Why Prices Don’t Respond Sooner
to a Prospective Sovereign Debt Crisis.∗R. Anton Braun†

Tomoyuki Nakajima‡

Federal Reserve Bank of Atlanta

Kyoto University

November 8, 2011

Abstract
We compare the dynamics of inflation and bond yields leading up to a sovereign debt
crisis in settings where asset markets are frictionless to other settings with financial frictions. As compared to the case with frictionless asset markets, an asset market structurewith financial frictions generates a significant delay in the response of prices to news about
a future debt crisis. With complete markets prices jump in response to news about thepossibility of a future debt crisis. However, when short selling of government bonds is
restricted some agents can’t act on their beliefs and prices don’t respond to the news.
Instead prices only move in periods immediately prior the crisis.
Keywords: sovereign debt crisis; deflation; fiscal risk; leverage; borrowing constraint.
JEL Classification numbers: E31, E62, H60.

∗

For helpful comments and suggestions, we thank Kosuke Aoki, Fumio Hayashi, Atsushi Kajii, Keiichiro

Kobayashi, Masao Ogaki, Kozo Ueda, and seminar participants at the Bank of Japan, CIGS, Federal Reserve
Bank of Atlanta, Hitotsubashi University (ICS), Kyoto University, and the University of Tokyo. Part of thisresearch was conducted while Nakajima was a visiting scholar at the Institute for Monetary and Economic
Studies at the Bank of Japan, whose hospitality is gratefully acknowledged. Nakajima thanks the financial
support from the JSPS.
†

Federal Reserve Bank of Atlanta. Email: r.anton.braun@gmail.com.

‡

Institute of Economic Research, Kyoto University, and the Canon Institute for Global Studies. Email:nakajima@kier.kyoto-u.ac.jp.

11Introduction

Recently, many developed countries have faced serious government debt problems in the midstof severe and persistent economic contractions. It is a politically difficult task for a government
to increase taxes or lower purchases and engineer a fiscal consolidation. For a governmentthat fails to generate a fiscal consolidation the alternative is default. Default can arise in oneof two ways. One way is to suspend payments on its debt. Most sovereign debt is nominallydenominated. This opens the door to a second form of implicit default. High inflation reduces
the real value of outstanding government debt. We refer to either of these forms of default asa sovereign debt crisis.
How does news about the possibility of a future debt crisis affect prices today? Under theassumptions of rational expectations and complete markets, prices will respond instantly tothe news. This result is at odds with what we see in some major economies. In Japan the
gross debt-GDP ratio has risen from 60 percent to over 200 percent and yet the yield curve isflat and the price level is falling. In the United States government debt has also risen sharply
and yet bond yields are low and inflation is muted.
The fact that prices have not responded is comforting to policy makers. On the one hand,
sharp increases in yields and/or inflation can be a powerful impetus for the fiscal authority
to get its house in order. On the other hand, if prices don’t respond, there is a tendency forpolicy makers to perceive that the problem is not severe and to kick the can down the road.
The fact that prices are not responding today does not mean that there is not a significant
risk of a future default. The objective of this paper is to use a model to make this point
explicit. We show that the asset market structure plays a crucial role in determining how
prices respond to an increase in the risk of a debt crisis. If markets are complete prices respondsharply and immediately to news about a prospective debt crisis. However, the response of
prices can be delayed when financial frictions are modeled. News about the prospect of afuture crisis has no impact on current prices. Instead prices increase sharply only shortly
before the crisis event.
We consider two models: one where default is implicit and another where default is explicit.
The particular model of implicit default we consider is a variant of the fiscal theory of the pricelevel (FTPL) studied, for instance, by Leeper (1991), Sims (1994), Woodford (1995), Cochrane(2001), and Bassetto (2002), among others. Following the convention of this literature, we
suppose that the government sets a sequence of real tax revenue as well as a sequence of
nominal interest rates. In equilibrium the price level adjusts to satisfy the government’s flow
budget constraint in every period. For simplicity, we consider a finite horizon model in which
the government collects taxes only in the last period. The amount of taxes collected in the2

last period are either high, TH , or low, TL . The equilibrium price level in the last period ishigher when the taxes collected in that period are smaller. We choose TL be so small that theequilibrium inflation rate is very large when the TL event occurs and refer to this outcome asa sovereign debt crisis.
Our model of beliefs and market structure builds on previous research by Geanakoplos(2003, 2010). Individuals have different beliefs about the probability of a debt crisis. Whena debt crisis occurs, the inflation rate rises and, as a result, the ex-post real rate of return on
government bonds falls. Thus, individuals who believe that the probability of a fiscal crisis is
low are relatively optimistic about the rate of return on government bonds.
We examine the dynamics of the inflation rate under two asset market structures. The firststructure is frictionless asset markets. Agents trade a complete set of contingent claims. In
the second market structure contingent claims are not traded. Agents can borrow to purchasegovernment debt, but short selling of government debt is ruled out. In the complete markets
specification both optimistic and pessimistic agents can freely bet on their beliefs. Relatively
optimistic agents buy the Arrow security which pays off in a state without a crisis and sell
the Arrow security which pays off in a state with a crisis. Relatively pessimistic agents makethe opposite trades.
In the second asset market structure, however, there is asymmetry between optimistic
and pessimistic agents. On the one hand, optimistic agents borrow as much as possible topurchase government bonds. This transaction is similar to purchasing the Arrow security
that pays off in the no crisis event. On the other hand, since government bonds cannot beshort sold by assumption, there is no transaction that allows one to mimic the purchase of
an Arrow security that pays off if there is a crisis. Pessimistic agents are not able to bet on
their beliefs and lend to optimistic individuals instead.
The two asset market structures have very different implications for the dynamics of theinflation rate. If asset markets are frictionless news about the possibility of a future debt
crisis gets immediately reflected in the price level and the inflation rate jumps up in theinitial period. In subsequent periods the inflation rate remains high along the path leading
to the debt crisis. In other words, there is a kind of inflation smoothing when asset markets
are frictionless.
In our setup with financial frictions the reaction of the initial price level to news about
the possibility of a future debt crisis is very small. In fact, in some of our examples the priceresponse is indistinguishable from zero. Instead the inflation rate only increases in states that
occur immediately prior to the crisis event. A final distinction is that the response of theinflation rate in the crisis state is much larger when there are financial frictions.

3

The intuition for this result is simple. When asset markets are frictionless both optimistic
and pessimistic agents can bet on their beliefs. In the presence of financial frictions, however,
only optimistic individuals can do so. This results in a higher real price of government bonds
and a lower inflation rate.
We also consider a setup with long-term debt and an exogenous price level. This is meant
to capture the situation for a country such as Greece that is a member of a currency union.
The results are very similar. In the complete markets specification with no financial frictions
the yield on long-term debt jumps immediately in response to news about a future default.
In the leverage specification, in contrast, the response of the yield curve to the same news isabout zero. The yield on long-term debt does eventually rise but only immediately before thecrisis event.

2

Motivation for the model

One of the most important implications of forward looking behavior is that news about future
events influences actions and the price system today. In the context of our model, news thatthe risk of a debt crisis has gone up should result in price increases today. In Section 3 wewill demonstrate this point with a formal model. Before discussing the model we first provide
some empirical evidence that suggests that price responses to news is delayed.
A variety of papers have provided evidence that sovereign debt crises are predictable.
Rheinhart and Rogoff (2010) find that the probability of a future sovereign debt crisis increases
for countries that experience banking crises and large increases in government and external
debt. Nieto Parra (2008) finds that investment banks demand higher underwriting fees asmuch as three years in advance of sovereign debt crises.
Bond yields, however, do not appear to respond contemporaneously. For intance, in Nieto
Parra’s (2008) sample of 29 emerging countries running from 1993 to 2006, bond risk premia
for crisis countries are somewhat elevated but stable in the 5 year period leading up to thecrisis and don’t respond to higher underwriting fees.
Inflation also does not appear to react contemporaneously to higher public deficits. Bassetto and Butters (2010) report empirical evidence on the contemporaneous correlation of
public deficits with inflation using a sample of OECD countries that extends from 19702009.1 . They find no evidence that high public deficits are associated with higher inflation
rates using their “corrected” measure of the deficit.2
1

Their sample excludes some countries that experienced crises such as Mexico and Turkey.

