Thesis for Master Degree

Financial Innovation News, Collateral Constraint,
and Boom-Bust Dynamics in the Housing Market

Satofumi Ushigami
Economic Policy, Graduate School of Public Policy,
University of Tokyo

December 13, 2012

Contents
1 Introduction

3

1.1

Financial Innovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.2

News: From Optimism to Pessimism . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.2.1

Optimism on the Financial Innovation . . . . . . . . . . . . . . . . . . . . . . .

8

1.2.2

Pessimism on the Financial Innovation . . . . . . . . . . . . . . . . . . . . . . .

11

Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.3

2 The Model

15

2.1

Patient Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.2

Impatient Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.3

Entrepreneurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.4

Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.5

Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.5.1

Transition Dynamics: Current Financial Innovation . . . . . . . . . . . . . . .

24

2.5.2

News Shock: News is Incorrect . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.6

Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.7

Constrained Entrepreneurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3 Conclusion

37

A Steady state system of the baseline model

39

B Steady state system of the extended model

40

2

Financial Innovation News, Collateral Constraint, andBoom-Bust Dynamics in the Housing Market∗
Satofumi Ushigami†
December 13, 20121

Introduction“But housing is housing; banks have been making mortgage loans for generations.Why wasanyone willing to believe that all the rules had changed?...So a bit of ﬁnancial engineering― with no new technology, no reason to think that the fundamentals of housing had changed― led to a gigantic ﬁnancial bubble. Wow. I guess hype really does spring eternal.” ―Paul

Krugman, “Mystery of the mortgage mess”, New York Times, 11/17/2007

The recent U.S. ﬁnancial crisis is originally characterized as a housing market boom and bust. As
Figure 1 shows, the U.S. housing market experienced prolonged boom that started around 1999. Afterthe peak of the price in early 2006, the price started to decline as the bad news about the housing
market continued on arriving. In fact, home prices dropped 18% on December 30, 2008 comparedwith the level in the previous year.1 This sharp drop came with the fact that there were tremendous∗ I thank Professor Kosuke Aoki and participants at Macro-Finance seminar. Especially, Professor Aoki provides mewith tremendous numbers of constructive comments in every aspect of this thesis. I also thank Masayo Hosono, Keichi
Ishiwata, Yuya Miyoshi, Kazushige Matsuda, Tomohito Hata and many friends for their heartfelt supports.
† Economic Policy, Graduate School of Public Policy, University of Tokyo
1 “Home prices oﬀ record 18% in past year, Case-Shiller says”, www.marketwatch.com.

3

numbers of borrowers who could not repay the money for the housing loan. The increment of theforeclosure rates frightened the marketers, making the turmoil deeper and deeper.

Figure 1: Source: Freddie Mac Housing Price Index, National

One of the key features of the crisis relates to the ﬁnancial innovation. Before the U.S. boom ended,lots of new ﬁnancial products and a set of policy reforms were introduced. For example, collateralized
mortgage obligation(CMO) was introduced in the mid 1990s, credit default swap were ﬁrst issued in
the late 1990s, and so on. It is also valuable to refer to a set of legal reforms. For instance, the 1995
New Community Reinvestment Act is a ingredient of the ﬁnancial innovation since it allowed Fannie
Mae and Freddie Mac to take stronger role in the mortgage market, which resulted in the deepeningof the mortgage market.
Figure 2 and Figure 3 represent one aspect of the proliferation of the ﬁnancial technology. Figure 2
shows that the market of the securitization and CMO tranching grew during the housing boom, whichin their peak amounted to trillions of dollars a year. Figure 3 presents the growing trend of the CDS
market when the economy was booming. Data is from mid 2000s because it was the time that CDS
technology were standardized for mortgages.
In this paper, the two facts shown in the previous discussion are connected. In other words, it is4

Figure 2: Amount of securitization and tranching, Data Source:IBS OTC Derivatives Market Statistics,
I borrow this graph from Geanakoplos(2012)

Figure 3: Outstanding National Amount of CDS Market, Data Source:IBS OTC Derivatives Market
Statistics, I borrow this graph from Geanakoplos(2012)5

shown that the ﬁnancial innovation can be a cause of the housing boom and subsequent bust, which
was typically observed in the recent U.S. housing market. The two speciﬁc information structures are
assumed. The ﬁrst is that the inﬂuence of the ﬁnancial innovation is inherently so diﬃcult to predictthat people could misunderstand that a set of events that do not indicate the substantial improvementof the ﬁnancial environment really matters. The second is that they believes that adaption to the
new ﬁnancial technology or environment takes time. These two assumptions make agents regard acurrent set of events, such as introduction of CMOs, CDSs, as well as new regulation acts, as thesignal that in a few years the ﬁnancial environment will improve. This paper proves that this type of
information shock, named as news shock, can generate boom and bust in the housing market. The
reason is as follows: In an economy where houses are used as a collateral to borrow, this shock induces
the future housing price increase because the cost of borrowing decreases when the innovation occurs
in the future. Anticipating the price increase, households can borrow more since the interaction of theprice increment and the accumulation of the housing allows them to do so, which leads to the pricehike even before the realization of the ﬁnancial innovation. This mechanism generates the prolonged
boom in the housing market as it was observed in the U.S. housing market. The boom will end whenpeople realize the news was incorrect. At this time, borrowers hold housing so much that they become
unwilling to possess them any more. This sudden disappearance of the demand collapses the housing
market, making the price precipitate.
In the following section, I ﬁrst survey the observed ”ﬁnancial innovation” before, and during theboom. Next, I qualitatively look through the movement of the attitude toward the innovation. Finally,
I move on to explain the model showing the related literature, and close the discussion in the conclusion
section.

