Exercise 13.8. Consider a world economy consisting of j = 1, ..., M economies. Suppose that each of those are closed and have access to the same production and R&D technology as described in Section 13.1. The only differences across countries are in the size of labor force, Lj , productivity of R&D, ηj , and discount rate ρj . You may also want to assume that one unit of R&D expenditure costs ζj units of final good in country j, and this varies across countries. There are no technological exchange across countries. (1) 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. (2) Characterize the world equilibrium. Show that in the world equilibrium, each country will grow at a constant rate starting at t = 0. Provide explicit solutionse for these growth rates. (3) Show that except in “knife-edge” cases, output in each country will grow at a different long-run rate. (4) 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 will lead to “infinitely large” differences in long-run income per capita. Does this resolve the empirical challenges discussed in those chapters? Does the environment in this exercise provide a satisfactory model for the study of long-run income per capita differences across countries? If yes, please elaborate how such a model can be mapped to reality. If not, explain which features of the model appear unsatisfactory to you and how you would want to change them.