2

They adjust the standard definition of the deficit to recognize savings that accure to the government when4

Figure 1 reports debt-GDP ratios, the inflation rate, and short and long-term interest
rates for 3 sets of country pairs. Korea is plotted with Japan, Canada is plotted with the U.S.
and Germany is plotted with Greece.3 The reason for these choices of parings can be seen bycomparing the evolution of the debt-GDP ratio in each country paring. Japan has seen thedebt-GDP ratio rise from 50 percent in 1988 to over 180 percent in 2009. This debt-GDP isvery high relative to other advanced economies. In Korea, in contrast, the debt-GDP ratio isvery low. It gradually rises during the sample from a low of 13 percent in 1988 but is less than
32 percent in 2010. Recent research by Imrohoroglu and Sudou (2010) and Braun and Joines(2011) suggests that the current trajectory of government debt in Japan is unsustainable. If
one accepts this view then there are four distinct ways that a fiscal consolidation could occur:
higher real growth, a fiscal consolidation produced by higher taxes and/or lower government
spending, higher inflation that devalues the real value of outstanding government liabilities
or a payments suspension. What is surprising about Japan is that in spite of a near triplingin the debt-GDP ratio, there is no evidence of an uptick in either inflationary pressure or theyield curve. Yields on long-term bonds are falling throughout that sample and the economy
has seen the inflation rate fall as the debt-GDP ratio increased and now finds itself in asituation with deflation. Observe also that both long-term bond yields and the inflation rate
are lower in Japan than Korea, a country with a much stronger fiscal situation.
A similar picture emerges if we compare the U.S. with Canada. Between 2007 and 2010
the U.S. has seen the debt-GDP ratio increase from 36 percent to 61 percent. Canada also
experiences an increase. The debt-GDP ratio rises from 25 percent in 2007 to 36 percent in2009 but then stabilizes at about that level in 2010. One might expect that the large increasein U.S. indebtedness in conjunction with difficulties in legislating either higher taxes or lower
government expenditures would lead agents to assign a higher probability to either an explicit
payment suspension or higher inflation in the U.S as compared to Canada. However, as theplots in Figure 1 reveal, the evolution of inflation and long-term bond yields in the U.S. in
Canada are virtually identical.
If markets assign very low or even zero probability to either explicit default or implicit
default via inflation in the U.S. and Japan prices may not respond. It is thus also useful toconsider an industrialized country which has experienced a debt crisis.
Consider the plots of Germany and Greece. Inflation rates are low in both countries after
they switch to the Euro and remain low through the end of 2010. Galati, Heemeijer anddebt is nominally denominated and there is inflation.
3

Our basic source for this data is OECD. In some cases this data has been supplemented with data from theIMF, the Bank of Japan and Bank of Greece. More details on our data sources can be found in the Appendix.

5

Mossner (2011) find evidence that inflation expectations respond to increases in sovereigndefault risk in 2009 and 2010. However, these beliefs cannot be seen in inflation rates in
Greece and Germany which remain low.
Between 2000 and 2007 the debt-GDP ratio in Greece was averaging a bit over 100 percent.
This compared with a debt-GDP ratio in Germany of about 40 percent. In spite of thisdifference, the risk premium on Greek long term bonds was only 25 basis points on average
between 2002 and 2007. Greece saw its debt-GDP ratio increase from 106 percent in 2007 to
148 percent in 2010. In December of 2008 the spread on long-term Greek debt over German
debt rose above 2 percent for the first time since the Euro was founded. A bit more than 16
months later Greece requested funds from the EU/IMF in April of 2010.
This concentration of the price increases in states close to the crisis event also occurredin Argentina. Chan-Lau (2003) estimates one year ahead default probabilities for Argentina
using credit default swap data and finds that the probability of default only starts rising in
October of 2000 which was about 2 months prior to the IMF package which, was approved in
December of 2000, and 15 months prior to its default in January of 2001.
Overall, this evidence raises the possibility of a disconnect between risk and price movements. In early periods as the risk of a future debt crisis increases price responses appear
to be muted or even zero. Even though there are early indications that the probability of adebt crisis is increasing, prices only appear to react at most 12-16 months before the crisis.
Our aim here is to formalize a specific explanation for these observations. We will next turnto illustrate that modeling financial frictions can have a first order impact on the timing andsize of inflation and yield curve responses to a debt crisis.

3

Two-period model

A central message of this paper is that the market structure has first order implications forthe dynamics of the inflation rate. To illustrate this point we will characterize the competitive
equilibrium under three different asset market structures. After that we will describe how theequilibrium inflation rate varies across them when agents begin to expect a debt crisis.
Our model is an exchange economy in which all agents consume a single consumption
good in each period. To make our arguments more transparent we follow the example of theliterature on the fiscal theory of the price level and abstract from money. The price levelis determined as the relative price of the consumption good and the (nominal) governmentbond.
To make the exposition of the model more transparent to the reader we begin by consider-6

ing a two-period version of the model. Then in Section 4 we generalize the model to allow for
an arbitrary number of time periods. Periods are indexed by t = 0, 1. There are two statesof nature in period 1, U and D. As discussed below, they are distinguished by the amount oftaxes collected by the government. In this two period model the event D will be associatedwith low taxes and a debt crisis. We let st denote the state of nature in period t, where s0 = 0
and s1 ∈ S ≡ {U, D}.
Individuals:

There is a continuum of agents indexed by h ∈ [0, 1]. Each agent receives anidentical endowment of yt units of the consumption good in period t = 0, 1. Note that theendowment does not depend on the state in period 1. Agents are also endowed with equalamounts of nominal government debt, B̄ > 0, in period zero. All agents have access to arisk-free storage technology that offers a gross rate of return denoted by R > 1.
Agents are identical except for their beliefs about the probability that state U occurs inperiod 1. Specifically, we assume that agent h believes that s1 = U with probability h. Thus,
agents with high h assign less probability to the debt crisis event. All agents have linear
preferences of the form:c0 +Xγ h (s1 )c(s1 )

(1)

s1 ∈S

where γ h (s1 ) denotes the subjective probability of agent h that state U occurs in period 1:(
γ h (s1 ) =Government:h,for s1 = U ,1 − h,

for s1 = D.

The government starts off with B̄ > 0 nominal liabilities to the private sector,

collects taxes and issues one-period bonds. We abstract from government consumption andassume that taxes are lump sum and identical across agents. Neither of these assumptions are
essential to our arguments. Let T0 and T (s1 ) denote the real amount of taxes in period 0 andin state s1 ∈ S of period 1, respectively. Let B0 be the nominal amount of bonds issued by
the government in period 0 and q0 ∈ (0, 1] be the price of those bonds. Given these definitions

the “flow budget constraints” of the government in period 0 and in state s1 ∈ S of period 1

7

are given by4
B̄ = P0 T0 + q0 B0 ,B0 = P (s1 )T (s1 ),(2)
s1 ∈ S,(3)

where P0 and P (s1 ) are the price levels in period 0 and in state s1 , respectively.
We will refer to the government’s choice of {T0 , {T (s1 )}s1 ∈S } as its fiscal policy, and its

choice of the one-period nominal interest rate 1/q0 as its monetary policy. Given a particular

choice of the nominal interest rate, the amount of government bonds issued in period 0, B0 ,is determined so as to satisfy demands from the private sector.
We assume that the government collects taxes in the following fashion:
T0 = 0,(T (s1 ) =

(4)
TH ,

if s1 = U ,TL ,if s1 = D,

(5)

where TH  TL > 0. We assume that the amount of taxes collected by the government isvery small in state D in period 1. This explains our use of the expression “debt crisis” to referto this state. Thus a government policy is given by (q0 , TH , TL ) ∈ (0, 1] × R2++ . We assume
that government policy is exogenous.

3.1

Complete markets

Let us begin by considering a setting with complete markets and no trading frictions. We willsubsequently refer to this as the complete markets specification. Suppose that a complete setof one period contingent claims (Arrow securities) are traded in period 0. Let q(s1 ) denotethe price of an Arrow security that pays off one unit of account in period 1 if and only if s1
occurs.
Each agent h ∈ [0, 1] maximizes utility (1) subject to the budget constraints:
c0 + k0 +X

q(s1 )s1 ∈S

c(s1 ) ≤ Rk0 +b(s1 )b0++ y1 − T (s1 ),P (s1 ) P (s1 )c0 , k0 , b0 , c(s1 ) ≥ 0,
4b0
B̄b(s1 )+ q0≤+ y0 ,P0
P0
P0

s1 ∈ S,(6)
s1 ∈ S,(7)(8)

Here we refer to (2)-(3) as the flow budget constraints of the government. But, as is well known, in thestandard formulation of the fiscal theory there is ambiguity about exactly what constraints the government
faces. Bassetto (2002) posits a game theoretic version of the fiscal theory that removes this ambiguity. It is
straightforward to rewrite our model in the same way as Bassetto (2002).

