Can cross-border ﬁnancial markets create good collateral ina crisis?∗
Makoto SAITO†, Shiba SUZUKI‡, Tomoaki YAMADA§
May 2010

Abstract
In this paper, we explore whether markets can create endogenously good collateral in a crisis by analyzing a simple model where a country-speciﬁc catastrophic
shock is shared between two countries in the presence of solvency constraints. In
this model, due to severe solvency constraints, realized catastrophic shocks cannot
be covered fully by ex ante arrangements. However, most uninsured shocks can be
ﬁnanced ex post by the collateral asset created endogenously through the simultaneous expansion of ﬁnancial balance sheets in both countries. A nondamaged country
ﬁnances risky loans by issuing relatively safe bonds to a damaged country. Suchsafe bonds in turn serve as high quality collateral on which a damaged country
issues risky bonds (loans) to ﬁnance uncovered losses without violating solvency
constraints. The above bilateral lending works to achieve almost perfect insurance
outcomes.
JEL classiﬁcation: F34, G12, G15.
Keywords: Solvency constraints, Collateral constraints, Dynamic optimal contract, Catastrophic shocks.

∗
The authors would like to acknowledge Toni Braun, Tomoyuki Nakajima, Hisashi Nakamura, Akihisa
Shibata, and seminar participants at Washington, Hitotsubashi, Osaka, and Tokyo universities, Bankof Japan for helpful and encouraging comments. The authors are thankful for a grant-in-aid from the
Ministry of Education and Science, Japan, and the Nomura Foundation.
†
Correspondence to Makoto Saito, Faculty of Economics, Hitotsubashi University, 2-1, Naka, Kunitachi, Tokyo, 186-8601, Japan, E-mail: makoto@econ.hit-u.ac.jp, phone: +81-42–580-8807, fax: +8142-580-8882.
‡
Faculty of Economics, Meisei University.
§
School of Commerce, Meiji University.

1

Introduction

In this paper, we explore whether cross-border ﬁnancial markets can create endogenously
good collateral when it is urgently needed in a crisis. If it is possible, then we examine
how high quality collateral is created from voluntary transactions among private agents.
We ﬁnally explore some policy implications from the theoretical exercise.
We often observe that collateral creation mechanisms are triggered by large-scaleinterventions of governments or central banks when a national economy suﬀers fromcatastrophic events such as natural disasters, political turmoils, economic depressions,and ﬁnancial crises. For example, the government bonds that are issued to ﬁnance reconstruction loans can serve as high quality collateral for agents in serious need of liquidity.
During a ﬁnancial crisis, a central bank extensively lends their owned government bonds
to private agents who are eager to hold high quality collateral. In addition, a central
bank actively swaps safe bonds at hand with the risky bonds in possession of private
agents in order to help their borrowing. To facilitate such operations on a large scale,central banks are required to aggressively expand their own ﬁnancial balance sheets.
In this paper, we step back for the moment from an economic environment where
public institutions play an active role in collateral provision. Instead, we construct a
simple model in which collateral creation emerges endogenously in equilibrium. If wethere ﬁnd some equilibrium phenomena quite similar to those associated with public
collateral provision, such as the lending of safe bonds, the exchange between safe bonds
and risky bonds, and the expansion of ﬁnancial balance sheets, then we ﬁnally inquire
into which kinds of economic functions are carried out by public institutions in terms ofcollateral provision.
Employing a simple two-country setup, we investigate theoretically how countryspeciﬁc catastrophic shocks are shared between countries in the presence of solvency
constraints. In this model, given severe solvency constraints, the solvency of an insurer
country is extremely limited. Accordingly, a realized catastrophic shock cannot be covered fully by ex ante arrangements such as catastrophe insurance. By simulation, we
demonstrate that good collateral assets (relatively safe bonds) are created endogenously
through the simultaneous expansion of ﬁnancial balance sheets of both damaged andnondamaged countries only in the aftermath, and that a damaged country can ﬁnance1uncovered shocks ex post on such collateral assets.
More concretely, a nondamaged country ﬁnances reconstruction loans (risky loans
to a damaged country) by issuing relatively safe bonds to a damaged country. With
relatively stable payoﬀs, the bonds issued by a nondamaged country in turn serve as
high quality collateral on which a damaged country issues risky bonds (loans) to a
nondamaged country for ﬁnancing uncovered losses. That is, the relatively safe bonds
issued by a nondamaged country is exchanged for the risky bonds issued by a damaged
country. In equilibrium, a damaged country as a net debtor can ﬁnance uninsured
losses, while a nondamaged country as a net creditor can smooth consumption. Without
violating solvency constraints, consequently, the above ex post bilateral lending between
countries in the aftermath works to achieve almost perfect insurance outcomes andreduce substantially equity premiums.1
The most essential element that underlies the above mechanism is that because ofbinding solvency constraints, the arbitrage opportunity associated with Lucas trees may
not be exploited completely. Consequently, the constrained competitive equilibrium maysustain the price of risky bonds (Lucas trees) that is slightly high relative to the pricedictated by arbitrage conditions. Given comparably expensive Lucas trees, a damaged
country can issue risky bonds on more favorable terms. Thanks to good collateral assets
issued by a nondamaged country, a damaged country can exploit the above arbitrage
opportunity to some extent by making large-scale short positions in Lucas trees. At thesame time, a nondamaged country can utilize a considerable investment in risky bonds
as an eﬀective instrument to smooth consumption.
The model employed in this paper is a simple two-country exchange economy withcountry-speciﬁc catastrophic shocks, in which markets are complete, but each country
is subject to solvency constraints, following Lustig and Nieuwerburgh (2005), Lustig(2007), and Chien and Lustig (2010). In this setup, contingent claims and Lucas trees
are traded between two countries in terms of either long or short positions. Here, thesolvency constraint is binding once the net position of ﬁnancial assets is non-positive fora certain future state. A country-speciﬁc catastrophic shock may be persistent on labor
1

It does not necessarily imply that solvency constraints never matter in either resource allocations orasset pricing. In Lustig (2007), solvency constraints have large impacts on equity premiums under theassumption that the volatility of idiosyncratic shocks is extremely large and counter-cyclical.

2

endowment in level terms, but not permanent.
The above type of solvency constraints have two potential impacts on a two-country
model embedded with complete markets. First, the solvency of an insurer country may
be limited by this constraint. With extremely severe solvency constraints, accordingly,
realized catastrophic shocks may not be insured fully by ex ante arrangements such as
catastrophe insurance. Accordingly, limited insurance is treated not exogenously, but
endogenously within the current framework.
Second, with solvency constraints satisﬁed, the ﬁnancial portfolio of each country
involves both long positions in one ﬁnancial instrument and short positions in another.
In this regard, the solvency constraint employed in our paper diﬀers substantially fromthe standard borrowing constraint that is often introduced into an incomplete market
setup. In the latter constraint, an agent (country) issues non-contingent bonds, andborrow resources on a part of human capital as collateral.2 Accordingly, there is not any
room for ﬁnancial instruments to be utilized as collateral at all. In the former constraint,on the other hand, collateral assets may be created endogenously through active ﬁnancialtransactions among countries.
One advantage of the two-country setup is that it is possible to make each country’s portfolio problem explicit. As in Lustig (2007) and others, we solve numerically aconstrained competitive equilibrium using a representative agent framework with timevarying Negishi weights. A portfolio problem is always implicit in representative agent
models. Unlike in the continuum agent setup of Lustig (2007), however, the two-country
setup allows us to easily recover optimal portfolio problems from the computed equilibrium asset prices and resource allocations.3 In so doing, we can investigate which
ﬁnancial transactions between countries would make the sharing of catastrophic shocks
quite eﬀective even in the presence of severe solvency constraints.

2

Using the theoretical framework constructed by Bewley (1986), many papers including Telmer(1993), Lucas (1994), and Aiyagari (1994) investigate how transitory idiosyncratic shocks are insured
against among agents in a setup where insurance markets do not exist, and borrowing constraints are
exogenous. With mild borrowing constraints, transitory shocks are insured against almost perfectly
through self-insurance, while a severe borrowing constraint reduces the self-insurance ability among
agents. On the other hand, as shown by Constantinides and Duﬃe (1996) and others, in a case where
idiosyncratic shocks are permanent, self-insurance does not work at all regardless of whether borrowing
constraints are present.
3
Using an extremely advanced technique, Kubler and Schmedders (2003) successfully solve portfolio
problems in the presence of collateral constraints in incomplete markets.

3The literature on international ﬁnance have intensively investigated the possible effects of solvency constraints because enforcement problems are quite serious in international ﬁnancial arrangements. They initially addressed whether sovereign debts are
sustainable in the presence of solvency constraints,4 and later investigated how enforcement constraints inﬂuence borrowing ability as well as risk-sharing capacity.5
One closely related paper is Kehoe and Perri (2002). They apply the framework
proposed by Alvarez and Jermann (2000) to a two-country model. The way solvency
constraints are endogenized diﬀers between Alvarez and Jermann (2000) and Lustig(2007). In the former model, once a country is in default, it is excluded forever frominternational ﬁnancial trading, while in the latter, even if a country defaults, it canremain in international ﬁnancial markets after its collateral is conﬁscated completely.
However, both models are similar to each other in that those ﬁnancial contracts are
made such that eﬀective default (insolvency) never occurs in equilibrium; in the lattermodel, for example, a borrowing country is indiﬀerent between default and repayment
even when solvency constraints are binding.
This paper is organized as follows. Section 2 constructs a two-country model withsolvency constraints, while Section 3 presents the calibration results. Section 4 oﬀers
concluding comments.

