The power allocation for multi-user sparse code multiple access (SCMA) downlink systems with finite- alphabet constraints is investigated. An explicit expression for the achievable rate for downlink SCMA systems with finite-alphabet inputs is derived, which is applicable to arbitrary number of users. Moreover, a novel power allocation scheme that can ensure users' fairness for multi-user SCMA downlink systems is proposed. In an effort to solve the formulated non-convex optimization problem, a low- complexity polynomial algorithm is proposed, which yields an optimal solution. Simulation results demonstrate that the proposed power allocation algorithm is capable of enhancing the performance significantly compared to the equal power allocation scheme.