R&D-based Calibrated Growth Models with Finite-Length Patents: A Novel Relaxation Algorithm for Solving an Autonomous FDE System of Mixed Type

The statutory patent length is 20 years in most countries. R&D-based growth models, however, often presume an infinite patent length. In this paper, finite-length patents are embedded in a non-scale R&D-based growth model, while allowing any patent’s effective life to be terminated prematurely, subject to two idiosyncratic hazards from imitation and creative destruction. This gives rise to an autonomous system of mixed-type functional differential equations (FDEs) that had never been encountered in the growth literature. Its dynamics are driven by current, delayed and advanced states. We present a relaxation algorithm to solve these FDEs by solving a sequence of standard boundary value problems for systems of ordinary differential equations. We use this algorithm to simulate a calibrated U.S. economy’s transitional dynamics by making discrete changes from the baseline 20 years patent length. We find that if transitional impacts are taken into account, the switch to the long-run optimal patent length can incur a welfare loss, albeit rather small.