Problem Set 4 Due: Tuesday, April 12 ECON 434: International Finance and Macroeconomics Penn State: Spring, 2016 1. Government Budget Constraints. This problem looks at the feasibility constraints facing the government and the sustainability of current-account balances. This will generalize some of the results we obtained in class. __ Consider an economy that lasts for N periods. In period t, the government purchases Gt dollars worth of goods, and it collects Tt dollars worth of taxes. The government can also purchase B g t bonds in period t; if it holds B g t bonds in period t, then it receives (1 + r)B g t bonds in period t + 1. (For simplicity, assume that the interest rate r is constant.) If B g t < 0, then it means the government is in debt. (a) What is the government's period-t budget constraint? (Your answer should contain Gt, Tt, B g t , and r.) (b) How do you compute the primary �scal de�cit in period t? How do you compute the secondary �scal de�cit in period t? (c) Combine the period budget constraints for t = 1, . . . ,N to show that: B g 0 + N∑ t=1 Tt (1 + r) t = N∑ t=1 Gt (1 + r) t + B g N (1 + r) N . (1) (d) Suppose that N is a �xed, �nite number. What condition does B g N have to satisfy, and why? (e) Individuals who pay taxes have �nite lives, but institutions, such as governments, can live a long time, possibly forever. This leads to the possibility that a government could, in principle, keep rolling over its debt, even as generations of citizens come and go.1 Mathematically, we model this by letting N →∞. What condition does B g N (1+r)N have to satisfy as N →∞? Explain your answer in words. (f) Suppose that the government starts out in debt, with B g 0 < 0. Is it possible for the government to run primary de�cits forever? Why or why not? (g) Now, suppose that the government starts out with positive assets B g 0 > 0. We'll look at the case of an in�nitely-lived government (N = ∞), and we'll assess whether it's possible for the government to make purchases while never taxing its citizens (i.e., Tt = 0, for t = 1,2, . . .). __ Consider the following plan. Before making purchases in period t, the government has (1 + r)B g t−1 dollars at its disposal from the interest it earned on its previous period's assets. The government decides to take a fraction δ of this money to spend on period-t government purchases Gt; the remaining fraction 1−δ is used to buy more bonds. i. Provide an expression for B g t in terms of B g t−1, and provide an expression for Gt in terms of B g t−1. (Both expressions will depend on δ and r.) ii. Provide an expression for B g t in terms of B g 0 and t, and provide an expression for Gt in terms of B g 0 and t. (Both expressions will depend on δ and r.) 1For an example, see �The Case of the Undying Debt� by François Velde: https://www.chicagofed.org/publications/working- papers/2009/wp-12. 1 iii. In each period t, does the government run a primary surplus or de�cit.

1. Problem Set 4
Due: Tuesday, April 12
ECON 434: International Finance and Macroeconomics
Penn State: Spring, 2016
1. Government Budget Constraints. This problem looks at the
feasibility constraints facing the
government and the sustainability of current-account balances.
This will generalize some of the results
we obtained in class.
__ Consider an economy that lasts for N periods. In period t, the
government purchases Gt dollars
worth of goods, and it collects Tt dollars worth of taxes. The
government can also purchase B
g
t bonds
in period t; if it holds B
g
t bonds in period t, then it receives (1 + r)B
g
t bonds in period t + 1. (For
simplicity, assume that the interest rate r is constant.) If B
g
t < 0, then it means the government is in

2. debt.
(a) What is the government's period-t budget constraint? (Your
answer should contain Gt, Tt, B
g
t ,
and r.)
(b) How do you compute the primary �scal de�cit in period t?
How do you compute the secondary
�scal de�cit in period t?
(c) Combine the period budget constraints for t = 1, . . . ,N to
show that:
B
g
0 +
N∑
t=1
Tt
(1 + r)
t
=
N∑
t=1
Gt
(1 + r)
t

3. +
B
g
N
(1 + r)
N
. (1)
(d) Suppose that N is a �xed, �nite number. What condition
does B
g
N have to satisfy, and why?
(e) Individuals who pay taxes have �nite lives, but institutions,
such as governments, can live a long
time, possibly forever. This leads to the possibility that a
government could, in principle, keep
rolling over its debt, even as generations of citizens come and
go.1 Mathematically, we model this
by letting N →∞. What condition does B
g
N
(1+r)N
have to satisfy as N →∞? Explain your answer
in words.
(f) Suppose that the government starts out in debt, with B
g
0 < 0. Is it possible for the government
to run primary de�cits forever? Why or why not?

