BIS Working Papers
No 885
Credit supply driven
boom-bust cycles
by Yavuz Arslan, Bulent Guler and Burhan Kuruscu
Monetary and Economic Department
September 2020
JEL classification: E21, E32, E44, E60, G20, G51.
Keywords: credit supply, house prices, financial crises,
household and bank balance sheets, leverage, foreclosures,
mortgage valuations, consumption and output.
BIS Working Papers are written by members of the Monetary and Economic
Department of the Bank for International Settlements, and from time to time by other
economists, and are published by the Bank. The papers are on subjects of topical interest
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necessarily the views of the BIS.
This publication is available on the BIS website ().
© Bank for International Settlements 2020. All rights reserved. Brief excerpts may be
reproduced or translated provided the source is stated.
ISSN 1020-0959 (print)
ISSN 1682-7678 (online)
Credit Supply Driven Boom-Bust Cyclesϑ
Yavuz Arslan Bulent Guler Burhan Kuruscu
September 17, 2020
Abstract
Can shifts in the credit supply generate a boom-bust cycle similar to the one observed in the US around
2008? To answer this question, we develop a general equilibrium model that combines a rich heterogeneous
agent overlapping-generations structure of households who make housing tenure decisions and borrow through long-
term mortgages, firms that finance their working capital through short-term loans from banks, and banks whose
ability to intermediate funds depends on their capital. Using a calibrated version of this framework, we find
that shocks to banks’ leverage can generate sizable boom- bust cycles in the housing market, the banking sector,
and the rest of the macroeconomy, which provides strong support for the credit supply channel. The deterioration
of bank balance sheets during the bust, the existence of highly leveraged households, and the general equilibrium
feedback from the credit supply to household labor income significantly amplify the bust. Moreover, mortgage
credit growth across the income distribution is consistent with recent findings that were otherwise argued to be
against the credit supply channel. A comparison of the model outcomes across credit supply, house price
expectation, and productivity shocks suggests that housing busts accompanied by severe banking crises are more
likely to be generated by credit supply shocks.
JEL Codes: E21, E32, E44, E60, G20, G51.
Keywords: Credit Supply, House Prices, Financial Crises, Household and Bank Balance Sheets, Leverage, Foreclo- sures,
Mortgage Valuations, Consumption, and Output.
ϑWe thank Pedro Gete and Gaston Navarro for their valuable discussions of our paper. We thank Stijn Claessens,
Dean Corbae, Victor Rios-Rull, Hyun Shin, Christian Upper, participants at NBER Summer Institute, SITE, “Hous- ing, Credit
and Heterogeneity: New Challenges for Stabilization Policies” conference co-organized by Swedish Central Bank, Saint Louis
FED, and CEPR, Macro-financial Issues Workshop at the Bank of Canada, and seminar par- ticipants at the Bank of
International Settlements, Indiana, Toronto, Zurich, and the Bank of Canada. Guler and Kuruscu acknowledge financial
support from the Bank of International Settlements, and Kuruscu thanks the Social Sciences and Humanities Research Council
of Canada. The authors also acknowledge the Indiana University Pervasive Technology Institute for providing computing resources
that have contributed to the research results reported within this paper. The views expressed here are those of the authors and not
necessarily those of the Bank for International Settlements. The paper has gone through different phases with different titles
before reaching its current state. The previous versions were presented under the title “Housing Crisis, Deterioration of Bank
Balance Sheets, and Macroprudential Policies” and “Bank Balance Sheets, Boom-Bust Cycles, and Macro-prudential
Policies.” Arslan: The Bank for International Settlements, @; Guler: Indiana University,
bguler@; Kuruscu: University of Toronto, @.
mailto:@
1 Introduction
The housing market in the US (and in many other countries) experienced a dramatic boom-bust cycle
during the last two decades. Real house prices increased by more than 30 percent between 1995 and 2006,
and then dropped by a similar amount between 2006 and 2011. Such a large decline in house prices pushed
many homeowners with mortgages into negative equity, which then increased quarterly foreclosure rates from 1
to 5 percent. Not only the housing market but also the financial sector and the rest of the macroeconomy
struggled: the losses in mortgage related assets weakened bank balance sheets and concerns about the value of
these assets made creditors withdraw from the wholesale funding market, disrupting the credit flow to non-
financial firms and GDP contracted by about 6 percent, employment and consumption declined
around 5 percent.
Several papers have studied the forces behind the boom and the subsequent collapse of the housing
market. One line of research has emphasized the role of the credit supply during the boom These
papers argue that an increase in the loan supply lowers interest rates and increases both credit and house
prices. Similarly, during the bust period, a decline in bank lending to firms has been effective on the
worsening of consumption and employment dynamics (Chodorow-Reich (2013) and Jensen and Johannesen
(2017)). However, another strand of literature argues that shifts in demand driven by changes in expectations
of house prices have been the main force behind the boom-bust cycle (Adelino et al. (2016) and Kaplan et
al. (2020)).
In this paper, we study how far shifts in the credit supply can generate boom-bust cycles in the housing
market, the banking sector, and the macroeconomy, as observed in the US around 2008. For this purpose,
we develop and study a quantitative general equilibrium model that combines three sectors of the
economy that played critical roles during the boom-bust episode: (i) a rich heterogeneous agent
overlapping-generations structure of households who face idiosyncratic income risk under incomplete markets
and make housing tenure decisions, (ii) banks that issue short- term loans to firms and long-term
mortgages to households and whose ability to intermediate funds depends on their capital, and (iii) firms that
finance part of their wage bill (working capital) through short-term loans from banks.
We explicitly model the housing tenure choices of households by allowing them to choose between owning
and renting a house of their desired size. Households can use long-term mortgages for their purchases and
have the option to prepay and refinance. Households can default on the mortgage in any period throughout
the life of the mortgage. As mortgage contracts internalize the default probabilities of households, each
mortgage is individual specific, and borrowing limits endogenously arise via limited commitment by
households.
The key theoretical contribution of our paper is to incorporate this rich mortgage structure into
1See Gertler and Gilchrist (2018) for an excellent review of the crisis and the literature, as well as for evidence on how the
disruption in the banking sector affected overall employment.
2Prominent examples are Mian and Sufi (2009), Shin (2012), Favara and Imbs (2015), Justiniano et al. (2017),
Landvoigt et al. (2015), Garriga et al. (2019), and Garriga and Hedlund (2020).
bank balance sheets. For this purpose, we assume a competitive banking industry with a continuum of
identical banks. Banks fund themselves through international investors and household deposits, and can lend
to firms, issue new mortgages, and invest in existing ones. We assume that bankers can steal a fraction of
assets and default. As a result, to avoid such behavior in equilibrium, lenders limit their funding to banks,
creating an endogenous constraint on bank leverage.
To study the role of shifts in the credit supply during the boom-bust episode, we assume that the
economy is initially in the steady state and calibrate the model to match several US data moments—
most importantly, regarding household and bank balance sheets—in 1995. We then give two subsequent
unexpected leverage shocks to bank balance sheets. First, in 1996, banks start increasing their leverage
gradually over , in 2008, however, the leverage constraint reverts back to its initial steady-state
level. We calibrate the size of the boom shock such that the changes in the banks’ book leverage matches
the data during the boom and study the transition of our model economy in response to these shocks.
The main driver of the boom-bust cycle is the changes in the equilibrium bank lending rate in
response to the credit supply shocks. With two unexpected and offsetting permanent shifts in bank
leverage, the bank lending rate first decreases gradually by percentage points until 2008 (and is
expected to stay at that level permanently) and then unexpectedly reverts back to its initial steady-state level
after a sharp jump (by percent) in 2008 due to a sharp deterioration of bank balance sheets.
The changes in the bank lending rate generate a large boom-bust cycle in the housing market and the
macroeconomy, and a slow recovery from the bust. During the boom, house prices increase around 12 percent
and price-rent ratio increases by 7 percent. As house prices increase and borrowing rates decline, households
borrow more by both lowering their down payments and tapping the refinancing option. As a result,
household debt increases around 35 percent. However, household leverage increases less because of higher
house prices. During the bust, house prices decline by percent on impact, price-rent ratio declines by
15 percent, and the foreclosure rate jumps by percentage points. On the real side of the economy,
output and consumption expand by 3 and 4 percent in the boom and decline by about 5 and 7 percent in
the bust, respectively.
The changes in the bank lending rate affect households both directly via borrowing costs and indirectly
through general equilibrium effects. Most importantly, household labor income increases 4 percent during the
boom and declines more than 9 percent during the bust as firms adjust their labor demand in response to the
changes in the cost of funding and in the aggregate capital stock. Overall, we find that this general
equilibrium effect accounts for about 50 percent of the house price and consumption dynamics, and the
direct effect of the bank lending rate accounts for the rest. These findings underline the importance of
modeling the feedback from the credit supply to labor income.
In the bust period, the credit supply declines not only because of the exogenous tightening of the
Figure 1: Linkages across sectors and amplification channels during the bust
Household Bank
Firm
bank leverage constraint but also because of the endogenous deterioration of bank balance sheets, which
further tightens the leverage constraint and significantly amplifies the bust. Two, sometimes reinforcing,
mechanisms drive the bank balance sheet amplification, as illustrated in Figure 1: (i) changes in mortgage
valuations and (ii) foreclosures. First, when banks cut credit in response to the tightening of the leverage
constraint, the equilibrium bank lending rate increases. But then, mortgage valuations decline and banks’
net worth deteriorates. Hence, banks cut back credit more, which further increases the bank lending rate.
Second, as house prices decline, a significant share of mortgage borrowers find themselves with negative
equity and default. As a result, bank balance sheets worsen because of the rise in foreclosures. We find that
the valuation losses account for more than two-thirds of the decline in bank net worth at the time of the
bust, while the increase in foreclosures accounts for the rest, which is consistent with the evidence
presented in IMF (2009). Overall, these two endogenous mechanisms cause a large but temporary spike in
the bank lending rate, which amplifies the drop in house prices, consumption, and output by 25, 44, and 64
percent, respectively.
The temporary spike in the bank lending rate particularly amplifies the drop in variables that depend on
short-term debt, such as output and labor It does not affect mortgage costs
3Gertler and Gilchrist (2018) provide evidence that the disruption in banking, as in our model, was central to the overall
employment contraction in the data.
