« Environmental Policy and Growth when Environmental Awareness

is Endogenous »

Karine CONSTANT Marion DAVIN

DR n°2016-08

Environmental Policy and Growthwhen Environmental Awareness is Endogenous �

Karine Constant: and Marion Davin;

Abstract

This paper examines the relationship between environmental policy and growth whengreen preferences are endogenously determined by education and pollution. We consideran environmental policy in which the government implements a tax on pollution andrecycles the revenue to fund pollution abatement activities and/or an education subsidy(influencing green behaviors). When the sensitivity of agents’ environmental preferencesto pollution and human capital is high, the economy can converge to a balanced growthpath equilibrium with damped oscillations. We show that this environmental policy canboth remove the oscillations, associated with intergenerational inequalities, and enhancethe long-term growth rate. However, this solution requires that the revenue from the taxrate must be allocated to education and direct environmental protection simultaneously.We demonstrate that this type of mixed-instrument environment policy is an effectiveway to address environmental and economic issues in both the short and the long run.

JEL Classification: I25; O44; Q58Keywords: Environmental Policy, Endogenous Growth, Environmental Awareness, Educa-tion.

1 Introduction

One of the main questions addressed by the literature on the relationship between eco-nomic growth and the environment is how environmental policy can be used to attain sustain-able development, in which economic growth is compatible with environmental conservation.1

Policy makers have at their disposal a number of economic policy instruments developed to�We would like to thank the editor and the two anonymous referees for their wise remarks. We are

also grateful to Carine Nourry, Thomas Seegmuller, Mouez Fodha, Oded Galor, Natacha Raffin, KathelineSchubert and Cees Withagen for their helpful comments and to participants at the following conferences:OLG Days (2013, Clermont-Ferrand), LAGV (2013, Marseille), PET (2013, Lisbon), SURED (2014, Ascona)and FAERE (2014, Montpellier).

:Université Paris Ouest Nanterre la Défense, EconomiX Building G, office 517b, 200 av. de la République,92001 Nanterre cedex, France. E-mail: constant.karine@gmail.com.

;University of Montpellier, LAMETA, avenue Raymond Dugrand - Site Richter C.S. 79606, F- 34960Montpellier Cedex 2, France. E-mail: marion.davin@umontpellier.fr

1See Brock and Taylor (2005) and Xepapadeas (2005) for literature reviews on this relationship.

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achieve this outcome. The most obvious are pollution taxation and public pollution abate-ment activities (e.g. water treatment, waste management, investment in renewable energy,conservation of forests, etc.), both of which have been designed in order to reduce the impactof economic activities on the environment. Yet governments may also invest in another typeof policy tool that aims to improve environmental outcomes only by modifying householdbehavior. Specifically, investing in education can be employed as an indirect intervention toprotect the environment, as educational attainment raises environmental conscientiousness.This idea is largely supported by international organizations. For example, the United Na-tions declared 2005-2014 to be the “UN Decade of Education for Sustainable Development”2,while the OECD (2008) refers to education as “one of the most powerful tools for providingindividuals with the appropriate skills and competencies to become sustainable consumers”.The European Commission (2005) recognizes also that education is a prerequisite for sus-tainable development and underlines the importance of combining a variety of policy toolsto meet this challenge. This point of view is shared by the OECD (2007, 2010), which em-phasizes that policy instruments are likely to reinforce each other. Moreover, it appears thatpollution taxation may not yield the expected impacts without the use of complementarypolicies that target household behavior. Despite the growing interest in such environmen-tal policy schemes, especially by international organizations, no theoretical studies have yetexamined this combination. Thus, the purpose of this paper is to investigate how an envi-ronmental policy that can combine pollution taxation, pollution abatement and educationalsupport affects economic activities and the environment.

In studying policies that target consumer behavior, it is especially relevant to considerthe role of agents’ preferences for the environment, which determine how agents respond topollution. In this regard, Bednar-Friedl (2012) shows that environmental preferences, and inparticular differences in these preferences between emerging and industrialized countries, arekey factors in determining the specific climate policy that maximizes welfare. Moreover, theempirical literature highlights that environmental awareness evolves over time and changesaccording to the economic and environmental context (see e.g. Dunlap and Scarce , 1991;European Commission , 2008 or Scruggs and Benegal , 2012). We therefore take into ac-count the endogeneity of individual green preferences in order to examine the consequencesof environmental policies on growth. More precisely, we consider the role of two major de-terminants of environmental awareness identified empirically: individual human capital andpollution level. With respect to human capital, the intuition is that the more educated anagent, the more she is informed about environmental issues, and the greater her potentialconcern about environmental protection (see e.g. Blomquist and Whitehead , 1998; Witzkeand Urfei , 2001 or European Commission , 2008). Pollution is thought to influence environ-

2See resolution 57/254 of United Nations General Assembly of 2002.

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mental awareness mainly due to the fact that environmental issues, such as climate changeand air pollution, harm welfare. High levels of pollution thus draw cognitive attention toenvironmental problems, compelling households to recognize the severity of these issues and,perhaps, to react (see e.g. Dunlap and Scarce , 1991; Tjernström and Tietenberg , 2008 orSchumacher , 2009).

With respect to endogenous environmental awareness, our analysis is related to the recentcontribution made by Prieur and Bréchet (2013), who are the first to consider the effect ofhuman capital on green preferences. The authors argue that public education may enablean economy to escape being caught in a steady state situation with no economic growth andinstead to achieve a long-term equilibrium with sustainable growth. In another recent article,Schumacher and Zou (2015) examine the case in which a pollution threshold determines thedegree to which agents value the environment. They show that the level of this threshold isa key determinant of the long-term behavior of the economy, in terms of both environmentalquality and economic dynamics. Here, we extend these papers by assuming that educationalchoices are not exogenous but stem from paternalistic altruism and that environmentalpreferences are driven by both human capital and pollution levels.

Studying the economic implications of an environmental policy, our analysis also con-tributes to the wide literature on the link between environmental policy and growth, and inparticular to the consideration of the role of human capital accumulation in this relationship.For example, Gradus and Smulders (1993) conclude that an improvement in environmentalquality can affect the long-term growth of the economy only when pollution directly affectshuman capital accumulation. More recently, Grimaud and Tournemaine (2007) or Pautrel(2015) find that a tighter environmental tax can favor education and hence growth in thelong run. This occurs because the tax makes polluting activities less attractive compared toa human-capital intensive sector. We depart from these papers in three major ways.

First, we analyze the effect of a “mixed-instrument” environmental policy in line with rec-ommendations made by international organizations regarding growth and the environment.More precisely, we model the implementation of a government tax on pollution whose revenuecan be recycled into two types of environmental interventions: public pollution abatementactivities and an education subsidy. The former represents an investment in environmentalprotection, and the latter aims to raise environmental awareness.

Second, in order to study the effect of such an educational intervention on green prefer-ences, we take into account agents’ endogenous preferences for the environment, as mentionedabove.

Third, given that the economic implications of environmental policies are generally stud-ied in the long run, we also investigate the consequences of this mixed-instrument environ-mental policy in the short run. Consequently, we follow the approach of contributions such

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as Zhang (1999) and Ono (2003), who recognize that short-term analysis is a crucial topicof study and that preferences for the environment play an important role in this type ofanalysis. In particular, Zhang (1999) shows that the economy may exhibit cyclical behaviorwhen green preferences are not sufficiently high and suggests that an environmental policymay be required in order to smooth convergence toward a sustainable equilibrium. In con-trast, Ono (2003) finds that high levels of concern for the environment may cause economicfluctuations and demonstrates the importance of examining both short- and long-term policyimplications by showing that an increase in a tax on pollution can mitigate these fluctuationsand have a non-monotonous impact on the long-term growth depending on the size of thetax. We contribute to this literature by examining the implications of a mixed-instrumentenvironmental policy, that employs both environmental and educational interventions, whenenvironmental awareness is endogenous.

We develop an overlapping generations model that incorporates environmental quality,and in which growth is driven by human capital accumulation. In this model, productioncreates a pollution flow, which damages environmental quality, whereas abatement activitiesimprove it. Using this model, we show that, the economy can converge to a sustainable long-term equilibrium, in which both environmental quality and human capital grow. We alsofind that the endogeneity of environmental preferences can cause the economy to experienceoscillations along the convergence path. More precisely, when household environmentalawareness is highly sensitive to levels of pollution and human capital, green preferencesfluctuate across generations, and this affects the household’s trade-off between education andabatement activities. This dynamics leads, therefore, to significant variations in the levels ofhuman capital and environmental quality across generations and produces intergenerationalinequalities that present a challenge for policy makers.

Furthermore, we show that an increase in pollution tax can eliminate these intergen-erational inequalities and improve the long-term growth rate.3 This win-win scenario isachieved when the tax revenue is allocated to both public pollution abatement activitiesand an education subsidy, in particular when the allocation enables households to investsufficiently in education while the provision of environmental maintenance is entirely public.When environmental protection is entirely done by public authorities, households can focuson education which makes their behavior less dependent on the economic and environmentalcontext so that intergenerational inequalities do not occur. This policy generates a level ofenvironmental quality that is high enough to induce agents to stop contributing to abate-ment activities (despite a joy of giving for it), while at the same time providing sufficientsupport to education. In this way, the policy results in a higher sustainable rate of growth.

3We do not conduct a formal welfare analysis because this is not analytically tractable in our setting, butwe do provide intuitions about the economic impacts of the environmental policy under consideration.

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Thus, we conclude that this mixed-instrument environmental policy can successfully addressboth short- and long-term environmental issues.

