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Endowments, Preferences, Technologies and Abatement:
growth-environment microfoundations
*
Alexander S.P. Pfaff
**
, Shubham Chaudhuri, and Howard L.M. Nye
Columbia University
Abstract
Will economic growth inevitably degrade the environment, throughout development?
We present a household-level framework emphasizing the tradeoff between consumption
that causes pollution and pollution-reducing abatement. Our model provides a simple
explanation for upward-turning, non-monotonic paths of environmental quality during
economic growth. Its innovation is sufficient conditions that simultaneously address
preferences and technologies. With standard preferences, an asymmetric endowment
(i.e., at zero income, consumption is also zero but environmental quality is positive)
leads low-income households not to abate, and further this condition is sufficient for an
EKC for a wide range of abatement technologies. Without such an endowment, however,
even strong economies of scale in abatement are, on their own, insufficient for an EKC.
Word Count: a little over 6000 (including footnotes)
JEL codes: D11, H41, O12, Q25
-----
*
We would like to thank for helpful comments Matt Kahn and participants in AERE/ASSA, NBER, NEUDC and
Harvard Environmental Economics and Policy seminars. We are of course responsible for any remaining errors.
**
Corresponding author: 420 W. 118
th
Street, Room 1306; New York, NY 10027
(212) 854-4190 (fax, -5765); ap196@columbia.edu
1. Introduc tio n
Whether en vironmental quality will inevitably fall during economic dev elopmen t has
spurred empirical, theoretical and policy debate. Early empirical analyses that used
countries as units of analysis suggested so-called En vironmental Kuznets Curves
(EKCs), i.e. non-monotonic U-shaped relationships between per-capita income and
en vironmental quality. Thus, environmental quality would rise during later stages of
development.
1
To caricature the early literature, such relationships seemed empiri-
cally robust, at a national level, but theoretically puzzling. The lac k of a predictive
framework helped to fuel ongoing tension c oncerning the appropriate interpretation
of empirical relationships of this type: do we believe and expect them, i.e. are they
and should they be robust? and are they in any way evidence that regulation is
unnecessary, or simply an implication of increasing regulation as incomes rise?
Any modeling framework must take a stand on whether external e¤ects among
households are internalized. If regulations do not exist and households do not fully
in ternalize their e¤ects, it is not surprising that environmen t can fall with rising
income.
2
But what is not as easy to explain in this setting is how environmental
quality could rise with increasing income. For that to happen, households would
ha ve to care enough about the environment to coordinate and aggregate preferences,
suc h as through voting. Thus, modeling of increasing regulation as incomes rise,
which many feel is central to explaining observed EKCs, requires an understanding
1
See, for instance, World Bank 1992, Selden and Song 1994, Sha…k 1994, Holtz-Eakin and
Selden 1995, Grossm an an d Krueger 1995, and more recently, specia l issues of both Environment
and Development E conom ics, in N ovember 19 97, and Ecol ogi cal Ec on omics, in May 1998.
2
Environmental econo mics tex tbook s feature environmentally d am aging external emiss ions that
rise with the scale of pro du ction of th e po ll utin g go od. If re gulations or Coasian bargaining d o not
lead to internalization, thes e emissions will sur e ly lower environmental quality as incomes rise.
of the evolution of household choices during economic growth.
3
This paper provides a household-level framework to explore a simple, powerful
reason why environmental quality may fall and then rise as incomes rise. It is rele-
vant as a background for voting models, but also relevant when degradation features
a signi…cant private component.
4
In addition to clarit y and transparency, this model
in particular yields su¢cient conditions for non-monotonic paths of environmental
quality. These permit easy evaluation of whether a given combination of preferences
and abatement technologies gives rise to such a path. While existing literature has
tended to fo cus either on preferences or on abatement technologies
5
, we allow the
e¤ect of a giv en technology to depend upon the preferences, and vice versa.
For each household, our model focuses upon the asymmetric endowment of con-
sumption and environment, i.e. positive environmental quality but zero consumption
at zero income. This is so natural an assumption as to appear obvious. We sho w
that it is nonetheless crucial. Given standard preferences for consumption and envi-
ronment, such that if households could purchase them separately and independen tly
both would be normal goods, an endowm en t is su¢cient for an EKC given a wide
3
Neoc lassical growth m odels th at consider pollution and growth have provided on e approach
when externalitie s a r e ass u me d to be interna liz ed, and c an provide results sim ilar to o urs (see
Plourde 1972, Ke eler et al. 1972, D ’Arg e a nd K og iku 1973, Forster 1973 , Gru ver 1976, S tephens
1976, Asako 1980 , Becker 1982, Tahvonen and Kuuluvainen 1993, J oh n and Pecchenino 1 994,
Selden and Song 1995, Jones a nd Ma nuelli 1995, Stokey 1998 , and Chi meli 200 1). But they wil l
not easi ly ex pl ain regula to ry choice g iven heterogen eous voters (note that while J on es and Ma nuelli
1995 fea tu res a represe ntative a gent at each point in time , th e paper co ns id ers t h e problem of
intertemporal co llective decisio nm a ki ng ). The dynamic r epre sentative agent framework lacks a
realisti c poli tic al econ omic mechanism through wh ich d e gradation might i n reality b e reversed. In
contrast, house hold models ca n yield insights withi n a sett ing of internalization a nd, as th e y c an
be applied in a multi-agent setting in the pres e nce of externalities, can permit explicit mode ling of
how environme ntal p ref eren ces mi ght be ag gregated throug h voting in order to produce regu lati on.
4
For example, Cha udhuri and P fa¤ 1998 an d 2002 conside r empirically how household fuel choice
in Pakistan changes with income , in light of fuels’ e¤ec ts on indoor air quality, a private good. Whi le
stove emissions have external e¤ects as well, private environmental quality is signi…cantly degraded.
