Problem set 4: Two - period consumption - saving framework (10 points). Consider the two - pen'od economy (with zero govenunent spending and zero taxation). in which the representative consumer has no control over real income. At the beginning of period 1. the consumer knows that real income in period 1 is 3'1 . but is unsure about real income in period 2. There are two different incomes a relatively low income. denoted as yf. or a relatively high income. denoted as 1? . with I15" :.~. I15 the consumer could receive at the beginning of peiiod 2. i The consumer believes that there is a 30% probability that income 1-" . 2 will be received at the beginning of period 2 and a 70% probability that income yf will be received at the beginning of period 2. The consumer's utility finiction. which is ir(cl}+0.30-,B-ir(c )+ 0.70-'uicf] takes into account the unceitainty about second-period income with the probability weights 0.30 (the chance of receiving low real income y; 31 and 0.70 {the chance of receiving high real income yf ) embedded in the utility function. The consumer begins period 1 with zero net wealth (i.e.._ : t} ). The period-l budget constraint is. in real terms. I1'l -r'} 1] = 0. and {keeping in mind that a: = 0) the TWO possible budget constraints for period 2 are if +(l +3001 ('' 2 0. which. from the very beginning of period 1. occurs with 30% probability. and I1_H +(l +1901 cf 2 0. which. from the veiy beginning of period 1. occurs with 70% probability. Denoting by [i e {0.1} the one-period-ahead subjective discount factor. the sequential Lagrange function for the consumer's utility maximization is L =H{('l }+ UGO-nk:]+O.70-,B-u[r2\"].AI[I1'lcla1] +0.30--121L1'2L+(l+r)01r]+0.?01112H-I'[1f+(Ier)alcf]- which. note. contain three Lagrange multipliers. which are it for the period-1 budget constraint. )5 for the period-2 budget constraint with relatively low real income. and 2.2\" for the period-2 budget constraint with relatively high real income.