8

where k0 is the amount goods stored in period 0, b0 is the amount of government bondspurchased in period 0, and {b(s1 )}s1 ∈S are the amounts of Arrow securities purchased inperiod 0. Notice that (8) imposes a short selling restriction on government debt. Given thatagents have access to a complete set of one period Arrow securities this restriction doesn’t
matter in this setting. Having described the agent’s problem we can now define a competitive
equilibrium.
Definition 1 (Complete Markets Competitive Equilibrium). Given a government policy(q0 , TH , TL ), a complete markets competitive equilibrium consists of an allocation {ch0 , [ch (s1 )]s1 ∈S ,k0h , bh0 , [bh (s1 )]s1 ∈S }h∈[0,1] , supply of government bonds B0 , and prices {P0 , [P (s1 )]s1 ∈S ,[q(s1 )]s1 ∈S } such that (i) for each agent h ∈ [0, 1], the allocation {ch0 , [ch (s1 )]s1 ∈S , k0h , bh0 ,[bh (s1 )]s1 ∈S } solves her utility maximization problem; (ii) the government flow budget con-straints (2)-(3) are satisfied, where taxes {T0 , [T (s1 )]s1 ∈S } are given by (4)-(5); and (iii) all
markets clear:

Z 1
0

Z 1

(ch0 + k0h ) dh = y0 ,

(9)
Z 1

h

c (s1 ) dh = y1 + R
0

0

Z 1
0

Z 1

k0h dh,s1 ∈ S,bh0 dh = B0 ,

(10)
(11)

bh (s1 ) dh = 0,s1 ∈ S.

0(12)

We now turn to provide a characterization of the complete markets equilibrium. In thissetting both storage and the government bond are redundant assets. Since the government
bond pays off one unit of account irrespective of the state in period 1, no-arbitrage requiresthatX

q(s1 ) = q0 .

(13)

s1 ∈S

Similarly, since one unit of goods put into storage in period 0 yields R units of goods in allstates in period 1, no-arbitrage also implies
X q(s1 )P (s1 )
s1 ∈S

P0

R = 1,(14)

when storage occurs in equilibrium.
It follows from these observations that the flow budget constraints (6)-(7) can be combined
to obtain the lifetime budget constraint: X
X q(s1 )P (s1 )B̄
q(s1 )P (s1 ) 
Rc0 +Rc(s1 ) ≤ R+ y0 +
R y1 − T (s1 )
P0
P0
P0
s1 ∈S

s1 ∈S

9

(15)

and the agent’s problem can be restated as maximizing (1) subject to (15). Since R > 1 ourpreference structure implies that no one consumes in period 0:
ch0 = 0,Hence

for all h ∈ [0, 1].

(16)

R1

h
0 k0 dh = y0 > 0 and thus there is storage in equilibrium. Define h0 as

h0 =q(U )P (U )
R.
P0

(17)

It follows from (14) that h0 is between 0 and 1, and that1 − h0 =q(D)P (D)R.
P0

Then we see that h0 is the marginal agent in the sense that agents h > h0 make different
choices than agents with h < h0 . Agents with h > h0 are optimistic and choose ch (U ) = 0.
Agents with h < h0 are pessimistic and choose ch (D) = 0.
Next consider the government. Equations (2) and (4) imply
B0 =

B̄
q0

(18)

and it follows that the price level in period 1 is
P (s1 ) =1 B̄,T (s1 ) q0

s1 ∈ S.

(19)

Using the no-arbitrage condition (13), we can combine the flow budget constraints (2)-(3) toderive the government’s present value budget constraint
X q(s1 )B̄=P (s1 )T (s1 ).
P0 s ∈s P0
1

Then use (17) to write this expression as
B̄
TH
TL
= h0
+ (1 − h0 ) .
P0
R
R

(20)

Note that this equation determines P0 given h0 .
Next we use the market clearing restrictions to determine the identity of the marginal
agent. Starting with the agent’s lifetime budget constraint (15) and using (20), its right handside becomes

R

B̄
+ y0
P0


+

X q(s1 )P (s1 ) R y1 − T (s1 ) = Ry0 + y1 .
P0

s1 ∈S

10

Thus (15) can be rewritten as
Rc0 + h0 c(U ) + (1 − h0 )c(D) ≤ Ry0 + y1 .
It follows that the consumption allocation is given by:(1
for h > h0 ,hh0 (Ry0 + y1 ),c (U ) =0,for h ≤ h0 ,(0,for h > h0 ,ch (D) =1for h ≤ h0 .
1−h0 (Ry0 + y1 ),

(21)

(22)

Market clearing implies
Z 1
Ry0 + y1 =ch (U ) dh =0

1 − h0(Ry0 + y1 ),h0which yields the identity of the marginal agent
1h0 = .
2
Finally, the Arrow security prices q(s1 ) are determined by solving q(s1P)P0 (s1 ) R = 12 for eachs1 ∈ S. The next proposition summarizes the result.
Proposition 1. In the two period model with complete asset markets, (i) the equilibrium
consumption allocation {ch0 , [ch (s1 )]s1 ∈S }h∈[0,1] is given by (16), (21), and (22), where themarginal agent h0 = 21 ; and (ii) the equilibrium price levels {P0 , [P (s1 )]s1 ∈S } are given by(19) and (20).

One implication of this proposition is that the consumption allocation is independent ofgovernment policy or in other words Ricardian equivalence obtains. In the other two market
structures we consider Ricardian equivalence fails.

3.2

No borrowing

Now suppose that Arrow securities are no longer traded, so that government bonds and storage
are the only assets available to agents. It follows that these assets are no longer redundant.
Since there are two assets (storage and government bonds) and two states (U and D) asset
markets are still complete. However, we now rule out borrowing. Moreover, our previous
assumption that short selling of government debt is prohibited is no longer innocuous. These
financial frictions imply that the competitive equilibrium in this specification is not Pareto
Optimal whereas the equilibrium considered in the previous subsection is Pareto Optimal.
For these reasons we will refer to this new setup as the no-borrowing specification.
11

Budget constraints for each agent are now given byB̄b0≤+ y0 ,P0
P0
b0c(s1 ) ≤ Rk0 ++ y1 − T (s1 ),P (s1 )

(23)

c0 + k0 + q0

c0 , k0 , b0 , c(s1 ) ≥ 0,

s1 ∈ S,s1 ∈ S.

(24)(25)

Each agent maximizes (1) subject to these constraints.
Definition 2 (No-borrowing Competitive Equilibrium). Given a government policy (q0 , TH ,TL ), a no-borrowing competitive equilibrium consists of an allocation {ch0 , [ch (s1 )]s1 ∈S , k0h ,bh0 }h∈[0,1] , supply of government bonds B0 , and prices {P0 , [P (s1 )]s1 ∈S } such that (i) for eachagent h ∈ [0, 1], {ch0 , [ch (s1 )]s1 ∈S , k0h , bh0 } solves her utility maximization problem; (ii) the

government flow budget constraints (2)-(3) are satisfied, where taxes {T0 , [T (s1 )]s1 ∈S } aregiven by (4)-(5); and (iii) all markets clear, i.e., equations (9)-(11) hold.

Consider the utility maximization problem of agent h ∈ [0, 1]. Once again no agent will

choose to consume in period 0: ch0 = 0 for all h ∈ [0, 1]. Each agent will hold only the asset
that offers her the highest expected return. Let h0 ∈ [0, 1] be the agent who is indifferent

between government bonds and storage. We refer to this agent as the marginal buyer. The
identity of h0 is given by the solution to


1
1
P0
h0 +
(1 − h0 ) = R,
q0 P (U )
P (D)

(26)

where the left-hand side is the real rate of return on government bonds, and the right-handside the real rate of return on storage. Agents with h > h0 have optimistic beliefs about
the return on government bonds and only hold government bonds. Agents with h < h0 , on
the contrary, are pessimistic about the return on government bonds and hold no government
bonds at all. Thus, the solution to the utility maximization problem can be summarized asfollows:
ch0 = 0,
(
k0h =
bh0 =

h ∈ [0, 1],

(27)

0,

h > h0 ,B̄P0 + y0 ,h ≤ h0 , P0 B̄ + y0 ,
q0

(28)

h > h0 ,P0 0,h ≤ h0 , 1 P0 B̄ + y0 + y1 − T (s1 ),P (sh1 ) q0 P0
c (s1 ) =
 R B̄ + y0 + y1 − T (s1 ),
P0
12

(29)
h > h0 ,h ≤ h0 .

