First-principles total-energy calculations on the tetragonal states of bulk elemental palladium have been made with a full-potential electronic structure program with both the local-density approximation (LDA) and the generalized gradient approximation (GGA). The unique path through tetragonal states produced by epitaxial strain on equilibrium states has been found. This path, called the epitaxial Bain path, shows that body-centered cubic Pd (axial ratio c/a=1) is unstable, but a shallow energy minimum exists for a body-centered tetragonal (bct) state with axial ratio c/a~0.9. Structure parameters and elastic constants of both the face-centered cubic (fcc) ground state and the bct state are determined and the fcc values are compared to experiment: the LDA results are better than the GGA results. Tests of the stability of the bct phase at c/a~0.9 show that this phase is unstable with respect to  shear in the (001) plane.