Asset-price collapse and market disruption– A model of ﬁnancial crises –(Incomplete and preliminary)Keiichiro Kobayashi∗
January 27, 2009 (First draft: October 30, 2008)

Abstract
We construct a search theoretic model à la Lagos and Wright (2005), that has
multiple steady-state equilibria, one of which may be interpreted as a state of ﬁnancial crisis. The key ingredient is the collateral-secured loan in the decentralized
matching market, in which the borrowers must put up their own land as collateral.
They borrow debt for intertemporal smoothing of consumption stream and also forfactor payment in production. In the crisis state, the land price is low and thedebt for factor payment, i.e., liquidity, dries up. Facing liquidity shortage, all sellers
choose not to participate in the matching market and the market is shut down due
to the search externality. This market disruption lowers the aggregate productivity,
while the low productivity justiﬁes the low asset price in turn.
We may be able to derive a policy implication that collective debt reduction
by the government intervention may solve the coordination failure and bring the
economy out of the crisis equilibrium.

Keywords: Collateral-secured loan, search frictions, liquidity, chain of productions.
JEL Classiﬁcations: E30, E60, G01, G12.
∗

Research Institute of Economy, Trade, and Industry (RIETI). E-mail: kobayashi-keiichiro@rieti.go.jp.

I thank seminar participants at Nihon University for helpful comments.

1

1

Introduction

In the recent ﬁnancial crises, asset-price collapses caused tightening of credit in theeconomy where collateral-secured loans were widespread. Examples are Japan’s lost
decade in the 1990s and the Asian currency crisis in 1997–1998, and the global ﬁnancial
crisis in 2008. Real estate, such as commercial properties and/or housing assets, are used
as collateral for loans all over the world, and once the prices of collateral assets collapse
credit and liquidity are severely tightened.
Macroeconomic theorists so far proposed two diﬀerent kinds of models of collateral
lending. In one strand of models collateral lending smooths consumption and investment
intertemporally (Kiyotaki and Moore 1997; Bernanke, Gertler, and Gilchrist 1999; Iacoviello 2005). In the other models collateral lending is modeled as an intratemporal debt
that is used as an instrument for factor payment for production (Carlstrom and Fuerst
1997, 1998; Jermann and Quadrini 2006; Mendoza 2006; Kobayashi, Nakajima and Inaba 2007; Kobayashi and Nutahara 2007). We call the instrument for factor payment
liquidity in this paper. Collateral lending works either as an intertemporal smootherof consumption and investment or as an intratemporal liquidity in respective strand ofexisting models.
In this paper, we propose a model in which collateral lending plays both roles of anintertemporal smoother and of intratemporal liquidity. We show that the interaction
between the two roles of collateral lending causes multiple steady-state equilibria toexist. There are two steady states, one of which is a bad equilibrium where the asset
price collapses and a decentralized matching market is disrupted due to shortage ofliquidity. In the bad equilibrium, the land price is low and the intratemporal debt, i.e.,liquidity, dries up because of the shortage of collateral. Consequently, the decentralized
matching market is disrupted: Since patient agents choose whether to participate in thematching market, when the land price is low all patient agents choose not to participate
and the market is shutdown. The participation choice by patient agents has an external
eﬀect on impatient agents through search probability, which makes the bad equilibriuma stable equilibrium. The market disruption lowers the aggregate productivity, while2the low productivity justiﬁes the low asset price in turn. We may be able to derive an
implication for “macroprudential policy” (Borio 2003, Bernanke 2008) or ﬁnancial crisis
management: Collective debt reduction by the impatient agents due to the government
intervention may solve the coordination failure and bring the economy out of the crisis
equilibrium.
Related literature:

This paper shows that an asset-price collapse can cause a poten-

tially persistent, maybe decade-long, recession due to liquidity shortage. The mechanism
proposed in this paper may be a possible explanation for persistent productivity declines
observed in various episodes of the “great depressions.” See Kehoe and Prescott (2002,2007) and references therein for neoclassical account for the great depressions in the 20th
century: In the most of the episodes, productivity declines were the primal factor thatcaused the depressions, while the root causes of productivity declines are left unexplainedin these research (see also Ohanian 2001). The mechanism of market disruption in ourmodel, that is, the disruption of transactions in the matching market, is close to thedisruption of chain of productions (or the division of labor among ﬁrms) in Blanchard
and Kremer (1997) and Kobayashi (2004, 2006). One contribution of this paper is thatthe model in this paper is able to analyze the relationship between the asset price and themarket disruption explicitly through adoption of the search theoretic framework of Lagos
and Wright (2005). Although the Lagos-Wright framework is intended to contribute tomonetary theory (see Rocheteau and Wright 2005 and Lagos and Rocheteau 2005 forapplication of this framework in the monetary theory literature), we use it to constructessentially a “real” model in the spirit of Carlstrom and Fuerst (1997, 1998).

In the next section, we present the model and show our basic results including policy
implication for ﬁnancial crisis management. In Section 3, we consider a generalized model
where multiple matching markets open sequentially in the day time. Section 4 concludes.
In Appendix, we brieﬂy describe a monetary version of our model.

3

2

Model

The model is a variant of the search theoretic model developed by Lagos and Wright(2005), in which we introduce collateral lending as a payment instrument just like Ferraris and Watanabe (2008). Unlike Ferraris and Watanabe’s model, in which lenders
give cash to the borrowers and the total amount of cash is exogenously given by thegovernment, banks in our model can costlessly create the payment instruments, which
may be interpreted as bank notes or promissory notes.

2.1

Setup

The model is a closed economy à la Lagos and Wright (2005), in which there are continuaof patient agents and impatient agents, who live forever. The measures of patient andimpatient agents are M and 1 respectively, while M is suﬃciently large:
M ≫ 1.

(1)

There is also a unit mass of banks that can create payment instruments (i.e., bank notes)
costlessly and lend them to the agents as collateral-secured loans. Time is discrete andcontinues from 0 to inﬁnity: t = 0, 1, · · · , ∞. The numeraire is the consumption good.
In each date t, market opens twice: The day market and the night market. Agents canconsume the consumption good only in the night market. The consumption goods are
not storable, that is, the consumption goods produced in the day market or the nightmarket of date t must be consumed in the date-t night market, otherwise they perish
before date t + 1 begins. The day market is a decentralized search market in which
patient agents and impatient agents trade intermediate goods when they meet. A tradein the day market is quid pro quo and no trade credit is available. The night market isa centralized Walrasian market in which patient and impatient agents and banks tradeconsumption goods, land, bonds issued by impatient agents, and bank notes.
In the day market, when a patient agent and an impatient agent meet each other,
the patient agent produces the intermediate good and sells it to the impatient agent at acompetitive market price. The impatient agent pays for the intermediate good by giving
4the bank notes that she borrowed from a bank in the previous night market. Patient
agents need to pay a ﬁxed cost for participating in the day market and they choose
whether they participate in the day market or not at the beginning of each period t.
In the night market, impatient agents issue intertemporal bonds and patient agents buy
the bonds in exchange for the consumption good. Impatient agents also borrow banknotes from banks and can hold productive asset, i.e., land. For simplicity we assume
that patient agents cannot hold land. Impatient agents have incomplete commitment
technology. Therefore, they cannot precommit to redemption of their bonds nor repayment of their bank loans. However, the impatient agents can use land as collateral for
their bonds and bank loans. Banks have complete commitment technology: They cancommit themselves to that in the night market they pay one unit of consumption goodin exchange for one unit of bank note that they issued in the previous night market.
Therefore, bank notes are circulated as a payment instrument in the day market.

