Chapter 13

General equilibrium analysis of

public and foreign debt
This chapter reviews long-run aspects of public and foreign debt in the light
of the continuous time OLG model of the previous chapter. Section 13.1
reconsiders the Ricardian equivalence issue. In Section 13.2 we extend this
enquiry to a general equilibrium analysis of budget decits and debt dynamics
in a closed economy. Section 13.3 addresses general equilibrium aspects of
public and foreign debt of a small open economy, including the so-called
twin-decits issue. The assumption of lump-sum taxes is replaced by income
taxation in Section 13.4 in order to examine the relationship between debt and
distortionary taxation. Optimal debt policy is addressed in Section 13.5 and
the concluding Section 13.6 discusses the time-inconsistency problem faced
by government policy when outcomes depend on private sector expectations.
13.1 Reconsidering the issue of Ricardian equiv-
alence
Ricardian equivalence is the claim that it does not matter for aggregate con-
sumption and saving whether the government nances its current spending
by taxes or borrowing. As we know from earlier chapters, the OLG approach
and representative agent approach lead to dierent conclusions regarding the
validity of this claim. Among prominent macroeconomists there exist dier-
ing opinions about which of the two approaches ts the real world best.
In representative agent models (the Barro and Ramsey dynasty models)
Ricardian equivalence holds. In these models there is a given number of
innitely-lived forward-looking families. A change in the timing of (lump-
sum) taxes does not change the present value of the innite stream of taxes
465
466
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
imposed on the individual dynasty. A cut in current taxes is oset by the
expected higher future taxes. Though public saving has gone down, private
saving goes up. And the latter goes up just as much as taxation is reduced,
since this is exactly what is needed for paying the higher taxes in the future.
Thus, consumption is not aected and aggregate saving in society as a whole
stays the same (higher government dissaving being matched by higher private
saving).
It is dierent in OLG models (without a Barro-style bequest motive).
Diamonds discrete time OLG model, for instance, reveals how taxes levied
at dierent times are levied on dierent sets of agents. In the future some of
those people alive today will be gone and there will be newcomers to bear part
of the higher tax burden. Therefore a current tax cut make current tax payers
feel wealthier and this leads to an increase in their current consumption.
So current aggregate saving in the economy ends up lower. The present
generations thus benet and future generations bear the cost in the form of
smaller national wealth than otherwise.
Because of the more rened notion of time in the Blanchard OLG model
from Chapter 12 and its capability of treating wealth eects more aptly, let
us see what this model has to say about the issue. To keep things simple,
we ignore retirement (` = 0) and assume that the given public consumption
ow, G
t
. does not aect marginal utility of private consumption. A simple
book-keeping exercise will show that the size of the public debt does matter.
By aecting private wealth, it aects private consumption.
To avoid confusion of the birth rate and the debt-income ratio, the birth
rate will here be denoted ,. Otherwise notation is as in the previous chapters:
real net government debt is denoted 1
t
and net tax revenue is 1
t


1
t
A
t
.
where

1
t
is gross tax revenue and A
t
is transfers. We assume that the interest
rate is in the long run higher than the output growth rate. Hence, to remain
solvent the government has to satisfy its intertemporal budget constraint.
Presupposing the government does not plan to procure more tax revenue than
needed to satisfy its intertemporal budget constraint, we have the condition
Z

t
1
c
c

o:
d: =
Z

t
G
c
c

o:
d: +1
t
. (GIBC)
where 1
t
is considered as historically given. In brief this says that the present
value of future net tax revenues must equal the sum of the present value of
future spending on goods and services and the current level of debt.
Given aggregate private nancial wealth,
t
. and aggregate human wealth,
H
t
, aggregate private consumption is
C
t
= (j +:)(
t
+H
t
). (13.1)
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.1. Reconsidering the issue of Ricardian equivalence 467
Because of the logarithmic specication of instantaneous utility, the propen-
sity to consume out of wealth is a constant equal to the sum of the pure rate
of time preference, j. and the mortality rate, :. Human wealth is the present
value of expected future net-of-tax labor earnings of those currently alive:
H
t
= `
t
Z

t
(n
c
t
c
)c

(v+n)o:
d:. (13.2)
Here, t
c
is a per capita (lump-sum) tax at time :. i.e., t
c
1
c
,`
c
(

1
c

A
c
),`
c
. where `
c
is population (here equal to the labor force, which in turn
equals employment). The discount rate is the sum of the risk-free interest
rate, :
c
and the actuarial bonus which is identical to the mortality rate, :.
To x ideas, consider a closed economy. In view of the presence of gov-
ernment debt, aggregate private nancial wealth in the closed economy is
t
= 1
t
+1
t
. where 1
t
is aggregate (private) physical capital. Thus, (13.1) can
be written
C
t
= (j +:)(1
t
+1
t
+H
t
). (13.3)
For a given 1
t
we ask whether the sum 1
t
+ H
t
depends on the size of 1
t
.
We will see that, contrary to the Ricardian equivalence hypothesis, a rise in
1
t
is not oset by an equal fall in H
t
brought about by the higher future
taxes. Therefore C
t
is increased. As an implication aggregate saving depends
negatively on 1
t
.
The argument is the following. Rewrite (13.2) as
H
t
= `
t
Z

t
n
c
`
c
1
c
`
c
c

(v

+n)o:
d: (from t
c
= 1
c
,`
c
)
=
Z

t
(n
c
`
c
1
c
)c
a(ct)
c

(v

+n)o:
d: (since `
t
= `
c
c
a(ct)
)
=
Z

t
(n
c
`
c
1
c
)c

(v

+a+n)o:
d: =
Z

t
(n
c
`
c
1
c
)c

(v

+o)o:
d:.
using that , = : +:. the birth rate. Therefore,
H
t
+1
t
=
Z

t
(n
c
`
c
1
c
)c

(v+o)o:
d: +1
t
=
Z

t
(n
c
`
c
G
c
)c

(v+o)o:
d: +1
t

Z

t
(1
c
G
c
)c

(v

+o)o:
d:. (13.4)
In view of (GIBC),
1
t
=
Z

t
(1
c
G
c
)c

vo:
d:. (13.5)
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
468
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
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Suppose 1
t
0. so that, loosely said, 1
c
G
c
is positive most of the time.
Comparing (13.5) and the last integral in (13.4), we see that

H
t
+1
t
is independent of 1
t
. if , = 0. while
H
t
+1
t
depends positively on 1
t
. if , 0.
The rst case corresponds to a representative agent model and here there
is Ricardian equivalence. In the second case the birth rate is positive, im-
plying that the higher tax burden in the future is partly shifted to new
generations. So when bond holdings are higher, the current generations do
feel wealthier. The discount rate relevant for the government when discount-
ing future tax receipts and future spending is just the market interest rate,
:. But the discount rate relevant for the households currently alive is : +,.
This is because the present generations are, over time, a decreasing fraction
of the tax payers, the rate of decrease being larger the larger is the birth
rate. In the Barro and Ramsey models the birth rate is eectively zero in
the sense that no new tax payers are born. When the bequest motive (in
Barros form) is operative, those alive today will take the tax burden of their
descendents fully into account.
This takes us to the distinction between new individuals and new de-
cision makers, a distinction related to the fundamental dierence between
representative agent models and overlapping generations models.
It is not nite lives or population growth
It is sometimes believed that nite lives or the presence of population growth
are basic theoretical reasons for the absence of Ricardian equivalence. This
is a misunderstanding, however. The distinguishing feature is whether new
decision makers continue to enter the economy or not.
To sort this out, let

, be a constant birth rate of decision makers. That
is, if the population of decision makers is of size `. then `

, is the inow
of new decision makers per time unit.
1
Further, let : be a constant and
age-independent death rate of existing decision makers. Then :

, : is
the growth rate of the number of decision makers. Given the assumption of
a perfect credit market, we claim:
there is Ricardian equivalence if and only if

, = 0. (13.6)
Indeed, when

, = 0. the current tax payers are also the future tax payers.
With perfect foresight and no credit market imperfections rational agents
1
In view of the law of large numbers, we do not distinguish between expected and
actual inow.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.1. Reconsidering the issue of Ricardian equivalence 469
respond to decit nance (deferment of taxation) by increasing current saving
out of the currently higher after-tax income. This increase in saving matches
the extra taxes in the future. Current private consumption is unaected by
the decit nance. If

, 0, however, decit nance means shifting part of
the tax burden from current tax payers to future tax payers whom current tax
payers do not care about. Even though representative agent models like the
Ramsey and Barro models may include population growth in a demographic
sense, they have a xed number of dynastic families (decision makers) and
whether the size of these dynastic families rises (population growth) or not
is of no consequence as to the question of Ricardian equivalence.
Another implication of (13.6) is that it is not the nite lifetime that is
decisive for absence of Ricardian equivalence in OLG models. Indeed, even
if we imagine the agents in a Blanchard-style model have a zero death rate,
there is still a positive birth rate. New decision makers continue to enter the
economy through time. When decit nance occurs, part of the tax burden
is shifted to these newcomers.
With ,. :. and : denoting the birth rate, death rate, and population
growth rate, respectively, in the usual demographic sense, we have in Blan-
chards model

, = ,. : = :. and : = :. In the Ramsey model, however,

, = : = : = 0 : = , :. With this interpretation, both the Blanchard

and the Ramsey model t into (13.6). In the Blanchard model every new
generation consists of new decision makers, i.e.,

, = , 0. In that setting,
whether or not the population grows, the generations now alive know that
the higher taxes in the future implied by decit nance today will in part fall
on the new generations. We therefore have : 0,

, = : + : : 0, and
in accordance with (13.6) there is not Ricardian equivalence. In the Ramsey
model where, in principle, the new generations are not new decision makers
since their utility were already taken care of through bequests by their fore-
runners, there is Ricardian equivalence. This is in accordance with (13.6),
since

