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Physical Capital, Human Capital, and the Health Effects of Pollution in an OLG Model
Sichao Wei and David Aadland∗
August 25, 2019
Abstract
Pollution reduces longevity and impedes learning through negative health effects, thus
channeling its damages on physical and human capital. In a standard overlapping generations
(OLG) model, we show that the accumulation differential between physical and human
capital imposed by pollution matters for policy analyses on the Balanced Growth Path (BGP)
and for the transitional dynamics. Two cases derived from our model are of particular interest.
One case is that two stable BGPs emerge with a boundary demarcating the two. One BGP is
desirable featuring high economic growth and low pollution, whereas the other should be
avoided because it is associated with low economic growth and high pollution. Another case
is that economic and environmental cycles may emerge, implying inequality between
generations. These theoretical results can be related to the empirical evidence revealed by
cross-sectional and time-series data. Government interventions can steer the economy towards
the desirable BGP and eliminate the cycles. We contribute to the literature by connecting the
pollution health effects with the capital ratio, and by identifying the capital accumulation
differential caused by pollution as a new source of economic and environmental cycles.
JEL Classification: C61; I15; I25; O44
Keywords: Endogenous Growth; Overlapping Generations; Pollution; Health Effects
∗Sichao Wei: School of Economics and Trade, Hunan University, Changsha, Hunan Province, China. Email:
weisichao@hnu.edu.cn. David Aadland: Corresponding author. Department of Economics, University of Wyoming,
1000 E. University Avenue, Laramie, WY 82070, USA. Email: aadland@uwyo.edu. Phone: (307) 766-4931.
We would like to thank Jason Shogren, Thorsten Janus, Sasha Skiba, Benjamin Rashford, Hangtian Xu, and partici-
pants at the 8th Congress of the EAAERE (Beijing 2019) for their helpful comments and suggestions. Sichao Wei also
appreciates the support of the Fundamental Research Funds for the Central Universities (No. 023400/531118010241).
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1 Introduction
Developing countries are often plagued by severe environmental problems. Based on mean
annual exposure to PM2.5 (particulate matter of less than 2.5 microns of diameter) from 2008 to
2017, the 100 most polluted cities in the world are located in developing countries (World Health
Organization, 2018). Pollution leads to negative health consequences and has received extensive
media coverage, some of which are rather shocking. Air pollution, particularly PM2.5, is
responsible for premature deaths of 3 million people a year around the globe (The Economist,
2017). It is reported that because of lead pollution, about 1/3 of Chinese children’s blood lead
levels are above the normal (The New York Times, 2011). By damaging children’s nervous
systems, lead pollution hampers learning and affects behavior (Reuters, 2012). These news
reports motivate our focus on two health effects imposed by pollution, namely, reducing longevity
and impeding learning.1
Why should we care about these two health effects of pollution? The reason is that
pollution limits the accumulation of physical capital by reducing longevity and limits the
accumulation of human capital by impeding learning, which has been well established both
empirically and theoretically. The empirical literature shows that pollution negatively affects
savings and hence the accumulation of physical capital because pollution reduces longevity (Wen
and Gu, 2012; Ebenstein et al., 2015) and a decreased longevity in turn lowers savings (Bloom
et al., 2003; Zhang and Zhang, 2005; De Nardi et al., 2009). Also see Pautrel (2008) for a
research summary of pollution’s impact on longevity. Motivated by the empirical evidence,
several theoretical papers incorporate the longevity effect into their models, such that pollution
endogenously modifies people’s incentive to save (Pautrel, 2009; Jouvet et al., 2010; Varvarigos,
2010, 2013a; Raffin and Seegmuller, 2014; Fodha and Seegmuller, 2014). The empirical evidence
also shows that pollution impedes children’s learning in various ways, thus negatively affecting
1Besides the two health effects of pollution we emphasize, the extant literature also highlights other health effects, such
as increasing morbidity (Gutiérrez, 2008; Wang et al., 2015) and decreasing labor supply (Hanna and Oliva, 2015;
Jhy-hwa et al., 2015). We acknowledge that potentially interesting results may arise from the interactions of these
pollution health effects, but reserve this idea for future research.
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the accumulation of human capital. Pollution increases school absences (Currie et al., 2009;
Mohai et al., 2011; Chen et al., 2018a), reduces years of schooling (Nilsson, 2009), enters and
damages human brains (Maher et al., 2016), causes a significant decline in cognitive performance
(Ebenstein et al., 2016; Zhang et al., 2018), and jeopardizes mental health (Zhang et al., 2017;
Kim et al., 2017; Chen et al., 2018b). The health impact of pollution on learning and
consequently on human capital has also inspired other theoretical studies (see, for example,
Raffin, 2012; Aloi and Tournemaine, 2013; Sapci and Shogren, 2017).
However, the literature cited above often deals exclusively with one aspect of pollution
health effects, and thus focuses only on the effect of pollution on physical capital or on the effect
of pollution on human capital. To the best of our knowledge, the literature remains silent about
the joint impact of the pollution health effects on the ratio of physical to human capital. But the
growth literature points out that the capital ratio is a key indicator for economic growth (see, for
example, Mulligan and Sala-i Martin, 1993; Ladrón-de Guevara et al., 1997; Barro, 2001;
Duczynski, 2002, 2003). The consequence of this research gap is straightforward. If pollution
only negatively affects the accumulation of physical capital, the physical-to-human-capital ratio
unambiguously decreases in pollution. In contrast, if we only analyze the negative effect of
pollution on human capital, the physical-to-human-capital ratio unambiguously increases in
pollution. It is reasonable, however, to postulate that if pollution imposes negative effects on
physical and human capital in an unbalanced way, the ratio of physical to human capital may
increase, decrease, or stay the same in pollution, and the subsequent dynamics and policy
implications may differ from past studies. By allowing pollution to have adverse impacts on both
types of capital, our model closes the gap between the two strands of literature that analyze only
one aspect of pollution health effects on capital accumulation. The modeling modification can
lead to interesting dynamics in terms of economic and environmental consequences, and these
results are supported by empirical evidence.
The model we employ is an otherwise standard overlapping generations (OLG) model.
What is novel in our model is that the presence of pollution health effects has a differential impact
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on the accumulation of physical and human capital, which in turn affects the
physical-to-human-capital ratio. Since physical capital is associated with pollution when
employed in production, whereas human capital provides solutions to alleviating pollution, the
physical-to-human-capital ratio also conversely influences pollution. Therefore, a feedback loop
between pollution and the physical-to-human-capital ratio is closed. The feedback loop works in
two plausible ways depending on the capital accumulation differential caused by pollution.
In the first case, the capital accumulation differential is positive. We derive analytical
results that on the Balanced Growth Path (BGP), pollution and the physical-to-human-capital
ratio are positively correlated, and the BGP can be stable. The intuition goes as follows. A
positive capital accumulation differential implies that pollution harms the accumulation of human
capital more than that of physical capital, and the effect is worsening in pollution. If the stock of
pollution is low, human capital is abundant relative to physical capital. A low ratio of physical to
human capital leads to a low stock of pollution, which in turn maintains the low capital ratio.
Thus, a virtuous circle continues with a high economic growth rate. In contrast, if pollution is
high, human capital becomes scarce relative to physical capital. A high ratio of physical to human
capital generates high pollution, which conversely reinforces the high capital ratio. So a vicious
circle is at work with a low economic growth rate because high pollution severely limits the
accumulation of both types of capital. Interestingly, two extreme BGPs and a middle one
separating the former two can simultaneously arise due to reasons similar to the intuition just
mentioned. The two extreme BGPs are “sinks”. One is desirable in the sense that it features a
high economic growth rate and a low stock of pollution, whereas the other should be avoided as it
is associated with a low economic growth rate and a high stock of pollution. The BGP in the
middle exhibits saddle stability, and it gives rise to a separatrix demarcating the two “sink”
regions. In terms of a policy implication, we show that the government can steer the economy
away from the inferior BGP towards the desirable one.
The theoretical results can be supported by the cross-sectional data at the provincial level
in China and at the country level across the world. We collect industrial waste gas emission data
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from the China Statistical Yearbook on Environment (2013-2016), and economic and
demographic data from the China Center for Human Capital and Labor Market Research (2018)
to construct proxies for pollution, capital ratio, and economic growth. Pollution is represented by
the logged values for population-weighted emission of industrial waste gas. The capital ratio is
calculated by dividing the monetary value for physical capital by that for human capita in
per-capita terms. The values for physical and human capital are measured in the Chinese currency
unit based on the price in 1985. Economic growth is the annual growth rate of real GDP per
capita. Figure 1 exhibits the scatter plots of 30 Chinese provinces for the years from 2012 to 2015.
Pairwise relations among pollution, the capital ratio, and economic growth are visualized, and the
slopes of trend lines are consistent with our theoretical results that arise from the case with a
positive capital accumulation differential. We also collect panel data on air stock pollutants (mean
annual exposure to PM2.5 and PM10 at the national level), population, and real GDP per capita
from the World Bank database (World Bank Group, 2018a,b).2 Pollution is represented by the
logged values of population-weighted PM2.5 and PM10, and economic growth is measured by
the annual growth rate of real GDP per capita. We exhibit the relationship between pollution and
economic growth with scatter plots of the countries for each year (see Figure 2 for 6 years of data
on the two stock pollutants).3 The negatively sloped trend lines indicate that some countries may
experience both robust economic growth and more favorable environmental quality, while other
countries may suffer from both lower economic growth and less favorable environmental quality.4
[Insert Figure 1 and Figure 2 here]
In the second case, the capital accumulation differential is negative. Our model shows that
2Compared with the Chinese data, the World Bank database is more comprehensive, but lacks data on physical and
human capital that can be used to calculate the capital ratio. The world panel data cover a wide range of countries,
including both developing and developed ones. The data on PM2.5 cover 190 countries and regions in 11 years (1990,
1995, 2000, 2005, 2010-2016). The data on PM10 cover 177 countries and regions in 22 years (1990-2011). The data
on population and real GDP per capita go with those on PM2.5 and PM10 air pollutants.3Due to space limit, we do not show all of the years with data available. For PM2.5, negatively sloped trend lines
appear in 9 years, accounting for 81.8% of the total years. For PM10, negatively sloped trend lines appear in 11 years,
accounting for 50% of the total years.4Note that a BGP describes the tendency of how a country will eventually operate. The empirical evidence we gather
does not necessarily imply the countries are standing on their BGPs, but only illustrates that the countries can operate
in a way following the tendencies characterized by their BGPs.
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the economy and the environment may exhibit cyclical movements. The intuition behind the
cyclical movements can be explained as follows. A negative capital accumulation differential
implies that pollution adversely affect the accumulation of physical capital more than that of
human capital, and the effect is worsening in pollution. Suppose initially as the stock of pollution
increases, the ratio of physical to human capital decreases. Because the “clean” human capital
becomes abundant relative to the “dirty” physical capital, less pollution is discharged into the
environment, which in turn leads to a higher ratio of physical to human capital. Then the stock of
pollution rises again. The back-and-forth dynamic relationship between capital ratio and pollution
thus leads the economy and the environment to move cyclically. These cycles represent inequality
between generations (Schumacher and Zou, 2008, 2015), and government policy can eliminate
the cycles.
The extant empirical research has documented cyclical movements of economic and
environmental variables. For example, there exists a cyclical correlation between mortality and
the economy (Tapia Granados, 2005; Rolden et al., 2014) as well as evidence on cycles of urban
air pollutants (Mayer, 1999). Also based on the time-series data on pollution and economic
growth in each country and region from the World Bank database (World Bank Group, 2018a,b),
we provide additional empirical support for the joint cycles of the economy and the environment.
We check the time paths of pollution and economic growth combinations for each country and
region, and depict some observed cycles with red continuous lines in Figure 3.
[Insert Figure 3 here]
The rest of the paper is organized as follows. Section 2 presents a literature review.
Section 3 sets up the model and shows that the market equilibria can be divided into three regimes
by policy parameters. In each of the three regimes, Sections 4-6 lay out difference equations
representing the dynamic interactions between the economy and the environment, and derive
analytical results about the BGP and the transitional dynamics. Section 7 provides numerical
examples to complement the previous analytical results. Section 8 concludes.
