-
Bankruptcy Equilibrium: Secured and Unsecured
assets
Aloisio Araujo a,b and J. Mauricio Villalba a
aIMPA, b FGV
Rio de Janeiro - Brazil
February, 2016Preliminary version
Abstract
Economies with lack of commitment have been extensively studied
us-ing instruments like collateral to back promises and market
exclusion toavoid default. We study an economy with secured and
unsecured assets,where the last ones play an important role in
allowing interstate and in-tertemporal transfers. Bankruptcy
punishment is the seize of a fractionof agent’s income, this leads
to lack of convexity in the budget set. If thereturn matrix has ex
ante full rank we can implement the Arrow Debreuallocation (AD)
with a large enough seizure rate, indirectly proving equi-librium
existence.
Key words: Bankruptcy, seizure rate, debt constraint,
contagion.
1 Introduction
Debt is an important financial tool to transfer wealth among
dates and to insur-ance among different states of nature. We can
separate debt into two categories:Secured debt, as mortgage loan,
is backed by some collateral; it played an im-portant role during
the last financial crisis [14]. Unsecured debt as credit carddebt
and student loans, is not backed by collateral, this does not mean
thereis no default punishment, actually its punishment (established
by a bankruptcylaw) might be hard.Although most debt is secured,
the role of unsecured debt is still important inthe economy,
especially for people who lack of collateral. It also depends
ondemographic variables. As can be seen in [12], unsecured debt is
more impor-tant for people under 35 years old. Also, in the case of
firm debt [9] showsthat unsecured debt moves procyclically and
tends to lead the GDP, while thesecured debt is acyclical.
1
-
For secured debts, the collateral constraint imposes a debt
limit; individualscannot take large amounts of secured debt because
at the same time they needto constitute the collateral requirement,
which is some durable commodity withlimited supply. For unsecured
debt, we need to introduce a maximum amountof debt. Consider an
individual who goes to a financial institution to ask fora loan,
the bank usually makes a credit analysis of the individual in order
todetermine the agent’s ability to repay the debt, and determines a
credit limit asthe maximum amount of money that the bank should
lend the individual [21].For simplicity, we do not model the way
this limit is estimated and we assumethat this limit is exogenously
established by some intermediate agent or a reg-ulator.1
When the regulator defines the limit of credit, he might not
have all individual’srelevant information, or might have incomplete
information about future eventsand fails to determine a limit of
credit below which the debtor always honorsher debts. So in certain
scenarios the debtor agent might not be able to repayher debt, in
this situation we say that she goes bankruptcy.Allowing for
bankruptcy brings some technical difficulties, agent’s feasible
set(including consumption and portfolio) is no longer a convex set,
so it is possiblethat the best response correspondence is not
convex, this prevents us to use theclassical fixed point theorems
to prove equilibrium existence. To overcome thisdifficulty, two
seminal papers on the subject [6] and [25] work with a
non-atomicset of agents2. In a dynamical economy, this approach has
been used to studysome policy implications of a change in the
Bankruptcy legislation [13].Bankruptcy is idiosyncratic; each agent
decides how much to delivery, thiscould create a cascade effect.
Agents might default by contagion. Portfoliosdefine a lending
network among agents, creating a framework to study sys-temic risk.
There are several studies about network stability (see for exam-ple
[1–3, 11, 16, 19, 27]) and its dependence on shocks and network
structure.Most of these works assume that links and portfolios are
exogenously given,implying that a more stable network is always
preferred. Allowing a greaternumber of defaults (less stability)
might be better when prices and portfoliosare endogenous. In an
exogenous framework, defaulting asset is overvalued ifthe shock
that leads to default is large enough (lender is paying too much
foran asset that defaults). An endogenous price reflects default
probabilities, solender pays a fair price.In this paper, we address
an economy with finite agents (or finite types). Thereare two dates
and agents can transfer resources among dates and states of
natureby negotiating a set of secured and unsecured assets. In
bankruptcy, the seizurerate defines how much of the income is
confiscated. This hybrid model combinesthe collateral model studied
in [7, 17] and the model of unsecured claims and
1In [23] this credit limit is endogenous. It is defined as an
optimal choice for the lendermanager. Because of private
information, bankruptcy probability might be positive.
2When introducing a set of continuous agents, the economy turns
out to be convex, in thesense that the aggregate best response
correspondence is convex valued, and, except for sometechnical
details, equilibrium existence can be proved in a similar way as in
an Arrow-Debreueconomy.
2
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bankruptcy studied in [6, 25]. Assuming that there is a finite
type of agentswe can prove equilibrium existence when agents choose
a convex combination ofoptimal strategies. This can be interpreted
in two ways: 1) there is a continuumof agents of each type, meaning
that in equilibrium identical agents can takedifferent choices, or
2) agents choose mixed strategies. In either case, the proofis
quite standard, we only need to replace the best response
correspondence byits convex hull.Without the convexification
approach, we can prove equilibrium existence byimplementing the
Arrow Debreu allocation. We are interested in study condi-tions
that ensure its implementation with a mix of secured and unsecured
assets.In [20] with permanent exclusion, equilibrium might be
Pareto ranked. In a sim-ilar context, [8] shows that the Arrow
Debreu allocation can be implementedwith partial exclusion and
strong risk aversion. In our setting, if the vector ofindividual
state transference is contained in the span of the payoff matrix,
thenwe need to check if the collateral constraint is satisfied. In
case it is not, wecan add unsecured assets without changing the
matrix span. Next, in orderto ensure that in equilibrium this span
is not changed, we need to consider aseizure rate high enough to
avoid bankruptcy in equilibrium; this does not ruleout the
possibility of default on some of the secured assets.A necessary
condition to implement the Arrow Debreu allocation is a
mono-tonicity relation3 similar to [5]. This condition is necessary
for the collateralconstraint and it is sufficient if we implement
with only one secured asset. Inthe pure secured model [5,17], the
collateral constraint implies that individualscannot make any
temporal transfers, they are net savers. Introducing unse-cured
assets relaxes this and individuals can transfer resources between
datesand states of nature without worrying about the collateral
constraint. This isvery important when there is lack of collateral;
also, it is important for individ-uals with low income, some of
them probably still young and looking for studentloans that does
not require collateral.Paper is organized as follows: next we study
the hydrid model of secured andunsecured assets are characterize
conditions to implement the Arrow DebreuAllocation without
bankruptcy.