2

Model

In this section, we construct a two-country exchange economies with solvency constraints
following the framework proposed by Kehoe and Perri (2002), Lustig (2007), and Chienand Lustig (2010). Our goal in this paper is to analyze how eﬀectively catastrophic
shocks are shared between countries in the presence of solvency constraints. In particular,
our simulation result based on the two-country model demonstrates that the expanded
4
Eaton and Gersovitz (1981) analyze how sovereign debts are sustained by a reputation mechanism,
while Bulow and Rogoﬀ (1989) present a case in which a reputation mechanism does not necessarily
make sovereign bonds sustainable when a borrowing country is allowed to make savings. More recently,
Hellwig and Lorenzoni (2009) demonstrate that sovereign bonds are still sustainable given a generalequilibrium eﬀect of limited commitment on interest rates. Eaton and Fernandez (1995) survey the issue
regarding sovereign bonds.
5
Using a small country setup, Atkeson and Rı́os-Rull (1996) construct a model with exogenous constraints on borrowing from foreign countries, while Caballero and Krishnamurthy (2001) explore thepossible eﬀects of collateral constraints on international borrowing.

4

ﬁnancial balance sheet of a nondamaged country helps a damaged country to ex post
ﬁnance catastrophic losses with solvency constraints satisﬁed. That is, a nondamaged
country ﬁnances loans to a damaged country by issuing ﬁnancial assets, while those assets
issued by a nondamaged country in turn serve as collateral for a damaged country.
In this setup, the labor endowment of each country is subject to country-speciﬁc
catastrophic shocks; country-speciﬁc catastrophic shocks on the level of the labor endowment may be persistent, but not permanent. On the other hand, dividends on Lucas
trees are proportional to the world endowment. In terms of market structures, markets
are complete with respect to country-speciﬁc catastrophic shocks. However, each country is subject to solvency constraints in the sense that net ﬁnancial positions cannot be
negative in every possible future state. That is, either Lucas trees or contingent claims
can be used as collateral in cross-border ﬁnancial transactions; however, nonﬁnancial
assets (human capital) cannot be.

2.1

Setup

Labor endowment and Lucas trees A world economy consists of inﬁnite-horizon
exchange economies of country i (i = 1 or 2) in a discrete time setup. It is assumed thatthe labor endowment is homogeneous within each country, but heterogeneous between
the two countries. Each country receives labor endowments subject to country-speciﬁc
catastrophic shocks. There is a ﬁxed supply of Lucas trees whose dividends are subject
to world common shocks. The quantity of Lucas trees is standardized to one.
A set of states of country-speciﬁc labor endowment is deﬁned as y ∈ Y = {y1 , . . . , ym },while a set of states of dividends on Lucas trees is denoted as z ∈ Z = {z1 , . . . , zn }. A
combination of country-speciﬁc and world common states is expressed by st = (yt , zt ),where st is in S = Y × Z. In addition, st = (y t , z t ) denotes a history from time 0 up to
time t, while sj ≽ st represents a continuation history from st .
Furthermore, we assume that the dividend on Lucas trees d(z) is proportional to
the total labor endowment (e1 (y) + e2 (y)). Therefore, a combination of country-speciﬁcshocks constitutes aggregate states. Hereafter, ei (yt ) denotes the labor endowment ofcountry i, and d(zt ) denotes the dividend on Lucas trees. The transition probability of
the above state variables π(y ′ , z ′ |y, z) evolves according to the following Markov process:5π(z ′ |z) =∑′ ′y ′ ∈Y π(y , z |y, z), ∀z ∈ Z, ∀y ∈ Y. Given the above processes of labor en-

dowment and dividends, optimal policy functions of consumption and portfolios dependon state zt .
Preferences and resource constraints Country i maximizes expected lifetime utility with respect to consumption at state st (ci (st )) as follows:
∞ ∑
∑
[
]
{
}∞
β t π(st |s0 )u ci (st ) , i ∈ {1, 2},
U ( ci (st ) t=0 )(s0 ) =t=0 st ∈S t

where a preference is characterized as utility with constant relative risk aversion, or[]i t )1−γ
u ci (st ) = c (s1−γ, γ denotes the degree of relative risk aversion, and β represents therate of time preference.
The resource constraint of the world economy is given by:e(zt ) = e1 (yt ) + e2 (yt ) + d(zt ).
Hereafter, α denotes the ratio of dividends to total labor endowment:d(zt ).
e(zt )α=Complete markets In this economy with complete markets, both the shares of Lucas
trees and one period contingent claims are traded between the two countries. θi (st )denotes the share of Lucas trees held by country i in time t, while ai (st , s′ ) represents
the holding of claims on one unit of goods at state s′ in the next period given the current
state st . p(z t ) is the price of Lucas trees, and q(st , s′ ) is the price of a contingent claim.
The market clearing conditions hold as follows:θ1 (st ) + θ2 (st ) = 1,
a1 (st , s′ ) + a2 (st , s′ ) = 0, for all s′ ∈ S.

(1)
(2)

The budget constraint of country i in time t is expressed by:ci (st ) + p(z t )θi (st ) +∑q(st , s′ )ai (st , s′ ) ≤ wi (st ),

s′ ∈S

6

(3)

where wi (st ) denotes the initial wealth in time t, which is described below.
Collateral constraints

In this economy, the outstanding liability in short positions

is enforceable up to the value of ﬁnancial assets as collateral. That is, net positions ofﬁnancial assets cannot be negative in every possible one-period ahead state:[ t+1]
p(z ) + d(zt+1 ) θi (st ) ≥ −ai (st , s′ ), ∀s′ ∈ S.

(4)

We call the above enforcement constraint a collateral constraint. This formulationof enforcement constraints implies that when equation (4) is binding, a debtor country
is indiﬀerent between default with conﬁscation and full repayment at maturity.
If a collateral constraint or equation (4) is binding for country i, then the possessedﬁnancial assets are exhausted for repayment (or they are conﬁscated), and the next
period’s wealth (wi (st+1 )) consists of only the labor endowment:wi (st+1 ) = ei (yt+1 ),otherwise, it is equal to:[]wi (st+1 ) = ei (yt+1 ) + p(z t+1 ) + d(zt+1 ) θi (st ) + ai (st , st+1 ).
We make some remarks on the above type of collateral constraints. First, an insurer
country has to back catastrophe insurance payments (−ai (st , s′ )) by his holdings of[]Lucas trees ( p(z t+1 ) + d(zt+1 ) θi (st )). In other words, a country can oﬀer catastrophe
insurance capacity only up to the value of Lucas trees at hand. Second, equation (4) does
not impose any upper limit on short positions in contingent claims. A country can issue
contingent bonds as long as he can repay bond obligations by Lucas trees as collateral.
Third, equation (4) does not exclude any short position in Lucas trees. A country canmake short positions as long as he carries long positions in contingent bonds. Here,
we assume that any short position is settled by cash or netting; that is, any deliveryof Lucas trees is not involved in trading short positions. In this regard, making shortpositions in Lucas trees may be interpreted as issuing risky bonds whose repayment is
proportional to the price of Lucas trees.
7In a two-country setup, the last two aspects of collateral constraints may work together to simultaneously expand the ﬁnancial balance sheets of the two countries inequilibrium. That is, one country may make large long positions in Lucas trees andshort positions in contingent bonds, while the other country may make large-scale reverse positions in Lucas trees and contingent bonds.
Asset pricing in collateral-constrained economy

Once a collateral constraint isbinding at a certain state in time t + 1 (st+1 ), then[][]q(st , st+1 )u′ ci (st ) − βπ(st+1 |st )u′ ci (st+1 ) > 0(5)holds as intertemporal eﬃciency conditions. These inequalities are often called Euler
inequalities.
u′ [ci (st+1 )]The Euler inequality implies that the stochastic discount factor (βπ(st+1 |st ) u′ [ci (st )] ,hereafter, SDF) of a country subject to a collateral constraint at a certain state in timet + 1 becomes irrelevant to the asset pricing behavior in time t. From equation (5), we
have

[
]
[ i t+1 ]′ ci′ (st+1 )uc (s )
βπ(st+1 |st ) ′ i t< βπ(st+1 |st ) ′ i′ t
= q(st , st+1 )u [c (s )]u [c (s )]u′for a constrained country (country i) and a unconstrained country (country i′ ). Thus,
the SDF of a unconstrained country is larger than that of a constrained country inequilibrium.
One thing to be noticed is that the equilibrium price of Lucas trees may not be evaluated precisely according to arbitrage conditions in the presence of collateral constraints.
That is, we may have the following inequality in equilibrium:p(z t ) ̸=∑[]q(st , st+1 ) p(z t+1 ) + d(zt+1 ) .

st+1 ≽st

In particular, when it is impossible to make ﬂexibly short positions in Lucas trees
due to binding collateral constraints, the arbitrage opportunity such asp(z t ) >∑[]q(st , st+1 ) p(z t+1 ) + d(zt+1 )st+1 ≽st8

may not be exploited to a full extent upon the realization of fundamental shocks.
Competitive equilibrium with collateral constraints

In sequential trading, eachcountry maximizes expected lifetime utility subject to budget constraint (3) and collateral constraint (4):max{ci },{θi },{ai }

∞ ∑
∑

[
]
β t π(st |s0 )u ci (st ) ,(6)

t=0 st ∈S ts.t. ci (st ) + p(z t )θi (st ) +∑q(st , s′ )ai (st , s′ ) ≤ wi (st ),

s′ ∈S

[ t+1
]
p(z ) + d(zt+1 ) θi (st ) ≥ −ai (st , s′ ), ∀s′ ∈ S.
A collateral constrained competitive equilibrium is deﬁned as follows. Given theinitial wealth {w01 , w02 }, the trading strategy {ai (st , s′t )}, {ci (st )}, {θi (st )}, the pricing
function {q(st , s′ )} and {p(z t )}, each country maximizes (6) subject to equations (3) and(4), and market clearing conditions (1) and (2) are satisﬁed.

