Discussion Papers In Economics
And Business

Wealth Heterogeneity and Escape from the
Poverty-Environment Trap
Masako Ikefuji Ryo Horii

Discussion Paper 05-09

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWealth Heterogeneity and Escape from the
Poverty-Environment Trap
Masako Ikefuji Ryo Horii

Discussion Paper 05-09
May 2005
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWealth Heterogeneity and Escape from the
Poverty-Environment Trap∗
Masako Ikefuji†

Ryo Horii‡

Osaka University
May 2005

Summary. A mutual link between poverty and environmental degradation is examined in an overlapping generations model with environmental externality, human capital, and credit constraints. Environmental quality aﬀects labor productivity and thus wealth dynamics, whereas wealth distribution determines the degree
to which agents rely upon natural resources and therefore the evolution of environmental quality. This interaction creates a ‘poverty-environment trap,’ where adeteriorated environment lowers income, which in turn accelerates environmental
degradation. We show that greater wealth heterogeneity is the key to escaping thepoverty-environment trap, although it has negative eﬀects both on the environmentand output when not in the trap.
Keywords and Phrases: Poverty trap, Environmental degradation, Wealth distribution, Human capital.
JEL Classification Numbers: O11, O13, O15.

∗

We are grateful to Koichi Futagami, Tokuo Iwaisako, Ryoji Ohdoi, Tetsuo Ono, Akira Yakita,

and seminar participants at Hitotsubashi University and Meiji Gakuin University for their helpfulcomments and suggestions. This study is partly supported by a JSPS Grant-in-Aid for Scientific
Research (No. 16730097). All remaining errors are naturally our own.
†

Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka 560-0043,

Japan. E-mail: cg043im@srv.econ.osaka-u.ac.jp‡

Correspondence: Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toy-

onaka 560-0043, Japan. E-mail: horii@econ.osaka-u.ac.jp1

1

Introduction

During the last decade, a substantial number of both empirical and case studies
have pointed out the mutual link between poverty and environmental degradation.
Poverty forces people to rely heavily on natural resources for agriculture, fisheries,
and timber production.1 However, excessive use of natural resources, such as overgrazing, excessive fuelwood harvesting and unsustainable farming practice, is a primary cause of environmental degradation. As pointed out by Barbier (1997), poor
rural households with limited access to capital tend to extract short-term rents fromexisting agricultural land, and such practices often result in land degradation, abandonment of existing land, and the expansion of agricultural activity into frontier
forests.2 Partly as a consequence of such exploitation, low-income countries lost
about 73 million hectares (i.e., about 8 percent) of their forest in the 1990s while
high-income countries reforested about 8 million hectares of forest in the same period(World Bank, 2004).
Environmental degradation, in turn, results in low productivity and makes poor
households poorer. Deforestation is a major source of human-induced soil degradation in developing regions, accounting for around 40 percent of erosion in Asia andSouth America (Oldeman et al. 1990).3 With reduction in land productivity, agricultural income of the poor falls and the risk of undernourishment rises. Moreover,poor households living in a degraded environment are confronted with a high possibility of water-related diseases from drinking untreated water (World Bank, 2004).
In this way, a degraded environment lowers agricultural and other incomes of the1

Cavendish (2000) empirically analyzes the poverty-environment relationship in Zimbabwe andshows that environmental resources make a significant contribution to average rural incomes, andthat poorer households are more resource dependent than are the rich.
2

Barbier and Burgess (1997) show that the demand for forest conversion is negatively correlatedwith income per capita. Fuelwood harvesting by the poor is also responsible for deforestation in
Peru (Swinton and Quiroz, 2003).
3

Deforestation also makes fuelwood collection by the poor more time-consuming (Dasgupta,

1998).

2

Figure 1: Education and Deforestation. The horizontal axis shows the average annual
deforestation rate between 1990 and 2000; the vertical axis shows the net enrollment ratio ofprimary school in the 2001/02 period. Data source: World Bank (2004).

poor, which forces them to rely on natural resources more than ever, furthering theprocess of environmental degradation and increasing poverty.4 This mutual link between poverty and environmental degradation creates a ‘poverty-environment’ trap,from which the poor find it hard to emerge.
This paper presents a model that formalizes this mutual link in an overlapping
generations setting with environmental externality, human capital accumulation, andcredit constraints. While the poverty-environment trap has been described largely
in verbal terms in the literature, our model clarifies its properties by showing theexistence of two stable steady states. In particular, one steady state is characterized
by a combination of a high level of wealth, a high level of human capital investment
and a good environment, whereas the other is characterized by poverty, little or no
human capital investment, and a degraded environment. Once the economy settles
into the latter steady state, it cannot escape from the poverty-environment trap.
A key specification of our model is that technologies that intensively use human4This ‘feedback loop’ has been pointed out by Duraiappah (1998) and Borghesi and Vercelli(2003).

3

capital are less reliant on natural resources and therefore have smaller impacts on
the environment than primitive technologies. A number of pieces of evidence areconsistent with this specification. First, an international comparison among 126 developed and developing countries shows a clear negative relationship between the
net enrollment ratio of primary school and the annual deforestation rate (see Figure 1). Second, Swinton and Quiroz (2003) show that human capital significantly
promotes the use of sustainable agricultural practices (e.g., fallowing) and reduces
the likelihood of tree felling. Torras and Boyce (1998) also report that, in lowincome countries, higher literacy rates improve environmental quality. Since only
wealthy households can aﬀord education when credit markets are imperfect, theseenvironmental eﬀects of human capital imply that higher average wealth improves
the environment, or, equivalently, that poverty causes environmental degradation.5
To complete the mechanics of the poverty-environment trap, the model also incorporates the fact that agents’ capacity to work depends on environmental quality.
Combined with the environmental eﬀects of human capital, the productivity eﬀect of
the environment creates intergenerational feedback loops. With a degraded environment, agents’ low productivity enables them to leave only small amounts of transfer
for their children, who therefore have to rely on technologies that deteriorate theenvironment even more, falling into the poverty-environment trap. Conversely, whenthe environment is favorable, agents receive suﬃcient amounts of assets from theirparents to invest in human capital out of received assets, improving the environment
for future generations. We show that the fate of an economy depends on both theinitial level of environmental quality and the initial distribution of wealth.
Poverty and the quality of the environment are often considered to be associatedwith inequality. Using the model explained above, we examine the dynamic consequences of distributional policies in both of the two steady states. Following Loury
5

Bahamondes (2003) finds this causation in a panel data study examining how asset levels aﬀect

the choice of agricultural practices and how those practices aﬀected natural resource status in arid
central Chile. He concludes that the impressive recovery of a fragile natural resource is based on
rising incomes.