, = 0, whereas : 0.
Note that the assumption in the Blanchard model that : (= :) is inde-
pendent of age might be more acceptable if we interpret : not as a biological
mortality rate but as a dynasty mortality rate.
2
Thinking in terms of dynas-
ties allows for some intergenerational links through bequests. In this inter-
pretation : is the approximate probability that the family dynasty ends
within the next time interval of unit length (either because members of the
family die without children or because the preferences of the current mem-
bers of the family no longer incorporate a bequest motive). Then, : = 0
corresponds to the extreme Barro case where such an event never occurs,
2
This interpretation was suggested already by Blanchard (1985, p. 225).
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
470
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
i.e., that all existing families are innitely-lived through intergenerational
bequests.
Yet, even in this limiting case we can interpret statement (13.6) as telling
us that if, in addition, newfamilies enter the economy (

, 0), then Ricardian

equivalence does not hold. How could new families enter the economy? One
could imagine that immigrants are completely cut o from their relatives in
their home country or that a parent only loves the rst-born. In that case
children who are not rst-born, do not, eectively, belong to any preexisting
dynasty, but may be linked forward to a chain of their own descendants (or
perhaps only their rst-born descendants). So in spite of the innite horizon
of every family alive, there are newcomers; hence, Ricardian equivalence does
not hold.
Statement (13.6) also implies that if

, = 0, then : 0 does not destroy
Ricardian equivalence. It is the dierence between the public sectors future
tax base (including the resources of individuals yet to be born) and the future
tax base emanating from the individuals that are alive today that accounts
for the non-neutrality of variations over time in the pattern of lump-sum
taxation. This reasoning also reminds us that it is immaterial for the validity
of (13.6) whether there is productivity growth or not.
Further aspects stressed by critics of the Ricardian equivalence hypothesis
are:
1. Short-sightedness. As emphasized by behavioral economists and ex-
perimental economics, many people do not seem to conform to the
assumption of full intertemporal rationality. Instead, short-sightedness
is prevalent and many people do not respond to a tax cut by a fully
osetting increase in savings out of after-tax income.
2. Failure to leave bequests. Though the bequest motive is certainly of
empirical relevance, it is operative for only a minority of the population
(primarily the wealthy families)
3
and it need not have the altruistic
form hypothesized by Barro.
3. Imperfections on credit markets. There are imperfections in the credit
markets and those people who are credit rationed eectively face a
higher interest rate than that faced by the government. Then, even if
people expect higher taxes in the future, the present value of these is
less than the current reduction of taxes.
4. The Keynesian view. Contrary to what the theory of Ricardian equiv-
alence presupposes, output and labor markets often do not function
3
Wolf (2002).
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.2. Dynamic general equilibrium eects of lasting budget decits 471
as in the classical or Walrasian theory. They do not clear like auc-
tion markets but are instead often characterized by a kind of excess
supply. Then, by stimulating aggregate demand, a government budget
decit stimulates consumption demand and possibly even investment.
So there tends to be real eects. Whether these are desirable or not
depends on the state of the economy. In a boom they are not, because
they may cause overheating of the economy, crowd out private invest-
ment, and increase the tax burden in the future. In a recession caused
by slack demand, however, stimulation of aggregate demand is what is
needed. A tax cut will raise consumption and possibly also investment
through a positive demand spiral: 1 C 1 1 1 ,
where in the second round the increased 1 raises C further, and so on.
A similar multiplier process takes o as a result of a decit-nanced in-
crease in government spending on goods and services: G 1 1
1 . In these ways otherwise unutilized resources are activated by
a budget decit. So in a recession there is neither debt neutrality, nor
crowding out of private capital, but possibly crowding in.
4
To sum up, there are good reasons to believe that Ricardian equivalence
fails. Of course, this could in some sense be said about nearly all theoretical
abstractions. But it seems fair to add that most macroeconomists are of the
opinion that Ricardian equivalence systematically fails in one direction: it
over-estimates the osetting reaction of private saving in response to budget
decits. Some empirical observations supporting this view were given in
chapters 6 and 7.
13.2 Dynamic general equilibrium eects of
lasting budget decits
The above analysis is of a partial equilibrium nature, leaving 1, :. and n
unaected by the changes in government debt. To assess the full dynamic
eects of budget decits and public debt we have to do general equilibrium
analysis. When aggregate saving changes in a closed economy, so does 1
and generally also : and n. This should be taken into account.
4
As Keynes recommended president Franklin D. Roosevelt in 1933: Look after un-
employment, and the budget will look after itself. This terse dictum alludes to the role
of the automatic stabilizers for the government budget, and should not be interpreted as
if scal policy can ignore the possibility of runaway debt dynamics. It is the task of the
structural or cyclically adjusted budget decit to look after that problem.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
472
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
Let us also here apply the Blanchard OLG model from Chapter 12. To
simplify, we ignore technological progress, population growth, and retirement
all together. Therefore q = : = ` = 0, so that birth rate = mortality rate
= :. and 1
t
= ` (a constant) for all t. Let public spending on goods
and services be a constant

G 0. assumed not to aect marginal utility
of private consumption. Suppose all this spending is (and has always been)
public consumption. There is thus no public capital. Let taxes and transfers
be lump sum so that we need keep track only of the net tax revenue, 1.
We consider a closed economy described by

1
t
= 1(1
t
. `) o1
t
C
t

G. 1
0
0. given, (13.7)

C
t
= (1
1
(1
t
. `) o j)C
t
:(j +:)(1
t
+1
t
). (13.8)

1
t
= [1
1
(1
t
. `) o] 1
t
+

G1
t
. 1
0
0. given, (13.9)
where we have used the equilibrium relation :
t
= 1
1
(1
t
. `) o. We assume
o 0 and j 0.
5
Here (13.7) is essentially just accounting for a closed econ-
omy; (13.8) describes changes in aggregate consumption, taking into account
the generation replacement eect; and (13.9) describes how budget decits
give rise to increases in government debt. All government debt is assumed to
be short-term and of the same form as a variable-rate loan in a bank. Hence,
at any point in time 1
t
is historically determined and independent of the
current and future interest rates.
As we shall see, the long-run interest rate will exceed the long-run output
growth rate (which is nil). We know from Chapter 6 that in this case, to
remain solvent, the government must satisfy its No-Ponzi-Game condition
which, as seen from time zero, is
lim
t
1
t
c

0
[1

(1,.)c]oc
0. (13.10)
For ease of exposition, let the aggregate production function satisfy the Inada
conditions, lim
10
1
1
(1. `) = and lim
1
1
1
(1. `) = 0.
So far the model is incomplete in the sense that there is nothing to pin
down the time prole of 1
t
. except that ultimately the stream of taxes should
conform to (13.10). Let us rst consider a permanently balanced government
budget.
5
We know from Appendix D of Chapter 12 that when j 0, the transversality condi-
tions of the households will automatically be satised in the steady state of the Blanchard
OLG model.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.2. Dynamic general equilibrium eects of lasting budget decits 473
Dynamics under a balanced budget
Suppose that from time 0 the government budget is balanced. Therefore,

1
t
= 0 and 1
t
= 1
0
for all t 0. So (13.9) is reduced to
1
t
= (1
1
(1
t
. `) o)1
0
+

G. (13.11)
giving the tax revenue required for the budget to be balanced, when the debt
is 1
0
. This time path of 1
t
is determined after we have determined the time
path of 1
t
and C
t
through the two-dimensional system

1
t
= 1(1
t
. `) o1
t
C
t

G. (13.12)

C
t
= [1
1
(1
t
. `) o j]C
t
:(j +:)(1
t
+1
0
). (13.13)
This system is independent of 1
t
. The implied dynamics can usefully be
analyzed by a phase diagram.
Phase diagram Equation (13.12) shows that

1 = 0 for C = 1(1. `) o1

G. (13.14)
The right-hand side of (13.14) is the vertical distance between the 1 =
1(1. `) curve and the 1 = o1 +

G line in Fig. 13.1. On the basis of
this we can construct the

1 = 0 locus in Fig. 13.2. We have indicated two
benchmark values of 1 in the gure, namely the golden rule value 1
G1
and
the value

1. These values are dened by
1
1
(1
G1
. `) o = 0. and 1
1

1. `

o = j.
respectively.
6
We have

1 1
G1
. since j 0 and 1
11
< 0.
From equation (13.13) follows that

C = 0 for C =
:(j +:) (1 +1
0
)
1
1
(1. `) o j
. (13.15)
Hence, for 1

1 from below we have, along the

C = 0 locus, C . In
addition, for 1 0 from above, we have along the

C = 0 locus that C 0.
in view of the lower Inada condition.
Fig. 13.2 also shows the

C = 0 locus. We assume that

G and 1
0
are of
modest size relative to the production potential of the economy. Then the
6
In this setup, where there is neither population growth nor technical progress, the
golden rule capital stock is that 1 which maximizes C = 1(1, ) c1

1 subject to
the steady state condition

1 = 0.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
474
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES

( , ) F K N
K

K

GR
K
Y
G
O
K G
N constant
Figure 13.1: Building blocks for a phase diagram.

C = 0 curve crosses the

1 = 0 curve for two positive values of 1. Fig. 13.2
shows these steady states as the points E and

E with coordinates (1

. C

)
and (

1

.