6

1.1

Financial Innovation

The recent ﬁnancial advance has occurred since around 1970. The government agencies Fannie Maeand Freddie Mac began the securitization of mortgage in earnest at that period. Salomon and First
boston ﬁrst made tranches by purchasing and slicing Fannie Mae and Freddie Mac pools in 1986.
The issuance of the ﬁrst collateralized debt obligations(CDOs) happened in 1987. By the mid 1990s,
the investment bank Kidder Peabody, one of the greatest mortgage buyers, sliced tons of mortgage
pools into tens of kinds of tranches called collateralized mortgage obligations(CMOs), named suchas ﬂoater, inverse ﬂoater Pac, Tac and so on. Credit default swaps(CDSs), kinds of insurance on anasset or bond, were invented in 1997 by JP Morgan for corporate bonds and government bonds. This
instrument was commonly used for mortgages in mid 2000s.
The ﬁnancial innovation also refers to the several policy reform regarding the ﬁnancial environment.
Examples of the Acts introduced around the end of the 20th century were three: the 1995 New
Community Reinvestment Act, the 1999 Gramm-Leach-Bliley Act, and the 2000 Commodity Futures
Modernization Act according to the selection of Mendoza(2012). The ﬁrst act was designed to empower
Fannie and Freddie in mortgage market in order to activate the regional ﬁnance, leading the deepeningof the securitization market . The second was created to partly abolish the Glass-Steagall Act thatprohibited the bank holding companies to take charge of other types of banking roles, aiming at
banning the excess risk taken by the ﬁnancial sector. Commodity Futures Modernization Act mitigated
the regulations on the over-the-counter derivatives like CDSs, clearly diﬀerentiating itself with theCommodity Exchange Act that stipulates the regulation of the trading of commodity futures.
Figure 4 provides the timeline of the selected ﬁnancial innovations. On the basis of the strand of
ﬁnancial innovations, it must be no harm to assume that it induces the prediction that it actually
reduces the transaction cost and risk itself. In the next section, I investigate the stance on thoseﬁnancial innovation upon which the marketers lean. From the survey it can be understood that theposition is not always consistent along the period. Rather, the systematic alteration of the mode7

Figure 4: Chronicle of the selected ﬁnancial innovation, timeline from Mendoza(2012)exists: from optimism to the pessimism.

1.2

News: From Optimism to Pessimism

1.2.1

Optimism on the Financial Innovation

Here I survey the news pressed in the US and look through the time-varying attitude toward ﬁnancial
innovation. The investigation qualitatively clariﬁes movement of the mode from the optimistic one tothe pessimistic. The story dates back around 20 years.
In the early 90s, the ﬁnancial innovation was taken as a plausible advance that made the way tothe complete ﬁnancial market. Even though there were some worries regarding its possible role ofampliﬁcation of the market stemming from the inherent diﬃculty and information asymmetry, themain attitude on the innovation was ”optimistic”. The statement by Professor Robert Cox Merton,
the Nobel Prize winner in 1997, represented his positive viewing on the innovation at that time:“...the dramatic changes over the past two decades are consistent with development toward a more eﬃcient ﬁnancial market and intermediation system. Indeed, such changes
can be seen as part of a ﬁnancial innovation

spiralthat proceeds as follows: The prolif-

eration of new trading markets in standardized securities such as futures makes possible the8creation of new custom-designed ﬁnancial products that improve

market completeness.

Next, volume in the new markets further expands as the producers themselves―typically,ﬁnancial intermediaries―trade simply to hedge their own exposures. Such increased volume
in turn reduces marginal transaction costs and thereby makes possible further implementation of new products and trading strategies―which in turn leads to still more volume.
Success of these trading markets then encourages investment in creating additional markets, and so on it goes, spiraling toward the theoretically limiting case of zero marginaltransactions costs and dynamically complete markets.”ー Robert C. Merton, “Financial
Innovation and Economic Performance”, Journal of Applied Corporate Finance, 1992

This intriguingly strong positive stance continued, or even strengthened through the array of policy
reforms and invention of new seemingly favorable ﬁnancial instruments such as CMO, and CDS by themid 2000s. The following sentences exist in the article that was written during the crisis, repenting
the past attitude of the economy.
“The apparent triumph of that (the transfer of risk)2 process came after the technology
stock bubble burst and the American economy went into recession in 2001. A lot of money
was lost because of huge overinvestment in things like ﬁber optic cable, but the losses weredispersed. No signiﬁcant banks failed or even got into trouble.The system worked, theregulators told themselves as they praised the ﬁnancial advances brought on by the derivative
revolution.”“Who s Going to Take the Financial Weight?”, New York Times,10/26/2007

The interesting thing is the fact that in the early 2000s, US experienced the burst of the IT
bubble and Enron Scandal that were obviously related to the ﬁnancial innovation. Nevertheless, theinterpretations of the events were against what would be expected ﬁrst: the above news indicatesthat the marketers in the US thought the burst was more modest than the case where there had not
2 Parenthesis added to mean the pronoun by the author of this paper.

9

been any ﬁnancial innovation, and therefore regarded the observed size of shock as the proof of thewinning of the ﬁnancial innovation. Doubt might come that this view was singular at the time andwas shown only in this article as the exaggeration during the crisis. However, it does not seem so.
In fact, the mood observed in the next article also illuminates the intact optimism when the Enronscandal astonished the market :“Home mortgages, car loans, credit card debt, student loans, equipment leases – allthese assets, and more exotic ones like future receipts from British pubs and the box-oﬃcetake for Steven Spielberg’s future movies, have been remodeled into asset-backed securities. Those securities can now be found generating handsome returns for pension funds,
mutual funds, banks, brokerage houses, insurance carriers, hedge funds and even some individual investors. A result, bond market experts say, has been the expansion of credit toconsumers, greater liquidity and ﬂexibility for lenders and the modulation of risk for corporations....Companies that want to use complicated structured-ﬁnancing techniques should
be prepared to explain them completely, he (Ronald Gilson, a law professor at both Stanford
and Columbia)3 said....But it would be a shame, he (Ronald Gilson) said, if the tools of
structured ﬁnance came to be seen as the equivalent of skeleton keys and safe-cracking implements. Like electric jigsaws and battery-powered screwdrivers, they can be used to create
shelter from hardships, safe exits from risky places, room to grow and a stable foundation.
But like all tools, he (Ronald Gilson) said, they are only safe if used at arm’s length in the
daylight.”,”The Brick Stood Up Before. But Now?”, New York Times, 2/10/2002

That is, the cause the author of the article, or possibly the readers had recognized behind thelatter scandal was neither the inherent weakness of ﬁnancial market that were activated by ﬁnancial
innovation nor new ﬁnancial instruments’ possible role as the accelerator of the ampliﬁcation; It was
the misuse of the innocuous technology by the bad guys. This way of looking at the event led the
3 Parenthesis added to mean the pronoun by the author of this paper.

10

point in question toward the transparency and disclosure of the information. This “misspeciﬁcation”
allowed the notorious exotic ﬁnancial product to be created, which dragged the world into the recent
great depression.