The paper is organized as follows. In Section 2, we introduce the theoretical model.Section 3 focuses on the long-term equilibrium and the transitional dynamics. In Section 4,we examine the short- and long-term implications associated with the environmental policyin question. Finally, Section 5 concludes. Technical details are provided in the Appendix.

2 The model

We consider an overlapping generations economy, with discrete time indexed by t �

0, 1, 2, ...,8. Households live for two periods, childhood and adulthood, and make all deci-sions during the second period. At each date t, a new generation of N identical agents isborn (N ¡ 1). We assume no population growth.

2.1 Consumer behavior

Individuals born in t� 1 care about their levels of consumption ct and the level of envi-ronmental quality Qt when they are adults. Through paternalistic altruism, they value thehuman capital of their children ht�1, such that parents finance their children’s education(as e.g. in Glomm and Ravikumar , 1992). Finally, agents also have altruistic preferencesfor the environment. Given that environmental quality is a public good, we refer to thebehavioral economics literature in order to identify the motives for private provision of thistype of collective good. In this literature, both theoretical and empirical studies suggest thatsuch contributions, including investment in environmental protection, arise from an “impurealtruism” (e.g. Ribar and Wilhelm , 2002; Menges et al. , 2005 or Crumpler and Grossman, 2008).4 We thus consider two motives that agents may have for private provision of envi-ronmental protection: a “pure altruism” for the level of environmental quality bequeathedto their children (i.e. the future environmental quality Qt�1) and a “joy of giving” associ-ated with the act of contributing itself (i.e. environmental maintenance mt). In addition toproperly representing environmental preferences, the formalization of impure altruism is im-portant for studying environmental policy. This is because the “joy of giving” motivationalaspect means that public and private contributions are not perfect substitutes (see Andreoni, 1990).

The preferences of a representative agent, born in t� 1, are represented by the followingutility function:

Upct,mt, ht�1, Qt�1q � ln ct � γ1t lnpε1 mt � ε2 Qt�1q � γ2 ln ht�1 � γ3 lnQt (1)4This formalization is not relevant for human capital, as it is a private good.

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with γ1t, γ2, γ3, ε1 and ε2 ¡ 0.The parameters γ3 and γ2 capture an agent’s usual taste for current environmental quality

and the preference for her child’s human capital, respectively.5

The weight γ1t captures environmental awareness. We assume that these environmentalpreferences are positively affected by the level of pollution and of individual human capital, assupported by the literature.6 Pollution has an impact on environmental behaviors through itseffect on welfare. The lower the level of environmental quality, the greater the opportunity foran individual to realize the severity of the situation and therefore the greater her incentiveto protect the environment, as Dunlap and Scarce (1991) and Schumacher (2009) note.Higher environmental quality, in contrast, may reduce an agent’s willingness to improvethe environment, as doing so appears less necessary under such conditions. The empiricalbehavioral economics literature also identifies education as a key determinant of contributionsmade to improving environmental quality (see Blomquist and Whitehead , 1998 or Witzkeand Urfei , 2001). The economic intuition behind this assertion is that the higher an agent’seducation level, the higher the likelihood that she is informed about environmental issuesand their consequences, and thus the greater her potential environmental concern. We thusassume that γ1t � γ1pht, Qtq where γ1 is increasing and concave with respect to humancapital h, and decreasing and convex with respect to environmental quality Q.

For tractability reasons, we consider the following functional form, in line with the formcommonly used to represent endogenous longevity, defined as the weight of future argumentsin the utility function (see e.g. Blackburn and Cipriani , 2002):

γ1t �βht � ηQtht �Qt

(2)

with parameters β, η P r0, 1s and β ¥ η.7 The parameters β and η embody the weight ofhuman capital and of environmental quality in green preferences, respectively. We note thatwhen β � η, environmental awareness is constant.

During childhood, the agent does not make any decisions. She is reared by her parentsand benefits from education.8 After reaching adulthood, the agent provides an inelasticsupply of one unit of labor remunerated at wage wt according to her level of human capitalht. She allocates this income to consumption ct, education per child et and environmental

5Current environmental quality is not related to altruism concerns and is taken as given by agents, asagents have no impact on the current level of environmental quality.

6In this paper, we consider pollution as the opposite of the environmental quality index Q. Thus, we willuse both concepts interchangeably.

7The assumption β ¡ η is required in order to ensure a positive effect of human capital on environmentalawareness.

8As is the standard in overlapping generations models, we assume that there is neither work nor explicitconsumption in the first period and that a child’s consumption is included in that of the parents.

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maintenance mt.9 The fact that environmental awareness depends on human capital impliesthat, public authorities can use educational support as an environmental policy tool. In thissense, we suppose that the government can subsidize education at rate 0 ¤ θet   1 usingthe revenue derived from a pollution tax that we will present later. This policy reduces theprivate cost of education, such that the budget constraint for an adult with human capitalht is:

ct �mt � etp1� θet q � wtht (3)

A child’s level of human capital is determined by the private education expenditure andhuman capital level of her parents, so that for a child born in t, her level of human capitalas an adult is:

ht�1 � ε etµht

1�µ (4)

where ε ¡ 0 is the efficiency of human capital accumulation. The parameter 0   µ   1is compatible with endogenous growth and captures the elasticity of human capital withrespect to private education, while 1 � µ represents the share of human capital resultingfrom intergenerational transmission within the family.

The index of environmental quality we consider in this paper reflects an agent’s perceptionof her local environment and corresponds to the amenity value derived from, for example,resource availability, the quality of air, water and soil, or the quality of national parks. Thelaw of motion for the environmental quality index is defined as:10

Qt�1 � p1� αqQt � aYt � bpmt �Mt �NGmt q (5)

where α ¡ 0 is the natural degradation of the environment and Yt represents the pollutionflow due to production in the previous period. The parameter a ¡ 0 corresponds to theemission rate of pollution, while b ¡ 0 represents the efficiency of environmental mainte-nance. These pollution abatement activities are represented by a Cournot-Nash equilibriumapproach. Each agent determines her own level of environmental maintenance activity (mt),taking the maintenance level provided by others (Mt) as given. The government can alsouse the revenue of the pollution tax to directly improve environmental quality, by investingpublic funds in environmental maintenance activities NGmt ¥ 0. Given our definition ofthe environment, public maintenance activities refer in particular to the implementation oflocal public projects aimed at protecting the environment.11 This expenditure corresponds,

9See Kotchen and Moore (2008) for empirical evidence relating to the private provision of environmentalpublic goods.

10Following the seminal contribution of John and Pecchenino (1994), Q is an environmental quality indexwith an autonomous value of 0 in the absence of human intervention. Moreover, we consider Q ¡ 0, which isa standard assumption (see e.g. Ono , 2003 or Mariani et al. , 2010).

11According to our definition of the environment, we do not address policies that tackle issues of global

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for example, to water treatment, waste management or the conservation of biodiversity andlandscape. As a result, both public and private efforts to protect the environment coexistand have an impact on the level of environmental quality. To avoid arbitrarily favoring onetype of pollution abatement over another, we assume that the efficiency of public and privateinvestments in environmental maintenances are equal.

The consumer program is summarized by:

maxet,mt

, Upct,mt, ht�1, Qt�1, Qtq � ln ct � γ1t lnpε1 mt � ε2 Qt�1q � γ2 ln ht�1 � γ3 lnQt (6)

s.t ct �mt � etp1� θet q � wtht

ht�1 � ε etµht

1�µ

Qt�1 � p1� αqQt � aYt � bpmt �Mt �NGmt q

with mt ¥ 0.

2.2 Production

The production of the consumption good is carried out by a single representative firm.Output of this good is produced according to a constant returns to scale technology:

Yt � AHt (7)

where Ht is the aggregate stock of human capital and A ¡ 0 measures a technology param-eter.

Because pollution is a by-product of the production process, human capital exerts twoopposite effects on the environment: a positive effect through increasing an agent’s level ofenvironmental awareness and thus her motivation for environmental protection, and a nega-tive effect through an increase in production that raises the amount of waste and pollutionemissions. Indeed, even if human capital is not a highly polluting input per se, it is a deter-mining factor in the scale of production and hence of emissions.12 In order to maintain thetractability of the analysis as well as a focus on the mechanisms linked to human capital,we assume that the only input in the production function is aggregate human capital andthat the share of polluting factors in the production process is constant and represented bythe parameter A.13 Therefore, the polluting production process is composed of an index

pollution (e.g. greenhouse gas emission limits).12This duality is supported by the fact that individuals in high-income economies generally have higher

green preferences but also a larger ecological footprint per capita than those in low-income countries (fivetimes higher, see WWF , 2014).

13Introducing physical capital accumulation, for example, would render the analysis much more complex,as it adds an additional dimension to the dynamics.

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of pollution intensity A as well as the level of human capital, and together these elementsdetermine the pollution emissions.

Defining yt � YtN as the output per worker and ht � Ht

N as the human capital per worker,we have the following production function per capita:

yt � Aht (8)

The government collects revenues through a tax rate 0 ¤ τ   1 on production, the sourceof pollution.14 The firm chooses input to maximize its profit p1� τqYt � wtHt, such that :

wt � Ap1� τq (9)

2.3 The government

The design of environmental policies represents a major challenge for governments. OECD(2007, 2008), among others, recommends recycling pollution tax revenues as a way for gov-ernments to mitigate the externalities generated by polluting activities. This type of policyhas precedent in several countries. In France, for example, the government implements a gen-eral tax on polluting activities (TGAP) and transfers the revenues of this tax to the FrenchEnvironment and Energy Management Agency (ADEME) that funds projects to improveenvironmental quality.

