Gain and threshold current density are analyzed for quantum-box lasers where electrons are confined in quantum well three-dimensionally, based on the density-matrix theory of semiconductor lasers with relaxation broadening. The electronic dipole moment and its polarization dependence are first analyzed, and it is shown that the gain becomes maximum when the electric field of light is parallel to the longest side of the quantum box. Calculated gain is about 10 times that of bulk crystal for 100 Å × 100 Å × 100 Å GaAs/Ga0.8Al0.2As quantum box, and 15 times for Ga0.47In0.53As/InP quantum box with the same size, respectively. The threshold current density are 45 A/cm2and 62 A/cm2for GRINSCH GaAs/(Ga0.8Al0.2As-Ga0.4Al0.6As) and Ga0.47In0.53As/(Ga0.28In0.72As0.6P0.4-InP), respectively, where for the GaInAs/ GaInAsP/InP system the intervalence band absorption and nonradiative recombinations have been assumed to be the same as those obtained for bulk crystals experimentally. These results show the possibility of remarkable reduction in the laser threshold by the quantum-box structures.