4. (g) Now, suppose that the government starts out with positive
assets B
g
0 > 0. We'll look at the case of
an in�nitely-lived government (N = ∞), and we'll assess
whether it's possible for the government
to make purchases while never taxing its citizens (i.e., Tt = 0,
for t = 1,2, . . .).
__ Consider the following plan. Before making purchases in
period t, the government has
(1 + r)B
g
t−1 dollars at its disposal from the interest it earned on its
previous period's assets.
The government decides to take a fraction δ of this money to
spend on period-t government
purchases Gt; the remaining fraction 1−δ is used to buy more
bonds.
i. Provide an expression for B
g
t in terms of B
g
t−1, and provide an expression for Gt in terms of
B
g
t−1. (Both expressions will depend on δ and r.)
ii. Provide an expression for B
g

5. t in terms of B
g
0 and t, and provide an expression for Gt in terms
of B
g
0 and t. (Both expressions will depend on δ and r.)
1For an example, see �The Case of the Undying Debt� by
François Velde:
https://www.chicagofed.org/publications/working-
papers/2009/wp-12.
1
iii. In each period t, does the government run a primary surplus
or de�cit? In each period t, does
the government have positive or negative asset holdings?
iv. Does this plan satisfy the government feasibility constraint?
(Hint: Compute
∑N
t=1 (1 + r)
−t
Gt,
and check whether (1 + r)
−N
B
g
N satis�es the condition from part (d) as N →∞. That will

6. allow you to check wither the present-value budget constraint
you derived in part (c) is
satis�ed.)
2. A Square-root Utility Function and Lump-Sum Taxes. In this
problem, we'll return to the
square-root utility function we used in Problem Set 3. The
advantage of this utility function is that it
makes it easy to compute marginal utilities. I suggest that you
make sure you're comfortable with the
solutions to question 2 from Problem Set 3 before proceeding.
__ Assume that:
U (C1,C2) =
√
C1 + β
√
C2, (2)
where β is a number that represents how much the household
likes period-two consumption, relative
to period-one consumption. In Problem Set 3, you established
that the marginal utilities of period-one
consumption and period-two consumption are, respectively:
U1 (C1,C2) =
1
2×
√
C1
, U2 (C1,C2) =
β

7. 2×
√
C2
. (3)
__ Consider a two-period small open economy in which there's a
representative household that has
the utility function given in equation (2). In period t, the
household's endowment is Qt, and it has
to pay Tt in lump-sum taxes. The household chooses
consumption Ct and private bond holdings B
p
t .
The household takes the constant world interest rate r as given.
Assume B
p
0 = 0.
__ In period t, the government collects Tt in taxes and
purchases Gt. The government can also have
bond holdings B
g
t , also at the interest rate r. Assume B
g
0 = 0.
(a) How do you compute the trade balance in this environment?
(b) Derive the household's present-value budget constraint.
(This should contain C1, C2, T1, T2, Q1,
Q2, and r, but not Gt nor B

8. p
t for any t.)
(c) With C1 on the horizontal axis and C2 on the vertical axis,
sketch the household's budget con-
straint. Label the horizontal and vertical intercepts. What is the
slope of the budget line? Sketch
the indi�erence curve that passes through the optimal
consumption bundle.
(d) The slope of the indi�erence curves will be given by the
negative marginal rate of substitution
U1(C1,C2)
U2(C1,C2)
. For the optimal consumption bundle, the slope of the
indi�erence curve will be equal
to the slope of the budget constraint. Given this fact, and the
special form of the utility function,
what is the optimal ratio C1/C2 chosen by the household, as a
function of r and β? (You can
consult the answers to the previous problem set, but you have to
understand what does or does
not change now that we've added �scal policy.)
(e) Derive the government's present-value budget constraint.
(This should contain G1, G2, T1, T2,
and r, but not Ct, Qt, nor B
g
t for any t.)
(f) Now, let's go the aggregate level and see how consumption
and trade balances depend on �scal

