中文摘要：

假定在完全竞争市场中，厂商的生产函数具有先凸后凹的性质，即：边际资本产出在前一阶段里递增，而在后一阶段里递减；厂商的成本是投资的凸函数, 厂商的目标是选择最优的投资策略，以使得动态约束下的总利润达到最大, 首先构造了在经济学上具有一定代表性并且在数学上易于处理的凸凹生产函数；然后证明了厂商在前后两个阶段均衡状态的存在性、唯一性、稳定性，并据此利用相平面勾勒出了满足最优性必要条件的轨线；最后对处于前后不同阶段中两个鞍点解的最优性进行了分析, 同时，将动态结果与已有的静态结果进行了比较.

英文摘要：

Consider a finn in a competitive market. The production function is of a convex-concave shape, i.e., the marginal output of capital is increasing at an early stage and decreasing at a later stage. The cost function is a convex function of investment. The firm＇s objective is to select an optimal investment path so as to maximize the total profit subjected to a dynamic constraint. We first construct a convex-concave production fimction which is reasonable in economics and tractable in mathematics; Second, based on this kind of production function, we prove existence, uniqueness, and stability for the steady states of the two different stages and accordingly, the trajectories meeting the necessary optimal condition are depicted through a phase plane; Finally, the optimality is briefly analyzed for the two saddle solutions corresponding to the two different stages, and a brief comparison is made between our dynamic result and the existing static one.