  We study the short-time dynamics of systems that develop ``quasi long-range
order'' after a quench to the Kosterlitz-Thouless phase. With the working
hypothesis that the ``universal short-time behavior'', previously found in
Ising-like systems, also occurs in the Kosterlitz-Thouless phase, we explore
the scaling behavior of thermodynamic variables during the relaxational process
following the quench. As a concrete example, we investigate the two-dimensional
$6$-state clock model by Monte Carlo simulation. The exponents governing the
magnetization, the second moment, and the autocorrelation function are
calculated. From them, by means of scaling relations, estimates for the
equilibrium exponents $z$ and $\eta$ are derived. In particular, our estimates
for the temperature-dependent anomalous dimension $\eta$ that governs the
static correlation function are consistent with existing analytical and
numerical results and, thus, confirm our working hypothesis.