1.2.2

Pessimism on the Financial Innovation

The doubt began to come with the weak indicators in the housing market. One is the stagnationof the housing price growth, in the comparison of that in the booming. Other is the news of themonetary tightening. Another is the slight increase of the delinquency rate in the housing market. It
was, however, not enough for the market participants to turn over their belief on the underlying mythin the housing market: the prolonged belief that the risk sharing through the new ﬁnancial productactually lessened the risk and therefore cost of borrowing, and that it was irrelevant to the potential
vulnerability of the ﬁnancial sector. Looking back from now, the recognized uncertainty was actually
the symptom of the bust and should have more seriously examined as a omen of the coming crisis.
It is worth noting that the following news clearly pointed out what actually happened after the twoyears from the publication.

“At issue is whether ﬁnancial innovations that have made it easier for Americans tobuy homes have also made the system less stable and more subject to shocks that could
drive many from their homes...For now, all the talk about bubbles may be just talk. In thelate 1990’s stock market, those who warned that prices were too high were discredited by
the ﬁnal huge rise in 1999 and early 2000. Only after most bears were silenced did they
become right...THE housing bears now are anything but silent, and there is little sign ofthe economic news – soaring interest rates or recession or both – that would bring downhome prices. But if and when a fall comes, watch the volume of home sales, particularly of
existing homes. In the early 1980’s recession, annual sales of existing homes fell 50 percent,11

to two million homes....When that (prices do start to fall)4 does happen, watch the volume
of sales of existing homes. If it falls rapidly, that will be an indication that not much haschanged, and the damage is likely to be limited. But if sales volume stays high, that could
indicate that the mortgage innovations are hurting. Then we could see rising numbers offoreclosures as homeowners discover they cannot sell their homes for what they owe but
also cannot pay their suddenly higher monthly mortgage bills.”,“Easy Credit In Mortgages
May Backﬁre.”,New York Times, 8/26/2005

The sad thing is that this prediction actually turned out to be the prophecy: the housing bubble
bursted due to the excess lending, securitization was nominated as one of the criminal of the crisis,
the number of foreclosure rate skyrocketed, and the housing market collapsed two years after the
publication. The news below pressed amid the turmoil of the crisis stated the confusion of the market
stemming from the inherent diﬃculty of the identiﬁcation of the location of the risk in the securities.

“If it was no longer the banks that would suﬀer when a ﬁnancial crisis came around, who
would? Could it be the insurance companies? Perhaps pension plans? Would hedge funds
that gambled and lost be unable to meet their obligations and bring on a systemic failure?In
this ﬁnancial crisis, the one that started with subprime loans, we are learning the answer
to that question. The risks that banks would have taken on under the old system ― whenbanks made loans and proﬁted only as they were paid back ― had been transferred througha bewildering wilderness of options, swaps, swap options, specialized investment vehicles,
collateralized debt obligations, variable interest entities and who knows how many other
instruments.”“Who s Going to Take the Financial Weight?”, New York Times,10/26/2007

At this time, the myth on the ﬁnancial innovation as the promising risk-reducing instrumentscompletely collapsed; on the contrary, it was notoriously criticized as the origin of the crisis.
4 Parenthesis added to mean the pronoun by the author of this paper.

12While it is impossible to tell what we viewed is all true and represent the general viewing at thattime because these set of news is just a small fraction of the news pressed in the world, the strand ofnews at least indicates the movement of the attitude: from the optimistic toward the pessimistic withrespect to the ﬁnancial innovation. This ﬁnding rationalizes the ﬁrst stage of the hypothesis that thisboom-bust-style change of the viewing generated the housing bubble and its burst.

1.3

Related literature

In the next section, I start constructing the model. This part is allocated to show how the following
discussion is related to other literatures. The structure of the model presented in this paper have three
speciﬁc features: information on the ﬁnancial innovation, collateral constraint and the news shock.
First of all, the model has many common features with Mendoza(2012). In the model of thementioned article, agents know that the ﬁnancial innovation comes and that possibly changes theborrowing cost. However, they do not know whether the good innovation actually happens since
they do not have enough information or knowledge about the outcome on the new environment andnew technology, so that they observe the realized data and update their prediction in the Baysianstyle. The clear diﬀerence is that my model assumes the news shock and Mendoza(2012) assumes thelearning from the market. While learning can also explain the boom and bust, the clear optimism on
the ﬁnancial innovation that we see in the previous news survey provides a slight advantage on thenews shock, for agents in the market already believe the innovation will reduce the transaction costand allow household to borrow money much more easily.
It is proﬁtable to compare with Adam and Marcet(2012a) and Marcet, et. al.(2012b) with regard to the focus on the information and asset pricing. These two papers examine, in a traditional
consumption-based asset pricing model, how the adaptive learning from the observation of the data
about the price growth generates boom and bust in the asset price. The similarity results from thefeedback rule in their model. The key feature of them is that the shock on the belief immediately

13

induces current price increase. This data feedbacks into their belief, strengthens it, and creates further
price increase. This mechanism makes the ampliﬁcation in the market. One of the diﬀerence regarding
the ampliﬁcation is that my model does not assume the backward feed back. Rather it clariﬁes other
path of the ampliﬁcation that the promised shock generate the prolonged boom in the market.
Moreover, my model relates the general role of the ﬁnancial innovation in the US crisis. One of
the important papers on this aspect is Geanakoplos (2012). He explains the boom comes with thetechnology that allows the asymmetric betting: optimistic agents can borrow money from the relatively
pessimistic agents using assets as a collateral and buy the asset more while the pessimistic have nomeans to bet on their belief. The bust, as he has shown, also stems from the ﬁnancial innovation thatallows pessimistic agents to bet on the worse case using money as a collateral. The mechanism on his
discussion crucially relies on the speciﬁc sequence of the innovation during the boom-bust period thatmakes the speciﬁc transition of the leverage: leverage by the optimistic agents followed with the one by
the pessimistic. my model does not identify the speciﬁc role of the ﬁnancial instrument. Furthermore,
it identiﬁes the information structure as the key role of the boom and bust, rather than the speciﬁcleverage as the cause of the dynamics.
The model also focuses on the change of the belief. More and more literatures start analyzing therelationship between belief dynamics and asset pricing. Eichenbaum et. al(2012) and Piazzesi andSchneider(2009) focus on the fact that during the US housing bubble the consumer sentiment comoves
with the housing price, and show the belief dynamics regarding the housing fundamentals can generateboom and bust in the housing market. Shiller(2003) and Shiller(2007) empirically suggest the diﬃcultyin the contributing the dynamics of the housing price to the change of the fundamentals and point out
the importance of the phycological factor. The boom and bust in Eichenbaum et.al(2012) result fromthe dynamics of the number of the optimistic agents. The sequential increase of this number generates
prolonged price rise. The two clear diﬀerences can be found: the ﬁrst is they assume the belief
dynamics on the housing fundamentals, and I will assume the dynamics on the ﬁnancial innovation.

