The phase transitions, elastic and thermodynamic properties of MnPd are investigated using first-principles calculations. The elastic constant calculations indicate that the paramagnetic cubic structure is unstable to external strain and the softening of C11-C12 triggers the paramagnetic cubic phase transformation. The paramagnetic tetragonal structure is mechanically and dynamically stable. When considering the magnetic property, a antiferromagnetic tetragonal structure is lower in energy than the paramagnetic tetragonal structure. So we can conclude that the structural behaviors of MnPd alloys change from paramagnetic cubic to paramagnetic tetragonal, and then to antiferromagnetic tetragonal structure. Moreover, we employ quasiharmonic approximation to calculate the heat capacity and the Debye temperature.