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).

1

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.

2

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

3

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

4

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.

5

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.

6

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

7

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

8

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,

9

∆ ∈ [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.”

10

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.

11

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

12

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).

13

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)

14

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

Agénor, P.-R. (2011). Schooling and public capital in a model of endogenous growth.

Economica, 78(309):108–132.

Agénor, P.-R. and Neanidis, K. C. (2011). The allocation of public expenditure and economic

growth. The Manchester School, 79(4):899–931.

Aloi, M. and Tournemaine, F. (2013). Inequality, growth, and environmental quality trade-offs in

a model with human capital accumulation. Canadian Journal of Economics/Revue

canadienne d’économique, 46(3):1123–1155.

Barro, R. J. (1990). Government spending in a simple model of endogeneous growth. Journal of

Political Economy, 98(5):S103–S125.

Barro, R. J. (2001). Human capital and growth. The American Economic Review, 91(2):12–17.

Bloom, D. E., Canning, D., and Graham, B. (2003). Longevity and life-cycle savings.

Scandinavian Journal of Economics, 105(3):319–338.

Buiter, W. H. and Kletzer, K. M. (1995). Capital mobility, fiscal policy, and growth under

self-financing of human capital formation. The Canadian Journal of Economics / Revue

canadienne d’Economique, 28:S163–S194.

Chakraborty, S. (2004). Endogenous lifetime and economic growth. Journal of Economic Theory,

116(1):119–137.

Chay, K. Y. and Greenstone, M. (2003). The impact of air pollution on infant mortality: Evidence

from geographic variation in pollution shocks induced by a recession. Quarterly Journal

of Economics, 118(3):1121–1167.

Chen, S., Guo, C., and Huang, X. (2018a). Air pollution, student health, and school absences:

Evidence from China. Journal of Environmental Economics and Management,

92:465–497.

Chen, S., Oliva, P., and Zhang, P. (2018b). Air pollution and mental health: Evidence from China.

Retrieved from SSRN: https://ssrn.com/abstract=3178649 or

49

http://dx.doi.org/10.2139/ssrn.3178649.

China Center for Human Capital and Labor Market Research (2018). China human capital

measurement and human capital index project. Retrieved from

http://humancapital.cufe.edu.cn/en/Human_Capital_Index_Project.htm.

Constant, K. and Davin, M. (2019). Environmental policy and growth when environmental

awareness is endogenous. Macroeconomic Dynamics, 23(3):1102–1136.

Currie, J., Hanushek, E. A., Kahn, E. M., Neidell, M., and Rivkin, S. G. (2009). Does pollution

increase school absences? Review of Economics and Statistics, 91(4):682–694.

de la Croix, D. and Doepke, M. (2003). Inequality and growth: Why differential fertility matters.

The American Economic Review, 93(4):1091–1113.

De Nardi, M., French, E., and Jones, J. B. (2009). Life expectancy and old age savings. The

American Economic Review, 99(2):110–115.

Devarajan, S., Swaroop, V., and Zou, H.-f. (1996). The composition of public expenditure and

economic growth. Journal of Monetary Economics, 37(2):313–344.

Duczynski, P. (2002). Adjustment costs in a two-capital growth model. Journal of Economic

Dynamics and Control, 26(5):837–850.

Duczynski, P. (2003). On the empirics of the imbalance effect. International Journal of Business

and Economics, 2(2):121–128.

Ebenstein, A., Fan, M., Greenstone, M., He, G., Yin, P., and Zhou, M. (2015). Growth, pollution,

and life expectancy: China from 1991-2012. American Economic Review,

105(5):226–231.

Ebenstein, A., Lavy, V., and Roth, S. (2016). The long-run economic consequences of high-stakes

examinations: Evidence from transitory variation in pollution. American Economic

Journal: Applied Economics, 8(4):36–65.

Economides, G. and Philippopoulos, A. (2008). Growth enhancing policy is the means to sustain

the environment. Review of Economic Dynamics, 11(1):207–219.

Fodha, M. and Seegmuller, T. (2014). Environmental quality, public debt and economic

50

development. Environmental and Resource Economics, 57(4):487–504.