More generally, other forms of degradation of the environment also feature private components, and
there exists signi…cant private provision of environmental abatement in the absence of regulations.
5
For instance, Stokey 1998 e m p h asi zes th e role of elastic ity of preferenc es, while Andreon i a nd
Levinson 2001 fo cuses up o n a role for a very par ticul ar typ e of increasing returns to abatement.
2
range of abatement technologies encompassing …xed costs and decreasing returns.
6
Thus, the model can account for the initial decrease in environmental quality as
income rises and for increasing en vironmental quality as income contin ues to grow.
The key in tuition is that, given an endowment of environmental qualit y that
is degraded by consumption, and given con vex preferences for consumption and
environment, at low incomes the marginal rate of substitution implies that the
household prefers not to spend to abate the e¤ects of consumption. This results in
a corner solution where no resources are expended on abatement, but consumption
will rise with income. This causes en vironmental quality to fall with income.
However, as income continues to rise, and the environment is degraded, the mar-
ginal rate of substitution between environmental quality and consumption increases
in favor of the environment, until it is desirable to abate. This moves the household
to an interior solution, in which it both consumes and abates, and for a wide range
of abatement technologies environmental quality will increase with income because
both goods are normal. We provide a condition which determines whether as in-
come gets high enough environmental quality will rise with income. This compares
the change in the marginal rate of substitution as incomes rise to the change in the
marginal rate of transformation implied by the abatemen t technology.
7
As noted,
such a general condition simpli…es consideration of the implications for EKCs of
6
Such an endowment as su mption (which we argu e is hard to ref u te ) is im plicit i n some existin g
papers (e.g., John and Pecchenino 1994). However, its truly central role has not b een highlighted.
Further, such endowments can b e thought of more broadly if we consider not only preferences and
the MRS but also techn ologies and the MRT (again, our framework easily perm its their comp aris on ) .
For instance, Chimeli 200 1 sugge sts that an o ¤ -equi librium -pa th ‘endowment’ of capital may exist
for e con omie s in tran sition and, given that, tra ces the optimal path of the MRT as income rises.
7
This part of the pa per s ign i… cantly ge neralizes ou r re lated work in Chaud huri and Pf a¤ (1997 a
and b) and Pfa¤, Ch au d huri and Nye 2002. The theo retica l analyses they p resent d evelop in d etail
the case of input substitution as an abatement technology. That case, in turn, corresponds to the
empirical work on “ househ old EKCs” for indoor air quality i n Chaudhuri and Pf a¤ 1998 a nd 2002.
3
combinations of whatever preferences and abatement technologies are of interest.
Finally, consider that while with externalities a fall in environmental quality is
the default and a rise is harder to explain, without externalities rising environmental
quality is the default and why it might fall with income is unclear. For most tech-
nologies, if the en vironment is a normal good then without further assumptions the
Engel curv es for environment ought to be, apriori, positively sloped at all incomes.
The need for an explicit explanation of a fall in environmental quality in this
setting of internalization is easy to overlook. The main theorem in Andreoni and
Levinson 2001, for instance, simply assumes a fall: [paraphrasing – assuming con-
sumption and environment are normal goods, and a particular increasing returns
abatement technology] ... “for any combination of utility and abatement technol-
ogy that yields positive pollution for some lev el of income, optimal pollution will
eventually decline back to zero for some su¢ciently large income.” In contrast to
this assumption of initially falling environmental quality, we show that for stan-
dard conv ex preferences without an asymmetric endowment, neither the abatement
technologies we consider nor the increasing returns technology in Andreoni and
Levinson 2001 accounts for a range of income in whic h en vironmen tal quality in fact
falls. Hence, these abatement technologies alone cannot generate EKCs.
Belo w, Section 2 presents our simple model and, retaining its generality, works to
our su¢cient conditions for an EKC. Section 3 adds intuition through speci…c cases
showing that our endowmen t-based is robust for a range of abatement technologies.
It also explores abatement tec hnologies without an environmental endowment, …nd-
ing that even increasing returns is not su¢cient for an EKC. Section 4 concludes
with a brief discussion and implications for further research.
4
2. Household Income and Environmental Quality
2.1. Preferences, Abatement Technology and Env ironmental Qualit y
A household gets utility from two goods, a marketed consumption good, denoted by
c, and environmental quality, denoted q, so that utility can be written as:
U = U(c; q) (2.1)
where U
c
> 0, U
q
> 0,andU is concave in c and q. Households enjoy an initial
endowment of environmental quality (q
0
¸ 0) that is degraded by pollution, which
as a byproduct of consumption rises with c. However, the household can choose
to expend resources to “abate” the e¤ects of pollution on the en vironmen t, i.e. to
make consumption less damaging, for instance by using cleaner but more expensive
inputs or by cleaning up pollution already generated. Denoting such expenditures
on environmen tal investment as e, we write environmental quality as:
q = q(c; e) (2.2)
where q
c
< 0–environmental quality falls with rising consumption–and q
e
> 0.
The general household problem, then, is to choose c and e to maximize (2.3)
subject to the budget constraint (2.4) and, since a household can choose to expend
zero resources on either c or e, also the non-negativity constraints (2.5):
U = U(c; q(c; e)) (2.3)
5
p
c
c + p
e
e = y (2.4)
c ¸ 0;e¸ 0 (2.5)
where y is household income, and p
c
and p
e
are, respectively, the prices of c and e.
