Risky Mortgages in a DSGE Model∗
Chiara Forlati†

Luisa Lambertini‡

EPFL

EPFL
August 30, 2010

Abstract
This paper develops a DSGE model with housing, risky mortgages and endogenousdefault. Housing investment is subject to idiosyncratic risk and some mortgages aredefaulted in equilibrium. An unanticipated increase in the volatility of housing investment
produces a credit crunch where delinquencies and mortgage interest rates increase, lendingis curtailed, and aggregate demand for non-durable goods falls. The economy experiencesa recession as a consequence of the credit crunch. We compare economies that differ onlyin the riskiness of housing investment. Economies with lower risk are characterized bylower mortgage default rates, higher loan-to-value and leverage ratios. The macroeconomic
effects of an unexpected increase in mortgage risk are amplified in high-leverage economies.
Monetary policy plays an important role in the transmission of mortgage risk, as moreinertial interest rate rules deepen the short-run contraction of output.
Keywords: Financial crisis; Housing; Mortgage default
JEL Codes: E32, E44, G01, R31
Address of corresponding author: Luisa Lambertini, EPFL CDM SFI CFI, ODY 2 05,
Station 5, CH-1015 Lausanne, Switzerland. E-mail: luisa.lambertini@epfl.ch∗

We would like to thank Matteo Iacoviello, Andrew Levin and Eileen Mauskopf for their comments andsuggestions and Yankai Shao for excellent research assistance. We are also grateful to seminar participants at
Universidad Autonoma de Barcelona, UNIL, the conference on Monetary and Fiscal Policy for Macroeconomic
Stability, the workshop on Heterogeneous Nations and Globalized Financial Markets and the 2010 EEA Meetingfor useful feedbacks.
†École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, chiara.forlati@epfl.ch.
‡École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, luisa.lambertini@epfl.ch.

1

1Introduction

The global financial crisis that began in August 2007 has its roots in increased mortgage delinquencies that put financial institutions into distress. The bursting of the housing bubble in the
United States put many borrowers in a difficult financial position with mortgages they could
not pay in the long run and larger than the value of the houses against which they were underwritten. As a result the rate of seriously delinquent mortgages1 increased from 2% in the third
quarter of 2006 to 10% by the first quarter of 2010. Banks were forced to write down several
hundred billion dollars in bad mortgages. These losses combined with a high degree of opacity
surrounding mortgage-backed securities and a complicated web of interconnected obligations
among financial institutions triggered a severe liquidity crisis in the interbank market. A credit
crunch followed that caused failure of several financial institutions, brought many others close
to it, raised interbank rates and drastically reduced household access to borrowing. The fall in
housing prices and tightened credit conditions forced many borrowers to quickly reduce leverage, cut consumption and housing investment. The turmoil that started in the mortgage market
amplified to the rest of the economy to spark the great recession.
Several factors contributed to the depth of the 2008 recession and no single factor in isolation
is likely to explain it. In particular, certain pre-crisis trends in the banking industry have been
key in amplifying the effects of an increase in mortgage delinquencies to the second largest
recession in U.S. history.
This paper focuses on one of these factors: the increase in mortgage delinquencies and its
transmission to the rest of the economy. It introduces endogenous default on mortgages in a
DSGE macroeconomic model with housing. The goal of the paper is to understand how aggregate shocks and variations in macroeconomic variables affect the rate of default on mortgages.
We are also interested in understanding how an increase in the rate of default on mortgages
transmits to the rest of the economy. We are interested in identifying qualitative effects, but
we also want to get an idea of how important this channel is and how much it can contributeto a fall in GDP. Driven by recent events, we focus on an unexpected increase in mortgage
risk, which we model as an unanticipated increase in the volatility of the idiosyncratic risk of1
According to the National Delinquency Survey of the Mortgage Banker Association, seriously delinquent
mortgages are those more than 90 days past due or in foreclosure.

2

housing investment.
Figure 1 shows the impulse responses from a VAR with de-trended seriously delinquent mortgages (DELHP), nominal short-term interest rate (RR), change in the log of the implicit pricedeflator for the nonfarm business sector (DP), de-trended real house prices (QQHP), de-trended
real per capita consumption (CCHP), and de-trended real per capita residential investment(IHHP) from 1980Q1 to 2009Q4.2 The top part of Figure 1 illustrates the impulse responses ofinterest rate, inflation, housing prices, consumption and residential investment to an innovationin serious delinquencies. It suggests a significant negative response of real house prices, private
consumption and residential investment to a positive disturbance to mortgage delinquencies.
The nominal interest rate and inflation respond positively, but with a delay. This evidence
points to a significant transmission mechanism of shocks from the mortgage market to the restof the economy. The bottom part of Figure 1 shows the impulse responses of seriously delinquent mortgages to innovations to all other variables. These responses suggest that mortgage
delinquencies react significantly to developments in the macro-economy. This VAR evidence
suggests that mortgage delinquencies are an endogenous variable that affects and is affectedby other fundamental macroeconomic variables. Our goal is to build a general equilibrium
macroeconomic model that captures these relationships.
Our model features two households that differ in terms of their discount factor. Savers havea higher discount factor than Borrowers and in equilibrium lend to Borrowers. Preferences
are specified over consumption of non-durable goods and housing services and hours worked.
Borrowers use their houses as collateral for mortgages and experience idiosyncratic housing
investment shocks that can only be observed by the household itself. Lenders must pay amonitoring cost to observe Borrower’s realized housing return. Borrowers experiencing low
realizations of the idiosyncratic shock default on their mortgages; Borrowers who repay theirmortgages pay a state-contingent adjustable mortgage rate that is above the risk-free one.
Hence, our model is characterized by endogenous default on mortgages and an external finance,or mortgage, premium.
We study the dynamic response to an unanticipated increase in the idiosyncratic risk tohousing investment. Our model produces a credit crunch that generates a recession in the nondurable sector. Mortgage default rates as well as the mortgage premium, namely the spread
2

See Appendix A for data definition and sources. De-trending is done using the Hodrick-Prescott filter.

3

Response of DELHP to DELHP

Response of RR to DELHP.20

Response of DP to DELHP

.0025

.16

.0012

.0020

.0008

.12
.0015
.08

.0004

.04

.0010

.00

.0005

.0000

-.04

-.0004.0000-.08-.0008-.0005-.12-.16-.0010
2

4

6

8

10

12

14

16

18

20

-.0012
2

4

6

Response of QQHP to DELHP

8

10

12

14

16

18

20

2

4

6

Response of CCHP to DELHP

.006

.0010

.004

8

10

12

14

16

18

20

16

18

20

16

18

20

16

18

20

Response of IHHP to DELHP
.015
.010

.0005

.002

.005
.0000

.000

.000
-.0005

-.002

-.005
-.0010

-.004

-.010
-.0015

-.006

-.015

-.0020

-.008
2

4

6

8

10

12

14

16

18

20

-.020
2

4

6

8

10

12

14

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20

2

4

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8

10

12

14

Impulse Responses to Cholesky One Standard Deviation Innovation to Delinquencies
Response of DELHP to DELHP

Response of DELHP to RR.20

Response of DELHP to DP

.08

.12

.06

.08

.04

.04

.02

.00

.00

-.04

-.02

-.08

.16
.12
.08
.04
.00
-.04
-.08
-.12
-.16

-.04
2

4

6

8

10

12

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16

18

20

-.12
2

4

6

Response of DELHP to QQHP

8

10

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18

20

2

4

6

Response of DELHP to CCHP

.12

10

12

14

Response of DELHP to IHHP

.12

.08

8

.15

.10

.08

.04

.05
.04

.00

.00
.00

-.04

-.05
-.04

-.08

-.10

-.08

-.12
2

4

6

8

10

12

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18

20

-.15
2

4

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8

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18

20

2

4

6

8

10

12

14

Impulse Responses of Delinquencies to Cholesky One Standard Deviation Innovations in
DELHP, RR, DP, QQHP, CCHP and IHHP
Figure 1: VAR Evidence
Notes: VAR estimated from 1980Q1 to 2009Q4. The dashed lines indicate the +/- one standard error bands.
The Choleski ordering is DELHP, RR, DP, QQHP, CCHP, IHHP. Vertical axis: percent deviation frombaseline.

