1. Introduction
Limiting global warming in the long run requires a substantial reduction in emissions in the coming years (IPCC, 2022). Taxation is a key instrument for mitigation because it forces agents to internalize their impact on the environment through market prices. It also provides a fiscal revenue which can be used to invest in the green transition, or to compensate losers from such policies, low income households among others.Footnote 1 While high-income households pollute more (Chancel, Reference Chancel2022), lower-income ones are the most impacted by climate policies as they spend a higher share of their disposable income on polluting goods and services (Grainger and Kolstad, Reference Grainger and Kolstad2010). Because climate and inequality issues are closely linked, it has been emphasized that climate policies should aim to reduce emissions while simultaneously reducing the burden on the most vulnerable individuals.Footnote 2 In the case of a regressive environmental policy, reducing emissions works against inequality reduction: this is the so-called equity-efficiency tradeoff. Understanding the distributional effect of environmental tax reforms is thus key when trying to reduce both emissions and inequality at the same time, while making environmental taxes socially acceptable. Recycling the tax revenue can be used to reconcile both goals.
This paper aims at theoretically analyzing the impact of environmental taxation and revenue recycling on pollution and inequalities when households have a level of subsistence consumption for pollution. Rather than studying an optimal climate policy, we focus on the impact of an increase in polluting commodity taxation when it is coupled with targeted redistribution. More precisely, we study whether pollution mitigation and inequality reduction are compatible in a dynamic setting with two goods, a green and a polluting one, and non-homothetic preferences.
Although wealth inequality has been attributed to disparities in labor earnings in the literature, models based solely on unequal earnings fail to replicate the wealth concentration observed in the data (De Nardi and Fella, Reference De Nardi and Fella2017). This inequality partly arises from differences in asset portfolios across groups. Among others, Kuhn et al. (Reference Kuhn, Schularick and Steins2020) show that in the U.S., only the wealthiest (top 10%) hold significant financial assets alongside a house and a small mortgage. Blanchet and Martinez-Toledano (Reference Blanchet and Martinez-Toledano2023) find a similar evidence. Saez and Zucman (Reference Saez and Zucman2016) also show that wealth is held by a small fraction of rich people. Kaplan et al. (Reference Kaplan, Violante and Weidner2014) document that around one third of people are hand-to-mouth, even if two thirds of them are wealthy. These different works show that a relevant representation of the population is to consider that only a share holds financial assets, while the rest of the population is financially constrained. A relevant way to consider such a distribution of assets and wealth is to introduce heterogeneous discount factors, which induces financial constraints. Indeed, as it is shown by Epper et al. (Reference Epper, Fehr, Fehr-Duda, Kreiner, Lassen, Leth-Petersen and Rasmussen2020), the relationship between wealth inequality and time discounting highlights that the wealthiest are the more patient. Heterogeneity in discount factors allows for differences in propensity-to-save and hence to recover hand-to-mouth agents who only receive a labor income and lump-sum transfers.
Based on this evidence, we build a two-sector Ramsey model with heterogeneous households and a subsistence level of consumption for the polluting good. Pollution negatively affects households’ utility function. A tax is set on polluting consumption, and its revenue is given back to households through individualized lump-sum transfers. There is no income tax in the economy: the only tax is polluting consumption. At the equilibrium, households spend a constant share of their expenditures on each good, augmented by the subsistence level of consumption for the polluting commodity. Heterogeneity in discount factors leads to inequality: the most patient household holds all the capital in a neighborhood of the steady state.
We analyze the impact of redistribution and environmental taxation on polluting consumption and inequality in the steady state. Agents have identical expenditure shares formalized by the same aggregator homogeneous of degree one over the two goods, net of the possible subsistence level. Therefore, decreasing income inequality through lump-sum transfers has no impact on aggregate polluting consumption: redistribution only reduces inequality, and the tax on polluting consumption is required to reduce pollution.Footnote 3 At the aggregate level, increasing the tax rate reduces pollution and increases clean consumption. However, the individual impact depends on the level of polluting subsistence consumption and the redistribution rate. Redistributing enough to workers pushes up their consumption of clean good, as well as polluting consumption provided that the level of subsistence consumption is sufficiently high. Two mechanisms are at play: a price effect and a redistribution effect. The tax increases the relative price of polluting consumption, and redistribution counteracts this negative effect by increasing disposable income. The strength of the redistribution effect depends on the redistribution rate and the subsistence level of consumption. If the redistribution effect is strong enough, it offsets the price effect, allowing an increase in polluting consumption for one household even if overall emissions are falling. Hence, the environmental tax reform might reduce pollution and inequality simultaneously. The welfare impact of the reform also depends on the environmental externality on utility. If households are highly affected by environmental quality, the positive effect of lower emissions offsets any potential loss in consumption and welfare improves for all.
Finally, we study the dynamics of the model using a local analysis. Due to some market imperfections, the steady state can lose its saddle-path stability. Identifying the conditions under which this occurs is crucial. If the steady state is locally indeterminate, fluctuations due to self-fulfilling prophecies can emerge. In this case, there is no convergence to the steady state and there is a cost associated to fluctuations. If the steady state is unstable, the dynamic path of the economy is explosive and may be unsustainable as soon as one deviates from the steady state. Both situations are undesirable. We study in detail how preferences and policy parameters affect the emergence of instability and indeterminacy. Focusing on the neighborhood of the steady state, we find that the level of subsistence consumption and the externality play a key role. A high level of subsistence consumption favors indeterminacy and instability, while a high environmental externality promotes stability. The environmental fiscal policy also matters: a higher tax rate and greater redistribution to workers extend the intervals in which the steady state is unstable or indeterminate. This suggests that the taxation and redistribution of polluting good must be treated with caution when taking into account a subsistence level of polluting consumption, both for the effects of environmental tax reform and for the stability of the steady state. While a low subsistence of polluting consumption allows to reduce pollution and leads to a stable steady state, a high level jeopardizes the stability of the economy.
Our paper is linked to three strands of the literature. First, part of the environmental literature analyzes the link between redistribution and pollution. Empirically, mixed or no effects are found, as for example in Lin and Li (Reference Lin and Li2011). Berthe and Elie (Reference Berthe and Elie2015) gather this literature and explain the differences in models, both theoretically and empirically, leading to this ambiguous effect. From a macroeconomic point of view, Oueslati (Reference Oueslati2015) shows that in a two-sector endogenous growth model, lump-sum transfers have no impact on aggregate variables, and hence on pollution. Rausch and Schwarz (Reference Rausch and Schwarz2016) find that heterogeneity and non-homotheticity of preferences matter when it comes to the impact of redistribution on aggregate variables. Along the literature, we find that there is no effect of redistribution on pollution under Stone-Geary preferences. Yet, redistribution has a role to play in the impact of taxation on individual consumption and welfare through the impact on disposable income, as well as on stability properties of the economy.
A second strand of interest studies the link between environmental taxation and inequality. Scalera (Reference Scalera1996) and Hofkes (Reference Hofkes2001) look at the long-term effect of environmental taxation in endogenous growth models. Part of this literature focuses on taxation and revenue recycling, and emphasizes the role of recycling schemes in the progressivity of environmental taxes. Klenert and Mattauch (Reference Klenert and Mattauch2016) analyze the importance of subsistence consumption in the distributional effect of environmental tax reforms. Klenert et al. (Reference Klenert, Schwerhoff, Edenhofer and Mattauch2018) show the importance of lump-sum transfers in order to reduce inequality. Chiroleu-Assouline and Fodha (Reference Chiroleu-Assouline and Fodha2014); Fried et al. (Reference Fried, Novan and Peterman2021); Eydam and Diluiso (Reference Eydam and Diluiso2022) analyze the welfare impact of redistribution schemes to environmental taxes, and emphasize the role of tax income rebates. Our paper focuses on individualized lump-sum transfers. More broadly, this paper falls within the literature on environmental taxation with heterogeneous households (Cremer et al. Reference Cremer, Gahvari and Ladoux2003; Jacobs and van der Ploeg, Reference Jacobs and van der Ploeg2019; Cronin et al. Reference Cronin, Fullerton and Sexton2019; Goulder et al. Reference Goulder, Hafstead, Kim and Long2019; Känzig, Reference Känzig2021; Benmir and Roman, Reference Benmir and Roman2022; Douenne et al. Reference Douenne, Hummel and Pedroni2024; van der Ploeg et al. Reference van der Ploeg, Rezai and Reaños2025). While part if this literature studies the normative side on environmental taxation, we focus on the positive side of it, i.e. its impact on pollution and inequality. Like the literature on taxation finding a positive relationship between higher taxation and lower pollution (Bovenberg and de Mooij, Reference Bovenberg and de Mooij1997; Bosquet, Reference Bosquet2000; Li et al. Reference Li, Lin, Du, Feng and Zuo2021; Wolde-Rufael and Mulat-Weldemeskel, Reference Wolde-Rufael and Mulat-Weldemeskel2023), we find that the tax reduces emissions. This suggests that the level of subsistence consumption has no impact on the effectiveness of an environmental tax reform at the aggregate level. However, this level matters for individual consumption patterns. Inequality is reduced as long as the government redistributes more towards workers.
Finally, our paper is linked to the literature concerned by the stability properties of models with an environmental component. Part of the literature analyzes the role of pollution on stability properties of OLG economies (Seegmuller and Verchère, Reference Seegmuller and Verchère2004, Reference Seegmuller and Verchère2007; Cao et al. Reference Cao, Wang and Wang2011; Raffin and Seegmuller, Reference Raffin and Seegmuller2017). Koskela and Puhakka (Reference Koskela and Puhakka2006) show the possibility of two-period bifurcations in an OLG model with Stone-Geary preferences. Antoci et al. (Reference Antoci, Galeotti and Russu2005) show that, in a growth model with two goods, consumption choices can lead to indeterminacy. Itaya (Reference Itaya2008) analyzes the impact of environmental taxation on long-run growth in a model with a representative agent and an environmental externality. He finds that the impact of taxation on growth depends on the indeterminacy of the balanced growth path. Close to our paper, Raffin and Seegmuller (Reference Raffin and Seegmuller2017) show that the fiscal policy aimed at reducing pollution has an impact on stability properties of the economy depending on how does pollution affect longevity. We analyze the impact of the environmental fiscal policy when households have a certain level of subsistence consumption for the polluting good. To the best of our knowledge, our paper is the first analyzing the stability properties of a Ramsey economy with both externalities and non-homothetic preferences. We find that non-homothetic preferences can lead to instability and indeterminacy, while the externality restores stability. The fiscal policy plays a key role in the occurrence of instability and indeterminacy. Increasing the tax rate promotes instability while redistributing more towards workers promotes the occurrence of endogenous cycles.
The rest of the paper is organized as follows. In Section 2, we present our framework. Sections 3 and 4 define the equilibrium and the steady state of the economy. Section 5 studies the impact of a change in taxation and in redistribution of the tax revenue on pollution and welfare. In Section 6, we analyze the local stability properties of our economy. Section 7 provides concluding remarks. All proofs are relegated to the Appendix.
2. The model
We consider a infinite-horizon two-sector model with an environmental externality. The economy is composed of households, firms and a government. Households are infinitely lived and have heterogeneous discount factors. Firms produce either a clean or a polluting good consumed by the households. Government intervention, through taxation and redistribution, aims to reduce environmental damage as well as inequalities in income and consumption.
2.1 Firms
There are 2 sectors (
$j = p,g$
) in the economy, each composed of a representative firm. The clean sector produces output
$Y_{gt}$
and the polluting sector produces output
$Y_{pt}$
at each period
$t$
. Each sector uses capital
$K_{jt}$
and labor
$N_{jt}$
to produce the good
$j$
using a Cobb-Douglas technology:
\begin{gather} Y_{jt} = A_jK_{jt}^\eta N_{jt}^{1-\eta }, \end{gather}
with
$A_j\gt 0$
a productivity parameter,
$A_g \neq A_p$
, and
$\eta \in (0,1)$
.
In the polluting sector, output is a pure consumption good,
$c_{pt}$
, say energy. The clean sector produces a final good that is either consumed,
$c_{gt}$
, or invested through capital. This distinction is in line with the literature on environmental taxation (Chiroleu-Assouline and Fodha, Reference Chiroleu-Assouline and Fodha2014; Klenert et al. Reference Klenert, Schwerhoff, Edenhofer and Mattauch2018; Douenne et al. 2023). Capital refers here to immaterial and non-polluting inputs used in the production process, such as R&D investment and human capital.
In each sector (
$j = p,g$
), a representative firm maximizes profits:
\begin{equation*} \pi _{jt} = p_{jt} Y_{jt} -r_t K_{jt} - w_t N_{jt} \end{equation*}
with
$p_{gt} = p_t$
,
$p_{pt} = 1$
,
$r_t$
the rental rate of capital and
$w_t$
the wage rate.
Assuming that capital and labor are perfectly mobile across sectors, first-order conditions for profit maximization give:
\begin{align} r_t = \eta p_t A_g K_{gt}^{\eta -1} N_{gt}^{1-\eta } = \eta A_p K_{pt}^{\eta -1} N_{pt}^{1-\eta } \end{align}
\begin{align} w_t = (1-\eta ) p_t A_g K_{gt}^{\eta } N_{gt}^{-\eta } = (1-\eta ) A_p K_{pt}^{\eta } N_{pt}^{-\eta } \end{align}
2.2 Pollution
We consider pollution as a flow due to consumption of the polluting good. One can think for example of gases emitted such as sulfur dioxide and carbon monoxide, which have a short lifetime and hence can be considered as pollutant flows (Liu and Liptak, Reference Liu and Liptak2000).
Environmental quality decreases with these pollutant flows and is therefore given by:
\begin{gather} E_t = \frac {1}{\gamma c_{pt}} \end{gather}
with
$\gamma \gt 0$
the emission factor.Footnote
4
Environmental quality positively affects households’ utility, while it has no direct effect on the production in the two sectors. It allows us to account for the effect of the environmental policy on households’ welfare through the modification of environmental quality. This external effect of pollution on the utility will also affect the intertemporal choices of households and, therefore, the dynamics of the economy, especially the stability properties of the steady state.
