On Balanced Growth Path Solutions of a Knowledge Diffusion and Growth Model

In this paper, we study a Boltzmann-type mean field game model proposed in Achdou et al. [Philos. Trans. A, 372 (2014), pp. 1-19] for knowledge diffusion and economic growth, where knowledge diffusion results from imitation by searching and learning and from innovation subject to Brownian noises. Largely inspired by Dai et al. [SIAM J. Control Optim., 48 (2009), pp. 1134-1154; J. Econom. Theory, 146 (2011), pp. 1598-1630; J. Differential Equations, 246 (2009), pp. 1445-1469], where the marginal value function has been used directly to study portfolio selection with transaction costs, we transform the original partial integro-differential equation system into an equivalent one by also studying a representative agent's marginal value function. We show that a necessary condition to generate a sustained growth is that innovation cannot dominate imitation. In particular, when learning technology is sufficiently inefficient or discount rate is sufficiently low, either of which leads individuals to put no effort into imitation, sustained economic growth then disappears. Further, if there exists a balanced growth path solution, a continuum of such solutions indeed exists and there is a special one with the form conjectured in Achdou et al. [Philos. Trans. A, 372 (2014), pp. 1-19]. Finally, we propose a new method to conduct an extensive numerical analysis.