Environmental health and Education: towards sustainable growth 

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The Model

In this section, we present the setup of the model and discuss the main assump-tions.

Basic Framework

In this model, we consider an infinite-horizon economy where time is discrete, t = 0 1 2 ∞ and we assume no demographic growth. The representative agent lives for three periods: childhood, adulthood and the old age. During childhood, agents do not derive any utility although they educate; during adulthood, agents supply their human capital to the market and all decisions are taken; finally, when old, agents benefit from the environment and care about the level of human capital reached by their offspring. Therefore, the representative agent born at date t − 1 maximizes its lifetime utility, defined over consumption when adult (ct), environmental quality when old (Et+1) and the level of human capital attained by their children (ht+1): Ut−1 = ln ct + γ(ln Et+1 + ln ht+1) (1.1) where γ > 0 is a discount factor that determines the weight given to both future level of human capital and environmental quality. It is assumed that children are endowed with one unit of time, dedicated to education that is privately funded. Here, parents invest in the education of their children through a warm glove effect. In line with de la Croix & Doepke (2003), human capital evolves according to: ht+1 = [µ + vtΛt]1−θ htθ (1.2)
The stock of human capital depends on two main elements: human capital inherited from parents, ht, and vt being the education expenditure. Notice that 0 < θ < 1 measures the relative importance of human capital transmission versus education expenditure in the production of human capital.5 In this framework, education ex-penditure, vt, could be regarded as the potential quantity of education provided by parents to their offspring while vtΛt embodies the effective quantity of education received by a child. Hence, (1 − Λt) ∈ (0 1] represents the fraction of education lost, due to illness, or the time spent at home instead of school etc. Broadly speaking Λt captures children’ health outcomes and can be regarded as the effectiveness of education expenditure. Yet, the determinants of this variable will be deeply dis-cussed later on (see section 1.2.2). Moreover, notice that agents are also endowed with « basic skills » (µ > 0) that allow their human capital to be still positive, even if parents do not invest in education.
In this setting, we consider that agents value the amenities provided by the environment, in particular when old. Agents care about environmental quality and both the use and non-use values of these amenities are taken into account: for instance, individuals may be concerned by the neighbouring environment (like green spaces, forests, water and air quality), as well as by more global environmental issues (like global warming, natural resources, biodiversity etc.). Following John & Pecchenino (1994), the law of motion of environmental quality writes as: Et+1 = (1 − η)Et − Pt + σmt (1.3) where 0 < η < 1 is the natural depreciation rate of the environment, Pt are pollution flows, mt is environmental maintenance, while σ > 0 accounts for its efficiency.
Let us underline that this maintenance represents all actions engaged by agents in order to preserve or improve the environment. Besides, equation (1.3) implies that, in order to improve future environmental quality, the effort of maintenance must be high enough to compensate for both natural depreciation and pollution flows, otherwise the environment deteriorates.
During adulthood, adults supply inelastically ht units of human capital and earn wtht, with wt being the wage rate. Using (1.2), the labour income depends, at least partially, on the parental investment in education, but also on the efficiency of such expenditure at date t − 1. In the end, we consider here that agents’ productivity is not directly affected by environmental conditions, but, alternatively that the supply of human capital depends on the environment: the labour income indirectly relies on the environment suffered during childhood. Income can be used for three alterna-tive purposes: consumption, education investment and environmental maintenance. Hence, the budget constraint when adult writes as: wtht = ct + mt + vt (1.4)
One good is produced in the economy and the production technology requires only one input, human capital. The production function can be expressed as: Yt = ωht, where ω is an index of productivity. As the production function exhibits constant returns to scale, it follows directly that the wage rate is given by the average pro-ductivity of human capital: wt = ω. In addition, we consider that pollution flows derive from consumption6 : Pt = βct (1.5) where β ∈ [0 1] represents the cleanness degree of consumption. Here, a high value of β induces that consuming implies more pressure on the environment. Otherwise said, even when quantitatively consumption is low, its harmful impact on the environment can be heavy, including for instance polluting emissions, wastes, or even natural resources depletion, etc..Moreover, in this set-up, consumption represents an inter-generational externality. Indeed, adults’ consumption deteriorates the future state of the environment suffered by the next generation when adult and the following generation when young, while this effect is not taken into account by agents when maximising their lifetime utility.

