the role of consumer leverage in financial crises...the role of consumer leverage in financial...

The Role of Consumer Leveragein Financial Crises

Dilyana Dimova∗

Oxford University

Abstract

Consumer leverage can contribute to nancial crises such as the subprimemortgage crisis characterized by increased bankruptcy prospects and tight-ened credit access. This paper embeds nancial frictions in the mortgagecontracts of home-buyers within a two-sector economy to show that althoughhouseholds are not production agents and may not start as highly leveraged asnancial institutions, their worsening debt levels can generate a lasting nan-cial downturn. Using two seemingly positive disturbances that contributed tothe subprime mortgage crisis - increased housing supply and relaxed borrow-ing conditions - the model demonstrates that the subprime downturn was nota precedent but the natural consequence of nancial frictions. The surplusof available housing leads to lower asset prices that in turn reduce the valueof the mortgaged houses relative to the loan held. This worsens the leverageof indebted consumers and raises their bankruptcy prospects. A relaxationof borrowing conditions turns credit-constrained households into a potentialsource of disturbance themselves when market optimism allows them to in-crease their indebtedness with relatively little downpayment. In both cases,the increased debt, along with higher repayment rates due to the larger defaultlikelihood, impairs household access to credit and plunges mortgage-holdersinto a lasting recession. Adding credit constraints to the nancial sector thatprovides housing mortgages creates opportunities for risk sharing where banksshift some of the downturn onto indebted consumers in order to hasten theirown recovery. This consequence is especially evident in the case of relaxedcredit access for banks. Financial institutions repair their debt position rela-tively fast at the expense of consumers whose borrowing ability is squeezed fora long period despite the fact that they are not the source of this disturbance.The outcome mirrors the recent subprime mortgage crisis characterized bya sharp but brief decline for banks and a protracted recovery for mortgagedhouseholds.

JEL Classication: E21, E27, E44, G21, G33

Keywords: Financial Frictions, Consumer Leverage, Pro-cyclical Risk Premium, Subprime Mort-

gage Crisis, Risk Sharing

∗Department of Economics, University of Oxford, Manor Road Building, Oxford, OX1 3UQ, U.K.; e-mail:

[email protected].

Contents

1 Introduction 3

2 The Consumer Mortgage Contract 7

2.1 Description of the Consumer Mortgage Contract . . . . . . . . . . . . . . . . . . . . . . 72.2 The Loan Contract of Credit-Constrained Households . . . . . . . . . . . . . . . . . . . 92.3 The Share Contracts of the Consumption and Housing Sectors . . . . . . . . . . . . . . 13

3 The Complete Model with Consumer Mortgage 15

3.1 Consumption Capital Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Consumption Producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Housing Capital Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Housing Producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5.1 Ricardian Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5.2 Credit-Constrained Consumers . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.6 Market Clearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 The Consumer Mortgage Model Results 22

4.1 The Housing Oversupply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 The Temporary Financial Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 The Chained Loan Contracts 29

5.1 Description of the Chained Loan Contracts . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 The Chained Loan Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 The Complete Model with Chained Loan Contracts 34

6.1 Complete Model Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 The Chained Loan Contracts Results 36

7.1 The Housing Oversupply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.2 The Temporary Financial Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Conclusion 45

References 47

Appendix A: Complete Consumer Mortgage Contract Model 52

Appendix B: Log-Linearization of Consumer Mortgage Contract Model 54

Appendix C: Complete Chained Contracts Model 57

Appendix D: Log-Linearization of the Chained Contracts Model 60

Appendix E: Derivatives of the Log-Normal Distribution 63

2

1 Introduction

In 2004, Edmund Andrews, an economic reporter for The New York Times, joinedmillions of American home-buyers in purchasing a house at the peak of the real estateprice bubble (Andrews 2009). The fact that he had regularly reported from the FederalReserve, covered the Asian nancial crisis of 1997, the Russian meltdown of 1998 andthe dot-com collapse of 2000, did not prepare him for what was in store at the subprimemortgage party. I had just come up with almost a half-million dollars, and I had barelylifted a nger. It had been so easy and fast, Andrews said of obtaining his mortgagedespite having modest disposable income and putting down very little downpayment.His mortgage was a classic subprime loan. The monthly payment rst jumped from$2,500 to $3,700. If he kept the mortgage for two years, the interest rate could jump ashigh as 11.5 percent, and the monthly payments could ratchet up to $4,500. After hiswife lost her job, he fell behind on all payments from the mortgage to the electricitybill. When he nally defaulted, he was far from being the only one. In fact, he hadoutlived two of his three mortgage lenders. The rst one collapsed overnight when thenancial markets rst froze up in August 2007 and the second one was forced out ofthe mortgage business by federal regulators.

Andrews' story is just one of the hundreds of recent foreclosure experiences thatcharacterize the mortgaged homeowner's experience with subprime loans. The sub-prime mortgage crisis was a seemingly unusual one since it was not precipitated byproduction rms or entrepreneurs and it was not triggered by adverse circumstances.The unraveling of the mortgage market began when two seemingly positive factors - in-creased housing supply and easy nancing conditions - allowed borrowers to excessivelyleverage with debt to disastrous consequences. At the heart of the downturn were notproduction agents but heavily indebted consumers and the nancial institutions thatsecuritized their mortgages. Nevertheless, the collapse of the highly leveraged mortgagemarket was sucient to trigger a signicant downturn.

The asset bubble that preceded the subprime mortgage crisis increased housing in-ventories to record high numbers (Coleman, LaCour-Little and Vandell, 2008). Afterthe bubble burst, the oversupply of real estate reduced both the sale price of real es-tate and the value of the houses held by homeowners (Duca, Muellbauer and Murphy,2010; Ellis, 2010). As a result, many mortgage-holders owed more on their housing loanthan their residence was worth. Furthermore, at the peak of the asset bubble, manyinvestors eager to tap into mortgage prots underwrote housing loans secured with verylittle downpayment and with a variable repayment rate, the so-called subprime loans(Duca, Muellbauer, and Murphy, 2009a and 2009b). These lax borrowing conditionsallowed consumers to hold signicant amounts of debt and to be at the mercy of theadjustable interest rate. Once the rst signs of trouble raised the repayment rate, theirrisk premiums rose worsening their indebtedness (Demirgüç-Kunt, Evano and Kauf-man, 2011; Laeven, Igan and Dell'Ariccia, 2008; Mian and Su, 2009). This causeda urry of short sales and defaults as mortgage-buyers attempted to deleverage or de-clared bankruptcy. The result was a depressed housing market with even lower assetprices and sharply tightened credit access (Dennis, 2010; Duke, 2012; Madigan, 2012).

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The housing market downturn also exported the worsened nancial conditions beyondthe aected sector and depressed demand for goods in the rest of the economy.

Glick and Lansing (2010) demonstrate that household leverage in the United Statesand in many industrial countries increased dramatically in the decade prior to 2007.Countries with the largest increases in household leverage underwent the fastest risein house prices over the same period. These same countries tended to experience thebiggest decline in household consumption once house prices started falling. House-hold leverage growth and the dependence on borrowing also explain a large fraction ofthe overall consumer default, house price, unemployment, residential investment anddurable consumption patterns during the recession (Mian and Su, 2010). Kim andIsaac (2010) show that consumer debt can aect macroeconomic dynamics and can con-tribute to nancial fragility as strongly as corporate leverage. The authors demonstratethat looser consumer credit can also make the system more vulnerable to changes inthe state of condence, the interest rate and the saving propensity of renters. Paperslike these motivate a microfounded modeling of the consequences of sharp decline inhouse prices and of improvements in borrowing condence on the leverage of indebtedhouseholds.

This paper develops a model of nancial frictions in the mortgage market to demon-strate how downturns such as the subprime mortgage crisis occur. It does not attemptto explain the the housing bubble and all aspects of the subsequent collapse. Rather, itfocuses on two major factors that contributed to the unraveling of the mortgage market- the excess housing supply that led to a decline in housing prices and the lax nancialconditions that permited increased borrowing - on the leverage of mortgage-buying con-sumers. For this purpose, the paper extends the one-sector model of credit constraintsof Bernanke, Gilchrist, Gertler (1999) to two sectors and shifts nancial frictions fromproducers to homeowner consumers. A two-sector model can generate a pro-cyclicalrisk premium that widens following improvements in the housing market. This riskpremium is function of the default risk that in turn depends on the size of the collateraland the value of the mortgage. So a shock that reduces housing prices makes real estatemore aordable but also creates a Fisher-type eect where the price decrease worsensthe value of the housing collateral of indebted households and increases their leverage.The higher leverage triggers an increase in the adjustable mortgage interest rate andthe risk premium widens. A credit relaxation on the other hand impacts the leverageof mortgage-buyers directly but creating the incentive to borrow excessively. Hence,with credit constraints in the consumer sector, both a technological expansion in thehousing sector and a nancial easing imply a downturn for mortgaged consumers thusbringing the model closer to the subprime crisis.

It order to describe the behavior of leveraged consumers, the paper departs fromthe traditional modeling of house purchasing decisions such as Kiyotaki and Moore(1997), Aoki, Proudman and Vlieghe (2004) and Iacoviello (2005) in favor of a micro-founded model of consumer credit constraints. The housing model of Aoki, Proud-man and Vlieghe (2004), which belongs to the Kiyotaki and Moore (1997) collateralconstraint type of models, imposes an exogenously determined limit on borrowing de-pending on the availability of collateral. Such models were a fairly common way of

4

modeling housing loans prior to the nancial crisis. However, the emergence of sub-prime loans requiring little to no collateral and instead have an adjustable repaymentrate makes such approaches unsuitable for modeling the subprime recession. To capturebetter the subprime mortgages, the model developed here avoids exogenously imposedloan-to-value ratios that cap borrowing and limit the resulting volatility. Instead, thispaper continues to use the Bernanke, Gilchrist and Gertler (1999) microfounded loancontract to model frictions in the mortgage market. Such a setup imposes no upperrestrictions on borrowing but introduces an endogenous adjustable mortgage rate thatmakes loans costlier as the leverage increases. A subprime loan within this frameworkcould be secured with modest downpayment but at a higher interest rate that adjustsupwards following unfavorable macroeconomic disturbances. This approach allows arobust study of the role of leveraged consumers in the subprime crisis.

Most of the works on nancial constraints overlook the role of consumer leveragein the recession since households are generally not as leveraged as nancial institu-tions. Bernanke, Gilchrist and Gertler (1999) and Luk and Vines (2011) show thathigh rm or nancial institutions leverage can be instrumental in precipitating a nan-cial downturn. Similarly, Fernández-Villaverde (2010) and Gertler and Kiyotaki (2010)demonstrate that a disruption to nancial intermediation can induce a crisis that af-fects real economic activity. The majority of models either omits the role of consumerleverage or only attempts to t models to empirical data. One of the few studies onconsumer mortgage decisions, Mian and Su (2009), establishes that consumers can ex-perience a worsening in their leverage as a result of deteriorating lending practices butdoes not study the consequences of this increase in indebtedness for their borrowing andthe economy. This paper augments our understanding of credit frictions by consideringexplicitly the role of household leverage in creating and propagating downturns. It alsodemonstrates that although households do not start very indebted and do not produceoutput, changes in their leverage can have lasting consequences for their credit accessand for the economy as a whole as mentioned in Kim and Isaac (2010).

At the heart of the nancial frictions setup is the inability of credit-constrainedconsumers to fully nance their housing purchase so they need to borrow externalfunds from risk-neutral investors. This borrowing is complicated by the presence of anidiosyncratic risk of default on the part of mortgage-buyers that is known to them butis unknown to lenders (Townsend, 1979). Unlike Cúrdia and Woodford (2009) whereborrowers can choose whether to default, here bankruptcy is involuntary but dependson factors both endogenous (the size of the mortgage) and exogenous (macroeconomicshocks) to consumer decisions. If credit-constrained households default, investors mustpay an auditing fee to assume possession of any remaining assets. Since investors cannotfully diversify away this risk, they charge borrowers a risk premium that would osetthe expenses associated with eventual bankruptcy. Investors obtain their funds from thedeposits of Ricardian consumers who have signicant non-wage income from rm protsand share ownership. In contrast, credit-constrained consumers earn only labor incomeand have no savings. The economy is completed by two production sectors: one thatproduces conventional consumption and one that manufactures housing. Housing is amulti-period durable good whose manufacture follows the setup of Iacoviello and Neri(2010). Both production sectors are perfectly competitive and there is no idiosyncratic

5

uncertainty in their returns.

The model demonstrates the eects of two shocks that contributed to precipitatingthe subprime mortgage crisis: the oversupply of housing and the relaxation of borrowingconditions. The overproduction of housing depresses the price of housing and lowersthe return on housing. As a result, the value of the collateral of credit-constrainedconsumers, primarily composed of their housing equity, decreases by both the price andthe return to housing. The drop in their housing equity value is proportionally largerthan the fall in the mortgage value which falls only due to the price reduction so theleverage of mortgaged home-buyers increases. The higher leverage signals increased riskof default since lenders that provide loans to credit-constrained consumers are exposedto bigger risks. To compensate for this, investors require a higher return from indebtedhouseholds so the risk premium rises. The higher risk premium further depresses creditaccess that can remain lower for a long time while consumers attempt to repair theirdebt position by gradually improving their real estate holding.

A lower housing price is not the only disturbance that can induce a credit downturnfor indebted consumers. A relaxation of credit access improves borrowing conditionsso mortgage loans attract a lower risk premium and require a smaller downpaymentto secure (Duca, Muellbauer, and Murphy, 2009a and 2009b; Mian and Su, 2009).The lax credit access prior to the crisis reduced interest rates and lured many potentialhomeowners into obtaining mortgages with little collateral. However, the lower col-lateral implies higher indebtedness for mortgage-buyers that can easily lead to higherprobability of default. Borrowers attempt to amass more collateral in order to improvetheir mortgage position but since they cannot achieve that overnight, their increasedbankruptcy prospects prompt investors to raise the repayment interest rate. The higherrepayment rate makes improving housing equity even harder for indebted consumers.As a result, households experience a lasting recession with tight credit access.

In an extension to the main model, the paper considers credit-constrained nan-cial institutions that serve as intermediaries in the loan contract between investors andhouseholds. Before the nancial crisis, the nancial sector did not receive much atten-tion in the literature on credit frictions. Papers like Kiyotaki and Moore (1997) andIacoviello (2005) focus primarily on the demand side of credit and abstract from anactive role of nancial intermediation. The rst models to consider the role of banksare Goodfriend and McCallum (2007) and Christiano, Trabandt and Walentin (2007).Both of these estimate the quantitative importance of the banking sector for centralbank policy and for business cycles. Christiano, Motto and Rostagno (2010), Gerali,Neri, Sessa and Signoretti (2009), Gertler and Karadi (2011), Gertler, Kiyotaki andQueralto (2011) and Luk and Vines (2011) develop a nancial sector to explain spe-cic precedents of the nancial crisis such as excessive volatility, the proliferation ofrisk accumulation or the popularity of unconventional monetary policy. The existingliterature overwhelmingly focuses on credit frictions in only one sector, namely thebanking sector, and abstract from constraints in other sectors of the economy, includ-ing households. This makes them unsuitable to studying the extent of risk sharing thatcan occur with more than one type of nancially-constrained agents. Only Hirakata,Sudo and Ueda (2009) introduce more than one leveraged sector modeling constraints

6

in the production sector along with the nancial sector but they also overlook the roleof indebted consumers.

