  We evaluate the charge and longitudinal spin response functions of a
two-dimensional electron gas with $e^2/r$ interactions in an arbitrary state of
spin polarization, using a structurally self-consistent approach to treat
exchange and correlations. From the results we assess the nature of the
magnetic order in the electronic ground state in zero magnetic field as a
function of electron density. We find that states of partial spin polarization
are thermodynamically unstable at all values of the coupling strength and that
a first-order phase transition occurs with increasing coupling strength from
the magnetically disorderd (paramagnetic) phase to the fully spin-polarized
(ferromagnetic) phase. This behavior is in qualitative agreement with diffusion
Monte Carlo data, although the location of the phase transition is
underestimated in our calculations.