In this model, we consider the following policy scheme: since pollution is a by-productof the production process, the government taxes production output at rate τ and the publicbudget generated by the revenue from this tax is spent on public environmental maintenanceNGmt and/or on education subsidies θet .15 The government’s budget is balanced at eachperiod, such that:

Npθet et �Gmt q � τYt (10)

To study possible policy mixes in a simple way, we define the constant share of publicexpenditure that is devoted to public environmental maintenance as 0 ¤ σ ¤ 1, and theshare that is devoted to an education subsidy as p1� σq, with

σ �NGmtτYt

; 1� σ �Nθet etτYt

(11)

such that the fiscal policy is summarized by two instruments: the pollution tax τ andthe allocation of public revenue σ, both taken as given by consumers. With this general

14Our main results remain valid if we consider, alternatively, a tax on consumption.15We consider a public investment in environmental maintenance rather than a subsidy for private spending

because the latter is not analytically tractable. We do, however, model an education subsidy because ourformalization precludes an analysis of public education spending.

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formalization, we examine the specific effects of each policy. We accomplish this by setting σto zero in order to model the case in which the government supports only education, and bysetting σ to one in order to examine the case in which the government finances only publicmaintenance activities. Finally, we also model a mixed-instrument scenario in which publicfunds are allocated to a combination of environmental maintenance and education subsidies.

2.4 Equilibrium

The maximization of the consumer program (6) leads to optimal choices regarding ed-ucation and maintenance in two regimes : an interior solution, in which both individualsand the government invest in environmental protection mt ¡ 0 (hereafter pm) and a cornersolution in which the government is the only contributor to the public good i.e. mt � 0(hereafter npm). The intertemporal Nash equilibria are given by:

mt �

$&%

γ1tc1Ahtp1�τq�ε2p1�γ2µqrp1�αqQt�ANhtpbστ�aqsγ1tc1�c2�ε2bNp1�γ2µq

pm

0 npm(12)

et �

$&%

γ2µrc3Ap1�τqht�ε2pp1�αqQt�ANhtpbστ�aqqsγ1tc1�c2�ε2bNp1�γ2µq

� p1� σqτAht pmAhtrγ2µp1�τq�τp1�σqp1�γ2µqs

1�γ2µnpm

(13)

where c1, c2 and c3, three positive constants defined by c1 � ε1 � ε2b ; c2 � ε1p1� γ2µq andc3 � ε1 � ε2bN .16

Education spending depends positively on environmental quality. The greater the healthof the environment, the lower the optimal amount of maintenance activities. As a result,education becomes more attractive and individuals are able to devote more resources toeducating their children.

Moreover, public policy instruments shape education and abatement spending in differentways. An increase in the pollution tax implies a negative income effect (wage decreases) butcontinues to favor education spending as long as the share of public expenditure in educationsubsidies is sufficiently high (σ low). However, an increase in the pollution tax always has anegative effect on environmental maintenance activities. In addition to the negative incomeeffect, the tax increases the amount of public pollution abatement activities, which crowds outprivate maintenance activities. Nevertheless, public spending on environmental maintenanceis only a partial substitute for private spending due to the joy-of-giving motive in the utilityfunction.

Remark 1 Below we note the implications of the model without a joy-of-giving motive forthe environment (i.e. ε1 � 0):

16See details in Appendix 6.1.1.

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• If all public expenditure were devoted to pollution abatement (σ � 1), there would bea total crowding-out of private maintenance such that the environmental policy wouldhave no effect on the environment.

• If the budget were allocated to both types of expenditure (σ   1), the fall in privatemaintenance would outweigh the increase in public maintenance. Consequently, theoverall level of abatement activity would decrease with the implementation of the envi-ronmental policy.

This scenario contradicts the empirical and experimental literature, which demonstrates onlya partial crowding-out of private contributions by government expenditure.17 Thus, the joy-of-giving motive (ε1) is necessary for a meaningful policy analysis in our setting.

To study endogenous growth, we write the environment in intensive form, by consid-ering environmental quality per unit of human capital Qt

ht� Xt. This ratio enables us to

capture the evolution of environmental quality relative to the development of economic ac-tivities. For the remainder of the analysis, we define Xt as a green development index. Thus,environmental awareness, given by (2), can be rewritten:

γ1t �β � ηXt

1�Xt(14)

Using equations (12) and (14), we emphasize that households invest in environmentalprotection for:

Xt ¤A�pβ�ηXtqc1p1�τq

1�Xt � ε2Np1� γ2µqpbστ � aq�

ε2p1� γ2µqp1� αq(15)

From this inequality, we deduce that there exists a critical threshold Λpσ, τq ¥ 0 suchthat environmental protection is privately supported as long as:18

Xt ¤ Λpσ, τq (16)

When this inequality is unsatisfied (Xt ¡ Λpσ, τq), the level of environmental quality is suffi-ciently high enough, and/or the level of human capital sufficiently low enough, to reduce theprivate provision of abatement activities to zero. The critical threshold Λpσ, τq is decreasingin the tax on pollution τ and in the share of public spending devoted to environmental main-tenance σ. Therefore, when the government intervenes, the economy is more likely to becharacterized by no private maintenance activities, since public maintenance partially sub-stitutes for private maintenance. This result continues to hold if the recycled revenues are

17See e.g. Ribar and Wilhelm (2002), Menges et al. (2005) or Crumpler and Grossman (2008).18See technical details in Appendix 6.1.2.

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entirely devoted to education, since agents replace some abatement activities with educationas the latter becomes relatively less costly. On the other hand, the higher an agent’s greenpreferences, the higher the level of environmental quality at which the agent decides to stopsupporting environmental protection (the higher the critical threshold Λpσ, τq).

Using the human capital accumulation (4), the environmental quality process (5), andthe first order conditions (12) and (13), we write the dynamic equation characterizing theequilibrium paths.

Definition 1. Given the initial condition X0 �h0Q0

¡ 0, the intertemporal equilibrium is thesequence pXtqtPN which satisfies, at each t, Xt�1 � FpXtq, with:

FpXtq �

$'&'%

p1�αq Xt rγ1t c1�c2s�AN rγ1t c1bp1�τq�pγ1c1�c2qpbτσ�aqs

εrγ2µA c3p1�τq�γ2µε2rp1�αqXt�ANpbστ�aqs�p1�σqτApγ1t c1�p1�γ2µqc3qsµrγ1t c1�p1�γ2µqc3s

1�µ pm

p1�αqXt�ANpbστ�aq

ε�γ2µAp1�τq�p1�σqτAp1�γ2µq

1�γ2µ

�µ npm

(17)

3 Balanced growth path and transitional dynamics

In this section, we examine the existence of a balanced growth path (hereafter BGP).

Definition 2. A balanced growth path (BGP) satisfies Definition 1 and has the followingadditional properties: the stock of human capital and environmental quality grow at the sameconstant rate gi, with subscripts i � tpm, npmu denoting respectively the regime with PrivateMaintenance and the regime with No Private Maintenance. This equilibrium path is suchthat the green development index Xt is constant and defined by Xt�1 � Xt � X̄i.

From Definitions 1 and 2 and equations (14) and (16), we explore the properties of thedynamic equation F and deduce the existence of a BGP X̄i corresponding to the solutionsof equation X̄i � FpX̄iq:

Proposition 1. When β ¡ η and σMinpτq   σ   σMaxpτq for all τ P r0, 1q, there exists aunique positive BGP pX̄iq, such that, according to a critical threshold σ̂pτq:

When σ ¡ σ̂pτq, the BGP occurs in the regime without private maintenance (npm).

When σ   σ̂pτq, the BGP occurs in the regime with private maintenance (pm).

where σMinpτq �apβc1�c2q�bβc1p1�τq

bτpβc1�c2q, σMaxpτq �

Apγ2µ�τq�ε�1{µp1�γ2µqp1�αq1{µAτp1�γ2µq

and σ̂pτq is adecreasing function of τ , with limτÑ0 σ̂pτq � �8 and limτÑ1 σ̂pτq � a{b.

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Proof. See Appendix 6.2.

Such a balanced growth path corresponds to a long-term equilibrium with sustainabledevelopment, in which both human capital and environmental quality improve across gen-erations. From Proposition 1, we identify two possible long-term regimes - with or withoutprivate provision of environmental maintenance. When the share of public spending devotedto environmental protection is sufficiently large (σ ¡ σ̂pτq), households stop privately in-vesting in abatement in the long run despite their willingness to protect the environment.This occurs due to the trade-off the agent makes between education and maintenance spend-ing. On the one hand, low public investment in education leads individuals to increase theirprivate efforts to finance education. On the other hand, the improvements in environmen-tal quality made possible by the policy reduce the need for private provision of abatementactivities. We note, however, that the greater the degree of environmental awareness (γ1),the more public spending on the environment is necessary in order to facilitate the long-runequilibrium in the no private maintenance regime (σ̂pτq increases with green preferences).

Whatever the regime, the existence of a long-term sustainable development depends onseveral conditions, summarized in Proposition 1 by σMinpτq   σ   σMaxpτq for all τ P r0, 1q.These conditions pertain to specific economic parameters and policy instruments employed.

With respect to economic parameters, the efficiency of human capital accumulation (ε)and of environmental maintenance (b) must be sufficiently high. In particular, a prerequisiteto obtaining sustainable development is that the environmental benefit resulting from oneunit of abatement is larger than the environmental damage resulting from one unit of produc-tion (b ¡ a). This seems a reasonable condition in our model, as the environmental qualitythat we consider is the quality that is perceived by agents. Since individuals preferentiallyinvest in abatement activities that target the environmental issues that matter most to themand pollution is simply a side-effect of the aggregate production process, then the marginalbenefit of abatement can be assumed to be relatively higher than the marginal damage ofproduction.

