Optimal Fiscal Policy in a Climate-Economy Model with Heterogeneous Households

Layer 1: Overview

This paper asks whether inequality and redistributive taxation should make climate policy more or less ambitious, and how optimal carbon taxes interact with optimal income taxes when households differ in productivity, wealth, and energy demand. The motivation is twofold: equity considerations belong at the center of normative climate analysis, and the distributional consequences of environmental policies are increasingly recognized as critical for their political feasibility — as illustrated by the Yellow Vests episode in France. The paper extends Barrage (2020)’s representative-agent dynamic climate-Ramsey model to a heterogeneous-agent setting, using the Werning (2007) technique to characterize the Ramsey optimum in terms of aggregate variables. The government maximizes utilitarian social welfare choosing linear taxes on labor income, capital income, energy, and pollution plus a uniform lump-sum transfer. The climate module is calibrated to DICE 2016 (Nordhaus, 2017). Household heterogeneity is calibrated to US data: ten productivity groups from SCF 2013 hourly wages ranging from $6.44 (bottom decile) to $101.35 (top decile), yielding a model consumption Gini of 0.33, very close to the empirical value of 0.32 (Heathcote et al., 2010). Tax rates are set at effective US rates from Trabandt and Uhlig (2012): capital income tax of 41.1% and labor income tax of 25.5%. The model period is five years beginning in 2015, and the discount factor follows DICE at beta = 1/(1.015) per year, with inverse IES sigma = 1.45. The main quantitative exercise compares optimal policy to a climate-skeptic planner who sets carbon taxes to zero. Key findings: (i) Tax distortions have a negligible effect on the optimal carbon tax in the heterogeneous-agent setting. The second-best carbon tax is initially only 0.5% below the social cost of carbon (SCC) and subsequently fluctuates within about 0.2% above or below it — in sharp contrast to Barrage (2020), who finds tax distortions reduce optimal carbon taxes by 8% in the representative-agent setting. The key mechanism is that, with heterogeneous agents, the government optimally levies distortionary taxes for redistributive purposes (not merely to finance public spending), so the marginal cost of public funds (MCF) averages to 1 over time and its temporal deviations are quantitatively trivial. (ii) Income inequality only slightly reduces the optimal carbon tax: residual consumption inequality after optimal income-tax redistribution lowers the SCC by 3.9% in the baseline. The mechanism is that inequality raises the average marginal utility of consumption (because the marginal utility function is convex), increasing the opportunity cost of abatement; this effect dominates when IES < 1 (sigma > 1 in the calibration). (iii) The optimal carbon tax path starts at $21.7/tCO2 in 2020 and reaches $229.2/tCO2 one century later — levels consistent with Barrage (2020) and Nordhaus (2017/2018) but insufficient to achieve the Paris +2°C target under baseline damages. (iv) Comparing optimal policy to the climate-skeptic baseline, the additional carbon tax revenue is split nearly equally: the present value of labor taxes falls by 0.7% of GDP, while transfers rise by 0.8% of GDP. This violates the weak double-dividend hypothesis, which prescribes using carbon tax revenue exclusively to cut distortionary taxes. (v) The optimal policy has progressive welfare effects in the 21st century, because increased tax progressivity benefits lower-income households. The average discounted welfare gain is 5.8% of consumption under baseline damages. In the long run, gains become regressive because richer households (with IES < 1) are willing to pay proportionally more in consumption to avoid temperature increases. By contrast, a representative-agent double-dividend policy — using all carbon revenue to cut labor taxes — is regressive from the outset, with low-income households bearing a net cost even in the short run. The 3.9% inequality effect on the SCC is robust to changes in fiscal pressure and damage calibration but is sensitive to sigma: with sigma = 2, inequality reduces optimal carbon taxes by 16.2% rather than 3.9%. Extensions with wealth heterogeneity, heterogeneous energy demand (calibrated to CEX), and heterogeneous environmental damage sensitivity confirm that the MCF remains negligible and the inequality effect on carbon taxes remains small in quantitative terms.

Layer 2: Deep Dive

What is the core theoretical result on the optimal carbon tax and why does it differ from Barrage (2020)?

The optimal carbon tax is approximately Pigouvian — set equal to the social cost of carbon — because the MCF averages to 1 over time with balanced-growth preferences when households are heterogeneous and the government can optimize a uniform lump-sum transfer. In Barrage (2020)’s representative-agent model, the government cannot choose the level of lump-sum taxes or transfers because there is no redistribution motive, so distortionary taxes are the only way to finance public spending and the MCF exceeds 1, reducing optimal carbon taxes by 8%. With heterogeneous agents, the government optimally provides lump-sum transfers for redistribution, so the constraint on transfers is barely binding and the MCF is close to 1 even when the ability to adjust transfers is removed.