2 Implementing the Arrow-Debreu Equilibrium
with a mix of secured and unsecured assets.
2.1 The Model
We consider an economy with two periods t = 0, 1 and S possible
states ofnature s ∈ S = {1, · · · , S} in t = 1.In each state s ∈
S∗ = {0, 1, · · · , S} (s = 0 represents period 0), there are
spotmarkets for two commodities l = 1, 2. Commodity l = 1 is
perishable whilel = 2 is durable, this means that consuming one
unit of the durable good in
3The value of net consumption should be larger in states with
higher relative prices.
3
-
t = 0 guarantees a consumption of one unit of this commodity in
the next pe-
riod. The vector of prices is p = (p0, p1, · · · , pS) ∈
R2(S+1)++ , where psl is the
price of commodity l in state s for l = 1, 2 and s ∈ S∗.There is
a set I = {1, · · · , I} of agents with strictly increasing,
concave and
continuous utility functions ui : R2(S+1)+ → R. They are also
characterized by
their endowments wi = (wi0, · · · , wiS) ∈ R
2(S+1)++ .
There is a set of assets that can be negotiated in t = 0 and
promise returns int = 1. In this set of assets we consider secured
and unsecured assets.Let Jsec = {1, · · · , J
sec} be the set of secured assets. A secured asset j promisesrjs
units of the perishable commodity in state s. They differ on the
collateralrequirement Cj to back the promises. An agent selling one
unit of the securedasset j should constitute Cj > 0 units of the
durable good. We assume thatdebtor holds the collateral and it can
not be used to back other promises. Agentdefaults if the value of
the debt is greater that the value of the collateral. So, asecured
asset with collateral requirement Cj returns min
{
ps1rjs; ps2C
j}
in states.On the other hand, there is a set Junsec = {1, · · · ,
J
unsec} of unsecured as-sets. Their promises are not backed by
collateral, which does not imply thatagents can default without
suffering any penalty. Creditors are protected by abankruptcy law
that establishes the bankruptcy procedure.We assume that all
unsecured assets are numeraire, they pay in units of theperishable
good, a unit of the unsecured asset j promises to pay r′js units of
theperishable commodity. If an agent i holds portfolio z ∈ RJ
unsec
, the total debtin state s is4
∑
j(zj)−ps1r
′js . Debtor might file for bankruptcy, in that case all
debt is forgiven (excluding secured debt), but the judge can
seize some of thedebtor income to partially repay the
creditors.Debtor’s income is composed by non financial and
financial income. Non finan-cial income is the value of the
endowment psw
is and the value of the durable good
consumed in t = 0, ps2x02. If the agent holds a portfolio (θ, ϕ)
∈ RJsec
+ ×RJsec
+ (θrepresents long positions and ϕ short selling) of secured
assets, then her financialincome is:
Jsec∑
j=1
(θj − ϕj)min{
ps1rjs; ps2C
j}
+
Junsec∑
j=1
(zj)+κjsps1r
′js .
Where κjs ∈ [0, 1] is the delivery rate, defined in equilibrium
according todebtors’ deliveries.The bankruptcy procedure states
that secured debt have senior priority up tothe value of the
collateral, after paying the secured debt up the collateral
value,all financial income can be garnished. But only a fraction γ
∈ [0, 1] of the
4For any number a we call (a)− = max{0,−a} and (a)+ = max{0, a}
the negative andpositive part of a.
4
-
endowment can be garnished. So, in case of bankruptcy, judge can
seize5:
γpswis + ps2x02 +
Jsec∑
j=1
(θj − ϕj)min{
ps1rjs; ps2C
j}
+
Junsec∑
j=1
(zj)+κjsps1r
′js ,
so, the delivery of unsecured debts will be:
Dis = min{
Junsec∑
j=1
(zj)−ps1r
′js ;
γpswis + ps2x02 +
Jsec∑
j=1
(θj − ϕj)min{
ps1rjs; ps2C
j}
+Junsec∑
j=1
(zj)+κjsps1r
′js }.
Let (qsec, qunsec) ∈ RJsec
++ × RJunsec
++ be the vector of asset prices in t = 0, agenti’s budget
constraint in t = 0 is
p0(x0 − wi0) + q
sec(θ − ϕ) + qunsecz ≤ 0 (1)
and the collateral constraint for secured assets
x02 ≥Jsec∑
j=1
ϕjCj (2)
In t = 1, state s, the budget constraint is
ps(xs−wis−(0, x02))+D
is ≤
Jsec∑
j=1
(θj−ϕj)min{
ps1rjs; ps2C
j}
+Junsec∑
j=1
(zj)+κjsps1r
′js
(3)So, given commodities and asset’s prices and delivery rate,
the budget setfor agent i, Bi(p, qsec, qunsec, κ) consists on all
vectors of consumption andportfolio satisfying (1),(2), (3) and a
bounded short selling constraint for un-secured assets zj ≥ −m with
m ≥ 0 for all j ∈ Junsec. We say that aconsumption bundle x is
affordable if there is a portfolio (θ, ϕ, z) such that(x, θ, ϕ, z)
∈ Bi(p, qsec, qunsec, κ)6.
