In this paper, we study the resource allocation design for downlink multicarrier non-orthogonal multiple access systems with short-packet communications (MC-NOMA-SPC). In contrast to long- packet communications in conventional wireless systems, SPC suffers from a transmission rate degradation and a significant decoding error rate. Thus conventional resource allocation design based on the Shannon capacity assuming infinite blocklength is no longer optimal. In this paper, we employ the effective-throughput as the performance metric to evaluate the tradeoff between the transmission rate and the decoding error rate. Then, we jointly optimize the subcarrier assignment, transmission power allocation, and transmission rate adaptation of each user to maximize the total weighted effective-throughput subject to various practical constraints. Since the problem formulated belongs to a non-convex mixed integer non-linear programming (MINLP) problem, we develop an efficient algorithm based on the dynamic programming (DP) recursion framework to obtain its optimal solutions. In addition, we analyze the complexity of the proposed algorithm theoretically. Finally, simulation results show that the proposed optimal algorithm outperforms the suboptimal baseline schemes significantly.