C

). respectively, where

1

< 1

<

1.
The direction of movement in the dierent regions of Fig. 13.2 are de-
termined by the dierential equations (13.12) and (13.13) and indicated by
arrows. The steady state E is seen to be a saddle point, whereas

E is a
source.
7
We assume that

G and 1
0
are modest not only relative to the
long-run production capacity of the economy but also relative to the given
1
0
. This means that

1

< 1
0
. as indicated in the gure.
8
The capital stock is predetermined whereas consumption is a jump vari-
able. Since the slope of the saddle path is not parallel to the C axis, it
follows that the system is saddle-point stable. The only trajectory consistent
with all the conditions of general equilibrium (individual utility maximiza-
tion for given expectations, continuous market clearing, perfect foresight) is
the saddle path. The other trajectories in the diagram violate the TVCs of
the individual households. Hence, initial consumption, C
0
, is determined as
the ordinate to the point where the vertical line 1 = 1
0
crosses the saddle
path. Over time the economy moves along the saddle path, approaching the
steady state point E with coordinates (1

. C

).
Although our main focus will be on eects of budget decits and changes
7
A steady state point with the property that all solution trajectories starting close to
it move away from it is called a source or sometimes a totally unstable steady state.
8
The opposite case,

1

1
0
, would reect that G
0
and 1
0
were very large relative
to the initial production capacity of the economy, so large, indeed, as to crowd out any
saving and bring about a shrinking capital stock so that starvation were in prospect.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.2. Dynamic general equilibrium eects of lasting budget decits 475
0
.
K
E
0
.
C
K
C
E

K
GR
K
*
C
G
*
K
0
K
*
K

Figure 13.2: Phase diagram under a balanced budget.
in the debt, we start with the simpler case of a tax-nanced increase in

G.
Tax-nanced shift to a higher level of public consumption Suppose
that until time t
1
( 0) the economy has been in the saddle-point stable
steady state E. Hence, for t < t
1
we have zero net investment and : =
1
1
(1

. `) o :

. Moreover, as 1

<

1. :

j ( 0).
At time t
1
an unanticipated change in scal policy occurs. Public con-
sumption shifts to a new constant level

G
0


G. Taxes are immediately
increased by the same amount so that the budget stays balanced. We as-
sume that everybody rightly expect the new policy to continue forever. The
change to a higher G shifts the

1 = 0 curve downwards as shown in Fig.
13.3, but leaves the

C = 0 curve unaected. At time t
1
when the policy shift
occurs, private consumption jumps down to the level corresponding to the
point A in Fig. 13.3. The explanation is that the net-of-tax human wealth,
H
t
1
. is immediately reduced as a result of the higher current and expected
future taxes.
As Fig. 13.3 indicates, the initial reduction in C is smaller than the
increase in G and 1. Therefore net saving becomes negative and 1 decreases
gradually until the new steady state, E, is reached. To nd the long-run
multipliers for 1 and C we rst equalize the right-hand sides of (13.14) and
(13.15) and then use implicit dierentiation w.r.t.

G to get
J1

J

G
=
:

j
C

11
(:+:

)(j +::

)
< 0;
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
476
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES

0
.
K
E
0
.
C
K
C
GR
K
G
*'
K
' E
A
' G
*
K
Figure 13.3: Tax-nanced shift to higher public consumption.
next, from (13.14), by the chain rule we get
JC

J

G
=
JC

J1

J1

J

G
= :

J1

J

G
1 < 1.
where :

= 1
1
(1

. `) o.
9
In the long run the decrease in C is larger than
the increase in G because the economy ends up with a smaller capital stock.
That is, a tax-nanced shift to higher G crowds out private consumption and
investment. Private consumption is in the long run crowded out more than
one to one due to reduced productive capacity. In this way the cost of the
higher G falls relatively more on the younger and as yet unborn generations
than on the currently elder generations.
10
Higher public debt
To analyze the eect of higher public debt, let us rst see how it might come
about.
A tax cut Assume again that until time t
1
( 0) the economy has had
a balanced government budget and been in the saddle-point stable steady
9
For details, see Appendix B.
10
This might be dierent if a part of G were public investment (in research and educa-
tion, say), and this part were also increased.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.2. Dynamic general equilibrium eects of lasting budget decits 477
state E. The level of the public debt in this steady state is 1
0
0 and tax
revenue is, by (13.11),
1 = (1
1
(1

. `) o)1
0
+

G 1

.
a positive constant in view of 1
1
(1

. 1) o = :

j 0.
At time t
1
the government unexpectedly cuts taxes to a lower constant
level,

1, holding public consumption unchanged. That is, at least for a while
after time t
1
we have
1
t
=

1 < 1

. (13.16)
As a result

1
t
0. The tax cut make current generations feel wealthier,
hence they increase their consumption. They do so in spite of being forward-
looking and anticipating that the current scal policy sooner or later must
come to an end (because it is not sustainable, as we shall see). The prospect of
higher taxes in the future does not prevent the increase in consumption, since
part of the future taxes will fall on new generations entering the economy.
The rise in C combined with unchanged

Gimplies negative net investment
so that 1 begins to fall, implying a rising interest rate, :. For a while all
the three dierential equations that determine changes in C. 1. and 1 are
active. These three-dimensional dynamics are complicated and cannot, of
course, be illustrated in a two-dimensional phase diagram. Hence, for now
we leave the phase diagram.
The scal policy (

G.

1) is not sustainable By denition a scal policy
(G. 1) is sustainable if the government stays solvent under this policy. We
claim that the scal policy (

G.

1) is not sustainable. Relying on principles
from Chapter 6, there are at least three dierent ways to prove this.
Approach 1. In view of 1

<

1 < 1
G1
. we have :

= 1
1
(1

. 1) o
1
1
(

1. 1) o = j 0. After time t
1
1
t
is falling, at least for a while.
So 1
t
< 1

and thus :
t
= 1
1
(1
t
. `) o :

0. Thereby the scal

policy (

G.

1) implies an interest rate forever larger than the long-run growth
rate of output (income). We know from Chapter 6 that in this situation a
sustainable scal policy must satisfy the NPG condition
lim
t
1
t
c

1
voc
0. (13.17)
This requires that there exists an 0 such that
lim
t

1
t
1
t
< lim
t
:
t
. (13.18)
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CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
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i.e., the growth rate of the public debt is in the long run bounded above by
a number less than the long-run interest rate .
The scal policy (

G.

1) violates (13.18), however. Indeed, we have for
t t
1

1
t
= :
t
1
t
+

G

1 (13.19)
:

1
0
+

G

1 :

1
0
+

G1

= 0.
where the rst inequality comes from 1
t
1
0
0 and :
t
= 1
1
(1
t
. 1) o
:

= 1
1
(1

. 1) o, in view of 1
t
< 1

. This implies 1
t
for t .
Hence, dividing by 1
t
in (13.19) gives

1
t
1
t
= :
t
+

G

1
1
t
:
t
for t . (13.20)
which violates (13.18). So the scal policy (

G.

1) is not sustainable.
Approach 2. An alternative argument, focusing not on the NPG condi-
tion, but on the debt-income ratio, is the following. We have, for t t
1
.
1
t
< 1

so that 1
t
< 1

= 1(1

. `) at the same time as 1

t
for
t . by (13.19). Hence, the debt-income ratio, 1
t
,1
t
. tends to innity
for t . thus conrming that the scal policy (

G.

1) is not sustainable.
Approach 3. Yet another way of showing absence of scal sustainability is
to start out from the intertemporal government budget constraint and check
whether the primary budget surplus,

1 G. which rules after time t
1
, satises
Z

t
1
(

1 G)c

0
voc
dt 1
t
1
. (13.21)
where 1
t
1
= 1
0
0. Obviously, if

1G 0. (13.21) is not satised. Suppose

1 G 0. Then
Z

t
1
(

1 G)c

1
voc
dt <
Z

t
1
(

1 G)c
v

(tt
1
)
dt =

1 G
:

< 1
0
= 1
t
1
.
where the rst inequality comes from :
t
:

. the rst equality by carrying

the integration
R

t
1
c
v

(tt
1
)
dt out, and, nally, the second inequality from
the equality in the second row of (13.19) together with the fact that

1 < 1

.
So the intertemporal government budget constraint is not satised. The
current scal policy is unsustainable.
Fiscal tightening and thereafter To avoid default on the debt, sooner or
later the scal policy must change. This may take the form of lower of public
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.2. Dynamic general equilibrium eects of lasting budget decits 479
consumption or higher taxes or both.
11
Suppose that the change occurs at
time t
2
t
1
in the form of a tax increase so that for t t
2
there is again a
balanced budget. This new policy is announced to be followed forever after
time t
2
and we assume the market participants believe in this and that it
holds true.
The balanced budget after time t
2
implies
1
t
= (1
1
(1
t
. `) o)1
t
2
+

G. (13.22)
The dynamics are therefore again governed by a two-dimensional system,

1
t
= 1(1
t
. `) o1
t
C
t

G. (13.23)

C
t
= [1
1
(1
t
. `
t
) o j]C
t
:(j +:)(1
t
+1
t
2
). (13.24)
Consequently phase diagram analysis can again be used.
The phase diagram for t t
2
is depicted in Fig. 13.4. The new initial 1
is 1
t
2
. which is smaller than the previous steady-state value 1

because of
the negative net investment in the time interval (t
1
. t
2
). Relative to Fig. 13.2
the

1 = 0 locus is unchanged (since

G is unchanged). But in view of the new
constant debt level 1
t
2
being higher than 1
0
. the

C = 0 locus has turned
counter-clockwise. For any given 1 (0.

1). the value of C required for

C = 0 is higher than before, cf. (13.15). The intuition is that for every given
1. private nancial wealth is higher than before in view of the possession of
government bonds being higher. For every given 1. therefore, the generation
replacement eect on the change in aggregate consumption is greater and so
is then the level of aggregate consumption that via the operation of the
Keynes-Ramsey rule is required to oset the generation replacement eect
and ensure

C = 0 (cf. Section 12.2 of the previous chapter).
The new saddle-point stable steady state is denoted E in Fig. 13.4 and it
has capital stock 1
0
< 1

and consumption level C

0
< C

. As the gure is
drawn, 1
t
2
is larger than 1
0
. This case represents a situation where the tax
cut did not last long (t
2
t
1
small). The level of consumption immediately
after t
2
. where the scal tightening sets in, is found where the line 1 = 1
t
2
crosses the new saddle path, i.e., the point A in Fig. 13.4. The movement
of the economy after t
2
implies gradual lowering of the capital stock and
consumption until the new steady state, E, is reached.
Alternatively, it is possible that 1
t
2
is smaller than 1
0
so that the new
initial point, A, is to the left of the new steady state E. This case is illustrated
in Fig. 13.5 and arises if the tax cut lasts a long time (t
2
t
1
large). The
low amount of capital implies a high interest rate and the scal tightening
11
We still assume seigniorage nancing is out of the question.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
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CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
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K
A
' E
E
*' K
K
GR
K
C C=0 new


0 K

* K
1
t
K
Figure 13.4: The adjustment after scal tightening at time t
2
, presupposing t
2
t
1
small.
must now be tough. This induces a low consumption level so low that net
investment becomes positive. Then the capital stock and output increase
gradually during the adjustment to the steady state E.
Thus, in both cases the long-run eect of the transitory budget decit is
qualitatively the same, namely that the larger supply of government bonds
crowds out physical capital in the private sector. Intuitively, a certain feasible
time prole for nancial wealth, = 1 +1. is desired and the higher is 1.
the lower is the needed 1. To this stock interpretation we may add a ow
interpretation saying that the budget decit oers households a saving outlet
which is an alternative to capital investment. All the results of course hinge
on the assumption of permanent full capacity utilization.
To be able to quantify the long-run eects of a change in the debt level
on 1 and C we need the long-run multipliers. By equalizing the right-
hand sides of (13.14) and (13.15), with 1
0
replaced by

1. and using implicit
dierentiation w.r.t.

