MIT LIBRARIES
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3 9080 02898 1394
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
FUTURE RENT-SEEKING AND CURRENT
PUBLIC SAVINGS
Ricardo
J.
Pierre
Caballero
Yared
Working Paper 08-20
October 9, 2008
RoomE52-251
50 Memorial Drive
Cambridge,
02142
MA
paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
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httsp://ssrn.com/abslract=]'-2-852T^
Future Rent-Seeking and
Current Public Savings
Ricardo
J.
Caballero and Pierre Yared*
This draft: October
9,
2008
Abstract
The conventional wisdom
debt and
This
deficits.
is
is
because myopic pohticians face pohtical risk and prefer to
An
extract pohtical rents as early as possible.
governments
pubhc
that pohticians' rent-seeking motives increase
imphcation of
this
argument
is
under-save during a boom, leaving the economy unprotected
will
event of a downturn. This view motivates a
number
of fiscal rules which are
that
in
the
aimed
at cutting deficits
and constraining borrowing so
distortion. In this
paper we study the determination of government debt and deficits
in a
dynamic model of debt which characterizes
as to limit the size of this political
political distortions.
We find that
in
our model the conventional wisdom always applies in the long run, but only does so
in the short
is
run when economic
high, a rent-seeking
volatility
is
low. Instead,
when economic
volatility
government over-saves and over-taxes along the equilibrium
path relative to a benevolent government. Paradoxically, the over-saving bias can
also be solved in this case by a rule of capping deficits, although the
operates through
its eff'ect
mechanism
on expectations of future rent extraction rather than
though the contemporary constraint. However, these
rules are ineffective in solving
the high taxation problem caused by the political friction, which in the short run
more acute
in the
JEL Codes:
Keywords:
high income volatility scenario.
E6, H2,
H6
Public debt, politicians, economic and political
precautionary savings, starve-the-beast,
'MIT and NBER, and Columbia
Marco
is
risk, rent-seeking,
fiscal rules
University, respectively.
We
are grateful to Stefania Albanesi,
Simon Johnson, Narayana
Kocherlakota, Jose Tessada, and Aleh Tsyvinski for comments. Caballero thanks the NSF for financial
support. First draft: November 2007. This paper replaces NBER
#13379 which circulated under
Battaglini, Patrick Bolton, V.V. Chari, Tito Cordelia, Michael Golosov,
WP
the
title of
"Inflating the Beast: Political Incentives
1
under Uncertainty."
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/futurerentseekinOOcaba
Introduction
1
The conventional wisdom
debt and
This
deficits.
is
is
that the rent-seeking motives of pohticians increase pubHc
because myopic pohticians face
An implication of this argument
rents as early as possible.
save during a boom, leaving the
view
is
political risk
economy unprotected
not only of theoretical interest, but
world which are aimed at cutting
deficits
it
is
and
prefer to extract
that governments will under-
in the event of a
motivates a number of
downturn.^ This
fiscal rules in
and constraining borrowing so as
the
to limit the
size of this political distortion."
In this paper
we study the determination
model that characterizes
wisdom always
volatility
is
of government debt
we
political distortions.'^ In a nutshell,
and
deficits in
find that the conventional
when economic
applies in the long run, but only does so in the short run
In contrast, the conventional
low.
when economic
volatility
is
a dynamic
wisdom does not hold
short run
in the
high since politicians choose public debt and deficits which
are too low. Paradoxically, the over-saving bias can also be solved in this case by a rule of
capping
deficits,
although the mechanism operates through
on expectations of
its effect
future rent extraction rather than through the contemporary constraint. However, these
rules are ineff'ective in solving the high taxation
which
in the short
More
run
specifically,
face political risk
is
problem caused by the
political friction,
moi'e accute in the high income volatility scenario.
we study an economy managed by a sequence
and who care about household welfare and
previous work on the political economy of debt,
we
of politicians
In contrast to the
rents.
consider the interrelated implications
of three important features: economic uncertainty, incomplete markets,
dynamics. The economy begins
at
in a
boom, and
who
this
boom
and transitional
can come to a permanent end
any date. Throughout the length of the boom, the benevolent government gradually
reduces
its
debt
in
We
order to prepare for the potential downturn.
compare
this
optimal
behavior to that of a rent-seeking government managed by politicians.
Our
first result is
ginning of the boom,
that while a rent-seeking government reduces
it
stops reducing
its
debt
if
the
boom
is
its
debt at the be-
sufficiently prolonged.
'See Battaglini and Coate (2008) and the survey article of Alesina and Perotti (1994) for a discussion
of this view.
^
Chile provides a recent example which has become a reference for
commodity producing economies more
terms of trades) surplus of 0.5 percent
boom, the
state runs very large fiscal
broadly.
The
fiscal
fiscal rule establishes
reforms in Latin America and
a structural
GDP. Thus, when terms of trade rise as a
surpluses (the sum of the structural surplus
of
income due to high commodity prices).
Acemoglu, Golosov, and Tsyvinski (2007a, 2007b) also study the effect
on taxes, though they do not consider the effect on government debt.
(i.e.,
at "normal"
result of a
commodity
target plus the excess
fiscal
*
of political
economy
distortions
This
because beyond a certain date, government resources become so abundant that
is
come
rent-seeking considerations
to
dominate intertemporal smoothing considerations.
rent-seeking government realizes that
if it
A
were to save more, then a future replacement
government would use the additional funds
for rent-seeking
(which only benefits incum-
bent politicians) as opposed to tax-cutting (which benefits households), and the govern-
ment
therefore restrains
its
savings in order to starve the future government of funds.
Therefore, in the long run, a prolonged
hold more assets and to tax
less
boom always
leads a benevolent
government to
than a rent-seeking government. This result
with that emphasized by Battaglini and Coate (2008). Our main contribution
that
it
consistent
is
is
to show
while this characterization applies to the long run fairly generally, whether or not
applies to the transitional dynamics of the
economy depends on the
level of
economic
volatility.
Our second
result
is
that
economic
if
volatility
is
sufficiently
low relative to
politi-
then the rent-seeking government over-borrows and under-taxes along
cal uncertainty,
the equilibrium path relative to a benevolent government. This insight-which
tent with the conventional
wisdom-emerges because low economic
is
consis-
volatility implies that
poHticians are biased toward extracting rents today versus in the future since political
risk
is
high and the cost of leaving the economy exposed in the downturn
causes governments to over-borrow and under-taxes at later stages of the
is
driven
down
and the prospect
sufficiently
early stages of the
boom
for rent-seeking
is
low. This
boom when
approaches.
debt
Politicians at
anticipate this behavior of politicians in the future, and for this
reason, they choose to over-borrow and to under-tax themselves.
Thus the prospect
of
future rent-seeking reinforces over-borrowing and under-taxation in the present.
Our
third
and most important result-which stands
wisdom-is that
if
economic
volatility
is
in contrast to the conventional
sufficiently high relative to political uncertainty,
then the rent-seeking government over-saves and over-taxes along the equilibrium path
relative to a benevolent government.
are less likely to
consume
rents today
this simultaneously protects the
the event of a
boom
Whenever economic
and more
likely to
volatility is high, politicians
consume them tomorrow
economy while providing them with potential
since
rents in
during which they are not replaced. In anticipation of these rents
in the future, the rent-seeking
government actually over-saves relative to a benevolent
government since the marginal value of additional funds
in the future
boom due
to rent-
seeking exceeds the marginal value of additional funds for a benevolent government
would instead use the additional savings to cut
save and over-tax at later stages of the
taxes. This causes
boom when
debt
is
driven
who
governments to over-
down
sufficiently
prospect for rent-seeking approaches. Politicians at early stages of the
boom
and the
anticipate
this behavior of politicians in the future,
and
they choose to over-save
for this reason,
and to over-tax themselves. The prospect of future rent-seeking therefore reinforces oversaving and over-taxing in the present.
Our
the popular
last result is that
>i^
capping
fiscal rule of
,:
deficits brings deficits
pluses closer to those of the benevolent government, although the
in the under-saving
would
and over-saving
do
save
this, so
mechanism
is
and
that
more and
different
In the under-saving region, the government
it
must
necessarily bind
more
to behave
it
However, the rule does not permit the government
rents.
and
it
forces the rent-seeking
government
to
a benevolent government. In the over-saving region,
like
the rule works through expectations by reducing the value of future public funds.
specifically,
sur-
save less in order to starve the future government of resources which
like to
would otherwise squander on
to
cases.
.
:
More
unconstrained governments over-save because they look forward to squander-
ing public funds in the future
however makes
it
boom
the
if
persists for sufficiently long.
The
fiscal rule
impossible to squander these public funds in the future since
it
forces
a future government to save more. Therefore, the rule reduces the value of future funds
from today's perspective, and
this induces today's
government to save
Part of this
less.
reduction in savings comes not from deep tax cuts but from earlier and higher levels of
rent extraction relative to the
on
its
own, the
sufficiently
fiscal rule
economy
absence of
in the
fiscal rules.