What is the mechanism by which inequality affects the optimal carbon tax, and what is the sign?

Inequality reduces the optimal carbon tax when IES < 1 (sigma > 1). The mechanism operates through the Pigouvian tax formula: pollution abatement reduces aggregate consumption, and the welfare cost of this reduction depends on the social marginal utility of consumption (Vc,t). With inequality, Vc,t is affected by two opposing forces. First, the average marginal utility of consumption is higher because of Jensen’s inequality (convex marginal utility function), increasing the opportunity cost of abatement and pushing the pollution tax down. Second, additional consumption goes disproportionately to richer households with lower marginal utilities, reducing Vc,t and pushing the tax up. When IES < 1, the first (higher average marginal utility) effect dominates, so inequality unambiguously reduces the SCC and hence the optimal pollution tax. When IES = 1, the two effects exactly cancel and inequality has no effect.

What is the MCF and why does it average to 1 in the heterogeneous-agent setting?

The MCF is defined as the ratio of the public (planner’s Lagrange multiplier on the resource constraint) to the private (aggregate welfare-weighted) marginal utility of consumption. It measures the social cost of transferring resources from the private to the public sector. The MCF averages to 1 because the first-order condition for the uniform lump-sum transfer implies that the sum of the Lagrange multipliers on agents’ implementability constraints is zero. With balanced-growth preferences, this implies the welfare-weighted average MCF equals 1 from period 0. The temporal covariance between type-specific shadow costs (theta_i) and the type-specific implementability term (I_{c,i,t}) averages to zero over time, so while the MCF can deviate temporarily from 1, it is 1 on average.

What is the double-dividend hypothesis and how does the paper’s optimal policy relate to it?

The weak double-dividend hypothesis holds that it is optimal to use carbon tax revenue exclusively to reduce distortionary taxes, yielding both environmental and efficiency dividends. The paper shows this does not hold with heterogeneous agents: at the optimum, the welfare gain from a marginal reduction in tax distortions equals the welfare loss from increased inequality, so the government splits carbon revenue between cutting distortionary taxes and increasing redistribution. In the baseline quantification, the split is roughly equal: present-value labor taxes fall by 0.7% of GDP and lump-sum transfers rise by 0.8% of GDP. By contrast, following the double-dividend prescription — using all carbon revenue to reduce labor taxes without raising transfers — generates a strongly regressive policy in which low-income households bear net welfare costs even in the short run.

What is the calibration strategy and how does the model match US inequality data?

The economic side is calibrated to the US, while the climate side uses DICE 2016. The discount factor follows DICE (beta = 1/(1.015) per year), and sigma = 1.45 (IES = 1/1.45). Household productivity is calibrated using SCF 2013 hourly wage deciles, yielding ten equal-sized groups with hourly wages from $6.44 (bottom) to $101.35 (top), normalized so that the productivity-weighted average is 1. Although productivity inequality is directly targeted rather than moments of the consumption distribution, the model correctly predicts the consumption Gini of 0.33, close to the empirical 0.32 (Heathcote et al., 2010). Capital and labor income tax rates are from Trabandt and Uhlig (2012): 41.1% and 25.5% respectively. Government debt-to-GDP is approximately 111% (average 2011-2015, IMF). The Frisch elasticity of labor supply is targeted at 0.75 (Chetty et al., 2011). Production in both sectors is Cobb-Douglas with energy share nu = 0.04 from Golosov et al. (2014).

What happens to optimal income taxes in the model?

The optimal labor income tax roughly doubles from its calibrated level of 25% to about 50% in the first period and stabilizes there. Revenue from these taxes is rebated via the uniform lump-sum transfer, achieving most of the desired redistribution. Because optimal labor income taxes are approximately constant over time, the associated intertemporal distortions are small, and the optimal capital income tax converges to zero quickly after the second period. The mechanism is that, with access to lump-sum transfers, the only reason to tax capital income is to mitigate intertemporal distortions created by labor income taxation; when labor taxes are roughly constant, this motive is weak.

What does the sensitivity analysis reveal about the robustness of the 3.9% inequality effect?

The effect of inequality on optimal carbon taxes is robust along several dimensions but sensitive to sigma. Under the high-damage scenario (cubic rather than quadratic damage function, yielding an SCC about four times larger), the inequality effect falls to 2.6% rather than 3.9%, because higher carbon taxes reduce warming and thus the share of utility (rather than production) damages. The effect is roughly proportional to the degree of productivity inequality: half the inequality implies about half the effect on the carbon tax. The effect changes more than proportionally with sigma: with sigma = 2 (IES = 0.5), inequality reduces carbon taxes by 16.2%, versus 3.9% with the DICE value of sigma = 1.45. With sigma = 1, the effect is exactly zero. Government expenditure levels and fiscal pressure have negligible effects on the results. The share of damages entering utility directly matters: if only 10% of damages affect utility directly (versus the baseline 26%), the inequality effect falls to 1.8%; if 40% affect utility directly, it rises to 5.2%.