2.2

Optimization problem for patient agentsThe state variable for a patient agent who enters the day market is the amount of bondsthat she hold from the previous period. She holds bonds issued by impatient agentsas her asset. When she enters the night market the bonds and the revenue she earnedin the day market are the state variables. We denote the value function for a patient
agent when she enters the day market by V p (dt−1 ) and that for the night market by
W p (nt , dt−1 ), where dt−1 is bond holding and nt is the revenue (i.e., bank notes) she getsin the day market. Both dt−1 and nt are measured in the unit of date-t consumption
good, that is, one unit of bond or bank note issued in date-(t − 1) night market is a claim
to one unit of date-t consumption good.
At the beginning of the day market, a patient agent chooses whether she participate
in the day market by paying ﬁxed cost κ for market participation. The Bellman equation

5

for the patient agent is
V p (dt−1 )
}
{
p
p
p
= max −κ + α(µt ) max[−γ(qt ) + W (pt qt , dt−1 )] + {1 − α(µt )}W (0, dt−1 ), W (0, dt−1 )
qt

{
}
p
p
= max −κ + α(µt ) max[−γ(qt ) + W (pt qt , dt−1 ) − W (0, dt−1 )], 0 + W p (0, dt−1 ),

(2)

qt

(3)
where qt is the quantity of the intermediate good produced; γ(qt ) is the cost for production, which satisﬁes γ ′ (q) > 0, γ ′′ (q) > 0, and γ(0) = 0; pt is the competitive price of theintermediate good; α(µt ) is the matching probability for a patient agent with which shemeets an impatient agent in the day market, where µt (0 ≤ µt ≤ M ) is the measure of
the patient agents who participate in the day market. The expected gain for a patient
agent from participating in the day market, π, is
π ≡ −κ + α(µt ) max{−γ(qt ) + W p (pt qt , dt−1 ) − W p (0, dt−1 )}.
qt(4)Given the aggregate measure of participating patient agents, µt , a patient agent decides
to participate if π > 0; not to participate if π < 0; and is indiﬀerent if π = 0. The
Bellman equation for the night market is
W p (nt , dt−1 ) = max ct − ht + βV p (dt ),ct ,ht ,dt

subject to

ct ≤ dt−1 −dt+ nt + h t ,1 + rt(5)(6)

where ct is the consumption, ht is the labor supply, rt is the real interest rate, and β(0 < β < 1) is the intertemporal discount factor. To simplify the analysis, we makethe assumption that patient agents directly gain ct units of utility from consuming ct ,which simplify the form of collateral constraint in the problem of impatient agents. As
is standard in the Lagos-Wright framework, we assume that the labor input in the nightmarket gives linear disutility and is transformed into the consumption good linearly. As
shown below, this convention simpliﬁes the analysis greatly by making all agent choose6the same value of dt and degenerating the heterogeneity among agents with respect torevenues in the period-t day market. Substituting ht in the budget constraint into (5),
we havedt+ nt + βV p (dt ).
1 + rt(7)

1.
1 + rt(8)

Wnp (nt , dt−1 ) = 1,

(9)

Wdp (nt , dt−1 ) = 1.

(10)W p (nt , dt−1 ) = max dt−1 −dt

The ﬁrst-order condition (FOC) for (7) is
βVdp (dt ) =The envelope conditions for (7) are

These conditions imply that W p (nt , dt−1 ) = nt + dt−1 + W0p , where W0p ≡ W p (0, 0).
Therefore, the Bellman equation for the day market can be rewritten as
V p (dt−1 ) = max{−κ + α(µt ) max[−γ(qt ) + pt qt ],qt0} + dt−1 + W0p .

(11)

Since the price of the intermediate good is determined competitively,1 FOC for qt is
pt = γ ′ (qt ).

(12)

The envelope condition, i.e., Vdp (dt−1 ) = 1, and (8) imply thatβ(1 + rt ) = 1.

(13)

The measure of the patient agents who participate in the day market, µt , is determined
endogenously by participation decision of individual agents. The gain from participation
π(µt ) can be rewritten asπ(µt ) = −κ + α(µt ){−γ(qt ) + pt qt }.
1(14)

Although we assume for simplicity price-taking behavior of all agents in the matching market, we

can easily modify our model such that the trading scheme in the day market is a bilateral bargaining orcompetitive search, following Rocheteau and Wright (2005).

7

There exist multiple equilibria corresponding to diﬀerent values of µt : µt = 0 can be
an equilibrium since π(0) = −κ < 0 and an impatient agent chooses not to participate
when π < 0; and the solution to π(µt ) = 0 can be an equilibrium, too. Therefore, µt is
determined by the following equation in equilibrium:µt [α(µt ){pt qt − γ(qt )} − κ] = 0.

2.3

(15)

Optimization problem for impatient agentsThe state variables for an impatient agent who enters the decentralized day market are
the amount of bonds she issued in the previous night market (bt−1 ); the amount of bankloans she borrowed in the previous night market (lt−1 ); and land she perchased in theprevious night market (kt−1 ). The impatient agent borrowes lt−1 from a bank in the formof bank notes, that is, she is given lt−1 units of bank notes when she borrows from thebank in the previous night market. Note that bt−1 and lt−1 are measured in the unit ofdate-t consumption good. The state variables when she enters the current night market
are bt−1 , lt−1 , kt−1 , the remaining bank notes (n′t ), and the output she produced in the
day market (yt ). In the date-t night market, the impatient agent must repay Rt−1 lt−1
units of the date-t consumption good to the banks, where Rt−1 is the gross rate of returnon the bank loan. The Bellman equation for the day market is
V (lt−1 , bt−1 , kt−1 ) = max α(µt )µt W (yt , n′t , lt−1 , bt−1 , kt−1 )yt ,qtsubject to+ {1 − α(µt )µt }W (0, lt−1 , lt−1 , bt−1 , kt−1 ),(16)n′t = lt−1 − pt qt(17)n′t ≥ 0,(18)

θyt = Akt−1
qt1−θ ,

(19)

where V ( · ) is the value function for an impatient agent who enters the day market, W ( · )is that for those who enters the night market, and α(µt )µt is the matching probability
for an impatient agent to meet a patient agent in the day market.2
2

We assume a standard constant-returns-to-scale matching function, Λ(µp , µi ), for the day market,

where µp and µi are the measures of patient and impatient agents who participate in the day market. The

8

The Bellman equation for the night market is
W (yt , n′t , lt−1 , bt−1 , kt−1 ) = ′ max′

ct ,ht ,lt ,bt ,kt

subject toU (c′t ) − h′t + β ′ V (lt , bt , kt ),(20)

c′t + at kt + bt−1 + Rt−1 lt−1
≤ yt + h′t + ωt kt−1 + n′t +bt+ at kt−1 ,1 + rt(21)

bt + Rt lt ≤ at+1 kt ,(22)

lt ≥ 0,(23)

where at the beginning of the night market one unit of land generates the dividend, ωt ,which is exogenously given by nature; at is the date-t land price; c′t is the consumption;
U (c) is the utility from the consumption, where U ′ (c) > 0, U ′′ (c) < 0, and U ′ (0) = +∞;h′t is the labor supply; and β ′ is the intertemporal discount factor that satisﬁes
0 < β ′ < β < 1.

