Finite-length patents and functional differential equations in a non-scale R&D-based growth model

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 relax- ation algorithm to solve the FDEs by solving a sequence of standard BVPs (boundary value problems) for systems of ODEs (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.