In order to minimize compression power and analyze the cause of exergy loss for a dry ice production cycle with 3-stage compression, the variation of compression power was investigated and the exergy analysis was peformed for the cycle. In this cycle, $CO_2$, is used both as a refrigerant and as a raw material for dry ice. The behavior of compression power and irreversibility in the cycle were examined as a function of intermediate pressure. From this result, the conditions for the minimum compression power were obtained in terms of the first stage or the third stage pressure. In addition, the irreversibilities for the cycle were investigated with respect to the efficiency of compressor. Result shows that the optimum pressure is not consistent with the conventional pressure obtained from the equal-pressure-ratio assumption. This is mainly due to the change in mass flow rate of the intermediate stage compressor by the flash gas evaporation from the flash drums. Most important is that the present exergy analysis enabled us to find bad performance components for the cycle and informed us of methods to improve the cycle performance.