Introduction to Modern Economic Growth We next specify the technology for producing machines of different qualities and the innovation possibilities frontier of this economy First, new machine vintages are invented by R&D The R&D process is cumulative, in the sense that new R&D builds on an existing machine type For example, consider the machine line ν that has quality q (ν, t) at time t R&D on this machine line will attempt to improve over this quality If a firm spends Z (ν, t) units of the final good for research on this machine line, then it generates a flow rate ηZ (ν, t) /q (ν, t) of innovation The innovation advances the knowhow on the production of this machine to the new rung of the quality ladder, thus creates a machine of type ν with quality λq (ν, t) Note that one unit of R&D spending is proportionately less effective when applied to a more advanced machine This is intuitive, since we expect research on more advanced machines to be more difficult It is also convenient from a mathematical point of view, since the benefit of research is also increasing with the quality of the machine (in particular, the quality improvements are proportional, with an innovation increasing quality from q (ν, t) to λq (ν, t)) Note that the costs of R&D are identical for the current incumbent and new firms We assume that there is free entry into research, thus any firm or individual can undertake this type of research on any of the machine lines As in the expanding varieties models of the previous chapter, the firm that makes an innovation has a perpetual patent on that new machine that it creates However, note that the patent system does not preclude other firms undertaking research based on the product invented by this firm We will discuss below how different patenting arrangements might affect incentives in this model Once a particular machine of quality q (ν, t) has been invented, any quantity of this machine can be produced at the marginal cost ψq (ν, t) Once again, the fact that the marginal cost is proportional to the quality of the machine is natural, since producing higher-quality machines should be more expensive One noteworthy issue here concerns the identity of the firm that will undertake R&D and innovation In the expanding varieties model, this was irrelevant, since machines could not be improved upon, so there was only R&D for new machines, and who undertook the R&D was not important Here, in contrast, existing machines can be (and are) improved, and this is the source of economic growth We have 612