14

The second is that mechanism of the protracted boom is due to the gradual increase of the demand in
Eichenbaum et.al(2012), and in my model it owes to the interaction between the collateral constraint
and the news shock.
In the model the news shock is adopted as the shock of the economy. The news shock is the signal
the agents get beforehand and believe some factors such as productivity will change in the future.
The news shock was ﬁrst introduced in Beaudry and Portier(2004).Beaudry and Portier(2004) showed
this news shock can generate the consumption and investment boom at the same time even when theeconomy does not experience any observed fundamental shock. The important assumption is that allagents believe the news is true. If this news shock were to happen many times, it would be diﬃcult
to justify that every time agents should hold ”incorrect” belief in the ex-post meaning. In this sense,
this news shock and the learning is closely related.

2

The Model

In order to explore the hypothesis that news generates boom and bust, the model is basically constructed following the spirit of Iacoviello(2005). To be speciﬁc, the nominal structure observed in theoriginal model is shaved oﬀ and the real housing system in the model is abstracted because the focus
is on the role of the information on the housing price. The model considers a discrete time, inﬁnite
horizon production economy with three type of agents. The ﬁrst type is the patient agents, who are
relatively able to wait compared with the other agents. They work using their labor and obtain theirwage. They can allocate the income into three: to consume, to buy houses, and to lend (or borrow).
The second type is the impatient agents, who has less patience: that is, they have smaller discountrate than the patient agents. They also work, gain income, allocate them into the consumption goods,
housing, and borrowing (or lending). The smaller discount rate naturally assures them to be borrowers. Let us assume the borrowing is limited by the amount of housing as a collateral. This feature is15

consistent with Kiyotaki and Moore(1997) in that the collateral constraint is tied to the real estatevalue. The last type of the agents is entrepreneurs. They make use of the two types of the labor force
and produce consumption goods, paying the wages for each households. In the model it is assumed
that the supply of the housing is ﬁxed. This assumption is consistent with the related literature suchas Iacoviello(2005), and Eichenbaum et. al(2011). Below it is shown that this standard setting can
generate the non-negligible ampliﬁcation eﬀect in the housing market.

2.1

Patient Households

The problem of the households is fairly standard. The only diﬀerence from the normal model is that it
is assumed the housing improves the utility of the household. The households maximize the following
discrete lifetime utility function given their budget constraint:

maxE0 Σ(β p )t (lnCtp + jt lnhpt −(ltp )η
)η

s.t.Ctp + qt (hpt − hpt−1 ) + (1 + rt−1 )bpt−1 = bpt + wtp ltp

(1)

(2)

E0 represents the expectation operator given the information set at period 0, β p ∈ 1 is the patients’discount factor, andCtp , hpt , ltp denoted as consumption, housing, and hours of working, respectively.
The subscript p means the patients’ variable. The housing is multiplied by jt so as to distinguish theeﬀect of housing from consumption. The structure of instant utility gain from the housing is diﬀerent
from the assumption on Eichenbaum et. al(2011) that assumes the housing provides the utility gainin the next period. η expresses the disutility of working. They can obtain money by working andborrowing (lending). The price of the house, the interest from the ﬁnance is denoted with qt , rt ,respectively. The household maximize the utility given the prices by choosing consumption, housing,amount of borrowing, and hours of work. Let us assume that the house will not depreciate, so that16

the households can fully carry the housing over to the next period. By solving this problem, threeﬁrst order condition are obtained as follows:

qt
jt
qt+1
= p + Et β p ( p )
Ctp
ht
Ct+1

(3)

1
1
= Et (1 + rt )β p p
Ctp
Ct+1

(4)

wtp
= (ltp )(η−1)
Ctp

(5)

The ﬁrst equation shows the marginal utility from the consumption at period t is the same with the
expected marginal utility when they allocate money into the housing at period t and sell it and buy
consumption goods in the next period. The second equation represents the case when the households
allocate the consumption at period t into the bond market. The left hand side describes the marginalutility of consumption at t and the right hand side shows the expected marginal utility from theconsumption at period t + 1 with the return from the bond. The third shows the intra-temporal
relationship between the consumption and the working. The equation shows that the disutility fromthe work equates with the consumption using the wages from the work.

2.2

Impatient Households

The basic structure of the problem of the impatient households is the same with that of the patients’:maxE0 Σ(β i )t (lnCti + jt lnhit −(lti )η
)η

s.t.Cti + qt (hit − hit−1 ) + (1 + rt−1 )bit−1 = bit + wti lti

17

(6)

(7)

bit ≤ mt Et (qt+1 hit)
1 + rt(8)

The clear diﬀerence is that the problem contains the collateral constraint. The borrowing is limited
to certain fraction of the expected discount value of the housing in the next period. mt represents
the degree of the ﬁnancial development, which is the main target of the analysis in this paper. If theborrowers choose not to repay money, the lender can seize the housing that the borrowers hold as
the collateral, paying the transaction cost (1 − mt )Et (qt+1 hit ) to liquidate the collateral. In total, thelender can at least obtain mt Et (qt+1 hit ) without risk, which makes the contract as the risk-free one.
In the setting, the ﬁnancial innovation that results in the reduction of the transaction cost ofﬁnancing is deﬁned as the increment of mt , so that impatient households are allowed to borrow more.
In order to understand the setting suppose that all agents believe housing price will increase in thenext period, even if the price today does not change. This belief change permits them to make more
use of the ﬁnancing. This stems from the description of the constraint that the decision of the lending
in the housing market is forward-looking.
It is worth pointing out that this constraint is introduced into the household sector, not into theﬁrms. This is because the impatient households is regarded as the ones that have less credibility in theﬁnancial market, such as sub-prime-layered people. They are not always allowed to borrow as muchas they want due to the past experience of arrears, default, and so on. Since the ﬁnancial innovation
shown in the US amid 2000 mainly aﬀected these people, introducing this constraint in the impatient
households is plausible as the analysis of the US experience.
One assumption β p > β i is made in order to represent the diﬀerence of the patience. This assumption makes the impatient as the borrowers. As it is observed later the case where this constraint isalways binding is investigated in this paper.
The ﬁrst order conditions consist of the three equations:

18

−

qt
jt
qt+1
qt+1
+ i + mt λt Et+ β i Et i = 0
1 + rt
Cti
ht
Ct+1

(9)

1
1
− λt − β i Et (1 + rt ) i = 0
Cti
Ct+1

(10)

wti
= (lti )(η−1)
Cti

(11)

λt ≥ 0, λt (mt Et (qt+1 hit) − bit ) = 0
1 + rt(12)

Here let us deﬁne λt as the Lagrange multiplier at period t. λt expresses the lifetime utility
improvement from the borrowing: the gain derived from the increase of the consumption at period t
or investment on the housing, subtracted by the repayment at period t + 1. This multiplier makes thediﬀerence from the ﬁrst order conditions of the patient households.
The ﬁrst line equates the marginal utility of the consumption at period t with the utility fromhousing at period t and consumption at t + 1 when the households sell the housing into consumption
goods at period t + 1. The second describes the equality of the consumption at period t with the oneat period t + 1 using the bond. The third line shows the disutility from working is the same with theutility gain from the consumption increment at period t. The last line describes KKT conditions.
The question is whether the inequality of the constraint will bind or not. For the case without
uncertainty, it is proved it will. The steady-state Euler equation for patient households is arranged
to be 1 + r = β1p . Substituting this into the steady-state Euler equation of the impatient households,)
λ = (1−βC(1+r))= (βC −β> 0 by the assumption that β p > β i . From this fact, KKT shows the
i
i βi
i

p

i

constraint is binding at the steady state. In the following part, the situation around the steady stateis investigated and the equations are log-linearized. While it is true that the it cannot be assumed the19constraint is globally binding, the analysis of the binding case is rationalized by assuming the shockis small enough to assure our focus.5 Thus, the KKT conditions degenerate to be the following:

bit = mt Et (2.3

qt+1 hit)
1 + rt(13)

Entrepreneurs

The problem of the entrepreneurs is a standard RBC type model. The entrepreneurs have production
technology that allows them to produce consumption goods using two type of the labor force.

maxπtf = yt − wtp ltp − wti lti(14)s.tyt = At (ltp )α (lti )1−α(15)

The technology is Cobb-Douglas constant returns to scale one that uses two labor force as the
input.At is the productivity parameter. They maximize their proﬁt given the wage and technology,changing the inputs. One interpretation why the two labor forces are distinguished stems from thediﬀerence of the skill of the two. The patient households and the impatient households can be regarded
as the wealthy people and the poor people, and they have diﬀerent level of skill due to the heterogeneity
stemming from various reasons such as education. This diﬀerence allows us to assume the two labor
forces are in the imperfect substitution relationship in terms of the production of the goods.

yt
α p = wtp
lt

(16)

5 Iacoviello(2005) also adopts the same assumption and ascertain the harmlessness by investigating the non-linear
simulation.

20

yt(1 − α) i = wti
lt

(17)

The two equations above are demand on the labor p and i, respectively. α represents the wage
share of the patient households. Because of the form of the production function and the structure of
the problem, the entrepreneurs naturally gain zero proﬁt, so that it is not necessary to consider theowner of the ﬁrms.

2.4

Equilibrium

The equilibrium of the model is an allocation of quantities (Ctp , Cti , hpt , hit , bpt , bit , ltp , lti , Yt )∞t=0 and sequence of prices (qt , wtp , wti , rt , λt ) where the agents maximize their object function under their speciﬁc
constraints, and where the markets clear. In fact, in addition to the key equations that are discussed
in the previous section, the market clearing conditions are needed:

Cti + Ctp = Yt

(18)

hit + hpt = H(19)

bit + bpt = 0(20)

These market clearing conditions are for goods, housing, and loan market, respectively. It is
assumed the Supply for housing is ﬁxed. In order to examine the transition dynamics around thesteady state, these equations are log-linearized using the methods described by Uhlig(1998). Let us
denote the percent deviation from the steady state as the hatted variables: xˆt ≡ lnxt − lnx, whichmakes the following systems:

21

qˆt − Ĉtp =q̂t = (

yˆt = Ât + αˆltp + (1 − α)ˆlti(21)

ŵtp + ˆltp = yˆt(22)ŵti + ˆlti = yˆt(23)

jC p ˆ jC p ˆppjt − p ht + β p Et q̂t+1 − β p Et Ĉt+1
hp q
h q

(24)

p
Ĉtp = −r̂t + Et Ĉt+1(25)

ŵtp − Ĉtp = (η − 1)ˆltp

(26)

m
m
jC i
m
m
i
− β i m)m̂t + (1 −)Ĉti − i ĥit −r̂t + (
− β i m + β)Et q̂t+1 + β i (m − 1)Et Ĉt+1
1+r
1+r
hq
1+r
1+r
(27)

b̂it = Et q̂t+1 + ĥit − r̂t + m̂t(28)

ŵti − Ĉti = (η − 1)ˆlti

(29)

22

Cp p Ci i
Ĉ −
Ĉ
y t
y t

(30)

b̂pt = −b̂it(31)

hp ĥpt = −hi ĥit(32)

C p Ĉtp + qhp (ĥpt − ĥpt−1 ) + bp (1 + r)r̂t−1 + bp (1 + r)b̂pt−1 = −bp b̂pt + wp lp (ŵt p + ˆltp )

(33)

ŷt =

Equation (21) is the production function. Equation (22) and (23) describe the labor demand.
Equation (24) is the Euler equation for the consumption and housing for the patient households.
Equation (25) denotes the Euler equation for consumption using the loan. Equation (26) is the labor
supply by the patient households. Equation (27) is the Euler equation for impatient households andequation (28), and equation (29) is the loan demand and labor supply by the impatient households.
Equation (30),equation (31), and equation (32) are market clear conditions for consumption goods,
loans, and housing, respectively. The ﬁnal equation (33) expresses the ﬂow of fund of the patient
households.

