A Poisson regression model with an offset assumes a constant baseline rate after accounting for measured covariates, which may lead to biased estimates of coefficients in an inhomogeneous Poisson process. To correctly estimate the effect of time-dependent covariates, we propose a Poisson change-point regression model with an offset that allows a time-varying baseline rate. When the non-constant pattern of a log baseline rate is modeled with a non-parametric step function, the resulting semi-parametric model involves a model component of varying dimensions and thus requires a sophisticated varying-dimensional inference to obtain the correct estimates of model parameters of a fixed dimension. To fit the proposed varying-dimensional model, we devise a state-of-the-art Markov chain Monte Carlo-type algorithm based on partial collapse. The proposed model and methods are used to investigate the association between the daily homicide rates in Cali, Colombia, and the policies that restrict the hours during which the legal sale of alcoholic beverages is permitted. While simultaneously identifying the latent changes in the baseline homicide rate which correspond to the incidence of sociopolitical events, we explore the effect of policies governing the sale of alcohol on homicide rates and seek a policy that balances the economic and cultural dependencies on alcohol sales to the health of the public.