(24)

We also assume the Lagos-Wright convention that h′t gives linear disutility and is transformed into the consumption good linearly. Because the impatient agent cannot precommit to repay bt to the creditor (i.e., a patient agent) nor to repay Rt lt to the bank in
the date-(t + 1) night market, she must put up her land as collateral for bt + Rt lt when
she borrow bt and lt in the date-t night market. When the impatient agent repudiate
her debt, the creditors seize the collateral and sell it oﬀ in the date-(t + 1) night market.
Since there is no further penalty for repudiation, the debtor surely diverts and defaults
on the excess amount of debt which exceeds the value of collateral. Therefore, there is no
reason for the creditors to lend more than the value of collateral.3 The Bellman equation
matching probability for a patient agent is Λ(µp , µi )/µp = Λ(1, µi /µp ), while that for an impatient agentis Λ(µp , µi )/µi = Λ(1, µi /µp )µp /µi . Since µp = µ and µi = 1 in our model, the matching probability fora patient agent is α(µ) and that for an impatient agent is α(µ)µ, where α(µ) ≡ Λ(1, 1/µ).
3
Although we do not consider stochastic shocks to the asset prices in this paper, our model can beeasily generalized to stochastic model. In such a case, the collateral constraint, (22), is rewritten as
bt + Rt lt ≤ Et [at+1 ]kt , where Et [ · ] is the mathematical expectations based on the information available
at date t. This is the risk neutrality of the patient agent that leads her to evaluate the value of collateral9(20) can be rewritten as
W (yt , n′t , lt−1 , bt−1 , kt−1 )= ′ max U (c′t ) − c′t + yt − at kt + (ωt + at )kt−1 − Rt−1 lt−1 + n′t +ct ,lt ,bt ,ktbt− bt−1
1 + rt+ β ′ V (lt , bt , kt ),subject to (22) and (23).
The FOCs are
U ′ (c′t ) = 1,

(25)

1
− ξt + β ′ Vb (lt , bt , kt ) = 0,
1 + rt(26)

β ′ Vk (lt , bt , kt ) = at − ξt at+1 ,(27)

β ′ Vl (lt , bt , kt ) + ηt − Rt ξt = 0,(28)

where ξt and ηt are the Lagrange multipliers for (22) and (23), respectively. The envelope
conditions are
Wy (t − 1) = 1,
Wb (t − 1) = −1,

Wl (t − 1) = −Rt−1 ,Wn′ (t − 1) = 1,

Wk (t − 1) = ωt + at ,where Wz (t − 1) is the derivative of W (yt , n′t , lt−1 , bt−1 , kt−1 ) with respect to z (=yt , n′t , lt−1 , bt−1 , or kt−1 ). These conditions imply that W (yt , n′t , lt−1 , bt−1 , kt−1 ) = yt −Rt−1 lt−1 + n′t − bt−1 + (ωt + at )kt−1 + W0 , where W0 ≡ W (0, 0, 0, 0, 0). The Bellman
equation for the day market can be rewritten asθV (lt−1 , bt−1 , kt−1 ) = max α(µt )µt {Akt−1
qt1−θ − pt qt } + (1 − Rt−1 )lt−1 − bt−1
qt

subject to+ (ωt + at )kt−1 + W0 ,(29)

pt qt ≤ lt−1 ,(30)

as the mathematical expectation of the value of the asset in the next period. If the utility function of
the patient agents is not linear, the collateral constraint in the stochastic model is no longer as simple
as the above one.

10

The FOC ispt =θ q −θ
(1 − θ)Akt−1t,
1 + xt(31)

where xt is the Lagrange multiplier for (30). The envelope conditions areVl (lt−1 , bt−1 , kt−1 ) = α(µt )µt xt ,(32)

Vb (lt−1 , bt−1 , kt−1 ) = −1,(33)

θ−1 1−θVk (lt−1 , bt−1 , kt−1 ) = ωt + α(µt )µt θAkt−1
qt + at .

(34)

Note that the collateral constraint (30) becomes binding with probability α(µt )µt if itbinds at all, since it becomes relevant only when the impatient agent successfully matcheswith a patient agent. Conditions (26)–(28) and (32)–(34) imply the following system of
equations that determines the dynamics of the models:

2.4

β ′ α(µt+1 )µt+1 xt+1 + ηt = Rt ξt ,(35)

1 = (1 + rt ){ξt + β ′ },[]1−θ
at = ξt at+1 + β ′ ωt+1 + α(µt+1 )µt+1 θAktθ−1 qt+1
+ at+1 .

(36)
(37)

Banks

A bank lends lt units of bank notes to an impatient agent in the date-t night market.
The bank has the commitment technology that she can give one unit of the consumption
good in exchange for one unit of the bank note in the date-(t + 1) night market. The
borrower repays Rt lt in the form of the consumption good to the bank in the date-(t + 1)
night market, where Rt is determined competitively. If the borrower repudiate her debt,
the bank can seize collateral from the borrower but cannot impose any further penality
on the borrower. The bank’s proﬁt in the unit of the date-(t + 1) consumption good is
Rt lt − lt . In the date-t night market, the bank chooses lt to maximize Rt lt − lt , where
there is no restriction on the supply of lt . Therefore, in equilibrium,
Rt = 1.

11(38)

This result parallels to that about the intraperiod debt in Carlstrom and Fuerst’s (1998)
model. Note that Rt = 1 is guaranteed by the fact that the bank can create costlessly anunlimited amount of the bank notes. If the bank needs to give cash to the borrower andthe total supply of cash is ﬁxed exogenously, Rt may exceed 1 when the total amount ofcash is small. See Ferraris and Watanabe (2008) for this case.

2.5

Equilibrium

The total supply of land is ﬁxed in this economy:
kt = K.

(39)

M d t = bt .

(40)

The bond market clears:∀t,

The equilibrium of this economy is deﬁned as follows.
Deﬁnition 1 A competitive equilibrium consists of prices, {pt , Rt , rt , at }, and quantities,{ct , c′t , ht , h′t , qt , kt , dt , bt , lt , µt }, that satisfy that (i) given the prices, the quantities satisfy(12) and (15), which are the optimality conditions for patient agents; (ii) given the pricesand {µt }, quantities solve impatient agents’ optimization problems, (20) and (29), andbanks’ optimization problem; and (iii) equilibrium conditions, (39) and (40), are satisfied.
The equilibrium path of this economy is described by the prices, {pt , Rt , rt , at , xt , ξt , ηt },and the quantities, {c′t , qt , kt , bt , dt , lt , µt }, where they are the solution to the following
system of equations: (12), (13), (15), (22), (25), (30), (31), (35), (36), (37), (38), (39),(40), andηt pt qt = 0.

(41)Equations (13) and (36) imply that 1 + rt = 1/β and ξt = β − β ′ . We can show thefollowing proposition:

12

Proposition 1 In equilibrium, either (qt , µt ) = (qh , µh ) or (qt , µt ) = (0, 0), where qh =q(µh ) and µh is the solution to Π(µ) = κ; where Π(µ) ≡ α(µ){γ ′ (q(µ))q(µ) − γ(q(µ))}
and q(µ) is an increasing function defined byγ ′ (qt )qtθ =α(µt )µt (1 − θ)AK θα(µt )µt + ββ′ − 1

.