4

(1981) and Owen and Weil (1998), we introduce heterogeneity in individuals’ ability
so that the distribution of wealth evolves over successive generations. It is shown thatthe eﬀect of inequality on environment and development is completely opposite forthe two steady states. If wealth inequality widens in the poverty trap, some wealthy
households start or increase investment in human capital whereas poor households do
not invest at all ab initio. The aggregate amount of human capital investment thusincreases, improving the quality of the environment in the economy. The improved
environment raises everyone’s productivity and income, and the wealth of households
over generations. It encourages poorer households to invest in human capital, further
improving the quality of the environment.6 Under a certain condition, this virtuous
spiral allows the economy to escape from the poverty trap to a better steady state.
It can be viewed as a particular case of the more general and abstract model of Galor
and Tsiddon (1997), who demonstrate that inequality of human capital distribution
may serve as a vehicle in the development of less-developed economies.
In the better steady state, conversely, our model shows that an appropriate levelof redistribution is beneficial both for aggregate output and for the environment:a narrower wealth distribution lets poor households rely less on natural resources
whereas the rich are always fully investing in human capital. Those contrasting
results in the two steady states are consistent with an empirical finding by Barro(2000) that the relationship between inequality and growth is positive for low-income
countries whereas it is negative for high-income countries.
In the literature of poverty traps, a number of theoretical studies examine the roleof wealth distribution on economic development. In those studies, multiple steadystates occur because of imperfections of credit markets combined with some nonconvexities in the model. Nonconvexities are usually introduced in terms of a discrete
choice being faced by agents between being uneducated or educated (Galor and Zeira,
1993; Owen and Weil, 1998); a discrete choice between exerting a fixed amount of
6This process is similar to the trickle-down mechanism of Aghion and Bolton (1997). They

consider the trickle-down process in terms of physical capital and interest rate, whereas ours isdriven by human capital and the quality of the environment.

5eﬀort or not (Piketty, 1997); or a discrete choice between occupations (Banerjee andNewman, 1993). A high-yielding choice typically requires agents to possess at leasta certain threshold level of wealth. Thus, agents below the threshold cannot escapefrom the poverty trap where a low-yielding choice is reinforced in subsequent periods. Our model diﬀers from these in allowing agents to choose arbitrary amounts ofinvestment in human capital. Rather, the obvious fact that the amount of humancapital must be nonnegative generates a convex correspondence between wealth andinvestment.7 Agents with assets lower than a certain threshold level choose not toinvest at all since consumption is their urgent priority, while those with larger assets
invest according to their wealth levels. Thus, from Jensen’s inequality, a meanpreserving spread in the distribution of wealth clearly increases the average level ofinvestment and hence promotes economic development, whereas the growth eﬀect ofwealth heterogeneity is less obvious in other models where nonconvexities arise froma discrete choice.8
The remainder of this paper is organized as follows. Section 2 presents the modeland derives the interrelated evolutions of the distribution of wealth and environmental quality. As a benchmark, Section 3 considers a representative agent economy anddemonstrates the existence of multiple steady states, one of which is characterized
as a poverty-environment trap. Heterogeneity across agents is introduced in Section 4, where the eﬀects of a redistributive policy on the environment and wealthdistribution are analyzed. Section 5 concludes.

2

The Model

We consider a discrete time model of an overlapping generations economy, where eachindividual lives for two consecutive periods. Agents in their first and second periods
7

Similarly, Moav (2002) obtains a convex bequest function by assuming that agents cannot leave

negative bequests.
8

In fact, Galor and Zeira (1993) and Banerjee and Newman (1993) conclude that equality in theinitial distribution of wealth is important for economic growth.

6are called young and adult individuals, respectively, and each adult is assumed tobear a single child (a young agent), implying that the total population is stationary.
The size of each generation is normalized to unity, with each individual of generationt being indexed by j ∈ [0, 1] and t.
Evolution of individuals’ wealth over generations
Individuals are endowed with a certain amount of labor for both periods of theirlives. The amount of labor, or their ability to learn and work, depends on thequality of the environment at the time of their birth.9 Let Qt ≥ 0 denote the qualityof the environment in the economy, whose evolution is explained later. The laborendowment for each of an individual’s periods is specified by10
`j,t = Qt εj,t .

(1)

In (1), εj is a random variable with mean 1, distributed independently, identicallyand uniformly within [1 − σ, 1 + σ], where σ is a constant representing the degreeof variations in ability. Inclusion of random variable εj enables us to capture theheterogeneity of agents within an economy, as in Loury (1981).
From parents, young agent j inherits bj,t units of goods, which are subject to an
inheritance tax of rate τ ∈ [0, 1]. The government revenue from the inheritance tax
is distributed equally among young agents, so that an agent’s total wealth at birthiswj,t = (1 − τ )bj,t +Z 1
0

τ bj 0 ,t dj 0 .

(2)

Those goods are divisible and nonstorable. A young individual uses a part of his/her
wealth, ej,t ≥ 0, as an input for human capital investment. It is combined with9

As mentioned in the introduction, the risk of disease and malnutrition in developing countries

increases with environmental degradation. Also in industrial countries, polluted air impairs thelabor productivity of those with health problems such as asthma and bronchitis.
10

We assume that an individual is endowed with the same amount of labor for each period,

implying that his/her ability to work in adulthood depends on the quality of the environment in
his/her youth. This assumption is not critical since the quality of the environment, Qt , changes
only gradually.