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2 Related Literature
We argue that the accumulation differential between physical and human capital caused by
pollution through the health effects is a new mechanism that gives rise to economic and
environmental cycles. There is a strand of literature that theoretically demonstrates the emergence
of cycles and explores the mechanisms behind the cycles. So we carefully review the mechanisms
already illustrated in the literature, which include the following. First, cycles arise due to the
relative magnitudes of technical parameters. Zhang (1999) and Seegmuller and Verchère (2004)
utilize similar models to that developed by John and Pecchenino (1994) where the agent engages
in environmental maintenance, and derive similar conditions that a sufficiently large
environmental degradation rate relative to environmental maintenance efficiency gives rise to
cycles. In addition, Varvarigos (2013b) finds that the emission rate of pollution above a threshold
level results in dampened cycles. Second, the emergence of cycles can originate from the
representative agent’s subjective factors. Schumacher and Zou (2008) introduce behavioral
economics into an otherwise standard OLG model, and show that the deviation of people’s
perceived level of pollution from the actual level of pollution can generate cycles. Schumacher
and Zou (2015) and Constant and Davin (2019) highlight the role of endogenous environmental
preference. In the former model, a threshold environmental quality alters generations’ preferences
of the environment over consumption, which consequently leads to cycles. In the latter, high
sensitivity of environmental preference to pollution and human capital causes cycles. Third,
cycles can be attributed to government interventions. Palivos and Varvarigos (2017) compare
models with and without public pollution abatement, and show that cycles arise in the absence of
public pollution abatement. Goenka et al. (2017) study a second-best optimal taxation scheme
that is contingent on physical capital. Interestingly, this optimal taxation scheme can be a source
of cycles. Fourth, cycles can stem from the health impacts of pollution relative to health
expenditures. Raffin and Seegmuller (2017) develop an OLG model where longevity is jointly
determined by pollution, as well as private and public health expenditures. The authors show that
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if the damaging effect of pollution on longevity outweighs the effect of health expenditures,
cycles can emerge. Similar to this strand of literature, our paper is an application of the OLG
model to study the dynamic interplay of the economy and the environment. However, the major
modeling departure from the literature lies in the analyzed types of productive capital. As
mentioned earlier, we study both physical and human capital, whereas the literature exclusively
focuses on one type of capital. This modeling assumption enables us to contribute to the literature
by further identifying the capital accumulation differential imposed by pollution as another source
of economic and environmental cycles.
It is also worth mentioning that one exception in the literature is Motoyama (2016), who
also touches on the dynamic interactions of pollution and the ratio of physical to human capital
and therefore merits a careful comparison with our paper. Motoyama (2016) shows that multiple
equilibria may emerge in an OLG model with physical capital being the source of pollution. If the
ratio of physical to human capital is less than a threshold value, the productivity of education is
moderately damaged by pollution. Households invest in education, and both physical and human
capital accumulate. The economy converges to a low ratio of physical to human capital. However,
if the ratio of physical to human capital surpasses the threshold, the productivity of education is
reduced. Households stop investing in education and only physical capital accumulates through
savings. The economy converges to a high ratio of physical to human capital. The key difference
between Motoyama (2016) and this paper is twofold. First, in Motoyama (2016) pollution only
negatively affects human capital. As pollution increases, the ratio of physical to human capital
unambiguously rises. In contrast, we emphasize the interaction of health effects of pollution on
physical and human capital. We not only derive multiple BGPs similar to Motoyama (2016), but
also show that if the ratio of physical to human capital decreases in pollution, a possibility absent
from Motoyama (2016), economic and environmental cycles may emerge. We also show that the
government has a role in eliminating the cycles. Second, in Motoyama (2016) the government
does not invest in education and pollution is implicitly modeled as being associated with physical
capital. In our model, however, the government provides public education even if agents may not
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invest in private education. We also explicitly model the dynamics for the stock of pollution
where unlimited growth of pollution can be checked by pollution abatement financed by
government spending. Our model thus allows us to discuss policy implications in terms of
government expenditures.
3 The Model
3.1 Firms
The production factors in this model are physical capital Kt , and labor Lt augmented by human
capital Ht . Denote rt as the rental price of physical capital, and wt as wage rate paid per unit of
labor. The production function that a typical competitive firm employs to produce a final good is
Yt = AKαt (HtLt)
1−α , where A > 0 is a production scalar, α ∈ (0,1) is physical capital’s share in
production, and 1−α is augmented labor’s share in production. The price of the final good is
normalized to 1. The firm pays a proportional tax, τ , on the final good to the government (Barro,
1990; Devarajan et al., 1996; Agénor and Neanidis, 2011). The representative firm hires physical
capital Kt and labor Lt to maximize its profits πt . The profit-maximization problem remains the
same in each period:
maxKt ,Lt
πt = (1− τ)AKαt (HtLt)
1−α− rtKt −wtLt .
Define kt = Kt/Ht as the ratio of physical to human capital. Since each input is paid its marginal
product, the first-order conditions are
rt = (1− τ)αAkα−1t L1−α
t , (1a)
wt = (1− τ)(1−α)Akαt HtL
−αt . (1b)
3.2 Government
The government collects fiscal revenues through the proportional tax on the final good,
τAKαt (HtLt)
1−α = τAkαt HtL
1−αt . The government allocates a portion of the fiscal revenues,
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∆ ∈ [0,1], to finance pollution abatement at , and the remaining portion, 1−∆ ∈ [0,1], to finance
public education mt .5 The government runs a balanced budget in each period, which requires
at = ∆τAkαt HtL
1−αt , (2a)
mt = (1−∆)τAkαt HtL
1−αt . (2b)
3.3 Stock of Pollution
The stock of pollution increases due to production activities but decreases due to public pollution
abatement. Because it is “too good to be true” that emissions cease to grow (Economides and
Philippopoulos, 2008), we assume that the unabated flow of pollution, ρKt , is proportional to
physical capital, where ρ > 0 represents the polluting capacity of physical capital. Thus as long
as physical capital accumulates, unabated pollution grows. We also specify that the abated flow of
pollution is the ratio of unabated pollution to public pollution abatement, ρKt/at (see, for example,
Gradus and Smulders, 1993; Smulders and Gradus, 1996; Pautrel, 2009, 2012).6 An implicit
assumption is at > 1, such that the abated flow of pollution cannot surpass the unabated flow. The
stock of pollution, zt , evolves according to
zt+1 = (1−θ)zt +ρKt
at, (3)
5The government may also allocate fiscal revenues to other public uses, such as infrastructure that would enhance the
physical capital (see, for example, Agénor, 2011). This additional use of fiscal revenues diverts resources away from
pollution abatement and public education, both of which support the accumulation of human capital. So introducing
government spending on infrastructure will increase the ratio of physical to human capital relative to the extant model.
Because the primary focus of this paper is on how the health effects of pollution influence the ratio of physical to
human capital, we abstract from the public expenditures on infrastructure.6There are two advantages associated with this specification. First, Gradus and Smulders (1993) show that even when
investment activities (e.g., the use of cleaner fuels, which allows for a reduction in the amount of pollution per unit
of capital in the production process) and abatement activities (e.g., “end-of-pipe measures”, which aim at cleaning up
existing pollution) are distinguished, this function for net emissions still qualitatively holds. So although we use the
term “abatement”, we cover both cases of reducing the flow of pollution and the existing stock of pollution. Second,
Pautrel (2012) argues that the linear specification of the net emissions (for example, ρkt − at ) is “not constant along
the Balanced Growth Path (BGP), and therefore the stock of pollution explodes in the long run.”
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where θ ∈ (0,1) represents the dissipation rate of pollution. The dynamics for the stock of
pollution (3) implies that the absence of human activities enables the stock of pollution to
converge towards zero in the long run. The stock of pollution adversely affects the economy by
inflicting two types of health effects on the representative agent, which will be fully explained in
Section 3.4.
3.4 Agents
The time length in each period is 1. The representative agent lives three periods, i.e., childhood,
adulthood, and elderhood. In childhood, the agent receives education to accumulate human
capital. In adulthood, the agent gives birth to one child, inelastically supplies one unit of labor to
earn wage income, and makes decisions in terms of consumption during adulthood, savings, and
private education expenditures on her child to maximize lifetime utility. In elderhood, the agent
enjoys the fruits due to the decisions made during adulthood, i.e., elderly consumption financed
by her savings and her child’s human capital as a result of her private education expenditures.
The agent lives the entirety of her childhood and adulthood, but lives only a fraction of her
elderhood, φ ∈ (0,1]. The representative agent born at the beginning of period t −1 thus has a
lifetime equal to 2+φt+1. The representative agent treats her longevity as given. Similar to
Varvarigos (2013b) and Fodha and Seegmuller (2014), the fraction of elderhood that an agent
lives depends on the stock of pollution.7 For the representative agent born at the beginning of
period t −1, her elderly longevity depends on the stock of pollution during her adulthood, i.e.,
φt+1 = φ(zt) ∈ [φ ,φ ], where φ > 0 and φ ≤ 1 are the lower and upper bounds of longevity. We
assume φ(0) = φ , φ(zt)→ φ for zt →+∞, and φ ′(zt)< 0 . The specification of longevity
depending on the stock of pollution captures the first type of health effect caused by pollution.
7This specification allows us to derive succinct analytical results that highlight the role of pollution health effect that
shortens longevity and further undermines capital accumulation. The longevity function can also include other positive
factors, such as per-capita income or healthcare expenditures. Recall the feedback loop between the environment and
the economy described in Introduction. Pollution affects the economy by damaging health. If factors such as per-capita
income or health expenditures are introduced to improve health, the negative effects of pollution on the economy are
weakened, but the feedback loop that drives our results still works. Therefore, the introduction of other factors in
longevity adds additional complexity to the results, but does not qualitatively alter the basic intuition.
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Education expenditures are necessary for the accumulation of human capital in childhood.
Denote et as private education expenditures by the agent. Total education expenditures are
µmt + et , where µ > 0 measures the relative strength of public to private education expenditures
(Buiter and Kletzer, 1995; Osang and Sarkar, 2008). One can think of total education
expenditures as the sum of what the government pays for compulsory education, plus what the
parents choose to pay in the form of college tuition and fees for her child. However, pollution
impedes learning because a worsened environmental quality reduces schooling time due to
absenteeism, undermines cognitive ability, or damages a child’s mental health and nervous
system. Consequently, each dollar spent on education becomes not so effective in the
accumulation of human capital as it would be without pollution. We follow Raffin (2012) and
introduce the effectiveness of education expenditures λt = λ (zt) ∈ [λ ,λ ], where λ ≥ 0 and λ ≤ 1
are the lower and upper bounds of the function. We assume λ (0) = λ , λ (zt)→ λ for zt →+∞,
and λ ′(zt)< 0. The effectiveness of education expenditures as a function of pollution thus
captures the second type of pollution health effect that hampers learning.
With λ (zt) adjusting total education expenditures, effective education expenditures thus
become λ (zt)(µmt + et). Besides education expenditures, the evolution of human capital also
depends on parents’ human capital (e.g., parental example and guidance). Both effective
education expenditures and parents’ human capital are subject to constant returns to scale in
human capital formation. As the agent born at the beginning of period t −1 has human capital Ht
in period t and gives birth to a child at the beginning of period t, the child born at the beginning of
period t has human capital in period t +1 equal to
Ht+1 = B [λt (µmt + et)]β
H1−βt , (4)
where B > 0 is a scalar, β ∈ (0,1) is the share of effective education expenditures in the formation
of human capital, and 1−β ∈ (0,1) is the share of parents’ human capital.
Taking her longevity φt+1 and human capital Ht as given, the agent born at the beginning
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of period t −1 makes decisions at the beginning of period t. The agent derives utility from her
adulthood consumption ct , elderhood consumption dt+1, and her child’s stock of human capital
Ht+1 due to altruism (Osang and Sarkar, 2008). Because the agent cares about her elderhood
consumption and her child’s human capital, the agent has motives to save and invest in private
education (de la Croix and Doepke, 2003). Assuming a logarithmic function that is additively
separable, the agent’s lifetime utility is
Ut−1 = lnct +φt+1 [lndt+1 +χ lnHt+1] , (5)
where the parameter χ > 0 represents the agent’s altruism towards her child’s human capital.
During adulthood, the representative agent uses her wage income wt to cover her
adulthood consumption ct , savings st , and private education expenditures for her child et . When
the agent is old, she uses the remunerated savings rt+1st to finance her elderly consumption dt+1.
In adulthood and elderhood, the agent follows the same budget principle that equates her income
per unit of time length to her total expenditures in that period.8 The budget constraints for
adulthood and elderhood are
wt = ct + st + et , (6a)
rt+1st
φt+1= dt+1. (6b)
The representative agent’s problem is to maximize (5) by choosing ct , st , et , and dt+1 subject to
(4), (6a), (6b), as well as an additional non-negative constraint et ≥ 0.
The agent may or may not invest in private education based on the Kuhn-Tucker
8Longevity φt+1 can be interpreted in two equivalent ways, but the elderhood budget constraints are the same. First,
φt+1 can be interpreted as the living time length in elderhood, so the entire lifetime of the representative agent is
2+φt+1. We employ the first interpretation. Second, φt+1 can be interpreted as the survival probability in elderhood,
so the life expectancy of the representative agent is still 2+φt+1. In the second interpretation, an assumed mutual fund
is called in. The mutual fund operates in a perfectly competitive annuities market, receives savings from the agent
paying return rt+1, and invests the savings in physical capital with return rt+1. Perfect competition in the annuities
market implies rt+1 = rt+1/φt+1. Similar details can be found in Chakraborty (2004, p. 122). It can be revealed that the
two interpretations of longevity φt+1 does not alter the elderhood budget constraint (6b).
