For a range of lattice parameters the Kohn–Sham equations were solved self-consistently in the muffin-tin and local spin density approximations for Li, Na, and K metals. The solutions were used to evaluate the spin susceptibilities χp via the variational principle of Vosko and Perdew for two choices of χh, the spin susceptibility of an interacting homogeneous electron gas: (i) the RPA result due to Shastry, and (ii) the Keiser and Wu (KW) result. For Li and K the volume dependence and absolute values of χp are in excellent agreement with experiment whereas for Na the RPA seems to be better. Calculations for K over an extended range of volumes show that non-linearity in χp sets in for V/V0 = 0.9.