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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 1032, October 2011 --- Screen Reader
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NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

We study housing and debt in a quantitative general equilibrium model. In the cross-section, the model matches the wealth distribution, the age profiles of homeownership and mortgage debt, and the frequency of housing adjustment. In the time-series, the model matches the procyclicality and volatility of housing investment, and the procyclicality of mortgage debt. We use the model to conduct two experiments. First, we investigate the consequences of higher individual income risk and lower downpayments, and find that these two changes can explain, in the model and in the data, the reduced volatility of housing investment, the reduced procyclicality of mortgage debt, and a small fraction of the reduced volatility of GDP. Second, we use the model to look at the behavior of housing investment and mortgage debt in an experiment that mimics the Great Recession: we find that countercyclical financial conditions can account for large drops in housing activity and mortgage debt when the economy is hit by large negative shocks.

Keywords: Housing, housing investment, mortgage debt, life-cycle models, income risk, homeownership, precautionary savings, borrowing constraints

JEL classification: E22, E32, E44, E51, D92, R21

This paper studies the business cycle and the life-cycle properties of housing investment and household mortgage debt in a quantitative general equilibrium model. To this end, we modify a life-cycle model with uninsurable individual income risk to allow for aggregate uncertainty and for an explicit treatment of housing. We introduce housing by modeling its role as collateral, its lumpiness, and the choice of renting versus owning; these features have, to a large extent, eluded existing business cycle models of housing.

At the cross-sectional level, our model accurately reproduces the U.S. wealth distribution, and replicates the life-cycle profiles of housing and nonhousing wealth. The young, the old and the poor are renters and hold few assets; the middle-aged and the wealth-rich are homeowners. For a typical household, the asset portfolio consists of a house and a large mortgage. The model also reproduces frequency and size of individual housing adjustment: because of nonconvex adjustment costs, homeowners change house size infrequently but in large amounts when they do so; renters change house size often, but in smaller amounts. Over the business cycle, the model replicates two empirical characteristics of housing investment: its procyclicality and its high volatility. In addition, the model matches the procyclical behavior of household mortgage debt. To our knowledge, no previous model with rigorous micro-foundations for housing demand has reproduced these regularities in general equilibrium.

We use the model to look at the role of the housing market in two events of the recent U.S. macroeconomic history: the Great Moderation and the Great Recession.

**Debt and Housing in the Great Moderation.** We study how higher household income risk and lower downpayments
affect the sensitivity of debt and housing to macroeconomic shocks.
Higher risk and the reduction in downpayments occurred around the
1980s, around the beginning of the Great Moderation,^{1} and
are potentially important determinants of housing demand and
housing tenure: higher risk should make individuals reluctant to
buy large items that are costly to sell in bad times; lower
downpayments should encourage and smooth housing demand. Their role
could be relevant given two observations on the post-1980s period
(see Figure 1 and Table 1). First, the volatility of housing
investment has fallen more than proportionally relative to GDP;
second, the correlations between mortgage debt and GDP and mortgage
debt and aggregate consumption have roughly halved, from
0.78 to 0.43 and from
0.72 to 0.37
respectively.^{2} In line with the data, we find that
lower downpayments and larger idiosyncratic risk reduce the
volatility of housing investment, and reduce the correlation
between mortgage debt and economic activity. Lower downpayments
provide a cushion to smooth housing demand; increase homeownership
rates, raising the number of people who do not change their housing
consumption over the cycle (relative to an economy with a large
number of renters who can become first-time buyers); lead to higher
debt, creating a mechanism that weakens the correlation between
output and hours. Higher idiosyncratic risk makes wealth-poor
individuals more cautious: these individuals adjust consumption,
hours, and housing by smaller amounts in response to aggregate
shocks. This mechanism is pronounced for housing purchases, since a
house is a large item that is costly to purchase and sell; and is
reinforced by low downpayments, since low downpayments allow people
to borrow more, increasing the utility cost of buying and selling
when net worth is lower. Together, lower downpayments and higher
risk can explain about 15 percent of the reduction in
the variance of GDP, 60 percent of the reduction in
the variance of housing investment, and the decline in the
correlation between debt and GDP.