5We could consider a more general punishment, but in order to
study the seizure rate γwe’ve chosen this form. Also notice that by
the collateral constraint this term is non negative,with a more
general punishment rule we can not assure this, so we should
consider only thepositive part.
6The budget set Bi(p, qsec, qunsec, κ) could be written as the
union of 2S convex sets, eachof them defines the states of nature
in which agent i defaults in the unsecured debt. Oneof such sets is
the non bankruptcy region in which for all states of nature, the
followingconstraints hold:
Dis =Junsec∑
j=1
(zj)−ps1r
′js .
5
-
If agent i hold a positive unsecured debt∑Junsec
j=1 (zj)−ps1r
′js > 0, the delivery
for each asset is proportional to the value of the debt:
Dis,j′ = τis(zj′ )
−ps1r′j′
s ,
with
τ is =Dis
∑Junsec
j=1 (zj)−ps1r
′js
The proceedings are distributed proportionally among all
creditors. Thedelivery rate adjusts to match the total delivery of
debtors with creditors returnsfor each unsecured asset j ∈ J unsec
and state s:
κjsps1r′js
∑
i∈I
(zij)+ =
∑
i∈I
Dis,j (4)
Definition 2.1 An equilibrium7 for the economy with secured and
unsecuredassets is a vector of prices p, qsec, qunsec, delivery
rates κ and an allocation ofconsumption and portfolios
(
xi, θi, ϕi, zi)
i∈I, such that:
• For each i, xi is affordable with the portfolio (θi, ϕi, zi)
and is optimal inthe set of all affordable consumption bundles in
Bi(p, qsec, qunsec, κ).
•∑
i∈I
(
xi0 − wi0
)
= 0.
• For each s ∈ S:∑
i∈I
(
xis − wis − (0, w
i02))
= 0.
•∑
i∈I zi = 0
•∑
i∈I(θi − ϕi) = 0
• (4) holds.
We would like to compare this equilibrium with the Arrow-Debreu
equilibrium,so we proceed to define it.
Definition 2.2 An Arrow-Debreu equilibrium (AD) is a vector of
allocations(xi)i∈I and a vector of prices p such that:
• For each i, xi is optimal in the set of all y ∈ R2(S+1)+
with
∑S
s=0 ps(ys −
wis)−∑S
s=1 ps2y02 ≤ 0.
•∑
i∈I
(
xi0 − wi0
)
= 0.
• For each s ∈ S:∑
i∈I
(
xis − wis − (0, w
i02))
= 0.
7We refer to the appendix for a proof of equilibrium existence
when each i ∈ I chooses aconvex combination of optimal strategies.
This could be interpreted as an equilibrium withrandom strategies
or a type of continuum agents such that in equilibrium identical
agentsmight choose different strategies.
6
-
2.2 Some Implementation Results
A first question to ask: When is AD allocation implementable for
the economywith secured and unsecured assets? Even though it is
hard to find conditionsthat are necessary and sufficient, we can
find a necessary condition associatedto the seizure rate.
Proposition 2.1 Let(
(xi)i∈I , p)
be the AD equilibrium. If xi is affordable inthe non bankruptcy
region for all agents, then seizure rate satisfies:
γ ≥ γ∗ = max(s,i)∈S×I
[
1−psx
is
pswis
]+
Proof Let mi =(
ps(xis − w
is)− ps2x
i02
)
s∈S∈ RS be the vector of transferences
between states of nature. Since xi is affordable in the non
bankruptcy region,then there are an unsecured and secured
portfolios zi and (θi − ϕi) such that:
mis =
Jsec∑
j=1
(θij − ϕij)min
{
ps1rjs; ps2C
j}
+
Junsec∑
j=1
zijps1r′js ,
where we have set the delivery rate κ equal to one. Also for xi
to be affordablein the non bankruptcy region:
−mis ≤ γpswis + ps2x
i02,
so, for all agents i and states s:
γ ≥
(
1−psx
is
pswis
)
implying that γ ≥ γ∗.
If Junsec = 0, in [17] it is showed that a necessary condition
to implement theAD equilibrium with secured assets is
ps(xis − w
is) ≥ 0 (5)
This condition implies that individuals can not transfer
endowments from onestate to another, the value of consumption must
be always at least equal to thevalue of the endowment. As in [5],
we say that J CC ⊂ Jsec is a set of completecollaterized assets if
for each state s, there is some secured asset j ∈ J CC withps1 =
ps2C
j (assuming that rjs = 1 for all secured assets). In [5] it is
shown anecessary condition to implement an AD equilibrium with
secured assets J CC :
Proposition 2.2 Given an AD equilibrium (p, (xi)i∈I), a
necessary conditionfor there to exists an equilibrium with complete
collaterized assets with the sameallocation is that for all agents
i and all pairs of states s and s′ with ps2
ps1> ps′2
ps′1:
1
ps1ps(x
is − w
is) ≥
1
ps′1ps′(x
is′ − w
is′ ) ≥ 0. (6)
7
-
It turns out that this condition is also sufficient for two
states of nature and ifagents preferences does not depend on the
durable commodity in t = 0.One of the main issues of the pure
secured model is that the collateral constraintis too hard for
debtors, actually if we consider the durable good as another wayof
saving, then all agents are net savers.The existence of the set of
secured assets J CC is guarantee if we assume that allagents have
identical homothetic preferences. With this, equilibrium
commodityprices depends only on aggregate terms, so if the AD
allocation is implementable,then prices should be the same. After
computing AD prices we can computethe collateral requirements
necessary to construct the set of secured assets J CC .Generically,
the S×S return matrix will be of full rank8, so all transfers
ps(x
is−
wis − (0, xi02)) are attainable for some secured portfolio.