1, we get
J1

J

1
=
:(j +:)
D
< 0. (13.25)
where D C

11
(: +:

)(j + ::

) < 0.
12
Next, by using the chain
12
For details, see Appendix B.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.2. Dynamic general equilibrium eects of lasting budget decits 481

K
A
' E
E
*' K
K
GR
K
C C=0 new


0 K

* K
1
t
K
Figure 13.5: The adjustment after scal tightening at time t
1
, presupposing t
1
t
0
large.
rule on (13.14), we get
JC

J

1
=
JC

J1

J1

J

1
= :

:(j +:)
D
< 0.
The multiplier J1

,J

1 tells us the approximate size of the long-run eect on
the capital stock, when a temporary tax cut causes a unit increase in public
debt. The resulting change in long-run output is approximately J1

,J

1
= (J1

,J1

)(J1

,J

1) = (:

+o):(j +:) ,D < 0.

Time proles It is also useful to consider the time proles of the variables.
Case 1: t
2
t
1
small (expeditious scal tightening). Fig. 13.6 shows
the time prole of 1 and 1. respectively. The upper panel visualizes that
the increase in taxation at time t
2
is larger than the decrease at time t
1
.
As (13.22) shows, this is due to public expenses being larger after t
2
because
both the government debt 1
t
and the interest rate, 1
1
(1
t
. `
t
)o. are higher.
The further gradual rise in 1
t
towards its new steady-state level is due to the
rising interest service along with a rising interest rate, caused by the falling
1.
The middle panel in Fig. 13.6 is self-explanatory. As visualized by the
lower panel, the tax cut at time t
1
results in an upward jump in consumption.
This implies negative net investment, so that 1 begins to fall. Hereby the
marginal product of labor is gradually reduced, implying also a fall in human
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CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
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1
t
2
t
t
T
T
0
1
t
2
t
t
B
2
t
B
0
B
0
1
t
2
t
t
0
K
C
K C,
* T
*' T
*' K
* K
*' C
* C
Figure 13.6: Case 1: t
2
t
1
small (expeditious scal tightening).
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.3. Public and foreign debt: a small open economy 483
wealth H. The implied falling household wealth induces falling consumption.
Because the exact time and form of the scal tightening were not anticipated,
a sharp decrease in the present discounted value of after-tax labor income
occurs at time t
2
, which induces the downward jump in consumption. Al-
though the fall in consumption makes room for increased net investment, the
latter is still negative so that the fall in 1 continues after t
2
. Therefore, also
the real wage continues to fall, implying falling H. hence further fall in C,
until the new steady-state level is reached.
If the time and form of the scal tightening were anticipated, consumption
would not jump at time t
2
. But the long-run result would be the same.
Case 2: t
2
t
1
large (deferred scal tightening). In this case the tax rev-
enue after t
2
has to exceed what is required in the new steady state. During
the adjustment the taxation level will be gradually falling which reects the
gradual fall in the interest rate generated by the rising 1, cf. Fig. 13.5. And
private consumption will at time t
2
jump to a level below the new (in itself
lower) steady state level, C
0
.
13.3 Public and foreign debt: a small open
economy
Now we let the country considered be a small open economy (SOE). Our
SOE is characterized by perfect substitutability and mobility of goods and
nancial capital across borders, but no mobility of labor. The main dierence
compared with the above analysis is that the interest rate will not be aected
by the public debt of the country (as long as its scal policy seems sustain-
able). Besides making the analysis simpler, this entails a stronger crowding
out eect of public debt than in the closed economy. The lack of an osetting
increase in the interest rate means absence of the feedback which in a closed
economy limits the fall in aggregate saving. In the open economy national
wealth equals the stock of physical capital plus net foreign assets. And it is
national wealth rather than the capital stock which is crowded out.
The model
The analytical framework is still Blanchards OLG model with constant pop-
ulation. As above we concentrate on the simple case: q = ` = 0 and birth
rate = mortality rate = : 0. The real interest rate is given from the world
nancial market and is a constant : 0. Table 13.1 lists key variables for an
open economy.
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CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
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Table 13.1. New variable symbols

a
t
=
t
1
t
= 1
t
+
)
t
= national wealth
1
t
= government (net) debt = government nancial wealth

)
t
= net foreign assets (the countrys net nancial claims on the rest of the world)
1
t
=
)
t
= net foreign debt

t
= 1
t
+1
t
+
)
t
= private nancial wealth

t
= o
j
t
= private net saving

1
t
= o
j
t
= 1
t
G:1
t
= government net saving = budget surplus

a
t
=

1
t
= o
j
t
+o
j
t
= o
a
t
= aggregate net saving
`A
t
= net exports

)
t
=

1
t

1
t
= `A
t
+:
)
t
= Co
t
= current account surplus
C1
t
= Co
t
= :1
t
`A
t
= current account decit
In view of prot-maximization the equilibrium capital stock, 1

. satises
1
1
(1

. `) = : + o and is thus a constant. The equilibrium real wage is

n

= 1
1
(1

. `). The increase per time unit in private nancial wealth is

t
= :
t
+n

` 1
t
C
t
= :
t
+ (n

t
t
)` C
t
. (13.26)
where t
t
1
t
,` is a per capita lump-sumtax. The corresponding dierential
equation for C
t
reads

C
t
= (: j)C
t
:(j +:)
t
. However, to keep track
of consumption in the SOE, it is easier to focus directly on the level of
consumption:
C
t
= (j +:)(
t
+H
t
). (13.27)
where H
t
is (after-tax) human wealth, given by
H
t
= `
Z

t
(n

t
c
)c
(v+n)(ct)
d: =
`n

: +:
`
Z

t
t
c
c
(v+n)(ct)
d:.
(13.28)
Suppose that from time 0 the government budget is balanced, so that 1
t
is constant at the level 1
0
and 1
t
= :1
0
+

G 1

. Consequently,
t
t
=
1

`
=
:1
0
+

G
`
t

. (13.29)
Under normal circumstances t

< n

. that is, 1
0
and

G are not so large
as to leave non-positive after-tax earnings. Then, in view of the constant per
capita tax,
H
t
=
n

: +:
` H

0. (13.30)
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.3. Public and foreign debt: a small open economy 485
Consequently (13.26) simplies to

t
= (: j :)
t
+ (n

)` (j +:)
n

: +:
`
= (: j :)
t
+
: j
: +:
(n

)`. (13.31)
This linear dierential equation has the solution (if : 6= j +:)

t
= (
0

)c
(vjn)t
+

. (13.32)
where

is the steady-state national wealth,

=
(: j)(n

)`
(: +:)(j +::)
. (13.33)
(For economic relevance of the solution (13.32) it is required that
0
H

.
since otherwise C
0
would be zero or negative in view of (13.27).) Substitution
into (13.27) gives steady-state consumption,
C

=
:(j +:)(n

)`
(: +:)(j +::)
. (13.34)
It can be shown by an argument similar to that in Appendix D of Chapter 12
that the transversality conditions of the individual households are satised
along the path (13.32).
By (13.31) we see that
t
is asymptotically stable if and only if
: < j +:. (13.35)
Let us consider this case rst. The phase diagram describing this case is
shown in the upper panel of Fig. 13.7. The lower panel of the gure illustrates
the movement of the economy in (. C) space, given
0
<

. The

= 0
line represents the equation C = :+ (n

)`. which in view of (13.26)

must hold when

= 0. Its slope is lower than that of the line representing
the consumption function, C = (j +:)(+H

). The economy is always at

some point on this line.
13
A sub-case of (13.35) is the following case.
Medium impatience: : : < j < :
As Fig. 13.7 is drawn, it is presupposed that

0. which, given (13.35),

requires : : < j < :. This is the case of medium impatience. Imagine
that until time t
1
0 the system has been in the steady state E.
13
If we (as for the closed economy) had based the analysis on two dierential equations
in and C, then a saddle path would arise and this path would coincide with the C
= (j +:)(+H

) line in Fig. 13.7.

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A
C
m
r
* A
0
A O
* C
0 A

E
A
A

O *
A
( )( *) C m A H
( * *) w N
Figure 13.7: Dynamics of a SOE with medium impatience, i.e., : : < j < :
(balanced budget).
A scal easing At time t
1
an unforeseen tax cut occurs so that at least for
some spell of time after t
1
we have 1 =

1 < 1

. hence t = t

1,` < t

.
Since government spending remains unchanged, there is now a budget decit
and public debt begins to rise. We know from the partial equilibrium analysis
of Section 13.1 that current generations will feel wealthier and increase their
consumption. This is so even if they are aware that sooner or later scal
policy will have to be changed again, because at that time new generations
have entered the economy and will take their part of the tax burden.
We assume this awareness is present but in a vague form in the sense that
the households do not know when and how the scal sustainability problem
will be remedied. As an implication, we can not assign a specic value to
the new after-tax human wealth, even less a constant value. A simple phase
diagram as in Fig. 13.7 is thus no longer valid. So for now we leave the phase
diagram.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.3. Public and foreign debt: a small open economy 487
From absence of Ricardian equivalence we know that H. and therefore C.
will increase somewhat. As we shall see, the rise in consumption at time
t
1
will be less than the fall in taxes. So there will be positive private saving,
hence rising private nancial wealth . for a while.
It is easiest to see this provisional outcome if we imagine that the agents
expect the new lower tax level to last for a long time. In analogy with
(13.30), if taxation is at a constant level, 1. forever, then human wealth is
H = (n

` 1),(: +:). From C

t
= (j +:)(
t
+H) we then get
C
t

JC
t
J1
1 = (j +:)
JH
J1
1 =
j +:
: +:
1 < 1. (13.36)
in view of 1 =

1 1

< 0 and : j. To the extent that the households

expect the new tax level

1 to last a shorter time, the boost to H and C will
be less than indicated by (13.36). This forties the rise in saving and the
resulting growth in .
Fiscal tightening at a higher debt level As hinted at, the scal policy
(

G.