The extension to the main model borrows from the chained contracts model ofHirakata, Sudo and Ueda (2009) but the nal borrowers are not entrepreneurs butcredit-constrained households. It demonstrates that when there are two types of credit-constrained agents, the chained loan contracts create opportunities for risk sharing. Asintermediaries of the two mortgage agreements, nancial institutions have the ability totransfer some of the burden of a downturn to nal borrowers consumers. Furthermore,banks are often more leveraged so they experience a proportionally larger deteriorationof their leverage ratio which is an added impetus for them to unload the leverage fast.The higher initial leverage and the sharper increase necessitate quick deleveraging inorder to maintain solvency. That is the primary reason why banks resort to shiftingpart of the burden of nancial tightening onto households. Two characteristics of thenancial system allow them to do so. First, credit-constrained consumers to haveno recourse to alternative funding so they will participate in the mortgage contracteven if the terms deteriorate. Second, the arrangement between banks and mortgage-buying households is such that the return from borrowing to consumers varies withthe realization of the idiosyncratic uncertainty going up in times of a downturn. Thisallows nancial institutions to extract a larger share from households to repair theirown balance sheet during a deterioration of nancial conditions. Consequentially, theycan recover relatively fast at the cost of inducing a lasting recessions for mortgagedhouseholds.

2 The Consumer Mortgage Contract

2.1 Description of the Consumer Mortgage Contract

This section develops the consumer mortgage contract in a partial equilibrium frame-work, taking as given the price of the collateral and the risk-free rate of interest. Thesubsequent section endogenizes these variables as part of a general equilibrium solu-tion. The model developed here extends the one-sector model of nancial frictions ofBernanke, Gertler and Gilchrist (1999) and adds credit constraints in the consumer sec-tor to demonstrate the role of household mortgage leverage in contributing to nancialcrises.

The economy consists of two sectors: one that produces generic consumption andone that manufactures housing (Figure 1). The consumption sector produces its outputusing sector-specic capital, labor and own technology as inputs. Following Iacovielloand Neri (2010), housing is a durable multi-period good. Housing manufacturers usehousing capital, labor and sector-specic technology along with land, a nite resource.Each period only a fraction of housing deteriorates. The rest survives for the subsequentperiod. In each period, new housing production only replaces the depreciated housing.

7

This approach to housing as a multi-period good brings a more staggered responseto disturbances since newly produced output is only a fraction of total housing onthe market. Both nal good rms face no risk and they can borrow funds for capitalnancing at the risk-free rate as part of a xed share arrangement. In addition to thetwo nal goods, there are capital producing rms that supply nal good manufacturerswith sector-specic capital. Investment in both types of capital is the output of theconsumption sector. The relationship between capital rms and nal goods producersis a two-way one in the sense that capital rms buy used capital from goods producersand along with investment transform it into new capital.

In order to motivate credit frictions in the household sector, this model formallyseparates consumers into two types following Iacoviello and Neri (2010). Ricardianconsumers possess no intrinsic risk of default and can borrow at the risk-free rate.They have signicant non-wage income in the form of revenue from owning shares innal good rms and from absorbing the prots of capital producers. Their wealthallows them to nance their own housing purchase entirely. Ricardians save any unusedincome as deposits that are lent to investors and to nal good rms. Credit-constrainedconsumers, on the other hand, receive only wage income so their net worth is not enoughto nance their housing acquisition. Therefore, they must obtain external nancingfrom investors.

The housing purchase of credit-constrained consumers is nanced by a mortgagecontract. The mortgage contract is necessitated by the existence of an agency problemthat makes external borrowing for house purchase more expensive than own funds. Thereason for this discrepancy is that credit-constrained consumers possess an inherent riskof default that is known to them but unobservable to lenders. In this case, investorscannot perfectly observe the borrower's ability to repay and must pay an auditingcost in order to learn the bankruptcy prospect of the mortgage-buyer. When indebtedconsumers go bankrupt, investors pay the monitoring fee and take possession of all ofthe borrower's remaining assets. In the household context, these auditing costs can beinterpreted as the costs of legal proceedings to the value the borrower's assets and theadministrative costs of selling the house to realize its collateral value. The presence ofthese nancial frictions motivates the need for a loan (mortgage) contract as opposedto a share contract that usually takes place in the absence of idiosyncratic uncertainty.Additionally, the idiosyncratic uncertainty implies that credit-constrained consumerscannot borrow directly from their Ricardian counterparts. The intermediary servicesof investors are necessary not only to disburse the loan but also to conduct monitoring.

Credit-constrained households obtain a loan from investors who pool the depositsof Ricardian consumers, paying them the risk-free rate, and lend to credit-constrainedhouseholds at a markup consumer interest rate. The dierence between the borrowingand the lending rates, known as the external nance (risk) premium, exists in orderto oset the expenses associated with a potential bankruptcy of mortgage-buyers. Thegreater the default prospect, the more likely the lender will have to pay the auditingcost so the risk premium will be higher. Hence the external nance premium is afunction of the default rate, the collateral (net worth) of credit-constrained consumersand the value of the mortgage. Investors do not make any prot on the loan to indebtedhouseholds. The risk premium is exactly sucient to oset the costs associated with a

8

Credit-Constrained

Consumers

Consumption

Capital

Firms

Len

din

g: C

ostly

State V

erificatio

n

Labor

Consumption

Purchase

Consumption Purchase

Housing

Firms

Labor

Consumption

Firms

Housing Purchase

Housing

Purchase

Investors New Capital

Ricardian

Consumers

Used Capital

Housing

Capital

Firms

New Capital

Used Capital

Labor

Labor

Dep

osit

Share

Purchase

Share

Purchase

Figure 1: Model with Financial Frictions in the Consumer Sector

potential borrower default.

2.2 The Loan Contract of Credit-Constrained Households

The mortgage loan contract begins when at time t, credit-constrained household iobtains a mortgage from investors to purchase housing at price pt for use at t + 1.To improve the model tractability, the paper assumes that mortgage contacts last oneperiod only and are renegotiated every period. This is not a signicant departurefrom reality since it captures the adjustable interest rate phenomenon of the subprimedownturn. In this sense, this loan contract can be thought as a multi-period mortgagewith an adjustable rate. A multi-period mortgage with a xed rate would not beproperly microfounded since it will not update according to the most recent informationon the value of the housing stock and the default risk and hence would not provide afair return to lenders.

The quantity of housing purchased is denoted HCi,t with the subscript denoting

the household that obtains it and the period in which the housing is purchased andthe superscript indicating the type of consumer that purchases it (C denotes credit-constrained households and R refers to Ricardian consumers). Each credit-constrainedhousehold faces an idiosyncratic shock ωCi to their return to housing. In each period,homeowner i draws from a distribution of ωCi where ωCi follows a log-normal distribu-

tion with mean one and variance(σC)2. The variance

(σC)2

can also be thought as

a measurement of the volatility of the default probability ωCi - a lower variance wouldimply safer loan returns. The ex-post gross return on housing is ωCi R

C , where ωCi isan idiosyncratic disturbance to household i's return and RC is the ex-post aggregatereturn to housing for credit-constrained consumers (the markup consumer interest ratepaid on the mortgage). The idiosyncratic shock ωCi is also independent across time and

9

across credit-constrained households. Lenders can diversify away the idiosyncratic riskof the borrowers by investing in a lot of borrowers.

This setup follows the nancial accelerator approach of Bernanke, Gertler and Gilchrist(1999). This approach assumes that there is costly state verication on the lender's partthat motivates nancial frictions. If investors want to observe the idiosyncratic shockωCi for a specic homeowner, they need to pay a monitoring cost which is a fractionµC of the homeowner's total housing stock. Due to the monitoring cost, it would betoo costly to monitor every contract. Therefore, there is a cuto value of the shockωCi above which investors do not monitor and below which they do. Above the cuto,the credit-constrained homeowner gets a return sucient to pay investors a xed rateof interest ZC and keeps what is left for themselves; investors have no incentive tomonitor the homeowner' true return. Below the cuto, credit-constrained householdsare bankrupt, and the lenders pay the monitoring cost and take their entire wealth.

The investors who make loans to credit-constrained households are risk-neutral andthe opportunity cost of their funds is the risk-free interest rate. They will agree tolend to credit-constrained households only if they can break even on their investment.This implies that at the time of the loan repayment, the interest rate RC charged toborrowers must be such that the expected gross return on the loan must equal theopportunity cost of lending:

ωCi RCpHC

i = ZCi (pHCi −NC

i )

The left hand side is the gross revenue from housing to credit-constrained homeowneri whose risk ωCi is just at the cuto point. The right hand side is the amount theconsumer needs to repay the investors: the total value of the loan pHC

i minus thenet worth of the borrower NC

i . The household is just able to repay the loan at thecontractual rate.

The idiosyncratic shock is continuous and has a cumulative distribution functionFC(ωCi ). To be perfectly insured against the idiosyncratic shocks of households, eachinvestor signs contracts with an innite number of households. The solution of the loancontract denes how the investment revenue is split between the lenders (investors)and the borrowers (credit-constrained households). Each lender gets a gross shareΓC(ωCi ) that is the sum of the share (before monitoring) in case the household defaults´ ωCi

0ωCi dF

C(ωCi ) and the xed interest payment the lender gets if the debtor does not

default with probability (1 − FC(ωCi )). Let the defaulting share´ ωCi

0ωCi dF

C(ωCi ) bedenoted GC(ωCi ), then the gross share ΓC(ωCi ) is:

ΓC(ωCi ) =

ωCiˆ

0

ωCi dFC(ωCi ) + (1 − FC(ωCi ))ωCi = GC(ωCi ) + (1 − FC(ωCi ))ωCi

The borrower receives the remaining share (1 − ΓC(ωCi )). After taking into accountthe monitoring cost µCG(ωCi ), the net share that goes to the lender is:

ΓC(ωCi ) − µCGC(ωCi )

By assumption, the investors make a zero prot on their loans to credit-constrainedhouseholds. Investors can diversity away the loan risk so their opportunity cost is the

10

risk-free rate Rt. Hence the zero-prot condition for lending to a particular borrowerat time t to be repaid at time t+ 1 is:

Et((

ΓC(ωCi,t+1) − µCGC(ωCi,t+1))RCt+1

)ptH

Ci,t = Rt

(ptH

Ci,t −NC

i,t

)(1)

The left hand side is the net return on the loan to investors (the gross returnEtΓ

C(ωCi,t+1) minus the monitoring cost µCEtGC(ωCi,t+1)) at the time of the repayment.

The right side is the opportunity cost of lending the funds (the value of the housingpurchase ptH

Ci,t minus the consumer's net worth NC

i,t) valued elsewhere at the risk-freerate Rt that prevails at the time of the loan agreement.

Since the price of housing, the risk-free interest rate and the homeowner's net worthNCi,t are predetermined, at time t the credit-constrained household gets to choose a pair

of (EtωCi,t+1, H

Ci,t) according to the zero-prot condition. Given the other variables,

this is equivalent to the lenders oering a schedule of loans EtBCi,t+1 and a non-default

interest rate Rt to the household.

In this partial equilibrium setting, the credit-constrained homeowner i is faced with amenu of loan EtB

Ci,t+1 and interest rate EtR

Ct+1 where both the loan and the interest rate

are related by the participation constraint of the investors. At time t, borrowers choosethe optimal pair of housing and the expected cuto risk (HC

i,t, EtωCi,t+1) to maximize

their expected share of the loan at time t+ 1 of the loan repayment:

max Et((

1 − ΓC(ωCi,t+1))RCt+1

)ptH

Ci,t (2)

subject to the zero prot condition of investors (1).

Households optimize their mortgage from the perspective of time t in terms of EtRCt+1

but repay it at time t + 1 at the realized rate RCt+1. The mortgage is state-contingent

so its return depends on the realization of the ex-post return on housing RCt+1. It is at

this rate that the loan contract is repaid, rather than at the expected return on housingEtR

Ct+1. The expected return on housing EtR

Ct+1 is all that is known at the time the

contract is negotiated but it does not account for unexpected shocks that can aectthe repayment ability of credit-constrained consumers at the mortgage maturity date.Hence to ensure a fair return, investors negotiate the loan contact at EtR

Ct+1 but require

repayment at the realized ex-post return RCt+1 that is formed only after all disturbances

have occurred.

Credit-constrained consumers maximize their share of the loan subject to the par-ticipation constraint of investors:

LCt = Et((

1 − ΓC(ωCi,t+1))RCt+1

)ptH

Ci,t +

+λt[Et((

ΓC(ωCi,t+1) − µCGC(ωCi,t+1

)RCt+1

)ptH

Ci,t −Rt

(ptH

Ci,t −NC

i,t

)]Solving the maximization problem gives the external nance (risk) premium dened

as the relationship between the risk-free rate and the markup consumer interest rateEtRCt+1

Rtas a function of the cuto default risk Etω

Ci,t+1:

EtRCt+1

Rt= Et

ΓCω (ωC

i,t+1)

(ΓCω (ωC

i,t+1)−µCGC

ω (ωCi,t+1

))(1 − ΓC(ωCi,t+1)

)+

ΓCω (ωC

i,t+1)

(ΓCω (ωC

i,t+1)−µCGC

ω (ωCi,t+1

))

(ΓC(ωCi,t+1) − µCGC(ωCi,t+1)

) (3)

11

This equation implies that unless the monitoring cost µC is zero, investors wouldrequire the aggregate return to the their investment EtR

Ct+1 to be larger than the risk-

free rate Rt.

An interesting feature of the risk premiumEtRCt+1

Rtis that it does not depend on the

amount of the loan since the monitoring fee µC is scale independent. It depends only onthe cuto risk Etω

Ci,t+1. This implies that a higher cuto risk will adjust the consumer

repayment rate EtRCt+1 upward for a given risk-free rate. Similarly, the interest rates

ratio is proportional to the variance(σC)2

(the volatility parameter) of EtωCi,t+1 so

a lower risk volatility would reduce the gap between the consumer rate and risk-freeinterest rate.

Since every credit-constrained consumer solves the same loan contract, they choosethe same expected cuto risk Etω

Ci,t+1 to maximize their share of the loan. Each house-

hold is essentially a representative home-buyer so equation (3) holds for every mortgage-buyer. Hence it is possible to aggregate across all credit-constrained households so thatthis equation is valid for the whole economy. The ratio of the realized return to housingto the expected return on the right hand side is determined macroeconomically, andgiven that every borrower chooses the same expected cuto default risk, then the ag-gregate cuto risk is Etω

Ci,t+1 = Etω

Ct+1 Hence equation (3) holds on the aggregate level

and determines the macroeconomic cuto risk EtωCt+1:

EtRCt+1

Rt= Et

ΓCω (ωC

t+1)

(ΓCω (ωC

t+1)−µCGC

ω (ωCt+1

))(1 − ΓC(ωCt+1)

)+

ΓCω (ωC

t+1)

(ΓCω (ωC

t+1)−µCGC

ω (ωCt+1

))

(ΓC(ωCt+1) − µCGC(ωCt+1)

) (4)

Similarly, the aggregate amount of net worth held by indebted households in theeconomy is NC

t =∑NCi,t and the aggregate amount of housing purchased by credit-

constrained consumers is HCt =

∑HCi,t.

The rst order condition (4) and the participation constraint (1) hold at any pointin time in which there have not been any unanticipated shocks. However, even after ashock occurs at time t+ 1, the borrowed share ptH

Ct −NC

t is already xed so investorsmust continue to receive the risk-free rate Rt. Taken together, these two equationsdetermine the realized cuto value ωCt+1 that adjusts after an unexpected disturbanceat t+1. Hence although consumers optimize using the expected participation constraintof lenders, the model utilizes the realized participation constraint that incorporates therealized default risk along with the expected rst order condition:(

ΓC(ωCt ) − µCGC(ωCt ))RCt pt−1H

Ct−1 = Rt−1

(pt−1H

Ct−1 −NC

t−1

)(5)

A positive productivity shock reduces the price of housing pt. This causes the networth of indebted households to decrease. To satisfy their participation constraint,the investors will either reduce lending or raise the interest rate they charge to thehouseholds or both which implies that the participation constraint contracts. However,in the general equilibrium context, the supply of housing is predetermined. To maintainequilibrium with the original level of housing, the rst-order condition moves to a higherlevel. This means that the ratio of the realized return to housing to the risk-free rate

12

increases raising the external nance premium. Observing this, households choose ahigher cuto risk ωC and an unchanged prot-maximizing level of housing. This is notthe long-run equilibrium. Over time, consumers reduce their housing stock and returnto the previous prot-maximizing level of ωC .

