
  - ℓ_a = [ Ω_{s v ξ} ˆα_s k^{ˆκ_s}_v / (w(1−τ_a)(r+δ_a) γ^{ˆα} ˆδ_a) ]^{1/(1−ˆα_s)} (50)
  - where ˆδ_a = δ^{1/σ σ−1 −(η+λ) −1}_a and steady-state expression a = (ℓ_a γ)^{α/δ_a}
- Relation between ℓ_a and k_v:
  - ˆκ_s ˆα_s ℓ_a k_v^{δ_a} = (r+δ_k) q_v / (r+δ_a) w(1−τ_a)
  - ⇐⇒ w ℓ_a = (r+δ_k) δ_a q_v k_v / (r+δ_a)(1−τ_a) α_s κ_s (51)
- Equating (51) with previous yields closed-form for k_v:
  - k_v = [ κ_s (r+δ_a) w(1−τ_a) / α_s (r+δ_k) δ_a q_v ]^{1−ˆα_s)/(1−ˆκ_s−ˆα_s)} [ Ω_{s v ξ} ˆα_s / w(1−τ_a)(r+δ_a) γ^{ˆα} ˆδ_a ]^{1/(1−ˆκ_s−ˆα_s)} (52)

### Vintage decision
- Firms can upgrade to a better vintage of capital; decision compares value of upgrading to value of continuing with old vintage.
- Steady-state value of an incumbent that keeps its old vintage:
  - v(a,k,v,γ,ξ) = 1/(r(1+r)) Ω_{s v ξ} k^{ˆκ_s}_v a^{1/σ σ−1 −(η_s+λ_s)} − 1/r q_v δ_k k_v − 1/r w(1−τ_a) ℓ_a
  - = 1/r q_v [ 1/(1+r) (r+δ_k) ˆκ_s − δ_k − (r+δ_k) δ_a/(r+δ_a) α_s κ_s ] k_v (53)
  - = 1/r q_v − ˆκ_s /(1−ˆκ_s−ˆα_s) v [ ... ] × [ κ_s (r+δ_a) w(1−τ_a) / α_s (r+δ_k) δ_a q_v ]^{1−ˆα_s)/(1−ˆκ_s−ˆα_s)} [ Ω_{s v}(v,γ,ξ,τ_e) ˆα_s w(1−τ_a)(r+δ_a) γ^{ˆα} ˆδ_a ]^{1/(1−ˆκ_s−ˆα_s)}
- Value of upgrading to the best available vintage V_s (firms retire old vintage and recover only a fraction of its value):
  - v_up(a,k_v,v,γ,ξ) = q^{−ˆκ_s/(1−ˆκ_s−ˆα_s)}_v V × { 1/r [ 1/(1+r) (r+δ_k) ˆκ_s − δ_k − (r+δ_k) δ_a/(r+δ_a) α_s κ_s ] }^{−1} × [ κ_s (r+δ_a) w(1−τ_a) / α_s (r+δ_k) δ_a ]^{1−ˆα_s)/(1−ˆκ_s−ˆα_s)} [ Ω_{s v}(v,γ,ξ,τ_e) ˆα_s w(1−τ_a)(r+δ_a) γ^{ˆα} ˆδ_a ]^{1/(1−ˆκ_s−ˆα_s)} + χ q_v k_v (1−δ_k) (54)
- Decision rule:
  - A firm upgrades iff v_up(a,k_v,v,γ,ξ) > v(a,k,v,γ,ξ).

### Lemma 2 (Vintage decision)
- Statement: Assume that the elasticity of z_v to v is strictly higher than −1.
  1. If a firm updates its vintage, it upgrades to the best vintage V_s.

*Italic source attribution: wpiea2023242-print-pdf - 5.  The final goods (20), bonds, capital goods (21), and labor markets clear (22).*

### 2.  There exists a unique v

### 2.  There exists a unique v

### Existence and monotonicity of the upgrade threshold
- There exists a unique v_s such that a firm upgrades to V_s if and only if v_s < v_s.
- Sufficient condition for monotonicity: last value function is increasing in v. After simplification, the first component of the value function is proportional to (v/q_v)^(−ˆκ_s/(1−ˆκ_s−ˆα_s)). Since ˆκ>0 by assumption, a sufficient condition for the value function to be increasing in v is that v/q_v increases in v.
- Interpretation: the elasticity of z_v to v must be strictly higher than −1 for the monotonicity condition to hold.
- Second sufficient condition (for the threshold property): the difference between the value function when upgrading and not upgrading must be decreasing in v. The difference decomposes into two terms:
  - The first term is independent of the current vintage v.
  - The second term is strictly increasing in v under the same assumption that v/q_v is strictly increasing in v.
- Conclusion: if v/q_v is strictly increasing in v, the difference is strictly decreasing in v, yielding a well-defined upgrade threshold.
- Additional note: the decision to upgrade is independent of γ and ξ.