(30)

As in the previous case, B0 = B̄/q0 , and hence the price level in state s1 , P (s1 ), is
determined by (19). From (29) it follows that the market clearing condition for government
bonds isq0
B0 = (1 − h0 )
P0




B̄
+ y0 ,
P0

which implies that the price level in period 0, P0 , is given by
P0 =

h0 B̄
.
1 − h0 y0(31)

The next proposition summarizes these results.
Proposition 2. The two period no-borrowing equilibrium has the following properties: (i) theequilibrium consumption allocation/portfolio, {ch0 , k0h , bh0 , [ch (s1 )]s1 ∈S }h∈[0,1] , is given by (27)-(30); (ii) the identity of the marginal buyer of government bonds, h0 , is given by (26); and(iii) the equilibrium price levels {P0 , [P (s1 )]s1 ∈S } are given by (19) and (31).

3.3

Leveraged purchases of government bonds

We now consider a market structure where agents can borrow and lend to each other. Shortsales of government debt though are still prohibited. We refer to this specification as theleverage specification. The leverage competitive equilibrium is also not Pareto Optimal but
produces different allocations from the no-borrowing specification.
In this market structure optimistic agents, who believe that the rate of return on government bonds is greater than the borrowing rate, want to borrow as much as possible andpurchase government bonds. Assume further that a borrower is required to post government
bonds as collateral in order to obtain a loan. How much can an agent borrow with one unit ofgovernment bonds as collateral? One way to proceed would be to impose an exogenous ad hoc
constraint as in e.g. Kiyotaki and Moore (1997). We pursue an alternative avenue that allows
us to determine the collateral constraint endogenously. Geanakoplos (2003, 2010) allows for abroad array of loan/default schemes and determines which ones trade in equilibrium. In our
setting this can be summarized by a “no-default constraint,” that requires that the amountof repayments not exceed the value of the collateral in any state. We derive this result in the
Appendix and directly impose the no-default constraint here.
Since there is no default on loans, loans are risk-free. Thus the interest rate on loans isequal to R in equilibrium (as long as the storage technology is used). Consider an agent whoborrows φ0 and purchases government bonds b0 in period 0. She must repay Rφ0 in periodb0, for all s1 ∈ S. Thus the budget
1. The no-default constraint requires that Rφ0 ≤ P (s1)

13constraints for each agents becomeb0
B̄≤+ y0 + φ0 ,P0
P0
b0c(s1 ) ≤ Rk0 ++ y1 − T (s1 ) − Rφ0 ,P (s1 )b0
Rφ0 ≤,s1 ∈ S,P (s1 )
c0 + k0 + q0

c0 , k0 , b0 , c(s1 ) ≥ 0,(32)
s1 ∈ S,s1 ∈ S.

(33)
(34)
(35)

Definition 3 (Leverage Competitive Equilibrium). Given a government policy (q0 , TH , TL ), aleverage competitive equilibrium consists of an allocation {ch0 , [ch (s1 )]s1 ∈S , k0h , bh0 , φh0 }h∈[0,1] ,supply of government bonds B0 , and prices {P0 , [P (s1 )]s1 ∈S } such that (i) for each agenth ∈ [0, 1], {ch0 , [ch (s1 )]s1 ∈S , k0h , bh0 , φh0 } solves her utility maximization problem; (ii) the

government flow budget constraints (2)-(3) are satisfied, where taxes {T0 , [T (s1 )]s1 ∈S } are
R1
given by (4)-(5); and (iii) all markets clear, i.e., equations (9)-(11) hold and 0 φh0 dh = 0.
To characterize this equilibrium start with the utility maximization problem. No one
consumes in period 0: ch0 = 0 for all h ∈ [0, 1]. Observe that P (U ) < P (D) in equilibrium0. In this setting there
and thus the collateral constraint (34) can be expressed as φ0 ≤ RPb(D)will be a marginal purchaser h0 who is indifferent between storage and borrowing and using
the proceeds to purchase government debt. This indifference relationship is given by
1
1
P (U ) − P (D)
h0 q0
=R
1
P0 − RP (D)(36)

and is identical to (26). Optimistic agents h > h0 want to borrow as much as possible topurchase government bonds. On the other hand, pessimistic agents h < h0 do not want tohold government bonds. However, they are perfectly willing to lend to optimistic agents at
the interest rate R. Thus there will be leverage in equilibrium.
Given these results the solution to the utility maximization problem is summarized by:
ch0 = 0,h ∈ [0, 1],
 
−1 

 q0 − 1B̄+y,h > h0 ,0P0P0
RP (D)bh0 = 0,h ≤ h0 ,


−1 

q0
 − 1
1
B̄
h > h0 ,hh
P0 − RP (D)P0 + y0 ,RP(D)
k 0 − φ0 =
 B̄ + y ,h ≤ h0 ,
0
P
 01


1
− P (D)B̄
1)
 Pq(s+y+ y1 − T (s1 ),h > h0 ,01
0
P0−
P0 RP (D) ch (s1 ) = R B̄ + y0 + y1 − T (s1 ),h ≤ h0 .
P0
14

(37)
(38)

(39)

(40)

Since storage and lending are perfect substitutes, only k0h − φh0 is determined for agents withh ≤ h0 . Agents with h > h0 set storage to zero, k0h = 0, and borrow as much as they can:bh

φh0 = RP 0(D) . As before, B0 and P (s1 ), s1 ∈ S, are determined by (18) and (19). Then it

follows from (38) that the market clearing condition for government bonds is expressed asB̄= (1 − h0 )
q0



q0
1
−
P0 RP (D)

−1 


B̄
+ y0 .
P0(41)

Given P (s1 ), s1 ∈ S, the initial price level P0 and the marginal agent h0 are determined as
the solution to (36) and (41). The following proposition summarizes these results.

Proposition 3. In the two period leverage equilibrium, (i) the equilibrium consumption allocation/portfolio, {ch0 , k0h − φh0 , bh0 , [ch (s1 )]s1 ∈S }h∈[0,1] , is given by (37)-(40); (ii) the marginal
buyer of government bonds, h0 , is given by (36); and (iii) the equilibrium price levels {P0 ,[P (s1 )]s1 ∈S } are given by (19) and (41).

3.4

Numerical example

We now turn to consider a particular thought exercise that is aimed to help us understandhow the inflation rate evolves in each of these specifications when agents begin to expect that
there is a possibility of a debt crisis.
We set the parameter values as B̄ = 1, y0 = y1 = 1, q0 = 1, R = 1.02, TH = R2 , andTL = TH /2. The choice of a gross nominal interest rate of one is chosen to reflect the current
situation in Japan and the United States.
Suppose that in all periods prior to period zero the gross nominal interest rate is one andeveryone believes that s1 = U with probability one. That is, prior to period 0, no one believes
that a debt crisis will occur. Under these assumptions it follows that P−1 = 1. Since q−1 = 1
the inflation rate in period -1 is given by π−1 = q−1 /R − 1 = −1.96 percent. Suppose nextthat in period zero agents become worried that a debt crisis may occur in period one and thatthey have different degrees of confidence about this possibility.
Table 1 shows the inflation rate at t = −1, t = 0 and at s1 = D for the “complete markets,”“no borrowing,” and “leverage” specifications, respectively. The inflation rate in period zero,

π0 , is highest in the model with complete markets and lowest in the model with leverage.
This ranking is reversed in period one if state D is realized and the debt crisis occurs. π(D)is highest with leverage and lowest with complete markets.
In the complete markets specification when the news about a future crisis arrives in periodzero these beliefs are instantly reflected in the price level and the inflation rate jumps up toa value of 30.72 percent. The inflation rate also jumps up in the leverage specification. But
15

the size of the jump in period zero is much smaller as compared to the complete markets
specification. Instead there is a much larger jump in the inflation rate in the state where thedebt crisis occurs.
There is simple intuition behind this result. In the model with complete markets, every
agent can freely bet on her beliefs: optimistic agents buy the Arrow security U and sell Arrowsecurity D, and pessimistic agents do the opposite. A unique feature of the complete markets
specification is that by betting against the perceived low probability state agents can commit
to deliver their future endowment if that state occurs. In the other two specifications theycannot make this promise. In the no-borrowing equilibrium agents can only choose how tosave in the form of bonds or storage. In the leverage equilibrium optimistic agents can also
undertake collateralized borrowing which makes it possible for them to promise to deliver the
return on government bonds in period one and thereby mimic a purchase of Arrow security U .
However, they are not able to mimic a sale of Arrow security D. For pessimistic agents the
problem is more severe as there is no way for them to mimic either selling the Arrow security
U or purchasing the Arrow security D.
The inability of agents to promise their period 1 endowment results in a higher valueof h0 in the no-borrowing and leverage specifications. This translates into a lower inflation
rate π0 compared to the case with complete markets. The asymmetry between optimists andpessimists in the leverage specification enhances these effects and that specification delivers
the lowest inflation rate in period zero.

