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Governments around the world are seeking an effective mechanism to cope with air pollution and climate change. The allocation of emission allowances, which is a key mechanism in the cap-and-trade system, is an important and intricate puzzle faced by environmental agencies. In this paper, we build a Stackelberg model to explore the emission allowance allocation mechanism design from an operations perspective. We demonstrate the feasibility and effectiveness of a linear emission allowance allocation mechanism. The results show that the emission allowance allocated by the government should always be insufficient to satisfy the ex-post emission demand at the industry level, even with low-carbon investment. To analyze the impacts on firms' decision-makings, we explore a scenario in which two firms in the same industry sell a homogenous product to the market. The optimal low-carbon investment and production decisions are significantly affected by these market and carbon-related factors. Numerical examples are presented to further demonstrate the results that our paper has derived and investigate the optimal operational decisions of the two firms. Several meaningful management insights on allocation mechanism design and low-carbon operations of firms are obtained. [ABSTRACT FROM AUTHOR]