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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
DYNAMIC
EFFICIENCY, THE RISKLESS RATE
PONZI
AND
DEBT
GAMES UNDER UNCERTAINTY
Blanchard
Philippe Weil
Olivier
Working Paper 01-41
November 2001
RoomE52-251
50 Memorial Drive
Cambridge, MA 021 42
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssrn.com/abstract= 290840
MASSACHUSETTS E.
OF TECHNOLOGY
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
DYNAMIC
EFFICIENCY, THE RISKLESS RATE
PONZI
AND
DEBT
GAMES UNDER UNCERTAINTY
Blanchard
Philippe Weil
Olivier
Working Paper
01 -41
November 2001
RoomE52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssm.com/abstract= xxxxx
Dynamic Efficiency, the Riskless Rate
and Debt Ponzi Games
Under Uncertainty
Olivier
J.
Blanchard
Final version:
Philippe Weil*
November 2001
Abstract
In a dynamically efficient
and avoid the need
economy, can
this
even
feasible
The paper then
The average
when
roll its
debt forever
paper shows that such Ponzi games
be infeasible even when the average
may be
government
In a series of examples of production
to raise taxes?
economies with zero growth,
a
rate of return
on bonds
the average rate of return on
reveals the structure
is
negative,
bonds
is
may
and
positive.
which underlies these examples.
realized real rate of return
on government debt
for
major
OECD
countries over the last 30 years has been smaller than the growth rate. Does this
imply that governments can play
a
Ponzi debt game, rolling over their debt without
ever increasing taxes?
If
only economies were both non stochastic and in steady
this question
would be
a
state,
the answer to
simple one. All interest rates would be the same, and
if
and NBER (email: blanchar@mit.edu); and Universite Libre de Bruxelles
NBER (email:Philippe.Weil@ulb.ac.be). The first draft of this paper was
written in November 1990, and later circulated as NBER working paper 3992. For various reasons,
* Respectively:
MIT
(ECARES), CEPR and
mainly our inability to derive
efficiency,
mitted
it
we
left
a necessary
and
sufficient condition for
the paper aside until recently, when,
to this journal.
We
Ponzi games under dynamic
the suggestion of the editors,
we
sub-
decided to stay close to the earlier structure, but to add a section on
developments since the paper was written.
We thank Andrew Abel, Willem
Buiter,
John Campbell,
for helpful discussions.
Fumio Hayashi, Lawrence Summers and
We thank anonymous referees and the editor, David Romer,
We
acknowledge support from the National Science Foundation.
Mathias Dewatripont, Stanley Fischer, Giampaollo
Oved Yosha
at
for useful suggestions.
gratefully
Galli,
,
the interest rate were less than the growth rate, the
inefficient. In this case, the
government could
economy would be dynamically
issue debt
and
never increasing taxes, and covering interest payments by
would grow
but the ratio of debt to
at the interest rate,
roll it
over forever,
new
debt issues. Debt
GNP
would eventually
tend to zero. Such a policy would crowd out the excessive capital accumulation
characteristic of dynamically inefficient
in general be Pareto
improving.
economies and, as we
Actual economies however are stochastic.
many
growth
would
it
in actual
economies, there are
some which are on average above the growth rate,
has been shown in particular that stochastic economies
are below.
It
rate.
2
But
this brings
and thus play
First, that
namic
in
the issue
is
below the
initial
ques-
economies that are not plagued
we
take
roll
up
over riskless debt,
in this paper.
constructed using the Socratic method. This means two things.
is
our objective in
this
paper
is
not to derive
and Ponzi games. Instead, we
efficiency
existing theoretical results
ours) to study an issue
baffled.
feasibility
we
results
and unabashedly
on dyrely
in contexts quite different
—
that often leaves
on
from
of Ponzi debt games in dynamically
safe interest rates
Second, that
new theoretical
explicitly
(most often developed
— the
economies with low
somewhat
(i.e.,
can governments issue and
Ponzi debt game? This
a
Our paper
real rate
us back, with an additional twist, to our
dynamically efficient economies
capital overaccumulation),
cient
And
be dynamically efficient while having an average riskless
tion. In
by
know,
different interest rates,
some which
may
also
1
effi-
macroeconomists
use an unorthodox exposition
style.
Instead
we rely on a series of simple, but
up common misconceptions on Ponzi games
use these examples to move from the specific to the
of proceeding from the general to the particular,
increasingly richer, examples to clear
in stochastic economies.
general,
and
We
to reveal the theoretical structure that
and why Ponzi games can
exist in
we need
to understand
when
dynamically efficient economies under uncer-
tainty.
Sections
1
features. First,
and thus
to 4 of the paper present our four examples. All share the following
all
characterize economies which are subject to technological shocks
to uncertainty, so that assets
ferent rates of return. Second,
all
with different risk characteristics have
characterize overlapping generation economies
'These results require various conditions, both technical and substantive, to be
such substantive condition
is
that the interest rate
be equal to the
an assumption which
dynamic
efficiency
social
satisfied.
One
marginal product of capital,
fails for example if there are external returns to capital. For an analysis of
and Ponzi games in an endogenous growth model with external returns to
capital, see Saint-Paul (1992).
2
dif-
See Abel, Mankiw,
Summers and Zeckhauser
(1989).
—
.
with capital accumulation, thus economies for which capital overaccumulation
dynamic
relying
—
inefficiency
is
not ruled out a priori. But,
third.,
in each case,
on the criterion derived by Zilcha (1991), underlying
parameters such the economy
economies,
we then
ask what
would happen
ernment were to issue debt and
And
the existing debt.