**Debt and Housing in the Great Recession.** During the 2007-2009 period, changes in financial conditions are
likely to have made the recession worse. In particular, the housing
market appears to have been held back - more than other sectors -
by tighter credit conditions and higher borrowing costs. In
hindsight, it looks like housing did not stabilize the economy
during the recession. We use the model to determine the extent to
which housing can smooth regular business cycle shocks but amplify
extremely negative ones, by defining "Normal Recessions" as
periods of low aggregate productivity, and "Great Recessions"
periods of low aggregate productivity coupled with tight credit
conditions. When we do so, we find an interesting nonlinearity:
higher risk and lower downpayments can make housing and debt more
stable in response to small positive and negative shocks (as in the
Great Moderation), but can make it more fragile in response to
large negative shocks (as in the Great Recession).

**Previous Literature.** Two strands of literature study the role of housing in the
macroeconomy. On the one hand, business cycle models with housing -
Greenwood and Hercowitz (1991), Gomme, Kydland and Rupert (2001),
Davis and Heathcote (2005), Fisher (2007) and Iacoviello and Neri
(2010) - match housing investment well, but abstract from a
detailed modeling of the microfoundations of housing demand; these
models feature no wealth heterogeneity, no distinction between
owning and renting, and unrealistic transaction costs. On the other
hand, incomplete markets models with housing - Gervais (2002),
Fernandez-Villaverde and Krueger (2004), Chambers, Garriga and
Schlagenhauf (2009), and Díaz and Luengo-Prado (2010) - have
a rich treatment of the microfoundations of housing demand, but
ignore aggregate shocks: however, because these papers model
individual heterogeneity, they are better suited to study issues
such as debt, risk, and wealth distribution.

Our model combines both strands of literature. Others have also
done so, albeit with a different focus. Silos (2007) studies the
link between aggregate shocks and housing choice, but does not
model the own/rent decision and assumes convex costs for housing
adjustment.^{3} Fisher and Gervais (2007) find that
the decline in housing investment volatility is driven by a change
in the demographics of the population together with an increase in
the cross-sectional variance of earnings. Their approach sidesteps
general equilibrium considerations. Kiyotaki, Michaelides and
Nikolov (2011) use a stylized life-cycle model of housing tenure to
study the interaction between borrowing constraints, housing
prices, and economic activity. Favilukis, Ludvigson and Van
Nieuwerburgh (2009) use a two-sector RBC model with housing that
also considers the interaction between borrowing constraints and
aggregate activity, but address a different set of questions than
we do. Finally, Campbell and Hercowitz (2005) study the impact of
financial innovation on macroeconomic volatility in a model with
two household types. In their model, looser collateral constraints
weaken the connection between constrained households' housing
investment, debt accumulation and labor supply through a mechanism
that shares some features with ours; however, their model does not
study the interaction between life cycle, risk and housing demand,
which are important elements of our story.

Our economy is a version of the stochastic growth model with overlapping generations of heterogeneous households, extended to allow for housing investment, collateralized debt and a housing rental market. Aggregate uncertainty is introduced in the form of a shock to total factor productivity. Individuals live at most periods and work until age Their labor endowment depends on a deterministic age-specific productivity and a stochastic component. After retirement, people receive a pension. Each period, the probability of surviving from age to is . Each period a generation is born of the same measure of dead agents, so that the total population, which we normalize to 1, is constant. When an agent dies, he is replaced by a descendant who inherits his assets.

At each point in time, agents differ by their age and
productivity; moreover, we assume that agents differ in their
degree of impatience. We do so for two reasons: first, a large
literature (see Guvenen, 2011) suggests that preference
heterogeneity may be an important source of wealth inequality. For
example, Venti and Wise (2001) study wealth inequality at the onset
of retirement among households with similar lifetime earnings and
conclude that the dispersion must be attributed to differences in
the amount that households choose to save.^{4} Second, we want a model
that generates average debt and wealth dispersion as in the data,
and a model with discount factor heterogeneity works remarkably
well in this regard (our robustness analysis discusses the
properties of the model with a single discount factor).

**Household Preferences and Endowments.** Households receive utility from consumption ,
leisure
(where
is the time endowment), and
service flows from housing, which are proportional to
the housing stock owned or rented. The momentary utility function
is:

. | (1) |

Above, if (the
individual owns), while if
(the individual rents). The assumption for
implies that a household experiences a utility gain when
transitioning from renting to owning, as in Rosen (1985) and
Poterba (1992). We also assume that homeowners need to hold a
minimum size house __, and that rental units
may come in smaller sizes than houses, allowing renters to consume
a smaller amount of housing services, as in Gervais (2002). The log
specification over consumption and housing services follows Davis
and Ortalo-Magné (2011) who find that, over time and across
cities, the expenditure share on housing is constant.__