2.3 Households
The economy is composed by two types of infinitely lived households (
$i = \{1,2\}$
). For simplification purposes, we assume there is one household of each type. At every period
$t$
, each household consumes a clean good
$c_{git}$
at price
$p_t$
and a polluting good
$c_{pit}$
which is the numeraire and taxed at a rate
$\tau$
. There exists a subsistence consumption level
$c_0$
of the polluting good, so that households always consume a positive amount of it, as in Ballard et al. (Reference Ballard, Goddeeris and Kim2005) and Klenert et al. (Reference Klenert, Schwerhoff, Edenhofer and Mattauch2018). This can be seen for instance as the minimum requirement of energy consumption (electricity, heating, gas). Households also derive utility from environmental quality
$E_t$
, which acts as an externality. Finally, they can invest
$a_{it+1}$
in the capital good (so that
$a_{1t}+a_{2t}=K_t$
), supply
$n_i=\frac {1}{2}$
of labor at the wage rate
$w_t$
, and receive a lump-sum transfer
$T_{it}$
from the government. The intertemporal utility of household
$i$
writes:
\begin{gather} \sum _{t=0}^\infty \beta _i^t E_t^\mu \frac {((c_{pit}-c_0)^{\alpha } c_{git}^{1-\alpha })^{1-\sigma }}{1-\sigma }, \end{gather}
with
$\sigma \in (0,1)$
the inverse of the elasticity of intertemporal substitution in consumption. Consumption can be summarized by a basket of good
$C_{it}$
, with
$C_{it} = (c_{pit}-c_0)^{\alpha } c_{git}^{1-\alpha }$
and
$\alpha \in (0,1)$
the share of consumption expenditures devoted to the polluting good. Pollution is taken as given by households as it plays the role of an externality in the utility function. We further note
$\beta _i \in (0,1)$
the discount factor of household i and we consider that:
Assumption 1.
$\beta _1 \gt \beta _2$
,
i.e. household 1 is more patient than household 2. This assumption takes into account the relationship between inequality and time discounting, highlighting that the wealthier are the most patient. As it is shown by Epper et al. (Reference Epper, Fehr, Fehr-Duda, Kreiner, Lassen, Leth-Petersen and Rasmussen2020), this positive association of patience and wealth provides a support to study models where heterogeneity in discounting explains wealth inequality. This is also a way to consider that only a share of the population holds financial assets, as it is argued in the introduction and is supported by empirical evidence (Kaplan et al. Reference Kaplan, Violante and Weidner2014; Saez and Zucman, Reference Saez and Zucman2016; Kuhn et al. Reference Kuhn, Schularick and Steins2020; Blanchet and Martinez-Toledano, Reference Blanchet and Martinez-Toledano2023).
The budget constraint at time
$t$
writes:
\begin{gather*} c_{pit}(1+\tau )+ c_{git} p_t+p_t(a_{it+1}-(1-\delta )a_{it}) = \frac {w_t}{2} + r_t a_{it}+ T_{it}. \end{gather*}
Households maximize their discounted lifetime utility facing, at each period, this budget constraint and the following borrowing constraint:
$a_{it+1} \geq 0$
. We deduce the first-order conditions:
\begin{gather} c_{git} = \frac {1-\alpha }{p_t}(P_{it}C_{it}-c_0(1+\tau )) \end{gather}
\begin{gather} c_{pit} = \frac {\alpha }{1+\tau }P_{it}C_{it}+c_0(1-\alpha ) \end{gather}
\begin{gather} \frac {C_{it}^{-\sigma }}{P_{it}}E_t^\mu p_t \geq \beta _i \frac {C_{it+1}^{-\sigma }}{P_{it+1}}E_{t+1}^\mu (p_{t+1}(1-\delta )+r_{t+1}) \end{gather}
\begin{gather} P_{it} = \frac {(1+\tau )^{\alpha } p_t^{1-\alpha }}{(1-\alpha )^{1-\alpha } \alpha ^{\alpha }}+\frac {(1+\tau ) c_0}{C_{it}} = (1+\tau )^{\alpha } P_t +\frac {(1+\tau ) c_0}{C_{it}} \end{gather}
with (8) holding as an equality when
$a_{it}\gt 0$
.
$P_{it}$
is the price of the bundle
$C_{it}$
for each household. Because households consume a positive minimal amount of the polluting good, this aggregate price depends on individual consumption and hence on households’ types.
Equations (6) and (7) allow to rewrite the budget constraint of household
$i$
as:
\begin{gather} P_{it} C_{it} +p_t(a_{it+1}-(1-\delta )a_{it}) = \frac {w_t}{2} + r_t a_{it}+ T_{it} \end{gather}
Since we have instantaneous utility functions which are Cobb-Douglas and homogeneous of degree one, we note, from (6) and (7), that agents spend a constant share of their consumption expenditures in each good, augmented by the subsistence consumption level for the polluting commodity and reduced by it for the clean good. Equation (8) states that households either smooth consumption using their savings or are prevented from borrowing and are financially constrained.
2.4 The Government
The government levies a tax on the polluting good. The tax revenue can be used to reduce income inequality: a share
$\varepsilon \in [0,1]$
is redistributed to household 1 and the rest to household 2. The government faces a balanced budget rule:
\begin{gather} T_t = T_{1t}+T_{2t} = \tau (c_{p1t}+c_{p2t}) = \tau c_{pt}, \end{gather}
with
$T_{1t} = \varepsilon T_t$
and
$T_{2t} = (1-\varepsilon ) T_t$
transfers given to households 1 and 2 respectively.
3. Equilibrium
Because we assume that household 1 is more patient than household 2, we prove in the next section that at the steady state, the most patient household holds all the capital and smooths consumption according to a binding Euler equation. On the contrary, the impatient household does not smooth consumption and holds no asset, so that
$a_{1t} = K_t \gt 0 = a_{2t}$
. This is a standard result in the literature (Becker, Reference Becker1980; Becker and Foias, Reference Becker and Foias1987; Sorger, Reference Sorger1994), supported by empirical evidence (Kaplan et al. Reference Kaplan, Violante and Weidner2014; Kuhn et al. Reference Kuhn, Schularick and Steins2020; Blanchet and Martinez-Toledano, Reference Blanchet and Martinez-Toledano2023). We focus on equilibria around the steady state, i.e. in which this result holds.
Using equations (2) and(3), and market-clearing conditions, we obtain:
\begin{align} K_t = \frac {K_{pt}}{N_{pt}} = \frac {K_{gt}}{N_{gt}} \end{align}
\begin{align} K_t = K_{pt}+K_{gt} \end{align}
\begin{align} N_{pt}+N_{gt} = 1 \end{align}
\begin{align} p_t \equiv p = \frac {A_p}{A_g} \end{align}
\begin{align} \end{align}
Substituting (12) and (15) in (2) and (3) yields:
\begin{align} r_t = \eta A_p K_t^{\eta -1} \end{align}
\begin{align} w_t = (1-\eta ) A_p K_t^{\eta } \end{align}
Substituting now (17) and (18) into (8) and (10), and using the market-clearing condition on the capital market for each household gives:
\begin{align} \frac {C_{1t}^{-\sigma }}{P_{1t}}E_t^\mu = \beta _1 \frac {C_{1t+1}^{-\sigma }}{P_{1t+1}} E_{t+1}^\mu \left ( 1-\delta + \frac {r_{t+1}}{p} \right ) \end{align}
\begin{align} P_{1t} C_{1t} &=\left ( \frac { (1+\eta )}{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) A_p K_t^\eta -\left(1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right) p(K_{t+1}-(1-\delta ) K_t)\nonumber \\& \quad +\frac {2 \varepsilon \tau (1+\tau )(1-\alpha )}{1+\tau (1-\alpha )} c_0 \end{align}
\begin{gather} \frac {C_{2t}^{-\sigma }}{P_{2t}} E_t^\mu \gt \beta _2 \frac {C_{2t+1}^{-\sigma }}{P_{2t+1}} E_{t+1}^\mu \left ( 1-\delta + \frac {r_{t+1}}{p} \right ) \end{gather}
\begin{gather} P_{2t} C_{2t} = \frac { (1-\eta )(1+\tau )}{2(1+\tau (1-\alpha (1-\varepsilon )))} A_p K_t^\eta +(1-\varepsilon )\tau \left ( \frac {\alpha P_{1t}C_{1t}+2 c_0 (1+\tau )(1-\alpha )}{(1+\tau (1-\alpha (1-\varepsilon ))} \right ) \end{gather}
Equations (20) and (22), obtained with the budget constraints of each household, show that consumption expenditures
$P_{it}C_{it}$
depend on
$(i)$
capital,
$(ii)$
the subsistence level of consumption, and
$(iii)$
fiscal tools used by the government, the tax rate on polluting consumption and the redistribution rate. Equations (19) and (21) illustrate the intertemporal decisions of households. As we focus on equilibria around the steady state, the distribution of capital among households is given by
$\beta _i$
and
$1-\delta +r_{t+1}/p$
. Household 1 being the most patient, the ratio of her marginal utility of consumption today relative to tomorrow equals her return on savings. As household 2 is less patient, she favors the present more than the future and hence her marginal utility of consumption today will always be higher than her marginal utility of consumption tomorrow. Since both the rental rate of capital and the asset distribution are shaped by the difference in discount factors, introducing endogenous labor supply would not change the intertemporal arbitrage for both households as it is defined by discount factors around the steady state.Footnote
5
As we focus on equilibria around the steady state, we use equations (20) and (22) evaluated at the steady state and look for conditions on subsistence consumption under which we have
$C_{1t} \geq 0$
and
$C_{2t} \geq 0$
. In Appendix A.2 where the critical values
$c_{01}$
and
$c_{02}$
are defined, we prove that this is the case if the following assumption is satisfied:
Assumption 2.
$c_0 \leq c_{01}$
and
$c_0 \leq c_{02}$
for any
$\varepsilon \in [0,1]$
.
Households’ income depends in part on the redistribution rate
$\varepsilon$
. When very few of the tax revenue is given back towards one household, a too high level of subsistence consumption for the polluting good would make the constraint impossible to maintain as the minimal level of consumption expenditures would be too high compared to the disposable income.
Using
$E_t = \frac {1}{\gamma c_{pt}}$
and substituting (17), equations (19) and (20) give the dynamic equations of the model:
\begin{align} K_{t+1}= \frac {\left ( \frac {1+\eta }{2}+ \frac { \alpha \varepsilon \tau }{(1+\tau (1-\alpha ))} \right ) A_p K_t^\eta + \frac {2\varepsilon \tau (1+\tau )(1-\alpha )}{1+\tau (1-\alpha )}c_0- P_{1t} C_{1t}}{p \left ( 1+ \frac {\alpha \varepsilon \tau }{(1+\tau (1-\alpha ))}\right )} + (1-\delta ) K_t \end{align}
\begin{align} C_{1t+1}^{\sigma } &= \beta _1 \frac {C_{1t}^{\sigma }\left (P+\frac {(1+\tau ) c_0}{C_{1t}}\right )}{P+\frac {(1+\tau ) c_0}{C_{1t+1}}}\left (\frac {\alpha P C_{1t} + \frac {\alpha (1-\eta )}{2}A_pK_{t}^\eta +(1+\tau )(2-\alpha )c_0}{ \alpha P C_{1t+1} + \frac {\alpha (1-\eta )}{2}A_pK_{t+1}^\eta +(1+\tau )(2-\alpha )c_0}\right )^{\mu }\nonumber \\[5pt] &\quad \times \left ( 1-\delta + \eta A_g K_{t+1}^{\eta -1} \right ) \end{align}
Definition 1.
Under Assumptions
1
and
2
, an equilibrium of the economy is a sequence
$(K_t,C_{1t})$
such that equations (
23
) and (
24
) are satisfied, given
$K_0 \geq 0$
.Footnote
6
Equations (6), (7), (20), (22) and market-clearing conditions allow to recover aggregate consumption for both goods:
\begin{gather} c_{pt} = \sum _i c_{pit} =\frac {\alpha }{1+\tau (1-\alpha )} \left (A_p K_t^\eta - p(K_{t+1}-(1-\delta ) K_t) \right ) +\frac {2(1+\tau )(1-\alpha )}{1+\tau (1-\alpha )}c_0 \end{gather}
\begin{gather} c_{gt} = \sum _i c_{git} = \frac {(1-\alpha )(1+\tau )}{p(1+\tau (1-\alpha ))} \left (A_p K_t^\eta - p(K_{t+1}-(1-\delta ) K_t) -2 c_0\right ) \end{gather}
4. Steady state
In this section, we show the existence of a unique steady state in which the most patient household holds a positive amount of capital, while the impatient one does not save at all. In the following, steady state variables are denoted with a star.
The Euler equation for agent
$i$
writes
$1 \geq \beta _i (1-\delta +\frac {r}{p})$
, holding with equality if
$a_i \gt 0$
. As we assume
$\beta _1 \gt \beta _2$
, we obtain
$\beta _1 (1-\delta +\frac {r}{p}) \gt \beta _2 (1-\delta +\frac {r}{p})$
, which leads to
$a_1^* = K^* \gt 0 = a_2^*$
. Rewriting the left hand-side of the inequality gives
\begin{gather} \frac {r^*}{p} = \frac {1-\beta _1}{\beta _1}+\delta . \end{gather}
As we found that
$p$
is a constant, we have
$P_i^* =P(1+\tau )^{\alpha }+(1+\tau )\frac {c_0}{C_i^*}$
, with
$P = \frac { p^{(1-\alpha )}}{\alpha ^{\alpha } (1-\alpha )^{1-\alpha }}$
. Using (17) and (18), capital-labor ratios remain constant, with
$K^* = \frac {K_p^*}{N_p^*} = \frac {K_g^*}{N_g^*} = \left ( \frac {\eta A_g}{\frac {1}{\beta _1} - (1-\delta )} \right )^{\frac {1}{1-\eta } }$
, and the wage rate writes
$w^* = (1-\eta ) A_p \left ( \frac {\eta A_g}{\frac {1}{\beta _1} - (1-\delta )} \right )^{\frac {\eta }{1-\eta }}$
.