4

between the adjustable mortgage rate and the risk-free rate, increase significantly. Householdswith mortgages are particularly hurt. Borrowers experience a deterioration of their financial
situation and tighter credit conditions, which force them to reduce their mortgages and cut
non-durable consumption and housing investment. Aggregate non-durable consumption andtotal output in the economy fall. The relative price of houses falls but it quickly rebounds,
driven by the fall in non-durable prices. An increase in idiosyncratic housing investment risk
generates a recession but it fails to generate a large and persistent fall in housing prices.
We consider two economies that differ only in mortgage risk, namely the volatility of idiosyncratic housing investment risk. The economy with the lower volatility has a lower steady-state
rate of default on mortgages. As a result, mortgages are larger and the economy is more leveraged. Economies with higher leverage ratios display more redistribution and more polarized
responses to aggregate shocks, as the two households stand at the opposite sides of the lending
contract. When mortgage risk increases, Borrowers are hurt more and the effects of the credit
crunch are amplified. Hence, more leveraged economies suffer deeper recessions.
We analyze the dynamic response to an increase in mortgage risk under alternative specifications of the interest rate rule that governs monetary policy. More inertial rules feature lessaggressive interest rate reductions in response to a mortgage risk shock that ultimately lead todeeper output contractions in the non-durable sector and in the economy in the short run. A
non-inertial interest rate rule is successful in smoothing out the contraction in the non-durable
sector. Hence, interest rate flexibility is important for responding to a mortgage risk shock.
The rest of the paper is organized as follows. Section 2 discusses the related literature andsection 3 presents the model. Section 4 presents two calibrations, benchmark and low leverage,
and analyzes the difference in the steady state of the two economies. Section 5 analyzes thetransmission mechanism in response to a mortgage risk shock in the benchmark and in the
Low-leverage economy. Section 6 considers monetary policy; section 6.1 analyzes the dynamic
response of the model to a monetary shock and section 6.2 analyzes the role of interest rate
flexibility. Section 7 summarizes and suggests future directions for research.

5

2

Related Literature

A growing literature has been incorporating housing in economic models. Iacoviello (2005)
builds on Kiyotaki and Moore (1997) to model an economy with two households with different
discount factors and housing as a durable good. To ensure the existence of an equilibrium,
Iacoviello (2005) features an exogenous borrowing constraint according to which impatient
agents can borrow a fraction of the expected discounted future value of their houses. Loans arealways repaid in this model. Iacoviello and Neri (2010) extend the work of Iacoviello (2005)
and write a DSGE model with housing that is estimated using U.S. data for the period 1965Q1
to 2006Q3. Calza, Monacelli and Stracca (2009) analyze how the transmission mechanism ofmonetary shocks in a housing model á la Iacoviello is affected by alternative values of the downpayment rate and the interest rate mortgage structure. Monacelli (2009) documents positive
co-movement in durable and non-durable consumption in response to a monetary policy shockand shows that a DSGE model with an exogenous borrowing constraint is consistent with theempirical evidence. Our housing model draws a number of features from these contributions.
As in Iacoviello (2005) and Iacoviello and Neri (2010), we build a model with two householdgroups and housing as a durable good. The novelty of our paper is to introduce idiosyncraticrisk in housing investment and the possibility for loans to be defaulted on, which results in an
endogenous borrowing constraint.
The literature on the financial accelerator is vast. Starting with Bernanke, Gertler andGilchrist (1999) and then Carlstrom and Fuerst (1997), many papers have introduced thiscredit friction in DSGE models to analyze its effects on the transmission of shocks. We do notpresent an exhaustive review of this literature here but rather focus on some recent applications
that are most relevant for our study. Christiano, Motto and Rostagno (2009) augment a standard monetary DSGE model to include financial markets and a financial accelerator and fit themodel to European and U.S. data. Cohen-Cole and Martinez-Garcia (2008) consider a modelwith a financial accelerator as in Bernanke et al. (1999) and introduce systemic risk, namely anaggregate variable that affects the variance of idiosyncratic risk, and banking regulation. Our
paper is the first, to our knowledge, to introduce a financial accelerator originating from default
on mortgages.
Dellas, Diba and Loisel (2010) add a banking sector to an otherwise standard New-Keynesian6model and consider some financial shocks, among them an increase in the exogenous rate ofdefault of firms on bank loans. They argue that price stability is all that matter for monetary
policy, even when financial factors are present. Our work differs from Dellas et al. (2010) ina number of ways. Our model features a housing sector but no financial sector; default on
loans is endogenous in our model while it is exogenous in Dellas et al. (2010). Iacoviello (2010)
introduces a banking sector in a model with housing and studies an exogenous shock to howmuch borrowers repay. This repayment shock is exogenous and different from default because
borrowers do not lose their houses following a negative repayment shock.
We analyze the dynamic response of the model to an increase in idiosyncratic housing investment risk. Other papers model an increase in risk. Christiano et al. (2009) introduce a financial
accelerator as in Bernanke et al. (1999) in a model of the financial sector and analyze an increase in the standard deviation of idiosyncratic risk in loans to entrepreneurs. In our setting,idiosyncratic risk is in mortgage loans. Cohen-Cole and Martinez-Garcia (2008) also consider
idiosyncratic risk in loans to entrepreneurs and introduce systemic risk as a correlation between
the mean of the idiosyncratic risk and the aggregate shock, as in Faia and Monacelli (2007).
We do not consider systemic risk but allow for exogenous shocks to the standard deviation of
the distribution of idiosyncratic risk.

3

The Model

Our starting point is a model with patient and impatient households that consume non-durable
goods and housing service and work. Many features of our model draw from the housing modelof Iacoviello (2005), Iacoviello and Neri (2010) and Monacelli (2009). Our focus, however, ison the mortgage contract and on how its features matter for the transmission of shocks. We
do not rely on an exogenous borrowing constraint but rather derive it endogenously from thelenders’ participation constraint after explicitly introducing idiosyncratic risk and default.

3.1

Households

The economy is populated by a continuum of households distributed over the [0, 1] interval.
A fraction ψ of identical households has discount factor β while the remaining fraction 1 − ψ

7has discount factor γ > β. We are going to refer to the households with the lower discountfactor as Borrowers, as these households value current consumption relatively more than theother agents and therefore want to borrow. We are going to refer to households with the higher
discount factor as Savers.

Borrowers
Borrowers have a lifetime utility function given by

max

∞
X

β t E0 {U (Xt , NC,t , NH,t )}

0<β<1

(1)

t=0

where NC,t is hours worked in the non-durable sector, NH,t is hours worked in the housing
sector, and Xt is an index of non-durable and durable consumption services defined as
h
i η
η−1
η−1 η−111
Xt ≡ (1 − α) η Ct η + α η Ht+1 η,(2)where Ct denotes consumption of non-durable goods, Ht denotes consumption of housing services, α is the share of housing in the consumption index and η ≥ 0 is the elasticity of substitution between housing and non-durable services. We assume that housing services in periodt are equal to the housing stock at the beginning of period t. Assuming that services are a
fraction of the stock is not going to change our results qualitatively.
We assume the following utility function:
1+ϕ

U (Xt , Nt ) ≡ ln Xt −iν h 1+ξ
1+ξ 1+ξNC,t + NH,t,1+ϕϕ, ξ ≥ 0.

(3)

Our specification for the disutility of labor follows Iacoviello and Neri (2010) in allowing thathours in the non-durable and housing sector are imperfect substitutes, as consistent with theevidence found by Horvath (2000). For ξ = 0 hours in the non-durable and housing sector are
perfect substitutes. On the other hand, positive values of ξ result in wages not being equalized
in the two sectors and the substitution of hours across sectors in response to wage differentials
being reduced. The parameter ϕ is the inverse of the Frisch labor supply elasticity.

8Borrowers are subject to the sequence of budget constraints:
PC,t Ct + PH,t Ht+1 + [1 − F (ω̄t )](1 + RZ,t )Lt = Lt+1 + WC,t NC,t + WH,t NH,t +(1 − δ) [1 − G(ω̄t )] PH,t Ht ,(4)

where PC,t is the price of non-durable goods, PH,t is the price of housing, Lt+1 are the loans
taken from Savers at t to be repaid in period t + 1. RZ,t is the state-contingent interest ratethat non-defaulting Borrowers pay at t on the loans Lt taken at time t − 1 and Wj,t is thenominal wage in sector j = C, H. In equilibrium some loans are going to be defaulted on.
The term [1 − G(ω̄t )] represents Borrowers’ housing stock at the end of period t and after afraction of the loans has been defaulted on; [1 − F (ω̄t )] is the fraction of loans that is repaid tolenders. We explicitly derive and explain these two terms later. The housing stock depreciates
at the rate δ. Each household decides non-durable good consumption, housing investment (andconsumption), working hours in the two sectors and loans.
It is worth noticing at this point that the state-contingent interest rate RZ,t is determinedat time t after the realization of shocks and in order to satisfy the participation constraint oflenders, which we explain in detail later. Hence, our mortgage contract is characterized byadjustable interest rates.
Each household consists of many members. The household decides total housing investment
Ht+1 , the state-contingent mortgage rates to be paid next period on the contracts signed thisR ii
to the i-th member, where i Ht+1
di = Ht+1 .
period, and then assigns the housing stock Ht+1i
The i−th member finalizes the mortgage contract connected to the housing stock Ht+1
following

the instructions of the household and manages his housing stock. All members are ex-ante
identical. After the mortgage contract is finalized, the i-th household member experiences aniii
idiosyncratic shock ωt+1
such that ex-post housing is ωt+1
Ht+1
. This assumption captures theiidea that housing investment is risky. The random variable ωt+1
is i.i.d. across members of thesame household and log-normally distributed with a cumulative distribution function Ft (ω),
which obeys standard regularity conditions.3 The mean and variance of ln ω are chosen so that
3

The c.d.f. is continuous, at least once-differentiable, and it satisfies∂ωh(ω)
> 0,
∂ω

9

i
) = 1 at the steady state. This implies that while there is idiosyncratic risk at the
Et (ωt+1iihousehold-member level, there is no risk at the household level and Et (ωt+1
Ht+1) = Ht+1 . We

are going to assume that housing investment riskiness can change over time, namely that thevariance σω,t of ln ω is subject to an exogenous shock and therefore displays time variation. Thei
is observed by the household but can only be observed by lenders after
random variable ωt+1

paying a monitoring cost.
After idiosyncratic shocks are realized, the household decides which loans to repay and which
to default. Loans connected to housing stocks that experienced high realizations of the idiosyni
cratic shock ωt+1
are repaid while loans connected to housing stocks with low realizations aredefaulted on. Let ω̄t+1 be the threshold value of the idiosyncratic shock for which the member
is able to repay the loan at the contractual rate RZ,t+1 , namely
ω̄t+1 (1 − δ)PH,t+1 Ht+1 = (1 + RZ,t+1 )Lt+1 .