Nevertheless, those
portfolios might violate the collateral constraint and in an
equilibrium withpure secured assets some assets might not be traded
and equilibrium will notbe Pareto efficient (although it might be
Pareto constraint).One way to weaken condition 5 is to introduce
unsecured assets. They donot require collateral and allow agents to
transfer endowments among states ofnature. If the return matrix of
the unsecured assets is equal to the J CC securedassets, then we
might be able to implement an AD equilibrium with a mix ofsecured
and unsecured assets. As the combined return matrix is already of
fullrank, in order to keep the full rank property the seizure rate
established by thebankruptcy law should be large enough to avoid
default on unsecured assets.The next result formalizes our
observation for the case of two states of nature
Proposition 2.3 Let (p, (xi)i∈I) be the AD equilibrium when S =
2. Suppose,without lost of generality, that β = p12
p22− p11
p21> 0. If the seizure rate satisfies
γ ≥ γ∗ = max(s,i)∈S×I
(
1−psx
is
pswis
)+
.
Then, there is a debt constraint m such that the AD allocation
is feasible for allagents if, and only if, one of the following
conditions holds:
a) For all i:
1
p11p1(x
i1 − w
i1)−
1
p21p2(x
i2 − w
i2) ≥ 0
b) For all i:
1
p11p1(x
i1 − w
i1)−
1
p21p2(x
i2 − w
i2) ≤
β
p11
(
p2(xi2 − w
i2) + p22x
i02
)
.
Proof The claim on the seizure rate is due to proposition
2.1.
8Generically the price ratios ps1/ps2 are different for all s,
we concentrate in this case.
8
-
a) We implement the AD equilibrium using the secured asset with
collateralrequirement C1 = p11/p12 and the unsecured asset with
returns (p11, p21).The collateral constraint is equivalent to:
xi02 + zi1C
1 ≥ 0
Let mi =(
ps(xis − w
is)− ps2x
i02
)
s∈S∈ RS be the vector of transferences
between states of nature.
Define
τ =1
p11p1(x
i1 − w
i1)−
1
p21p2(x
i2 − w
i2),
then,
τ =1
p11(mi1 + p12x
i02)−
1
p21(mi2 + p22x
i02)
The secured and unsecured portfolios span all transfers, so
mi1 = p11zi1 + p11z
i2
mi2 = p22p11p12
zi1 + p21zi2,
so
τ =p12p21 − p11p22
p11p21(xi02 + z
i1C
1)
Then, τ ≥ 0 if, and only if, the collateral constraint is
satisfied.
b) In this case we implement the AD equilibrium with the secured
asset withcollateral requirement C2 = p21/p22 and the unsecured
asset with returns(p11, p22p11/p12). The collaretal constraint is
equivalent to
xi02 + zi2C
2 ≥ 0
Define
τ∗ =1
p11p1(x
i1 − w
i1)−
1
p21p2(x
i2 − w
i2)−
β
p11
(
p2(xi2 − w
i2) + p22x
i02
)
,
The return matrix is the same as in the previous item, so
τ∗ =p12p21 − p11p22
p11p21(−zi2C
2 − xi02)
then, τ∗ ≤ 0 if, and only if, the collateral constraint is
satisfied.
The first condition is similar to (6), it preserves the order,
but it doesn’t requirenon negativeness. With this, the AD
allocation is feasible with the securedasset with the lowest
collateral requirement and the unsecured asset. The secondcondition
is new to our knowledge and it says that even if the monotone
relation(6) fails, the AD allocation is still feasible if the
failure of (6) is not too large. Inthis case, the AD allocation is
feasible using the secured asset with the largest
9
-
collateral requirement.Compared to the pure collateral economy,
Proposition 2.3 enlarges the sets ofeconomies (endowments) for
which the AD allocation is implementable. In thisproposition at
least one asset is secured, even if those conditions are not
satisfied,we still can implement AD, but we will need both assets
to be unsecured.Proposition 2.3 shows conditions that guarantee
feasibility of the AD allocation,but it does not say anything about
optimality. Recall that debtors who holdsthe unsecured asset might
file for bankruptcy and deliver less than the promise.The budget
set can be decomposed as the union of convex sets, one of themis
the set of consumption bundles and portfolios such that agent does
not filefor bankruptcy. Proposition 2.3 actually shows that
consumption bundles inthis sets are contained in the Arrow Debreu
budget set, but it is still possibleto find another consumption
bundle in the other sets (sets in which agent fillsfor bankruptcy
at least in one state) that improves her utility (see the
2x2example).Next proposition establishes that if the AD allocation
is feasible, then it isoptimal, provided a high enough seizure.
Proposition 2.4 If the Arrow Debreu Allocation (xi)i∈I is
feasible in the nonbankruptcy region and if for all agents,
preferences are represented by a C1
utility function ui that satisfy Inada condition
limmin{xsl}→0
u′i(x) = +∞.
Then, there is some γ∗∗ < 1 such that the Arrow Debreu
allocation is an equi-librium for the non commitment economy for
all γ ≥ γ∗∗.