Writing (6) and (7) at the steady state gives the individual consumptions. Equations (20) and (22)–(26) allow to recover consumption expenditures for each household, as well as aggregate consumption in both sectors:
\begin{gather} P_1^*C_1^* = K^* p \left ( \frac {r^*/p}{\eta } \left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right ) - \delta \left ( 1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) \right )+\frac {2\varepsilon \tau (1+\tau )(1-\alpha )}{1+\tau (1-\alpha )}c_0 \end{gather}
\begin{align} P_2^*C_2^* = K^*p \left (\left ( \frac {r^*/p}{\eta }-\delta \right ) \frac {\alpha (1-\varepsilon ) \tau }{1+\tau (1-\alpha )} + \frac {r^*/p}{\eta }\frac {1-\eta }{2} \right )+\frac {2(1-\varepsilon )\tau (1+\tau )(1-\alpha )}{1+\tau (1-\alpha )}c_0 \end{align}
\begin{align} c_p^* = \frac { \alpha }{1+\tau (1-\alpha )} K^*p \left ( \frac {r^*/p}{\eta } -\delta \right )+\frac {2(1+\tau )(1-\alpha )}{1+\tau (1-\alpha )}c_0 \end{align}
\begin{align} c_g^* = \frac { (1-\alpha ) (1+\tau ) }{(1+\tau (1-\alpha ))} K^* \left ( \frac {r^*/p}{\eta } -\delta \right ) -\frac {2(1+\tau )(1-\alpha )}{p(1+\tau (1-\alpha ))}c_0 \end{align}
and environmental quality writes
$E = \frac {1}{\gamma c_p^*}$
. We further assume:
Assumption 3.
$c_0 \lt \frac { K^*p \left ( \frac {r^*/p}{\eta } -\delta \right )}{2}$
.
This ensures positive clean consumption.
Proposition 1.
Under Assumptions
1
–
3
, there exists a unique steady state in the economy
$(K^*,C_1^*)$
solution to (
23
) and (
24
), and characterized by equations (
28
)–(
31
). At the steady state, household 2 is constrained and household 1 holds all the capital.
From now on, we will call household 1 the capitalist and household 2 the worker.
5. Public policy
The government has two tools: individualized lump-sum transfers and polluting commodity taxation. This part aims at analyzing the impact of both tools on pollution and inequality. More precisely, we first study whether a higher tax rate on polluting consumption decreases emissions, as well as its impact on welfare and inequality. Second, we analyze the impact of redistribution on emissions and welfare.
5.1 Taxation
At the individual level, there are two effects at play when increasing the tax rate: a price effect and a redistribution effect. By increasing the consumer price for the polluting commodity, purchasing power of both households is lowered so that for the same income, they consume less. Increasing the consumer price also leads to a substitution between the two goods: agents increase their consumption in the clean good as its relative price decreases. Increasing the tax rate increases the government revenue, and hence the amount received through lump-sum transfers. Everything else equal, consumption expenditures increase with the rise in disposable income: this is the redistribution effect.
5.1.1 Effect on consumption and environmental quality
We first investigate the effect of an increase in
$\tau$
on aggregate consumptions,
$c_p^*$
and
$c_g^*$
.
Proposition 2. Under Assumptions 1 – 3 , an increase in polluting commodity taxation always leads to an increase in clean consumption and a decrease in polluting consumption.
Proof. see Appendix A.3.
The increase in clean consumption comes from the substitution effect, as an increase in taxation makes the clean good relatively cheaper than the polluting one. The decrease in aggregate polluting consumption comes from a price effect, as the tax makes the polluting good relatively more expensive. Indeed, aggregate capital is not affected by the tax rate. Therefore, the increase of the tax rate has a redistributive effect between clean and polluting aggregate consumptions. The level of subsistence consumption cannot modify the sign of the effects on the consumptions at the aggregate level. Yet, it is important for the impact of the tax rate at the individual level, as for the redistribution rate.
Regarding individual behaviors and using equation (6), a change in the tax rate on polluting consumption at the steady state gives:
\begin{gather*} \frac {d c_{gi}^*}{d \tau } = \frac {(1-\alpha )}{p} \left (\frac {d (P_i^*C_i^*)}{d \tau } -c_0 \right ) \end{gather*}
for any household
$i$
. This leads to the following proposition:
Proposition 3.
Under Assumptions
1
–
3
, there exist
$\varepsilon ^1 \in (0,1) \mbox{ and } \varepsilon ^2 \in (\varepsilon ^1,1)$
such that
$c_{g1}$
increases with the tax rate if
$\varepsilon \gt \varepsilon ^1$
and
$c_{g2}$
increases with the tax rate if
$\varepsilon \lt \varepsilon ^2$
. Clean consumption increases for both households whenever
$\varepsilon \in (\varepsilon ^1,\varepsilon ^2)$
.
Proof. see Appendix A.4.
Clean consumption increases at the individual level for each household if the redistribution rate is between
$\varepsilon ^1$
and
$\varepsilon ^2$
. If this is not the case, clean consumption increases for one household and decreases for the other. This result is quite intuitive: if the redistribution to the capitalist is very high (
$\varepsilon$
close to 1), the clean consumption of the capitalist increases, while the one of the worker decreases. In contrast, if the redistribution to the worker is very high (
$\varepsilon$
close to 0), the clean consumption of the capitalist decreases while the one of workers increases. This highlights the crucial role of redistribution in the effect of a higher tax rate at the individual level. However, as aggregate clean consumption increases, the increase in clean consumption of household
$i$
always offsets the decrease in clean consumption of household
$j$
.
The effects of a higher tax rate on individual polluting consumptions are summarized in the following proposition:
Proposition 4.
Let
$\varepsilon ^*= \frac {\frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta }{\frac {r^*/p}{\eta }-\delta }$
. Under Assumptions
1
–
3
, following an increase in
$\tau$
:
-
• For
$\varepsilon \lt \varepsilon ^*$
,
$c_{p1}^*$
always decreases.
$c_{p2}^*$
decreases for
$c_0 \lt \tilde {c_0^2}$
and increases otherwise;
-
• For
$ \varepsilon = \varepsilon ^*$
, both
$c_{p1}^*$
and
$c_{p2}^*$
decrease;
-
• For
$ \varepsilon \gt \varepsilon ^*$
,
$c_{p2}^*$
always decreases. The impact on
$c_{p1}^*$
is mixed for
$c_0 \lt \tilde {c_0^1}$
and depends on the level of
$\alpha$
, and is positive for
$c_0 \gt \tilde {c_0^1}$
.
Proof. see Appendix A.5.
From Proposition4, three variables are key when analyzing which effect dominates the other: the redistribution rate, the size of subsistence consumption and the spending share on the polluting good. These three variables play a role on the redistribution effect, as the transfer received depends on both
$c_0$
and
$\alpha$
, and by definition on
$\varepsilon$
. When
$\varepsilon \lt \varepsilon ^*$
, the increase in price for the capitalist is not offset by the transfer received:
$c_{p1}^*$
decreases. For the worker, the impact of an increase in the tax rate depends on the level of subsistence consumption. When it is low, the price increase dominates and consumption decreases. When it is high, the redistribution effect is pushed up and offsets the price increase: polluting consumption increases. When
$\varepsilon \gt \varepsilon ^*$
, the price effect dominates for the worker and their polluting consumption decreases. For the capitalist, the impact depends on the level of subsistence consumption, and the spending share on the polluting commodity. Whenever the level of subsistence consumption is relatively high, the redistribution effect is pushed up by both
$c_0$
and
$\varepsilon$
, and dominates the price effect. When the level of subsistence is very low, polluting consumption always increases if
$\alpha$
is high as it pushes up the redistribution effect. If
$\alpha$
is low, the impact on
$c_{p1}^*$
depends on the initial tax rate: it increases if it is low, and decreases otherwise, the price effect offsetting any effect of redistribution. The reverse occurs when
$c_0$
is a little bit higher but still lower than
$\tilde {c_0^1}$
.
Increasing the tax rate has also an impact on income inequality:
Proposition 5.
Under Assumptions
1
–
3
,
$\frac {\partial (P_2^*C_2^*)}{\partial \tau } \gt \frac {\partial (P_1^*C_1^*)}{\partial \tau }$
whenever
$\varepsilon \lt \frac {1}{2}$
.
Proof. see Appendix A.6.
Recall that
$P_i^*C_i^*$
is equal to disposable income net of capital holdings. At the steady state, labor and capital incomes depend only on the capital-labor ratio, which is independent of
$\tau$
(and so are savings). The only change in disposable income happens through redistribution: a change in consumption expenditures reflects a change in disposable income. Increasing the tax rate decreases inequality if and only if the government redistributes more to the worker.
5.1.2 Effect on welfare
Changing the commodity tax rate affects individual welfare through consumption and environmental quality:
\begin{gather} \frac {d U_i^*}{d \tau } = \mu \frac {d E^*}{d \tau }E^{*\mu -1}\frac {C_i^{*1-\sigma }}{1-\sigma }+\frac {d C_i}{d \tau }C_i^{*-\sigma } E^{*\mu } \end{gather}
The environmental effect on utility is always positive as increasing the tax rate decreases aggregate polluting consumption. The consumption effect depends on both the mixed impacts on green and polluting consumptions. If consumption
$C_i^*$
increases, welfare increases for household
$i$
. If it decreases, the impact on welfare depends on the size of the environmental externality,
$\mu$
.
Using (6) and (9), basket consumption can be rewritten as a function of clean consumption:
\begin{gather*} C_i^* = \frac {p}{(1-\alpha )(1+\tau )^\alpha P}c_{gi}^* \end{gather*}
The impact on raising the tax rate on overall consumption for each household hence depends on the negative impact on prices and the ambiguous impact on clean consumption. More precisely, the impact is given by:
\begin{gather} \frac { \partial C_i^*}{\partial \tau } = \frac {p}{(1-\alpha )(1+\tau )^\alpha P^*} \left ( \frac {\partial c_{gi}^*}{\partial \tau } - \alpha \frac {c_{gi}^*}{1+\tau }\right ) \end{gather}
which is positive when
$\epsilon _{c_{gi}} \equiv \frac {\partial c_{gi}^*}{\partial \tau } \frac {1+\tau }{c_{gi}^*} \gt \alpha$
. Consumption increases for household
$i$
whenever the cross-price elasticity of clean consumption is higher than the expenditure share on polluting consumption. This assumes that clean consumption increases with the tax rate. If not, consumption always decreases with the tax rate and the welfare impact of the environmental tax reform depends on
$\mu$
. If clean consumption increases along the tax rate, its reaction to the tax reform must be high enough for basket consumption to increase. Results are summarized as follows:
Proposition 6.
Under Assumptions
1
–
3
, there exist
$\mu _i^*$
(
$i=1,2$
) such that increasing the tax rate on the polluting commodity:
-
• Increases welfare for household
$i$
when
$\epsilon _{c_{gi}} \gt \alpha$
; -
• Increases welfare for household
$i$
when
$\epsilon _{c_{gi}} \lt \alpha$
and
$\mu \gt \mu _i^*$
.
When clean consumption increases and the share of polluting consumption in the utility is low enough, the welfare increases whatever the level of the externality. Using Proposition3, it is interesting to see that
$\epsilon _{c_{g1}}$
and
$\epsilon _{c_{g2}}$
can both be positive for
$\varepsilon \in (\varepsilon ^1,\varepsilon ^2)$
. In such a case, welfare might increase for all households. Otherwise, it will depend on the degrees of the externalities
$\mu _1^*$
and
$\mu _2^*$
.
When consumption decreases for household
$i$
, an increase in welfare is still possible as long as
$\mu$
is high enough, as the positive impact on environmental quality then offsets the negative impact on consumption: household
$i$
gains from the policy not by increasing their consumption, but from enjoying a higher environmental quality.
As long as
$\mu \gt \max (\mu _1^*,\mu _2^*)$
, welfare increases for both households when increasing the tax rate. The government can also play on the level of
$\varepsilon$
such that inequality is also reduced. In this case, the environmental tax reform allows to reduce pollution and inequality, as well as increasing welfare for both households. If
$\mu \lt \min (\mu _1^*,\mu _2^*)$
, increasing the tax rate might leave one household worse off through lower consumption and a relatively low effect of reducing pollution. When analyzing the welfare impact of such tax reform, the level of subsistence consumption, the redistribution rate and the externality intensity are of importance. In particular, if
$\varepsilon \gt \varepsilon ^2$
, a higher tax rate may increase the welfare inequality between capitalists and workers by increasing the gap between their consumptions of clean good.
Consumption taxation is hence an efficient tool to reduce emissions. Coupled with a specific redistribution rate, it is also effective in reducing inequalities at the individual level. Yet, assuming that only the environmental fiscal policy affects emissions and individual consumptions is quite restrictive. Adding more policy tools could shade results at the individual level by creating new tradeoffs. For example, the tax revenue could be partly used for public abatement. In this case, the price effect of the tax would be the same on individual consumption, while the redistribution effect would be lowered. On the welfare impact, this would modify both the environmental impact and the consumption impact of taxation.Footnote 7
5.2 Redistribution
We consider now the fiscal pressure (
$\tau$
) as given and study the effects of a redistribution of the tax revenue among the worker and capitalist.
5.2.1 Effect on consumption and emissions
The effect of redistribution through a change in the redistribution rate
$\varepsilon$
is straightforward. From equations (30)–(31), redistribution has no impact on aggregate consumptions
$c_p^*$
and
$c_g^*$
. Capital is given by the modified golden rule, so that redistribution has no impact on it either.
The only impact of
$\varepsilon$
is hence on individual variables, i.e on individual consumption. As consumption of both goods are functions of overall consumption, the only thing we have to look at is the impact of a change in redistribution on
$C_1^*$
and
$C_2^*$
respectively. From (28)–(29), increasing transfers given to the worker has a positive effect on her consumption, while it has a negative effect for the capitalist. This is a pure revenue effect: increasing redistribution towards the worker raises her disposable income, and hence her consumption in both goods proportionally, while it reduces the capitalist’s consumption. Yet, both impacts perfectly compensate each other so that aggregate consumption, and more broadly all aggregate variables, do not change, environmental damages included. Hence, it is possible for the government to reduce income and consumption inequalities by increasing redistribution towards the worker, while not harming environmental quality. Rausch and Schwarz (Reference Rausch and Schwarz2016) show that when preferences are homothetic and identical, aggregate behaviors are similar to a single-agent behavior. Thus, changing the redistribution pattern has no impact on aggregate consumption as it is similar to giving back everything to a single agent. Heterogeneous preferences should make it possible to have a redistribution effect. Having households with different marginal propensities to consume, i.e. heterogeneous
$\alpha$
’s, the impact on aggregate polluting consumption would depend on which household spends a higher expenditure share on this good, as redistribution effects on both households would not offset each other. Changing the redistribution rate hence would have an impact on pollution.