More
generally,
cannot force governments to cut taxes when resources become
abundant, and
in the
long run, additional increases in savings are used purely
for rent-seeking.
This paper builds on the literature on optimal
fiscal
policy and debt
dating back to the classical work of Barro (1979) and Lucas and Stokey
management
(1983).''
We depart
from this work by relaxing the assumption of a benevolent government and by assuming
that the
who
economy
is
managed by
politicians
who
derive partial utility from rents
face potential replacement. In this regard, this paper
literature
on the
political
economy
of Battaglini and Coate (2008).
governments face economic
risk
of debt.
As
and
in
More
political risk.
exceeds that of the benevolent government.
economy
specifically,
most
closely related to the
our work complements that
our work, they consider a setting
risk implies that in the long run, a rent-seeking
implications of political
is
We
They show
in
which current
that the presence of political
government holds a
level of
debt which
depart from their work by focusing on the
distortions along the equilibrium path and
away from
steady state. In the process, we describe a novel over-saving mechanism. Our work
related to that of Song, Storesletten,
and
Zilibotti (2007)
'See also Aiyagari, Marcet, Sargent, and Seppala (2002),
1993b).
and
Bohn
who show
(1990),
is
also
that intergenerational
and Chari and Kehoe (1993a,
conflict in a
dynamic model can cause a government
to the social
optimum.
We
depart from their work by abstracting from intergenerational
and considering instead the impact of
conflict
over-saving result
is
to under-save or over-save relative
related to the
political
and economic
work of Yared (2008) who argues that prescribing
high levels of savings in the presence of rent-seeking politicians
is
Finally, our
risk.^
associated with the anticipation of future rents.
distortionary since
is
In contrast, in the current
paper we
explain these high savings as an endogenous mechanism to extract future rents
effective
economic uncertainty
This introduction
is
is
it
when
high.
followed by five sections and an appendix. Section 2 describes the
environment and Section 3 describes the corresponding equilibrium under a benevolent
Section 4 describes the equilibrium under a rent-seeking government and
government.
compares
it
to that of a benevolent government. Section 5 describes a simulation of our
economy and
The Appendix contains
discusses policy implications. Section 6 concludes.
the proofs and additional material.
Model
2
2.1
:;:.'"';;
,;"'
'^\/''y'"
Economic Environment
—
{0, ...,oo}
households with the following period
welfare:
There are discrete time periods
for
\
t
:,,
•
Q >
which represents consumption and
0, v! (0)
=
oo,
and
lump sum taxes
There
is
=
0.
it
number
for
u {) which
<
e,
of potential
and
which represents socially wasteful
their
identical politicians
rents.''
utility
6*
>
'For additional work on the political economy of debt, see
Alesina and Perotti (1994), Alesina and Tabellini (1990),
and Svensson
•
(1)
,
—u"
{)
>
0,
e
,
v!"
(•)
>
they pay
budget so that q
= e— r^.
can also be interpreted as negative of public spending.
u (q) when out of power and who derive the flow
>
of identical
,
satisfies u' {)
and they balance
1
..
Households hold a constant endowment
to the government r^
a large
and a continuum of mass
Eolf^P'uic)], Pe{0,l),
v! (oo)
Since r^ can be negative,
Xj
;:./:^''.//',\':v'-"V^
who
u (q)
and we
for
Amador
derive the flow utility
Qxt
-f-
when
in
power
for
refer to the special case
example Aghion and Bolton (H
(2003), Lizzeri (1999), and Persson
(1989).
''While the linearity of rents in the utility function
is
important
for
the
full
characterization of the
model, the over-saving mechanism we describe depends on the existence of a region
zero. Specifically
if
v (i) represents the flow utility of rents,
we
require v' (0)
<
in
which rents are
oo. Details available
upon
of ^
=
government since
as a benevolent
it
corresponds to the case in which incumbent
poUticians have the same preferences as households. Levels of 9 which exceed
captures
the inverse cost of rent-seeking for the politician so that higher levels of 9 are associated
with
less costly rent-seeking.
A politician in
an identical
power
politician
-
in period
from
i-|- 1
is
t
,
,
permanently removed from
office
onward with exogenous probability
1
—
and replaced with
g G
incumbent
q represents the survival rate of a politician.^ Therefore, the welfare of the
i
=
so that
(0, 1),
at
can be written as
/?'
{u
{ct)
+
q'9xt)
(2)
,
J
where we have taken into account that a politician
probability
in period zero survives to period
Tt
<
e
and borrowing
ment's dynamic budget constraint
is
(5ht+i
a given
6o
—y
(L)
=
follows a
(7
>
first
0.
yt is
stochastic
The government
order
and debt
by raising
^
6f
at a price
endowment shock
yt-^
6
/?
(0, 1).
The govern-
.
^bt + xt-
subject to limj_oo P^bt+i
The endowment
>
from international markets
^
6(+]
In addition, the government experiences an exogenous
for
with
g*.
In every period, the government finances rents Xt
revenue
t
<
{Tt
+
yt)
0.
with y (H)
=
smooth household's consumption.
St
and depends on the
therefore exists to
Markov process and
(3)
-.
is
state
St
€ {L,
H}
independent of the political replacement shock.
= H. We simplify our discussion by assuming that Pr {sj = L\st-i — L} = I and
that Pr {s( = H\st_i = H} — a E {0, 1). We refer to state H as the boom and state L
as the downturn. We will focus on the path of the economy with Sq = H. Therefore, the
Let So
economy
is
probability
experiencing a temporary
I
boom which may permanently end
any date with
at
— a^
request.
We
could instead allow politicians to return to power, and none of our results would change
probability of holding power at any
**There
is
t
if
the
is i.i.d.
no difference between letting the government or the households experience this endowment
shock.
"This formulation allows
for tractability.
We
have numerically simulated economies
in
which
S(
=
I, is
not absorbing and achieved similar characterization to our analytical results here. Details available upon
request.
Political
2.2
The order
Environment
of events at every period
1.
Nature determines
2.
The period
3.
Markets open and
t
yt
as follows:
is
t
and potentially replaces the period
(
—
1
incumbent.
politician chooses policies {Tt,xt,bt+\}.
clear.
Given that there are many potential equilibria which can emerge
consider the symmetric
economy with
T
Markov
we
Perfect Equilibrium which coincides with the limit of our
T —
periods as
in this setting,
In this equilibrium, the incumbent poHtician-
oo.'"
>
independently of identity and of past political shocks-chooses policies as a function of
the state
chooses
St
C(,
and the
level of
debt
bt.
V^
in choosing Tt, the
incumbent effectively
we
will refer to c {b, s),
x
so that without loss of generahty,
the politician's choices of q, Xt, and
Define
Note that
(6, s)
and V'^
(6, s)
s.
The
set of policies {c
an Markov Perfect Equilibrium
given b and
s
on
bt
s)
and
—
as the continuation value of being out of office
respectively, with debt b in state
stitutes
respectively, conditional
fe(+i,
[b,
{c
if
{b, s)
,
x
(6, s)
(6, s)
,
,
x
b' {b,
(6, s)
s)}
,
b' [b,
b
b'
(6, s) as
and
and
s.
in office,
s)}^^^
maximizes
—
St
^
con-
V^ {b, s)
and subject to the government's dynamic budget constraint.
Benevolent Government Benchmark
3
We
begin by considering the policies of the benevolent government which corresponds to
a special case of our
and to
economy with
facilitate future discussion,
and the
policies of the benevolent
^
=
we
0.
let
In this circumstance,
the superscript
B
V^
(6, s)
equals
V^
{b, s),
denote the continuation value
government. The problem of the government in the
downturn can be written as
F^(6,L)
,.
= maxu(c)+/3y^(6',L)
";"
s.t.
\
Pb'
(4)
;
^b + x-{e-a), /
(5)
Since households are always better off consuming more, the solution to this problem
assigns
x^
"That
is,
(6,
L)
=
0.
Conditional on
subject to the constraint that
b',
/?
the politician
fcr+i
<
0.
is
always better
off
taxing
less
versus
extracting more rents. Therefore, the problem
consumption problem
e
-a -
6 (1
-
b'^
/?),
mathematically equivalent to a personal
which smoothing consumption
in
{b,
is
L)
=
6,
V^
and
{b,L)
=
is
optimal. Thus, c^ [b, L)
u{e - a - b(l
-
-
P)) / (1
=
/3).
Using this characterization, we can now consider the government's problem during the
preceding boom:
V'^ {b,H)
= maxu{c)+PE,V^
{b',s)
(6)
c,x,b'
/~'^
s.t.