7his/her endowed labor `j,t to produce
hj,t+1 = γej,t `j,t(3)

units of human capital, where γ > 0 is a productivity parameter. As its subscript
shows, this human capital hj,t+1 becomes available in adulthood. We assume thatcredit markets are imperfect so that an individual can neither borrow nor lend.11 It
means that the amount of human capital investment is restricted by ej,t ∈ [0, wj,t ]and that the remainder is used for an agent’s young period consumption
cyj,t = wj,t − ej,t ≥ 0.

(4)

In the second period of life, the agent produces goods employing two kinds oftechnology. One is the primitive technology that produces goods from labor andnatural resources according top= ap min{rj,t+1 , `pj,t+1 },yj,t+1(5)where constant ap > 0 denotes the marginal productivity of labor and natural resources, `pj,t+1 ∈ [0, `j,t ] is the amount of labor used for the primitive technology outof his/her second period endowment `j,t , and rj,t+1 ≥ 0 is the amount of natural
resources that he/she exploits without private costs.12 In equilibrium, an individual
uses exactly the same amount of rj,t+1 as `pj,t+1 since the additional marginal productof natural resources is zero. Although each agent does not incur private costs, theaggregate usage of natural resources deteriorates the quality of the environment, as
will be specified later.
The other production technology, which we call the sustainable technology, produces goods from labor and human capital according tos= as min{hj,t+1 , `sj,t+1 },yj,t+1
11

The assumption that agents cannot save is made only for simplicity. It does not aﬀect theequilibrium outcomes other than the consumption profile of individuals.
12(6)

The cost of exploitation can be interpreted as being included in `pj,t+1 .

8

where constant as > ap denotes the marginal productivity of labor and human capital, `sj,t+1 ∈ [0, `j,t ] is the amount of labor used for the sustainable technology, andhj,t+1 is the amount of human capital accumulated through (3). This production
technology does not require natural resources, and in that sense is good for theenvironment.
Subject to the time constraint, `pj,t+1 +`sj,t+1 = `j,t , each adult agent allocates laborps+ yj,t+1. The latter can be written
so as to maximize the total output yj,t+1 = yj,t+1

as yj,t+1 = ap (`j,t − `sj,t+1 ) + as min{hj,t+1 , `sj,t+1 }, since substituting rj,t+1 = `pj,t+1

into (5) shows that the output from the primitive technology is ap `pj,t+1 . Note thatthe assumption as > ap implies that the marginal product of labor is higher in thesustainable technology as long as there is a suﬃcient amount of human capital to be
used. Thus, it is optimal to set `sj,t+1 = min{hj,t+1 , `j,t }. Consequently, an agent’s
output is described as a function of human capital investment and labor endowment:y(hj,t+1 , `j,t ) ≡ (as − ap )min{hj,t+1 , `j,t } + ap `j,t .

(7)

An individual’s objective throughout his/her life is given by a lifetime utility
function,
Uj,t = ln cyj,t + α ln caj,t+1 + β ln bj,t+1 ,(8)where cyj,t and caj,t+1 denote the amount of consumption in youth and adulthood, respectively, while bj,t+1 represents the amount of transfer to oﬀspring out of altruism.13
An agent chooses the amounts of human capital investments ej,t , transfer to his/herchild bj,t+1 , and the agent’s consumption schedule cyj,t , caj,t+1 so as to maximize (8).
The maximization is done subject to human capital production (3), the nonnegativity constraint on investment ej,t ≥ 0,14 the first-period budget constraint (4), and
13

The third term in (8), β ln bj,t+1 , shows that we employ the joy-of-giving formulation of altruism.

Alternatively, we can replace it with β ln(1 − τ )bj,t+1 so that the parents receive utility only fromthe amount of transfer net of the inheritance tax. This reformulation does not aﬀect equilibriumin any way because β ln(1 − τ )bj,t+1 = β ln bj,t+1 + constant.
14

Since the utility function satisfies the Inada conditions, the nonnegativity constraints on cyj,tand caj,t+1 never bind in the optimal solution and therefore can be omitted. We also omit theinequality constraint ej,t ≤ wj,t since it is equivalent to cyj,t ≥ 0 from (4).

9that of the second-period caj,t+1 + bj,t+1 = y(hj,t+1 , `j,t ), where function y(hj,t+1 , `j,t )is defined by (7).
This problem can be solved using the Kuhn-Tucker method as detailed in Appendix A. The optimal amount of investments in human capital is found as a functionof wj,t .
0if wj,t ≤ w,ej,t = e(wj,t ) ≡ (α + β)(1 + α + β)−1 (wj,t − w) if wj,t ∈ (w, w),





1/γ
if wj,t ≥ w,

(9)where w ≡ ap ((α+β)γ(as −ap ))−1 and w ≡ as ((α+β)γ(as −ap ))−1 +1/γ. Investments
in human capital increase with wealth because agents with insuﬃcient wealth are
liquidity constrained. Specifically, when wj,t ≤ w agents cannot aﬀord to invest in
human capital since consumption in youth is quite low and therefore is their firstpriority. Agents with wj,t ∈ (w, w) are also liquidity constrained, but they can aﬀord
to invest in human capital, although less than the desirable amount 1/γ.
The procedure in Appendix A also finds the optimal amount of transfer that isleft for his/her oﬀspring as a function of wj,t and `j,t . In particular, it can be written
as bj,t+1 = b(wj,t )`j,t , where
β(α + β)−1 ap
if wj,t ≤ w,
b(wj,t ) ≡ β(1 + α + β)−1 (ap + γ(as − ap )wj,t ) if wj,t ∈ (w, w),





β(α + β)−1 asif wj,t ≥ w.

(10)

The amount of transfer b(wj,t )`j,t depends on wj,t because initial wealth determines
the amount of human capital investment in youth and hence an agent’s income inadulthood. This creates an intergenerational linkage in wealth. The dynamics ofwealth over generations are obtained by substituting this result into (2) and eliminating `j,t by (1):µwj,t+1 = Qt (1 − τ )b(wj,t )εj,t + τ
10

Z 1
0

b(wj 0 ,t )dj

0

¶

.

