This paper presents a poverty trap model at the firm level, which is driven by time capacity constraints, and involves the optimal allocation of time between a basic and a more-productive business. To operate the more-productive business requires additional time input but enhances the ability to process information in the future, which in turn, determines the firm's time capacity. As a result, a firm with a low ability may encounter challenges in expanding time capacity for further growth. We highlight the importance of passing proficiency thresholds in operating the more-productive business to achieve sustainable growth. The model can explain why aid that increases the ability to process information generates heterogeneous effects on firms, in terms of both short-run and long-run growth.