5.2.2 Effect on welfare
Knowing that changing the redistribution rate does not affect pollution, the impact on welfare is only done through consumption. More precisely:
\begin{gather} \frac {d U_i^*}{d \varepsilon } = \frac {d C_i}{d \varepsilon }C_i^{*-\sigma } E^{*\mu } \end{gather}
Redistributing more to the worker decreases consumption for the capitalist and increases consumption for the worker due to the revenue effect. Hence, decreasing income inequality through a lower
$\varepsilon$
decreases welfare for the capitalist and increases welfare for the worker.
Putting taxation and redistribution altogether, there always exists a way to reduce both pollution and inequality by playing on redistribution when the government taxes the polluting commodity and redistributes the tax revenue through lump-sum transfers. Coupled with a strong impact of environmental quality on the utility function, this policy mix is welfare improving for both households. At the aggregate level, the environmental tax reform is always effective in the long run: pollution is reduced, and the economy benefits from a better environmental quality. At the individual level, the level of subsistence consumption and the redistribution rate matter, both for consumption and income inequality. The environmental externality is key for the welfare impact: the higher it impacts agents, the greater the scope for a welfare improving tax reform.
6. Local stability
The level of subsistence consumption and the pollution externality play important roles in the long-run impact of environmental fiscal policies at the individual level. Characterizing the stability properties of the dynamic system (23)–(24) around the steady state allows us to analyze the effect of these two parameters in the short run, i.e. whether the steady state loses its saddle-path stability. We also study the role played by the fiscal policy to obtain the conditions under which instability and indeterminacy emerge.
To do so, we look at the log-linearized system evaluated at the steady state:
\begin{align*} \begin{bmatrix} \tilde {K_{t+1}} \\[5pt] \tilde {C_{t+1}} \end{bmatrix} = \begin{bmatrix} \Omega _1 & \Omega _2 \\[5pt] \Omega _3 & \Omega _4 \end{bmatrix} \begin{bmatrix} \tilde {K_{t}} \\[5pt] \tilde {C_{t}} \end{bmatrix} \end{align*}
with
\begin{gather} \Omega _1 = \frac {\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}}{\left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )}\left ( \frac {1}{\beta _1}-(1-\delta ) \right ) + (1 -\delta ) \end{gather}
\begin{gather} \Omega _2 = -\frac {P(1+\tau )^\alpha C_1^*}{K^*p\left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )} \end{gather}
\begin{gather} \Omega _3 = \frac {\frac {\mu \alpha \frac {(1-\eta )}{2}r^*K^*\left (1 -\Omega _1 \right )}{c_p^*(1+\tau (1-\alpha (1-\varepsilon )))}+ K^* f''(K^*) \Omega _1}{\sigma - \frac {(1+\tau )c_0}{P_1^*C_1^*}+ \frac {\mu \alpha P (1+\tau )^\alpha C_1^*}{c_p^*(1+\tau (1-\alpha (1-\varepsilon )))}} \end{gather}
\begin{gather} \Omega _4 = 1+\frac {f''(K^*)K^* \Omega _2-\frac {\mu \alpha \frac {(1-\eta )}{2}r^*K^*\Omega _2}{c_p^*(1+\tau (1-\alpha (1-\varepsilon )))}}{\sigma - \frac {(1+\tau )c_0}{P_1^*C_1^*}+ \frac {\mu \alpha P (1+\tau )^\alpha C_1^*}{c_p^*(1+\tau (1-\alpha (1-\varepsilon )))}}. \end{gather}
and
$f''(K^*) = (\eta -1) \eta A_p K^{*\eta -2} \lt 0$
. The trace and the determinant of the associated Jacobian matrix are given by:
\begin{align} Tr = \Omega _1+\Omega _4 \end{align}
\begin{align} D = \Omega _1 - \frac {\mu \alpha \frac {(1-\eta )}{2}r^*K^* \Omega _2}{c_p^*\left (1+\tau (1-\alpha (1-\varepsilon ))\right ) \left ( \sigma - \frac {(1+\tau )c_0}{P_1^*C_1^*} \right )+ \mu \alpha P (1+\tau )^\alpha C_1^*} \end{align}
We explore the role of the externality
$\mu$
and subsistence consumption
$c_0$
on the stability properties of the economy. We first analyze the role of subsistence consumption by setting
$\mu = 0$
, and then relax this assumption. Following the geometrical method of Grandmont et al. (Reference Grandmont, Pintus and De Vilder1998), we use the characteristic polynomial
$Pol(\lambda ) =\lambda ^2-\lambda Tr+D$
evaluated at
$\lambda = -1$
,
$0$
and
$1$
that we locate in the (Tr,D) plane (see Figure 1). Along (AB),
$Pol({-}1)=0$
(one eigenvalue is equal to −1) and along (AC),
$Pol(1) = 0$
(one eigenvalue is equal to 1). On [BC], the two eigenvalues are complex conjugates with modulus one. The equilibrium is a sink (locally indeterminate) inside the triangle ABC, meaning that stochastic endogenous fluctuations can occur around the steady state. It is a saddle on the right side or left side of both (AB) and (AC), and it is a source otherwise, i.e. the equilibrium is unstable. A flip bifurcation occurs when (Tr,D) crosses (AB), leading to two-period cycles. A Neimark-Sacker bifurcation occurs when (Tr,D) crosses [BC], giving rise to endogenous periodic or quasi-periodic fluctuations.
Figure 1. Geometrical method.
6.1 Local dynamics when
$c_0 \gt 0$
and
$\mu =0$
When the externality is not taken into account, the trace and the determinant simplify to:
\begin{gather} Tr = 1+ \Omega _1 -\frac { f''(K^*) C_1^* P(1+\tau )^\alpha }{p\left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right ) \left (\sigma -\frac {(1+\tau )c_0}{P_1^*C_1^{*}} \right )} = Tr(c_0) \end{gather}
\begin{gather} D = \Omega _1 \end{gather}
where
$C_1^*$
is a function of
$c_0$
. Making values of
$c_0$
vary between
$0$
and
$c_{01}$
, we analyze the stability properties by looking at how the trace and the determinant move on the plane described in Figure 1 . Since
$D$
does not depend on
$c_0$
, they describe an horizontal line. Starting at
$c_0=0$
, the point
$(Tr(0),D)$
lies on the right side of (AC). If
$ \eta \gt \overline {\eta }$
,Footnote
8
i.e. production is relatively capital intensive,
$(Tr(0),D)$
lies above point C. Otherwise, it is below. Increasing
$c_0$
leads to drawing a horizontal line going to the right on the
$(Tr,D)$
plane. At
$c_0=c_0^F$
, the horizontal line crosses (AB), leading to a source or a sink for
$c_0\gt c_0^F$
depending on the level of capital intensity. In the limit case where
$c_0$
tends to
$c_{01}$
, the point
$(Tr(c_{01}),D)$
lies on (AC).
When accounting for subsistence consumption but not for the externality, some instability can emerge depending on the level of
$c_0$
and
$\eta$
. As long as
$c_0 \lt c_0^F$
, saddle-path stability is ensured. When
$c_0 \gt c_0^F$
, local instability arises when
$\eta \gt \overline {\eta }$
, and local indeterminacy occurs whenever
$\eta \lt \overline {\eta }$
. We deduce that:
Proposition 7. Under Assumptions 1 – 3 , a steady state with no environmental externality and a positive level of subsistence consumption has the following stability properties:
-
• when
$c_0 \lt c_0^F$
, the steady state is a saddle;
-
• when
$c_0 \gt c_0^F$
, the steady state is a source for
$\eta \gt \overline {\eta }$
and a sink otherwise.
A flip bifurcation occurs at
$c_0=c_0^F$
, and a Neimark-Sacker bifurcation arises when
$c_0\gt c_0^F$
and
$\eta$
crosses the value
$\overline {\eta }$
.
Proof. see Appendix A.8, where we also define the critical values
$c_0^F$
and
$\overline {\eta }$
.
As both
$c_0^F$
and
$\overline {\eta }$
depend on
$\tau$
and
$\varepsilon$
, there is a role to play for both taxation and redistribution in maintaining the stability and determinacy of the equilibrium. To analyze this role, we need the following lemma:
Lemma 1.
Under Assumptions
1
–
3
and
$\mu =0$
,
$\overline {\eta }$
decreases with
$\varepsilon$
and
$\tau$
.
$c_0^F$
decreases with
$\tau$
when redistributing almost all the tax revenue to the worker.
$c_0^F$
increases with
$\varepsilon$
when the initial redistribution rate and
$\sigma$
are relatively high.
Proof. see Appendix A.9.
Increasing the tax rate promotes instability whenever the government redistributes almost all the tax revenue towards the worker, as both
$c_0^F$
and
$\overline {\eta }$
decrease with the tax rate. Increasing the share of the tax revenue redistributed towards the worker can promote expectation-driven fluctuations associated with the indeterminacy of the steady state. This assumes that redistribution favors capitalists and that relative risk aversion is relatively high, as
$c_0^F$
decreases and
$\overline {\eta }$
increases.
Figure 2. Local dynamics with
$\mu =0$
.
To give economic intuition for expectation-driven fluctuations, recall that the dynamics are driven by equations (19) and (20). Equation (20) can be rewritten as
$\frac {U'(C_{1t})}{P_{1t}} = \beta _1 \frac {U'(C_{1t+1})}{P_{1t+1}}\big( \frac {r_{t+1}}{p}+ 1 - \delta \big)$
. Expecting a higher
$C_{t+1}$
leads to a higher
$\frac {U'(C_{1t+1})}{P_{1t+1}}$
when
$c_0$
is high through the negative effect on
$P_{1t+1}$
. To maintain equality,
$r_{t+1}$
must decrease, which means an increase in capital holdings
$K_{t+1}$
. As capital and labor incomes are positive functions of capital, they increase at period
$t+1$
, so that next period consumption increases. Expectations are then self-fulfilling and there are oscillations because a higher investment in capital
$K_{t+1}$
also implies a lower current consumption
$C_{1t}$
. Indeterminacy and cycles are possible only because subsistence consumption establishes interactions between prices and quantities.Footnote
9
Without a sufficiently high subsistence level of consumption, the aggregate price does not depend on consumption, and
$\frac {U'(C_{1t+1})}{P_{1t+1}}$
always decreases when expecting a higher consumption level.
Increasing the tax rate and redistributing more towards the worker has a positive long-run impact by reducing pollution, income and welfare inequalities. Yet, opposite forces are playing in the shorter run: redistribution promotes indeterminacy and endogenous cycles, while increasing taxation on the polluting good has a mixed effect at the individual level depending on how much is redistributed to the worker. A high subsistence consumption leads the fiscal policy to have a mixed effect in the short run: increasing the tax rate decreases pollution and does not necessarily favors stability or determinacy. Redistributing more to the worker decreases inequality but promotes the emergence of cycles when increasing the fiscal pressure.
6.2 Local dynamics when
$c_0\gt 0$
and
$\mu \gt 0$
When pollution affects the utility, the trace and the determinant are given by (39)–(40). We depart from the case without pollution externality, and analyze what happens when
$\mu$
increases. Doing so, the point
$(Tr,D)$
draws a half line of slope
$S(c_0)=\frac {\frac {1-\eta }{2}r^*\left (\sigma -\frac {(1+\tau )c_0}{P_1^*C_1^*}\right )}{ \frac {1-\eta }{2}r^*\left (\sigma -\frac {(1+\tau )c_0}{P_1^*C_1^*}\right )+ P(1+\tau )^\alpha C_1^*f''(K^*)}$
. The origin lies on
$(Tr(c_0),D)$
for
$\mu =0$
, which belongs to the horizontal line of the previous case. As
$c_0$
moves from
$0$
to
$c_{01}$
, the origin
$(Tr(c_0),D)$
moves right to the limit point
$(Tr(c_{01}),D)$
which is on the line (AC). At the same time,
$S(c_0)$
decreases (resp. increases) when it is downward (resp. upward) sloping. Whenever
$c_0\lt c_0^F$
, the whole half line lies in the saddle region, (below (AB) and above (AC) when
$c_0 \gt c_0^*$
, below (AC) and above (AB) otherwise): local indeterminacy and instability never occur, regardless of the size of
$\mu$
. When
$c_0 \gt c_0^F$
, one lies above (AB) and above (AC), such that for
$\mu =0$
, the steady state is a sink for
$\eta \lt \overline {\eta }$
and a source when
$\eta \gt \overline {\eta }$
. An increase in
$\mu$
causes the half-line drawn by
$(Tr,D)$
to move downward to the left.
When
$\eta \gt \overline {\eta }$
, the half line crosses
$(AB)$
and/or
$[BC]$
when
$\mu$
increases, depending on the value taken by
$c_0$
. More precisely, the steady state is a source for any
$\mu \lt \mu _H$
, and a sink for any
$\mu \in (\mu _H,\mu _F)$
. A Neimark-Sacker bifurcation arises when
$\mu =\mu _H$
. For any
$\mu \gt \mu _F$
, the steady state lies back in the saddle region, with a flip bifurcation occurring at
$\mu =\mu _F$
. When
$\eta \lt \overline {\eta }$
, the half line crosses (AB) when
$\mu = \mu _F$
: a flip bifurcation occurs. The steady state is hence a sink for any
$\mu \lt \mu _F$
and a saddle otherwise. These results are summarized in the following proposition:
Proposition 8.
Under Assumptions
1
–
3
and
$\mu \gt 0$
, there exists
$\mu _H$
for
$\eta \gt \overline {\eta }$
and
$\mu _F$
, such that the steady state is:
-
• a source for
$c_0 \gt c_0^F$
and
$\mu \lt \mu _H$
; -
• a sink for
$c_0 \gt c_0^F$
and
$\mu \in (\mu _H, \mu _F)$
; -
• a saddle otherwise.
A Neimark-Sacker bifurcation occurs when
$\mu$
crosses
$\mu _H$
, and a flip bifurcation arises for
$\mu =\mu _F$
.
Proof. see Appendix A.10.
Saddle-path stability is ensured by a sufficiently important externality (
$\mu \gt \mu _F$
) and a relatively low subsistence level of consumption. When the externality is relatively weak and the subsistence level of consumption relatively high, the steady state can lose its stability through the occurrence of two-period cycles. However, the fiscal policy through a modification of
$\tau$
and/or
$\varepsilon$
can affect the stability properties of the economy modifying the critical values of
$\mu$
. The results are summarized in the following lemma:
Lemma 2.