What is the role of initial wealth inequality?

Initial wealth inequality (studied in Section 6.1) creates an additional motive for deviating from Pigouvian taxation in period 0 only. Because the planner cannot use the period-0 capital tax to expropriate initial wealth (it is fixed at 41.1%), higher damages would reduce interest rates and thereby partially mitigate wealth inequality (a subtle indirect redistribution mechanism), calling for lower pollution taxes in period 0. Quantitatively, this produces a significant reduction in the initial-period optimal carbon tax. However, from period 1 onward, the optimal tax rules are unaffected by initial wealth heterogeneity, and the effects of MCF and income inequality remain very similar to the baseline. Welfare gains from carbon taxation in the wealth-heterogeneity extension are U-shaped with income but strictly increasing in initial wealth.

How does energy-demand heterogeneity (Stone-Geary extension) affect the results?

The extension introduces a second dirty consumption good with Stone-Geary preferences, calibrated using CEX data to match the average energy expenditure share of 10.8% and the observed distribution of energy budget shares across and within income groups. Target emissions share from household energy consumption is 30%. The optimal pollution tax formula remains a modified Pigouvian rule (the MCF structure is unchanged), and the MCF effect remains negligible. The inequality effect on carbon taxes stays near 3.9%, rising marginally to 4.1% with identical energy necessity and 4.1% with heterogeneous energy necessity. Theoretically, the optimal excise tax on the energy good is zero when energy preferences are homogeneous; with heterogeneous necessity levels calibrated to the US, the optimal energy excise tax is quantitatively tiny: about -0.4% of energy prices (a small subsidy). The negative sign arises because within-income-group heterogeneity in energy needs means that energy-intensive households (who are valued more by the planner on average) can be partially targeted via a subsidy. Under the double-dividend scenario with energy inequality, regressive effects are magnified: the poorest, most energy-intensive households actually lose in welfare terms even accounting for long-run climate mitigation benefits.

What does the paper establish theoretically about heterogeneous environmental damages?

Proposition 6 (Section 6.3) shows that with additively separable environmental utility and a utilitarian planner, heterogeneous marginal utility damages from pollution have no effect on the optimal pollution tax: they enter the welfare criterion symmetrically and cancel in the aggregate. The pollution tax increases relative to the utilitarian benchmark only if the planner’s welfare weights are positively correlated with marginal utility damages — that is, if the planner cares relatively more about the households that are more exposed. A Rawlsian planner would set a higher pollution tax if and only if the least-well-off household is also more sensitive to environmental degradation.

What are third-best policy results when either income tax is fixed?

The paper analyzes policies where either the labor or capital income tax is fixed at its current calibrated level (studied in Appendix E, with results referenced in the main text). These constraints introduce an additional fiscal interaction effect on the optimal carbon tax — the carbon tax is pushed below its second-best Pigouvian level when the fixed tax is set at a sub-optimally low level, and above it when the fixed tax is sub-optimally high. The roles of the MCF and income inequality remain similar to the second-best baseline under these third-best constraints.

How does the paper relate to and differ from the double-dividend and pollution taxation literatures?

The paper builds on three earlier pillars. First, Pigou (1920) established first-best Pigouvian taxation. Second, a large literature (Sandmo, 1975; Bovenberg and de Mooij, 1994; Bovenberg and Goulder, 1996) showed that in representative-agent second-best settings the MCF exceeds 1 and optimal pollution taxes fall below the Pigouvian level. Barrage (2020) is the closest dynamic general-equilibrium predecessor, finding the 8% reduction from tax distortions. Third, Jacobs and de Mooij (2015) and Jacobs and van der Ploeg (2019) showed in static models with heterogeneous agents and a uniform lump-sum transfer that the MCF equals 1. This paper extends this insight to a fully dynamic climate-economy framework with general equilibrium and a rich model of household heterogeneity. The key innovation relative to Barrage (2020) is agent heterogeneity, which both provides microfoundations for distortionary taxation and significantly changes the quantitative implications for optimal carbon taxes. Relative to Jacobs and de Mooij (2015), the contribution is the dynamic setting, the linkage to the DICE climate module, and the full quantitative characterization including distributional welfare analysis and multiple sources of heterogeneity.

What are the policy implications and their scope conditions?