-
Pb'
As
in the
requires c^
downturn, the solution to
[b,
H)
u, [c^
Lemma
b'^ (6,
1 c^
H) <
{b,
[b,
is
this
(7)
problem yields x^ [b,H)
=
0,
and optimality
to be defined by the following Euler equation:
H))
H)
-
au, [c^
is strictly
[b' (6,
H) H)) +
,
decreasing in
b,
(1
-
b'^ (6,
a) u, (c^
H)
(6' (6,
is strictly
H)
,
L))
(8)
.
increasing in
b,
and
b.
The government
economy
^b + x-{e + a).
taxes
more and saves
relatively poor.
infinity since the
when government debt
The government always
preparation for the downturn and
Note that as the boom
less
it
continues to drive
persists, the size of the
is
high since the
boom in
boom ends.
raises its savings in the
down
its
debt until the
government asset position approaches
government always benefits from saving more
in preparation for the
downturn.
4
Rent- Seeking Government
We now
consider the behavior of a government more generally for
all
^
>
0.
Here we
write the problem of the government recursively (Section 4.1), characterize the dynamics
of
consumption and debt (Section
government (Section
4.3).
4.2),
and compare these
policies to those of
a benevolent
Recursive Program
4.1
Conditional on entering a downturn, the incumbent politician solves the following problem:
7^
(6,
L)
= max {u
+
(c)
Ox
+P
(qV''
{b'
,
L)
+
(1
-
V'^ {b\ L))
g)
c,x,b'
Pb'
The government
>
and
s.t.
X
=
+ x-{e-a).
b
(10)
smooth consumption, though
clearly wishes to
rent-seeking which provides a marginal utility of 9 and sets a lower
utility of
it is
also interested in
bound
for the
- min{e-a-
x^{b,L)
= max
^ 0,
I'P /L
—
< o,
r\
L)
max
)
-^)6,n,-^(^)}
(1
^^^^^4^ 1-/3
e
u
-
cr
-
6
it^M^)
1-/3
rent-seeking government follows the
same smooth
policies with zero rent-seeking
as those of a benevolent government as long as its initial stock of debt b
{e
—a—
uj^
{6)) / (1
any additional reductions
—
In this case, the government
/3).
in b are
used
for
is
poor and
reducing taxes on households as opposed to
below this threshold, then the government
rents, they tax
above a
is
relatively
raising rents (since the marginal benefit of cutting those taxes exceeds
If b is
fol-
P denotes the policies of a rent-seeking government:
c^{b,L)
(6,
threshold
marginal
consumption. This means that during the downturn, politicians choose the
lowing policies, where the superscript
The
(9)
}
is rich.
6.).
Politicians extract positive
households more than the benevolent government, and they borrow more
than the benevolent government. More
specifically,
consumption
is
held at u~^
{9),
so that
the marginal benefit of rent-seeking equals the marginal benefit of consumption. Moreover,
debt
is
held at (e
used only
—
cr
—
u~^ {9)) /
for rent-seeking as
{1
—
P).
Therefore, any additional reductions in b are
opposed to tax or debt reduction. By following
this strategy,
the incumbent politician
who may be
extraction and leaves
future politicians with zero rents. Note that the threshold which
all
replaced in the future chooses to frontload
all rent-
separates the zero rent region from the positive rent region rises with the rent-seeking
bias
9.
Given these
policies,
we can
characterize V'^
(6,
L) and
V^
(6,
L).
Lemma
V^
1.
2.
L)
{b,
mb forb>
V^
L)
{b,
V^
I/j,^
{b,
decreasing in
is strictly
tiable
with
3.
=
2 The following conditions hold for j
is
{b,
L)
[e-cj - uj^
linear in b
L)
=
{6)) / (1
-
N
•
concave in
and continuously differen-
b,
/3),
<
[e
— a — u~^
[6]) / (1
—
P)
bforb <
(e
—
(6'))
/ {I
—
P)
and continuously
differentiable in b for b
and continuously
differentiable in
-e,
linear inb
is
strictly
b,
P,
cr
—
u"-^
'.
with
,
lim
V,,^
{b,L)
= -9 and
V^''{b,L)=0.
lim
6-[(e-<r-ur'
6_[(e-a-ur'W)/(l-/3)]^
The important feature of Lemma 2 is that V'^ (6,
point {e — a — u~^ {6)) / {1 - P) where rent-seeking
L)
is
,
(e))/(l-/3)]
not differentiable at the cutoff
begins. This
is
because additional
resources are no longer used for cutting taxes and are instead used for raising rents which
does not benefit society.
We
an analogous
will see that
result to
Lemma
2 holds in the
boom.
Given the behavior of the economy
the downturn, we characterize the policy of
in
the benevolent government in the boom.
The incumbent
politician solves the following
problem:
V^{b,H)=ma.x{u{c) + 9x + PEs{{qV''{b',s) + {l-q)V''{b',s))}}
-
(11)
s.t.
_
.
.
.
Pb'
To
facilitate discussion,
6
We
will
=
(e
+
we
{9)
,
b represents the
converges during a sustained boom.
on the
level of volatility a,
Specifically, define a* as
x
+ c-{e +
a)
(12)
define the following cut-off point:
a - max {u;'
show that
= b+
and
this
2a
+
u^'
{0 (1
-
aq)
/ (1
-
a))})
/ (1
-
/3)
.
(13)
steady state level of debt to which the economy
Note that the exact characterization of
is
b
depends
important since there are two cases two consider.
Note that
survival q goes to
a* increases.
that a*
if
<
>
0.
The
a* goes to
0.
Moreover, as the persistence of the
since g
1,
Finally,
political risk
As we
will
low, the
is
2.
is
Lemma
2 holds
strictly decreasing
is
m
tiable
m b for b > e + a — u~^ {9)
V^
H)
(6,
linear in b
is
=
V^
{b,H)
i,
with
...-•
.
.
and we can characterize
b,
+
—
strictly
cr
-
likely to
— u'^^
(6,
-
high.
Thus,
(^)
(6,
H).
N
concave in
b,
differentiable in b
i,
and continuously
differentiable
:
,1
/.,•
-.
:
'..-''
V^^ {b,H)
and continuously dijjeren-
<
forb
e
+ a — u~^
{0)+/3b
'
--'
...
inbforb < e+a — u~^ (^)+/9^
,'
:
= -9 and
,
lim
'
.'
•
_
V^^
^
//)
(fe,
=
.
if
b'^ {b,
H) >
e
+
_,.''..;"'
>
^
au, (c^
'
6-.[e+o--ur^(6»)+/3b]
then
H))
is
exceed a*
+ PkV^ (6, H) and V^
order conditions and the envelope condition imply that
lij^ (e) -h/36,
Ur (c^
P,
and continuously
lim
first
more
>
j3b,
6-»[e+CT-u^'(6l)+/3fc]'^
The
increases,
-9,
linear in b
is
boom a
temporary, and the rent-seeking bias 6
3 The following conditions hold for j
with V^P {b,H)
3.
boom
is
show, rent-seeking begins at levels of debt below e + a
V^ {b,H)
so that as pohtical
q,
can be shown by implicit differentiation given that ^i"'()
it
an analogous result to
1.
cutoff value a* decreases in
decreasing in the rent-seeking bias 9}^ Therefore, a
is
Lemma
1, ct*
,
(r (6, H),H)) + [l-a) u,
(c^ (6'^
(6,
//)
,
L))
,
(14)
so that the Euler equation holds with equality as under a benevolent government. Moreover,
if b'^
{b,H)
<e + a-
u"^
u,{c^{b,H))
.:
These two equations
(9)
+ /36,
then
.
=aq9 + {l-a)u,{c^
_
{b'^{b,H),L)).
relate the marginal cost of public funds
...
(15)
today to the expected
marginal cost of public funds tomorrow. They show that the marginal cost of public funds
tomorrow depends on whether
"Formally,
^
< \
({u,^ (."i (0)))"^
^Savings are never high enough
this is
suboptimal
for
or not rent-seeking takes place during the boom.^^
- (u„ ^u^^
for rent-seeking to
today's government.
10
j^^lz^) jy'^j <
occur both
in
the
If
no
o.
boom and
in the
downturn since
rent-seeking takes place, the marginal cost of public funds equals the marginal utihty of
consumption since additional resources are used to boost consumption (equation.
In contrast,
if
rent-seeking takes place, the marginal cost of public funds
today's politician maintains power with probability q and extracts rents in the
which provide marginal benefit 9 (equation
We
since
future
(15)).''^
Dynamics
Transitional
4.2
(14)).
qO
is
begin by describing the transitional dynamics of policies under a rent-seeking govern-
ment.
Proposition
(dynamics)
1
H) =
b if b
<b,
Policies satisfy the following properties for
H) <
b if b
>b, and
2.
Ifa<a%
thenc^{b,H) < {=)u-^
(9)
and
3.
If a
>
thenc^{b,H) < {=) u-^
{9)
andx^{b,H) - (>)0
1
and 2 display
as a function of 6 for
ment, the rent-seeking government
lets
,
Much
respectively.
like
b.
