Introduction to Modern Economic Growth ∗ From equation (13.12), this implies that total output grows at the rate gN + n But now there is population growth, so consumption per capita grows only at the rate (13.37) ∗ gC∗ = gN n = 1−φ We can then use the consumer Euler equation, equivalent of (11.4) incorporating the fact that the discount factor is ρ − n instead of ρ, to determine BGP interest rate as ∗ +ρ−n r∗ = θgN φ − (1 − θ) = n + ρ > 1−φ The most noteworthy feature is that this model generates sustained and exponential growth in income per capita in the presence of population growth More interestingly, in order to achieve this growth rate, it allocates more and more of the labor force to R&D The reason for this is that the technology for creating new ideas, (13.33), only features limited spillovers, thus to maintain sustained growth, more resources need to be allocated to R&D Proposition 13.5 In the above-described expanding input-variety model with limited knowledge spillovers as given by (13.33), starting from any initial level of technology stock N (0) > 0, there exists a unique balanced growth path equilibrium ∗ as given by in which, technology and consumption per capita grow at the rate gN ∗ + n (13.36), and output grows at rate gN This analysis therefore shows that sustained equilibrium growth of per capita income is possible in an economy with growing population Intuitively, instead of the linear (proportional) spillovers in the baseline Romer model, the current model allows only a limited amount of spillovers Without population growth, these spillovers would affect the level of output, but would not be sufficient to sustain long-run growth Continuous population growth, on the other hand, steadily increases the market size for new technologies and generates growth from these limited spillovers While this pattern is referred to as “growth without scale effects,” it is useful to 592