2.2

Construction of a representative agent model with time-varying
Negishi weights

2.2.1

Time 0 cost minimization problem

It is not possible to directly solve the above sequential trading problem because of thepresence of collateral constraints. Following Lustig (2007), we thus construct a representative agent model with time-varying Negishi weights (Negishi, 1960), and computethe constrained competitive equilibrium using stochastic discount factors derivable fromthe representative agent model. For this end, we ﬁrst translate the sequential trading
problem into the time 0 problem. More precisely, we convert the sequential trading utility maximization problem to its dual problem or cost minimization in time 0 together
with a single promise-keeping constraint, and the solvency constraints, both of which
are deﬁned below.
The construction of the time 0 problem greatly simpliﬁes the computation procedure of the constrained equilibrium for the following reasons. First, the value function
represented by a promise-keeping constraint can summarize a history of the realized9

states, and serve as a state variable; consequently, the space of state variables is reduced
substantially. Second, Negishi weights can be computed from the cumulation of the Lagrange multipliers associated with solvency constraints. Because the Lagrange multiplieris positive at a default state and zero otherwise, Negishi weights become time-varying
depending on whether solvency constraints are binding.
We rewrite a collateral constraint (4) as the following solvency constraint. If the net
ﬁnancial asset is zero upon default at state st , then consumption from state st on has
to be ﬁnanced by only the current and future labor endowment. Accordingly, lifetime
consumption is equal to lifetime labor endowment at state st :6∑ ∑( ) ∑ ∑Q(st , sj )ci sj =Q(st , sj )ei (yj ) ,j≥t sj ≽stj≥t sj ≽st()
where q st−1 , st corresponds to a stochastic discount factor between state st−1 and
(
) (
)
(
)
state st , and Q(s0 , st ) = q s0 , s1 · q s1 , s2 · · · q st−1 , st . If collateral constraints are
not binding and the net ﬁnancial asset is still positive, then:∑ ∑( ) ∑ ∑Q(st , sj )ci sj >Q(st , sj )ei (yj ) .

j≥t sj ≽stj≥t sj ≽st

Employing the above solvency constraints, we can translate the sequential trading
problem into the time 0 problem:
max
{ci }

s.t.
∑ ∑

∞ ∑
∑

[
]
β t π(st |s0 )u ci (st ) ,t=0 st ∈S t

∑ ∑

[
]
Q(s0 , st ) ci (st ) − ei (yt ) ≤ w0i ,t≥0 st ∈S t( ) ∑ ∑Q(st , sj )ci sj ≥Q(st , sj )ei (yj ) , ∀st ∈ S t , t ≥ 0.

j≥t sj ≽stj≥t sj ≽st

If a country is in default at a certain state, then the last constraint (solvency constraint)
is binding.
The dual problem to the above time 0 problem, that is, the cost minimization problem

6

See Proposition 3.1 in Lustig (2007).

10

to attain lifetime utility v0i in time 0, is characterized as follows:
C i (s0 ) = inf

Q(s0 , st )ci (st ),

(7)

[
]
β t π(st |s0 )u ci (st ) = υ0i ,

(8)

{ci }

s.t.
∑ ∑

∑ ∑

∞ ∑
∑
t=0 st ∈S tt≥0 st ∈S t( ) ∑ ∑Q(st , sj )ci sj ≥Q(st , sj )ei (yj ) , ∀st ∈ S t , t ≥ 0.

j≥t sj ≽st(9)

j≥t sj ≽st

The second equation is called a promise-keeping constraint in the sense that the
optimal solution allows a consumer to attain at least lifetime utility v0i . As mentioned
before, v0i can summarize the history of realized states, and economize the space of statevariables.
2.2.2

Time-varying Negishi weights

In the above cost minimization problem, the Lagrange multiplier µi0 is assigned to thepromise-keeping condition (8), while the multipliers τ i (st ) are associated with the solvency constraints (9) state by state. The multiplier τ i (st ) may be either zero or positive
depending on whether a solvency constraint is binding. Using these multipliers, werewrite the cost minimization problem (7) as:






∑∞ ∑
t=0

t i t
st ∈S t Q(s0 , s )c (s )







[
[
]]
∑
∑
+µi0 υ0i − t≥0 st ∈S t β t π(st |s0 )u ci (st ).
{c } 
[
[∑
]] 


(
)
∑
∑
∑
∑
∑


t j i
t
i t
t j i
j
 + ∞

j≥t
sj ≽st Q(s , s )e (yj ) −j≥t
sj ≽st Q(s , s )c s
t=0
st ∈S t Q(s0 , s ) τ (s )C i (s0 ) = infiExploiting a technique presented by Marcet and Marimon (1999),7 we deﬁne thecumulative multiplier χi (st ) as χi (st ) ≡ χi (st−1 ) − τ i (st ) given χi−1 = 1,8 and further
rewrite the above cost minimization problem as:

C i (s0 ) = inf

{ci }

 ∑ ∑
∞

t
t=0



s ∈S t[] ∑ ∑Q(s0 , st ) χi (st )ci (st ) − τ i (st ) j≥t sj ≽st Q(st , sj )ei (yj ) [.
[ i t ]]
∑ ∑

i
t
t
i
+µ0 υ0 − t≥0 st ∈S t β π(s |s0 )u c (s )7Marcet and Marimon (1992) used the same technique. Messner and Pavoni (2004) presented somecases in which Marcet and Marimon (1999) may not work.
8
If Lucas trees are equally endowed in time 0, then χ10 = χ20 = 1.

11

The ﬁrst order condition with respect to ci (st ) in the above cost minimization problem is]µi0 ′ [ iQ(s0 , st )
u
c
(s
)
=
.
t
χi (st )β t π(st |s0 )Because the right hand side of the above equation is independent of i, we have
[
]
[
]
µ10
µ20
′ 1
′ 2
u
c
(s
)
=
u
c
(s
)
,
t
t
χ1 (st )χ2 (st )
or

[
]
[
]
ζ 1 (st )u′ c1 (st ) = ζ 2 (st )u′ c2 (st ) ,

(10)

µi

where ζ i (st ) is deﬁned as χi (s0 t ) .
In a representative agent framework (a planner’s problem), Negishi weights are assigned to each lifetime utility, and correspond to the ratio of period marginal utility
between the two agents (countries in our context). Thus, as equation (10) implies,ζ 1 (st ) and ζ 2 (st ) can be used as Negishi weights. As Lustig (2007) demonstrates, thetime 0 planner’s objective (a representative agent model) is formulated as:
max{c1 ,c2 }

∞ ∑
∑

[
[
]
[
]]
β t π(st |s0 ) ζ 1 (st )u c1 (st ) + ζ 2 (st )u c2 (st ) .

t=0 st ∈S t
1−γ

Since a period preference is u(c) = c1−γ ,
1

[ 2

1

c (st )
c1 (st )

]γ

2

t

)
= ζζ 1 (sholds. Given that h(st ) ≡
(st )
i

1
t γ

(s )
ζ 1 (st ) γ + ζ 2 (st ) γ , the consumption of country i is derived as ci (st ) = ζ h(st ) e(zt ). In
1()
i (st ) γ
addition, ω i st ≡ ζ h(scorresponds to the consumption share of each country.
t)Without any solvency constraint, Negishi weights are constant over time. Accordingly, the cross-country consumption share does not change dynamically at all. Withsolvency constraints, however, the consumption share between the two countries may
ﬂuctuate. The Negishi weight ζ i (st ) is constant unless country i is in default at state st ,but otherwise, it is revised upward as a result of positive τ i (st ) (the multiplier associated with a solvency constraint). Therefore, the consumption share of country i increases
when country i is in default at state st . One country subject to a solvency constraint
at state st cannot transfer resources from state st to any state which is realized earlier.
Consequently, the consumption share of the corresponding country at state st becomes12

large relative to the share of a previous state where solvency constraint is not binding.
The country in constraint yields higher consumption growth toward a state in which asolvency constraint is binding.
11Here, g(st+1 ) denotes the growth of h(st ) = ζ 1 (st ) γ +ζ 2 (st ) γ (the sum of nonlinearlytransformed Negishi weights) from state st to state st+1 , orh(st+1 ).
h(st )g(st+1 ) ≡By construction, g(st+1 ) is one or higher. A higher g(st+1 ) implies that either of the twocountries face severer solvency constraints between time st and time st+1 . Lustig (2007)
called g(st+1 ) liquidity shocks.
As demonstrated by Lustig (2007),9 thanks to a complete markets setup, a stochastic
discount factor between state s and state s′ can be deﬁned as a function of the aggregate
endowment and the above liquidity shock, or′π(s |s)(e(z ′ )e(z))−γ

g(s′ |s)γ .

(11)

Without any solvency constraint (g(s′ |s) = 1), a stochastic discount factor reduces( ′ )−γ)
to a standard one or π(s′ |s) e(z
.
e(z)

2.3

Numerical procedures

The construction of a representative agent model with time-varying Negishi weights helps
to substantially simplify the numerical computation procedure. In particular, once therevision rule of Negishi weights is established, it is possible to compute a stochastic
discount factor between state st and state s0 (Q(s0 , st )) by equation (11). Then, we
can pin down the equilibrium path of the consumption share of each country and asset
pricing without solving any individual optimization problem including optimal portfolio
problems.
Thus, the derivation of the revision rule of Negishi weights plays an essential role in
9

In Lustig (2007), when the aggregate endowment declines (that is,(e(z ′ )e(z))−γ

is larger), more con-sumers face solvency constraints as a result of more volatile idiosyncratic shocks (that is, g(s′ |s)γ is
larger). Accordingly, stochastic discount factors tend to correlated heavily negatively with dividends on
Lucas trees in a future recession state; this is a source of a larger risk premium in his model.