(11)

Evolution of the quality of the environment
Both nature and the activity of individuals drive the evolution of the quality ofthe environment Qt . The environment gradually deteriorates when people exploit
natural resources to use as an input for their primitive technology. However, natural
resources have the ability to recover their pristine abundance if exploitation ceases.15
Specifically, the change in the quality of the environment over one generation is givenbye − Qt ) − η∆Qt+1 ≡ Qt+1 − Qt = δ(QZ 1

rj,t+1 dj.

(12)

0

Parameter δ ∈ (0, 1) measures the speed with which the environment recovers to-e It is natural to assume that the initial level of the enviwards pristine quality Q.

e The last term of (12),ronment does not exceed the pristine level; i.e., Q0 ∈ (0, Q].
R1
η 0 rj,t+1 dj, shows that the quality of the environment deteriorates in proportion
to the aggregate amount of exploitation, where parameter η ∈ (0, 1) measures themagnitude of this eﬀect.
Individual usage of natural resources rj,t+1 can be found from preceding arguments. From the Leontief property of primitive technology (5), rj,t+1 coincides withlabor input `pj,t+1 , which is from the time constraint equal to labor endowment `j,tless the amount used for the sustainable technology `sj,t+1 . The labor input for thesustainable technology `sj,t+1 in turn coincides with the amount of human capital
hj,t , again from the Leontief property of (6). Human capital is produced throughhj,t = γej,t `j,t as in (3), where ej,t and `j,t are respectively given by (1) and (9). In
sum, we obtainrj,t+1 = (1 − γe(wj,t )) Qt εj,t .

(13)

Recall that human capital investment function e(wj,t ) is increasing with theagent’s initial wealth wj,t , as shown by (9), due to credit market imperfections.
Therefore, (13) describes the tendency of poorer households to rely more on natural
resources. In particular, those with initial wealth wj,t below w do not invest in hu15This assumption is appropriate when we consider renewable kinds of resources, such as soil,

forests, water and atmosphere.

11man capital at all and produce goods entirely through the primitive technology that
exploits natural resources, whereas those with wj,t above w fully invest in human capital and make use of the sustainable technology without harming the environment.
Agents with wj,t ∈ (w, w) also exploit some amount of natural resources, but not
as much as poorer agents do. Interestingly, (13) also implies that, ceteris paribus,
agents utilize natural resources more when Qt is larger since better environments
make agents more active.
Aggregating (13) over agents and substituting the result into (12) gives the evolution of Qt over generations:
µ
¶
Z 1
e
e(wj,t )dj Qt .
Qt+1 = δ Q + 1 − δ − η + γη

(14)

0

Equation (14) shows that the evolution of an environmental state is determined by
the average level of human capital investments, which is derived from the distributionof wealth through the investment function (9). In the other direction, the evolutionof the distribution of wealth (11) is dependent on environmental quality since thelatter determines the ability of workers. Distribution of wealth and environmental
quality are thus interdependent so that dynamics are simultaneously determined.
The following section demonstrates that this interdependency gives rise to multiple
steady states.

3

Homogeneous Economy

As a benchmark, this section examines the working of an economy when all individuals are identical in their abilities to work and in their initial wealth. Specifically,
we assume that variation in ability, defined by σ, is zero so that agents’ labor endowments are given by `j,t = Qt for all j (see equation 1). If both ability and wealthare the same across agents, they leave the same amount of wealth for their oﬀspring.
Thus, given that agents in the initial generation start from the same amount ofwealth, wj,t ’s are the same among individuals for all subsequent periods, and therefore can simply be written as wt . Since any redistributive policy (e.g., the inheritance12

Figure 2: Dynamics of wealth of the representative individual. Parameters: as = 0.5,
ap = 0.3, α = 0.4, β = 0.6, γ = 6, δ = 0.3, η = 0.9, σ = 0. The values for Qt are 1 (small), 1.7(intermediate), and 2.5 (large).

tax) has no eﬀect in this case, the dynamics of wealth (11) simplifies to
wt+1 = b(wt )Qt .

(15)

Figure 2 depicts the evolution of the representative agent’s wealth wt for given
levels of Qt ’s. As shown by the figure, there is an intersection between (15) and the
wt+1 = wt line. The point of intersection represents the fixed point of the dynamics,
toward which wt converges over generations. For each Qt , the fixed point is uniquesince the elasticity of b(wt )Qt with respect to wt is always less than unity (see equation10). A straightforward calculation gives the fixed point in terms of Qt ,β(α + β)−1 ap Qt ≤ w
if Qt ≤ Q,w∗ (Qt ) ≡ β(α + β + 1 − Qt /Q)−1 ap Qt ∈ (w, w) if Qt ∈ (Q, Q),





β(α + β)−1 as Qt ≥ w
if Qt ≥ Q,

(16)

where Q ≡ (βγ(as − ap ))−1 and Q ≡ Q + (α + β)(βγas )−1 . A larger Qt shifts thewhole schedule upward, and thus also raises w∗ (Qt ).
Even before analyzing the dynamics of Qt , Figure 2 and equation (16) are alreadysuggestive of the existence of multiple steady states. If the environment is suﬃciently
13good (Qt > Q), the wealth of representative agent wt converges to a high level above
w so that everyone uses the sustainable technology, maintaining a good environment.
With a bad environment (Qt < Q), conversely, wt converges to a low level below w sothat everyone uses the primitive technology, further deteriorating their environment.
The latter situation corresponds to what we call the poverty-environment trap.
To obtain the precise nature of multiple steady states, we now examine the simultaneous movements of wt and Qt . From equation (14), it is straightforward toshow that Qt+1 R Qt if and only if
Qt Q Q∗ (Et ) ≡where Et ≡

R1
0

e
δQ
,
δ + η(1 − γEt )(17)e(wj,t )dj represents the aggregate amount of human capital invest-

ment. Since Et = e(wt ) in the representative agent setting, substituting (9) into (17)
gives

±


e (δ + η)δQ
if wt ≤ w,



±
¡
¢
e (δ + η w − w)−1 (w − w) if wt ∈ (w, w),Q∗ (e(wt )) = δ Q





e
Q
if wt ≥ w.

