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• 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).

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)=AKtαLt

1- 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=1

F(K,L)

E

β ∈ (0,1)

β>1

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 (1,+∞) having positive slope lower than 1∈

and the half-line Ω ≡ (Tr(J),Det(J)) starting from Δ₁ parameterized by β and with slope η

2L

L*

*

ΔΩΩ

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.

• 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

What does it mean?

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).

• β= 8 period 4

Complex dynamics via flip bifurcation

• β= 8.5 period 4

Complex dynamics via flip bifurcation

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 equilibriaDefined by

Thank you