These
boom
The
it
b
b.
In contrast,
H)
the benevolent govern-
rise forever.
Beyond
minimum
if cr
>
ct*,
6,
a
point
a <
a*,
then rent-seeking
b.
a.
If
cr
<
a*, then starting
from
never extracts rents along the path. Once debt
chooses
b'^ (6,
H) — bso
that the
government never saves beyond
by a
>
{b,
implied dynamics of consumption and rents depend crucially on the degree of
economic uncertainty
and
they depict b'^
figures also depict the rent-seeking regions for different levels of a. If
when debt drops below
b,
> {<)b.
if b
causes the government to stabilize tomorrow's debt at a
then rent-seeking begins when debt goes below
begins
b:
debt decline monotonically throughout the boom,
but unlike the benevolent government, government assets do not
prolonged
>
H) = {>)0 ifb> {<)b, and
x'" {b,
this proposition graphically. Specifically,
a < a* and a > a*
b
weakly increases in
b'^ (6,
Figures
b'^ {b,
H)
1.
a*,
b'^ {b,
some
likely
6o
>
6 first
economy reaches the steady
b since politicians
know
b,
the governments saves
reaches
6,
the government
state with zero rents.
The
that rents would be extracted
replacement government, and the additional benefit of making these savings
^^Note that if 6'-f° [h,H) = e + a-u-^ [0) + fib, then 1/^ (b,H) is not differentiate, though Uc (c^ {b,H))
must be in the range between the right hand side of (15) and the right hand side of (14). Specifically,
Uc (c^
(fa,
H)) € [aqO
-H (1
-
q) u, [c^ {b'^
{b,
H)
11
,L)) .aO
+
[l
-
a) u^ (c^ {b""
{b,
H)
,
L))]
.
available for a do^'viiturn do not outweigh the cost of leaving additional rents for a replace-
ment government
boom. For the same reason,
in a
the government chooses c^
{bo, H) =
u~^
environment leads debt to
b
Figure
b,
(9),
b,
in
boom
in
b
—
bo,
and
b'^ {bo,
H)
summary, a prolonged
In
and to zero rent-seeking.
H)
b'^ {b,
1:
Posituf Rents
h"' [b,
<
=
x^ {bo, H) —
order to starve the future government of resources.
this
the economy starts from 60
if
Z«o
'
vs. 6 for
ct
<
a*
Rpni%
H)
45';f4nt
These dynamics are
zero initial rents, and
b.
Once debt
uj-* {9),
x^
{b,
b
different
b
—
b,
fe,
starts
future
the
if
from zero
boom
and
to too
much
>
Starting from 69
b'^ {b,
is
H) —
b.
boom
6,
the government chooses
until debt eventually reaches
in contrast to the
a
<
The
Thus, even
a* case.
rents, there is a possibility that rents
may be
H) =
if
the
positive in the
current politician does not want
government of rents since he knows that
volatility,
{b,
Therefore, the government reaches a steady
persists for sufficiently long.
to fully starve the future
economy
.
the government chooses positive rents so that c^
state with positive rents, which
economy
a > a*
gradually saves during the
it
drops below
H) =
\i
it
would expose the
and he may as well postpone rent-seeking given that he
has a sufficiently high survival probability and
12
is
likely to
consume these
rents himself.
Figure
6'^
2:
(6,
H)
vs. 6 for
a > o*
b"'{h.Hi
4Sltme
Comparison to Benevolent Government
4.3
we compare the path
In this section,
of debt
and consumption under a rent-seeking gov-
ernment to that under a benevolent government.
of the equihbrium
if
the
boom
is
We begin by considering the impHcations
prolonged. Let {c^J'^q and
[bf^-^
j^^^ correspond to the
equilibrium sequence of consumption and debt, respectively, conditional on a
sisting forever
{cf
}r!.Q,
under a benevolent government starting from some
and {6^i}j_„ analogously
for
initial
debt
boom
bo.
per-
Define
a rent-seeking government.
Proposition 2 (long run)
hm
6f+i
= — oo <
oo
lim cf
t—>oo
>
lim
boom
more debt than a benevolent government and
of the
boom,
it
Though a
Ci
=
u^
(9)
t—^oo
Proposition 2 implies that a prolonged
olent government.
anc
lim b (+1
to
leads a rent-seeking government to hold
consume
less (tax
rent-seeking government reduces
stops reducing
its
debt
if
the
boom
is
its
more) than a benev-
debt at the beginning
sufficiently prolonged.
This
is
be-
cause beyond a certain date, government resources become so abundant that rent-seeking
13
come
considerations
government
realizes that
rent-seeking
were to save more, then a future replacement government
if it
would use the additional funds
as
A
dominate intertemporal smoothing considerations.
to
(which only benefits incumbent politicians)
for rent-seeking
opposed to tax-cutting (which benefits households), and the government therefore
strains
savings in order to starve the future government of funds.
its
boom
long run, a prolonged
and to tax
Therefore, in the
always leads a benevolent government to hold more assets
than a rent-seeking government. This result
less
re-
run
is
to
level of
em-
show that while
whether or not
fairly generally,
economy depends on the
the transitional dynamics of the
consistent with that
Our main contribution
phasized by Battaglini and Coate (2008).
this characterization applies to the long
is
economic
applies to
it
volatility.
Next we consider the dynamics of public debt and taxes along the equilibrium path.
With some abuse
of notation, let Uc (c^
(6,
H;
a)) represent the value of Uc (c^
{b,
H))
for
a benevolent government facing uncertainty a. Define a and a as the unique solutions to
the following two equations:
^"y
•
a
^
,H;ajj=q9
fTTfl
\
,
.
^
„e-a-u-'{d{l-aq)l{l-a))
^,,,,
b( +^-a-u:'[e)^-P
^"
uAc"[e
'-^,H-a\\^qe
^ _ ^
^
,
:
Lemma
(Hi)
\\
f e-a-u:'{6il-aq)/il-a))
f
-
4
Q
(i)
<
<
cj*
a_
a and a approach
< a
,
'
^
m
a and a are decreasing
(ii)
as q approaches 1, (iv) Uc
(^c^ [b,
<
i/))
q
and increasing in a,
qd
iff
a
> a
,
and
(v)
Uc{c'^{b,H))<qeiffa>a.
The lemma
q
and increase
states that like a*, the cutoff points
in the persistence
zero as q approaches
q approaches
Uc (c^
(6,
H)
1.
\a^
so that
1,
parameter
Uc [c^
qO.
{b,
Moreover,
a.^'*
H)
;
a^ decreases in
to cause the benevolent government's
marginal
level of
The
If
(J
<
utility of this
debt
6,
where
ct
ct
and
it
decreases by an
q6.
at b to rise.
Uc[c^
the level of volatility such that o
{b,
H)^ > qd
for 6
G
[6,6],
exceeds steady state debt and in which rent-seeking
'Comparative
statics with respect to d are
ambiguous.
14
> a
> a
which
is
implies
amount
implies Uc (c^
is
>
large
Eventually, the
Analogous arguments hold
interpretation of these cutoff points for economies with a
£, then
converge to
since as economic volatihty a
consumption
consumption goes below
is
like a*, these
the level of volatility for which a
is
increases, the steady state level of debt b decreases,
enough
in political risk
any positive value of a must necessarily exceed a as
The parameter a
<
a and a decrease
a*
for the
(b,
H^^ <
is
as follows:
q6.
the region in which debt
positive. Therefore, the marginal
boom
value of public funds for a benevolent government in the
marginal value of public funds
who
with probability q and
then Uc [c^
{b,
H)) < qO
government in the boom who survives
for a rent-seeking
values marginal rents with weight
for 6
G
[6, 6]
a benevolent government in the
.
exceeds the (expected)
9.
In contrast,
>
if cj
ct,
In this case, the marginal value of public funds for
boom
is
below the (expected) marginal value of public
funds for a rent-seeking government in the boom.
As we
show, whether the marginal value of public funds for a benevolent gov-
will
ernment exceeds or
below qO
is
<
g_
We show
more than a benevolent government.
rent-seeking government saves less or
economies with a
whether or not the
in the rent-seeking region affects
that
feature over-borrowing along the equilibrium path (Section 4.3.1),
and we show that economies with a > a feature over-saving along the equilibrium path
In the Appendix,
(Section 4.3.2).
we
consider economies with a £
(ct, ct),
and
that both over-borrowing or over-saving can occur along the equilibrium path,
depends on
4.3.1
initial
condition
Low Economic
and
6o-
this
•
Volatility
We begin by showing that
is
we show
the rent-seeking government over-borrows
if
economic volatility
low.
Proposition 3 (starve the beast)
c^ {b,H)
If a
<
a, then h'^
yb>b.