Foreclosure Channel
Bank
Lending
Rate r*
Short-term
debt
Capital
Investment
Income
Foreclosures
Labor
Demand
House
Prices
Bank Net
Worth
significantly since mortgages are long-term. However, it reduces housing demand indirectly by lowering
firms’ labor demand and hence household labor income. Households reduce their savings, hence investment,
in response to the decline in income at the time of the bust. The capital stock recovers slowly and the
decline in income persists despite the quick recovery of the banking sector. The persistent decline in income
amplifies the decline in house prices. This analysis suggests that firms’ short-term liability structure is the
key mechanism that translates the temporary spike in the bank lending rate to a significant and persistent
decline in house prices.
The dynamics of interest rates and bank loans implied by our credit supply shock benchmark are
supported by the empirical findings in the literature. Interest rates on firm loans and mortgages have declined
during the boom (Glaeser et al. (2012a) and Justiniano et al. (2017)). On the effects of deregulation on
interest rates, Jayaratne and Strahan (1997) and Favara and Imbs (2015) find significant declines in
lending interest rates after the branching deregulation in the US. For the crisis period, Ivashina and
Scharfstein (2010) document a more than 50 percent decline in bank real investment loans to
In parallel, Adrian et al. (2013) find that real investment loans to firms have declined substantially, while
interest rates on loans more than quadrupled during the Gilchrist and Zakrajšek (2012) show that
credit spreads spike during downturns, predicting significant declines in subsequent economic activity.
Together, these papers provide evidence for the distruption in the bank credit supply during the 2008 crisis.
The model’s cross-sectional implications are also consistent with the recent evidence from de- tailed
micro-level data analysis, some of which is argued to be inconsistent with the credit supply mechanism. In
particular, we find that credit grows similarly across different income quantiles in our model over the boom
episode, as shown to be the case in the data (Adelino et al. (2016) and Foote et al. (2016)). Consistent
with the findings of Albanesi et al. (2017), our model implies that credit growth has been stronger for
consumers with faster income growth. We also find that the higher leverage during the boom and the
decline in income during the bust are the major factors that increased foreclosures. Taken altogether, these
results provide support for our framework and the credit supply channel.
The rise of highly leveraged households during the boom causes a deeper contraction during the bust.
To quantify its importance during the bust, we keep the aggregate debt constant but redistribute some
part of the debt of households who fall into negative equity to the rest of the households. In this
counterfactual economy, foreclosures do not increase during the bust, and as a result, house prices decline
less: 15 percent with redistribution instead of percent in the benchmark. Consumption and output
also decline less, by about 1 percentage point.
We compare the model’s dynamics across credit supply, productivity, and house price expectation
4Real investment loans include capital expenditure and working capital loans.
5Adrian et al. (2013) also report that non-financial US corporations counteracted the decline in the loan supply by
increasing bond issuances. However, total credit (both loans and bonds) has declined. Thus, financial conditions must have
tightened for non-corporate businesses, which do not have access to the bond market.
shocks. While we find many similarities, there are also several important differences. For example, with
house price expectation shocks, households reduce capital accumulation, and thus output and labor income
decline during the boom, and consumption barely rises in the short run and declines in the long run. In
addition, the equilibrium bank lending rate does not increase significantly during busts with productivity and
house price expectation shocks. This is because, in contrast to credit supply shocks, these shocks primarily
reduce the credit demand. While increases in foreclosures cause losses in bank balance sheets and reduce
the credit supply, the bank lending rate does not increase significantly at the time of the bust under these
shocks unless they generate unrealistically high foreclosures. As a result, relative to the credit supply shock,
mortgage valuations and, hence, bank net worth decline by significantly less. This result suggests that
housing busts accompanied by severe banking crises are more likely to be generated by credit supply
shocks rather than by house price expectation or productivity shocks.
Finally, our model allows us to study effects of both the ex ante and ex post policies on both
household and bank balance sheets. For example, tighter LTV restrictions mitigate the increase in house
prices by constraining household leverage, which subsequently reduces the fallout in the bust. Banks also
become less vulnerable to declines in mortgage valuations and increases in foreclosures since the fraction of
mortgages in bank portfolios are lower to start with and do not increase as much during the boom. Thus,
overall, we find that stricter LTV requirements significantly reduce fluctuations in house prices,
consumption, and output. Comparing capital injections to banks and household bailouts in a revenue-
neutral fashion shows that capital injections to banks are more effective in eliminating the drop in bank
net worth at the time of the bust and hence more effective in the short run, especially on variables that depend
on short-term financing. The household bailout, on the other hand, is more effective in mitigating the drop
in all variables in the longer run, the relative effectiveness appearing earlier on variables, such as house
prices, that depend on long-term debt.
Related Literature
Our paper contributes to the literature that studies the dynamics of the housing market and the
macroeconomy around the 2008 financial Justiniano et al. (2017) and Greenwald (2016), using
representative borrower and savers, and Huo and Rios-Rull (2013), Sommer et al. (2013), and Favilukis
et al. (2017), using heterogeneous agent frameworks, show that credit conditions such as changes in
maximum LTV or payment-to-income (PTI) ratios, and/or in credit supply can generate significant
changes in house prices and However, Kaplan et al. (2020)
6For excellent surveys, see Davis and Van Nieuwerburgh (2015), Piazzesi and Schneider (2016), and Guerrieri and Uhlig
(2016).
7In Huo and Rios-Rull (2013), there is a feedback from household balance sheets to aggregate output because of
good market frictions. In our model, feedback from household balance sheets to aggregate output goes through bank balance
sheets that deteriorate because of higher foreclosures, which reduces the bank credit supply.
argue that the absence of the rental market and/or long-term defaultable mortgages are critical for obtaining
large effects of credit conditions on house prices since, with rental markets, households can rent a house
of their desired size if they are constrained in purchasing one. So, LTV and PTI constraints—even if
they bind for some households—do not significantly affect the aggregate housing demand. Furthermore,
defaultable mortgages generate endogenous borrowing limits that make the LTV constraint less relevant.
With these extensions, Kaplan et al. (2020) argue that shifts in household demand due to shocks to house
price expectations, rather than changes in credit conditions, were the driving force behind the boom-bust
cycle in the housing market.
In this paper, similar to Kaplan et al. (2020), we model the rental market and long-term de- faultable
mortgages. However, in contrast to Kaplan et al. (2020), we find large effects of credit supply shocks
because of two differences in our analysis. First, we consider permanent changes in bank leverage that
essentially translate into permanent changes in the bank lending rate rather than the LTV, PTI, or temporary
interest rate shocks considered in Kaplan et al. (2020). Second, the credit supply shock in our framework
is not an isolated shock to households since we model the interaction between the bank credit supply and
firms’ production. Consequently, the permanent changes in the bank lending rate create large income and
wealth effects on households, which then create boom-bust cycles in the housing market and the rest of the
macroeconomy.
The degree of segmentation between owner-occupied and rental units matters for how far the changes
in credit conditions (such as LTV limits) move house prices. For example, while Favilukis et al. (2017)
assume a perfectly segmented housing market by assuming a fixed homeownership rate, Kaplan et al.
(2020) assume a frictionless housing market where rental and owner-occupied units can be converted to
each other without any cost. Partly because of the stark difference in this modeling choice, two papers
reach opposing results. In a recent paper, Greenwald and Guren (2020) document empirical evidence that
housing market is close to being fully segmented. In our model, the housing market is partially segmented as
converting owner-occupied units to rental units is costly. We calibrate this cost parameter so that the
degree segmentation in our benchmark is lower than the estimates of Greenwald and Guren (2020).8 By
doing so, we make sure that our results are not driven by a high degree of market segmentation.
Garriga and Hedlund (2018) also find that lower interest rates can account for the boom in house prices and
consumption. There, the bust is generated through tighter down payment constraints and higher left tail
income risk (see also Garriga and Hedlund (2020)). In our framework, as well, the credit supply expansion
lowers the bank lending rate and creates a boom. The reversal of the credit supply shock by itself generates
a deep bust in our model. The endogenous change in credit due to changes in bank balance sheets and firms’
dependence on bank credit are the two key features of our framework that amplify the bust. Finally, all the
aforementioned papers abstract from the
8We experimented with higher degree of housing market segmentation. The key difference in that case is the decline in
the price-rent ratio becomes larger during the bust. The dynamics of other variables remain very similar.
bank balance sheet effects. By connecting the banking sector with the real sector, we can study housing
and banking crises jointly. We can also compare the effectiveness of household versus bank bailout policies in
a revenue-neutral fashion.
Our paper is also related to the literature that combines a banking sector that faces balance sheet
constraints with household and/or production sectors. Landvoigt (2016) and Ferrante (2019) argue that
credit supply shocks, along with shocks to house price uncertainty, play important roles for house prices
changes. These papers assume within-sector perfect risk sharing so that each sector is represented by a single
Compared to these papers, our paper’s richer heterogeneity in the household sector allows us to
compare our model’s implications with cross-sectional facts that were argued to be against the credit supply
channel. We also model the rental market for housing, which is important for analyzing house prices, as
shown in Kaplan et al. (2020).
Our framework combines key elements from two strands of literature. On the one hand, an active
literature has studied the pricing of default risk in the context of unsecured or mortgage debt. Prominent
examples for unsecured credit are Chatterjee et al. (2007) and Livshits et al. (2010, 2007), and for
mortgage debt are, Jeske et al. (2013), Corbae and Quintin (2015), Chatterjee and Eyigungor (2015),
Arslan et al. (2015), Guler (2015), Hatchondo et al. (2015), Kaplan et al. (2020), and Garriga and
Hedlund (2018, 2020). In this literature, banks are modeled as risk-neutral and zero-profit making
competitive financial intermediaries. On the other hand, the literature on bank balance sheets has studied
how depletion of a bank’s capital reduces its ability to intermediate funds (Mendoza and Quadrini (2010),
Gertler and Kiyotaki (2010, 2015), Gertler and Karadi (2011), He and Krishnamurthy (2012, 2013),
Brunnermeier and Sannikov (2014), Bianchi and Bigio (2014), Boissay et al. (2016), and Navarro
(2016)). However, in this literature, banks’ asset structure typically takes a simple form such as one-
period bonds or lacks the rich heterogeneity observed in banks’ portfolios. By combining these two strands
of the literature, our model allows us to study the rich interactions among households, firms, and banks.
2 Quantitative Model
The model economy is composed of five different sectors: (i) a unit measure of finitely lived house- holds,
(ii) a continuum of all-identical financial intermediaries, called banks, (iii) rental companies,
(iv) final good producing firms, and (v) the government. We consider bankers as separate households in the
economy.
