A complete numerical solution of the evolution equation for parton multiplicities in quark and gluon jets with initial conditions at threshold is presented. Data on both hadron and jet multiplicities in e + e \Gamma annihilation are well described with a common normalization, giving further support to the picture of Local Parton Hadron Duality. Predictions for LEPII energies are presented. Furthermore we study the sensitivity to the cutoff parameter Q 0 and the scale of ff s . 1 to be published in the Proceedings of the XXVII International Symposium on Multiparticle Dynamics, Frascati, Italy, September 1997, to appear in Nucl. Phys. B (Proc. Suppl.) Perturbative QCD Description of Mean Jet and Particle Multiplicities in e + e \Gamma annihilation Sergio Lupia and Wolfgang Ochs a a Max-Planck-Institut fur Physik, (Werner-Heisenberg-Institut) Fohringer Ring 6, 80805 Munchen, Germany A complete numerical solution of the evolution equation for parton multiplicities in quark..