We calculated the elastic properties of Mg2SiO4 spinel, using the plane-wave pseudopotential method. The athermal elastic constants were calculated directly from the stress-strain relations up to 30 GPa, which encompasses the experimentally observed stability field of spinel. The calculated elastic constants are in very good agreement with Brillouin scattering data at zero pressure. We calculated the isotropically averaged elastic wave velocities and the anisotropy from our single crystal elastic constants. We find that the elastic anisotropy is weak (azimuthal and polarization anisotropy of S-waves: 5%, azimuthal P-wave anisotropy: 2.5%, at zero pressure) compared to other silicates and oxides. The anisotropy decreases initially with increasing pressure, changing sign at 17GPa before increasing in magnitude at higher pressures. At typical pressures of the earth's transition zone (20--25 GPa), the elastic anisotropy is 1% and 2% for P- and S-waves respectively. ¿ 1997 American Geophysical Union