2.5

Calibration

This section begins with the calibration of the model in order to understand the transition mechanism.6
The values of the parameters in the model follow Iacoviello(2005). Especially the important thing isthat it is assumed the patient households have larger discount factors than that of the impatient
households. The speciﬁc value of the preference is derived from the analysis of Emily Lawrance(1991)

6 I denote the system of the steady state in the appendix.

23

for poor (or impatient) households and Samwick(1998) for patient (or wealthy) households, whichallow us to set β p = 0.99 and β i = 0.95.
First the transition dynamics when the persistent ﬁnancial innovation actually occurs now making
use of these parameters is shown. Then the discussion moves on to the case where the news shock
occurs now.
Table 1-Parameters in the model
Description
Parameter
Patient households
βp
Impatient households
βi
Weight on housingj
Labor supply aversion
ηProductivity
A
Degree of ﬁnancial developmentm
Patient households wage shareα
Autocorrelation of ﬁnancial innovation shock ρm

2.5.1

Value
0.99
0.95
0.1
1.01
1
0.55
0.64
0.99

Transition Dynamics: Current Financial Innovation

In the calibration one percent deviation shock on the degree of ﬁnancial development m is introduced.
The timing of perception and realization of the shock are both current period so that the role of thenews shock in the economy can be clearly understood . Figure 5 and Figure 6 describe the responseof each variables.
When the shock occurs at period zero, the housing price jumps to increase and returns to thesteady state level as the eﬀect of the ﬁnancial innovation disappears as the time goes. The jump comesfrom the fact that the shock reduces the cost of borrowing, which leads the virtual price of housing todecrease from the impatient households’ point of view. Thus, the demand of housing rises and therefore
housing price increases when the shock realizes at period zero. While the amount of the housing thatthe impatient households holds decreases at ﬁrst and it seems to contradict the discussion above, thisis true because the housing price increases so much that it is beyond the hand of the impatient at
ﬁrst. However, after accumulating enough housing as collateral by reducing consumption and working
24

Figure 5: Response to current ﬁnancial innovation

Figure 6: Response to current ﬁnancial innovation(cont.)25

more, the amount of housing holding turns to be positive because the impatient can eventually borrow
enough to ﬁnance the housing purchase. In fact, under the given parameters the consumption of theimpatient households decreases and hours of work increases in order to ﬁnance the housing demand.
Moreover, the amount of borrowing sharply increases after the shock. This increment is permitted by
the construction of the collateral constraint in two ways: because of the accumulation of the collateral,and because of the housing price increase. These two paths simultaneously ampliﬁes the economy.
A few things about the patient households and entrepreneurs are worth the attention. The changesof the actions come only through the movements of the prices and are straightforward to understand.
The patient households increase their consumption because of the substitution from the increase of
the housing price. They also increase the housing because they simply aﬀord to purchase it.

2.5.2

News Shock: News is Incorrect

Now let us start investigating the eﬀect of the news shock. To be speciﬁc, it is assumed that at period
zero, the agents obtains the signal that in 5 years, the ﬁnancial innovation will make fruits to reducethe borrowing cost. In setting the length of time by the realization from the acquisition of the shock,
it is supposed implicitly that it was about 5 years from the set of the events that indicated the changeof the regime to the occurrence of the crisis. The structure of the shock follows the spirit of Beaudryand Portier(2004):

lnmt = ρm lnmt−1 + ϵnews
t−n + ϵt(34)

The degree of the ﬁnancial development evolves according to the stationary AR(1) process in whichthe ρm is strictly smaller than 1. ϵnews
t−n represents the news shock that agents in the economy acquirein period t and that indicates in n years the ﬁnancial innovation happens. In our simulation let us
set n = 5 to target the US experience. ϵt is the shock which could possibly reverse the news: thatis, if the ϵnews
t−n = 1 and ϵt = −1, the degree of the ﬁnancial development m does not change at all26throughout the periods. However, this set of two shocks aﬀect the economy because at time k wheret − n ≤ k < t, all agents believe that the ﬁnancial innovation must occur at period t, so that they
react and take actions optimally so as to adapt themselves to the ”promised” shock.
In this paper, the focus is on the case where the news is incorrect.

7

This is because the key

component of the optimism on the ﬁnancial innovation stems from the risk sharing that at the sametime reduces the risk of borrowing and lending, which was proved false by the harsh experience during
the crisis. Figure 7 describes the transition of the economy when the news shock occurs and it turns
out to be false.

Figure 7: Response to future ﬁnancial innovation and news is incorrect

Figure 7 shows the housing price jumps as the news arrives, and continues increasing until the news
turns out to be false at period 5. When the belief collapses, the price immediately goes below thesteady state. In the following period, the housing price converges to the steady state level as the time
7 Let us refer to Nutahara(2007) very much in making the code.

27

goes by. The reason why the news shock generates boom and bust results from the forward-looking
collateral constraint. When the ﬁnancial innovation actually loosens the borrowing limit, the housing
price increases due to the reduction of the cost of borrowing as is seen in the previous simulation.
Given this expectation, the lenders in the previous period allow the impatient households to borrow
more because they anticipate the capital gain from the housing as collateral. Since borrowers use
some fraction of their borrowing to purchase houses, the housing price actually rises at the time.
Repeating this anticipation, surges of borrowing, and actual increment of demand, the housing market
experiences the boom as soon as the news arrives in the market. The interesting thing is that eventhough the fundamentals does not change before the news, the economy is in the prosperous times.
Regarding other variables, at ﬁrst due to the high price the amount of holding of the housing by
the impatient households decreases, but they accumulate it as the years pass by. They ﬁnance themoney to purchase houses by working more, and reducing their consumption.
The bust happens because impatient households realizes that housing is not worth holding as they
thought. They do want to sell houses in the market and this sudden increase in the ”conceptual”
supply of the housing in the market loosen the market so that makes the need of the huge priceadjustment.
The sudden fall of the housing price immediately tighten the loan market and therefore the borrowing by the impatient households drops as the news proves wrong. After this fall the borrowing
gradually reduces to the steady state level along the decrease of the holding of the houses.
The consumption by the patient households jumps above the steady state level owing to thesubstitution eﬀect. They understand that the housing price is now so high that they want to allocate
their money into the consumption. This increment of the consumption reduces the hours of workby them since the increase dampens the incentive to work through the intra-temporal relationshipbetween consumption and working.
The action taken by the patient households is as straightforward as is discussed in the previous

28section. Since the housing price increases they have more incentive to allocate their income toward
consumption. As for housing, even though at ﬁrst amount of housing increase due to the high price,as the impatient become eligible to purchase lots of housing, the holdings by the patient households
decreases. The transition of the consumption explains the movement in the hours of working by thepatient.