H) >
{b,
b'^ (6,
if
economic
volatility
is
the benevolent government for levels of debt which exceed
path starting from
bo
>b
wisdom
features over-spending
in the political
intuition for this result
is
6.'^
it
consumes more than
Therefore, the transition
and over-borrowing, which
economy
is
the cost of leaving the
sufHciently
the
boom
and the prospect
with
that low economic volatility implies that politicians are
economy exposed
and over-consume
in line
literature.
biased towards extracting rents today versus in the future, since political risk
to over-borrow
H) >
low, then the rent-seeking govern-
ment always borrows more than the benevolent government, and
The
{b,
.
This proposition states that
the conventional
H) V6 and c^
in the
downturn
at later stages of the
for rent-seeking
is
low. This causes
boom when
approaches.
debt
is
is
high and
governments
driven
down
Politicians at early stages of
anticipate this behavior of politicians in the future,
and
for this reason,
they
choose to over-borrow and to over-consume. The prospect of future rent-seeking therefore
and over-consumption
reinforces over-borrowing
^^Whenever
ff
<
£, there
case, this cutoff point
is
is
some
below
in the present.
cutoff level of debt below which
b.
15
c'^ (b,
H) < c^
(b,
H). For the a < a*
More
formally, imagine
of debt which exceed b
and only
if
Since
6'-^
if
b'^ (6,
ib,H) >
if
H) from
b'^ {b,
H)) exceeds the marginal
which equals u^ (c^
=
//)
V6
>
6,
then c^
(6,
H) > c^
{b,
H)
the dynamic budget constraint of the economy.
(6,
>
steady state, the government over-borrows and over-consumes,
in
and the marginal cost of public funds
{b,
(6,
levels
{b,H), the rent-seeking government must be choosing c^ {b,H)
b'^
c^{b,H). Therefore,
Uc [c^
under
so low that rents are never extracted
is
a < a*). Since x^
(i.e.,
H) >
volatility
=
//))
under a benevolent government which equals
at b
cost of public funds under a rent-seeking
This affects savings decisions
9.
government
for all levels of
debt above
Consider the Euler conditions of the benevolent and rent-seeking government, (8) and
b.
(14), respectively, for 6
€
Since b
\b,'b\.
>
c^
b,
{b,
—
L)
c^
(6,
L) because debt is never
low in the downturn to induce rent-seeking. Therefore, satisfaction of (8) and
sufficiently
(14) implies that b'^
{b,
H) <
=
{b,H)
b'^
since the benevolent government perceives
b,
a higher marginal cost of public funds in the future than the rent-seeking government.
Thus, Uc (c^
b
(6,
H)) < Uc [c^
H)) so that the marginal cost of public funds
{b,
is
higher at
under a benevolent government. Forward iteration of this argument implies that
all rent-
seeking governments perceive a lower marginal cost of public funds in the future than the
benevolent government, and they consequently save
An
under
x^
(6,
only
the
analogous argument holds
levels of
H) >
if
b'^ (6,
boom
debt that exceed
for
some
H) >
b'^
and
b
{b,
if
instead volatility
b
and are below
is
it
less
is
than the benevolent government.
low,
though rents are extracted
b (i.e., a*
no longer the case that c^
< a <
{b,
a).
H) > c^
In this case,
{b,
H)
and
if
H). Nonetheless, note that the marginal cost of public funds
for the rent-seeking
government
G
for 6
[6, b]
equals qO since the government
expects to survive with probability q and to extract rents which provide marginal utility
However, given the definition of
a, Uc (c^
(6,
i/))
>
in
6.
qO in this region so that the benevolent
government values public funds more on the margin than the rent-seeking government.
Therefore, analogous arguments to the previous case comparing (8) and (15) imply that for
6
>
c^
6 for
which
b'^ {b,
H) €
[6,
H)
(since
x^
H) —
0) so that the rent-seeking
(6,
(6,
consumes. Since Uc [c^
public funds at
occurs for
b
{b,
//))
fe]
,
<
it is
the case that b'^
u^ (c^
//))
(6,
and forward iteration on
all 6.
.
,
H) >
b'^ {b,
H) and
c^
High Economic
The previous
argument implies that over- borrowing
.
Volatility
picture changes dramatically for high levels of economic volatility.
16
H) >
government over-borrows and over-
_
4.3.2
(6,
the rent-seeking government under-values
this
,
{b,
,-
,
,,
,
Proposition 4 (feed the beast)
c^ib,H) <c^{b,H)
ernment saves more than
it
consumes
The
tlie
less (taxes
if
is less
the event of a
>
and
h
boom
high, then the rent-seeking gov-
is
6,
more) than the benevolent government.
is
consume
likely to
economic volatihty
rents today
and more
likely to
volatility
is
high, the
consume them tomorrow
economy while providing him with potential
during which he
not replaced.
is
may
government
in the future, the rent-seeking
Whenever economic
as follows.
since this simultaneously protects the
in
\/b
benevolent government for levels of debt which exceed
intuition for this result
politician
a > a, then b"^{b,H) < b'^{b,H)
V6.
This proposition states that
and
If
rents
In anticipation of these rents
actually over-save relative to a benev-
olent government since the marginal value of additional funds in the future
boom due
to rent-seeking exceeds the marginal value of additional funds for a benevolent govern-
ment who would instead use the additional savings to
governments to over-save and under-consume at
driven
down
stages of the
sufficiently
boom
and the prospect
increase consumption. This causes
later stages of the
boom when debt
for rent-seeking approaches. Politicians
is
at early
anticipate this behavior of politicians in the future, and for this reason,
they choose to over-save and to under-consume themselves. The prospect of future rentseeking therefore reinforces over-saving and under-consumption
governments are not cutting taxes during the
boom
in
the present.
Future
in response to additional savings-the
natural response of a benevolent government-and this provides additional incentives for
savings today.
More
government at values of debt
formally, consider the
6
6
\b,b\.
In this region,
the government chooses positive rents, and the marginal value of public funds for a rent-
seeking government
who may be
Moreover, by the definition of
that
is
its
the benevolent government
marginal value of public funds Uc
H)
(6,
is
below
qO.
extracting rents and also over-taxing in order to do so.
which b'^{b,H) e
if
ct,
potentially replaced prior to entering the
and only
b'^ (6,
H) <
if
b'^{b,H)
b'^ (6,
In this region,
[b,b].
<
H) and
b'^{b,H).
c^
(6,
over-saves and under-consumes.
Given
H) <
c^
(6,
(8)
H)
and
at b
rent-seeking government
consider values of 6
so that c'"{b,H)
(15),
q6.
so wealthy in this region
The
Now
is
it
>
6 for
< c^{b,H)
must be the case that
so that the rent-seeking government
Since Uc[c^{b,H))
government over-values public funds
that over-borrowing occurs for
x^ [b,H)
=
is
boom
and forward
>
u^ [c^
{b,
H)), the rent-seeking
iteration
on
this
argmnent imphes
all b.
Note that even though the rent-seeking government over-saves along the equilibrium
path, in steady state
it
over-borrows relative to a benevolent government
17
who
instead
drives
its
In a sense then,
asset position to infinity.^''
and over- borrowing
it is
the prospect of rent-seeking
which induces poHticians to over-save in the present.
in the future
This induces the rent-seeking government to over-tax both when
when
rent-seeking and also in steady state
anticipating future
rent-seeking takes place.
Policy Implications and Discussion
5
A
it is
central implication of our
model
place for distortions to emerge.
expectations. For example,
that rent-extraction does not actually have to take
is
The main mechanism
when debt
is
in our
framework operates through
sufficiently high, there are
no rents independently
and high
of the regime. However, there are important distortions in both the low
volatility
scenarios.
In the low volatihty scenario there
too
little,
since the
everything. That
is,
government
is
is
a wedge pushing the government to tax and save
worried that
the current government
too
much. Here,
benefit from cutting taxes
fiscal policy is
and saving
is
In
less.
what
follows,
we
illustrate these scenarios
fiscal rules.
The Two Scenarios
Consider an economy with u{c)
=
log (c)
equal to 10. Consider two economies: q
current incumbent has an
80% chance
=
and {P,e,a,a,6}
where we have chosen 9 such that the long run
is
a wedge pushing the government to tax
actually too contractionary, and society would
and conclude by analyzing the impact of standard
5.1
potential replacement will squander
too expansionary and borrows too much.
is
In contrast, in the high volatility scenario, there
and save
its
~
.2
level of
and q
debt in a
—
and
incumbent has virtually no chance of being replaced. Under
low q case corresponds to an economy with
cr
<£,
boom
in the
a < a* case
that in one economy, the
.99, so
of being replaced
{.95, 100, 1.5, .95, .001},
in the
other economy the
this parameterization, the
so that the
government under-taxes
and over-borrows, and the high q case corresponds to an economy with a > a so that the
government over-taxes and over-saves.