(5)iLoans connected to ωt+1
∈ [ω̄t+1 , ∞] are repaid and lenders receive (1 + RZ,t+1 )Lt+1 . On theiother hand, loans connected to ωt+1
∈ [0, ω̄t+1 ] are defaulted. For the defaulted loans lenders

pay a monitoring cost to assess and seize the collateral. It is exactly the presence of monitoringthat induces Borrowers to truthfully reveal their idiosyncratic shock and justifies the incentive
compatibility constraint (5). As in Bernanke et al. (1999), we assume that the monitoring costi
is equal to the fraction µ of the housing value, ωt+1
PH,t Ht+1 , and that the defaulting household

member looses his housing stock. Regarding the defaulting household members we follow theliterature on matching and assume that there is perfect insurance among household members
so that consumption of non-durable goods and housing services are ex-post equal across all
members. Hence, Borrower household members are ex-post identical.
Following Bernanke et al. (1999) we consider a one-period mortgage contract that guarantees
lenders a pre-determined rate of return on their total loans. At time t Savers make total loans
Lt+1 to Borrowers and demand the gross rate of return 1 + RL,t . This rate of return is predetermined at t and non-state contingent. Hence, the time t participation constraint of lenders
where h(ω) is the hazard rate.

10

is given by:Z ∞Z ω̄t+1(1+RZ,t+1 )Lt+1 f (ω)dω, (6)ωt+1 (1−µ)(1−δ)PH,t+1 Ht+1 f (ω)dω +(1+RL,t )Lt+1 =

ω̄t+1

0

where f (ω) is the probability density function of ω. The return on total loans is equal tothe housing stock net of monitoring costs and depreciation of defaulting Borrower members(the first term on the right-hand side of (6)) and the repayment of non-defaulting members(the second term on the right-hand side of (6)). After idiosyncratic and aggregate shocks haverealized, the threshold value ω̄t+1 and the state-contingent mortgage rate RZ,t+1 are determinedso as to satisfy the participation constraint above. Other things equal, an aggregate shockthat raises ω̄t+1 and the rate of default on mortgages generates an increase in the adjustable
rate RZ,t+1 paid by non-defaulting members in order to satisfy the participation constraint (6).
This implies that periods characterized by rising mortgage default rates are also accompanied
by rising mortgage interest rates in our model.
We use one-period debt contracts as in Bernanke et al. (1999) for tractability reasons. In
reality, conventional U.S. mortgages typically have a fixed 30-year term and about 70% of
mortgages have fixed rates, even though this percentage has changed a lot in recent years.
Moreover, subprime mortgages with nontraditional ARM features were at the heart of the recent
crisis. Our model does not consider these alternative mortgage instruments and therefore it
cannot capture their role at the onset of the crisis. Nevertheless, our general equilibrium model
captures the effect of a fall in housing prices on mortgage default rates and its transmission tothe other sector of the economy.
We simplify the Borrower problem as follows. LetZ ω̄t+1G(ω̄t+1 ) ≡ωt+1 f (ω)dω(7)

0

be the expected value of the idiosyncratic shock conditional on the shock being less than orequal to the threshold value ω̄t+1 and let
Z ∞Γ(ω̄t+1 ) ≡ ω̄t+1

f (ω)dω + G(ω̄t+1 )

(8)

ω̄t+1

be the expected gross share of housing value that goes to lenders. Then the participation
11

constraint in terms of non-durable consumption prices can be written more compactly as(1 + RL,t )lt+1 = (Γ(ω̄t+1 ) − µG(ω̄t+1 ))(1 − δ)pH,t+1 πC,t+1 Ht+1 ,(9)where pH,t+1 is the relative price of houses in terms of non-durable consumption and lt+1 arereal loans, namely Lt+1 /PC,t . The loan-to-value ratio is given byΓ(ω̄t+1 ) − µG(ω̄t+1 ),(10)

and it measures the size of the loan (principal plus interests) as a fraction of the housing value
net of depreciation. The loan-to-value ratio also measures the net share of the housing valuethat goes to the lender for repayment.
Following the decision to default at time t, Borrowers are left with the following stock of
housing:Z ∞ωt+1 (1 − δ)PH,t Ht+1 f (ω)dω = [1 − G(ω̄t+1 )] (1 − δ)PH,t Ht+1 ,

(11)

ω̄t+1

where the second equality makes use of the fact that Et (ωt+1 ) = 1. This is the expression usedin the Borrowers budget constraint (4).
Using the relationship between ω̄t+1 and RZ,t+1 we can eliminate RZ,t from the Borrowerbudget constraint and rewrite it in real terms as
Ct + pH,t Ht+1 + (1 + RL,t−1 )lt= lt+1 + (1 − δ) [1 − µG(ω̄t )] pH,t Ht + wC,t NC,t + wH,t NH,t , (12)πC,twhere πC,t is non-durable-good inflation and wC,t , wH,t are real wages in the C and H sector,
respectively, in terms of PC,t . Borrowers maximize (1) subject to the budget constraint (12)and the participation constraint (9) with respect to the variables Ct , Ht+1 , NC,t , NH,t , lt+1 , ω̄t+1 .
The respective first-order conditions are:
UC,t − λBC,t = 0,(13)

UH,t+1 − λBC,t pH,t + β(1 − δ)Et {[1 − µG(ω̄t+1 )] pH,t+1 λBC,t+1 +(14)

λP C,t+1 pH,t+1 πC,t+1 [Γ(ω̄t+1 ) − µG(ω̄t+1 )]} = 0,12

UNj ,t + λBC,t wj,t = 0 j ∈ [C, H]
λBC,t+1
λBC,t − (1 + RL,t )Et λP C,t+1 + β= 0,πC,t+1−βλBC,t+1 µG0 (ω̄t+1 ) + λP C,t+1 πC,t+1 [Γ0 (ω̄t+1 ) − µG0 (ω̄t+1 )] = 0,

(15)
(16)
(17)

where λBC,t is the Lagrangian multiplier on Borrowers budget constraint (12) and λP C,t+1 is the
Lagrangian multiplier on the participation constraint (9). Notice that the first-order condition
with respect to ω̄t+1 is state-by-state and not in expected terms.

SaversWe denote Savers’ variables with a ˜, except for loans. Savers maximize lifetime utility
∞
n
o
X
t
e
e
e
max γ E0 U (Xt , NC,t , NH,t )0<β<γ<1(18)

t=0

et is defined similarly to (2). We assume for Savers the same utility function as forwhere X
Borrowers. Savers maximize lifetime utility subject to the sequence of budget constraints:eet + pH,t H
e t+1 + e
e t + (1 + RL,t−1 ) lt + wC,t N
eC,t + wH,t N
eH,t + ∆e t,
Clt+1 = (1 − δ)pH,t H
πC,t

(19)

e t are profits in the intermediate goods sector, which are taken as given.
where ∆et , He t+1 , N
eC,t , NeH,t , eSavers maximize (18) subject to the budget constraint (19) with respect to C
lt+1 .
The first-order conditions, respectively, areeUC,te − λBC,t = 0,

(20)

h
i
eBC,t pH,t + γ(1 − δ)Et λeBC,t+1 pH,t+1 = 0,
UH,t+1−λe(21)j ∈ [C, H],"#eBC,t+1
λeBC,t + γ(1 + RL,t )Et−λ= 0,
πC,t+1(22)

eBC,t wj,t = 0
UNe ,j,t + λeBC,t is the Lagrangian multiplier on Savers budget constraint (19).
where λ

13

(23)

3.2

Firms and Technology

Both the non-durable C and the housing H sector have intermediate and final good producers.

Final Good Producers
Final good producers are perfectly competitive and produce Yj,t , j = C, H. The technology inthe j−th final good sector is given by
Z 1
Yj,t (i)Yj,t =εj −1
εjj ε ε−1
j

di

,

(24)

0

where εj > 1 is the elasticity of substitution among intermediate goods in sector j. Standard
profit maximization implies that the demand for intermediate good i is given by
Yj,t (i) =Pj,t (i)Pj,t−εjYj,t , ∀i(25)where the price index isZ 1
Pj,t (i)Pj,t =

1−εj

1
 1−ε

j

di

.