In order to complete the partial equilibrium setting, it is necessary to determinethe evolution of the credit-constrained households' net worth. In any given period, theequity of the credit-constrained households, V C

t , is the remaining share of the mortgageafter replaying back investors:

V Ct =(1 − ΓC(ωCt )

)RCt pt−1H

Ct−1 (6)

Consumers can spend this dividend income on new housing. When house prices fall- and therefore the equity of the households V C

t - households face the following decisionproblem. If they decrease housing demand today, current household utility would fall.But, if demand were kept constant, net worth would decrease, increasing the futureexternal nance premium. Thus households face a trade-o between current housingpurchase and future borrowing.

It is also necessary to make sure that credit-constrained consumers do not eventuallygrow out of their nancial constraints. This paper assumes that every period a constantfraction 1−νC of households retire. When they retire, they spend their remaining equityon consumption. The retirement consumption CC

E,t of credit-constrained consumers is:

CCE,t =(1 − νC

) (1 − ΓC(ωCt )

)RCt pt−1H

Ct−1 (7)

Over time, the number of credit-constrained consumers decreases but their individualnet worth increases as they cycle through many periods of mortgage contracts. Hencemacroeconomically, the value of the net worth remains the same.

Credit-constrained consumers need to get started on their net worth with someincome not devoted to purchasing consumption goods and housing. This is equivalentto establishing a savings account dedicated to the initial mortgage downpayment. Themodel assumes they provide one unit of labor inelastically to the production of housingthat generates a wage wCt . This labor supply is solely for the purposes of startingtheir net worth accumulation and is weighted heavily so that it does not distort theoverall labor supply. Having determined the period equity and the start-up net worthof credit-constrained consumers, is it easy to describe the evolution of their net worth.The evolution of the credit-constrained households' net worth is the sum of equity ofnon-retiring households plus their income from work:

NCt = νCV Ct + wCt (8)

2.3 The Share Contracts of the Consumption and Housing Sectors

The contracts between investors on one side and producers of consumption and of

housing on the other hand are share contracts. As equity holders of nal goods rms

in both sectors, investors nance their capital purchase and absorb their prots and

13

losses. No monitoring takes place. The manufacturers and investors split the revenue

according to the shares of their investments, regardless of the idiosyncratic shocks to

consumption producers ωF and to housing producers ωH . To diversify away from the

rm-specic idiosyncratic risk, each consumer will invest in an innite number of rms.

Investors nance the capital purchase xFt KFt of consumption rms, which occurs one

period prior to production. The aggregate return to capital purchased by the consump-tion rms is the risk-free rate Rt. In the following period, the share of consumptionrms' revenue that goes to investors is ΓFt .

ΓFt RtxFt K

Ft =

xFt KFt −NF

t

xFt KFt

∞

0

ωFRtxFt K

Ft dF (ωF )

Since lenders can diversify over an innite number of food producers and thereis no upper cuto amount of ωF ,

´∞0ωFdF (ωF ) = 1 This implies that the share is

independent of the idiosyncratic shock ωF :

ΓFt =xFt K

Ft −NF

t

xFt KFt

(9)

Firms in the consumption sector accrue prots which they split with investors ac-cording to their share contract. In any given period, the equity of consumption rmsis their part of the share agreement in the previous period:

V Ft =(1 − ΓFt

)Rt−1x

Ft−1K

Ft−1 = Rt−1N

Ft−1 (10)

In order to prevent consumption rms from growing out of the nancial constraints,the model assumes that at the end of every period, a constant fraction 1 − νF exitsthe market. Exiting rms consume immediately their remaining equity. In this caseconsumption producers' consumption CF

E,t on exit is:

CFE,t =(1 − νF

)Rt−1N

Ft−1 (11)

Similarly to consumers, rms need to start o with some net worth so this setupassumes they provide inelastically one unit of labor to the production of their outputfor which they receive a wage wFt . The labor contribution of consumption producersis signicantly discounted to avoid crowding out regular household labor supply. Theevolution of rms' net worth is the sum of equity of surviving rms plus their incomefrom work:

NFt = νFV Ft + wFt (12)

Investors also nance the capital purchase xHt KHt of housing rms. Similarly to the

consumption sector, there is no idiosyncratic risk in the housing sector and rms canborrow at the risk-free rate. The share purchase in the housing sector ΓHt equals:

ΓHt =xHt K

Ht −NH

t

xHt KHt

(13)

Housing producers also accrue prots which they split with investors according to

14

their share contract. In any given period, the equity of housing rms is:

V Ht =(1 − ΓHt

)Rt−1x

Ht−1K

Ht−1 = Rt−1N

Ht−1 (14)

Housing producers' consumption CHE,t on exit is:

CHE,t =(1 − νH

)Rt−1N

Ht−1 (15)

Housing rms also provide inelastically one unit of labor to the production of theiroutput for which they receive a wage wHt . The labor contribution of housing producersis also discounted. The evolution of rms' net worth is the sum of equity of survivingrms plus their income from work:

NHt = νHV Ht + wHt (16)

3 The Complete Model with Consumer Mortgage

This section embeds the partial equilibrium of the loan contract derived in the previoussection into a general equilibrium framework that endogenizes the risk-free rate Rt andthe price of housing pt. The economy consists of two production sectors: consumptionand housing. Capital producers supply sector-specic capital to both types of nalgood rms. Households consume both consumption and housing.

3.1 Consumption Capital Sector

Firms that produce capitalKFt for the consumption sector own technology that converts

nal goods into capital. They purchase depreciated capital from nal good rms andmake investments to produce new capital. The investment IFt is consumption. Thenewly produced capital is sold back to consumption producing rms.

There are standard quadratic adjustment costs to producing capital. The capitaladjustment costs for the consumption capital are:

KFt = (1 − δ)KF

t−1 + J

(IFtKFt−1

)KFt−1 (17)

The function J is such that J ′ > 0 and J” < 0. New capital is produced within theperiod and sold to nal good producing rms at the price xFt . The optimal conditionfor investment is:

xFt J′(

IFtKFt−1

)= 1 (18)

15

3.2 Consumption Producers

Firms in the consumption sector use capitalKFt , labor L

Ft and sector-specic technology

AFt to produce their output. Consumption producing rms buy capital one period inadvance. They borrow funds for the purchase of capital at the risk-free rate Rt whichis equal to the expected return on capital. In order to do so, these rms issue claims toRicardian consumers at the prevailing price of consumption capital xFt . At the end ofeach production period, they sell the remaining capital back to capital producing rms.The production function of consumption rms is:

Y Ft = AFt (KFt−1)αF (LFt )(1−αF ) (19)

According to the share purchase setup, consumption rms supply inelastically oneunit of labor to their own production in order to start the accumulation of their networth. Factoring in this labor supply in the production function, the total labor supplyin the consumption sector is:

LFt = (LFF,t)ΩF (LFH,t)

(1−ΩF ) = (LFH,t)(1−ΩF ) (20)

where LFH,t is the regular labor supply by both types of consumers and LFF,t is thelabor supply by consumption producers.

Recasting the production function only in terms of household labor yields:

Y Ft = AFt (KFt−1)αF (LFH,t)

(1−αF )(1−ΩF ) (21)

The rms in the sector are perfectly competitive so they maximize prots subjectto input costs. The rst-order conditions for optimal capital and labor are:

wt = (1 − αF ) (1 − ΩF )

(Y FtLFH,t

)(22)

Rt =αF

Y Ft+1

KFt

+ (1 − δ)xFt+1

xFt(23)

The wage consumption rms receive for their labor supply is:

wFt = (1 − αF ) ΩFYFt (24)

3.3 Housing Capital Sector

Firms that produce capital KHt for the housing sector own technology that converts

goods into capital. They purchase depreciated capital from nal goods rms in thesame sector and obtain investments to produce new capital. The investment IHt isconsumption. The newly produced capital is sold back to housing producers.

Housing capital is subject to the same adjustment costs as consumption capital. Thehousing capital production equation is:

KHt = (1 − δ)KH

t−1 + J

(IHtKHt−1

)KHt−1 (25)

16

New capital is produced within the period and sold to nal good producing rms atthe price xHt . The optimal level of investment in housing capital:

xHt J′(

IHtKHt−1

)= 1 (26)

3.4 Housing Producers

Housing is a multi-period good that survives for more than one period, unlike consump-tion which is not durable beyond the period in which it is produced. The production ofhousing follows closely Iacoviello and Neri (2010). Housing producers use capital KH

t ,labor LHt , land Xt and sector-specic technology AHt to produce new houses. Housingrms also buy capital one period earlier. In this version, housing rms have no aggre-gate uncertainty so they can borrow funds for the purchase of capital at the risk-freerate Rt which is equal to the expected return on capital. In order to do so, these rmsissue claims to Ricardian consumers at the prevailing price of housing capital xHt . Atthe end of each production period, they sell the remaining capital back to housing rms.The production function of housing rms is:

Y Ht = AHt (KHt−1)αH (Xt−1)ε(LHt )(1−αH−ε) (27)

The amount of land is xed and normalized to one. Furthermore, both the sharepurchase setup and the loan contract of credit-constrained consumers assumed thathousing producers and indebted households supply inelastically one unit of labor inorder to start the accumulation of their respective net worth. Factoring in that laborsupply, the total labor supply in the production of housing by origin is:

LHt = (LHF,t)ΩH (LHCC,t)

ΩC (LHH,t)(1−ΩH−ΩC) = (LHH,t)

(1−ΩH−ΩC) (28)

Where LHH,t is the labor supply of both types of consumers for the purpose of nancing

their regular consumption, LHCC,t is the labor supply of credit-constrained consumers in

order to start their net worth accumulation and LHF,t is the labor supply of housingproducers.

Recasting the production function only in terms of household labor and factoring inthe xed supply of land yields:

Y Ht = AHt (KHt−1)αH (LHH,t)

(1−αH−ε)(1−ΩH−ΩC) (29)

The price of housing is pt. Housing rms in the sector are perfectly competitive sothey maximize prots subject to input costs obtaining the following optimal conditionsfor housing capital and labor:

wt = (1 − αH − ε) (1 − ΩH − ΩC)

(ptY

Ht

LHH,t

)(30)

Rt =αH

pt+1YHt+1

KHt

+ (1 − δ)xHt+1

xHt(31)

17

The wage housing rms receive for their labor supply is:

wHt = (1 − αH − ε) ΩHptYHt (32)

The wage credit-constrained consumers receive for the purpose of starting their networth is:

wCt = (1 − αH − ε) ΩCptYHt (33)

3.5 Consumers

Both Ricardian and credit-constrained consumers have the same preferences. House-holds choose consumption Ci

t , housingHit and labor L

it subject to their respective budget

constraints. Here the superscript i denotes the type of consumers: R for Ricardian andC for credit-constrained. Housing is purchased one period in advance. This approachmatches empirical reality better where acquiring a house involves transactional delaysthat involve search time, time spent with real estate agents and time to process escrow,payment and home insurance. Furthermore, there are nancial motivations for theadvance purchase. Since credit-constrained consumers purchase housing in advance ofusing it and they do not have non-wage income like Ricardians, they need the mort-gage arrangement to facilitate the housing acquisition. Furthermore, the mortgage isintertemporal where consumers optimize their expected share of the loan at the time ofthe housing purchase but repay it only in the subsequent period after using the housing.As a result, they are exposed to unexpected shocks at the time of repayment so themortgage is an inherently risky undertaking.

Each household seeks to maximize its lifetime expected utility:

U = Et∞∑t=0

βtU(Cit , Hit−1, L

it) (34)

The period utility of each household is given by:

U(Cit , Hit−1, L

it) = log(Cit) + κlog(Hi

t−1) − γLi(1+ϕ)t

1 + ϕ(35)

The period utility function is separable in consumption Cit , housing H

it−1 and labor

Lit. Housing is purchased one period in advance and consumed the following period.At the end of the period, the remaining housing minus depreciation is sold back onthe market. Following Iacoviello and Neri (2010), housing enters the utility functionadditively, rather than as part of a consumption aggregator in order to demonstrateits direct eect on consumer decisions. The additive nature of the utility functionalso facilitates housing to be be purchased both directly (by Ricardians) and via amortgage (by credit-constrained consumers). There is a taste parameter κ that reectsthe relative preference for consumption and housing.

18

3.5.1 Ricardian Consumers

Ricardian consumers purchase consumption and housing. Each period, they lend anamount Bt at the risk-free rate. Their lending covers the mortgage loan to credit-constrained households as well as nances the capital purchase of consumption rms andhousing producers through their respective share arrangements. Ricardian consumersalso absorb the prots Πt of both capital sectors. The budget constraint of Ricardianhouseholds is:

CRt + ptHRt +Bt = wtL

Rt + (1 − δ)ptH

Rt−1 +Rt−1Bt−1 + Πt (36)

Ricardian consumers maximize their utility function subject to this budget con-straint. The left hand side reects their consumption and housing purchase as well astheir lending, while the right hand side represents their income from wages and fromreselling the non-depreciated housing from the previous period as well as their returnsfrom lending and from capital rms prots. The Lagrangian for Ricardian consumersyields three rst-order conditions for consumption, housing and leisure. The rst-ordercondition for the consumption-labor tradeo is fairly standard:

γ(LRt )ϕCRt = wt (37)

The relationship between consumption and housing reects the fact that housingis purchased one period in advance so the tradeo between housing and consumptiondepends on both the current and on the future price of housing as well as on theintertemporal consumption substitution and the depreciation rate of housing:(

ptβEt

(CRt+1

CRt

)− (1 − δ)pt+1

)HRt = κEtC

Rt+1 (38)

Ricardian consumers also have a standard Euler equation:

Et

(CRt+1

CRt

)= βRt (39)

Combing the last two equations yields a simpler expression for the consumption-housing substitution:

(Rtpt − (1 − δ)pt+1)HRt = κEtC

Rt+1 (40)

3.5.2 Credit-Constrained Consumers

Just like Ricardian households, credit-constrained households also consume both con-sumption and housing. They earn income only from labor and do not own any shares.Since their income is not sucient to allow them to purchase housing in full, they mustobtain a mortgage from investors. Their mortgage is subject to an idiosyncratic risk ofdefault so their borrowing is not riskless. Due to this probability of bankruptcy, theycannot borrow at the risk-free rate Rt and can do so only at the consumer interestrate RC

t . As a result, buying a house is costly for them. Credit-constrained householdsalso cannot optimize intertemporally their purchase of consumption since they have noaccess to risk-free nancing. Hence their consumption needs must be met solely withtheir wages after their mortgage is repaid. The budget equation for credit-constrainedhouseholds is:

CCt +RCt pt−1HCt−1 = wtL

Ct + (1 − δ)ptH

Ct−1 (41)

19

The left hand side reects their consumption purchase as well as their housing mort-gage, while the right hand side represents their income from wages and returns fromreselling the non-depreciated housing from the previous period. Credit-constrained con-sumers maximize their utility function subject to this budget constraint. The optimiza-tion problem yields two rst-order conditions for housing and leisure. The rst-ordercondition for consumption-labor tradeo is identical to that of Ricardian households:

γ(LCt )ϕCCt = wt (42)

The relationship between consumption and housing however depends on the con-sumer interest rate RC

t instead of the risk-free rate Rt:(RCt pt−1 − (1 − δ)pt

)HCt−1 = κCCt (43)

The rst-order condition for housing-consumption tradeo for credit-constrainedconsumers is lagged, unlike that for Ricardian households which is forward-looking.This is due to the fact that the return rate at which credit-constrained consumers re-pay their mortgage is state-contingent and depends on the realization of the ex-postreturn on housing RC

t that incorporates all shocks at the time of the repayment. Therealized return on housing RC

t depends on the past purchasing price pt−1 and the currentselling price pt of housing. It does not depend on the expected future price of housingwhich is unknown at the time of the loan contract. Hence equation (43) is a functionof the ex-post return on housing RC

t , while the corresponding equation for Ricardianhouseholds (40) is not lagged since Ricardians borrow at the risk-free rate which is notstate-contingent.

