max ixf exp(-pt)u(c(t))dt, subject to initial capital k(0) and law of motion of capital given by f(k(1))-8k(t)- c(t) k(r) = { if f(k(t))-c(1) ≥ k if f(k(1)) - c(t) 8k* for k*= f(p+8) is a minimum investment size requirement. Suppose that k(0)= k*. Show that there does not exist a solution to this optimal control problem. Explain why and relate your answer to Theorem 7.15. (Hint: Show that k(t) = k* for all t, which would have been the optimal policy without the minimum investment size requirement, is not feasible. Then show that the value function that would obtain for k(t)=k* can be approximated arbitrarily closely by a policy that alternates between f(k(1)) - c(t)=0 for an interval of length A₁ >0 and f(k(t)) - c(t) = k for an interval of length A₂ >0 so as to keep k(t) close to k* on average. Then argue that any admissible pair (k(1), c(t)) can always be improved by a policy of this kind.]

max ixf exp(-pt)u(c(t))dt, subject to initial capital k(0) and law of motion of capital given by f(k(1))-8k(t)- c(t) k(r) = { if f(k(t))-c(1) ≥ k if f(k(1)) - c(t) 8k* for k= f(p+8) is a minimum investment size requirement. Suppose that k(0)= k. Show that there does not exist a solution to this optimal control problem. Explain why and relate your answer to Theorem 7.15. (Hint: Show that k(t) = k* for all t, which would have been the optimal policy without the minimum investment size requirement, is not feasible. Then show that the value function that would obtain for k(t)=k* can be approximated arbitrarily closely by a policy that alternates between f(k(1)) - c(t)=0 for an interval of length A₁ >0 and f(k(t)) - c(t) = k for an interval of length A₂ >0 so as to keep k(t) close to k* on average. Then argue that any admissible pair (k(1), c(t)) can always be improved by a policy of this kind.]

Economics (MindTap Course List)

13th Edition

ISBN:9781337617383

Author:Roger A. Arnold

Publisher:Roger A. Arnold

Chapter20: Consumer Choice: Maximizing Utility And Behavioral Economics

Section20.1: Utility Theory

Problem 3ST

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