We assume that total housing stock in the economy is fixed at H̄ , but the homeownership rate
is not. This becomes possible as part of the housing stock is owned by homeowners and the rest
9Elenev (2017), Elenev et al. (2016), and Elenev et al. (2018) also use an approach similar to these papers to address
different questions from ours. Elenev (2017) studies the effectiveness of large-scale asset purchases during busts. Elenev et
al. (2016) and Elenev et al. (2018) study the incentive effects of government guarantees on financial sector risk taking and
fragility.
j
ε
J
j=1
i j j
is owned by rental companies who rent it to the households. There is perfect competition in all markets.
There is no aggregate uncertainty in the model. Boom-bust transitions are generated by two
unexpected shocks, both of which are perceived as permanent shocks. Other than the periods that the
shocks hit, there is perfect foresight. Since households are ex post heterogeneous in several dimensions, all the
endogenous prices, value functions, and policy functions depend on the aggregate state of the economy and the
distribution of households. For notational convenience, we suppress these dependencies.
Households
At the heart of the model economy is a rich household sector with realistic housing tenure and mortgage
We assume that households work until the mandatory retirement age Jr and live up to age J
after the retirement. Working-age households are subject to idiosyncratic income uncertainty: before
retirement, log labor income consists of a deterministic component f (j), which only depends on age, and a
stochastic component zj , which is an AR(1) process. Thus, a household’s income process y(j, zj) can be
summarized by
y(j, z ) =
w (1 −τ ) exp(f (j) + zj), if j ≤Jr (1)
wyR(zJr ), if j > Jr
zj = ρzj−1 + εj , εj ϑ. N (0, σ2),
where w is the wage per efficiency units of labor, τ is the tax rate, and yR(zJr ) is a function that
approximates the US retirement system, as in Guvenen and Smith (2014). Households supply labor
inelastically. However, the wage w depends on aggregate labor utilization rate as discussed in section .
We assume that there are two types of households: capitalists (K) and depositors (D). The key
distinction between capitalists and depositors arises from the difference in savings options. Depositors
can only save at the risk-free deposit rate r, while capitalists own the final good producing firm and the rental
company, as we elaborate later, which give the same rate of return r̃ .
Households receive utility from consumption and housing services and can choose between rent- ing or
owning a house of their desired size. Capitalists and depositors also have different discount factors. Thus,
the preferences of a household of type i ϑ{K, D}takes the following form:
E0[
Σ
βj−1u(ci , si )],
10The household sector builds on the ones in Arslan et al. (2015) and Guler (2015) but is extended in some important ways, such
as flexible housing and rental sizes, and refinancing options.
where βi is the discount factor, ci is consumption, and si is the housing services at age j for a type-i
j j
household.
Housing Choices: Households enter the economy as active renters and can stay as renters by renting a
house at the desired size at the price pr per unit of housing service. However, they can also purchase a
house and become homeowners at any time. Purchasing a house is costly, especially for young
households who do not have sufficient wealth to afford it. Although we do not allow unsecured borrowing in
the model, we do allow households to have access to the mortgage market to finance their housing
purchases. An important element of our model is that the terms of mortgage contracts, down payment and
mortgage pricing, are endogenous and depend on household characteristics. Homeowners can choose to stay as
homeowners or become renters again, by either selling their houses or defaulting on mortgage loans.
Homeowners can refinance their houses at any point in time. Refinancing is the same as obtaining a mortgage
at the time of purchase. Households also have the option of upgrading or downgrading the house size by
selling the current house and buying a new one.
Several transaction costs are associated with owning a house. The purchase price of a house is ph per
unit of housing. To finance the purchase, the household can obtain a mortgage from banks. However,
mortgages involve three types of costs. First, there is a fixed cost by the bank, ϑf , for originating a
Second, banks charge a variable cost of origination for mortgages. This cost is ϑm fraction of
the mortgage debt at the origination. Selling a house is also costly. A seller has to pay ϑs fraction of the
selling Lastly, since mortgages are risky, lenders charge a premium for the risk of defaulting. This
premium shows up in the origination price of the mortgage.
Defaulting on a mortgage is possible, but it is costly. The cost is that after default, households become
inactive renters; that is they temporarily lose access to the housing market. Inactive renters become
active renters with probability π. Therefore, agents have three statuses regarding their housing decision:
homeowner, active renter, or inactive renter.
Mortgage Payments: To keep the tractability in the model, we assume that mortgages are due by the end
of life, which is deterministic, so that the household’s age captures the maturity of the mortgage contract.
We also allow for only fixed rate mortgages. The mortgage contract can be characterized by its maturity,
the periodic mortgage payment m. We assume that the mortgage
payments follow the standard amortization formula computed at the bank lending rate rϑ. Thus, the
relation between mortgage debt d and mortgage payment m in a period is given as
11Some examples of these costs are attorney fees, appraisal fees, and title company fees. These costs are fixed and do not
depend on the size of the mortgage.
12Fees paid to real estate agents are the main part of these costs.
. Σ
.
1 1 1
Σ
rϑ (1 + rϑ)J−j
d = m 1 +
1 + rϑ
+
(1 + rϑ)2
+ ... +
(1 + rϑ)J−j ϑ m = d (1 + rϑ)J −j+1 −1
(2)
The remaining mortgage debt in the following period will be (d −m) (1 + rϑ).
The mortgage interest rate differs across households since ex post households are heterogeneous.
In principle, this should imply that the amortization schedule should be computed at the individual
mortgage interest rate instead of rϑ. However, to save from an additional state variable, we assume that
mortgage amortization is computed at the risk-free mortgage rate, as in Hatchondo et al. (2015)
and Kaplan et al. (2020). As will be clear later, individual default risk will show up in the pricing of the
mortgages at the origination rather than in the mortgage interest rate. Thus, essentially all
households pay points at the origination to reduce the mortgage interest rate to rϑ.
Household’s problem
Active Renters
An active renter has two choices: to continue to rent or purchase a house, that is, V r = max V rr, V rh where V
rr is the value function if she decides to continue renting and V rh is the value function if she decides to
purchase a house. If she decides to continue to rent, she chooses rental unit size s at price pr per unit,
makes her consumption and saving choices, and remains as an active renter in the next period. After
purchasing a house, she begins the next period as a homeowner. The value function of an active renter who
decides to remain as a renter is given by
V rr(a, z) = max
.
u(c, s) + βiEV r (aJ, z J)
Σ
(3)
subject to
ij
c,s,a ≥0
aJ
j+1
c +
1 + ri
+ prs = w (1 −τ ) y(j, z) + a,
where a is the beginning-of-period financial wealth, prs is the rental payment, ri is the return to savings,
and w is the wage rate per efficiency unit of labor. Remember that capitalists have rate of return rK = r̃
and depositors have rate of return rD = r. The expectation operator is over the
income shock z J.
If an active renter chooses to purchase a house, she can access the mortgage market to finance
her purchase. She chooses a mortgage debt level d that determines qm(d; a, h, z, j), the price of the
mortgage at the origination, which will be a function of the current state of the household (current wealth a,
income realization z, and age j), house size h, and the amount of debt d. Then the value function of an
active renter who chooses to buy a house is given by
J
. Σ
ij c,aJ≥0 ij+1
V rh(a, z) = max
,
u(c, h) + βiEV h (aJ, h, d, z J)
,
(4)
subject to
ij
c,d,h,a ≥0
aJ
ij+1
c + phh + δhphh + ϑf +
1 + ri
= w (1 −τ ) y(j, z) + a + d
.
qm(d; aJ, h, z, j) −ϑm
Σ
dJ ≤ phh (1 −q) ,
where ph is the housing price, δh is the proportional maintenance cost of housing, ϑm is the variable cost of
mortgage origination, ϑf is the fixed cost paid at the origination if the individual gets a mortgage, and φ
is the minimum down payment required to get a mortgage.
Inactive Renters
Inactive renters are not allowed to purchase a house because of their default in previous periods. However,
they can become active renters with probability π. Since they cannot buy a house, they only make rental
size, consumption, and saving decisions. The value function of an inactive renter is given by
V e(a, z) = max
.
u(c, s) + βi
Σ
πEV r (aJ, z J) + (1 −π)EV i (aJ, z J)
ΣΣ
(5)
ij
subject to
c,s,aJ≥0
aJ
j+1 ij+1
Homeowners
c +
1 + ri
+ prs = w (1 −τ ) y(j, z) + a.
The options of a homeowner are: 1) stay as a homeowner, 2) refinance, 3) sell the current house (become
a renter or buy a new house), or 4) default. The value function of an owner is given as the maximum of these
four options, that is, V h = max V hh, V hf , V hr, V he , where V hh is the value of staying as a homeowner, V
hf is the value of refinancing, V hr is the value of selling, and V he is the value of defaulting (being excluded
from the ownership option).
A stayer makes a consumption and saving decision given his income shock, housing, mortgage debt, and
assets. Therefore, the problem of the stayer can be formulated as follows:
V hh (a, h, d, z) = max
,
u (c, h) + βiEV h
.
aJ, h, dJ, z J
Σ,
(6)
J
ij ij
ij
c,s,aJ≥0 ij+1 ij+1
subject to
aJ
c + δhphh + 1 + r + m = w (1 −τ ) y (j, z) + a
dJ = (d −m) (1 + rϑ) ,
where m is the mortgage payment following the amortization schedule determined in equation 2.
The second choice for the homeowner is to refinance, which also includes prepayment. Refinanc- ing
requires paying the full balance of any existing debt and getting a new mortgage. We assume that
refinancing is subject to the same transaction costs as new mortgage originations. So, we can formulate the
problem of a refinancer as
V hf (a, h, d, z) = max
,
u(c, h) + βiEV h (aJ, h, dJ, z J)
,
(7)
subject to
c,dJ,aJ≥0
aJ
ij+1
c + d + δhphh + ϑf +
1 + ri
= w (1 −τ ) y(j, z) + a + dJ
.
qm(dJ; a, h, z, j) −ϑm
Σ
dJ ≤ phh (1 −q) .
The third choice for the homeowner is to sell the current house and either stay as a renter or buy a new
house. Selling a house is subject to a transaction cost that equals fraction ϑs of the selling price.
Moreover, a seller has to pay the outstanding mortgage debt, d, in full to the lender. A seller, upon selling the
house, can either rent a house or a buy a new one. Her problem is identical to a renter’s problem. So, we
have
V hr (a, h, d, z) = V r (a + phh(1 −ϑs) −d, z) .