2.2.1 2× 2 Model
We borrow a parametric version of an example given in [5] to
point out how toimplement the AD allocation with a mix of secured
and unsecured assets andthe relevance of the latter. AD allocation
might not be implementable with apure secured asset market when
durable good is unevenly distributed, we needto introduce at least
one unsecured asset. The unsecured asset used dependson durable
preferences and distribution. Also, the interstate shock plays an
im-portant role when the interstate shock gets larger, agents need
larger portfoliosto attain these hedging.There are two agents with
identical homothetic preferences and two equal prob-ably states of
nature
α log(x01) + (1− α) log(x02) +1
2
2∑
s=1
(α log(xs1) + (1− α) log(xs2)) ,
where α denotes the preference for the perishable good.
Endowments are asfollow
wi = (wi01, ηiw02, w
i11, 0, w
i21, 0).
10
-
Agents receive an endowment of the durable commodity only in t =
0, ηi rep-resents the fraction of the durable commodity that
belongs to agent i.Since utilities are homothetic, commodity prices
are independent on the endow-ment’s distribution. Arrow Debreu
prices are
p =
(
1, 2(1− α)
α
w01w02
,w012w11
,(1− α)
α
w012w02
,w012w21
,(1− α)
α
w012w02
)
,
where wsl =∑
i wisl.
Notice that durable price is the same in both states of nature.
This equality isdue to equal probabilities and because agents
receive durable endowments onlyin t = 0.Endogenous collateral
requirements are
Cs =α
1− α
w02ws1
.
The return matrix for the J CC secured assets (if w11 > w21,
asset 1 is thesub-prime and asset 2 is prime)9,10is
R =w012w11
(
1 11 w11
w21
)
,
which is clearly a full rank matrix, so all transfers are
possible, but the neededportfolio might not meet the collateral
constraint.The collateral constraint depends on durable
consumption:
xi02 = αw022
(
wi01w01
− 2ηi +wi112w11
+wi212w21
)
+ ηiw02.
We can also compute portfolios for assets 1 and 2:
zi1 =α
2w11
(
wi01w01
+ 2(1− α)
αηi +
wi112w11
+wi212w21
)
− w11wi11 − w
i21
w11 − w21,
zi2 =wi11w21 − w
i21w11
w11 − w21= −
p11wi11 − p21w
i21
p11 − p21
The collateral constraint:
∆i = xi02 − C1(zi1)
− − C2(zi2)− ≥ 0,
its sign is independent of durable endowment. Both durable
consumption andcollateral requirement are proportional to durable
endowment, so the sign of thelast expression will be independent of
it. To fix ideas lets fix some numericalvalues:
w1 = (4, 4(1− η), 4, 0, 4, 0)
9The case w11 < w21 is analogous and if there is equality,
the return matrix is singular.10Sub-prime asset is the one with
highest default probability. In our context (as in [5] ) the
sub-prime asset is the one with the lowest collateral
requirement.
11
-
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
α
η
η − 1 +α(1+9α)15(1−α)2 ≤ 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
α
η
η ≥35
α
1−α
a) b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
α
η
00.2
0.40.6
0.81
00.2
0.40.6
0.810
0.1
0.2
0.3
0.4
αη
γ
c) d)
Figure 1:
w2 = (2, 4η, 6, 0, 2, 0),
agent 1 endowment is deterministic, she has a long position on
the sub-primeasset and a short position on the prime
asset.Collateral constraint for agents 1 and 2 are, respectively,
equivalent to:
1− η −α(1 + 9α)
15(1− α)2≥ 0 (7)
η −3
5
α
1− α≥ 0. (8)
If at least one asset is unsecured, the seizure rate necessary
to implement theAD allocation must be at least:
γ ≥ γ∗ = max
{
0;1
10−
3
2
(1− α)
α(1− η);
1
3−
5
3
(1− α)
αη
}
(9)
γ∗ is a non decreasing function of α and it is concave on the
durable distribution.Given a preference parameter, the seizure rate
needed to implement AD attainsits minimum when the durable is more
fairly distributed.Figure 1 (above) shows the values of α and η for
which the collateral constraintare satisfied. The left (right) one
corresponds to collateral constraint when onlythe prime (sub-prime)
asset is secured. c) shows the values of α and η for whichthe AD
allocation can be implemented using secured assets only. In d) we
showthe minimum seizure rate that implements the AD allocation when
at least oneasset is unsecured.
12
-
2.2.2 Feasibility does not imply optimality
If η = 0 and α = 0.2, agent 1 holds all the durable, the AD
allocation is
x1 = (5.52, 3.68, 9.20, 3.68, 5.52, 3.68)and x2 = (0.48, 0.32,
0.80, 0.32, 0.48, 0.32).
The required portfolios are z1 = (9.20,−4.00) and z2 = (−9.20,
4.00). Agent 2is poor in the first date and borrows heavily on
asset 1, but she does not satisfythe collateral constraint: x202 −
(z
21)
−C1 = −0.6. The AD allocation is notfeasible with two secured
assets. The trade off agents face is choosing betweenincreasing
consumption today and reducing risk in t = 1. Clearly asset 2
isbetter for risk sharing, but since agent 2 is too poor and have
high preferenceson consuming the durable good, then she prefers to
choose asset 1 with thelowest collateral requirement.If asset 1
were unsecured, agents 1 and 2 would be able to choose portfolios
z1
and z2, respectively. Agent’s 2 total debt in both states is
(z21)−ps1r
1s = 2.76.