In our
first
taste
actually dynamically efficient.
is
roll
it
debt-to-GNP
to the
over time, issuing
we choose,
and technology
In each of these
ratio if the gov-
new debt to pay interest on
each of the four examples gives us a very different answer.
example, the average
riskless rate
is
negative; nevertheless, the ex-
pected value of the debt-to-GNP ratio under a rollover strategy explodes, and thus
the government cannot play a debt Ponzi game. In the second, the average riskless
rate
ratio
is
again negative, and, furthermore, the expected value of the debt-to-GNP
under rollover goes to
in expected value
But the
does not imply that
strictly positive probability,
debt-to-GNP
zero.
ratio
and
is
it is
fact that a
feasible.
Ponzi game appears viable
Indeed, in this example, with
debt rollover leads to an arbitrarily large value of the
thus again infeasible. By then, the reader
may
suspect
that dynamic efficiency rules out Ponzi games, no matter what the behavior of
the average riskless rate may be. But the last two examples show this guess to be
wrong. In our third example, the riskless rate is always negative and under a Ponzi
debt game, the debt-to-GNP ratio goes to zero with certainty, so that rollover
indeed
first.
feasible.
In that
Our fourth example
example the average
offers a nearly perfect counterpoint to the
riskless rate
over two-period bonds, the government can
We
is
positive; yet
still
by issuing and
play a Ponzi debt game.
rolling
.
spend section 5 of the paper to reveal the structure behind the
why Ponzi games
is
results,
two examples but not in the
first two, and to acknowledge our many debts to the literature up to 1990, when
this paper was first written. The literature on debt Ponzi games has made much
to explain
are feasible in the last
progress since then (in the direction pointed out by this paper!): the
last
section
of the paper reviews these contributions.
We
do not want
to give the answers
away
reader, they are restated in the conclusion.
in the introduction.
For the busy
A first example: a Diamond model with logarithmic
1
preferences
Our
example
first
uncertainty.
consumers
and
in
3
is
a straightforward extension of the
Diamond
(1965) model to
Consider an overlapping generation economy in which two-period
inelastically
when young and retire when old,
and normalized to one. 4 Assume that con-
supply one unit of labor
which population
constant
is
sumers have time- and state-additive logarithmic preferences. An individual repborn
resentative of the generation
at
time
(l-/?)lnClit
t
therefore maximizes
+ /?E lnC2it+1
(1.1)
t
subject to
W
C + Kt+1 =
lt
=
Qm+i
(1.2)
,
Rt+iKt+i,
t
ual
t
W
t
t,
is
K
t
and the discount
Output
t
measures subjective time preference.
fl G (0, 1)
produced according to a Cobb-Douglas technology. Output
factor
at
time
given by:
t is
Y=T
t
where
K
t
of capital
denotes capital per worker
in
output.
at
t
K°,
time
t,
(1.4)
and a G
the constant share
(0, 1)
The logarithm of the productivity shock
independently and identically normally distributed, with
a2
(1-3)
and C^t+i denote first and second period consumptions of an individthe capital stock
born at t,
and R the wage and capital rental rates at t,
where C\
at
t
T
mean
is
assumed
to
be
zero and variance
Capital fully depreciates in production.
.
Equilibrium capital accumulation and dynamic efficiency
1.1
Solving for consumption from utility maximization, and replacing wages and
rental rates
by their values from
Kt+1
profit
maximization,
gives:
= {l-a)Fr K?,
CM = (l-P)(l-a)T K?,
C2 = aTt K?.
t
t
,
3
4
This model
We
shall
is
by
now
t
(1.5)
(1.6)
(1.7)
standard. See for example Blanchard and Fischer (1989).
throughout consider economies with no growth. Introducing non stochastic popu-
lation or productivity
growth would be straightforward, and only complicate notation.
Note
—
this will
young and
—
be relevant
later
that, at
the old are perfectly correlated.
any time
Our
the consumptions of the
t,
moment
interest for the
is
how-
ever in the behavior of capital. Denote, hereafter, the logarithm of an uppercase
variable
by
lowercase counterpart.
its
=
k t+ i
And
the behavior of output
y
t
is
=
ln[(l
We then
-
have
+ ak +
ot)0\
t
vt
(1.8)
.
given by
aln[(l
-
a)/3]
+ ay
t
+
-\
vt
(1.9)
.
Capital accumulation leads to serial correlation of capital and output in response
to white noise shocks.
We can now ask under what parameter values this economy is dynamically efficient.
The
criterion
dynamic
natural extension of the aggregative Cass (1972)
under certainty
that the
is
economy
is
dynamically efficient
not exist another feasible sequence of capital which provides
and
efficiency
there does
if
much
at least as
ag-
consumption at all dates and in all states,
consumption in at least one date or state. Zilcha (1991) has derived a necessary
and sufficient condition for the efficiency of stationary economies such as the one
gregate
we
consider. In this
economy with constant population,
higher aggregate
strictly
the condition
is
that un-
conditional expectation of the logarithm of the gross marginal product of capital,
Er
t
,
be nonnegative. Here,
Er = \na + (a-
l)Efc t
t
a
= In
(1
so that the
economy is dynamically efficient
8
the
economy
is
1 -
1
dynamically
efficient.
(1.10)
<x)0.
if
>0.
(1.11)
a)(3
we assume
In the rest of this section,
and only
if
a
=
-
1
'
that condition (1.11)
We now turn
is
satisfied
and that
to the determination of the
riskless rate.
5
Note that the condition
condition,
is
for
dynamic
aT K", exceed gross investment, K
t
if
a > 0(1 -
efficiency of Abel et
—
often not conclusive in particular models
a),
i.e., if
and only
if 6
t
+\
>
=
0.
0(1
is
— a)T K^,
t
al.