The fourth possible choice for a homeowner is to default on the mortgage, if she has one. A defaulter
has no obligation to the bank. The bank seizes the house, sells it on the market, and returns any
positive amount from the sale of the house, net of the outstanding mortgage debt and transaction costs,
back to the defaulter. For the lender, the sale price of the house is assumed to be (1 − ϑe) phh.
Therefore, the defaulter receives max {(1 −ϑe) phh −d, 0} from the lender. The defaulter starts the next
period as an active renter with probability π. With probability (1 −π), she stays as an inactive renter.
The problem of a defaulter becomes the following:
V hi (a, d, z) = max
.
u (c, s) + βiE
Σ
πV r
.
aJ, z J
Σ
+ (1 −π) V i
.
aJ, z J
ΣΣΣ
(8)
ij
i
u
Kt,Nt,ut
t t+1
subject to
aJc +
1 + ri
+ prs = a + w (1 −τ ) y (j, z) + max {(1 −ϑe) phh −d, 0}.
The problem of a defaulter is different from the problem of a seller in two ways. First, the de- faulter
receives max {(1 −ϑe) phh −d, 0} from the housing transaction, whereas a seller receives (1 −ϑs) phh −d.
We assume that the default cost is higher than the sale transaction cost, that is, ϑe > ϑs, the defaulter
receives less than the seller as long as (1 −ϑs) phh −d ≥ 0 (., the home equity net of the transaction
costs for the homeowner is positive). Second, a defaulter does not have access to the mortgage in the next
period with some probability. Such an exclusion lowers the continuation utility for a defaulter. In sum, since
defaulting is costly, a homeowner will choose to sell the house instead of defaulting as long as (1 −ϑs) phh
− d ≥ 0 (., net home equity is posi- tive). Hence, negative equity is a necessary, but not sufficient,
condition for default in the model. Therefore, in equilibrium, a defaulter gets nothing from the lender.
Firms
A perfectly competitive firm produces final output by combining capital Kt and number of workers
Nt. The firm can also choose the utilization rate per worker ut. The wage per efficiency units
of a worker is assumed to depend on the utilization rate, that is, w (w̄ , u ) = w̄ 1+ψ+ ϑ t , where
t t t 1+ψ
w (w̄t, ut) is the efficiency units of labor, same as w in previous sections, ϑ and ψ are constants, and
w̄t and ut are determined in equilibrium. A household’s labor income is given by y (z, j) w (w̄t, ut) .
The firm has to finance a fraction µ of the wage payment in advance from banks and pay interest on that
portion. Then, the firm’s problem is given by
max ZtKα (Ntut)1−α −(r̃t + δ)Kt −
.
1 + µrϑ
Σ
w (w̄t, ut) Nt,
where r̃t is the rate of return to capital and δ is the depreciation rate. Since labor supply is
exogenous, a worker’s labor income depends on the firm’s labor utilization rate. The basic idea
behind this formulation is that the firm reduces labor utilization in response to an increase in bank lending
rate rϑ, which in turn reduces
13We could have achieved the same effect without labor utilization but endogenous labor supply. In that case, the firm would
reduce labor demand, which would reduce wages. Since households would reduce labor supply, aggregate output would decline.
However, our formulation is easier to handle computationally.
.
u
t−1
t
t
t−1
.
−
t t t
t t t−1 t t t t t−1 t t
t−1
t t+1 t t t−1
t t t−1 t t 2 t t t−1 t t
t−1
Σ
Σ ,
2 t t
The firm’s first-order conditions are given as
αZt
Kt
Ntut
Σα−1
= r̃t + δ
(1 −α) Ztut
Kt α
Ntut
.
Kt
Σα
=
.
1 + µrϑ+1
. ϑ
Σ
.
w̄t + ϑ
Σ ψ
1+ψ
t
1 + ψ
(1 −α) Zt
Rental Companies
Ntut
= 1 + µrt+1 ϑut .
The rental company enters period t with (1 −δh) Hr units of rental housing stock where δh
is the depreciation rate of rental housing. Then it chooses Hr. In that period, the company receives
net rent (pr −κ) Hr where pr is the rental price per unit of housing and κ is the per-period
maintenance cost and pays dividend xr = ph (1 −δ) Hr −phHr −η ph
.
Hr −Hr
Σ2 +(pr −κ) Hr
to shareholders. η ph
.
Hr −Hr
Σ2 is the quadratic adjustment cost of changing the rental supply
(. converting rental and owner-occupied units to each other). A higher value of η implies a
more segmented housing market. Since both capital and rental company shares are riskless in a
deterministic equilibrium, (., in the steady state and along the transition path except for the
unanticipated shock periods), both assets have to pay the same rate of return in equilibrium, which implies
1 + r̃t =
.
xr + V rc (Hr)
Σ
/V rc
.
Hr
Σ
,
where Vt+1 (Hr) is the post-dividend market value of the company at the end of period
The objective of the company is to maximize its total market value Vt
.
Hr
Σ
:
V rc
.
Hr
Σ
= max
1 .
xr + V rc (Hr)
Σ
t t−1 rt
.
1 + r̃t t t+1 t
η . Σ2
xr = ph (1 −δ) Hr −phHr − ph Hr −Hr + (pr −κ) Hr.
The first-order condition to the above problem gives the rental price as functions of the house price
14At the time of an unexpected shock, capital and the rental housing return could be different. Then, the realized return of the
capitalists, which will be different from the contracted return, would be given by
J Kt ∫ Kt
,
xr + V rc (Hr)
1 + r̃t = A
(1 + r̃t) + 1 −
A t t+1V Hrwhere At is the total assets of the capitalists, which is equal to the Kt + V rc (Hr 1), and r̃t = αZtKα−1N 1−α −δ.
The aggregate income of the capitalists is r rc r t t
rc r
t t r : ther̃tKt + xt + Vt+1 (Ht ) Vt (Ht−1) , which in steady state is r̃K + xreturn to capital plus the dividend from the rental company.
H
t t t
t
−
.
2
t+1 .
t t
t t t t t−1 1 + r̃t+1
h t+1 t+1 t+1 t
t 1 + ̃ t t t t−1
and rental housing stocks in periods t −1, t, and t + 1.
pr = κ + ph + ηph
.
Hr −Hr
Σ
−
1 .
(1 −δ ) ph + ηph
.
Hr −Hr
ΣΣ
. (9)
This is the supply equation for the rental housing. The demand for rental housing comes from
households’ housing choices.
In order to see how pr is affected by ph and homeownership rate, first consider the case where
t t
η = 0, which corresponds to the frictionless housing market explored in Kaplan et al. (2020).
Equation 9 in this case becomes
pr = κ + ph − (1 −δh) p
h
t t 1 + r̃t+1
This equation implies that, for a given ph, a higher future house price ph reduces pr. This is the
t t+1 t
main mechanism in Kaplan et al. (2020) that generates an increase in the price-rent ratio. However, the
homeownership rate does not have any effect on rental price in this case. So, policies, such as relaxation of
LTV limits that affect homeownership rate, does not move the price-rent ratio.
Next, consider a one-time permanent increase in homeownership rate in period t. Since home-
ownership rate increases in the current period, rental supply should decline for housing market to
clear. As a result, we have Hr < Hr and Hr = Then, we can write equation 9 as
t t−1 t+1 t
pr = κ +
r̃ + δh ph + ηph
.
Hr −Hr
Σ
.
This equation shows that holding ph fixed, an increase in the homeownership reduces pr (since
Hr −Hr < 0) if η > 0, and thus, the price-rent ratio increases. The higher the value of η, the
t t−1
higher is the increase in the price-rent ratio in response to a change in the homeownership rate. As it turns out,
the leverage-shock-driven boom in our benchmark analysis does not affect homeownership rate significantly.
Consequently, this mechanism is not significant during the boom. However, the decline in homeowneship
during the bust is significant due to the increase in foreclosures and then, the price-rent ratio declines more
with higher degree of market segmentation (see section ).
15We would like to make a disclaimer here. Owner-occupied housing demand is not equal to homeownership rate because of
differences in housing size per household. However, since homeownership rate and owner-occupied housing demand typically move
together, we have chosen to explain this equation in terms of homeownership rate.
∞
t
t+1
t+1
t+1
t+1
t+1
t+1
t=0
L t
t t+1
Banker
We assume a competitive banking industry with a continuum of identical banks that are risk-averse and
maximize the discounted lifetime utility
Σ
βt−1 log
.
cB
Σ
,
where cB is the banker’s consumption. There is no entry to the banking sector. Banks fund their
operations from their net worth ωt and by borrowing Bt+1 in the international market at a risk-
free interest rate rt+1, lend Lk to the firm at rϑ+1 , and issue mortgages and purchase existing
mortgages.
Similar to Gertler and Kiyotaki (2015) and Gertler and Karadi (2011), we assume that banks can
walk away at the beginning of a period without paying back their creditors. In that case, the bank can steal
a fraction ξ of its assets but is excluded from banking operations in the future and can invest those assets
at rate rt. Knowing this, creditors lend to the bank to the extent that the bank does not walk away. Since
the bank’s outside option depends on its assets in this case, we need to keep track of assets and debt
separately.
Letting θ = (d; a, h, z, j) define the type of a mortgage, ωt be the bank’s net worth, and At+1 (θ) be the
amount of investment in mortgage type θ (which includes any newly issued as well as existing mortgages), the
budget constraint of the bank is given by
cB + Lk +
∫
pt (θ) At+1 (θ) = ωt + Bt+1.
The bank’s net worth evolves according to the following law of motion:
ωt+1 =
θ
∫
θJ
l
t+1 .θ J
Σ
Π
.
θ J |θ
Σ
At+1 (θ) + Lk
.
1 + rϑ+1
Σ
−Bt+1 (1 + rt+1) ,
where vl (θ J) = mt+1 (θ J)+ pt+1 (θ J) and Π (θ J |θ) is the endogenous transition probability governed
by exogenous household characteristics as well as endogenous choices.
If the bank defaults, it can steal a fraction ξ of its assets next period and save at interest rate
r. We denote its value of default by Ψ̃D
.
ξLt+1
Σ
, where
Lt+1 =
.∫ ∫
θJ
l
t+1 .
θ J
Σ
Π
.
θ J |θ
Σ
At+1 (θ) + Lk
.
1 + rϑ+1
Σ
Σ
.
