Pró-Reitoria de Pós-Graduação e Pesquisa
Programa Stricto Sensu em Economia
Trabalho de Conclusão de Curso
BANKS AND CREDIT DERIVATIVES IN A GENERAL
EQUILIBRIUM MODEL WITH INCOMPLETE MARKETS AND
DEFAULT
Brasília - DF
2010
ADRIANO CAMPOS MENEZES
BANKS AND CREDIT DERIVATIVES IN A GENERAL EQUILIBRIUM MODEL WITH INCOMPLETE MARKETS AND DEFAULT
Tese apresentada ao Programa de Pós-Graduação Stricto Sensu em Economia da Universidade Católica de Brasília, como requisito parcial para obtenção do Título de Doutor em Economia.
Orientador: Dr. Jaime José Orrillo Carhuajulca
Ficha elaborada pela Biblioteca Pós-Graduação da UCB
06/04/2010
M543b Menezes, Adriano Campos
Banks and credit derivatives in a general equilibrium model with incomplete markets and default. / Adriano Campos Menezes. – 2010.
59f. 30 cm
Tese (doutorado) – Universidade Católica de Brasília, 2010. Orientação: Jaime José Orrillo Carhuajulca
1. Sistema bancário. 2. Créditos. 3. Finanças. 4. Contratos. 5. Economia. I. Carhuajulca, Jaime José Orrillo, orient. II. Título.
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•
Tese de autoria de Adriano Campos Menezes, intitulada "BANKS AND CREDIT DERIVATIVES IN A GENERAL EQUILIBRIUM MODEL WITH INCOMPLETE MARKETS AND DEFAULT", requisito parcial para obtencao do grau de Doutor em Economia de Empresas, defendida e aprovada em 26 de marco de 2010, pela banca examinadora constituida por:
Prof. Dr. Jaime Jose Orrillo Carhuajulca Orientador
Universidade Cat6lica de Brasflia
ＡｕｾｩｦＺ
Prof. Dr-ｊｾｾ･ ａｮｧｾｳｴ｡ do Amor Divino
ｾｮ｡､ｯｲ Intemo
Universidade Cat6lica de Brasilia
Prof. Dr. Paulo Cesar Coutinho Examinador Extemo Universidade de Brasilia
Referência: MENEZES, Adriano Campos Menezes. Banks and Credit Derivatives in a General Equilibrium Model with Incomplete Markets and Default. 59. Tese (Doutorado em Economia)-Universidade Católica de Brasília, Brasília, 2010.
Na primeira parte, desenvolve-se um modelo de equilíbrio geral com mercados incompletos e default para estudar em que condições os bancos se formam. Para que os bancos se formem endogenamente, nosso modelo integra os trabalhos de Zame (2007) e Dubey, Geanakoplos e Shubik (2005). A dinâmica do modelo é a seguinte: os bancos ou intermediários financeiros que são constituídos, os arranjos contratuais que prevalecem, as funções dos agentes nos bancos, os preços praticados pelos bancos no processo de intermediação financeira, os preços das commodities, os incentivos e a previsão da taxa de default são determinados endogenamente no equilíbrio. Os agentes escolhem em quais bancos trabalhar, quais funções irão exercer, e qual o nível de esforço será realizado. Além disso, eles também escolhem os níveis de consumo, aplicações e empréstimos no sistema financeiro, bem como os pagamentos efetuados no segundo período. O modelo acomoda perigo moral e seleção adversa dentro da estrutura organizacional do banco. Nós provamos que o equilíbrio existe. Na segunda parte, desenvolve-se um novo tratamento para uma economia com problemas de direitos de propriedade sobre ativos colaterizados. Para lidar com esta questão, introduzimos derivativos de crédito na estrutura de um modelo GEI. Neste novo mercado provamos a existência de equilíbrio, combinando a técnica de abordagem da demanda e preços livres de arbitragem.
We developed a general equilibrium model with incomplete markets and default to study in which conditions banks could appear/be formed. To the banking system be formed endogenously, our model integrate the works of Zame (2007) and Dubey, Geanakoplos e Shubik (2005). The dynamic of the model is the following: the set of Banks or Financial Intermediaries that form, the contractual arrangements that appear (wages and payments contractual), the assignments (roles) of agents to bank, the intermediation prices (lending/borrowing money) faced by banks for trading, the price of commodities, the incentives to agents and forward looking of default rate are all determined endogenously at equilibrium. Agents choose which banks to work, which roles to occupy in those banks, and which actions to take in those roles. Beside they also choose consumption, how much to borrow or lend in the banking system, and deliveries at second period. The model accommodates moral hazard and adverse selection within the inner structure of bank. We proved the equilibrium exists. In second work, we developed a new approach to an economy with problems of the ownership rights on assets backed by collaterals. To deal with this issue, we introduce credit derivatives structure in a GEI model. We proof the existence of equilibrium by combining the demand approach and no-arbitrage prices in this new market.
1 Introduction p. 7
2 Endogenous Formation of Banking System in the GEI model with
Default p. 9
2.1 Motivation and Review of Literature . . . p. 9
2.2 The Model . . . p. 12
2.2.1 Time and Uncertainty . . . p. 12
2.2.2 Commodities . . . p. 12
2.2.3 Banking System . . . p. 13
2.2.4 Agents . . . p. 15
2.2.4.1 Roles . . . p. 15
2.2.4.2 Investments, Consumption and Deliveries . . . p. 16
2.2.4.3 Choice Set . . . p. 17
2.2.4.4 Payoffs and Utilities . . . p. 17
2.2.4.5 Skills . . . p. 19
2.2.4.6 The Space of Agent Characteristics . . . p. 19
2.2.5 The Economy . . . p. 19
2.3 Budget Set . . . p. 20
2.4 Beliefs and Expected Payoff . . . p. 22
2.4.1 Job Markets Clearing . . . p. 23
2.4.2 Aggregation . . . p. 25
2.4.2.1 Distribution on Skill and Behaviors . . . p. 25
2.4.2.2 Aggregation of Individual Choices . . . p. 26
2.4.2.3 Aggregation of the Portfolio of Banking System . . . . p. 28
2.5 Equilibrium . . . p. 28
2.6 Main Results . . . p. 31
2.7 Proofs . . . p. 31
3 Credit Derivatives in the GEI model p. 42
3.1 Motivation and Review of the Literature . . . p. 42
3.2 The Model . . . p. 44
3.2.1 Time and Uncertainty . . . p. 44
3.2.2 Commodities Market and Credit Derivatives . . . p. 44
3.2.3 The Economy . . . p. 45
3.3 Budget Set . . . p. 45
3.4 Equilibrium . . . p. 45
3.5 Arbitrage . . . p. 46
3.6 Results . . . p. 49
1
Introduction
The objective of this PhD thesis is to offer contributions in the fields of general
equilibrium theory, more specifically, focused in endogenous formation of a banking system
and development of new financial instruments. The subjects studied here required finance
and microeconomic analysis. In banking context, the next chapter presents a general
equilibrium model with incomplete markets and default to study in which conditions
banks could appear/be formed. From the point of view of the economic literature, this
analysis takes steps toward integrating production in the context of GEI models as Zame
(2007) and GEI models with default from Dubey, Geanakoplos e Shubik (2005). However,
instead production of goods, we are interested to present a structure where the financial
intermediaries or firms called banks are endogenously formed at equilibrium.
