Use case analysis has been widely adopted in modern software engineering due to its strength in capturing the functional requirements of a system. It is often done with a UML use case model that formalizes the interactions between actors and a system in the requirements elicitation iteration, and with architectural alternatives explored and user interface details specified in the following analysis and design iteration. On the other hand, to better support decision making in software management, effort estimation models are required to provide estimates about the required project effort at the very early stage of a project, which, however, provides little information for accurately evaluating system complexity. To solve this dilemma, an incremental approach of integrating information available throughout the early iterations to provide multiple effort estimations is preferred in keeping the balance between utility and accuracy. In this paper, we proposed an effort estimation model that incorporates two sub-models to provide two points of effort estimation during the early iterations of a use case driven project. Our proposed model is lightweight due to the fact that its size metrics are defined to be countable directly from the artifacts of the early iterations. To better calibrate the model, especially in considering the situation of having limited data points available, we also introduced a normalization framework in our model calibration process to reduce noise from the effort data. By calibrating the proposed sub-models with the data points collected from 4 historical projects, we demonstrated that the sub-models fit the data set well, and the later-phase model is superior to the early-phase model for it fits the data set better and shows less uncertainty in the calibrated parameters.