(42)

Proof: In equilibrium either ηt = 0 or ηt > 0. In the case where ηt > 0, the nonnegativity
condition (23) is binding. Since lt = pt qt = 0, qt = 0 in equilibrium. Equation (15)implies that since the gain from participating the day market is zero for a patient agent,
all patient agents decide not to participate and µt = 0 in this case. Therefore, in
equilibrium where ηt = 0, (qt , µt ) = (0, 0). In the case where ηt = 0, it must be thecase that µt ̸= 0, since otherwise (35) implies that ξt = 0, which contradicts the factthat ξt = β − β ′ > 0. Therefore, µt > 0 when ηt = 0. With µt > 0, (35) implies thatxt = (β − β ′ )/{α(µt )µt }. Therefore, (12), (31), and (39) imply that qt satisﬁes (42).
Note that (42) determines qt as an increasing function of µt , that is, qt = q(µt ), since
the left-hand side of (42) is monotonically increasing in qt and the right-hand side ismonotonically increasing in µt . Since µt > 0 and qt = q(µt ) in the case where ηt = 0,(15) implies that µt is determined by Π(µt ) = κ. Since (42) implies that q(0) = 0,
Π(0) = 0. Since the matching probability for a patient agent, α(µ), converges to zero asµ → ∞, it must be the case that limµ→∞ Π(µ) = 0. Since α(µ) is decreasing in µ andγ ′ (q(µ))q(µ) − γ(q(µ)) is increasing in µ, Π(µ) has only one peak as shown in Figure 1 iffunctional forms for α( · ) and γ( · ) are given appropriately.4
Figure 1
We assume that κ is suﬃciently small and functional forms for α( · ) and γ( · ) are
given appropriately such that there exists unique µ such that Π(µ) is increasing in µ for0 ≤ µ ≤ µ and decreasing for µ ≥ µ. And we assume that κ is suﬃciently small suchthat Π(µ) = κ has exactly two solutions, µl and µh , where 0 < µl < µh < M . Notethat we assumed that the total measure of patient agents, M , is suﬃciently large. Π(µ)
4For example we can set that α(µ) = min{1, µ−δ } and γ(q) = q ϵ , where 0 < δ < 1 < ϵ.

13is increasing in the neighborhood of µl and decreasing in the neighborhood of µh . The
equilibrium with µ = µl is unstable because a slight deviation from µl is ampliﬁed byparticipation decisions of individual agents. The equilibrium with µh is stable, since a
deviation from µh is corrected by participation decisions by individual agents. So we
can focus on the solution µt = µh . We have shown that for appropriate parameters andfunctional forms, (qt , µt ) = (qh , µh ) in equilibrium where ηt = 0. Q.E.D.

Since there is no technological innovation nor capital accumulation in this model,
there are no time-varying state variables relevant to the aggregate dynamics of the economy. So any sequence of {qt , µt }∞t=0 that satisﬁes ∀t, (qt , µt ) ∈ {(qh , µh ), (0, 0)}, can be∞
an equilibrium path: Given any such sequence of {qt , µt }∞t=0 , the sequence of {at , bt }t=0

is determined by (37) andbt = at+1 K − ph qh

µt+1
,
µh

(43)

where ph = γ ′ (qh ). Therefore, an equilibrium path of this model is a sunspot equilibriumin the sense that any path that satisﬁes Proposition 1 can be the equilibrium outcome
solely depending on the agents’ expectations.

2.6

Steady States

In what follows in this paper we focus on the steady-state equilibria. Steady-state valuesof the asset price, a, and bonds, b, are determined bya=β′{ω + α(µ)µθAK θ−1 q 1−θ },1−βb + pq ≤ aK,

(44)
(45)
(46)

The welfare of patient and impatient agents are measured by the values of their valuefunctions at the beginning of the daytime in the steady state:
b
,
M
α(µ)µ
ωK
1−β
V =
{AK θ q 1−θ − pq} +
−
b,
′
′
1−β
1−β
1 − β′
Vp =d=

14

(47)
(48)

where V p and V are the welfare of patient and impatient agents, respectively. If wedeﬁne the total welfare of the economy in the steady state, E, as the sum of the patientand impatient agents’ welfare, it can be written as
E ≡ MV p + V =

α(µ)µ
ωK
β − β′
θ 1−θ
{AK
q
−
pq}
+
+
b.
1 − β′
1 − β′
1 − β′(49)

There exist two types of steady state in our model: One with (q, µ) = (qh , µh ) and theother with (q, µ) = (0, 0).
2.6.1

Normal state

We call steady state with (q, µ) = (qh , µh ) “normal state.” The asset price and theamount of bond are determined byan =β ′ {ω + α(µh )µh θAK θ−1 qh1−θ },1−βbn = an K − ph qh ,

(50)
(51)

where qh = q(µh ) and ph = γ ′ (qh ). In the equilibrium where µ is positive, collateralis used to issue intertemporal bonds for consumption smoothing and also it is used toborrow bank notes for factor payment in the decentralized day market. The day market
in which agents form chains of productions is operative in the equilibrium where µ > 0,
while it is shutdown in the equilibrium we describe in the following section.
2.6.2

Crisis state

We call the steady state with (q, µ) = (0, 0) the “crisis state,” because the decentralized
day market is shutdown in this steady state. The asset price and bonds are determined
by
ac =

β′ω
,
1−β

bc = ac K.

(52)
(53)

Obviously from (50) and (52), the asset price is lower in the crisis state than in thenormal state: an > ac . The debt-asset ratio, b/(aK), is higher in the crisis state than in15

the normal state:
bn
1
bc
=
(an K − ph qh ) < 1 =
.
an K
an K
ac K

(54)

It is shown as follows that the total welfare in the normal state, En , is higher than thatin the crisis state, Ec . Deﬁne ∆ ≡ (1 − β ′ )(En − Ec ). Then (49) implies{ ′}β∆ = α(µh )µh {AK θ qh1−θ − γ ′ (qh )qh } + (β − β ′ )α(µh )µh θAK θ qh1−θ − γ ′ (qh )qh .
1−β(55)
Since (42) implies γ ′ (q)q = (1 − θ)β ′ α(µ)µAK θ q 1−θ /{β ′ α(µ)µ + β − β ′ } and {α(µ)µ +β − β ′ }/{β ′ α(µ)µ + β − β ′ } < 1/β ′ , this equation implies{}(β − β ′ )β ′∆> 1+θα(µh )µh AK θ qh1−θ > 0.
1−β

(56)

Therefore, the total welfare, E, is higher in the normal state than in the crisis state.