Zame (2007) in his seminal paper begins an integration of firm theory in the spirit of
Alchian e Demsetz (1972) and Grossman e Hart (1986), contract theory in the spirit of
Holmstrom (1979), and general equilibrium theory in the spirit of Arrow e Debreu (1954)
and McKenzie (1959). He developed an model in which agents interact anonymously with
the large market, but strategically within productive groups called firms. As he regarded,
the literature largely ignores the interaction between firm and the market, taking given
the set of firms that form, the organization structure of these firms, the assignments of
agents to firms, and the other things. With the pioneering works of Dubey, Geanakoplos
e Shubik (2005), default was introduced in GE model with incomplete markets. They
of equilibrium1. The main result of the next chapter is to present one necessary condition
to form a banking system at equilibrium. This condition claims that the population
of agents supports common beliefs compatible with the distribution of skill and actions
required by banking system. Those beliefs are formed by rational expectations about the
inner structure of bank in spirit of issues raised by the modern theory of the firm and
contract theory. Another interesting point is that the Dubey, Geanakoplos e Shubik (2005)
model goes out as a special case from our model. It happens when the idiosyncratic risks
are not present inside banks, and also does not exists moral hazard and adverse selection in
the organization structure. Further, our model also accommodates moral hazard, adverse
selection, signaling, and insurance in the same manner as shown by Zame (2007) model.
In the third chapter, we developed a new approach to an economy with problems of
the ownership rights on assets backed by collaterals. To deal with this issue, we introduce
credit derivatives structure in a GEI model. Credit Derivatives have been applied in
situations in which it is necessary to protect the ownership rights and transfer risks to
other part.
The general equilibrium literature has suggested two mechanisms to enforce standard
debt contracts: penalization in terms of utility (see Dubey, Geanakoplos e Shubik (2005));
or the requirement of collateral from borrowers (see Geanakoplos e Zame (2007)).
How-ever, until now nothing was done in terms of applications of the credit derivatives in
earlier works.
2
Endogenous Formation of
Banking System in the GEI
model with Default
2.1
Motivation and Review of Literature
WHY Do Financial Intermediaries exist? In the ideal world of frictionless and
com-plete markets, both investors and borrowers would be able to diversity perfectly and
obtain optimal risk sharing. Moreover, with small indivisibilities and nonconvexities in
transaction technologies, the things never are what they seem. So, Financial
Intermedi-aries (or banks) appears to solve the problem (see Fama (1980), Gurley e Shaw (1960),
Benston e Smith (1976) Pyle (1971) )
But it is not complete. As presented by our model, it also depends on the agents’
beliefs about the banking system. More specifically, those beliefs analyze the capable,
abilities or skills that a particular agent has about itself and the other agents with whom
they are matched in the various banks. From this point of view, many issues can be
exploited as, for example, confidence degree, security, and so.
Our model provides new approach in this subject/literature. First, we include firms
with features of intermediation structure called banks. Secondly, we introduce default as
endogenous variable in the model that allows the agents do not keep all their promises.
The dynamic of the model is the following: the set of Banks or Financial Intermediaries
the roles/assignments of agents to bank, the intermediation prices (lending/borrowing
money) faced by banks for trading, the price of commodities, the incentives to agents and
forward looking of default rate are all determined endogenously at equilibrium. Agents
choose which banks to join/work, which roles to occupy in those banks, and which actions
to take in those roles. Beside they also choose consumption, how much to borrow or
lend in the banking system, and deliveries at second period. The model accommodates
moral hazard and adverse selection within the inner structure of bank. We proved the
equilibrium exists.
We focus in the endogenous formation of the banking system. It means that the set
of possible bank types is viewed as given exogenously, but the set of observed bank types
is determined at equilibrium. This approach is parallel to taken in Zame (2007), Dubey,
Geanakoplos e Shubik (2005), Dubey e Geanakoplos (2002), Geanakoplos e Zame (2002).
In this model, the sources of uncertainty in this virtual economy came from two kinds
of risks. First, agents face the stochastic nature of the finance intermediation process and
the endogenously determined behavior of other agents with whom they are matched in
the various banks. We called it as idiosyncratic or individual risks. Secondly, there exist
macroeconomic risks or, more specifically, exogenous collective risks connected with each
possible state of nature at date 2.
The modern theory about general equilibrium began with Arrow e Debreu (1954)
and McKenzie (1959), when this theory is applied to an economy with uncertainty and
complete contingent markets. With recent developments, this theory has explored the
implications of incomplete markets, where Diamond (1967) is responsible by the seminal
work. This paper was the first to consider which agents have limited ability to transfer
income across time in relation to the possible probabilistic states that an economy might
find itself in. We follow the line of Radner (1972). Ever since, many authors have
From another standpoint, default was considered as a sign of disequilibrium in terms
of GE models. Then agents keep all their promises by assumption. However, Dubey,
Geanakoplos e Shubik (2005) showed the contrary. They introduced default in the context
of GE model with incomplete markets, allowing agents do not honor their commitments.
As mentioned before, the framework of the present chapter builds on Zame (2007) and
Dubey, Geanakoplos e Shubik (2005). Thus, it may be useful to note some of differences
from the present paper and Zame (2007) and Dubey, Geanakoplos e Shubik (2005):
• First, in Zame (2007), he considers a production economy, not banks. There is not
finance market to transfer wealth across time and his model is static. We work with
a two-period economy where the sell and buy of securities in the market is made
through banks. There are not firms in sense of production economy.
• Second, in Dubey, Geanakoplos e Shubik (2005), the expected delivery rate is
mea-sured for each state and asset. In the present paper, this payment rate is meamea-sured
for each state and bank, incorporating the aggregate risk of its portfolio of securities.
• The uncertainty in Zame model came from the idiosyncratic risk inside the inner
or organizational structure of firms in which agents chose to match or join. DGS
model consider just collective risks (i.e, it affects all agents) in usual sense of the GEI
literature. Our model works with both risks in sense adopted in the two models.
The remainder of this paper is organized as follows. Sections 2.2, 2.3, 2.4 present the basic
structure of model. Section 2.5 defines conditions for existence of equilibrium. Section
2.2
The Model
2.2.1
Time and Uncertainty
We consider a two-period economy, where agents know the present but face an
un-certain future. There are a finite set of agents H and a finite number of bank types
b∈B ={1,2, . . . , B}. The sources of uncertainty in this virtual economy came from two
kinds of risks. First, we model a finite set of states of nature or collective risks as usual
in general equilibrium literature since Arrow e Debreu (1954). In period 0 (the present)
there is just one state of nature (called state 0) In the second period (date 2), there are
ξ states of nature to be revealed. These are represented by a set
Ξ ={1, . . . , ξ, . . . ,Ξ}.
Latter, following Zame (2007), idiosyncratic risks are introduced inside the organizational
structure of banks that could be form. In period 0, agents face the uncertainty due to the
stochastic nature of the finance intermediation process and the endogenously determined
behavior of other agents with whom they are matched in the various banks where they
decide to work. We model this uncertainty by
Ω = Ω1×. . .×ΩB
where Ωb are the consequences originated into the bank type b.
2.2.2
Commodities
There are L commodities traded on competitive markets, so the commodity space is
RL
2.2.3
Banking System
Again, we assume that there are a finite number of bank types b∈B ={1,2, . . . , B}.
Eachbank type b characterizes by its technology and financial contracts.
Banking Technology
Following Zame (2007) a banking technology can be described as a tupleTb = (F
b, Sfb, Abf,Ω,R, π˜ )
where:
• Fb = {1, . . . , Fb} represents a finite set of position (jobs) inside of bank b. ∀f ∈ b
∃v(f) that represents the number of vacancies to an specific functionf inside of the
bank b.
• For each job f ∈ Fb, we associate two finite sets: Sfb and Abf. Sfb is the set of skills
required for the execution of the job f; and Af is the set of actions realized by the
agents that occupied the vacancies of the job f.
• Ωb ={1, . . . ,Ω}is a finite set of consequences originated by the interaction of agents
into bankb. ω ∈Ω summarizes everything that is observable or verifiable hence what
is contractible.