1) is not sustainable. It generates a growth rate of government debt
which approaches :. whereas income and net exports are clearly bounded in
the absence of economic growth.
14
To end the runaway debt spiral a scal
tightening sooner or later is carried into eect. Suppose this happens at time
t
2
t
1
. Let the scal tightening take the form of a return to a balanced
budget with unchanged

G. That is, for t t
2
the tax revenue is
1 = :1
t
2
+

G 1
0
1

.
where the inequality is due to 1
t
2
1
0
. The corresponding per-capita tax is
t
0
1
0
,` t

.
Since the budget is now balanced, a phase diagram of the same form as
in Fig. 13.7 is valid and is depicted in Fig. 13.8. Compared with Fig. 13.7
the

= 0 line is shifted downwards because n

t
0
is lower than before t
1
.
For the same reason the new H. which is denoted H
0
. is lower than the old,
H

. So the line representing the consumption function is also shifted down

compared to the situation before t
1
. Immediately after time t
2
the economy
is at some point like P, where the vertical line =
t
2
(

) crosses the
new line representing the consumption function. The economy then moves
along that line and converges toward the new steady state E. At E we have
=
0
<

and C = C
0
< C

.
14
Indeed, as in the analogue situation for the closed economy,

1

,1

= r+(

G

T),1

r for t . Because we ignore economic growth, lasting budget decits indicate an
unsustainable scal policy.
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CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
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new ( )( *') C m A H
A
C
m
r
2
0
( )
new A
after t

* A *' A
O
* C
P
2
t
A
1
0
( )
old A
before t

E
A
A

O * A
old ( )( *) C m A H
*' C
( * *') w N
*' A
' E
( ) * m H
Figure 13.8: The adjustment after time t
2
showing the eect of a higher level of
government debt.
As a consequence national wealth goes down more than one to one with
the increase in government debt when we are in the medium impatience case.
Indeed, for a given level

1 of government debt. long-run national wealth is

1. (13.37)
An increase in government debt by

1 increases national wealth by

a
(J

,J

1 1)

1 <

1. since J

,J

1 < 0 when : : < j < :.
The explanation follows from the analysis above. On top of the reduction
of government wealth by

1 there is a reduction of private nancial wealth

due to the private dissaving during the adjustment process. This dissaving
occurs because consumption responds less than one to one (in the opposite
direction) when 1 is changed, cf. (13.36).
To nd the exact long-run eect, in (13.29) replace 1
0
by

1 and substitute
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.3. Public and foreign debt: a small open economy 489
into (13.33) to get

=
(: j)(n

` :

1

G)
(: +:)(j +::)
. (13.38)
Hence, the eect of public debt on national wealth in steady state is
J
a
J

1
=
(: j):
(: +:)(j +::)
1. (13.39)
This gives the size of the long-run eect on national wealth when a temporary
tax cut causes a unit increase in long-run government debt. In our present
medium impatience case, : : < j < : and so (13.39) implies J
a
,J

1
< 1.
15
Very high impatience: j :
Also this case with high impatience is a sub-case of (13.35). When j :
(13.39) gives 1 < J
a
,J

1 < 0. This is because such an economy will have
0 < J

,J

1 < 1. In view of the high impatience,

< 0. That is, in the long

run the SOE has negative private nancial wealth reecting that all physical
capital in the country and some of the human wealth is essentially mortgaged
to foreigners. This outcome is not plausible in practice. Credit constraints as
well as politically motivated government intervention will presumably hinder
such a development long before national wealth is in any way close to zero.
Very low impatience: j < : :
When j < ::. a steady state no longer exists since that would, by (13.34),
require negative consumption. In the lower panel of the phase diagram the
slope of the C = (j + :)( + H

) line will be smaller than that of the

= 0 line and the two lines will never cross for a positive C.
16
With initial
total wealth positive (i.e.,
0
H

). the excess of : over j + : results

in sustained positive saving so as to keep growing forever along the C
= (j +:)( +H

) line. That is, the economy grows large. In the long run
the interest rate in the world nancial market can no longer be considered
independent of this economy the SOE framework ceases to t.
15
In the knife-edge case j = r, we get

= 0. In this case 0

,0

1 = 1.
16
In the upper panel of the phase diagram the line representing

as a function of will
have positive slope. The stability condition (13.35) is no longer satised. There is still a
mathematical steady-state value

< 0, but it can not be realized, because it requires

negative consumption.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
490
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
As long as the country is still relatively small, however, we may use the
model as an approximation. Though there is no steady state level of national
wealth to focus at, we may still ask how the time path of national wealth,

a
t
. is aected by a rise in government debt caused by a temporary tax
cut. We consider the situation after time t
2
. where there is again a balanced
government budget. For all t t
2
we have
a
t
=
t

1. where

1 = 1
t
2
and, in analogy with (13.32),

t
= (
t
2

)c
(vjn)(tt
2
)
+

.
with

dened as in (13.38) (now a repelling state). For a given

t
2
H
0
we nd for t t
2
J
a
t
J

1
=
J
t
J

1
1 =

1 c
(vjn)(tt
2
)

J

1
1
=

1 c
(vjn)(tt
2
)

(: j):
(: +:)(j +::)

1. (13.40)
by (13.38).
17
Since j < : :. this multiplier is less than 1 and over time
rising in absolute value though bounded. In spite of the lower private saving
triggered by the higher taxation after time t
2
, private saving remains positive
due to the low rate of impatience. Thus nancial wealth is still rising and
so is private income. But the lower saving out of a rising income implies
more and more forgone future income. This explains the rising (although
bounded) crowding out envisaged by (13.40).
Current account decits and foreign debt
Do persistent current account decits in the balance of payments signify
future borrowing problems and threatening bankruptcy? To address this
question we need a few new variables.
Let `A
t
denote net exports (exports minus imports). Then, the output-
expenditure identity reads
1
t
= C
t
+1
t
+G
t
+`A
t
. (13.41)
Net foreign assets are denoted
)
t
and equals minus net foreign debt, 1
t
=
t
1
t
1
t
. Gross national income is 1
t
+:
)
t
= 1
t
:1
t
.
18
The current
17
The condition

2
H
0
is needed for economic relevance since otherwise C

2
0.
The condition also ensures
2

, since

< H
0
when j < r :.
18
In a more general setup also net foreign worker remittances, which we here ignore,
should be added to GDP to calculate gross national income.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.3. Public and foreign debt: a small open economy 491
account surplus at time t is
Co
t
=

)
t
=

1
t

1
t
= :
)
t
+`A
t
(13.42)
= C
t
+1
t
+G
t
(1
t
+:
)
t
).
by (13.41). The rst line views Co from the perspective of changes in
assets and liabilities. The second line views it from an expenditure-income
perspective, that is, the current account surplus is the excess of home ex-
penditure over and above gross national income. Gross national saving, o
t
.
equals, by denition, gross national income minus the sum of private and
public consumption, that is, o
t
= 1
t
+ :
)
t
C
t
G
t
. Hence, the current
account decit can also be written as the excess of gross investment over and
above gross national saving: Co
t
= o
t
1
t
. Of course, the current account
decit, CAD, is C1
t
= Co
t
= 1
t
o
t
.
In our SOE model above, with constant : 0 and no economic growth,
the capital stock is a constant, 1

. Then (13.41) gives net exports as a

residual:
`A
t
= 1(1

. `) C
t
o1

G. (13.43)
where C
t
= (j +:)(
t
+H
t
). In the steady state ruling for t < t
1
. 1
t
= 1
0
.
H
t
= H

. and
t
=

. as given in (13.30) and (13.33), respectively. Thus,

1
t
=
)
t
=

1
0
1

=
)
= 1

so that Co
t
= 0. Then, by
(13.42),
`A
t
= :
)
= :1

= `A

. (13.44)
This should also be the value of net exports we get from (13.43) in steady
state. To check this, we consider
`A
t
= 1(1

. `)C

o1

G = 1
1
(1

. `)1

+1
1
(1

. `)`C

o1

G.
where we have used Eulers theorem on homogeneous functions. By (13.26)
in steady state, this can be written
`A
t
= (: +o)1

+n

` (:

+ (n

)`) o1

G
= :(1

) +t

`

G = :(1

1
0
) = :1

.
where the third equality follows from the assumption of a balanced budget.
Our accounting is thus coherent.
We see that permanent foreign debt is consistent with a steady state if
net exports equal the interest payments on the debt. That is, in an economy
without growth a steady state requires not trade balance, but a balanced
current account. In a growing economy, however, not even a balanced current
account is required, as we will see below. Before leaving the non-growing
economy, however, a few remarks about the current account out of steady
state are in place.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
492
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
Emergence of twin decits Consider again the scal easing regime ruling
in the time interval (t
1
. t
2
). The higher C
t
resulting from the scal easing
leads to a lower `A
t
than before t
1
, cf. (13.43). As a result C1
t
0.
That is, a current account decit has emerged in response to the government
budget decit. This situation is known as the twin decits. As we argued,
the situation is not sustainable. Sooner or later, the incipient lack of solvency
will manifest itself in diculties with continued borrowing something must
be changed.
From mere accounting we have that the current account decit also can be
written as the dierence between aggregate net investment, 1
a
t
. and aggregate
net saving, o
a
t
. So
C1
t
= 1
t
o
t
= 1
t
o1
t
(o
t
o1
t
) = 1
a
t
o
a
t
= 1
a
t
(o
j
t
+o
j
t
) = 1
a
t
o
j
t
+

1
t
. (13.45)
since public saving, o
j
t
. equals

1
t
. the negative of the budget decit. Gener-
ally, whether, starting from a balanced budget and balanced current account,
a budget decit tends to generate a current account decit, depends on how
net investment and net private saving respond. In the present example we
have 1
a
t
= 0 for all t. And for t < t
1
. o
j
t
= :

+ (n

)` C

= 0
together with

1
t
= 0. In the time interval (t
1
. t
2
).

