The equal share proportional solution in a permit sharing problem

A permit sharing problem is described by a list of countries, each of which owns a certain amount of emission permits and has a unique technology that requires permits to produce output. We consider the solutions of sharing the optimal global surplus generated beyond the autarky economy output. We divide countries into two groups based on the types of contribution, the technology contributors and the permit contributors. Suppose that the division of the total surplus between the two groups is fixed at an arbitrary ratio \(\alpha \in [0,1]\) (Separation Principle of parameter \(\alpha \)). The fixed amount of surplus assigned to each group is distributed based on the contributions of the members of the group. By requiring that no subgroup of countries can increase their share by reallocating the total amount of their contributions among themselves (Permit Reallocation-Proofness and Technology Reallocation-Proofness), we characterize a family of solutions indexed by parameter \(\alpha \in [0,1]\), called the proportional solution of parameter \(\alpha \). By further requiring that each country receives at least the level of output it can produce with its own technology and permit (Voluntary Participation), we show that only the solution with \(\alpha =1/2\), called the equal share proportional solution, meets this requirement. Under this solution, the technology contributors and the permit contributors are treated equally.

1. In the cap and trade system, the distribution of permits are mostly determined by the developed countries who set their voluntary cap at their free access equilibrium level of emissions. In this case, the permits are distributed by countries’ ability to pollute. Countries with a higher total output or GDP have a higher share of the pollution permits.

   Unlike the current distribution of the total permits based on the ability to pollute, treating the clean atmosphere as a global common good to everyone, the permits may be distributed proportionally to the populations of countries. In this case, even if a country does not own any GHG emitting technology, it owns a certain amount of permits. See Baer (2002) for more on the equal per capita allocation of the GHGs.

2. Refer to the papers by Li et al. (2020); Suh (2001); Shin and Suh (2007) that have the same model and also treat the permits as a factor input.

3. The work of this paper is closely related to the literature on the axiomatic analysis of cost or surplus sharing problems, where economic agents contribute inputs and share the jointly produced output. The main difference between the models of this paper and the papers on the cost sharing problem is the specification of the utility functions and the technology. While the utility functions in the cost sharing problem play a crucial role, in this paper the utility is trivially determined by the level of output each country receives. Hence the richness of the model in this paper comes from their technologies, not from the utility functions of the countries. Refer to Moulin (2002) for a survey of the cost sharing problem. For the axiomatic investigation of the proportional solution, see Moulin (1987, 1990); Roemer and Silvestre (1993); Banker (1981); O’Neill (1982); Chun (1988, 1999); De Frutos (1999); Ju et al. (2007); Bergantiños et al. (2010).

4. We are not dealing with the two important questions in this paper, about the optimal level of pollution permits and the issue of “fair” division of the permits. If we treat the clean air as a common resource for all human beings regardless of nationality, the equal division of permits per capita may be the answer. But currently, those, who have technologies and hence can pollute, claim the ownership of the permits. This “free access equilibrium” may explain why the pollution level is so high. But if every individual in the world owns the equal amount of permits, then a developed country would have to pay to produce beyond the level of output possible from the permits it owns. Such an increase in cost would reduce both the global output and the level of GHGs, and would reduce global income inequality.

5. Since a technology is not a one dimensional object but is a function, it is difficult to find a solution that is responsive to the different production. Instead we require solutions to be responsive to the different outputs at which the total permit is distributed.

6. This problem is similar to the bankruptcy problem considered by O’Neill (1982); Chun (1988); De Frutos (1999); Ju et al. (2007); Bergantiños et al. (2010). For a survey, see Thomson (2003); Moulin (1985) introduced the condition Reallocation-Proofness. Ju et al. (2007) also used the condition as a main axiom extending Chun (1988) results in the bankruptcy problem.

7. In determining the ownership of permits, assuming an equal right to each individual, we may think of distributing the total permits equally per individual. But currently, whichever country emits more GHGs has a higher level of initial permits.

8. Refer to Suh (2001) for the manipulability of the price solution in the model.

9. Since the total amount of pollution is fixed, the model here does not deal with the issues related to allocating resources under externality.