She will honor her debt as long as the seizure rate is large
enough:
mins=1,2
{
γps1w2s1 + ps2x
202 + z
22ps1
}
≥ 2.76
then agent 2 does not default in the unsecured asset and the
Arrow Debreuallocation is attainable.For this example γ∗ = 0.33. We
also need a constraint for the maximum amountof unsecured debt,
otherwise agent 2 could take an arbitrarily large amount ofdebt in
the unsecured asset and attain an unbounded utility. For this
examplewe take a debt constraint m = 9.2, that is just enough to
allow portfolio z2.For this example, condition b) of proposition
2.3 holds, so the AD allocation issupported with the secured asset
with the largest collateral constraint (asset 2)and an unsecured
asset with the same payoff of secured asset 1.As we mentioned, γ∗
only guarantees that the AD allocation is feasible andis optimal in
the non bankruptcy subset. We should check whether there isanother
consumption bundle in the bankruptcy region that improves the
utilityof agent 2 (the seller of the unsecured asset).Budget set
can be written as the union of four convex sets:
1. The Non bankruptcy region.
2. Bankruptcy only in state s = 1.
3. Bankruptcy only in state s = 2.
4. Bankruptcy in both states.
Table (1) shows optimal consumption bundles and portfolios for
each case. No-tice that in order to keep her promise agent 2 must
buy some amount of thesecured asset. In the bankruptcy regions she
could reduce her long position onthe secured asset, this will
reduce her income in t = 1 and she won’t be ableto keep her
promise, her consumption in the bankruptcy state will reduce,
butshe increases her consumption in t = 0. If the reduce of her
income (associated
13
-
Case x01 x02 x11 x12 x21 x22 z θ ϕ Utility1 0.48 0.32 0.80 0.32
0.48 0.32 -9.20 4.00 0.00 -3.07292(s = 1) 0.63 0.28 0.80 0.32 0.51
0.34 -9.20 5.20 0.82 -3.05753(s = 2) 0.56 0.47 0.93 0.37 0.26 0.17
-3.20 0.00 2.80 -3.17384 1.50 0.84 0.80 0.32 0.27 0.18 -9.20 0.00
5.01 -2.6643
Table 1: Agents optimize in the full bankruptcy region.
to γ) is small enough, then her extra consumption in t = 0 more
than offsether losses in t = 1, this way she can get an utility
greater than with the ADbundle. As can be seen in the table her
best choice is to file for bankruptcy inboth states, so the AD
allocation (even being feasible) is not an
equilibrium.Nevertheless, as proposition 2.4 says, if we increase
further the seizure rate,then agent 2 income losses for bankruptcy
have a greater (negative) effect thanthe consumption increases in t
= 0. Our numerical experiments shows that itsuffices to take a
seizure rate γ ≥ 0.46.This example shows the relevance of unsecured
assets when collateral con-
straints are too tight. Unsecured debt provides a way of
transferring endow-ments between states without worrying about the
correct amount of durableconsumption. Whats more, to avoid
unsecured default, the seizure rate doesnot need to take too large
values, avoiding draconian penalties of default.
2.2.3 A Numerical Exercise with social mobility
Through this example, we show how the seizure rate needed to
implement theAD allocation reduces as the social mobility
increases. Agents in the lowersquartiles borrows on unsecured
assets, if in the next period there is a reasonablechance to end up
in one of the uppers quartiles, then they have more chances torepay
their debts. The reduction on the default probability reduces the
seizurerate (default punishment) needed to implement the AD
allocation.As in [5], there are 4 agents, endowments roughly match
US income and wealthdistribution [15]. Data on social mobility [18]
shows that agents in the lowersquartiles jump to the upper ones
with positive probability. To keep thingssimple we consider 3
states of nature:
1. Agents remain in the same quartile.
2. Agent 1 jumps to quartile 2; agent 3 jumps to quartile 4 and
vice verse.
3. An state with more social mobility: agents in quartiles 3-4
jump to quar-tiles 1-2 and vice verse.
State 3 represent the state with more social mobility, its
probability is ǫ > 0.State 1’s probability is π.After
normalizing, endowments distribution is as follow:
14
-
Agent 1 agent 2 agent 3 agent 4w01 0.61 0.22 0.12 0.05w02 0.84
0.12 0.04 0w11 0.63 0.21 0.11 0.05w12 0 0 0 0w21 0.21 0.63 0.05
0.11w22 0 0 0 0w31 0.11 0.05 0.63 0.21w32 0 0 0 0
Agents preferences are identical and homothetic.
α log(x01) + (1− α) log(x02) +
3∑
s=1
πs (α log(xs1) + (1− α) log(xs2)) .
In [28] they estimate α = 0.8.In the AD equilibrium each agent
consumes xi = αpwi/2 of each commodity ineach state of nature;
where:
pwi = 1 + 21− α
αwi02 + π(w
i11 − w
i21) + w
i21 + ǫ(w
i31 − w
i21),
is increasing with more social mobility for poor agents (3 and
4).The vector of inter temporal and interstate transfers for each
agent is:
mi = (πs(xi − wis1))s=1,2,3.
Consider 3 Arrow securities paying on the perishable commodity.
Return matrixwill be:
R =
π 0 00 1− π − ǫ 00 0 ǫ
.
Portfolios are:zi = xi1− wi1,
where 1 is a column vector of ones and wi1 is agent i’s
endowment of the per-ishable commodity in t = 1.Asset 3 plays an
important role for poor agents, they would like to short sell itand
pay in the state in which they have more endowment. If it were
securedagents would not be able to meet the collateral
requirements.Following [5], the endogenous collateral requirement
is C = α/(1− α) (it is thesame for the three assets because there
is no aggregate risk).Numerical exercises11 shows that assets 2 and
3 need to be unsecured to imple-ment the AD allocation.