• For each type of bankb ∈B, there areM assets1 (credit products) andN liabilities2
available for trading in the first period. These securities are characterized by their
payoffs ˜Rj : Ξ → RL, j = 1, . . . M +N which are summarized by the following
matrix
˜
R= [I P]
whereI ∈RΞ×M×L
+ is the investment-payoff matrix andP ∈RΞ+×N×Lis the
liability-payoff matrix
˜
R = [I1 . . . IM P1 . . . PN]
1what the bank purchases
• Each bank purchases and sells securities. The purchasing and selling portfolios are
represented by the vector (α, δ) : Ω →RM
+ ×RN+
• There is a family of conditional probabilities
πb :Sb×Ab →P(Ω) where
Sb =×f∈FbS
b
f, and A
b
=×f∈FbA
b f
e.g, π(ω|s,a) is the conditional probability that ω will occur/happen if the various
roles are filled by individuals whose skill profile is s= (s1, . . . , sFb) and who choose
action profile a= (a1, . . . , aFb).
Contracts
Given the technology Tb, the menu of contracts is described by the following
func-tion
C :F ×RL
++×Ω×Ξ→R
Each bank with technology Tb enters in the economy by purchasing and selling
secu-rities in the first period. In the second period the following condition must be satisfied.
Hence, for each consequenceω ∈Ω one has
X
f∈F
Cb(f, p, ω, ξ) =p
ξ·[Iξαb(ω)−Pξδb(ω)], ξ ∈Ξ (1)
says (1) that after the state nature ξ is solved in the second period, the bank b spends
in payments of contracts to employees and payments of the borrowing made in the first
period. All these are financed of the receipts from purchasing assets (granting of credits).
2.2.4
Agents
2.2.4.1 Roles
Roles inside each individual bank
We call of role to the job-action pair (f, a)∈Fb×Af.
For the sake of simplicity some times we call of role to the job without mentioning
the action. The set of all roles into bankb is denoted byRb
o. More precisely,
Rb
o ={(f, a) :f ∈Fb, a∈Af}
is the set of all roles inside of bank of typeb which consists of positions (jobs) f ∈F and
actionsa∈Af taken by the agents who occupy such position f and
Rb =Rb0∪ {0}
to be the set of roles augmented which allows agents do not have any role inside bank b.
That is, an agenth∈ Rb may not have any participation inside bankb so that he/she do
not receive income from wages for each job chosen plus contractual payments payed by
the bankb.
Roles inside banking system
In a similar way, define the sets
R0 =×b∈BRb0
to be the set of all roles inside any bank of typeb; and
R=×b∈BRb
That is, an agenth∈ Rb may not have any participation inside any bank so that he/she
do not receive income from contracts payed by any banks.
A bank-role-action choice is an element γ = (γ1, . . . γB)∈ R. That is,
γb ∈ Rb =Rb
o∪ {0}={(f, a) :f ∈Fb, a∈Af}
Thus if γb 6= 0 then γb = (f, a) ∈ F
b ×Af; by convention, γb = 0 represents the choice
not to belong to bank type b.
The role function
For each bankb define the mapping
(ρb, τb) :R → Rb
to be
(ρb, τb)(γ) := (ρb(γ), τb(γ)) =γb = (f, a)∈F
b×Af
This implies that
ρb(γ) =f ∈Fb and τb(γ) =a ∈Af with f ∈Fb
Notice that if γb = 0 then no job has been chosen and therefore no action will be taken.
That is,ρb(γ) = 0 =τb(γ)
2.2.4.2 Investments, Consumption and Deliveries
Agents choose consumption-investment-deliveries plans. These choices depend on the
realization of the uncertainty they faced.
An consumption plan is
x= (xo,x˜)∈R+ΩL×(RΩ+L)Ξ =R
A portfolio plan is
(θ, ϕ)∈RΩ+N ×RΩ+M
B
A delivery plan is
D∈RΩΞ+ BL
2.2.4.3 Choice Set
Define the correspondence
X : 2R →→RΩ+L(Ξ+1)×
RΩ+N ×RΩ+MB×RΩΞ+ BL×Γ
such that for Γ⊂ Ra nonempty set of bank-role-action choices assigns the corresponding
choice set
X(Γ) = {((x, θ, ϕ, D);γ)∈RΩ+L(Ξ+1)×
R+ΩN ×RΩ+MB×RΩΞ+ BL×Γ :
γb = 0⇒(x, θ, ϕ, D) is independent ofωb}
2.2.4.4 Payoffs and Utilities
Define the sets:
S=×b∈BSb and A=×b∈BAb
to be skill and action sets respectively.
Each agent h is characterized by payoff function
Vh :X(Γ)×S×A →R
and an initial endowment ((e,0,0,0); 0) ∈ X(Γ). We require the following assumption:
This payoff is defined by
Vh(x, θ, ϕ, D, γ,s,a) =Uh(x, γ,s,a)−Λ([Iϕ−D]+,s,a)
where the first term is the expected utility for consumption and bank-role-action plan;
and the second is the expected disutility Λ felt by delivering D∈(RM×Ξ
+ )B less than the
promisedIϕ, andI ∈RM×Ξ
+ Notice that the utility penalty is the only incentive to agent
delivers anything on his promises.
Utility used to define the expected utility above is a mapping
uh :RL+(1+Ξ)× R ×S×A×Ω→R
so thatuh(x, γ,s,a, ω) is the utility obtained if the agent h consume x∈RL(1+Ξ)
+ chooses
the bank-role-action profileγ,faces bank members with skill-action profile (s,a),and the
profile of consequencesω occurs. We assume uh,for simplicity, separable among states of
natureξ. Thus
uh(x, γ,s,a, ω) =uho(xo, γ,s,a, ω) +
X
ξ
π(ξ)vh(xξ,s,a, ω)
where uh
o : RL+ × R ×S× A×Ω → R and vh : RL+ ×S ×A× Ω → R are utilities
satisfying the usual properties (concave, continuous, and strictly increasing). We require
the following assumptions:
u(0, γ,s,a, ω)< u(e,0,s,a, ω) for all γ,s,a, ω (H2)
lim
|x|→∞u(x, ǫ,s,a, ω) = ∞, for every s,a, ω (H3)
There is a z >0 such that u(x, ǫ,s,a, ω)≤z(1 +|x|) for every ǫ,s,a, ω. (H4)
The function Λh :R
+× R ×S×A×Ω→R is the penalty for defaulting.
1. The definition of uh(x, γ,s,a, ω) implicitly require that each agent believes that in
any bank she/he joins, all vacancies for jobs will be filled.
2. It is required that
γb = 0⇒uh is independent of ωb, γb = (f, a)⇒uh is independent of sb
f,a
b
f.
That is, utility is independent of skills and actions in banks to which the agent h
does not belong, and utility is independent of skills and actions of others in the
banking roles that the agent h fills.
2.2.4.5 Skills
Each agent h is endowed with an array of skills:
sh ∈Sh =×b∈BShb=×b∈B[×f∈FbS
hb
f ]
2.2.4.6 The Space of Agent Characteristics
Write Z for the space of characteristics
(Γ; (U,Λ);e;s)∈2R× V × X(Γ)×S.
Any agenth∈H is modelled by its characteristics (Γ; (Uh,Λh);eh;sh). That is, the agent
h∈His characterized by choice set (common for everyone); his/her payoff which depends
on utility function and penalty; his/her initial endowment and his/her array of skills.
2.2.5
The Economy
An economy is defined by
E =
(Γ; (Uh,Λh, Qh);eh;sh)
h∈H,(T
b)
b∈B
where Tb represents the technological arrangement which is used by a bank b and Qh is
the bound on short sale issued by each agent.