1
t
0 and o
j
t
0, but
the budget decit dominates and results in C1
t
0.
As before let taxation be increased at time t
2
so that the government
budget is balanced for t t
2
. Then again

1
t
= 0. Yet for a while C1
t
0
because now o
j
t
< 0 as reected in

t
< 0. The decit on the current ac-
count is, however, only temporary and certainly not a signal of an impending
default. It just reects that it takes time to complete the full downward ad-
justment of private consumption after the scal tightening.
19
Let us consider a dierent scenario, namely one where the scal easing
after time t
1
takes the form of a shift in government consumption to

G
0


G
without any change in taxation. Suppose the household sector expects that
a scal tightening will not happen for a long time to come. Then, H
t
and
C
t
are essentially unaected, i.e., C
t
= C

and H
t
= H

as before t
1
. So
also remains at its steady-state value

from before t
1
. given in (13.33).
Owing to the absence of private saving, the government decit must be fully
nanced by foreign borrowing. Indeed, by (13.45),
C1
t
=

1
t
0
in this case. Here the two decits exactly match each other. The situation
19
By construction of the model, agents in the private sector are never insolvent.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.3. Public and foreign debt: a small open economy 493
is not sustainable, however. Government debt is mounting and if default is
to be avoided, sooner or later scal policy must change.
It is the absence of Ricardian equivalence that suggests a positive rela-
tionship between budget and current account decits. On the other hand,
the course of events after t
2
in this example illustrates that a current account
decit need not coincide with a budget decit. The empirical evidence on
the relationship between budget and current account decits is mixed. A
cross-country regression analysis for 19 OECD countries with each countrys
data averaged over the 1981-86 period pointed to a positive relationship.
20
In fact, the attention to twin decits derives from this period. Moreover,
time series for the U.S. in the 1980s and rst half of the 1990s also indi-
cated a positive relationship. Nevertheless, other periods show no signicant
relationship. This mixed empirical evidence becomes more understandable
when short-run mechanisms, with output determined from aggregate demand
rather than supply, are taken into account.
The current account in a growing economy The above analysis ig-
nored growth in GDP and therefore steady state required the current ac-
count to be balanced. It is dierent if we allow for economic growth. To see
this, suppose there is Harrod-neutral technological progress at the constant
rate q and that the labor force grows at the constant rate :. Then in steady
state GDP grows at the rate q +:. From (13.42) follows, in analogy with the
analysis of government debt in Chapter 6, that the law of movement of the
foreign-debt/GDP ratio d 1,1 is

d = (: q :)d
`A
1
. (13.46)
A necessary condition for the SOE to remain solvent is that circum-
stances are such that the foreign-debt/GDP ratio does not tend to explode.
For brevity, assume `A,1 remains equal to a constant, r. Then the linear
dierential equation (13.46) has the solution
d
t
= (d
0
d

)c
(vja)t
+d

.
where d

= r,(: q :). If : q +: 0. the SOE will have an exploding

foreign-debt/GDP ratio and become insolvent vis-a-vis the rest of the world
unless r (: q :)d
0
. Note that the right-hand-side of this inequality is
an increasing function of the initial foreign debt and the growth-corrected
interest rate.
20
See Obstfeld and Rogo (1996, pp. 144-45).
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
494
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
Suppose d
0
0 and r = (: q :)d
0
. Then d remains positive and
constant. The SOE has a permanent current account decit in that foreign
debt, 1. is permanently increasing. But net exports continue to match the
growth-corrected interest payments on the debt, which then grows at the
same constant rate as GDP. The conclusion is that, contrary to a presump-
tion prevalent in several introductory textbooks and the media, a country
can have a permanent current account decit without this being a sign of
economic disease and mounting solvency problems. In this example the per-
manent current account decit merely reects that the country for some
historical reason has an initial foreign debt and at the same time a rate of
time preference such that only part of the interest payment is nanced by
net exports, the remaining part being nanced by allowing the foreign debt
to grow at the same speed as production.
The required net exports-income ratio, (:q:)d
0
. measures the burden
that the foreign debt imposes on the country. The higher this ratio, the
greater the likelihood that the debtor country will face nancial troubles. If
the foreign debt directly or indirectly is public debt, the additional problem
of levying sucient taxation to service the debt arises.
A worrying feature of the U.S. economy is that its foreign debt has been
growing since the middle of the 1980s accompanied by a permanent trade
decit. The triple decits characterizing the U.S. economy in the new millen-
nium (government budget decit, current account decit, and trade decit)
indicate an unsustainable state of aairs.
The debt crisis in Latin America in the 1980s In the early 1980s, the
real interest rate for Latin American countries rose sharply and net lending
to corporations and governments in Latin America fell severely, as shown in
Fig. 13.9. The solid line in the gure indicates the London Inter-Bank Of-
fered Rate (LIBOR) deated by the rate of change in export unit prices; the
LIBOR is the short-term interest rate that the international banks charge
each other for unsecured loans in the London wholesale money market. In-
terest rates charged on bank loans to Latin American countries were typically
variable and based on LIBOR.
21
A debt crisis ensued. Mexico suspended its
payments in August 1982. By 1985, 15 countries were identied as requiring
coordinated international assistance. The average debt-exports ratio (our
d,r) peaked at 384 per cent in 1986 (Cline, 1995).
21
The correlation coecient between the two variables in Fig. 13.9 is -0.615. The growth
rate of total external debt is based on data for the following countries: Argentina, Bolivia,
Brazil, Chile, Columbia, Costa Rica, Cuba, Dominican Republic, Ecuador, El Salvador,
Guatemala, Haiti, Honduras, Mexico, Nicaragua, Panama, Paraguay, Peru, Uruguay, and
Venezuela.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.4. Government debt when taxes are distortionary* 495
Figure 13.9:
13.4 Government debt when taxes are distor-
tionary*
So far we have, for simplicity, assumed that taxes are lump sum. Now we
introduce a simple form of income taxation. We build on the same version
of the Blanchard OLG model as was considered in Section 13.1. That is,
the economy is closed, there is technological progress at the rate q 0, and
the population grows at the rate : 0. whereas retirement is ignored (i.e.,
` = 0). In addition to income taxation we bring in specic assumptions about
government expenditure, namely that spending on goods and services as well
as transfers grow at the rate q +:. The focus is on capital income taxation.
Two main points of the analysis are that (a) capital income taxation results
in lower capital intensity and consumption in the long run (if the economy
is dynamically ecient); and (b) a higher level of government debt requires
higher taxation and tends thereby to increase the excess burden of taxation.
Elements of the model
The household sector Assume there is a at tax on the return on nancial
wealth at the rate t
v
. That is, an individual, born at time and still alive at
time t 0. with nancial wealth c
t
has to pay a tax equal to t
v
:
t
c
t
per time
unit, where t
v
is a given constant capital-income tax rate, 0 t
v
< 1. The
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
496
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
actuarial bonus is not taxed since it does not represent genuine income. There
is symmetry in the sense that if c
t
< 0. then the tax acts as a subsidy (tax
deductibility of interest payments). Labor income and transfers are taxed at
a at time-dependent rate, t
&t
< 1. Only in steady state is the labor-income
tax rate constant. Because labor supply is inelastic in the model, t
&t
acts
like a lump-sum tax and is not of interest per se. Yet we include t
&t
in the
analysis in order to have a simple tax instrument which can be adjusted to
ensure a balanced budget when needed.
The dynamic accounting equation for the individual is
c
t
= [(1 t
v
):
t
+:] c
t
+ (1 t
&t
)(n
t
+r
t
) c
t
. c
0
given,
where r
t
is a lump-sum per-capita transfer. The No-Ponzi-Game condition,
as seen from time t
0
, is
lim
t
c
t
c

0
[(1t

)v

+n]oc
0.
and the transversality condition requires that this holds with strict equality.
With logarithmic utility the Keynes-Ramsey rule takes the form
c
t
c
t
= (1 t
v
):
t
+:(j +:) = (1 t
v
):
t
j.
where j 0 is the rate of time preference and : 0 is the actuarial bonus,
which equals the death rate. The consumption function is
c
t
= (j +:)(c
t
+/
t
). (13.47)
where
/
t
=
Z

t
(1 t
&c
)(n
c
+r
c
)c

[(1t

)v

+n]o:
d:. (13.48)
At the aggregate level changes in nancial wealth and consumption are:

t
= (1 t
v
):
t

t
+ (1 t
&t
)(n
t
+r
t
)`
t
C
t
. and

C
t
= [(1 t
v
):
t
j +:] C
t
,(j +:)
t
.
respectively, where , is the birth rate.
Production The description of production follows the standard one-sector
neoclassical competitive setup. The representative rm has a neoclassical
production function, 1
t
= 1(1
t
. T
t
1
t
). with constant returns to scale, where
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.4. Government debt when taxes are distortionary* 497
T
t
(to be distinguished from the tax revenue 1) is the exogenous technol-
ogy level, assumed to grow at the constant rate q 0. In view of prot
maximization under perfect competition we have
J1
t
J1
t
= ,
0
(

/
t
) = :
t
+o.

