The elastic shear constants C=C44 and C'=12(C11-C12) of lithium have been measured as a function of pressure at room temperature. The measured values of dlnCdlnr and dlnC'dlnr are -4.1 and -2.8, respectively. This inequality of the two pressure variations is contrary to the situation found in sodium, where the two coefficients were equal. Theoretically, the two coefficients are expected to be the same, if only the electrostatic interaction between the electrons and the ions is responsible for the elastic stiffiness of metal. The different behavior in lithium can be understood in terms of the extra contribution to the shear constants arising due to the change in the Fermi energy of the electrons on shearing.