(1989)
—which, being
satisfied here.
at all dates
Gross
and
in
all
a sufficient
=
profits,
RtKt
states if
and only
The
1.2
riskless rate
The equilibrium
R t+1
riskfree rate of return
on
one-period bond (which
a safe
pays one unit of the consumption good in every state at date
t
+
1) satisfies the
first-order condition for utility maximization:
(l-P)C^=PR{+1 E
C^,
t
(1.12)
Evaluating the equilibrium risk free rate at a zero net supply of bonds, and thus re-
and second-period consumption
and ( .7), we get
placing
first-
tions
.6)
( 1
in
( 1
.
1
by their values from equa-
2)
1
R{+
,
—^r«^ +
Q
=
d-13)
i\
which, using our distributional assumptions about v, implies that
f
r t+i == In
Using
(1.5), (1.14)
and
a+
-
(a
- a 2 /2.
l)k t+1
(1.11), the unconditional
(1.14)
mean and
variance of the
logarithm of the riskfree rate are thus given by
Er/+1
Varr/+1
so that,
= ln(l+0)-a 2 /2,
=
(1.15)
Z
\1 +^o\
Q
(1.16)
finally,
ER{+1 =
(l
+
T2
9)e-°"
^ 1+a K
(1.17)
Were there no uncertainty, the net riskless rate would be equal to 9, and thus
would be positive under dynamic efficiency. But if the variance of the technological
shocks
may be
less
1 .3
is
large
enough, the average gross
than one, and the net rate
may be
riskless rate
under uncertainty
negative.
Debt Ponzi games
Assume
that the underlying parameters are such that the
efficient
and
riskless rate
Our
that the average net riskless rate
imply that the government can
is
negative.
roll its
economy
is
dynamically
Does the average negative
debt forever?
strategy in assessing whether or not Ponzi debt
games
are feasible in each
of our examples will be to characterize the behavior of debt using the interest rates
corresponding to a zero net supply of debt, thus corresponding to the case where
consumption and
capital
dynamics are given by
(
1.5) to (1.7). If
Ponzi games are not feasible under such interest
were we to do the same exercise
we can show that
be true
rates, this will a fortiori
corresponding to a positive
at the interest rates
supply of debt (an exercise however considerably more difficult analytically): under our assumption on the utility function which implies that saving
fraction of labor income, Ponzi
games crowd out
(1987)) and thus raise, ceteris paribus,
that Ponzi
games are
implodes over time under
issues a small
Let
under a rollover
at
time
amount of one-period
the
instead
we can show
debt-to-GNP ratio
remain true if the government
bonds issued
ratio at time
t
time
at
0.
Then,
follows
(1.18)
t
f
/Y ).
[R t /(Yt /Yt _ l )}...[R{/(Y1 /Y )}(B
Using the characterization of output and the
and
If
{R f ...R{)B /Yu
=
=
Bt/Yt
a constant
0.
riskless
debt-to-GNP
strategy, the
is
accumulation (see Weil
interest rates, that the
rollover, then this will
enough amount of debt
Bq be
interest rates.
all
under such
feasible
capital
riskless rate given in
(1.19)
equations (1.9)
(1.14), this implies:
t
bt-Vt
=
(bo
-
yd)
+ t[ln(l + 9) -
a
2
/2]
-
£>
s
(1.20)
.
s=l
The logarithm of the debt-to-GNP
ratio follows a
random walk with
drift,
with in-
novations equal to the technological shocks. This in turn implies that the expected
value of the debt-to-GNP ratio follows:
E[Bt /Yt
}
=
[Bo/Yo](l
+ ey.
(1.21)
Thus, the behavior of the expected debt-to-GNP ratio
rate 6, irrespective of the value of the average riskless rate.
Ponzi game
is
not
feasible: in expectation,
proportional to
GNP— an impossibility.
What
is
is
at
work here
the riskless rate from
time.
6
t
is
t
is
known
at
This implies that a
The behavior of debt
at rate 6.
is
time
As
t,
at
is
clear
from equation
(1.14),
but varies stochastically through
of the product of the
— as opposed to the debt-to-GNP ratio—
itself
it
debt becomes larger than saving, which
Jensen's inequality.
+1
simple:
6
What matters for the behavior of debt is the expectation
average riskless rate
grows
to
grows
is
is
more complex.
negative, expected debt initially decreases. Asymptotically however,
If the
it
also
riskless rates,
not the product of the expected riskless
average riskless rate
This
first
is
rates.
This
is
why a
negative
consistent with an exploding expected debt-to-GNP ratio.'
example gives a
ative riskless rate (or, if we
clear warning.
were to allow
An economy may have an average neg-
for growth,
an average
riskless rate
below
the average growth rate) but this does not imply that the government can rollover
economy
debt forever; in this example, under the condition that the
expected debt-to-GNP ratio grows
cally efficient, the
positive, regardless of the riskless rate.
if
One may
at a rate
ask however
people were more risk averse than implied by logarithmic
further decrease the average riskless rate and reintroduce
We
up
take the issue
in the next
In this example,
we maintain
dynami-
necessarily
what would happen
utility.
room
Wouldn't
for a
this
Ponzi game?
Allowing for more risk aversion
the assumption that consumers
mic intertemporal preferences, but no longer
risk aversion to equal unity.
is
is
example.
A second example:
2
which
The
still
have logarith-
restrict their coefficient
of relative
rationale for adopting this specification
while the assumption of a unit elasticity of intertemporal substitution
is
is
that,
neces-
sary to preserve the simplicity of the model, the ability to choose the degree of
risk aversion allows us to
risk aversion
examine the comparative dynamic
on equilibrium returns and on the asymptotic
effects
rate of
of increased
growth of debt
Ponzi games.
we assume
Thus, following Selden (1978) and Weil (1990),
that
consumers
now maximize
(l-^lnd. + ^ln^C^]
where 7 >
(7
^
1)
is
1
^ -^,
1
the coefficient of relative risk aversion.