4

T -period model

In this section we shall see that the effect of the asset market structure on the dynamics of
the inflation rate becomes more apparent when the time horizon is extended.
Suppose that there are T +1 periods indexed by t = 0, 1, . . . , T . In each period t = 1, . . . , T ,a shock st ∈ S = {U, D} is realized. For t = 1, . . . , T , let st = (s1 , . . . , st ) ∈ S t denote thehistory of shocks, and also let s0 ≡ 0 and S 0 ≡ {0}. In each period t, regardless of the history
st , agent h believes that st+1 = U with probability h and st+1 = D with probability 1 − h.
Then her expected utility is expressed as

T X
X

γ h (st )c(st ),(42)

t=0 st ∈S t

where γ h (st ) denotes the subjective probability that agent h assigns to history st in period 0.

16That is,(γ h (st ) =hγ h (st−1 ),if st = U ,(1 − h)γ h (st−1 ),if st = D,with γ h (s0 ) ≡ 1.

In each period t, all agents are endowed with equal amounts of the consumption good,y(st ) for st ∈ S t . For simplicity, we assume that y0 ,ty(s ) =
0,


 y ,
T

for t = 0,
for all st with t = 1, . . . , T − 1,for all sT .

In addition, at the beginning of period 0, all agents are endowed with equal amounts ofgovernment debt B̄ > 0.
As in the two-period model, the government specifies a fiscal policy which consists of astate-contingent path of lump-sum taxes {T (st ) : st ∈ S t , t = 0, . . . , T }. To be specific, fiscal

policy takes the following form:



 0,
T (st ) =

TL ,


 T ,
H

for all st with t = 0, . . . , T − 1,for sT = DT ,(43)

for all sT 6= DT .

The government also chooses a monetary policy which is a state-contingent path of the priceof government bonds, {q(st ) : st ∈ S t , t = 0, . . . , T }. Given these government policies, thegovernment debt B(st ) evolves as

B̄ = q0 B0 ,(44)

B(st−1 ) = q(st )B(st ),st ∈ S t , t = 1, . . . , T − 1,

B(sT −1 ) = P (sT )T (sT ),sT ∈ S T .

(45)
(46)

Figure 2 illustrates the implications of these assumptions using an event tree for the special
case where T = 2. A debt crisis only occurs when the government collects TL taxes in the
final period. Under our assumption (43), it occurs only if st = D for all t = 1, . . . , T . In
the case of T = 2 illustrated in the figure this corresponds to the bottom outcome where thehistory is {D, D}. In the other three histories in the last period there is no crisis. For general

T there will only be one history in the final period that produces a crisis. From this it follows
that extending the horizon acts to lower the perceived probability of a crisis.

17

4.1

Complete markets

Let q(st+1 |st ) denote the price of the Arrow security traded at st that pays off one unit ofaccount if and only if st+1 occurs in the next period. With a complete set of Arrow securities,
the flow budget constraints for each agent are given by
c0 + k0 +X
s1 ∈S

q(s1 |s0 )c(st ) + k(st ) +X
st+1 ∈Sb0
B̄
b(s1 |s0 )+ q0≤+ y0 ,P0
P0
P0

q(st+1 |st )tb(st+1 |st )t b(s )+q(s)
P (st )P (st )b(st |st−1 ) b(st−1 )++ Rk(st−1 ),t = 1, . . . , T − 1, st ∈ S t ,P (st )P (st )b(sT |sT −1 ) b(sT −1 )++ Rk(sT −1 ) + yT − T (sT ),sT ∈ S T ,c(sT ) ≤P (sT )
P (sT )

(47)
(48)

≤

c(st ), k(st ), b(st ) ≥ 0,st ∈ S t , t = 0, . . . , T,

(49)
(50)

where b(st+1 |st ) denotes the quantities of the Arrow securities purchased at st , and b(st )denotes the quantities of government bonds purchased at st .

Given a policy ({q(st )}, TH , TL ), a competitive equilibrium is defined as in the two-period
case.
It is also interesting to understand how the identity of the marginal buyer evolves along
the path that results in a crisis. It can be shown that her identity evolves according to thefollowing rule:h(Dt ) =1,
t+2

t = 1, 2, . . . , T,with h0 = 1/2. A proof of this result can be found in the Appendix.

18

(51)

4.2

Leveraged purchases of government bonds

Now suppose that Arrow securities are no longer traded, and borrowing is limited by the
no-default condition. The flow budget constraints for agents areb0
B̄≤+ y0 + φ 0 ,P0
P0b(st )c(st ) + k(st ) + q(st )P (st )t−1b(s )≤+ Rk(st−1 ) − Rφ(st−1 ) + φ(st ),t = 1, . . . , T − 1, st ∈ S t ,P (st )b(sT −1 )c(sT ) ≤+ Rk(sT −1 ) + yT − T (sT ) − Rφ(sT −1 ),sT ∈ S T ,P (sT )b(st )Rφ(st ) ≤,t = 0, . . . , T − 1, st ∈ S t , st+1 ∈ S,
P (st , st+1 )c0 + k0 + q0c(st ), k(st ), b(st ) ≥ 0,

4.3

st ∈ S t , t = 0, . . . , T,

(52)

(53)
(54)
(55)
(56)

Two additional numerical examples

We now consider two further numerical examples that illustrate how the properties of themodel change when the time horizon is extended. In the first example all uncertainty is
resolved at the end of period two. In the second example the economy ends at the end of
period five.
We consider the same parameterization of the model as before. Let B̄ = 1, y0 = yT = 1,q(st ) = 1 for all st and t, R = 1.02, TH = RT +1 , and TL = TH /2. Also, assume that in period-1, everyone believes that Pr(sT = DT ) = 0, that is, everyone believes that the government
collects taxes of amount TH in period T for sure. It follows that P−1 = 1. Recall also that
π−1 = −1.96 percent.

Table 2 contains results for the three period model. The first two columns report π−1 , theinflation rate in period -1, and π0 , the inflation rate in period zero. Columns three and four
report the inflation rate in particular states. Column three reports its value in period one
when the realization is D and column four reports the value of the inflation rate in period
two when the realization is {D, D}. The final two columns of this table report the identity ofthe marginal purchaser in period zero and in period one when the state is D.

Consider first the results for complete markets. For this parameterization of the model
the inflation rate jumps from -1.96 percent to 17.65 percent in period zero when individuals
realize that there is a possibility of a debt crisis. As compared to Table 1, the size of thejump in the inflation rate in period zero is now smaller. For all agents the possibility of a
fiscal crisis is lower when the model has three periods. It is now (1 − h)2 as compared to19

(1 − h) before. The size of the jump in period zero at 17.65 percent is still very large. Noticealso that the jump if the crisis state is realized in the final period, π(D2 ), is smaller when thehorizon is extended by one period. Extending the horizon tends to smooth out the inflation
response in the complete markets specification.
It is also interesting to consider the dynamics of the marginal buyer. In the two period
model the marginal buyer is h0 = 1/2 or the midpoint of the interval. This continues to be
the case in the three period model. In period 1 when the state D is realized the net worth ofthe optimistic individuals with h > h0 falls to zero. The remaining individuals with positive
net worth enter into new agreements and it follows from equation (51) that the new marginal
purchaser is h(D) = 1/3.
Observe next that the properties of the model are very different when we use the market
structure with leverage. The size of the inflation rate response is now very small (-1.09 percent)when the news arrives in period zero. For purposes of comparison, in the two period model
inflation increased to 9.46 percent in period zero. This response is also qualitatively quite
different from the complete markets specification. In the specification with leverage there is
still deflation immediately after the news arrives whereas with complete markets the inflation
rate is very large and positive. A further distinction between the two period and three period
model is that the inflation rate is now slightly lower in the final period. In the two period
model it is 75.62 percent in the final period in the three period model it falls to 71.89 percent.
However, the gap between the inflation rate in the final period and the inflation rate in periodzero is now larger.
A further important distinction between the two market structures relates to the dynamics
of the marginal buyer. For complete markets we know from (51) that the marginal buyer inperiod zero is always h0 = 1/2 regardless of the number of periods in the model. In theleverage specification the opposite is the case. The identity of the marginal buyer in periodzero increases as the number of periods in the model is increased. Suppose next that we follow
the path leading to the crisis state and consider the period prior to the crisis. Along this legof the event tree we see that the identity of the marginal buyer in the three period model,
0.75, is quite close to the identity of the marginal buyer in the two period model, 0.79.
Taken together these results suggest that increasing the horizon acts to reduce the impact
response of inflation at the time news arrives about the possibility of a future crisis andconcentrate its response in the crisis state.
To make this point more transparent we report results for the model with a six period
planning horizon in Figure 3. Figure 3 plots the inflation rate for the complete markets specification and the leverage specification. Observe that for the complete markets specification

