We have employed a multiband envelope function method to study wurtzite [0001] InN nanowires of cylindrical cross section. The electronic subband structure and optical absorption spectrum are calculated as a function of the nanowire radius. The energies of the Γ-point conduction band states show a monotonous increase with decreasing radius. On the other hand, the size dispersion of the valence band states is more involved, showing various crossing and anticrossing effects due to the interplay between the confinement and band mixing effects. Thus, for small (R<2nm) nanowires the highest valence band state has dominant pz-symmetry, but for R>2nm the highest state has dominant p⊥-symmetry, although it is optically inactive. The optical absorption spectra polarized along or perpendicular to the nanowire axis are different on account of the different transitions involved. The spectra of the smaller nanowires display a distinguishable sequence of peaks reflecting the one-dimensional density-of-states. The absorption peaks become denser for larger nanowires, and for radii larger than 10nm the spectrum starts to ressemble that of bulk InN.