Finally, credit-constrained consumers cannot optimize intertemporally their con-sumption purchase since they cannot borrow at the risk-free rate Rt. Their demand forconsumption must be met by their income once all housing loans are repaid:

CCt = wtLCt + ((1 − δ)pt −RCt pt−1)HC

t−1 (44)

Any disturbances that can increase the mortgage repayment, would crowd out reg-ular consumption and the resulting diminished demand might impact negatively theconsumption sector.

3.6 Market Clearing

Market clearing requires that the output of consumption must cover household con-sumption, consumption by conventional good producers on exit and consumption bycredit-constrained consumers on retirement as well as investment in the two sectors:

Y Ft = CRt + CCt + CFE,t + CCE,t + IFt + IHt (45)

Housing is a multi-period good and each period a fraction δ of the housing availableon the market depreciates. The remaining non-depreciated housing along with newproduction constitutes the available housing in the subsequent period. In each period,

20

the sum of new production and leftover housing must meet the housing needs of con-sumers as well as the consumption of housing rms on exit and the monitoring costs ofthe mortgage contract:

ptYHt + (1 − δ)pt(H

Rt−1 +HC

t−1) = pt(HRt +HC

t ) + CHE,t + µCGC(ωCt )RCt pt−1HCt−1 (46)

The labor that both types of households supply equals the demand by housing andconsumption rms:

LFH,t + LHH,t = LRt + LCt (47)

Finally, Ricardian consumer lending must equal the loan to credit-constrained house-holds as well as the share purchase of consumption rms and of housing rms:

Bt = ptHCt −NC

t + xFt KFt −NF

t + xHt KHt −NH

t (48)

All the equations for this model are in Appendix A.

3.7 Model Calibration

The calibrated model is quarterly so four periods correspond to a year. The modelparameters are chosen to be as close as possible to standard literature values whilereecting a number of key model features. The parameters governing the loan contractare calibrated to match those in Bernanke, Gertler and Gilchrist (1999) and satisfyseveral steady state conditions:

1. The steady state rate of the external risk premium is 0.5%. (Bernanke, Gertlerand Gilchrist, 1999).

2. The steady state housing rm leverage, i.e. value of housing capital stock to networth ratio is xHKH

NH = 2 (Bernanke, Gertler and Gilchrist, 1999).3. The failure rate of housing manufacturers FH(ωH) (i.e. the number of rms that

exit the market each period) is 2% (Bernanke, Gertler and Gilchrist, 1999).The remaining model parameters are chosen to be as close as possible to a repre-

sentative two-sector economy. (Table 1). Both sectors are identical in their productionfunction. Since housing and consumption enter the utility function additively, the pa-rameter κ determines the consumption-housing tradeo for consumers taking as giventhe interest rate and the depreciation rate of housing. Housing depreciates at the samerate as capital, δ. For the temporary volatility shock, the persistence parameter %has a value of 0.95, a conventional choice for the frequency of the model (Fernández-Villaverde, 2010).

Table 1: General Parameters in the Consumer Mortgage Contract Economy

Parameter αF αH ε β γ ϕ κ J” δ Ωi

Value 0.35 0.25 0.10 0.99 0.03 0.33 0.41 −10 0.0025 0.01

Solving the rst-order condition of credit-constrained consumers, the participationconstraint of investors and the cumulative distribution function FC(ωC) in the steady

21

state gives the parameters µC , νC , νH , νF , the steady state value of the cuto risk ωC

and the standard deviation σC . The credit contract parameters are given in Table 2.

Table 2: Loan Contract Parameters in the Consumer Mortgage Contract Economy

Parameter νF νH νC µC

Value 0.99 0.99 0.98 0.06

The complete model is solved for the deterministic steady state and then log-linearizedaround that steady state. The steady state values of the model variables are in Table3.

Table 3: Steady State Variables in the Consumer Mortgage Contract Economy

Variable Y F Y H CR CC HR HC LR LC IF IH

Value 100 10 32.8 30.2 210 169 15.5 19.9 24.9 3.28

Variable KF KH LF LH p w R RC NF CFE

Value 997 131 29.9 5.48 1.84 2.15 1.01 1.015 498 5.68

Variable NH CHE NC CCE ωC σC

Value 65.7 0.78 156 3.10 0.5 0.31

Variable ΓC GC ΓCω GCω ΓCσ GCσ ΓCωω GCωω ΓCωσ GCωσ

Value 0.50 0.01 0.98 −0.02 −0.02 0.16 −0.31 2.02 −0.36 1.89

Appendix B outlines the log-linearization of the complete model. The derivations ofthe log-normal distribution of ωC are in Appendix E.

4 The Consumer Mortgage Model Results

This section demonstrates the eect of two disturbances that were the major contribu-tors to the subprime mortgage crisis - the oversupply of housing depressed the return onhousing and the nancial easing permitted credit-constrained consumers to amass toomuch debt. The oversupply of housing is modeled as a 10% technological innovationin the production of housing. This stylistically simple setup allows generating excesssupply of houses without unnecessarily complicating the model with discussions of thepreceding housing bubble. Nevertheless, this approach successfully captures the hous-ing inventory buildup that amassed during the boom. This disturbance is simulated asa permanent shock since the excess housing availability began at the peak of the assetbubble and continued during the subsequent recession as the housing demand shrank(Duke, 2012; Madigan, 2012).

22

The relaxation of borrowing conditions is introduced as a 10% reduction in thevolatility of the default risk that made loans seem safer to lenders. The easing permit-ted borrowers to secure loans with less collateral and as a result to increase their debtholdings. The reduction in nancial volatility is a temporary shock corresponding tothe brief period of lax lending practices that prevailed on the cusp of the subprime crisis.There are empirical and theoretical underpinnings for the temporary nature of the dis-turbance. Empirically, the lax access to credit was quickly reversed as the proliferationof defaults signaled the beginning of an economic downturn (Dennis, 2010; Duke, 2012).Theoretically, the model assumes that lenders update the loan terms after observing theborrower risk. A permanent shock would be at odds with the microfounded nature ofthe loan contract and would imply that the loan terms remain relaxed despite the wors-ened bankruptcy prospects of borrowers following a leverage buildup. Taken together,the two disturbances demonstrate that positive shocks can have negative consequencesfor credit-constrained consumer borrowers and that indebted agents can experience adownturn even if they are not production rms or nancial institutions.

4.1 The Housing Oversupply

A permanent expansion of the housing stock depresses the price of all houses on themarket including the value of the housing collateral held by indebted consumers. Sincethe value of their housing stock falls relative to their mortgage, households enter arecession characterized by a worsening in their leverage and an increased likelihood ofdefault. Seeing this, lenders require a higher risk premium restricting household accessto house nancing for a prolonged period of time.

The oversupply of housing aects the leverage of indebted consumers by reducing thevalue of the mortgage collateral (Figure 2). A permanent positive technological shockin the production of housing raises the supply of houses (Figure 2.2) and creates excessreal estate inventory. In the absence of concurrent changes in housing demand, theequilibrium price of housing decreases (Figure 2.17) to accommodate the extra supply.The lower price of housing, however, applies not only to newly produced houses, but alsoto the existing housing stock that includes the housing equity held by credit-constrainedconsumers. The housing equity of borrowers consists of their share of housing boughtwith the loan contract after repaying investors. It depends both on the price of housingand on the realized return on housing that is the mortgage borrowing rate RC

t . Theunanticipated fall in the housing price reduces not only the value of the housing stockbut also the return to housing in the period in which the shock occurs (Figure 2.20),i.e. the adjustable rate of the subprime-like loans decreases. Hence the lower price ofhousing aects the net worth of credit-constrained consumers both directly through theconsumer share of real estate holdings and indirectly via the ex-post return on housingRCt . In contrast, the value of the loan diminishes only by the drop in the housing price.

As a result, the net worth of indebted households (Figure 2.21) falls proportionallymore than the value of the mortgage so the leverage of borrowers increases (Figure2.24). Higher leverage signals increased probability of default for credit-constrainedconsumers (Figure 2.23).

23

The aggregate net worth of credit-constrained consumers falls not only because therealized return of housing is low relative to the expected return evaluated one periodbefore, but also because a higher cuto default risk increases bankruptcy prospects andreduces the share of the mortgage returns that goes to credit-constrained consumersin the loan contract. The investors are now exposed to bigger risks of default onthe part of indebted households. To compensate for the rise in the monitoring costs,the investors require a higher return from borrowers, forcing up the external nancepremium (Figure 2.29), i.e. the adjustable mortgage rate increases. Consequentially,the falling equilibrium price of housing generates a pro-cyclical risk premium. In bothcases, the shock reduces the equilibrium price of housing. Borrowing consumers see thevalue of their housing collateral fall so they experience a worsening of their bankruptcyprospects.

The worsened indebtedness levels prompt mortgaged households to attempt to repairtheir net worth but the higher risk premium implies that borrowers can increase theirhousing stock very gradually. The inability of credit-constrained consumers to easilyaord housing after the deterioration of their leverage is the reason housing outputmakes a dip from its initially high levels (Figure 2.2). The innovation in the housingsector also aects demand for consumption so output decreases (Figure 2.1). Due tothe lower price of housing, Ricardian consumers switch away from consumption towardsreal estate (Figure 2.3). Indebted households initially enjoy a lower repayment rateon their existing loan so they can initially aord slightly more consumption (Figure2.5). This rise, however, is a token amount so it is not sucient to compensate forthe reduced demand by Ricardians. Furthermore, the higher consumption demandby credit-constrained households is short-lived. The negative eects of the worseningleverage and the widening risk premium soon catch up with them and credit-constrainedconsumers have to switch away from consumption and toward repairing their net worth(Figure 2.21) and their borrowing abilities. Over time, indebted consumers graduallyincrease their housing stock (Figure 2.6) which translates into improving net worth(Figure 2.21) and decreasing default prospects (Figure 2.23).

The leverage and probability of default of credit-constrained consumers eventuallyreturns to pre-shock levels owing to their diverting resources away from consumptiontoward increasing their housing stock (Figures 2.21 and 2.23). However, the recoveryof household leverage is a protracted process. Their improvement is relatively slow andtakes indebted households about 100 quarters (25 years) to return to their pre-crisisnancial position. The simulations demonstrate that the interim period is characterizedby a subprime-like recession with higher leverage and increased bankruptcy prospectsfor credit-constrained households.

4.2 The Temporary Financial Relaxation

Temporarily easing access to house nancing for credit-constrained consumers allowsthem to obtain a mortgage with less downpayment. As a result, their debt holding

24

are higher implying a rise in their bankruptcy prospects. Since the external nancepremium reects the solvency of borrowers, it rises with the increased risk. A higherrisk premium tightens the previously relaxed access to credit plunging consumers intoa recession despite the short-lived nature of the initial shock.

Leading up to the subprime mortgage crisis, the housing market was characterizedby exceptionally favorable borrowing conditions with low interest rates and little down-payment requirements. In the model, this lax lending atmosphere is captured by a

negative temporary shock on(σC)2, the variance of the consumer default risk ωC . A

lower variance implies narrower swings in the bankruptcy prospects ceteris paribus andsignals safer loans (Figure 3). For borrowers, it can translate into a smaller risk pre-mium on loans, less required collateral, or both. The reduction in volatility is modeledas an autoregressive temporary disturbance with moderate persistence to reect thatthis phenomenon was relatively short-lived and to demonstrate the worsening of re-nancing possibilities as the initial shock dies o. A permanent credit easing, on theother hand, would imply that borrowing terms remain permanently favorable althoughconsumers have raised their leverage, a contradiction to the microfounded nature ofthe model. The temporary shock reects that relaxed borrowing conditions promptmortgagers to raise their leverage leading to a deterioration of their debt position ascredit access tightens subsequently.

Reduced nancial volatility implies safer loan contracts that require a smaller down-payment and that attract a reduced risk premium (Figure 3.29). The lower risk pre-mium is passed on to credit-constrained consumers as a decreased mortgage interest rateRCt (Figure 3.20). Faced with a more favorable borrowing rate, indebted households

increase their housing demand (Figure 3.6), driving marginally up the price of housing(Figure 3.17). Housing output also increases being demand driven by consumers (Fig-ure 3.2). The higher housing price raises the value of their housing stock and positivelyinuences their net worth immediately after the shock. However, the downward eectof the reduced consumer rate is quantitatively larger than the upward inuence of thehousing price so the net worth of borrowers eventually diminishes below pre-shock levels(Figure 3.21). The falling net worth, coupled with the subsequent credit tightening asthe temporary shock wears out, worsen the leverage of credit-constrained households(Figure 3.24). Higher leverage signals increased probability of bankruptcy despite theimproved nancial conditions (Figure 3.23).

Seeing their indebtedness increase, credit-constrained consumers attempt to delever-age. They reduce their housing demand after the initial increase in an attempt to im-prove their net worth (Figures 3.6 and 3.21). The lower demand for housing reduceshousing production shortly after the initial increase (Figure 3.2). It takes 100 quar-ters (25 years) of gradual improvement for their debt position to recover and for theirbankruptcy prospects to return to pre-shock levels (Figures 3.23 and 3.24). The recov-ery is staggered since the higher renancing rate restricts the ability of debtors to easilyimprove their housing stock which in turn maintains a higher renancing rate. Overall,the eects of reduced default risk volatility are far from benecial to indebted con-sumers. The initial benets of relaxed borrowing conditions are quickly overshadowedby the tightening of credit access as the shock wears out. The result is rising consumer

25

leverage and increased bankruptcy prospects. Even a temporary easing of lending haslong-lasting consequences for the debt position of credit-constrained households. Thevolatility shock, just as the production shock, shows that positive disturbances can havenegative eects on the indebtedness of households since they permit them to increasetheir leverage. It also demonstrates that non-production agents, such as consumers,can be the source of nancial disturbances when they are leveraged.

