This article examines the conditions under which endogenous fluctuations and periodic welfare inequality can emerge in OLG economies having an environmental dimension.Endogenous fluctuations; Environmental quality; Intergenerational equity

Alban Verchere

Thomas Seegmuller

English

Research Papers in Economics

Pollution as a source of endogenous ￿uctuations
and periodic welfare inequality in OLG economies
Thomas Seegmuller∗ and Alban Verchere† ‡
June 18, 2002
Abstract
This article examines the conditions under which endogenous ￿uc-
tuations and periodic welfare inequality can emerge in OLG economies
having an environmental dimension.
RØsumØ : cet article examine les conditions sous lesquelles des ￿uc-
tuations endogŁnes et une inØquitØ pØriodique du bien-Œtre peuvent
Ømerger dans des Øconomies ￿ GI prenant en compte une dimension
environnementale.
Keywords: Endogenous ￿uctuations; Environmental quality; Intergenera-
tional equity
JEL classi￿cation: E32, Q20
1 Introduction
Since Howarth and Norgaard (1992) one knows that in OLG economies, the
risk that present generations sacri￿ce the welfare of future ones is high, given
that the former are engaged in polluting activities which e￿ects go on for
a long time after their death. Moreover as it was shown the integration of
an e￿cient tax scheme and intergenerational transfers would not guaran-
tee a sustained development for future generations because of the recurring
problem of the social discount rate.
Within a framework very close to that of John and Pecchenino (1994),
we show that another type of intergenerational inequity can appear, however
∗BETA-Theme, Louis Pasteur University, 61, avenue de la ForŒt-Noire 67085
Strasbourg Cedex France, Tel: (33)-3-90-24-20-98, Fax: (33)-3-90-24-20-71, e-mail:
seegmu@cournot.u-strasbg.fr.
†Corresponding author: BETA-Theme, Louis Pasteur University, 61, avenue de la
ForŒt-Noire 67085 Strasbourg Cedex France, Tel: (33)-3-90-24-20-98, Fax: (33)-3-90-24-
20-71, e-mail: verchere@cournot.u-strasbg.fr.
‡We want to thank Isabelle Maret and ThØophile Azomahou for helpful comments. All
remaining errors are own.
1it is not possible to hold present generations responsible for it. Indeed, in
John and Pecchenino (1994) it is shown that di￿erent dynamics of capital
accumulation and environmental preservation can emerge when generations
contribute to an environmental maintenance sector, and considering increas-
ing external returns or not. In the latter case, they highlight that there exists
a path of monotonous convergence up to the steady state depending on the
initial conditions and the length of which one observes, either a continuous
growth of capital and environmental quality or conversely a fall of them up
to the steady state. If dynamics would emphasize two non autarkic steady
states ￿ one stable, one unstable ￿ then according to the initial conditions
there also exists a monotonous convergence either to the autarkic equilibrium
or to the higher steady state. Within a framework very close to this one, we
demonstrate that endogenous ￿uctuations can emerge and that a new form
of intergenerational inequity, quali￿ed as cyclic or periodic, can appear.
The article is organized as follows. In section 2 we present the model and
the equilibrium conditions whereas in section 3 we study the local dynamics.
Section 4 establishes the welfare analysis and concludes the paper.
2 The model
We consider a perfectly competitive overlapping generations economy with
discrete time t = 0,1,...,∞ and perfect foresight. At each date a genera-
tion of consumers appears and lives two periods. The population of each
generation is constant and normalized to one. When young, each consumer
supplies one unit of labor inelastically and shares his income, wt, between the
purchase of productive capital kt+1 and the abatement of pollution emissions
dt. When old, he rents his capital to ￿rms, which is remunerated at the real
interest factor rt+1,1 and consumes the ￿nal good. We further consider that
consumers derive utility only at the second period of their life, from con-
sumption Ct+1 and environmental quality. More precisely we assume that
this latter decreases with the pollution stock Spt+1 which is determined at
end of period t as follows:
Spt+1 = (1 − m)Spt + akt − bdt (1)
where m ∈ (0,1) is the natural rate of pollution absorption, a > 0 the
emission rate of pollution by unit of capital and b > 0 the coe￿cient of the
linear abatement technology. Finally, Spt is the stock of pollution inherited
from the previous period.
Each consumer maximizes his utility under his two budget constraints
and the equation of pollution stock accumulation (1). The program is:
1We assume that capital totally depreciates after one period.
2max Ct+1 −
Sp2
t+1
2B
s.t.