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conditions. If the agent invests in private education, et > 0, the functions for adulthood
consumption, savings, and private education expenditures are
ct =Φt+1
φt+1(wt +µmt), (7a)
st = Φt+1(wt +µmt), (7b)
et = Ωt+1(wt +µmt)−µmt , (7c)
where Φt+1 = Φ(φt+1) =1
1+χβ+(1/φt+1)is the agent’s propensity to save because Φt+1 =
∂ st
∂wt, and
Ωt+1 = Ω(φt+1) =χβ
1+χβ+(1/φt+1)is the propensity to invest in private education because
Ωt+1 =∂et
∂wt. The propensity to save satisfies Φ
′(φt+1)> 0 and Φt+1 ∈
[φ
(1+χβ )φ+1, φ
(1+χβ )φ+1
]
,
and the propensity to invest in private education satisfies Ω′(φt+1)> 0 and
Ωt+1 ∈
[χβφ
(1+χβ )φ+1, χβφ
(1+χβ )φ+1
]
.
However, if the following condition holds:
χβφt+1
µmt<
1
wt − st, (8)
the agent does not invest in private education, i.e., et = 0. Condition (8) says that if the marginal
utility gained from the first dollar invested in private education is smaller than the utility lost due
to the foregone young consumption, the agent does not invest in private education. The stock of
pollution plays a role in modifying the agent’s decision to invest in private education because
pollution reduces longevity, rendering the marginal utility gained from the first dollar invested in
private education even smaller. As the representative agent does not invest in private education,
human capital accumulation only depends on public education expenditures. The functions for
adulthood consumption and savings become
ct =Φt+1
φt+1wt , (9a)
st = Φt+1wt , (9b)
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where Φt+1 = Φ(φt+1) =1
1+(1/φt+1)is the agent’s propensity to save when et = 0. The propensity
satisfies Φ′(φt+1)> 0 and Φt+1 ∈
[φ
1+φ ,φ
1+φ
]
. All else equal, if the agent lives for a longer time,
she increases savings and cuts back on adulthood consumption.
3.5 Market Equilibria
For ease of exposition, we assume that there is no population growth and normalize the labor size
to unity, i.e., Lt = 1 for all t, and we interchangeably use per-capita and aggregate variables.
Because this period’s savings become next period’s physical capital, we have st = Kt+1.
The evolution of the economy varies due to the representative agent’s decision on private
education expenditures. Whether or not the agent invests in private education alters the
accumulation of physical and human capital. We hereafter denote PE (Private Education) as the
regime where private education expenditures are positive, et > 0, and NPE (No Private
Education) as the regime where et = 0. The combination of policy parameters, τ and ∆, dictates
the representative agent’s incentive to invest in private education.
Proposition 1. (The Three Regimes) The policy set, consisting of the tax rate τ ∈ (0,1) and the
composition of fiscal revenues ∆ ∈ (0,1), is divided into the following three regimes:
(i) The PE regime. Given a value for τ , as long as a value for ∆ satisfies
∆ ≥ 1− (1−α)φ
1+φχβµ
1−ττ ≡ f1(τ), where f1(τ) satisfies f ′1(τ)> 0, f ′′1 (τ)< 0, and solving
f1(τ) = 0 yields τ = (1−α)φ
1+φχβµ
/[
1+(1−α)φ
1+φχβµ
]
, for any stock of pollution
zt ∈ [0,+∞), the agent’s investment in private education is always positive, et > 0.
(ii) The NPE regime. Given a value for τ , as long as a value for ∆ satisfies
∆ ≤ 1− (1−α) φ
1+φ
χβµ
1−ττ ≡ f2(τ), where f2(τ) satisfies f ′2(τ)> 0, f ′′2 (τ)< 0, and solving
f2(τ) = 0 yields τ = (1−α) φ
1+φ
χβµ
/[
1+(1−α) φ
1+φ
χβµ
]
, for any stock of pollution
zt ∈ [0,+∞), the agent’s investment in private education is always zero, et = 0.
(iii) Both the PE and NPE regimes. Given a value for τ , as long as a value for ∆ satisfies
f2(τ)< ∆ < f1(τ), there exists a threshold stock of pollution zo(τ,∆) that is implicitly defined by
φ(zo)1+φ(zo) =
1χβ
µτ(1−∆)(1−τ)(1−α) . When zt ∈ [0,zo], the agent’s investment in private education is positive,
15
et > 0; when zt ∈ (zo,+∞), the agent’s investment in private education is zero, et = 0.
Proof. See Appendix A.
Proposition 1 is more clearly revealed in Figure 4. Region I corresponds to the PE regime,
where the combinations of τ and ∆ lead the representative agent to invest in private education
irrespective of pollution. Region II corresponds to the NPE regime, where the agent never invests
in private education. Region III corresponds to both the PE and NPE regimes, where the agent’s
decision on private education expenditures is determined by the threshold stock of pollution,
zo(τ,∆).
[Insert Figure 4 here]
The evolution of pollution takes the same form throughout the three regimes in
Proposition 1. Substituting (2a) into (3) gives the difference equation for the stock of pollution:
zt+1 = (1−θ)zt +ρ
∆τAk1−α
t . (10)
From Equation (10), setting zt+1 = zt and rearranging yields the zz locus:
−θzt +ρ
∆τAk1−α
t = 0. (11)
The zz locus defines all the combinations of kt and zt where the stock of pollution is in steady
state. The slope of the zz locus reflects how the economy affects the environment. On the zz locus,
the stock of pollution increases in the ratio of physical to human capital because human capital is
“clean” and physical capital is “dirty” in production, which implies that an economy with
abundant physical capital relative to human capital tends to have a higher stock of pollution.
In each of the three regimes, we will characterize the Balanced Growth Path (BGP) and
the transitional dynamics towards the BGP. Along the BGP under regime i, where i = PE,NPE,
the physical-to-human-capital ratio kt , the stock of pollution zt , longevity φ(zt), and the
effectiveness of education expenditures λ (zt) remain constant. We denote the BGP capital ratio as
16
k∗i = kt+1,i = kt,i and the BGP stock of pollution as z∗i = zt+1,i = zt,i. Also along the BGP under
regime i, where i = PE,NPE, physical capital Kt , human capital Ht , final output Yt , pollution
abatement expenditures at , public education expenditures mt , young consumption ct , and elderly
consumption dt grow at the same rate g∗i for any t. We define the growth rate along the BGP as
g∗i = ln
(Kt+1
Kt
)∣∣∣∣i
= ln
(Ht+1
Ht
)∣∣∣∣i
= ln
(Yt+1
Yt
)∣∣∣∣i
= ln
(at+1
at
)∣∣∣∣i
= ln
(mt+1
mt
)∣∣∣∣i
= ln
(ct+1
ct
)∣∣∣∣i
= ln
(dt+1
dt
)∣∣∣∣i
, where i = PE,NPE. (12)
4 The Private-Education (PE) Regime
Under the PE regime, the agent invests in private education for any stock of pollution.
Substituting (1b) and (2b) into the savings function (7b), setting st = Kt+1, and rearranging yields
the difference equations for physical capital:
PE :Kt+1
Kt= A [(1− τ)(1−α)+µτ(1−∆)]Φ(zt)k
α−1t . (13)
The term Φ(zt) captures the damage inflicted by pollution on physical capital accumulation. The
propensity to save increases in longevity, and longevity decreases in pollution, implying the
propensity to save decreases in pollution, Φ′(zt)< 0. Pollution thus reduces the agent’s savings
that transform into physical capital. So we establish the link describing how pollution damages
physical capital by reducing longevity. Also as is evident from (13), other things equal, the
growth of physical capital decreases convexly in the physical-to-human-capital ratio kt .
Substituting (1b), (2b), and (7c) into (4) yields the difference equations for human capital:
PE :Ht+1
Ht= BAβ [(1− τ)(1−α)+µτ(1−∆)]β [Ω(zt)λ (zt)]
βk
αβt . (14)
The terms Ω(zt) and λ (zt) reveal that pollution undermines human capital accumulation.
Pollution reduces longevity, which in turn decreases the agent’s propensity to invest in private
17
education, and thus Ω′(zt)< 0. Pollution also decreases the effectiveness of education
expenditures, and thus λ ′(zt)< 0. Because pollution reduces the effective education expenditures
in human capital formation, we establish the link describing how pollution damages human
capital. In addition, all else equal, the growth of human capital increases concavely in the
physical-to-human-capital ratio kt .
From equations (13) and (14), we derive the difference equation for the ratio of physical to
human capital. Because kt+1/kt = (Kt+1/Kt)/(Ht+1/Ht), dividing (13) by (14) yields
PE : kt+1 =A1−β
B[(1− τ)(1−α)+µτ(1−∆)]1−β Φ(zt)
[Ω(zt)λ (zt)]β
kα−αβt . (15)
Equation (15) describes the evolution of capital ratio over time. The capital ratio in period t +1
depends on four terms. The first term A1−β/B consists of the scalars in the production function and
human capital formation function. The second term [(1− τ)(1−α)+µτ(1−∆)]1−βreveals the
economic sources of physical and human capital accumulation. Wage income corresponds to
(1− τ)(1−α), which contributes to physical capital through savings and to human capital
through private education expenditures. Fiscal revenues allocated to public education
expenditures correspond to µτ(1−∆), which contributes to physical capital by boosting savings
(see equation 7b) and to human capital by providing public education. Because 0 < 1−β < 1, the
capital ratio kt+1 concavely increases in economic sources. The reason is that physical capital
accumulation exhibits constant returns to total savings by (13), while human capital exhibits
diminishing returns to total education expenditures by (14). The third term Φ(zt)/[Ω(zt)λ (zt)]β
captures the pollution effects on the capital ratio. The numerator shows that pollution reduces
physical capital through the propensity to save Φ(zt), such that other things equal, the
physical-to-human-capital ratio declines in pollution. The denominator reveals that pollution
decreases human capital through the propensity to invest in private education Ω(zt) and through
the effectiveness of education expenditures λ (zt), such that all else equal, the
physical-to-human-capital ratio increases in pollution. The third term stresses the primary
18
departure from the literature. We focus on the whole fraction, while the literature examines either
the numerator or the denominator. The fourth term shows the relationship between the capital
ratios in periods t and t +1. Because 0 < α −αβ < 1, the capital ratio in the next period
concavely increases in the capital ratio in the previous period.
From (15), setting kt+1 = kt yields the kk locus under the PE regime:
PE :A1−β
B[(1− τ)(1−α)+µτ(1−∆)]1−β Φ(zt)
[Ω(zt)λ (zt)]β
kα−αβt − kt = 0. (16)
The kk locus defines all the combinations of kt and zt where the ratio of physical to human capital
is in steady state. The slope of the kk locus reflects how pollution affects the economy, but the
introduction of pollution health effects renders the slope less straightforward.
To check the kk locus slope, we define three elasticities that will be used later. First,
EΦt+1,zt= Φ
′(zt)Φ(zt)
zt < 0 is the elasticity of the propensity to save with respect to pollution. Second,
EΩt+1,zt= Ω
′(zt)Ω(zt)
zt < 0 is the elasticity of the propensity to invest in private education with respect
to pollution. These two elasticities originate from the pollution health effect on longevity. Third,
Eλt ,zt= λ ′(zt)
λ (zt)zt < 0 is the elasticity of the effectiveness of education expenditures with respect to
pollution. This elasticity stems from the pollution health effect on learning. We further define the
capital accumulation differential for any zt > 0 under the PE regime:
Ψt,PE = EΦt+1,zt︸ ︷︷ ︸
physical capital effect
−β (EΩt+1,zt+Eλt ,zt
)︸ ︷︷ ︸
human capital effect
. (17)
Equation (17) summarizes and compares the adverse effects of pollution on the accumulation of
physical and human capital. The term EΦt+1,zt< 0 reflects how sensitive the propensity to save is
to changes in pollution. As the propensity to save determines savings and thus physical capital,
we call EΦt+1,ztthe physical capital effect. The term EΩt+1,zt
< 0 reflects how sensitive the
propensity to invest in private education is to changes in pollution. The term Eλt ,zt< 0 reflects
how sensitive the effectiveness of education expenditures is to changes in pollution. The
19
propensity to invest in private education and the effectiveness of education expenditures together
determine the accumulation of human capital. So we call β (EΩt+1,zt+Eλt ,zt
) the human capital
effect. The human capital effect is adjusted by the parameter β , which is the share of effective
education expenditures in human capital formation. By subtracting the human capital effect from
the physical capital effect, we get the capital accumulation differential caused by pollution Ψt,PE .
The capital accumulation differential Ψt,PE captures the asymmetry of pollution health
effects on the accumulation of physical and human capital. This asymmetry leads the
physical-to-human-capital ratio kt to increase or decrease in pollution zt , such that on the kk locus,
kt may increase or decrease in zt . We summarize the relationship between Ψt,PE and the kk locus
slope in the following proposition.