26

Figure 2: Housing Oversupply in Economy with Credit-Constrained Consumers

0 50 100−0.004

−0.002

0.0001. Consumption Output

0 50 1000

0.1

0.22. Housing Output

0 50 100−0.004

−0.002

0.000

3. Cons. Good Demand ofRicardian Consumers

0 50 1000

0.1

0.2

4. Housing Demand ofRicardian Consumers

0 50 100

0.000

0.002

0.004

5. Cons. Good Demand of Credit−Constrained Consumers

0 50 100−0.1

0

0.1

6. Housing Demand of Credit−Constrained Consumers

0 50 100−0.004

−0.002

0.0007. Consumption Capital

0 50 1000

0.02

0.048. Housing Capital

0 50 100−0.01

0

0.01

9. Labor Demand inConsumption Sector

0 50 1000

0.05

0.1

10. Labor Demand inHousing Sector

0 50 1000

0.01

0.02

11. Labor Supply ofRicardian Consumers

0 50 100−0.001

0.000

0.001

12. Labor Supply of Credit−Constrained Consumers

0 50 100−0.01

−0.005

0

13. Investmentin Consumption

0 20 40 600

0.05

0.1

14. Investmentin Housing

0 50 100−0.004

−0.002

0.000

15. Price of Consumption Capital

0 50 100

0

0.02

0.04

16. Price of Housing Capital

0 50 100−0.2

−0.1

017. Price of Housing

0 50 100−0.001

0.000

0.001

0.00218. Wage

0 50 1000

0.5

119. Risk−Free Rate

0 50 100−1000

−500

0

20. Consumer Interest Rate

0 50 100−0.2

−0.1

0

21. Net Worth of Credit−Constrained Consumers

0 50 100−0.2

−0.1

0

22. Retirement Consumption ofCredit−Constrained Consumers

0 50 1000

0.05

0.1

23. ConsumerDefault Risk

0 50 1000

0.05

0.124. Consumer Leverage

0 50 1000.000

0.002

0.004

25. Net Worth of Consumption Firms

0 50 1000.000

0.002

0.004

26. Consmption on Exit byConsumption Firms

0 50 1000

0.005

0.01

27. Net Worth ofHousing Firms

0 50 1000

0.005

0.01

28. Consmption on Exitby Housing Firms

0 50 1000

10

20

29. ConsumerRisk Premium

0 50 100

−0.004

−0.002

0.00030. Ricardian Lending

27

Figure 3: Reduced Volatility in Economy with Credit-Constrained Consumers

0 50 100−0.001

0.000


0 50 100−0.01

0


0 50 100−0.001

0.000

0.001


0 50 100−0.02

0

0.02


0 50 100−0.001

0.000

0.001


0 50 100−0.05

0

0.05


0 50 100−0.001

0.000


0 50 100−0.001

0.000


0 50 100−0.001

0.000

0.001


0 50 100−0.01

0

0.01


0 50 100−0.005

0.000

0.005


0 50 100−0.001

0.000

0.001


0 50 100

−0.002

0.000

0.002


0 50 100−0.005

0.000

0.005


0 50 100−0.001

0.000

0.001

15. Price of Consump. Capital

0 50 100−0.001

0.000

0.001

16 Price of Housing Capital

0 50 100−0.005

0.000

0.00517. Price of Housing

0 50 100−0.001

0.000

0.00118. Wage

0 50 100

0

0.5


0 50 100−20

−10

0


0 50 100−0.02

0

0.02


0 50 100−0.02

0

0.02

22. Retirement Consumption of Credit−Constrained Consumers

0 50 1000

0.02

0.04


0 50 1000

0.02


0 50 1000.000

0.001

0.002


0 50 1000.000

0.001

0.002


0 50 1000.000

0.001

0.002


0 50 1000.000

0.001

0.002


0 50 100−20

−10

0


0 50 1000

0.005

0.0130. Ricardian Lending

28

5 The Chained Loan Contracts

5.1 Description of the Chained Loan Contracts

The model in the previous sections considered credit frictions in the loan contract be-tween investors and credit-constrained consumers. This extension adds nancial insti-tutions to the mortgage contract. The model assumes that nancial institutions act asintermediaries that borrow funds from investors and in turn lend to credit-constrainedhouseholds. Both banks and borrowing consumers are subject to idiosyncratic uncer-tainty and participate in loan contracts. Taken together, the two transactions constitutetwo joined mortgage contracts similar to the ones in Hirakata, Sudo and Ueda (2009).

Ricardian

Consumers

Credit-Constrained

Consumers

Housing

Firms

Consumption

Capital

Firms

Consumption

Firms

Consumption Purchase L

end

ing

: Co

stly

Ve

rification

Labor

Dep

osit

Labor

Consumption

Purchase

Housing

Purchase

Investors

New Capital

Used Capital

Housing

Capital

Firms

Housing Purchase

New Capital

Used Capital

Labor

Labor

Financial

Intermediaries

Len

din

g: C

ostly

Ve

rification

Share

Purchase Share

Purchase

Figure 4: Model with Financial Frictions in the Consumer and Financial Sectors

In this extension, banks borrow from investors and in turn lend to households. Unlikeinvestors whose sole role in the model is to facilitate the lending of Ricardian consumersand who do not possess intrinsic risk, in this version banks have an inherent probabilityof default just as credit-constrained consumers do but their risk is separate from that ofborrowers (Figure 4). The distinction between investors and banks is not purely a modelcomplication but mirrors real life nancial markets. Investors can be characterized asrepresenting safe mutual funds institutions that possess no idiosyncratic risk, whilenancial institutions correspond to investment banks that are highly leveraged. Theemergence of such highly leveraged banks may be traced back to the desire for largerprots from riskier investments which safe nancial agents such as the investors in thismodel would be unwilling to nance directly.

The presence of idiosyncratic uncertainty in the returns of banks and householdsnecessitates two loan contracts that provide state-contingent returns to investors. At

29

the beginning of the nancial intermediation, investors lend to nancial institutionsin a loan arrangement that ensures them a fair return regardless of the realizationof the idiosyncratic shock of banks. At the tail end of the nancial intermediation,credit-constrained consumers borrow from nancial institutions as part of another loancontract that ensures their participation. Banks are the intermediaries that participatein both loan contracts so they optimize both arrangements at the same time. Thisis also consistent with empirical observations of real life banks. Hence the two loancontracts must be chained in the sense that they are linked in the same optimizationproblem. The remainder of the paper explores the role of leveraged nancial institutionsin magnifying the eect of exogenous disturbances.

5.2 The Chained Loan Contracts

Financial institutions (i.e banks) borrow from investors one period in advance and lendto mortgage-buying consumers to nance their housing purchase. Both banks andcredit-constrained consumers are borrowing-constrained in the sense that they eachhave a distinct probability of default that is known to them but unknown to otherparticipants in the nancial market who know only the distribution of the bankruptcylikelihood. Hence there are nancial frictions both in the contract between investorsand nancial institutions and in the contract between banks and credit-constrainedhouseholds. The presence of credit constraints necessitates that both arrangementsare loan contracts where the returns are contingent on the realization of the two id-iosyncratic shocks. Since nancial institutions are intermediaries and participate inboth contacts, the contracts are chained so that the returns from the rst contract canprovide sucient funds to cover the lending in the second one.

In this version, there is an idiosyncratic shock ωB associated with lending to nancialinstitutions and a distinct idiosyncratic shock ωC associated with lending to credit-constrained consumers. Similar to the nancial institutions' loan contract, investorshave to pay an auditing fee µB to learn the realization of ωB. This makes lending tobanks risky so nancial institutions have to pay a premium on external funds. Financialinstitutions on their part, must pay a monitoring cost µC to learn the realization of theconsumer default probability ωC .

Since the individual optimization problem of each bank can be aggregated to holdfor the whole economy in the same way as before, the chained contracts setup proceedsdirectly on the aggregate level. Banks borrow funds from investors and in turn lend tocredit-constrained households. Every period, they choose the optimal pair of cuto riskωB and housing HC to maximize their next period expected share 1 − ΓB(ωBt+1) of thetotal value of the contact that consists of the housing stock ptH

Ct minus the net worth

of credit-constrained consumers NCt :

max Et((

1 − ΓB(ωBt+1))RBt+1

) (ptH

Ct −NC

t

)(49)

Banks lend to households at a state contingent markup rate EtRCt+1 that is dierent

from the rate EtRBt+1 at which they repay their loans. The dierence accounts for the

30

distinct probabilities of default of both types of indebted agents. The expected earningsof nancial institutions from lending to credit-constrained consumers equal the shareEtΓ

C(ωCt+1) they will receive tomorrow from the loan made today to households minusthe auditing fee on insolvent consumer loans µCEtG

C(ωCt+1):

Et((

ΓC(ωCt+1) − µCGC(ωCt+1))RCt+1

)ptH

Ct

The earnings of banks must equal the opportunity cost of lending to credit-constrainedconsumers, which is the value of the loan to banks at their markup interest rate EtR

Bt+1.

Hence the gross return on the banks' loan to credit-constrained consumers is:

Et((


)ptH

Ct = EtR

Bt+1

(ptH

Ct −NC

t

)(50)

The left hand side is the banks' share of the loan to borrowing households aftermonitoring and the right hand side is the gross return (the value of the housing purchaseptH

Ct minus the consumers' net worth NC

t valued at the bank interest rate EtRBt+1) from

the housing purchase to nancial institutions.

Credit-constrained households will participate in the chained loan contracts onlyif their participation constraint is met. Instead of taking part in the chained loancontracts, credit-constrained consumers can purchase housing using their own net worthNCt . In this alternative case, the ex-post return to their investments equals EtR

Ct+1N

Ct .

Hence credit-constrained consumers will participate in the chained contracts only iftheir share of the loan is at least equal to the value of their net worth:

Et((

1 − ΓC(ωCt+1))RCt+1

)ptH

Ct ≥ EtR

Ct+1N

Ct (51)

The rst part of the chained contracts consists of investors lending to banks. Finan-cial institutions split the gross prot from their loan to credit-constrained consumerswith investors. This contract has the same costly state verication structure as the sin-gle loan contract. Banks own the net worth NB

t and invest the amount ptHCt −NC

t inthe loan to credit-constrained consumers. They borrow the rest ptH

Ct −NC

t −NBt from

investors and repay the loan using their prots from lending to credit-constrained house-

holds. Investors receive a net share of the loan to banks Et(ΓB(ωBt+1) − µBGB(ωBt+1)

)that includes the monitoring fees paid on failed banks. Financial institutions are sub-ject to an idiosyncratic bankruptcy shock ωB and their ex-post gross return on theloans to credit-constrained consumers is ωBRB. Investors would participate in thechained contracts only if they get a fair return on their lending to nancial institu-tions. Like in the single contract model, the lender's share of the prot in the contract

with nancial institutions Et(ΓB(ωBt+1)RB

t+1

) (ptH

Ct −NC

t

)after paying the monitor-

ing fee µBEtGB(ωBt+1) must at least equal the opportunity cost of the investors' funds

Rt

(ptH

Ct −NC

t −NBt

)valued at the risk-free rate Rt. Hence the zero prot partic-

ipation constraint for investors must specify the amount of funds that banks borrowfrom investors pHC−NC−NB, the cut-o value of the idiosyncratic shock ωB and thereturn rate of the loan to non-defaulting banks RB. Since banks borrow at time t andrepay the funds at t+ 1, the participation constraint of investors is:

Et((

ΓB(ωBt+1) − µBGB(ωBt+1))RBt+1

) (ptH

Ct −NC

t

)≥ Rt

(ptH

Ct −NC

t −NBt

)(52)

Lenders sign contracts with a lot of banks, to diversify away the idiosyncratic riskof nancial institutions.

31

Substituting equation (50) into (52) and noting that both equations (51) and (52)bind at the optimum, eliminates the bank interest rate EtR

Bt+1 and reduces the condi-

tions that nancial institutions must satisfy to two:

Et((

ΓB(ωBt+1) − µBGB(ωBt+1)) (


)ptH

Ct = (53)

= Rt(ptH

Ct −NC

t −NBt

)Et(1 − ΓC(ωCt+1)

)ptH

Ct = NC

t (54)

In the previous loan contract, credit-constrained consumers maximized the arrange-ment. The superior knowledge of borrowers about their own possibility of defaultallowed them to push lenders to their participation constraint and extract maximumreturns from the loan. The chained loan contracts, however, are linked by the presenceof nancial institutions. Hence it is the intermediaries (nancial institutions) whichmaximize their prots subject to satisfying the participation constraints of both credit-constrained consumers and of investors. In this case banks, which are borrowers inone part of the contract and lenders in the other part, optimize the two sides of thecontract. The reason for banks maximizing their prots is twofold. First, it is empiricalfact that nancial institutions are often the ones which dictate both the lending andthe borrowing terms. The reason for that may be informational asymmetry. Banksuse monitoring technology to collect information on borrowers and lenders that wouldreduce the agency cost associated with lending. This process is costly and location-specic (a bank would not be willing to lend outside of its geographic and sectoralarea of expertise). On the other hand, investors, who are geographically dispersed andconsumers who lack the means nd the cost of this monitoring technology prohibitive.Second, nancial institutions, as the intermediaries that participate in both parts of theloan contract, have a transactional advantage. It is easier for them to optimize the twoloan contracts together rather than optimizing only their borrowing arrangements withinvestors and leaving the second loan contract to consumers. The subsequent analysisdemonstrates that the results from the chained contracts model are not analyticallydierent from those of two separate loan arrangements.

In the chained contracts model, the presence of two sectors with idiosyncratic un-certainty has important implications for risk sharing between the indebted banks andhouseholds. As intermediaries in the two mortgage contracts, nancial institutions arethe rst to experience a deterioration in their balance sheet. However, since they alsomaximize the chained contracts, they can shift some of the burden of the increased riskof default onto credit-constrained consumers. As end participants in the mortgage ar-rangement, households have no control over the leverage distribution between indebtedsectors. In times of a downturn characterized by worsened leverage for nancial insti-tutions, they can improve their own position by forcing consumers to bear a share ofthe deteriorating leverage. Banks achieve this by extracting a larger return from theloan contract with households in times when the default probabilities of both borrowersare higher. The participation constraint of households, unlike the share agreement inthe single loan contract of nancial institutions, depends on changes in bankruptcyprospects. The increased bankruptcy prospects allow nancial institutions to demand

32

a higher repayment thus shifting the participation constraint (54) of consumers to theleft. This can work because, in contrast to the share contract, consumers in the chainedmortgage contracts have no recourse to funds outside of the borrowing arrangement dueto their idiosyncratic uncertainty. Hence credit-constrained consumers have no choicebut to bear some of the deterioration in indebtedness. Oftentimes, this aords banks aspeedier recovery at the expense of indebted households who are faced with a protractedrecession.

The chained loan optimization problem involves banks simultaneously maximizingboth loan arrangements. Let for simplicity of expression ΓB(ωBt )−µBGB(ωBt ) = ΨB(ωBt )and ΓC(ωCt ) − µCGC(ωCt ) = ΨC(ωCt ). Banks choose the optimal level of ωC , ωB andHC by solving the following Lagrangian:

LBt = Et((

1 − ΓB(ωBt+1))

ΨC(ωCt+1)RCt+1

)ptH

Ct +

+λ1,t

[Et(ΨB(ωBt+1)ΨC(ωCt+1)RCt+1

)ptH

Ct −Rt

(ptH

Ct −NC

t −NBt

)]+λ2,t

[Et(1 − ΓC(ωCt+1)

)ptH

Ct −NC

t

]The rst order conditions yield two equations for the Lagrange multipliers:

λ1,t = Et

(ΓBω (ωBt+1)

ΨBω (ωBt+1)

)(55)

λ2,t = Et

[[(1 − ΓB(ωBt+1)

)+

ΓBω (ωBt+1)ΨB(ωBt+1)

ΨBω (ωBt+1)

]RCt+1

ΨCω (ωCt+1)

ΓCω (ωCt+1)

](56)

Here λ1,t is the marginal value of the internal funds to nancial institutions, and λ2,t

is the marginal increase in the prots of nancial institutions per unit increase in thenet worth of credit-constrained consumers.

Substituting both into the rst order condition for HCt and rearranging gives the

consumer risk premiumEtRCt+1

Rt:

EtRCt+1

Rt= Et

(ΓCω (ωCt+1)(

1 − ΓC(ωCt+1))

ΨCω (ωCt+1) + ΓCω (ωCt+1)ΨC(ωCt+1)

)× (57)

Et

(ΓBω (ωBt+1)(

1 − ΓB(ωBt+1))

ΨBω (ωBt+1) + ΓBω (ωBt+1)ΨB(ωBt+1)

)

Equation (57) along with the two participation constraints determines the realizeddefault probabilities ωB and ωC . Even after a shock occurs, investors continue to receivethe risk-free rate Rt so the model implements the realized participation constraints:

(ΓB(ωBt ) − µBGB(ωBt )

) (ΓC(ωCt ) − µCGC(ωCt )

)RCt pt−1H

Ct−1 = (58)

= Rt−1

(pt−1H

Ct−1 −NC

t−1 −NBt−1

)(1 − ΓC(ωCt )

)pt−1H

Ct−1 = NC

t−1 (59)

33

along with the rst order condition that contains the expectation of what will occurat t+ 1.