Time supplied in the labor market is paid at the wage
. The productivity endowment of an agent
at age is given by
where is a
deterministic age-specific component and is a
shock to the efficiency units of labor,
.
The shock follows a Markov process with transition matrix
and stationary distribution
. The total
amount of labor efficiency units
and of age-specific productivity values
are constant and normalized to one. From
onwards labor efficiency is
zero () and agents live off their pension
and their accumulated wealth. Pensions are
fully financed through the government's revenues from a lump-sum
tax paid by workers.^{5} Total net income
at age in period is denoted by
. Then:

if if . | (2) |

Households start their life with endowments and the accidental bequests left
by a dead agent. They can trade a one-period bond
which pays a gross interest rate of . Positive
amounts of this bond denote a debt position.^{6} Households cannot
borrow more than a fraction of their
housing stock and a fraction of their
expected earnings:

. | (3) |

Above,
approximates the present discounted value of lifetime labor
earnings and pension.^{7} The motivation for this borrowing
constraint is realism: we want to study mortgage debt and we want
to have a constraint which prevents the elderly from borrowing too
much late in life (when the present discounted value of earnings is
low), as in the data. The constraint is also consistent with
typical lending criteria in the mortgage market that take into
account minimum downpayments, ratios of debt payments to income,
current and expected future employment conditions.^{8}
Finally, we assume that an owner incurs a transaction cost whenever
he adjusts the housing stock:
if
.
This assumption captures common practices in the housing market
that require, for instance, fees paid to realtors to be equal to a
fraction of the value of the house being sold. Summing up,
households maximize expected lifetime utility:

(4) |

where denotes expectations at age , is a deterministic preference shifter that mimics changes in household size, and is a household-specific discount factor. In the calibration, we assume that households are born either impatient (low ) or patient (high ).

**Financial Sector and Housing Rental Market.** A competitive financial sector collects deposits from households
who save, lends to firms and households who borrow, and buys
capital to be rented in the same period to tenants. The financial
sector can convert the final good into housing and capital at no
cost. This assumption ensures that the consumption prices of
housing and capital are constant. Let be the
price of one unit of rental services. Then a no-arbitrage condition
holds such that the net revenue from lending one unit of financial
capital must equal the net revenue from renting one unit of housing
capital,

(5) |

at any where
is the depreciation rate of the
housing stock.^{9}

**Production.** The goods market is competitive and characterized by constant
returns to scale, so that we consider a single representative firm.
Output is produced according to

(6) |

where and are total capital and labor input; is the capital share, and is a shock to total factor productivity. This shock follows a Markov process with transition matrix . The aggregate feasibility constraint requires that production of the good equals the sum of aggregate consumption investment in the stock of aggregate capital investment in the stock of aggregate housing and total transaction costs incurred by homeowners for changing housing stock, denoted by :

(7) |

with and denoting the depreciation rates of housing and capital, respectively.

**The Household Problem and Equilibrium.** Denote with
the distribution of households over earnings shocks, asset
holdings, housing wealth, discount factors and ages in period
Without aggregate uncertainty, the economy
would be in a stationary equilibrium, with an invariant
distribution and constant prices. Given aggregate
volatility, this distribution will change over time. When solving
their dynamic optimization problem, agents need to predict future
wages and interest rates. Both variables depend on future
productivity and aggregate capital-labor ratio, which in turn are
determined by the overall distribution of individual states. As a
consequence, the distribution - and its
law of motion - is one of the aggregate state variables that agents
need to know in order to make their decisions (together with total
factor productivity). This distribution is an infinite-dimensional
object, and its law of motion maps an infinite-dimensional space
onto itself, which imposes a crucial complication for the solution
of the model economy. To circumvent this problem, we adopt the
strategy of Krusell and Smith (1998) and let agents use one moment
of the distribution - the aggregate capital
stock - in order to forecast future prices. As
documented in Appendix A, using one moment only allows us to obtain
a fairly precise forecast, as measured by the of the forecasting equations, which are between
0.99 and 1.^{10}

We write the household optimization problem recursively. The individual states are productivity debt and housing wealth . We assume that agents observe beginning of period capital and approximate the evolution of aggregate capital and labor with linear functions that depend on the aggregate shock Denote the vector of individual and aggregate states. The dynamic problem of an age household is:

(8) |

where and are the value functions if the agent owns and rents, respectively, and corresponds to the decision to own. The value of being a homeowner solves:

(9) | |

s.t. | |

. |

Here and are linear functions in whose parameters depend on the . They denote the law of motion of the aggregate state, which agents take as given.

The value of renting a house is determined by solving the problem:

(10) | |

s.t. | |

At the agent's last age, for any .