2.6

Extension

One clear problem observed in the transition is the movement of the consumption by the impatient
household. Figure 8 and Figure 9 show the comovement of the housing and the other total averageannual expenditure by the highest 20 quantile and lowest 20 quantile income households in US.8 The
income is deﬁned by the income before taxes. The horizontal axis is the year and the vertical axis is
the annual growth rate. These graphs show that two kind of expenditures have positive relationship.
In fact, the number 0.70 and 0.34 is found as the correlation between the two expenditure for wealthy
and poor people, respectively. Especially, during 2005 and 2006, the time the housing market isat their best, both expenditure signiﬁcantly increases. This observation motivates to modify mymodel to provide the positive movement in the consumption by the impatient households. The key
strategy is to provide the impatient households with more money to purchase. It is hypothesizedthat if the constrained households earn enough money, they start allocating some fraction of it toward
consumption, which will lead to the positive movement of the consumption of the impatient households.
The room for entrepreneurs to react to the shock to make more goods is considered so as to make thisdynamics,. In order to represent the situation where entrepreneurs are willing to make more goods
when they expect the shock and price increase of the housing, the constraint on the entrepreneurs is
assumed as Iacoviello(2005) has adapted.

8 Housing expenditure includes housing related expenditure such as purchase of housing, public services, housekeeping

supplies and so on. The other expenditure includes expenditure on things like food, health care, entertainment, reading,
education and so on.

29Figure 8: Housing and other expenditure by the highest 20 quantile income households, Data source:
Consumer Expenditure Survey, Bureau of Labor Statistics

Figure 9: Housing and other expenditure by the lowest 20 quantile income households, Date source:
Consumer Expenditure Survey, Bureau of Labor Statistics

30

2.7

Constrained Entrepreneurs

The problem of the entrepreneurs is modiﬁed. They maximize their lifetime utility rather than proﬁt
itself.
maxE0 Σ(γ t lnCtf )(35)

s.t.Ctf + qt (hft − hft−1 ) + (1 + rt−1 )bft−1 + wtp ltp + wti lti = bft + Yt

(36)

bft ≤ mft Et (qt+1 hft)
1 + rtYt = At (hft )v (ltp )α(1−v) (lti )(1−α)(1−v)

(37)

(38)

They have production technology to create the ﬁnal goods. They use two types of labor and realestate as the input and pay wages and housing price. Let us assume today’s real estate can be usedas the input. Assumption behind the modiﬁcation is that entrepreneurs easily utilize their real estate.
The timing is modiﬁed because the late utilization only slightly disturbs the transition and it does notshow any meaningful diﬀerence. In fact, that the change does not matter on the implication of themodel is assured by examining the both transitions. The expenditure share of the housing is denotedas v as in Iacoviello (2005). They are eligible to borrow money from households. However, they areat the same time constrained by the collateral constraint. The patience is smaller than that of thepatient households, that is, γ < β p , which simpliﬁes the analysis. The ﬁrst order conditions are allstraightforward:

1
Ctf

(

vyt
hft

− qt ) +γqt+1µt mft Et qt+1
+ Et
=0
1 + rt
Ct+1

31

(39)

wtp ltp = (1 − α)(1 − v)Yt(40)

wti lti = α(1 − v)Yt

(41)

1
Ctf

− µt − Etµt ≥ 0, µt (mft Et (γ(1 + rt )
f
Ct+1

=0

qt+1 hft) − bft ) = 0
1 + rt(42)(43)

The ﬁrst equation is the Euler equation between consumption and housing. µt describes the
Lagrange multiplier of the collateral constraint. The second, and the third are the labor supply
functions. The fourth equation shows the intertemporal relationship using the loan. The last line is
the KKT conditions. Let us allow to focus around the steady state where the constraint is binding
because of γ < β p as in the problem of the impatient households. Thus, the following loan demandfor entrepreneurs is obtained:

bft = mft Et (qt+1 hft)
1 + rt(44)

The problem of the households are unchanged. Only the market clearing conditions need slight
modiﬁcation:

Cti + Ctp + Ctf = Yt

(45)

hit + hpt + hft = H

(46)

32

bit + bpt + bft = 0(47)

The new equations are ﬁrst log-linearized as follows:1yˆt = Ât + v ĥft + α(1 − v))ˆltp + (1 − α)(1 − v)ˆlti(48)

ŵtp + ˆltp = yˆt(49)

ŵti + ˆlti = yˆt(50)

qt + (−
mf

1− 1+r

+

mf
1+r
mf
1− 1+rmf
1+r
mf
1− 1+rff
qm+ vy +(1−mf )q )m̂t − (
hf

mf
1+r
mf
1− 1+rf)q)Et q̂t+1 ++ vy(1−m+(1−mf )q

(51)

hf

vy

f
f
hf
+ vy +(1−mr̂t − Ĉtf + Et Ĉt+1f )q (−Et ŷt+1 + ĥt ) = 0

(52)

hf

ŷt =

Cp p Ci i Cf f
Ĉ −
Ĉ +
Ĉ
y t
y t
y t

(53)

bp b̂pt + bi b̂it + bf b̂ft = 0(54)

hp ĥpt + hi ĥit + hf ĥft = 0(55)b̂ft = Et q̂t+1 + ĥft − r̂t + m̂ft

(56)

33

C f Ĉtf + qhf (ĥft − ĥft−1 ) + bf (1 + r)r̂t−1 + bf (1 + r)b̂ft−1 = bf b̂ft − wp lp (ŵtp + ˆltp ) − wi li (ŵti + ˆlti ) + y ŷt (57)

The ﬁrst equation is the production function. The second and the third are labor demand functions. The fourth equation describes the Eular equation between the consumption and housing for theentrepreneurs. The ﬁfth to seventh show the market clearing conditions and the last two indicate loan
demand by the entrepreneurs and the ﬂow of fund for them, respectively.
Now let us investigate how the transition of the economy has changed compared with the baselinemodel.9 Since there are two paths that the ﬁnancial innovation could possibly inﬂuence on the market,
both, that is, shocks on the impatient households and on the constrained entrepreneurs are considered.
Table 2-Added Parameters in the extended model
Description
Parameter
Patience of entrepreneursγ
Degree of ﬁnancial development for entrepreneursmf
Housing expenditure sharev
Autocorrelation of ﬁnancial innovation shock on the entrepreneurs ρmf

Value
0.98
0.89
0.03
0.99

The parameter values added in the model are summed up in Table 2. These values are also in
Iacoviello(2005). The important thing is that the patience of the entrepreneurs is strictly smaller thanthat of the patient households, which allows us to concentrate on the binding case in terms of thecollateral constraint.
Figure 10 and 11 present the transition to the shock on the households and entrepreneurs, respectively. The boom and bust is still in the graphs. Regarding the purpose, it is found that the result
is on the way: At period zero when the news arrives, the consumption of the impatient households
actually rises. This jump is derived from the increment of the total production named as GDP inthe ﬁgure. In fact, compared with the baseline case, it is ten times more at the peak of the tran9 I put the system of steady state of the extended model in the Appendix.