Under Assumptions
1
–
3
,
$\mu \gt 0$
and denoting
$\frac {\partial C_1^*}{\partial x}\frac {x}{C_1^*} = \epsilon _{C_1^*,x}$
(
$x=\{\tau , \varepsilon \}$
):
-
•
$\mu _H$
and
$\mu _F$
increase in
$\varepsilon$
when
$\epsilon _{C_1^*,\varepsilon } \lt \overline {\epsilon _{C_1^*,\varepsilon }}$
; -
•
$\mu _H$
and
$\mu _F$
increase in
$\tau$
when
$\epsilon _{C_1^*,\tau } \lt \overline {\epsilon _{C_1^*,\tau }}$
.
Proof. see Appendix A.11, where
$\overline {\epsilon _{C_1^*,\varepsilon }}$
and
$\overline {\epsilon _{C_1^*,\tau }}$
are also defined.
The fiscal policy plays again a key role in the occurrence of instability and indeterminacy: it increases the range for
$c_0$
under which the steady state is unstable or indeterminate, and the range for
$\mu$
under which the steady state is unstable or a sink as long as consumption is not responding much to the tax system. Even if the externality might restore some stability, using the fiscal tools to fight polluting emissions and inequality harms stability properties of the economy.
A higher tax rate reduces pollution, redistributing more to the worker reduces income inequality, and the combination of both might reduce welfare inequality. Yet, the fiscal policy promotes instability and indeterminacy in the short-run. Under a high subsistence consumption, both fiscal tools promote again instability and indeterminacy in the short-run. This suggests that when people have a high level of subsistence consumption for the polluting good, taxation might not be the best tool as it destabilizes the economy. However, a higher impact of pollution on utility promotes stabilization. In this case, the fiscal policy goes also in favor of welfare improvement. The extent to which people are affected by pollution is hence of importance because it can help to understand what is the best tool to reduce emissions without harming households in another way, through a higher cost of fluctuations and a lower level of stationary welfare.
Figure 3. Local dynamics with
$\eta \gt \overline {\eta }$
.
Figure 4. Local dynamics with
$\eta \lt \overline {\eta }$
.
7. Concluding remarks
In this paper, we investigate the effect of environmental taxation and redistribution on pollution, income and welfare inequalities. To address these questions, we consider a two-sector Ramsey model with heterogeneous households, an environmental externality affecting their utility and a subsistence consumption for the polluting good.
In this framework, there exists a unique steady state in which the most patient household holds all the capital. The population endogenously splits in two classes, the wealthier who holds financial assets, the capital, and workers who are financially constrained. We then discuss the impact of taxation and redistribution on pollution, income and welfare inequalities. A higher redistribution to workers reduces income inequality and a higher tax rate on the polluting good reduces aggregate polluting consumption. However, the most interesting results concern the individual consumptions and the welfare. The effect of a higher tax rate strongly depends on the level of redistribution and subsistence consumption. There exist configurations where welfare inequality can increase, whereas it could decrease for a different level of redistribution. The redistribution between the different households could therefore be fixed taking into account the effect of taxation on welfare inequality. The size of the externality also matters: if households are highly impacted by pollution, an increase in taxation will always be welfare improving for both types. Analyzing local dynamics, we show that a sufficiently high subsistence consumption can lead to instability and indeterminacy. Increasing polluting commodity taxation while redistributing more to the worker affects instability and the occurrence of indeterminacy. The environmental externality has a stabilizing role in the economy, because taking it into account restores stability. In this case, the environmental fiscal policy plays a key role in the occurrence of indeterminacy. Hence, policy makers must carefully handle environmental taxation and redistribution to avoid instability and indeterminacy, and must take into account subsistence consumption when implementing an effective environmental tax reform that reduces inequalities.
Extensions of this framework deserve further investigation. In particular, interactions with other fiscal tools such as income taxation in an endogenous labor supply framework, or abatement policies on the production side might interact with the environmental fiscal reform and lead to new insights on the role of subsistence consumption. We leave these aspects for further research.
Acknowledgements
We thank the associate editor and the two referees for their suggestions and comments which helped to significantly improve this paper. We are grateful to Emmanuelle Taugourdeau, Peter Claeys, Andreas Schaefer, Nahid Masoudi, Alain Venditti, Katheline Schubert, participants at the FIRE 2021 and LORDE 2022 workshops and AMSE PhD seminar 2021, as well as participants at ASSET 2021, ICMAIF 2022, SURED 2022, EAERE 2022 and FAERE 2022 conferences for helpful discussions and comments. This project has been funded by the French government under the ”France 2030” investment plan managed by the French National Research Agency Grant ANR-17-EURE-0020 and by the Excellence Initiative of Aix-Marseille University - A*MIDEX, and by the European Union (ERC, GRETA, 101116659). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. All remaining errors are ours.
A. Appendix
A.1 Household’s Problem
The household’s problem writes:
\begin{align*} \begin{aligned} & \underset {a_{it+1},c_{pit},c_{git}}{\text{max}} & & \sum _{t=0}^\infty \beta _i^t U_i(c_{pit},c_{git}, E_t) \\[5pt] & \text{subject to} & &c_{pit}(1+\tau )+ c_{git} p_t +p_t(a_{it+1}-(1-\delta )a_{it}) = w_t n_i + r_t a_{it}+ T_{it}\\[5pt] & & & a_{it+1} \geq 0 \end{aligned} \end{align*}
Using the following Lagrangian to solve the optimization problem:
\begin{align*} \mathbb{L} &=\sum _t \beta _{i}^t \bigg[ E_t^\mu \frac {(c_{pit}^{\alpha } c_{git}^{1-\alpha })^{1-\sigma }}{1-\sigma } + \lambda _{it}^{BC} (w_t n_i + r_t a_{it}+ T_{it} - c_{pit}(1+\tau )\\&\quad - c_{git} p_t-p_t(a_{it+1}-(1-\delta )a_{it})) + \lambda _{it}^{FC} a_{it}\bigg] \end{align*}
Optimality conditions:
\begin{gather*} \frac {1-\alpha }{c_{git}}C_{it}^{1-\sigma }E_t^\mu = \lambda _{it}^{BC} p_t \\[5pt] \frac {\alpha }{c_{pit}-c_0}C_{it}^{1-\sigma }E_t^\mu = \lambda _{it}^{BC} (1+\tau ) \\[5pt] \lambda _{it}^{BC} p_t = \beta _i \lambda _{it+1}^{BC} (p_{t+1}(1-\delta )+r_{t+1}) + \beta _i \lambda _{it+1}^{FC} \end{gather*}
Dividing the two first FOCs and rearranging gives
$c_{git} p_t = \frac {1-\alpha }{\alpha } (c_{pit}-c_0) (1+\tau )$
. Denote
$I_{it} = \frac {w_t}{2} + r_t a_{it} -(a_{it+1}-(1-\delta ) a_{it}) + T_{it}$
. Plugging the previous equation for
$c_{git} p_t$
in the budget constraint of agent i yields:
\begin{gather*} c_{pit} = \frac {\alpha }{1+\tau }I_{it}+c_0(1-\alpha ) \\[5pt] c_{git} = \frac {1-\alpha }{p_t}(I_{it}-c_0(1+\tau )) \end{gather*}
The ex post budget constraint can be written
$P_{it} C_{it} = I_{it}$
.
Recall
$C_{it} = (c_{pit}-c_0)^{\alpha } c_{git}^{1-\alpha }$
, so that
$C_{it} = \left ( \frac {\alpha }{1+\tau }(I_{it}-c_0(1+\tau )) \right )^{\alpha } \left (\frac {1-\alpha }{p_t}(I_{it}-c_0(1+\tau )) \right )^{1-\alpha }$
.
Plugging this into the ex post budget constraint and rearranging yields
$P_{it} = \frac {(1+\tau )^{\alpha } p_t^{1-\alpha }}{(1-\alpha )^{1-\alpha } \alpha ^{\alpha }}+\frac {(1+\tau )c_0}{C_{it}}$
.
A.2 Computations for Assumption2
We have
\begin{align} C_{1t} &= \frac {1}{P(1+\tau )^\alpha } \bigg( \left ( \frac { (1+\eta )}{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) A_p K_t^\eta -\left(1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right) p(K_{t+1}-(1-\delta ) K_t)\nonumber \\[5pt] &\quad -\frac {(1+\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )}{1+\tau (1-\alpha )} c_0\bigg) \end{align}
\begin{align} C_{2t} &= \frac {1}{P(1+\tau )^\alpha } \bigg( \left ( \frac { (1-\eta )}{2}+\frac {\alpha (1-\varepsilon ) \tau }{1+\tau (1-\alpha )} \right ) A_p K_t^\eta -\left(\frac {\alpha (1-\varepsilon )\tau }{1+\tau (1-\alpha )}\right) p(K_{t+1}-(1-\delta ) K_t)\nonumber \\[5pt] &\quad -\frac {(1-\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )}{1+\tau (1-\alpha )} c_0\bigg) \end{align}
\begin{gather} P = \frac {p^{1-\alpha }}{(1-\alpha )^{1-\alpha } \alpha ^{\alpha }} \end{gather}
$C_{1t}$
and
$C_{2t}$
are positive whenever, respectively:
\begin{align} c_0 &\leq \left ( \frac { (1+\eta )(1+\tau (1-\alpha ))}{2(1+\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )}+\frac {\alpha \varepsilon \tau }{(1+\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )} \right ) A_p K_t^\eta\nonumber \\[5pt]& \quad -\left (\frac {1+\tau (1-\alpha )}{(1+\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )}+\frac {\alpha \varepsilon \tau }{(1+\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )} \right ) p(K_{t+1}-(1-\delta ) K_t)\nonumber \\[5pt] &\equiv c_{01,t} \end{align}
\begin{align} c_0 &\leq \left ( \frac { (1-\eta )(1+\tau (1-\alpha ))}{2(1-\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )}+\frac {\alpha (1-\varepsilon ) \tau }{(1-\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )} \right ) A_p K_t^\eta \nonumber\\[5pt]& \quad -\frac {\alpha (1-\varepsilon )\tau }{(1-\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )} p(K_{t+1}-(1-\delta ) K_t) \equiv c_{02,t}, \end{align}
which, evaluated at the steady state, yields:
\begin{align} c_0 &\leq \left ( \frac { (1+\eta )(1+\tau (1-\alpha ))}{2(1+\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )}+\frac {\alpha \varepsilon \tau }{(1+\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )} \right ) A_p K^{*^\eta }\nonumber \\ &\quad -\left (\frac {1+\tau (1-\alpha )}{(1+\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )}+\frac {\alpha \varepsilon \tau }{(1+\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )} \right ) p \delta K^* \equiv c_{01} \end{align}
\begin{align} c_0 &\leq \left ( \frac { (1-\eta )(1+\tau (1-\alpha ))}{2(1-\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )}+\frac {\alpha (1-\varepsilon ) \tau }{(1-\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )} \right ) A_p K^{*^\eta }\nonumber \\ &\quad-\frac {\alpha (1-\varepsilon )\tau }{(1-\tau (1-\alpha )(1-2 \varepsilon )) (1+\tau )} p \delta K^* \equiv c_{02} \end{align}
when
$\varepsilon \lt \frac {1}{2 \tau (1-\alpha )} + \frac {1}{2}$
for agent 1 and any value of
$\varepsilon$
for agent 2. As
$\frac {1}{2 \tau (1-\alpha )} + \frac {1}{2} \geq 1$
, the condition for household 1 always holds in this setting.
A.3 Proof of Proposition2
\begin{gather*} \frac {d c_p^*}{d \tau } = \frac {-\alpha (1-\alpha )}{(1+\tau (1-\alpha ))^2}\left [ K^*p \left ( \frac {r^*/p}{\eta }-\delta \right )-2 c_0\right ] \\[5pt] \frac {d c_g^*}{d \tau } = \frac {\alpha (1-\alpha )}{p(1+\tau (1-\alpha ))^2}\left [ K^*p \left ( \frac {r^*/p}{\eta }-\delta \right )-2c_0\right ] \end{gather*}
It is straightforward to deduce that
$\frac {d c_g^*}{d \tau } \gt 0$
and
$\frac {d c_p^*}{d \tau } \lt 0$
.
A.4 Proof of Proposition3
\begin{gather} \frac {\partial (P_1C_1)}{\partial \tau } = K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \frac {\alpha \varepsilon }{(1+\tau (1-\alpha ))^2}+ \frac {2c_0 \varepsilon (1-\alpha )(1+2\tau +\tau ^2(1-\alpha ))}{(1+\tau (1-\alpha ))^2} \end{gather}
\begin{gather} \frac {\partial (P_2C_2)}{\partial \tau } = K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \frac {\alpha (1-\varepsilon )}{(1+\tau (1-\alpha ))^2}+ \frac {2c_0 (1-\varepsilon ) (1-\alpha )(1+2\tau +\tau ^2(1-\alpha ))}{(1+\tau (1-\alpha ))^2} \end{gather}
Clean consumption for the capitalist increases along the tax rate for:
\begin{gather*} \frac {\partial c_{g1}}{\partial \tau } \gt 0 \Longleftrightarrow K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \frac {\alpha \varepsilon }{(1+\tau (1-\alpha ))^2}+ \frac {2c_0 \varepsilon (1-\alpha )(1+2\tau +\tau ^2(1-\alpha ))}{(1+\tau (1-\alpha ))^2} - c_0 \gt 0\\[5pt] \Longleftrightarrow K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha \varepsilon \gt c_0 \left ( (1+\tau (1-\alpha ))^2 (1-2\varepsilon )+2\alpha \varepsilon \right ) \\[5pt] \Longleftrightarrow \varepsilon \gt \frac { c_0 (1+\tau (1-\alpha ))^2}{ K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha + 2c_0((1+\tau (1-\alpha ))^2-\alpha )} \equiv \varepsilon ^1 \end{gather*}
Clean consumption for the worker increases along the tax rate for:
\begin{align*} \frac {\partial c_{g2}}{\partial \tau } \gt 0& \Longleftrightarrow K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \frac {\alpha (1-\varepsilon )}{(1+\tau (1-\alpha ))^2}\nonumber \\&+ \frac {2c_0 (1-\varepsilon ) (1-\alpha )(1+2\tau +\tau ^2(1-\alpha ))}{(1+\tau (1-\alpha ))^2} - c_0 \gt 0\nonumber \\[5pt] & \Longleftrightarrow \varepsilon \lt \frac { K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha + c_0 ((1+\tau (1-\alpha ))^2-2\alpha )}{ K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha + 2c_0((1+\tau (1-\alpha ))^2-\alpha )} \equiv \varepsilon ^2 \end{align*}
Note that
$\varepsilon ^2 \gt \varepsilon ^1$
. When
$c_0=0$
,
$\varepsilon ^1 = 0$
and
$\varepsilon ^2 = 1$
, so clean consumption increases for both households. When
$c_0= \overline {c_0}$
,
$\varepsilon ^1 = \varepsilon ^2 = \frac {1}{2}$
, so clean consumption cannot increase for both types.
