Intermediate Macroeconomics, Sciences Po, 2014
Zsófia Bárány
Answer Key to Problem Set 9
1. Ricardian Equivalence Consider a two-period economy in which the representative consumer maximizes the utility function U (c1 , c2 ) = ln(c1 ) +
β ln(c2 ) subject to the life-time budget constraint c1 + c2 /R = W , where
0 < β < 1, ci is consumption in period i = 1 or 2, W is the present value
of after-tax life-time income and R = 1 + r, where r is the interest rate.
(a) Derive the level of optimal consumption in the two periods. Provide economic intuition for your derivations.
Answer:
As in the lecture slides, we maximize a general utility function subject to an inter-temporal budget constraint. First, substitute the budget constraint into the objective function to eliminate c2 and then
maximize with respect to c1 :
max u(c1 ) + β u [R (W − c1 )]
Using the chain rule, the first-order condition for c1 is
u0 (c1 ) − β u0 [R (W − c1 )] R = 0
Note that the expression in the squared bracket is simply c2 , so we
can rewrite it as:
u0 (c1 ) = β R u0 (c2 )
Intuitively, we can express this condition in terms of the marginal
rate of substitution:
M RS =
u0 (c1 )
=R
β u0 (c2 )
which says that consumer maximize their utility when their desired
allocation satisfies the condition that the internal relative value of c1
in terms of c2 equals to the market exchange value of these goods.
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Using the utility function u(c) = ln(c), we have
1
βR
=
c1
c2
or c2 = β R c1 . Finally, using this condition along with the budget
RW
W
and c2 = β1+β
.
constraint, we can derive c1 = 1+β
(b) Suppose the consumer receives Y1 and Y2 and pays taxes T1 and T2
in periods 1 and 2. Use this model to explain the Ricardian equivalence of the timing of taxes. For simplicity assume that the number
of consumers N = 1.
Answer:
Write down the government’s budget constraint:
T1 +
G2
T2
= G1 +
R
R
and the consumer’s life-time income as
T2
Y2
− T1 +
W = Y1 +
R
R
If the present value of government spending, G1 + G2 /R, remains
constant then any change of taxes in the current period must be
eventually compensated by a proportional change of taxes in the
future period to satisfy the government’s budget constraint.
Therefore consumer’s wealth W is unaffected by the change in taxes
and the optimal consumption levels remain the same because they
are a function of the present discounted value of wealth.
Therefore, the timing of taxes does not matter if the present value of
government spending remains the same.
Mathematically, this can be seen by combining the above two equations and showing that the budget constraint of consumers doesn’t
depend of taxes:
G2
Y2
− G1 +
W = Y1 +
R
R
Since the timing of taxes does not affect W , c1 =
remain unchanged.
2
W
1+β
and c2 =
βRW
1+β
When the government changes taxes in the first period by ∆T1 , it
will have to change taxes in the second period by ∆T2 = −R∆T1
to satisfy its inter-temporal budget constraint. For example, a cut
in taxes in the first period by ∆T1 = −1, means an increase in
government debt by 1, ∆B = −∆T1 = 1 (a change in government savings by ∆S g = −∆B = −1) and requires higher taxation in the second period by ∆T2 = −R∆T1 = R. Households do
not change their consumption, so the change in private savings is
∆S p = ∆Y1 − ∆T1 − ∆c1 = 0 − (−1) − 0 = 1. National saving
S = S p + S g remain therefore unchanged.
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(c) Suppose the consumer cannot borrow. Show that what you derived
in part (b) might not hold.
(c) Answer:
Suppose the consumer cannot borrow. Show that what you derived in part (b)
might not hold.
We have to distinguish two cases: first, a consumer is a lender (i.e.