0

Intermediate Good Sectors
There are two intermediate good sectors j ∈ [C, H] and in each intermediate sector there is acontinuum of firms each producing a differentiated good i ∈ [0, 1]. These firms are monopolistically competitive. We assume that intermediate good firms readjust their price according toa Calvo-type mechanism. Hence, in any given period a firm in sector j may reset its price withprobability 1 − θj .
Intermediate good firm i produces according to the following production function
i ς
h 1
ς−1
ς−1 ς−11eςςςς+ (1 − ζ) Nj,t (i),Yj,t (i) = Aj,t ζ Nj,t (i)(26)

ej,t (i) are the two
where Aj,t is the stochastic level of technology in sector j and Nj,t (i) and N
labor types respectively supplied by Borrowers and Savers. ζ ∈ (0, 1) is the labor share of
Borrowers in the production function and ς > 0 is the elasticity of substitution across labor14inputs. For simplicity these two parameters are assumed to be equal across sectors.
∗
A firm i re-optimizing in period t chooses labor and its nominal price Pj,t(i) so as to maximize

the expected discount sum of nominal profits over the period during which its price remains
unchanged. Thus, the maximization problem for firm i is given by

max

Et

(∞
X

∗ (i),NePj,tj,t+k|t (i),Nj,t+k|t (i)θjk Λt,t+kh∗
fj,t+k N
ej,t+k|t (i)Pj,t(i)Yj,t+k|t (i) − Wj,t+k Nj,t+k|t (i) − W

k=t

(27)

+mcj,t+k|t (i)Pj,t


h 1
i ς
ς−1
ς−1 ς−11
ej,t+k|t (i) ς
Aj,t ζ ς Nj,t+k|t (i) ς + (1 − ζ) ς N− Yj,t+k|t (i),where Yj,t+k|t (i) denotes output in period t + k for a firm i that last changed its price in period
ej,t+k|t (i). mcj,t+k|t (i) is the real marginalt. A similar interpretation applies to Nj,t+k|t (i) and N
cost of firm i.
In (27) the demand and the stochastic discount are respectively given by
Yj,t+k|t (i) =

∗
(i)
Pj,t
Pj,t+k−εjΛt,t+k ≡Yj,t+k ,eBC,t+k
γkλ
.
eBC,t
λ

ej,t+k|t (i) are
The first-order condition relative to is Nj,t+k|t (i) and N
1

1

1−Wj,t+k + mcj,t+k|t (i)Pj,t+k Yj,t+k|t (i)− ς ζ ς Nj,t+k|t (i)− ς = 0,
1

1

(28)

1

ej,t+k|t (i)− ς = 0,
fj,t+k + mcj,t+k|t (i)Pj,t+k Yj,t+k|t (i)− ς (1 − ζ) ς N
−W

(29)

which state that the nominal marginal cost equals the ratio of the nominal wage to the marginal
product of each type of labor input. By rearranging (28) and (29) we obtain:
mcj,t+k|t (i) =

h
i 1
1
1−ς
f 1−ς 1−ς .
ζWj,t+k+ (1 − ζ)Wj,t+k
Aj,t+k Pj,t+k(30)

According to (30), mcj,t+k|t (i) = mcj,t+k . Marginal costs are equal across firms because wages
are the same across all firms.
The first-order condition relative to the price is given by

Et

(∞
X
k=t

θjk Λt,t+k

"

∗
(i)
Pj,t
Pj,t+k(−εj −1)Yj,t+k

15

∗
Pj,t
(i)
ε
−
mct+k
Pj,t+k
ε−1

#)
= 0.

(31)

Finally, it can be shown4 that, under Calvo price setting, the optimal price set by re-optimizing
firms is linked to the aggregate price behavior by the following condition:


∗
Pj,t
Pj,t

(1−ε)ε−1
1 − θj πj,t=,
1 − θj(32)

where πj,t denotes gross inflation in sector j prices.

3.3

Monetary Policy

We assume that monetary policy follows a Taylor-type rule for the nominal interest rate:
φr
hi1−φr  1 + R
1 + RL,tL,t−1
φπ,= AM,t πC,t
1 + RL
1 + RLφπ > 1, φr < 1.

(33)

where RL is the steady-state nominal interest rate, φπ is the coefficient on the inflation target,
φr is the coefficient on the lagged interest rate, and AM,t is a monetary policy shock. In ourbenchmark calibration monetary policy targets inflation in the non-durable goods sector andimplements interest-rate smoothing. Assuming that monetary policy targets inflation in both
sectors does not affect our results.

3.4

Market Clearing

Equilibrium in the non-durable goods market requires that production of the final non-durable
good net of adjustment costs equals aggregate demand:
et .
YC,t = ψCt + (1 − ψ)C(34)

Similarly, equilibrium in the housing market requireshi
e t+1 − (1 − δ)Het .
YH,t = ψ [Ht+1 − (1 − δ)(1 − µG(ω̄t ))Ht ] + (1 − ψ) H
4

For a formal proof see for instance Woodford (2003).

16

(35)

Equilibrium in the labor market requires
Z 1
eC,t ,
LC,t (i)di = ψNC,t + (1 − ψ)N(36)

eH,t ,
LH,t (i)di = ψNH,t + (1 − ψ)N

(37)

0

Z 1
0

while the equilibrium in the credit market requiresψlt = (1 − ψ)e
lt .

(38)

Yt = YC,t + ph,t YH,t .

(39)

We define total output as

Notice that our measurement of total output reflects variations in the relative price of housing.
National account statistics, on the other hand, measure GDP at constant relative prices.

3.5

Exogenous Shocks

There are five exogenous shocks in our model. Aggregate productivity in the two sectors andthe monetary policy shock evolve according to the following first-order autoregressive processesln AC,t = ρC ln AC,t−1 + C,t ,ρC ∈ (−1, 1),(40)

ln AH,t = ρH ln AH,t−1 + H,t ,ρH ∈ (−1, 1),(41)

ln AM,t = ρM ln AM,t−1 + M,t ,ρM ∈ (−1, 1),(42)

where C , H , M are i.i.d. innovations with mean zero and standard deviation σC , σH , σM , respectively, and ρC , ρH , ρM are persistence parameters.
As for the idiosyncratic risk in the housing sector, we follow Bernanke et al. (1999) and assume
that ω is distributed log-normally:σ2ln ω ∼ N (− ω , σω2 ).
2

17

(43)

As stated earlier, the mean of the distribution is chosen so that E(ω) = 1 at the steady state.
Later we are going to analyze the case where idiosyncratic risk exogenously increases. To dothis, we assume that the standard deviation of ln ω is itself an exogenous shock subject to afirst-order autoregressive process:lnσω,tσω,t−1= ρσ ln+ σω,t ,
σω
σω

(44)

where σω is an i.i.d. shock with mean zero and finite standard deviation σσω and ρσ is theserial correlation coefficient.

4

Steady-State Analysis

4.1

Benchmark Calibration

The parameters values for our benchmark calibration are specified in Table 1. We follow
Monacelli (2009) in choosing the values for the discount factors for Borrowers and Savers, therate of depreciation for housing and the elasticity of substitution between non-durable goods
and housing services. The Saver’s discount factor γ is set equal to 0.99 and Borrower’s β isset equal to 0.98. We choose an annual depreciation rate for housing of 4 percentage points,
implying δ = 0.01. The elasticity of substitution between non-durable consumption and housingis η = 1, which implies a Cobb-Douglas specification for the composite consumption index Xt .
U.S. private fixed investment in structures, residential and nonresidential, has been on average
5 percent of GDP from 1960 to 2009, but during the period 2000 to 2007 it averaged 8 percent
of GDP. We set the parameter α that measures the share of housing in the consumption bundle
equal to 0.16 so that the housing sector represents 8 percent of total output at the steadystate. The Saver discount factor pins down the steady-state interest rate at RL = 0.0101 on aquarterly basis. This implies an annual interest rate equal of 4.1 percentage points. The inverseof the Frisch elasticity of labor supply ϕ is set equal to one, as in Barsky, House and Kimball(2007) and as typical in the macro literature. As for the parameter ξ that measures the degreeof substitutability between hours worked in the two sectors, we set it equal to 0.871. This is
the appropriate weighted average of the ξ for Borrowers and Savers estimated by Iacoviello and18

Parameter
γ
β
ψ
δ
εC
εH
ς
ζ
ξ
α
ν
η
ϕ
θC
θH
φπ
φr
ρC
ρH
ρM
σω
µ

Value
0.99
0.98
0.5
0.01
7.5
7.5
3
0.5
0.871
0.16
2.5
1
1
0.67
0
1.5
0.9
0.9
0.9
0
0.20
0.12