The derivatives with respect to the tax rate are increasing in
$c_0$
for the capitalist and decreasing in
$c_0$
for the worker, which means that the value of
$\varepsilon ^1$
(resp.
$\varepsilon ^2$
) is increasing (resp. decreasing) in
$c_0$
. In other words, since
$c_0 \lt \frac {K^*p \left (\frac {r^*/p}{\eta }-\delta \right )}{2}$
, there always exists a
$\varepsilon \in (\varepsilon ^1, \varepsilon ^2)$
such that both individual clean consumption increase at the same time.
A.5 Proof of Proposition4
A.5.1 For the capitalist
The derivative of
$c_{p1}^*$
with respect to
$\tau$
is given by:
\begin{align} & \frac {\tau ^2}{(1+\tau (1-\alpha ))^2} \!\left ( 2c_0 \varepsilon (1-\alpha ) -K^*p\! \left ( \frac {r^*/p}{\eta } - \delta\! \right ) \alpha \varepsilon (1-\alpha ) - K^*p\! \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right )\! (1-\alpha )^2 \right ) \nonumber \\[5pt] & \quad + \frac {\tau }{(1+\tau (1-\alpha ))^2} \left ( 4c_0\varepsilon (1-\alpha ) - 2K^*p \left (\frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha )\right )\nonumber \\[5pt] & \quad + \frac {K^*p \left (\left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha \varepsilon - \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) \right ) + 2c_0\varepsilon (1-\alpha )}{(1+\tau (1-\alpha ))^2} \end{align}
where the numerator if a second degree polynomial, which discriminant is positive as long as
\begin{gather*} \left (K^*p\left ( \frac {r^*/p}{\eta } - \delta \right ) - 2c_0 \right ) \left ( K^*p\left ( \frac {r^*/p}{\eta } - \delta \right )\varepsilon - K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right )\right ) 4\alpha ^2\varepsilon (1-\alpha )\gt 0 \end{gather*}
The first component of this inequality is always positive as it is the upper bound of
$c_0$
stated in Assumption3. Hence, one needs
\begin{gather} \varepsilon \gt \frac {\frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta }{\frac {r^*/p}{\eta } - \delta } \equiv \varepsilon ^* \end{gather}
for it to be positive.
If
$\varepsilon \lt \varepsilon ^*$
, the discriminant is negative, there is no root and whether
$c_{p1}$
decreases or not depends on the sign of
$2c_0 \varepsilon (1-\alpha ) -K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha \varepsilon (1-\alpha ) - K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha )^2$
. This yields a condition on
$c_0$
:
\begin{gather} c_0 \gt \frac {K^*p}{2\varepsilon } \left ( \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha \varepsilon +\left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha )\right ) \equiv \tilde {c_0^1} \end{gather}
which is never satisfied on
$(0,\tilde {\varepsilon })$
. This means that
$2c_0 \varepsilon (1-\alpha ) -K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha \varepsilon (1-\alpha ) - K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha )^2 \lt 0$
and that
$c_{p1}$
decreases along the tax rate.
If
$\varepsilon = \varepsilon ^*$
, the discriminant is equal to 0 and
$2c_0 \varepsilon (1-\alpha ) -K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha \varepsilon (1-\alpha ) - K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha )^2 \leq 0$
, so
$c_{p1}$
decreases along
$\tau$
as the unique root is negative.
If
$\varepsilon \gt \varepsilon ^*$
, the discriminant is positive and has two roots,
$\tilde {\tau }^1_1$
and
$\tilde {\tau }^2_1$
, given by:
\begin{gather} \tilde {\tau }^1_1 = \frac { 2 K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha ) - 4c_0\varepsilon (1-\alpha ) - \sqrt {\Delta _{c_{p1}}}}{2 \left ( 2c_0 \varepsilon (1-\alpha ) -K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha \varepsilon (1-\alpha ) - K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha )^2\right )} \end{gather}
\begin{gather} \tilde {\tau }^2_1 = \frac { 2 K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha ) - 4c_0\varepsilon (1-\alpha ) + \sqrt {\Delta _{c_{p1}}}}{2 \left ( 2c_0 \varepsilon (1-\alpha ) -K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha \varepsilon (1-\alpha ) - K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha )^2\right )} \end{gather}
where
$\Delta _{c_{p1}}$
is the discriminant associated to the polynomial in equation (52).
The denominator of these roots is positive if and only if
$c_0 \gt \tilde {c_0^1}$
. It is negative for any
$c_0 \in [0,\tilde {c_0^1})$
and positive for any
$c_0 \in \left (\tilde {c_0^1}, \frac {K^*p \left ( \frac {r^*/p}{\eta } - \delta \right )}{2} \right ]$
.
For
$c_0 \in \big[0,\tilde {c_0^1}\big)$
. As the denominator is negative,
$\tilde {\tau }^1_1$
and
$\tilde {\tau }^2_1$
are positive if
$2 K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha ) - 4c_0\varepsilon (1-\alpha ) \lt \sqrt {\Delta _{c_{p1}}}$
and
$-\left (2 K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha ) - 4c_0\varepsilon (1-\alpha ) \right ) \gt \sqrt {\Delta _{c_{p1}}}$
respectively.
If
$2 K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha ) - 4c_0\varepsilon (1-\alpha ) \lt 0$
,
$\tilde {\tau }^1_1 \in (0,1)$
for
$\alpha \lt \overline {\alpha }$
, and
$\tilde {\tau }^1_1 \gt 1$
otherwise, and
$\tilde {\tau }^2_1 \lt 0$
. This means that when
$\alpha \gt \overline {\alpha }$
,
$\frac {\partial c_{p1}}{\partial \tau } \gt 0$
for any
$\tau$
. When
$\alpha \lt \overline {\alpha }$
,
$\frac {\partial c_{p1}}{\partial \tau } \gt 0$
for any
$\tau \lt \tilde {\tau }^1_1$
and
$\frac {\partial c_{p1}}{\partial \tau } \lt 0$
for any
$\tau \gt \tilde {\tau }^1_1$
.
If
$2 K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha ) - 4c_0\varepsilon (1-\alpha ) \gt 0$
, then
$\tilde {\tau }^2_1 \lt 0$
always, and
$\tilde {\tau }^1_1 \gt 0$
for
$\alpha \gt \tilde {\alpha }$
. For
$\alpha \lt \tilde {\alpha }$
, both tax rates are negative and polluting consumption always decreases with the tax rate. For
$\alpha \gt \tilde {\alpha }$
,
$\frac {\partial c_{p1}}{\partial \tau } \gt 0$
for any
$\tau \lt \tilde {\tau }^1_1$
. Note that
$\tilde {\tau }^1_1 \gt 1$
for
$\varepsilon \gt \underline {\varepsilon }$
: in this case,
$\frac {\partial c_{p1}}{\partial \tau } \gt 0$
.
For
$c_0 \in \Big(\tilde{c_0^1}, \frac{K^*p \left( \frac{r^*/p}{\eta} - \delta\right)}{2}\Big]$
. The denominator of both roots is positive, which means that
$\tilde {\tau }^1_1$
and
$\tilde {\tau }^2_1$
are positive if
$2 K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha ) - 4c_0\varepsilon (1-\alpha ) \gt \sqrt {\Delta _{c_{p1}}}$
and
$-\left ( 2 K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha )- 4c_0\varepsilon (1-\alpha ) \right ) \lt \sqrt {\Delta _{c_{p1}}}$
respectively.
$2 K^*p \left ( \frac {r^*/p}{\eta } \frac {(1+\eta )}{2} - \delta \right ) (1-\alpha ) - 4 c_0\varepsilon (1-\alpha ) \lt 0$
in this case, and both tax rates are negative: polluting consumption increases with the tax rate.
A.5.2 For the worker
The derivative of
$c_{p2}^*$
with respect to
$\tau$
is given by:
\begin{align} &\frac {\tau ^2}{(1+\tau (1-\alpha ))^2}\nonumber \\&\quad \times \bigg( 2c_0 (1-\varepsilon )(1-\alpha ) -K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha (1-\varepsilon ) (1-\alpha ) - K^*p \frac {r^*/p}{\eta } \frac {(1-\eta )}{2} (1-\alpha )^2 \bigg) \nonumber \\[5pt]&\quad + \frac {\tau }{{(1+\tau (1-\alpha ))^2}} \left ( 4c_0(1-\varepsilon )(1-\alpha ) - 2K^*p \frac {r^*/p}{\eta } \frac {(1-\eta )}{2} (1-\alpha )\right ) \\[5pt] & \quad + \frac {K^*p \left (\left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha (1-\varepsilon ) - \frac {r^*/p}{\eta } \frac {(1-\eta )}{2} \right ) + 2 c_0(1-\varepsilon )(1-\alpha )}{(1+\tau (1-\alpha ))^2} \nonumber \end{align}
As for the capitalist, the numerator of the derivative is a polynomial of degree 2, which discriminant is positive as long as
$\varepsilon \gt \varepsilon ^*$
.
When
$\varepsilon \lt \varepsilon ^*$
, the discriminant is negative and of the sign of
$2c_0 (1-\varepsilon )(1-\alpha ) -K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha (1-\varepsilon ) (1-\alpha ) - K^*p \frac {r^*/p}{\eta } \frac {(1-\eta )}{2} (1-\alpha )^2$
. This yields a condition on
$c_0$
:
\begin{gather} c_0 \gt \frac {K^*p}{2(1-\varepsilon )} \left ( \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha (1-\varepsilon ) +\frac {r^*/p}{\eta } \frac {(1-\eta )}{2} (1-\alpha )\right ) \equiv \tilde {c_{0}^2} \end{gather}
For
$c_0 \lt \tilde {c_{0}^2}$
,
$\frac {\partial c_{p2}}{\partial \tau } \lt 0$
. For
$c_0 \gt \tilde {c_{0}^2}$
,
$\frac {\partial c_{p2}}{\partial \tau } \gt 0$
.
When
$\varepsilon = \varepsilon ^*$
,
$c_0 \leq \tilde {c_{0}^2}$
and
$\frac {\partial c_{p2}}{\partial \tau } \leq 0$
.
When
$\varepsilon \gt \varepsilon ^*$
, the discriminant is positive and there are two roots,
$\tilde {\tau }^1_2$
and
$\tilde {\tau }^2_2$
, given by:
\begin{gather} \tilde {\tau }^1_2 = \frac { 2 K^*p \frac {r^*/p}{\eta } \frac {(1-\eta )}{2} (1-\alpha ) - 4c_0(1-\varepsilon )(1-\alpha ) - \sqrt {\Delta _{c_{p2}}}}{2 \left ( 2c_0 (1-\varepsilon )(1-\alpha ) -K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha (1-\varepsilon ) (1-\alpha ) - K^*p\frac {r^*/p}{\eta } \frac {(1-\eta )}{2} (1-\alpha )^2\right )} \end{gather}
\begin{gather} \tilde {\tau }^2_2 = \frac { 2 K^*p \frac {r^*/p}{\eta } \frac {(1-\eta )}{2} (1-\alpha ) - 4c_0(1-\varepsilon )(1-\alpha ) + \sqrt {\Delta _{c_{p2}}}}{2 \left ( 2c_0 (1-\varepsilon )(1-\alpha ) -K^*p \left ( \frac {r^*/p}{\eta } - \delta \right ) \alpha (1-\varepsilon ) (1-\alpha ) - K^*p\frac {r^*/p}{\eta } \frac {(1-\eta )}{2} (1-\alpha )^2\right )} \end{gather}
where
$\Delta _{c_{p2}}$
is the discriminant associated to the polynomial in equation (57). In this case,
$ \tilde {\tau }^2_2 \lt 0$
and
$\tilde {\tau }^1_2 \lt 0$
, so
$\frac {\partial c_{p2}}{\partial \tau } \lt 0$
.
A.6 Proof of Proposition5
From equations (50) and (51), is straightforward that
$\frac {\partial (P_2^*C_2^*)}{\partial \tau } \gt \frac {\partial (P_1^*C_1^*)}{\partial \tau }$
for any
$\varepsilon \lt \frac {1}{2}$
.
A.7 Proof of Proposition6
A.7.1 Impact on overall consumption
For the capitalist:
Taking equation (28), one can rewrite:
\begin{align*} P_1^*C_1^* &= (1+\tau )^\alpha P^*C_1^*+ (1+\tau ) c_0\nonumber \\ &= K^* p \!\left ( \frac {r^*/p}{\eta } \!\left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right ) - \delta\! \left ( 1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )\!\! \right )+\frac {2\varepsilon \tau (1+\tau )(1-\alpha )}{1+\tau (1-\alpha )}c_0 \\[5pt] \Longleftrightarrow C_1^* &= \frac {K^* p}{(1+\tau )^\alpha P^*} \left ( \frac {r^*/p}{\eta } \left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right ) - \delta \left ( 1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) \right )\\ &\quad +\frac {2\varepsilon \tau (1+\tau )(1-\alpha )}{(1+\tau )^\alpha P^*(1+\tau (1-\alpha ))}c_0 - \frac {(1+\tau )}{(1+\tau )^\alpha P^*}c_0 \end{align*}
From equation (6) at the steady state, one can rewrite:
\begin{gather} C_1^* = \frac {p}{(1-\alpha )(1+\tau )^\alpha P^*}c_{g1}^* \end{gather}
Taking the derivative of
$C_1^*$
with respect to
$\tau$
yields:
\begin{gather} \frac { \partial C_1^*}{\partial \tau } = \frac {p}{(1-\alpha )(1+\tau )^\alpha P^*} \left ( \frac {\partial c_{g1}^*}{\partial \tau } - \alpha \frac {c_{g1}^*}{1+\tau }\right ) \end{gather}
The first component of the derivative is always positive, hence the sign of
$ \frac { \partial C_1^*}{\partial \tau }$
depends on the sign of
$\frac {\partial c_{g1}^*}{\partial \tau } - \alpha \frac {c_{g1}^*}{1+\tau }$
.
