On the Environmental Kuznets Curve: A Real Options Approach

Masaaki Kijima, Katsumasa Nishide and Atsuyuki Ohyama

Tokyo Metropolitan UniversityYokohama National University

NLI Research Institute

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 economic growth; but then starts decreasing to settle down – Kuznets (1955，1973)；Robinson (1976); Barro (1991); Deininger and Squire (1996); Moran (2005), etc.

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Literature Review

• Environmental Kuznets Curve Similar 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)

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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.

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. the production.

Selden and Song (1995)Representative agent in a dynamic setting (no uncertainty)- utility from consumption and disutility from pollution- if the abatement function satisfies some property, the agent switches the strategy when the pollution touches a certain 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 is large enough, the agent pays a more amount of abatement 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

Micro’sperspective

Macro’s 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 processesSwitchings produce ‘on’ and ‘off’ alternately with iid lifetimes – Ross (1996), etc.

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Optimal Environmental Policy

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Model Setup: A Real Options Approach • From the micro’s 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 an

environmental policy: one under regulations L and the other under de-regulations H (including no regulation). Of course, it is possible to switch the regimes.

ku

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where A is a constant,

where B is a constant,

• The country i’s 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 two thresholds , , and the coefficients and . Therefore, we can obtain the solution at least numerically.

Smooth-pasting ConditionValue-matching Condition

Why Does the Kuznets Curve Present ?

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Model Setup: Alternating Renewal Process

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

【Assumption】Instead of , we investigate the shape of .

Therefore, we consider the following stochastic process.

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

itPlogi

tP

＜＜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 switch the states. The sequences , are independent and identically distributed (iid) except . Suppose that , .

RegulationDe-regulation

【Thresholds】

offon on on

onoff

off

offoff

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iL

iH μμ >> 0

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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withand also

Since and are independent, we denote

Also, we denote

where is the convolution operator.

The sequence is called a (delayed) alternating renewal process.

Density FunctionDuration

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K,3,2=n

K,2,1=n

【Delayed renewal processes】

【Renewal functions】

＜＜Renewal densities＞＞

By 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, afterTime=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 t by .Note that, because , we have

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H0 =S

Case 1： , that is,

Case 2： and that is,

Case 3： and that is,

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

These 3 cases are mutually exclusive and exhaust all the events.

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【Case 1】

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【Case 2】

In this case, the event to hit at some time s has occurred.

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【Case 3】

Transition density is given by

Time

Density

State

From the basic renewal theory, as , we have

Hence, when and , we obtain

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0<Lμ

The Inverse-U-Shaped Pattern

【A Model for the Aggregated Level】Consider the sum of each country’s log-stock pollutant

where with

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subject to the switching at ii xx >

【Assumptions】Because each country’s economic scale is different, its

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

independent, because each country executes environmental policy non-cooperatively.

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

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The switching thresholds are the same over the countries.

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

Hence, in principle, we can calculate the transition probability 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

ii ywy

1

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Numerical Examples

We are interested in the shape of

with respect to t with

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Monte Carlo Simulation

An inverse-U-shaped pattern

The Environmental Kuznets Curve

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GDP per capita also grows in average exponentially in time.

][log tPE

][log tGDPE

Conclusion

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We describe a simple real options (switching) model to explain why the environmental Kuznets curve presents for various pollutants when each country executes its environmental policy optimally.

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

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

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

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

Thank you for your attention

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