(2.1)
8
Equilibrium capital accumulation and dynamic efficiency
2. 1
Because of the assumption of logarithmic intertemporal preferences, consumers
choose to save
7
A
a
constant fraction
similar point has
been made
/5
in Galli
of their lifetime wealth regardless of the coefand Giavazzi (1992)
in the context of a
Ramsey econ-
omy.
8
Note
that [E ( C'
2
1 /<
7+ 1
'
]
1_ "
'
r
'
is
simply the certainty equivalent of second period consumption
for an individual with constant relative risk aversion 7.
One can
that the preferences represented in (2.1) reduce to the timein (1.1)
when 7
—
>
1.
also verify, using L'Hospital's rule,
and
state- additive
logarithmic form
ficient
tion
out.
of relative risk aversion: attitudes towards risk are irrelevant for consump-
and savings decisions when income and substitution effects cancel each other
As a consequence, the equilibrium consumption allocation and capital accu-
mulation process are the same as in the previous section, and the condition for
dynamic
efficiency
The
2.2
>
as in (1.1 1), 9
is still,
0.
riskless rate
While the value of 7 does not matter
for capital
accumulation when intertemporal
preferences are logarithmic, attitudes towards risk are of course a crucial determi-
nant of the implicit
Adapting the argument
riskless interest rate.
section, the equilibrium gross return
on
a safe claim
in the previous
on consumption
at
t
+ 1 must
satisfy
?/>7
-77-
Cu
where the notation
on
R{'+\
is
LJt
PKt+1
~=V*
°2,i+l
E C}~
2,t+]
t
adopted to highlight the dependence of the
riskless rate
7.
Evaluating this expression along the no debt path, one finds that:
Bl^ =
^^aK^.
Under the lognormality assumption,
R{?1
where R{+\
is
(2.3)
> t+i
jt
it
=Rfi
t
1
follows that:
eW
2
(2.4)
,
the riskless rate for the unit risk aversion case
computed
(in loga-
rithmic form) in (1.14).
As a consequence,
ER&
=
=
ER&e^*
(
1+ ^ e [-(«/(i+-))]-
The
effect of increased risk aversion
rate
by the same
factor in
tivity shocks, the
risk averse.
all
2
states
is
2
e
[(i-7)]a 2
(2.5)
_
thus to decrease proportionately the riskless
and
at all dates.
For a given variance of produc-
average net riskless rate can be very negative
if
agents are very
Debt Ponzi games
2.3
we
Following the same logic as in the previous section,
debt-to-GNP
under
ratio
rollover,
We
output characterized above.
time
at
derive the behavior of the
using the processes for the riskless rate and for
get that the logarithm of the
debt-to-GNP
ratio
under Ponzi finance follows:
t
t
bt
-y =
-
(b
t
y
+
)
i[ln(l
+ 6) +
(1
2
-
2 7 )a /2]
-
£
v..
(2.6)
s=l
As before, the debt-to-GNP
ratio follows a
may now be
a given variance of shocks, the drift
Equation
(2.6) in turn implies:
random walk with
+
t ]
The unconditional expected debt-to-GNP
the constant rate (1
dynamic
under
ratio
enough
Ponzi
the
to
and negative.
7
'
.
6)* e
(1
ratio will
1
this
all states.
exceed any
(2.7)
.
ratio thus
game
grows (or declines.
The answer
feasible?
9
.
.
)
at
What
is
is
no.
required
It is
is
not
that the
Equation (2.6) shows that the logarithm of
random walk with
finite limit,
drift.
It
follows that the debt
such as the ratio of saving to GNP, with
expected debt-to-GNP ratio
the expected debt to
2t
Provided that agents are sufficiently risk averse,
Ponzi
ratio follows a
if the
-^
need not imply an exploding expected debt-to-GNP
0)
is
feasible in
debt-to-GNP
if
>
arbitrarily large
that the expected value of the ratio go to zero.
GNP
even
9) e'
So
rollover.
game be
probability
if
+
efficiency (9
1-
However, for
.
E[B /Y = [B /y ](l
t
drift.
rises,
and with positive probability
GNP ratio decreases. Thus, with strictly positive probability,
GNP ratio goes to zero, the Ponzi game will prove
the expected debt to
infeasible.
This second example shows that, in an
the average riskless rate
may be
economy which
negative, the
expected value under rollover, and yet there
Ponzi
this
game cannot be played
the initial
ratio
may go
efficient,
to zero in
a strictly positive probability that the
The proximate cause of the result is that, in
follows a random walk with drift: no matter
amount of debt
issued, there
is
a positive probability that the
(non-stationary) debt-to-GNP ratio eventually exceeds
particular the
dynamically
forever.
example, the debt-to-GNP ratio
how small
is
debt-to-GNP
is
wage income of the young
—an event
some given bound, and
in
inconsistent with equilibrium.
This naturally raises another question. Could the government issue bonds with
different risk characteristics so that,
This
is
under some conditions, the debt
indeed the main point raised by Bohn (1995).
10
to
GNP
ratio
would instead be stationary around
some
value be
critical
made
we
to Section 6 below; there
a
mean and
the probability that
arbitrarily small or even zero?
show
shall
that the answer
is
We
it
reaches
defer the question
no.