Second, the condition described above depends on how the government recycles pollutiontax revenue. To examine this issue further, we depict σminpτq and σmaxpτq in Figure 1. Asillustrated in this figure, extreme allocations of public funds are excluded for high levels ofthe tax (shaded area). The reason for this is that when the tax rate is high, the share of anagent’s income that is available for investment is low. When the share of the tax revenueused for education subsidies or environmental maintenance is relatively too great, one ofthe two actions that drive sustainable development is highly disadvantaged (too expensiveand not sufficiently supported through public funds). Thus, in order to achieve a state inwhich both environmental quality and the economy improve, the government must allocate

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its public budget judiciously (white area).

Figure 1: Environmental policy conditions that lead to sustainable development

0 τ

σMinpτq ab

σMaxpτq

σ 1

For the remainder of the paper, we assume the condition of Proposition 1 to be true:19

Assumption 1. For all τ P r0, 1q, we assume that σMinpτq   σ   σMaxpτq.

From the dynamic equation, we derive the stability properties of the BGP presented inProposition 2. Note that when the BGP occurs in the no private maintenance regime, weobtain an explicit solution whose dynamics is easily deduced. However, in order to analyzethe stability of the equilibrium in the private maintenance regime, we normalize X̄pm to oneusing the scaling parameter ε.

Proposition 2. Under Assumption 1 and β ¡ η:

• The BGP in the regime without private maintenance, X̄npm (i.e X̄ ¡ Λpσ, τqq, isglobally and monotonously stable.

• The BGP in the regime with private maintenance, X̄pm (i.e X̄   Λpσ, τqq, is locallystable and for N ¡ N̄pτ, σq, there exists a β̃pτ, σq P p0, 1s such that:

– when β   β̃pτ, σq, the convergence is monotonous.

– when β ¥ β̃pτ, σq, the convergence is oscillatory.

where Λpσ, τq is the critical threshold of Xt at which the economy is in the no private main-tenance regime (see eq. (16)).

Proof. See Appendix 6.3.

Figure 2 provides an illustration of the cases identified in Proposition 2.19If this condition does not hold, either no balanced growth path exists, or the growth rate of human capital

is always negative, causing the economy to collapse.

14

(a) when σ ¡ σ̂pτq

(b) when σ   σ̂pτq

Figure 2: Dynamics when N ¡ N

15

As underlined in Proposition 2, the economy may display damped oscillations due toendogeneity of green preferences.20 The oscillatory dynamics emerges when an agent’s envi-ronmental awareness γ1 is highly sensitive to the green development index X (β ¥ β̃) andwhen the population size is not too low (N ¡ N̄), such that individuals behavior has asignificant impact on environmental quality.21

The mechanism responsible for the emergence of this dynamics consists of a feedback ef-fect between the green development index and environmental awareness, which impacts thetrade-off between education and environmental maintenance spending. Indeed, in the ab-sence of private maintenance (i.e. X̄ ¡ Λpσ, τq), this trade-off does not exist, as householdsfocus solely on education, and the dynamics is therefore always monotonous. Reversely,when an agent invests in environmental protection (i.e. X̄   Λpσ, τq), cyclical convergencemay occur and can be described as follows. An increase in environmental awareness γ1t en-courages private investment in the environment at the expense of investment in education.This generates a fall in human capital ht�1, a rise in environmental quality Qt�1, and thusa decline in green preferences γ1t�1 for the next generation. These changes impact privatechoices in several ways. Each change negatively affects private investment in environmentalmaintenance mt�1, whereas their impact on private investment in education et�1 is ambigu-ous. Indeed, education spending is positively affected by the fact that private provision ofenvironmental quality becomes both less necessary (from the raise in Qt�1) and less desirable(from the decrease in γ1t�1), while it is negatively affected by an income effect due to thefall in human capital ht�1. The opposite variation of human capital ht�1 acts therefore as abrake on oscillations and serves a stabilizing effect, such that cyclical variations are damped.As a result, the economy displays oscillations as long as the two positive impacts on educa-tion exceed the negative income effect. Note that this is possible only when environmentalawareness is highly responsive to the economic and environmental context (β ¡ β̃), whichenables the positive impacts to dominate through γ1.

This oscillatory dynamics implies that environmental awareness (γ1) undergoes varia-tions along the converging trajectory. Because human capital and environmental quality aretwo stock variables, there is some inertia in preferences that mitigates extreme variationsacross generations. Significant variations in green preferences are, however, consistent withempirical evidence. For example, Dunlap and Scarce (1991) remind us that, although publicconcern for the environment has increased globally since the late 1960s, it has also experi-enced fluctuations according to economic and environmental context. Likewise, Scruggs and

20In the extreme case in which β � η, γ1 is exogenous and the dynamics is always monotonous. The proofis available upon request.

21The term sensitivity of preferences to X refers to the convexity and the elasticity of γ1 to X: the secondderivative is given by 2pβ�ηq

p1�Xq3 and the elasticity by �pβ�ηqXp1�Xqpβ�ηXq

. At a given X, oscillations occur for highelasticity or convexity.

16

Benegal (2012) show that the decline of public concern about climate change observed since2008 in the United States is due in large part to the great recession. These studies demon-strate that a rise in environmental problems tends to lead to a substantial increase in publicsupport for environmental protection, while an intensification of economic issues tends tohave the reverse effect.

In this paper, we find that fluctuations in environmental awareness cause variations inenvironmental quality and human capital across generations. It follows that some generationsexperience higher levels and growth rates of human capital and environmental quality thanothers. As Seegmuller and Verchère (2004) show, cyclical convergence paths result in welfarevariations across generations and are associated with intergenerational inequalities. Theterm inequality refers here to the fact that, when these fluctuations occur, the high levelsof environmental quality that are enjoyed by some generations are obtained at the expenseof other generations. As a result, this complex dynamics translates into cyclical short-rungrowth and hence is closely related to the concept of volatility, as Varvarigos (2011) argues.This result underscores the importance of studying the short-term effects associated withenvironmental policies.

4 Environmental policy implications

In this section, we analyze the consequences of the environmental policy on intergenera-tional inequalities in the short-run and on growth in the long run.

4.1 The short-term effect of the environmental tax

In the preceding analysis, we showed that the economy may exhibit complex dynamicswhen environmental awareness is endogenous. In this section we model how a tighter envi-ronmental tax affects this short-term scenario and whether the use of a policy mix allows fora reduction in intergenerational inequalities.22 Focusing on the BGP in the private mainte-nance regime, where damped oscillations may occur, we examine the effect of an increase inthe environmental tax on transitional dynamics.23

Proposition 3. Under Assumption 1 and β ¡ η, for a sufficient increase in the tax onpollution, σ becomes higher than σ̂pτq. In this way, the BGP that occurs in the regime withprivate maintenance moves to the regime without private maintenance, in which there areno oscillations.

22 The effect of the policy on oscillations is analyzed through local dynamics. In this case, the economy isclose to the BGP but the convergence occurs across several periods, illustrating the short-run dynamics.

23A BGP that initially occurs in the npm regime cannot shift to the pm regime, following an increase inthe tax.

17

An increase in the environmental tax makes more likely that agents do not privatelyinvest in environmental protection at the BGP (as σ̂pτq declines). The main reason for thisis that a tighter tax allows for an increase in the government’s budget, and hence in theamount of public environmental maintenance. Moreover, as there is no longer a trade-offbetween private choices in this regime, transitional dynamics do not result in oscillations.Thus, the government may avoid intergenerational inequalities by fixing a sufficiently hightax on pollution and devoting a large enough share of public spending to environmentalmaintenance (such that σ̂pτq   σ), enabling the long-term equilibrium to move to the regimein which environmental maintenance is entirely publicly funded.

The tax needed to achieve the regime without private provision of environmental mainte-nance may, however, be very high. In these extreme cases, the policy appears unreasonableas it deprives agents of consumption and reduces their welfare. In this regard, it seemspertinent to study the possible effects of the policy when the economy remains in the regimewith private maintenance.

Remark 2 Under Assumption 1 and β ¡ η, an increase in τ which does not allow the BGPto reach the regime without private maintenance pσ   σ̂pτqq, there exists a σ̃ P p0, 1q suchthat :

• For σ   σ̃, the cases in which oscillations occur are less frequent.

• For σ ¡ σ̃, the cases in which oscillations occur may be more or less frequent.

Proof. See Appendix 6.4.

From Remark 2, we observe that the government intervention may either neutralize orgenerate oscillations and hence intergenerational inequalities. This dynamics arises from thefact that the policy shapes the trade-off between maintenance and education spending, andhence the mechanism driving oscillations. As previously mentioned, this mechanism relieson the fact that variations in spending on education stem from environmental quality Q,environmental awareness γ1 and human capital h. Changes driven by the former two factorsexacerbate oscillations while changes driven by the latter attenuate oscillations.

When σ   mintσ̃; σ̂pτqu, an increase in the environmental tax reduces the frequency ofcases in which oscillations arise. The reason for this is that as long as σ is low enough, publicspending sufficiently supports education so that a tighter tax reinforces the impact of humancapital on private education spending. Indeed, the fall in wages, caused by the pollution tax,is overcompensated by the increase in the education subsidy. Thus, spending on educationis mainly driven by government action and becomes less sensitive to variations in greenpreferences γ1. As a result, oscillatory trajectories are less frequent, i.e. the conditionsunder which we observe damped oscillations are more restricted.