2.2. Su¢cient Conditions for an EKC
Before providing speci…c results for particularly interesting cases (see Section 3), we
derive general conditions for the two parts of an EKC, i.e. environmen tal quality
falling with income at low incomes, and rising with income at higher incomes. For
these general results, we start with a few assumptions about preferences:
i) U
c
> 0 (ii) U
cc
< 0 (iii) U
q
> 0 (iv) U
qq
< 0 (2.6)
(v) U
qq
U
cc
¡U
2
cq
¸ 0 (vi) lim
c!0
U
c
(c; q)=+1 (vii) lim
q!0
U
q
(c; q)=+1
We assume further that preferences are such that the demand for both of these
goods, c and q, would be normal if these goods could both be purchased separately
and independently:
(i) U
c
U
cq
¡U
q
U
cc
> 0 (ii) U
q
U
cq
¡U
c
U
qq
> 0 (2.7)
We also make the following assumptions about the relatio nships between consump-
tion, en vironmental degradation and the abatement technology:
(i) q
e
> 0 (ii) q
ee
· 0 (iii) q
c
< 0 (iv) q
cc
· 0
(v) q
ce
¸ 0 (vi) lim
e!0
q
e
(c; e)=m<+1 (vii) q(0; 0) = q
0
> 0 (2.8)
6
To simplify the notation, we set p
c
= p
e
=1.
Given these conditions, we ask whether an asymmetric endowment (q
0
> 0)leads
to a low income range in which nothing is spent on abatement but consumption
occurs, such that environmental quality falls. Also, we examine whether such a
range is followed by one in which consumption and abatement occur, and both rise
with income such that environmen tal quality also rises, yielding an EKC. Here we do
so for the general model, while in Section 3 w e provide speci…c results (for instance
income ranges for environmental quality falling and rising) for some cases of interest.
2.2.1. No Abatemen t at Low Incomes
From (2.6, vi), we know the non-negativity constraint on c will never bind, and from
(2.8, i) we know the budget constrain t will alwa ys bind. Hence we can write the
…rst-order condition for maximization of (2.3) subject to (2.5) and the budget as:
U
c
(c; q(c; y ¡c)) + U
q
(c; q(c; y ¡ c))q
c
(c; y ¡ c) ¸ U
q
(c; q(c; y ¡ c))q
e
(c; y ¡ c) (2.9)
which holds with equality only if e = y ¡ c>0. On the left is the net marginal
utility from additional consumption, including the marginal disutility from the loss
of envir onmental qualit y brough t about by additional consumption. The term on
the right represents the marginal utility from additional abatement expenditures.
Let c
¤
(y ) and e
¤
(y) denote the optimal choices of c and e from the maximization
problem abo ve. Given the above assumptions regarding preferences and technology,
w e will show here that there exists
b
y>0 such that for all y<
b
y:
c
¤
(y )=ye
¤
(y )=0
dq
dy
= q
c
dc
¤
dy
+ q
e
de
¤
dy
= q
c
< 0
7
To see this, we can by start by de…ning:
g(y) ´ U
c
(y; q(y; 0)) + U
q
(y; q(y; 0))q
c
(y; 0)
l(y) ´ U
q
(y; q(y; 0))q
e
(y; 0) (2.10)
For income y, g(y) is the net marginal gain from devoting all income to consumption,
whereas l( y) is the marginal loss from doing so. Di¤erentiation of g and l shows that
(2.6), (2.7) and (2.8) imply that g(y) declines but l(y) increases with y.Further,
since from (2.6, vi) along with (2.8, vi and vii) we kno w that:
lim
y!0
g (y)=+1 lim
y!0
l(y)=K<+1
It follows that there exists
b
y>0 such that:
g(y) >l(y) 8y<
b
y
g(
b
y)=l(
b
y)
g(y) <l(y) 8y>
b
y
(2.11)
The result follows from (2.11) given (2.9). Note the crucial role here of the assump-
tion that q
0
> 0–without the en vironmental endowment the three ranges in (2.11)
may not exist. Given the endowment, when y<
b
y the household will not spend
on abatement because the net marginal utility of consumption, taking into account
environmental degradation, is greater than the gain from abatement spending. This
dictates the corner solution in which environmental quality must fall with income.
8
2.2.2. Rising Consumption and Abatement at Higher Incomes
Here we will show that under assumptions (2.6), (2.7) and (2.8), for all y>
b
y:
0 <c
¤
(y ) <y 0 <
dc
¤
dy
< 10<e
¤
(y) <y 0 <
de
¤
dy
< 1
By totally di¤erentiating (2.9) and rearranging terms, we can see that:
dc
¤
dy
=
!
c
!
e
+ !
c
where:
!
c
= U
q
(q
ee
¡q
ce
) ¡U
cq
q
e
¡U
qq
q
e
(q
c
¡q
e
) < 0 (2.12)
!
e
= U
cc
+ U
cq
q
c
+ U
qc
(q
c
¡q
e
)+U
qq
q
c
(q
c
¡q
e
)+U
q
(q
cc
¡q
ce
) < 0
Hence we can immediately see that:
0 <
dc
¤
dy
< 1
0 <
de
¤
dy
=(1¡
dc
¤
dy
)=
!
e
!
e
+ !
c
< 1
These expressions indicate that, with rising income, eventually the household will
want to spend on both consumption and environment. Further, we can see that
the expenditures on each will rise with income. The question, then, is whether
the simultaneous increases in pollution-causing consumption and pollution-reducing
abatement expenditures will permit env ironmental quality to rise with income.
9
2.2.3. Falling Then Rising Environmental Quality
That the abatement expenditures will rise with income once y>
b
y does not by itself
guarantee that en vironmental qualit y will rise with income bey ond the threshold
b
y. Because consumption is rising as well, the increase in e has to be large enough
to o¤set the additional pollution caused by increased consumption. Under what
combinations of preferences and abatement technologies is that likely to occur?