A third example: An economy with stochastic stor-
3
age
Assume
that, as in
our
first
example, consumers have time- and state-additive log-
arithmic preferences. Assume, however, that the technology
born
at
time
receive a non-stochastic first period
t
is
endowment
W>
0,
access to a stochastic constant returns to scale storage technology with
and have
random
The logarithm of R is assumed to be identically and
2
/i and variance a
Zilcha criterion, this economy is dynamically efficient if and only if
gross rate of return Rt+i.
10
t
independendy distributed with mean
Using the
People
different.
.
E\nR = V>0. u
t
Under those assumptions,
the
consumption of the young, the consumption of
the old and the capital stock are given by:
C
C
= (1-/?)W,
= PRtW,
= (3W.
lt
2t
K
t
Because the endowment
and
capital
is
non
stochastic, the
(3.1)
(3.2)
(3.3)
consumption of the young
accumulation are also non stochastic. Because storage
stochastic,
is
and independendy and identically distributed over time. In contrast to the previous examples, the consumption of
young and old are not perfectly correlated, a point to which we shall return bethe consumption of the old
is
stochastic,
low.
10
This model
equilibria,
model was
11
is also standard. It was used by Koda (1984) to study the existence of fiat currency
and was more recently analyzed by Gale (1990) in the context of public debt. A similar
also
used by Summers (1984).
Formally, our example, which has constant returns to storage does not satisfy the concavity
conditions required for the Zilcha criterion to apply.
those conditions, and has the
depreciation
random
is
same implications
stochastic so that output
is
An
alternative formalization,
produced according
variable.
11
which
for the existence of debt rollover
to
Y = K
t
t
a
is
— SK
t
satisfies
one where
where 5
is
a
Having characterized consumption, we can derive the implicit equilibrium
on
rate of return
sumer,
it is
a
one-period bond. From the
—1— = &~
Rf =
Thus
order conditions of the con-
first
given by:
if
the variance of
endowment and
R
a2 l 2
is
also
sufficiently large.
is
(3.4)
.
1
on one-period bonds
the rate of return
be negative
ER-
12
non
stochastic.
It
may
well
Since both the first-period
the riskless rate are constant, a necessary
and
for the feasibility of issuing at least a small quantity of debt
sufficient condition
and
rolling
over
it
is
simply
Rf <
Thus,
(3.5)
1.
sharp contrast to our previous examples, in this dynamically efficient
in
economy, debt Ponzi games may actually be
feasible.
And
the riskless rate plays a
crucial role in determining the feasibility of Ponzi finance. Ponzi finance
if
and only
if
riskless rate
the net riskless rate
may
feasible or not.
A
4
is
negative.
13
is
This suggests that, after
feasible
all,
the
be an important variable in assessing whether Ponzi finance
The
last
example shows that
fourth example:
this guess
would
also be
is
wrong.
Serially correlated returns in
storage
We
consider the same storage economy, but
turns on storage.
we assume
14
The reason
for
that the rate of return
doing
is
identically
and variance a 2 and with p
,
From
Or,
if
we had used
This result
monetary
'''This
is
p
(4.1)
and independently distributed with mean
economy
t
is
is
fi
and the unconditional expecta-
dynamically efficient
if
and only
if
= (l+p)Elne>0,
the preferences of section
2, if
consumers are
(4.2)
sufficiently risk averse.
closely related to Koda's (1984) characterization in this
equilibria with constant
example
a
e t et + i,
Zilcha's criterion, taking the logarithm
Elni?
13
More specifically,
geometric MA(1) process:
later.
> — 1.
tion of both sides of (4.1), this
12
allow for serially correlated re-
be clear
on storage follows
R t+1 =
where the logarithm of e
now
this will
money
supply.
a direct application of
Gale (1990).
12
model of the
existence of
or, equivalently, if
Consider
and only
>
if \x
0.
now the following debt
Ponzi game. At time
debt in the form of two-period bonds,
At time
the government buys back
1,
with the proceeds of
new
to
what
are
now
we
the
government
issues
later.
one-period bonds, and pays
and so on. To characterize
issues of two-period bonds,
the behavior of debt under this scheme,
0,
one unit of good two periods
titles
determine the equilibrium prices
first
of one- and two-period bonds.
The
is
price at time
t
of a two-period bond, issued
at
t
—
1
and maturing
at t
+ 1,
simply, from the first-order condition of the consumers
=
Vlt
1-Pa
E
*
{C2tt+1 j
= E Rt +1
= e p Ee-\
t
(4.3)
t
Note, for use below, that the average one-period gross riskless rate
E(l/pit).
15
The average net
-1
(Ee p )/(Ee
>
)
1,
riskless rate
or equivalently,
Notice, also for later use, that,
if
fi
p
is
(1
thus
is
-
non negative
2
p)a /2
>
when
one-period bond
is
0.
positive, the price of a
—
the current rate of return
P2t
[l-0
M
C2,t+2
1
1
as the
(4.4)
.
consider the dynamic behavior of debt (or
endowment,
low return on
i
t
to first period
a
Ci,t+i
= E (R t+ Rt+2)
= ^ P Ee~ (i+p) e r
Now
given by
if
on storage is low 16 because
storage today creates the expectation of a low future return.
The price at time t of a newly issued two-period bond is
high
is
and only
if
W,
endowment,
—
is
trivially
—the
ratio of debt
constant). At time
t,
debt
satisfies:
PitBt
= PltBtr-l,
(4-5)
so that
B =
>
15
By
arbitrage, the rates of return
to maturity
16
Given
must be
e t -i
,
a
eF^ b
->'
et
means
a
'
on one-period bonds, and on old two-period bonds one year
equal.
high
<4 6)
lowpi t and
a high
13
R
t
-
Thus, the rate of growth of the debt
Are there parameters such that
is
constant every period?