11Let µ = (µ1, µ2, µ3) such that µj = 1 if asset j is secured
and 0 otherwise. We can writethe collateral constraint for each
agent as a function of probabilities π, ǫ and µ:
∆i(π, ǫ, µ) = xi −α
1− αµ(zi)−.
15
-
00.1
0.20.3
0.40.5
0.6
00.2
0.40.6
0.810
0.2
0.4
0.6
0.8
1
επ
γ0.5 0.55 0.6 0.65 0.7 0.75 0.8
Figure 2: Minimum seizure rate for AD feasibility. The gray
region representscombinations of π and ǫ that satisfy the
collateral constraint.
The minimum seizure rate for AD allocation’s feasibility is:
γ ≥ γ∗ = max(s,i)∈{2,3}×I
[
1−pei
2wis1
]+
.
Which is decreasing with ǫ when it is low enough as can be seen
in figure 2.Since asset 1 is secured, we also need to check the
collateral constraint. Thecombinations of probabilities that
satisfies it are given by:
mini=1,··· ,4
[
1 + 21− α
αwi02 − (2− π)w
i11 + (1− π − ǫ)w
i21 + ǫw
i31
]
≥ 0.
3 Conclusions
We have studied an economy with bankruptcy, showed conditions to
implementthe AD allocation with a mix of secured and unsecured
assets and providedconditions to guarantee equilibrium existence.In
a pure secured economy AD allocation might not be implemented even
if the
For each µ we define the set:
Lµ ={
(π, ǫ) ∈ [0, 1]2 : π + ǫ ≤ 1,∆i(π, ǫ, µ) ≥ 0 for all i = 1, · ·
· , 4}
.
We estimate the set Lµ for the 8 possible values of µ and verify
that except for µ = (0, 0, 0)and µ = (1, 0, 0) the set is empty.
Since we try to use the lest quantity of unsecured assetsthen we
keep with µ = (1, 0, 0)
16
-
return matrix has full rank, individuals are not able to meet
the collateral con-straints. In this situation it is useful to
introduce unsecured without changingthe return matrix. Agents do
not need to worry with the collateral constraint,nevertheless to
guarantee that individuals do not default on unsecured
debt,preserving the return spam, the seizure rate should be large
enough. We alsoprovided and example with social mobility in which
today poor agents can bor-row using an unsecured asset and if the
probability of ending up in the firstquantiles (more social
mobility) is positive, then we can implement AD alloca-tion.
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4 Appendix A
Mixed Strategy EquilibriumWe will call a mixed equilibrium an
equilibrium in which each agent ı ∈ Ichooses a convex combination
of optimal strategies in the budget set.
To prove a mixed equilibrium we proceed as follows: first we
define a trun-cated economy, associated to it there is a
generalized game, whose equilibriumis proved using the Kakutani fix
point theorem after defining an adequate cor-respondence. Then we
show that this fixed point is an equilibrium for the trun-cated
economy as the boundaries get larger. Finally, by lower
hemi-continuityof the budget set correspondence we prove that the
sequence of truncated equi-librium converges to a mixed equilibrium
for the original economy.
4.1 The truncated Economy
For each n ∈ N we define a truncated economy.
Let
Kn = [0, n]2(S+1) ×
[
0,In
C
]Jsec
×[
0,n
C
]Jsec
× [−2m, 2Im]Junsec
,
with C = minj∈Jsec
Cj .
LetP = ∆2+j
sec+Junsec−1+ ×
(
∆2−1+)S,
be the space of prices. We define the truncated budget set
βin : P× [0, 1]SJunsec
⇒ Kn,
as βin(p, q, κ) = Bi(p, q, κ) ∩Kn.
Using standard arguments we prove the following12
Lemma 4.1 For n large enough, βin is continuous.
12To prove upper hemi-continuity its enough to notice that Bi
has a closed graphic andKn is compact. Since βin has no empty
interior, it is easy to prove that β̊
in is lower hemi-
continuous, then¯̊βin is lower hemi-continuous too. It remains
to prove that β
in =
¯̊βin, which
can be proved for n large enough as in [24]
19
-
4.1.1 A Generalized Game
In the generalized game we consider all agents i ∈ I, the market
and an addi-tional player that defines the equilibrium delivery
rates. We proceed to defineeach player optimization problem and
constraints.
Each agent ı ∈ I solves:
max(x,θ,ϕ,z,τ)∈βi
n×[0,1]S
U i(x)−1
2
∑
s∈S
τ is∑
j∈Junsec
ps1rsj(zij)
+ −Dis
2
(10)
Letψin : P× [0, 1]
SJunsec
⇒ Kn × [0, 1]S
be agent i’s optimal choice for 10. Consider the function w :
R2(S+1)+ ×R
Jsec
+ ×
RJsec
+ ×RJunsec × [0, 1]
S→ R
2(S+1)+ ×R
Jsec
+ ×RJsec
+ ×RJunsec
+ ×RJunsec
+ ×RSJunsec
+
as(x, θ, ϕ, z, τ) → (x, θ, ϕ, z+, z−, τ ◦ z),
with τ ◦ z = (τsz+j ; (s, j) ∈ S × Junsec).
Next we consider the compact set:
∇n = [0, n]2(S+1)
×
[
0,In
C
]Jsec
×[
0,n
C
]Jsec
×[0, 2Im]Junsec
×[0, 2m]Junsec
×[0, 2m]SJunsec
.
And define13:
Ψin = Conv w ◦ ψin : P× [0, 1]
SJunsec
⇒ ∇n, (11)
By the Berge maximum theorem [10] and the fact that w is a
continuous functionand convex combination preserves upper
hemi-continuity we have
Lemma 4.2 The correspondence Ψin is upper hemi-continuous, with
convex,compact and non empty values.
