Over the ongoing time we all are facing the impact of ‘Climate Change’, which is also known as ‘Global Warming’ or in other words “An unnatural weather change”. One of the significant reasons are increase in level of Carbon dioxide in the atmosphere. Now a day, the main concern of the researchers is to discover how to reduce the carbon discharge to decrease the unnatural weather change. In our study, we investigate how firms manage carbon footprint in lot sizing problem under ‘Carbon Emission Trading’ mechanism. So, we used the concept of Pochet and Wolsey’s capacitated lot sizing problem formulation and give a new formulation with carbon footprint constraints. The problems under consideration are single level, the single item production problem with a finite time planning horizon and the demand is also deterministic. We also consider the shortage cost. As per our knowledge in the literature, there is no such work has been done over single item capacitated lot sizing problem with shortage variable and with strict carbon cap constraint. We considered initial inventory and end period shortage is zero (I₀ = S_T.= 0) Also, for shortage cost, in earlier formulations are unable to eliminate them; however, our formulation is able to eliminate all shortage (We used the concept from Shrama et al (2017)). This is a solid theoretical contribution as it offers significant computational savings.