From our
is
derivations above,
The conditions on p, p and a are that p > 0,
These conditions are satisfied for example by p = p
it is
is
dynamically
efficient
answer
clear that the
(1
+
p)a
=
o
=
2
> p >
-
—
(1
is
p)a 2
.
\.
Thus, Ponzi finance using non-contingent two-period bonds
though the economy
p
as
economy is dynamically efficient, ii) the
non negative, (iii) debt decreases or stays
2
yes.
and of the same sign
the
i)
average one-period net riskless rate
deterministic,
is
feasible, al-
and the average one-period net
risk-
less rate is positive.
Dynamic
5
Why do
efficiency
results differ
between the
and Pareto-optimality
first
two and the
last
two examples? Under un-
certainty, overlapping generation models differ from Ramsey economies in two
ways.
The
first is
that people have finite
markets are incomplete.
In
all
economic horizons. The second
is
that
four examples, the condition that dynamic
effi-
ciency holds rules out the capital overaccumulation which
feature.
The
feature. In the first
equilibrium
ing.
two
difference between the
sets
may arise from
the
first
of examples comes from the second
two examples, incomplete markets do not matter, and thus the
last two, they do, and debt is Pareto improv-
Pareto optimal. In the
is
We consider these propositions in more detail.
In
all
four examples, the overlapping generation structure implies that the
young cannot enter insurance contracts with the old so as to share risk in the first
period of their life. But in the first two, the consumption of the young and the
consumption of the old are perfectly correlated, so that there is no room anyway
for further risk sharing. Indeed,
in
our
first
it is
easy to check that the decentralized allocation
two examples maximizes the following
social welfare function
oo
E [£(l + #r 5 ((l t
where 6
is
defined as earlier and
In line with previous literature,
optimal: there
hurting
17
some
is
no way
to
1
V)
is
''
lnCu+s +
PE t+s lnC2>(+s+1
therefore positive under
this
means
dynamic
that the market
(5.1)
)]
efficiency.
outcome
is
Pareto
improve the welfare of current generations without
future generation.
Thus the government cannot
See, for instance, Peled (1982), Aiyagari
play Ponzi games,
and Peled (1991), or Gottardi (1996).
14
no matter what the average
expected value of the debt-to-GNP
riskless rate or the
ratio.
This
cal
is
not the case in the
uncertainty
falls
last
two examples. In the third example, technologi-
only on the old
at
time
t,
while the consumption of the young
remains constant. This suggests room for debt to provide such insurance, and
indeed the case. This has been emphasized by Gale (1990)
in the context of that
who
has
shown
this
that
example, the government can not only issue and rollover
new
debt, but further issue
debt every period so as to maintain a constant debt-
ratio. Not only is such a policy feasible but it is Pareto improving,
amount of debt does not lead to a positive riskless rate. This can be
seen easily. Carrying a constant amount of debt at a negative riskless rate is equivalent to transferring a constant amount, r from the young to the old. It is easy
to-endowment
so long as the
to check that the condition for such a transfer to increase expected utility
the riskless rate be negative.
able to play a
Ponzi game
return. (This conclusion
equivalent
insurance
role of the
lated, a
that
why the government is
bonds provide insurance and require a low rate of
similar to that in Bertocchi (1991),
who
focuses
on the
phenomenon of bubbles.) The role for non contingent debt to provide
may however be quite limited (or not present at all). This explains the
debt in the
low
last
example.
When
returns to storage are positively corre-
realization of the rate of return
riskfree rate next
on
storage
is
associated with a lower
period and thus a high price of one-period bonds. Thus, issuing
two period bonds and buying them back
to the old.
is
Put another way, the reason
that
is
is
18
as
one-period bonds provides insurance
The price they get for their bonds
are low. This suggests that there
is
high
when
the returns to storage
may be an optimal maturity for bonds,
a line that
has been explored by Gale. Indeed, this suggests the issuance of explicitly con-
bonds to provide the needed insurance. And under those conditions, the
government may well be able to issue and rollover debt.
Should one conclude that Pareto suboptimality is a sufficient condition for the
tingent
feasibility
of debt Ponzi games, for some form of debt? Surely not. Competitive
economies may be Pareto suboptimal for reasons that have nothing
inadequate intergenerational transfers. To see
ple modification of
this,
to
do with
consider for example a sim-
our third example. Assume that preferences and the storage
technology facing each individual are the same as in that example, but that uncertainty
18
is
idiosyncratic so that, while the return
on individual storage
is
uncertain,
The proof is immediate and holds for a general utility function. Expected utility when an interr takes place is U = Eu\W — K(t) — r, RK(t) + t). Straightforward
generational transfer of size
algebra shows that
dU/dr >
as
long as
Rf <
1.
15
aggregate storage
not.
is
Assume
further that individual realizations are private in-
formation, preventing private contracts that pool away this individual uncertainty
from being written. The lack of infra-generational sharing makes the equilibrium
obviously Pareto suboptimal. Can the government play a Ponzi game? As aggregate variables are
non
we
stochastic,
debt games. The riskless rate
is
can, without loss of generality, look at riskless
same
the
condition for Ponzi games to be feasible
be negative, namely that
example, storage
is
\x
—
cr
2
/2
<
is
0.
as
it
was
sufficiently productive, there
examples considered
generally, for reasons that have
in this
condition
is
no scope
is
is
yes,
whenever the private marginal product of
(which
Ponzi
is itself
game
is
but
given by Saint-Paul (1992),
to capital. In his
''
Pareto-improving?
feasible, are
paper the answer
satisfied, if for
for Ponzi finance.
endogenously growing overlapping generations economy with
from external increasing returns
not
is
this
nothing to do with uncertainty per
An example
with second-best theory.
our example, so that the
again that the average net riskless rate
If this
Should one conclude that Ponzi games, when
In the four
in
se
is
not true
but rather
who studies an
a distortion
coming
model, Ponzi games are feasible
capital falls short of the
growth
rate
always lower than the social rate of return on capital). But a debt
never Pareto improving: since
it
lowers the growth
is
adversely affected.
rate,
it
crowds out
capital accumulation,
so that eventually the welfare of some future generations
Subsequent literature
6
The
literature has
made much
progress since this paper was
first
written in 1990.