Description
Discount factor of Savers
Discount factor of Borrowers
Relative size of Borrower group
Rate of depreciation for housing
Elasticity of substitution for C goods
Elasticity of substitution for H goods
Elasticity of substitution across labor inputs
Share of Borrower labor in the production function
Elasticity of substitution across labor types
Share of housing in consumption bundle
Disutility from work
Elasticity of substitution between C and H goods
Inverse of elasticity of labor supply
Calvo probability in C
Calvo probability in H
Taylor-rule coefficient on inflation
Taylor-rule coefficient on past nominal interest rate
Serial correlation of productivity shocks in C
Serial correlation of productivity shocks in H
Serial correlation of monetary policy shocks
Standard deviation of idiosyncratic shocks
Monitoring cost
Table 1: Benchmark Calibration

Neri (2010).
For the degree of price stickiness, we assume that housing prices are fully flexible and set
θH = 0, which is in line with the empirical estimation of Iacoviello and Neri (2010) and theempirical evidence on price stickiness for durable goods. For non-durable goods, θC is set equal
to 0.67 to imply that firms in the non-durable sector change their prices on average every ninemonths. For monetary policy, we set φπ = 1.5 and φr = 0.9, which are standard values in theliterature. The serial correlation of the monetary policy shock is ρM = 0. We assume that the
Borrower and Saver groups have equal size so that ψ = 0.5.
For technology, we follow Calza et al. (2009) and set the elasticity of substitution amongintermediate goods εj equal to 7.5 in each sector. Labor inputs are imperfect substitutes in
production and the elasticity of substitution across Borrower’s and Saver’s labor is ς = 3. We
also assume that the share of Borrower’s labor in the production function ζ is equal to 0.5. The
serial correlation of the productivity shocks in the non-durable and housing sectors are chosen

19to be ρC = 0.9 and ρH = 0.9, respectively.
Table 2 reports the steady-state values for the benchmark calibration. The loan-to-value ratio
is defined in equation (10) and the leverage ratio for Borrowers at the steady state is calculatedasLeverage Ratio =ll + wC Nc + wH NH,

which measures the fraction of total expenses financed by loans, namely consumption of C andH plus loan repayment over loans. The leverage ratio captures the dependence of Borrowers
from external funding.
We choose the variance of the distribution of ln ω to match the pre-crisis delinquency rates
for prime loans. In the fourth quarter of 2006 seriously delinquent mortgages represented 2.21
percentage points of all mortgages.5 We set σω = 0.2 and obtain an annual default rate of 2.34
percentage points. As for the shocks that raise the volatility of idiosyncratic housing risk, we
believe they are persistent but there is no previous work we can rely on. Christiano et al. (2009)
estimate the persistence of the idiosyncratic productivity shock for the United States to be 0.85.
We set ρσ = 0.9. The median foreclosure price for single-family residencies, condominiums, andtownhouses in California in the first half of 2006 was 12% lower than the median market price ofhomes sold within the same area without having been previously foreclosed – see Cagan (2006).
Hence, we set µ = 0.12 and monitoring costs are 12% of the housing value. Interestingly, thisis also the value for monitoring cost in Bernanke et al. (1999). At the steady state, the loanto-value ratio in our benchmark calibration is almost 60 percentage points. This is lower than
75.7 percentage points, which is the average U.S. loan-to-value ratio between 1973 and 2008.
We can raise the steady-state loan-to-value ratio by reducing σω , the riskiness of loans; however
this is going to reduce the steady-state rate of default. For this reason, we prefer to match thedelinquency rate but have a lower loan-to-value ratio than suggested by the data.
At the steady state, the quarterly mortgage rate rate paid by non-defaulting Borrowers is
RZ = 0.0111, which corresponds to an annual rate of 4.51 percentage points. We define theexternal finance premium at t as RZ,t − RL,t , namely the difference between the ex-post statecontingent rate paid by non-defaulting household members and the risk-free interest rate, whichin our setting is equivalent to the pre-determined rate received by lenders on aggregate loans.6
5
6

See the National Delinquency Survey conducted by the Mortgage Bankers Association.
Our definition of external finance premium differs from that in Bernanke et al. (1999), where the premium

20Variable
Consumption, Borrowers*
44.28
Housing Demand, Borrowers*
39.40
Hours Worked in C Sector, Borrowers* 54.30
Hours Worked in H Sector, Borrowers* 54.30
ph Output H/Total Output
8.09
Loans
2.1747
Loan-to-Value Ratio
59.17
Leverage Ratio
80.12
Default Rate on Mortgages†
2.34
External Finance Premium†
0.41
Mortgage Interest Rate†
4.51* Expressed as percentage of aggregate level, e.g. Consumption, Borrowers = ψC/(ψC + (1 − ψ)C̃).
†Annual, percentage points.
Note: The Leverage Ratio is calculated as l/(l + wC Nc + wH NH ).

Table 2: Steady State under the Benchmark Calibration
This premium captures the additional cost that Borrowers must pay for their risky mortgages
relative to risk-free borrowing. At the steady state, the external finance premium is equal
to 0.41 percentage points on an annual basis. We calculate the empirical counterpart to ourexternal finance premium as the difference between the 30-Year Conventional Mortgage Rate7
and the interest rate on the U.S. Treasury 30-Year bonds.8 The average difference between
these two annual interest rates between 1977 and 2009 was 1.5 percentage points. This makes
the finance premium of our benchmark model one percentage point lower than its empiricalcounterpart.

4.2

Low-Leverage Calibration

We additionally consider an economy characterized by higher idiosyncratic risk but otherwise
identical to the benchmark economy described above. For this economy the steady-state standard deviation of the distribution of idiosyncratic housing risk is set to σω = 0.6. All the otherparameters are as in Table 1. Higher idiosyncratic housing risk implies that more household
members are in the left tail of the distribution and that the steady-state rate of default isis the difference between the costs of funds raised externally and the opportunity costs of funds internal to thefirm.
7
See the H-15 Release of the Federal Reserve Economic Data
8
See the Economic Report to the President, Table B73.

21

Variable
Benchmark Low Leverage % Difference
Output C
0.54070.53990.15
Output H
0.14650.1419
3.24
Consumption, Borrowers
0.47890.4887-2.01
Consumption, Savers
0.6026
0.5912
1.93
Housing Demand, Borrowers
11.5421
10.5337
9.57
Housing Demand, Savers
17.7524
17.8431-0.51
Hours Worked, Borrowers in C Sector
0.58790.5789
1.55
Hours Worked, Borrowers in H Sector
0.16170.1549
4.41
Hours Worked, Savers in C Sector
0.49480.5019-1.41
Hours Worked, Savers in H Sector
0.13610.13431.37
Loans
2.17470.7980172.54
Loan-to-Value Ratio*
59.17
24.37
142.80
Leverage Ratio*
80.12
60.01
33.51
Default Rate on Mortgages†
2.36
8.21
-71.22External Finance Premium†
0.41
2.44
-83.20Mortgage Interest Rate†
4.51
6.54
-31.04* Percentage points.
†Annual, percentage points.

Table 3: Benchmark and Low-Leverage Economies: Steady-state Comparison
higher. This raises the external finance premium and reduces mortgage loans, the loan-tovalue ratio and leverage ratio relative to the benchmark economy. We label this calibration as“Low-Leverage” and compare it with the benchmark, “High-Leverage” calibration.
Table 3 reports the steady-state values of some endogenous variables in the benchmark andthe Low-Leverage economy. The last column of Table 3 reports the percentage point differenceof steady-state values between benchmark and Low-Leverage economies. Because loans arelarger when idiosyncratic housing risk is lower, steady-state loans are 173 percentage points
higher in the benchmark economy, the loan-to-value ratio is 143 percentage points higher andthe leverage ratio is 34 percentage points higher. At the same time, the mortgage interest rate
and the external finance premium are considerably lower in the benchmark economy. As aresult, overall economic activity is higher in economies with high leverage ratios.

22

5

Credit Crunch

5.1

Benchmark Calibration

This section analyzes the dynamic response of the model to an unexpected increase in σω,t , thestandard deviation of the distribution of idiosyncratic housing investment risk. This increasein risk wants to capture the situation in which loans are made on the basis of an expecteddistribution for idiosyncratic risk, but the actual distribution turns out to be characterized
by a higher standard deviation. In other words, the riskiness of mortgages changes over time
and these changes are persistent. More broadly, exogenous shocks to σω,t are an admittedly
reduced-form way to capture the entrance in the mortgage market of subprime, or higher risk,
debtors. Notice that an increase in σω,t has two effects on the distribution of ln ω: it increases
the variance and lowers the mean, which results in a leftward shift of the distribution. See
Figure 10 in Appendix B. Since the log-normal distribution does not take negative values, afall in the mean implies a thicker lower tail of the distribution. Thus, for the same value of ω̄,a higher standard deviation implies a higher cumulative distribution function and therefore ahigher default rate on mortgages.
From the fourth quarter of 2006 to the first quarter of 2010 the delinquency rate on U.S. realestate loans increased from 2.21 to 10.44 percentage points, an increase of 823 basis points.9
Figure 2 shows the impulse responses of six selected variables to a 40% increase in the standard
deviation σω,t . The size of the shock is chosen so as to generate an increase in the default rateof about 800 points. The impulse responses to the same shock but of a larger set of variables
are shown in Figure 6 in Appendix B. The default rate increases sharply and, together withit, the monitoring costs. Since the mortgage contract guarantees the risk-free interest rate
to aggregate loans, the state-contingent mortgage interest rate must increase to satisfy thelenders’ participation constraint, which implies an increase in the external finance premium
paid by Borrowers. Our model predicts a 150 basis points increase for the mortgage interest
rate and the external finance premium. Borrowers financial conditions worsen significantly.
More Borrower household members default on their loans and loose their housing stock, while
the non-defaulting household members pay a higher mortgage interest rate. The overall effect9

See the National Delinquency Survey of the Mortgage Bankers Association.