Answer:
consumes to the left of her endowment point). Then, the fact that
there are limits on borrowing doesn’t affect her consumption choices
We have to distinguish two cases: first, a consumer is a lender (i.e. consumes
attoall
equivalence
applies.
theand
left Ricardian
of her endowment
point).still
Then,
the fact that there are limits on
borrowing doesn’t affect her consumption choices at all and Ricardian equivalence still
applies.
Next,
assume
that a consumer actually wants to consume to the
right of her endowment point, so that she would be a borrower if
Next, assume
that aallowed.
consumer When
actually
wants
to consume
theconsumpright of her
market
conditions
she
cannot
borrow,toher
endowment
point,
so
that
she
would
be
a
borrower
if
market
conditions
tion will be c1 = Y1 − T1 and c2 = Y2 − T2 . A tax increase today willallowed. current
When she
cannot borrow,
herifconsumption
willknows
be c1 =that
Y1 −
T1 and
reduce
consumption
even
the consumer
future
c2 = Y2 − T2 . A tax increase today will reduce current consumption even if the
taxes will fall, so the timing of taxes matters and Ricardian Equivaconsumer knows that future taxes will fall, so the timing of taxes matters and
lence
doesn’t hold. Both cases are depicted in figure 1.
Ricardian Equivalence doesn’t hold. Both cases are depicted in figure 1.
c′
c′
IC1
IC1
IC2
BC
BC
c
c
Figure 1: Ricardian equivalence and credit constraints
Figure 1: Ricardian equivalence and credit constraints
For
constraint
consumer
= 1∆Y
1 − ∆T
1 = −∆T1 .
= 1∆Y
− ∆T
For the
the borrowing
borrowing constraint
consumer
∆c1 ∆c
1 = −∆T1 . If all
Ifconsumers
all consumers
are
borrowing
constraint
and
behave
like
the
are borrowing constraint and behave like this, the changethis,
in private
p
p
change
in∆S
private
is ∆S
= ∆Y11−(−∆T
− ∆T11 )−=∆c
− ∆T1taxes
−
savings is
= ∆Ysavings
0. 1So=if0current
1 −∆T1 −∆c
1 = 0−∆T
g
g
p
increase,
∆T0.1 >
= −∆B
= ∆T
saving=S −∆B
= S +=S g
(−∆T
So0,if∆S
current
taxes
increase,
∆Tnational
1 > 0, and
1) =
1 > 0, ∆S
p
g
increase.
∆T
1 > 0, and national saving S = S + S increase.
(d) Another
Another proposed
reason
whywhy
the Ricardian
equivalence
might notmight
hold isnot
that
(d)
proposed
reason
the Ricardian
equivalence
consumers
have
different
life
horizon
than
the
government.
How
could
you
use
hold is that consumers have different life horizon than the governthe two-period model to show this?
ment. How could you use the two-period model to show this?
Answer:
This can be illustrated by having
4 two different consumers living in each period
(consumer 1 is alive only in period 1 and consumer 2 only in period 2) and
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Answer:
This can be illustrated by having two different consumers living in
each period (consumer 1 is alive only in period 1 and consumer 2
only in period 2) and a single government with a planning horizon
that spans over the two periods. Assume that there is no bequest
motive so that consumer 1 doesn’t care about consumer 2 at all.
In terms of the utility function set β = 0 so that consumers only care
about current consumption and use two single-period budget constraints instead of one inter-temporal.
In this case, consumer will consume everything in the period when
they are “alive”. A tax cut in period 1 will increase consumption in
period 1 even if there is no borrowing constraint.
The consumer living in period 1 doesn’t internalize the greater tax
burden that falls on the consumer living in period 2, whose consumption will decrease.
2. The financial crisis: Consider a two-period consumption model with housing. Suppose there are no taxes and that housing is illiquid (i.e. it cannot
be sold in the first period). The lifetime budget constraint can be written
as:
c+
y 0 + pH
c0
≤y+
1+r |
1+r}
{z
W
Here total wealth (W ) consists of the present discounted value of income
and the presented discounted value of housing wealth.