20

the smoothing property is more pronounced. The inflation rate now jumps to 7 percent inperiod zero and then rises slowly but steadily to a maximum of about 14 percent.
The concentration of inflation in the crisis state that occurs in the leverage specification
is much more pronounced in the six period model. In fact, there is no longer a discernible
jump in the inflation rate in period zero.5 We previously saw in the three period model thatthe inflation rate turned positive in period 1. Here we can see very clearly that increasing thehorizon acts to delay the response of the inflation rate. In period 3, for instance, the inflation
rate is still very low at -1.01 percent. It is only in period four that we see a substantial increasein the inflation rate to 10.97 percent. The biggest increase is concentrated in the final period
when the crisis state is realized (71.57 percent). This is very close to the value of the inflation
rate in the crisis state in the three period model (71.89 percent).
This fact can be related to the identity of the marginal buyer in the period before thecrisis. For the three period model her identify was (about) 0.75. Interestingly, her identity
is also about 0.75 in the six period model. This is why the final period inflation rate is very
similar in the two simulations.

5

Explicit default

We now turn to consider a setting with explicit default and long-term debt. We will assume
that this is the only way default can occur. In particular, price levels {P (st )} are now

given exogenously and thus the price level is no longer determined by fiscal policy. Theseassumptions are designed to reflect the situation of a small country (or state) that is a memberof a currency union.
Let B̄ be the face value of government debt in period 0 and suppose that it does notissue any new debt in any other period. All government debt is long-term and matures in thelast period, T . Moreover, suppose that the government only collects taxes in the last period.
Under these assumptions the nominal outstanding value of government debt is B̄ in all but
the last period.
As before, a shock st ∈ {U, D} is realized in each period t. The government defaults inperiod T only if sT = DT ; it repays the full amount of B̄ otherwise. When the government

defaults, it repays only a fraction α ∈ (0, 1) of B̄. It follows that the amount of taxes collected
5

Using Matlab the increase in the size of the inflation rate between period -1 and zero is 2e-014 which is

close to the limits of machine precision.

21

in the last period, T (sT ), is given by
T (sT ) = α(sT )B̄,P (sT )

where α(sT ) is defined as(α(sT ) =α,

if sT = DT ,1,otherwise.

In the complete markets specification, the flow budget constraints for each agent are givenbyc0 + k0 +X
s1 ∈S

q(s1 |s0 )c(st ) + k(st ) +X
st+1 ∈Sb0
B̄
b(s1 |s0 )
+ q0
≤ q0
+ y0 ,P0
P0
P0

q(st+1 |st )tb(st+1 |st )t b(s )+q(s)
P (st )P (st )t−1 )b(st |st−1 )
t b(s
+
q(s
)
+ Rk(st−1 ),t = 1, . . . , T − 1, st ∈ S t ,P (st )P (st )b(sT |sT −1 )b(sT −1 )c(sT ) ≤+ α(sT )+ Rk(sT −1 ) + yT − T (sT ),sT ∈ S T ,T
P (s )
P (sT )≤c(st ), k(st ), b(st ) ≥ 0,st ∈ S t , t = 0, . . . , T,

The market clearing condition isZ 1
bh (st ) dh = B̄,
for all st ∈ S t and t = 0, . . . , T − 1,0
Z 1bh (st+1 |st ) dh = 0,
for all st ∈ S t , st+1 ∈ S, and t = 0, . . . , T − 1,0
Z 1(ch0 + k0h ) dh = y0 ,
0
Z 1
Z 1
h t
h t
(c (s ) + k (s )) dh =Rk h (st−1 ) dh,
for all st ∈ S t and t = 0, . . . , T − 1,
0
0
Z 1
Z 1
h T
c (s ) dh = yT +Rk h (sT −1 ) dh,
for all sT ∈ S T .
0

0

22

In the leverage specification, the flow budget constraints are given by
B̄
b0
≤ q0
+ y0 + φ 0 ,P0
P0b(st )c(st ) + k(st ) + q(st )P (st )b(st−1 )≤ q(st )+ Rk(st−1 ) − Rφ(st−1 ) + φ(st ),t = 1, . . . , T − 1, st ∈ S t ,P (st )b(sT −1 )c(sT ) ≤ α(sT )+ Rk(sT −1 ) + yT − T (sT ) − Rφ(sT −1 ),sT ∈ S T ,P (sT )b(st )Rφ(st ) ≤ q(st , st+1 ),t = 0, . . . , T − 2, st ∈ S t , st+1 ∈ S,
P (st , st+1 )b(sT −1 ),sT −1 ∈ S T −1 , sT ∈ S,
Rφ(sT −1 ) ≤ α(sT −1 , sT )P (sT −1 , sT )c0 + k0 + q0c(st ), k(st ), b(st ) ≥ 0,st ∈ S t , t = 0, . . . , T.

The market clearing conditions have a similar form to before.
Suppose that the nominal interest rate in the world is constant and given by 1/q̄. It follows
that the price level evolves as Pt+1 /Pt = 1/(Rq̄). In the following numerical example, we set
1/q̄ = R = 1.02; y0 = yT = B̄ = 1; α = 0.2. Let ρ(Dt ) denote the log yield of the government
debt in state Dt for each period t. That is,ρ(Dt ) ≡ −1
ln q(Dt ) ,T −t

t = 0, . . . , T − 1.

As before, we assume that prior to period 0, everyone believes that the probability of thegovernment defaulting is zero, that is, α(sT ) = 1 for all sT . It follows that the log yield in
period t = −1, ρ−1 , is equal to − ln(q̄).

Table 3 reports results for the complete markets and the leverage specifications for thecase where T = 4. Consider the complete markets specification first. When the news arrives
in period zero the prospect of a future default on the long-term bond produces a jump in
its yield from 1.98 percent to 6.34 percent. As before the marginal buyer h0 = 1/2. Alongthe path towards default h(Dt ) falls in exactly the same way as before, at the rate 1/(t + 2).
Along the path to default optimistic agents from the previous period get wiped out when theD event is realized. They have zero wealth and are inactive in subsequent periods along thispath. The result is that h(Dt ) falls quickly. A lower value of h(Dt ) implies successively lower
prices on government debt or alternatively higher yields.
In the leverage specification there is once again a delayed response. In period zero h0 is
virtually one and the yield on the long term bond does not respond to the news. Along thepath leading to default optimistic individuals experience a gradual tightening in the amount23

that they can borrow as the price of the long-term bond increases. However, optimistic agents
continue to be active and the declines in h(Dt ) are much smaller here as compared to thecomplete markets case. This results in a higher price (lower yield) on the long term bond in
all periods along this path relative to the complete markets benchmark.

6

Discussion

The results we have presented are based on a very stark model that is designed to capture
the essential features of the world we are attempting to model. We believe though that theseresults also apply in more general and complex environments. We now turn to discuss therobustness of our findings to some of our principal assumptions.
The single most important assumption driving our results is the restriction on short salesof government debt. It is well known that some institutional investors take short positions onsovereign debt. However, the data presented in Table 4 indicates that the financial sector asa whole has a very large long position in government debt in both the U.S. and Japan.
Table 4 provides data on ownership of government debt in these two countries. In the U.S.,
financial institutions hold 68 percent of total outstanding privately held debt.6 In Japan theratio is even higher. Almost 84 percent of privately held debt is held by financial institutions.
Another piece of evidence that suggests that financial frictions on short sales of government
debt are significant is the gap on mortgage and government bond rates. If it was cheap toshort sovereign debt this would be a much cheaper way to finance a mortgage. Table 5 reports
spreads of government bond yields and mortgage rates for the U.S. and Japan at alternative
maturities. U.S. spreads range from a high of 2.84 percent to 0.51 percent. In Japan theyrange from 1.22 percent to 0.46 percent. Loan origination costs increase the gap even further.
Finally it is not unusual for governments to introduce a state dependent form of restriction
on short sales that make it more difficult to short government debt in states of nature where
the risk of default is elevated. For instance, in August of 2011 as risk premia on their sovereign
debt shot up governments in Belgium, France, Italy and Spain all imposed bans on short sales.
According to the results in Section 5 these actions are an effective way to lower yields. In ourcurrent model though they have no implications for the probability of default.
A second issue relates to the specific motives of why particular individuals/sectors takelong leveraged positions in government debt when the risk of a crisis increases. The model
relies on heterogeneous beliefs to generate differences in demand for government debt. Our6

Our definition of privately held is based on the depositor. Thus we classify public pension plans and central

bank holdings as part of privately held debt.