23

24

10

15

5

10

15

-0.1

5

5

10

Default Rate

10Output

15

15

-50

-40

-30

-20

-10

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

5

5

10

Loans

10

Output H

15

15

Figure 2: Impulse Responses to a 40% Increase in σω,tNote: Default rate is annual and in percentage points. Loans are difference from steady state, multiplied by 100. All other variables are percentage
point deviation from steady state.

2

3

4

-0.2

5

0

6

7

8

0.2

0.1

9

10

0.3

0.4

Rel Price H

-1.2

-1

5

-1

-0.8

-0.6

-0.4

-0.2

0

-0.8

-0.6

-0.4

-0.2

0

Aggr Consumption

on the budget constraint of the Borrower household is large and negative because the mortgage
interest rate paid by non-defaulting household members raises enough to cover the increase inmonitoring costs paid by the lenders – see the participation constraint (6). In addition to this,Borrowers experience a tightening in credit conditions due to a reduction in the loan-to-value
ratio that reduces their capacity to borrow out of their housing stock. Worsening financial
conditions force Borrowers to cut consumption of non-durable goods, housing investment, andloans and to raise hours worked. Savers, on the other hand, are consumption smoothers. In
response to a lower real interest rate they reduce lending and consume more non-durable goods.
However, Savers’ increase in consumption is small relative to the large cut by Borrowers andaggregate consumption of non-durable goods falls.
The non-durable good sector experiences a sharp fall in demand that reduces output andprices. Because prices are sticky in the C sector and cannot fully adjust, the output reduction
is significant and more persistent relative to the H sector, where prices are fully flexible. Nevertheless, inflation of non-durable goods falls. Because monetary policy responds to inflation in
the C sector, lower inflation in C causes a reduction in the nominal interest rate. The housing
sector is also hit by a reduction in demand stemming from Borrowers’ housing downsizing,
which leads to a fall in housing output.10 Nominal housing prices are flexible and they adjust instantaneously, causing nominal deflation in housing. As for the relative price of houses,
namely the price of housing in terms of non-durable goods, it falls on impact because nominal
house prices fall more than nominal non-durable good ones. A lower relative price of housing
raises the demand for housing by Savers, who substitute out of loans into housing to smooth
their inter-temporal consumption profile. Housing prices bottom out instantaneously and growafterward returning to their steady-state value, whereas non-durable prices continue falling fortwo more periods. Two factors contribute to this difference. First, non-durable good prices are
sticky while housing prices are flexible; second, housing supply expands faster than the supply
of non-durable goods. Borrowers need to replenish their housing stock and this contributes toa quick rebound of housing demand and supply. In fact, housing output gross of monitoring
costs (not shown in the graphs) increases following the shock.
Because of imperfect substitutability of hours in the utility function and in the production
function, wage differentials emerge in our model. Wage for Borrowers in the housing sector
10Output H is measured net of monitoring costs.

25

falls but less than his wage in the non-durable sector. This wage differential encourages partial
switching of hours by Borrowers out of the non-durable into the housing sector and, as a result,
hours supplied by Borrowers in the housing sector increase. Wages fall in the C sector for bothgroups and remain below steady state. Borrowers, who are constrained, still increase hours in
the C sector, which lowers their wage in the sector well below those of Savers. As a result, theoverall wage index for Borrowers falls more than the wage index for Savers. Total output in theeconomy falls, independently of whether housing output is measured net or gross of monitoring
costs. For consistency, our graphs report output net of monitoring costs. The non-durable
sector represents more than 90% of the economy and its dynamics drives that of total output.
In the model of Barsky et al. (2007), a shock that brings down the non-durable sticky-pricesector generates an offsetting response in the durable flexible-price sector and GDP is left
unchanged, independently of the relative size of the durable good sector. Our model sharessome features with that of Barsky et al. (2007), in particular a durable flexible-price sector anda non-durable sticky-price one. The response to a monetary shock (analyzed in the next section)shows hints of a similar behavior. Nevertheless, our results differ. In our model, total output
and GDP fall11 in response to an increase in housing investment risk independently of whether
we measure output in the durable sector net or gross of monitoring costs. In our model labor
does not move across sectors so as to close wage differentials due to imperfect substitutability
both on the supply and demand side. In addition to that, the Borrower household in our model
is constrained and fails to behave as a consumption smoother. In particular, Borrower’s housing
demand is driven by the fact that housing can be used as collateral for borrowing, which makes
the shadow value of housing different between Savers and Borrowers.
Our model generates a 1.2 percentage point fall of total output from steady state and a smalland short-lived reduction in real housing prices following an increase in housing investment
risk. In both these dimensions the model fails to replicate the data. The rapid rebound in
housing demand due to the replacement of monitoring costs seems particularly counterfactual.
We speculate that adding a housing rental market may improve the fit of our model. We
also believe that the credit crunch that followed the liquidity crisis has been quantitatively
important in the recession of 2008. Our model, however, has no capital and therefore no role11

We do not report GDP measured as total output at steady-state relative prices. Nevertheless, GDP would
fall in our model.

26

for financial intermediation for providing capital to firms. Better modeling of the financial
sector and a financial accelerator in loans to firms are indeed interesting extensions of thisframework. Nevertheless, our model captures well the transmission of shocks across sectors, as
the non-durable sector is significantly affected by a risk shock in the durable sector.

5.2

Low-Leverage Calibration

Next we analyze how differences in steady-state leverage ratios imply different dynamic responses to a housing investment risk shock. Figure 3 plots the impulse responses of six selectedvariables to a 40% increase in σω,t for the Low-Leverage economy (red line with diamonds) andfor the benchmark economy (blue starred line). Figure 7 in Appendix B plots the responses
for a larger set of variables. The effects of an increase in the idiosyncratic risk in housing investment are amplified in the highly-leveraged benchmark economy, which is the economy withthe lower steady-state standard deviation and default rate. In particular, the credit crunch is
deeper and the adverse effects on Borrowers stronger. Loans fall more in the High-Leverage
economy, which implies that Borrowers must de-leverage more in these economies. As a result,
Borrowers consumption of non-durable goods and housing fall more in the economy with highleverage; aggregate consumption and prices of non-durable goods also fall more. The deeper
contraction in non-durable demand generates a deeper reduction in total output.

6

Monetary Policy

6.1

Monetary Policy Shock

Figure 4 illustrates the impulse responses of the model under the benchmark calibration in
response to a monetary shock, namely a 25 basis point increase in the nominal interest rate
RL,t . Figure 8 in Appendix B displays the responses of a larger set of variables. Savers, who are
consumption smoothers, reduce consumption of non-durable goods in response to higher interest
rates. Borrowers are adversely affected in two ways. First, they experience an increase in thecost of borrowing. Second, deflation in non-durable goods raises their real debt via the Fisher
effect. As a result, Borrowers reduce mortgage loans and consumption of non-durable goods
and housing services. Borrowers also raise internal funds by increasing their labor supply in the
27

28

10

15

5

10

15

10

-0.1

-0.3
0

5

15

0

-0.2

20

25

30

0.1

0.2

0.3

0.4

Rel Price H

-1.2

-1

5

-1

5

10

15

5

10

Default Rate

15

-50

-40

-30

-20

-10

-4

-3.5

-3

-2.5

-2

-0.6
-0.8

-1.5

-1

-0.5

0

5

5

10

Loans

10

Output H

15

15

Figure 3: Impulse Responses to a 40% Increase in σω : Low-Leverage Calibration

Note: Default rate is annual and in percentage points. Loans are difference from steady state, multiplied by 100. All other variables are percentage
point deviation from steady state.

σω=0.6σω=0.2

Output

-0.4

-0.2

0

-0.8

-0.6

-0.4

-0.2

0

Aggr Consumption

housing sector, which depresses their real wage in the housing sector fall and thereby the relative
price of houses. As the real housing prices falls, more Borrower household members default.
Technically, ω̄t , the threshold value of the idiosyncratic shock below which household members
do not repay their mortgages, goes up and with it the default rate and monitoring costs. Since
fewer Borrower household members repay their loans, the state-contingent mortgage interest
rate rises on impact. The nominal interest rate RL,t , which is the risk-free rate guaranteed
to Savers on outstanding mortgages to be repaid at time t, was set at time t − 1 and cannotchange on impact. As a result the external financial premium jumps up on impact. The
mortgage interest rate increases more than the nominal interest rate and the external financepremium remains above steady state. The negative wealth effect stemming from higher default,
higher mortgage rates and lower housing value makes Borrowers further cut their consumptionof non-durable goods. Aggregate consumption and production of non-durable goods fall. The
fall in relative housing prices raises housing demand by Savers, who substitute out of loans
into housing. Monitoring costs reduce the existing stock of housing, thereby raising housing
demand. As a result, production in the housing sector increases slightly.
A monetary policy shock causes changes of output of opposite sign in the two sectors, along
the lines of Barsky et al. (2007). However, the expansion in the construction sector is small
and the contraction in the non-durable sector more than compensates it and total output falls.

