An overlapping Generation
Model with Environment
Angelo Antoci, University of Sassari
Mauro Sodini, University of Pisa
Plan of Presentation
• Motivations of the work;
• Description of some characteristics of existing
literature on the theme (in overlapping
generations framework);
• Introduction of some informal ideas behind the
modeling;
• The mathematical model;
• The well-being problem;
• Dynamics of the model;
• Conclusions;
• Really preliminary results of a second model.
Motivations
• Develop an overlapping generations framework
to study the problem of environmental quality
(bounded rationality in allocation problem);
• Illustrate and clarify possible peculiar interplays
between environmental quality and consumption
pattern even in a simplified model;
• Why fluctuations arise? Only (imperfections in)
economic sectors matter?;
• Evaluate the overall well-being effects of
economic growth.
Bibliography (I): The widespread
view
• Jhon A., Pecchenino R., 1994, An Overlapping
Generations Model of Growth and the Environment. The
Economic Journal 104, 1393-1410.
• Jhon A., Pecchenino R., Schimmelpfennig D. and
Schreft S., 1995, Short-lived Agents and the Long-lived
Environment, Journal of Public Economics 58, 127-141.
• Zhang J., 1999, Environmental Sustainability, Nonlinear
Dynamics and Chaos, Economic Theory 14, 489-500.
• Seegmuller T., Verchère A., 2005, Environment in an
Overlaping Generations Economy with Endogenous
Labour Supply, Document de travail n. 2005-05, Bureau
d'économie théorique et appliquée (BETA), France.
Main characteristics
• The mechanism: Agents allocate their resources
between consumption, saving and
environmental defensive expenditures that
improve environmental quality by reducing the
negative effects of production processes;
• The consequences: A long run positive
correlation between well-being and economic
growth: that is, the increase of the production of
consumption goods is always a desirable
outcome because it leads to a more developed
country with a better defense against the
environmental degradation
Bibliography (II): An alternative
view of the same problem
•
•
•
•
•
•
•
Antoci A., Bartolini S., 1999, Negative Externalities as the Engine of Growth in an
Evolutionary Context, Working paper 83.99, FEEM, Milan.
Antoci A., Bartolini S., 2004, Negative Externalities and Labor Input in an Evolutionary
Game, Journal of Environment and Development Economics 9, 1-22.
Antoci A., Galeotti M., Russu P., 2005, Consumption of Private Goods as Substitutes
for Environmental Goods in an Economic Growth Model, Nonlinear Analysis:
Modelling and Control, 10, 3-34.
Antoci A., Galeotti M., Russu P., 2007, Undesirable Economic Growth via Economic
Agents' Self-protection Against Environmental Degradation, Journal of The Franklin
Institute 344, 377-390.
Antoci A., Borghesi S. and Galeotti M., 2008, Should we Replace the Environment?
Limits of Economic Growth in the Presence of Self-Protective Choices, International
Journal of Social Economics 35 (4), 283-297.
Hueting R., 1980, New Scarcity and Economic Growth. More Welfare Through Less
Production?, North Holland , Amsterdam.
Leipert C., Simonis U. E., 1988, Environmental Damage - Environmental
Expenditures: Statistical Evidence on the Federal Republic of Germany, International
Journal of Social Economics, 15 (7), 37-52.
Main characteristics of this
alternative approach to the problem
•
1)
2)
3)
4)
5)
The mechanism:
The environment creates free goods;
Private production causes environmental degradation;
Environmental degradation destroys free-goods;
No market for environmental defensive expenditures
(Environment is macro-level variable and the single
agent is an individual. The perception of a single agent
is that its value is given);
Each Individual defends himself from environmental
degradation by increasing his consumption of
produced private goods (substitution of public goods
with private goods).
Examples
• Mineral water may substitute spring water or tap water;
• Medicines may mitigate the effects of respiratory
diseases caused by air pollution;
• Individuals may react to the deterioration of the seaside
near home by going to a less deteriorated seaside area
by car or by boat, they may build a swimming pool in
their gardens, they may purchase houses in exclusive
areas at the seaside or buy holiday-packages in tropical
paradises;
• Individuals may defend themselves from external
sources of noise by installing (REALLY EXPENSIVE)
sound-proofing devices;
In general:
Urban life-styles in modern cities are often
characterized by the scarcity of free
access environmental resources and, at
the same time, they are able to supply a
considerable variety of private and
expensive consumption opportunities.
The consequences: only a change
in consumption pattern? NO
• Self-protection through private consumption
choices generate further environmental damage;
• Self-protection choices are usually enforced
beyond the socially optimal level (agents do not
coordinate themselves);
• Possible negative correlation between economic
growth and individuals' well-being (failure of the
promise of capitalism: ”Growth is good”);
The model
• Agent’s utility (C and E are substitute):
• where Et represents the value of a given
environmental quality index at time t; P is a
positive parameter; (1/(1+θ)) is the discount
factor; Ct is the private consumption at time t; L*
is the time resource at every t; Lt is the individual
labor supply at time t.
• Budget constraint:
Ct+1 = Lt Rt+1 Wt+1
where
• Wt is the wage at time t
• Rt is the interest factor at time t
• Time constraint:
Lt∈[0, L* ]
Maximization problem
Max Ui (Lt , Ct+1 , Et+1 )
s.t.
