Blazars are a sub-category of radio-loud active galactic nuclei with relativistic jets pointing towards to the observer. They are well-known for their non-thermal variable emission, which practically extends over the whole electromagnetic spectrum. Despite the plethora of multi-wavelength observations, the issue about the origin of the $gamma$-ray and radio emission in blazar jets remains unsettled. Here, we construct a parametric leptonic model for studying the connection between the $gamma$-ray and radio emission in both steady-state and flaring states of blazars. Assuming that relativistic electrons are injected continuously at a fixed distance from the black hole, we numerically study the evolution of their population as it propagates to larger distances while losing energy due to expansion and radiative cooling. In this framework, $gamma$-ray photons are naturally produced at small distances (e.g. $10^{-3}$ pc) when the electrons are still very energetic, whereas the radio emission is produced at larger distances (e.g. $1$ pc), after the electrons have cooled and the emitting region has become optically thin to synchrotron self-absorption due to expansion. We present preliminary results of our numerical investigation for the steady-state jet emission and the predicted time lags between $gamma$-rays and radio during flares.
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