6.2

The Importance of Interest Rate Flexibility

Our benchmark calibration features a large inertia term in the interest rate rule. Here weanalyze the implications of adopting less inertial interest rate rules.12 Figure 5 shows theimpulse responses of six selected variables to a 40% increase in the standard deviation σω,t , thesame shock analyzed in section 5, under alternative values for the inertial term in the interest
rate rule. The starred blue line is the benchmark specification where φr = 0.9; the red line withdiamonds is the specification for φr = 0.5; the light blue line with triangles features no interestrate smoothing, φr = 0. For all specifications we keep the coefficient on inflation constant andequal to φπ = 1.5. Figure 9 in Appendix B reports the response of a larger set of variables.
The negative contemporaneous response of the nominal interest rate is dampened down under
12

The previous version of this paper also considered interest rate rules that respond to output. Since thedifferences are small, we decided to drop the analysis of the response to output in the current version.

29

30

-1.2

-1.2

-3

-4
-4.5

2.6
2.5
2.4

-0.4

-0.6

-0.8

-1

15

-2.5

2.7

-3.5

-2

5

5

10

Loans

10

Output H

15

15

Figure 4: Impulse Responses to a 25 basis points Monetary Shock

Note: Default rate is annual and in percentage points. Loans are difference from steady state, multiplied by 100. All other variables are percentage
point deviation from steady state.

10

-1.5

2.8

-0.2

-0.5
-1

0

0

0

10

15

2.9

Default Rate

10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

3

5

5

Output

0.2

15

-1

-1

5

-0.8

-0.8

Rel Price H

-0.6

-0.6

15

-0.4

-0.4

10

-0.2

-0.2

5

0

0

Aggr Consumption

31

5

10

15

-0.2

0

0.2

0.4

0.6

5

10

Rel Price H

15

2

3

4

5

6

7

8

9

10

-1.2

-1

5

5

10

Default Rate

10Output

15

15

-50

-40

-30

-20

-10

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

5

5

10

Loans

10

Output H

15

15

Figure 5: Impulse Responses to a 40% Increase in σω and Interest Rate Inertia

Default rate is annual and in percentage points. Loans are difference from steady state, multiplied by 100. All other variables are percentage point
deviation from steady state.

φr=0

φr=0.5φr=0.9

-1

-0.8

-0.6

-0.4

-0.2

0

-0.8

-0.6

-0.4

-0.2

0

Aggr Consumptioninertial interest rate rules. While the interest rate falls less than 60 basis points when φr = 0.9,
it falls more than 300 basis points when φr = 0. The different response of the nominal interest
rate has important implications for non-durable consumption. Under inertial interest rate rules,
Borrowers cut their non-durable goods consumption more and Savers increase it by less, thereby
making the negative response of aggregate consumption stronger. Since the nominal interest
rate falls less under inertial rules, Borrowers enjoy a smaller reduction in their interest rate
payments and therefore reduce non-durable consumption more and raise hours worked more,
especially in the construction sector. Higher real interest rates under inertial rules relative
to non-inertial ones also make Savers raise their non-durable consumption less. As a result,output in the non-durable sector as well as total output display a deeper contraction underinertial interest rate rules. Because supply more hours in the housing sector when φr is high,wages in the sector fall more and housing prices are more depressed. This explains why thereal price of houses falls more with inertial rules. Interestingly, Housing demand, the default
rate, monitoring costs, the external finance premium, the loan-to-value ratio and loans display
almost identical responses under different degree of interest rate smoothing.
Interest rate rules featuring no inertia display a large degree of interest rate flexibility. The
nominal interest rate is cut aggressively in response to higher volatility of the idiosyncratic
housing investment risk so that the recession in the non-durable sector is significantly softened.
On the other hand, inertial interest rate rules feature small interest rate responses that deepen
the recession. In a sense, strong inertial rules mimic a zero bound scenario where interest ratecannot be lowered further and the negative effects of a risk shock are amplified.

7

Conclusions

This paper introduces endogenous default in DGSE model with housing and analyzes the dynamic response to an increase in the volatility of the distribution of idiosyncratic housing
investment risk, i.e. mortgage risk. We calibrate the size of the shock so as to generate anincrease in the default rate of the same magnitude as seen in the data. Under the benchmark
calibration, which features a inertial interest rate rule, our model predicts a smaller fall in output, both in the durable and non-durable sector, relative to what we have seen in the data. At
the trough of the recession, per capita U.S. real GDP was below trend by 3% while our model
32

predicts a fall of 1.25% below steady state. Our model predicts a very small reduction in real
housing prices followed by quick rebound. In the data, real house prices have been below baseline since 2008Q3 and were below baseline by more than 7 percentage points in 2009Q1. The
calibration without inertial monetary policy predicts even smaller output reductions, while therelative price of housing increases on impact and falls below steady state only after 5 quarters
after the shock. We speculate that some features and assumptions of our model play a key rolein this respect. We discuss them below.
If we measure output in the housing sector gross of monitoring costs, the housing sector expands following an increase in mortgage risk. This happens because monitoring costs cause
a reduction in the housing stock of Borrowers that they want to replenish quickly. A modelfeaturing a rental market may help reducing strong and counterfactual responses in the construction sector. Alternatively, the housing sector could be characterized by lags in productionor the use of capital that, together with investment adjustment costs, makes changes in output
smoother.
In standard DSGE models with a non-durable sticky price and a durable flexible price sector,
a disturbance that generates a recession in the non-durable sector causes an expansion in the
durable sector that leaves GDP unchanged. While our model is non-standard in a number ofways, the housing sector displays some tendency to move in the opposite direction from the nondurable sector. Imperfect substitutability of hours in the two sectors reduces. In fact, an earlier
version of our model that did not feature imperfect substitutability of hours displayed a large
positive response of the housing sector to an increase in mortgage risk, irrespective of whether
output in the sector was measured net or gross of monitoring costs. We believe, however, thatwage stickiness will also improve bringing the model closer to reality and dampening out theoutput response in the H sector.
Our model features financial intermediation and a financial accelerator only for the purpose of
housing investment. There is no capital in our simple model and therefore no financial intermediation for providing capital to firms. We believe that this specific financial intermediation has
been quantitatively important in the great recession. In particular, we believe that the credit
crunch, i.e. the tightening of lending conditions faced by firms, that followed the liquidity crisishas been an important element of the recent crisis. At the same time, introducing a formalbanking sector with capital requirements is likely to amplify the output effect of financial shocks
33

like the mortgage risk shock considered here.
The mortgage contract in our model is a one-period adjustable rate mortgage contract.
Lenders do not take any risk, as they are guaranteed the risk-free rate on the total amountof loans they give out to Borrowers. Mortgage contracts in reality are more complex than thecontract we analyze here. In particular, fixed-rate multi-year contracts and ARM contracts
with nonstandard features may have played a role in the persistence and amplification of thecrisis.
We believe these are important directions for future research.

34

References
Barsky, Robert B., Christofer L. House, and Miles S. Kimball, “Sticky-Price Models
and Durable Goods,” American Economic Review, 2007, 97 (3), 984–98.
Bernanke, Ben, Mark Gertler, and Simon Gilchrist, The Financial Accelerator in a
Quantitative Business Cycle Framework, Amsterdam: North Holland, 1999.
Cagan, Christopher, “A Ripple, Not A Tidal Wave: Foreclosure Prevalence and Foreclosure
Discount,” First American Real Estate Solutions, 2006, November, 1–17.
Calza, Alessandro, Tommaso Monacelli, and Livio Stracca, “Housing Finance andMonetary Policy,” ECB Working Paper 1069, 2009.
Carlstrom, Charles T. and Timothy S. Fuerst, “Agency Costs, Net Worth, and Business
Fluctuations: A Computable General Equilibrium Analysis,” American Economic Review,
1997, 87 (5), 893–910.
Christiano, Lawrence, Roberto Motto, and Massimo Rostagno, “Financial Factors in
Business Cycles,” Manuscript, 2009.
Cohen-Cole, Ethan and Enrique Martinez-Garcia, “Systemic Risk, Banking Regulation
and Optimal Monetary Policy,” Manuscript, 2008.
Dellas, Harris, Behzad Diba, and Olivier Loisel, “Optimal Policy Responses to Financial
Shocks,” Manuscript, 2010.
Faia, Ester and Tommaso Monacelli, “Optimal Interest Rate Rules, Asset Prices, andCredit Frictions,” Journal of Economic Dynamics and Control, 2007, 31 (10), 3228–54.
Horvath, Michael, “Sectoral Shocks and Aggregate Fluctuations,” Journal of Monetary Economics, 2000, 45 (1), 69–106.
Iacoviello, Matteo, “House Prices, Borrowing Constraints and Monetary Policy in the Business Cycle,” American Economic Review, 2005, 95 (3), 739–764.
, “Financial Business Cycles,” Manuscript, 2010.
35

and Stefano Neri, “Housing Market Spillovers: Evidence from an Estimated DSGE
Model,” American Economic Journal: Macroeconomics, 2010, (forthcoming).
Kiyotaki, Nobuiro and John Moore, “Credit Cycles,” Journal of Political Economy, 1997,(105(2)).
Monacelli, Tommaso, “New Keynesian Models, Durable Goods, and Collateral Constraints,”
Journal of Monetary Economics, 2009, 56 (2), 242–254.
Woodford, Michael, Interest and Prices: Foundations of a Theory of Monetary Policy,
Princeton University Press, 2003.

