  A model describing amplification and quantum statistics of the exciton
polaritons with k=0 in a non-resonantly excited semiconductor quantum well
embedded in a microcavity is presented. Exploiting the bottleneck effect for
exciton polaritons, it is assumed that the polaritons with k different from 0
act as a reservoir. The time evolution of the k=0 polaritons is described by a
master equation, in which two- and one-polariton transitions between the mode
with k=0 and the reservoir are accounted for. The k=0 mode exhibits a threshold
depending on the material parameters and on the injected exciton density. Below
threshold the quantum statistics of the emission is characteristic of an
incoherent process, while above threshold it approaches that of a laser.
