Consider the model of endogenous technological progress with limited knowledge spillover as discussed in Section 13.3.

(a) Characterize the transitional dynamics of the economy starting from an arbitrary N(0) > 0.

(b) Characterize the Pareto optimal allocation, and compare it to the equilibrium allocation in Proposition 13.5.

(c) Analyze the effect of the following two policies: first, a subsidy to research; second, a patent policy, in which each patent expires at the rate ι > 0. Explain why the effects of these policies on economic growth are different than their effects in the baseline endogenous growth model.

Proposition 13.5

Suppose that ρ >(1- φ - θ)n/(1- φ). Then in the expanding input variety model with limited knowledge spillovers, there exists a unique BGP in which technology and consumption per capita grow at the rate g∗N, as given by (13.37), and output grows at the rate g∗ N + n.This analysis therefore shows that sustained (and stable) equilibrium growth of per capita income is possible in an economy with a 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 not be sufficient to sustain long-run growth. ontinuous population growth 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 note that there are two senses in which there are limited scale effects in these models. First, a faster rate of population growth
translates into a higher equilibrium growth rate. Second, a larger population size leads to higher output per capita (see Exercise 13.20). It is not clear whether the data support these types of scale effects either. Put differently, some of the evidence suggested against the scale effects in the baseline endogenous technological change models may be inconsistent with this class of models as well. For example, there does not seem to be any evidence in the postwar data or from the historical data of the past 200 years that faster population growth leads to a higher equilibrium growth rate. It is also worth noting that these models are sometimes referred to as "semi-endogenous growth" models, because while they exhibit sustained growth, the per capita growth rate of the economy, (13.38), is determined only by population growth and technology and does not respond to taxes or other policies. The literature has also developed models of endogenous growth without scale effects, with equilibrium growth responding to policies, though this normally requires a combination of restrictive assumptions.