Efficiency of Education Expenditure

Let us now consider Λt, the efficiency of education expenditure, as a function of environmental quality, such that: Λt ≡ Λ(Et), where Λ(•) is increasing and concave and takes it values on the interval [0 λ ], with λ ≤ 1.
This assumption can be justified by a wide empirical economic literature deal-ing with the effects of environmental quality on human health. As mentioned in Introduction, it has been often highlighted that children’ health can be harshly affected by deteriorated environmental conditions (Pope (2000), Chay & Green-stone (2003), Evans & Smith (2005), Bleakley (2003), Dasgupta (2004), Currie & Schmieder (2008)). This may have, for instance, a direct outcome on the level of school attendance, so that when environmental quality is damaged, the level of ab-senteeism at school is higher. Yet, it is well established that health plays a crucial role in human capital production: healthier agents display better cognitive capaci-ties, more willingness to learn, less absenteeism etc. (see, for instance, Weil (2008), Grossman & Kaestner (1997)). Hence, everything goes as if the environment deter-mines education expenditure effectiveness, through its impacts on children’s health. Environmental quality becomes then a key factor for human capital production.
In order to obtain closed-form solutions, we assume that the effect of the environment on children’ health is described by: Λ(Et) = λEt (1.6) (1 + Et) where Λ(•) is increasing and concave in environmental quality and limEt→∞ Λ(Et) = λ. As environmental quality improves, children spend more and more time at school and present better cognitive capacities; nevertheless, it is supposed that this effi-ciency of educational spending is strictly bounded and smaller than one. Indeed, children health status depends on other factors, beyond environmental quality, which could explain why children may be absent of school. In particular, λ accounts for the health care effectiveness, the medicine technology or the access to health care ser-vices, meaning that if it is low, an improvement in environmental quality translates into a small enhancement of children’ health. Notice that through this function Λ(•), a two-way causality between human cap-ital and environmental quality occurs. On the one hand, the environment affects the productivity of education when young, and determines, in the end, the labour income earned when adult: a poor environment reduces education efficiency when young, and ceteris paribus, the level of human capital reached when adult.7 On the other hand, human capital, through the choices of consumption and maintenance, influences the evolution of environmental quality. This twofold relationship will be established formally in the following section which presents the optimal microeconomic choices of agents.

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Microeconomic Choices

In this section, we derive and discuss the optimal microeconomic choices from the maximisation of the Utility function (1.1) under the constraints (1.2), (1.3), (1.4) and (1.5). Solving the maximization program yields the following First Order Condition (FOC) on education expenditure: γ(1 − θ)Λ(Et)ct ≤ [µ + vtΛ(Et)] an equality holds if vt > 0 (1.7) We also obtain the FOC on maintenance: γσct ≤ Et+1 an equality holds if mt > 0 (1.8)
Notice that the presence of a minimal level of human capital, µ implies that it may be optimal for parents not to invest in education for their children. Sym-metrically, if environmental quality is sufficiently high, agents have no incentive to engage in environmental maintenance. Hence, the optimisation problem may induce the existence of corner solutions with respect to educational spending and mainte-nance. Hereafter, superscript i = {S U } and j = {C D } will denote respectively the Schooled solution when children are educated (vt > 0) and the Unschooled one (vt = 0), when it is optimal for parents not to invest in education. Symmetrically, we define the Clean solution when agents invest in maintenance or the Dirty one when no environmental preservation is engaged.
In order to describe clearly the dynamics of our model, the plan (Et h t) has been divided into four regions. As a result of utility maximization, the corner solution for education occurs if, according to equation (1.7): ht < µ[σ(1 + γ) + β] − γ(1 − η) ≡ Φ(Et) (1.9) γ(1 − θ)σωΛ(Et) ωσ where Φ(Et) being a downward sloping curve, meaning that the level of human capital required to invest in education reduces with environmental quality. Similarly, according to equation (1.8), the corner solution for maintenance occurs if: ht < (1 − η)[1 + γ(1 − θ)]Et − µ ≡ Ψ(Et) (1.10) ω(γσ + β) ωΛ(Et)
where Ψ(Et) is increasing and concave in environmental quality. It could also be the case that agents do not invest neither in education nor in maintenance: according to both equation (1.7) and (1.8), this would happen if both the following inequalities hold:
ht < µ ≡ φ(Et) (1.11)
γ(1 − θ)ωΛ(Et)
and
ht < (1 − η)Et ≡ ψ(Et) (1.12)
ω(γσ + β)
where φ(Et) is a downward sloping curve while ψ(Et) is increasing and linear. These boundary conditions (1.9)-(1.12) divide the state space (Et h t) into four sets, Sij , as depicted in Figure 1.1:
SSC = (ht E t) ∈ R+2 : ht > Φ(Et) and ht > Ψ(Et)
SUC = (ht E t) ∈ R+2 : ψ(Et) < ht < Φ(Et)
SSD = (ht E t) ∈ R+2 : φ(Et < ht < Ψ(Et)) (1.13)
SUD = (ht E t) ∈ R+2 : (Et h t) S SC ∪ SUC ∪ SSD
To summarize, for very low levels of development (human capital) agents will not invest neither in education, nor in maintenance, because income is too low. If the environment deteriorates, agents do not educate their children, as the return to education is depleted, but might invest in maintenance. Alternatively, if envi-ronmental conditions improve, agents will only start to invest in education, not in maintenance. Finally, an interior solution appears for high enough levels of human capital and as soon as environmental quality reaches intermediate values; parents invest in both education and environmental preservation.