Ct+1 = Lt Rt+1 Wt+1
Lt∈[0, L* ]
At each date, Et+1 is considered as given by the individual
Private market: perfect competition
among many little firms
• Cobb-Douglas specification
Yt=AF(Kt,Lt)=AKtαLt1- α =Aktα
where k=K/L
and perfect competition hypothesis lead to
Rt = A α ktα-1
Wt = A(1-α) ktα
Environmental dynamics
• Assumption: no accumulation of
environmental deterioration (quite
optimistic and makes result about non
desiderability of high growth more robust):
Et+1 = E- η[F(Kt,Lt)]β
Where the bar on Capital and Labour stands
for aggregate level variables
Production and environment
E
β>1
β=1
β ∈ (0,1)
F(K,L)
Equilibrium dynamics are defined
by
Equilibrium dynamics are defined
by
Ex-post equivalence of single decision and
macroeconomic variables and expectations
(perfect foresight)
A natural comparison
If η=0 we have the Reichlin model (1986) in which:
1. The decentralized solution coincides with the
centralized one (no externalities and no problems of
coordination between different generations (quite
strange result in overlapping generations model and
due essentially to the simple structure );
2. If we assume “regular” description of the economy
(Cobb-Douglas description) the steady state is a
saddle;
3. Only considering really strong assumptions on
elasticity of substitution (Leontieff) we have complex
dynamics of the equilibrium system;
Existence of steady states and
normalized steady state
Problem of the model
• Many parameters
• Steady states could not exist
We proceed to “create” a (normalized)
steady state (fixing a part of parameters).
So we can concentrate on an interesting
subset of parameters (standard technique
used in many OGM)
After some algebraic manipulations dynamical system
could be written as
For which, k=L=1 is a steady state for the whole range
of parameters and Es.s.=1
Notice that kt is a predetermined variable
meanwhile Lt is a jump variable
The Jacobian matrix, evaluated at the normalized fixed point, is
with
The next figure indicates, for each subset of the plane (Tr(J), Det(J)),
the corresponding stability regime.
We consider the half-line Δ ≡(Tr(J)|β=0,Det(J)| β=0) parameterized by
L*
∈(1,+∞) having positive slope lower than 1
*
L 2
and the half-line Ω ≡ (Tr(J),Det(J)) starting from Δ₁ parameterized by β
and with slope η
Ω
Ω
Δ
Local dynamics around the
normalized steady state
The steady state could be a saddle, the “normal” result:
one state variable kt (predetermined variable)
one jump variable Lt (decision variable)
There exists a one dimensional path converging to equilibrium and the
agents, given k0 chose the unique value of L to put the economy on
this path
……..But for a large set of parameters the steady state could be a sink
or in economic terms the equilibrium is indeterminate.
What does it mean?
• The expectations matter and drive economic
convergence to equilibrium (without the usually assumed
imperfections in the productive side of the economy);
• The implications? The medium term results are not
specified by economic fundamentals but stands on the
animal spirits of the agents;
But the impact of environmental degradation could create
other phenomena:
Cyclical behaviors could emerge around the steady state
when this is repulsive but even multiplicity of steady
states
Well-being vs economic growth (I)
Well being and economic activity, varying η in
the steady state
Well-being vs economic growth
Two steady states: convergence to normalized steady
state starting near a repulsive one
Complex dynamics via flip
bifurcation (α=0.1, η=0.41, L =7, θ=0.2 )
*
• β= 6.79 period 2
The normalized fixed point (1,1) loses its (two dimensional) stability becoming a saddle
and a period 2_cycle appears via a supercritical flip bifurcation (period doubling
bifurcation).
Complex dynamics via flip
bifurcation
• β= 8 period 4
Complex dynamics via flip
bifurcation
• β= 8.5 period 4
Subsequent increases of lead to further flip
bifurcations according to which cycles of periods
4,8,...,2ⁿ arise until the rise of a strange attractor
(period-doubling route to chaos).
Complex dynamics via flip
bifurcation
• β= 9 period 4
The evolution of Lyapunov
exponents
One dimensional bifurcation
diagrams
Complex dynamics via Hopf
bifurcations (α=0.1, η=0.41, L =7, θ=0.2 )
*
• β= 1.2
Complex dynamics via Hopf
bifurcations (α=0.1, η=0.41, L* =7, θ=0.2 )
• β= 2.4
β= 2.5
Complex dynamics can also occur via Hopf
bifurcations: the cycle breaks in several attracting
isolated islands
Conclusion for the first model
• Our work has highlighted a mechanism according to
which environmental degradation may lead to complex
dynamic behavior in an overlapping generation model
described by a two-dimensional discrete dynamical
system:
• Ceteris paribus, an increase in the environmental impact
of economic activity may lead to chaotic behavior.
• Differently from the mainstream literature concerning
overlapping generation models, indeterminacy and
chaotic dynamics don't occur in a context in which there
are positive externalities in the production process but in
a context where there are negative externalities
generated by the production process.
Second model
• Linear specification of the impact of
economic activity on environmental quality
Et+1 = E- η[AF(Kt,Lt)]
• But more articulated framework of
substitutability between private good and
environmental quality
Proceeding in a similar way we define
the following dynamical system (with a
normalized steady state)
Where ω=η/(1-α)
σ plays a fundamental role
σ ∈(0,1) complementarity
unique steady state (saddle)
σ >1 substitute goods:
multiple steady state and
complex behavior
Jacobian matrix:
With the following
Case: σ ∈(0,1)
Case: σ >1
Preliminary results
• The role of substitutability matters:
• If we assume complementarity between the
environmental good and private good (diffuse
hypothesis) the agents move their private and
environmental consumption in the same way
but….
• If substitutability effect prevails the results are
reversed through a perverse mechanism (in
terms of well-being) no registered by GDP.
• Enough high value of ε could create complex
dynamics
Multiplicity of equilibria
Defined by
Thank you