  We have extended previous kinetic results to compute the exchange correction
to the electrostatic electron susceptibility for arbitrary frequencies and
wavenumbers in the low temperature limit. This has allowed us to make a general
comparison with a much used hydrodynamic expression for exchange effects. It is
found that the hydrodynamic expression gives a useful approximation when the
phase velocity is roughly a factor $2.5$ larger than the Fermi velocity. For
low phase velocities, as for ion-acoustic waves, wave-particle interaction
leads to a strong enhancement of the exchange correction and the hydrodynamic
results is smaller by an order of magnitude.
