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On the Environmental Kuznets Curve:A Real Options Approach

Masaaki Kijima, Katsumasa Nishide and Atsuyuki Ohyama

Tokyo Metropolitan UniversityYokohama National University

NLI Research Institute

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I. Introduction

II. Optimal Environmental Policy

III. Why Does the Kuznets Curve Present ?

IV. Conclusions

1. Model setup : A real options approach2. Thresholds for stopping and restarting

1. Model setup : Alternating renewal processes2. Transition density of the pollution level3. The inverse-U-shaped pattern as expected pollution level4. Numerical example

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Introduction

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What is the Kuznets Curve ?

The Kuznets Curve reveals thatIncome differential first increases due to the economicgrowth; but then starts decreasing to settle down

Kuznets (19551973)Robinson (1976); Barro (1991);Deininger and Squire (1996); Moran (2005), etc.

t

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Literature Review

Environmental Kuznets CurveSimilar curves are observed in various pollution levels

Empirical studies

Grossman and Krueger (1995) Shafik and Bandyopadhyay (1992) Panayotou (1993)Many other empirical studies, while just a few theoretical research

Theoretical studies Lopez (1994) Selden and Song (1995)

Andreoni and Levinson (2001)5Page.

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Itaru Yasui, "Environmental Transition -A Concept to Show the Next Step of Development .Symposium on

Sustainability in Norway and Japan: Two Perspectives. April 26, 2007 NTNU, Trondheim, Norway 6Page.

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Lopez (1994)Macroeconomic model (no uncertainty)

- the production is affacted by the level of pollution- in the optimal path, pollution is U-shaped w.r.t. theproduction.

Selden and Song (1995)Representative agent in a dynamic setting (nouncertainty)- utility from consumption and disutility from pollution- if the abatement function satisfies some property, the

agent switches the strategy when the pollution touches acertain level.

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Andreoni and Levinson (2001)Representative agent in a static setting (no uncertainty)

- utility from consumption and disutility from pollution- if the elasticity of pollution w.r.t. the abatement effort islarge enough, the agent pays a more amount ofabatement cost as his income becomes larger.

In the previous literature,

uncertainty is not considered,

macroeconomic effect is not examined as the

aggregation of microeconomic behavior.

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Purpose

A real options approach

Alternating renewal processes

Microsperspective

Macros

perspective

Our purpose is to present a simple model to explain the

inverse-U-shaped pattern using a real options model.

What is the optimal management of stock pollutants?

Derive the thresholds of regulation and de-regulation.

As a result,

How will stock pollutants change in time

How about expected stock pollutants in total ?

An inverse-U-shaped pattern(Environmental Kuznets Curve)

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Two Ingredients

Real Options Approach

(strategic) switching model under uncertainty

Dixit and Pindyck (1997), etc.

We use the same framework as Dixit and Pindyck(1994, Chapter 7) and Wirl (2006)

Alternating renewal processes

Switchings produce on and off alternately with iid

lifetimes

Ross (1996), etc. A C

B D

0 tSystem on System off System on System off

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Optimal Environmental Policy

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Model Setup: A Real Options Approach From the micros perspective, we analyze each country i

Stock Pollutants :where k represents each regime as shown below.

Cost of external Effects

Benefit in regime k :

Government chooses alternative regimes for anenvironmental policy: one under regulations L and theother under de-regulations H (including no regulation). Ofcourse, it is possible to switch the regimes.

ku

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whereA

is a constant,

where B is a constant,

The country is problem

Under the de-regulation regime, the value function is

Under the regulation regime, the value function is

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Thresholds for Stopping and Restarting

We derive two thresholds: one for starting regulation ,and the other for de-regulation .

These equations have four unknowns; i.e. the twothresholds , , and the coefficients and .

Therefore, we can obtain the solution at least numerically.

Smooth-pasting ConditionValue-matching Condition

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Why Does the Kuznets CurvePresent ?

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Model Setup: Alternating Renewal Process

We calculate the transition density of the pollution levelusing the theory of alternating renewal processes, and then,illustrate the inverse-U-shaped pattern.

AssumptionInstead of , we investigate the shape of .Therefore, we consider the following stochastic process.

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Suppose that countries execute optimally the switchingoptions for regulating and de-regulating pollutions in time.

i

tPlogi

tP

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Alternating Renewal Process

Consider a system that can be in one of two states: on

(regulation) or off (de-regulation).

Let , be the sequences of durations to switchthe states. The sequences , are independent andidentically distributed (iid) except .

Suppose that , .

Regulation

De-regulation

Thresholds

offon on on

onoff

off

offoff

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iL

iH 0

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The transition probability density for country i:

To simplify our notation, we omit the superscript i for a

while.

Definition of the hitting times

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with

and also

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Since and are independent, we denote

Also, we denote

where is the convolution operator.The sequence is called a (delayed) alternatingrenewal process.

Density FunctionDuration

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,3,2n

,2,1n

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Delayed renewal processes

Renewal functions

Renewal densitiesBy the definition,

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via numerical inversion

Also, following the basic renewal theory, we obtain

Laplace Transform

Inverse Laplace TransformInverse Laplace Transform

Laplace Transform

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Renewal Functions: ,

Time

Time

Time

State

In this case, after

Time=300, then

Equal

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Transition Probability of the Pollution Level

Notation

In order to calculate , we define

and denote

These transition densities are known in closed form for the

case of geometric Brownian motions.Also, we denote the regime at time tby .

Note that, because , we have

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H0 S

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Case 1 , that is,

Case 2 and

that is,

Case 3 andthat is,

To calculate the transition probability density,we consider the following three cases

These 3 cases are mutually exclusive and exhaust all theevents.

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Case 3

Transition density is given by

Time

Density

State

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From the basic renewal theory, as , we have

Hence, when and , we obtain

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0L

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The Inverse-U-Shaped Pattern

A Model for the Aggregated LevelConsider the sum of each countrys log-stock pollutant

where with

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subject to the switching at ii xx

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Assumptions

Because each countrys economic scale is different, its

initial stock pollutant is distinct over countries.The uncertainties (Brownian motions) are mutually

independent, because each country executesenvironmental policy non-cooperatively.

Because environmental problems are the world-wide issue,technological transfers are smoothly performed; so that itis plausible to assume the parameters to be the sameover countries, i.e.

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The switching thresholds are the sameover the countries.

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Under these assumptions, is a weighted average ofindependent replicas with different initial states.

Hence, in principle, we can calculate the transitionprobability density of

However, when N is sufficiently large, the effect from

the law of large numbers (or the central limit theorem)

becomes dominant, and we are interested in the mean(or the variance) of . That is,

Moreover, as the first approximation, we consider

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N

i

i

iywy

1

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Numerical Examples

We are interested in the shape of

with respect to twith

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Monte Carlo Simulation

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An inverse-U-shaped pattern

The Environmental Kuznets Curve

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GDP per capita also grows inaverage exponentially in time.

][log tPE

][log tGDPE

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Conclusion

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We describe a simple real options (switching) model toexplain why the environmental Kuznets curve presents forvarious pollutants when each country executes itsenvironmental policy optimally.

The transition probability density of the pollution level isderived using the alternating renewal theory.

In particular, its mean is calculated numerically to showthe inverse-U-shaped pattern.

The assumption of GBM can be removed as far as theconstant switching thresholds and the Laplace transformof the first hitting time to the thresholds are known.

As a future work, our model can be applied to estimatewhen the peak of the curve will present.

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Thank you for your attention

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