This paper considers a dynamic lot-size model with make-or-buy decisions. In each period, the production level may not exceed a given capacity, but the purchase is unrestricted. The costs for production, purchase and inventory are concave. The problem is to find a minimum-cost production and purchase schedule that satisfies known demands over T periods. After characterizing the form of an optimal solution, we present an O(T5) algorithm when the production capacities are stationary. The algorithm is extended to the case where the capacities are integer multiples of a constant. The results are further extended to the cases of backlogging, bounded inventory, and capacitated purchase.