### Consistency with stylized facts (energy intensity and TFP)
- Energy intensity is decreasing with vintage and research efficiency (conditional on asset/size). Derived expressions:
  - n_y = 1/(m+τ_e v^(−ε_v)) η_s/(η_s+λ_s) C_y = 1/(m+τ_e v^(−ε_v)) η_s/(η_s+λ_s) Γ_s v_y^(1/(η_s+λ_s) − 1) s_i = 1/(m+τ_e v^(−ε_v)) η_s/(η_s+λ_s) × {Ω_s v ξ σ/(σ−1) [1/(λ_s+η_s−1)]}^{1−η_s−λ_s} Γ^{η_s+λ_s}_s v (a_k κ_s v)^{1−1/(σ+(1−σ)(λ+η))} = ... (expression chain as in source)
  - Used optimal share of intangibles over tangibles given by equation (55): a^{1/α_s} q_v/k_v = ℓ_a γ δ^{1/α_s} and the expression for operating costs C = Γ_s v ξ y^{1/(η_s+λ_s)} s_i.
  - Assumed τ_a = 0 in derivations.
- Log-linear expression for energy intensity:
  - ln n_y = ln c_s − ζ_v1 ln v + ζ_v2 ln(m+τ_e v^(−ε_v)) + ζ_γ ln γ − ζ_k ln k + ζ_ξ ln ξ_si
  - Coefficients:
    - ζ_v1 = (κ_s + α_s ε_{q v})/[σ+(1−σ)(η_s+λ_s)]
    - ζ_v2 = η_s/[σ+(1−σ)(η_s+λ_s)] − 1
    - ζ_γ = α_s/[σ+(1−σ)(η_s+λ_s)]
    - ζ_ξ = (1−η_s−λ_s)/[σ+(1−σ)(η_s+λ_s)]
    - ζ_k = (α_s + κ_s)/[σ+(1−σ)(λ_s+η_s)]
  - By assumption: ln e_n = −ε_v ln v. Combining yields the first result of the proposition.
- Total factor productivity (TFP) result:
  - From production function, pricing decision, and operating cost expressions (26) and (27), with TFP estimated from revenues:
  - p_y = TFP × C^{1−1/σ (λ_s+η_s)} × k^{κ_s(1−1/σ)} v^{...}
  - Log TFP expression:
    - ln TFP = (μ[1−1/σ]) [ln a + κ_s ln v + ρ ln A_s] + (1/σ) ln ξ_i + ln P_s + (1/σ) ln Y_s + (μ[1−1/σ])·[λ_s ln(μ λ_s/(λ_s+η_s)) + η_s ln(μ η_s/(λ_s+η_s)) − η_s ln(m+τ_e v^(−ε_v)) − λ_s ln w]
  - Substituting a from equation (55):
    - ln TFP = (μ[1−1/σ]) α_s ln(1/γ) + (μ[1−1/σ]) α_s ln k + (μ[1−1/σ]) κ_s ln v + (1/σ) ln ξ_i − (μ[1−1/σ]) η_s ln(m+τ_e v^(−ε_v)) + ln z_{s t}
  - Implication: firms with newer vintages and higher knowledge intensity have higher TFP.