18

When σ̃   σ   σ̂pτq, the policy may increase the occurrence of intergenerational inequal-ities. The intuition is the following. Public revenue devoted to abatement activities σ issuch that agents continue to invest in environmental maintenance and it is not low enoughto ensure that the level of the education subsidy prevents oscillations. Indeed, the environ-mental tax diminishes the influence of human capital on private investment in educationand oscillations may occur more frequently as a result. In this case, the conditions that arenecessary for damped oscillations are more easily satisfied.

Thus, our short-term analysis of the effect of the environmental policy reveals that thegovernment can play a role in avoiding the fluctuations in environmental preferences thatare costly in equities across generations. To accomplish this, policy makers should investsufficiently in education or in environmental protection in order to reduce the dependency ofhouseholds behavior on the economic and environmental context. Our study is one of the fewthat examine the implications of environmental policy in the short-run. A notable exceptionis Ono (2003), who emphasizes that a sufficient increase in the environmental tax may shiftthe economy from a fluctuating regime to a BGP in which capital and environmental qualityincrease perpetually. In this paper, we demonstrate that the way in which the revenue fromsuch a tax is used is a determining factor in producing such a change.

4.2 The long-term effect of the environmental tax

In accordance with the concept of sustainable development and the well-known definitiondeveloped by the Brundtland Commission (WCED , 1987), environmental policy attemptsto strike the right balance between economic and environmental interests in the long run. Inthis respect, we wish to explore what solutions the government can put in place in order toachieve higher long-term growth in both human capital and environmental quality. In thissection, we therefore examine how the environmental policy affects the long-term equilibriumand the corresponding growth rate.

Using equations (4) and (13), the expression of the long-term growth rate is given by:

1� gH �

$''&''%

ε

�γ2µrAp1�τqc3�ε2pp1�αqX̄pm�ANpbστ�aqqs�p1�σqτApγ̄1c1�c3p1�γ2µqq

γ̄1c1�p1�γ2µqc3

�µpm

ε�Arγ2µp1�τq�τp1�σqp1�γ2µqs

1�γ2µ

�µnpm

(18)A more strict environmental policy influences the long-term growth rate through severalchannels. First, it influences growth directly, by affecting the trade-off between educationand maintenance activities. Second, it influences growth indirectly by modifying the green

19

development index and environmental preferences.24 As a result, the global impact of thepolicy on growth is ambiguous in the long run. In the following proposition, we show howauthorities can improve the growth rate along the BGP:

Proposition 4. Under Assumption 1 and β ¡ η, following an increase in τ :

• When the BGP remains in the pm regime pσ   σ̂pτqq, there exists an interval pσpτq, σpτqqsuch that the growth rate increases for σpτq   σ   σpτq.

• When the BGP initially occurs in or moves to the npm regime pσ ¡ σ̂pτqq, the growthrate improves and is greater than in the pm regime for σ   1

1�γ2µ.

Proof. See Appendix 6.5

Considering a tighter tax, an economy in which agents initially invest in the private pro-vision of environmental maintenance may switch to the other regime, in which no privateinvestment occurs. When the BGP remains in the regime with private provision of main-tenance (σ   σ̂pτq), an increase in the tax favors both human capital and environmentalquality if the allocation between public spending σ is at an intermediate level. This occursbecause extreme allocations render one of the two private spending (maintenance and ed-ucation) too expensive relative to the other, which leads agents to neglect one of the twodrivers of growth. When σ is too low, the policy favors spending on education. Despite theimprovement in environmental consciousness it entails, an increase in the tax makes privatemaintenance too expensive, leading to a deterioration of the environment and of the growthrate. Conversely, when σ is too high, tax revenues contribute mostly to public maintenanceof the environment. Even if private investment in the environment diminishes in favor ofeducational spending, education costs remain too high, which weakens human capital ac-cumulation. Authorities can therefore increase growth only by supporting both educationand maintenance simultaneously. Note that the two thresholds σpτq and σpτq increase withthe parameters of green preferences β and η. It follows that the higher the environmentalawareness, the larger the public environmental support necessary to ensure an improvementof economic growth.

When households do not invest in private maintenance in the long run or when theystop investing in it following the government intervention (σ ¡ σ̂pτq), a tighter tax leadsto the highest growth rate as long as it is accompanied by sufficient support for human

24As in Prieur and Bréchet (2013), we have that the stronger the public environmental concern, the lowerthe growth rate. In their paper, an increase in environmental awareness always reduces physical capitalaccumulation. Here, γ1 affects growth through an additional channel, as an increase in environmental qualityresults in increased spending on education. Nevertheless, the negative direct impact of γ1 on educationspending more than offsets its positive impact through an improvement in environmental quality. The proofis available upon request.

20

capital accumulation (σ   1{p1� γ2µqq. The intuition is the following. On the one hand, inthe no private maintenance regime, maintenance is entirely publicly funded despite agents’willingness to contribute to pollution abatement activities. Environmental quality is thussufficiently high. On the other hand, when the government also allocates a large enoughportion of its budget to education, the negative effect of the tax on available income is morethan offset by the positive effect of the tax that operates through the education subsidy. Inthis way, human capital accumulation and the environment are both enhanced. Note thatwhen σ ¡ 1{p1�γ2µq, a tighter tax is growth-reducing when maintenance is initially entirelypublic. This result may also be observed when there is a regime switch, particularly if theincrease in τ when moving the BGP to the regime without private maintenance is significant.In this case, the negative income effect exceeds the positive impact of the education subsidyand hence human capital accumulation deteriorates.

Our results contribute to the literature examining the effects of environmental policieson sustainable development (e.g. Ono , 2003 ; Grimaud and Tournemaine , 2007 ; Prieur andBréchet , 2013 or Pautrel , 2015). However, we differ from the majority of this literature bycalling attention to the role of an environmental policy mix that is composed of support forboth environmental protection as well as education. Whatever the tax rate, we emphasizethat the recycling of the fiscal revenue can lead to a win-win situation in terms of improve-ments in environmental quality as well as economic growth. However, this result is onlyensured if the government invests in both education and environmental protection. Morespecifically, despite the positive effect of environmental quality on education, a policy focus-ing solely on public maintenance of the environment may reduce human capital and growth.In contrast, despite the fact that human capital contributes to environmental awareness, apolicy providing support for education alone may not be sufficient to enhance environmentalquality, and hence to favor a sustainable growth.

4.3 How can the government policy improve both the short- and long-term scenarios?

We have previously emphasized the role that the government can play in avoiding in-tergenerational inequalities in the short run and in enhancing economic and environmentalgrowth in the long run. In this section, we examine how a tighter tax can generate bothshort- and long-term benefits. To do so, we consider the properties of the function σ̂pτq,that defines the regimes with and without private maintenance, in conjunction with the con-ditions σMinpτq and σMaxpτq, that determine the possible policy schemes compatible withsustainable development (see Assumption 1). We summarize the results of Propositions 3

21

and 4 in Figure 3, which depicts the implications of a tighter tax for a given σ.25 Four areasare distinguished.

3

4

2

1

Figure 3: Short- and long-term implications of a tighter tax at a given σ, when ab  

11�γ2µ

0 τ̄0 τ

σMinpτqab

σ̂pτq

σ̃

11�γ2µ

σMaxpτq

σ τ̄1 1

In areas 1 and 2, the long-term equilibrium of the economy occurs in the regime withprivate maintenance. Turning to the short run, a tighter tax makes the occurrence of oscil-lations less likely when σ   σ̃ (area 1), while the opposite holds when σ ¡ σ̃ (area 2). Toimprove long-term growth in these areas, an increase in the pollution tax must be associ-ated with an intermediate level of σ (σpτq   σ   σpτq). Unfortunately, no clear conclusionemerges when comparing these values of σ with the value that corresponds to environmentaland economic benefits in the short-run.

As long as the tax rate is sufficiently high, the economy is able to achieve areas 3 and4, where the BGP is in the regime without private maintenance and hence contributionsto pollution abatement activities are entirely publicly funded. In this state, the short-termissue vanishes and the economy achieves a higher long-term growth rate when the educationsubsidy is sufficiently high (σ low, area 4). Furthermore, as stressed in Proposition 4, for agiven σ, the growth rate is higher in area 4 than in the other regime.

As a result, we identify the most favorable tax scheme as one in which environmentalprotection is funded exclusively by public investment and educational support is sufficientlyhigh (area 4). This scheme is recommended for two reasons. First, even if the efficiency of thepublic and private spending on environmental maintenance are similarly efficient, our find-ings indicate that the government should take full responsibility for environmental protection

25From Proposition 4, we deduce that a growth-enhancing policy exists in the regime npm if and only ifab  1

1�γ2µ, where the ratio a

brepresents the minimum level of the threshold σ̂pτq above which the economy

is in the regime without private maintenance.

22

so as to avoid intergenerational inequalities. In this way, households may focus on their chil-dren’s education and their spending on environmental maintenances is no longer vulnerableto potential fluctuations in green preferences. This second outcome is a particularly note-worthy result given empirical evidence demonstrating that environmental preferences canfluctuate significantly according to the economic and environmental context in which agentsare situated (see Dunlap and Scarce , 1991 or Scruggs and Benegal , 2012). Second, in thiscase, the level of environmental quality is sufficiently high so that agents do not privatelycontribute to the environment despite the increase in welfare they might receive from doingso. Moreover, human capital accumulation is sufficiently supported to achieve both higherenvironmental and economic growth. Thus, we conclude that a mixed-instrument policy isa promising tool for encouraging sustainable development.

