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

Problem 4: Fiscal Stimulus and Borrowing Constraints [20 marks] 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: C (CF)1G' max + [3 c.c',s l - C! l - 0' subject to 0+ 5 = y t, (l) c'zy't'+(1+r)s. (2) s 2 b, - (3) where o > 1, and [3' E (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. 2. Find the optimal level of consumption in the rst period (6*) 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. 11 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 (2* respond to the stimulus package if the constraint (3) is not binding? How does c* 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, 0') plane