Note that the assumptions we ha ve made thus far are not su¢cient to ensure
en vironmental quality rising with income. To see that this is the case, by way of
contrast consider …rst the familiar case from basic consumer theory, in which the
marginal rate of transformation (MRT) the consumer faces—i.e., the rate at which
the consumer is able to exchange one marketed commodity for another—is …xed by
exogenously given market prices and hence is independen t of the consumer’s income.
In that case simple restrictions on preferences, e.g. of the sort w e have imposed, do
su¢ce to guarantee that the demand for these marketed commoditiesisnormal.
We require further assumptions because environment is a non-marketed com-
modity. This implies that the relative shado w price of environmental quality, i.e.
the MRT along the c-q consumption possibilit y frontier, will generally (though not
always) depend on the household’s income. Whether non-marketed environmental
quality falls or rises with income will, therefore, depend not just on preferences, i.e.
how the marginal rate of substitution (MRS) of c for q changes, but instead on how
boththeMRSandtheMRTchangeaswemovebetweenoptimaasincomerises.
TheassumptionswehavemadesofarpindownthechangesintheMRSbothalong
an indi¤erence curve and in moving between indi¤erence curves within a shift to
a new optimum. They also pin down the c hange in MRT along a given consump-
10
tion possibilit y frontier. The proposition below determines what we need to assume
in addition, in order for EKCs to arise, about the change in the MRT in moving
from one consumption possibility fron tier to another within shifts to new optima,
conditional on and speci…cally relative to the change in the MRS.
Proposition 2.1.
Let:
MRS(c; q) ´
U
c
U
q
MRT(c; q) ´ q
e
¡q
c
If assumptions (2.6), (2.7) and (2.8) hold and there exists
e
y such that for all y>
e
y :
@M RS(c
¤
(y );q
¤
(y))
@c
¯
¯
¯
¯
q=q
¤
¡
@M RT(c
¤
(y );q
¤
(y ))
@c
¯
¯
¯
¯
q=q
¤
< 0 (2.13)
then:
dq
¤
dy
< 0 for all y<
b
y where
b
y is implicitly de…ned by g(
b
y)=l(
b
y)
dq
¤
dy
> 0 for all y>maxf
b
y;
e
yg
Proof: That en vironment decreases with rising income until
b
y ,givenq
0
> 0,follows
from Section 2.2.1. To see that adding (2.13) is su¢cient for there to exist an
income level beyond which en vironmental quality increases with income, note that
for income above
b
y, when the non-negativity constraint on e is no longer binding:
dq(c
¤
(y );e
¤
(y ))
dy
= q
c
dc
¤
dy
+ q
e
de
¤
dy
= q
c
!
c
!
e
+ !
c
+ q
e
!
e
!
e
+ !
c
=
q
c
!
c
+ q
e
!
e
!
e
+ !
c
11
where !
e
and !
c
arede…nedasin(2.12). Since(!
e
+ !
c
) < 0; we have that
dq
¤
dy
> 0
if and only if (q
e
!
e
+q
c
!
c
) < 0. Substituting (2.9) and (2.12) above and rearranging:
q
e
!
e
+ q
c
!
c
= U
q
[(
U
cc
U
q
¡
U
qc
U
c
U
2
q
)+
1
q
e
(q
e
(q
cc
¡q
ce
)+q
c
(q
ee
¡q
ce
))]
But w e also know that:
@M RS(c
¤
(y );q
¤
(y ))
@c
¯
¯
¯
¯
q=q
¤
=
U
cc
U
q
¡
U
qc
U
c
U
2
q
@M RT(c
¤
(y );q
¤
(y ))
@c
¯
¯
¯
¯
q=q
¤
=
1
q
e
(q
e
(q
ce
¡q
cc
)+q
c
(q
ce
¡q
ee
))
and thus we can see directly that:
q
e
!
e
+ q
c
!
c
= U
q
[
@M RS(c
¤
(y );q
¤
(y ))
@c
¯
¯
¯
¯
q=q
¤
¡
@M RT(c
¤
(y );q
¤
(y))
@c
¯
¯
¯
¯
q=q
¤
]
Clearly then,
dq
¤
dy
> 0 if and only if [
@M RS(c
¤
(y);q
¤
(y))
@c
¡
@M RT(c
¤
(y);q
¤
(y))
@c
] < 0,asthe
su¢cient condition in the proposition suggests. If
e
y<
b
y,then
dq
¤
dy
> 0 from the
moment that abatement expenditures are positive. If on the other hand
e
y>
b
y,then
even after households start to spend on abatement, en vironmen t al quality ma y fall
with rising income, although only up to the threshold level of income
e
y.Beyond
that income level, en vironmental quality will improve with increases in income.
This result completes the intuition for su¢ciency of an asymmetric endo wment
for an EKC (since Section 2.2.1 showed falling environmen t at low incomes, i.e. the
…rst part of an EKC). In light of (2.13), see that an endo w ment yields a falling MRS
as the scale of income and consumption rises, because with convex preferences the
marginal gain from consumption falls as consumption rises. Thus, even were the
12
MRT not to change with scale, given an endowment the conditions would exist for
rising q
¤
(y ) once y is high enough, i.e. for the second part of the EKC. In fact, for
a wide set of technologies the endo wm ent will be su¢cient for an EKC .
This result also permits the direct evaluation of whether a particular combina-
tion of preferences and abatement technologies can be expected to generate an EKC.
Constant returns (unchanging MRT) leaves matters to the preferences, such that an
asymmetric endowment yields an EK C . Increasing returns to abatement spending
(e.g. q
ee
> 0, q
cc
= q
ce
=0) should help the second part of the EKC, i.e. rising en-
vironment, because raising q through abatement is easier as scale rises with income.
In light of (2.13), note that this makes the change in MRT as scale rises positive.
Th us, as per Proposition 2.1 even if the MRS were unchanged with scale ev entually
en vironmental quality would rise with income, i.e. increasing returns abatement
technologies do help generate the second part of the EKC, rising environment.