### Calibration and estimation: internal calibration of non firm-specific parameters
- Sector definition: 2-digit SIC industry.
- Calibration steps and identifying moments:
  - β_s (elasticity of utility to consumption in each sector) calibrated to match sectoral share of sales: β_s = (Σ_{i∈Ω_s} p_i y_i) / (Σ_s Σ_{i∈Ω_s} p_i y_i). Shares reweighted to match coarser sectoral shares.
  - κ_s (elasticity of output to capital) identified using average mark-up via Sales = (σ/(σ−1)) COGS/(η_s+λ_s). With CRS, κ_s = 1 − λ_s − η_s − α_s.
  - Estimate κ_s + α_s = 1 − (σ/(σ−1)) Σ_{i∈Ω_s} COGS_i / Σ_{i∈Ω_s} Sales_i (σ externally calibrated).
  - η_s and λ_s from factor cost shares:
    - η_s = (1 − κ_s − α_s) × A/(A+B)
    - λ_s = (1 − κ_s − α_s) × B/(A+B)
    - A = expenditure share of COGS on energy; B = expenditure share of COGS on labor and other variable inputs.
  - α_s from first-order conditions for research (51) and capital (47), and operating profits (34):
    - α_s = (δ_a + r)/[δ_a(η_s+λ_s)] × (R&D spending)/COGS.
  - κ_s finally from κ_s = 1 − α_s − λ_s − η_s.
- Knowledge spillover parameter ρ: set ρ_s = α_s, consistent with literature (Griliches (1992), Bloom et al. (2013)).
- Productivity of vintages grows at constant rate g_v with relationship:
  - v(age of capital_i) = v_0 (1+g_v)^{−age of capital_i} (equation (56)).
  - g_v estimated by regressing firms’ productivity (TFPR) on age of capital and other controls; estimated g_v = 11.9% growth rate in the productivity of capital goods every year.
- Elasticity of emissions to capital vintage ε_v:
  - Start from equation (5): log(Emission/Energy)_i = log(φ_s) − ε_v log v.
  - Regression with sector, country, and time fixed effects:
    - log(Emission_{i t}/Energy_{i t}) = b_s + b_j + b_t + b_age × Age of Capital_{i t} + ε_{s j t i} (equation (57)).
  - Compute ε_v = ˆb_age / log(1+g_v) and φ_s = exp(b_s + b_j).

### Calibration and estimation: internal calibration of firm-specific variables
- Distribution of entrants G(γ,ξ):
  - Estimate pairs (γ_i, ξ_i) for each firm within a sector.
  - Define sample space {γ_i}_{i∈Ω_s} × {ξ_i}_{i∈Ω_s}.
  - Assume uniform probability over this set: G(γ,ξ) ∼ U( {γ_i} × {ξ_i} ).
- Cost of operating a firm κ_f: common to all firms in a country, calibrated so least profitable firm is indifferent between staying and exiting.
- Cost of starting a business ˆκ_e: common to all firms in a sector, calibrated so potential entrant is indifferent:
  - ˆκ_e = E_{ˆG_s}[ V_{s i}(k_V^s, V_s, γ, ξ) ] where expectation uses estimated G_s and V_{s i}(., ., ., .) from solving the model.
- Firm-level calibration of state variables v, γ, ξ:
  - Research-efficiency γ calibrated to match ratio of intangibles over tangibles:
    - γ = 1 / (Ratio of Intangibles over Tangibles) × (r+δ_k)(r+δ_a) δ^{1/α_s − 1} a^{α_s} w κ_s  (equation (58) and repeated in A2.3 and A2.4 proofs).
  - Vintage productivity v from age of capital: v = (1+g_v)^{−age of capital_i}.
  - Consumer taste ξ recovered from relative firm size and other parameters:
    - ln(ξ_{si}/ξ_{s j}) = ln(COGS_i/COGS_j) − α_s(σ_s−1) ln(γ_j/γ_i) + κ_s(σ_s−1) ln(1+g_v) (age of capital_i − age of capital_j) (equation (59)).
  - Normalize ξ so mean across firms within each sector is 1.

### Empirical/regression estimates and calibration outcomes
- Estimated g_v = 11.9% (growth rate in productivity of capital goods every year).
- Elasticity of emissions to capital vintage estimation (Table A1):
  - Regression: log(S1S2 / Energy) on Age of capital.
  - Coefficient on Age of capital = 0.04** (standard error 0.02).
  - N = 5,467.
  - R^2 = 0.00; Adj-R^2 = −0.02.
  - Industry + country + year fixed effects included.
- Table A2 (alternative industry classifications) key coefficient estimates for dependent variable log(emissions/energy):
  - Age of capital coefficients across specifications:
    - 0.10*** (SIC4), 0.10*** (SIC3), 0.10*** (SIC2), 0.03** (GICS4), 0.03** (GICS2), 0.09*** (NAICS4), 0.07*** (NAICS2), 0.14*** (HP).
    - Standard errors reported in parentheses (e.g., 0.02, 0.01, 0.04).
  - Share intangibles coefficients vary (some insignificant, some positive in specific specifications).
  - TFP coefficients vary in sign and significance across specifications.
  - Log(assets) coefficients vary; examples: −0.07** (SIC4), −0.07** (SIC3), −0.06** (SIC2), 0.02 (GICS4), 0.03** (GICS2), −0.05** (NAICS4), −0.03 (NAICS2), −0.08* (HP).
  - Sample sizes and R^2 values differ by specification (e.g., N = 2,690 for several SIC/GICS/NAICS variants; R^2 ranges from 0.29 to 0.51).
- Figure and table notes:
  - Industry classification: SIC-2 used for sector grouping in many analyses.
  - Sample coverage figures and kernel densities described in Figures A1–A6 (sample counts, firm size distributions, emissions by industry, heterogeneity in emission intensity, counterfactuals for age of capital and share of intangible capital). (Figures and panels referenced as in source.)