13the numerical procedure. While the appendix reviews the numerical method in detail,a key idea is conceptually simple. To begin with, we compute the consumption sharethat satisﬁes a solvency constraint or equation (9) for every one-period ahead state s′
for country i, denoted by ω i (s′ ). If the current consumption share ω i (s) is smaller thanω i (s′ ), then a solvency constraint is regarded as binding at state s′ , and the Negishi
weight for a constrained country is revised upward from state s onto state s′ . Moreconcretely, if ω i (s) < ω i (s′ ) at state s′ for country i, then the Negishi weight of country[]γ
11
i is revised upward as ζ i (s′ ) = ω i (s′ )h(s′ ) , where h(s′ ) ≡ ζ 1 (s′ ) γ + ζ 2 (s′ ) γ .10

2.4

Derivation of portfolio positions

In standard representative agent models, optimal portfolio problems are implicit in solving equilibrium paths, and are often considered as trivial issues. A major reason for thisis that a portfolio problem is reduced to a simple allocation of market portfolios andnon-contingent bonds when any constraint other than resource constraints is absent. As
mentioned in the previous subsection, portfolio problems are also implicit in solving ourplanner’s problem. However, they are potentially important when solvency constraints
are present, because there may emerge complicated ﬁnancial transactions between acountry damaged by catastrophic shocks and a nondamaged country.
Thanks to a two-country setup, it is possible to recover the portfolio positions ofcountry 1 and country 2 as follows. From the budget constraint (3), we obtain thefollowing system of equations to determine portfolio rules together with the market
clearing conditions (1) and (2):c1 (st )+p(z t )θ1 (st )+∑[]q(st , s′ )a1 (st , s′ ) = e1 (yt )+ p(z t ) + d(zt ) θ1 (st−1 )+a1 (st−1 , st ),

s′ ∈S

and
c2 (st )+p(z t )θ2 (st )+∑[]q(st , s′ )a2 (st , s′ ) = e2 (yt )+ p(z t ) + d(zt ) θ2 (st−1 )+a2 (st−1 , st ).

s′ ∈SNote that both consumption and asset prices are standardized by the total endowment.
10

In the numerical procedure described in the appendix, we use as Negishi weights ω i (s) instead ofζ (s) after all variables are standardized by the total world endowment e(zt ).
i

14

It is possible to identify from simulation results which state and which country faces
a solvency constraint. These identiﬁed facts simplify the above system of equations.
When a solvency constraint is binding on country 1 in state s′ in time t, country 1
repays up to:[]−a1 (st−1 , s′ ) = p(z t ) + d(zt ) θ1 (st−1 ).

When a solvency constraint is binding on country 2 in state s′ in time t, country 1 isrepaid by:[]a1 (st−1 , s′ ) = p(z t ) + d(zt ) (1 − θ1 (st−1 )).

After simplifying the system, we approximate portfolio rules by θi (st ) = ν0i + ν i ci (st )and ai (st , s′ ) = α0i + αi ci (st ). Given the simulated series of asset prices and consumption
shares, we identify the values of ν0i , ν i , α0i , and αi that minimize the sum of squared
residuals of the above system for a certain range of ci (st ). In so doing, we classify
current states (time t states) into three possible states, including (1) neither country 1
nor country 2 receives adverse shocks, (2) only country 1 receives shocks, and (3) only
country 2 receives shocks.
We may have a special and convenient case in which as of time t − 1, either country
would be subject to solvency constraints in any possible state of a one-period ahead
period (time t). In this case, binding solvency constraints can identify portfolio positions
precisely, and we can obtain exact positions without using any approximation. Indeed,
the calibration results presented in Section 3 do not require using any approximation.

3

Calibration Exercises

3.1

Setup

This section explores numerically how country-speciﬁc catastrophic shocks are shared
between the two countries, and to what extent solvency constraints matter in sharing
those shocks ex ante and ex post. As described in the previous section, we considertransitory or persistent country-speciﬁc catastrophic shocks on the labor endowment;
we introduce such shocks with the limited number of states z. For a technical reason,15

we exclude any permanent catastrophic shock.11
We determine the size of country-speciﬁc catastrophic shocks following the existing
empirical literature. Using US data for the period between 1869 and 1985, Cecchetti,
Lam, and Mark (1990) identify catastrophic shocks on GDP. In their estimation, total
annual output declines by 15.1% in the catastrophic regime, while it grows by 2.5% in thenormal regime. The normal regime moves to the catastrophic regime with probability of
1.8% per year. Once the economy enters the catastrophic state, the state repeats itself
with probability 51.0%.
On the other hand, Barro (2006) argues that the annual probability of catastrophic
states is around 1.7%, and that the loss amounts to 15% through 64% of total output
through intensively collecting data of developed and developing countries. These papers
ﬁnd that such catastrophic shocks permanently reduce the level of national output. But,
we refer to their ﬁndings in determining the size of transitory or persistent catastrophic
shocks relative to national outputs.
In addition, we set the degree of persistence of catastrophic states at extremely highlevels partly to capture the fact that some types of catastrophic shocks have permanent eﬀects on national output as discussed above, and partly to consider the fact thatcatastrophically damaged countries often experienced eventual recoveries, as discussed
in Gourio (2008).
Following the above ﬁndings, we assume that the labor endowment of a country
declines by 20% or 40% with probability 1.8% per year. Without the realization ofcatastrophic shocks, the labor endowment remains at a given level. When we consider
the case of extremely persistent catastrophic shocks, we assume that a catastrophic state
repeats itself with probability 80%, once a catastrophic state hits one country. That is,1) on average. A catastrophic shock isa catastrophic state continues for ﬁve years ( 1−0.8

assumed to be country-speciﬁc and uncorrelated between the two countries.
We treat cases where solvency constraints are severely binding by making the ratio ofdividends to the world labor endowment (α) rather low. In time 0, both labor endowment
and Lucas trees are equally distributed between the two countries. The rate of time11

It is possible to introduce permanent shocks with the small number of states for the endowment
process of each country, but the number of states that is required to characterize total world outputs orendowment shares between the two countries becomes extremely large.

16

preference is 5% (β = 0.95), and the degree of relative risk aversion is ﬁve (γ = 5).
When calibration results are reported below, all variables except for portfolio positions such as ai (s, s′ ) and θi (s) are standardized by the total world endowment e(zt ).
Thus, what is implied by ‘share’ is the ratio relative to the total endowment.

3.2
3.2.1

Calibration results
Purely transitory case

Almost perfect insurance outcomesWe ﬁrst consider a case with purely transi-

tory catastrophic shocks to understand how risk-sharing works between two countries
in the presence of collateral (solvency) constraints. More concretely, a country-speciﬁc
catastrophic shock reduces labor endowment by 20% with probability 1.8% per year,
but without any persistence. That is, the labor endowment share of a damaged country
1−0.2(1 − α) upon the realization of catastrophic shocks. Whendeclines from 0.5 to 1+(1−0.2)
1−0.2
α is close to zero, the labor endowment share declines by about 5.6% (= 0.5 − 1+(1−0.2))

due to a catastrophic event.
One of the most important observations about this case is that catastrophic shocks
are insured almost perfectly between the two countries. With extremely low α (= 0.1%),
Figure 1 plots the consumption share between a nondamaged country (country 1) anda damaged country (country 2); a catastrophic state takes place only in time 0. As
demonstrated by Figure 1, the consumption share of the damaged country declines only
by about 0.2% in time 0, although his labor endowment share declines by 5.6%. It
implies that the damaged country can ﬁnance 5.4% out of 5.6% losses immediately after
a catastrophic event. Even in a long term, the damaged country suﬀers from only 0.1%
permanent losses unless another catastrophic shock hits this country.
However, almost perfect insurance outcomes do not necessarily imply that collateral
constraints do not matter in the above case. Due to extremely low α (= 0.1%), thesolvency of country 1 as an insurer is indeed crucially limited as follows. Note that
all asset prices are standardized by the total endowment. As equation (4) implies,
country 1 (insurer) can oﬀer insurance payments to country 2 (insured) only up to(p(s0 ) + α)θ1 (s−1 ) where θ1 (s−1 ) = 0.5 in time 0. In equilibrium, (p(s0 ) + α) × 0.5
amounts to only 0.7% of the total world endowment.
17

Accordingly, the catastrophe insurance payment from country 1 to country 2 (0.7%)
is far short of the catastrophic loss borne by country 2 (5.6%). In other words, exante insurance arrangements between the two countries are not responsible for almost
perfect insurance outcomes. It thus follows that in the constrained competitive equilibrium, most of country 2’s uncovered losses are ﬁnanced from country 1 through ex postarrangements.
In which condition can the damaged country ﬁnance?

Let us ﬁrst explore inwhich condition the damaged country can ﬁnance a positive amount of resources fromthe nondamaged country in time 0 without violating solvency constraints at any oneperiod ahead state. As shown below, the most essential condition is that the price ofLucas trees is slightly expensive relative to the price dictated by arbitrage conditions
upon the realization of a catastrophic shock in time 0:p(z 0 ) =∑[]q(s0 , s1 ) p(z 1 ) + d(z1 ) + ϵ,(12)s1 ≽s0where ϵ is positive.
As mentioned in Section 2.1, the above arbitrage opportunity may not be exploitedcompletely in the presence of solvency constraints. More concretely, the damaged country cannot make enough short positions in Lucas trees due to binding solvency constraints. On the other hand, the nondamaged country that intends to spread relative
gains over time is not interested in making short positions in Lucas trees at all. Accordingly, the above arbitrage condition (12) may not be exploited to a full extent as of time0.
How much the damaged country (country 2) can ﬁnance by making simultaneously
short positions in Lucas trees (θ2 (s0 ) < 0) and long positions in relatively safe bonds(a2 (s0 , s1 ) > 0) is equal to:−p(z 0 )θ2 (s0 ) −∑s1 ≽s018

q(s0 , s1 )a2 (s0 , s1 ).

