Problem4: Fiscal Stimulus and Borrowing Constraints[20marks]
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:
c1−σ (c0)1−σ max +β −
c,c0,s 1−σ 1 σ
subject to
c +s = y −t, c0 = y0 −t0 +(1+r)s, (1) (2)
s ≥−b, (3)
where σ> 1, and β∈(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 household can only borrow up to b ≥ 0. In this problem, you will analyse how this constraint shapes the response of the household to a fiscal stimulus package.
1. Plot the lifetime budget constraint of the household (based on (1)-(2)), along with the borrowingconstraint(3). Useadiagramwithc onthehorizontalaxisandc0 onthevertical axis.
2. Find the optimal level of consumption in the first period (c∗) analytically, under two scenarios:
(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 constraints (1)-(2).
3. Show that, other things being equal, the borrowing constraint is more likely to bind when: i. (y0 −t0) is high. ii. (y −t) is low.
iii. b is low.
Provide an interpretation in each case.
4. Suppose that the government announces a fiscal stimulus package whereby it lowers taxes in the first period t by ∆> 0, and it increases taxes in the second period by ∆(1+r) (so as to keep the present value of taxes unchanged). How does c∗ 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,c0) plane.

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