Lt+1 = Lk +
∫
θ pt (θ) At+1 (θ) is the investment in t and Lt+1 is the value of that investment
in period t + 1 after returns are realized. Investors lend to the bank up to a point where the bank does
not steal in equilibrium. Denoting the value to the bank of honoring its obligations by
θ
θ
v
v
∫
t+1
∫
t+1
∫
v
θ t+1
pt(θ)
t
t
Ψt+1 (Lt+1, Bt+1) where Lt+1 is the bank’s asset portfolio, the enforcement constraint is given as
Ψt+1 (Lt+1, Bt+1) ≥Ψ̃D
.
ξLt+1
Σ
.
The bank does not face any uncertainty in its net worth even though each mortgage is a risky
investment. This is because we assume a continuum within each household type, which will trans- late into a
continuum within each mortgage type θ. Thus, even if a bank invests in a particular type of mortgage θ by a
tiny amount, its return is deterministic since a known fraction of θ-type house- holds default and the
remainder continue to pay their mortgages with certainty. The continuum assumption grants us tractability
while keeping the rich heterogeneity in the household sector.
Since the bank does not face any uncertainty, an important property of the bank’s problem is
that all assets have to generate the same rate of return, which is equal to rϑ+1. That is, the gross
return on a mortgage of type θ is
,
J vl (θJ)Π(θJ |θ) and has to be equal to the gross return on loans to
the firm 1 + rϑ+1. The price of the mortgage after that period’s mortgage payment has been made is then
given as
p (θ) =
1
1 + rϑ+1
l
t+1
θJ
.
θ J
Σ
Π
.
θ J |θ
Σ
for all θ.
Since vl (θ J) = mt+1 (θ J) + pt+1 (θ J) , the price of the mortgage is essentially the expected present
discounted value of mortgage payments. As we will illustrate, the no-arbitrage condition greatly
simplifies the problem of the bank. Since the bank is indifferent between investing in any asset, we do
not have to keep track of its asset distribution in the bank’s problem. Then, using pt (θ) =
1
1+rϑ+1
l
θJ t+1 (θ J) Π (θ J |θ), we can simply show that Lt+1 =
.
1 + rϑ+1
Σ
Lt+1 . Then, the bank’s
problem can be written as
Ψt (Lt, Bt) = max
.
log
.
cB
Σ
+ βLΨt+1 (Lt+1, Bt+1)
Σ
.
Bt+1,Lt+1
,cB
t
cB + Lt+1 = (1 + rϑ) Lt −(1 + rt) Bt + Bt+1
t
Ψt+1 (Lt+1, Bt+1) ≥ ˜Dt+1
t
.
ξ
.
1 + rϑ+1
Σ
Lt+1
Σ
,
where Ψ̃D (W ) = maxW J log (W −W J)+βLΨ̃D ((1 + rt+1)W J) . We can show that the enforcement
t
constraint of the bank can be written as
t+1
(1 −φt+1)
.
1 + rϑ+1
Σ
Lt+1 ≥(1 + rt+1) Bt+1
and implies an endogenous upper bound on bank This leverage constraint is essentially
16Appendix C provides characterization of the bank’s problem in detail.
v
Ψ
∫
1+rt+1−(1−φt+1)(1+rϑ+1) t t+1
a collateral constraint: it states that the bank can borrow up to a fraction of its assets and φt+1
reflects the haircut on its collateral, where φt is defined recursively as follows:
φt = ξ1−βL
.
(1 + rt+1) /
.
1 + rϑ+1
Σ
−(1 −φt+1)
Σ
. (10)
βL
If the bank was not able to steal, (., ξ = 0), then φt = 0 and rϑ+1 = rt+1. Thus, the collateral
premium rϑ+1 −rt+1 would be zero.
Finally, perfect competition among banks implies that at the time of the mortgage initiation, the
present value of mortgage payments should be equal to the loan amount
dqm(d; a, h, z, j) = m +
1
1 + rϑ+1 θJ
Given d and m, this equation solves for qm(d; a, h, z, j).
l
t+1
.
θ J
Σ
Π
.
θ J |θ
Σ
. (11)
Bank’s solution
Given the collateral constraint the bank is facing, we can explicitly solve for the bank’s problem, which is
summarized in the following proposition.
Proposition 1. The decision rules when the no-default constraint is binding (if rϑ+1 > rt+1) are:
(1 + rt+1)L = β ω
t+1
Bt+1
1 + rt+1 −(1 −φt+1)(1 + rϑ+1)
=
(1 −φt+1)(1 + rϑ+1)
1 + rt+1 −(1 −φt+1)(1 + rϑ+1)
L t
βLωt,
where ωt = (1 + rϑ)Lt −(1 + rt)Bt.
The decision rules when the no-default constraint is not binding (if rϑ+1 ≤rt+1) are:
ϑ
Σ
0, βL(1−φt+1)(1+r
ϑ
+1) ω
Σ
if rϑ = r
and
0 if rϑ+1 < rt+1
Lt+1 = Bt+1 + βL ((1 + rϑ)Lt −(1 + rt)Bt) .
Characterization of the Bank’s Problem in Stationary Equilibrium
We can further characterize the bank’s problem under stationarity. Throughout the paper, we will focus on
stationary equilibria where the capital requirement constraint is binding. If it were not, then bank balance
sheets would not have any impact on the economy. However, we do not rule out the case that there
might be some periods in the transition where this constraint becomes
t+1
v
B =
t+1
^
^−
^ ^
^ ^
t t t=1
slack. Using the general formula capturing both the exogenous and endogenous capital requirement constraint,
we have the following decision rules when the constraint is binding:
Lt+1 = βLλ̂tωt and Bt+1 = βL
.
λ̂t −1
Σ
ωt,
where (1 + rt+1)
λ̂t = 1 + rt+1 −(1 −φ t+1 )(1 + r
ϑ
+1
. (12)
)
Then the law of motion for net worth is given as
ωt+1 = Lt+1
.
1 + rϑ+1
Σ
−Bt+1 (1 + rt+1) .
Then, we can obtain the next period’s net worth as
ωt+1 = βL
.
λt
.
1 + rϑ+1
Σ
−
.
λt −1
Σ
(1 + rt+1)
Σ
ωt.
Imposing steady state ωt+1 = ωt and λt = λ gives
rϑ r =
1 −βL(1 + r) ,
λβL
where rϑ −r is the premium due to the bank capital constraint. If βL(1 + r) < 1 and λ < ∞, then rϑ
−r > 0 . Thus, the capital constraint will be binding in the stationary equilibrium. To understand this
point, assume that βL(1 + r) < 1 but the bank starts with a high net worth so that
the capital requirement constraint is not binding. In that case, rϑ+1 = r and the bank’s decision rule is
Lt+1 −Bt+1 = βLωt. Using that, we can show that ωt+1 = (1 + r) βLωt < ωt. Thus, the bank eats up its net
worth until the capital constraint starts to bind. Thus, the economy will converge to a stationary
equilibrium where it actually binds.
Symmetric Equilibrium
We focus on a symmetric equilibrium where each bank holds the market portfolio of mortgages. Thus, we
have a representative bank. The definition of equilibrium is straightforward: all economic agents maximize
their objectives given the exogenous price sequence {rt}t=1 and endogenous price
sequences
.
rϑ, r̃t, w̄t, ph, pr
Σ∞
. The labor market clears in all periods, . Nt = 1. We discuss the
Credit market: Letting Γt (θ) be the distribution of available mortgages after HH’s make their decisions
at time t, the credit market clearing conditions can be summarized by the following conditions:
credit and housing market equilibrium conditions and the government budget next.
∫
t+1 t
t
JR J
t t t t−1 1 + r̃t+1 t+1
t+1 t
j=1 z j=JR+1 z
1. The representative bank holds the mortgage portfolio
At+1(θ) = Γt(θ).
2. Two credit market equilibrium conditions are
Lt+1 = µw (w̄t, ut) + pt (θ) Γt(θ),
θ
and
At+1 = Kt+1 + V rc (Hr) .
The first one determines the equilibrium rϑ+1, and the second one determines the equilibrium r̃t+1.
Housing market: Remember that total housing supply is fixed at H. Thus, the total demand of owners and
renters should be equal to the supply, which determines house price ph(t). Given house prices ph (t) and pr
(t), households solve their optimal housing choices, which gives the demand for owner-occupied units Ho,D
and rental units Hr,D. The supply of rental housing units is given byt t
the first-order condition of the rental company, which is given as
pr = κ + ph + η
.
Hr,S −Hr,S
Σ
−
1 .
(1 −δ ) ph + η
.
Hr,S −Hr,S
ΣΣ
.
Then, the following two equilibrium conditions give the house price ph and rental prices pr:
r,S
t
H̄
t t
= Hr,D
= Hr,D + Ho, t
Government: The government runs a pay-as-you-go pension system. It collects social security taxes
from working-age households and distributes to retirees. We assume the pension system runs a balanced
budget:
Σ Σ
τy (j, z) πj (z) =
Σ Σ
yR (j, z) πj (z)
where πj (z) is the measure of individuals with income shock z at age j.
3 Calibration
Timing: The model period is two years. We assume that households start the economy at age 26 and
work until age 65. After that, households retire and live until age 85.
H
h
Y
Ȳ
1+µr
K α
Preferences: Households receive utility from consumption and housing services captured by the following
CRRA specification:
u(c, s) = c1−σ
1 −σ
s1−θ+ γ .
1 −θ
We set θ = σ = 2. We calibrate γ to match the share of housing services in aggregate income
(including imputed income from housing services) as 15 percent. We assume 20 percent of the
population is capitalist and the rest is depositor. These household types are drawn randomly at the
beginning of life and are permanent. We calibrate the discount factor for the capitalists, βk, to match a
capital-output ratio of 1 in our biannual model. Lastly, we calibrate the discount factor for the depositors, βD,
so that the share of aggregate wealth that belongs to capitalists is 80
Income Process: For the income process before retirement, we set the persistence parameter ρ =
and σε = , which correspond to an annual persistence of and a standard deviation of following
Storesletten et al. (2004). We approximate this income process with a 15-state first- order Markov process
using the discretization method, as in Tauchen (1986). Retirement income approximates the US retirement
system, as in Guvenen and Smith (2014). We adjust the retirement income level such that working age-
households pay 12 percent tax.
Production Sector:
We assume the capital share in the final good production is α = . Denoting Y as the final good or
output, we target a capital-output ratio of K = 1, which corresponds to a capital-output ratio of
2 in an annual We normalize N = 1, Z = 1, and target u = 1 at the steady state. Then, since Y
= ZKα (Nu)1−α, we get Y = K = 1.