Substituting equation (12) to the above equation, we obtain:−p(z 0 )θ2 (s0 )−∑q(s0 , s1 )a2 (s0 , s1 ) = −ϵθ2 (s0 )−s1 ≽s0∑[[]]q(s0 , s1 ) a2 (s0 , s1 ) + p(z 1 ) + d(z1 ) θ2 (s0 ) .

s1 ≽s0

(13)
[

[

]

]

As long as solvency constraints are satisﬁed as of time 0, a2 (s0 , s1 ) + p(z 1 ) + d(z1 ) θ2 (s0 )is non-negative for any state in time 1 (s1 ). More concretely, as described in footnote
[

[

]

]

13, a2 (s0 , s1 ) + p(z 1 ) + d(z1 ) θ2 (s0 ) is equal to zero for three cases (out of four) wheresolvency constraints are binding. Thus, the second term of the right hand side of equation (13) is negative, but quite close to zero. If the time-0 price of Lucas trees (p(z 0 )) isslightly high relative to arbitrage pricing,12 then there is still a chance for the left handside of equation (13) to be positive.
What we below demonstrate by simulation is the cases where the equilibrium priceof Lucas trees is slightly expensive relative to arbitrage pricing; then, the damaged
country eﬀectively ﬁnances resources from the nondamaged country. In those cases, thedamaged country utilizes the arbitrage opportunity (comparably expensive Lucas trees)
to some extent by building up short positions in Lucas trees until solvency constraints
are binding.
Expansion of ﬁnancial balance sheets in two countries Table 1 reports ﬁnancial transactions between the two countries when a catastrophe hits country 2 in time
0 without any further catastrophe in either country in the case of α = 0.1%. As mentioned above, in the aftermath (time 0), the damaged country (country 2) receives only
0.7% of the total endowment through insurance payments from the nondamaged country(country 1).
There are two means by which country 2 can cover uninsured losses in time 0. First,
the damaged country self-ﬁnances losses by savings in Lucas trees; the gross return frompreviously invested Lucas trees ((p(s0 ) + α)θ2 (s−1 ) where θ2 (s−1 ) = 0.5) amounts to
0.7% of the total endowment in time 0.
Second, more importantly, the damaged country ﬁnances uncovered losses by ag121
If ϵ is greater than − θ2 (s
0)

∑

0 1
s1 ≽s0 q(s , s )[ 2 0 1[]]a (s , s ) + p(z 1 ) + d(z1 ) θ2 (s0 ) , then the left

hand side of equation (13) turns out to be positive.

19

gressively expanding his ﬁnancial balance sheet. With collateral constraints satisﬁed forevery possible future state, country 2 can ﬁnance 3.9% of the total endowment by building intensively long positions in contingent-claim portfolios (3.6%) and short positions
in Lucas trees (−7.5%) in time 0. Together with insurance receipts (0.7%) and savingsin Lucas trees (0.7%), the damaged country can mitigate endowment losses substantiallyfrom 5.6% to 0.2% by such ex post ﬁnancing from country 1. Note that because any
delivery of Lucas trees is not required in this economy, making short positions in Lucas
trees can be regarded as issuing risky bonds whose repayments are proportional to theprice of Lucas trees.
On the opposite side, the nondamaged country also expands aggressively his ﬁnancialbalance sheet in time 0. That is, country 1 builds large long positions in Lucas trees(8.8%) and short positions in contingent-claim portfolios (−3.6%), while he satisﬁes
collateral constraints for every possible future state.13
Relatively safe assets as collateral

As shown above, the contingent contract issued

by the nondamaged country (country 1) to the damaged country (country 2) serves as
eﬀective collateral for country 2 to ﬁnance uninsured losses from country 1. Let us takea close look at the payoﬀ structure of the one-period contingent contract as collateral
assets. As shown in Table 2, the contingent contract issued by country 1 has a time-1
claim on 11.8% of the total world endowment in a state without any catastrophic event(Case 1), 9.5% in a case where a catastrophic shock hits on country 2 (Case 2), 8.0% ina case where a catastrophic shock hits on country 1 (Case 3), and 5.1% in a case where
catastrophic shocks hit on both countries (Case 4).
This type of contingent contract cannot be regarded as catastrophe insurance because
country 1 is obliged to make positive payments in any state of the next period (time13

For example, the portfolios that are constructed by both countries in time 0 can satisfy a collateral
constraint in each state of time 1 as follows. As to country 2, the time-1 receipt from long positions
in contingent bonds (a2 (s0 , s1 )) is 11.8% of the total world endowment in a case where no catastrophic
shock is realized in either country, 9.5% in a case where a catastrophic shock hits on country 2, 8.0% in acase where a catastrophic shock hits on country 1, 5.1% in a case where catastrophic shocks hit on bothcountries. On the other hand, the time-1 repayment on short positions in Lucas trees (−(p(s1 )+α)θ2 (s0 ))is 11.8%, 8.0%, 8.0%, and 5.1% respectively. Then, for country 2, the receipt dominates the repayment
in any state of time 1. Similarly, as to country 1, the time-1 receipt from long positions in Lucas trees(p(s1 + α)θ1 (s0 )) is 13.9%, 9.5%, 9.5%, and 6.1% respectively. On the other hand, the time-1 repayment
on short positions in contingent bonds (-a1 (s0 , s1 )) is 11.8%, 9.5%, 8.0%, and 5.1% respectively. Again,
for country 1, the receipt dominates the repayment at any state of time 1.

201). In particular, country 1 is forced to make positive payments even if he faces acatastrophic shock in time 1 (Cases 3 and 4). Thus, it may be instead considered as
relatively safe bonds in the sense that it has larger claims than Lucas trees in Case
2. With such relatively stable payoﬀs, country 2 can ﬁnance enough resources against
suﬃcient receipts from the contingent bonds at any future state. Consequently, the bondissued by country 1 can serve as high quality collateral on behalf of country 2 in theaftermath.
Closing expanded ﬁnancial balance sheets

As reported in Table 1, from time

1 onward, short positions in both Lucas trees disappear completely in terms of international capital ﬂows. In other words, the creation of relatively safe bonds as highquality collateral emerges only immediately after a catastrophic event. The net value∑of contingent-claim portfolios ( q(s, s′ )ai (s, s′ )) is close to zero in both countries fromtime 1 on. In addition, as shown in Table 2, contingent contracts between two countries
turn back to conventional insurance contracts with limited coverage. That is, a damaged
country receives insurance beneﬁts, though insuﬃcient, from a nondamaged country.
On the other hand, the number of invested Lucas trees (θi (st )) is between 0 and 1 forboth countries during the corresponding period. However, the allocation of Lucas trees
tilts heavily to the nondamaged country (country 1); the share of Lucas trees owned
by country 1 increases from 50% in time −1 to more than 90% from time 1 on. As aconsequence of the above ﬁnancial transactions, the damaged country is subject to only
0.1% permanent losses.
Note that even the nondamaged country beneﬁts from the above cross-border transactions. By standing on the opposite side of the damaged country, the nondamaged
country can smooth consumption by spreading a temporary relative gain (5.6% of thetotal endowment) in time 0 over time. Alternatively stating, thanks to the above transactions involved with uneven allocation of Lucas trees, the nondamaged country is able
to enjoy 0.1% permanent increases in consumption shares.14

14

The nondamaged country can enjoy permanently beneﬁts at an annuity rate equal to 1.8% (≈Since the implied annuity rate is lower than the average equilibrium risk-free rate (5.058%), theconsumption smoothing is not achieved to a full extent from the viewpoint of the nondamaged country.
0.1%
).
5.6%

21

Summary We again ask how the two countries with expanded ﬁnancial balance sheets
can achieve almost perfect insurance outcomes without violating severe collateral constraints. On the one hand, the nondamaged country ﬁnances loans to the damaged
country by issuing relatively safe bonds (contingent bonds) to the damaged country.
On the other hand, with relatively stable payoﬀs, the bonds issued by the nondamaged
country in turn serve as high quality collateral on which the damaged country can issue
risky bonds (short positions in Lucas trees) to ﬁnance uncovered losses.
The most essential factor that underlies the above ﬁnancial risk-sharing mechanismis that because of binding solvency constraints, the equilibrium price of risky bonds(Lucas trees) can be slightly expensive relative to arbitrage pricing. Thus, the damaged
country is able to utilize such arbitrage opportunities to some extent, thereby successfully
ﬁnancing positive amounts of resources even in the aftermath. In this way, the expansionof ﬁnancial balance sheets of both countries helps the damaged country to ﬁnance expost most of uncovered losses in spite of the presence of severe collateral constraints.
At the same time, such cross-border transactions assist the nondamaged country tosmooth consumption by spreading temporary relative gains over time. Note that inboth countries, the expansion of ﬁnancial balance sheets is cleared immediately in time0.
Because such ﬁnancial transactions between the two countries work to achieve almost
perfect insurance outcomes under severe collateral constraints, the asset pricing behavior of the collateral-constrained economy is quite similar to that of the full insurance
economy. Table 3 reports the average equity prices and equity premiums for various
dividend-to-endowment ratios (α); all pricing variables represent the ratio relative tothe total world endowment. For example, when α is 0.1%, the average equity premium(0.994%) is much closer to the perfect insurance premium that emerges when collateral
constraints are absent (0.937%) than to the closed economy premium that emerges whencross-border risk-sharing is absent (4.981%). When α is greater than 0.5%, collateral
constraints are not binding any longer, and the economy reduces to the full insurance
equilibrium.15
15

When α is 0.5%, the equity valuation ((p(s0 ) + 0.005) × 1) is equal to 6.3% of the total world
endowment (see Table 3), and it can cover completely country 2’s losses in labor endowment shares(5.6%). Accordingly, a solvency constraint is no longer binding in ex ante arrangements.

