We analyze the decay of the first- and second-order correlations, ${g}^{(1)}(t)$ and ${g}^{(2)}(t)$, using a Langevin equation of the order parameter for polaritons in a microcavity, pumped nonresonantly with a noise-free laser diode. We consider a coupling of the condensate to the excited states by two types of scattering mechanism both derived from the polariton-polariton interaction: (a) one particle is scattered from an excited state to the ground state, while another excited-state particle is scattered to a higher-energy state, and (b) two ground-state particles are scattered into low-lying excited states with opposite momenta. This nonresonant scattering rate increases with the condensate population. We calculate from the Langevin equation for the order parameter the temporal decay of ${g}^{(1)}(t)$ and the linewidth $\ensuremath{\kappa}$ analytically. Our results make close contacts with the linewidth enhancement factor well known from semiconductor lasers. A semiclassical evaluation for the polariton kinetics yields a first-order correlation function, which is in good agreement with experimental results and with the results of Schwendimann and Quattropani, although our formulation is more complete because it contains also the dispersive effects of both types of scattering processes and uses no fitting parameters. Our analysis also provides an understanding of the rather different pump dependencies of the linewidth reported in the literature. The calculated second-order correlation function ${g}^{(2)}(0)$ is shown to stay for larger pump values as observed on a plateau above the coherent limit again due to the nonresonant scattering processes.