34

sition. This movement results from the collateral constraint on the entrepreneurs. First, the future
price hike, which is derived from the action of the impatient households, loosens the constraint of theentrepreneur because it is forward-looking. The reduction of the cost of borrowing allows them topurchase more houses and so increase the total production. However, the impatient households reduce
their consumption to be below the steady state level. This is because the price rises so high that
entrepreneurs become unwilling to purchase houses more, letting the total production converge to theoriginal steady state level.

Figure 10: Response to future ﬁnancial innovation for households and news is incorrect; entrepreneurs
are constrained
The interesting thing is that the two diﬀerent shock present almost the same transition. One reason
is that in any case housing price due to the decrease of the cost of borrowing allow both households
and entrepreneurs to borrow more before the shock. The important thing for them is the fact thatthe credit limit is mitigated, and therefore given the future price hike they borrow more now and socurrent price moves to increase. This makes the transitions of the two shock look surprisingly similar.
35

There are two ways to construct the model that feature the desirable characteristics. One is toallow the impatient households’ income to grow continuously so that they will become willing to sharesatisfactory fraction into consumption. This is what the extended model is aiming at. The other way
is to change the degree of elasticity of substitution. In this direction, the diﬃculty is to keep theirincome at least increasing.
In conclusion, the introduction of the constraint on the entrepreneurs partly mitigates the contradicting transition of the impatient consumption in the baseline model. However, the new transition
is not satisfactory because it is still pulled back to the baseline even though at ﬁrst the consumption
actually increases due to the increment on the total production, which is mainly from the collateral
constraint on the entrepreneurs.

Figure 11: Response to future ﬁnancial innovation for entrepreneurs and news is incorrect; entrepreneurs are constrained36

3Conclusion

Throughout the discussion in this paper, the eﬀect of the news about the ﬁnancial innovation on the
housing price is examined. In both baseline and extended model, the boom-bust transition of the
housing price is clearly observed. The housing price increases when the ﬁnancial innovation occurs
whether it happens now or it will in the future. When the shock is about the future, the price hikebefore the realization is not from the change of the fundamentals. But it is due to the forwardlooking collateral constraint. Anticipation of the future price hike allows the impatient households toborrow more because lenders think even increased amount loan will be repaid taking the capital gainof the housing into account. Since the loan is used to purchase housing, the surge of the borrowing
actually increases the demand of housing, resulting in the price increment in the housing market. Thismechanism continues till the period when the news shock arrives in the market, and therefore generates
the boom even though the market does not virtually experience the change of the fundamentals. The
discovery of this path is my main contribution.
While the strength of the model can be stipulated, it is worth considering the ways for the further
research in order to enrich the analysis. Three promising topics are pointed out here: modiﬁcationof the consumption by the impatient agents, deepen the analysis regarding the information structure,and the degree of the ampliﬁcation.
The ﬁrst is to modify the consumption action by the impatient households to ﬁt the data. This
modiﬁcation gets us to understand the mechanism of the boom and bust by the news shock more.
The strategy for this way is discussed in the previous part.
The second way is to introduce other information structure. One important assumption of thenews shock is that people can misunderstand the signal. Even though this seems true and inevitablein the real world, it is interesting to allow agents to behave more cautiously. One direction is the
Baysian learning as Mendoza(2012) introduces.
The third is about the strength of the ampliﬁcation. It would be intriguing to think about the case37

where the market liquidity in the boom and bust is diﬀerent. If agents believe the ﬁnancial innovation
surely comes as it is assumed in the model, this optimistic mood reduces the transaction or searchcost because everyone believes housing market is booming and sellers can easily ﬁnd the buyers. This
mood fuels the economy. However, if the bust occurs, it is natural to assume people become more
pessimistic and dubious, which will incredibly aggravate the shock by increasing supply and reducing
demand. This is related to the ﬁre sales literature and it may be worth investigating the connection
with this ﬁeld.

38

Appendix
A

Steady state system of the baseline model−yα p = wpl(58)y
α i = wi
l

(59)

j
q
q
= p + βp( p )
p
C
h
C

(60)

1
1
= (1 + r)β p p
p
C
C

(61)

wp
= (lp )(η−1)
Cp

(62)

j
q
q
q
+ i + mλ+ βi i = 0
i
C
h
1+r
C

(63)

1
1
− λ − β i (1 + r) i = 0
Ci
C

(64)

wi
= (li )(η−1)
Ci

(65)

qhi
)
1+r

(66)

bi = m(39

B

y = A(lp )α (li )1−α(67)

Ci + Cp = Y(68)

hi + hp = H(69)

bi + bp = 0(70)

C p + rbp = wp lp(71)

Steady state system of the extended model(1 −yα(1 − v) p = wp
l

(72)

y
(1 − α)(1 − v) i = wi
l

(73)

vY
mf
)q = γ( f + (1 − mf )q)
1+r
h

(74)

bf = m(

qhf
)
1+r

40

(75)

−

q
j
q
= p + βp( p )
Cp
h
C

(76)

1
1
= (1 + r)β p p
Cp
C

(77)

wp
= (lp )(η−1)
Cp

(78)

q
j
q
q
+ i + mλ+ βi i = 0
Ci
h
1+r
C

(79)

1
1
− λ − β i (1 + r) i = 0
Ci
C

(80)

wi
= (li )(η−1)
Ci

(81)

qhi
)
1+r

(82)

bi = m(Y = A(hf )v (lp )α(1−v) (li )(1−α)(1−v)(83)

Ci + Cp + Cf = Y(84)

hi + hp + hf = H

(85)

41

bi + bp + bf = 0(86)

C p + rbp = wp lp(87)

Y = C f + rbf + wp lp + wi li(88)

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paper[3] Beaudry, P and Portier, F.(2004) “An exploration into Pigou’s theory of cycles”, Journal of
Monetary Economics, 51, 1183-1216.
[4] Boz, E and Mendoza, E G.(2012). “Financial Innovation, the Discovery of Risk, and the U.S.
Credit Crisis”, Working Paper[5] Burnside, C Eichenbaum, M and Rebelo, S.(2012). “Understanding Booms and Busts in Housing
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on Economic Activity, Economic Studies Program, The Brookings Institution, 34(2), 299-362.
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