As noted earlier, though, without an asymmetric endowment we lack an explicit
story for why envir onment falls in the low income range, i.e. for the …rst part of
an EKC. Thus, despite its role in raising environmental quality at higher incomes,
increasing returns shifting the abatemen t MRT is not su¢cient for an EKC.
3. Robustness and Su¢ciency
We now work through several illustrati ve examples in some detail, for two purposes:
…rst, to demonstrate that an environmental endo wmen t is su¢cient for an EKC
under a broad range of abatemen t technologies; and second, to show that even in-
creasing returns to abatement is not su¢cient, as without externalities an additional
explicit story is necessary for why envir onmental quality falls with income.
13
3.1. The Su¢ciency of Asymmetric Endowm ents
3.1.1. Constant Returns to Abatement
For a …rst simple but in many ways quite representative general example, we assume
Cobb-Douglas preferences for consumption and environmental quality:
U(c; q)=c
®
q
¯
® + ¯ =1 (3.1)
We assume an asymmetric endowmen t q
0
> 0, i.e. positive en vironmental quality
but zero consumption at zero income. This is a natural assumption (again, below we
argue that it is hard to see when it is not reasonable, for people who are able to sta y
alive and thus face this optimization problem). For simplicity and transparency, we
specify in (3.2) a class of simple constant-returns abatement functions
8
:
q = q
0
¡°c + ±e °; ± > 0 (3.2)
Given this expression for q, the household chooses c and e to maximize (3.1) subject
to the budget constraint (2.4) and the non-negativity constraints (2.5). This gives
rise to a non-linear programming problem, the …rst-order Kuhn Tuc ker conditions
of which lead one to consider the follow ing two cases: 1) c>0;e =0;and2)
c>0;e>0.
9
The …rst case corresponds to a corner solution in which the household
8
Note that the input-substitution technology in Pfa¤, Chaudhuri and Nye 2002 is constant
returns. As that paper argues, there are ma ny ca ses in w hi ch inp ut subs tituti on is the relevant
abateme nt techn o logy. Note also , inc luding as mo t ivation f or Section 3.1. 3 , th at the existence of a
cleanest in put may imply th a t at highe st inc omes e nvironme ntal quality will again fall with in c ome .
At the highest incom es only the cl ean e st input is used, and its us e rises with income.
9
In all, the K uhn-Tucker conditions allow for four cases: 1)
c>0;e =0
;2)
c>0;e > 0
;3)
c = e =0
;and4)
c =0;e > 0
. Given ou r assumptio n on prefere nces (2.7 , vi), as l ong as
y>0
the
non-negativity constraint on
c
will never b e binding, ruling out 3) and 4).
14
ch ooses not to abate, but does consume, and thus en vironment falls with income.
The e
¤
=0result is optimal for poorer households, i.e. those satisfying:
y ·
q
0
®p
c
p
e
°p
e
+ ¯±p
c
(3.3)
For a household in this income range, the optimal level of consumption will rise with
income (so that pollution will rise with income as well). Since nothing is spent on
abatement, the optimal level of environmental quality must fall with income:
c
¤
=
y
p
c
; e
¤
=0;q
¤
= q
0
¡
°y
p
c
;
dq
¤
dy
= ¡
°
p
c
< 0 (3.4)
While abatement is feasible, at lo w incomes it is not desirable. The household
devotes all of its resources to consumption (expenditure on consumption, p
c
c
¤
, equals
y). If there were no environmental endowment (q
0
=0), though, there would be
no income range in which abatement is zero. It is the asymmetric endowment that
leads to the boundary solution in which environmental quality falls with income.
The case where e
¤
> 0 is optimal for richer households, those satisfying:
y>
q
0
®p
c
p
e
°p
e
+ ¯±p
c
(3.5)
Under the linear technology in (3.2), the MRT faced by the household does not vary
with income. From Proposition 2.1, we know then that all that matters is whether
the MRS falls with increases in income (and consumption). But with Cobb-Douglas
preferences, which ensure that q is a normal good, this is guaranteed. Hence, ev en
though consumption (and pollution) will rise with income, the household spends
15
enough on abatement to ensure that environmen tal quality also increases:
c
¤
=
y®± + q
0
®p
e
±p
c
+ °p
e
; e
¤
=
y( °p
e
+ ¯±p
c
) ¡ q
0
®p
c
p
e
p
e
(°p
e
+ ±p
c
)
;
dq
¤
dy
=
±¯
p
e
> 0 (3.6)
The derivative of optimal environmental qualit y with respect to income in these
results conveys that the weigh t on the environment within the preferences matters.
These results (see 3.5) also con…rm that the asymmetric endowm ent is crucial. Were
q
0
=0(i.e., the standard, zero-endowmen ts case in which normal goods are de…ned),
the solution in (3.6) would always obtain. Thus, as normal goods, both consumption
and environmen tal qualit y would increase with income at all income levels.
3.1.2. Decreasing Returns
Since increasing returns to abatement spending was seen abo ve (see discussion of
Proposition 2.1) to support the second part of an EKC (en vironmental quality rising
with income at higher incomes), and since constant returns to abatement lea ves
things to the preferences, it is worth considering whether decreasing returns to
abatement prevents an env ironmental endowm ent from leading to an EKC. With
the preferences in (3.1), we know from Section 2.2.1 that the endowment will be
su¢cien t for the fall in environmental quality within the low income range. Th us, the
question is whether, with an endowment but also decreasing returns to abatement,
the quality of the en vironment can still rise with income at higher incomes.