*Source: wpiea2023242-print-pdf — 2.  There exists a unique v (IMF working paper content unit).*

### 500. All variables are standardized to have mean zero and standard deviation of one. All variables are standard-

### Mitigating Climate Change at the Firm Level: Mind the Laggards — Selected Appendix Tables and Figures (excerpt)

### Regression evidence: Emission intensity and firm characteristics
- Dependent variable: Emission Intensity (Log emissions / revenue(t-1)) or Emissions over Energy (Log emissions / energy). All variables standardized to mean zero and standard deviation of one.
- Common controls and design:
  - Industry×country×year fixed effects included in reported specifications.
  - Finance, public administration, and utilities sectors excluded from calculations.
  - Standard errors clustered at country×industry×year (tables A3–A9) except where noted otherwise.
- Key coefficient estimates (selected, exact values as reported):
  - Age (age of capital / age of capital stock):
    - Table A3: Age 0.04*** (columns 1–2).
    - Table A4: Age 0.05*** (columns 1–2).
    - Table A5 (robustness): Age 0.05**, 0.04*, 0.05***, 0.03**, 0.07***, 0.04***, 0.04***, 0.02* (across columns).
    - Table A9 (firm-level clustering): Age of capital 0.05**, 0.03, 0.10***, 0.10*** (columns 1–4 and 5–8).
    - Table A11 (OLS/2SLS using 5-yr growth rate assets): OLS Age of capital 0.0198*; First stage coefficient on 5-yr growth rate assets -1.336***; 2SLS Age of capital 0.112***; 2SLS coefficient on 5-yr growth rate assets -0.149***. N 4,262; R2 reported across regressions 0.849, 0.629, 0.849, 0.086.
    - Table A12 (using 5-yr growth rate sales): OLS Age of capital 0.0203*; First stage on 5-yr growth rate sales -1.401***; 2SLS Age of capital 0.135***; 2SLS coefficient on 5-yr growth rate sales -0.189***. N 4,254; R2 0.849, 0.627, 0.850, 0.068.
  - Share intangibles:
    - Table A3: Log(RD / assets) -0.07***; Share intangibles entries in robustness tables reported as negative and significant: e.g., Table A5: -0.16***, -0.13***, -0.23***, -0.20***, -0.17***, -0.14***, -0.15***, -0.12*** (across columns).
    - Table A8 (using alternative intensity definitions): Share intangibles -0.19***, -0.16***, -0.35***, -0.24*** (columns shown).
  - Productivity / TFPR / TFP:
    - Table A3/Table A4: Log(EBIT/assets) -0.05*** (Table A3); TFP/TFPR negative and significant across many specifications: Table A5 TFP -0.15***, -0.10***, -0.17***, -0.11***, -0.18***, -0.14***, -0.15***, -0.11***.
    - Table A8 TFPR -0.14***, -0.09***, -0.51***, -0.43*** (columns reported).
  - Firm size (Log(assets)):
    - Table A3: Log(assets) 0.03*, 0.03***, 0.04***, 0.01 (columns 1–4).
    - Table A4: Log(assets) -0.13***, -0.11***, -0.10***, -0.13*** (columns 1–4) for Emissions over Energy.
    - Table A5: Log(assets) reported with varying magnitudes depending on sample: e.g., 0.30***, 0.45***, 0.34***, 0.46*** and smaller positive coefficients 0.05***, 0.09***, 0.13***, 0.13***, 0.08***, 0.11***, 0.15***, 0.15*** across panels.
    - Table A9 (Emissions over Energy columns): Log(assets) entries negative and insignificant in some columns: -0.07, -0.06, -0.06, -0.07 (columns 5–8).
- Fit and sample sizes (examples):
  - Table A3: N 6,204; R2 0.79–0.80; Adj-R2 0.68.
  - Table A4: N 2,566; R2 0.52; Adj-R2 0.23.
  - Table A5: N ranges 2,515; 3,774; 3,541; 6,092 across blocks; R2 up to 0.85; Adj-R2 up to 0.77.
  - Table A8: N reduced from 13,950 to 6,537 (columns 1–4) and from 12,308 to 5,617 (columns 5–8); R2 ~0.79–0.82; Adj-R2 ~0.68–0.72.