Figures 3 and 4 illustrate the path of debt and consumption in the q
during a prolonged
boom
benevolent government.
starting from a level of debt 6o
The
—
—
.2
economy
30 for a rent-seeking and a
rent-seeking government over-borrows relative to the benev-
olent government. This difference can be substantial. For example, at
Formally, there exists a cutoff point in the range
over-borrows.
18
\b,h\
t
=
40, the rent-
below which the rent-seeking government
seeking government holds a level of debt equal to 10 whereas the benevolent government
holds a level of debt equal to -33, a difference equal to over
economy. The counterpart of the path of debt
is
40%
of the endovi^ment of the
not rent extraction (since a
but excessive consumption (low taxes) during the transition (Figure
fragihty during the
downturn (not shown).
Figure
Path
3:
-
Figure
4:
of
Bene\«lenl
Debt
(cr
<
ct)
Rent-Seeking
Path of Consumption {a < a)
-
BenewDlent
19
Rent-Seeking
4),
< a* <
a)
and economic
In contrast, Figures 5 and 6 consider the q
starting from a level of debt 6o
=
=
.99
economy during a prolonged
30. In this situation, the rent-seeking
saves early on relative to the benevolent government (Figure 5).
two governments can be substantial. For example,
holds level of debt equal to
at
i
=
government
10%
of the
over-
The difference between
the
40 the rent-seeking government
—46 whereas the benevolent government holds a
equal to —33, a difference equal to over
boom also
endowment
level
of debt
of the economy. Ea,rly on,
the high taxes are used to reduce debt but later on they finance government rents.
As a
lower than under the benevolent government throughout the
boom
result,
consumption
(Figure
6).
is
Early on,
when no
rents are extracted, the
economy gains
protection against the contraction. Later on, consumption
is
and the contraction.
Figure
Path of Debt [a > a)
5:
-
BenewDlent
20
Rent-Seeking
in
terms of extra
lower both during the
boom
Figure
6:
Path of Consumption [a
a)
Fiscal Rules
5.2
The conventional
view, captured in Figures 3 and
popular policy option of adopting
surpluses) during
booms
has given support to the increasingly
4,
rules that essentially
fiscal
(the budget, surplus or deficit rules).
the degree to which such
specifically, consider
ment would choose a sequence
an economy starting from
of
consumption {cf\
whereby the rent-seeking government
to the constraint that such a policy
in period
must
t
deficits (or require
A natural question concerns
This question
6.
commodity-economies which experience high economic
More
cap
economies in which over-saving occurs
fiscal rules are useful in
along the equilibrium path as in Figures 5 and
for
>
is
is
particularly relevant
volatility.
b^ in
in the
which a benevolent govern-
boom. Imagine a
fiscal rule
allowed to choose any policy subject
satisfy
Cj -f
X(
<
C(
;i6)
,
so that the government effectively cannot run a primary deficit above that of the benevolent government at any given date.
2.2
The
with the exception that (16) must be
political
satisfied
environment
is
as described in Section
by every government
in every period.
Since rents are zero under a benevolent government, (16) implies that the rent-seeking
government must save
at least as
much
as the benevolent government at every date.
next proposition characterizes the behavior of the economy under the
[cf^
and (xf }
fiscal rule
The
where
correspond to the path of consumption and rents, respectively.
21
during the
boom under
Proposition 5
a rent-seeking government subject to the
+ xf =
(fiscal rules) cf
cf at every
in the
t
fiscal rule.
economy under the
fiscal
and
rule
Cf
= min
ff
= me.x{0,cf-u-'{e)}.
Proposition 5 states that the
rule binds in
binds, and c^ and
fiscal rule (16)
Section 4 so that rents are only positive
economies in which a
if
< a
and
^17^ (6')}
{cf,
2f are chosen
the marginal value of consumption equals
6.
as in
The
government
since the unconstrained rent-seeking
has higher equilibrium path deficits than the benevolent government. Thus the
fiscal rule
reduces the government deficit along the equihbrium path and increases public saving.
More
surprisingly, the rule binds in
economies in which
cr
>
ct
so that the unconstrained
rent-seeking government has a lower equilibrium path deficit than the benevolent govern-
ment
on
in the early phase of the
deficits, it actually
boom. Therefore, even though the
imposes a cap
induces the rent-seeking government to borrow more than
were unconstrained. The reason
if it
fiscal rule
for this
is
in this region
unconstrained
governments over-save because they look forward to squandering public funds
to
if
the
boom
persists for suflficiently long.
squander these public funds
would
that the rule works through expectations by
reducing the value of future public funds. More specifically,
ture
it
The
fiscal rule
in the future since
it
however makes
forces a future
it
in
the fu-
impossible
government to save
more. Therefore, the rule reduces the value of future funds from today's perspective, and
this induces today's
Note that the
more
rents than
government to save
rule induces the
it
would
less.
government to consume more (tax
today, and
itself
it
is
levels of
debt
in
it
would
in the
absence of
comparison to an economy
lower, the current
will
same fashion
is
government
begin to extract rents at
rules, since rent-seeking
in the
absence of
as the benevolent government,
it
begins at higher
rules.
Finally, note that while a fiscal deficit rule can force a rent-seeking
in the
to extract
does so in the form of higher consumption
and higher rent-seeking. This means that the government
an earlier date than
and
were unconstrained along the equilibrium path.'^ This
if it
because since the marginal value of funds in the future
decides to use funds for
less)
government to save
cannot control the composition of
public spending. Specifically, the government continues to squander resources on rents as
'^More
specifically, the fiscal rules
induce more consumption at high levels of debt and more rent-seeking
at intermediate levels of debt.
22
opposed to cutting taxes
if tlie
boom
is
sufficiently prolonged or
if
initial
resources are
very abundant. This suggests that a deficit rule must be combined with a cap on taxes,
so as to achieve the social
Final
6
optimum.
Remarks
We
developed a dynamic political economy model of debt that characterizes public debt
and
deficits along the transitional
the conventional
path and in the long run. This allowed us to re-examine
wisdom regarding the nature
that in the short run phase of a
boom-when
of political distortions.
the level of public debt
whether the government faces high or low economic
wisdom
of under-saving holds in the latter case,
volatility
is
we intend
volatility environments.
The
properties of a broad class of
''"^See
for
mendment
is
high-it matters
While the conventional
does not in the former.
If
to extend our analysis of fiscal policy in high
economic
economic
natural next steps are to study the qualitative and welfare
fiscal rules
at aligning these different rules
regions.
is still
result
high, politicians over-save in the short run by keeping taxes too high.
In future work
aimed
it
volatility.
Our main
found
in practice,'*^
and to pursue empirical work
with the characteristics of different countries and
.
example Azzimonti, Battaglini, and Coate (2008)
to the
US
constitution.
23
for
an analysis of a balanced budget am-
Appendix
7
7.1
Proofs
Step
Lemma
Proof of
7.1.1
V^
1.
{b, s) is
-
e,
and
satisfies all constraints
is
a potential solution for 6
strictly decreasing in b since
arbitrarily small lets c^ {b
objective function
1
H) = c^
(6,
H) +
and
e
strictly raises welfare.
2.
First order conditions
—Uc[c^{b,H)^, which by step
imply that V^^
is
H) = aV^^
{b,
strictly increasing in
Step
from
3.
If b'^ {b,
this contradicts
7.1.2
Step
c^
(8) implies
6'-^
Proof of
[b'^
{b,H),H) + {l- a) V^
'
then from step
>
{b,L))
(6,//)
>
Lemma
constraint,
is
(fe'^ (6,
{b' {b,
Step
Step
H)))
bt
=
{e
kr+i
=
0.
r
so that b'^
also
[b,
H)
u, (c^
{b,
H)), which
(3),
•::
and the dynamic budget
policies in the text
'
f^^.
-.-„: W_,1„,
l^P
I
J
-.
2. All of the properties follow
1. In a
These
—
2
1-/3
Lemma
>
'
—p
u {mm{e~a-b{l-/3),u:'{6)})
Proof of
7.1.3
b.
{^-.H)
However, given the budget constraint
-(min(.-a-Kl-/3).v'(f»)})^^^^
=
,
Vj^^
'
I
K^(6,L)
since the
''
we can write
V^a.L) =
H) L)
Q.E.D.
6.
b
Differentiability
decreasing in
'
u^ (c^
2,
c^ {{b,H)).
Given the characterization of
1.
convex.
is
H) which
and Sheinkman (1979).
c^{b,H)
implies that
b.
b,
concave in
>
for e
e
b'^ (6,
and the envelope condition imply that
'"
H) >
[b'
1
H) =
e,
strictly
is
concave and the constraint set
strictly
follows from the standard arguments of Benveniste
Step
It
-
b'^ (6
—
from
Q.E.D.
this characterization.
'
'
3
period economy, define
+ a - max {u-^
{9),
2a
+
u-^ (9(1
-aq)/
.
.
24
(I
-a))})
I
J2^j ^i^T
and
6(
=
Step
2.