(a) Credit market imperfections: Suppose that, due to asymmetric information, consumers face different interest rates for borrowing (rB )
and for lending (rL ). The difference between the borrowing and
lending rates (rB − rL )) is known as the ”interest rate spread”. Assume that rB ≥ rL and suppose that an increase in credit risk causes
a rise in the interest rate spread. What is the effect on the consumption of lending households? What about households that borrow?
Answer:
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Firstly, we denote the lifetime budget constraint of a lending household (s ≥ 0) by:
c+
c0
≤ WL
1 + rL
where
WL = y +
y 0 + pH
1 + rL
And the lifetime budget constraint of a borrowing household (s < 0)
by:
c+
c0
≤ WB
1 + rB
where
WB = y +
y 0 + pH
1 + rB
In general, the effect of the increase in the interest rate spread on
lending (unconstrained) and borrowing (constrained) households
depends on the underlying cause of the change in spread. The
spread can change because the borrowing rate (rB ) rises and/or the
lending rate (rL ) falls. Suppose that financial institutions respond
to
the higher
risk
by increasing
rate; this
borrowing
rate credit
(rB ) rises
and/or
the lending the
rate borrowing
(rL ) falls. Suppose
that will
financause
the
consumption
of
borrowing
households
to
fall,
but
the
concial institutions respond to the higher credit risk by increasing the borrowing
sumption
of lending
be unaffected.
rate; this will
cause thehouseholds
consumptionwill
of borrowing
households to fall, but the
consumption of lending households will be unaffected.
C′
Slope = (1 + rL )
BC
E
Slope = (1 + rB )
ICB
C
Figure
2: Increase
theborrowing
borrowing rate
Figure
2: Increase
in in
the
rate
Recall that the slope of the budget constraint is given by (1 + r). Given the
Recall
thatinthe
slopeand
of borrowing
the budget
constraint
given byis(1kinked
+ r).at
differences
lending
rates,
the budgetisconstraint
Given
the
differences
in
lending
and
borrowing
rates,
the
budget
the endowment point (E), corresponding to the point on the budget line where
consumers neither save nor borrow. As shown in figure 2, the rise in the borrowing rate causes the budget constraint to become steeper to the right of the
6 region’). In contrast, the change in the borendowment point (the ‘borrowing
rowing rate has no effect on the budget constraint to the left of the endowment
point (the ‘saving region’). Borrowing households therefore end up on a lower
indifference curve after the change in the interest rate and so are worse off.
The saving households remain on the same indifference curve and hence their
welfare does not change.
constraint is kinked at the endowment point (E), corresponding to
the point on the budget line where consumers neither save nor borrow. As shown in figure 2, the rise in the borrowing rate causes
the budget constraint to become steeper to the right of the endowment point (the ‘borrowing region’). In contrast, the change in the
borrowing rate has no effect on the budget constraint to the left of
the endowment point (the ‘saving region’). Borrowing households
therefore end up on a lower indifference curve after the change in
the interest rate and so are worse off. The saving households remain on the same indifference curve and hence their welfare does
not change.
Alternatively, it is possible that financial institutions respond to the
greater credit risk by lowering the interest rates they offer to saving households. This would correspond to the budget constraint
becoming flatter in the saving region, but being unchanged in the
borrowing region. In this case, the saving households are worse off
while the borrowing households are unaffected. This case is represented in figure 3.
C′
Slope = (1 + rL )
ICL
E
Slope = (1 + rB )
BC
C
Figure
3: Decrease
ininthe
Figure
3: Decrease
thelending
lending rate
rate
Of course, we may also have some combination of the above in
which
case both groups of households are made worse off. In either
Note that there will also be substitution effects from the higher interest rate
case
theifincrease
in spread
the interest
spread
has a negative
‘income
spread;
the higher
is due rate
to higher
borrowing
rates, only
borroweffect’
on
both
types
of
household.
ers will be affected (less borrowing because current consumption is relatively
more expensive); if the higher spread is due to lower lending rates, only lenders
will be affected (less saving because
current consumption becomes relatively
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cheaper).
