2. (10 points) Should one prepared for old age? Cl John values consumption the same way every period, where abs) = 1;: and \"y = 2. He is impatient with I3 = 0.95. Something else that enters his lifetime utility is the chance of surviving to future periods, denoted by it. Say one period is 10 years. John starts to make plans when he turns 20; this is period 0. The maximum life span afterward is 6 more periods. 51 is John's chance of surviving to 30 while as is John's chance of surviving to 80. John's expected lifetime utility is Z: jjcj) where 51 = 0.987, n2 2 0.971, r33 = 0.940, a4 = 0.871, 55 = 0.743, and as = 0.506. Say John has stable income for the rst 5 periods of his life and no income afterwards. That is yo yl y; y3 '94 650, 000 and y5 2 ya = 0. The real interest rate, 'r, is 5%. Set up the problem with one intertemporal budget constraint as the following. ti 03' 6 93' Zn\")? Sinwv j=U i=0 (a) (6 points) Compute John's optimal consumption plan. (b) (4 points) John's wealth at period 3' is denoted by c,- and saving by 35,-. John has no inheritance; an 2 0. The law of motion for wealth is described as the following: s,=m_,+y,c, \"3+1 = \"j + 5;: Plot J ohn's consumption, saving, and wealth over his life time. Comment on your plots