Summary: Evaluating the efficiency and the performance of decision making units (DMUs) at different time periods is one of the most critical and important issues of managers. Data envelopment analysis (DEA) is a powerful non-parametric technique to measure the relative efficiency of a set of DMUs where each DMU consumes multiple inputs to produce multiple outputs. In many DEA applications, DMUs are considered as systems with a two-stage structure. In these situations, two-stage DEA models are used to measure the efficiencies of these systems. In many of such systems, the simultaneous presence of two stages is not necessary for the final product and the shortcoming of one stage is compensated by another stage. Therefore, this paper will use compensatory property of the sum operator and will propose the additive model to measure the multi-period efficiency of these systems under the constant returns to scale (CRS) assumption. In addition, based on the obtained efficiencies, the new efficiency changes Indexes (ECIs) related to the whole system and the first and second stages between two periods will be proposed that have circularity property. Furthermore, ECI of the whole system (and stages) for two periods is defined as the difference between the efficiencies in these periods. Moreover, positive changes (or negative changes), or unchanged in the efficiency of stages will be concluded by the positive changes (or negative changes), or unchanged of the whole system. Finally, the data of 21 non-life insurance industry in Taiwan are used to describe our suggested model that extracted from the extant literature.