2.3
Budget Set
In order to define the budget set we need to define the income from the job market
for those ones who decided to join a bank type. For that we follow Zame (2007). Thus we
interpret a role in a bank as a job and its price as a wage; and to adopt the convention
that wages are paid to the members of the bank. We formalize this in the following way:
2.3.1
Wages
Define
M={(b, f) :b∈B, f ∈Fb}
the set of memberships (or bank-role pairs). A wage structure, or wage for short, is a
function
w:M → R
such that
X
f∈Fb
v(f)w(b, f) = 0, for each bank b.
If w is a wage structure and γ ∈ R is a profile of bank-role-action choices, then we
write
w.γ = X
γb6=0
w(b, ρb(γ))
If we assuming thatγb = 0 then ρb(γ) = 0 therefore w(b, ρb(γ)) = 0. Thus we can write
w.γ= X
γb6=0
w(b, ρb(γ)) = X
b∈B
2.3.2
Budget Constraints
Given private good prices p, asset prices (q1, q2), payment rate K, and wages w, the
choice ((x, θ, ϕ, D);γ) is budget feasible for the agent h,if the choices are budget feasible
in each profile of consequences and states of nature (ω, ξ)∈Ω×Ξ
poxo(ω) +q1θ(ω)≤poeho +q2ϕ(ω) +w.γ (2)
pξx˜(ω, ξ) +pξD(ω, ξ)≤pξeh(ξ) +Kξ(pξPξθ(ω) +Cξ) (3)
ϕhb(ω)≤Q h
b ∈R
M
+,∀b∈B (4)
we have Ω(1 + Ξ +M B) budget constraints in total.
Notice that
Db(ω, ξ) =
M
X
m=1
Dbm(ω, ξ)
and
0≤Db(ω, ξ)≤pξIξϕb(ω) = pξ M
X
m=1
Iξmϕbm(ω)
Then,
D(ω, ξ) := X
b
Db(ω, ξ)∈RL
+ (5)
Notice that also
θ(ω) := X
b
θb(ω)∈RN
+ (6)
and thereforeq1Pbθb(ω)∈R+
q1 ∈RN+ independent of ω as in Zame (2007)
Similarly, define
ϕ(ω)∈RM+
Define
KξpξPξθ(ω) :=
X
b∈B
KξbpξPξθb(ω) (7)
and
KξCξ :=
X
b∈B
Kb
ξC
b(ρb(γ), ω, ξ) (8)
The budget set for the agent h is therefore
Bh(p, q1, q2, K) ={((x, θ, ϕ, D);γ)∈ X(Γ) : (2),(3) and (4) are satisfied}
2.4
Beliefs and Expected Payoff
Next we are going to put in explicit way with respect to what probability are taken
the expectation of the terms of the payoffs.
A system of beliefs is a probability measure β on
S×A=×b∈B
Sb ×Ab
the space of all skill-action profiles for all types of banks . Given the belief β, write βb
for the marginal β on Sb ×Ab, and write βb
f for the marginal β on Sbf ×Abf. Consider
an agenth with characteristicsζh = (Γ; (Uh,Λh);eh;sh), who holds beliefs β and chooses
((x, θ, ϕ, D);γ)∈ X(Γ).Fixω = (ω1, . . . , ωB)∈Ω and a skill-action profile (s,a)∈S×A
Ifγb 6= 0 andf =ρb(γ) is the job chosen in the bankb,then the probability that the agent
h assigns to observing the consequence ωb in bank type b when complementary agents
have skill action profile (s,a) is
ˆ
πb(ωb|γ,s,a) =πb(ωb|sb, τb(γ),sb
−f,a b
−f).
Ifγb = 0, define
ˆ
πb(ωb|γ,s,a) = πb(ωb|sb,ab).
complementary agents have skill-action profile (s,a) is
ˆ
π(ω|γ,s,a) = Πb∈Bπˆb(ωb|γ,s,a)
and the agenth′s expected payoff is
EVh(x, θ, ϕ, D, γ|β) =E
ˆ
πUh(x, γ|β)−EπˆΛh
[Iϕ−D]+|β
(9)
where the agent h′s expected utility is
EπˆUh(x, γ|β) = Eˆπuho(xo, γ|β) +
X
ξ∈Ξ
π(ξ)Eπˆ(vh(˜x(., ξ)|β)
=X
(s,a)∈S×Au
h
o(xo(ω), γ,s,a, ω)ˆπ(ω|γ,s,a)β(s,a)
+X
ξ∈Ξ
π(ξ)X
(s,a)∈S×Av(˜x(ω, ξ),s,a, ω)ˆπ(ω|γ,s,a)β(s,a)
and the agenth′s expected utility penalty is
EπˆΛh
[Iϕ−D]+|β=X
ξ
π(ξ)Eπˆ
h
Λh([pξI(ξ)ϕ−D(., ξ)]+|β)
i
=X
ξ∈Ξ
π(ξ)X(s,a)∈S×AΛh([pξI(ξ)ϕ−pξD(ω, ξ)]+,s,a, ω)ˆπ(ω|γ,s,a)β(s,a)
Remark: Notice that in the definition of expectation above we have assumed that the
probability of stateξ occurs, π(ξ), is independent of idiosyncratic risk.
2.4.1
Job Markets Clearing
Let γ :H → R be an allocation of bank-role-action choices. Let ρb :R → F
b be the
projection map on the set of roles in the bankb.
For each b ∈B define
Tb(f) = (ρb◦γ)−1(f) = {h∈H : (ρb◦γ)(h) =f}={h∈H :ρb(γh) = f}
which consists of all of agents who choose an specific role/jobf ∈Fb inside in individual
two different jobs or roles in the same bank. More precisely, one has: If f′, f ∈ F
b and
f′ 6=f then
Tb(f′)∩ Tb(f) =∅
Thus the set of all agents that fill the vacancies of the jobs in the bank b is
Tb =
X
f∈Fb
Tb(f)
where the right hand side is the disjoint union of sets Tb(f).
If let us denote by n(Tb) the number of agents that filled the vacancies of the jobs in
the bankb, then
n(Tb) =
X
f∈Fb
n(Tb(f)) =
X
f∈Fb
v(f)
The condition Tb(f) 6= ∅,∀f ∈ Fb is a necessary condition for the previous equality be
satisfied (that is, the bank b form), however it is not sufficiency. I may happen that
Tb(f) 6= ∅ but n(Tb(f)) 6= v(f). From this we can conclude a necessary and sufficiency
condition for the bankb form is
∀f ∈Fb, n(Tb(f)) =v(f).
More formally, one has the following result
Lemma 2.1. The bank of type b ∈B forms if and only if
∀f ∈Fb, n(Tb(f)) =v(f).
Definition 2.1. 1. We say that the job market clears if all banks have filled all their
vacancies. That is,
n(Tb) =
X
f∈Fb
v(f),∀b∈B
2. We say that the structure wage w:M → R clears the job market if
X
h∈Tb
Define the set
Ψ ={(γ1, . . . , γH)∈(Γ)H : X
ρb(γh)∈F
b
n(Tb(ρb(γh)) =
X
ρb(γh)∈F
b
v(ρb(γh)),∀b∈B} as all feasible configurations so that all banks have been filled by agents.