Define
+a-
e
il^^ (6)
+
/36,_^i
Let the economy begin in
boom
tlie
T=
in period 0. If
and 6^ >
the pohcies of the benevolent government are chosen since those entail c^ [br,
and x^
{br,
H) =
H) — u~^
c^ {br,
at date
If
0.
<
67
67-,
and x^
(9)
respect to
We
b.
H) >
{br,
p
Let V/
0.
let
{bt, s)
s)
V/j, (bt,
-uJcf
j\t
f
=
(bt,H))
^t^
))
10
all
3. If
decreasing in
T=
consider V^^
1,
H)
{bt,
since the constraint set
bt
bf {bt,H) —
To
^(+1-
-
cf{bt,H)
u-^
see why,
the fact that a
<
can hold and
6f' {bt,
<
if bt
if
a*.
and xf
bt
If b^'
if 6,
>
-t6,
if bt
<
(6^,
H) =
{bt,H)
for
is
H) <
{bt,
with
<
br
^^^
^^^^
br
=
i
given step 2 and
tighter
and
it is
{bt,
H)
If
>
<
ct*
bt
<
bf
then
If bt
equals V,^
is
{bt,
4. Successive application of
Define
1.
r
Step
_
~
2.
6 as
e
-
The
{bt,
H) >
=
if
0,
in
is
bt
then necessarily
x^ {bt,H) >
0,
then
b^'{bt,H)
>
b^^^,
6,+i. If
if fef'
{bt,H)
>
6(^j, neither (14)
>
bt,
then V,^ {bt,H)
is
strictly
nor (15)
H) >
lemma for
{bt,
concave in
bt
=
0.
strictly concave. Therefore, (17) holds at
H)
^
plus expected future rents, the arguments of
Step 3 taking
=
V,^ {bt,H) in this region.
T
to 00 yields the result.
+
2a))
Q.E.D.
1
.
+ {I \ e-u;^{9) + a + /3b
f
xf
otherwise. This implies the properties of the
Proof of Proposition
7.1.4
continuously differentiable
so that 6,+i
Benveniste and Sheinkman (1979) imply that V^^ {bt,H)
Step
Vj^ [bt,H)
then (15) cannot hold, where we have used
b^_^_^,
a*,
Lemma 2.
then (15) cannot hold. This implies that xf
bt_^,^,
H) =
ct
<
and the objective
Moreover, since V,^
Step
its derivativ^e
(•)
b-,
order conditions imply that
first if
instead a
If
{bt,H) and Vt^ {bt,H) for
xf
first
Consider
{9).
then (14) cannot hold.
since
correspond to
d
by the arguments of Benveniste and Sheinkman (1979).
V";^
correspond to the value of V^
of the the properties follow from this characterization at T.
Step
only
-u^^ (6)
can write
VS{bt,H)
and
H) <
then
order conditions and the budget constraint yield
first
economy and
in a finite period
t
then
b-^,
u-^ {au,
{9)
fact that b'^
{b,
a) u^ (uj'
H) =
{9)
+
a
+ /3b
Ha <
ii
b
\f
b
25
<
b
and property
(iii)
a*
a > a*
for
a > a" follows
Lemma
from step 3 of the proof of
(ii)
a < a*
for
=
V,''{b,H)
Step
Uc [c^
plies
Given
H) L) > c^
[b'^ (6,
H) <
(6,
Vl''{b,H)
= aVf
V,^ib,H)
=
must be that
b'^ {b,
(6,
b ii b
<
and property
b
b if b
>
is
(6,
H) =
ioi b
[b'^
>
>
[b,H) ,H))
This then im-
b.
a contradiction given the dynamic budget
b.
Substitute the envelope condition into (14) and (15) to achieve:
4.
whenever
H) =
then necessarily Uc[c'"
b,
H), but this
(6,
,
b'^ (6,
H) ^
H)
e
H),H) e
{b"'{b,H),H)
+ {I-
ag\/,^(6'^(6,/f),//)
hand
respectively. Since the right
V;^ (6'^
>
H)) from the envelope condition since x^
Step
{b,
(15) together with the envelope condition that
b'^{b,H)
if
(14),
constraints. Therefore, b'^
it
and
fact that h'^
-u,{c{b,H)).
3.
{b,
c^
follows from (14)
The
3.
+
is
side of (19)
single valued,
a -
u.;^ {9)
+
Vf{b"'ib,H),L)
s^nd
(19)
+ (l-a)\/,^(r(6,//),L),
and
If b'^
+a-
(20)
(20) are strictly decreasing in b'^
continuous, and
13b.
[-9, -q9] and V^^ {e
a)
=
{b,H)
u;^
(9)
+
is
e
/36,
(6,
H),
strictly increasing in b
+
L)
a - u-^
+
[9]
then
/36,
single-valued, which
is
implies that
+
(e
66
+
(e
so that 6'^
C7)
(6,
H)
is
Proposition
1.
bounded
>
it
<
implies that c^
(6,
b.
Step
2.
6^ > — oo
Step
1.
since
We
b,
implies that 6/^j
is
constant.
Q.E.D.
1
b''^ (6,
first
u'^
G
[6,
&f )
if b^
1
implies that 6,^j
;
{9)
-.
:,.
Lemma
b.
Since
bf_^-^
=
b\ft.
is
monotonic and
H) <
b
Finally, Proposition
1
b since 6'^ [b,
{9).
implies that 6f^j G (— oo,6f).
H) <
>
cannot converge to any point other than
It
then Proposition
H) =
Lemma
Proof of
7.1.6
1
must converge.
If 6o
for b
continuous in this region of b and
+ pb,
+ pb
[9)
Proof of Proposition 2
7.1.5
Step
+ l3) + u;' {aq9 + (1 - a) Vj" {e + a - u^' {9) + /36, L)) + Pu;'
(1 + /3) + u-i {a9 + (1 - a) V^^ (e + a - u^' {9) + /36, L)) + Pu-'
a){l
b for all
6.
Given
It
cannot be that limt^oo^^i
(3), this implies
that
lim(--,oo
cf
—
=
oo.
4
show that a and a
exist
and are uniquely
correspond to the equilibrium value of consumption at date
26
defined. Let
i
c^ for j
=
H, L
as a function of the shock
j for
an economy beginning with debt
and
bo
bt-A
Note that
Cq
= e — — 6o (1 —
ct
We
state Sq.
„
1
1
= e — — 6t
ct
=
Therefore cf
is
(1
—
Substitution
/3).
consumption
into the above equation then yields a difference equation for
cf
increasing in cf_j and decreasing in cflj.
downturn
in the
+ i2a(l-/3).
-^cf_i--^(l-/?)cf_i
write
(3) to
,
and more generally cf
P),
can manipulate
(21)
.
Substitution of this equation
into the Euler equation yields
u'{c^)^
Therefore c^
Step
6o (1
~
2.
is
^^
increasing in
The path
and decreasing
c^_-^
consumption follows
of
and
b
under the case
6 as
unchanged and
If
an increase in a reduces
(22) implies that cf
=
and
(21)
=
(22) subject to Cq
— a —
e
b.
<
3.
Properties
Step
4.
An
b
(iv)
5.
2,
with no
As
b
q
a
cf^
=
for all
=
9 [l
—
increase in a leaves
weakly declines then forward
(,
violating (23). Therefore c^
aq) /
—
{1
a)
— oo
approaches
on
is
Cq
If
6o
.
t,
=
6,
violating (23).
q9 under either bo
1
with no
b
and
—
b or
effect
2.
on
Cq
.
This raises
This reduces
Cq
An
approaches
(e
27
—
ct
—
u~^ {9))
/ (1
—
Cq
by the
increase in
by the arguments of step
unchanged. This establishes property
1, b
9.
so that Cq approaches oo, and Uc (c^)
from steps
and
>
for all
so that Uc (c^) decreases whereas q9 increases.
effect
>
and are uniquely defined.
exist
(v) follow
weakly decreases
then u^ (c^)
0,
approach
a and a
and
Uc [cq) decreases whereas q9
Step
and
oo, b
If
increase in q reduces b
arguments of step
c^ decline
An
b.
weakly declines then forward iteration on (21) and
If Cq
q9. Therefore,
Step
and
Cg.
=
bo
side of (23). If c^
then Uc (c^)
0,
(23)
.
Consider
.
weakly increases so that
As a approaches
approaches
raises b
=
cr
+ a)-6o.
J]/3'(e
a > a*
hand
strictly increases in a.
Cq
=
(22) implies that
strictly increases in a.
Therefore
for
raises the right
and
iteration on (21)
bo
in cfij.
oo
J]/3'cf
Cq
(22)
P) and Cq chosen to satisfy the present value budget constraint of the government
oo
Define
^.
^
^
2,
a
so that
(ii).
/?)
and
b
approaches
2a
+
-a -
(e
>
otherwise c^
Step
u~^
By Lemma
c^
To
6.