$C_1^*$
increases with the tax rate if and only if
\begin{gather*} \frac {\partial c_{g1}^*}{\partial \tau } \frac {1+\tau }{c_{g1}^*} \gt \alpha . \end{gather*}
For the worker:
Taking equation (29), one can rewrite:
\begin{align*} P_2^*C_2^* &= (1+\tau )^\alpha P^*C_2^*+ (1+\tau ) c_0 \\[5pt]&= K^*p \left [\left ( \frac {r^*/p}{\eta }-\delta \right ) \frac {\alpha (1-\varepsilon ) \tau }{1+\tau (1-\alpha )} + \frac {r^*/p}{\eta }\frac {1-\eta }{2} \right ]+\frac {2(1-\varepsilon )\tau (1+\tau )(1-\alpha )}{1+\tau (1-\alpha )}c_0 \\[10pt]\Longleftrightarrow C_2^* &= \frac {K^*p}{(1+\tau )^\alpha P^*} \left [\left ( \frac {r^*/p}{\eta }-\delta \right ) \frac {\alpha (1-\varepsilon ) \tau }{1+\tau (1-\alpha )} + \frac {r^*/p}{\eta }\frac {1-\eta }{2} \right ]\\[10pt]&\quad +\frac {2(1-\varepsilon )\tau (1+\tau )(1-\alpha )}{(1+\tau (1-\alpha ))(1+\tau )^\alpha P^*}c_0 - \frac {(1+\tau )}{(1+\tau )^\alpha P^*}c_0 \end{align*}
From equation (6) at the steady state, one can rewrite:
\begin{gather} C_2^* = \frac {p}{(1-\alpha )(1+\tau )^\alpha P^*}c_{g2}^* \end{gather}
Taking the derivative of
$C_2^*$
with respect to
$\tau$
yields:
\begin{gather} \frac { \partial C_2^*}{\partial \tau } = \frac {p}{(1-\alpha )(1+\tau )^\alpha P^*} \left ( \frac {\partial c_{g2}^*}{\partial \tau } - \alpha \frac {c_{g2}^*}{1+\tau }\right ) \end{gather}
The first component of the derivative is always positive, hence the sign of
$ \frac { \partial C_2^*}{\partial \tau }$
depends on the sign of
$\frac {\partial c_{g2}^*}{\partial \tau } - \alpha \frac {c_{g2}^*}{1+\tau }$
.
$C_2^*$
increases with the tax rate if and only if
\begin{gather*} \frac {\partial c_{g2}^*}{\partial \tau } \frac {1+\tau }{c_{g2}^*} \gt \alpha . \end{gather*}
A.7.2 Impact on welfare
An increase in the commodity tax rate leads to changes in utilities:
\begin{equation*} \frac {d U_i}{d \tau } = \mu \frac {d E^*}{d \tau }E^{*\mu -1}\frac {C_i^{*1-\sigma }}{1-\sigma }+\frac {d C_i}{d \tau }C_i^{*-\sigma } E^{*\mu } \end{equation*}
where
$\frac {d E^*}{d \tau } \gt 0$
. The result is straightforward:
-
• when
$ \frac {\partial c_{gi}^*}{\partial \tau } \frac {1+\tau }{c_{g2}^*} \gt \alpha$
,
$C_i$
increases and
$U_i$
increases; -
• when
$ \frac {\partial c_{g2}^*}{\partial \tau } \frac {1+\tau }{c_{g2}^*} \lt \alpha$
,
$C_i$
decreases.
$U_i$
increases if
$\mu \gt \mu _i^*$
and decreases otherwise.
with
$\mu _i^* = - \frac {\frac {d C_i^*}{d \tau }}{C_i^*} \frac {E^*}{\frac {d E^*}{d \tau }} (1-\sigma )$
.
A.8 Proof of Proposition7
Looking at the characteristic polynomial when
$\mu =0$
:
\begin{gather*} Pol({-}1) =2(1+ D) +\frac { f''(K^*) K^* \Omega _2}{\sigma - \frac {(1+\tau )c_0}{P_1^*C_1^*}}\\[5pt] Pol(1) = -\frac { f''(K^*) K^* \Omega _2}{\sigma - \frac {(1+\tau )c_0}{P_1^*C_1^*}} \end{gather*}
If
$c_0 \lt c_0^*$
, then
$\sigma P_1^*C_1^*-(1+\tau )c_0 \gt 0$
and we have a saddle as
$Pol({-}1) \gt 0$
and
$Pol(1) \lt 0$
. If
$c_0 \gt c_0^*$
,
$Pol(1) \gt 0$
so we must check the sign of
$Pol({-}1)$
in order to know whether there is indeterminacy of the equilibrium or not. If
$Pol({-}1)\gt 0$
, the equilibrium is either a sink or a source. This depends on whether
$D\gt 1$
, which happens when
$\eta \gt \overline {\eta }$
.
Setting
$D=1$
, we have
$\overline {\eta } = \frac {\delta }{\frac {1}{\beta _1}-1+\delta } - \frac {\left ( \frac {1}{\beta _1}-1 \right )}{\frac {1}{\beta _1}-1+\delta } \left ( 1+ \frac {2 \alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )$
.
$Pol({-}1)=0$
is a second degree polynomial in
$c_0$
which writes:
\begin{align*}& c_0^2 \left [ f''(K^*) \frac {(1+\tau )^2(1+\tau (1-\alpha )(1-2 \varepsilon ))}{p(1+\tau (1-\alpha ))}\left (\frac {(1+\tau (1-\alpha )(1-2 \varepsilon ))}{1+\tau (1-\alpha )} -1 \right ) \right ]\\[5pt] & \quad +c_0 \bigg[ 2(1+\tau )\left ( (2-\delta ) \left ( 1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right )+\frac {r^*}{p} \left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )\right )\\[5pt] &\quad \times \left (1-\sigma \frac {2 \varepsilon \tau (1-\alpha )}{1+\tau (1-\alpha )}\right ) -f''(K^*) (1+\tau )K^*\left ( \frac {r^*/p}{\eta } \left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )\right.\\[5pt] &\quad \left. - \delta \left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )\right ) \times \left (\frac {2(1+\tau (1-\alpha )(1-2 \varepsilon ))}{1+\tau (1-\alpha )} -1 \right ) \bigg ]\\[5pt] & \quad - 2\left ( (2-\delta ) \left ( 1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right )+\frac {r^*}{p} \left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )\right ) \sigma K^*p \\[5pt] & \quad \times \left ( \frac {r^*/p}{\eta } \left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) - \delta \left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )\right )\\[5pt] & \quad + f''(K^*)p \left ( K^*\left ( \frac {r^*/p}{\eta } \left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) - \delta \left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )\right ) \right )^2 =0 \end{align*}
which discriminant is always positive. The first root is always negative. The second root, denoted
$c_0^F$
, is positive and lower than
$c_{01}$
, which means that
$Pol({-}1) \lt 0$
whenever
$c_0 \in (c_0^*, c_0^F)$
, and is positive otherwise. When
$c_0 \rightarrow c_{01}$
,
$Pol({-}1) \rightarrow 2(1+D)$
and
$Pol(1) \rightarrow 0$
, which means that
$(Tr(c_{01}),D(c_{01}))$
lies on (AC).
$c_0^F$
is given by:
\begin{align} \frac {\begin{aligned} &-2(1+\tau )(1+\Omega _1) p\left ( 1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) \left ( 1-\frac {2\sigma \varepsilon \tau (1-\alpha )}{(1+\tau (1-\alpha ))}\right )+ f''(K^*)K^*(1+\tau )\\[8pt] & \left ( \frac {r^*/p}{\eta } (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}) - \delta (1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )})\right ) \left ( \frac {2(1+\tau (1-\alpha )(1-2\varepsilon ))}{1+\tau (1-\alpha )} -1\right )+ \sqrt {\Delta }\end{aligned}}{\dfrac { 2f''(K^*) (1+\tau )^2(1+\tau (1-\alpha )(1-2\varepsilon ))}{p(1+\tau (1-\alpha ))} \left (\dfrac {1+\tau (1-\alpha )(1-2\varepsilon )}{1+\tau (1-\alpha )}-1\right )} \end{align}
with
$\Delta$
the discriminant associated to
$Pol({-}1)$
.
A.9 Proof of Theorem 1
Using
$Pol({-}1)$
and the implicit function theorem,
$\frac {\partial c_0^F}{\partial \tau } = -\left .\frac { \partial Pol({-}1)/\partial \tau }{\partial Pol({-}1)/\partial c_0}\right |_{c_0=c_0^F}$
. Around
$c_0^F$
,
$\frac {\partial Pol({-}1)}{\partial c_0} \gt 0$
. The derivative of
$Pol({-}1)$
with respect to
$\tau$
is given by:
\begin{align*} &c_0^2 \frac {f''(K^*)}{p} \bigg [ \bigg ( \frac {2(1+\tau )(1+\tau (1-\alpha )(1-2\varepsilon ))}{1+\tau (1-\alpha )} - \frac {2 \varepsilon (1-\alpha )(1+\tau )^2}{(1+\tau (1-\alpha ))^2} \bigg ) \bigg ( \frac {1+\tau (1-\alpha )(1-2\varepsilon )}{1+\tau (1-\alpha )} -1\bigg ) \\[5pt] & - \frac {2 \varepsilon (1-\alpha )(1+\tau )^2}{(1+\tau (1-\alpha ))^2} \frac {(1+\tau (1-\alpha )(1-2\varepsilon ))}{1+\tau (1-\alpha )} \bigg ]\\[5pt] & +c_0 \bigg [ 2 \bigg (2-\delta + \frac {r^*}{p} \bigg ) \frac {\alpha \varepsilon (1+\tau )}{(1+\tau (1-\alpha ))^2}\bigg (1- \frac {2\sigma \varepsilon \tau (1-\alpha )}{1+\tau (1-\alpha )}\bigg ) \\[5pt] & + \bigg ((2-\delta )\bigg ( 1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \bigg ) +\frac {r^*}{p} \bigg ( \frac {1+\eta }{2}+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \bigg ) \bigg )\\[5pt] & \times \bigg (1-\frac {2\sigma \varepsilon \tau (1-\alpha )}{1+\tau (1-\alpha )}-\frac {2 \sigma \varepsilon (1+\tau )(1-\alpha )}{(1+\tau (1-\alpha ))^2} \bigg ) - f''(K^*)K^* \bigg [\bigg ( \frac {r^*/p}{\eta }\bigg (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \bigg )\\[5pt] & - \delta \bigg (1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\bigg ) \bigg ) \bigg (\frac {2(1+\tau (1-\alpha )(1-2 \varepsilon ))}{1+\tau (1-\alpha )} -1 \bigg ) \\[5pt] & + (1+\tau ) \bigg ( \frac {r^*/p}{\eta } - \delta \bigg ) \frac {\alpha \varepsilon }{(1+\tau (1-\alpha ))^2} \bigg ( \frac {2(1+\tau (1-\alpha )(1-2 \varepsilon ))}{1+\tau (1-\alpha )} -1\bigg ) \\[5pt] & - (1+\tau ) \frac {4\varepsilon (1-\alpha )}{(1+\tau (1-\alpha ))^2} \bigg ( \frac {r^*/p}{\eta }\bigg (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \bigg ) - \delta \bigg (1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\bigg ) \bigg ) \bigg ] \bigg ]\\[5pt] & - 2 \sigma K^*p \bigg [\bigg (2-\delta + \frac {r^*}{p} \bigg )\times \frac {\alpha \varepsilon }{(1+\tau (1-\alpha ))^2}\bigg ( \frac {r^*/p}{\eta }\bigg (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \bigg )\\ & - \delta \bigg (1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\bigg ) \bigg ) + \bigg ((2-\delta )\bigg ( 1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \bigg )\\[5pt] & + \frac {r^*}{p} \bigg ( \frac {1+\eta }{2}+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \bigg ) \bigg ) \frac {\alpha \varepsilon }{(1+\tau (1-\alpha ))^2} \bigg ( \frac {r^*/p}{\eta } - \delta \bigg )\bigg ]\\[5pt] & +2 f''(K^*)pK^{*2} \frac {\alpha \varepsilon }{(1+\tau (1-\alpha ))^2} \bigg ( \frac {r^*/p}{\eta } - \delta \bigg )\\[5pt] &\times \bigg ( \frac {r^*/p}{\eta }\bigg (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \bigg ) - \delta \bigg (1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\bigg ) \bigg ) \end{align*}
When
$\varepsilon \rightarrow 0$
,
$\frac {\partial Pol({-}1)}{\partial \tau } = c_0 \big((2-\delta )+\frac {r^*}{p}\frac {1+\eta }{2}\big)-f''(K^*)K^* \left ( \frac {r^*/p}{\eta } \frac {1+\eta }{2} - \delta \right ) \geq 0$
for any
$c_0$
, so
$\frac {\partial c_0^F}{\partial \tau } \lt 0$
.
