Weighted fair division problem

Abstract

The exact problem we are concerned with in this paper is of the following nature. There are a finite number of producers each equipped with a utility function of the standard variety, which converts and input into a producer specific output. An allocation of the input among the producers is sought which is Pareto efficient i.e. there is no reallocation which increases the output of one producer without decreasing the output of any other. This, as is very widely known, corresponds to maximizing the weighted sum of the utility functions subject to a resource constraint. Alternatively, the weights can be interpreted as exogenously specified prices of the separate outputs and then the problem reduces to maximizing the aggregate revenue subject to a resource constraint. Out analysis focuses on the relations between the optimal solutions and the price and aggregate resource pair. Further, we also study the effect on the former of varying the latter pair.