2.7

Coordination failure in the crisis state and policy implication

In date-t night market, impatient agents choose lt , given µt+1 ; and at the beginning of
date-(t + 1), the measure of participating patient agents, µt+1 , is determined by participation choices of individual patient agents, which are made taking µt+1 and lt asgiven.
In the crisis state, patient agents collectively set µt+1 = 0 and impatient agentschoose lt = 0 for all t. Given that lt = 0, a patient agent has no incentive to participate
in date-(t + 1) day market: If she participate, she can meet with an impatient agent withprobability 1 because µt+1 = 0. (Note that all impatient agents participate in the day
market.) But the impatient agent has no payment instrument, and therefore the patient
agent can get nothing in the day market, while she must pay κ when she enters the day
market. Thus the expected gain from participation is negative for a patient agent.
Given that µt+1 = 0, an impatient agent has no incentive to borrow from a bank in
date-t night market and bring bank notes into date-(t + 1) day market: Since she canmeet with a patient agent in the day market with zero probability, the bank notes are16

useless. Meanwhile, she wants to maximize the amount of bonds that she issues, bt , since
the market rate, rt = β −1 − 1 is cheaper than her subjective rate of time discount, i.e.,(β ′ )−1 − 1. Therefore, all impatient agents set bt at its maximum possible value, at+1 K,
and they set lt = 0.
Note that the coordination failure that causes the crisis equilibrium to exist occurs
due to the existence of both the intertemporal consumption loan, bt , and the loan for
intratemporal payment, lt . Both debts are necessary to make multiple equilibria. For
example, if we did not introduce consumption loan, bt , in our model, the crisis state
would be eliminated and the normal state would become the only steady-state equilibrium
because lt is always positive under the collateral constraint, lt ≤ at+1 kt . The novel featureof our model, that is, collateral-secured loan is used as both intertemporal consumption
loan and intratemporal liquidity, enables the multiple steady-state equilibria to exist.
Policy implication:

As we saw above, there is no incentive for impatient agents to

reduce their bonds individually in the crisis state. Therefore, there may arise eﬀectivenessof the government intervention to coordinate the expectations and eliminate the crisis
equilibrium. For example, if the government intervenes and imposes the restriction that
all impatient agents must set their intertemporal liabilities, bt , such that bt ≤ at+1 kt −ph qh , then the crisis state is eliminated and the economy can immediately jump to thenormal state. The government-coordinated debt restriction may be interpreted as asimpliﬁed model of the government policies during the episodes of ﬁnancial crises, suchas the 700 billion-dollar TARP (Troubled Assets Relief Program) scheme initiated by theTreasury in 2008, in which the US government will buy and dispose of the nonperforming
assets and will ultimately bailout the debt-ridden households and ﬁrms. In the past
episodes of ﬁnancial crises all over the world, the governments of the crisis aﬀected
countries undertook various crisis-management policies to urge ﬁnancial institutions todispose of nonperforming assets and to reduce the debt burdens in the private sector. In
light of our model, we can consider that the essence of these crisis-management policies
may be to restrict bt in order to restore the supply of liquidity, lt .

17

Another policy implication would be that if the binding collateral constraint forborrowers is crucial in a ﬁnancial crisis, policies targeting at lenders may not be eﬀective
to attain economic recovery. For example, rehabilitation of banking sector by capital
injections and liquidity provision to banks by the central bank may not be eﬀective
to resolve the market disruptions due to a ﬁnancial crisis unless the debt reduction ofborrowers are properly addressed.

3

Generalization – Chains of productions

There may be several directions for generalization of this model. One of the most intriguing generalizations would be to incorporate ﬁat money in this model in a meaningful
way, while it turns out to be impossible in our patient-and-impatient-agents framework.
Because of the diﬀerence of time discount factors, impatient agents never hold nominal money in a steady-state equilibrium. It is also easily shown that even when aninterest-bearing money, i.e., bank deposits, is introduced, the impatient agents do nothold deposit money in a steady state (See Appendix).
In this section, we consider a generalization that makes our stylized model a little
closer to the reality of chains of productions and business cycles by introducing multiple
matching markets which open distinct subperiods in the day time. We can assume that
impatient agents go through the matching markets sequentially during the day time,in each of which they produce the consumption good using land and the intermediate
good produced by patient agents. We assume a simple structure of chains of productions
that an impatient agent who could not produce in a matching market cannot enter
the subsequent matching markets and must go to the night market directly; and only
an impatient agent who successfuly produces in a matching market can enter the next
matching market. In equilibrium, several matching markets may be operative, while theother matching markets are shutdown. Although we have only two steady states in thebasic model, there arise naturally more than two steady-state equilibria in the generalized
model, each of which is distinguished by the number of open matching markets or,equivalently, by the level of land price. In a steady state with a higher land price,18

more matching markets are operative. The steady states in the generalized model may
be interpreted as representing various stages in the business cycles, such as a boom, ashallow recession, a deep recession, etc. Thus the generalized model may be potentially
useful to analyze both ordinary business cycles and extraordinary ﬁnancial crises in a
uniﬁed framework.
The generalized model shows a new theoretical possibility of expectations-driven
business cycles: Changes in asset prices may cause changes in the current and future
productivity through changes in the expectations on the number of operative matchingmarkets, while the productivity changes justify the ﬂuctuations in the asset prices. In
short, the generalized model show that the expectations on the operative matching markets may drive the business cycles, while the expectations are justiﬁed by the induced
productivity changes.5 The feature of our model is exactly the same as sunspot equilibrium models in that the number of operative matching markets, the asset price, and theaggregate productivity are endogenously selected from candidates of steady states, andthere is no selection mechanism that is based on the economic fundamentals.

3.1

Setup

The model is the same as the basic model described in the previous section except that
there are N subperiods in the day time, where N ≥ 2, and a distinct matching market
opens in each subperiod. We call the matching market in i-th subperiod D-i market,
where i = 1, 2, · · · , N .

3.2

Optimization of impatient agents

An impatient agent enters D-1 market in the ﬁrst subperiod of the day time of date t,bringing lt−1 units of bank notes which are borrowed from a bank, bt−1 units of debt
5This type of the expectations-driven business cycles may be interesting because in the existing modelsof the expectations-driven business cycles pioneered by Beaudry and Portier (2004), the expectations on
future productivity is exogenously given, while in our model the expectations change endogenously. Seefor examples of the existing models of the expectations-driven business cycles Christiano, Ilut, Mottoand Rostagno (2007), Jaimovich and Rebelo (2008), and Kobayashi, Nakajima, and Inaba (2007).

19

obligations, and kt−1 units of land from date-(t − 1) night market. The impatient agent
searches for a patient agent and if she successfully meets a patient agent she buys theintermediate good from the patient agent and produces the consumption good; and only
if she successfully produces in D-1 market, she enters D-2 market in the second subperiod,
bringing the output of D-1 market and the remaining bank notes. If she could not meet apatient agent in D-1 market, she cannot participate in any subsequent markets in the day
time and must go directly to the night market. This process is repeated N times goingfrom D-1 to D-N markets. In D-i market, an impatient agent meets a patient agent
(i)

(i)

(i)

with probability α(µt )µt , where µt

is the relative measure of patient agents who

participate in D-i market, that is, the measure of participating patient agents divided by
the measure of impatient agents who participate in D-i market.6 We denote variables in
D-i market by putting superscript (i) on those variables.
The Bellman equation for D-i market (1 ≤ i ≤ N ) is as follows. The state variables(i−1)

in the value function are yt(i−1)

beginning of D-i market; nt, the consumption good that the agent carries at the, the remaining bank notes that the agent carries at the

beginning of D-i market; lt−1 , the bank loan that the agent borrowed in date-(t − 1)night market; bt−1 , the bond; and kt−1 , the land.
(i−1)

V (i) (yt

(i−1)

, nt

, lt−1 , bt−1 , kt−1 ) =

(i)

max
y (i) ,q (i) ,n(i)

(i)

(i)

(i)

α(µt )µt V (i+1) (yt , nt , lt−1 , bt−1 , kt−1 )
(i)

(i)

(i−1)

+ {1 − α(µt )µt }W (yt

(i−1)

, nt

, lt−1 , bt−1 , kt−1 ),
(57)

(i)

(i−1)

subject to nt = nt(i) (i)− pt q t ,(58)n(i) ≥ 0,
(i)

(59)
(i)

(i−1)

yt = A(i) {kt−1 }θ {qt }1−θ + yt

,

(60)

where V (i) ( · ) is the value function for an impatient agent entering D-i market; V (N +1) ( · ) ≡(0)

W ( · ); yt
6(0)= 0; and nt= lt−1 . Note that the production technology given in (60)(i)

Using the relative measures, µt , the measure of impatient agents who participate in D-i marQ(j)(j)
and the measure of patient agents who participate in D-i market is
ket is written as i−1
j=1 α(µt )µt
Qi−1(j) (i)(j)[ j=1 α(µt )µt ]µt .