The period equity of credit-constrained consumers remains the same as in the con-sumer mortgage contract. The period equity of nancial institutions is amended toinclude the return to banks ΨC(ωCt ) from lending to households:

V Bt =(1 − ΓB(ωBt )

)RBt

(pt−1H

Ct−1 −NC

t−1

)= (60)

=(1 − ΓB(ωBt )

)ΨC(ωCt )RCt pt−1H

Ct−1

The consumption on retirement of borrowing households remains the same as inthe consumer mortgage contract. However, the consumption of banks CB

E,t on exitbecomes:

CBE,t =(1 − νB

) (1 − ΓB(ωBt )


Ct−1 (61)

6 The Complete Model with Chained Loan Contracts

6.1 Complete Model Changes

The production sectors and consumer behavior remain the same. The introduction ofnancial institutions necessitates some changes to accommodate their presence. Theseinvolve the labor supply of banks in the housing sector and the market clearing con-ditions for consumption and housing. The production functions in both sectors, thecapital development and the goods demand by consumers remain unchanged.

The labor supply equation in the housing sector must account for the unit laborsupply by nancial institutions denoted LHB,t. Factoring in that labor supply, the totallabor supply in the production of housing by origin is:

LHt = (LHF,t)ΩH (LHCC,t)

ΩC (LHB,t)ΩB (LHH,t)

(1−ΩH−ΩC−ΩB) = (LHH,t)(1−ΩH−ΩC−ΩB) (62)

where LHF,t is the labor supply by housing rms, LHCC,t the labor supply by credit-

constrained consumers dedicated to starting their net worth, LHB,t. The labor supply

by nancial institutions and LHH,t is regular household labor supply.

The production function for housing becomes:


(1−αH−ε)(1−ΩH−ΩC−ΩB) (63)

The optimal condition for housing capital remains the same while that for laborbecomes:

wt = (1 − ΩH − ΩC − ΩB) (1 − αH − ε)

(ptY

Ht

LHH,t

)(64)

The market clearing condition for consumption includes the consumption of nancialinstitutions on exit CB

E,t in addition to that of consumers CCE,t:

Y Ft = CRt + CCt + CFE,t + CCE,t + CBE,t + IFt + IHt (65)

34

The market clearing condition for housing is augmented by the monitoring fee paid

by investors on their loan to banks µBGB(ωBt )RBt

(pt−1H

Ct−1 −NC

t−1

)which is in addition

to the one investors paid on their lending to credit-constrained consumers:


Rt−1 +HC

t−1) = pt(HRt +HC

t ) + CHE,t + (66)

+[µCGC(ωCt ) + µBGB(ωBt )ΨC(ωCt )

]RCt pt−1H

Ct−1

A complete list of the model equations is in Appendix C.

6.2 Model Calibration

The parameters that govern the general equilibrium for the chained contracts modelare the same as those in the consumer mortgage contract. The chained contracts modelsatises the same steady state requirements as the respective single loan contracts forconsumers and for nancial institutions:

1. The steady state rate of the external consumer risk premium is 0.5% (Bernanke,Gertler and Gilchrist, 1999).

2. The steady state consumer leverage, i.e. value of housing stock to net worth ratio

is pHC

NC = 2 (Bernanke, Gertler and Gilchrist, 1999).

3. The steady state leverage of nancial institutions is pHC−NC

NB = 5 (Hirakata, Sudoand Ueda, 2009).

4. The failure rate of both credit-constrained consumers FC(ωC) and nancial in-stitutions FB(ωB) is 2% (Bernanke, Gertler and Gilchrist, 1999).

The loan parameters that satisfy the chained contracts model are in Table 4.

Table 4: Loan Contract Parameters in the Chained Contracts Economy

Parameter νF νH νC νB µC µB

Value 0.99 0.99 0.98 0.97 0.02 0.03

The complete model is solved for the deterministic steady state and then log-linearizedaround that steady state. The steady state values of the model variables are in Table5.

35

Table 5: Steady State Variables in the Chained Contracts Economy

Variable Y F Y H CR CC HR HC LR LC IF IH

Value 100 10 32.5 29.9 210 169 15.6 19.9 24.9 3.35

Variable KF KH LF LH RC RB NF CFE NH CHE

Value 997 134 30.0 5.53 1.015 1.0147 498 5.68 67.0 0.80

Variable NC CCE NB CBE ωC σC ΓC GC ΓCω GCω

Value 159 2.53 31.9 1.11 0.5 0.31 0.50 0.01 0.98 0.15

Variable ΓCσ GCσ ΓCωω GCωω ΓCωσ GCωσ ωB σB ΓB GB

Value −0.02 0.16 −0.31 2.02 −0.36 1.89 0.79 0.11 0.79 0.02

Variable ΓBω GBω ΓBσ GBσ ΓBωω GBωω ΓBωσ GBωσ p w

Value 0.98 0.45 −0.04 0.74 −0.57 10.8 −0.97 14.4 1.88 2.15

Variable ΨB ΨBω ΨB

σ ΨBωω ΨB

ωσ ΨC ΨCω ΨC

σ ΨCωω ΨC

ωσ

Value 0.79 0.97 −0.05 −0.83 −1.32 0.50 0.98 −0.02 −0.36 −0.41

The log-linearization of the additional equations is in Appendix D.

7 The Chained Loan Contracts Results

This section compares the eect of an oversupply of housing and of a reduction innancial volatility in a chained contracts economy with both single contract versionswhere either credit-constrained consumers or nancial institutions are the subject tocredit constraints.

7.1 The Housing Oversupply

Previously consumers experienced a downturn following an increase in housing inven-tory that reduced the market price of housing. Chained loan contracts provide morethan a single loan contract when they are the sole indebted borrowers. As a result, adisturbance that reduces the value of the housing stock generates a smaller improve-ment in their leverage than in the share contract version but nevertheless puts them ina more favorable position than before. When consumers participate in the chained loancontracts, intermediary nancial institutions transfer some of their leverage worseningonto households in order to repair their own debt position faster.

36

This version again simulates a permanent technological improvement in the pro-duction of housing. However, since both types of agents are credit-constrained thecombined leverage in the economy is larger contributing to more volatility. The pres-ence of credit frictions in both the nancial sector and the household sector does notchange the directional impact of the housing innovation shock on households and onproducers but creates opportunities for risk sharing driven by the larger joint leverage(Figure 5). The oversupply of housing still triggers a reduction in the housing pricethat raises consumer demand for housing (Figures 5.4 and 5.6). The return to hous-ing decreases and the value of the housing stock held as collateral falls (Figures 5.17and 5.18). However, the chained model version yields qualitatively and quantitativelydierent eects on the leverage of indebted agents. Credit-constrained consumers expe-rience a mild improvement in their debt position while leveraged nancial institutionsenter a considerable recession (Figures 5.25 and 5.26). The diverging experiences canbe attributed to the relative position in the chained contracts of both borrowers andtheir initial leverage ratios. As optimizers of the chained contracts, banks must meetthe participation constraint of both investors and households so they absorb most ofthe adverse eects of the shock. Their leverage deteriorates by as much as in theirsingle loan contract.

Consumers know that banks will honor their participation constraint so they arerelatively shielded from the adverse eects of the falling housing price on their equity.Their net worth still diminishes but by less than in the single loan contract when credit-constrained consumers had to meet the participation constraint of investors. Overall,the decrease in the net worth of credit-constrained consumers (Figure 5.19) is propor-tionally less than the fall in the value of their housing stock so households experiencea small improvement in their leverage position. The reduced leverage, however, is suf-cient to imply lower bankruptcy prospects despite the higher risk premium (Figure5.23). Hence the chained contracts model creates opportunities for risk sharing be-tween indebted agents. Consumers are not as insulated from the downturn despite thepresence of nancial intermediaries.

Financial institutions, on the other hand, experience considerable deterioration intheir leverage. Although the combined worsening of consumer and bank net worth islarger in the chained contracts model than in the single contract version, banks shiftsome of the burden of the downturn onto consumers. This is due to the fact thatas maximizers of the chained contracts, nancial institutions can force consumers tothe break-even point of their participation constraint and extract a higher return fromthem. Despite this, they still must satisfy the participation constraints of both lenders(investors) and nal borrowers (credit-constrained consumers) so their ability to sharerisk is limited. As a result, the net worth of banks bears most of the devaluation in thehousing equity and their leverage ratio deteriorates accordingly (Figures 5.20 and 5.26).Unlike in the single loan contract where nancial institutions are exposed only to theirown idiosyncratic risk, here they are exposed both to their own risk of the default andthat of credit-constrained households. Hence their bankruptcy prospects deterioratemore than in the single contract (Figure 5.24). However, due to the risk sharing, bankssee a smaller rise in their risk premium. This implies that despite the dual risks in thechained contracts, the worsening in the leverage position of nancial institutions is not

37

proportionally larger (Figure 5.26). The opportunity for risk sharing allows banks todampen their recession at the expense of mortgaged households.

Comparing the subsequent recovery of the optimizing agents in the two versions, itis clear that nancial institutions return to their pre-shock position faster than con-sumers. Banks start with a higher leverage which necessitates a speedy recovery inorder to maintain their solvency. Furthermore, they are the ones that maximize theloan contracts so they can recover faster than mortgaged households, who have no sayin the loan arrangement and no recourse to outside funds (Figures 5.26 and 5.25). Theleverage of nancial institutions in the chained contracts version takes about 60 quarters(15 years) to return to pre-shock levels while that of credit-constrained consumers in thesingle contract remains above its steady state value for almost 100 quarters (25 years).The default risk mirrors the respective leverage evolution of the two credit-constrainedagents (Figures 5.24 and 5.23).

The single loan contract models and the chained contracts version oer insight intothe contrasting experiences of credit-constrained households. In the single consumermortgage model they maximize the loan contract, and in the chained contracts extensionthey are nal borrowers whose participation constraint must be satised by nancialinstitutions. As maximizers in the rst version of the model, indebted households bearfully the negative consequences of the oversupply of houses, while in this case they arerelatively shielded by the binding of their participation condition or the share agreementwith nancial institutions. In the chained contracts version, they are part of the jointmortgage contract where they are exposed to bankruptcy prospects but the bindingof their participation constraint still guarantees them a degree of protection from therecession, albeit on a smaller scare than in the nancial institutions' loan contract.The chained contracts model is more favorable toward credit-constrained consumersthan the single mortgage version. However, it is not preferable since the worseningof the debt position of banks is quantitatively larger making the economy as a wholemore volatile. Its only redeeming quality is that it results in a relatively short-liveddownturn compared to the single loan contract economy since nancial institutions usetheir role as optimizers to repair their leverage position relatively fast. What used to bea moderate but protracted credit crunch in the consumer loan setup becomes a steepbut short-lived nancial downturn in the chained loans version. Nevertheless, in bothscenarios the falling housing price can act as a vehicle of shock propagation that turnsa positive innovation in the housing sector into the main cause of debt worsening forcredit-constrained agents that characterizes a downturn such as the subprime mortgagecrisis.

The models described so far, although, theoretical in nature, have important policyimplications. The three versions of credit frictions demonstrate the importance of soundnancial regulation. In order to protect consumers from an unduly high burden ofrisk sharing from the nancial system, policymakers should consider regulating themaximum permissible leverage ratio for borrowers. However, mandating a debt-to-income cap only for indebted households may be of limited usefulness when banksare also leveraged. As intermediaries in the mortgage contract, nancial institutionsusually enter the loan arrangement with much less equity and experience a more volatile

38

leverage than households. Their excessive indebtedness in the chained contracts versioncauses a far larger drag on economic performance than that of consumers in the singlemortgage model so it is important that there are sucient loan loss provisions andreserve requirements in place to guarantee the liquidity of the nancial system in adownturn.

7.2 The Temporary Financial Relaxation

In the chained contracts setting, both consumers and nancial institutions can expe-rience a credit easing. Relaxing borrowing access for households spreads the resultingleverage worsening across both types of credit-constrained agents but the combineddownturn is deeper than that of households in their sole borrowing arrangement dueto the larger combined leverage. However, a nancial relaxation for banks impliesstronger risk sharing that is more tilted toward consumers. Banks struggle to repairtheir debt positions after the credit easing in their own sector so they shift a large shareof the downturn onto households. Credit-constrained consumers suer the negativeconsequences of a reduction in bank volatility with the same magnitude as following adecrease in their own volatility, although in this case they are not the originators of thedisturbance. While the downturn is sharp but brief for banks, consumers are drawninto a prolonged recession.

Unlike the single contract model, the chained contracts setup allows to study bothversions of a relaxation in borrowing conditions - one that targets credit-constrainedconsumers and one that pertains to nancial institutions - within the same model. Areduction in the volatility of consumer borrowing in the chained contracts model yieldsqualitatively dierent results than the same disturbance in the single contract version(Figure 6). The decreased consumer volatility in the chained contracts version reducesnot only the household risk premium but also the risk premium of banks since the twoloan contracts are linked in the same maximization problem (Figures 6.27 and 6.28).The reduction in risk premium is passed from consumers to nancial institutions. As aresult, households experience a smaller reduction in the external nance premium thanin the consumer loan contract.

The improvements in consumer credit access also aect nancial institutions which,as intermediaries in the loan contract, need less downpayment to secure funding frominvestors on behalf of consumers (Figure 6.20). Less net worth implies higher indebted-ness both for nancial institutions and for households. However, as optimizing agentsin the chained loan contract, nancial institutions experience most of the consequencesof the collateral reduction on their own debt position (Figure 6.26). Their indebtednessrises more than that of consumers but less than the corresponding increase in the singleloan contract version. The presence of two chained loan contracts allows them to sharethe downturn with consumers. The leverage of credit-constrained consumers rises aswell but by only half as much as in their own mortgage arrangement described (Figure6.25). Overall, the risk sharing in the chained loan contract dampens the eect of the

39

shock, spreading it over both types of indebted agents while at the same time concen-trating a larger share in the intermediary sector that optimizes the dual contracts. Thedefault probability of both agents also increases but by less than before (Figures 6.23and 6.24).

Taken together, the worsening in the leverage of both credit-constrained consumersand banks is larger than that of only consumers in the single loan contract. This is dueto the higher initial indebtedness of banks which reects on both agents. Despite thelarge combined downturn in the chained contracts model, the debt position of nancialinstitutions recovers relatively fast and about 70 quarters (17.5 years) after the shock,their leverage ratio and bankruptcy prospects return to pre-shock levels (Figure 6.26).Bank recovery, however, is at the expense of the revival of the borrowing ability ofhouseholds. As maximizers of the chained contracts, nancial institutions can pushconsumers to their participation constraint so most of the subsequent improvements inhousing equity benet them at the cost of delaying the recovery of mortgaged house-holds. The indebtedness of credit-constrained consumers unwinds slowly and remainsabove its pre-shock level for close to 100 quarters (25 years) (Figure 6.25). Risk sharingimplies that the net worth of banks and their leverage position improve relatively fast,while consumers are faced with a protracted recovery.

A model with relaxed credit access for nancial institutions produces the deepestrecession of all scenarios discussed so far (Figure 7). Following a reduction in their ownvolatility, nancial institutions see a smaller increase in their leverage than when theyare the only borrowers since some of the increase is transferred onto consumers (Figures7.25 and 7.26). Unlike the single loan contract version where the arrangement betweenconsumers and nancial institutions is a share one, here households participate in thechained contract creating opportunities for risk sharing. Consumers have no ability toborrow outside of the loan contract due to their idiosyncratic uncertainty so banks canaord to transfer some of the downturn onto them without turning them away from themortgage purchase. Hence households are forced to absorb some of the deteriorationin the balance sheets of banks although they are not the source of the disturbance.Following a shock to the volatility of banks, the leverage of mortgaged households risesby as much as when their own volatility is reduced in the same chained contracts setting(Figure 7.25).