Given an allocation (xi, θi, ϕi, zi1, zi2, y
i)i∈I ∈ ∇In, the market player solves:
max(p,q)∈P
p0∑
i∈I
(xi0−ei0)+q
sec∑
i∈I
(θi−ϕi)+qunsec∑
i∈I
(zi1−zi2)+
∑
s∈S
ps∑
i∈I
(xis−eis).
(12)Delivery rate player solves:
maxκ∈[0,1]SJ
unsec−
∑
(s,j)∈S×Junsec
ps1rsj
[
κsj∑
i∈I
zi2j −∑
i∈I
yisj
]2
; (13)
13Conv refers to the convex hull of a set.
20
-
Let µ1 and µ2 be the best response correspondence for the market
and the de-livery rate players, respectively.
A mixed equilibrium for the generalized game is characterized by
a fix pointof the following correspondence:
µ =∏
i∈I
Ψin × µ1 × µ2 : ∇In × P× [0, 1]
SJunsec
⇒ ∇In × P× [0, 1]SJunsec
,
by Kakutani fix point theorem we have the following
Proposition 4.3 The Generalized game has a mixed
equilibrium.
4.1.2 Equilibrium for the truncated economy
Now we prove the following proposition
Proposition 4.4 For n large enough, a mixed equilibrium for the
generalizedgame is an equilibrium for the truncated economy.
Proof By definition all agents choice a convex combination of
optimal strate-gies, so we only need to check market clearing
conditions and the equilibriumdelivery rate.
Given a fix point(
(
xi, θi, ϕi, zi1, zi2, y
i)
i∈I, p, q, κ
)
. For each agent there are
indexes li = 1, · · · , N14 such that:
(
xi, θi, ϕi, zi1, zi2, y
i)
=
N∑
li=1
αli
(
xili , θili, ϕili ,
(
zili)+,(
zili)−, τ ili ◦ z
ili
)
,
with αli ≥ 0,∑
αli = 1 and(
xili , θili, ϕili , z
ili, τ ili)
∈ ψin(p, q, κ).
By 13 we have that κ satisfy the equilibrium delivery rate
equation 4:
ps1rsj
[
κsj∑
i∈I
∑
li
αli(zilij)− −
∑
i∈I
∑
li
αliτilis
(zilij)−
]
= 0.
To prove market clearing we first notice that by adding agents
budget con-straint in t = 0 we have:
p0∑
i∈I
(xi0 − ei0) + q
sec∑
i∈I
(θi − ϕi) + qunsec∑
i∈I
zi ≤ 0, (14)
and by the market optimization problem 12:(
∑
i∈I
(xi0 − ei0),∑
i∈I
(θi − ϕi),∑
i∈I
zi
)
≤ 0. (15)
14By Carathéodory’s theorem we can fix N =
2(S+1)+2Jsec+Junsec+S+1 for all agents
21
-
Adding up the budget constraint in t = 1, using 15 and the
delivery rate equa-tion:
ps(∑
i∈I
(xis − eis − Y e
is)) ≤ 0 (16)
and by 13 we have that
∑
i∈I
(xis − eis − Y e
i0) ≤ 0. (17)
15, 17, the collateral constraint and the short selling
constraint on unsecuredassets implies that consumption and
portfolios are uniformly bounded (inde-pendent of n). So, for n
large enough the optimal choice is in the interior ofKn, this
implies two things: 1) Prices are strictly positive and 2) the
budgetconstraints in t = 0, 1 hold with equality. This means that
14 and 16 holdswith equality, and since prices are strictly
positive 15 and 17 also holds withequality.
4.2 A mixed equilibrium
We proceed to prove:
Theorem 4.5 Under the hypothesis made on preferences, endowments
and as-set structure, there is a mixed equilibrium for the
economy.
Proof We have a bounded sequence of equilibria(
(
xin, θin, ϕ
in, z
in1, z
in2, y
in
)
i∈I, pn, qn, κn
)
,
with
(
xin, θin, ϕ
in, z
in1, z
in2, y
in
)
=N∑
li=1
αnli
(
xinli , θinli, ϕinli ,
(
zinli)+,(
zinli)−, τ inli ◦ z
inli
)
,
and(
xinli , θinli, ϕinli , z
inli, τ inli
)
∈ ψin(pn, qn, κn).
We can extract a sub-sequence, such that
(
xinli , θinli, ϕinli , z
inli, τ inli
)
→(
xili , θili, ϕili , z
ili, τ ili)
,
and(pn, qn, κn) → (p, q, κ).
Feasibility, market clearing conditions and the equilibrium
seizure rate equationholds (because it holds for each n), so to
prove equilibrium existence we haveto prove that
(
xili , θili, ϕili , z
ili, τ ili)
is optimal when prices are (p, q, κ).
Take any (x, θ, ϕ, z) ∈ Bi(p, q, κ).
For n large enough (x, θ, ϕ, z) ∈ βin(p, q, κ).
22
-
By lower hemi-continuity, there is a sequence (xm, θm, ϕm, zm,
pm, qm, κm)converging to (x, θ, ϕ, z, p, q, κ) with (xm, θm, ϕm,
zm) ∈ β
in(pm, qm, κm) for m
large enough. For m ≥ n large enough (xm, θm, ϕm, zm) ∈ βim(pm,
qm, κm), so
U i(ximli) ≥ Ui(xm),
taking limits we obtain:U i(xili) ≥ U
i(x),
which proves optimality.
23
