Properties of N-person axiomatic bargaining solutions if the Pareto frontier is twice differentiable and strictly concave

In this paper we discuss properties of N-person axiomatic bargaining problems, where the Pareto frontier of S can be described by a strictly concave and twice differentiable function. These type of problems are characteristic for the empirical policy coordination literature. In that literature the Pareto frontier of the bargaining problem coincides with the set of solutions a social planner finds, who maximizes a convex combination of N individual utility functions which are strictly concave and twice differentiable. We present an algorithm which determines the Nash bargaining solution much faster than the usual approach, in which one uses the standard optimization tools in order to maximize, straight away, the product of the players' benefits in relation to the gains of the disagreement point. Next, we show that it is possible to determine a subset of the Pareto frontier in which the Nash bargaining and Kalai-Smorodinsky solution will always fall. Furthermore, we consider effects of changes in the disagreement point d, for a fixed set S. If di increases, while for each j .ne. i; dj remains constant, than the corresponding Kalai-Smorodinsky solution has the property that player i is the only one who gains. This property is, however, not generally met for the Nash bargaining solution.