The extended Lagrangian method is applied to incorporate induced polarization effects in pure water, using molecular dynamics simulations, at a cost in computer time which is only twice that of modelling the corresponding pairwise additive system. Thermodynamic, structural and dynamical properties of the mean-field SPC and the polarizable PSPC models are compared. All simulations were performed in the canonical ensemble by use of the Nose thermostat method. We confirm that explicitly including induced polarization effects considerably improves the quality of the SPC model for the transport properties. We show that within some difference in simulation conditions, the method of the extended Lagrangian gives quite satisfactory results when compared to the more classical self-consistent iterative/predictive procedure previously used to calculate the non-additive effects. The latter part of this study assesses the importance of many-body effects in determining the properties of liquid water. This question is addressed with simulations, using a water model, where the induced dipole moment of the PSPC is represented by a single fixed point dipole positioned at the oxygen. It is shown that, whereas the many-body effects appear to have little influence on static properties and on the self-diffusion coefficient, their accurate and explicit representation significantly affects the orientational times.