We also target the share of housing services in aggregate income as . Since in our model
aggregate income (including the imputed income from housing) corresponds to Ȳ = Y + prH̄ ,
this results in Ȳ 1 and prH̄ =
. In the data, the ratio of non-residential investment to
aggregate income is . Since, at the SS, this ratio is δk K , this gives us a capital depreciation
rate of δk = . Given these targets, the model-implied biannual return to capital becomes
r̃ = α Y −δk = − = 11 percent. We set ψ = . Since at the steady state we target u = 1,
from the firm’s problem, we have ϑ =
.
1−α
α
r̃+δ Σ
(1−α) , which gives the calibrated value for ϑ.
Housing Market:
The probability of becoming an active renter, while the household is an inactive renter, is set to
to capture the fact that the bad credit flag remains, on average, for seven years in the credit
17The top 20 percent holds 80 percent of aggregate wealth in the US.
18This implies a capital-to-aggregate-income ratio (including the imputed income from housing) of . See the discussion
in the next paragraph.
=
Σ .
pr
Y
pr
1+r̃
Table 1: Externally Set Parameters
Parameter Explanation Value
σ risk aversion 2
α capital share
ρε persistence of income
σε std of innovation to AR(1)
ϑh selling cost for a household 7%
ϑe selling cost for foreclosures 25%
ζ fixed cost of mortgage origination 1
δh housing depreciation rate 3%
τ variable cost of mortgage origination
η rental adjustment cost 3
π prob. of being an active renter
q down payment requirement 0
history of the household. Consistent with the estimates of Gruber and Martin (2003), we set the selling
cost (ϑs) to 7 percent, and for foreclosed properties, we set it to 25 percent, consistent with the estimates
of Campbell et al. (2011). We set the fixed mortgage origination cost ζ = 1 percent of the aggregate
output, and the variable cost of mortgage origination τ = percent of the mortgage loan. We
assume that there is no down payment requirement, that is q = 0. We also assume that maximum
payment to income ratio is 45%, which does not bind.
In the US data, the ratio of the house price to annual rental payments is around 11. So, in our biannual
model, we target ph = . This moment, together with the fact that the ratio of housing
services to output is , implies phH̄
= × = percent. So, we set H̄ to match this ratio.
We set the biennial depreciation rate for housing units as δh = 3 percent. The steady-state relation
between the rental price and house price is given by pr = κ + +̃δh ph. This gives us an estimate of κ given our
target ph = . We also restrict the minimum house size for owner-occupied units to be h to match a
homeownership rate of 66 percent. Lastly, a higher level of the parameter governing the adjustment cost of
rental supply (η) implies a higher degree of housing market segmentation. Using regional variation in credit
supply, Greenwald and Guren (2020) find that rental markets are close to fully segmented. In our
benchmark, we choose η = 3, which implies an intermediate level of housing market segmentation. In
Appendix , we report sensitivity analysis with respect to η. The key difference in that case is the decline
in the price-rent ratio becomes larger during the bust. The dynamics of other variables remain very similar.
Table 1 lists our parameter choices.
Financial Sector:
Since not only banks but other institutions as well, such as GSEs, hold large amounts of mortgage- related
products, it is necessary to consider banks in our model as a collection of financial institutions that hold
mortgages. With these considerations in mind, we follow Shin (2009) and include deposit-
^
^
βL
is the haircut. We calibrate r to match a debt-
taking institutions (US chartered depository institutions and credit unions), issuers of asset backed securities,
GSEs, and GSE-backed pools from FED Z1 data in our bank definition. Then we match bank balance
sheets to the 1985-1994 average in the data. We use Tables and to obtain the total amount of
home and multifamily residential mortgages held by banks. Banks on average hold $ trillion of these
mortgages, which correspond to 86 percent of all mortgages. This 86 percent ratio is fairly constant from
1985 to 1994. To compute the amount of lending to non- financial firms, we use the balance sheets of
non-financial firms (Table ). We use total loans (loans from depository institutions, mortgages, and
other loans), which average to $ trillion and miscellaneous liabilities, which average to $ trillion.
Residential mortgages would constitute 49 percent of banks’ balance sheets if we include the loans only and
39 percent if we also includemiscellaneous liabilities as firms financing from banks. Thus, we chose
,
θ pt(θ)Γt(θ) (the
ratio of mortgages to banks’ total financial assets) as 45 φw(w̄t,ut)Nt+
,
θ pt(θ)Γt(θ)
In the steady state, we have rϑ −r = 1−βL(1+r) , where λ̂ = (1+r) is the endogenous
.
^
ΣβL 1+r−(1−φ)(1+r )output ratio of 40 percent (corresponding to an 80 percent ratio in an annual model), and we target
rϑ −r = 3 percent, representing the average biannual gap between the 30-year mortgage interest rate and
the Treasury rate in the data. We also target λ as . These two targets give us the
bank’s discount factor βL and the bank’s seizure rate ξ, which imply a steady state leverage ratio (βLλ) of
.
To summarize, overall we have 11 parameters that we calibrate internally: discount factor for
capitalists (βK ), discount factor for depositors (βD), minimum house size (h), deposit rate (r), weight
of housing services in utility (γ), housing supply H̄ , share of wage bill financed by banks (µ), bank’s
discount factor (βL), bank’s asset seizure rate (ξ), maintenance cost for rental units (κ), and capital
depreciation rate (δk). The last four of these parameters are identified directly through analytical moments
obtained through the model as discussed above. This leaves us with seven parameters that we calibrate
using the model simulated data to jointly match the following seven data moments (Tables 2 and 3): 66
percent average homeownership rate, 40 percent mortgage-debt- to-output ratio, capital-goods production
ratio of 1, house price-to-output ratio of , share of aggregate wealth that belongs to capitalists as 80
percent, share of mortgages in bank balance sheet as 45 percent, and share of housing services in GDP as 15
percent.
19There are obviously other items in banks’ balance sheets that we do not model and do not take into account in these
calculations. We will provide robustness of our results by including these residuals into banks’ balance sheets. We did not take
this approach as our benchmark since we do not model the demand for these residual assets and its dependence on the bank
lending rate. We will provide robustness results based on two cases: 1) the dependence of the residual demand to bank lending
rate is zero, so the residual demand is constant, and 2) the residual demand changes proportionally to mortgage and firms’
demand for loans.
leverage ratio and φ = ξ1−βL
ϑ
1+r
1+rϑ −(1 −φ)
Statistic
Table 2: Moments
Note: Flow variables (output and rental price) are measured biannually.
Table 3: Internally Calibrated Parameters
Parameter Value
βK discount factor–capitalist
βD discount factor–depositor
h minimum house size
r deposit rate
γ weight of housing services in utility
H̄ housing supply
µ share of wage bill financed from banks
βL bank discount factor
ξ bank seizure rate
κ rental maintenance cost
δk capital depreciation rate
Data Model
Capital-output ratio 1 1
Homeownership rate–aggregate 66 percent 66 percent
Share of wealth that belongs to capitalists 80 percent 80 percent
Debt-output ratio 40 percent 40 percent
House price-output ratio
Share of housing services in aggregate output 15 percent 15 percent
Ratio of mortgage loans to total loans in bank assets
Mortgage premium
Bank leverage ratio 10 10
House price-rental price ratio
Non-residential investment-output ratio 16 percent 16 percent
Leverage Shock:
Several shocks have been proposed for the boom and bust phases of the last housing cycle. For example,
optimistic expectations, improved labor income prospects, lower regulation, and relaxed lending conditions.
Even if all these shocks may have been important to some extent, we specifically aim to explore the credit
supply channel and therefore pay less attention to other possible To study the role of bank credit
supply, we study the following scenario. We assume that the economy is at steady state before 1996, but in
1996, unexpectedly, bank lending capacity gradually starts increasing. Each agent in the economy expects
that it will take 25 years to reach the next steady state where the banks will have a higher leverage ratio.
Unexpectedly, in 2008 however, the leverage reverts back. The parameter that controls the bank leverage
is ξ: the fraction of assets that a bank can steal. A lower value reflects higher trust for banks and
allows banks to have a higher leverage. To calibrate the changes in this parameter, we refer to two sources.
First, Federal Reserve Bank of New York (2020) documents that the leverage ratio of the consolidated US
banking organizations has increased by 25 percent from the first quarter of 1996 to the last quarter of 2007.
We use the leverage ratio of all institutions (see page 34 of the report). Second, the Financial
Stability Report by Federal Reserve Board (2019) documents that the leverage ratio of security brokers-
dealers has increased by 50 percent from the first quarter of 1995 to the first quarter of 2008 (see figure
3-5 in the report). Both studies report marked-to-book leverage. However, in our model bank assets, Lt+1,
and net worth, Nt, are in market values and the ratio Lt+1/Nt gives the marked-to-market leverage, which is
the same as the book leverage when the economy is in steady state. However, after unexpected shocks,
market and book values will no longer be equal. In order to compare to these two data sources, we compute
the book values of bank loans and net worth, and calculate the corresponding book leverage in our model.
We calibrate the changes in parameter ξ to have an increase in the financial system book leverage for 35
percent (from 1996 to 2006), which falls in the mid-range of 25 percent and 50 percent. We compare the
book leverage from our model
and from these sources in Figure 3.
While we consider changes in leverage constraints as the main driving force, an alternative scenario
would be a decline in haircuts during the boom period and an increase during the bust period. We do not
choose this path because of the limited availability of haircut data prior to the crisis. That said, available
data (CGFS (2010)) suggest that haircuts more than doubled for most mortgage-related securities after the
crisis. And for some nonprime products, the market ceased to exist. These changes in haircuts correspond to
the leverage dynamics that we outlined above since in our framework the leverage constraint and haircuts on
collateralized loans are equivalent.
Several changes in US legislation have deregulated the financial markets that provide explicit support
to the leverage shock (see Sherman (2009)). Most relevant for our case is that, starting in 1986, the
Federal Reserve gradually loosened the Glass-Steagall Act (the bill that strictly separates
20We compare our benchmark results with several alternative shocks in Section 7.
^
^
^−
lending business from retail investment banking clients) several times, eventually, in 1996, allowing bank
holding companies to earn up to 25 percent of their revenues in investment banking. The Gramm-Leach-
Bliley Act repealed the Glass-Steagall Act completely in 1999, meaning that all restrictions against the
combination of banking, securities, and insurance operations for financial institutions were removed.