Goenka, A., Jafarey, S., and Pouliot, W. (2017). Pollution, mortality and ramsey taxes. Retrieved

from https://admission.lums.edu.pk/sites/default/files/user376/gjp_16jan17_ag.pdf.

Gradus, R. and Smulders, S. (1993). The trade-off between environmental care and long-term

growth—Pollution in three prototype growth models. Journal of Economics, 58(1):25–51.

Gutiérrez, M. J. (2008). Dynamic inefficiency in an overlapping generation economy with

pollution and health costs. Journal of Public Economic Theory, 10(4):563–594.

Hanna, R. and Oliva, P. (2015). The effect of pollution on labor supply: Evidence from a natural

experiment in Mexico City. Journal of Public Economics, 122:68–79.

Jhy-hwa, C., Jhy-yuan, S., and Juin-jen, C. (2015). Environmental policy and economic growth:

The macroeconomic implications of the health effect. The B.E. Journal of

Macroeconomics, 15:223.

John, A. and Pecchenino, R. (1994). An overlapping generations model of growth and the

environment. The Economic Journal, 104(427):1393–1410.

Jouvet, P.-A., Pestieau, P., and Ponthiere, G. (2010). Longevity and environmental quality in an

OLG model. Journal of Economics, 100(3):191–216.

Kim, Y., Manley, J., and Radoias, V. (2017). Air pollution and long-term mental health. Retrieved

from SSRN: https://ssrn.com/abstract=3028930 or

http://dx.doi.org/10.2139/ssrn.3028930.

Ladrón-de Guevara, A., Ortigueira, S., and Santos, M. S. (1997). Equilibrium dynamics in

two-sector models of endogenous growth. Journal of Economic Dynamics and Control,

21(1):115–143.

Maher, B. A., Ahmed, I. A. M., Karloukovski, V., MacLaren, D. A., Foulds, P. G., Allsop, D.,

Mann, D. M. A., Torres-Jardón, R., and Calderon-Garciduenas, L. (2016). Magnetite

pollution nanoparticles in the human brain. Proceedings of the National Academy of

Sciences, 113(39):10797–10801.

Mayer, H. (1999). Air pollution in cities. Atmospheric Environment, 33(24-25):4029–4037.

51

Mohai, P., Kweon, B.-S., Lee, S., and Ard, K. (2011). Air pollution around schools is linked to

poorer student health and academic performance. Health Affairs, 30(5):852–862.

Motoyama, T. (2016). From physical to human capital accumulation with pollution. Retrieved

from http://www2.econ.osaka-u.ac.jp/library/global/dp/1603.pdf.

Mulligan, C. B. and Sala-i Martin, X. (1993). Transitional dynamics in two-sector models of

endogenous growth. The Quarterly Journal of Economics, 108(3):739–773.

Nilsson, J. (2009). The long-term effects of early childhood lead exposure: Evidence from the

phase-out of leaded gasoline. Retrieved from

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.212.3578.

Osang, T. and Sarkar, J. (2008). Endogenous mortality, human capital and economic growth.

Journal of Macroeconomics, 30(4):1423–1445.

Palivos, T. and Varvarigos, D. (2017). Pollution abatement as a source of stabilzation and

long-run growth. Macroeconomic Dynamics, 21(3):644–676.

Pautrel, X. (2008). Reconsidering the impact of the environment on long-run growth when

pollution influences health and agents have a finite-lifetime. Environmental and Resource

Economics, 40(1):37–52.

Pautrel, X. (2009). Pollution and life expectancy: How environmental policy can promote growth.

Ecological Economics, 68(4):1040–1051.

Pautrel, X. (2012). Pollution, private investment in healthcare, and environmental policy. The

Scandinavian Journal of Economics, 114(2):334–357.

Raffin, N. (2012). Children’s environmental health, education, and economic development.

Canadian Journal of Economics/Revue canadienne d’économique, 45(3):996–1022.

Raffin, N. and Seegmuller, T. (2014). Longevity, pollution and growth. Mathematical Social

Sciences, 69:22–33.