2.4.2
Aggregation
2.4.2.1 Distribution on Skill and Behaviors
A choice of banks, roles, and actions for each agent in the economy induces a
distri-bution of skills and behaviors within each bank. Remember that
Tb(f) = (ρb◦γ)−1(f)
where
ρb :R → F
b and γ :H → R
Now, define
(id, γ) :Tb(f)→ Tb(f)× R
and
η:H× R →Sfb ×A
b f
η(h, γ) = (sh
f, τ
b(γ))
Then
g = [η◦(id, γ)] :Tb(f)→Sfb ×A b f
g(h) = η(h, γh) = (sh
f, τb(γh))
Remember also that
∀f ∈Fb, n(Tb(f)) =v(f)
is a necessary and sufficiency condition for the bank b to form.
byςb
f =g∗n is a measure on Sfb ×Abf.Normalizing one has
ϑbf =
ςb f
||ςb
f||
is a distribution of skill and action in role f in the bank b given the choices γh ∈ R. In
addition
X
(s,a)∈Sb f×Abf
ϑbf(s, a) = 1
By definition
ςb
f(s, a) =n(g
−1(s, a)) =n{h∈T
b(f) :g(h) = η(h, γh) = (sh, τb(γh)) = (s, a)}
||ςb
f||=ς
b
f(S
b
f ×A
b
f) =n(Tb(f)) =v(f)
Because matching is random, the distribution of skills and actions over roles in the
bankb is
ϑb(.|γh) =ϑb1 ×. . .×ϑbFb =×f∈Fbϑ
b f
and the distribution of results across banks of typeb is given by
˜
π(ωb|γh) = X
Sb×Ab
π(ωb|s,a)ϑb(s,a|γ)
2.4.2.2 Aggregation of Individual Choices
Fix γ = (γ1, . . . , γB)∈ R an bank-role-action plan. Now, define the following sets
B(γ) ={b ∈B :γb 6= 0}
T(γ) ={h ∈H :ψh =γ}
Forb∈B(γ), define
Sb(γ) =
Sb, if b6∈B(γ)
Sb
−ρb(γ), if b∈B(γ) Ab(γ) =
Ab, if b 6∈B(γ)
Ab
Define also
S(γ) =S1(γ)×. . .×SB(γ) A(γ) =A1(γ)×. . .×AB(γ).
For (s,a)∈S(γ)×A(γ), b∈B, and ω = (ω1, . . . , ωB)∈Ω, define
ˆ
πb(ωb|γ,(s,a)),πˆ(ω|γ,(s,a)), and ϑ(.|γ) as before.
Set ϑ(.|γ) =ϑ1(.|γ)×. . . ϑB(.|γ) and let ϑ
γ(.|γ) be the marginal of ϑ(.|γ) on S(γ)×
A(γ).
ϑγ(.|γ) is the distribution of skills and actions in the all roles in all banks except those
inγ ∈ R.
Thus the expected consumption of h with characteristicsζh = (Γ; (Uh,Λh);eh;sh) is
E[xho(.)] =
X
ω∈Ω
X
(s,a)∈S(γ)×A(γ)
xho(ω)ˆπ(ω|γ,(s,a))ϑγ(s,a)
Thus the expected consumption of agents in T(γ) is
X
h∈T(γ)
E[xho(.)]
so, by an abuse of notation, the total expected consumption of all agents is defined as
X
h∈H
xho :=
X
γ∈R
X
h∈T(γ)
E[xho(.)]
Of course, some of the setsT(γ) may be empty, in which case, the corresponding
contri-bution to aggregate consumption will be zero. Similarly, we define
X
h∈H
θh,X
h∈H
ϕh, X
h∈H
xhξ,
X
h∈H
2.4.2.3 Aggregation of the Portfolio of Banking System
Using the same distribution of skills and behaviors described above, we can define the
expected loans of a bank typeb as
E[αb] =
1
v(Fb)
X
f∈Fb
n(Tb(f))
X
ωb∈Ωb
αb(ω)˜π(ωb) where ˜π(ωb) = P
s,a∈Sb×Abπˆb(ωb|s,a)ϑb(s,a). Thus, by an abuse of notation, the total of
loans in the economy is
X
b∈B
αb :=X
b∈B
E[αb] Similarly, one defines
X
b∈B
δb :=X
b∈B
E[δb]
2.5
Equilibrium
Definition 2.2.An equilibrium for economyE =
(Γ; (Uh,Λh, Qh);eh;sh)
h∈H,(T
b)
b∈B
consists of (p; (q1, q2);K;w;β) a commodity-price system, security-price system, payment
rate system, wage structure, and beliefs; an allocationγh; (xh, θh, ϕh, Dh)
h∈H of
individ-ual choices, such that:
1. Feasibility of choices:
For each hwith characteristicζh,the vectorγh; (xh, θh, ϕh, Dh)is a budget-feasible
at (p; (q1, q2);K;w).
2. Optimality
Given(p; (q1, q2);K;w)and beliefsβ one has
γh; (xh, θh, ϕh, Dh)maximizesEVh(x, θ, ϕ, D, γ|β
on the budget set of agent h.
(a) Job market:
(γ1, γ2, . . . , γH)∈Ψ (b) Commodity markets:
X h xh o = X h eh o X
h∈H
xhξ =
X
h∈H
ehξ, ξ∈Ξ
(c) Security markets:
X
h∈H
ϕh = X
b∈B
αb ∈RM
+
X
h∈H
θh = X
b∈B
δb ∈RN
+
4. Banking Payment rate is correctly anticipated:
Kb ξ = P
h∈Hpξ✵˜b(D h ξ)
P
h∈HpξIξ✵b(ϕ
h), if pξIξ✵b(ϕ
h)>0
arbitrary, if pξIξ✵b(ϕh) = 0
,∀b∈B
where
˜ ✵b : M
b∈B
RL
+→R+L
is the linear function defined to be
˜
✵b(d1+. . .+dB) =db and
✵b :M
b∈B
RM+ →R+M
is the linear function defined to be
✵b(v1+. . .+vB) =vb 5. Beliefs are correct for banks that form:
Remarks:
1. From the condition (4), like Dubey, Geanakoplos e Shubik (2005), it suffices to say
that each potential lender is correct in his expectation about the fraction of promises
that do in fact get delivered from certain pool of agents, in this case, clients to bank
typeb. Moreover, his expectation of the rate of delivery does not depend on anything
2.6
Main Results
We require the assumptions H1, H2, H3 andH4 defined previously.
Theorem 2.1. If individual endowments are bounded and all goods are available in the
aggregate, then an equilibrium exists.
2.7
Proofs
Proof of Theorem 2.1: The idea of the proof is to construct an artificial economy
for which prices in generic terms (It considers prices of commodities, securities, wages and
beliefs) are bounded. Like Dubey, Geanakoplos e Shubik (2005), consider an external ǫ
-agent who sells and buysǫ= (ǫb)b∈B ∈RM B++ for every banking securities, and fully delivers
on his promises. Letν(ǫ) be an equilibrium obtained with the ǫ-agent. This equilibrium
is called anǫ-boosted equilibrium. Then, an equilibrium ¯ν is called arefined equilibrium if
there exists a sequence ofǫ-boosted equilibriaν(ǫ) withǫ→0 andν(ǫ)→ν.¯ The argument
is developed in 6 steps. For each k > 0, Step 1 constructs a compact convex space of
prices, choices and banking payment rate. These choices refer to as soon
consumption-investment-deliveries plans as bank-role-action choices. These spaces are built so that
the commodity demands of agents are impossibly large when prices are on the boundary.
Step 2 constructs an aggregate excess demand correspondence from this truncated space
into itself. Step 3 shows that the excess demand in all markets are bounded. Hence, we
show that aggregate excess demand (including the external fictitious agent) goes to zero
ask →0. In Step 4, one proves that the commodities prices, securities prices and wages
in this artificial economy stay away from the boundary specified in Step 1. Step 5 shows
that the aggregate excess demand is bounded therefore has a convergent subsequence.
Finally, Step 6 shows that our artificial economy has a refined equilibrium.
λ∈RHΞB
+ , are finite.
