Suppose there are three periods (t = 0, 1, 2) and that an individual has discounting function D(t) =1/(t+1) (hyperbolic discounting) and utility function u(x) = ln x. Therefore, from theperspective of time zero, the individual assigns utility

091 P U (CO, T1, (2) = In co + In Toto the consumption stream {mm :51, 12}. For simplicity, we will assume the individual receives income ya. = 11 in period I] and no other income? and that the interest rate is R = I]. Therefore, the budget constraint is just In + 9:1 + $2 = 11. (a) Find MRS:,t+1 for t = [i and t = 1 (see the slides if you forgot the formula]. {b} Use the M RSilt+1 = 1 +R condition, along with the budget constraint? to solve for the optimal consumption stream (35, mi, $5] from the perspective of time i]. {c} Suppose that in period I] the individual consumes the quantityr if. that you found in {b}. In period 1I what is the new budget constraint? Use this budget constraint {along with the MRSt,t+l = 1+R condition) to solve for the optimal stream (1, 2) from the perspective of period 1. (d) How does (of, $5) from part (b) compare to (1, 3} from (o)? Intuitiveljr, what explains the difference (if any}? Suppose there are three periods (t = 0, 1, 2) and that an individual has discounting function D(t) = 1 (hyperbolic discounting) and utility function u(r) = Inc. Therefore, from the perspective of time zero, the individual assigns utility