Question

1. A simple dynamic programming model of capital accumulation Consider the following economy. Individuals have preferences U Blog (ct) and a constraint of the form (a) Write the Bellman equation for this economy. (b) Find the FOC(s) that must be satisfied for an optimal consumption and capital plan. (c) Show the following consumption policy is consistent with the condition(s) that you produced in (b). objective. Show that it takes the form and determine the values of γ and θ. (d) Derive the value function by substituting the optimal consumption policy into the (e) Show that the optimal policy maximizes log(ct) + Bv(ker) where v(ke+i) is the value function derived in pard (d). (t) Now suppose that there is an arbitrary value function of the for nm which appears on the right hand side of the Bellman equation. Show that takes the form and determine the coefficients Yn and B(k Un (kt)-max (log(q) + βι'n-i (kt +1)) in this manner (g) Show that the sequence of coefficients(r) and 1%) and 8 values that you determined in part (b) from any initial yo and 8o-

Answer 1

Solaubion Given-that oll man's equa- HenceCp on Kt Foc s