Within the self-consistent-field approximation, a quantum-mechanical derivation is given for the dielectric response function of an arbitrarily degenerate free-electron gas which is subjected to a drift field. Neglecting in the equation of motion for the one-electron density operator a convection term, significant in the classical-collision-dominated regime only, the dielectric response function and the acoustic gain factor for a piezoelectrically active sound wave are obtained for the quantum and semiclassical-microscopic regimes. The manner in which the theory can be extended to the collision-dominated regime is discussed. For a collision-free electron gas, the requirements of energy and momentum conservation in individual electron-phonon interactions lead to a cutoff in the acoustoelectric coupling when the acoustic wave number exceeds the characteristic electron wave number. The broadening of this cutoff due to collisions is investigated and compared with thermal broadening. A number of useful approximations for the acoustic gain factor are derived.