20

guarantees that the production in D-i market is independent from the output in D-jmarket for i ̸= j. Note also that the assumption of chains of productions that only those
who successfully produce in D-i market can enter D-(i + 1) market greatly simpliﬁes theanalysis by avoiding the curse of dimensionality. This is because this assumption guarantees that all impatient agents who participate in a matching market have identicaltrading history and have identical values of state variables.
Since the problem for an impatient agent in the night market is identical to the basicmodel, the Bellman equation for the night market is
W (yt , nt , lt−1 , bt−1 , kt−1 ) =subject toU (c′t ) − h′t + β ′ V (1) (0, nt+1 , lt , bt , kt ),
(0)

max
(0)
c′t ,h′t ,nt+1 ,lt ,bt ,kt(61)

c′t + at kt + bt−1
≤ yt + h′t + ωt kt−1 + nt − Rt−1 lt−1 +bt+ at kt−1 ,1 + rt(62)

bt + Rt lt ≤ at+1 kt ,(63)

lt ≥ 0,

(64)

(0)

(65)

nt+1 ≤ lt .

The same arguments as those in the previous section imply that the value function for
the night market can be written as
W (yt , nt , lt−1 , bt−1 , kt−1 ) = yt + nt − Rt−1 lt−1 − bt−1 + (ωt + at )kt−1 + W0 .

(66)

Using this expression, the reduced form of the value function V (1) ( · ) is derived, given
(i)

(i)

(N ) ( · ) is solved using (66); and V (i) ( · ) is solved backwardly{pt , µt }N
i=1 : First, V

from i = N − 1 to i = 1. Thus we have(0)

V (1) (0, nt , lt−1 , bt−1 , kt−1 ) =

N
∑



max 
(i)

i
∏

(i)

i=1 qt ,nt[](j) (j)(i)(i) (i)θα(µt )µt  A(i) kt−1{qt }1−θ − pt qt

j=1− Rt−1 lt−1 − bt−1 + (ωt + at )kt−1 + W0 ,subject to(i) (i)

(i)

(i−1)

pt qt + nt ≤ nt
n(i) ≥ 0,,

for i = 1, 2, · · · , N,

for i = 1, 2, · · · , N.
21

(67)
(68)
(69)

(i)

(i)

Given {pt , µt }Ni=1 , the FOCs are
θ {q }−θ
(1 − θ)A(i) kt−1
t
(i)

(i)

pt =
(i−1)

xt
(i)

(i)

where xt and ηt

(i)

,

(70)

1 + xt= β ′ α(µt )µt xt + ηt(i)(i) (i)(i−1)

for i = 1, 2, · · · , N,,

are the Lagrange multipliers for (68) and (69), respectively, and the= β − β ′ and ηt

(0)

(0)

FOCs for (61) and the envelope conditions for (67) imply that xt

3.3

(71)

= 0.

Optimization of patient agent

We assume that at the beginning of each date a patient agent choose whether she participate in a decentralized matching market in the day time. A patient agent can participate
in at most one market among D-1, · · · , D-N markets. Thus the optimization problem(i)for a patient agent is identical to the basic model. Therefore, µt , the relative measureof patient agents who participate in D-i market, is determined by
[
(i)

(i)

If µt > 0, qt](i)(i) (i)(i)α(µt ){pt qt − γ(qt )} − κ(i)µt = 0.

(72)

is determined by the FOC:
γ ′ (qt ) = pt .
(i)

3.4

(i)

(73)

Equilibrium

We can deﬁne a competitive equilibrium of the N -market model as follows.
(i)

Deﬁnition 2 A competitive equilibrium consists of prices, {{pt }N
i=1 , Rt , rt , at }, andquantities, {ct , c′t , ht , h′t , kt , dt , bt , lt , {qt , nt , µt }Ni=1 }, that satisfy that (i) given the prices,
(i)

(i)

(i)

the quantities satisfy (72) and (73), which are the optimality conditions for patient agents;(i)(ii) given the prices and {µt }Ni=1 , the quantities solve impatient agents’ optimization
problems, (67) and (61), and banks’ optimization problem; and (iii) equilibrium conditions, (39) and (40), are satisfied.
The equilibrium values of variables in D-i market are determined by solving (70), (71),(0)(72) and (73) forwardly, given xt= β − β ′ and ηt

(0)

22

(i)

= 0, where ηt = 0 if D-i market

is operative. There exist multiple steady-state equilibria. D-i market is either open or
shutdown for i = 1, 2, · · · , N in a steady-state equilibrium. Since the assumption of(i)chains of productions guarantees that if µt(j)= 0 then µt= 0 for all j ≥ i + 1, there

exist at most N + 1 steady-state equilibria: Each equilibrium is distinguished by the
number of the matching markets which is not shutdown. We can deﬁne I-equilibrium(I = 0, 1, · · · , N ) as a steady-state equilibrium where the ﬁrst I matching markets areoperative, that is, µ(i) > 0 for 1 ≤ i ≤ I and µ(i) = 0 for I + 1 ≤ i ≤ N . In Iequilibrium, the equilibrium variables in D-i market are determined by the following
system of equations, given x(0) = β − β ′ :p(i) =

θ {q (i) }−θ
(1 − θ)A(i) kt−1,
1 + x(i)(74)

x(i−1) = β ′ α(µ(i) )µ(i) x(i) ,(75)

α(µ(i) ){p(i) q (i) − γ(q (i) )} − κ = 0,(76)

γ ′ (q (i) ) = p(i) ,(77)

for i = 1, 2, · · · , I; and q (i) = µ(i) = 0 for i = I + 1, · · · , N .
The values of {p(i) , q (i) , µ(i) , x(i) }N
i=1 are determined by solving the system of equations(74)–(77) forwardly from i = 1 to i = N , given x(0) = β − β ′ . Note that if D-i market isopen in both I-equilibrium and I ′ -equilibrium, the values of variables in D-i market are
identical in I- and I ′ -equilibria for I ̸= I ′ .7
The asset price in I-equilibrium is determined bya(I) =

β ′ {ω +

∑I

i=1 [

∏i

(j) (j)(i) θ−1 {q (i) }1−θ }j=1 α(µ )µ ]θA K

1−β

,

(78)

β ′ ∏Ior, equivalently, a(I) = a(I − 1) + 1−β[ j=1 α(µ(j) )µ(j) ]θA(I) K θ−1 {q (I) }1−θ . Obviously,a(I) is increasing in I, that is, the asset price is higher in the equilibrium where more
matching markets are operative. The amount of bonds in I-equilibrium is determined
7

Note also that for a certain range of parameter values there may exists J(≤ N ) such that the above

system of equations (74)–(77) has no positive solution {p(i) , q (i) , µ(i) , x(i) } for i = J. In this case, D-J,
D-(J + 1), · · · , D-N markets are always shutdown in any equilibrium. In this case, there exist only J
steady-state equilibria: 0-, 1-, · · · , (J − 1)-equilibria.