Our analysis provides additional insights through an examination of extreme allocationsof the tax revenue between education and environmental maintenance. We conclude that theeffects of an environmental policy that consists in investing only in education (σ � 0) is notsatisfying. From Figure 3, such a policy scheme is always located in area 1. Despite the factthat this policy reduces the risk of experiencing oscillations (a source of intergenerationalinequalities), it does not guarantee a higher growth rate. Indeed, for the case in which greenpreferences are not sufficiently sensitive to human capital, this policy leads to a low level ofenvironmental protection, which damages economic growth. In contrast, when tax revenueis entirely devoted to maintenance (σ � 1), only areas 2 and 3 can be achieved. Thus, thepolicy can completely eliminate oscillatory behavior but this is not sufficient to guaranteea higher rate of growth. The negative direct effect of the policy on educational spendingexceeds the positive indirect effect that operates through the improvement in environmentalquality. The analysis of these two extreme cases confirms our suggestion that a policy mix isnecessary to achieve the best situation in both the short and long run, since it correspondsto a policy scheme in area 4 of Figure 3 which is not compatible with σ � 0 or σ � 1.Moreover, these two cases represent the most restrictive policy schemes in which the rangeof the pollution tax that guarantees sustainable development is limited by τ̄0 when σ � 0and τ̄1 when σ � 1.26

Finally, note that even if we conclude that environmental maintenance activities shouldbe entirely publicly funded, environmental awareness is important as it affects the policy toimplement. When green preferences are higher, for any given tax the government shouldorient further its policies towards environmental protection in order to both avoid intergen-erational inequalities and improve growth (area 4).

26τ̄0 is such that σMinpτ̄0q � 0 and τ̄1 is such that σMaxpτ̄

1q � 1.

23

5 Conclusion

In this paper, we examine the implications of an environmental policy mix on sustain-able development when environmental awareness is endogenously determined by both indi-vidual’s human capital and pollution. We model an environmental policy that consists ofimplementing a tax on pollution, the revenues of which can be allocated between two cate-gories of environmental expenditure: public pollution abatement activities and an educationsubsidy that aims to develop ecological consciousness. Public investment in education andenvironmental maintenance consequently affect agents’ choices regarding private investmentin these two areas. If public investment in environmental maintenance is sufficiently high,households may stop investing privately in maintenance activities despite the benefits theycould derive from this investment.

Using the present model, we show that the economy can converge to a long-term equi-librium characterized by sustainable growth. However, when environmental preferences arehighly sensitive to changes in human capital and pollution levels, this convergence may beoscillatory. This is due to the feedback effect of human capital and environmental quality onendogenous environmental awareness, which shapes the trade-off between private investmentchoices. This complex dynamics represents an important issue, as it can cause variations inwelfare across generations that lead to intergenerational inequalities.

Our study reveals that an environmental policy can eliminate these inequalities in twoways: by providing sufficiently high support for environmental maintenance activities suchthat private contributions are no longer made, or by sufficiently subsidizing education soas to make private investment decisions less dependent on green preferences. Additionally,our analysis indicates that a tighter tax can also improve the long-term economic and en-vironmental growth rate when the tax revenue is allocated between both types of publicexpenditures simultaneously. In this case, human capital accumulation is favored withoutdamaging the environment. We wish to emphasize that this type of policy can result in awin-win situation, by both avoiding intergenerational inequalities as well as fostering growthof in human capital and environmental quality. More specifically, our results indicate thatthe most favorable policy scheme is characterized by an allocation of the tax revenue that isshared between environmental maintenance and education. This research therefore providestheoretical support for the use of mixed-instrument policies that combines public investmentin environmental maintenance with other tools that aim to directly modify environmentalbehaviors.

As such, we have contributed a novel result concerning the effectiveness of using a com-bination of environmental policy instruments in order to generate both economic and envi-ronmental improvements. We note that further extensions will certainly provide additional

24

insights. It could be worthwhile, for example, to consider the more direct effects of pollutionon economic growth. In this regard, one could explore the role of the environment as aninput in the production function or as a determinant of worker productivity through healtheffects. Including both direct and indirect impacts of the environment on production wouldcall for a much more complex analysis but we contend that this constitutes an interesting lineof future research, enabling to take into account additional effects of our policy instrumentsas well to analyze other types of policy mixes.

6 Appendix

6.1 Equilibrium

6.1.1 First Order Conditions

The maximization of the consumer program (6) leads to the following first order condi-tions on education expenditure and on environmental maintenance:

BU

Bet� 0 ô 1� θet

ct�γ2µ

et(19)

BU

Bmt¤ 0 ô 1

ct¥

γ1tpε1 � ε2bq

ε1mt � ε2Qt�1(20)

From equations (3), (5) and the first order conditions (19) and (20), we deduce the optimalchoices in terms of education and maintenance in two regimes: pm (where mt ¡ 0) and npm(where mt � 0).

et �

$'&'%

�γ2µ

1�θet

�pε1�ε2bqwtht�ε2rp1�αqQt�aYt�bpMt�NGmt qs

p1�γ1t�γ2µqpε1�ε2bq

pm

wthtγ2µp1�γ2µqp1�θet q

npm

mt �

$&%

γ1tpε1�ε2bqwtht�ε2p1�γ2µqrp1�αqQt�aANht�bpMt�NGmt qsp1�γ1t�γ2µqpε1�ε2bq

pm

0 npm

At the symmetric equilibrium, Mt � mtpN � 1q, the wage equilibrium is wt � Ap1� τq, theproduction function is Yt � ANht and the government budget constraint is given by (10).The Nash intertemporal equilibria are thus given by:

et �

$&%

γ2µrc3Ap1�τqht�ε2pp1�αqQt�ANhtpbστ�aqqsγ1tc1�p1�γ2µqc3

� p1� σqτAht pmAhtrγ2µp1�τq�τp1�σqp1�γ2µqs

1�γ2µnpm

25

mt �

$&%

γ1tc1Ahtp1�τq�ε2p1�γ2µqrp1�αqQt�ANhtpbστ�aqsγ1tc1�p1�γ2µqc3

pm

0 npm

6.1.2 Condition for the regime without private maintenance

Using equation (12), we deduce the condition such that the regime without private envi-ronmental maintenance occurs:

Xt ¥A rγ1tc1p1� τq � ε2Np1� γ2µqpbστ � aqs

ε2p1� γ2µqp1� αq(21)

This condition can be written as:

PpXtq � X2t ε2p1� γ2µqp1� αq �Xtrε2p1� γ2µqp1� α�ANpbστ � aqq �Aηc1p1� τqs

�Arβc1p1� τq � pbστ � aqNε2p1� γ2µqs ¥ 0(22)

Given the expression of PpXtq, the polynomial PpXtq � 0 admits at most one positivesolution. We define Λ as the critical threshold of Xt over which the economy is in the noprivate maintenance regime, with Λ � maxt0, soltPpXtq � 0uu.

• When PpXtq � 0 does not admit a positive solution, the polynomial is positive or nul@ Xt ¥ 0. In this case, the economy is always in the npm regime (i.e. Λ � 0).

• When PpXtq � 0 admits a positive solution, the economy is in the npm regime onlywhen Xt is sufficiently high (i.e. Λ ¡ 0).

As a result, the economy is in the npm regime when Xt ¥ Λ, with Λ ¥ 0.

6.1.3 Growth rates

We define the endogenous growth rate of human capital gH , and environmental qualitygQ, using equations (4), (5), (13) and (12):

1�gHt �

$'&'%

ε�γ2µrAp1�τqc3�ε2pp1�αqXt�ANpbστ�aqqs�p1�σqτApγ1tc1�c3p1�γ2µqq

γ1tc1�p1�γ2µqc3

�µpm

ε�Arγ2µp1�τq�τp1�σqp1�γ2µqs

1�γ2µ

�µnpm

p18q

In the npm regime, human capital growth rate is constant, as education spending does notdepend on the environment. The pm regime is characterized by a human capital growth rateincreasing in the green development index, directly and indirectly though environmentalawareness γ1t.

26

1� gQt �

$&%

p1�αqXtpγ1tc1�c2q�ANrγ1t c1p1�τqb�pγ1tc1�c2qpbτσ�aqsXtpγ1tc1�c3p1�γ2µqq

pm

1� α� ANpbστ�aqXt

npm(23)

In the case with private maintenance, the green development index Xt has a direct negativeimpact on the growth rate of environmental quality and an indirect positive effect throughenvironmental awareness γ1t. In the corner solution, gQt is always negative without publicabatement. However, when the government intervenes, the growth of the environmentalquality at the corner may be positive for a share of policy devoted to public maintenance(σ) sufficiently high.

6.2 Proof of Proposition 1

6.2.1 Properties of the dynamic equation

From Definitions 1 and 2 and equations (14) and (16), we emphasize the properties ofthe dynamic equation characterizing equilibrium paths, FpXtq, defined on p0;�8q:

• When Xt P p0 ; Λq the function is given by equation (17 pm). We have

Fp0q � ANpβ c1p1� τqb� pβc1 � c2qpbτσ � aqq

ε rAγ2µ p c3p1� τq � ε2Npbστ � aqq � p1� σqτApβ c1 � p1� γ2µqc3qsµrβ c1 � p1� γ2µqc3s

1�µ

Finally, with equation (16), limXtÑΛ�

FpXtq �p1�αqΛ�ANpbστ�aq

ε�γ2µAp1�τq�p1�σqτAp1�γ2µq

1�γ2µ

�µ � v.

