Consider a world economy consisting of j = 1,...,J , economies. Suppose that each of these is closed and has access to the same production and R&D technology as described in Section 13.1. Countries differ in terms of labor force Lj, productivity of R&D ηj , and discount rate ρj . Also assume that one unit of R&D expenditure costs ζj units of final good in country j. There are no technological exchanges across countries.
(a) Define the "world equilibrium" in which each country is in equilibrium in analogy with the equilibrium path of the one-country economy in Section 13.1.
(b) Characterize the world equilibrium. Show that in the world equilibrium, each country grows at a constant rate starting at t = 0. Provide explicit solutions for these growth rates.
(c) Show that except in "knife-edge" cases, output in each country grows at a different long-run rate.
(d) Now return to the discussion in Chapters 3 and 8 regarding the effect of policy and taxes on long-run income per capita differences. Show that, in the model discussed in this exercise, arbitrarily small differences in policy or discount factors across countries lead to infinitely large differences in long-run income per capita. Does this property resolve the empirical challenges discussed in those chapters?