Raffin, N. and Seegmuller, T. (2017). The cost of pollution on longevity, welfare and economic

stability. Environmental and Resource Economics, 68(3):683–704.

Reuters (2012). China lead pollution poisons 160 children: Report. Retrieved from

52

https://www.reuters.com/article/us-china-lead-posion/china-lead-pollution-poisons-160-

children-report-idUSTRE82303F20120304.

Rolden, H. J. A., van Bodegom, D., van den Hout, W. B., and Westendorp, R. G. J. (2014). Old

age mortality and macroeconomic cycles. Journal of Epidemiology and Community

Health, 68(1):44–50.

Sapci, O. and Shogren, J. F. (2017). Environmental quality, human capital and growth. Journal of

Environmental Economics and Policy, 7(2):184–203.

Schumacher, I. and Zou, B. (2008). Pollution perception: A challenge for intergenerational

equity. Journal of Environmental Economics and Management, 55(3):296–309.

Schumacher, I. and Zou, B. (2015). Threshold preferences and the environment. Journal of

Mathematical Economics, 60:17–27.

Seegmuller, T. and Verchère, A. (2004). Pollution as a source of endogenous fluctuations and

periodic welfare inequality in OLG economies. Economics Letters, 84(3):363–369.

Smulders, S. and Gradus, R. (1996). Pollution abatement and long-term growth. European

Journal of Political Economy, 12(3):505–532.

Tapia Granados, J. A. (2005). Increasing mortality during the expansions of the US economy,

1900-1996. International Journal of Epidemiology, 34(6):1194–1202.

The Economist (2017). Airborne particles cause more than 3m early deaths a year. Retrieved

from https://www.economist.com/science-and-technology/2017/04/01/airborne-particles-

cause-more-than-3m-early-deaths-a-year.

The New York Times (2011). Lead poisoning in China: The hidden scourge. Retrieved from

https://www.nytimes.com/2011/06/15/world/asia/15lead.html.

Varvarigos, D. (2010). Environmental degradation, longevity, and the dynamics of economic

development. Environmental and Resource Economics, 46(1):59–73.

Varvarigos, D. (2013a). Endogenous longevity and the joint dynamics of pollution and capital

accumulation. Environment and Development Economics, 19(04):393–416.

Varvarigos, D. (2013b). Environmental dynamics and the links between growth, volatility and

53

mortality. Bulletin of Economic Research, 65(4):314–331.

Wang, M., Zhao, J., and Bhattacharya, J. (2015). Optimal health and environmental policies in a

pollution-growth nexus. Journal of Environmental Economics and Management,

71:160–179.

Wen, J., Luo, Y., and Liu, B., editors. China statistical yearbook on environment (2013-2016).

China Statistics Press, Beijing (in Chinese).

Wen, M. and Gu, D. (2012). Air pollution shortens life expectancy and health expectancy for

older adults: The case of China. Journals of Gerontology - Series A Biological Sciences

and Medical Sciences, 67(11):1219–1229.

World Bank Group (2018a). The world development indicators. Retrieved from

https://datacatalog.worldbank.org/dataset/world-development-indicators.

World Bank Group (2018b). The world development indicators database archives. Retrieved from

https://databank.worldbank.org/data/reports.aspx?source=1277&series=EN.ATM.PM10.MC.M3.

World Health Organization (2018). Global ambient air quality database. Retrieved from

https://www.who.int/airpollution/data/cities/en/.

Zhang, J. (1999). Environmental sustainability, nonlinear dynamics and chaos. Economic Theory,

14(2):489–500.

Zhang, J. and Zhang, J. (2005). The effect of life expectancy on fertility, saving, schooling and

economic growth: Theory and evidence. Scandinavian Journal of Economics,

107(1):45–66.

Zhang, X., Chen, X., and Zhang, X. (2018). The impact of exposure to air pollution on cognitive

performance. Proceedings of the National Academy of Sciences, 115(37):9193–9197.

Zhang, X., Zhang, X., and Chen, X. (2017). Happiness in the air: How does a dirty sky affect

mental health and subjective well-being? Journal of Environmental Economics and

Management, 85:81–94.

54