Problem Set 1
Oleg Itskhoki
Posted: January 20, 2021.
Due: January 30, 2021
Problem 1: Three-period consumption-savings problem
Consider a three-period decision problem:
max u(c0) + βu(c1) + β
2u(c2)
subject to the flow budget constraints for t = 0, 1, 2:
bt+1 = (1 + r)bt + yt − ct
,
where b0 is the initial wealth of the country.
1. Explain why in this three-period model it cannot be that b3 < 0 and it should not
be that b3 > 0.
2. Given this, use the flow budget constraints to derive the intertemporal budget constraint:
c0 +
c1
1 + r
+
c2
(1 + r)
2
= (1 + r)b0 + y0 +
y1
1 + r
+
y2
(1 + r)
2
.
Interpret this equation.
3. Show that the intertemporal budget constraint is equivalent to
(1 + r)b0 + nx0 +
nx1
1 + r
+
nx2
(1 + r)
2
= 0,
where nxt = yt − ct
. Explain why it is also equivalent to b0 + ca0 + ca1 + ca2 = 0,
where cat = rbt + nxt = bt+1 − bt
.
When is it possible to have nxt < 0 for every t = 0, 1, 2 and why? Which country
that we discussed might fit this description? Does it violate the logic that all trade
deficits must be compensated by trade surpluses?
1
4. Using your favorite method, derive the intertemporal optimality conditions for t = 0, 1:
u
0
(ct) = β(1 + r)u
0
(ct+1).
5. Assume b0 = 0 and β = 1 and r = 0. Solve for consumption c0, net exports nx0,
and current account ca0, by defining the concept of permanent income ¯y. Interpret
your results by providing examples for different y0 and ¯y.
Discuss intuitively how the result change when b0 < 0, β < 1, and r > 0.
Problem 2: Two-period model with investment
Consider a two-period economy facing the following budget constraints:
c1 + k + b ≤ y1,
c2 ≤ y2 + (1 + r)b,
where y1 is an exogenous endowment and second-period output
y2 = Akα
with 0 < α < 1 and productivity A. Note that k is both first period investment and
second period capital stock (implicitly assuming full depreciation, δ = 1). Also note that
initial b0 = 0, and hence b is both first-period current account and second-period net
foreign assets.
1. Explain how this special environment maps into the general framework of National
Income Accounts, and in particular why:
ca1 = y1 − c1 − k = b and ca2 = rb + y2 − c2 = −b.
2. Explain how to derive the intertemporal budget constraint:
c1 +
c2
1 + r
= y1 − k +
Akα
1 + r
.
3. Given this budget constraint, characterize the optimal capital investment k of the
country and interpret your results (how does optimal k depend on r and A, and why).
4. Explain why it is possible to determine optimal investment without characterizing
the optimal consumption-savings decision. In other words, why investment and
savings decisions separate and when would they not?
5. Why do we expect a country with a high A (relative to y1) to run a current account
deficit? What may be real-world examples of such countries?
2
Problem 3: “Current Account Deficits in the Euro Area”
Based on your reading of Blanchard and Giavazzi’s Brookings Papers article, evaluate as
being true, false or “it depends” the following statements (write 1–2 paragraphs for each):
1. Euro Zone integration in early 2000’s allowed Southern European countries to run
current account deficits because (1) they now faced lower interest rates and (2)
needed to borrow for investment to catch up to more developed countries of the
Northern Europe.
2. Policymakers should be acutely concerned about these current account deficits, as
they are driven by the private-sector borrowing as opposed to sovereign borrowing
by governments.
3. Closer financial and trade integration between countries allows for larger gaps between domestic investment and savings and should lead to larger current account
imbalances, with poorer developing countries running large current account deficits
against richer developed countries.