The leverage of banks, however, rises more than in the previous case with consumercredit easing. Since the default probability of nancial institutions is initially muchhigher than that of households, a reduction in their own volatility leads to a moresignicant tightening of their nancial conditions. In order to mitigate this threatto their own solvency, nancial institutions shift a considerable share of the burdenonto households. Despite the risk sharing, the higher pre-shock indebtedness of nan-cial institutions implies that their leverage worsens slightly more than when they arefaced with a reduction in consumer volatility. As a result, the combined worsening ofnancial positions of both credit-constrained agents is the most severe of all consid-ered scenarios. The subsequent recovery again benets nancial institutions more thancredit-constrained consumers. As optimizers of the chained loan contract, nancialinstitutions can repair their debt position faster at the expense of consumers who expe-

40

rience a smaller leverage worsening and despite not being the source of the disturbance,undergo a slower recovery.

When it comes to relaxing credit access, the single consumer loan contract producesthe sharpest deterioration in the leverage of credit-constrained households since thereis no risk sharing. The two chained contract simulations demonstrate that consumersshare some of the burden of higher indebtedness with nancial institutions so theirleverage does not deteriorate as much as before. Nevertheless, their subsequent recov-ery could be more protracted since banks can use their optimizing position to extractmore equity from consumers in order to improve their own debt position. This is espe-cially evident in the case with a decrease in nancial volatility. Since consumers are notthe source of the disturbance, they should be relatively protected by their participationconstraint. However, the deterioration in the balance sheets of banks is so considerablethat nancial institutions resort to extracting as much equity from households as pos-sible in order to repair their own debt position. Credit-constrained consumers suerthe negative consequences of a reduction in bank volatility with the same magnitude asfollowing a decrease in their own volatility. The nancial troubles of banks draw theminto a prolonged downturn and their leverage remains above its steady state value formore than 100 quarters (25 years). The last scenario describing a reduction in bankvolatility in a chained contracts setup oers the closest cautionary tale to the subprimerecession. Leading up to the subprime crisis, the balance sheets of banks were deemedsuciently low risk and fairly stable allowing them to accumulate excessive debt. Theresulting downturn was passed on to consumers who experienced signicant worseningin their debt positions that led to a prolonged tightening of credit access. Nevertheless,all three scenarios in the single and chained contracts context demonstrate the perilsof higher leverage that a temporary relaxation in borrowing conditions can create.

The policy recommendations for the scenario with decreased nancial volatility pointin the same direction as these for housing oversupply. In order to safeguard againstsuch sudden relaxations in borrowing conditions that reect animal spirits and arenot supported by macroeconomic fundamentals, it is important to have loan-to-valuerequirements and debt-to-income caps in place for mortgage-buyers. Both would ensurethat homeowners would not raise their leverage signicantly or secure loans with littledownpayment. Like in the case with housing oversupply, macroprudential regulationsshould be aimed at nancial institutions as well to ensure that they hold sucientreserves to weather unexpected downturns that would reduce the incentive to shift theburden of recessions onto consumers. The scenario with a reduction in bank volatilityalso implies that in times of bank distress due to excessive leverage, consumers shouldnot be forced to share the burden of repairing the balance sheets of nancial institutionsand may need alternative sources of credit.

41

Figure 5: Housing Oversupply in a Chained Contracts Model

0 50 100

−0.004

−0.002


0 50 1000

0.1


0 50 100

−0.004

−0.002

0.000


0 50 1000

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015. Price of Housing

0 50 1000

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0 50 100−1000

−500

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−500

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18. Bank Interest Rate

0 50 100−0.2

−0.1

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0 50 100−0.4

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20. Net Worth ofBanks

0 50 100−0.2

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22. Consmption on Exit byBanks

0 50 100−0.05

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0.05

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24. Bank Default Risk

0 50 100−0.1

0


0 50 1000

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26. Bank Leverage

0 50 1000

20

40


0 50 1000

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28. Bank Risk Premium

0 50 100−0.005

0.000

0.005


0 50 1000.000

0.002

0.004


0 50 1000

0.005

0.01


0 50 1000

0.005

0.01


Chained loan contracts model

Consumer mortgage contract model

42

Figure 6: Reduced Consumer Volatility in a Chained Contracts Model

0 50 100−0.001

0.000


0 50 100−0.01

0


0 50 100−0.001

0.000

0.001


0 50 100−0.02

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0 10 20 300

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0 50 100−20

−10

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0.01


0 50 1000

0.02

0.04


0 50 1000

0.005

0.01


0 50 1000

0.02


0 50 1000

0.02

0.0426. Bank Leverage

0 50 100−20

−10

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0 50 100−20

−10

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0 50 100−0.001

0.000

0.001


0 50 100−0.001

0.000

0.001


0 50 100−0.001

0.000

0.001


0 50 100−0.001

0.000

0.001


Chained loan contracts model


43

Figure 7: Reduced Bank Volatility in a Chained Contracts Model

0 50 100−0.001

0.000


0 50 100−0.01

0


0 50 100−0.001

0.000

0.001


0 50 100−0.02

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−10

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0 50 100−0.020

0.000

0.020


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0.02


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0.000

0.020

21. Retirement Consumption ofCredit−Constrained

0 50 100−0.02

0

0.02


0 50 1000

0.02

0.04


0 50 1000

0.005

0.01


0 50 1000

0.02


0 50 1000

0.02

0.0426. Bank Leverage

0 50 100−20

−10

0


0 50 100−20

−10

0


0 50 100−0.001

0.000

0.001


0 50 100−0.001

0.000

0.001


0 50 100−0.001

0.000

0.001


0 50 100−0.001

0.000

0.001


Chianed loan contracts model with a reduction in financial volatility

Chianed loan contracts model with a reduction in consumer volatility


44

8 Conclusion

The subprime mortgage crisis dashed the hopes of many for home ownership and seto a deep recession. Loan applicants saw their nancing prospects reduced for yearsahead as suddenly prudent banks struggled to improve their balance sheet positions.Many nancial institutions were brought to the brink of collapse or saved by the toobig to fail policy only to push through their recovery at the expense of credit-squeezedconsumers. The consequences of excessive leverage and excessive lending as subprimeloans had many calling for more stringent supervision of borrowing transactions. An im-proved regulation framework cannot emerge without understanding how the subprimemortgage crisis happened to be.

The straightforward setup of nancial frictions in the household and nancial sec-tors in this paper updates the nancial accelerator approach of Bernanke, Gertler andGilchrist (1999) to capture some of the causes and consequences of the subprime reces-sion. The model establishes that the subprime crisis is not a unique occurrence; on thecontrary, a nancial downturn can be triggered relatively easily when there are creditfrictions associated with lending. Using a two-sector economy, this model demonstratesthat a positive housing supply shock can have negative repercussions for mortgage-buyers since it reduces the value of the good used as collateral in the mortgage con-tract. A lower value of the collateral implies higher leverage of indebted home-buyers,increased default prospects and a higher risk premium. Such a pro-cyclical risk pre-mium that occurs in a two-sector economy is in stark contrast to Bernanke, Gilchristand Gertler (1999) where one-sector model shocks can generate only an anti-cyclicalexternal nance premium.

An improvement in the borrowing conditions also causes adverse consequences forleveraged agents and demonstrates that the credit-constrained sector can be the sourceof a crippling downturn even if it is not a production sector. A reduction in the riski-ness of the nancial market can impact indebted consumers directly by allowing themto excessively leverage using relatively little collateral. When as a result of the over-leveraging, the lending rate adjusts upwards, these households end up with subprimemortgages that they cannot service and are more likely to become bankrupt. Finally,as the chained contacts extension demonstrated, there are opportunities for risk shar-ing between banks and consumers when nancial institutions also participate in themortgage contract. Banks may shift some of the consequences of the downturn ontocredit-constrained consumers dragging them into a protracted recession. The scenariowith relaxed credit access for banks captures fairly well the consequences of the sub-prime mortgage crisis with a sharp but brief decline for banks and a protracted recoveryfor mortgaged-households.

While the presence of idiosyncratic default probabilities implies that nancial fric-tions cannot be easily eliminated, the consequences of excessive leverage hint at thetype of macroprudential regulation needed to safeguard a sound nancial market (Di-mova, Kongsamut and Vandenbussche, 2014). Debt-to-income caps and loan-to-valuerequirements for consumers would ensure that households obtain sensible mortgages

45

that they can service even in a downturn. Reserve requirements and loan loss provi-sions for nancial institutions would guarantee sucient liquidity in times of distress.Taken together, these measures would prevent a potential overleverage that can cripplecredit access for a prolonged time and can spill out of the mortgage market and distressthe whole economy.

The model, while aptly demonstrating the role of leverage in triggering a nancialdownturn, could be enriched further to oer a deeper understanding of credit mecha-nisms. When the crisis started, many banks attempted to salvage their debt positionby recalling loans to other nancial institutions, rather than merely bearing out theincreasing bankruptcy risk as described in this paper. The model can be augmentedby adding inter-bank relationships and loan networks that may trigger a domino-likeeect of rising default risk. Furthermore, the potential role of a bailout policy could beexplored by adding government to the existing setup. It may also be a cautionary taleto consider the possibility of endogenous steady state leverage ratio for both banks andconsumers. As history leading to the crisis demonstrates, leverage limits were poorlyregulated and enforced prior to the subprime mortgage crisis allowing instead nan-cial institutions to reach dangerously high debt to equity ratios. Finally, an importantcontribution could be to model the role of heterogeneous consumer expectations. Itmay be especially interesting to consider to what extent departures from the represen-tative agent theory could explain the collective failure of agents to foresee the subprimemortgage crisis.

46

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51

Appendix A

Complete Consumer Mortgage Contract Model

The participation constraint for investors:(ΓC(ωCt ) − µCGC(ωCt )

)RCt pt−1H

Ct−1 = Rt−1

(pt−1H

Ct−1 −NC

t−1

)The rst order condition of credit-constrained households' maximization:

EtRCt+1

Rt= Et

ΓCω (ωC

t+1)

(ΓCω (ωC

t+1)−µCGC

ω (ωCt+1

))(1 − ΓC(ωCt+1)

)+

ΓCω (ωC

t+1)

(ΓCω (ωC

t+1)−µCGC

ω (ωCt+1

))

(ΓC(ωCt+1) − µCGC(ωCt+1)

)

The consumption of credit-constrained households on retirement:

CCE,t =(1 − νC

) (1 − ΓC(ωCt )

)RCt pt−1H

Ct−1

The law of motion of the net worth of credit-constrained households:

NCt = νC

(1 − ΓC(ωCt )

)RCt pt−1H

Ct−1 + (1 − αH − ε)ΩCptY

Ht

The consumption on exit by consumption rms:

CFE,t =(1 − νF

)Rt−1N

Ft−1

The law of motion of the net worth of consumption rms:

NFt = νFRt−1N

Ft−1 + (1 − αF )ΩFY

Ft

The consumption on exit by housing rms:

CHE,t =(1 − νH

)Rt−1N

Ht−1

The law of motion of the net worth of housing rms:

NHt = νHRt−1N

Ht−1 + (1 − αH − ε)ΩHptY

Ht

The consumption capital accumulation equation:

KFt = (1 − δ)KF

t−1 + J

(IFtKFt−1

)KFt−1

Investment demand by consumption capital rms:

xFt J′(

IFtKFt−1

)= 1

The consumption production function:


(1−αF )(1−ΩF )

The labor demand by consumption rms:

wt = (1 − αF ) (1 − ΩF )

(Y FtLFH,t

)

52

The capital demand by consumption rms:

Rt =αF

Y Ft+1

KFt

+ (1 − δ)xFt+1

xFt

The housing capital accumulation equation:

KHt = (1 − δ)KH

t−1 + J

(IHtKHt−1

)KHt−1

Investment demand by housing capital rms:

xHt J′(

IHtKHt−1

)= 1

The housing production function:


(1−αH−ε)(1−ΩH−ΩC)

The labor demand by housing rms:

wt = (1 − ΩH − ΩC) (1 − αH − ε)

(ptY

Ht

LHH,t

)The capital demand by housing rms:

Rt =αH

pt+1YHt+1

KHt

+ (1 − δ)xHt+1

xHt

The labor supply of Ricardian consumers:

γ(LRt )ϕCRt = wt

The consumption housing tradeo of Ricardian consumers:


Rt+1

The Euler equation of Ricardian consumers:

Et

(CRt+1

CRt

)= βRt

The labor supply of credit-constrained consumers:

γ(LCt )ϕCCt = wt

The consumption-housing tradeo of credit-constrained consumers:(RCt pt−1 − (1 − δ)pt

)HCt−1 = κCCt

The consumption purchase equation of credit-constrained consumers:


t−1

Market clearing in the consumption sector:

Y Ft = CRt + CCt + CFE,t + CCE,t + IFt + IHt

Market clearing in the housing sector:


Rt−1 +HC

t−1) = pt(HRt +HC

t ) + CHE,t + µCGC(ωCt )RCt pt−1HCt−1

Market clearing in the labor market:

LFH,t + LHH,t = LRt + LCt

53

Appendix B

Log-Linearization of Consumer Mortgage Contract Model

This appendix derives the log-linearized form of the model equations as used in thesimulations. The steady state value for each variable appears without time subscriptand hats denote the percentage deviation of the variable from its steady state such thatZt = Zt−Z

Z.