(18)

Using (16) and (18), the evolution of wealth and environmental quality is described by the phase diagram in Figure 3, drawn in (w, Q) space. The wt+1 = wt
locus is given by wt = w∗ (Qt ), whereas the Qt+1 = Qt locus is from Qt = Q∗ (e(wt )).
e is
The figure is drawn by assuming that the pristine quality of the environment Q

within´³e ∈ Q, (1 + η/δ)Q ,Q(19)

for which case there exist three steady states.16 Intuitively, multiple steady states
emerge since the Qt+1 = Qt locus has an S-shape while the wt+1 = wt locus has an
16

The interval in the RHS of (19) is nonempty when η/δ is large. Recall that parameters η andδ represent the speed of the environment’s recovery and that of deterioration due to exploitation,
respectively. Therefore, multiple steady states occur only when the impacts of exploitation on theenvironment are considerably stronger than its ability to recover. We consider this case throughout
the paper.

14

Figure 3: Dynamics of wealth and environmental quality. The parameters are the samee
as in Figure 2. There exist three steady states since assumption (19) implies δ Q/(δ+ η) < Q ande
Q < Q.

inverted S-shape. These properties derive from the upper and lower bounds of theinvestments in human capital. On one hand, the Qt+1 = Qt locus is upward sloping
if an increase in wealth raises the amount of human capital investment, which occurs
only when wt ∈ (w, w). On the other hand, the slope of the wt+1 = wt locus isflatter when wt is within (w, w) than otherwise since only in this range a change
in the environmental quality aﬀects not only the ability of workers but also their(descendants’) human capital investment behavior, resulting in greater variation in
the steady state wealth.
e
The phase diagram has two stable steady states, denoted by B (one with Qt = Q

and wt ≥ w) and T (where Qt < Q and wt ≤ w), and one saddle point S (whereQt ∈ (Q, Q) and wt ∈ (w, w)). There is a downward sloping saddle path, i.e., the setof (w0 , Q0 ) from which (wt , Qt ) converges to saddle point S. However, convergence
to S is unlikely since the initial (w0 , Q0 ) pair is given historically and therefore it isan event of measure zero that the pair happens to be exactly on the saddle path.
The economy converges to a better steady state B if it starts from the upper
right side of the saddle path. Steady state B is better in that both wealth of the15

representative agent and the quality of the environment are higher than the otherstable steady state T. In this steady state, the productivity of every agent is highthanks to good environments. High productivity enables agents to leave large transfers to their oﬀspring, who use the received wealth to obtain human capital. Highlevels of investment in human capital in turn contribute to maintaining good environments because human capital enables people to adopt the sustainable technology.
This virtuous cycle makes both high wealth and good environment sustainable overgenerations.
When the initial pair of (wt , Qt ) is on the other side of the saddle path, the economy converges to steady state T. In contrast to steady state B, here a deteriorated
environment reduces agents’ ability to work, and also the amount of wealth that canbe left to their oﬀspring. Agents of subsequent generations cannot aﬀord to invest in
human capital, and have no other choice than to use the primitive technology that
exploits natural resources, and as a consequence degrade the environment more andmore. Once the economy falls into this poverty-environment trap, no one changes
their practices of environmental degradation since they are equally poor.
The last point suggests that the existence of the poverty-environment trap depends on the assumption of homogeneity among agents. If we allow heterogeneity intheir ability and wealth, some relatively rich agents may be able to aﬀord to invest in
human capital even though most agents are trapped in the vicious cycle of poverty
and bad environment. The next section examines this possibility and shows that
wealth heterogeneity is in fact the key to escaping the poverty-environment trap.

4

Economy with Heterogeneous IndividualsThis section introduces heterogeneous individuals diﬀering in their ability (i.e., in
the amount of labor endowment) and also in the initial wealth inherited from theirparents. Our first task is to establish a way to find the steady states in which boththe level of environmental quality and the distribution of wealth are stationary. Thenwe examine the eﬀect of the redistribution of wealth in the two stable steady states.
16

Let Ft (w) denote the cumulative distribution function of wealth among generation t. Given Ft (w) and environmental quality Qt , equation (11) generates the
distribution of wealth among generation t + 1. Let this process be denoted by
Ft+1 (·) = TQt Ft (·),

(20)

where TQt is an operator that maps a wealth distribution to that of the next period
e for allunder a given level of Qt . In Appendix B, we show that Qt evolves within (0, Q]

e mapping TQ has an invariant distribution. Let us denotet and that for any Q ∈ (0, Q]

by F ∗ (·; Q) the cumulative distribution function of wealth that solves the functional

equation TQ F ∗ (·; Q) = F ∗ (·; Q). Then the condition that the distribution of wealthamong agents does not change over generations is written as Ft (·) = F ∗ (·; Qt ).
In steady states, the quality of the environment must also be stationary. Recall
from condition (17) that this is the case if and only if Qt = Q∗ (Et ), where Et
is the aggregate amount of human capital investments. The aggregate amount of
human capital investment, Et , is in turn determined by the distribution of wealthand the individual investment function. Specifically, this relationship can be writtenR
as Et = e(w)dFt (w) ≡ E[Ft (·)], where E[·] is a functional that maps a wealthdistribution to the aggregate human capital investments.

In sum, the triple {Ft (·), Qt , Et } constitute a steady state when the following
simultaneous equations are satisfied:Ft (·) = F ∗ (·; Qt ),Qt = Q∗ (Et ),Et = E[Ft (·)].

