Due to its minimal energy requirement the PoS consensus protocol has become an attractive alternative to PoW in modern cryptocurrencies. In this protocol the chance of being selected as a block proposer in each round is proportional to the current stake of any node. Thus, nodes with higher stakes will achieve more block rewards, resulting in the so-called rich-getting-richer problem. In this paper, we introduce a new block reward mechanism called the FRD (Fair Reward Distribution) mechanism, in which for each block produced, in addition to a major reward given to the block proposer, a small reward is given to all other nodes. We prove that this reward mechanism makes the PoS protocol fairer in terms of concentration of wealth by developing on the Bagchi-Pal urn model.