24

results would also go through if these differences in demand were produced by alternative
differences in preferences such as differences in risk aversion. Government debt is risky andsafe storage is not. With differences in risk aversion less risk averse individuals will borrowfrom the high risk averse types to purchase more government debt.
In the real world leveraged purchases of government debt are central to financial intermediation. To see why this is the case consider the following strategy that optimistic individuals
undertake in our economy: borrow from pessimistic individuals and use these resources topurchase government debt. In the real world this is a central role of the financial services
sector: it accepts deposits and purchases government debt. Institutional restrictions such as
capital requirements provide specific incentives for banks to behave in this way. If a bank uses
any new deposits it receives to purchase government bonds, this increases tier one capital andthereby helps a bank to meet its capital requirements. Another institutional constraint applies
to public pension plans which are required to take long leveraged positions on governmentdebt.
We have assumed in the numerical examples that q(st ) = 1 for all st or that the nominalinterest rate is zero in all histories. That assumption was made to capture the current situations of Japan and the U.S. However, this assumption is not essential to our results. Our
results would also go through though if 1/q > 1. Setting 1/q > 1 would just scale up the
inflation rate.

7

Conclusion

In this paper we have described the evolution of inflation in an economy that faces a surpriseincrease in the risk of a sovereign debt crisis. A debt crisis is a situation where taxes remain
low and the government budget constraint can only be satisfied by a large increase in the pricelevel which deflates the real value of government liabilities. We have found that if markets
are complete the inflation rate jumps on the date that the risk of a crisis is perceived toincrease and remains high. We also described an alternative market structure with short sale
restrictions. Our principal result is that under this alternative there is no discernible impact
response of the inflation rate to an increase in the risk of default. Inflationary pressure canremain low for many periods and becomes concentrated in the states of nature where thecrisis occurs. In short, the message of our paper is that the fact that the inflation rate is low
today does not mean that the risk of a future crisis is low.
We also considered a setting with long-term government debt and explicit default. Our
results suggest that actions by governments to rule out short selling of government debt are,25

in principal, an effective way to reduce the response of prices. However, in our model thedefault event is not effected by these actions. In our future work we plan to consider settings
where the probability of default is endogenous such as in Bi (2011) or Davig, Leeper andWalker (2011).

8

Appendix

8.1

Equilibrium level of leverage

Here we follow Geanakoplos (2003, 2010) and discuss how the equilibrium level of leverageis determined in our model. Consider the two period model in which a variety of loans aretraded in period zero. The unit of each loan is normalized so that one unit of each loanrequires one unit of collateral (government debt). One unit of type j loan is a promise to pay
j units of goods in period one. All loans are assumed to be non-recourse. It follows that theactual repayment made in period one by an agent who obtains a unit of type j loan repays
in period zero is {j, 1/P (s1 )} for each s1 ∈ S. Let J denote the set of loans traded in period
0, which is assumed to be finite. We assume thatj ∗ ≡ 1/P (D) ∈ J.
Note that the loan of type j ∗ corresponds to the loan considered in the main text. Here weshow that only loans of type j ∗ are traded in equilibrium.
Let ρj,0 be the price of a unit of type j loan in period 0, and aj,0 be the amount of type jloan obtained in period 0. Then the budget constraints for each agent can be expressed as
X
B̄
b0
≤
+ y0 +
ρj,0 aj,0c0 + k0 + q0
P0
P0
j∈J
X
max(aj,0 , 0) ≤ b0
j∈J



X
1
b0
c(s1 ) ≤ Rk0 +− T (s1 ) + y1 −aj,0 min j,
P (s1 )
P (s1 )
j∈J

Define h0 be the identity of the agent who is indifferent between government debt and storage:


q0
1
1
1
=
h0
+ (1 − h0 )P0R
P (U )P (D)Without loss of generality, we assume that j < 1/P (U ) for all j ∈ J. We use the no-arbitrage condition to determine ρj,0 . First, since loans of type j ≤ 1/P (D) are risk-free, wehaveρj,0 =

j,
R

for j ≤
26

1
.
P (D)

Next consider loans of type j ∈ (1/P (D), 1/P (U )). One unit of such a loan yields j in state

U and 1/P (D) in state D. It is thus equivalent to the following portfolio: (i) obtain x unitsof type j ∗ loan and purchase x units of government debt; and (ii) make one unit of type j ∗loan, wherex=1
j − P (D)1
1 .
P (U ) − P (D)It follows thatρj,0 = xq0− ρj ∗ ,0P0

This equation can be rewritten as11
ρj,0 =
h0 j + (1 − h0 )
,
R
P (D)

+ ρj ∗ ,0 .


for j ∈


1
1,.
P (D) P (U )Now let us examine the demand and supply of each type of loans. Let us start with loansof type j ≤ j ∗ = 1/P (D). These loans are risk-free and their rate of return is R. Thus, agentsh ≤ h0 are indifferent across these types of loans. For agents h > h0 , the interest rate on theseloans is less than their expected rate of return on government debt. If agent h obtains one
unit of type j ≤ j ∗ loan and purchase one unit of government debt, she obtains 1/P (s1 ) − j

in state s1 in period one. Thus, the expected rate of return on such a portfolio for her is




1
1
1
1
−
j
+
(1
−
h)
−
j
h P (Uh P (U)P (D)) + (1 − h) P (D) − j
=R
1
1
q0
j
h0 P (U) + (1 − h0 ) P (D) − j
P − R
0

≤R

1
1
∗
h P (U) + (1 − h) P (D) − j

11∗
h0 P (U) + (1 − h0 ) P (D) − j,

for j ≤ j ∗ and h > h0 , where the inequality is strict for j < j ∗ . It follows that agents h > h0

strictly prefers loans of type j ∗ to loans of type j < j ∗ .

Next consider loans of type j > j ∗ . If agent h provide such a loan, the expected rate of
return is
1
ρj



1
hj + (1 − h)
P (D)


=R

1
hj + (1 − h) P (D)1h0 j + (1 − h0 ) P (D)< R,

for h ≤ h0 and j > j ∗ .

That is, agents h ≤ h0 strictly prefers providing loans of type j ∗ to providing loans of typej > j ∗ . On the other hand, if agent h obtains one unit of type j > j ∗ loan and purchase one
unit of government debt, the expected rate of return isno1h P (U
−
j
)
h
=R ,
q0
h0
P0 − ρj27

which is independent of j. Thus, agents h > h0 are indifferent among loans of type j ≥ j ∗ .

The above argument proves that only type j ∗ loans are traded in equilibrium. This

completes the proof.

8.2

Marginal agents in the complete-markets specification

Here we prove that the marginal buyers at history Dt , h(Dt ), is 1/(t + 2) in the completemarkets specification. With complete markets, we can write the present-value budget constraint for agent h at history st as:
T X
X
u=t su |stq(su |st )P (su ) uc(s ) ≤ W h (st ),P (st )

where W h (st ) is the beginning-of–period wealth of agent h in history st :W h (st ) ≡T Xbh (st |st−1 ) bh (st−1 )h t−1
T t P (s )
+
+
Rk
(s
)
+
q(s
|s
)
yT − T (sT ) ,t)
P (st )P (st )
P
(s
T t
s |s

and q(su |st ) is defined asq(su |st ) = q(st+1 |st ) × · · · × q(su |su−1 ).
Note that at each history st , W h (st ) is either 0 or some constant W̄ (st ) > 0, which does notdepend on h.
Let us restrict attention to the histories of the form st = Dt , t = 0, . . . , T − 1. Note that(0,for h > h(Dt ),W h (Dt+1 ) =W̄ (Dt+1 ),for h ≤ h(Dt ),(0,
for h ∈/ [h(Dt ), h(Dt−1 )],W h (Dt , U ) =W̄ (Dt , U ),for h ∈ [h(Dt ), h(Dt−1 )].
Furthermore, we haveh(Dt )W̄ (Dt+1 ) = h(Dt−1 ) − h(Dt ) W̄ (Dt , U )= Rt+1 y0 + R−T +t+1 yT .

