Some anisotropy in both mechanical and thermodynamical properties of bismuth is expected. A combination of density functional theory total energy calculations and density functional perturbation theory in the local density approximation is used to compute the elastic constants at 0 K using a finite strain approach and the thermal expansion tensor in the quasiharmonic approximation. The overall agreement with experiment is good. Furthermore, the anisotropy in the thermal expansion is found to arise from the anisotropy in both the directional compressibilities and the directional Grüneisen functions.