Authors looking for existence conditions for fiat-currency equilibria (another
name
for debt Ponzi
games) have confirmed the essential themes of
the irrelevance of the average riskless rate,
in
providing intergenerational insurance.
and the
Some
this paper:
crucial role of debt Ponzi
games
papers have gone beyond the
ter-
and have provided the necessary and sufficient condition for
the existence of monetary equilibria (or debt Ponzi games) we were unable to deritory explored here,
rive ten years ago.
Manuelli (1990) shows,
omy (in which
19
in a stochastic
overlapping generations exchange econ-
the very issue of dynamic productive efficiency that preoccupied us
Okuno and Zilcha (1980) have shown, under certainty, that the often-asserted connecbetween the Pareto suboptimality of "non-monetary" equilibria and the existence of "monetary" equilibria does not generally hold: their results about fiat currency translate almost directly
Cass,
tion
to debt Ponzi
games, and are
valid, a fortiori
under uncertainty.
16
is
moot)
that the sign of the expected riskfree interest rate puts
the existence of
they
exist, often
no
restriction of
currency equilibria. In addition, monetary equilibria,
fiat
do not look
like
simple deterministic transfers from the young
to the old, but instead provide partial insurance
(1990) constructs an example in which
explains the value of money"
when
it is
between generations. Manuelli
indeed "this 'insurance' aspect that
(p. 278). All this
is
of course in direct
line, albeit in
an exchange economy, with the message of our four Socratic examples.
Also in a stochastic exchange economy, Chattopadhyay and Gottardi (1999)
provide a necessary and sufficient for conditional Pareto optimality.
Closer to our framework,
Wang
(1993) analyzes the existence and unique-
ness of stationary equilibria in overlapping generations economies with stochastic
production.
He does not
deal
Diamond
Ponzi games. In a stochastic
fully analyzes safe
economy with
(1965) economy, Bertocchi (1994) care-
capital similar to
might be unable to remove
sharing between generations.
size, as
we
inefficiency,
capital
because of inadequate
risk-
based on stochastic multipliers. They empha-
crucial for understanding
ulative bubbles... [E]ven
bles are sustainable
example, she shows
maximum principle for Markov controls
between dynamic efficiency and Pareto
did, that a "careful distinction
is
first
to derive a complete characterization of Pareto optimality for
economies with
optimality
our
Furthermore, Bertocchi and Kehagias (1995) de-
velop a stochastic version of Pontryagin's
which enables them
stochastic
efficiency, optimality, or
debt Ponzi games in the dynamically ^efficient case. Using a
log-linear stochastic
that safe debt
however with dynamic
if
phenomena such
when dynamic
as
Ponzi games and spec-
efficiency obtains, Ponzi
games and bub-
they can provide insurance and thus improve, in the Pareto
sense, the allocation of aggregate risk" (pp. 321-322).
This
is
also the
ize the results
duction.
message of Barbie, Hagedorn and Kaul (2000),
of Chattopadhyay and Gottardi (1999) to an
They provide
a single necessary
and
who
sufficient condition for conditional
Pareto optimality with dynamically complete markets. This criterion
eluded us
sum
when we wrote our paper
—
general-
economy with pro-
—which
alas
involves checking that a properly weighted
of inverse contingent prices does not converge over any possible history of
the economy. Chattopadhyay and Gottardi (1999) also
show
whenever the
competitive equilibrium is Pareto suboptimal, Pareto optimality can always be restored by a scheme that closely resembles a Ponzi game, and rolls over an insurance
that
insurance contract in exchange for contributions. The lesson that "sophisticated"
Ponzi games improve intergenerational risk sharing
our example
is
indeed the one taught by
4.
Last but not least,Barbie,
Hagedorn and Kaul (2001) use data from the matu17
.
rity structure of U.S.
ically efficient (this
ally
government debt
was the
gist
Pareto suboptimal. Barbie et
for a
"dynamic
fiscal
to argue empirically that the U.S.
of the Abel
al.
(2001
)
et al.
conclude that there
policy" to insure against
economy under
uncertainty.
is,
macroeconomic
running (sophisticated) debt Ponzi games
icy involves
is
dynam-
(1989) paper) but also conditiontherefore,
This pol-
risks.
dynamically
in a
room
efficient
.
Conclusion
7
Arguments as to whether governments can rollover debt are often cast in terms of
a comparison of the average growth rate and average riskless rate. In a series of
examples, we have shown that this may be misleading. The average riskless rate
may be less than the growth rate while Ponzi games are infeasible, or it may be
greater than the growth rate, while Ponzi games are feasible.
Turning to the structure underlying those examples, we have shown that, even
in economies in which equilibrium is dynamically efficient, Pareto suboptimality
may also
lead to the feasibility of Ponzi schemes.
In thinking about the implications of Pareto suboptimality,
this
we have focused
in
paper on the suboptimality which comes naturally from the incompleteness of
markets under uncertainty
other reasons
may
why
actual
in
overlapping generation models. But there are
many
economies may not be Pareto optima. Missing markets
be missing for reasons ranging from asymmetric information to transaction
costs, leading for a potential role
Ponzi games.
between
20
Distortions,
from
of public debt, and opening the possibility of
externalities to taxation,
risk adjusted interest rates
Thus, Ponzi games
may be
feasible.
and the
And
if
social
may
also create
marginal product of
they are, they
may
wedges
capital.
