Deriving the Initial Mass Function and the Star For- mation Rate (hereafter IMF and SFR) history of disc stars from local data implies assumptions concerning the vertical structure of the disc. Conversely, an accurate description of the vertical density of the stellar disc would require a precise knowledge of the stellar formation history. In this paper, the constraints avail- able from local data on these two aspects are reviewed, and the interdependence between the SFR, the secular heating of the disc stars, and the shape of the density prole in the z direction is quantied. As a consequence, the validity of exponential density mod- els to represent the vertical structure of the disc is questioned and the uncertainties that affect the IMF derivation from the local LF are reevaluated. We show that under realistic assump- tions on the SFR history, the observed local luminosity function gives little support for a bimodal IMF. Comparison of the ob- served local LF with synthetic LF brings tighter contraints on acceptable SFR and their corresponding IMF between 1 and 3 solar masses. These include decreasing SFR with IMF index x 1.5, constant SFR with IMF index x 2.0, or increasing SFR with IMF index x 2.5.