(21)

Although equations in (21) well describe the mutual relationship among wealth distribution, environmental quality and human capital, we need to eliminate one of themto obtain a two-dimensional figure of the steady state. Let us eliminate Ft (w) by substituting the first equation into the RHS of the third and define E ∗ (Q) ≡ E[F ∗ (·; Qt )].
Function E ∗ (Q) gives the amount of human capital investment when wealth distribution is stationary under environment Qt . Using this, (21) can be simplified toEt = E ∗ (Q),

Qt = Q∗ (Et ).
17

(22)

The pair {Et , Qt } constitutes a steady state if and only if simultaneous equations(22) are satisfied. Once such a pair is found, the stationary wealth distribution Ft (·)
can be calculated by the first equation in (21).
The advantage of the representation of steady states by (22) is that both Et and
Qt are scalar variables and therefore their relationships can be drawn as two curves in(Et , Qt ) space. We numerically calculated the curves for the conditions Et = E ∗ (Q)and Qt = Q∗ (Et ), and the results are shown in Figure 4. To examine the eﬀects ofredistribution, calculations are performed for four diﬀerent rates of inheritance tax
while keeping other parameters fixed.
The upper left panel of Figure 4 shows the extreme case of τ = 1, which serves
as a benchmark. In this case all transfers from parents are expropriated by theauthority and then equally distributed among all young agents. That is, there is noheterogeneity in wealth at birth, and, from (9), educational investment is also thesame among individuals. Thus, the steady states should coincide with ones derived
in Section 3. In fact, we see that there are three steady states; again denoted as B,
S and T. Note that the aggregate levels of human capital investment are exactly
1/γ (the maximum) and zero at steady states B and T, respectively, because theEt = E ∗ (Qt ) curve is vertical at these two values. That is, when Q ≤ Q no agent
invests in human capital and therefore the Et = E ∗ (Qt ) curve is vertical at Et = 0.
The curve is also vertical at Et = 1/γ since all agents fully invest in human capitalwhenever Qt ≥ Q.
When the magnitude of redistribution is not so large (τ < 1), the initial wealth ofagents diﬀers, depending on their parents’ ability and wealth. In this case, the Et =E ∗ (Qt ) locus no longer has vertical segments because the choice of technology among
agents is not uniform. As the tax rate τ decreases, wealth distribution becomes
more heterogeneous. Then the choice of technology becomes more mixed and theEt = E ∗ (Qt ) curve gets smoother. Given that the upward sloping Qt = Q∗ (Et )curve does not depend on the degree of heterogeneity, this change implies that both
Et and Qt decreases at steady state B whereas the opposite holds at steady state T.
Although heterogeneity among individuals deteriorates the environment and reduces
18

(i) τ = 1(ii) τ = 0.5(iii) τ = 0.3(iv) τ = 0

Figure 4: Steady States with Heterogeneity. Parameters: as = 0.5, ap = 0.3, α = 0.4,
e = 4. The poverty-environment trap disappears when τ ≤ 0.3.
β = 0.6, γ = 5.8, η/δ = 2, σ = 0.7, Q

output when the economy is in the better steady state, it mitigates environmental
degradation and poverty when the economy is in the poverty-environment trap.
Moreover, our numerical example shows that the poverty-environment trap disappears when the rate of inheritance tax is below a certain threshold.17 Figure 5
illustrates the relationship between the level of τ and the steady state values of Qt .
17

The threshold level of τ varies considerably depending on parameters. Under diﬀerent sets ofparameters, it is possible that reductions in τ cause the better steady state to disappear before thepoverty-environment trap vanishes.

19

Figure 5: Rate of inheritance tax and the steady state levels of environmental quality.
Parameters are the same as figure 4.

Thick curves represent stable steady states while the dotted curve represents a saddle
point. Arrow a shows that when the degree of wealth heterogeneity is increased by
reductions in τ to a certain extent, an economy initially in the poverty-environment
trap escapes from the trap and shifts to the better steady state. However, a further
heterogeneity in wealth in the better steady state causes environmental deterioration,
as shown by arrow b.
The result obtained in this section can be interpreted in terms of an intergenerational linkage in wealth and a trickle-down eﬀect. Consider an economy trapped in
the poverty-environment trap and suppose that the heterogeneity of wealth among
agents is increased by a reduction in the rate of inheritance tax. Then, even whenthe environment is bad, agents with relatively high ability may be able to leave sufficient wealth for their children to aﬀord to invest in human capital. Those receiving
suﬃcient wealth to invest in human capital employ the sustainable technology, which
is more productive than the primitive technology. Thus, they can leave more wealth,giving their oﬀspring an even greater chance to invest in human capital. In thisway, the intergenerational linkage in wealth enables fortunate households to get outof the poverty trap. In addition, the wealth of rich households trickles down to thepoor through their adoption of the sustainable technology. As rich households switch
to the sustainable technology, the aggregate environmental load of production de20

creases. The quality of the environment gradually improves, and the ability of every
individual increases. When redistribution is reduced considerably, even the poorest
may eventually obtain enough wealth to invest in human capital. In that case, theshrinkage in redistribution lets the economy escape from the poverty-environment
trap to the better steady state.
However, this result is reversed if the economy is already in the better steadystate, where a majority of individuals employ the sustainable technology and thequality of the environment is relatively good. In this case, greater inequality lets someindividuals use the primitive technology and deplete natural resources through theirproduction activities. These practices result in environmental degradation, which
gradually erodes the ability and wealth of every individual. Thus, redistribution ofincome or wealth is desirable in this case because it not only reduces inequality but
also improves the environment and the productivity of all individuals.

5

Conclusion

We model an overlapping generation economy with heterogeneous agents, whose
ability depends on environmental quality. The relationship between wealth heterogeneity, the environment, and economic development are examined in that setting.
The dynamics of wealth distribution are aﬀected by environmental quality throughthe ability of agents, whereas wealth distribution aﬀects the evolution of environmental quality through the use of primitive technology that exploits natural resources.
This interaction creates a “poverty-environment trap”, where environmental degradation lowers the productivity of the poor and reduces the amount of wealth left for
their oﬀspring, which, in turn, deprives them of the opportunity to invest in humancapital and thereby accelerates deterioration in environmental quality.
We show that increased heterogeneity in the amount of wealth at birth, resulting
from a reduction in the rate of inheritance tax, has a positive eﬀect on both theenvironment and aggregate output in a trapped economy. When relatively capable
individuals can leave a large amount of wealth, their children can employ the sus21

tainable technology and leave more wealth to their oﬀspring. In addition, the wealthof fortunate households trickles down to the poor since these households’ adoptionof the sustainable technology improves environmental quality and therefore the productivity of every individual. Accumulation of those eﬀects may eventually enable
the economy to escape from the poverty-environment trap to a better steady state.
However, it should be noted that wide heterogeneity has a negative eﬀect on boththe environment and aggregate output when the economy is already in the better
steady state.