23

byb(I) = a(I)K −I∑

p(i) q (i) .

(79)

i=1

The welfare of an agent can be deﬁned as the value of her value function at the beginningof the daytime in the steady state. Thus the welfare of a patient agent in I-equilibrium,
V p (I), and that of an impatient agent, V (1) (I), areV p (I) = d(I) =V (1) (I) =

b(I)
,
M


1 
ωK − (1 − β)b(I) +1 − β′


I ∏
i
∑
i=1



α(µ(j) )µ(j)

j=1


(


(80)


)
A(i) K θ {q (i) }1−θ − p(i) q (i)  .
(81)

We can deﬁne the total welfare of the economy in I-equilibrium by E(I) ≡ M V p (I) +V (1) (I), which is written as




I ∏
i
(
)
∑
1 
E(I) =
ωK + (β − β ′ )b(I) +α(µ(j) )µ(j)A(i) K θ {q (i) }1−θ − p(i) q (i)  .

1 − β′
i=1

j=1

(82)
It is shown as follows that E(I) is increasing in I, that is, the welfare of the economy
increases as the number of operative matching markets increases. Deﬁne ∆(I) ≡ (1 −β ′ ){E(I) − E(I − 1)}.
[{}](β − β ′ )β ′ θ
1−θA(I) K θ {q (I) }1−θ , (83)∆(I) =
1+
Γ(I) − {Γ(I) + β − β ′ }1−β1 + x(I)∏I(i) (i)(I) = (β − β ′ )/[(β ′ )I Γ(I)] and {Γ(I) + β −where Γ(I) =i=1 α(µ )µ . Since x
β ′ }(β ′ )I /{(β ′ )I Γ(I) + β − β ′ } < 1, it is shown that{}(β − β ′ )β ′∆(I) > 1 +θΓ(I)A(I) K θ {q (I) }1−θ > 0,
1−β

(84)

which implies that E(I) > E(I − 1) for all I ≥ 1.
Policy implication:

Multiple steady-state equilibria arise in this generalized modeldue to the same coordination failure that we discussed in the previous section. This24

model is essentially a sunspot equilibrium model and the equilibrium selection depends
on the macroeconomic expectations. Government policy or regulation that restricts theamount of debt issued by impatient agents may improve the coordination failure andmay change the equilibrium. For example, if the government imposes the restriction∑that bt ≤ at+1 kt − Ji=1 p(i) q (i) , then the candidates of the steady-state equilibrium thatcan be realized become restricted to J-, (J + 1)-, · · · , N -equilibria. In other words, thesteady states from 0-equilibrium to (J − 1)-equilibrium are eliminated from the candidates of realizable steady-state equilibrium. Therefore, this model implies that monetary
policy and/or ﬁnancial regulations that restrict the aggregate level of debt may raise theaggregate productivity and the asset prices through increasing liquidity that enhances
economic transactions in decentralized matching markets. This policy implication may
be consistent with the historical episodes of ﬁnancial crises. Japan and Sweden, for example, experienced collapses of land prices and emergences of nonperforming loans problems
almost simultaneously in the early 1990s. Sweden disposed of bad loans aggressively in
1992–1994, and then attained a V-shaped economic recovery in the middle of the 1990s,
while Japan postponed the disposal of nonperforming loans and its debt-ridden economy
experienced decade-long slow growth.

4Conclusion

In this paper, we propose a model in which collateral-secured loan is used to smoothconsumption intertemporally and also used as a payment instrument, i.e., liquidity, intratemporally. We show that the interaction between the two roles of collateral lending
causes multiple steady-state equilibria to exist. The role of liquidity to mitigate thesearch friction in the decentralized market plays the key role to generate the multiple
equilibria. In our basic model, there are two stable steady states, one of which is abad equilibrium where the asset price collapses and the decentralized matching market
is disrupted due to shortage of liquidity. In the bad equilibrium, the land price is low
and the intratemporal debt, i.e., liquidity, dries up because of the shortage of collateral.
Consequently, the decentralized matching market is shutdown, since sellers choose not25

to participate in the market facing buyers’ liquidity shortage. This market disruption
lowers the aggregate productivity, while the low productivity justiﬁes the low asset prices
in turn. The asset price is lower and the debt/asset ratio, bt /(at+1 K), is higher in thebad equilibrium. Therefore, a policy implication for ﬁnancial crisis management is that
if the government imposes the restriction that all impatient agents must set their debt
burden, bt , such that bt ≤ at+1 kt − ph qh , then the crisis state is eliminated and theeconomy may jump to the normal state. This result seems to support the eﬀectivenessof debt reduction policy as ﬁnancial crisis management. Thus our analysis in this paper
may shed some light on assessment of policies for ﬁnancial crisis management that are
explicitly concerned with the asset prices and the aggregate amount of debt.

References
Beaudry, P. and F. Portier (2004). “An exploration into Pigou’s theory of cycles.”Journal of Monetary Economics 51(6): 1183–1216.

Bernanke, B. S. (2008) “Reducing systemic risk.” Speech at the Federal Reserve Bank ofKansas City’s Annual Economic Symposium, Jackson Hole, Wyoming, August 22, 2008.

Bernanke, B. S., M. Gertler, and S. Gilchrist (1999). “The ﬁnancial accelerator in aquantitative business cycle framework.” In J. B. Taylor and M. Woodford eds., Handbookof Macroeconomics 1C. 1341–1393.

Blanchard, O., and M. Kremer (1997). “Disorganization.” Quarterly Journal of Economics 112(4): 1091–1126.

Borio, C. (2003). “Towards a macroprudential framework for ﬁnancial supervision andregulation?” BIS Working Papers No.128. Bank for International Settlements.

26

Carlstrom, C. T., and T. S. Fuerst (1997). “Agency costs, net worth, and businessﬂuctuations: A computable general equilibrium analysis.” American Economic Review
87(5): 893–910.

Carlstrom, C. T., and T. S. Fuerst (1998). “Agency costs and business cycles.” Economic
Theory 12: 583–597.

Christiano, L. J., C. Ilut, R. Motto, and M. Rostagno (2007). “Monetary policy andstock market boom-bust cycles.” European Central Bank Working Paper Series 955.

Ferraris, L., and M. Watanabe (2008). “Collateral secured loans in a monetary economy.”
Journal of Economic Theory (forthcoming).

Iacoviello, M. “House prices, borrowing constraints, and monetary policy in the businesscycle.” American Economic Review 95 (3): 739–764.

Jaimovich, N. and S. Rebelo (2008). “Can news about the future drive the businesscycle?” American Economic Review (forthcoming).