The participation constraint for investors:

RCt = Rt−1 +NC

pHC −NC

(pt−1 + HC

t−1 − NCt−1

)−

−ΓCω (ωC) − µCGCω (ωC)

ΓC(ωC) − µCGC(ωC)ωC ˆω

Ct − ΓCσ (ωC) − µCGCσ (ωC)

ΓC(ωC) − µCGC(ωC)σC σCt−1

The rst-order condition of credit-constrained households' maximization:

0 =ΓCω (ωC)

ΓCω (ωC) − µCGCω (ωC)(EtR

Ct+1 − Rt) −

−(1 − ΓC(ωC))RC

R

(ΓCωω(ωC)

ΓCω (ωC)− ΓCωω(ωC) − µCGCωω(ωC)

ΓCω (ωC) − µCGCω (ωC)

)ωCEt ˆω

Ct+1 +

+RC

R

(−ΓCσ (ωC) +

ΓCω (ωC)(ΓCσ (ωC) − µCGCσ (ωC)

)ΓCω (ωC) − µCGCω (ωC)

)σC σCt −

−RC

R

((1 − ΓC(ωC))

(ΓCωσ(ωC)

ΓCω (ωC)− ΓCωσ(ωC) − µCGCωσ(ωC)

ΓCω (ωC) − µCGCω (ωC)

))σC σCt


CCE,t = RCt + pt−1 + HCt−1 −

ΓCω (ωC)

1 − ΓC(ωC)ωC ˆω

Ct − ΓCσ (ωC)

1 − ΓC(ωC)σC σCt−1


NCt = νC(1 − ΓC(ωC))

RCpHC

NC(RCt + pt−1 + HC

t−1) − νCRCpHC

NCΓCω (ωC)ωC ˆω

Ct −

−νCRCpHC

NCΓCσ (ωC)σC σCt−1 + (1 − αH − ε)ΩC

pY H

NC

(pt + Y Ht

)The consumption on exit by consumption rms:

CFE,t = Rt−1 + NFt−1


NFt = νFR(Rt−1 + NF

t−1) + (1 − αF )ΩFY F

NFY Ft


CHE,t = Rt−1 + NHt−1


NHt = νHR(Rt−1 + NH

t−1) + (1 − αH − ε)ΩHpY H

NH

(pt + Y Ht

)54


KFt = KF

t−1 + δ(IFt − KF

t−1

)Investment demand by consumption capital rms:

xFt = J”δ(KFt−1 − IFt

)The consumption production function:

Y Ft = AFt + αF KFt−1 + (1 − αF )(1 − ΩF )LFH,t


wt = Y Ft − LFH,t


Rt + xFt =R− (1 − δ)

R(Y Ft+1 − KF

t ) +(1 − δ)

RxFt+1


KHt = KH

t−1 + δ(IHt − KH

t−1

)Investment demand by housing capital rms:

xHt = J”δ(KHt−1 − IHt

)The housing production function:

Y Ht = AHt + αHKHt−1 + (1 − ΩH − ΩC) (1 − αH − ε) LHH,t


wt = pt + Y Ht − LHH,t

The capital demand by housing rms:

Rt + xHt =R− (1 − δ)

R(pt+1 + Y Ht+1 − KH

t ) +(1 − δ)

RxHt+1


ϕLRt + CRt = wt


CRt+1 = HRt + pt+1 +

R

(R− 1 + δ)

(Rt + pt − pt+1

)The Euler equation of Ricardian consumers:

EtCRt+1 − CRt = Rt


ϕLCt + CCt = wt

55

The consumption housing tradeo of credit-constrained consumers:

CCt = HCt−1 + pt +

RC

(RC − 1 + δ)

(RCt + pt−1 − pt

)The consumption purchase equation of credit-constrained consumers:

CCt =wLC

CC(wt + LCt ) +

(1 − δ)pHC

CCpt −

RCpHC

CC(RCt + pt−1) +

(1 − δ −RC)pHC

CCHCt−1


Y Ft =CR

Y FCRt +

CC

Y FCCt +

CFEY F

CFE,t +CCEY F

CCE,t +IF

Y FIFt +

IH

Y FIHt


Y Ht =δ(HR +HC) − Y H

Y Hpt +

HR

Y HHRt +

HC

Y HHCt − (1 − δ)

Y H(HRHR

t−1 +HCHCt−1) +

+CHEpY H

CHE,t + µCGC(ωC)RCHC

Y H

(RCt + pt−1 + HC

t−1

)+

+µCGCω ωC ˆω

Ct

RCHC

Y H+ µCGCσ σ

C σCt−1

RCHC

Y H


LFH LFH,t + LHH L

HH,t = LRLRt + LCLCt

56

Appendix C

Complete Chained Contracts Model


ΨB(ωBt )ΨC(ωCt )RCt pt−1HCt−1 = Rt−1

(pt−1H

Ct−1 −NC

t−1 −NBt−1

)The participation constraint for credit-constrained consumers:

NCt−1 =

(1 − ΓC(ωCt )

)pt−1H

Ct−1

The rst order condition of nancial institutions' maximization:

EtRCt+1

Rt= Et

(ΓCω (ωCt+1)(

1 − ΓC(ωCt+1))

ΨCω (ωCt+1) + ΓCω (ωCt+1)ΨC(ωCt+1)

)×

Et

(ΓBω (ωBt+1)(

1 − ΓB(ωBt+1))

ΨBω (ωBt+1) + ΓBω (ωBt+1)ΨB(ωBt+1)

)

The consumption on exit by banks:

CBE,t =(1 − νB

) (1 − ΓB(ωBt )


Ct−1

The law of motion of the net worth of banks:

NBt = νB

(1 − ΓB(ωBt )


Ct−1 + (1 − αH − ε)ΩBptY

Ht


CCE,t =(1 − νC

) (1 − ΓC(ωCt )

)RCt pt−1H

Ct−1


NCt = νC

(1 − ΓC(ωCt )

)RCt pt−1H

Ct−1 + (1 − αH − ε)ΩCptY

Ht

The consumption on exit by consumption rms:

CFE,t =(1 − νF

)Rt−1N

Ft−1


NFt = νFRt−1N

Ft−1 + (1 − αF )ΩFY

Ft


CHt =(1 − νH

)Rt−1N

Ht−1


NHt = νHRt−1N

Ht−1 + (1 − αH − ε)ΩHptY

Ht


KFt = (1 − δ)KF

t−1 + J

(IFtKFt−1

)KFt−1

Investment demand by consumption capital rms:

xFt J′(

IFtKFt−1

)= 1

57

The consumption production function:


(1−αF )(1−ΩF )


wt = (1 − αF ) (1 − ΩF )

(Y FtLFH,t

)


Rt =αF

Y Ft+1

KFt

+ (1 − δ)xFt+1

xFt


KHt = (1 − δ)KH

t−1 + J

(IHtKHt−1

)KHt−1

Investment demand by housing capital rms:

xHt J′(

IHtKHt−1

)= 1

The housing production function:


(1−αH−ε)(1−ΩH−ΩC−ΩB)


wt = (1 − ΩH − ΩC − ΩB) (1 − αH − ε)

(ptY

Ht

LHH,t

)


Rt =αH

pt+1YHt+1

KHt

+ (1 − δ)xHt+1

xHt


γ(LRt )ϕCRt = wt



Rt+1

The Euler equation of Ricardian consumers:

Et

(CRt+1

CRt

)= βRt


γ(LCt )ϕCCt = wt

The consumption-housing tradeo of credit-constrained consumers:(RCt pt−1 − (1 − δ)pt

)HCt−1 = κCCt

58

The consumption purchase equation of credit-constrained consumers:


t−1


Y Ft = CRt + CCt + CFE,t + CCE,t + CBE,t + IFt + IHt



Rt−1 +HC

t−1) = pt(HRt +HC

t ) + CHE,t +

+[µCGC(ωCt ) + µBGB(ωBt )ΨC(ωCt )

]RCt pt−1H

Ct−1


LFH,t + LHH,t = LRt + LCt

59

Appendix D

Log-Linearization of the Chained Contracts Model


RCt = Rt−1 +1

pHC −NC −NB

((NC +NB

) (pt−1 + HC

t−1

)−NCNC

t−1 −NBNBt−1

)−

−ΨCω (ωC)

ΨC(ωC)ωC ˆω

Ct − ΨB

ω (ωB)

ΨB(ωB)ωB ˆω

Bt − ΨC

σ (ωC)

ΨC(ωC)σC σCt−1 −

ΨBσ (ωB)

ΨB(ωB)σBσBt−1

The rst-order condition of nancial institutions' maximization of the chained con-tracts:

EtRCt+1 − Rt =

=

[ΓCωω(ωC)

ΓCω (ωC)−(1 − ΓC(ωC)

)ΨCωω(ωC) + ΓCωω(ωC)ΨC(ωC)

(1 − ΓC(ωC)) ΨCω (ωC) + ΓCω (ωC)ΨC(ωC)

]ωCEt ˆω

Ct+1 +

+

[ΓCωσ(ωC)

ΓCω (ωC)−(1 − ΓC(ωC)

)ΨCωσ(ωC) − ΓCσ (ωC)ΨC

ω (ωC) + ΓCω (ωC)ΨCσ (ωC) + ΓCωσ(ωC)ΨC(ωC)

(1 − ΓC(ωC)) ΨCω (ωC) + ΓCω (ωC)ΨC(ωC)

]σC σCt +

+

[ΓBωω(ωB)

ΓBω (ωB)−(1 − ΓB(ωB)

)ΨBωω(ωB) + ΓBωω(ωB)ΨB(ωB)

(1 − ΓB(ωB)) ΨBω (ωB) + ΓBω (ωB)ΨC(ωB)

]ωBEt ˆω

Bt+1 +

+

[ΓBωσ(ωB)

ΓBω (ωB)−(1 − ΓB(ωB)

)ΨBωσ(ωB) − ΓBσ (ωB)ΨB

ω (ωB) + ΓBω (ωB)ΨBσ (ωB) + ΓBωσ(ωB)ΨB(ωB)

(1 − ΓB(ωB)) ΨBω (ωB) + ΓBω (ωB)ΨB(ωB)

]σBσBt

The nancial institutions' interest rate:

RBt = RCt +NC

pHC −NC

(NCt−1 − pt−1 − HC

t−1

)+

ΨCω (ωC)

ΨC(ωC)ωC ˆω

Ct +

ΨCσ (ωC)

ΨC(ωC)σC σCt−1

The participation constraint of credit-constrained consumers:

NCt−1 = pt−1 + HC

t−1 −ΓCω (ωC)


Ct − ΓCσ (ωC)


The consumption on exit of nancial institutions:

CBE,t = RCt + pt−1 + HCt−1 −

ΓBω (ωB)

1 − ΓB(ωB)ωB ˆω

Bt − ΓBσ (ωB)

1 − ΓB(ωB)σBσBt−1+

+ΨCω (ωC)

ΨC(ωC)ωC ˆω

Ct +

ΨCσ (ωC)

ΨC(ωC)σC σCt−1

The law of motion of the net worth of nancial institutions:

NBt = νB(1 − ΓB(ωB))ΨC(ωC)

RCpHC

NB(RCt + pt−1 + HC

t−1) + (1 − αH − ε)ΩHpY H

NB

(pt + Y Ht

)+

−νBRCpHC

NBΨC(ωC)

(ΓBω (ωB)ωB ˆω

Bt + ΓBσ (ωB)σBσBt−1

)+

+νBRCpHC

NB

(1 − ΓB(ωB)

) (ΨCω ω

C ˆωCt + ΨC

σ σC σCt−1

)The consumption of credit-constrained households on retirement:

CCE,t = RCt + pt−1 + HCt−1 −

ΓCω (ωC)


Ct − ΓCσ (ωC)


60


NCt = νC(1 − ΓC(ωC))

RCpHC

NC(RCt + pt−1 + HC

t−1) − νCRCpHC

NCΓCω (ωC)ωC ˆω

Ct −

−νCRCpHC

NCΓCσ (ωC)σC σCt−1 + (1 − αH − ε)ΩC

pY H

NC

(pt + Y Ht

)The consumption on exit by consumption rms:

CFE,t = Rt−1 + NFt−1


NFt = νFR(Rt−1 + NF

t−1) + (1 − αF )ΩFY F

NFY Ft


CHE,t = Rt−1 + NHt−1


NHt = νHR(Rt−1 + NH

t−1) + (1 − αH − ε)ΩHpY H

NH

(pt + Y Ht

)The consumption capital accumulation equation:

KFt = KF

t−1 + δ(IFt − KF

t−1

)Investment demand by consumption capital rms:

xFt = J”δ(KFt−1 − IFt

)The consumption production function:

Y Ft = AFt + αF KFt−1 + (1 − αF )(1 − ΩF )LFH,t


wt = Y Ft − LFH,t


Rt + xFt =R− (1 − δ)

R(Y Ft+1 − KF

t ) +(1 − δ)

RxFt+1


KHt = KH

t−1 + δ(IHt − KH

t−1

)Investment demand by housing capital rms:

xHt = J”δ(KHt−1 − IHt

)The housing production function:

Y Ht = AHt + αHKHt−1 + (1 − ΩH − ΩC − ΩB) (1 − αH − ε) LHH,t

61


wt = pt + Y Ht − LHH,t


Rt + xHt =R− (1 − δ)

R(pt+1 + Y Ht+1 − KH

t ) +(1 − δ)

RxHt+1


ϕLRt + CRt = wt


CRt+1 = HRt + pt+1 +

R

(R− 1 + δ)

(Rt + pt − pt+1

)The Euler equation of Ricardian consumers:

EtCRt+1 − CRt = Rt


ϕLCt + CCt = wt

The consumption housing tradeo of credit-constrained consumers:

CCt = HCt−1 + pt +

RC

(RC − 1 + δ)

(RCt + pt−1 − pt

)The consumption purchase equation of credit-constrained consumers:

CCt =wLC

CC(wt + LCt ) +

(1 − δ)pHC

CCpt −

RCpHC

CC(RCt + pt−1) +

(1 − δ −RC)pHC

CCHCt−1


Y Ft =CR

Y FCRt +

CC

Y FCCt +

CFEY F

CFE,t +CCEY F

CCE,t +CBEY F

CBE,t +IF

Y FIFt +

IH

Y FIHt


Y Ht =δ(HR +HC) − Y H

Y Hpt +

HR

Y HHRt +

HC

Y HHCt − (1 − δ)

Y H(HRHR

t−1 +HCHCt−1) +

+CHEpY H

CHE,t +[µCGC(ωC) + µBGB(ωB)ΨC(ωC)

] RCHC

Y H

(RCt + pt−1 + HC

t−1

)+

+RCHC

Y H

[µC(GCω (ωC)ωC ˆω

Ct +GCσ (ωC)σC σCt−1

)+ µBG(ωB)

(ΨCω ω

C ˆωCt + ΨC

σ σC σCt−1

)]+

+µBΨ(ωC)RCHC

Y H

[GBω ω

B ˆωBt +GBσ σ

BσBt−1

]Market clearing in the labor market:

LFH LFH,t + LHH L

HH,t = LRLRt + LCLCt

62

Appendix E

Derivatives of the Log-Normal Distribution

The distribution of ω is log-normal such that lnω ∼ N(

12σ2, σ2

)follows the Normal

distribution and where σ2 is the variance of the cuto idiosyncratic risk. The cumulativedistribution function F (ω), the investors' share of the loan contract from non-defaultingborrowers Γ (ω) and from defaulting debtors G (ω) are dened as:

F (ω) = Φ

(lnω + 1

2σ2

σ

)

G (ω) = Φ

(lnω − 1

2σ2

σ

)

Γ (ω) = Φ

(lnω − 1

2σ2

σ

)+

[1 − Φ

(lnω + 1

2σ2

σ

)]ω

The derivatives of the distribution terms, namely Fω, Fσ, Γω, Γσ, Gω, Gσ, Γωω, Γωσ,Gωω and Gωσ are derived below. The derivatives use the normal distribution propertythat φ′(x) = −xφ(x) where Φ and φ are the cumulative distribution function and theprobability distribution function of the standard normal distribution.

Fω =1

σωφ

(lnω + 1

2σ2

σ

)> 0

Fσ = − 1

σφ

(lnω + 1

2σ2

σ

)(lnω − 1

2σ2

σ

)> 0

Gω =1

σωφ

(lnω − 1

2σ2

σ

)> 0

Gσ = − 1

σφ

(lnω − 1

2σ2

σ

)(lnω + 1

2σ2

σ

)> 0

Γω = 1 − F (ω) = 1 − Φ

(lnω + 1

2σ2

σ

)> 0

Γσ = Gσ − ωFσ =

= − 1

σφ

(lnωC − 1

2σ2

σ

)(lnω + 1

2σ2

σ

)+ω

σφ

(lnω + 1

2σ2

σ

)(lnω − 1

2σ2

σ

)

Γωω = −Fω = − 1

σωφ

(lnω + 1

2σ2

σ

)< 0

Γωσ = −Fσ =1

σφ

(lnω + 1

2σ2

σ

)(lnω − 1

2σ2

σ

)< 0

63

Gωω = − 1

σω2φ

(lnω − 1

2σ2

σ

)+

(1

σω

)2

φ′(lnω − 1

2σ2

σ

)(lnω − 1

2σ2

σ

)=

= − 1

σω2φ

(lnω − 1

2σ2

σ

)−(

1

σω

)φ

(lnω − 1

2σ2

σ

)(lnω − 1

2σ2

σ

)

Gωσ = − 1

σ2ωφ

(lnω − 1

2σ2

σ

)− 1

σωφ′(lnω − 1

2σ2

σ

)(lnω − 1

2σ2

σ2

)=

= − 1

σ2ωφ

(lnω − 1

2σ2

σ

)+

1

σ2ωφ

(lnω − 1

2σ2

σ

)(lnω − 1

2σ2

σ

)(lnω + 1

2σ2

σ

)

64

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