Stoichiometric Nickel-Titanium alloys show the so called shape memory effect. There are numerous applications for materials showing this effect, ranging from the automobile industry to medical technology. The shape memory effect is based on a martensitic phase transition during cooling such materials. By alloying Hafnium to the compound it is possible to raise the transition temperature to higher values. In this work the elastic constants of different phases appearing in the system are studied. The elastic constants may help to improve the understanding of the martensitic phase transition and in addition to that they may be used as an input for further micro-mechanical simulations of this transition. The elastic constants are computed within the framework of density functional theory (DFT), using the exciting- and the Wien2k code. First the elastic constants of the pure elemental solids are calculated, whereat convergence tests for the most important parameters of such simulations, concerning the bulk modulus, are done. Furthermore the results of the bulk modulus for the two used codes and different approximation to the exchange-correlation-potential are compared. Next these calculations are done for the binary phases taking part at the martensitic phase transformation. Finally those ternary structures of the NiTiHf system are computed, which turn out to be energetically favorable during so called cluster-expansion calculations. The achieved results of all these calculations are compared, if present, to experimental or other theoretically computed values found in the literature. Moreover the influence of the hafnium concentration on different elastic constants is discussed.