### Heterogeneity: Advanced Economies (AEs) vs Emerging Markets and Developing Economies (EMDEs)
- Table A7 highlights differences by country group:
  - Age of capital:
    - AEs: 0.04***, 0.02* (columns for advanced economies).
    - EMDEs: 0.18***, 0.17*** (columns for emerging markets).
  - Share intangibles:
    - AEs: -0.16***, -0.13***.
    - EMDEs: -0.22***, -0.14***.
  - TFP:
    - AEs: -0.18***, -0.15***.
    - EMDEs: -0.32***, -0.27***.
  - Log(assets) reported across AEs and EMDEs with exact coefficients for each column (e.g., AE columns: 0.04***, 0.08***, 0.11***, 0.13***; EMDE columns: 0.02, 0.07, 0.22***, 0.22***).
  - Sample sizes: N reported (e.g., 6,050; 5,876; 5,988; 5,791 for AE columns; 799; 766; 795; 743 for EMDE columns).
  - R2 and Adj-R2 reported per column (R2 ~0.79–0.81; Adj-R2 ~0.63–0.70).

### Management practices and emission intensity
- Model: Residualized log emission intensities regressed on residualized overall management score and residualized log assets (Equation 61 described).
- Table A10 estimates:
  - Management score coefficient -0.465*** (column 1) and -0.532*** (column 2).
  - STD log(assets) coefficient 0.107 (column 2; not significant at conventional levels as no stars).
  - N 92 in each column.
  - R2 0.191 and 0.193; Adj-R2 0.182 and 0.175.
- Notes: Management data limited — World Management Survey data available for a small subset (23 US manufacturing firms noted in footnote), and residualizing procedures applied as described.

### Robustness and alternative specifications
- Robustness checks reported across Tables A5–A9 and A11–A12:
  - Variants include: largest 50% of firms, adding financial controls (lagged liquidity, leverage, capitalization ratios, market share), removing the Covid period (defined differently across tables: one note defines Covid period as 2020 and afterwards; another defines as 2022 and afterwards), restricting to Scope 1 emissions, using age of capital, using alternative dependent variables (Log emissions / assets(t-1), Log emissions / value added(t-1)), and clustering at firm level.
  - Results indicate consistent negative associations of share intangibles and productivity measures (TFP/TFPR) with emission intensity, and a generally positive association of age of capital with emission intensity across many specifications.
  - Instrumental variable evidence (Tables A11–A12) uses 5-year growth rates of assets or sales as instruments for age of capital with strong first-stage F-statistics reported (e.g., first-stage F-statistic 368.06 in Table A11; 357.42 in Table A12).

### Counterfactual policy simulations (figures, descriptive summaries)
- Figures explore country-level distributions and cross-country heterogeneity when calibrating policy instruments to achieve a 15% reduction in emissions relative to baseline:
  - Figure A7: Histograms show the number of countries requiring given policy values (US dollars or percentage points) for Carbon tax, Feebate, New vintage subsidy, and Intangible subsidy to reach a 15% emissions reduction target.
  - Figure A8: Binscatter of cross-country average capital elasticity of output against Change output (in pp) after a capital subsidy calibrated for 15% emissions reduction.
  - Figure A9: Comparison of net present value of consumption (NPVC) costs for a carbon tax and a capital subsidy under varying planner discount rates, given a market discount rate of 4%. X-axis: Discount rates (in pp); Y-axis: Change NPVC (in pp).
  - Figure A10: Scatter comparing average energy intensity and changes in NPVC across countries between simulations allowing firms to upgrade capital vintages versus those that do not; axes labeled Upgrade vs no upgrade (in pp) and Energy intensity.
  - Figure A11 (panels a–d): Scatter plots relating average age of capital stock across countries to:
    - (a) Upgrade Share (in percentage points),
    - (b) Long-Run Consumption (in percentage points) relative to actual economy,
    - (c) Transition Costs (in percentage points),
    - (d) Net Present Value of Consumption: Upgrade vs no upgrade (in percentage points).
- Notes accompanying figures: each policy calibrated to achieve 15% emissions reduction; figures illustrate cross-country heterogeneity in required policy values, macroeconomic impacts, and welfare measures.

*Mitigating Climate Change at the Firm Level: Mind the Laggards — Working Paper No. WP/2023/242*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023242-print-pdf.pdf_