Therefore, it is reasonable to expect that banks could have projected to increase their leverage to the levels
of investment banks, which was around 40 before the crisis. On the securitization side, from 1995 to 2005,
the volume of private-label mortgage backed securities increased dramatically from negligible levels to $
trillion, but disappeared with the
One could also consider that after the bust period, policy makers became wary of banks and had the
will and power to regulate the banks. Indeed, in the US, the Dodd-Frank Act (a federal law that passed in
2007) and the Federal Reserve’s stress tests imply tighter regulation than the ones seen before 1995. At the
global level, as well, the increase in the use of macroprudential policies and Basel III standards imply tighter
regulation. All these developments suggest that the bank leverage ratio may have become even lower than it
was in the pre-boom period. Therefore, the bust episode dynamics implied by our model can be thought of as
a lower limit.
4 Results
Before turning to the analysis of transition dynamics, it is useful to check the model’s performance in matching
some key life-cycle statistics that may be important for the soundness of the quantitative exercise. The life-
cycle implications of the model closely match the data (see Figure 4). The homeownership rate
increases over the life cycle, similar to the data. Mortgage debt relative to housing value declines with
age in both the data and the model. But it declines more in the model compared to the data. Average
consumption and housing consumption in the model more than double over the life cycle and are very close
to the values reported in the literature, such as Aguiar and Hurst (2013).
Transmission of the Shock and Banking Sector Dynamics
It will be instructive to illustrate how the leverage shock translates into changes and amplification in the
bank lending rate. Focusing first on the steady state, an increase in bank leverage decreases the collateral
premium
rϑ r =
1 −βL(1 + r) .
λβL
Thus, a permanent increase in λ will eventually lead the economy to a steady state with a lower interest
rate. Moreover, when the bank net worth effects are absent, changes in λ will translate
21The ease of securitization increased the liquidity in the housing market and led lenders to extend credit to marginal
borrowers, increasing the credit supply (Keys et al. (2012)).
^
∫
t+1
^
t t
t
Figure 2: Life-Cycle Properties: Model vs. Data
100
80
60
40
20
0
30 40 50
Age
60 70
1
0
30 40 50
Age
60 70
30 40 50 60 70
Age
Notes: The graph shows the life-cycle properties of housing and mortgage debt. The left panel plots the homeown- ership rate.
The middle panel plots mortgage debt relative to housing value. The data come from 1995 Survey of Consumer Finances.
The right panel plots the log difference of housing expenditure at a given age from its level at age 25.
into changes in rϑ during the transition, as given by this equation. As a result, the equilibrium interest
rate gradually falls during the boom and reverts back to the steady state level after the
bust. However, changes in bank net worth amplify the changes in the collateral premium, which turns out
to be significant during the bust. We will explain this amplification mechanism next.
Although all variables of interest affect each other simultaneously, we will proceed with an iterative
approach in demonstrating the amplification mechanism. For this purpose, remember that the bank net worth
in period t is given as
ωt =
∫ ∫ .
mt
.
θ J
Σ
+ pt
.
θ J
ΣΣ
Π
.
θ J |θ
Σ
Γt−1 (θ) + Lk (1 + rϑ) −Bt (1 + rt) .
The shock that generates the bust is essentially a decrease in λt back to its steady-state level, which reduces
the loan supply through Lt+1 = βLλ̂ As a result, the equilibrium bank lending rate rϑ+1 increases.
However, a higher rϑ+1 reduces the bank’s net worth today by lowering mortgage valuations since
p (θ) =
1
1 + rϑ+1
l
t+1
θJ
.
θ J
Σ
Π
.
θ J |θ
Σ
for all θ,
where vl (θ J) = mt+1 (θ J) + pt+1 (θ J). In response, loan supply Lt+1 declines further and rϑ+1
increases more. With higher rϑ+1, mortgage valuations and bank net worth declines further, which
generates further increases in rϑ+1 and future bank lending rates. This is the key mechanism through
which the deterioration of bank balance sheets amplifies the transmission of a shock to bank leverage. Figure 3
shows the dynamics of the banking sector variables. In the top left panel of Figure
3, we report the evolution of the bank lending rate rϑ; and in the top middle panel, we report
evolution of the bank’s net worth, which is intimately linked to rϑ. Remember that rt (the bank
22
λt is an endogenous object determined by equations 10 and 12. The parameter that goes back to its steady-state level is ξ,
which decreases ^t to its steady-state level.
θ
Model
Data
Model
Data
Consumption
Housing
v
θJ
funding rate) is constant. The amplification arising from the bank balance sheet deterioration is the
difference ∆rϑ −∆rt > 0. Due to the decline in mortgage valuations (since rϑ is higher) as well as the
increase in foreclosures, bank net worth declines sharply at the time of the bust. However, the spike in rϑ
and the sharp drop in bank net worth turn out to be short-lived. This is because the amplification
mechanism that creates the sharp drop works exactly the opposite way in the recovery. When rϑ starts
coming back, the market value of the bank’s mortgage portfolio starts recovering, which increases the
bank’s net worth, which in turn allows the bank to further extend credit, reducing rϑ’s even more. As a
result, bank net worth recovers very quickly.
The model generates a 20 percent rise in bank assets in the boom. Since the loan supply increases with the
extended leverage possibilities, the equilibrium bank lending rate declines. The value of the mortgage pool
that the financial system holds increases. However, with a lower equilibrium bank lending rate, bank net
worth declines during the boom. Overall, the banking sector supports more credit with lower bank net worth
but with higher debt. Crisis occurs as the leverage constraint reverses to the initial steady-state level: the bank
lending rate jumps to around 9 percent (Figure 3), and mortgage valuations and bank net worth sink
substantially. However, as we discussed earlier, banks recover quickly. As mortgages are long-term assets,
banks cannot flexibly adjust their balance sheets by issuing fewer mortgages. Therefore, they reduce their
lending to firms (about percent lower than the peak of the boom).
The dynamics of interest rates and bank loans are consistent with the findings in the literature. During
the boom period, interest rates on firm loans and mortgages declined (Glaeser et al. (2012a) and Justiniano
et al. (2017)). On the effects of deregulation on the interest rates, both Jayaratne and Strahan (1997)
and Favara and Imbs (2015) find significant declines in lending interest rates after the branching
deregulation in the US. For the crisis period, Ivashina and Scharfstein (2010) document a more than 50
percent decline in bank real investment loans to In parallel, the lending interest rate on
loans more than quadrupled (Adrian et al. (2013)) and credit spreads spiked during the financial crisis
(Gilchrist and Zakrajšek (2012)).
The model implied bank leverage dynamics (based on both book and market values) are consis- tent with
the data. The lower left panel shows that the marked-to-market bank leverage (Lt+1/Nt) increases during the
boom, spikes at the time of the bust as asset valuations decline and bank net worth sinks, and declines
afterward. Consistent with our model’s implications, Begenau et al. (2018) find that the market leverage of
listed banks increased during the boom and spiked during the We also construct book values of bank
assets and equity, and compute marked-to-book leverage, which we have data for. The lower-middle panel
compares the percentage change in book leverage in the model with those of commercial banks and security
brokers-dealers in the
23Real investment loans include capital expenditure and working capital loans.
24See also He et al. (2010) for the role of asset valuations on bank leverage during the bust.
25Consistent with what we report here, there is broad agreement that marked-to-book leverage is procyclical (Adrian and Shin
(2010), Nuno and Thomas (2017), and Coimbra and Rey (2017)).
During the boom, the leverage increases by 35 percent in the model as it is calibrated. The corre- sponding
changes are 25 percent and 50 percent for commercial banks and securities brokers-dealers. The leverage does
not change significantly at the time of the bust shock (2008), declines by 52 per- cent in the period after
(2010), and recovers slowly. The leverage of commercial banks and security brokers-dealers decline by 27
and 71 percent from 2008 till 2010.
The share of mortgages in bank assets (the lower right panel) increases by 6 percent–matching the data
counterpart–due to the rises in house prices, refinancing, and the decline in equilibrium down payment
amounts. Finally, another implication of the leverage shock is an increase in capital inflow during the boom
and a sharp decline in the bust. In the data too, the net capital inflow to GDP ratio increased from about 1
percent of GDP in 1996 to 5 percent of GDP in 2007 and declined to − percent by the first quarter of
2008. Since then, it has been hovering around 1 percent of GDP.
Figure 3: Bank Balance Sheet Dynamics
1996 2008 2018
0
-20
-40
-60
-80
1996 2008 2018
30
20
10
0
1996 2008 2018
140
120
100
80
60
40
20
1996 2008 2018
60
40
20
0
-20
-40
1996 2008 2018
0
1996 2008 2018
Notes: The graph plots the dynamics of key banking variables during the boom-bust episode. The shock is an unexpected
ease and then an unexpected reversal of borrowing-lending constraints of the banks where bank leverage increases from 9 to
from 1995 to 2007. Total data for bank loans include home and multi-family residential mortgages, and firm loans and
miscellaneous liabilities. The data for book leverage of banks is from Federal Reserve Bank of New York (2020) and of
security brokers-dealers is from Federal Reserve Board (2019). For the mortgage share in bank loans, we report changes
relative to 1990-1995 average.
Security Brokers-Dealers
Commercial Banks
Benchmark
Output Dynamics
The strong macroeconomic environment in the US during the boom period was partly seen as the driving
force behind the boom in the housing market. For instance, per capita output was 6 percent and per capita
labor income was 8 percent higher than their linear trends. Our model also features a strong macroeconomic
outlook during its boom phase as a response to the relaxation of the bank leverage constraint (Figure 4).
Output increases around 3 percent during the boom period. Capital, labor utilization, and wages increase
around 2-4 percent.
With the crisis, macroeconomic conditions reverse sharply: output, and labor income decline steeply
(5-10 percent). Even though the shock’s impact on the banking sector is short-lived, sur- prisingly the real
sector recovers very slowly. Overall, the credit supply shock can account for 40-60 percent (depending on
the variable) of the increase during the boom period and more than 50 percent of the decline during the
bust period.
In the data, too, there has been very slow recovery. Motivated by this slow recovery, the “secular
stagnation” view suggests that some structural changes may have happened, and it may not be possible to
reach the earlier trend. The findings in our paper present the possibility of an alternative view that builds
on the sharp and temporary decline in capitalists’ income. With a large loss in income, the capitalists reduce
their investment by about 30 percent. As a result, the capital stock declines by 8 percent and recovers
slowly. Even after 10 years, the capital stock is still more than 1 percent below its steady-state level. This
persistent decline of capital is key for generating the persistent decline in output and wages.