Uc (c^
Step
1.
b'^ (6,
H) yb<b.
>
imagine
(i),
^
u^ [c^ (p^^))
Imagine
a.
qO since 6'^
=
a
if
H)
(6,
= b^b <
Since b'^ {b,H)
Step
Imagine
2.
=
/f )
if
L
from Proposition
(fe,
Since c^
{b,
Step
a <
2,
(6,
G
If 6
=
L)
3.
c^
b'^ {b,
6
If 6
c^ (6'^
H)
Step
(6,
b,
H) <
b'^
<
q9.
=
<bhy Lemma
a -
+
{e
Therefore, a*
3.
-
u~'^ (6*)) / (1
<
and
/3)
Q.E.E).
a.
H) =
(6,
Imagine
4.
must be that
Step
5.
x^
(6,
If
> c^
H) and
H)
Step
6'^ (6, if )
>
u;^ {6)
Lemma
H)
c^
{b,
1,
b'^{b,H)
+ (l-Q)u,
c^
>
(6'^ (6,
(6,
c^
//)
,
L) but c^
b'^ (6,
<
H) and
For any
a*.
H) H) >
{b,H) >
1, b'^
since
x^
=
(6,
H) =
and
b,
since
H) V6 >
(6,
(c^(6,L)).
H)
in this region.
1 b'^ {b,
(6'^ (6, /f )
b'^ (6,
(24)
order that (8) hold given (24),
(6,
H) <
,
H) e
i^)
> c^
b'^ [b,
=
,
€
<
if )
and from
\b,b\,
(6'^ (6, if)
H) and
c^
,
if).
H) <
(6,
and
c-^ (6,
if)
{b,
b,b'^~'
{b,
H)\
<
c^
1.
H)
Vfe
>
':
b.
'
'
b'^{b,H) €
Since c^ (6'^
[6,6),
and
H) L) =
(6,
,
,
if)
must be that
<
c^
b'^
{b,
(6'^ (6, if )
H) <
b'^
,
if )
then
,
{b,H) and
.
(6,
if )
V6
s.t.
can be applied then to show that
6'^ (6, if
)>
6.
{9)
<
6'^
(6,
if)
Q.E.D.
Proof of Proposition 4
7.1.8
1.
it
ii) in this region.
(6,
(6'^ (6, if)
it
6.
if )< c^
c^
g^, then in order that (8) hold given (15)
then since c^
6,
H) <
(6,
from Proposition
,
6'^ (6, if)
c^
6
6. Successive applications of step 4
and c^
in
H) < c^
(6,
must be that
it
(6'^ (6, ii)
6'^ (6, if)
(6,
then
(6, //),
then from Proposition
,
c^ (6'^
H) <
if ct
6'^ (6, if)
if)
given (14) and (15), in order that (8) hold
H) <
then from
1,
Successive applications of this argument until the natural debt
c^ [b'^ {b,H),L) but c^
Step
ct
then from Proposition
(6,
b'^ (6,
(b,
H),L) =
in this region.
(15) holds since
(6,
so that Uc [c^ (6,//))
b
from Proposition
b
[6,6],
L) but c^
(6,
limit implies that b'^
c^
>
9, since
which from step 3 implies that a >
Then
a*.
Then c^
a*.
Therefore, in order that (8) hold,
c^
For any a, Uc [cq)
the Euler equation implies that
0,
must be that
step
if cr
9^)
u,(c^(6,/f)) <an,(c^(6,i7))
it
<
(iii).
Proof of Proposition 3
7.1.7
x^
<
Uc [c^ ik,H))
3,
establishing property
/?),
yielding a contradiction.
,
H))>e>
{b,
-
establish property
Therefore, a
a.
{9)) / (1
Given the definition of
a, c^
(6,
H) =
28
u'^
u;^ {q9)
<
c^
(6,
H)
for 6
<
6.
<
step
For any
2.
>
6,6'^"'
G
b
{b,H)], b""{b,H)
from Proposition
6
and
[b,b),
holds since
(15)
H),L) = c^
B)
x^
(6'^ (6, //),//)
but
H),H) > q9, then in order that (8) hold it must be that 6'^ {b, H) >
{b, H) and c^ (6, H) > c^ {b, H) in this region.
Step 3. If b'P {b, H) = b, then since uf {b'^ [b, H) H) < 9, then given (14) and (15),
b'P
Since c^ [b""
1.
{b,
[b'^ (6,
,
L)
c'^ [b'^ {b,
,
in order that (8) hold
V6
>
b'^ {b,
H) >
b'^
{b,H) and c^ {b,H) > c^ {b,H)
6.
Step
c^
must be that
it
H) > c^
(6,
(b,
H) V6
H) >
s.t. b'P {b,
{b,
H) >
b'^ {b,
H)
Q.E.D.
b.
Given
1.
(3), (16)
< 6^i
implies that 6^j
along the equilibrium path, where 6^j
corresponds to the equilibrium level of debt under a politician constrained by the
and
6f^j
Step
2.
rule
and
Proof of Proposition 5
7.1.9
Step
Successive apphcations of Step 2 then imply that b'^
4.
deficit
corresponds to the equilibrium level of debt under a benevolent government.
Consider an economy
bind, then this implies that 6^^^
<
period
in final
6f_^j
=
T
in
which 6^ < 6f
.
If (16)
does not
implying that the rent-seeking government
0,
can strictly raise welfare by raising c^ or x^ and increasing t^+j. Therefore, (16) binds
at T.
•
Step
b^
<
If (16)
binds for
all
government
k
4.
>
does not bind at
for all k
By
In this section,
>
t
[b,b]
sider b
e
s.t.
we
(^)
Now
which
(16) binds for all k
then this imphes that 6^j
<
bf^j
—
0.
<
bf_^-^.
>
Given that
all
t
and as
if
(16)
T
^ oo.
t
< T.
Q.E.D.
(o;,^)
which we do
a, there exists a cutoff point
qB so that u,{c^{b,H))
<
{>) q9
if 6
<
{>)b.
Con-
application of step 2 in the proof of Proposition 4 im-
"^^^^
b"^
{b,H) and c^ (6,//) > c^ {b,H)
H) <
t
by raising cf or xf and increasing 6^j, leaving cf and
Given the definitions of a and
j,
>
This implies that the rent-seeking
)•
ib'^~^ (b) ,b'^~^ (b)
{b,
in
briefly describe the region of intermediate volatility
{b,H)
implies that b'^
< T
since this increases fe^+j. Therefore, (16) binds at
uJc^(b,H)) =
b.b'^'^^
plies that b'^
G
t
forward induction, (16) binds for
not consider in the text.
b
t,
period
Intermediate Volatility: a G
7.2
be
in
this implies that 6^_^j
t,
strictly raise welfare
x^ unchanged
Step
,
Consider an economy
3.
^f
•
in this region.
Moreover,
for
then the application of step 4 in the proof of Proposition 3
b'^ {b,
H) and
consider the region for which
c^
b'-^
(6,
H) < c^
ib,H)
29
=
b.
{b,
H)
in this region.
Equation
(15) holds
with equality
at a
minimum
c^ {b,H)
<
value of b in this region, which means that b'^
Since
at this point.
H) ^
6'^ (6,
m
b
increasing, there exists a cutoff point b
H) >
(6,
this region
which
b'^ (6,
H) and
in this region
then
b'^ {b,
H) >
b'^ (6,
H) and c^
b'^
Analogously, we can find a cutoff b such that
(6,
H) >
of regions
c^
(6,
(6,
(6,
H) >
c^
c^
{b,
(6,
H), and
H) > c^
b'^ {b,
H)
,
H
{b,H) <
b'^
if
6
if 6
b
we can apply
limit in
<
and
b is
> ^ and
b is
6
b's
for
which
{b,H) and c^(b,H) < c^ {b,H).
step 2 in the proof of Proposition
e (b,bj and show that
and the natural debt
H)
H).
b'^ (6,
H) >
H). Forward iteration on this argument implies that there
between
(6,
monotonically
is
step 4 in the proof of Proposition 3 to the set of
b""{b,H) e (6,6) and and show that
which
H)
H) <
H) <
6's for
b'^ {b,
(b,
c^
b'^ (6,
4 to the set of
and since
c^
the region such that
then
we can apply
H) and
splits
in this region
Therefore,
b'^ (6,
H) and
maximum
b'^ (6,
c^ {b,H) at this point. Equation (14) holds with equality at the
point in this region, which means that b'^
c^
H) <
(6,
which there
is
b'^ (6,
is
H)
and
a sequence
either over-borrowing
and over-spending or over-saving and under-spending.
Thus, the path taken by the economy depends on the region in which
6o is located. If
6o is in
the over-borrowing region, then over-borrowing occurs along the equilibrium path
until b
is
passed and
if 6o is
equilibrium path until b
is
in the over-saving region,
passed.
,
30
,
then over-saving occurs along the
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