We say that beliefs β ∈ △(S×A) and the distribution ϑb
f(.|γh), on skills and actions
Sb
f ×Abf, induced byγh for each agent are compatible if
βfb =ϑ b
f(.|γ
h
)⇐n(Tb(f)) = v(f)>0 (10)
Notice that βb
f is the marginal of βb onSfb ×Abf. Define
H ={(β,(γh)
h∈H)∈ △(S×A)×Ψ}
For each k >0, define
△k ={(p, q1, q2)∈RΞ(+L+1)×RN+ ×RM+ :pξ ∈ △L−1, ξ∈Ξ∗,
pξl≥k∀(ξ, l)∈Ξ∗×L, and 0≤q1n≤
1
k,0≤q2m ≤
1
k,∀n ∈N, m∈M}
Wk={w∈RM :|w(b, f)| ≤
1
k,∀b, f}
The assumption H3 implies that
Uh(x)> Uh(2 X
h′∈H
eh′
),∀h∈H
Define the truncated budget set of agenth ∈H to be
Bh
✷ ={((x, θ, ϕ, D);γ)∈ X(Γ) :||x||∞≤M , ϕˆ ≤Q
h, θ≤Q˜h,||D||
∞≤Qˆh, γ ∈Γ}
Let
B :=×h∈HB✷h.
Define the following sequence
ν := (p, q1, q2, w, K, β,((xh, θh, ϕh, Dh);γh)h∈H)∈ △k× Wk×[0,1]Ξ×B×Π1(H)× B
K(ǫb) = min{1,
pξIξǫb+Ph∈Hpξ✵˜b(Dξh)
pξIξǫb+Ph∈HpξIξ✵b(ϕh)
}
Consider the map
Kk :△k× Wk×[0,1]Ξ×B×Π1(H)× B →[0,1]Ξ×B
defined by
Kk(ν)bξ =
(K(ǫ
b), if Iξ 6= 0
1, if Iξ = 0
Step 2: Next, we define the following excess of demand in the markets of commodities
and securities
Z(p, q1, q2, w, K, β) := (Zo,(Zξ)ξ∈Ξ)
where
Zo(p, q1, q2, w, K, β) :=
X
h∈H
(xh
o −e
h
o)
Zξ(p, q1, q2, w, K, β) :=
X
h∈H
(xh
ξ −e
h
ξ), ξ ∈Ξ
Y(p, q1, q2, w, K, β) = (Y1, Y2)
where
Y1 =
X
h∈H
θh− X
b∈B
δb, and Y2 =
X
h∈H
ϕh− X
b∈B
αb
Define also the excess of demand in job markets
Y(p, q1, q2, w, K, β) = (Yfb(p, q1, q2, w, K, β))(b,f)∈M
where
Yb
f(p, q1, q2, w, K, β) =n(Tb(f))−v(f)
Thus
w.Y(p, q1, q2, w, K, β) =
X
(b,f)∈M
Next, consider the correspondence
P :△k× Wk×[0,1]Ξ×B×Π1(H)× B → △k× Wk×Π1(H)
defined as
P(ν) = arg max
(p,q1,q2,w,β){po·Zo(p) +q1·Y1(p) +q2·Y2(p)−
−w· Y(p) +X
ξ
pξ·[Zξ(p)−
X
b∈B
(1−Kk(ν)bξ)Iξǫ]}
where
p= (p, q1, q2, w, K, β).
Finally, for eachh∈H, define the correspondence
Φh
k :△k× Wk×[0,1]Ξ×B×Π1(H)× B →Bh✷
to be
Φh
k(ν) = arg max
((x,θ,ϕ,D);γ)
n
EVh(x, θ, ϕ, D, γ|β) : ((x, θ, ϕ, D);γ)∈Bh∩Bh
✷
o
whereEVh(x, θ, ϕ, D, γ|β) is defined in (9) Notice that Φh
kis nonempty valued and
convex-valued, since V is a continuous and concave function.
Claim 1 The correspondenceB∩Bh
✷ is lower and upper semi-continuous.
Demonstra¸c˜ao. It is of routine.
Claim 1 implies, from maximum principle, the correspondence Φh
k is upper
semi-continuous. Now, define the total correspondence
Φk :△k× Wk×[0,1]Ξ×B×Π1(H)× B ←֓
Φk(ν) =P(ν)×
×b∈B,ξ∈ΞKk(ν)bξ
××h∈HΦhk(ν)
By Kakutani’s Theorem, Φk has a fixed point
νk:= (pk, qk1, qk2, wk, Kk, βk,((xhk, θhk, ϕhk, Dhk);γhk)h∈H)
Step 3: Claim 2: The sequence {νk}
k is bounded therefore has a convergent
subse-quence ν.
It is useful to note that νk depends on ǫ ∈ RM×B
++ and therefore its limit as k goes to
zero.
To avoid notational clutter, we suppress thek. First, note that the excess demand in the
markets in state 0,
poZo(p) +q1Y1(p) +q2Y2(p)−w.Y(p) = 0.
Given the monotonicity of utility, this equality holds for eachhindividually in his budget
set. It implies that the ”price player”or ”fictitious auctioneer”could not make the value
of excess demand (across commodities and assets) positive in period 0. Suppose for
some (b, f) ∈ M, (n(Tb(f))−v(f)) > 0. By taking w(b, f) = 1/k and w(b′, f′) = 0 for
(b′, f′)6= (b, f), q
1n= 0,∀n ∈N, q2m = 0,∀m∈M, then one has
1
k (n(Tb(f))−v(f)) +
X
l∈L
˜
p0l
X
h∈H
(xh
0l−eh0l)≤0
for all ˜p∈ Ck≡ {pˆ∈R+L : ˆpl≥k for all l ∈L,PLl=1pˆl = 1}. AsPl∈Lp˜0lPh∈H xh0l ≥0, we
can claim that
1
k(n(Tb(f))−v(f))≤
1
k (n(Tb(f))−v(f)) +
X
l∈L
˜
p0l
X
h∈H
xh
0l ≤
X
l∈L
˜
p0l
X
h∈H
eh
0l
1
k (n(Tb(f))−v(f))≤
X
l∈L
˜
p0l
X
h∈H
eh0l
1
k(n(Tb(f))−v(f))≤
X
l∈L
˜
p0l
X
h∈H
max
h e
h
0l
1
k (n(Tb(f))−v(f))≤
X
l∈L
˜
p0l
X
h∈H
Hence, givenP
l∈Lp˜0l = 1, we have
(n(Tb(f))−v(f))≤kke0k∞
where ke0k∞ =Ph∈Hkeh0k∞. Now, let suppose for some n ∈N, Ph∈Hθhn−
P
b∈Bδnb >0.
Consider also that ˜q1n = 1/k and ˜q1n′ = 0, ∀n′ =6 n, and q2m = 0,∀m ∈ M, w(b, f) = 0 for all (b, f)∈ M, we get
1
k
X
h∈H
θh
n−
X
b∈B
δb n
+
X
l∈L
˜
p0l
X
h∈H
(xh
0l−e h
0l)≤0.
1
k
X
h∈H
θnh−
X
b∈B
δnb
≤ 1 k X
h∈H
θhn−
X
b∈B
δnb
+
X
l∈L
˜
p0l
X
h∈H
xh0l ≤
X
l∈L
˜
p0l
X
h∈H
eh0l.
1
k
X
h∈H
θh
n−
X
b∈B
δb n
≤
X
l∈L
˜
p0l
X
h∈H
eh
0l.