Demonstrating the utility of Proposition 2.1, w e can simply check whether a
particular combination of preferences and an abatement technology satisfy the con-
ditions provided there for environmental quality rising with income, once income is
16
above a given level. Consider, then, (3.1)’s preferences and (3.7)’s technology:
q = q
0
+(1¡ exp[°c]) + (1 ¡ exp[¡±e]) °>±>0 (3.7)
For these speci…cs, all of (2.6), (2.7) and (2.8) hold. In terms of (2.13), w e ha ve:
MRS(c; q) ´
®
(1 ¡ ®)
q
c
MRT(c; q) ´ ± exp[¡±e]+° exp[°c]
@M RS
@c
¡
@M RT
@c
=
¡®
(1 ¡ ®)
q
c
2
¡° exp[°c](° ¡±) < 0 (3.8)
Th u s, given (3.1), the asymmetric environmen t al endo w ment remains su¢cient for
an EKC ev en for the decreasing returns to abatemen t tec hnologies in (3.7).
3.1.3. Extreme Decreasing Returns to Abatement
Consider again the constant-returns abatement function (3.2), except now add an
extreme diminishing returns componen t, such that actual abatement, denoted a,
rises with abatement expenditures e only up to e
max
. After that poin t, actual
abatement a equals e
max
no matter how high the abatement exp enditures e:
a(e)=
8
>
>
>
<
>
>
>
:
e;
e
max
;
if e · e
max
if e>e
max
(3.9)
Going from a level of abatement spending that is below e
max
to one abov e it, the
marginal abatement per unit of spending decreases discretely from 1 to 0.
The household then maximizes (3.1) subject to (2.4) and (2.5), given the tech-
17
nology q = q
0
¡ c + a(e), where a(e) is as de…ned in (3.9) (and the ° and ± from
(3.2) are dropped to avoid unnecessary clutter). The optimization problem yields
threeactivecases:1)c>0;e=0;2)c>0; 0 <e<e
max
;and3)c>0;e= e
max
.
The …rst two cases are essentially identical to the two cases in Section 3.1.1, with
households in the low income range (as in (3.3)) spending nothing on environmental
in vestment and lowering the quality of the environment as income rises. Those with
higher incomes (as in (3.5), though in this case also bounded above b y the expression
in (3.10)) spend on both consumption and environmental investment, and impro ve
the quality of the environment as income rises. Thus, the basic EKC result from
Section 3.1.1 is seen to hold with this decreasing returns abatemen t tech nology.
The new feature is case 3), which is optimal for the richest households:
y ¸
e
max
p
e
(p
e
+ p
c
)+q
0
®p
c
p
e
p
e
+ ¯p
c
(3.10)
Although environmental quality is still normal, households cease investing in the en-
vironment through abatement spending because the marginal abatement from envi-
ronmental investment is zero after e exceeds e
max
. However, consumption continues
to increase with income, such that pollution increases and environmental quality
must fall with income, as seen in the following optimal values for this income range:
c
¤
=
y ¡ p
e
e
max
p
c
; e
¤
= e
max
; q
¤
= q
0
¡
y
p
c
+
(p
c
+ p
e
)
p
c
e
max
;
dq
¤
dy
= ¡
1
p
c
< 0 (3.11)
Thus with decreasing returns to abatement, both poor and rich households can
arrive at corner solutions where environmen tal quality falls with income because of
a lack of additional abatemen t e¤ort to o¤set rising consumption. The relationship
18
between income and environmental quality can then become an “inverted N” or
“sideways S”, as quality decreases, increases, and then decreases again with income.
This is an interesting result at the least because of related …ndings in the empir-
ical literature on EK Cs, where some …tted aggregate relationships take this shape.
10
Also, such an empirical …nding might even be expected, given a …nite set of feasible
abatement technologies to choose from (as opposed to a technology within which
one can in vest continuously in abatement without limit), such that the rich, upon
using only the “cleanest” technology, ma y not have further scope for abatement.
11
3.2. The Insu¢ciency of Increasing Returns
3.2.1. Fixed Costs of Abatemen t
Now we modify (3.2) again, but instead of facing decreasing productivit y of abate-
men t spending on the margin as in (3.9) now a household can c hoose from two types
of en vironmental investment : e
1
, with no …xed cost but a relatively high marginal
cost p
1
;ande
2
, with a …xed cost, f>0, but a relativ ely lower marginal cost p
2
.
12
Together, these abatement choices
¡!
e =(e
1
;e
2
) form the simple increasing returns
abatement technology in (3.12), the last part of a q = q
0
¡c + a(e) technology:
a(
¡!
e )=e
1
+ e
2
(3.12)
10
See, for e xample, G rossma n and Krueg er 1995 (pag e 361, Figu res 1,3 a nd 4, an d pag e 369),
Torras and B oyce 1998 (pages 152-3 , 157) and Hill and Magn an i 2001 (Table 1).
11
See Pfa¤, Chaudhuri an d Nye 2 002 for a formalized theo retical resul t. One exam p le they
mention is switching amon g a …n ite set of f uel s in order to s hift the cons umpt ion - air quality MRT.
Note also the discussi on in Jones and Manuelli 1995 an d Torras and Boyce 1998.
12
Fixed costs may well e x ist. Further, lower-marginal -cost options m ay have higher … xed costs.
Andreon i and Levin son 2001 provides usef u l evidence that abatement t echnologies with higher …xed
costs may have lower m argina l co sts. They cit e EPA studies of the emission control from large coal-
…red burners, and they also regress pollution abatement op erating costs by industry and by U.S.
state on a measure of the size of the industry’s contribution to gross state product.