We are now ready to define the equilibrium for this economy.

A recursive competitive equilibrium consists of value functions policy functions for each age and period , prices , and aggregate quantities and for each taxes and pensions and laws of motion and such that at any :

Agents optimize: Given , and the laws of motion and , the value functions solve the individual's problem, with the corresponding policy functions.

Factor prices and rental prices satisfy:

(11) | |

(12) | |

. | (13) |

Markets clear:

(labor market), | (14) |

(goods market) | (15) |

where and are defined as:

, | (16) |

(17) |

The government budget is balanced:

. | (18) |

The laws of motion for the aggregate capital and aggregate labor are given by

. | (19) |

Appendix A provides the details on our computational strategy.

Our calibration is summarized in Table 2. One period is a year. Agents enter the model at age 21, retire at age 65, and die no later than age 90. The survival probabilities correspond to the survival probabilities for men aged 21-90 from the U.S. Decennial Life Tables for 1989-1991. Each period, the measure of those who are born is equal to the measure of those who die. The age polynomial , which captures the effect of demographic variables in the utility function, is taken from Cagetti (2003) and approximated using a fourth-order polynomial (see Figure 2). After normalizing the household size to 1 at age 21, the household size peaks at 2.5 at age 40, and declines thereafter.

We take the deterministic profile of efficiency units of labor
for males aged 21-65 from Hansen (1993) and
approximate it using a quadratic polynomial (see Figure 2). Upon
retirement, an agent receives a pension equal to 40
percent of the average labor income.^{11} The idiosyncratic
shock to labor productivity is specified as:

, | (20) |

which we approximate with a three-state Markov process following Tauchen (1986). There is a vast literature on the nature and specification of a parsimonious yet empirically plausible income process: the bulk of the studies (see Guvenen, 2011) look at earnings (rather than wages) and estimate persistence coefficients ranging from 0.7 to 0.95. Exception are Floden and Lindé (2001), who use PSID data to estimate an AR(1) process for wages similar to ours and find an autocorrelation coefficient of 0.91; and Card (1991), who finds an AR(1) coefficient of 0.89. Based on this evidence, we set and conduct robustness analysis in Section 8, based on evidence from other studies that we review in Appendix B. The standard deviation of the labor productivity process is set at (see Appendix B). Later, we increase to 0.45 to capture the increased earnings volatility of the 1990s, and to study the consequences for macroeconomic aggregates of increased risk at the household level, as emphasized by Moffitt and Gottschalk (2008) and Dynan, Elmendorf and Sichel (2007).

We assume that there are two classes of households, a "patient" group with a discount factor of 0.999 (one third of the population) and an "impatient" group with a discount factor of 0.941 (two thirds of the population). The high discount factor pins the average real interest rate down to 3 percent. The low discount factor is in the range of estimates in the literature (see, for instance, Hendricks, 2007). The gap between discount rates and the relative population shares deliver a Gini coefficient for wealth around 0.75, close to the data. In Section 8 we discuss the properties of the model when we assume that all people have identical discount rates. We set and the endowment of time ; these parameters imply that time spent working is 40 percent of the agents' time.

We set the weight on housing in utility at and the depreciation rate for housing
. These parameters yield
average housing investment to private output ratios around
7 percent, and a ratio of the housing stock
to output 1.4. These values are in accordance with
the National Income and Product Accounts and the Fixed Assets
Tables.^{12} Finally, the housing transaction
cost is set at based on estimates from the
National Association of Realtors (2005).^{13} Section 8 conducts
robustness analysis for alternative values of and
.

We set and These values yield an average capital to output ratios around 2.2 and average business investment to output ratios around 20 percent. The aggregate shock is calibrated to match the standard deviation of output in the data for the period 1952-1982. We use a Markov-chain specification with seven states to match the following first-order autoregression for the log of total factor productivity:

. | (21) |

We set and . After rounding, the first number mimics a quarterly autocorrelation rate of productivity of 0.979, as in King and Rebelo (1999). The second number is chosen to match the standard deviation of model output to its data counterpart.

Our baseline calibration sets the maximum loan-to-value ratio
at 0.75 We increase
to 0.85 in the
calibration for the late period. The value of is set at 0.25 in the baseline and
raised to 0.5 in the late period: with these
numbers, the income constraint only binds late in life, preventing
old homeowners from borrowing. Aside from this, our choice for
is of small importance for the model
dynamics. Lastly, the minimum-size house available for purchase
(__)costs 1.5 times the average annual pre-tax household
income.__^{14} Together with the minimum house
size, the parameter that has a large impact on homeownership is the
utility penalty for renting
. We set
to obtain a homeownership rate
of 64 percent, as in the data for the period
1952-1982.