Proposition 2. (Slope of the kk Locus) Under the PE regime, the capital accumulation
differential Ψt,PE dictates the slope of the kk locus in (zt ,kt) space. The kk locus slopes up if
Ψt,PE > 0 and slopes down if Ψt,PE < 0.
Proof. See Appendix B.
Proposition 2 indicates that the kk locus slope depends on the capital accumulation
differential Ψt,PE . If Ψt,PE > 0, the human capital effect is larger than the physical capital effect,
i.e., 0 > EΦt+1,zt> β (EΦt+1,zt
+Eλt ,zt). Pollution adversely affects human capital more than
physical capital, and thus physical capital becomes relatively abundant. The
physical-to-human-capital ratio rises in pollution and the kk locus slopes up in (zt ,kt) space. In
contrast, if Ψt,PE < 0, the physical capital effect is larger than the human capital effect, i.e.,
EΦt+1,zt< β (EΦt+1,zt
+Eλt ,zt)< 0. Pollution damages physical capital more than human capital,
and thus human capital becomes relatively abundant. The physical-to-human-capital ratio
declines in pollution and the kk locus slopes down in (zt ,kt) space. It is also possible that the kk
locus first slopes up and then slopes down. In a numerical example to be illustrated in Figure 7,
the kk locus under the PE regime first rises and then declines in pollution, indicating the capital
accumulation differential may switch its sign as pollution changes.
20
So far we have completed the description of the model building blocks under the PE
regime, and we turn our attention to the BGP. Graphically, a BGP is represented by the
intersection of the kk locus and the zz locus. Mathematically, the three key BGP variables, k∗PE ,
z∗PE , and g∗PE , can be solved from (13), (14), and (11). Taking natural logs on both sides and
applying the definition of BGP growth rate (12) gives the following three equations:
g∗PE = lnA+ ln [(1− τ)(1−α)+µτ(1−∆)]+ lnΦ(z∗PE)− (1−α) lnk∗PE , (18a)
g∗PE = lnBAβ +β ln [(1− τ)(1−α)+µτ(1−∆)]+β ln [Ω(z∗PE)λ (z∗PE)]+αβ lnk∗PE , (18b)
lnθ + lnz∗PE = lnρ
∆τA+(1−α) lnk∗PE . (18c)
Equations (18a) and (18b) together solve for k∗PE = k(g∗PE) and z∗PE = z(g∗PE). Inserting k(g∗PE)
and z(g∗PE) into (18c) yields
lnθ + lnz(g∗PE) = lnρ
∆τA+(1−α) lnk (g∗PE) , (19)
which implicitly determines the BGP growth rate g∗PE . With g∗PE , the capital-to-human-capital
ratio k∗PE and pollution z∗PE can be further determined. We assume equation (19) satisfies g∗PE > 0,
a mild and reasonable assumption that is commonly adopted by the literature.
In the following proposition, we describe the conditions on whether and how the BGP can
be reached. Substituting the BGP value for pollution z∗PE into (17) yields
Ψ∗PE = EΦ∗
PE ,z∗PE−β
(
EΩ∗PE ,z
∗PE+Eλ ∗
PE ,z∗PE
)
, which is the capital accumulation differential
evaluated on the BGP. The magnitude and sign of Ψ∗PE determine the dynamic properties
surrounding the BGP.
Proposition 3. (Dynamic Properties around the BGP) Under the PE regime, the dynamic
properties around the BGP depend on the capital accumulation differential evaluated on the BGP
Ψ∗PE . Notice that the parameters satisfy
α(1−β )(1−θ)−1
θ(1−α) <−[α(1−β )−(1−θ)]2
4θ(1−α) < 0 < 1−α+αβ1−α < (2−θ)(1+α−αβ )
θ(1−α) .
21
(i) The kk locus slopes up on the BGP and Ψ∗PE > 0. The BGP is locally stable if
0 < Ψ∗PE < 1−α+αβ
1−α . The BGP exhibits locally saddle stability if
1−α+αβ1−α < Ψ
∗PE < (2−θ)(1+α−αβ )
θ(1−α) . The BGP is locally unstable if(2−θ)(1+α−αβ )
θ(1−α) < Ψ∗PE .
(ii) The kk locus slopes down on the BGP and Ψ∗PE < 0. The BGP is locally stable if
−[α(1−β )−(1−θ)]2
4θ(1−α) < Ψ∗PE < 0. The BGP features locally dampened cycles if
α(1−β )(1−θ)−1
θ(1−α) < Ψ∗PE <−
[α(1−β )−(1−θ)]2
4θ(1−α) . The BGP features locally outward cycles if
Ψ∗PE < α(1−β )(1−θ)−1
θ(1−α) .
Proof. See Appendix C.
Proposition 3 states that the capital accumulation differential Ψ∗PE characterizes the
attainability of the BGP and how the BGP is reached. The examination of BGP attainability, or
divergence versus convergence, is important because any policy discussion revolving around a
BGP that turns out be unattainable is in vain. The BGP attainability depends on the magnitude of
the absolute value for Ψ∗PE . Proposition 3 implies that failure of the system’s convergence towards
the BGP (instability and outward cycles) arises from a sufficiently large absolute value for the
capital accumulation differential Ψ∗PE , whereas convergence towards the BGP (stability, saddle
stability, and dampened cycles) occurs due to a sufficiently small absolute value for Ψ∗PE . In the
following analyses, we assume that the condition for convergence is always satisfied, such that the
BGP is attainable. It is equally important to understand how the BGP is reached. The transitional
dynamics leading to the BGP may entail policy implications. Among the transitional dynamic
patterns that may arise, dampened cycles of the economy and the environment are of policy
interest. The emergence of cycles depends on the sign of Ψ∗PE . Proposition 3 implies that no cycle
arises when Ψ∗PE > 0, whereas cycles emerge when Ψ
∗PE < 0. To understand the intuition behind
BGP attainability and cycles, note that the dynamic interactions between pollution and the
physical-to-human-capital ratio are at work. Pollution influences the capital ratio based on the
capital accumulation differential, and the capital ratio conversely affects pollution depending on
the abundance of “dirty” physical capital relative to “clean” human capital. Further, the
movement of the physical-to-human-capital ratio is checked by the mechanisms built in equations
22
(13) and (14). As the capital ratio increases, the growth of physical capital convexly decreases,
whereas the growth of human capital concavely increases. So the checking force on the capital
ratio is strong for a lower capital ratio and weak for a higher capital ratio. In line with the
exposition of Proposition 3, we break the detailed explanation of intuition into two scenarios.
In the first scenario where the capital accumulation differential is positive, pollution
causes the physical-to-human-capital ratio kt to move in the same direction as pollution. Suppose
initially pollution is above its BGP value. Through the positive capital accumulation differential,
pollution tends to generate a larger value for kt . Whether kt deviates from or goes back to the
BGP depends on the capital accumulation differential relative to the checking force on kt . If the
capital accumulation differential is sufficiently large, the increase in kt cannot be checked, and
thus kt becomes even larger, which in turn generates higher pollution. Then the positive capital
accumulation differential renders the previous process to repeat, and thus causing monotonic
divergence away from the BGP. In contrast, if the capital accumulation differential is sufficiently
smaller, the checking force is strong enough to pull kt back towards BGP. The value for kt
becomes smaller, which in turn generates lower pollution. Then the positive capital accumulation
differential initiates monotonic convergence towards the BGP.
In the second scenario where the capital accumulation differential is negative, pollution
causes the physical-to-human-capital ratio kt to move in the opposite direction of pollution. A
pollution level that is initially larger than its BGP value, through the negative capital accumulation
differential, generates kt lower than its BGP value. This lower kt drives pollution down below its
BGP value, but pollution escalates kt through the negative capital accumulation differential. Then
the increased kt is followed by higher pollution. The back-and-forth movements of capital ratio
and pollution repeat and cycles emerge. Still, the magnitude of absolute value for the capital
accumulation differential determines the type of cycles. If the absolute value for the capital
accumulation differential is so large that the checking force on kt cannot pull it back towards the
BGP, outward cycles emerge. In contrast, if the absolute value for the capital accumulation
differential is small enough that the checking force on kt dominates each time kt reciprocates
23
around the BGP, dampened cycles that eventually converge to the BGP emerge.
Next, we derive comparative statics around the BGP to reveal the relations among k∗PE ,
z∗PE , and g∗PE , as well as the policy effects on these three BGP variables. From equations (18a) and
(18b), we derive the following partials:
∂k∗PE
∂g∗PE
g∗PE
k∗PE
=g∗PE
β
Ψ∗PE
Λ∗PE
and∂ z∗PE
∂g∗PE
g∗PE
z∗PE
=g∗PE
β
1−α +αβ
Λ∗PE
< 0, (20)
where Λ∗PE = αEΦ∗
PE ,z∗PE+(1−α)(EΩ∗
PE ,z∗PE+Eλ ∗
PE ,z∗PE)< 0. As is evident from (20), the
economic growth rate g∗PE is negatively correlated with pollution z∗PE because pollution hampers
the accumulation of physical and human capital by damaging health. However, the ambiguity of
∂k∗PE
∂g∗PE
g∗PE
k∗PEarises from the capital accumulation differential Ψ
∗PE , which can be positive or negative.
Suppose Ψ∗PE > 0,
∂k∗PE
∂g∗PE
g∗PE
k∗PE< 0, implying when human capital is scarce relative to physical
capital, the growth rate is lower. The reason is that pollution negatively affects human capital
more than physical capital, the physical-to-human-capital ratio increases in pollution, and thus
∂k∗PE
∂ z∗PE
z∗PE
k∗PE> 0. Because the relationship between pollution and the growth rate is unambiguously
negative, g∗PE and k∗PE must exhibit a negative relationship. Therefore, the sign∂k∗PE
∂g∗PE
g∗PE
k∗PE< 0 is
confirmed. In contrast, suppose Ψ∗PE < 0, pollution negatively affects physical capital more than
human capital. Then from (20), we have∂k∗PE
∂ z∗PE
z∗PE
k∗PE< 0 and
∂k∗PE
∂g∗PE
g∗PE
k∗PE> 0. Empirically, the pairwise
relations among pollution, capital ratio, and economic growth with scatter plots of Chinese
provinces in Figure 1 match the case where Ψ∗PE > 0. Moreover, we find no economic and
environmental cycles in Chinese provinces, which is consistent with Proposition 3 that cycles
cannot emerge when Ψ∗PE > 0.
The policy parameters in the model are the tax rate τ and the share of fiscal revenues
devoted to pollution abatement ∆. The policy effects on the BGP are reported in the form of
elasticities of the BGP fundamental variables with respect to τ and ∆. We define two new
notations for simplicity in later expressions, ΘPE = (1−α)−µ(1−∆)(1−τ)(1−α)+µτ(1−∆) and
ΠPE = µτ(1−τ)(1−α)+µτ(1−∆) > 0. The sign of ΘPE is undetermined. Recall the second term of (15),
24
the numerator of ΘPE reflects how an increase in τ affects the economic sources of physical and
human capital accumulation. An increase in τ contributes to public education expenditures with
the marginal increase represented by µ(1−∆), but transfers wage income away from the agent
with the marginal decrease represented by 1−α . The sign of ΘPE depends on 1−α relative to
µ(1−∆). From (18a), (18b), and (18c), we derive the effects of τ on k∗PE , z∗PE , and g∗PE :
dk∗PE
dτ
τ
k∗PE
=Ψ
∗PE +(1−β )τΘPE
(1−α)Ψ∗PE − (1−α +αβ )
,dz∗PE
dτ
τ
z∗PE
=
(+)︷ ︸︸ ︷
(1−α +αβ )+(1−α)(1−β )τΘPE
(1−α)Ψ∗PE − (1−α +αβ )
< 0,
dg∗PE
dτ
τ
g∗PE
=β
g∗PE
Λ∗PE + τΘPE(1+Λ
∗PE −EΦ∗
PE ,z∗PE)
(1−α)Ψ∗PE − (1−α +αβ )
. (21)
And the effects of ∆ on k∗PE , z∗PE , and g∗PE are
dk∗PE
d∆
∆
k∗PE
=Ψ
∗PE +(1−β )∆ΠPE
(1−α)Ψ∗PE − (1−α +αβ )
,dz∗PE
d∆
∆
z∗PE
=(1−α +αβ )+(1−α)(1−β )∆ΠPE
(1−α)Ψ∗PE − (1−α +αβ )
< 0,
dg∗PE
d∆
∆
g∗PE
=β
g∗PE
Λ∗PE +∆ΠPE(1+Λ
∗PE −EΦ∗
PE ,z∗PE)
(1−α)Ψ∗PE − (1−α +αβ )
. (22)
We focus on the BGP that can be automatically reached,9 so by Proposition 3,
α(1−β )(1−θ)−1
θ(1−α) < Ψ∗PE < 1−α+αβ
1−α , such that the denominators in (21) and (22) are negative,
(1−α)Ψ∗PE − (1−α +αβ )< 0. The policy effects on pollution are clear. An increase in τ
unambiguously reduces pollution because more tax revenues contribute to more pollution
abatement, which reduces pollution. An increase in ∆ also unambiguously reduces pollution
because given the total taxes, a larger share of fiscal revenues devoted to pollution abatement
decreases pollution.
