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6. (25 marks) Consider the following dynamic optimization problem alen (mm) + 1113711] , subject to $t+1= 5c; (1 at) at 6 (0,1) for all t 0 1. T.1, {at}:=Ol :30 > 0 given and we require 13:1 > O. (a) (2 marks) Dene the Hamiltonian for the problem. (b) (4 marks) For given (13,53, Ag), solve for a; from maximizing the Hamiltonian in part (a)- e) (2 marks) Derive the rst order conditions for the Hamiltonian H in part (a). ( (d) (15 marks) To solve for optimal triple (m*,u*, X") , list results derived in parts (b) and (c), and state equation as follows = derived in part (b) Atl = H; (part ((2)) $t+1 = g (t,:r:,u) (state equation) :50 given and conditions on AT (part (0)) . - a: A: BIG 1' (10 marks) Fmd AT: 11.}, it}, AT11 u}15 3'}13 and ATm \"EH2! m}2' ii. (5 marks) Based on what you have derived in part (i), try to nd general expres- sions for A1114, 11.3.4, 51:23.4 for all k = U, 1, 2, . . . ,T. (e) (2 marks) Verify if (s*,u*) derived in part (d) is the optimal solution