Appendix
Appendix A
Using (3), we eliminate hj,t+1 from (7) to obtain¡¢ (as − ap )γej,t + ap `j,ty(γej,t `j,t , `j,t ) =ah `j,t

for ej,t ∈ [0, 1/γ],(23)

for ej,t > 1/γ.

Equation (23) shows that any investment in human capital above 1/γ has no eﬀect
on the output in the second period. Since the marginal utility of consumption in thefirst period is always positive, it cannot be optimal to invest more than 1/γ.
Considering this fact, the current problem is restated as follows: to maximize (8)with respect to cyj,t , caj,t+1 , ej,t and bj,t+1 undercyj,t + ej,t = wj,t¡¢caj,t+1 + bj,t+1 = (as − ap )γej,t + ap `j,t ,ej,t ≥ 0, ej,t ≤ 1/γ.

(24)
(25)
(26)

The Lagrangian for this problem is£¤= ln cyj,t + α ln caj,t+1 + β ln bj,t+1 + λ1 wj,t − ccj,t − ej,t£¡¢¤+ λ2 (as − ap )γej,t + ap `j,t − caj,t+1 − bj,t+1 + λ3 ej,t + λ4 [1/γ − ej,t ] .
22

The first-order conditions with respect to cyj,t , caj,t+1 , ej,t and bj,t+1 are respectively
given byλ1 = 1/cyj,t ,(27)

λ2 = α/caj,t+1(28)

λ2 = β/bj,t+1(29)−λ1 + λ2 (as − ap )γ`j,t + λ3 − λ4 = 0.

(30)

Since (26) is a pair of inequality constraints, complementary slackness conditions arealso required:λ3 ej,t = 0,

λ3 ≥ 0,λ4 [1/γ − ej,t ] = 0,

λ4 ≥ 0.

(31)(32)

The optimal solution is characterized by {cyj,t , caj,t+1 , ej,t , bj,t+1 , λ1 , λ2 , λ3 , λ4 } that solve(24)-(32), the latter consisting of eight equality and four inequality conditions.
The solution is found by a guess-and-verify method. The first guess is thatej,t = 0. Then from (24) and (27) λ1 = 1/cyj,t = 1/wj,t . From (25), (28) and (29)λ2 = α/caj,t+1 = β/bj,t+1 = (α + β)/(ap lj,t ). From (32), λ4 = 0. Substituting λ1 , λ2
and λ4 into (30) gives λ3 = 1/wj,t − (α + β)γ(as − ap )/ap . If λ3 ≥ 0, the remaining
condition (31) is also satisfied, implying that the guess ej,t = 0 is correct. This lastcondition is met if and only if wj,t ≤ ap ((α + β)γ(as − ap ))−1 ≡ w.
The next guess is that ej,t ∈ (0, 1/γ). Then (31) and (32) show λ3 = λ4 = 0.

From (24) and (27), λ1 = 1/cyj,t = (wj,t − ej,t )−1 . From (25), (28) and (29), λ2 =¤£α/caj,t+1 = β/bj,t+1 = (α + β)/ ((as − ap )γej,t + ap )`j,t . Substituting λ1 , λ2 , λ3 andλ4 into the remaining condition (30) gives ej,t = (α + β)(1 + α + β)−1 (wj,t − w). If

the latter value is within (0, 1/γ) then our guess is correct. This is the case if andonly if w < wj,t < wj,t + (1 + α + β)(α + β)−1 γ −1 ≡ w.

Finally, we guess that ej,t = 1/γ. Then from (24) and (27) λ1 = 1/cyj,t = 1/(wj,t −1/γ). From (25), (28) and (29) λ2 = α/caj,t+1 = β/bj,t+1 = (α + β)/(as lj,t ). From(31), λ3 = 0. Substituting λ1 , λ2 and λ3 into (30) gives λ4 = −1/(wj,t − 1/γ) +(α + β)γ(as − ap )/as . If λ4 ≥ 0, the remaining condition (32) is also satisfied,23

implying that the guess ej,t = 1/γ is correct. This last condition is met if and only
if wj,t ≥ as ((α + β)γ(as − ap ))−1 + 1/γ ≡ w.
It can be confirmed that an agent with any arbitrary wj,t ≥ 0 fits (exactly) into
one of the above three cases.
Appendix B
e for all t ≥ 0. Since e(wj,t ) ∈ [0, 1/γ] from (9),We first prove that Qt ∈ (0, Q]

equation (14) implies that

Qt+1 ≤e − (1 − δ)(Qe − Qt ),Q

e − Qt ) + (1 − η)Qt ,Qt+1 ≥ δ(Q

(33)
(34)

e then (33) and (34) gives Qt+1 ∈ (0, Q].
ewhere δ, η ∈ (0, 1). Note that if Qt ∈ (0, Q],

e by induction we
Since the initial level of Qt is assumed to be within Q0 ∈ (0, Q],e for all t ≥ 0.
obtain Qt ∈ (0, Q]

We next show that wj,t is within a bounded interval for all j and t ≥ 1. Notethat equation (10) shows b(wj,t ) ∈ [(α + β)−1 βap , (α + β)−1 βas ]. Also recall thateεj,t ∈ [1 − σ, 1 + σ] and τ ∈ [0, 1]. Substituting these and the property Qt ∈ (0, Q]
as proved above into equation (11) gives wj,t ∈ (0, W ] for all j and t ≥ 1, whereeW ≡ (α + β)−1 β(1 + σ)as Q.

