We consider an overlapping generations model with environment and an elastic labor supply. In this framework, consumers have to choose between consumption, environmental quality, and leisure. We show the existence of both deterministic cycles and indeterminacy. In contrast to previous results, the emergence of endogenous fluctuations does not require a high emission rate of pollution.Environment ; labor supply ; overlapping generations ; indeterminacy ; endogenous cycles

Alban Verchère

Thomas Seegmuller

English

Research Papers in Economics

A NOTE ON INDETERMINACY IN OVERLAPPING
GENERATIONS ECONOMIES WITH ENVIRONMENT
AND ENDOGENOUS LABOR SUPPLY1
Thomas Seegmuller
CNRS and CES
106-112 Boulevard de l’H￿pital 75647 Paris Cedex 13, France
tel: (33) 1 44 07 81 99
fax: (33) 1 44 07 82 31
e-mail: seegmu@univ-paris1.fr
and
Alban VerchŁre
PREACTIS
Saint-Etienne University,
6, rue Basse des Rives 42023 Saint-Etienne Cedex 2, France
tel: (33) 4 77 42 19 07
fax: (33) 4 77 42 19 50
e-mail: alban.verchere@univ-st-etienne.fr
1We would like to thank an associate editor and two anonymous referees for helpful
comments and suggestions. This paper also bene￿ted from comments of participants
to CEPET workshop at Udine June 2004, XIXth EEA Annual Congress August 2004,
JournØe ￿ GØnØrations ImbriquØes at La Rochelle November 2004, and T2M Conference
at Lyon January 2005. All remaining errors are of our own responsibility.
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Author manuscript, published in "Macroeconomic Dynamics 11, 3 (2007) 423-429"
 DOI : 10.1017.S1365100506060093INDETERMINACY IN OLG WITH ENVIRONMENT
Corresponding author:
Thomas Seegmuller
CNRS and CES
106-112 Boulevard de l’H￿pital 75647 Paris Cedex 13, France
tel: (33) 1 44 07 81 99
fax: (33) 1 44 07 82 31
e-mail: seegmu@univ-paris1.fr
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7Abstract
We consider an overlapping generations model with environment and an
elastic labor supply. In this framework, consumers have to choose between
consumption, environmental quality, and leisure. We show the existence of
both deterministic cycles and indeterminacy. In contrast to previous results,
the emergence of endogenous ￿uctuations does not require a high emission
rate of pollution.
JEL classi￿cation: C62, E32, Q50.
Keywords: Environment, labor supply, overlapping generations, indetermi-
nacy, endogenous cycles.
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71 INTRODUCTION
The link between environment and economic dynamics has been studied in a
lot of contributions.1 In this literature, overlapping generations models have
been used in order to clearly analyze intergenerational problems (John and
Pecchenino (1994), John et al. (1995), Pezzey and Toman (2002)). However,
most of these papers deal with the question of sustainability. This is why
they are only interested in steady state analysis and monotonic convergence.
Recently, some papers have enlightened that more complex dynamics can
emerge when one considers the interaction between environment and eco-
nomic activity. Indeed, considering overlapping generations models where
consumers have to choose between consumption and abatement, Seegmuller
and VerchŁre (2004) and Zhang (1999) have shown the existence of deter-
ministic cycles and chaos.
However, in these contributions, labor supply is inelastic and indetermi-
nacy is ruled out. Nevertheless, elastic labor supply and the consumer choice
between leisure and consumption often play key roles on the emergence of
endogenous ￿uctuations.2 In this paper, we introduce an elastic labor supply
in an overlapping generations economy with environment. Then, consumers
have to choose between consumption, environmental quality, and leisure.
Moreover, they share their labor income between savings and abatement,
because pollution negatively a￿ects their preferences.
Studying local dynamics, we prove that a ￿ip bifurcation as well as local
indeterminacy can occur. It means that not only deterministic cycles can
appear, but also endogenous ￿uctuations due to self-ful￿lling expectations.
Moreover, these ￿uctuations occur under weaker conditions than previous
existing works. Indeed, contrary to several contributions, the existence of
indeterminacy and cycles needs neither a weak substitution between pro-
duction factors (Grandmont et al. (1998), Reichlin (1986)), nor increasing
returns or imperfect competition (see Benhabib and Farmer (1999) for a
survey). We can further notice that, in contrast to Seegmuller and VerchŁre
(2004) and Zhang (1999), the occurrence of such ￿uctuations does not require
a high level of the pollution emission rate. Consequently, this paper shows
more generally that non monotonic dynamics can easily occur in economies
where an environmental dimension is taking into account.
The paper is organized as follows. In the next section, we present the
model. In section 3, we establish the existence of a steady state. Finally,
in section 4, we study local dynamics and provide interpretations of our
￿ndings.
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72 THE MODEL
We consider a perfectly competitive overlapping generations model with dis-
crete time t = 1,2,...,∞ and a constant population size normalized to one.
A generation of consumers is born in each period and households live two
periods. When young, the representative consumer supplies labor lt, which
is remunerated at the real wage wt. He shares his wage earnings between
savings, through the purchase of aggregate capital kt, and environmental
maintenance dt.3 When old, he rents capital to the ￿rms, earns the real
interest factor rt+1 and consumes the ￿nal good ct+1.4 The two budget con-
straints faced by the consumer can be written:
kt + dt = wtlt (1)
ct+1 = rt+1kt (2)
In this economy, the environmental quality decreases with respect to the
pollution. At period t + 1, the level of pollution is given by:
Pt+1 = αkt−1 − dt ≡ P(kt−1,dt) , with α > 0 (3)
The pollution, which is always strictly positive, linearly increases with
respect to the capital stock inherited from the previous period and is a
decreasing function of the environmental maintenance.5
Consumers derive utility from consumption, leisure and environmental
quality. Assuming additively separable preferences, the utility function of
the representative household is given by:
U(ct+1/B) − vlt − P(kt−1,dt)1+φ/(1 + φ) (4)
where B > 0 and v > 0 are two scaling parameters, and φ > 0. We can notice
that as usually in dynamic macroeconomic models, the disutility of labor is
linear (see Hansen (1985)), while the disutility of pollution is increasing and
convex. Furthermore, we assume:
Assumption 1 The function U(x) is continuous for all x ≥ 0, Cn for x > 0
and n large enough, with U0(x) > 0, U00(x) ≤ 0 and U0(x) + xU00(x) > 0.
The representative consumer maximizes his utility function (4) under the
constraints (1), (2) and (3). We deduce the two following equations:
U0(ct+1/B)rt+1wt/B = v (5)
(αkt−1 − dt)φwt = v (6)
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7These two expressions de￿ne the consumer choice between leisure, envi-
ronmental maintenance and future consumption.
The ￿nal good is supplied by a representative ￿rm using a constant re-
turns to scale technology. The production is given by yt = f(at)lt, where
at = kt−1/lt denotes the capital-labor ratio and f(at) the intensive produc-
tion function. In what follows, we assume:
Assumption 2 The intensive production function f(a) is continuous for
a ≥ 0, positively valued and di￿erentiable as many times as needed for a > 0,
with f0(a) > 0 and f00(a) < 0.
The producers maximize their pro￿ts. Since the economy is perfectly
competitive, we obtain:
rt = f0(at) ≡ r(at) (7)
wt = f(at) − atf0(at) ≡ w(at) (8)
Substituting equations (1), (2), (7) and (8) into (5) and (6), we can de￿ne
an intertemporal equilibrium as follows. An intertemporal equilibrium is a
sequence (at,kt−1)t≥1 which satis￿es:
U0(r(at+1)kt/B)r(at+1)w(at)/B = v (9)
kt = w(at)
kt−1
at
− αkt−1 +

