Governments and organizations design performance-based research funding systems (PBRFS) for strategic aims, such as to selectively allocate scarce resources and stimulate research efficiency. In this work, we analyze the relative change in research productivity of Italian universities after the introduction of such a system, featuring financial and reputational incentives. Using a bibliometric approach, we compare the relative research performance of universities before and after the introduction of PBRFS, at the overall, discipline and field levels. The findings show convergence in the universities' performance, due above all to the remarkable improvement of the lowest performers. Geographically, the universities of the south (vs central and northern Italy) achieved the greatest improvement in relative performance. The methodology, and results, should be of use to university management and policy-makers.