econ question

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\f8. Consumption/ savings. (a) Irene solves the following optimization problem: max In co + In c1, Co , C1 subject to her lifetime budget constraint C1 co + y1 1 tr - = yo+ 1tr Solving the budget constraint for c1, we find c1 = (yo-co) (1+r) +y1 and can rewrite this problem as max In co + In [(yo - co) (1 + r) + y1]. co The first-order condition to this problem is 1 1+20 - c1 = co(1 +r). Co C1 Substituting the last expression for c1 directly above back into the lifetime budget constraint and then using this expression again, you can find co = =[yo + y1/(1 + r)] and c1 = 2 [yo(1 + r) + yi] (b) Irene would be a net borrower when co - yo = = [y1/ (1 +r) - yo] > 0, which is more likely when y1/yo is larger and when r is smaller. (c) Never. From the first-order condition, we have c1 = co(1 + r); given r > 0, c1 > co always holds. The intuition here is that Irene does not discount utility from future consumption relative to utility from current consumption. Put differently, Irene is quite patient, willing to sacrifice some of her consumption today pro- vided there is some reward (however small) in terms of additional future consumption from doing so (i.e., r > 0).8. Suppose Irene values current and future consumption (respectively de noted by co| and cl) according to the following lifetime utility function: U(co, cl) : lnco | In cl. Her initial lifetime wealth is yo + yl/(l | 7\"), where yo denotes her current disposable income, yl denotes her future disposable income, and r > 0 denotes the interest rate. (a) Find Irene's utility-maximizing choices of co and 01 as functions of 7', yo and y1. (b) Under what conditions is Irene more likely to be a net borrower (today)? Be as precise as possible. (c) Would Irene ever consume more today than in the future? Why or why not