kt+1 + dt = wt
Ct+1 = rt+1kt+1
Spt+1 = (1 − m)Spt + akt − bdt
(2)
where B > 0 is a scaling parameter.
We deduce the following ￿rst order condition:
rt+1 =
b
B
Spt+1 (3)
We assume that the ￿nal good is produced by a representative compet-
itive ￿rm with a constant returns to scale technology yt = f(kt)lt, where
f(kt) is the intensive production function and lt the labor. We further as-
sume that f(kt) is a continuous function and has continuous derivatives of
all required orders for kt > 0, with f0(kt) > 0, f00(kt) < 0 and f(1) = 1.
Firm maximizes its pro￿ts and we obtain the two usual following conditions:
rt = f0(kt) ≡ r(kt) and wt = f(kt) − ktf0(kt) ≡ w(kt) (4)
Before determining the intertemporal equilibrium, it is useful to de-
￿ne the following relationships. First, we note the capital share of in-
come s(k) = f0(k)k/f(k) ∈ (0,1). Moreover we de￿ne the elasticities
of wage and interest rate with respect to k: εw(k) = w0(k)k/w(k) and
εr(k) = r0(k)k/r(k). Finally, the elasticity of capital-labor substitution σ(k)
is determined by:
1
σ(k)
=
dln(w(k)/r(k))
dlnk
= εw(k) − εr(k) (5)
From f(k) = kr(k) + w(k), we deduce that w0(k) = −kr0(k), and using this
last relation, we ￿nally obtain:
εw(k) =
s(k)
σ(k)
and εr(k) = −
1 − s(k)
σ(k)
(6)
We can now determine the intertemporal equilibrium. Substituting the
￿rst budget constraint and relation (3) into equation (1) and using (4), we
have2:
B
b
f0(kt+1) − bkt+1 = (1 − m)
B
b
f0(kt) + akt − b[f(kt) − ktf0(kt)] (7)
This equation is satis￿ed at each period and entirely rules the dynamic.
2We notice that labor market clears with lt = 1 since the labor supply is inelastic and
equal to 1.
33 Local dynamics
As we study local dynamics we ￿rst analyze the existence of a steady state.
From (7), we deduce that a steady state is a solution of the following equa-
tion:
B
b
mf0(k) = (a + b)k − b[f(k) − kf0(k)] (8)
Existence can be established by appropriately selecting the scaling parameter
B so as to get a normalized steady state k = 1. Indeed, there is a unique
solution B∗ to (8) such that k = 1:
B∗ =
a + bs
s
b
m
(9)
where s = f0(1) is the capital share of income evaluated at the steady state
k = 1.
In order to study local dynamics we di￿erentiate equation (7) around the
steady state k = 1:
dkt+1
dkt
=
(1 − m)(a + bs)1−s
σ + m
 
bs1−s
σ − a

(a + bs)1−s
σ + bm
(10)
In this model local endogenous ￿uctuations can only emerge through the
occurrence of a ￿ip bifurcation, i.e. if
dkt+1
dkt ≤ −1. This last condition is
equivalent to:
b(2s(1 − s) + mσ) ≤ a((m − 2)(1 − s) + mσ) (11)
This relation is satis￿ed only if σ > 2−m
m (1 − s). Under this condition,
inequality (11) can be rewritten:
a
b
≥
mσ + 2s(1 − s)
mσ − (2 − m)(1 − s)
≡
a
b