• When Xt P rΛ ; �8q, the function is given by equation (17 npm). F is increasing andlinear in X, FpΛq � v and limXÑ�8FpXq � �8.

As limXtÑΛ

FpXtq � FpΛq, the function is continue on p0;�8q.

6.2.2 Existence and Unicity of the Balanced Growth Path

npm solution. A BGP in the npm regime is characterized by X̄npm � FpX̄npmq.Using (17 npm), we obtain:

X̄npm �ANpbτσ � aq

ε�Arγ2µp1�τq�p1�σqτp1�γ2µqs

1�γ2µ

�µ� p1� αq

(24)

To exists X̄npm has to be positive. Following (18 npm), if the denominator of (24) is negative,so does the growth rate in the npm regime. Therefore, the case where the denominator andthe numerator of (24) are negative is meaningless. The existence conditions are thus σ ¡ a

bτ

and A1 ¡ 0 with A1 ��Arγ2µp1�τq�p1�σqτp1�γ2µqs

1�γ2µ

�µ�p1�αq. Note that the condition A1 ¡ 0

implies that limX�8

FpXqX   1.

27

Then, we have to check the admissibility of the steady state, i.e. if it effectively belongsto the npm region. To do this, we examine the sign of FpΛq � Λ. Under condition A1 ¡ 0and σ ¡ a

bτ , X̄npm is admissible if FpΛq ¥ Λ, which is equivalent to X̄npm ¥ Λ.

pm solution. A BGP in the pm regime is characterized by X̄pm � FpX̄pmq. As we focuson X ¡ 0, we determine the solutions X̄pm which satisfy FpX̄pmq{X̄pm � 1. Using equa-tions (14) and (17 pm.), it corresponds to the following equation. For the sake of simplicity,subscripts on X are removed.

ε�γ2µpAc3p1� τq � ε2pp1� αqX �ANpbτσ � aqqq �Aτp1� σq

�c1β�ηX1�X � p1� γ2µqc3

�µ�β�ηX1�X c1 � c3p1� γ2µq

�1�µ� p1� αq

�β�ηX1�X c1 � c2

�

AN�β�ηX1�X c1pbp1�τq�bτσ�aq�c2pa�bτσq

X

We define D1pXq and D2pXq, respectively the term on the left and on the right hand side.Their properties are:

• D1 is decreasing and then increasing with X. Moreover, D1pXq ¡ 0 for all X,limXÑ0

D1pXq � C ¡ 0, with C a constant and limXÑ�8

D1pXq � �8.

• Concerning D2 we have that if A2 � βc1pbp1 � τq � bτσ � aq � c2pa� bτσq ¡ 0 (resp.  0), lim

XÑ0D2pXq ¡ 0 (resp.   0). Moreover, lim

XÑ�8D2pXq � p1� αqpηc1 � c2q ¡ 0.

The condition A2 ¡ 0 guarantees that the curves D1pXq and D2pXq cross at least oncein the positive area. Thus, a positive solution exists. From equation (17 pm.) and A2 ¡ 0,we have dFpX̄pmq{dX   1, hence the positive solution X̄pm is unique.

We check the admissibility of the steady state, i.e. if it effectively belongs to the pmregion. Under condition A2 ¡ 0, X̄pm is admissible if lim

XÑΛFpXq   Λ, which is equivalent

to X̄pm   Λ. As X̄pm ¤ X̄npm, the condition X̄npm   Λ guarantees that X̄pm is admissible.The case where both X̄pm and X̄npm are admissible is excluded. Indeed, when X̄pm exists

(condition A2 ¡ 0 verified) and is admissible, then FpΛq   Λ. But under condition A1 ¡ 0the slope of F in the npm regime is lower than one, which means that F cannot cut thebisector in this regime. Reversely, when Xnpm exists (conditions A1 ¡ 0 and σ ¡ a

bτ verified)and is admissible then FpΛq ¥ Λ. Under condition A2 ¡ 0, this implies that F does not cutthe bisector in the pm area.

To sum up, under these conditions it is not possible to have X̄pm   Λ ¤ X̄npm, andhence to have cases with multiple steady states: when X̄npm   Λ, the BGP is in the regimepm, while when X̄npm ¥ Λ, the BGP is in the regime npm.

For the rest of the paper, we rewrite the conditions A1,A2 ¡ 0, as σMinpτq   σ  

σMaxpτq with

σMinpτq �apβc1 � c2q � bβc1p1� τq

bτpβc1 � c2qand σMaxpτq �

Apγ2µ� τq � ε�1{µp1� γ2µqp1� αq1{µ

Aτp1� γ2µq

28

For τ � 0, the condition σMinpτq   σ   σMaxpτq is equivalent to:

apβc1 � c2q � bβc1p1� τq   0   Apτ � γ2µq � p1� γ2µq

�1� α

ε

1µ

Given this, σMinpτq is increasing in τ from limτÑ0 σMin � �8 to limτÑ1 σMin � a{b, whileσMaxpτq is decreasing in τ from limτÑ0 σMax � �8 to limτÑ1 σMax � 1� 1

ε1{µA.

Condition for the nature of the BGP. The analysis of admissibility conditions comesdown to X̄npm   or ¡ Λ.

From equation (22) in Appendix 6.1.2, X̄npm ¥ Λ is equivalent to PpX̄npmq ¥ 0. Wedefine PpX̄npmq � J pτ, σq, where the function J is increasing in τ and σ. Under A1 ¡ 0:

• For τ � 0, X̄npm   0, J   0 and does no longer depend on σ.

• For τ � 1, we get J ¡ 0 @ σ P r0, 1s.

We depict a representation of J at given τ :J pσ, τq

0

τ � 1

τ P p0, 1q

τ � 0

σσcpτq

6

-

Figure 4: Function J at given τ

We deduce that there exists a σcpτq decreasing in τ such that J � 0, with limτÑ0

σcpτq �

�8 and σcp1q   0. Thus a minimum level of the tax is required to make the npm

regime possible. When σ   σcpτq, we get PpX̄npmq   0, meaning that the equilibriumis in the pm regime. Respectively, when σ ¥ σcpτq we get PpX̄npmq ¥ 0, and fromequations (22) and (24) we have X̄npm ¥ Λ if and only if σ ¡ a

bτ . Thus, the BGPis in the npm regime when σ ¡ Maxtσcpτq; a

bτ u � σ̂pτq and in the pm regime whenσ   σ̂pτq. When σ � σ̂pτq, the BGP is in the npm regime if σ̂pτq � σcpτq and in thepm regime if σ̂pτq � a

bτ . Note that σ̂pτq ¡ σMinpτq and limτÑ1 σ̂pτq   limτÑ1 σMaxpτq.

6.3 Proof of Proposition 2

pm solution. We use the scaling parameter ε in order to normalize the steady stateX̄pm to one. There is a unique solution ε� such that X̄pm � 1 and from equation (17 pm),

29

the expression of the normalization constant is given by:

ε�pX̄pmq ��p1� αq X̄pm rγ̄1c1 � c2s �ANpγ̄1c1bp1� τq � pγ̄1c1 � c2qpbτσ � aqq

���

γ2µA c3p1� τq � γ2µε2�p1� αqX̄pm �ANpbστ � aq

�� p1� σqτApγ̄1 c1 � p1� γ2µqc3q

��µrγ̄1 c1 � p1� γ2µqc3s

µ�1

Then, by differentiating equation (17 pm.) and analyzing it around the steady state X̄pm � 1and ε � ε�pX̄pmq, we obtain:

dXt�1 �pp1�αqpγ̄1c1�c2�γ̄11c1q�ANγ̄11c1pbp1�τq�bτσ�aqqB1B2�B3pµB2pγ2µε2p1�αq�p1�σqAτγ̄11c1q�p1�µqγ11c1B1q

B3B1B2dXt

(25)

with γ̄1 � β�η2 , γ̄1

1 � pη � βq{4, B1 � γ2µA c3p1 � τq � γ2µε2 rp1� αq �ANpbστ � aqs �

p1 � σqτApγ̄1 c1 � p1 � γ2µqc3q, B2 � γ̄1 c1 � c3p1 � γ2µq and B3 � p1 � αqpγ̄1 c1 � c2q �

ANpγ̄1c1bp1� τq � pγ̄1c1 � c2qpbτσ � aqq.From (25), we get dXt�1{dXt   1. Thus, when dXt�1{dXt ¡ 0, transitional dynamics

is monotonous and the BGP equilibrium is locally stable. Using equation (25), we havedXt�1{dXt ¡ 0 if and only if:

B2p1� αq rp1� σqτAB2 � γ2µpAε1p1� τq � ε2p1� µqqs pγ1c1 � c2q

� B2p1� αqAε2bNp1� τqpγ1c1p1� µq � c2q

� γ̄11c1B5 rγ2µp1� α�ANpb� aq � bNτpσ � 1qqpbNp1� γ2µqε2p1� µq � µB2qs

� γ̄11c1B5ANbp1� γ2µqpγ2µε1p1� µq � p1� σqτB2q ¡ 0

with B4 � 1�α�ANpbp1�τq�bτσ�aq and B5 � Ac3p1�τq�ε2p1�α�ANpbτσ�aqq. Rewritingthis expression, we have dXt�1{dXt ¡ 0 if and only if the following polynomial is positive:

Rpβq � a1β3 � a2β

2 � a3β � a4

with a4 ¡ 0 and expressions for a1, a2 and a3 given by:

a1 �c3

18 p1� σqp1� αqτA ¡ 0

a2 �c21p1�αq

4�γ2µp1� µqB5 � p1� σqτAr2c3p1� γ2µq � c2s � γ2µ

2ε1Ap1� τq�

�c218�3ηc1p1� αqp1� σqτA� B5

�γ2µ

2B4 � bANp1� γ2µqτp1� σq��

a3 �c31η

2p1�σqp1�αqτA8 �

2ηc214 p1� αq

�γ2µp1� µqB5 � p1� σqτAr2c3p1� γ2µq � c2s � γ2µ

2ε1Ap1� τq�

� c1p1�αq2

�3p1� σqτAc2c3p1� γ2µq � p1� µqγ2µB5pc3p1� γ2µq � c2q � 2γ2µ

2Ap1� τqε1c3p1� γ2µq�

� c1B54

�γ2µ

2c3p1� γ2µqB4 � bNp1� γ2µqpγ2µp1� µqB5 � p1� σqτAc3p1� γ2µqq�

We have Rp1q which is a polynomial of degree three in N . When N � 0, we haveRp1q ¡ 0, while when N tends to 8, Rp1q   0. As N3 intervenes positively in Rp1q, there

30

exists a critical threshold N̄ over which the dynamics is oscillatory for β � 1. Given thata1 ¡ 0 and a4 ¡ 0, we can conclude that for N ¡ N̄ there exists a β̃ P p0, 1s over which thedynamics is oscillatory.

