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 $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] (Separation Principle of parameter $$\alpha $$ α ). 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 $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] , called the proportional solution of parameter $$\alpha $$ α . 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 $$\alpha =1/2$$ α = 1 / 2 , called the equal share proportional solution, meets this requirement. Under this solution, the technology contributors and the permit contributors are treated equally.