(b) Limited commitment: Suppose now that there is no asymmetric information
so that consumers face the same interest rate for borrowing and lending (r =
r B = r L ). But now suppose that consumers are only allowed to borrow a
certain fraction of the value of their housing wealth (θ) where 0 ≤ θ ≤ 1.
Note that there will also be substitution effects from the higher interest rate spread; if the higher spread is due to higher borrowing
rates, only borrowers will be affected (less borrowing because current consumption is relatively more expensive); if the higher spread
is due to lower lending rates, only lenders will be affected (less saving because current consumption becomes relatively cheaper).
(b) Limited commitment: Suppose now that there is no asymmetric information so that consumers face the same interest rate for borrowing
and lending (r = rB = rL ). But now suppose that consumers are
only allowed to borrow a certain fraction of the value of their housing wealth (θ) where 0 ≤ θ ≤ 1. We can refer to θ as the ”loan-tovaluation ratio”. The collateral constraint becomes:
−s(1 + r) ≤ θpH
Now suppose that the increase in credit risk causes a tightening in
lending standards, which we can represent as a fall in the loan-tovaluation ratio (θ). What impact will this have on the consumption
of households that lend? What about households that borrow?
Answer:
Note that because housing is illiquid the first period budget constraint is given by:
c+s≤y
But the collateral constraint says:
−s(1 + r) ≤ θpH
Combining these two equations we obtain:
c≤y+
θpH
1+r
In figure 4, the endowment point (E) (where consumers neither save
nor borrow) corresponds to the point on the x-axis where c = y. But,
importantly, the kink in the budget constraint does not occur at this
endowment point because of the collateral constraint. While consumers cannot consume the full value of their home in the first period, the collateral constraint implies that they can borrow against
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Combining these two equations we obtain:
c≤y+
θpH
1+r
In figure 4, the endowment point (E) (where consumers neither save nor borrow) corresponds to the point on the x-axis where c = y. But, importantly,
the kink in the budget constraint does not occur at this endowment point because of the collateral constraint. While consumers cannot consume the full
value
of their
home home
in the first
period,
collateral
that
the
value
of their
in the
first the
period
(i.e. constraint
they are implies
not comthey
can
borrow
against
the
value
of
their
home
in
the
first
period
(i.e.
they
pletely credit rationed). This implies that the kink in the budget
are not completely
credit
rationed).
This
that
constraint
occurs at
point
A where
c =implies
y + θpH
. the kink in the budget
1+r
constraint occurs at point A where c = y + θpH
.
1+r
C′
BC
E
B
A
ICB
C
Figure 4: Tighter bank lending standards
Figure 4: Tighter bank lending standards
The tightening in lending standards, as represented by a lower θ, causes a leftwardtightening
vertical shiftinin lending
this budget
constraint ‘kink’
from A to B.by
The
the
The
standards,
as represented
a slope
lowerof θ,
budget aconstraint
unaffected
as the
rate does
not change
in this
case.
causes
leftwardisvertical
shift
in interest
this budget
constraint
‘kink’
from
A to B. The slope of the budget constraint is unaffected as the interIn effect,
the tightening
in lending
est
rate does
not change
in thisstandards
case. causes the consumption of (credit-
constrained) borrowing households to fall. In contrast, it has no impact on the
In effect, the tightening in lending standards causes the consump7
tion of (credit-constrained) borrowing households to fall. In contrast, it has no impact on the consumption of lending households.
Basically, for any given value of their housing wealth, borrowing
households are forced to borrow less when lending standards become tougher.
Note: Incidentally, there is significant evidence that changes in bank lending standards played an important role in the recent US subprime mortgage crisis.
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