Numerical simulations for scroll, two-stage twin rotary, and two-cylinder reciprocating compressors have been carried out to understand the effectiveness of each type compressor for heat pump water heater application using $CO_2$ as refrigerant. For suction pressure of 3.5 MPa and discharge pressure of 9 MPa, clearance volume ratio of the reciprocating compressor needs to be about 5% or less to have the volumetric efficiency comparable to that of the scroll compressor with tip clearance of $5;{mu}m$. Volumetric efficiency of the scroll compressor is quite sensitive to tip clearance. Adiabatic efficiency of the twin rotary compressor was calculated to be the lowest among the three types, and the most severe drawback of the $CO_2$ scroll compressor was a significant increase in the mechanical loss at the thrust surface supporting the orbiting scroll member. While the scroll compressor showed very smooth torque load variation, peak-to-peak torque variations of the twin rotary and two-cylinder reciprocating compressors were about 50% and 250%, respectively.