The derivative of
$Pol({-}1)$
with respect to
$\varepsilon$
is given by:
\begin{align*} & - c_0^2 f''(K^*) \frac {2\tau (1-\alpha )(1+\tau )^2}{p(1+\tau (1-\alpha ))} \left (\frac {2(1+\tau (1-\alpha )(1-2 \varepsilon ))}{1+\tau (1-\alpha )} -1 \right )\\[5pt] &\quad +c_0 \left [ 2(1+\tau )\left (2-\delta + \frac {r^*}{p}\right ) \frac {\alpha \tau }{1+\tau (1-\alpha )} \left ( 1- \frac {2\varepsilon \sigma \tau (1-\alpha )}{1+\tau (1-\alpha )} \right )\right .\end{align*}
\begin{align*} & \left . -f''(K^*) (1+\tau ) K^* \left (\frac {\alpha \tau }{1+\tau (1-\alpha )} \left ( \frac {r^*/p}{\eta } - \delta \right ) \left (\frac {2(1+\tau (1-\alpha )(1-2 \varepsilon ))}{1+\tau (1-\alpha )} -1 \right ) \right .\right . \\[5pt] & - \left.\left. \frac {4(1-\alpha ) \tau }{1+\tau (1-\alpha )} \left ( \frac {r^*/p}{\eta }\left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) - \delta \left (1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right ) \right ) \right ) \right ]\\[5pt] & -2 \sigma K^* p \frac {\alpha \tau }{1+\tau (1-\alpha )}\! \left [ \!\left (2-\delta + \frac {r^*}{p} \right ) \!\left ( \frac {r^*/p}{\eta }\!\left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\! \right ) - \delta \!\left (1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right ) \right ) \right . \\[5pt] & \left . + \left ((2-\delta ) \left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) + \frac {r^*}{p} \left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )\right ) \left ( \frac {r^*/p}{\eta } - \delta \right ) \right ] \\[5pt] & + \frac {2f''(K^*)K^{*2} p\alpha \tau }{1+\tau (1-\alpha )} \left ( \frac {r^*/p}{\eta } - \delta \right ) \left ( \frac {r^*/p}{\eta }\left (\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right ) - \delta \left (1+ \frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}\right ) \right ) \end{align*}
This derivative is negative as long as
$\varepsilon \gt \frac {1}{4}+\frac {1}{4\tau (1-\alpha )}$
and
$\sigma \gt \frac {\alpha (1+\tau (1-\alpha ))}{(1-\alpha )(4\varepsilon \tau +(1+\eta )(1+\tau (1-\alpha )))}$
. In this case, using the implicit function theorem,
$\frac {\partial c_0^F}{\partial \varepsilon } \gt 0$
, which means that redistributing more towards the worker destabilizes the economy.
A.10 Expressions for critical values of
$\mu$
The trace and determinant of the characteristic polynomial when
$c_0\gt 0$
and
$\mu \gt 0$
are given by
\begin{align} Tr(c_0)=\frac {\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}}{\left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )}\frac {r^*}{p} + (1 -\delta ) +1+\frac {f''(K^*)K^* \Omega _2-\frac {\mu \alpha \frac {(1-\eta )}{2}r^*K^*\Omega _2}{c_p^*(1+\tau (1-\alpha (1-\varepsilon )))}}{\sigma - \frac {(1+\tau )c_0}{P_1^*C_1^*}+ \frac {\mu \alpha P (1+\tau )^\alpha C_1^*}{c_p^*(1+\tau (1-\alpha (1-\varepsilon )))}} \end{align}
\begin{align} D(c_0) = \frac {\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}}{\left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )}\frac {r^*}{p} + (1 -\delta ) - \frac {\mu \alpha \frac {(1-\eta )}{2}r^*K^* \Omega _2}{c_p^*\left (1+\tau (1-\alpha (1-\varepsilon ))\right )\!\! \left ( \!\sigma - \frac {(1+\tau )c_0}{P_1^*C_1^*} \right )+ \mu \alpha P (1+\tau )^\alpha C_1^*} \end{align}
$\mu _H(\tau ,\varepsilon )$
is such that
$D=1$
, which yields
\begin{equation} \mu _H(\tau ,\varepsilon ) = \frac { \left [\frac {\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}}{\left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )}\frac {r^*}{p} -\delta \right ]c_p^*\left (1+\tau (1-\alpha (1-\varepsilon ))\right ) \left ( \frac {(1+\tau )c_0}{P_1^*C_1^*} - \sigma \right ) }{\alpha P(1+\tau )^\alpha C_1^* \left (\frac {r^*}{p} - \delta \right )} \end{equation}
$\mu _F(\tau ,\varepsilon )$
is such that 1+Tr+D=0:
\begin{align}& \mu _F(\tau ,\varepsilon )\nonumber \\ &= \frac {- c_p^*\left (1+\tau (1-\alpha (1-\varepsilon ))\right ) \left (2 \left [\frac {\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}}{\left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )}\frac {r^*}{p} + (2 -\delta ) \right ] \left ( \sigma - \frac {(1+\tau )c_0}{P_1^*C_1^*} \right ) + f''(K^*)K^* \Omega _2 \right )}{2\alpha P(1+\tau )^\alpha C_1^* \left (\frac {r^*}{p} + 2-\delta \right )} \nonumber \\[1pt] & = \frac { 2c_p^* \left (1+\tau (1-\alpha (1-\varepsilon ))\right ) \!\left [\frac {\frac {1+\eta }{2}+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )}}{\left (1+\frac {\alpha \varepsilon \tau }{1+\tau (1-\alpha )} \right )}\frac {r^*}{p} + (2 -\delta ) \right ]\!\left (\frac {(1+\tau )c_0}{P_1^*C_1^*} - \sigma\! \right )}{2\alpha P(1+\tau )^\alpha C_1^* \left (\frac {r^*}{p} + 2-\delta \right )} + \frac {f''(K^*)c_p^*(1+\tau (1-\alpha ))}{2 \alpha p\left (\frac {r^*}{p} + 2-\delta \right )} \end{align}
A.11 Proof of Lemma2
The derivatives of
$\mu _H$
and
$\mu _F$
with respect to
$\varepsilon$
are given by:
\begin{align*} \frac {\partial \mu _H}{\partial \varepsilon }& = \frac {\alpha P(1+\tau )^\alpha \left ( \frac {r*}{p}-\delta \right )c_p^*}{\left (\alpha P(1+\tau )^\alpha C_1^* \left (\frac {r^*}{p} - \delta \right )\right )^2}\\&\quad \times \left [ \left (\frac {\partial \Omega _1}{\partial \varepsilon }(1+\tau (1-\alpha (1-\varepsilon )))+ (\Omega _1 -1) \alpha \tau \right )C_1^*\left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right ) \right . \\[5pt] &\left .\quad - \frac {\partial C_1^*}{\partial \varepsilon } (\Omega _1-1)(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {P(1+\tau )^{\alpha +1}c_0C_1^*}{(P_1^*C_1^*)^2} + \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\right ]\\[5pt] \frac {\partial \mu _F}{\partial \varepsilon } & = \frac {c_p^*\alpha P(1+\tau )^\alpha \left ( \frac {r^*}{p}+2-\delta \right )}{\left (\alpha P(1+\tau )^\alpha C_1^* \left (\frac {r^*}{p} + 2-\delta \right ) \right )^2}\\ & \quad \times \left [\left (\alpha \tau (1+\Omega _1) + \frac {\partial \Omega _1}{\partial \varepsilon } (1+\tau (1-\alpha (1-\varepsilon )))\right )C_1^*\left (\frac {(1+\tau )c_0}{P_1^*C_1^*} - \sigma \right ) \right . \\[5pt] & \left . \quad - \frac {\partial C_1^*}{\partial \varepsilon }(1+\tau (1-\alpha (1-\varepsilon ))) (1+\Omega _1) \left (\frac {P(1+\tau )^{\alpha +1}c_0 C_1^*}{(P_1^*C_1^*)^2} + \left (\frac {(1+\tau )c_0}{P_1^*C_1^*} - \sigma \right )\right )\right ] \end{align*}
Both thresholds increase with the redistribution rate when the elasticity of consumption of the capitalist with respect to their redistribution rate is low enough, i.e. when consumption does not react strongly to more redistribution. As the threshold for the increase in
$\mu _H$
is greater than the one on
$\mu _F$
, a condition for both thresholds to increase with
$\varepsilon$
is:
\begin{gather} \frac {\partial C_1^*}{\partial \varepsilon }\frac {\varepsilon }{C_1^*} \leq \frac {\varepsilon \left (\alpha \tau (1+\Omega _1) + \frac {\partial \Omega _1}{\partial \varepsilon } (1+\tau (1-\alpha (1-\varepsilon )))\right )\left (\frac {(1+\tau )c_0}{P_1^*C_1^*} - \sigma \right )}{(1+\tau (1-\alpha (1-\varepsilon ))) (1+\Omega _1) \left (\frac {P(1+\tau )^{\alpha +1}c_0 C_1^*}{(P_1^*C_1^*)^2} + \left (\frac {(1+\tau )c_0}{P_1^*C_1^*} - \sigma \right )\right )} \equiv \overline {\epsilon _{C_1^*,\varepsilon }}. \end{gather}
Derivatives with respect to
$\tau$
yield:
\begin{align*} \frac {\partial \mu _H}{\partial \tau } &= \frac {\alpha P(1+\tau )^\alpha \left ( \frac {r*}{p}-\delta \right )}{\left (\alpha P(1+\tau )^\alpha C_1^* \left (\frac {r^*}{p} - \delta \right )\right )^2}\\[5pt]&\quad \times \left [\left (\frac {\partial \Omega _1}{\partial \tau }c_p^*(1+\tau (1-\alpha (1-\varepsilon )))+ (\Omega _1 -1) \frac {\partial c_p^*}{\partial \tau }(1+\tau (1-\alpha (1-\varepsilon ))) \right .\right . \\[5pt]& \left . \quad+\, (\Omega _1 -1) c_p^* (1-\alpha (1-\varepsilon ))\Bigg)C_1^*\left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right ) \right . \\[5pt] & \left . \quad+\, (\Omega _1 -1)C_1^*c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {P_1^*C_1^*c_0-(1+\tau )c_0\frac {\partial P_1^*C_1^*}{\partial \tau }}{(P_1^*C_1^*)^2}\right ) \right .\\[5pt] & \left . \quad- \left ( \alpha \frac {C_1}{1+\tau } + \frac {\partial C_1^*}{\partial \tau }\right ) \left ((\Omega _1-1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\right )\right ] \end{align*}
\begin{align*} \frac {\partial \mu _F}{\partial \tau } &= \frac {\alpha P(1+\tau )^\alpha \left ( \frac {r*}{p}+2-\delta \right )}{\left (\alpha P(1+\tau )^\alpha C_1^* \left (\frac {r^*}{p}+2 - \delta \right )\right )^2}\\ & \quad\times \left [\left (\frac {\partial \Omega _1}{\partial \tau }c_p^*(1+\tau (1-\alpha (1-\varepsilon )))+ (\Omega _1 +1) \frac {\partial c_p^*}{\partial \tau }(1+\tau (1-\alpha (1-\varepsilon ))) \right .\right . \\[5pt] & \left .\left . \quad+\, (\Omega _1 +1) c_p^* (1-\alpha (1-\varepsilon ))\right )C_1^*\left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right ) \right . \\[5pt] & \left . \quad+\, (\Omega _1 +1)C_1^*c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {P_1^*C_1^*c_0-(1+\tau )c_0\frac {\partial P_1^*C_1^*}{\partial \tau }}{(P_1^*C_1^*)^2}\right ) \right . \\[5pt] & \left . \quad- \left ( \alpha \frac {C_1}{1+\tau } + \frac {\partial C_1^*}{\partial \tau }\right ) \left ((\Omega _1+1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\right )\right ]\\[5pt] &\quad + \frac {f''(K^*)c_0(1-\alpha )}{ \alpha p \left ( \frac {r^*}{p}+2-\delta \right )} \end{align*}
Both thresholds increase with the tax rate when the elasticity of consumption of the capitalist with respect to
$\tau$
is low enough, i.e. when consumption does not react strongly to higher taxation:
$\frac {\partial C_1^*}{\partial \tau }\frac {\tau }{C_1^*} \leq \overline {\epsilon _{C_1^*,\tau }}$
, with
\begin{align}&\overline {\epsilon _{C_1^*,\tau }} \equiv min \left ( \frac {\begin{aligned} &\tau \left(\frac {\partial \Omega _1}{\partial \tau }c_p^*(1+\tau (1-\alpha (1-\varepsilon )))+ (\Omega _1 -1) \frac {\partial c_p^*}{\partial \tau }(1+\tau (1-\alpha (1-\varepsilon ))) + (\Omega _1 -1) c_p^* (1-\alpha (1-\varepsilon ))\right)\left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\end{aligned}}{ \begin{aligned}&\left( \left ((\Omega _1-1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\right ) \right. \left.\quad + (\Omega _1 -1)C_1^*c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \frac {(1+\tau )^{\alpha +1} P c_0}{\left(P_1^*C_1^*\right)^2}\right)\end{aligned}} \right . \nonumber \\[5pt]& \left . + \frac { \begin{aligned}& \tau \left((\Omega _1 -1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {P_1^*C_1^*c_0-(1+\tau )c_0 \left (\alpha (1+\tau )^{\alpha -1}PC_1^* + c_0 \right )}{(P_1^*C_1^*)^2}\right )\right. \left. - \left ( \frac {\alpha }{1+\tau } \right ) \left ((\Omega _1-1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\right )\right)\end{aligned}}{\begin{aligned} &\left( \left ((\Omega _1-1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\right ) \right. \left. + (\Omega _1 -1)C_1^*c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \frac {(1+\tau )^{\alpha +1} P c_0}{(P_1^*C_1^*)^2}\right)\end{aligned}},\right. \nonumber\\& \left . \frac {\begin{aligned} &\tau \left(\frac {\partial \Omega _1}{\partial \tau }c_p^*(1+\tau (1-\alpha (1-\varepsilon )))+ (\Omega _1 +1) \frac {\partial c_p^*}{\partial \tau }(1+\tau (1-\alpha (1-\varepsilon ))) + (\Omega _1 +1) c_p^* (1-\alpha (1-\varepsilon ))\right)\left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\end{aligned}}{\begin{aligned} &\left( \left ((\Omega _1+1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\right ) + (\Omega _1 +1)C_1^*c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \frac {(1+\tau )^{\alpha +1} P c_0}{(P_1^*C_1^*)^2}\right )\end{aligned}} \right . \nonumber \\[5pt]& \left . + \frac {\begin{aligned} & \tau \left((\Omega _1 +1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {P_1^*C_1^*c_0-(1+\tau )c_0 \left (\alpha (1+\tau )^{\alpha -1}PC_1^* + c_0 \right )}{(P_1^*C_1^*)^2}\right ) - \left ( \frac {\alpha }{1+\tau } \right ) \left ((\Omega _1+1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\right ) +\frac {f''(K^*)c_0(1-\alpha )}{ C_1^*\alpha p \left ( \frac {r^*}{p}+2-\delta \right )}\right)\end{aligned}} { \begin{aligned} &\left( \left ((\Omega _1+1)c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \left ( \frac {(1+\tau )c_0}{P_1^*C_1^*}-\sigma \right )\right ) + (\Omega _1 +1)C_1^*c_p^*(1+\tau (1-\alpha (1-\varepsilon ))) \frac {(1+\tau )^{\alpha +1} P c_0}{(P_1^*C_1^*)^2}\right)\end{aligned}} \right ) \end{align}
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