36

A

Data and Sources

CC : Aggregate Consumption. Real Personal Consumption Expenditure (seasonally adjusted,billions of chained 2005 dollars, Table 1.1.6), divided by the Civilian Noninstitutional
Population (CNP16OV, source: Bureau of Labor Statistics). Source: Bureau of Economic
Analysis (BEA).
IH : Residential Investment. Real Private Residential Fixed Investment (seasonally adjusted,billions of chained 2005 dollars, Table 1.1.6.), divided by CNP16OV. Source: BEA.
DP : Inflation. Quarter on quarter log differences in the implicit price deflator for the nonfarm
business sector, demeaned. Source: Bureau of Labor Statistics (BLS).
RR : Nominal Short-term Interest Rate. 3-month Treasury Bill Rate (Secondary Market
Rate), expressed in quarterly units. (Series ID: H15/RIFSGFSM03 NM). Source: Boardof Governors of the Federal Reserve System.
QQ : Real House Prices. Census Bureau House Price Index (new one-family houses soldincluding value of lot) deflated with the implicit price deflator for the nonfarm businesssector. Source: Census Bureau, http://www.census.gov/const/price sold cust.xls.
DEL : Seriously delinquent mortgages, not seasonally adjusted, percentage of total mortgages.
Source: Mortgage Bankers Association, National Delinquency Survey.

B

Figures

37

38

10

10

15

15

15

10

10

10

15

15

15

5

10

15

Loan-to-Value

5

Output5

Infl H

5

Aggr Consumption

-10
-20
-30
-40
-50

2

4

6

8

10

-2

-1

0

-15

-10

-5

0

10

10

10

5

10

Loans

5

Default Rate

5

Wage B

5

15

15

15

15

Hous Demand B

3.6

3.8

4

4.2

0.2

0.4

0.6

-1.5

-1

-0.5

0

0

5

10

10

10

10

15

15

15

5

10

15

Nom Int Rate

5

Monit Cost

5

Wage S

5

Hous Demand S

Figure 6: Impulse Responses to a 40% Increase in σω,t : Large Set of Variables

Note: Default rate, inflation in the H and C sector, external finance premium, mortgage interest rate and nominal interest rate are annual and inpercentage points. Monitoring costs and loan-to-value ratio are in percentage points. Loans are difference from steady state, multiplied by 100. Allother variables are percentage point deviation from steady state.

10

4.5

0.5
5

54

5

1

58
56

60

-1

-0.5

0

5.5

15

10

Mortgage Int Rate

5

15

-3

-2

-1

0

1.5

10

10

Output C

5

15

-1

-0.5

0

6

5

10

Infl C

5Cons S

2

Ext Fin Prem

-1

0

-3

-2

-1

0

-3

10

15

15

-0.5

5

Output H

5

Rel Price H

5

-0.5

0

0.5

-2

-1

-0.2

0

0.2

0.4

-3

-2

-1

0

Cons B

39

10

5

10

5

10

10

10

15

15

15

5

10

15

Mortgage Int Rate

5

Output C

5

Infl C

5Cons S

20

30

40

50

60

-1

-0.5

0

-3

-2

-1

0

-1

-0.5

0

10

10

10

15

15

15

5

10

15

Loan-to-Value

5

Output5

Infl H

5

Aggr Consumption

-10
-20
-30
-40
-50

0

10

20

30

-2

-1

0

-15

-10

-5

0

10

10

10

5

10

Loans

5

Default Rate

5

Wage B

5

15

15

15

15

Hous Demand B

3.6

3.8

4

4.2

0.2

0.4

0.6

-1.5

-1

-0.5

0

0

5

10

10

10

10

15

15

15

5

10

15

Nom Int Rate

5

Monit Cost

5

Wage S

5

Hous Demand S

Figure 7: Impulse Responses to a 40% Increase in σω,t : Large Set of Variables, Low-Leverage Economy

Note: Default rate, inflation in the H and C sector, external finance premium, mortgage interest rate and nominal interest rate are annual and inpercentage points. Monitoring costs and loan-to-value ratio are in percentage points. Loans are difference from steady state, multiplied by 100. Allother variables are percentage point deviation from steady state.

0

10

5

15

15

-1

-0.5

0

-3

-2

-1

0

10

Ext Fin Prem

5

15

15

-0.5

0

0.5

15

-4

-3

-2

-1

Output H

105σω=0.6

0

10

Rel Price H

5σω=0.2

-0.2

0

0.2

0.4

-3

-2

-1

0

Cons B

40

10

10

10

5

10

Ext Fin Prem5Output H

5

Rel Price H

5Cons B

15

15

15

15

10

4.4

4.6

4.8

5

-1

-0.5

0

10

15

5

10

15

Mortgage Int Rate

5

Output C

60.2
60
59.8
59.6
59.4
59.2

-1

-0.5

0

-6

-3
15

-4

-2

10

-2

-1

5

15

-1

-0.5

0

0

Infl C

5Cons S

0

0.4
0.2
0
-0.2
-0.4
-0.6
10

10

10

15

15

15

5

10

15

Loan-to-Value

5

Output5

Infl H

5

Aggr Consumption

-4

-3

-2

-1

0

2.4

2.6

2.8

3

-2

-1

0

-2

-1.5

-1

-0.5

10

10

10

5

10

Loans

5

Default Rate

5

Wage B

5

15

15

15

15

Hous Demand B

4

4.2

4.4

4.6

0.16

0.17

0.18

0.19

0.2

-2

-1

0

1.2
1
0.8
0.6
0.4
0.2
10

10

10

15

15

15

5

10

15

Nom Int Rate

5

Monit Cost

5

Wage S

5

Hous Demand S

Figure 8: Impulse Responses to a 25 basis points Monetary Shock: Larger Set of Variables

Note: Default rate, inflation in the H and C sector, external finance premium, mortgage interest rate and nominal interest rate are annual and inpercentage points. Monitoring costs and loan-to-value ratio are in percentage points. Loans are difference from steady state, multiplied by 100. Allother variables are percentage point deviation from steady state.

0.45

0.5

0.55

0

0.1

0.2

0.3

0.4

-1

-0.5

0

-2

-1.5

-1

-0.5

0

41

5

10

15

10

15

3
2

0.5
5

10

15

54

56

4

1

58

5

1.5

15

15

-1

-0.5

0

-3
10

15

10

15

5

10

15

Loan-to-Value

5

Output

-10
-20
-30
-40
-50

2

4

6

8

10

-2

0

-2

5

15

-15

-10

-5

0

-1

6

10

10

Mortgage Int Rate

5

15

10

Infl H

5

Aggr Consumption

-1

0

60

5

10

Output C

5

15

-1

-0.5

0

2

Ext Fin Prem

-1

-3

0
-0.5

5

Output H

-3

-2

-1

-0.2

-2

0

10

Infl C

5Cons S

10

10

10

5

10

Loans

5

Default Rate

5

Wage B

5

15

15

15

15

Hous Demand B

1

2

3

4

0.2

0.4

0.6

-1.5

-1

-0.5

0

0

5

10

10

10

10

Figure 9: Impulse Responses to a 40% Increase in σω and Interest Rate Inertia: Large Set of Variables

15

15

15

5

10

15

Nom Int Rate

5

Monit Cost

5

Wage S

5

Hous Demand S

Note: Default rate, inflation in the H and C sector, external finance premium, mortgage interest rate and nominal interest rate are annual and inpercentage points. Monitoring costs and loan-to-value ratio are in percentage points. Loans are difference from steady state, multiplied by 100. Allother variables are percentage point deviation from steady state.

φr=0

φr=0.5φr=0.9

-1

0

0.2

0.4

0.6

Rel Price H

-0.5

-3
15

0

-2

10

0.5

-1

5

1

1.5

0

Cons B42

Probability Distribution

0

0.5

1

1.5

2

2.5

0

0.2

0.6

0.8

ω

1

1.2

Figure 10: Probability Distribution: An Increase in σω,t

0.4

1.4

1.6

1.8

ω

σ =0.28σω=0.22