تخصیص مجوز در انتشار تجارت با استفاده از توزیع بولتزمن

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.

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] and [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] and [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.

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 multiple agents [45]. In this paper, we applied entropy maximization to the problem of permit allocation in emissions trading. When brought to initial permit allocation, the Boltzmann distribution provides the most probable allocation among multiple countries. The concept of most probable in the physical sciences may be translated into fair in permit allocation, as the distribution provides a natural and undistorted allocation among participating countries. We showed that the Boltzmann distribution includes one of the fairness notions, Egalitarian, when the ββ value approaches 0. Throughout this paper, we developed a mathematical description of permit allocation using the Boltzmann distribution. Then, through an empirical data analysis, we demonstrated how initial permit allocation can be performed over eight countries using the Boltzmann distribution. If emissions permits are allocated to the participating countries free of charge, the Boltzmann allocation is more like the grandfathering method, in which emissions permits are distributed freely based on the historical output of each country. If emissions permits are allocated to the countries at their expense (for example, proportional to the amount of allocated permits), then the Boltzmann allocation is similar to auctioning. However, in the Boltzmann allocation, the cartel of bidders and speculative behaviors observed in auctioning become more limited. Further studies are needed to develop a proper price model for the Boltzmann allocation. In addition, more work is needed to find a sophisticated definition of the allocation potential energy per capita that reflects a variety of political and economical parameters. Note that the permit allocation using the Boltzmann distribution described here is a simple yet versatile, flexible method. In other words, it can be modified and adjusted for a variety of different purposes in addition to permit allocation. Two fundamental facts, limited resources and unlimited human wants, provide a foundation for the field of economics [46]. Where resources are limited or scarce and all human wants are virtually unlimited or insatiable, the Boltzmann distribution might be applicable. Such situations include various economic and environmental problems, such as the tradable permits approach to protect the commons (e.g., allowances of water and air pollution control, rights in water supply management, and quotas in fisheries management). In addition, the Boltzmann distribution might be applicable to fair division, also known as the cake-cutting problem. There are numerous methods for obtaining a fair division, or dividing a resource in such a way that all recipients believe they have received a fair amount (e.g., proportional fair division, envy-free division, exact division, Pareto optimal division, and equitable division) [47], [48], [49], [50], [51], [52] and [53]. In the simplest two-player case, the well-known procedure of “the divide and choose method” leads to a proportional and envy-free division [54] and [55]. However, this case may not consider other essential factors such as the weight or the daily required caloric intake of players. Furthermore, fair division with three or more players is considerably more complex than the two-player case [47], [48], [51], [56], [57] and [58]. For example, let’s assume a cake is to be divided and distributed among three players: two adults, one of whom weighs 100 kg and the other 55 kg, and a child who weighs 20 kg. If this cake is cut into three identical pieces, is it a fair division for the players? The division of a single homogeneous good (for example, a cake) among multiple players is easily accomplished using the Boltzmann distribution as described in this paper. In this case, an individual player’s weight or other essential factors (for example, daily required caloric intake) can be considered his or her allocation potential energy. For example, the allocation potential energy can be defined as the daily required caloric intake per unit weight. The cake can then be distributed to the players, taking into consideration their total weight and following the Boltzmann distribution probability. In summary, the Boltzmann distribution can be applied not only to permit allocation in emissions trading but also to other economic (e.g., fair division) and environmental (e.g., air and water pollution control, water and food supply management, and fisheries management) problems by replacing the allocation potential energy with the proper economic concepts. In this sense, the Boltzmann distribution based on entropy maximization may become a useful method for a variety of economic and environmental applications.