It follows thatW̄ (Dt+1 )h(Dt−1 ) − h(Dt )=.
h(Dt )W̄ (Dt , U )28

On the other hand,q(U |Dt )P (Dt , U )W̄ (Dt , U )P (Dt )
q(D|Dt )P (Dt+1 )=W̄ (Dt+1 )P (Dt )W̄ (Dt ) =It follows thatq(U |Dt )P (Dt ,U )h(Dt−1 ) − h(Dt )P (Dt )=.
tt+1
q(D|D )P (D)
h(Dt )
P (Dt )The identity of the marginal agent, h(Dt ), satisfiesh(Dt ) P (D1t ,U )q(U |Dt )P (Dt )=

(1 − h(Dt )) P (D1t+1 )q(D|Dt )P (Dt ),

which can be rewritten asq(U |Dt )P (Dt ,U )h(Dt )
P (Dt )= q(D|Dt )P (Dt+1 ) .
t
1 − h(D )t
P (D )

Thus we obtainh(Dt )
h(Dt−1 ) − h(Dt )=.
1 − h(Dt )h(Dt )For t = 0, this becomesh0
1 − h0=,1 − h0h0which implies h0 = 1/2. As a result, we obtain h(Dt ) = 1/(t + 2). This completes the proof.

References[1] Bassetto, Marco. 2002. “A game-theoretic view of the fiscal theory of the price level.”
Econometrica, 70, 2167-2195.
[2] Bassetto, Marco and Andrew Butters. 2010. “What is the relationship between large
deficits and inflation in industrialized countries?” Economics Perspectives: Federal Reserve Bank of Chicago, 83-98.
[3] Braun, R. Anton and Douglas Joines. 2011. “The implications of a greying Japan forgovernment policy.” Unpublished manuscript.
29

[4] Bi, Huixin. 2011. “Sovereign default risk premia, fiscal limits and fiscal policy.” Unpublished Manuscript.
[5] Cochrane, John H. 2001. “Long term debt and optimal policy in the fiscal theory of theprice level.” Econometrica, 69, 69-116.
[6] Davig, Troy, Eric M. Leeper and Todd B. Walker. 2011. “Inflation and the fiscal limit.”
European Economic Review, 55, 31-47.
[7] Fostel, Ana, and John Geanakoplos. 2008. “Leverage cycles and the anxious economy.”
American Economic Review, 98, 1211-1244.
[8] Galati, Gabriele, Peter Heemeijer and Richhild Moessner. 2011.“How do inflation expectations form? New insights from a high-frequency survey.” Bank for International
Settlements Working Papers, No. 349.
[9] Geanakoplos, John. 2003. “Liquidity, default, and crashes: Endogenous contracts in general equilibrium.” In Advances in Economics and Econometrics: Theory and Applications, Eighth World Conference, Vol. 2, 170-205. Econometric Society Monographs.
[10] Geanakoplos, John. 2010. “The leverage cycle.” In NBER Macroeconomics Annual 2009,ed. Daron Acemoglu, Kenneth Rogoff, and Michael Woodford, 1-65. University of Chicago
Press.
[11] Imrohoroglu, Selahattin and Nao Sudo. 2010. “Productivity and fiscal policy in Japan:
Short term forecasts from the standard growth model.” IMES Discussion Paper Series
10-E-23, Institute for Monetary and Economic Studies, Bank of Japan.
[12] Chan-Lau, Jorge“Anticipating credit events using credit default swaps, with an application to sovereign debt crises.” IMF Working Paper, WP/03/106.
[13] Leeper, Eric M. 1991. “Equilibria under ‘active’ and ‘passive’ monetary policies.” Journalof Monetary Economics, 27, 129-147.
[14] Nieto Parra, Sebastián. 2008. “Who saw sovereign debt crises coming?” OECD Development Centre Working Paper No. 274.
[15] Reinhart, M. Carmen and Kenneth Rogoff. 2010. “From financial crash to debt crisis.”
NBER Working Papers, No. 15795.
[16] Sims, Christopher A. 1994. “A simple model for study of the determination of the pricelevel and the interaction of monetary and fiscal policy.” Economic Theory, 4, 381-399.
30

[17] Woodford, Michael. 1995. “Price level determinacy without control of a monetary aggregate.” Carnegie-Rochester Conference Series on Public Policy, 43, 1-46.

31

Table 1: Inflation rates (%) and marginal buyers in the two-period model
π−1

π0

π(D)

h0

(1) complete markets

-1.96

30.72

47.06

0.5

(2) no borrowing

-1.96

25.57

53.09

0.56

(3) leverage

-1.96

9.46

75.62

0.79

Table 2: Inflation rates (%) and marginal buyers in the three-period model
π−1

π0

π(D)

π(D2 )h0h(D)(1) complete markets

-1.96

17.65

22.55

30.72

0.50

0.33

(2) leverage

-1.96

-1.09

10.86

71.89

0.94

0.75

Table 3: Log yields in the five-period model (%)
ρ−1

ρ0

ρ(D)

ρ(D2 )

ρ(D3 )(1) complete markets

1.98

6.34

14.84

34.68

104.15(2) leverage

1.98

1.98

2.05

5.04

38.70

32

Table 4
Holdings of U.S. Government Debt
End of Calendar year 2010

Total
Government
Individuals and non-financial companies
Domestic Financial Institutions
PrivatePublicCentral Bank
Foreign sector

Amount(trillion $)
14.03
6.17
1.41
5.38
1.82
2.44
1.11
4.44

Fraction(%, net of govt)

12.4
47.2
16.0
21.4
9.8
38.9

Holdings of Japanese Government Debt
End of fiscal year 2008

Total
Government
Individuals and non-financial companies
Domestic Financial Institutions
PrivatePublicCentral Bank
Foreign sector

33

Amount(trillion yen)936.63114.0475.88
687.45354.27268.06
65.12
59.26

Fraction(%, net of govt)

9.2
83.6
43.1
32.6
7.9
7.2

Table 5
Spreads on Mortgage Rates over
Government Debt in U.S. and Japan
Data collected on October 24, 2011*
U.S.
Japan
(Percentage) (Percentage)
1- Year ARM
2.84
0.86
5/1-year ARM
1.92
1.22
15-Year Fixed
0.5125n.a.
20-Year Fixed
n.a.
0.47
30-Year Fixed
1.06
0.46
*Government debt yields andU.S. Mortgage rates are from Bloomberg.
Japan Mortgage rates are from Shinsei Bank.

34

Govt. Debt/GDP

Govt. Debt/GDP

Govt. Debt/GDP

70
60

100
50

50

Percentage

150

Percentage

Percentage

200

40
30
20
10

0

0

Korea

USA

10

8

4

2
0

USA

5.000

35

12

30

10

25

8
6
4

20
15
10

2

5

0

0

Korea

USA

Core Inflation

Germany

Canada

10

6

8

20

5

4
2

3
2

0

1

-2

0

Japan

Percentage

25

Percentage

74

GreeceCore Inflation

Core Inflation

12

6

GreeceShort-Term Interest Rate

14

0.000

Percentage

Germany

CanadaShort-Term Interest Rate

Percentage

Percentage

10.000

4

0

Short-Term Interest Rate
15.000

6

2

Korea

20.000Japan

Percentage

106GreeceLong-term Bond Yields

128Percentage

Percentage

Japan

GermanyLong-term Bond Yields

Long-term Bond Yields
18
16
14
12
10
8
6
4
2
0

CanadaPercentage

Japan

160
140
120
100
80
60
40
20
0

15
10
5
0

Korea

USA

CanadaGermany

GreeceFigure 1: Pairwise comparisons of countries with large and increasing deb-GDP ratios andcountries without.

35

U
h

T1 = 0
U
h

D
1−h

D
1−h

U
h

T2 = THT2 = THT0 = 0T1 = 0D
1−h

T2 = THT2 = TL

Figure 2: The event tree for the case where T = 2

36inflation rate

8070

leveraged
complete

60
50
40
30
20
10
0
-10
-1

0

1

2

3

4

5

Figure 3: The inflation rate for the complete markets and leverage specifications when T = 5.

37