  The evolution of the particle distribution functions inside a relativistic
jet containing an e^-e^+ pair plasma and of the resulting $\gamma$-ray and
X-ray spectra is investigated. The first phase of this evolution is governed by
heavy radiative energy losses. For this phase, approximative expressions for
the energy-loss rates due to inverse-Compton scattering, using the full
Klein-Nishina cross section, are given as one-dimensional integrals for both
cooling by inverse-Compton scattering of synchrotron photons (SSC) and of
accretion disk photons (EIC). We calculate instantaneous and time-integrated
$\gamma$-ray spectra emitted by such a jet for various sets of parameters,
deducing some general implications on the observable broadband radiation.
Finally, we present model fits to the broadband spectrum of Mrk 421. We find
that the most plausible way to explain both the quiescent and the flaring state
of Mrk 421 consists of a model where EIC and SSC dominate the observed spectrum
in different frequency bands. In our model the flaring state is primarily
related to an increase of the maximum Lorentz factor of the injected pairs.