Hence,
X
h∈H
θhn−
X
b∈B
δnb ≤kke0k∞ (11)
we have the same result. Similarly, we can use the same proceeding toq2Y2. That is,
X
h∈H
ϕh
m−
X
b∈B
αb
m ≤kke0k∞ (12)
Finally, if P
h∈H(xh0l −eh0l) > 0 for some l , then by taking q1n = 0∀n ∈ N, q2m =
0∀m ∈M and ˜p0l = 1−(L−1)k and ˜p0i =k for all i6=l ,w(b, f) = 0 for all (b, f)∈ M,
we obtain
X
l∈L
˜
p0l
X
h∈H
(xh0l−e h
0l) +
X
i6=l
˜
p0i
X
h∈H
(xh0i−e h
0i)≤0
Given thatP
i6=lp˜0iPh∈Hxh0i ≥0, one has
˜
p0l
X
h∈H
(xh0l−e h
0l)≤p˜0l
X
h∈H
(xh0l−e h
0l) +
X
i6=l
˜
p0i
X
h∈H
xh0i ≤
X
i6=l
˜
p0i
X
h∈H
˜
p0l
X
h∈H
(xh0l−e h
0l)≤
X
i6=l
˜
p0i
X
h∈H
eh0i
X
h∈H
(xh
0l−e h
0l)≤
(L−1)kke0k∞
1−(L−1)k
From the fact that agents have optimized and that ¯Kk fixed K so that pξ✵˜b(Dξh) ≤
pξIξ✵b(ϕh), whenever Iξ>0 one has
Kb
ξ =
pξIξǫb+Ph∈Hpξ✵˜b(Dξh)
pξIξǫb+Ph∈H pξIξ✵b(ϕh)
≤1.
Hence,
X
h∈H
pξ✵˜b(Dhξ) =
X
h∈H
Kb
ξpξIξ✵b(ϕh)−(1−Kξb)pξIξǫ (13)
From optimization of monotonic utilities in the budget set in the second period, using
(5), (6), (7) and (8), we get
pξ(xhξ −e h
ξ) =
X
b∈B
[Kξb(pξPξθh,b+C h,b
ξ )−pξD
h,b
ξ ]
Adding over agentsh∈H,
pξ
X
h∈H
(xhξ −e h
ξ) =
X
h∈H
X
b∈B
[Kξb(pξPξθh,b+Cξh,b)−pξDh,bξ ] (14)
By abuse notation, we call
X
h∈H
X
b∈B
pξDξh,b :=
X
b∈B
X
h∈H
pξ✵˜b(Dξh)
and
X
h∈H
X
b∈B
ϕh,bξ :=
X
b∈B
X
h∈H
✵b(ϕh) Thus, replacingDh,bξ in (14) by (13), we obtain
pξ
X
h∈H
(xh
ξ −e
h
ξ) =
X
h∈H
X
b∈B
[Kb
ξ(pξPξθh,b+Cξh,b)−K
b
ξpξIξϕh,bξ +
X
b∈B
(1−Kb
ξ)pξIξǫb]
pξ
X
h∈H
(xh
ξ −e
h
ξ) =
X
b∈B
(1−Kb
ξ)pξIξǫb +
X
b∈B
X
h∈H
Kb
ξ[pξPξθh,b+Cξh,b−pξIξϕh,b]
pξ
X
h∈H
(xh
ξ −ehξ) =
X
b∈B
(1−Kb
ξ)pξIξǫb+
X
b∈B
X
h∈H
Kb
+X
b∈B
Kξb
X
h∈H
Cξh,b−
X
b∈B
X
h∈H
KξbpξIξϕh,b
Now, from definition of contracts, remember that
X
h∈H
Cξh,b = X
f∈Fb
Cb(f, p, ξ) = X
h∈H
Ch,b(ρb(γh), ξ)
Notice thatP
h∈H Ch,b(ρb(γh), ξ) does not depend on the idiosyncratic risk,ω. Therefore,
we can write
pξ
X
h∈H
(xhξ −e h
ξ) =
X
b∈B
(1−Kξb)pξIξǫb+
X
b∈B
X
h∈H
KξbpξPξθh,b+
+X
b∈B
Kξb
X
f∈Fb
Cb(f, p, ξ)−X
b∈B
X
h∈H
KξbpξIξϕh,b (15)
Then, using the accounting identity described in (1) when uncertainty is solved to
substitute for P
f∈FbCb(f, p, ξ) in (15) , one has pξ
X
h∈H
(xh
ξ −e
h
ξ) =
X
b∈B
(1−Kb
ξ)pξIξǫb+
X
b∈B
X
h∈H
Kb
ξpξPξθh,b+
+X
b∈B
Kb
ξpξ·[Iξαb −Pξδb]−
X
b∈B
X
h∈H
Kb
ξpξIξϕh,b
Rearranging the terms, we get
pξ
X
h∈H
(xhξ −e h
ξ) =
X
b∈B
(1−Kξb)pξIξǫb+
X
b∈B
KξbpξPξ
X
h∈H
θh,b+ +X
b∈B
Kξbpξ·[Iξαb−Pξδb]−
X
b∈B
KξbpξIξ
X
h∈H
ϕh,b pξ
X
h∈H
(xhξ−e h
ξ) =
X
b∈B
(1−Kξb)pξIξǫb+
X
b∈B
KξbpξPξ[
X
h∈H
θh,b−δb]−X
b∈B
KξbpξIξ[
X
h∈H
ϕh,b−αb] LettingkP˜k∞= maxPξ, kI˜k∞= maxIξ , and Kξb ≤1, we obtain
pξ
X
h∈H
(xh
ξ −e
h
ξ)≤
X
b∈B
(1−Kb
ξ)pξIξǫb+BkP˜k∞[
X
b∈B
X
h∈H
θh,b−X
b∈B
δb]−
−BkI˜k∞[
X
b∈B
X
h∈H
ϕh,b−X
b∈B
αb] (16)
We also know that
X
b∈B
X
h∈H
θh,b = X
h∈H
and
X
b∈B
X
h∈H
ϕh,b = X
h∈H
ϕh
Hence, plugging these definitions into (16), one has
pξ
X
h∈H
(xhξ −e h
ξ)≤
X
b∈B
(1−Kξb)pξIξǫb +BkP˜k∞[
X
h∈H
θh− X
b∈B
δb]−BkI˜k∞[
X
h∈H
ϕh−X
b∈B
αb] Now, applying the results of (11) and (12)
pξ
X
h∈H
(xh
ξ−e
h
ξ)≤
X
b∈B
(1−Kb
ξ)pξIξǫb+BkP˜k∞[
X
h∈H
θh−X
b∈B
δb]−BkI˜k
∞[
X
h∈H
ϕh−X
b∈B
αb]≤
≤ X
b∈B
(1−Kb
ξ)pξIξǫb+BkP˜k∞ˆ1kke0k∞−BkI˜k∞ˇ1kke0k∞
where
ˆ
1= (1, . . . ,1)∈RN and ˇ1= (1, . . . ,1)∈RM
Hence,
pξ
X
h∈H
(xhξ −e h
ξ)≤
X
b∈B
(1−Kξb)pξIξǫb+BkP˜k∞ˆ1kke0k∞−BkI˜k∞ˇ1kke0k∞
Like Dubey, Geanakoplos e Shubik (2005), suppose P
h∈H(xhξl − ehξl) −
P
b∈B(1 −
Kb
ξ)Iξlǫb > 0 for some b ∈ B. Since we are at a fixed point, the price player cannot
increase the value of excess demand in stateξ by taking ˜pξl = 1−(L−1)k and ˜pξi =k
for all i6=l. Hence,
pξl[
X
h∈H
(xhξl−e h
ξl)−
X
b∈B
(1−Kξb)Iξlǫb] +
X
i6=l
˜
pξi[
X
h∈H
(xhξi−e h
ξi)−
X
b∈B
(1−Kξb)Iξiǫb]≤
≤BkP˜k∞ˆ1kke0k∞−BkI˜k∞ˇ1kke0k∞
Then,
pξl[
X
h∈H
(xh
ξl−e
h
ξl)−
X
b∈B
(1−Kb
ξ)Iξlǫb]≤pξ[
X
h∈H
(xh
ξl−e
h
ξl)−
X
b∈B
(1−Kb
ξ)Iξlǫb]+
X
i6=l
˜
pξi
X
h∈H
xh
≤X
i6=l
˜
pξi[
X
h∈H
ehξi +
X
b∈B
(1−Kξb)Iξiǫb] +BkP˜k∞ˆ1kke0k∞−BkI˜k∞ˇ1kke0k∞
pξl[
X
h∈H
(xhξl−e h
ξl)−
X
b∈B
(1−Kξb)Iξlǫb]≤
≤X
i6=l
˜
pξi[
X
h∈H
ehξi +
X
b∈B
(1−Kξb)Iξiǫb] +BkP˜k∞ˆ1kke0k∞−BkI˜k∞ˇ1kke0k∞
Hence,
X
h∈H
(xh
ξl−e
h
ξl)−
X
b∈B
(1−Kb
ξ)Iξlǫb ≤
≤ 1
1−(L−1)k
(L−1)k[ke0k∞+kI˜k∞
X
b∈B
ǫb] +BkP˜k∞ˆ1kke0k∞−BkI˜k∞ˇ1kke0k∞
Thus aggregate excess demand (including the external agent) goes to zero as k →0.
