The growth process for a technological leader is different from that of a follower. While followers can grow through imitation and capital deepening, a leader must undertake original research. This suggests that as the gap between the leader and the follower narrows, the follower must undertake more formal R&D and possibly face a slower overall growth rate. This paper examines these ideas by discussing some simple models of technological catch-up and convergence and then applying them to the relative growth experiences of US and Japanese manufacturing. We construct measures of relative total factor productivity for eleven Japanese manufacturing industries and test whether a smaller productivity gap leads to slower growth, and whether R&D takes over as the engine of growth as Japan approaches the technological frontier. Our results suggests that Japanese and US productivity have been growing at similar rates since the mid-1970s, and that some of the Japanese growth slowdown is attributable to the exhaustion of imitation possibilities. Furthermore, since Japanese total factor productivity growth is faster than US growth before the mid-1970s, our results cast doubt on much of the cross-section convergence literature that assumes similar technology parameters across countries.