9. policy.
i. Combine the household budget constraint with the
government budget constraint to eliminate
T1 and T2.
ii. Combine your answer to part i. with your answer to part (d)
to obtain an expression for C1
as a function of Q1, Q2, G1, G2, r, and β.
iii. Combine your answer to parts i. and ii. to obtain an
expression for C2 as a function of Q1,
Q2, G1, G2, r, and β.
iv. What's the period-one trade balance, as a function of Q1,
Q2, G1, G2, r, and β?
2
v. Suppose that the government cuts taxes in period t = 1 and
compensates by raising taxes in
period t = 2. Government spending is unchanged. What is the
e�ect on the trade balance in
each period?
vi. If G1 increases, will the period-one trade balance increase or
decrease? If the change in G1 is
one dollar, will the change in TB1 be more than a dollar or less
than a dollar?
3. A Square-root Utility Function and Distortionary Taxes. This
problem will look at the e�ects
of a consumption tax, or sales tax. Again, we'll consider a two-
period small open economy with an

10. endowment. Like before, the household has a square-root utility
function.
__ The di�erence is that now, the government levies a
consumption tax. That is, Ct units of con-
sumption in period t costs the household (1 + τt)Ct dollars, with
τtCt going to the government in the
form of taxes. The household still gets an endowment Qt in each
period t, and the household can buy
private bonds B
p
t that earn the world interest rate r. Assume B
p
0 = 0.
__ We'll also assume that the government doesn't purchase any
goods, even though it's collecting tax
revenues. Instead, it gives the household Et dollars as part of an
entitlement program. This is a lump-
sum payment to the household, and it doesn't depend on
anything the household does. (Equivalently,
you can think of it as the negative of a lump-sum tax.) The
government can buy bonds B
g
t that earn
the world interest rate r, and assume B
g
0 = 0.
(a) How do you compute the trade balance in this setting? (Hint:
Taxes and within-country transfers
represent the reallocation of goods within the country, but they
don't entail the movement of

11. goods between countries.)
(b) Derive the household's present-value budget constraint.
(This should contain C1, C2, τ1, τ2, Q1,
Q2, E1, E2, and r, but not B
p
t for any t.)
(c) With C1 on the horizontal axis and C2 on the vertical axis,
sketch the household's budget con-
straint. Label the horizontal and vertical intercepts. What is the
slope of the budget line? Sketch
the indi�erence curve that passes through the optimal
consumption bundle.
(d) Again, the slope of the indi�erence curves will be given by
the negative marginal rate of substi-
tution −U1(C1,C2)
U2(C1,C2)
. For the optimal consumption bundle, the slope of the
indi�erence curve will
be equal to the slope of the budget constraint. Given this fact,
and the special form of the utility
function, what is the optimal ratio C1/C2 chosen by the
household, as a function of r, β, τ1, and
τ2? (Hint: This will not be the same as the condition you
derived in question 2.)
(e) Derive the government's present-value budget constraint.
(This should contain C1, C2, τ1, τ2, E1,
E2, and r, but not B

12. g
t for any t.)
(f) Now, let's go the aggregate level and see how consumption
and trade balances depend on �scal
policy.
i. Combine the household budget constraint with the
government budget constraint to eliminate
E1 and E2.
ii. Combine your answer to part i. with your answer to part (d)
to obtain an expression for C1
as a function of Q1, Q2, τ1, τ2, r, and β.
iii. Combine your answer to parts i. and ii. to obtain an
expression for C2 as a function of Q1,
Q2, τ1, τ2, r, and β.
iv. What's the period-one trade balance, as a function of Q1,
Q2, τ1, τ2, r, and β?
v. Suppose that the government cuts taxes in period t = 1 and
compensates by raising taxes in
period t = 2. Entitlement spending is unchanged. What is the
e�ect on the trade balance in
period one?
3