Education Expenditure

Taking into account (1.7) and (1.8), we derive the optimal choice of education expenditure, when mt = 0 and when mt > 0:
for (Et h t) ∈ SSD
Λ(Et )[1+γ(1−θ)]
vt = γ(1−θ)Λ(Et )ωht −µ (1.14)
γ(1−θ)Λ(Et )[(1−η)Et +σωht ]−µ[σ(1+γ)+β] SC
Λ(Et )[σ(1+γ(2−θ))+β] for (Et h t) ∈ S
In both configurations, a clean environment boosts education expenditure. In fact, when environmental conditions improve, the return on the investment in human cap-ital rises, thus encouraging parents to engage in educational spending. In addition, in the interior regime, a substitution effect arises: if less maintenance is needed, ceteris paribus, parents will be able to educate more their offspring.

Maintenance

Similarly, substituting (1.14) into (1.8) yields the expression for optimal mainte-nance, when vt = 0 and when vt > 0:
ωht ( t for (Et h t) ∈ SUC
σ(1+γ)+β
mt = γσ+β)−(1−η)E (1.15)
(γσ+β)[Λ(Et )ωht +µ]−Λ(Et )(1−η)(1+γ(1−θ))Et SC
Λ(Et )[σ(1+γ(2−θ))+β] for (Et h t) ∈ S
The above expression reproduces the standard results found in the literature (see for instance John & Pecchenino (1994); Ono (2002)), according to which both pollution flows and income have a positive impact on environmental actions, while improved environmental conditions tend to reduce maintenance. However, in the interior regime, introducing human capital accumulation reinforces the second effect: for a clean environment, agents are more prone to invest in education since investment in human capital is more efficient. Then, a substitution effect arises thus reducing green expenditure. This additional effect vanishes obviously when parents do not invest in education.

Dynamics

Substituting optimal choices into the dynamic equations describing the evolu-tion of human capital (1.2) and environmental quality (1.3) yields a bi-dimensional dynamical system that illustrates the co-evolution of ht and Et:
ht+1 = [µ + v(Et h t)Λ(Et)]1−θ hθt (1.16)
Et+1 = (1 − η)Et − P (Et h t) + σm(Et h t)
with given initial conditions (E0 h 0). Here the functions v(Et h t), m(Et h t) and P (Et h t) describe respectively the optimal choices of education and maintenance, and the resulting level of pollution, derived from optimal choices of consumption (see equation (1.5)). In order to assess the dynamic properties of the setting, we draw a phase diagram that depicts the overall dynamics. To do this, we start by presenting separately the human capital stationary locus (hh locus) and that of the environment (EE locus).

Table of contents :

Introduction 
1 Environmental health and Education: towards sustainable growth 
1.1 Introduction
1.2 The Model
1.2.1 Basic Framework
1.2.2 Efficiency of Education Expenditure
1.3 Microeconomic Choices
1.3.1 Education Expenditure
1.3.2 Maintenance
1.4 Dynamics
1.4.1 The hh Locus
1.4.2 The EE Locus
1.4.3 Global Dynamics
1.4.4 Environmental Kuznets Curve
1.4.5 Escaping the Environmental Poverty Trap
1.5 Conclusion
1.6 Appendices
A Proof of Lemma 1
B Proof of Lemma 2
C Proof of Proposition 1
2 Life expectancy and the environment1 
2.1 Introduction
2.2 Stylized facts
2.3 The benchmark model
2.3.1 Structure of the model
2.3.2 Optimal choices
2.3.3 Dynamics
2.3.4 An analytical illustration
2.3.5 In and out of the trap
2.3.6 Welfare analysis
2.4 Introducing human capital accumulation
2.4.1 Structure of the model
2.4.2 Optimal choices
2.4.3 Dynamics
2.4.4 Welfare analysis
2.5 Conclusions
2.6 Appendices
A Optimality of the dynamics
1.This chapter is a joint work with Fabio Mariani and Agustin Perez-Barahona.
B Physical capital
C Proof of Proposition 4
3 Education and the Political Economy of Environmental Protection 
3.1 Introduction
3.2 The model
3.2.1 The basic framework
3.2.2 Microeconomic choices
3.3 Dynamics
3.3.1 Perfect foresight dynamics
3.3.2 Indeterminacy and self-fulfilling equilibria
3.4 Policy implications
3.4.1 Transition phase
3.4.2 Long-run dynamics
3.4.3 Out of the trap
3.5 Conclusion
3.6 Appendices
A Appendix A
B Proof of proposition 6
C Proof of Proposition 7
D Dynamic implications of the public policy
E Proof of Proposition 8
F Substitutability vs Complementarity
Conclusion 
Bibliography

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