The results of numerical simulations of red sprite discharges, namely the temporal evolutions of optical emissions, are presented and compared with observations. The simulations are done using the recently recalculated runaway avalanche rates. The temporal evolution of these simulations is in good agreement with ground-based photometer and CCD TV camera observations of red sprites. Our model naturally explains the hairline of red sprites as a boundary between the region where the intensity of optical emissions associated with runaway breakdown has a maximum and the region where the intensity of optical emissions caused by conventional breakdown and ambient electron heating has a maximum. Other important characteristics of the simulated sprite such as color, shape, altitude, and intensity are in agreement with the observations as well. We also present for the first time simulations of red sprites with a daytime conductivity profile. ¿ 1999 American Geophysical Union