/
t
1
t
,(T
t
1
t
). (13.49)
J1
t
J1
t
=
h
,(

/
t
)

/
t
,
0
(

/
t
)
i
T
t
= n
t
. (13.50)
where o 0 is the constant capital depreciation rate and , is the production
function in intensive form, given by 1,(T 1) = 1(

/. 1) ,(

/). ,
0

0. ,
00
< 0. We assume , satises the Inada conditions. In equilibrium, 1
t
=
`
t
. so that

/
t
= 1
t
,(T
t
`
t
). a pre-determined variable.
The government sector Government spending on goods and services,
G, and transfers, A. grow at the same rate as the work force measured in
eciency units. Thus,
G
t
= T
t
`
t
. A
t
= T
t
`
t
. . 0. (13.51)
Gross tax revenue,

1
t
. is given by

1
t
= t
v
:
t

t
+t
&t
(n
t
+r
t
)`
t
. (13.52)
Budget decits are nanced by bond issue whereby

1
t
= :
t
1
t
+G
t
+A
t

1
t
(13.53)
= (1 t
v
):
t
1
t
+T
t
`
t
+ (1 t
&t
)T
t
`
t
t
v
:
t
1
t
t
&t
n
t
`
t
.
where we have used (13.51) and the fact that in general equilibrium
t
=
1
t
+ 1
t
. We assume parameters are such that in the long run the after-
tax interest rate is higher than the output growth rate. Then government
solvency requires the No-Ponzi-Game condition
lim
t
1
t
c

0
(1t

)v

oc
0.
It is convenient to normalize the government debt by dividing with the
eective labor force, T `. Thus, we consider the ratio

/
t
1
t
,(T
t
`
t
). By
logarithmic dierentiation w.r.t. t we nd

/
t
,

/
t
=

1
t
,1
t
(q +:). so that

/
t
=

1
t
T
t
`
t
(q+:)

/
t
= [(1 t
v
):
t
q :]

/
t
++(1t
&t
)t
v
:
t

/
t
t
&t
n
t
.
where n
t
n
t
,T
t
. Note that the tax t
v
redistributes income from the wealthy
(here the old) to the poor (here the young), because the old have above-
average nancial wealth and the young have below-average wealth.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
498
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
General equilibrium
Using that : , :. we end up with three dierential equations in

/.
c C,(1`). and

/:

/
t
= ,(

/
t
) c
t
(o +q +, :)

/
t
. (13.54)

c
t
=
h
(1 t
v
)(,
0
(

/
t
) o) j q
i
c
t
,(j +:)(

/
t
+

/
t
). (13.55)

/
t
=
h
(1 t
v
)(,
0
(

/
t
) o) q (, :)
i

/
t
+ + (1 t
&t
)
t
v
(,
0
(

/
t
) o)

/
t
t
&t
n(

/
t
). (13.56)
where n(

/
t
) ,(

/
t
)

/
t
,
0
(

/
t
). cf. (13.50). Initial values of

/ and

/ are
historically given and from the NPG condition of the government we get the
terminal condition
lim
t

/
t
c

0
[(1t)()
0
(

I)c)j(on)]oc
= 0. (13.57)
assuming that the NPG condition is not over-satised.
Suppose that for t 0 the growth-corrected budget decit is balanced
in the sense that the growth-corrected debt is constant. Thus,

/
t
=

/
0
for all
t 0. This requires that the labor income tax t
&t
is continually adjusted so
that, from (13.56),
t
&t
=
1
+ n(

/
t
)
nh
(1 t
v
)(,
0
(

/
t
) o) q (, :)
i

/
0
+ + t
v
(,
0
(

/
t
) o)

/
t
o
.
(13.58)
Then (13.55) simplies to

c
t
=
h
(1 t
v
)(,
0
(

/
t
) o) j q
i
c
t
,(j +:)(

/
t
+

/
0
).
which together with (13.54) constitutes an autonomous two-dimensional dy-
namic system. Note that only the capital income tax t
v
enter these dynam-
ics. The labor income tax t
&t
does not. This is a trivial consequence of the
models simplifying assumption that labor supply is inelastic.
To construct the phase diagram for this system, note that

/ = 0 for c = ,(

/) (o +q +, :)

/. (13.59)

c = 0 for c =
,(j +:)(

/ +

/
0
)
(1 t
v
)(,
0
(

/) o) j q
. (13.60)
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.4. Government debt when taxes are distortionary* 499

c
k

*
k

( ) ( ) c f k g b m k


0 k

0 new c

*'
k

O
'
k

GR
k

*'
c
E
' E
0 old c

P

0
k

Figure 13.10: Phase diagram illustrating the eect of a fully nanced reduction
of capital income taxation.
There are two benchmark values of the capital intensity. The rst is the
golden rule value,

/
G1
. given by ,
0
(

/
G1
) o = q + :. The second is that
value at which the denominator in (13.60) vanishes, that is, the value,

/.
satisfying
(1 t
v
)(,
0
(

/) o) = j +q.
The phase diagram is shown in Fig. 13.10. We assume

/
0
0. But at the
same time

/
0
and are assumed to be modest, given

/
0
. such that the
economy initially is to the right of the totally unstable steady state close to
the origin.
We impose the parameter restriction j :. which implies

/ /
G1
for
any t
v
[0. 1) . thus ensuring

/

< /
G1
. in view of

/

<

/. That is,
,
0
(

) o ,
0
(

/) o =
j +q
1 t
v

q +:
1 t
v
q +:.
It follows that (13.57) holds at the steady state, E.
22
At time 0 the economy
will be where the vertical line

/ =

/
0
crosses the (stippled) saddle path. Over
22
And so do the transversality conditions of the households. The argument is the same
as in Appendix D of Chapter 12.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
500
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
time the economy moves along this saddle path toward the steady state E
with real interest rate equal to :

= ,
0
(

) o. Further, in steady state the

labor income tax rate is a constant,
t

&
=
h
(1 t
v
)(,
0
(

) o) q :
i

/
0
+ + t
v
(,
0
(

) o)

+ n(

)
. (13.61)
from (13.58).
The capital income tax drives a wedge between the marginal transfor-
mation rate over time faced by the household, (1 t
v
)(,
0
(

/) o). and that

given by the production technology, ,
0
(

/) o. The implied eciency loss

is called the excess burden of the tax. A higher t
v
implies a greater wedge
(higher excess burden) and for a given

/
0
, a lower

/

, cf. (13.60). Similarly,

for a given t
v
. a higher level of debt,

/
0
. implies a lower

/

and a higher :

(and a corresponding adjustment of t

&
).
23
Finally, if for some reason (of a
political nature, perhaps) t

&
is xed, then a higher level of the debt may
imply crowding out of

/

for two reasons. First, there is the usual direct

eect that higher debt decreases the scope for capital in households port-
folios. Second, there is the indirect eect, that higher debt may require a
higher distortionary tax, t
v
. which further reduces capital accumulation and
increases the excess burden.
We may reconsider the Ricardian equivalence issue from the perspective
of both these eects. The Ricardian equivalence proposition says that when
taxes are lump-sum, their timing does not aect aggregate consumption and
saving. In the rst section of this chapter we highlighted some of the reasons
to doubt the validity of this proposition under normal circumstances. En-
compassing the fact that most taxes are not lump sum casts further doubt
that debt neutrality should be a reliable guide for practical policy.
A fully nanced reduction of capital income taxation
Now, suppose that until time t
1
. the economy has been in its steady state
E. Then, unexpectedly, the tax rate t
v
is reduced to a lower constant level,
t
0
v
. The tax rate is then expected by the public to remain at this lower
level forever. The government budget remains balanced in the sense that
taxation of labor income is immediately increased such that (13.58) holds for
t
v
replaced by t
0
v
. This shift in taxation policy does not aect the

/ = 0
locus, but the

c = 0 locus is turned clockwise. At time t
1
. when the shift
23
We can not say in what direction t

has to be adjusted. This is because it is theoret-

ically ambiguous in what direction ()
0
(

) c)

moves when /

goes down.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.4. Government debt when taxes are distortionary* 501
in taxation policy occurs, the economy jumps to the point P and follows the
new saddle path toward the new steady state with higher capital intensity.
(As noted at the end of the previous chapter, such adjustments may be quite
slow.)
We see that the immediate eect on consumption is negative, whereas the
long-run eect is positive (as long as everything takes place to the left of the
golden rule capital intensity

/
G1
). The positive long-run eect on

/ is due to
the higher saving brought about by the initial fall in consumption. But what
is the intuition behind this initial fall? Four eects are in play, a substitution
eect, an income eect, a wealth eect, and a government budget eect. To
understand these eects from a micro perspective, the intertemporal budget
constraint of the individual is helpful:
Z

t
1
c
t
c

1
[(1t
0

)v+n]oc
dt = c
t
1
+/
t
1
. (IBC)
The point of departure is that the after-tax interest rate immediately rises.
As a result:
1) Future consumption becomes relatively cheaper as seen from time t
1
.
Hence there is a negative substitution eect on current consumption c
t
1
.
2) For given total wealth c
t
1
+/
t
1
. it becomes possible to consume more
at any time in the future (because the present discounted value of a given
consumption plan has become smaller, see the left-hand side of (IBC)). This
amounts to a positive income eect on current consumption.
3) At least for a while the after-tax interest rate, (1 t
0
v
): +:. is higher
than without the tax decrease. Everything else equal, this aects /
t
1
nega-
tively, which amounts to a negative wealth eect.
On top of these three standard eects comes the fact that:
4) At least initially, a rise in t
&
is necessitated by the lower capital income
taxation if an unchanged

