Mechanism design for a social utility being the sum of agents' utilities (SoU) is a well-studied problem. There are, however, a number of problems of theoretical and practical interest where a designer may have a different objective than maximization of the SoU. One motivation for this is the desire for more equitable allocation of resources among agents. A second, more subtle, motivation is the fact that a fairer allocation indirectly implies less variation in taxes which can be desirable in a situation where (implicit) individual agent budgetary constraints make payment of large taxes unrealistic. In this paper we study a family of social utilities that provide fair allocation (with SoU being subsumed as an extreme case) and derive conditions under which Bayesian and Dominant strategy implementation is possible. Furthermore, it is shown how a simple modification of the above mechanism can guarantee full Bayesian implementation. Through a numerical example it is shown that the proposed method can result in significant gains both in allocation fairness and tax reduction.