An overlapping Generation Model with Environment

1
An overlapping Generation Model with Environment

• Angelo Antoci, University of Sassari
• Mauro Sodini, University of Pisa

2
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.

3
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.

4
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.

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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

6
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.

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Main characteristics of this alternative approach
to the problem

• 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).

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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

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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.

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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)

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The model

• Agents 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?)) 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.

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• Budget constraint
• Ct1 Lt Rt1 Wt1
• where
• Wt is the wage at time t
• Rt is the interest factor at time t
• Time constraint
• Lt?0, L

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Maximization problem

• Max Ui (Lt , Ct1 , Et1 )
• s.t.
• Ct1 Lt Rt1 Wt1
• Lt?0, L

At each date, Et1 is considered as given by the
individual
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Private market perfect competition among many
little firms

• Cobb-Douglas specification
• YtAF(Kt,Lt)AKtaLt1- a Akta
• where kK/L
• and perfect competition hypothesis lead to
• Rt A a kta-1
• Wt A(1-a) kta

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Environmental dynamics

• Assumption no accumulation of environmental
  deterioration (quite optimistic and makes result
  about non desiderability of high growth more
  robust)
• Et1 E- ?F(Kt,Lt)ß
• Where the bar on Capital and Labour stands for
  aggregate level variables

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Production and environment
E
ßgt1
ß1
ß ? (0,1)
F(K,L)
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Equilibrium dynamics are defined by
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Equilibrium dynamics are defined by
Ex-post equivalence of single decision and
macroeconomic variables and expectations (perfect
foresight)
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A natural comparison

• If ?0 we have the Reichlin model (1986) in
  which
• 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
  )
• If we assume regular description of the economy
  (Cobb-Douglas description) the steady state is a
  saddle
• Only considering really strong assumptions on
  elasticity of substitution (Leontieff) we have
  complex dynamics of the equilibrium system

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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)

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After some algebraic manipulations dynamical
system could be written as
For which, kL1 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
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The Jacobian matrix, evaluated at the normalized
fixed point, is
with
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The next figure indicates, for each subset of the
plane (Tr(J), Det(J)), the corresponding
stability regime.We consider the half-line ?
(Tr(J)ß0,Det(J) ß0) parameterized by
?(1,8) having positive slope lower
than 1and the half-line O (Tr(J),Det(J))
starting from ?1 parameterized by ß and with
slope ?
O
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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.
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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

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Well-being vs economic growth (I)
Well being and economic activity, varying ? in
the steady state
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Well-being vs economic growth
Two steady states convergence to normalized
steady state starting near a repulsive one
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Complex dynamics via flip bifurcation (a0.1,
?0.41, L 7, ?0.2 )

• ß 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).
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Complex dynamics via flip bifurcation

• ß 8 period 4

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Complex dynamics via flip bifurcation

• ß 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).
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Complex dynamics via flip bifurcation

• ß 9 period 4

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The evolution of Lyapunov exponents
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One dimensional bifurcation diagrams
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Complex dynamics via Hopf bifurcations (a0.1,
?0.41, L 7, ?0.2 )

• ß 1.2

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Complex dynamics via Hopf bifurcations (a0.1,
?0.41, L 7, ?0.2 )

• ß 2.4

• ß 2.5

Complex dynamics can also occur via Hopf
bifurcations the cycle breaks in several
attracting isolated islands
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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.

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Second model

• Linear specification of the impact of economic
  activity on environmental quality
• Et1 E- ?AF(Kt,Lt)
• But more articulated framework of
  substitutability between private good and
  environmental quality

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Proceeding in a similar way we define the
following dynamical system (with a normalized
steady state)
s ?(0,1) complementarity unique steady state
(saddle)
Where ??/(1-a)
s plays a fundamental role
s gt1 substitute goods multiple steady state and
complex behavior
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Jacobian matrix

With the following
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Case s ?(0,1)
41
Case s gt1
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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 e could create complex
  dynamics

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Multiplicity of equilibria
Defined by
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• Thank you