We examine the stability of the equilibrium when the dynamics is oscillatory. Fromequation (25), we have dXt�1{dXt ¡ �1 if and only if:

rp1� αqpγ̄1c1 � c2 � γ̄11c1q �ANγ̄1

1c1pbp1� τq � bτσ � aqsB1B2 � B1B2B3

� rp1� µqc1γ̄11B1 � B2µ pγ2µε2p1� αq � p1� σqAτγ̄1

1c1qsB3 ¡ 0

Replacing expressions B1, B1 and B3, we finally obtain:

c3p1� γ2µqc2p1� αqpB6 � ε2bNµq � γ2µpγ̄1c1q2pB6p1� αq � B5B4q

� γ̄1c1p1� γ2µqrp1� αqε1pB6 � ε2bNµq � γ2µε1p1� α�ANpbτσ � aqq � c3pB4 � B6qs

� γ2µγ̄11c1p1� γ2µqpB4c3 � p1� µqε1p1� α�ANpbτσ � aqqq

� γ2µ2γ̄1

1γ̄1c21B5B4 � B5c3p1� γ2µqc2p1� α�ANpbτσ � aqq

� p1� σqτApγ̄1c1 � c3p1� γ2µqqγ11c1bNp1� γ2µqB5

� p1� σqτApγ̄1c1 � c3p1� γ2µqq2rγ̄1c1B4 � c2p1� α�ANpbτσ � aqq � p1� αqpγ̄1c1 � c2qs

with B6 � c2 � ε2bNp1� γ2µq ¡ 0.As �γ̄1

1   γ̄1 and c3B4 ¡ bNB5, we easily see that this term is always positive. TheBGP equilibrium is always locally stable.

npm solution. The npm BGP is obtain from (17 npm) and given in Appendix 6.2.We differentiate equation (17 npm) and obtain:

dFpXtq

dXt�

p1� αq

ε�Arγ2µp1�τq�p1�σqτp1�γ2µqs

1�γ2µ

�µ

Under Assumption 1, the slope of FpXtq in the npm regime is always positive and lowerthan one, the npm BGP is thus monotonously stable.

6.4 Proof of Remark 2

The condition to observe oscillatory cases given in Appendix 6.3 (i.e. N ¡ N̄) dependson policy instruments. Moreover, by examining the terms a2 and a3 given Appendix 6.3,we conclude that if a2 is positive, a3 is positive as well. A necessary condition is thusrequired to have N ¡ N̄ : the term a2 has to be negative. An analysis of the impact of thepolicy instruments on the polynomial Rpβq is not analytically tractable, but the analysis ofSign

Ba2Bτ

(gives us interesting intuitions. Using the expression of a2 given in Appendix 6.3,

31

we obtain:

Sign Ba2Bτ

(� �p1� σqbANp1� γ2µp1� µqqpS2 � τAε2bNp1� σqqpτp1� σqS3 � p1� τqγ2µAε1q

�p1� σqpbτp1� σqANp1� γ2µp1� µqq � S1qpε2ANbS4 � S3S2q � 3p1� σqηc1AS4

�Aε1µ3p1� α�ANpb� aqqpS1 � bτp1� σqANp1� γ2µp1� µqqq

with S1 � µ2γ2p1�α�ANpb�aqq, S2 � Ap1� τqε1� ε2pANpb�aq�1�αq, S3 � Ap3ε1p1�γ2µq�

2ε2bN � γ2µε2bNp1� µqq and S4 � γ2µpε2p1� µqp1� α�ANpb� aqq �Aε1q

We can define Sign Ba2Bτ

(as a polynomial of degree three in σ, with Ba2

Bτ   0 whenσ � 1 and Ba2

Bτ ¡ 0 when σ � 0. Since Sign Ba2Bτ

(is decreasing in σ for σ P r0, 1s, there

exists a critical value σ̃ P p0, 1q such that: for 0   σ   σ̃, Ba2Bτ ¡ 0 and for σ̃   σ   1,

Ba2Bτ   0. Moreover, from Assumption 1, limτÑ0 σMinpτq   σ̃   limτÑ1 σMaxpτq. When σ issufficiently low, a tighter tax tightens the condition to observe oscillatory cases, while whenσ is high enough the condition to observe oscillatory dynamics may be relaxed.

6.5 Proof of Proposition 4

We examine the impact of taxation on the growth rate along the BGP.

pm solution. Using equation (18 pm) with Xt � X̄pm we have:

Sign�BgpmBτ

� V2

�p1� σqpγ̄1c1 � c3q � σγ2µε1 � γ2µε2p1� αq

BX̄pmBτ

� c1pβ�ηq

p1�X̄pmq2BX̄pmBτ γ2µ pV1 � τp1� σqAV2q

with V1 � γ2µAc3p1� τq� γ2µε2�p1� αqX̄pm �ANpbστ � aq

��p1�σqτApγ̄1c1 �p1� γ2µqc3) and

V2 � γ̄1c1 � p1� γ2µqc3. From the implicit function theorem and equation (17), we have:

BX̄pmBτ �

�V2X̄pmA rpγ̄1c1pσ � 1q � σc2qV1N � µV3p�σε1γ2µ� p1� σqpγ̄1c1 � c3qqs

�c1X̄pmpβ�ηq

p1�X̄pmq2

�V1V2pX̄pmp1� αq �ANpbp1� τq � bτσ � aqq

�µV2V3p1� σqτA� p1� µqV1V3q � µV2V3X̄pmγ2µε2p1� αq �ANV1V2V4

��1

with V3 � p1 � αqX̄pm rγ̄1c1 � c2s � AN rγ̄1c1bp1� τq � pγ̄1c1 � c2qpbτσ � aqs and V4 � c2pbτσ �

aq � γ̄1c1bp1� τp1� σqq

Thus, substituting BX̄pmBτ in Sign

�BgpmBτ

, we finally obtain:

Sign�BgpmBτ

�

�γ2µε2p1� αqV2X̄pmAN �

c1X̄pmANpβ�ηq

p1�X̄pmq2 rV1 � τp1� σqAV2spγ̄1c1pσ � 1q � σc2q

��c1X̄pmpβ�ηq

p1�X̄pmq2

�V2pX̄pmp1� αq �ANpbp1� τq � bτσ � aqq � V3

�� V2V4AN

p�σε1γ2µ� p1� σqpγ̄1c1 � c3qqV1

32

Under Assumption 1, policy improves the BGP growth rate when the following sufficientcondition is satisfied:

f1pσq   σ   f2pσq

with f1pσq �γ̄1c1

γ̄1c1�c2  1 and f2pσq �

γ̄1c1�c3γ̄1c1�c2�γ2µε1

  1. These two functions are increasingin γ̄1, and as Bγ̄1

BX̄pm  0 and BX̄pm

Bσ ¡ 0, they are decreasing in σ. As a result, there exists aunique range of value [σpτq;σpτq] which satisfies this condition. Moreover, under Assump-tion 1, limτÑ0 σMinpτq   σpτq   σpτq   limτÑ1 σMaxpτq.

npm solution. We use equation (18 npm) with Xt � X̄npm and deduce:

Sign�BgcBτ

� 1� σp1� γ2µq

A tighter tax is growth promoting as long as σ̂pτq   σ   1{p1� γ2µq.

Regime switch. We consider the case in which an increase in τ leads the economyfrom a pm regime to a npm regime. The opposite switch cannot be observed as σ̂pτq isdecreasing in τ . For a given σ, we compare equations (18 npm) and (18 pm), by consideringa higher tax rate in the npm regime (τN ) than in the pm one pτP q. The growth rate in thepm regime is higher than in the npm if and only if:

γ2µp1� γ2µq�c3ApτN � τP q � ε2pp1� αqX̄pm �ANpσbτP � aqq

��p1� σqApγ1c1 � p1� γ2µqc3qpτN � τP q �Aγ2µp1� τN qγ1c1 ¡ 0

This expression is increasing in σ and from (16) is never satisfied when σ � 1{p1 � γ2µq.And according to Appendix 6.2, the npm regime exists only if a

b   σ. Thus, for a givenσ P pa{b ; 1{p1� γ2µqq, the growth rate in the npm regime is higher than in the pm one.Moreover, under Assumption 1, σMinpτq   1{p1� γ2µq   σMaxpτq.

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Contact :

Stéphane MUSSARD : mussard@lameta.univ-montp1.fr