Consider the baseline AK model in Section 19.4. Suppose that production and allocation decisions within each country are made by a country-specific social planner (who maximizes the utility of the representative household within the country).
(a) Show that the allocation in the text is no longer an equilibrium. Explain.
(b) Characterize the equilibrium in this case and show that all of the qualitative results derived in the text apply. Provide generalizations of Propositions 19.11 and 19.12.
(c) Show that world welfare is lower in this case than in the equilibrium in the text.
(d) Do you find the equilibrium in this exercise or the one in the text more plausible? Justify your answer.
Proposition 19.11
In the general model with labor, the world equilibrium is characterized by (19.35) for each j and t, as well as the following two additional equations:
Proposition 19.12
In the general model with labor, there exists a unique steady-state world equilibrium. In this equilibrium, capital stock and output in each country grow at the constant rate g∗ as in (19.37), and the world steady-state growth rate g∗ is the unique solution to (19.38). This unique steady-state equilibrium is globally stable.