223.2.2Extremely large and persistent caseOne-time occurrence of a catastrophic event

We next move to a case with ex-tremely large and persistent catastrophic shocks. That is, a country-speciﬁc catastrophic
state reduces labor endowment by 40% with probability 1.8% per year, while it repeats
itself with probability 80%. When α is close to zero, the labor endowment share of
1−0.4
the damaged country declines by 12.5% from 50% to 37.5% (= 1+(1−0.4)) upon theoccurrence of a catastrophic event.
As demonstrated below, such huge losses in labor endowment cannot be covered by
the insurance transfer from the nondamaged country to the damaged country under severe collateral constraints (extremely small α). Nevertheless, the same ex post ﬁnancing
mechanism as in the purely transitory case works eﬀectively to achieve almost perfect
insurance outcomes even in this case. Table 4 reports ﬁnancial transactions between thetwo countries when a catastrophe hits country 2 in time 0 without any further catastrophe in either country in the case of α = 0.2%. Figure 2 plots the time series of theconsumption share in the corresponding case.
In the aftermath (time 0), the damaged country receives only 4.6% of the total
endowment as insurance payments from the nondamaged country.16 However, the damaged country ﬁnances 1.5% by expanding his ﬁnancial balance sheet; country 2 builds
intensively long positions in contingent-claim portfolios (46.5%) and short positions in
Lucas trees (−48.0%) in time 0. Together with the insurance payment (4.6%) and thegross return from previously invested Lucas trees (4.6%), the damaged country mitigates current endowment losses substantially from 12.5% to 1.9%. At the same time,
the nondamaged country expands his ﬁnancial balance sheet; country 1 builds large net
ﬁnancial positions (10.4%) by having long positions in Lucas trees (56.9%) and shortpositions in contingent-claim portfolios (−46.5%).
Again, let us take a close look at the payoﬀ structure of the one-period contingent
bonds as collateral assets. As shown in Table 5, the contingent bonds issued by country
1 have time-1 claims on 86.5% of the total world endowment in a state without any
catastrophic event (Case 1), 54.2% in a case where a catastrophic shock hits on country

16

The equity valuation is much larger in the case with persistent shocks than in the case with purelytransitory shocks as a result of stronger precautionary demand for Lucas trees.

232 (Case 2), 45.7% in a case where a catastrophic shock hits on country 1 (Case 3), and21.3% in a case where catastrophic shocks hit on both countries (Case 4). As in theprevious case, country 1 is obliged to make positive payments in any state of the next
period, though the payoﬀ of the contingent contract carries a little less stable proportion
to the total endowment than in the previous case. Thus, it may be still regarded as
relatively safe bonds in the sense that it has larger claims than Lucas trees in Case 2.
Also in this case, such relatively safe bonds serve as high quality collateral for country
2 to ﬁnance uncovered losses in the aftermath.
As in the purely transitory case, from time 1 onward, short positions in both Lucas
trees disappear completely in terms of international capital ﬂows. Again, the creation ofrelatively safe bonds as high quality collateral emerges only immediately after a catastrophic event. More precisely, the net value of contingent-claim portfolios is close to zeroin both countries from time 1 on. In addition, contingent contracts between two countries become standard insurance contracts with limited coverage. On the other hand,the number of invested Lucas trees (θi (st )) is between 0 and 1 for both countries during
the corresponding period. However, the allocation of Lucas trees tilts heavily to thenondamaged country (country 1); the share of Lucas tress owned by country 1 increases
from 50% in time −1 to around 90% from time 1 on. Consequently, the damaged country
is subject to only 1.4% permanent losses.
By standing on the opposite side of the damaged country, the nondamaged country can smooth consumption by spreading relative gains over time. More concretely,
the nondamaged country enjoys 1.4% increases in consumption shares forever.17 Consequently, such intensive cross-border transactions help to restore almost perfect insurance
outcomes without violating the collateral constraints imposed on either country.
As reported in Table 6, the average equity premium is quite close to the perfect
insurance premium because almost full insurance is achieved through the above ex post
17
When catastrophic shocks are persistent, the relative gain acquired by the nondamaged country
reﬂects not only an increase in current labor endowment shares (12.5%), but also a substantial improvement in lifetime income shares. When α = 0.2%, the average equity premium is 7.657% per year. If therisk premium on labor endowment is equal[ to the equity premium, then ]the expected gain in lifetime

income shares approximately equals

∑∞

t=0

t

(0.8 (1 + 0.07657)) × 12.5% = 90.1% . Accordingly, the

1.4%
nondamaged country receives permanently beneﬁts at an annuity rate equal to 1.6% (≈ 90.1%). Unlikein the purely transitory case, the implied annuity rate is higher than the average equilibrium risk-freerate (−3.549%).

24

ﬁnancial transactions between the two countries. For example, with α = 0.2%, the average equity premium (7.657%) is much closer to the perfect insurance premium (7.397%)
than to the closed economy premium (25.076%). When α = 7.4%, the aftermath equity
valuation (82.366%) is large enough to cover the required insurance transfer between
the two countries,18 and the average equity premium is accordingly exactly equal to theperfect insurance premium.
Consecutive and alternate occurrence of catastrophic events To demonstratethat the above risk-sharing mechanism is fairly powerful, we present two more examples
with extremely large and persistent shocks; one case where a catastrophic state hits on
1
country 2 consecutively for the expected length of time (ﬁve periods, 1−0.8) (Table 7

and Figure 3), and the other case where a catastrophic shock in turn hits country 1 intime 1 (Table 8 and Figure 4).
In the former case, the maximum loss of the consumption share of the damaged
country is at most 3.2% in time 5 (much lower than 12.5%), while the permanent loss
falls to 1.5%. As Table 7 illustrates, the large long (short) position in Lucas trees heldby the nondamaged (damaged) country facilitates a large-scale transfer between thetwo countries, as long as catastrophic shocks hit on country 2; the damaged country
ﬁnances uncovered losses by building up large-scale long positions in contingent bonds
and short positions in Lucas trees; θ2 (st ) is negative from time 0 to time 4. In otherwords, relatively safe bonds are again created as high quality collateral endogenously
while catastrophic events take place consecutively.
In the latter case, on the other hand, the newly damaged country (country 1) suﬀers
from 1.1% permanent loss in its consumption share. There, country 1 instead ﬁnances
uncovered losses by borrowing resources from country 2 on the relatively safe bonds
issued by country 1 in time 1. On the opposite side, the previously damaged country(country 2) ﬁnances reconstruction loans to country 1 by issuing relatively safe bonds intime 1, and enjoys a long-run beneﬁt from consumption smoothing from time 1 onward.
18
When catastrophic shocks are persistent, the damaged country has to ﬁnance not only current losses,
but also a substantial decline in lifetime income shares. As described in footnote 17, the expected loss
in lifetime income shares approximately equals around 90%. Hence, the equity valuation (82.4% with
α = 0.5%) needs to be comparable to that magnitude to make solvency constraints free in ex antearrangements.

25

Also in this case, contingent contracts between two countries revert to usual insurance
contracts from time 2 on.

4

Conclusion

In this paper, we have examined how catastrophic shocks, possibly persistent, are shared
between two countries in the presence of solvency constraints. Given severe collateral
constraints, the solvency of an insurer country is extremely limited, and realized catastrophic shocks cannot be covered fully by ex ante arrangements such as catastrophe
insurance. However, the damaged country can ﬁnance ex post most uninsured shocks
by intensive cross-border ﬁnancial transactions without violating severe collateral constraints. At the same time, the nondamaged country can smooth consumption by spreading a temporary gain over time.
A key element is that high quality collateral is created endogenously through thesimultaneous expansion of ﬁnancial balance sheets of both countries. More concretely,
the nondamaged country ﬁnances risky loans (long positions in Lucas trees without any
delivery) by issuing relatively safe bonds (short positions in contingent-claim portfolios)
to the damaged country. With relatively stable payoﬀs, the bonds issued by the nondamaged country in turn serve as good collateral assets on which the damaged country
can issue risky bonds (short positions in Lucas trees without any delivery). Thanks tothe above bilateral transactions, the damaged country can ﬁnance eﬀectively uncovered
losses from the nondamaged country.
The most essential factor that underlies the above ﬁnancial risk-sharing mechanism isthat because of binding solvency constraints, the equilibrium price of risky bonds (Lucas
trees) can be slightly expensive relative to arbitrage pricing upon the realization of catastrophic shocks. Thus, the damaged country is able to utilize such arbitrage opportunities
to some extent, thereby successfully ﬁnancing positive amounts of resources even in theaftermath. In this way, the simultaneous expansion of ﬁnancial balance sheets in bothcountries would emerge as a powerful risk-sharing scheme for catastrophic shocks. Suchintensive cross-border transactions works eﬀectively to achieve almost perfect insurance
outcomes and reduce substantially equity premiums. Surprisingly, almost perfect risksharing still emerges even if one country faces severe catastrophic shocks in a consecutive
26

manner. In any case, the expansion of ﬁnancial balance sheets is closed immediately after
catastrophic events go away.
Coming back to the policy question raised in the introduction, we ﬁnally inquire into
which kinds of economic functions are carried out by public institutions, in particular
central banks, in terms of collateral provision during ﬁnancial crises. We may broadlyinterpret that in actual ﬁnancial crises, a central bank assumes the role that is playedby the nondamaged country in our model. A central bank generously lends his owned
safe assets such as government bonds to private ﬁnancial institutions in serious need ofliquidity, while with fairly positive evaluation of risky bonds such as low-grade corporate
bonds, he is willing to swap safe bonds at hand for risky bonds in possession of private
agents. To facilitate such operations on a large scale, a central bank is required to expand
his own ﬁnancial balance sheets. Thus, such operations conducted by central banks may
be broadly justiﬁable as long as these operations are closed immediately after ﬁnancial
crises go away.

