PS1 Solution.pdf - Tutorials in International...

Tutorials in International Macroeconomics
PS1: Intertemporal approach to the current account
1 Model solution with growth shocks
Consider the baseline model from the lecture which assumes quadratic consumption utility
and (1 + r ) β = 1. Assume that output follows a growth process, i.e. { Δ y
t
} is an AR(1)
process with decay coefficient 0 < ρ < 1.
1. Set up the household’s problem and derive the Euler equation.
2. State the transversality condition.
3. Derive the policy function for consumption c
t
= f ( y
t
, y
t - 1
, d
t - 1
) by following the same
steps as in the lecture. Why does the policy function for consumption depend on y
t - 1
?
4. Also state the policy function for debt d
t
and for the current account ca
t
.
2 Guess and verify
Consider the same set-up as in Problem 1 on this Problem Set. As in the lecture, you had
obtained the policy function for consumption by using the intertemporal resource constraint
of the economy. Another way of deriving f is called the “guess and verify” approach, which
works as follows.
1. Guess that the policy function is of the form
c
t
= ζ
0
+ ζ
1
y
t
+ ζ
2
y
t - 1
+ ζ
3
d
t - 1
,
for a set of unknown coefficients { ζ
0
, ζ
1
, ζ
2
, ζ
3
} .
2. Insert the guess into the model equilibrium conditions (the one-period resource con-
straint and the Euler equation)
3. Choose coefficients { ζ
0
, ζ
1
, ζ
2
, ζ
3
} such that the model equilibrium conditions are always
satisfied. You should obtain the same coefficients as in the policy function of Problem
1, which tells you that you have obtained the correct solution of the model.
3 Anticipated endowments
Consider again the same model as in Problems 1-2 on this Problem Set, but now assume that
{ y
t
} follows the process
y
t
= ρy
t - 1
+ ε
t - 1
,
that is, an AR(1) but the shock entering from period t - 1. This specification implies that
endowment shocks y
t
are known one period in advance!