F
(12)
Proposition 1 If σ > 2−m
m (1−s), then a cycle of period two emerge through
the occurrence of a ￿ip bifurcation when a
b =
 a
b

F.
This proposition shows that endogenous cycles can occur if the emission
rate of pollution a is su￿ciently high regarding the coe￿cient of abatement
technology, b.3 The emergence of such ￿uctuations can be explained as fol-
lows: if the capital increases from its steady state value, the net pollution
emissions as well as the next period stock of pollution increase. From the ￿rst
order condition of the consumer program, it requires an higher expected in-
terest factor. Consequently, the next period capital stock will decrease which
3The emergence of endogenous ￿uctuations also requires that capital and labor are not
weak substitutes.
4is the source of endogenous cycles in this economy. Moreover, remark that  a
b

F decreases with the natural rate of pollution absorption m. This means
that a greater natural rate of pollution absorption promotes the appearance
of a cycle of period two, by facilitating the occurrence of a ￿ip bifurcation.
It is an interesting fact insofar that the greater the natural factor of assim-
ilation is, the more the risk to observe endogenous ￿uctuations of activity
is high. Indeed, when m is high, the pollution stock mainly depends on the
new net pollution emissions which reinforce the opposite e￿ects described
above.
4 Welfare analysis
Contrary to the case where the economy is at the steady state, all generations
do not have the same level of welfare along the cycle. We can further establish
that some generations have a greater utility than some others, revealing a
new kind of intergenerational inequity that we could qualify as "cyclic" or
"periodic".
Proposition 2 Assume that it exists a cycle of period two. It is described
by two values k0 and k1 such that k1 > 1 > k0 > 0. If σ(kt) > 1 − s(kt) for
all t, then U1 > U0, where Ui corresponds to the utility of a generation born
when kt = kj with i,j = {0,1} and i 6= j.
Proof. Using equations (2), (3) and (4), the utility of the generation who
arises at period t can be rewritten:
Ut = Ct+1 −
Sp2
t+1
2B
= f0(kt+1)kt+1 −
B
2b2f0(kt+1)2 ≡ V (kt+1) (13)
Hence, we have:
V 0(kt+1) = f0(kt+1)

1 −
1 − s(kt+1)
σ(kt+1)

−
B
b2f00(kt+1)f0(kt+1)
> 0, because σ(kt) > 1 − s(kt) for all t by assumption
(14)
This last inequality means that utility increases with next period capital
stock and this concludes the proof. 
So, all along the cycles generations with weak utility periodically alter-
nate with generations with a greater one. This is the main result of our quite
simple model. It puts in evidence that OLG economies with environmental
dimension could endogenously encounter a cyclical intergenerational inequity
with loser and winner generations. The former encounter both low capital
and high pollution stocks4 whereas the latter enjoy the preferable opposite
4Observe with (3) and (4) that Spt+1 =
B
b f
0(kt+1), and so Sp1 < Sp0 since k1 > k0.
5situation with a higher level of welfare. As we noticed in introduction, even
if it is an intergenerational inequity, one can not allot the responsibility for
it to any generation in particular. Nevertheless, such a characteristic added
to the risk that such a situation occurs, with then a periodic but permanent
inequality of welfare between some generations and some others, will have
to be seen by public authorities as an incentive to implement the means so
that it actually never appears.
References
Howarth, R., and R. Norgaard (1992): ￿Environmental Valuation un-
der Sustainable Development,￿ AEA Papers and Proceedings, American
Economic Review, 82(2), 473￿477.
John, A., and R. Pecchenino (1994): ￿An Overlapping Generation Model
of Growth and the Environment,￿ The Economic Journal, 104, 1393￿1410.
6

Pollution as a source of endogenous fluctuations and periodic welfare inequality in OLG economies.

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Pollution as a source of endogenous fluctuations and periodic welfare inequality in OLG economies.

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