Jermann, U., and V. Quadrini (2006). “Financial innovations and macroeconomic volatility.” Mimeo. University of Southern California.

Kehoe, T. J., and E. C. Prescott (2002). “Great depressions of the 20th century.” Review
of Economic Dynamics 5 (1): 1–18.

Kehoe, T. J., and E. C. Prescott eds.(2007). Great depressions of the twentieth century
Federal Reserve Bank of Minneapolis.

Kiyotaki, N., and J. Moore (1997). “Credit cycles.” Journal of Political Economy 10527

(2): 211–48.

Kobayashi, K. (2004). “A theory of debt disorganization.” Journal of Restructuring
Finance 1 (1): 1–11.

Kobayashi, K. (2006). “Payment uncertainty, the division of labor, and productivity
declines in great depressions.” Review of Economic Dynamics 9 (4): 715–741.

Kobayashi, K., T. Nakajima, and M. Inaba (2007). “Collateral constraint and newsdriven cycles.” RIETI Discussion Paper Series 07-E-013.

Kobayashi, K. and K. Nutahara (2007). “Collateralized capital and news-driven cycles.”Economics Bulletin 5 (17): 1–9.

Lagos, R., and G. Rocheteau (2005). “Inﬂation, output and welfare.” International
Economics Review 46: 495–522.

Lagos, R., and R. Wright (2005). “A uniﬁed framework for monetary theory and policy
analysis.” Journal of Political Economy 113 (3): 463–484.

Mendoza, E. G. (2006). “Endogenous sudden stops in a business cycle model withcollateral constraints: A Fisherian deﬂation of Tobin’s q.” NBER Working Paper 12564.

Ohanian, L. E. (2001). “Why did productivity fall so much during the Great Depression?” American Economic Review 91 (2): 34–38.

Rocheteau, G., and R. Wright (2005). “Money in search equilibrium, in competitive
equilibrium, and in competitive search equilibrium.” Econometrica 73 (1): 175–202.

28

A

Appendix

In this Appendix, we conﬁrm that our results in the previous section hold even if we introduce money in our model. As is the case for Carlstrom and Fuerst’s (1997) model, thesetting of our model that there are two diﬀerent types of agents, patient and impatient,implies that impatient agents do not hold cash for payment in a steady-state equilibrium.
In this Appendix, we consider a slightly stronger form of money than cash. We conﬁrm
that money does not matter for our results in a modiﬁed version of our model in which
we introduce deposit money that can earn interest.
We assume that agents can hold intertemporal bank deposits that earn interest withthe rate rtd , instead of bonds. Impatient agents can use the principal and interest of theirdeposits as a payment instrument in the decentralized market in addition to the banknotes that they borrow from banks.

A.1

Optimization of patient agents

Since patient agents hold bank deposit, dt , instead of bonds, bt , we get the Bellman
equations for our monetary model by substituting dt and rtd for bt and rt in the previous
section. Note that dt is measured in the unit of date-(t + 1) consumption good.
V p (dt−1 ) = max {−κ + α(µt )[−γ(qt ) + W p (pt qt , dt−1 ) − W p (0, dt−1 )],0} + W p (0, dt−1 ),
(85)

and
W p (lt , dt−1 ) = max dt−1 −dt

dt+ lt + βV p (dt ).
1 + rtd(86)

Similar arguments as those in the previous section implyβ(1 + rtd ) = 1.

A.2

(87)

Optimization of impatient agents

Since impatient agents can hold bank deposits, d′t , as their assets, we need to include d′t
as the state variables. In this modiﬁed model, impatient agents borrow intertemporal29

loan, bt , from banks instead of borrowing from patient agents directly. The interest rate
for bt is rt , where rt ≥ rtd . The Bellman equation for the day market is
V (lt−1 , bt−1 , d′t−1 , kt−1 ) =α(µt )µt max ′′ W (yt , n′t , lt−1 , bt−1 , kt−1 )yt ,qt ,lt ,dt

subject to+ (1 − α(µt )µt )W (0, lt−1 , lt−1 + d′t−1 , bt−1 , kt−1 ),(88)n′t = d′t−1 + at kt−1 − pt qt ,(89)

n′t ≥ 0,(90)θyt = Akt−1
qt1−θ ,

(91)

where n′t is the total of bank notes and deposits that the impatient agent brings into thenight market. The Bellman equation for the night market is
W (yt , n′t , lt−1 , bt−1 , kt−1 ) = ′ max′

ct ,lt ,bt ,dt ,kt+U (c′t ) − c′t + yt − at {kt − kt−1 } + ωt kt−1 + n′t −bt− bt−1 + β ′ V (lt , bt , d′t , kt ),1 + rtd′t− Rt l t
1 + rtd(92)

subject to bt + Rt lt ≤ at+1 kt + d′t . FOCs and envelope conditions for bt and d′t imply
1 = (1 + rt )ξt + β ′ (1 + rt )(1 + α(µt+1 )µt+1 xt+1 ),(93)

1 ≥ β ′ (1 + rtd ){1 + α(µt+1 )µt+1 [Rt+1 + Rt+1 xt+1 − 1]},(94)

where d′t = 0 if (94) holds with strong inequality. Therefore, equations (93) and (94)
imply that d′t = 0 if Rt+1 = 1 and rtd < rt , which, we will show below, is the case in theequilibrium.

A.3

Optimization of banks

Banks accept deposits dt from patient agents and d′t from impatient agents, and invest
them into intertemporal lending bt and the real cash reserve mt under a technologicalconstraint that total deposits8 cannot exceed ϕmt where ϕ (ϕ > 1) is a technological
8

Here we make a slightly problematic assumption that the intraperiod bank notes, lt , that the bankissues at the day market are not constrained by cash holding of the bank. Otherwise the constraint (97)
would be dt + d′t + lt ≤ ϕmt and the intraperiod rate of interest would exceed one, that is, Rt > 1. In
this case, the analysis would be complicated and our results hold only for a certain range of parametervalues. For simplicity, we assume in this paper that the constraint for the bank is (97).

30

parameter for production of payment services. The Optimization problem of bank inperiod t is written as follows:mt− (dt + d′t ) + (Rt − 1)lt ,1 + πtlt ,bt ,dt ,dt
bt
dt + d′t
subject to+ mt + lt ≤+ lt ,
1 + rt1 + rtd
dt + d′t≤ ϕmt ,
1 + rtdmax ′ bt +

(95)
(96)
(97)

where πt is the inﬂation rate. Note that lt denotes the bank notes lent to impatient
agents. The bank’s problem can be rewritten as follows: Given rt , rtd and Rt ,max[(rt − rtd )ϕ + (1 + πt )−1 − (1 + rt )]mt + (Rt − 1)lt .
lt ,mt(98)

Since lt < ∞ and mt < ∞ in the equilibrium,
Rt = 1,
1 + rt =(99)
β −1 ϕ − (1 + πt )−1
ϕ−1

.

(100)

As long as 1 + πt ≥ β, rt exceeds rtd = β −1 − 1, and therefore, d′t = 0 from (93) and (94).
It is easily shown that qualitatively the same results as those in Section 2 holds for thesteady-state equilibria of the model in this Appendix as long as 1 + π ≥ β, since d′t = 0
in this case.

31

Π(µ)

κ

0

µ

µl

µ
h

Figure 1: Determination of the measure of participating patient agents in Normal State

32