TwoPeriod LifeCycle Model Without Savings 0 Without any savings, model just becomes two static problems: max U1 ([1, C1) 11,61 subject to: ll 2 T hl max u2(12, C2) 12,62 subject to: I2 = T h2 0 That is not terribly interesting so let's introduce some savings! Formal Model TwoPeriod LifeCycle Model With Savings 0 Let 5 denote savings (which can be negative) 0 Then consumption in periods one and two are given by: c1=w1h1+n1s C2=W2h2+n2+5 o where all notation is as in static model a ct: consumption in period t 0 wt: wage in period t o ht: hours worked in period t 0 nt: nonlabour income in period t o It is easy to see that your choice of s, h1 and h2 completely determine c1 and C2 0 So we have three choice variables to solve for o (and these choices implicity solve for c1 and c2) Formal Model Setting up TwoPeriod LifeCycle Model With Savings 0 Individual maximization problem: $33522; u1(I1, c1) + U202, C2) subject to: [1 = T h1 [2 = T h2 c1 = w1h1ln1s C2 = w2h2 ln2 +5 0 This looks really complex, but do not worry it will simplify! a sub those budget constraints in! a (there are some inequality constraints in there as well that c1 2 0 and C2 2 0 but ignore those for this class) Setting up Two-Period Life-Cycle Model With Savings . Individual maximization problem subbing in budget constraints: max u1 ( T - h1, wih1 + n1 - s) + u2( T - h2, wah2 + n2 + s) h1, h2,s . FOCs: a Ou1 = 0 + W1- Oh1 al ac a = 0 + W2 ac Oh2 al ac as ac . If u1 = u2 and w1 = w2 it is actually easy to see from FOCs individual will set /1 = /2 and c1 = C2 . individual equates marginal utilities of leisure and consumption across periodsExample 0 Bill knows he will live two periods. Bill has 16 hours a period to work or do leisure. His utility function in each period is given by: u(ct, It) 2 0.25!n(lt) l 0.75ln(ct). Bill gets an hourly wage of $20 in period 1 and knows that in period 2 he will be paid $30 an hour since he is more experienced. Bill gets a nonlabour income of $10 in each peod. o (a) How many hours will Bill work in each period and how much will he save (after period 1)? o (b) How many hours would Bill work in each period if the wage increase from $20 to $30 for period was unanticipated? (i.e., he expects in period 1 to also have a wage of $20 in period 2)