19
where the household is faced with the piecewise de…ned budget constraint,
y =
8
>
>
>
<
>
>
>
:
p
c
c + p
1
e
1
p
c
c + p
1
e
1
+ p
2
e
2
+ f
if e
2
=0
if e
2
> 0
(3.13)
where p
2
<p
1
. The household is also faced with the non-negativity constraints:
c ¸ 0;e
1
¸ 0;e
2
¸ 0 (3.14)
and picks c and
¡!
e to maximize (3.1) subject to (3.13), (3.14) and (3.15):
q = q
0
¡c + e
1
+ e
2
(3.15)
Assuming that the …rst t ype of abatement investment (i.e., e
1
) is not dominated
13
,
the optimization problem leads one to consider three cases
14
:
1) c>0;e
1
= e
2
=0;2)c>0;e
1
> 0;e
2
=0;3)c>0;e
1
=0;e
2
> 0 (3.16)
13
Speci…cally, this is the assumption that:
f>
q
0
®p
2
c
¯(p
1
¡ p
2
)
p
1
+ ¯p
c
14
Conside r two n on-linear programmi ng proble ms, o ne for
e
2
=0
and one for
e
2
> 0
.The
e
2
=0
prob lem yields four cases: 1)
c>0;e
1
= e
2
=0
;2)
c>0;e
1
> 0;e
2
=0
;3)
c = e
1
= e
2
=0
;4)
c =0;e
1
> 0;e
2
=0
. However, g iven (2 .7, vi), such that when
y>0
the non-negativity constraint
on
c
will not be binding, 3) and 4) are ruled out. The
e
2
> 0
problem also y iel ds four case s: 1)
c = e
1
=0;e
2
> 0
;2)
c =0;e
1
> 0;e
2
> 0
;3)
c>0;e
1
=0;e
2
> 0
;4)
c>0;e
1
> 0;e
2
> 0
.As
above, 1) and 2) are ruled out by (2.7, vi) wh en
y>0
. Also, it is easil y shown t hat onc e
e
2
> 0
,
i.e. if th e … x ed cost has been incu rred , given
p
1
>p
2
case 4) is ruled out. Fro m both problem s
together, then, we are left with the th ree cases considered in the text.
20
The c>0;e
1
= e
2
=0result is optimal for the poorest households:
y ·
q
0
®p
c
p
1
p
1
+ ¯p
c
(3.17)
For these households, the optimal v alues c
¤
and q
¤
arelikethoseforthepoorer
households in Section 3.1.1, and so
dq
¤
dy
here is equal to ¡
1
p
c
< 0. Th us, this is again
an income range in which environmen tal quality falls with increasing income (and,
as above, this is an income range which does not exist if q
0
=0).
The c>0;e
1
> 0;e
2
=0result is optimal for middle incomes
15
:
q
0
®p
c
p
1
p
1
+ ¯p
c
<y·
q
0
®p
c
p
2
p
2
+ ¯p
c
+ f (3.18)
The optimal values c
¤
, q
¤
,ande
¤
1
for households in this income range are like those
for the richer households in Section 3.1.1 (substituting e
1
for e and p
1
for p
e
).Thus,
much as in that setting,
dq
¤
dy
=
¯
p
1
> 0, i.e. environmental quality rises with income.
Lastly, the c>0;e
1
=0;e
2
> 0 result is optimal for the richest households:
y>
q
0
®p
c
p
2
p
2
+ ¯p
c
+ f (3.19)
This is mu ch like just above (but no w substitute e
2
and p
2
for e and p
e
in Section
3.1.1). Thus,
dq
¤
dy
=
¯
p
2
> 0, and en vironmental quality rises with income. While
q rises in both the middle and the highest income ranges, because p
2
<p
1
the
15
The assumpti on of th e con di tion under which
e
1
is not domi na ted, speci…ed earl ier, ensures
that this in c om e range exists, i.e. that:
q
0
®p
c
p
1
p
1
+ ¯p
c
<
q
0
®p
c
p
2
p
2
+ ¯p
c
+ f
21
derivative of environmental qualit y with respect to income is greater for the higher
income range. Note, then, that the transition between en vironmental inv estmen ts,
which raises the …xed costs but lowers the marginal cost of abatement, discretely
increases the rate at which environmental quality rises with income.
In any case, these results further demonstrate the robustness of the endowment-
based EKC result, for an increasing returns technology. More importantly, though,
they sho w the insu¢ciency of the increasing returns abatement technology on its
own. If q
0
=0, the income range in (3.17) vanishes, and environmental quality
always rises with income, as the middle income range in (3.18) becomes simply
y<f, and the high income range in (3.19) becomes y ¸ f .As
dq
¤
dy
> 0 for
both ranges, we can see that without the asymmetric endowment the quality of the
en vironment will rise with income for all incomes, i.e. there will not be an EKC .
To consider the validity of the asymmetric endo wment, note the results when
even e
1
has a …xed cost, but there is no endowment. If a household is rich enough
(given this …xed cost) to both consume and abate, then outcomes are as just de-
scribed: the income range in (3.17) vanishes and en vironmen tal quality rises with
income. However, until that point, the household neither abates nor consumes.
Th us, a starving household will choose not to consume because of the implications
for the environmen t. In our m inds this is so generally unrealistic, when thinking
of actual lo w-income households, as to lead us to seek the source of the lack of
relevance of the result. Our conclusion is that households would be dead at q =0.
Thus, the almost-starving household (low c due to low y)inwhichpeoplecanstay
alive and consume (giv en that many die of, e.g., lack of potable water) clearly has
an endowment of environmental quality, e.g. water to drink and air to breathe.
22
3.2.2. ‘Explosive’ Increasing Returns to Abatement
Andreoni and Levinson 2001 posit a particular increasing returns abatement technol-
ogy which depe nds upon consumption directly. Their speci…cation of the technology
q = q
0
¡c + a assumes q
0
=0and an a(c; e) where a
c
> 0;a
e
> 0,anda is homoge-
nous of degree k where k>1. We call this ‘explosive’ increasing returns because
as the scale of income and consumption rise, the returns to abatement investments
in e increase ad in…nitum. Their motivating example, however, is small-scale: a
broom can for the same level of e¤ort accomplish more abatement when sweeping
up a quarter inch of dust, e.g., than when sweeping up an eigh t h of an inc h . It may
not be appropriate to generalize from this small scale to unlimited scale.
