An individual lives for three periods (t = 1; 2; 3). There is one non-storable good. In period t, the individual receives utility u(xt) = ln(xt); where xt denotes period-t consumption of the good. The individual chooses x = (x1; x2; x3) to maximize the expected present value of their lifetime utility ow, with discount factor 2 (0; 1] applied between periods. The individual has income m at the beginning of period 1. Income not spent in one period can be saved (at zero interest) to the next period. Denote by pt the period-t price of the consumption good. Assume that p1 = p2 = 1, and that p3 is a random variable that takes on two di erent values: pL3 < 1 with probability 2 (0; 1), and pH3 > 1 with probability 1 . (a) Find the individual's utility maximizing choice x in the following cases: (i) 7 points The value of p3 is observed before period 1. (Thus, the individual can choose x1, x2, and x3 knowing p3.)