Therefore, to represent the wealth distribution for t ≥ 1, it is suﬃcient to consider

the family of distribution on a closed and bounded (i.e., compact) interval [0, W ]. In
addition, since the number of agents in each generation is 1 for all t, we can further
restrict our attention to distributions such that F (W ) = 1. Let us denote by Λ thee operator TQ defined byfamily of such distributions. Then, whenever Q ∈ (0, Q],(20) maps Λ into itself, i.e., TQ : Λ → Λ. For later use, we order elements in Λ
by first-order dominance and denote the ordering by operator <. That is, Fa < Fbmeans Fa (w) ≤ Fb (w) for all w ∈ [0, W ].
The aim of this appendix is to show that mapping TQ : Λ → Λ has a fixed point.
The proof is based on Theorem 1 of Hopenhayn and Prescott (1992).18
18

Aghion and Bolton (1997) and Owen and Weil (1998) rely on Theorem 2 of Hopenhayn and24

Theorem 1 (Hopenhayn-Prescott) Let Λ be a compact subset of(S) and T :Λ → Λ an increasing map. Then T has a fixed point if and only if there exists a
measure µa in Λ such that T µa < µa .

In this theorem,(S) refers to the space of finite measures on (S,), whereisthe Borel σ-algebra of subsets of S. In our context, S = [0, W ] and Λ is a closed setof uniformly bounded measures, satisfying the requirements in Theorem 1.19 Let δ0represent the distribution where all the mass is concentrated at 0 (i.e., δ(w) = 1 for
all w ∈ [0, W ].) Then δ0 ∈ Λ is the minimum element in Λ in that F < δ0 for all

F ∈ Λ and therefore TQ δ0 < δ0 .

To apply Theorem 1 for TQ , then, we only need to prove that TQ is an increasing
map. Consider two distributions of wealth Fa , Fb ∈ Λ and assume that Fa first-order

dominates Fb ; i.e. Fa < Fb . By definition, TQ is an increasing map if TQ Fa < TQ Fb

for any such Fa and Fb . To show this, we use the fact that the wealth of the next
generation, given by (11), is the sum of individual transfer, Qt (1 − τ )b(wj,t )εj,t ,R1
and the amount of redistribution, Qt τ 0 b(wj 0 ,t )dj 0 . Note that function b(wj,t ) is
increasing in the previous generation’s wealth wj,t , as shown by (10), and εj,t isdistributed independently of wj,t . These imply that the distribution of individualtransfer, when the distribution of the previous generation’s wealth is given by Fa ,first-order dominates that when the distribution of the previous generation’s wealth
is Fb . In addition, the amount of redistribution is proportional to the average amountof individual transfer, which is again higher when the distribution of the previous
generation’s wealth is Fa than when it is Fb . Thus, the distribution of wealth of thenext generation under Fa first-order dominates that under Fb ; i.e., TQ Fa < TQ Fb .
We have shown that Theorem 1 can be applied for mapping TQ : Λ → Λ, which
Prescott (1992), which is simpler but requires the mapping T to be linear. Specifically, the lattertheorem is applicable when individual wealth follows an independent Markov Process since in thatcase T (Fa + Fb ) = T Fa + T Fb . This is not the case for our model, since the amount of redistribution
depends on the entire wealth distribution in the economy and therefore the evolution of individual
wealth is interrelated.
19

Set Λ conforms to example (c) of Hopenhayn and Prescott (1992, p. 1390).

25

assures the existence of an invariant distribution.

26

Reference
Aghion, P., Bolton, P.: A theory of trickle-down growth and development. Review
of Economic Studies 64, 151-172 (1997)
Bahamondes, M.: Poverty-environment patterns in a growing economy: farming
communities in arid central Chile, 1991-99. World Development 31, 1947-1957(2003)
Banerjee, A., Newman, A.F.: Occupational choice and the process of development.
Journal of Political Economy 101, 274-298 (1993)
Barbier, E.B.: The economic determinants of land degradation in developing countries. Philosophical Transactions: Biological Sciences 352, 891-899 (1997)
Barbier, E.B., Burgess, J.C.: The economics of tropical forest land use options.
Land Economics 73, 174-195 (1997)
Barro, R.J.: Inequality and growth in a panel of countries. Journal of Economic
Growth 5, 5-32 (2000)
Borghesi, S., Vercelli, A.: Sustainable globalisation. Ecological Economics 44, 77-89(2003)
Cavendish, W.: Empirical regularities in the poverty-environment relationship of rural households: evidence from Zimbabwe. World Development 28, 1979-2003 (2000)
Dasgupta, P.: The economics of poverty in poor countries. Scandinavian Journal of
Economics 100, 41-68 (1998)
Duraiappah, A.K.: Poverty and environmental degradation: a review and analysisof the nexus. World Development 26, 2169-2179 (1998)
Galor, O., Tsiddon, D.: The distribution of human capital and economic growth.
Journal of Economic Growth 2, 93-124 (1997)27

Galor, O., Zeira, J.: Income distribution and macroeconomics. Review of Economic
Studies 60, 35-52 (1993)
Hopenhayn, H.A., Prescott, E.C.: Stochastic monotonicity and stationary distributions for dynamic economies. Econometrica 60, 1387-1406 (1992)
Loury, G.C.: Intergenerational transfers and the distribution of earnings. Econometrica 49, 843-867 (1981)
Moav, O.: Income distribution and macroeconomics: the persistence of inequality
in a convex technology framework. Economics Letters 75, 187-192 (2002)
Oldeman, L.R., van Angeles. V., Pulles, J.: The extent of human-induced soil
degradation. In: Oldeman L.R., Hakkeling, R.T.A., Sombroek W.G. (eds.) Worldmap of the status of human-induced soil degradation: an explanatory note, 2nd ed.
Wageningen: International Soil Reference and Information Centre 1990
Owen, A.L., Weil, D.N.: Intergenerational earnings mobility, inequality and growth.
Journal of Monetary Economics 41, 71-104 (1998)
Piketty, T.: The dynamics of the wealth distribution and the interest rate withcredit rationing. The Review of Economic Studies 64, 173-189 (1997)
Swinton, S.M., Quiroz, R.: Is poverty to blame for soil, pasture and forest degradation in Peru’s Altiplano? World Development 31, 1903-1919 (2003)
Torras, M., Boyce, J.K.: Income, inequality, and pollution: a reassessment of theenvironmental Kuznets curve. Ecological Economics 25, 147-160 (1998)
World Bank: World Development Indicators 2004. Washington, D.C.: World Bank
200428