Step 4: Next we will show that (p, q1, q2) does not belong to the boundary of the price
simplex △k. To see this, note first that if p is on the boundary, then some commodity
l has price pl = k. If pξi/pξl became unbounded as k → 0, some agent with ehξl > 0
could have consumed ˆM units of commodity i in state ξ, obtaining more utility than
Uh(2P
h′∈Heh
′
), for all small k (i.e, kxk∞ >Mˆ). Furthermore, since xh ≤2Ph′∈Heh ′
as
k→0, this contradicts thath has optimized. In relation to banking prices, we argue that
(q1, q2) must remain bounded as k → 0. If Qhm = 0 for all h, then q2m = 1. Otherwise,
if q2m → ∞, any agent h with Qhm > 0 could replace his entire action by selling a tiny
amount ∆ ofm, buying ˆM(≤∆/L) units of each period 0 good. Sinceeh
ξ 6= 0 for allξ, and
commodity price ratios are bounded in each state, agent h can do this without incurring
any default. But this gives him utility that exceedsUh(2P
h′∈Heh
′
),which is more that he
can possibly be getting at the fixed point, a contradiction. Similar proceeding/rationing
can be adopted byq1n. Hence, all asset prices are bounded.
Finally, we also have to show that w does not belong to the boundary of the wage
spaceWk.Ifw is on the boundary of Wk, then some wage has absolute value 1/k. Then,
when k → 0, some agent who receive w could have consumed ˆM units of commodity
with the cheapest price, obtaining more utility thanUh(2P
h′∈Heh
′
kxk∞ > Mˆ). Hence, since xh ≤ 2Ph′∈Heh ′
as k → 0, this contradicts that h has
opti-mized.
Step 5: Thus, since the sequence {νk(ǫ)}
k is bounded as it was shown above, we
can obtain convergent subsequences with ¯ν(ǫ) as a limit point. Taking the limit of all
inequalities derived above, we conclude that aggregate excess demand for commodities,
banking system, job markets is less than or equal to zero in {νk(ǫ)}
k. Since price ratios
¯
pξi/p¯ξl are bounded in the two periods, the limiting ¯p ≫ 0, and all agents have positive
endowments in every state in ¯ν(ǫ), then the bounds in Bh
✷ imposed on ((x, θ, ϕ, D);γ)
are not binding in ¯ν(ǫ).Therefore, because theEVh(x, θ, ϕ, D, γ|β) is concave, agents are
optimizing in ¯ν(ǫ) on theirs budgets sets. Finally, if ¯p≫0, there cannot be excess supply
in any commodity in ¯ν(ǫ), otherwise the ”fictitious auctioneer”could be making negative
profits. Also there cannot be excess supply of any security traded by banks in ¯ν(ǫ), unless
(q1, q2) = (0,0). In this case, no bank trades securities, except when Pξ = 0 for all ξ∈Ξ.
Without loss of generality, no agent would be interested to buy securities, i.e,θh = 0 and
ϕh = 0, for allh.
Step 6: Claim 3: ν(ǫ) constitute an ǫ− boosting equilibrium.
Now by making ||ǫ|| → 0 we have that ν(ǫ) → ν therefore we have shown that ν is
an refined equilibrium our original economyE.This prove the theorem for finite penalties
λ. If some penalties are infinite, we have to take limits of equilibria with increasing
penalties. Because all actions need to stay bounded along the sequence (because ˜Qh <∞
and ˆQh < ∞), any cluster point of these equilibria will serve as the desired refined
3
Credit Derivatives in the GEI
model
3.1
Motivation and Review of the Literature
In global terms, the market for credit derivatives is growing rapidly, even though it is
still quite small compared with other derivatives markets. By definition, credit derivatives
are contracts that transfer an asset’s risk and return from one counterpart to another
with-out transferring ownership of the underlying asset. In banking system, credit-line
manage-ment and ”regulatory arbitrage”are the two most applications of credit derivatives
moti-vating market participants to purchase protection against credit risk. Credit-line
manage-ment is particularly relevant for dealing with situations where a bank is over-concentrated
in loans to companies in specific sectors of the economy, for example, because it has a
comparative advantage in originating loans in those sectors. While concentration risk can
be mitigated by other means (such as selling loans in the secondary market or originating
loans in non-traditional sectors), there are advantages to using credit derivatives for this
purpose. To begin with, loan sales can potentially damage valuable client relationships
(i.e., clients may resent the fact that their bank is reducing its exposure to them, seeing
this as a signal that the bank has diminished faith in their creditworthiness). Second, the
origination of loans in non-traditional sectors can expose the bank to new risks. On the
other hand, credit derivatives can help banks to diversify their loan portfolios more cost
effectively, without damaging client relationships. Credit derivatives can also be used for
imposed by national regulators according to the rules set out in the Bank for
Interna-tional Settlements ”Capital Accord”(BIS 1988). The general equilibrium literature has
suggested two mechanisms to enforce standard debt contracts: penalization in terms of
utility (see Dubey, Geanakoplos e Shubik (2005)); or the requirement of collateral from
borrowers (see Geanakoplos e Zame (2007)). However, until now nothing was done in
terms of applications of the credit derivatives in earlier works. In a collateral model, each
lender expects to receive only the minimum between the claim and the market value of
the depreciated collateral1. However, even one supposes that the borrower is rational by
assumption, this kind of default is not taken by agents’ natural choices and, therefore,
this situation could be considered as non-strategy default. Indeed, because the law
ar-rangements, the ability of lenders to foreclose the mortgage or pledge is extremely limited
in many places. In some jurisdictions, foreclosure and sale can occur quite rapidly, while
in others, foreclosure may take many months or even years. Our model provides a new
approach in which it is not necessary the lender worries about the delivery of ownership
of collateral. In this model, it is the lender who purchases the collateral and passes it on
to the borrower so that he may use it without the lender losing the ownership rights. The
borrower pays the premium of a call option, issued by the lender, on underlying asset, in
this case, the collateral whose exercise price is the return of the collateralized loan. The
ownership, however, remains in hands of the lender. Similar to Orrillo (2006), we proof
the existence of equilibrium by combining the demand approach and no-arbitrage prices in
this new market. Technically, other advantage of our approach is not affected by standard
critics imputed, in general, to GEI model such as: i) the equilibrium exists generically
(see Duffie e Sahafer (1985), Duffie e Sahafer (1986), and Hirsch, Magill e Mas-Colell
(1990)); ii) necessity to express all payoffs in terms of a numeraire good (see Geanakoplos
e Polemarchakis (1986)); and iii) restrictions on asset returns (see Cass (1984), Werner
(1985), Duffie (1987) and Zhang (1996)). The paper is organized as follows. Section 3.2
and 3.3 present the basic structure of model. In Sections 3.4 and 3.5, we defined the