The response of the firms’ labor demand to the changes in the bank lending rate is the key driver of the
boom-bust in the production sector. Our model generates changes in total per capita hours worked, labor
income, and firm loans qualitatively similar to the There is extensive evidence that financial
conditions indeed affect firm labor decisions, providing evidence for the mechanisms in our model. For
example, Chodorow-Reich (2013) finds that firms that worked with weaker banks prior to the crisis, reduced
employment more. Benmelech et al. (2019) find similar evidence from the depression era, and Popov and
Rocholl (2015) bring evidence from Germany during the 2008 crisis. Finally, Ivashina and Scharfstein
(2010) document a more than 50 percent decline in bank real investment loans to corporations, and Adrian
et al. (2013) find that real investment loans to firms have declined substantially, while interest rates on
loans more than quadrupled during the crisis.
Housing Market Dynamics
Figure 5 illustrates the dynamics of housing market variables. In response to the increase in bank lending,
house prices increase around percent in the model. With the reversal of the shock, it
26We compare the labor utilization from the model to the hours per worked in the data.
Figure 4: Macroeconomic Dynamics
6
4
2
0
-2
-4
-6
40
20
0
-20
-40
1996 2008 2018
1996 2008 2018
6
4
2
0
-2
-4
10
0
-10
-20
-30
1996 2008 2018
1996 2008 2018
4
2
0
-2
1996 2008 2018
1996 2008 2018
5
0
-5
-10
1996 2008 2018
10
5
0
-5
-10
-15
-20
1996 2008 2018
30
20
10
0
-10
1996 2008 2018
Notes: The graph plots the dynamics of production sector and consumption during the boom-bust episode. Con- sumption,
output, investment, labor income, and firm loans data are percentage deviations from their linear trends obtained from 1985-
2006 period.
Figure 5: Housing Market Dynamics
30
20
10
0
-10
1996 2008 2018
40
30
20
10
0
-10
1996 2008 2018
1996 2008 2018
80
60
40
20
0
1996 2008 2018
1996 2008 2018
4
3
2
1
0
-1
1996 2008 2018
Notes: The graph plots the dynamics of housing market variables during the boom-bust episode. House price data is the
percentage deviation from the linear trend obtained from 1985-2006 period.
undershoots and declines by percent. Afterward, it slowly converges to its initial steady-state value.
Overall, the model’s implications regarding house prices are in line with the US housing price dynamics.
Quantitatively, a credit supply shock by itself can generate more than one third of the boom and almost all
of the bust in house
The model generates a modest rise in the homeownership rate compared to the data during the boom
but generates a significant decline during the bust. The main reason is that while the decline in mortgage
rates makes owning more affordable, the rise in the price-rent ratio increases the cost of owning a house
relative to renting. We could have generated a further increase in the homeownership rate by imposing and
then relaxing borrowing constraints on households (., LTV or PTI or both). However, this extension
would blur the effects of the leverage shock. Therefore, we choose not to have them in our benchmark.
The model generates a qualitatively similar but quantitatively smaller changes in the price-rent ratio: the
rise in the price-to-rent ratio is 7 percent during the boom accounting for about 20 of the rise in the data.
Several mechanisms are important for the dynamics of the price-to-rent ratio.
27As we have already mentioned, several factors might have contributed to the boom-bust in house prices: cheap borrowing
conditions (Favara and Imbs (2015), Glaeser et al. (2012b), Garriga and Hedlund (2018), and Garriga et al. (2019)),
securitization and subprime lending (Mian and Sufi (2009)), and optimistic expectations (Kaplan et al. (2020)). Therefore, it
would be unrealistic to expect and/or force the changes in bank lending rate to account for all the movements in the house prices.
First, as can be seen from Equation 9, a rise in house prices in the following periods, a lower
financing cost (r̃t+1), and higher current homeownership rate decrease equilibrium rental prices in the current
period. During the boom period, since house prices are not steep after the initial jump and the
homeownership rate barely moves, the price-rent ratio does not increase as much as in the data. A lower
financing cost helps to keep rental costs lower and contributes to the increase in the price-to-rent ratio.
During the bust, the increase in foreclosures significantly lowers homeownership rate and thus, mitigates the
decline in rental prices. As a result, the price-rent ratio declines by 15 percent, which accounts for almost
half of the decline in the data.
Household debt increases 35 percent in the model during the boom period. Consistent with the data,
household leverage increases less since house prices also increase. During the bust, both debt and leverage
gradually converge to their steady-state levels. The rise of home equity extraction and refinancing
activity during the boom period in the US were partly responsible for the rise of household debt and
leverage (Mian and Sufi (2011)). In the model, we do not have home equity extraction. However,
households can refinance and withdraw some cash from their home equity. During the initial boom period,
refinancing activity jumps to 20 percent from percent and returns to low levels in the following
periods. Both higher house prices and lower interest rates cause an increase in refinancing volume. Unlike
in the data, however, refinancing does not stay high for the whole boom episode, since after the initial shock
there is perfect foresight.
The foreclosure rate has been very low in the data (on average, 1 percent annually) before the crisis.
With the crisis, it increased by 4 percentage points (annual). The foreclosure rate in the model stays low
during the boom and jumps by percentage points in the bust period as the more than percent
decline in house prices, combined with 9 percent decline in income, during the bust pushes many households
to negative equity, which makes default an attractive option.
Consumption Dynamics
The model generates a significant boom-bust in consumption: it increases by more than 4% during
the boom and contracts by about 10% during the bust (Figure 4). Like many other macro variables in the
model, the recovery takes a long time.
The declines in house prices and labor income are two important channels that derive con- sumption
drop in the bust. To disentangle the role of each one, we ask how much the aggregate consumption would
drop if we fix all prices at their boom levels and feed only equilibrium house prices or only equilibrium
wages. The first analysis implies that the decline in house prices by itself generates 25% of the drop in
consumption while the second one implies that the decline in wages by itself generates 74% of the total
decline in However, there is an indirect effect of labor income on consumption that this
exercise does not capture because house prices are also
28The implied elasticity of consumption with respect to house prices is , which is at the lower end of the estimates reported in
Berger et al. (2018).
endogenously affected by labor income. We analyze its total effect on consumption in the next section.
5 The Drivers of the Results
In this section, we explore three mechanisms that are relevant for the boom-bust in house prices and
consumption: general equilibrium feedback from credit supply to household labor income, the
amplifications arising from the deterioration of bank balance sheets during the bust, and the existence of
highly leveraged households.
The Roles of Labor Income and the Bank Lending Rate
The changes in bank leverage influence model dynamics through their effects on the bank lending rate.
The changes in the bank lending rate affect households both “directly” via borrowing costs and “indirectly”
through affecting labor income. The 4 percent increase in labor income during the boom and the 9 percent
decline during the bust, as firms adjust their labor demand in response to the changes in the cost of
funding, affect households’ consumption and housing demand. To isolate these direct and indirect effects,
we solve two versions of our model where we keep wages and the bank lending rate constant separately at
their initial steady-state levels and analyze how the boom-bust cycles differ from our benchmark economy.
We present the results of this analysis in Figure 6.
Our results suggest that the changes in labor income have large effects on the dynamics of house prices
and consumption. With labor income, house prices increase 6 percent during the boom and decline 8 percent
at the bust, which is about half of the size of the boom-bust in house prices in the benchmark economy. The
boom-bust in consumption is significantly reduced when we hold wages constant. The direct effect of the
bank lending rate is equally important for house prices; however, its direct effect on consumption is limited:
a 1 percent increase in consumption during the boom followed by around a 1 percent decline during the
bust.
Kaplan et al. (2020) study a similar framework with aggregate uncertainty and reach the conclu- sion that
credit conditions (LTV , PTI, mortgage origination, and temporary interest rate shocks) cannot generate a
significant boom-bust cycle in house prices. Our analysis differs from theirs in three important aspects.
First, the credit supply shock in our framework is not an isolated shock to the household borrowing rate; it
also affects labor income. Our findings in this section suggest that about half of our results are driven by this
channel. The second major difference is the persistence of the shocks. The shocks in our framework
correspond to the changes in banking regulation that are more likely to be permanent. Not surprisingly,
the boom-bust cycle gets amplified when the shock is more Finally, there is a critical
difference between LTV and PTI shocks and
29We have experimented with temporary shocks to bank leverage. The effects on housing market as well as on the rest of the
economy are much smaller with temporary leverage shocks.
Figure 6: The Role of Labor Income and Bank Lending Rate rϑ
4
2
0
-2
-4
-6
1996 2008 2018
1996 2008 2018
10
5
0
-5
1996 2008 2018
4
2
0
-2
-4
1996 2008 2018
Notes: The graph plots the dynamics of wages, house prices, and consumption. For the “Fixed Wages” exercise, we impose the
same bank lending rate dynamics that arise in the benchmark economy. For the “ rϑ Effect” exercise, we
keep the wages at the steady-state level, as shown in top left panel, and shock the economy with rϑ boom and bust
sequences of the benchmark economy (top right panel).
permanent changes in the bank lending rate. Relaxation of LTV and PTI constraints shift housing demand
from renting to owning. Since households can rent a house of the desired size, these shocks do not
significantly affect aggregate housing demand. On the other hand, a permanently lower bank lending rate
creates a significant income effect—since mortgage payments decline for a given debt amount–and a wealth
effect—since labor income permanently increases—, and thus, increases the total housing demand.
The Role of Bank Balance Sheet Deterioration in Amplifying the Bust
In the bust period, the credit supply declines not only because of the exogenous tightening of the bank
leverage constraint but also because of the endogenous deterioration of bank balance sheets, which further
tightens the banks’ leverage constraint and significantly amplifies the bust. In this section, to quantify the
role of the deterioration of bank balance sheets on the aggregates during the bust, we eliminate the decline in
bank net worth in the bust and analyze the equilibrium The red line in Figure 7 shows the model
dynamics for this exercise, and the blue line shows the
30Essentially, we solve the transition of the economy starting with the bust distribution but with bank net worth fixed at the
boom level in the first period of the bust. We focus on the bust period only, since the bank balance sheet channel has a small
effect during the boom period.
dynamics for the benchmark. The difference between the blue and red lines indicates the role of bank
balance sheet deterioration in amplifying the bust, which we report under the “BBS” column in Table 4.
Overall, we find that the deterioration of the bank balance sheet significantly amplifies the bust. For
example, we