21
—but need not— be
Pareto improving.
How do we then
establish, short of identifying the sources of failure of Pareto
optimality, whether there
section,
we
is
scope for Ponzi games in a given economy? In the
derive a necessary condition for Ponzi games.
The
last
condition, which
involves the limit of the expectation of a product of marginal rates of substitution,
can in principle be
tested.
If rejected, it
matter what the behavior of riskless
20
implies
no scope
no
For instance, Woodford (1986), Scheinkman and Weiss (1986) and Kocherlakota (1992) pro-
vide examples of economies with infinitely-lived agents in an unbacked
to a debt Ponzi
21
for Ponzi games,
rates.
game
The paper by
—may
fiat
—
currency
equivalent
substitute for an exogenously closed market.
Saint-Paul (1992)
we have
already referred to earlier explores that route.
References
Abel, Andrew, N. Gregory Mankiw, Lawrence H.
Summers, and Richard J. Zeck-
hauser, "Assessing dynamic efficiency: theory and evidence," Review of Eco-
nomic
Aiyagari, S.
Studies, 1989,56, 1-20.
Rao and Dan
timality
Peled,
"Dominant Root Characterization of Pareto Op-
and the Existence of Monetary Equilibria
in Stochastic
Generations Models," Journal of Economic Theory, 1991, 54
Barbie,
Marcus
Martin,
and
ficiency
and
Production
Hagedorn,
Optimality
Pareto
Social
and Ashok Kaul,
Stochastic
a
in
Security,"
IZA
(1),
Overlapping
69-83.
"Dynamic
Ef-
OLG Model
with
paper
209
Discussion
(ftp://repec.iza.org/RePEc/Discussionpaper/dp209.pdf), October 2000.
,
and
,
surance
"Dynamic
Macroeconomic
,
against
Fiscal
Risk,"
Policy
In-
as
IZA
mimeo,
(http://www.iza.org/personnel/papers/hagedorn3.pdf), October 2001.
Bertocchi, Graziella, "Bubbles and Inefficiencies," Economics Letters, 1991, 35,
117-122.
,
"Safe Debt, Risky Capital," Economica,
and Athanasios Kehagias,
"Efficiency
November
1994, 61 (244), 493-508.
and Optimality
in Stochastic
Mod-
els with Production," Journal of Economic Dynamics and Control, JanuaryFebruary 1995, 19 (1-2), 303-325.
Blanchard, Olivier Jean and Stanley Fischer, Lectures on macroeconomics,
bridge, Massachusetts:
Bohn, Henning, "The
MIT Press,
Cam-
1989.
sustainability of budget deficits in a stochastic economy,"
Journal of Money, Credit and Banking, 1995,27, 257-271.
Cass, David,
"On
capital
overaccumulation
in the aggregative neoclassical
model
of economic growth: a complete characterization," Journal of Economic Theory,
,
1972,4,200-223.
Masahiro Okuno, and Itzhak Zilcha, "The
role of fiat
ing the Pareto optimality of competitive equilibrium in
type models," 1980.
N. Wallace,
in
eds., Federal
money
in support-
consumption loan
"Models of monetary economies" (Kareken
Reserve Bank of Minneapolis.
19
J.
and
Chattopadhyay, Subir and Piero Gottardi, "Stochastic
OLG
Models, Market
Structure and Optimality," Journal of Economic Theory, 1999, 89, 21-67.
Diamond,
Peter, "National debt in a neoclassical
growth model," American Eco-
nomic Review, 1965,55, 1126-1150.
Gale, Douglas,
"The
Mario Draghi,
efficient design of public debt," in
eds., Public
Rudiger Dornbusch and
debt management: Theory and history, Cambridge
University Press, 1990.
Galli,
Giampaolo and Francesco Giavazzi,
blico negli anni ottanta:
"Tassi di interesse reali e debito
pub-
Interpretazioni, prospettive e implicazioni per la
politica di bilancio," in "II disavanzo pubblico in Italia: natura strutturale e
politiche di rientro," Bologna:
II
Mulino, 1992.
mod-
Gottardi, Piero, "Stationary monetary equilibria in overlapping generations
els
with incomplete markets," Journal of Economic Theory, October 1996, 71
75-89.
(1),
Kocherlakota, Narayana
R.,
"Bubbles and Constraints on Debt Accumulation,"
Journal of Economic Theory, 1992, 57, 245-256.
Koda, Keiichi, "A note on the existence of monetary equilibria
erations
models with
storage," Journal of
in overlapping gen-
Economic Theory, 1984,
34,
388-
395.
Manuelli, Rodolfo,
"Existence and Optimality of Currency Equilibrium in
Stochastic Overlapping Generations Models:
The Pure Endowment
Case,"
Journal of Economic Theory, August 1990, 51 (2), 268-94.
Peled, Dan, "Informational diversity over time
and the optimality of monetary
equilibria," Journal of Economic Theory, 1982, 28,
255-274.
Endogenous Growth Model," Quarterly
Journal of Economics, November 1992, CVII (4), 1243-1260.
Saint-Paul, Gilles, "Fiscal Policy in an
Scheinkman, Jose and Lawrence Weiss, "Borrowing constraints and aggregate
economic activity," Econometrica, 1986, 54 (1), 23-46.
Selden, Lawrence, "A
tain'
New
Consumption
Representation of Preferences over 'Certain x Uncer-
Pairs:
The 'Ordinal Certainty
Econometrica, 1978,46, 1045-1060.
20
Equivalent' Hypothesis,"
MIT LIBRARIES
3 9080 02240 4666