However, the introduction of pollution health effects blurs the policy effects on the capital
ratio and economic growth. To see this, suppose the pollution health effects are gone for the
moment, such that Ψ∗PE = Λ
∗PE = EΦ∗
PE ,z∗PE
= EΩ∗PE ,z
∗PE
= Eλ ∗PE ,z
∗PE
= 0. From (21), the effects of τ
9A BGP that is stable or features dampened cycles can be automatically reached, while the achievement of a BGP
associated with a saddle path requires government intervention. Further, we will see later in Proposition 5 and in
Figure 8 that when both a stable BGP and a BGP with a saddle path exist, the stable BGP is superior based on its
higher economic growth rate and lower pollution.
25
on k∗PE and g∗PE only depend on the sign of ΘPE . Suppose ΘPE > 0 and 1−α > µ(1−∆). The
marginal decrease in wage income is greater than the marginal increase in public education
expenditures, so the total economic resources contributing to capital accumulation decline in τ .
As a result, the economic growth rate g∗PE decreases in τ . But physical capital decreases more
than human capital, so k∗PE also decreases in τ . The similar logic also applies to the case where
ΘPE < 0. However, reintroducing the pollution health effects makes the above analyses less
clear-cut. The ambiguity of signingdk∗PE
dττ
k∗PEarises not only from ΘPE , but also from the capital
accumulation differential Ψ∗PE . Even if we know the sign of ΘPE , pollution can either reduce
physical capital more or reduce human capital more, such that Ψ∗PE can be positive or negative,
which confounds the sign ofdk∗PE
dττ
k∗PE. The sign of
dg∗PE
dττ
g∗PEis even more complicated. Although
Λ∗PE < 0, the interactive term of τΘPE(1+Λ
∗PE −EΦ∗
PE ,z∗PE) is undetermined. From (22), without
the pollution health effects, an increase in ∆ unambiguously reduces the growth rate. The reason
is that a larger share of fiscal revenues devoted to pollution abatement translates into fewer
economic resources that contribute to physical and human capital accumulation, which in turn
leads to a lower growth rate. The decrease in total economic resources causes a larger decrease in
physical capital than human capital, and thus k∗PE also decreases in ∆. Again, if pollution resumes
the health effects, the signs ofdk∗PE
d∆
∆
k∗PEand
dg∗PE
d∆
∆
g∗PEmay become reversed. Therefore, the
pollution health effects modify the way of how the BGP variables respond to policy changes.
Equations (21) and (22) thus demonstrate the importance of understanding both the absolute and
relative effects of pollution on physical and human capital accumulation when considering policy
changes.
5 The No-Private-Education (NPE) Regime
In this section we focus on the NPE regime where the agent’s private education expenditures are
zero for any stock of pollution. We also carefully compare the analyses in the NPE and PE
regimes. Substituting (1b) into (9b), setting st = Kt+1, and rearranging yields the difference
26
equation for physical capital:
NPE :Kt+1
Kt= A(1− τ)(1−α)Φ(zt)k
α−1t . (23)
Again, pollution lowers the agent’s propensity to save as Φ′(zt)< 0, thus hampering the
accumulation of physical capital. Substituting (2b) into (4) and setting et = 0 yields the difference
equation for human capital:
NPE :Ht+1
Ht= BAβ [µτ(1−∆)]β [λ (zt)]
βk
αβt . (24)
Contrary to the PE regime, the agent does not invest in private education under the NPE regime,
and thus there is no propensity to invest in private education in (24). The stock of pollution
reduces the accumulation of human capital by weakening the effectiveness of education
expenditures, and education expenditures are funded only by the government.
Dividing (23) by (24) yields the difference equation in terms of the
physical-to-human-capital ratio:
NPE : kt+1 =A1−β
B
(1− τ)(1−α)
[µτ(1−∆)]βΦ(zt)
λ (zt)βk
α−αβt . (25)
Equation (25) describes how kt+1 is determined under the NPE regime. Similar to the PE regime,
the first term contains function scalars and the fourth term contains the physical-to-human-capital
ratio in period t. Different from the PE regime, the second term (1−τ)(1−α)/[µτ(1−∆)]β shows that
the public education expenditures do not influence the agent’s savings by (9b). The only source of
physical capital accumulation is the agent’s wage income represented by (1− τ)(1−α), and the
only source of human capital accumulation is public education expenditures represented by
µτ(1−∆). The third term Φ(zt)/λ (zt)β reveals the effects of pollution on the capital ratio when
private education expenditures are zero. Pollution damages physical capital by reducing the
propensity to save Φ(zt), and undermines human capital by decreasing the effective education
27
expenditures λ (zt).
From equation (25), setting kt+1 = kt gives the kk locus under the NPE regime:
NPE :A1−β
B
(1− τ)(1−α)
[µτ(1−∆)]βΦ(zt)
λ (zt)βk
α−αβt − kt = 0. (26)
Thus far we have laid out the model components under the NPE regime. Similar to the PE
regime, the slope of the kk locus and the dynamic properties around the BGP under the NPE
regime are also determined by the capital accumulation differential. But different from (17) under
the PE regime, the capital accumulation differential for any zt > 0 under the NPE regime is
Ψt,NPE = EΦt+1,zt
−βEλt ,zt, (27)
where EΦt+1,zt
= Φ′(zt)
Φ(zt)zt < 0 is the elasticity of the propensity to save with respect to pollution
when the agent’s private education expenditures are zero. In (27), the physical capital effect is
EΦt+1,zt
because pollution decreases the propensity to save and thus savings by reducing longevity,
such that physical capital is damaged. The human capital effect is βEλt ,ztbecause pollution
undermines the effectiveness of education expenditures by impeding learning, such that human
capital is damaged. The capital accumulation differential (27) under the NPE regime differs from
that under the PE regime in that the agent does not invest in private education, thus modifying the
propensity to save and leaving out the propensity to invest in private education. Evaluated on the
BGP, the capital accumulation differential becomes Ψ∗
NPE = EΦ∗
NPE ,z∗NPE
−βEλ ∗NPE ,z
∗NPE
.
Propositions 2 and 3 also carry over to the NPE regime only after the capital accumulation
differentials are updated to Ψt,NPE and Ψ∗
NPE , so under the NPE regime the slope of the kk locus
and the dynamic properties around the BGP can be determined. The formal proofs of the kk locus
slope and the dynamic properties are relegated to the second parts of Appendixes B and C.
Now we characterize the BGP under the NPE regime. Graphically, the BGP is
represented by the interaction of the kk locus (26) and the zz locus (11). Mathematically, taking
natural logs on both sides of (23), (24), and (11), and applying the definition of BGP growth rate
28
(12) yields the following three equations:
g∗NPE = lnA+ ln(1− τ)(1−α)+ lnΦ(z∗NPE)− (1−α) lnk∗NPE , (28a)
g∗NPE = lnBAβ +β ln µτ(1−∆)+β lnλ (z∗NPE)+αβ lnk∗NPE , (28b)
lnθ + lnz∗NPE = lnρ
∆τA+(1−α) lnk∗NPE . (28c)
Equations (28a) and (28b) together yield k∗NPE = k(g∗NPE) and z∗NPE = z(g∗NPE). Substituting
k(g∗NPE) and z(g∗NPE) into (28c) implicitly defines the BGP growth rate g∗NPE . The relations
among k∗NPE , g∗NPE , and z∗NPE are
∂k∗NPE
∂g∗NPE
g∗NPE
k∗NPE
=g∗NPE
β
Ψ∗
NPE
Λ∗
NPE
and∂ z∗NPE
∂g∗NPE
g∗NPE
z∗NPE
=g∗NPE
β
1−α +αβ
Λ∗
NPE
< 0. (29)
where Λ∗
NPE = αEΦ∗
NPE,z∗NPE
+(1−α)Eλ ∗NPE ,z
∗NPE
< 0. Similar to the PE regime, pollution and the
economic growth rate are negatively correlated. The relationship between the capital ratio and the
economic growth rate, and the relationship between the capital ratio and pollution also depend on
the capital accumulation differential evaluated on the BGP.
From (28a), (28b), and (28c), the policy effects of τ on k∗NPE , z∗NPE , and g∗NPE are
dk∗NPE
dτ
τ
k∗NPE
=Ψ
∗
NPE +(β + τ
1−τ
)
(1−α)Ψ∗
NPE − (1−α +αβ ),
dz∗NPE
dτ
τ
z∗NPE
=β + 1−α
1−τ
(1−α)Ψ∗
NPE − (1−α +αβ )< 0,
dg∗NPE
dτ
τ
g∗NPE
=β
g∗NPE
EΦ∗
NPE ,z∗NPE
+ 1−α1−τ Eλ ∗
NPE ,z∗NPE
− 1−τ−α1−τ
(1−α)Ψ∗
NPE − (1−α +αβ ). (30)
And the policy effects of ∆ on k∗NPE , z∗NPE , and g∗NPE are
dk∗NPE
d∆
∆
k∗NPE
=Ψ
∗
NPE −β ∆
1−∆
(1−α)Ψ∗
NPE − (1−α +αβ ),
dz∗NPE
d∆
∆
z∗NPE
=(1−α +αβ )− (1−α)β ∆
1−∆
(1−α)Ψ∗
NPE − (1−α +αβ ),
dg∗NPE
d∆
∆
g∗NPE
=β
g∗NPE
Λ∗
NPE +(1−α) ∆
1−∆
(
1−EΦ∗
NPE ,z∗NPE
)
(1−α)Ψ∗
NPE − (1−α +αβ ). (31)
Again, we focus on the scenario where the BGP can be automatically reached, so the
29
denominators of the above comparative statics satisfy (1−α)Ψ∗
NPE − (1−α +αβ )< 0. From
(30) and (31), how policy changes influence pollution is independent of the pollution health
effects. An increase in the tax rate unambiguously reduces pollution anddz∗NPE
dττ
z∗NPE< 0. The
reason is that more taxes finance more public pollution abatement, which contributes to less
pollution. But the termdz∗NPE
d∆
∆
z∗NPEcannot be signed. The ambiguity of signing
dz∗NPE
d∆
∆
z∗NPEoriginates
from two conflicting effects on pollution imposed by an increase in ∆. One effect is to directly
reduce pollution through public abatement, while the other is to indirectly increase pollution
through a larger capital ratio due to fewer public education expenditures. The sign ofdz∗NPE
d∆
∆
z∗NPE
depends on which effect on pollution is larger.
Similar to the PE regime, ambiguities of signing the remaining terms in (30) and (31)
arise from the health effects of pollution revealed by the terms EΦ∗
NPE ,z∗NPE
, Eλ ∗NPE ,z
∗NPE
, Ψ∗
NPE , and
Λ∗
NPE . Suppose the pollution health effects are absent, the policy effects become clear-cut. For the
tax rate τ ,dk∗NPE
dττ
k∗NPE< 0 because more taxes divert savings away from the agent to contribute to
more public education expenditures, such that the physical-to-human-capital ratio decreases. At
the same time, more taxes lead to a larger increase in human capital growth than the decrease in
physical capital growth, thus boosting the economic growth rate anddg∗NPE
dττ
g∗NPE> 0. For the share
of fiscal revenues dedicated to pollution abatement ∆, when taxes are given, a larger share of
pollution abatement in the fiscal revenues is equivalent to a smaller share of public education
expenditures. Because the agent does not invest in private education, the
physical-to-human-capital ratio increases in ∆,dk∗NPE
d∆
∆
k∗NPE> 0, and the economic growth rate
declines in ∆,dg∗NPE
d∆
∆
g∗NPE< 0. However, the resumption of pollution health effects in (30) and (31)
may reverse the policy effects of τ and ∆ on k∗NPE and g∗NPE . That the pollution health effects blur
the policy effects happens even if the agent does not invest in private education under the NPE
regime. The interactions between private actions and government policies are gone as revealed by
the comparison of the agent’s savings functions (7b) and (9b), and thus the policy considerations
under the NPE regime are not so complicated as under the PE regime. Therefore, these analyses
again demonstrate the importance of understanding the pollution health effects when deciding
30
government policy changes.
6 The PE and NPE Regimes
As is stated in Proposition 1, when certain conditions are satisfied, both the PE and NPE regimes
exist and a threshold stock of pollution arises and distinguishes the two regimes. From (16) and
(26), the kk loci are different in the two regimes. What happens to the kk loci when the PE regime
switches to the NPE regime? To answer this question, we check the continuity and slopes of the
two kk loci under the PE and NPE regimes at the threshold stock of pollution zo(τ,∆), provided
that this threshold exists based on the conditions established in Proposition 1. The results are
summarized in the following proposition.
