In a recent note, the idea was proposed that red-shift excess observed in quasars, could be interpreted as due to direct Compton effect caused by relativistic electrons. The purpose of the present paper is to give a physical basis justifying that, for which the Compton reddening formula and the cross-section of the process is needed. In order to derive first the Compton effect for moving electrons and secondly the effectivity of the process, we use the relativistic conservation equations (Section I) describing an elastic collision of a photon with a moving electron then (Secion II) the results from quantum electrodynamics for the cross section finally a model is proposed in Section III.