Consider the following variant of the two-period endowment of the household studied in class (we apply the same notation used in the lectures). The household solves the following problem:

Consider the following variant of the two-period endowment of the household studied in class (we apply the same notation used in the lectures). The household solves the following problem: Clor (cr)1U + [3335 10' 10' subjectto c+s=yt, (1} c'=y't'+(1+r)s, (2} 5219, (3) where a > 1, and 6 (0, 1). Unlike the benchmark model studied in class, the household faces the borrowing constraint (3), which might or might not be binding for the household. Due to this constraint, the house hold can only borrow up to b 2 0. In this problem, you will analyse how this constraint shapes the response of the household to a scal stimulus package. 1. Plot the lifetime budget constraint of the household (based on (1)-(2)), along with the borrowing constraint (3). Use a diagram with c on the horizontal axis and c' on the vertical axis. 2. Find the optimal level of consumption in the rst period (c*) analytically, under two sce narios: (a) Assuming that the constraint (3) does not bind (i.e., holds with a strict inequality). (b) Assuming that the constraint (3) binds (i.e., holds with equality). Hint: Under (a), we can just ignore the constraint and write a Lagrangian using only con straints (1)-(2). 3. Show that, other things being equal, the borrowing constraint is more likely to bind when: i. (y' t") is high. ii. (y t) is low. iii. 19 is low. Provide an interpretation in each case. 4. Suppose that the government announces a scal stimulus package whereby it lowers taxes in the rst period t by A > 0, and it increases taxes in the second period by A(1 + r) (so as to keep the present value of taxes unchanged). How does 6* respond to the stimulus package if the constraint (3} is not binding? How does 0* respond to the stimulus package if the constraint (3) is binding? Explain. 5. Show the effect of the stimulus package when the constraint (3} binds, and when it does not bind using a diagram on the (c, 6') plane