This paper investigates the tradeoff between the average age of information (AoI) and the transmission cost for networks with stochastic packet arrival and random erasure channel. Specifically, we model the resource-constrained AoI minimization problem as a constrained Markov decision process (CMDP) and propose a low-complexity single threshold packet scheduling policy for it. The key advantage of the proposed policy is its tractability and convenience for implementation. The AoI distribution and long-term average transmission cost of the proposed policy are derived as closed-form functions of the selected threshold. Furthermore, we show that the proposed policy reduces to the optimal policies under special settings and achieves close-to-optimal performance under general settings.