16
The point here is that this technological assumption cannot by itself generate an
EKC. It implies that as income and c rise, marginal productivity of e also rises. A
given investment in e yields more a. That supports the upward-sloping part of an
EKC, as per Proposition 2.1, but does not substitute for the asymmetric endowment
in explaining (as in Section 2.2.1) the downward-sloping part of an EK C.
Consider a(c; e)=ce.Herea
c
> 0;a
e
> 0,anda is homogenous of degree k
where k>1. The household’s problem is to pick c and e to maximize (3.1) subject
to (2.4), (2.5) and, of course, this speci…cation of a(c; e) and thus also of q(c; e).As
in some of the problems above, the cases to consider are: 1) c>0;e =0;and2)
16
As in the bro om exam ple, it may of ten be the case that rising
c
increases
da
de
near
c =0
:no
matter how hard you try (
e>0
), vacuumin g a c lean (
c =0
) rug accomplishes nothing (
a =0
).
However, often a cap aci ty constraint (g iven
e
) may arise well within the relevant scale of
c
.Consider
a single broom, thought of as a single unit of
e
spend ing. A sweep of a clean ‡oor a ccom plishes
nothing, wh il e a sweep of a ‡oor with a half-inch of di rt accomplishes more t han a sweep o f a ‡oor
with a quarter-inch. But then consider a ‡o or with two feet of dirt, a scale likely to be beyond the
capacity of a single sweep of the broom. At four inches per sweep, e.g., simple division suggests
that it wil l t ake six sweeps to e liminate the dir t . But simple division is pr e c isely a state ment of
capacity an d , by impl ica tion , consta nt return s over large scales. T hus, for a scal e of
c
well beyond
the capacity of the
e
in question, a ba tement will b ecom e e¤ectively constant returns to sc ale.
23
c>0;e>0.Thee
¤
=0result is optimal for poorer households, satisfying:
y ·
p
p
2
e
+4¯q
0
®p
c
p
e
¡p
e
2¯
(3.20)
The key point here can already be made, with reference to this expression: with
no environmen tal endowment (q
0
=0), this income range in whic h environmen-
tal quality will fall with income (as in (3.4) and (3.11)) simply vanishes. Since
elsewhere environmen tal quality rises with income (as discussed above, increasing
returns mak es this more likely), lackin g an endowment this technology does not
generate an EK C. Formally, the e
¤
> 0 case is optimal for richer households:
y>
p
p
2
e
+4¯q
0
®p
c
p
e
¡p
e
2¯
(3.21)
so that if en vironmental quality is in fact rising with income within this range, then
for the q
0
=0case it will alw ays rise with income. And in fact
17
:
e
¤
=
2yp
e
¯ + p
e
y + p
2
e
¡
p
p
2
e
(y ¡ p
e
)
2
+ q
0
®p
3
e
p
c
[¯ + p
2
e
]
2(2 ¡®)p
2
e
dq
¤
dy
=
p
©+D
£
4yp
e
(2 ¡®)¯ +2yp
e
¯ + yp
e
+ p
2
e
¤
+ p
2
e
¯(y ¡ 2p
e
)
2
+ D
2(2 ¡ ®)p
c
p
©+D
> 0
where ©=p
2
e
(y ¡p
e
)
2
and D = q
0
®p
3
e
p
c
[¯ +p
2
e
] are used to simplify. Without an en-
vironmental endowment, ev en this ‘explosive’ increasing returns to scale technology
explains only environmental quality increasing with income, not an EKC.
17
This is one of two roots of a quadratic equation. It is the one in which a higher environmental
endowment implies lower op tim al abatement expendi tures (as m akes intuitive sense, given the e¤ ect
of “free” environment on the MRS and given our previous results, e.g. in Sect ion 3.1.1).
24
4. Conclusion
Using a household-choice framework, this paper provided a simple explanation for
non-monotonic, upward-turning paths of environmental quality during economic
growth . The very natural assumption of an asymmetric endo wmen t (i.e., positive
en vironmental qualit y but zero consumption at zero income) is su¢cien t. The intu-
ition is that, given the endowmen t, the MRS between consumption and environment
at lo w incomes mak es abatement undesirable. As income and consumption increase,
though, and the endowment is degraded by consumption, this corner solution gives
way to interior solutions in which both consumption and abatement expenditures
rise with income. We provide su¢cient conditions, involving the details of both
preferences and abatement technologies, that ensure for this interior solution that
en vironmental quality also rises with income, i.e. that the abatement increase is
large enough to o¤set the e¤ect of the increased consumption on pollution.
This endowment-based result is robust to a wide range of abatement technologies,
including …xed costs of and decreasing returns to abatement. Our decreasing-returns
case stimulates further empirical examination (national level or more disaggregate)
of the growth-environment relationship for results other than “U shapes”. We also
sho w that even relatively extreme abatement technologies do not on their own gen-
erate such “EK C -like” paths of environmental quality. The reason is that they do
not generate an income range in which environmental qualit y falls with income.
This work suggests more formalized analysis of microfoundations of national-
level EKCs, i.e. adding formal aggregation of heterogeneous household preferences
(given relevant abatement technologies) to the literature on dynamic optimization
by national social planners. Our household approach could be applied in a setting
25
of externalities and mu ltiple agents whose voting for taxation and environmental
spending evolves with income if they care about environmental outcomes (as as-
sumed here, and see micro-level empirical evidence in Chaudhuri and Pfa¤ 1998).
We plan to pursue this application in future research.
26
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