Consider the following optimal control problem: subject to max x 1 e-p c(t) "/dt A=rA(t) = c(t) + w A(0) = 0 and lim A(t) = " 8047 where 0 < y < 1, I is the steady state value of A(t), w = 10 is a lifetime income, p = 0.05 and r = 0.02. a) (4 points) Write down the Hamiltonian. b) (4 points) Write down the first order conditions for a maximum. c) (4 points) Find the system of differential equations (for and A) that we need to solve to obtain the optimal c(t) and A(t). d) (8 points) Using the boundary conditions, find the optimal c(t) function. e) (5 points) Plot the solution c(t) and A(t) on a graph with time on the x-axis. Discuss the economic meaning of the solution. If you could not solve it, try to guess what the solution should be.