v
w(at)
1/φ
(10)
αkt−1 > w(at)kt−1/at − kt ≥ 0 (11)
where k0 > 0 is given.
Note that inequalities (11) ensure a strictly positive pollution and a pos-
itive environmental maintenance. Taking as given these conditions, equa-
tions (9) and (10) de￿ne a two-dimensional dynamic system with one pre-
determined variable, the capital. One can further notice that equation (10)
means in fact that kt = w(at)lt −dt at equilibrium, where dt = αkt−1 −Pt+1
and Pt+1 = (v/w(at))1/φ. Note that in the limit case without any pollu-
tion (α = 0), the trade-o￿ between investments in productive capital and
environmental maintenance disappears (dt = 0), and equation (10) becomes
kt = w(at)lt.
Before studying steady states, we de￿ne the following relationships. We
note s(a) ≡ r(a)a/f(a) ∈ (0,1) the capital share in income and further-
more, if σ(a) represents the elasticity of capital-labor substitution, 1/σ(a) =
dlnw(a)/dlna − dlnr(a)/dlna. Since w0(a) = −ar0(a), we obtain the two
following expressions:
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7w0(a)a/w(a) = s(a)/σ(a) and r0(a)a/r(a) = −(1 − s(a))/σ(a) (12)
3 EXISTENCE OF A STEADY STATE
A stationary solution of the dynamic system de￿ned by (9), (10) and (11) is
given by (a,k) such that:
U0(r(a)k/B)r(a)w(a)/B = v (13)
k

