3. Within the context of the Ramsey-Cass-Koopmans model of optimal growth, assume that agents have instantaneous utility given by: e de(t) " (c(t)) = - where # > 0. In the economy, there is no technological progress, no labor growth, and assume that capital does not depreciate. Agents discount future utility at the subjective rate of p > 0. Technology is represented by Y (t) = F (K (t), L(t)) where F (.) exhibits constant returns to scale. Agents are endowed with one unit of labor. (a) Express the social planner's problem as the associated present-vlane Hamiltonian and derive the associated necessary conditions. (b) Construct the phase diagram associated with this economy. 4. Suppose that there is no population growth and that the per-capita production function is linear: y (t) = rk (t) Also assume that agents' instantaneous preferences are given by " (c) = 16:071 Inc ; 0 = 1 Capital depreciates at the rate of & and there is no technological progress. Do the following: (a) Express the social planner's problem in the form of a present-value Hamiltonian and derive the associated necessary conditions. Prof. Seungduck Lee Graduate Macroeconomic Theory 1 (b) Define the steady-state in this economy. (c) Suppose the economy is not at the steady-state. For this class of preferences and technology, we can solve for the policy function c(t) = ck (t)] . Conjecture that the policy function is linear: c(t) = nk (t) where , is a constant (to be determined). Show that this conjecture is verified and solve for n. Source: Romers' Advanced Macroeconomics and Kevin Salyer's notes