Exercise 14.23. Set up the social planner’s problem (of maximizing the utility of the representative household) in Section 14.3. (1) Show that this maximization problem corresponds to a concave current-value Hamiltonian and derive the unique solution to this problem. Show that this solution involves the consumption of the representative household growing at a constant rate at all points. (2) Show that the social planner will tend to increase growth because she avoids the monopoly markup over machines. (3) Show that the social planner will tend to choose lower entry because of the negative externality in the research process. (4) Give numerical examples in which the growth rate in the Pareto optimal allocation is greater than or less than the decentralized growth rate.