Gamma-ray burst prompt emission is believed to originate from electrons accelerated in a highly relativistic outflow. Internal shocks, which are a result of collisions between shells ejected by the central engine, are the leading candidates for electron acceleration. While synchrotron radiation is generally invoked to interpret prompt gamma-ray emission within the internal shock model, the synchrotron self-Compton (SSC) process is also considered as a possible candidate for the radiation mechanism. In this case, we would expect a synchrotron emission component at low energies, and the “Naked-Eye Burst” GRB 080319B is considered to be such an example. Considering that the gamma-ray light curve of GRB 080319B is much more variable than its optical counterpart, in this paper, we study the relative variability between the synchrotron and SSC components. We develop a ‘top-down’ formalism using observed quantities to infer the physical parameters, and subsequently to study the temporal structure of the synchrotron and SSC components of a GRB. We complement the formalism with a ‘bottom-up’ approach, where the synchrotron and SSC light curves are calculated using Monte Carlo simulations of the internal shock model. Both approaches lead to the same conclusion. Small variations in the synchrotron light curve can only be moderately amplified in the SSC light curve. Therefore, the SSC model cannot adequately interpret the gamma-ray emission properties of GRB 080319B.