/ is to be maintained, cf. (13.58). Everything else
equal, this also aects /
t
1
negatively and gives rise to a further negative eect
on current consumption through what we may call the government budget
eect.
24
To sum up, the total eect on current consumption of a permanent de-
crease in the capital income tax rate and a concomitant rise in the tax on
labor income and transfers consists of the following components:
substitution eect + income eect + wealth eect
+ eect through the change in the government budget = total eect.
24
The proviso everything else equal both here and under 3) is due to the fact that
counteracting feedbacks in the form of higher future real wages and lower interest rates
arise during the general equilibrium adjustment.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
502
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
From the consumption function c
t
= (j + :)(c
t
+ /
t
). cf. (13.47), we see
that the substitution and income eects exactly cancel each other (due to
the logarithmic specication of the utility function). This implies that the
negative general equilibrium eect on current consumption, visible in the
phase diagram, reect the inuence of the two remaining eects.
The conclusion is that whereas a tax on an inelastic factor (in this model
labor) obviously does not aect its supply, a tax on capital or on capital
income aects saving and thereby capital in the future. Yet such a tax may
have intended eects on income distribution. The public nance literature
studies, among other things, under what conditions such eects could be
obtained by other means (see, e.g., Myles 1995).
13.5 Debt policy*
Main text for this section not yet available. See instead Elmendorf and
Mankiw, Section 5 (Course Material).
A proper accounting of public investment
As noted by Blanchard and Giavazzi (2004), public investment as a share of
GDP has been falling in the EMU countries since the middle of the 1970s,
in particular since the run-up to the euro 1993-97. This later development is
seen as in part induced by the decit rule of the Maastrict Treaty and the SGP
which, like the standard government budget accounting we have described in
Chapter 6, attributes government net investment as a cost in a single years
account instead of the depreciation of the public capital. Blanchard and
Giavazzi and others propose to exclude government net investment from the
denition of the public decit.
To see the implications of this proposal, we partition G into public con-
sumption, C
j
. and public investment, 1
j
. that is, G = C
j
+ 1
j
. Public in-
vestment produces public capital (infrastructure etc.). Denoting the public
capital 1
j
we may write

1
j
= 1
j
o1
j
. (13.62)
where o is a (constant) capital depreciation rate. Let the annual nancial
return per unit of public capital be :
j
. This is the sum of the direct nancial
return from user fees and the like and the indirect nancial return deriving
from the fact that infrastructure tends to reduce the costs of public services
and increase productivity in the private sector and thereby the tax base.
Net government revenue, 1. now consists of net tax revenue, 1
0
. plus the
nancial return :
j
1
j
. In that now only interest payments and the capital
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.6. Credibility problems due to time inconsistency 503
depreciation, o1
j
. enter the decit account as true costs, the true budget
decit is :1 +C
j
+o1
j
1. where 1 = 1
0
+:
j
1
j
.
We may now impose a rule requiring balanced budget in the sense that
1 = :1 +C
j
+o1
j
(13.63)
should hold on average over the business cycle. Debt accumulation still obeys
(DGBC) so that

1 = :1+C
j
+1
j
1. Substituting (13.63) into this, we get

1 = 1
j
o1
j
=

1
j
. (13.64)
by (13.62). If public capital keeps pace with trend GDP, 1

. so that 1
j
,1

equals a positive constant, say /. we have

1
j
,1
j
= q + :. Then (13.64)
implies

1 = (q +:)1
j
= (q +:)/1

.
So the debt-to-trend income ratio,

/ = 1,1

. satises the linear dierential

equation

/ =
1

1 1

1

(1

)
2
= (q +:)/ (q +:)

/.
For q +: 0. this has the solution

/
t
= (

/
0

)c
(j+a)t
+

/

. where

/

= /.
We see that

/
t
/ for t . Run-away debt dynamics is precluded.
The ratio 1,1
j
. which equals

/,/. approaches 1 so that eventually the entire
public debt is backed by public capital. Fiscal sustainability is ensured in
spite of a positive structural budget decit, as traditionally dened, equal
to

1
j
. This result holds even when :
j
< :. which perhaps is the usual case.
Still, the public investment may be worthwhile in view of a non-nancial
return in the form of the utility contribution of public goods.
13.6 Credibility problems due to time incon-
sistency
When outcomes depend on expectations in the private sector, government
policy may face a time-inconsistency problem.
As an example consider the question: What is the position stated of a
government about negotiating with hostages? The ocial line, of course, is
that the government will not negotiate. ...
Text not yet available.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
504
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
13.7 Bibliographic notes
For very readable surveys about how important empirically the depar-
tures from Ricardian equivalence are, see for example Symposium on the
Budget Decit in Journal of Economic Perspectives, vol. 3, 1989, Himarios
(1995), and Elmendorf and Mankiw (1999).
13.8 Appendix
A. A growth formula useful for debt arithmetic
Not yet available.
B. Long-run multipliers
We show here in detail how to calculate the long-run crowding-out eects of
increases in government consumption and debt in the closed economy model
of Section 13.2. In steady state we have

1
t
=

C
t
=

1
t
=

1
t
= 0, hence
1(1

. `) o1

= C

+

G. (13.65)
(1
1
(1

. `) o j)C

= :(j +:)(1

+

1). (13.66)
1

= (1
1
(1

. `) o)

1 +

G. (13.67)
We consider the level

1 of public debt as exogenous along with public con-
sumption

G and the labor force `. The tax revenue 1

in steady state is
endogenous.
Assume (realistically) that 1

+

1 0. Now, at zero order in the causal
structure, (13.65) and (13.66) simultaneously determine 1

and C

as im-
plicit functions of

Gand

1. i.e., 1

= 1(

G.

1) and C

= C(

G.

1). Hereafter,
(13.67) determines the required tax revenue 1

at rst order as an implicit

function of

G and

1. i.e., 1

= 1(

G.

1).
To calculate the partial derivatives of these implicit functions, insert C

= 1(1

. `)o1

Gfrom(13.65) into (13.66) and take the total dierential:

(1
1
oj)

(1
1
o)d1

+C

1
11
d1

= :(j+:)(d1

+d

1). i.e.,
D d1

= (1
1
o j)d

G+:(j +:)d

1. (13.68)
where
D C

1
11
+ (1
1
o j)(1
1
o) :(j +:). (13.69)
and the partial derivatives are evaluated in steady state.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
13.8. Appendix 505
We now show that in the interesting steady state we have D < 0. As
demonstrated in Section 13.2, normally there are two steady-state points in
the (1, C) plane.
25
The lower steady-state point, that with 1 =

1

in
Fig. 13.2, is a source, i.e., completely unstable. The upper steady-state
point, that with 1 = 1

. is saddle-point stable. The latter steady state is

the interesting one (when

G and

1 are of moderate size). In that state the

C = 0 locus crosses the

1 = 0 locus from below. Hence
JC
J1
|

C=0
1
1
o. i.e.,
:(j +:)
1
1
o j (1

+

1)1
11
(1
1
o j)
2
1
1
o
:(j +:) :(j +:)
(1

+

1)
:

j
1
11
(:

j):

:(j +:) C

1
11
(:

j):

0 C

1
11
+ (:

j):

:(j +:) = D,
(13.70)
where the rst implication arrow follows from 1
1
= 1
1
(1

. `) o = :

. the
second from (13.66), and the third by rearranging. A convenient formula for
D is obtained by noting that
(:

j):

:(j+:) = :
2
+::

::

j:

:(j+:) = (:

+:)(:

(j+:)).
Hence, by (13.70),
D = C

1
11
(:

+:)(j +::

) < 0.
So the implicit function 1

= 1(

G.

1) has the partial derivatives, also
called the long-run or steady-state multipliers,
1
G
=
J1

J

G
=
:

j
D
< 0. (13.71)
1
1
=
J1

J

1
=
:(j +:)
D
< 0. (13.72)
using (13.68) and :

= 1
1
o j. As to the eect on 1

of balanced
changes in

G. it follows that 1

(J1

,J

G)

G = (:

j)

G,D < 0
for

G 0. This gives the size of the long-run eect on the capital stock,
when public consumption is increased by

G (

G small), and at the same

25
This is so, unless

G and

1 are so large that there is only one (a knife-edge case) or no
steady state with 1 0.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011
506
CHAPTER 13. GENERAL EQUILIBRIUM ANALYSIS OF
PUBLIC AND FOREIGN DEBT ISSUES
time taxation is increased so as to balance the budget and leave public debt
unchanged in the indenite future.
As to the eect on 1

of higher public debt, it follows that 1

(J1

,J

1)

1 = :(j +:)

1,D < 0 for

1 0. This formula tells us

the size of the long-run eect on the capital stock, when a tax cut implies, for
some time, a budget decit and thereby a cumulative increase,

1. in public
debt; afterwards the government increases taxation to balance the budget
forever.
26
Similarly, :

1
11
(1

. `)1

1
11
(1

. `) (J1

,J

1)

1
0, for

1 0.
The long-run or steady-state multipliers associated with the implicit func-
tion C

= C(

G.

1) are now found by implicit dierentiation in (13.65) w.r.t.

G and

1. respectively. We get JC

,J

G = (1
1
(1

. `)o)J1

,J

G1 < 1
and JC

,J

1 = (1
1
(1

. `) o)J1

,J

1 < 0.
Similarly, from (13.67) we get J1

,J

G = 1
11
(1

. `)(J1

,J

G)

1 +1
1 and J1

,J

1 = 1
11
(1

. `) (J1

,J

1)

1 +1
1
(1

. `) o 0 (since
1
11
< 0).
13.9 Exercises
13.1 To the notation given in Section 13.1 we add: `
t
is the monetary
base, D
t
nominal government debt, 1
t
the price level for goods and services,
:
t
the ination rate, and q
Y
is a constant growth rate of output. Then

D
t
+

`
t
= i
t
D
t
+1
t
(G
t
1
t
).
a) Interpret this equation.
b) Show that

1 = :
t
1
t
+G
t
1
t

`
t
,1
t
and

/
t
= (:
t
q
Y
)/
t
+(G
t
1
t
),1
t

`
t
,(1
t
1
t
).
c) Show that if the velocity of base money and the growth rate of base
money are constant, then the seigniorage-income ratio is constant.
d) In theoretical studies of government debt dynamics for a modern econ-
omy the seigniorage-income ratio is not seldom ignored. Why? Follow-
ing this lead, derive the time path for the debt-income ratio under the
assumption that :
t
= : q
Y
. G
t
,1
t
= (0. 1). and 1
t
,1
t
= t 0
(hint: the dierential equation r+cr = /. where c and / are constants,
c 6= 0. has the solution r
t
= (r
0
r

)c
ot
+r

. where r

= /,c.) Com-
ment.
26
We assume that t
2
t
1
, hence

1, is not so large as to not allow existence of a

saddle-point stable steady state with 1 0 after t
2
.
C. Gr ot h, Lect ur e not es i n macr oeconomi cs, ( mi meo) 2011