α + 1 −
w(a)
a

=

v
w(a)
1/φ
(14)
α + 1 > w(a)/a > 1 (15)
We can notice that in section 4 we are interested in ￿uctuations around a
steady state. Since environmental maintenance has to be positive at each pe-
riod, we assume a strictly positive environmental maintenance at the steady
state, i.e. w(a)/a > 1.
Following Cazzavillan et al. (1998), we ensure in what follows the ex-
istence of a normalized steady state (a,k) = (1,1) by choosing appropriate
values of the two scaling parameters B > 0 and v > 0.6 In order to do that,
we assume:
Assumption 3 α + 1 > w(1) > 1.
Under Assumptions 1-3, there exists a unique solution v∗ > 0 to the
equation:
v∗ = (α + 1 − w(1))φw(1) (16)
Assume that limx→0xU0(x) < v∗/w(1) < limx→+∞xU0(x). Then, under
Assumption 1 and taking v∗ as given, there is a unique B∗ which satis￿es:
U0(r(1)/B∗)r(1)/B∗ = v∗/w(1) (17)
Proposition 1 Assuming that limx→0xU0(x) < v∗/w(1) < limx→+∞xU0(x)
and Assumptions 1-3 are satis￿ed, then (a,k) = (1,1) is a steady state of the
dynamic system (9)-(10) if and only if v∗ and B∗ are the unique solutions
of (16) and (17).
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74 LOCAL DYNAMICS
In this section, we analyze the occurrence of local indeterminacy and endoge-
nous cycles. Furthermore, interpreting our results, we put in evidence that
the consumer choice between leisure, environmental maintenance and future
consumption has a key role on the occurrence of endogenous ￿uctuations.
In order to do that, we study local dynamics in the neighborhood of the
steady state (a,k) = (1,1). If we note s = s(1), we assume:
Assumption 4 s < 1/2 and U00(x) = 0.
It means that the capital share in income is smaller than one half, which is
usually assumed in macroeconomic dynamic models and veri￿ed by empirical
studies, and the utility for consumption is linear. We introduce this last
assumption for simpli￿cation.7 We obtain the following result:
Proposition 2 Assuming that there exists a steady state (Proposition 1)
and Assumptions 1-4 are satis￿ed, the steady state is locally indeterminate
for α < w(1) + 1, a ￿ip bifurcation occurs for α = w(1) + 1 and the steady
state is a saddle for α > w(1) + 1.
Proof. We ￿rst di￿erentiate the dynamic system (9)-(10) around (a,k) =
(1,1). Using (12) and after some computations, we obtain the trace T and
the determinant D of the associated Jacobian matrix, which respectively
represent the sum and the product of the two eigenvalues of the characteristic
polynomial Q(λ) ≡ λ2 − Tλ + D = 0:
T =
s
1 − s
+ w(1) − α (18)
D =
s
1 − s
(w(1) − α) (19)
One can easily deduce that the two eigenvalues are de￿ned by λ1 =
s/(1−s) and λ2 = w(1)−α. Using Assumptions 3 and 4, we have λ1 ∈ (0,1)
and λ2 < 1. Moreover, λ2 > −1 for α < w(1)+1, λ2 = −1 for α = w(1)+1
and λ2 < −1 for α > w(1) + 1. This concludes the proof. 
This proposition establishes that both deterministic cycles and local in-
determinacy, i.e. ￿uctuations due to self-ful￿lling prophecies, can occur. In
contrast to previous results analyzing the existence of cycles in dynamic mod-
els with environment (see Seegmuller and VerchŁre (2004), Zhang (1999)),
here the emission rate of pollution (α) has not to be too high. Furthermore,
endogenous ￿uctuations can occur when capital and labor are not weak sub-
stitutes as it is often required (see among others Grandmont et al. (1998)
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7or Reichlin (1986)). Our results do not depend any more on the existence
of increasing returns or imperfect competition (see Benhabib and Farmer
(1999) for a survey).
We now interpret the occurrence of endogenous ￿uctuations by explaining
how non monotonic and cyclical trajectories can emerge due to consumer
choices between consumption, environmental quality and leisure.
Assume that one deviates from the steady state by an increase of the cap-
ital stock kt−1. Then, taking into account equation (3), young consumers
expect that future pollution Pt+1 will increase and reduce the level of their
utility. Since from relation (6) the real wage decreases, they increase their
labor supply lt, and they also reallocate their savings from capital accumu-
lation kt to environmental maintenance dt (see equation (1)). Following this
decrease of the capital stock, the next generation of consumers will expect a
decreased pollution ￿ow Pt+2 and then, by the reverse mechanism, both lt+1
and dt+1 will go down and kt+1 will raise; and so on successively along the
￿uctuations.
Finally, one can observe that the greater is α, the more volatile is dt,
which can be a source of instability of the steady state and promote the
determinacy of the equilibrium. It explains why ￿uctuations due to self-
ful￿lling expectations emerge only if α is not too high.
NOTES
1. For a recent survey see Xepapadeas (2003).
2. See among others Benhabib and Farmer (1994), Cazzavillan (2001) or
Grandmont (1985).
3. As in John and Pecchenino (1994), we consider positive environmental
maintenance or pollution abatement (dt ≥ 0).
4. We assume that capital totally depreciates after one period of use.
5. Note that the pollution Pt+1 can be interpreted as a ￿ow, or a stock
determined by Pt+1 = (1−m)Pt+αkt−1−dt, with a natural rate of absorption
m equal to 1. Since, in overlapping generations models, the length of period
is assumed to be long, the assumption m = 1 does not seem to be too
restrictive.
6. In order to be as short as possible, we do not discuss uniqueness or
multiplicity of steady states. For such an analysis, the reader can refer to
Seegmuller and VerchŁre (2005).
7. The reader can refer to Seegmuller and VerchŁre (2005) who relax this
condition.
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7REFERENCES
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A Note on Indeterminacy in Overlapping Generations Economies with Environment and Endogenous Labor Supply

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A Note on Indeterminacy in Overlapping Generations Economies with Environment and Endogenous Labor Supply

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