Download On the Environmental Kuznets Curve: A Real Options Approach

On the Environmental Kuznets Curve:
A Real Options Approach
Masaaki Kijima, Katsumasa Nishide and Atsuyuki Ohyama
Tokyo Metropolitan University
Yokohama National University
NLI Research Institute
I.
Introduction
II.
Optimal Environmental Policy
1. Model setup : A real options approach
2. Thresholds for stopping and restarting
III. Why Does the Kuznets Curve Present ?
1.
2.
3.
4.
Model setup : Alternating renewal processes
Transition density of the pollution level
The inverse-U-shaped pattern as expected pollution level
Numerical example
IV. Conclusions
2Page.
Introduction
3Page.
What is the Kuznets Curve ?
• The Kuznets Curve reveals that
Income differential first increases due to the economic
growth; but then starts decreasing to settle down
– Kuznets (1955，1973)；Robinson (1976); Barro (1991);
Inequality
Deininger and Squire (1996); Moran (2005), etc.
Income per Capita
t
4Page.
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)
5Page.
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.
7Page.
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.
8Page.
Purpose
• Our purpose is to present a simple model to explain the
inverse-U-shaped pattern using a real options model.
Micro’s
What is the optimal management of stock pollutants?
perspective
A real options approach
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 ?
Alternating renewal processes
Macro’s
perspective
An inverse-U-shaped pattern
(＝Environmental Kuznets Curve)
9Page.
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
A
C
– Ross (1996), etc.
B
D
0
t
System on
System off
System on System off
10Page.
Optimal Environmental Policy
11Page.
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 : u 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.
12Page.
• The country i’s problem
• Under the de-regulation regime, the value function is
where A is a constant,
• Under the regulation regime, the value function is
where B is a constant,
13Page.
Thresholds for Stopping and Restarting
• We derive two thresholds: one for starting regulation
and the other for de-regulation .
Value-matching Condition
,
Smooth-pasting Condition
These equations have four unknowns; i.e. the two
thresholds ,
, and the coefficients
and .
Therefore, we can obtain the solution at least numerically.
14Page.
Why Does the Kuznets Curve
Present ?
15Page.
Model Setup: Alternating Renewal Process
Suppose that countries execute optimally the switching
options for regulating and de-regulating pollutions in time.
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】
i
i
Instead of Pt , we investigate the shape of log Pt .
Therefore, we consider the following stochastic process.
16Page.
＜＜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
i
i


0


, H
L .
【Thresholds】
off
Regulation
De-regulation
on
off
on
off
off
on
on
off
17Page.
The transition probability density for country i:
To simplify our notation, we omit the superscript i for a
while.
【Definition of the hitting times】
with
and also
18Page.
Since
and
are independent, we denote
Duration
Density Function
n  1,2, 
n  2,3, 
Also, we denote
where
is the convolution operator.
The sequence
renewal process.
is called a (delayed) alternating
19Page.
【Delayed renewal processes】
【Renewal functions】
＜＜Renewal densities＞＞
By the definition,
20Page.
Also, following the basic renewal theory, we obtain
Laplace Transform
Inverse Laplace Transform
Laplace Transform
Inverse Laplace Transform
via numerical inversion
21Page.
Renewal Functions:
,
In this case, after
Time=300, then
Time
State
Equal
Time
22Page.
Time
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 S0  H , we have
23Page.
【To calculate the transition probability density,
we consider the following three cases】
Case 1：
, that is,
Case 2：
that is,
and
Case 3：
that is,
and
These 3 cases are mutually exclusive and exhaust all the
events.
24Page.
【Case 1】
【Case 2】
In this case, the event to hit
at some time s has occurred.
25Page.
【Case 3】
Transition density is given by
State
Density
26Page.
Time
From the basic renewal theory, as
Hence, when
, we have
and  L  0 , we obtain
27Page.
The Inverse-U-Shaped Pattern
【A Model for the Aggregated Level】
Consider the sum of each country’s log-stock pollutant
where
with
subject to the switching at
xi  xi
28Page.
【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.
The switching thresholds
over the countries.
are the same
29Page.
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
N
y   wi y i
i 1
30Page.
Numerical Examples
We are interested in the shape of
with respect to t with
31Page.
Monte Carlo Simulation
32Page.
The Environmental Kuznets Curve
E[log Pt ]
E[log GDPt ]
An inverse-U-shaped pattern
GDP per capita also grows in
average exponentially in time.
33Page.
Conclusion
34Page.
 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.
35Page.
Thank you for your attention
36Page.