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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. Introduction Optimal Environmental Policy Why Does the Kuznets Curve Present ? Conclusions.

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## On the Environmental Kuznets Curve: A Real Options Approach

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**On the Environmental Kuznets Curve: A Real Options Approach**Masaaki Kijima,Katsumasa Nishide andAtsuyuki Ohyama Tokyo Metropolitan University Yokohama National University NLI Research Institute**Introduction**Optimal Environmental Policy Why Does the Kuznets Curve Present ? Conclusions • Model setup : A real options approach • Thresholds for stopping and restarting • Model setup : Alternating renewal processes • Transition density of the pollution level • The inverse-U-shaped pattern as expected pollution level • Numerical example**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); Deininger and Squire (1996); Moran (2005), etc.**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)**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**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.**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.**Purpose**• 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) Micro’s perspective A real options approach Alternating renewal processes Macro’s perspective**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.**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.**The country i’s problem**• Under the de-regulation regime, the value function is • Under the regulation regime, the value function is where A is a constant, where B is a constant,**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 Condition Value-matching Condition**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. Suppose that countries execute optimally the switching options for regulating and de-regulating pollutions in time.**＜＜Alternating Renewal Process＞＞**Consider a system that can be in one of two states: on (regulation) or off (de-regulation). Regulation De-regulation off on off off on on on off off • Let , be the sequences of durations to switch the states. The sequences , are independent and identically distributed (iid) except . • Suppose that , . 【Thresholds】**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**Duration**Density Function Sinceandare independent, we denote Also, we denote whereis the convolution operator. The sequenceis called a (delayed) alternating renewal process.**【Delayed renewal processes】**【Renewal functions】 ＜＜Renewal densities＞＞ By the definition,**Also, following the basic renewal theory, we obtain**Laplace Transform Laplace Transform Inverse Laplace Transform Inverse Laplace Transform via numerical inversion**Renewal Functions: ,**In this case, after Time=300, then Time State Equal Time 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 , we have**【To calculate the transition probability density,**we consider the following three cases】 Case 1： , that is, Case 2： and that is, Case 3： and that is, These 3 cases are mutually exclusive and exhaust all the events.**【Case 1】**【Case 2】 In this case, the event to hit at some time s has occurred.**【Case 3】**Transition density is given by State Density Time**From the basic renewal theory, as , we have**Hence, when and , we obtain**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**【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 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**Numerical Examples**We are interested in the shape of with respect to t with**The Environmental Kuznets Curve**• An inverse-U-shaped pattern GDP per capita also grows in average exponentially in time.**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.

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