Permit allocation in emissions trading using the Boltzmann distribution

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Abstract

In emissions trading, the initial allocation of permits is an intractable issue because it needs to be essentially fair to the participating countries. There are many ways to distribute a given total amount of emissions permits among countries, but the existing distribution methods, such as auctioning and grandfathering, have been debated. In this paper we describe a new method for allocating permits in emissions trading using the Boltzmann distribution. We introduce the Boltzmann distribution to permit allocation by combining it with concepts in emissions trading. We then demonstrate through empirical data analysis how emissions permits can be allocated in practice among participating countries. The new allocation method using the Boltzmann distribution describes the most probable, natural, and unbiased distribution of emissions permits among multiple countries. Simple and versatile, this new method holds potential for many economic and environmental applications.

Highlights

► We describe a new method for permit allocation using the Boltzmann distribution. ► We demonstrate how permits can be practically allocated among multiple countries. ► The method presents the most probable distribution of permits among countries. ► It is a natural and unbiased distribution of permits among multiple countries. ► The new method has potential for various economic and environmental applications.

Introduction

Scientists have warned that global warming of more than 1 °C would constitute a dangerous climate change based on the likely effects on sea levels and the extermination of species [1]. Furthermore, climate change that occurs as a result of increases in CO2 concentration is largely irreversible for 1000 years, even after CO2 emissions cease [2]. According to the Stern Review, prompt, decisive action is clearly warranted, and, because climate change is a global problem, the response to it must be international [3].

Various ideas have been proposed for slowing global warming and reducing CO2 emissions into the atmosphere, including reflecting solar radiation with small particles in the stratosphere, putting deflectors in space, growing trees and other biomass to remove CO2 from the atmosphere, fertilizing oceans with iron to remove CO2, and reducing CO2 emissions through carbon taxes or emissions trading [4]. Among these, numerous studies found that emissions trading lowers the cost of reaching the commitments of the Kyoto Protocol [5].

The basic concepts of emissions trading were established in the past decades [6], [7], [8], [9], [10], [11], [12], [13]. As with any trading system, in the emissions trading system, the flow and value of what is traded depends on its initial allocation, its supply, and the demand for it [14]. There are many possible ways to distribute a given total of emissions permits among participants [15]; traditionally, grandfathering and auctioning have been suggested for initial permit allocation [16], [17].

Permit allocation is one of the most intractable issues to resolve in designing emissions trading systems. A permit-allocation rule should be simple, should be based in part on historical data, and should be perceived as fair [18]. Because the flow and value of emissions permits depends on their initial allocation, the fair allocation of a limited number of permits among countries or firms is not only important but also controversial.

In this paper, we introduce an alternative method for initial permit allocation using the Boltzmann distribution. We first describe the basic concept of the Boltzmann distribution and then develop its mathematical formula for the allocation of emissions permits. Next, through empirical data analysis, we demonstrate how this allocation method can be used in practice for initial permit allocation.

Section snippets

The Boltzmann distribution

In the physical sciences, the Boltzmann distribution yields the equilibrium probability distribution of a physical system in its energy substates [19], [20]. The description is valid as long as each physical particle of the system is identical to but distinguishable from the others and as long as the interaction among the particles can be taken to be negligible. Based on the Boltzmann distribution, the probability (Pi) that a particle can be found in the ith substate is inversely proportional

The Boltzmann distribution for permit allocation

Suppose that n countries are participating in the permit allocation for international emissions trading. Consider that the total available emissions permits (αN) are allocated to the countries and that a country i has the allocation potential energy per capita of Ei and a population Ci (where i=1,2,3,,n;α is the unit emissions permit; and N is the total number of unit emissions permits). The number of unit emissions permits that are allocated to the jth individual in a country i is Nij.

We

The β value in the Boltzmann distribution

The next step is to develop a guideline for determining the β value in the Boltzmann distribution. It is assumed that the allocation potential energy per capita (Ei) of a country i is negatively proportional to the CO2 emissions per capita of a country i.

With a β value in the positive regime (β>0, which corresponds to the positive temperatures of a physical system), emissions permits tend to be allocated to the countries that have relatively lower allocation potential energy. In this situation,

Empirical data analysis

To demonstrate permit allocation using the Boltzmann distribution, we selected eight countries. Table 2 shows the CO2 emissions for the eight countries in 2007 and 2008, the CO2 emissions per capita in 2008, and the total population in 2008, the latest years for which annual CO2 emissions data are available, when the following empirical data analysis was carried out. It is assumed that the global (in this case, 8-country) target of CO2 emissions in 2008 is a 3% reduction of the total CO2

Discussions and conclusions

The Boltzmann distribution, originating in the physical sciences, is based on entropy maximization and thus provides the most probable distribution of a physical system at equilibrium. Banerjee and Yakovenko (2010) observed that social and economic inequality is ubiquitous in the real world, and showed that the common theme of three specific cases (e.g., the distribution of money, income, and global energy consumption) is entropy maximization for the partitioning of a limited resource among

Acknowledgments

We thank Timothy Mount and Kieran P. Donaghy (Cornell University) for thoughtful comments.

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    Professor Isard passed away before the submission of this manuscript at the age of 91.

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