Appendix: The numerical computation methods
As mentioned in Section 2, it is not possible to directly solve the sequential trading
problem characterized by equation (6) because of the presence of solvency constraints.
Following Lustig (2007), we instead solve the time 0 cost minimization problem dual tothe utility maximization problem. We omit the time subscript t because the problem isformulated in a recursive manner.
We below standardize all endogenous variables except for asset volume by the total
world endowment.19 Accordingly, we transform the stochastic discount factors as follows:( )π̂ s′ |s = ∑π(s′ |s)s′ ∈Sβ̂ (s) = β∑(e(z ′ )e(z)π(s′ |s)′π(s |s)

s′ ∈S

19

(

(

)1−γ

e(z ′ )e(z)e(z ′ )e(z))1−γ ,

)1−γ

.

(14)

(15)

Alvarez and Jermann (2000) adopted the same transformation to make endogenous variables stationary in the case where the total endowment is growing.

27

1
( )
i (st ) γ
We can use as a state variable the stationary consumption share (ω i st = ζ h(s∈t)[0, 1]) instead of the Negishi weight ζ i (st ). As the Negishi weight is revised upward upondefault, the consumption share is revised upward based on a cutoﬀ rule as described
below.
There are two steps in ﬁnding the equilibrium pricing and allocation. Given theinitially guessed liquidity shocks g guess (s′ |s), the ﬁrst step consists of solving the cost
minimization problem given the sequence of prices, and of deriving optimal policy functions. In the second step, the sequence of consumption and asset pricing is computedfrom the simulation based on the derived policy functions; it is possible to map from liq( ′ )−γ)
uidity shocks g(s′ |s) to stochastic discount factors β e(zg(s′ |s)γ , and to computee(z)equilibrium asset pricing. We repeat this two-step procedure until the initially guessedliquidity shocks g guess (s′ |s) coincide with the newly generated liquidity shocks g new (s′ |s).
In solving the cost minimization problem, the current history is replaced by a truncated history z k . Here, the control variable is not current consumption, but a consumption share ω i in a detrended version of the cost function (7), and it is rewritten in arecursive manner:
[
i

k

i

Ĉ(ω (s), s, z ) = min ω + β̂ (s)
ωi

∑

]
′

′

γ

i′

′

k′

π̂(s |s)g(s |s) Ĉ(ω , s , z ) ,

(16)

s′ ∈S

where π̂(s′ |s) and β̂ (s) are deﬁned in equations (14) and (15). Note that π̂(s′ |s)g(s′ |s)γin the cost function corresponds to a stochastic discount factor or a pricing kernel; as a′)result of detrending, ( e(ze(z) ) is always equal to one.

Similarly, a detrended version of the present value of the endowment sequence iswritten as follows:Ĉ e (s, z k ) = êi (s) + β̂ (s)∑′π̂(s′ |s)g(s′ |s)γ Ĉ e (s′ , z h ),s′ ∈Swhere êi (s) is the share of individual labor endowment to the aggregate endowment. Because ω i is bounded from below upon default, the lower bound of ω i or ω(s) is determinedby:Ĉ(ω(s), s, z k ) = Ĉ e (s, z k ).

28

Lustig (2007) ﬁnds that ω i (s′ ) is bounded from ω(s′ ) as a result of binding solvency
constraints, and constructs the following cutoﬀ rule to revise a state variable ω i upward:
i

′

ω (s)
ω(s )
′
i ′
i
that is, if ω i (s) > ω(s′ ), then ω i (s′ ) = g(s′ |s) , and if ω (s) ≤ ω(s ), then ω (s ) = g(s′ |s) .

Given the exogenous endowment process (s, z k ), the social planner solves the above cost
minimization problem together with the cutoﬀ rule by adjusting the current consumption
share ω i (s) and the share allowed in the next period ω i (s′ ).
Because equation (16) is a standard dynamic programming problem, we can solve it
by a policy function iteration procedure. For this purpose, the cost function is approximated by a cubic spline interpolation with 100 grids for state ω i ∈ [0, 1]. In addition, it
is assumed that k = 3 for the history parameter. It is possible to derive a policy function
ω ′ = f (ω, s, z k ) from the computed cost function. It is also possible to obtain the shareof consumption of the two countries from the sequence of promised consumption shares
ωi.
In simulation, we ﬁrst generate the sequence of aggregate and idiosyncratic shocks
31,000for 31,000 periods while omitting the initial 1000 periods. Given this gener{st }t=1ated sequence, we derive the sequence of consumption shares from the computed policy
function, and then compute asset pricing and liquidity shocks. As mentioned above, we
repeat this procedure until the generated liquidity shocks converge.

References[1] Aiyagari, S. R., 1994, “Uninsured Idiosyncratic Risk and Aggregate Saving,” Quarterly Journal of Economics 109, 659–684.
[2] Alvarez, F. and U. F. Jermann, 2000, “Eﬃciency, Equilibrium, and Asset Pricing
with Risk of Default,” Econometrica 68, 775–797.
[3] Atkeson, A. and J. V. Rios-Rull, 1996, “The Balance of Payments and Borrowing
Constraints: An Alternative View of the Mexican Crisis,” Journal of International
Economics 41, 331–349.
[4] Barro, R. J., 2006, “Rare Disasters and Asset Markets in the Twentieth Century,”
Quarterly Journal of Economics 121, 823–866.

29

[5] Bewley, T. F., 1986, “Stationary Monetary Equilibrium with a Continuum of Independently Fluctuating Consumers,” in W. Hildenbrand and A. Mas-Colell, eds.
Contributions to Mathematical Economics in Honor of Gerard Debreu, Amsterdam:
North-Holland.
[6] Bulow, J. and K. Rogoﬀ, 1989, “Sovereign Debt: Is to Forgive to Forget?” American
Economic Review 79, 43–50.
[7] Caballero, R. and J. A. Krishnamurthy, 2001, “International and Domestic Collateral Constraints in A Model of Emerging Market Crisis,” Journal of Monetary
Economics 48, 513–548.
[8] Cecchetti, S. G., P. Lam, and N. C. Mark, 1990, “Mean Reversion in Equilibrium
Asset Prices,” American Economic Review 80, 398–418.
[9] Cecchetti, S. G., P. Lam, and N. C. Mark, 1993, “The Equity Premium and the Risk
Free Rate: Matching the Moments,” Journal of Monetary Economics 31, 21–45.
[10] Chien, Y. and H. Lustig, 2010, “The Market Price of Aggregate Risk and the WealthDistribution,” Review of Financial Studies 23, 1596–1650.
[11] Constantinides, G. M. and D. Duﬃe, 1996, “Asset Pricing with Heterogeneous
Consumers,” Journal of Political Economy 104, 219–240.
[12] Eaton, J. and E. Fernandez, 1995, “Sovereign Debt,” in G. M. Grossman and K. Rogoﬀ, eds. Handbook of International Economics 3, 2031–2077, Amsterdam: NorthHolland.
[13] Eaton, J. and M. Gersovitz, 1981, “Debt with Potential Reputation: Theoreticaland Empirical Analysis,”Review of Economic Studies 48, 289–309.
[14] Gourio, F., 2008, “Disasters and Recoveries: A Note on the Barro-Rietz Explanationof the Equity-risk Premium Puzzle,” American Economic Review 98, 68–73.
[15] Hellwig, C. and G. Lorenzoni, 2009, “Bubbles and Self-Enforcing Debt,” Econometrica 77, 1137–1164.

30

[16] Kehoe, P. and F. Perri, 2002, “International Business Cycles with Endogenous
Incomplete Markets,” Econometrica 70, 907–928.
[17] Kubler, F. and K. Schmedders, 2003, “Stationary Equilibria in Asset Pricing Models
with Incomplete Markets and Collateral,” Econometrica 71, 1767–1793.
[18] Lucas, D., 1994, “Asset Pricing with Undiversiﬁable Income Risk and Short Sales
Constraints: Deepening the Equity Premium Puzzle,” Journal of Monetary Economics 34, 325–341.
[19] Lustig, H., 2007, “The Market Price of Aggregate Risk and the Wealth Distribution,” NBER Working Paper Series No. 11132.
[20] Lustig, H. N. and S. G. van Nieuwerburgh, 2005, “Housing Collateral, Consumption
Insurance, and Risk Premia: An Empirical Perspective,” Journal of Finance 60,
1167–1219.
[21] Marcet, A. and R. Marimon, 1992, “Communication, Commitment, and Growth,”
Journal of Economic Theory 58, 219–249.
[22] Marcet, A. and R. Marimon, 1999, “Recursive Contract,” mimeo., Pompeu Fabra
University.
[23] Messner, M. and N. Pavoni, 2004, “On the Recursive Saddle Point Method,” IGIER
Working Paper Series No. 225.
[24] Negishi, T., 1960, “Welfare Economics and Existence of an Equilibrium for a Competitive Economy,” Metroeconomica 12, 92–97.
[25] Rietz, T. A., 1988, “The Equity Premium A Solution,” Journal of Monetary Economics 22, 117–131.
[26] Telmer, C. I., 1993, “Asset-Pricing Puzzles and Incomplete Markets,” Journal of
Finance 48, 1803–1832.

31

Table 1: Portfolio transaction behavior with one-time shock (shock size: 20%, persistence: i.i.d., α=0.1%)
labor endowment sharerealized
tree value(p(z) + α)θi (s−1 )time -1time 0time 1time 2time 3

0.500
0.556
0.500
0.500
0.500

0.011
0.007
0.139
0.020
0.020

time -1time 0time 1time 2time 3

0.500
0.444
0.500
0.500
0.500

0.011
0.007
-0.1180.002
0.001

insurance consumpreceipt
tion shareai (s−1 , s)nondagmaged country
0.000
0.500
-0.0070.502-0.118
0.502
0.002
0.501
0.001
0.501
damaged country
0.000
0.500
0.007
0.498
0.118
0.498
-0.0020.499-0.001
0.499

investedtrees
θi (s)

investedtrees valuep(z)θi (s)contingent claims∑q(s, s′ )ai (s, s′ )invested

0.500
6.540
0.920
0.940
0.950

0.010
0.088
0.019
0.019
0.019

0.000
-0.036
0.000
0.000
0.000

0.500
-5.540
0.080
0.060
0.050

0.010
-0.075
0.002
0.001
0.001

0.000
0.036
0.000
0.000
0.000

Note: All variables except for the number of labor endowment share and invested Lucas trees (θi )represent the ratio relative to the total world endowment. The labor endowment share in the second
column represents the ratio relative to the total labor endowment.

