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A central decision maker (CDM), who seeks an efficient allocation of scarce resources among a finite number, n, of players, often has to incorporate fairness criteria to avoid unfair outcomes. Indeed, the price of fairness (POF), a term coined in Bertsimas et al. (2011), refers to the efficiency loss due to the incorporation of fairness criteria into the allocation method. Quantifying the POF would help the CDM strike an appropriate balance between efficiency and fairness. In this paper we improve upon existing results in the literature, by providing tight bounds for the POF for the proportional fairness criterion for any n, when the maximum achievable utilities of the players are equal or are not equal, and we provide as well a tight bound for the max-min (Kalai - Smordinsky) fairness criterion, when the maximum achievable utilities of the players are not equal.