Proposition 4. (Continuity and Slopes of the kk Loci When the Regime Switches) Suppose the
threshold stock of pollution zo(τ,∆) exists according to Proposition 1. At the threshold stock of
pollution,
(i) the ratios of physical to human capital determined by the two kk loci under the PE and
NPE regimes are equal, so there is no discontinuity between the two kk loci;
(ii) the slope of the kk locus under the PE regime is larger than that of the kk locus under
the NPE regime.
Proof. See Appendix D.
The first part of Proposition 4 is a mathematical fact. The second part highlights the
difference in the slopes of the kk loci when the regime switches. The reason is that compared with
the NPE regime, the adverse effect of pollution on physical capital accumulation is smaller under
the PE regime because the decline in the propensity to save is smaller for a same increase in
pollution. Further, the adverse effect of pollution on human capital accumulation is larger under
the PE regime because pollution additionally reduces the propensity to invest in private
education. Thus, when the regime switches from PE to NPE, the ratio of physical to human
capital declines, and the slope of the kk locus becomes smaller in (zt ,kt) space.
31
How about the BGP and the surrounding dynamic properties when both the PE and NPE
regimes exist? First, the zz locus may intersect with the kk locus under the PE regime (see Figures
6 and 7 to be shown in Section 7). The BGP and local dynamics are dictated by difference
equations (10) and (15) described in Section 4. Second, the zz locus may intersect with the kk
locus under the NPE regime. The BGP and local dynamics are dictated by difference equations
(10) and (25) described in Section 5. Third, the zz locus may simultaneously intersect with both
of the kk loci under the PE and NPE regimes, and an interesting case with multiple BGPs
emerges. We will explore this interesting case in Figure 8 to be shown in Section 7. But before we
move onto the next section, a question to policymakers naturally arises as to which BGP to pick
up when multiple ones emerge. We summarize the guidelines in the following proposition.
Proposition 5. (Ranking of BGPs) When multiple BGPs emerge, a BGP that features a lower
stock of pollution and a higher economic growth rate is preferred by policymakers. Policymakers
rank BGPs in the following two scenarios.
(i) For BGPs under the same regime, the BGP with a lower stock of pollution is preferred.
(ii) For BGPs under different regimes, the BGP under the PE regime is preferred.
Proof. See Appendix E.
Proposition 5 states that a BGP beats another one from both the environmental and
economic perspectives. Pollution hampers the accumulation of physical and human capital
through the health effects, so a lower stock of pollution leads to a higher economic growth rate. If
two candidate BGPs lie under the same regime, policymakers would pick up the BGP with a
lower stock of pollution. But if two BGPs respectively fall into the PE and NPE regimes, the
comparison of pollution stocks associated with each BGP is not straightforward. Recall that under
the NPE regime, the agent does not invest in private education because pollution higher than the
threshold renders the first dollar invested in private education not worthwhile. The stock of
pollution associated with the BGP under the PE regime must be lower than the threshold, while
pollution associated with the BGP under the NPE regime must be higher than the threshold.
32
Policymakers thus prefer the BGP under the PE regime. The policy implication is that the
government should steer the economy to converge towards the BGP with a lower stock of
pollution, which merits further discussion in the next section.
7 Numerical Examples
In this section, we provide numerical examples to complement the analytical results in the
previous sections. The purpose of these numerical analyses is threefold. First, we investigate the
curvature of the functions representing the pollution health effects. We have so far only presented
the basic properties of these functions. Second, we present a possible case where cycles emerge
and illustrate how the government policy can eliminate the cycles. Third, we exhibit another
interesting case where multiple BGPs arise when both regimes exist. We then illustrate how the
government policy can avoid the inferior BGP and achieve the desirable one.
To proceed, we specify the functions reflecting the health effects of pollution, φ(zt) and
λ (zt). Empirical evidence has documented non-linearity of the pollution health effects (Chay and
Greenstone, 2003; Chen et al., 2018a), but there is a lack of empirical research on how pollution
reduces longevity and the effectiveness of education expenditures. Therefore, we focus on the
curvature of the pollution health effects. We adopt a flexible functional form that satisfies the
basic properties and encompasses different shapes of φ(zt) and λ (zt). The functional form we
utilize is j(zt) =(
j+ jzc jt
)
/(
1+zc jt
)
, where j = φ ,λ . In this functional form, j and j are the upper
and lower bounds of the functional values, and for simplicity, we set j = 1 and j = 0. By
adjusting the curvature parameters c j > 0 ( j = φ ,λ ), this functional form is general enough to
allow for a wide range of possibilities for the negative health effects imposed by pollution. Figure
5 exhibits that how longevity φ(zt) and the effectiveness of education expenditures λ (zt) decrease
in the stock of pollution zt depends on the curvature parameter c j.
[Insert Figure 5 here]
To systematically illustrate how our model responds to changes in the curvature
parameters, we introduce the differential between the two curvature parameters ε = cλ − cφ . We
33
fix the curvature of the longevity function cφ and vary the value for ε , such that we can
experiment with the curvature for the effectiveness of education expenditures cλ . We also present
comprehensive numerical examples, making sure that the threshold stock of pollution arises, and
both the PE and NPE regimes are present. The benchmark parameters used in the following
numerical examples are listed in Table 1.
[Insert Table 1 here]
Example 1, ε = 0. The longevity function is φ(zt) =1
1+z3t
and the education expenditures
effectiveness function is λ (zt) =1
1+z3t. Figure 6 shows that there exists a threshold stock of
pollution that separates the PE and the NPE regimes. Consistent with Proposition 4, the kk loci
are continuous when the regime switches, and the slope of the kk locus under the PE regime is
larger than that of the kk locus under the NPE regime. The intersection of the zz locus with the kk
locus under the PE regime determines the BGP. By Proposition 3, the kk locus slopes up along
the BGP, the capital accumulation differential is positive, and pollution more heavily damages
human capital than physical capital. In this numerical example, the eigenvalues associated with
the BGP are 0.55 and −0.22, indicating the BGP is locally stable.
[Insert Figure 6 here]
One point is worth highlighting. Even if the two functions reflecting the health effects of
pollution, φ(zt) and λ (zt), are identical, the capital accumulation differential imposed by pollution
still exists. Thus, the slope of the kk locus is upward under the PE regime but is downward under
the NPE regime. This example demonstrates that it is not the health effects of pollution that
directly drive the transitional dynamics. Rather, by undermining health, pollution modifies
people’s behaviors of savings and private education expenditures,10 thus creating an accumulation
10Recall from (7b), (7c), and (9b) that the propensities to save and invest in private education under the PE regime,
Φ(φt+1) and Ω(φt+1), depend on how pollution reduces longevity, φt+1 = φ(zt). The propensity to save under the
NPE regime, Φ(φt+1), also relies on φt+1 = φ(zt). In addition, equation (8) says that φt+1 = φ(zt) is decisive in
ascertaining whether or not the agent invests in private education, which in turn distinguishes the PE regime and the
NPE regime.
34
differential between physical and human capital. So it is equally important to understand the
health effects of pollution as well as people’s responses to pollution health damages.
Example 2, ε =−2.5. The longevity function is φ(zt) =1
1+z3t
and the education
expenditures effectiveness function is λ (zt) =1
1+z0.5t
. Recall that Proposition 2 establishes the
connection between the slope of the kk locus and the sign of capital accumulation differential
caused by pollution. In Figure 7 below the threshold, as the stock of pollution rises, the kk locus is
hump-shaped because the capital accumulation differential switches its sign from positive to
negative. More specifically, the kk locus first rises for lower pollution because the negative effect
of pollution on human capital is larger than on physical capital. The kk locus then falls for higher
pollution because the negative effect of pollution on physical capital becomes larger than on
human capital. That the capital accumulation differential switches its sign can be explained by the
relative shapes of pollution health effect functions. For a lower stock of pollution, λ (zt) is steeper
than φ(zt). As the stock of pollution increases, however, φ(zt) eventually becomes steeper than
λ (zt). Thus, as pollution rises, the propensity to save initially declines slower and physical capital
is relatively abundant. But then the propensity to save declines faster and physical capital
becomes relatively scarce.
[Insert Figure 7 here]
The upper panel of Figure 7 shows the phase diagram and local dynamics of the unique
BGP that lies under the PE regime. The zz locus intersects with the kk locus when the kk locus
slopes down. The eigenvalues associated with the BGP are 0.167±0.169i, indicating that the
BGP features locally dampened cycles. We have presented the empirical relevance of these
economic and environmental cycles in Figure 3. As mentioned earlier, the cycles represent
inequality between generations, and government policy is required to remove the cycles.
The lower panel of Figure 7 shows the effects of government intervention that aims at
eliminating the cycles. By raising the tax rate on the output to finance more pollution abatement
and public education, the government is able to eliminate the inter-generational inequality
associated with the cycles. For example, if the government raises the tax rate from τ = 0.05 to
35
τ = 0.06, the zz locus rotates counter-clockwise and intersects with the kk locus when the kk locus
slopes up. The eigenvalues associated with the new BGP become 0.25 and 0.08, and the new BGP
turns locally stable without cycles. The reason why the government policy works to eliminate
cycles also rests on the capital accumulation differential imposed by pollution and on the dynamic
interactions between kt and zt . As the government shifts the BGP to the area where pollution
harms human capital more than physical capital, the physical-to-human-capital ratio kt changes
with pollution zt in the same direction. In addition, as kt measures the abundance of “dirty”
physical capital relative to “clean” human capital, zt moves in the same direction as kt changes.
Therefore, the two-way interactions of kt and zt are in the same direction, and no cyclical
convergence happens.
Example 3, ε = 2.5. The longevity function is φ(zt) =1
1+z3t
and the education
expenditures effectiveness function is λ (zt) =1
1+z5.5t
. In Figure 8 below the threshold stock of
pollution, the kk locus convexly increases because human capital is more severely damaged and
the severity is increasingly intensified by higher pollution. The convex shape of the kk locus can
be revealed by the shapes of longevity function φ(zt) and education expenditures function λ (zt)
in Figure 5, in which λ (zt) decline faster than φ(zt) in zt . Above the threshold stock of pollution,
the kk locus becomes flatter because after the regime switches, the negative effect of pollution on
physical capital becomes larger and that on human capital becomes smaller relative to below the
threshold. This observation is consistent with Proposition 4. The shapes of kk loci under the PE
and NPE regimes give rise to the possibility that multiple BGPs may emerge.
[Insert Figure 8 here]
The upper panel of Figure 8 shows that below the threshold stock of pollution, the zz locus
intersects with the kk locus twice at A and B. BGP A is locally stable with eigenvalues 0.40 and
−0.07. BGP B exhibits locally saddle stability with eigenvalues 1.28 and −0.95. Above the
threshold stock of pollution, the zz locus intersects with the kk locus again at C. BGP C is also
locally stable with eigenvalues 0.72 and −0.39. Due to the local dynamic property, BGP B gives
rise to a separatrix. This separatrix along with the threshold stock of pollution serve as a boundary
36
that demarcates the first quadrant into two “sink” regions. The points to the left of the boundary
will converge to BGP A, whereas the points to the right of the boundary will converge to BGP C.
Among the three BGPs, BGP A features the highest economic growth rate and the lowest stock of
pollution, whereas BGP C lies at the opposite extreme. This result is consistent with Proposition
5. To policymakers, BGP A is desirable while BGP C should be avoided. But what if the economy
lies to the right of the boundary, such that the economy will eventually converge to the
undesirable BGP C? Policymakers should steer the economy to the left of the boundary and the
economy will converge to the desirable BGP A.
The lower panel of Figure 8 illustrates how the policy interventions work. The dashed
lines represent the old loci when τ = 0.05, and the solid lines represent the new loci when
τ = 0.052 (a 4% increase in the tax rate). As the BGP in the middle shifts from B to B′, a new
separatrix is generated. Point D initially lies to the right of the old separatrix and to the left of the
threshold stock of pollution. If there were no government intervention, the local dynamics are
dictated by (15) and (10), and zt will increase until it jumps over the threshold stock of pollution.
As the local dynamics are dictated by (25) and (10) instead, the economy converges to the
undesirable BGP C. If the government raises the tax rate, however, point D lies to the left of the
new separatrix and the economy will converge to the desirable BGP A. As an alternative measure,
the government can decrease the ratio of physical to human capital by encouraging agents to
increase private education expenditures, decrease savings, or both. Point D will jump over the old
separatrix to reach point E. The economy will converge to the desirable BGP A without the
government modifying the tax rate τ .
8 Conclusions
Pollution reduces the accumulation of physical and human capital through negative health effects.
Although the existing research has fully explored the pollution health effects either on physical
capital or on human capital, little has been said about the simultaneous effects of pollution on
both types of capital, thus ignoring the consequences of pollution on the capital ratio, an
37
important indicator for economic growth emphasized by the literature. By incorporating the
health effects of pollution that influence both physical and human capital, we establish a link
connecting the two strands of literature that focus either on physical capital or on human capital.