32

Table 2: Receipts from or repayments on contingent contracts at maturity with one-time
shock (shock size: 20%, persistence: i.i.d., α=0.1%)
shares ofinvested trees
θi (s−1 )no cat.
shock realizedai (s−1 , s)time 0time 1time 2time 3time 4time 5

0.500
6.540
0.920
0.940
0.950
0.960

0.000
-0.118
0.002
0.001
0.001
0.001

time 0time 1time 2time 3time 4time 50.500-5.540
0.080
0.060
0.050
0.040

0.000
0.118
-0.002-0.001-0.001-0.001cat. shock
cat. shockon country 2 on country 1
ai (s−1 , s)ai (s−1 , s)nondagmaged country-0.0070.007-0.095-0.080-0.0130.001-0.0140.001-0.0140.001-0.0140.001
dagmaged country
0.007-0.007
0.095
0.080
0.013
-0.0010.014-0.0010.014-0.0010.014-0.001cat. shockon countries 1 and 2
ai (s−1 , s)
0.000
-0.051
0.001
0.001
0.000
0.000
0.000
0.051
-0.001-0.0010.000
0.000

Note: ai (s−1 , s) is standardized by the total world endowment.

Table 3: Averages of equity prices and dividend ratios (shock size: 20%, persistence:
i.i.d., unit: %)dividend ratio (α)
0.1
0.2
0.3
0.4
0.5

average equity priceaverage price-dividend ratio
normal state shock state normal stateshock state
2.024
1.353
2024.31352.7
3.931
2.580
1965.71290.05.879
3.857
1959.81285.87.819
5.130
1954.81282.59.680
6.351
1936.11270.2
perfect insurance equity premium
closed economy equity premium

average equity premium

Note: All pricing variables represent the ratio relative to the total world endowment.

33

0.994
0.984
0.975
0.967
0.937
0.937
4.981

Table 4: Portfolio transaction behavior with one-time shock (shock size: 40%, persistence: 80%, α=0.2%)
labor endowment sharerealized
tree value(p(z) + α)θi (s−1 )time -1time 0time 1time 2

0.500
0.625
0.500
0.500

0.076
0.046
1.025
0.135

time -1time 0time 1time 2

0.500
0.375
0.500
0.500

0.076
0.046
-0.8650.017

insurance consumpreceipt
tion shareai (s−1 , s)nondagmaged country
0.000
0.500
-0.0460.519-0.865
0.516
0.017
0.514
damaged country
0.000
0.500
0.046
0.481
0.865
0.484
-0.0170.486

investedtrees
θi (s)

investedtrees valuep(z)θi (s)invested
contingent
claims
∑

0.500
6.390
0.890
0.900

0.075
0.569
0.141
0.135

0.000
-0.465
0.003
0.002

0.500
-5.390
0.110
0.100

0.075
-0.480
0.017
0.015

0.000
0.465
-0.003-0.002q(s, s′ )ai (s, s′ )Note: All variables except for the number of labor endowment share and invested Lucas’ trees (θi )represent the ratio relative to the total world endowment. The labor endowment share in the second
column represents the ratio relative to the total labor endowment.

Table 5: Receipts from or repayments on contingent contracts at maturity with one-time
shock (shock size: 40%, persistence: 80%, α=0.2%)
shares ofinvested trees
θi (s−1 )no cat.
shock realizedai (s−1 , s)time 0time 1time 2time 3

0.500
6.390
0.890
0.900

0.000
-0.8650.017
0.015

time 0time 1time 2time 30.500-5.390
0.110
0.100

0.000
0.865
-0.017-0.015cat. shock
cat. shockon country 2 on country 1
ai (s−1 , s)ai (s−1 , s)nondagmaged country-0.0460.046-0.542-0.457-0.0810.010-0.0820.009
dagmaged country
0.046-0.046
0.542
0.457
0.081
-0.0100.082-0.009Note: ai (s−1 , s) is standardized by the total world endowment.

34

cat. shockon countries 1 and 2
ai (s−1 , s)
0.000
-0.213
0.004
0.004
0.000
0.213
-0.004-0.004

Table 6: Averages of equity prices and dividend ratios (shock size: 40%, persistence:
80%, unit: %)dividend ratio (α)
0.2
1.0
2.0
3.0
4.0
5.0
6.0
7.0
7.4

average equity priceaverage price-dividend ratio
normal state shock state normal stateshock state
14.9628.904
7481.04452.047.14726.4654714.72646.562.65131.9823132.51599.185.11742.5522837.21418.4107.270
53.2102681.81330.3126.450
62.2052529.11244.1147.41072.3142456.91205.2
162.130
78.7812316.21125.4169.760
82.3662294.11113.1perfect insurance equity premium
closed economy equity premium

average equity premium
7.657
7.826
8.134
8.068
7.916
7.740
7.633
7.433
7.397
7.397
25.076

Note: All pricing variables represent the ratio relative to the total world endowment.

Table 7: Portfolio transaction behavior with ﬁve consecutive shocks (shock size: 40%,
persistence: 80%, α=0.2%)
labor endowment sharerealized
tree value(p(z) + α)θi (s−1 )time -1time 0time 1time 2time 3time 4time 5time 6time 7time 8

0.500
0.625
0.625
0.625
0.625
0.625
0.500
0.500
0.500
0.500

0.076
0.046
0.542
0.864
0.831
0.814
1.521
0.102
0.118
0.129

time -1time 0time 1time 2time 3time 4time 5time 6time 7time 8

0.500
0.375
0.375
0.375
0.375
0.375
0.500
0.500
0.500
0.500

0.076
0.046
-0.457-0.779-0.746-0.729-1.361
0.050
0.033
0.023

insurance consumpreceipt
tion shareai (s−1 , s)nondagmaged country
0.000
0.500
-0.0460.519-0.5420.526-0.8640.529-0.8310.531-0.8140.532-1.361
0.532
0.050
0.522
0.033
0.517
0.023
0.515
dagmaged country
0.000
0.500
0.046
0.481
0.542
0.474
0.864
0.471
0.831
0.469
0.814
0.468
1.361
0.468
-0.0500.478-0.0330.483-0.0230.485

investedtrees
θi (s)

investedtrees valuep(z)θi (s)invested
contingent
claims
∑

0.500
6.390
10.190
9.800
9.600
9.480
0.670
0.780
0.850
0.890

0.075
0.569
0.844
0.811
0.795
0.785
0.106
0.117
0.127
0.133

0.000
-0.465-0.745-0.717-0.702-0.693
0.021
0.012
0.006
0.003

0.500
-5.390-9.190-8.800-8.600-8.480
0.330
0.220
0.150
0.110

0.075
-0.480-0.761-0.728-0.712-0.702
0.052
0.033
0.022
0.016

0.000
0.465
0.745
0.717
0.702
0.693
-0.021-0.012-0.006-0.003q(s, s′ )ai (s, s′ )Note: All variables except for the number of labor endowment share and invested Lucas’ trees (θi )represent the ratio relative to the total world endowment. The labor endowment share in the second
column represents the ratio relative to the total labor endowment.

35

Table 8: Portfolio transaction behavior with alternate shocks (shock size: 40%, persistence: 80%, α=0.2%)
labor endowment sharerealized
tree value(p(z) + α)θi (s−1 )time -1time 0time 1time 2

0.500
0.625
0.375
0.500

0.076
0.046
0.542
-1.630time -1time 0time 1time 2

0.500
0.375
0.625
0.500

0.076
0.046
-0.4571.791

insurance consump- invested
receipttion sharetrees
ai (s−1 , s)θi (s)
initially nondagmaged country
0.000
0.500
0.500
-0.0460.519
6.390-0.4570.489-10.160
1.630
0.489
0.050
initially damaged country
0.000
0.500
0.500
0.046
0.481
-5.3900.457
0.511
11.160-1.6300.511
0.950

investedtrees valuep(z)θi (s)invested
contingent
claims
∑

0.075
0.569
-0.8410.008

0.000-0.465
0.811
0.002

0.075
-0.480
0.924
0.151

0.000
0.465
-0.811-0.002q(s, s′ )ai (s, s′ )Note: All variables except for the number of labor endowment share and invested Lucas’ trees (θi )represent the ratio relative to the total world endowment. The labor endowment share in the second
column represents the ratio relative to the total labor endowment.

36

Figure 1: Consumption shares with one-time shock (shock size: 20%, persistence: i.i.d.,α=0.1%)
0.51
country1, shock in period 1
country2, shock in period 1
0.508

0.506

consumption share

0.504

0.502

0.5

0.498

0.496

0.494

0.492

0.49
-2

0

2

4

6

8

10

periodFigure 2: Consumption shares with one-time shock (shock size: 40%, persistence: 80%, α=0.2%)
country1, shock in period 1
country2, shock in period 1

0.53

0.52

consumption share

0.51

0.5

0.49

0.48

0.47
-2

0

2

4

6
period

37

8

10

12

14

Figure 3: Consumption shares with ﬁve consecutive shocks (shock size: 40%, persistence: 80%,α=0.2%)
country1, shock in period 1
country2, shock in period 1

0.53

0.52

consumption share

0.51

0.5

0.49

0.48

0.47
-2

0

2

4

6

8

10

12

14

periodFigure 4: Consumption shares with alternate shocks (shock size: 40%, persistence: 80%,α=0.2%)
country1, shock in period 1
country2, shock in period 1

0.53

0.52

consumption share

0.51

0.5

0.49

0.48

0.47
-2

0

2

4

6
period

38

8

10

12

14