10. For example, refer to Moulin (1990), where a technology is commonly owned and participants’ utility functions are defined over the input–output space. The richness of the model comes from the utility functions.

11. If a country has no technology, it is assumed that its production function is the zero function. The zero production function, which is not allowed in our model, can be considered as the limit of an infinite sequence of production functions. Since our purpose in this example is to motivate the Separation Principle, we ignore the technical issue raised by the zero production function.

12. The cooperative bargaining requires such information about what each group can achieve independently and what they achieve together. There is no additional information about the measure of each group’s relative contribution to facilitate solving the problem. In our problem, it is not possible to directly measure the relative contributions of the two groups in generating the surplus. Hence the sharing of the surplus between groups A and B is well represented by the model of the cooperative bargaining problem.

13. Kalai (1977) considered a general bargaining game, where the bargaining set is convex, compact and comprehensive, and characterized a family of proportional solutions. According to Kalai (1977)’s proportional solution, the distribution of the total output between the technology contributors and the permit contributors maintains a fixed ratio \(\alpha \in (0,1)\). The value \(\alpha \) may be considered as a weight that represents the relative bargaining power of the technology contributors to that of the permit contributors. Other bargaining solutions including the Nash bargaining solution with weights \((\alpha , 1-\alpha )\) would also maintain a fixed ratio \(\alpha \in (0,1)\) in distributing the surplus for the problem \(S(\lambda )\).

14. The idea of the Separation Principle may be explained simply as follows:

    1. (1)

       The total surplus is divided into the permit share and the technology share in a fixed proportion \(\alpha \).

    2. (2)

       The permit share of a country \(p_i\), for \(i\in S^+\) for example, is assumed to be a certain function of the total permit share and the permit contributions of the countries in \(S^+\). There is no restriction imposed on the function, \(p_i\) except that the sum of permit shares must be equal to \(\Delta P^-\). It is yet to be shown in Theorem 1 that the function \(p_i\) is a proportional solution.

    3. (3)

       A country’s total share is the sum of its permit share and technology share.

15. For the work related to the issue of manipulation via technologies in this model, refer to Suh (2001); Shin and Suh (2007).

16. Null Consistency is similar to null claims consistency in claims problems. See Thomson (2003).

17. The case for \(n=2\) is derived from the results for \(n\ge 3\) by applying Null Consistency.

18. Although our model is different from the claim problems in Chun (1988); Ju et al. (2007), the basic structure of the problem of sharing the total technology share \(\Delta T^+\) among the countries in \(S^+\) is the same as those of Chun (1988); Ju et al. (2007). In our model, the total technology share \(\Delta T^+\) corresponds to the asset shared among the claimants and \(\Delta y_i\) corresponds to the claim by agent i.

    The model in Ju et al. (2007) allows agents having multi-dimensional characteristics in bankruptcy problems and other resource allocation problems. Their model includes many prior existing results in bankruptcy problems, cost and surplus sharing problems, claim problems, etc. For example, the proportional solution in claim problems is a special case of their model. Here for the t function (also the p function), we can use Corollary 1 in Ju et al. (2007) to show that it is a proportional solution. Note that in Corollary 1 they use “no award for null” instead of Null Consistency. Here Null Consistency implies no award for null.

19. Note that \(\sum _{i\in S^+}\Delta x_i=-\sum _{j\in S^-}\Delta x_j\).

20. We can find such a problem that the condition (21) on p is satisfied in the price solution of the problem.

We are very grateful to the editor, Clemens Puppe, and Justin Leroux, Youngsub Chun, and an anonymous associate editor for their very helpful comments and suggestions on the previous versions of the manuscript. This is a revised version from the paper originally titled, “The Proportional Solution in a Permit Sharing Problem.” The authors declare that no funding was received for conducting this study. The authors have no competing interests to declare that are relevant to the content of this article.

Authors and Affiliations

1. Department of Economics, University of Windsor, Windsor, ON, N9B 3P4, Canada

   Sang-Chul Suh & Yuntong Wang

Corresponding author

Correspondence to Yuntong Wang.

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