The primary policy implication is that a carbon tax should be set approximately equal to the SCC (Pigouvian level) and the associated revenue should be split roughly equally between increasing lump-sum transfers and reducing distortionary labor taxes — rather than following the double-dividend prescription of using all revenue to reduce distortionary taxes. This combination is both more efficient (the MCF argument) and more equitable (progressive in the short run). The scope conditions are: (a) the result applies under a utilitarian welfare criterion with linear income taxes and a uniform lump-sum transfer; (b) it requires that the government can optimize the level of lump-sum transfers for redistribution; (c) the approximately Pigouvian result is quantitatively robust to alternative damage functions, fiscal pressure, and energy demand heterogeneity, but the degree to which inequality lowers the carbon tax depends sensitively on the IES/inequality aversion parameter sigma; (d) the calibration is designed to capture US conditions assuming that the US internalizes the full global impact of its emissions (strategic considerations are abstracted away); (e) heterogeneous environmental damage sensitivity does not affect the utilitarian optimum, but would increase the optimal carbon tax under a more inequality-averse social planner.

Key Concepts

Marginal Cost of Public Funds (MCF): The ratio of the public (planner’s shadow price on the resource constraint) to the private (aggregate welfare-weighted) marginal utility of consumption. In this paper, it captures the divergence between second-best and first-best pollution taxes due to fiscal distortions. With heterogeneous agents and an optimized uniform lump-sum transfer, the MCF averages to 1 over time under balanced-growth preferences, implying that tax distortions do not systematically push the carbon tax below the Pigouvian level — unlike in the representative-agent setting where the MCF exceeds 1.

Pigouvian tax (second-best): In this paper’s context, the Pigouvian tax refers to the pollution tax equal to the social cost of pollution (the discounted present value of marginal production and utility damages), evaluated at the second-best allocation rather than the first-best. When the MCF equals 1 (as it approximately does in the heterogeneous-agent setting), the second-best optimal pollution tax is equal to this second-best Pigouvian level, which may itself differ from the first-best Pigouvian level due to residual consumption inequality.

Social Cost of Carbon (SCC): The present discounted value of marginal climate damages (both production and utility losses) from emitting one additional ton of CO2, converted into consumption units using the social marginal utility of consumption. In the paper, the SCC corresponds to the case where the MCF is set to 1 in every period, and it is affected by consumption inequality through its effect on the social marginal utility of consumption. With sigma > 1, residual inequality raises the opportunity cost of abatement, reducing the SCC by 3.9% in the baseline calibration.

Double-dividend hypothesis (weak): The claim that it is optimal to use the entire proceeds of a carbon tax to reduce existing distortionary taxes, yielding both an environmental dividend (less pollution) and an efficiency dividend (lower tax distortions). The paper shows this does not hold with heterogeneous agents: because distortionary taxes serve a redistributive purpose, reducing them at the margin has a welfare cost (increased inequality), so the planner optimally splits revenue between tax reduction and increased transfers.

Ramsey problem (climate-economy): The government’s optimization problem in this paper: maximizing utilitarian social welfare over an infinite horizon by choosing paths for linear taxes on labor income, capital income, energy, and pollution, plus a uniform lump-sum transfer, subject to households’ optimality conditions (implementability constraints), resource constraints, climate dynamics from DICE, and abatement technology constraints. The approach extends Werning (2007) to a dynamic climate-economy context.

Implementability condition: The constraint in the Ramsey problem that captures each household’s lifetime budget constraint in terms of aggregate variables and market weights. It requires that the present value of a household’s consumption minus labor income equals its initial assets plus its share of the present value of lump-sum transfers, evaluated using the social marginal utilities implied by the planner’s choice of taxes. The shadow cost of this constraint for each household type (theta_i) determines the MCF through its covariance with a fiscal externality term.

Residual inequality: The level of inequality that remains after the planner has optimally set all income taxes and the lump-sum transfer — i.e., the inequality that cannot be eliminated because individualized lump-sum transfers are not feasible and only linear instruments are available. In the paper, it is this residual inequality (not total inequality) that affects the optimal carbon tax: the carbon tax responds to the inequality that income-tax policy cannot address, not to the underlying productivity or wealth dispersion per se.

Balanced-growth preferences: A preference specification of the form u(c, h, Z) = [c(1 - varsigma*h)^gamma]^(1-sigma)/(1-sigma) + u_hat(Z), with 1/sigma the intertemporal elasticity of substitution. This specification ensures that the economy admits a balanced growth path and plays a key role in the paper’s theoretical results: under balanced-growth preferences, the welfare-weighted average MCF equals 1 from period 0, and when IES = 1 (sigma = 1) the MCF is exactly 1 in every period.