Further, economic and environmental cycles are an empirical reality, and the literature has found
mechanisms explaining the cycles. Cyclical movements also arise in our model. We contribute to
the literature on cycles by identifying the accumulation differential between physical and human
capital caused by pollution as a new source of economic and environmental cycles.
Our analysis is based on a standard overlapping generations (OLG) model that depicts a
decentralized economy. We investigate two types of pollution health effects. One is pollution
reducing longevity and the other is pollution impeding children’s learning. Longevity as a
function of pollution is directly built into the agent’s lifetime utility, and the idea of pollution
hampering learning is equivalently modeled as pollution reducing the effectiveness of education
expenditures. The introduction of these two pollution health effects creates an accumulation
differential between physical and human capital. Thus, pollution influences the ratio of physical to
human capital in two possible scenarios. One scenario is that the capital accumulation differential
is positive, pollution more negatively affects human capital, physical capital accumulates faster,
and thus an increase in pollution raises the capital ratio. The other scenario is that the capital
accumulation differential is negative, pollution more adversely affects physical capital, the
accumulation of human capital becomes faster, and thus an increase in pollution reduces the
capital ratio. Conversely, the capital ratio affects pollution due to the nature of physical and
human capital. As physical capital generates pollution while human capital does not, an increase
in the physical-to-human-capital ratio elevates pollution. The above-described dynamic
interactions between pollution and the capital ratio portray the basic operation of our model.
We characterize the Balanced Growth Path (BGP) and the associated transitional
dynamics. Meanwhile, we highlight the role of capital accumulation differential initiated by
pollution. We show that the capital accumulation differential modifies the way that fundamental
variables respond to policy changes on the BGP. We also show that the capital accumulation
38
differential matters for the dynamic properties around the BGP. When the capital accumulation
differential is positive, the movement directions of pollution and the capital ratio remain the same
in their dynamic interactions, pollution and the capital ratio reinforce each other, and the dynamic
property around the BGP is stable. In line with this intuition, a numerical example reveals the
emergence of two extreme BGPs separated by a boundary. One BGP is strictly preferred over the
other because the superior one features lower pollution and higher economic growth. In contrast,
when the capital accumulation differential is negative, the movements of pollution and the capital
ratio do not reinforce each other in their dynamic interactions. Instead, pollution causes the
capital ratio to move in the opposite direction of pollution, whereas the capital ratio causes
pollution to move in the same direction of the capital ratio. Thus, cyclical movements in the
economy and environment may arise. Empirical evidence based on the panel data from China and
the world lends support in favor of the theoretical results. We have also discussed policy
interventions that can navigate the economy towards the desirable BGP and eliminate the
economic and environmental cycles.
As a last note, our theoretical results highlight the importance of understanding precisely
how the accumulation of physical capital is negatively affected by pollution relative to that of
human capital. However, there is a lack of empirical evidence documenting the relative pollution
health effects. Thus, we call for future research that estimates the relative health effects of specific
pollutants.
39
Beijing
Tianjin
Hebei
Inner Mongolia
LiaoningJilin
HeilongjiangShanghai
Jiangsu
Zhejiang
Anhui
FujianJiangxi
ShandongHenan Hubei
Hunan
Guangdong
Guangxi
Hainan
Sichuan Guizhou
Yunnan
ShaanxiGansu
Qinghai
Ningxia
Xinjiang
0.05 0.10 0.15g
1.8
2.0
2.2
2.4
2.6
k
2012. R2=0.0919
Tianjin
HebeiShanxi
Inner Mongolia
LiaoningJilin
HeilongjiangShanghai
Jiangsu
Zhejiang
Anhui
Fujian
Jiangxi
ShandongHenan
Hubei
Hunan
Guangdong
Guangxi
Hainan
Chongqing
Sichuan Guizhou
Yunnan
Shaanxi
Qinghai
Ningxia
Xinjiang
0.05 0.10 0.15g
1.8
2.0
2.2
2.4
2.6
k
2013. R2=0.2078
Beijing
Tianjin
HebeiShanxi
Inner Mongolia
Liaoning Jilin
Heilongjiang
Shanghai
Jiangsu
Zhejiang
Anhui
Fujian
Jiangxi
HenanHubei
Hunan
Guangdong
Guangxi
Hainan
Chongqing
Sichuan Guizhou
Yunnan
ShaanxiGansu
Qinghai
Xinjiang
0.02 0.04 0.06 0.08 0.10 0.12g
1.8
2.0
2.2
2.4
2.6
2.8
k
2014. R2=0.1162
Tianjin
HebeiShanxi
Inner Mongolia
LiaoningJilin
Heilongjiang
Shanghai
Jiangsu
Zhejiang
Anhui
FujianJiangxi
Henan Hubei
Hunan
Guangdong
Hainan
Chongqing
Sichuan Guizhou
Yunnan
ShaanxiGansu
Qinghai
Ningxia
Xinjiang
0.05 0.10 0.15g
1.8
2.0
2.2
2.4
2.6
2.8
k
2015. R2=0.1147
Beijing
Tianjin
HebeiShanxi Inner Mongolia
Liaoning
Jilin
Heilongjiang
Shanghai Jiangsu
Zhejiang
AnhuiFujian
Jiangxi
Shandong
HenanHubei
Hunan
Guangdong
Guangxi
Hainan
ChongqingSichuan
Guizhou
Yunnan
Shaanxi
Gansu
Qinghai
Ningxia
Xinjiang
0.05 0.10 0.15g
0.2
0.4
0.6
0.8
1.0
1.2
z
2012. R2=0.0298
Beijing
Tianjin
HebeiShanxi
Inner Mongolia
Liaoning
Jilin
Heilongjiang
ShanghaiJiangsu
Zhejiang
AnhuiFujian
JiangxiHenan
Hubei
Hunan
Guangdong
Hainan
Chongqing
Sichuan
Guizhou
Yunnan
ShaanxiGansu
Qinghai
Ningxia
Xinjiang
0.05 0.10 0.15g
0.2
0.4
0.6
0.8
1.0
1.2
z
2013. R2=0.0931
Beijing
Tianjin
HebeiShanxi
Inner Mongolia
Liaoning
Jilin
Heilongjiang
Shanghai
Jiangsu
Zhejiang
Anhui
FujianShandong
Henan Hubei
HunanGuangdong
Guangxi
Hainan Chongqing
Sichuan
Guizhou
Yunnan
ShaanxiGansu
Qinghai
Ningxia
Xinjiang
0.02 0.04 0.06 0.08 0.10 0.12g
0.2
0.4
0.6
0.8
1.0
1.2
z
2014. R2=0.0981
Beijing
Tianjin
Hebei
Shanxi
Inner Mongolia
Liaoning
Jilin
Heilongjiang
Jiangsu
Zhejiang
AnhuiFujian
Jiangxi
Shandong
HenanHubei
Hunan
Guangdong Hainan
Chongqing
Sichuan
Guizhou
Yunnan
Shaanxi
Gansu
Qinghai
Ningxia
0.05 0.10 0.15g
0.2
0.4
0.6
0.8
1.0
1.2
z
2015. R2=0.1492
Beijing
Tianjin
Hebei
Shanxi
Inner Mongolia
Liaoning
Jilin
Shanghai
Jiangsu
Zhejiang
Anhui
FujianJiangxi
Shandong
Henan
Hubei
Hunan
Guangdong
Guangxi
Hainan
Chongqing
Sichuan Guizhou
Yunnan
ShaanxiGansu
Qinghai
Ningxia
Xinjiang
0.4 0.6 0.8 1.0 1.2z
1.8
2.0
2.2
2.4
2.6
k
2012. R2=0.2491
Beijing
Tianjin
Hebei Shanxi
Inner Mongolia
Liaoning
Jilin
Heilongjiang
Shanghai
Jiangsu
Zhejiang
Anhui
FujianJiangxi
Shandong
HenanHubei
Hunan
Guangdong
Guangxi
Hainan
Chongqing
Sichuan Guizhou
Yunnan
ShaanxiGansu
Qinghai
Ningxia
Xinjiang
0.4 0.6 0.8 1.0 1.2z
1.8
2.0
2.2
2.4
2.6
k
2013. R2=0.2531
Beijing
Tianjin
Hebei
Shanxi
Inner Mongolia
Liaoning
Jilin
Shanghai
Jiangsu
Zhejiang
Anhui
Fujian
Jiangxi
Shandong
HenanHubei
Hunan
Guangdong
Guangxi
Hainan
Chongqing
Sichuan Guizhou
Yunnan
Gansu
Qinghai
Ningxia
Xinjiang
0.4 0.6 0.8 1.0 1.2z
1.8
2.0
2.2
2.4
2.6
2.8
k
2014. R2=0.2953
Beijing
Tianjin
HebeiShanxi
Inner Mongolia
LiaoningJilin
Heilongjiang
Shanghai
Jiangsu
Zhejiang
Anhui
Fujian
Jiangxi
ShandongHenan Hubei
Hunan
Guangdong
Guangxi
Hainan
Chongqing
Sichuan
Guizhou
Yunnan
ShaanxiGansu
Qinghai
Ningxia
Xinjiang
0.4 0.6 0.8 1.0 1.2z
2.0
2.2
2.4
2.6
2.8
k
2015. R2=0.2794
Notes. (1) The relations among pollution (z), capital ratio (k), and economic growth (g) are visualized pairwise. The
first row of figure cells plots economic growth (g) against capital ratio (k), the second row plots economic growth (g)
against pollution (z), and the third row plots pollution (z) against capital ratio (k).
(2) The trend lines are drawn based on least-squares fits to scatters of 30 Chinese provinces from 2012-2015, and
values for R2 are reported above each figure cell.
(3) Tibet is excluded as an outlier, but does not significantly alter the slope of the trend lines.
