Overlapping generations models and Environment

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					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
• The mathematical model;
• The well-being problem;
• Dynamics of the model;
• Conclusions;
• Really preliminary results of a second model.
• 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
• 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
•    The mechanism:
1)   The environment creates free goods;
2)   Private production causes environmental degradation;
3)   Environmental degradation destroys free-goods;
4)   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);
5)   Each Individual defends himself from environmental
     degradation by increasing his consumption of
     produced private goods (substitution of public goods
     with private goods).
• 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
• 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

• 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 )
                   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



       β ∈ (0,1)
Equilibrium dynamics are defined
 Equilibrium dynamics are defined

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
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

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             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
  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
     Complex dynamics via flip
• β= 8 period 4
           Complex dynamics via flip
• 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
• β= 9 period 4
The evolution of Lyapunov
One dimensional bifurcation
    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
• 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 ∈(0,1) complementarity
                             unique steady state (saddle)
σ plays a fundamental role
                                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
• 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
    Multiplicity of equilibria
Defined by
Thank you