Figure 1: The Pairwise Relations among Pollution, Capital Ratio, and Economic Growth in China
40
PM2.5
DZA
AND
AGO
ATG
ARG
BHS
BHR
BRB
BLR
BMU
BTN
BWABRA
TCDCHN
COL
COG
HRV
DMAERI
ETH
FRA
GEO
DEU
GRC
GRD
HTI
ISL
IND
IDN
JAM
JPN
JORKIR
KWT
KGZ
LAO
LVA
LSO
MDG
MDV
MHL
MEXFSM
MMR
NZL
MNP
PAK
PRY
PERPHL
QAT
ROU
STP
SYC
SGP
SVK
ESP
LCAVCT
SUR
THA
TLS
TUR
ARE
GBR
USA
URY
VUT
VEN
VIR
PSE
ZWE
-2 2 4 6z
-0.10
-0.05
0.05
0.10
0.15
g
2010. R2=0.0974
AFG
DZAASM
ANDATG
ARG
AZE BHR
BLR
BEN
BMU
BTN
BRA CPVCAN
TCD
CHN
COL
COM
CIV
CYP
DNK
DMA
ECUERI
ETH
FRA
GMB
GEO
DEU
GHA
GRC
GRL
GRD
GNB
HTI
IND
IDN
ITAJPN
JOR
KAZ
KIR
LAO
LVA
LBN
LSO
MDG
MDV
MHL
MEX
FSM
MNG
MNPOMN
PAN
PRT
RUS
SAU
SYCSLB
ESP
LKASDN
SUR
THA
TLS
TUN
TUR
TKM
USA
VUT
VIR
ZWE
-2 2 4 6z
-0.10
-0.05
0.05
0.10
0.15
g
2011. R2=0.0222
AFG
ASM
ATG
ARG
ARM
AUS
BGD
BMU
BTN
BIH
BWA
BRA
KHM
CAF
CHN
COM
CIV
CYP
DMA
GNQ
ETH
FIN
FRADEU
GHA
GRC
GRD
GNB
GUY
IND
IDN
IRN
IRQ
ITA
JAM
JPN
KOR
LBN
LSO
LUX
MKD MDV
MLI
MHL
MNG
MNE
MMR
NLD
NER
MNP
PRY
PHL
PRT
QAT
RUS
STP
SRB SYC
SLE
SVNESP
LKA
SDN
THA
TKM
UKR
USA
VNM
PSE
ZWE
-2 2 4 6z
-0.10
-0.05
0.05
0.10
0.15
g
2012. R2=0.0017
ASM
ATGAUT
BHS
BHR
BGD
BMU
BTN
BWA
BRA
BRN
CPV
CHN
COG
CIV
CYP
DMA
GNQ
SWZ
ETH
FJI
GMB
DEU
GRC
GRD
GNB
GUYIND
IDN
IRN
ITA
JPN
JOR
KWT
KGZ
LAO
LBN
LBR
MDV
MLI
MHL
FSM
MDA MNG
MMR
NGAMNP
OMN
PRY
PHL
RUS
SYC
SVNESP
LCA
TON
TTO
TUR
TKM
USA
UZB
VUT
VNM
VIR
PSE
-2 2 4 6z
-0.10
-0.05
0.05
0.10
0.15
g
2013. R2=0.0382
ANDATG
ARG
AZE
BHR
BGD
BLZ
BTN
BRA
BRN
KHM
CHNCOD
COG
HRV
GNQ
ETH
FJI
FIN
FRA
GMB
GEO
DEU
GRL
GRD
GNB
IND
IDN
IRQ
IRL
JPN
JOR
KWT
LAO
LBN
LUX
MDG
MLT
MHL
MEX
FSM
MMR
NGAMNP
OMN
PNG
PHL
QATRUS
STP
LCA
VCTSWE
TTO
TKM
UKR
USA
UZB
VEN
VNM
VIR
PSE
YEM
-2 2 4 6z
-0.10
-0.05
0.05
0.10
0.15
g
2014. R2=0.0154
AFG
ANDARG
AUS
BGD
BLR
BLZ
BEN
BTN
BWABRA
BDI
CPVCAN
CAF
CHN
COL
CIV
DMA
ECU
ETH
FRAGRL
GRD
GNB
HTI
ISL
IND
IDN
IRN
JPN
JOR
KIR
KWTLBN
LBR
LBY
MLT
MHL
MEX
FSM
MMR
MNP
PHL
QAT
RUS
RWA
SYC
ESP
SUR
TKM
UKR
ARE
USA
UZBVNM
VIR
PSE
-2 2 4 6z
-0.10
-0.05
0.05
0.10
0.15
g
2015. R2=0.0044
PM10
DZA
AND
AGO
ATG
ARG
ARM
AUS
BHR
BGD BRB
BLR
BLZBEN
BRA
CMR
TCD
CHN
COMCIV
CUB
DMA
ERI
EST
FRA
GAB
GMB
DEU
GRD
GIN
GNB
ISL
IND
IDN
JAMJPN
LVA
MAC
MLT
MRT
MUS MNE
MMR
PANQATRUS
SMRESP
KNA
LCA
VCT
TTO
USA
VUT
VEN
ZWE
-2 2 4 6z
-0.15
-0.10
-0.05
0.05
0.10
0.15
0.20
g
2006. R2=0.0015
AFG
DZA
AND
AGO
ATG
ARG
ARM
AUS
BGD
BRB
BLR
BLZ
BTN
BRA
BRN
CHN
COM
COG
CIV
CYP
DMA
SLV
GNQ
FJI
GEO
DEU
GUYISL
IND
IDN
IRQ
ITA
JPN
LVA
LBN
LSO
LTU MAC
MEX
MNG
MMR
NER
PAN
PHL
RUS
SMR
STP
SAU
SVK
KNAVCT
TGO
ARE
GBRUSAVUT
ZWE
-2 2 4 6z
-0.15
-0.10
-0.05
0.05
0.10
0.15
0.20
g
2007. R2=0.0117
ALB
AND
AGO
ATG
ARMAZE
BGD
BRB
BLR
BIH
BWABRA
CHN
COM
CYP
DMA
GNQ
ERI
EST
ETH
GAB
DEUGRD
IND
IDN
IRN
IRL
ITAJPN
KENKWT
LVA
LBN
LSO
MRT
MNGMNE
NZL
NOR
PAN
PHL
ROU
RUS
SMR
SGP
KNA
THA
TKM
UGA
ARE
GBRUSA-2 2 4 6
z
-0.15
-0.10
-0.05
0.05
0.10
0.15
0.20
g
2008. R2=0.0051
AFG
ALB
DZA
AND
AGO
ATG
ARG
ARM
AZE
BHR
BGD
BRB
BTN
BWA
BRA
CHN
HRV
DMA
EST
ETH
FJI
FIN
FRA
GRDISL
IND
IDN
IRLITAJPN
KWT
LAO
LVA
LBN
LTU
LUXMDG
MWI
MDV
MEXMDA
MOZ
MMR
PANPHL
RUS
SMR
SVN
SLB
ZAF
ESPKNA
LCA
VCT
SUR
TTO
UKRARE
USA
UZB
VUT
VNM
ZMB
ZWE
-2 2 4 6z
-0.15
-0.10
-0.05
0.05
0.10
0.15
0.20
g
2009. R2=0.0533
ALB
AND
ATG
ARG
AUS
BLR
BLZ
BTN
BWABRA
BDI
TCDCHN
CYP
DMA
GNQ
ETH
FRA
GEO
DEU
GRC
GRD
HTI
ISL
IND
JAM
JPN
JOR
KOR
KWT
KGZ
LSO
MDG
MLT
MNG
MMR
PAK
PRY
PHLQAT
SMR
SGP
SLB
KNALCA
VCT
THA
ARE
GBR
USA
VUT
VEN
ZWE
-2 2 4 6z
-0.15
-0.10
-0.05
0.05
0.10
0.15
0.20
g
2010. R2=0.0725
AND
AGO
ATG
AZE BHR
BGD
BLZBEN
BTN
BWABRA
CAN
TCD
CHN
COD
CIV
CYP
DMA
ETH
GMB
GEO
DEU
GHA
GRC
IND
ITAJPN
LVA
LBN
MAC
MDV
MNG
OMN
PAN
PRT
QAT
RUS
SMR
SAU
SVN
SLB
ESP
LKA
KNA
SUR
TUN
TUR
TKM
UKR
GBR
USA
VUT
VNM
YEM
ZWE
-2 2 4 6z
-0.15
-0.10
-0.05
0.05
0.10
0.15
0.20
g
2011. R2=0.0084
Notes. (1) In each figure cell, pollution (z) on the horizontal axis is represented by air stock pollutants, which are
the logged values of population-weighted PM2.5 (the first two rows, 2010-2015) and PM10 (the last two rows, 2006-
2011). Economic growth (g) on the vertical axis is the annual growth rate of real GDP per capita.
(2) The country codes can be found in World Bank Group (2018a).
(3) The trend lines are drawn based on least-squares fits to scatters of countries and regions in each year, and values
for R2 are reported above each figure cell.
Figure 2: The Negative Relationship between Pollution and Economic Growth in the World
41
PM2.5
1990
1995
2000
2005
2010
2011
2012
2013
2014 2015
2016
1990
1995
2000
2005
2010
2011
2012
2013
2014 2015
2016
-0.3 -0.2 -0.1 0.1 0.2 0.3z
-0.02
-0.01
0.01
0.02
0.03
0.04
0.05
g
Algeria
1990
1995
2000
2005
2011
2012
20132014
2015
2016
1990
1995
2000
2005
2011
2012
20132014
2015
2016
4.0 4.1 4.2 4.3 4.4z
-0.04
-0.02
0.02
0.04
g
Barbados
1990
1995
2000
2005
2010
2011
2012
2013
2014
2015
2016
1990
1995
2000
2005
2010
2011
2012
2013
2014
2015
2016
2.2 2.3 2.4 2.5 2.6 2.7 2.8z
-0.04
-0.02
0.02
0.04
0.06
0.08
0.10
g
Botswana
19901995
2000
2005
2010
2011
2012
2013
2014
2015
2016
19901995
2000
2005
2010
2011
2012
2013
2014
2015
2016
1.7 1.8 1.9 2.0z
0.005
0.010
0.015
0.020
0.025
0.030
0.035
g
El Salvador
1990
1995
2000
2005
2010
2011
2012
2013
2014
20152016
1990
1995
2000
2005
2010
2011
2012
2013
2014
20152016
0.05 0.10 0.15 0.20 0.25 0.30z
-0.08
-0.06
-0.04
-0.02
0.02
0.04
g
Greece
1990
1995
2000
2005
2010
20122013
2014
20152016
1990
1995
2000
2005
2010
20122013
2014
20152016
-1.40 -1.35 -1.30 -1.25 -1.20 -1.15 -1.10z
-0.03
-0.02
-0.01
0.01
0.02
0.03
0.04
g
Italy
PM10
1990
19911992
19931994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011 1990
19911992
19931994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
1.6 1.8 2.0 2.2 2.4z
-0.05
0.05
g
Hong Kong SAR, China
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
0.5 0.6 0.7 0.8 0.9 1.0 1.1z
-0.02
0.02
0.04
0.06
0.08
0.10
0.12
g
Iran, Islamic Rep.
19911992
1993
19941995
1996 1997
19981999
2000
2001
2002
2003
2004
20052006
2007
2008
2009
2010
2011
19911992
1993
19941995
1996 1997
19981999
2000
2001
2002
2003
2004
20052006
2007
2008
2009
2010
2011
3.7 3.8 3.9 4.0 4.1z
-0.08
-0.06
-0.04
-0.02
0.02
0.04
0.06
g
Macedonia, FYR
19961997
19981999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
20102011 1996
1997
19981999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
20102011
4.1 4.2 4.3 4.4 4.5 4.6z
-0.1
0.1
0.2
g
Maldives
1995
1996
1997
1998
1999
2000
200120022003
2004
2005
200620072008
2009
20102011
1995
1996
1997
1998
1999
2000
200120022003
2004
2005
200620072008
2009
20102011
0.65 0.70 0.75 0.80 0.85 0.90 0.95z
-0.15
-0.10
-0.05
0.05
0.10
g
Tajikistan
1990
19911992
1993
1994
1995
1996
19971998
1999
2001
2002
2003
2004
2005
2007
2008
2009
2011
1990
19911992
1993
1994
1995
1996
19971998
1999
2001
2002
2003
2004
2005
2007
2008
2009
2011
0.1 0.2 0.3 0.4 0.5 0.6 0.7z
-0.2
-0.1
0.1
g
Ukraine
Notes. (1) In each figure cell, pollution (z) on the horizontal axis is represented by air stock pollutants, which are the
logged values of population-weighted PM2.5 (the first two rows) and PM10 (the last two rows). Economic growth (g)
on the vertical axis is the annual growth rate of real GDP per capita.
(2) The cycles are colored red after some points are smoothed if necessary. For example, to highlight the cycle of
PM2.5 and economic growth in Algeria (the upper-left corner), we smooth the cycle by omitting the points for 2011
and 2014.
Figure 3: The Economic and Environmental Cycles in Selected Countries and Regions
42
0 τ˜ τ 1
1
τ: Proportional Tax Rate on the Output
Δ:ShareofFiscalRevenuesforPollutionAbatement
I
(PE)III
(PE and NPE)II
(NPE)
f1(τ) f2(τ)
Figure 4: Three Regimes in the Policy Set
43
0.0 0.5 1.0 1.5 2.0 2.5zt0.0
0.2
0.4
0.6
0.8
1.0
ϕ(zt),λ(zt)
1
1+z0.5
1
1+z3
1
1+z5.5
Figure 5: Functions Simulating the Health Effects of Pollution
44
the kk locus under NPE
the threshold
stock of pollutionzο
the kk locus under PE
the zz locuskt
zt
Note. In this and the following phase diagrams, we plot the stock of pollution zt on the horizontal axis, and the ratio
of physical to human capital kt on the vertical axis.
Figure 6: The Benchmark Phase Diagram, ε = 0
45
the zz locus
the threshold
stock of pollutionzo
the kk locus under NPE
the kk locus under PE
kt
zt
kt
the old threshold
stock of pollution
the new threshold
stock of pollution
the old zz locusthe new zz locus
the new kk locus the old kk locus
z2oz1o
zt
Figure 7: The Phase Diagram and the Effects of Government Policy, ε =−2.5
46
kt
the threshold
stock of pollutionzo
the kk locus
under PE
the kk locus
under NPE
the zz locusseparatrix
A
B
C
zt
kt
the old threshold
stock of pollution
the new threshold
stock of pollution
the old separatrixthe new separatrix
B
B'
A
C
C'
D
E
z1o
z2o
zt
Figure 8: The Phase Diagram and the Effects of Government Policies, ε = 2.5
47
Table 1: The Benchmark Parameters
Category Description Parameter Value
ProductionProduction function scalar A 10
Physical capital’s share in production α 0.33
EnvironmentThe dissipation rate of the pollution stock θ 0.8
The polluting capacity of physical capital ρ 0.2
Longevity∗Lower bound of longevity φ 0
Upper bound of longevity φ 1
Curvature of the longevity function cφ 3∗∗∗
Education spending
effectiveness∗∗
Lower bound of the effectiveness λ 0
Upper bound of the effectiveness λ 1
Curvature of education spending effectiveness cλ 3∗∗∗
Human capital
Scalar in the evolution of human capital B 5
Education expenditure’s share in human capital β 0.6
The relative strength of public to private education µ 1
Utility The agent’s altruism χ 0.65
GovernmentProportional tax on final output τ 0.05
Pollution abatement’s share in fiscal revenues ∆ 0.5
Notes. ∗The longevity function is φ(zt) =φ+φz
cφt
1+zcφt
.
∗∗The education expenditures effectiveness function is λ (zt) =λ+λ z
cλt
1+zcλt
.
∗∗∗The differential between the two curvature parameters, ε , is equal to 0 in the benchmark because
ε = cλ − cφ = 3−3 = 0.
48
References
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