**Household Behavior.** At each stage in the life, the household chooses consumption,
saving, hours, and housing investment by taking into account
current and expected income, and liquid assets and housing position
at the beginning of the period. Here, we mostly focus on housing
decisions, since other features of the model are in line with
existing models of life-cycle consumption and saving behavior. We
defer illustrating labor supply behavior to the next section, when
we discuss the model dynamics in response to aggregate shocks.

It is simple to characterize the behavior of agents depending on whether they start the period as renters or homeowners. For renters, the housing choice is as follows: given the initial state, there is a threshold amount of liquid assets ( in our notation) such that, if assets exceed the threshold, renters become homeowners. Also, the larger initial liquid assets are, the less likely a household is to borrow to finance its housing purchase.

Homeowners can stay put, increase house size, downsize or switch
to renting. Figure 3 plots optimal housing
choice as a function of initial house size and liquid
wealth.^{15} The downward sloping line plots the
borrowing constraint that restricts debt from exceeding a fraction
of its housing stock. As the figure
illustrates, larger liquid assets trigger larger housing. In
addition, buying and selling costs create a region of inaction
where the household keeps its housing constant. If liquid wealth
falls, the household either downsizes or switches to renting. One
feature of the model is that, for a household with very small
liquid assets, the housing tenure decision is non-monotonic in the
initial level of housing wealth. Consider, for instance, a
homeowner with liquid assets equal to about one. If the initial
house size is small, the homeowner does not change house size,
since, given the small amount of assets, the house size is closer
to its optimal choice. If the initial house is medium-sized, the
homeowner pays the adjustment cost and, because of his low liquid
assets, switches to renting. If the initial house size is large, it
is optimal to downsize, and to buy a smaller house.

**Life-Cycle Profiles.** Figure 4 plots a typical individual life-cycle
profile in our model. We choose an agent with a low discount factor
since the behavior of an agent with low assets and often close to
the borrowing constraint best illustrates the main workings of the
model. The agent starts life as a renter, with little assets and
low income. At the age of 22, he is hit by a
positive income shock, saves in order to afford the downpayment and
buys a house a year after. Prior to buying a house, the individual
works more: the positive income shock raises the incentive to work;
and such incentive is reinforced by need to set resources aside for
the downpayment. Following a series of above average income shocks
beginning at the age of 32, the agent buys a larger
house at the age of 39. This time, in order to
afford the larger house, the individual is much closer to his
borrowing limit. In particular, while he owns and is close to the
borrowing limit, hours move in the opposite direction to wage
shocks, rising in bad times (age 42), falling in good
times (age 45): such mechanism is explained in
detail in the next Section. As retirement approaches, the agent
pays back part of the mortgage, and works more. After retirement,
at the age of 70, he switches to a small rental unit,
before dying at the age of 90.

One dimension where it is illustrative to compare the model with
the data is the frequency of housing adjustment for
homeowners.^{16} Using the 1993 Survey of Income and
Program Participation, Hansen (1998) reports that the median
homeowner stays in the same house for about 8 years. Anily, Hornik,
and Israeli (1999) estimate that the average homeowner lives in the
same residence for 13 years. The corresponding number for our model
is 15 years.^{17}

Figure 5 compares the age profiles of housing, debt and homeownership with their empirical counterparts. Like the data, the model is able to capture the hump-shaped profiles of these variables. There are two discrepancies: as for mortgage debt, the model slightly underpredicts debt early in life, and overpredicts debt later in life. The model also underpredicts homeownership later in life: we believe that, late in life, the absence of any bequest motive and the need to finance consumption expenditure by selling the house more than offset the adjustment costs, thus generating a sharp decline in homeownership.

**The Wealth Distribution.** Our model reproduces the U.S. wealth distribution quite well.
The Lorenz curves for the U.S. economy and for our model economy
are reported in figure 6. The Gini coefficient for
wealth in the model is 0.73, and is about the same
as in the data (equal to 0.79). Our model still
underpredicts wealth inequality at the very top of the
distribution, both for housing and for total wealth. However, the
model does well at matching the fraction of wealth (both housing
wealth and overall wealth) held by the poorest 40
percent of the U.S. population, which has essentially no assets and
no debt. Instead, a model without preference heterogeneity would do
much worse: in Section 8 we show that the Gini
coefficient for wealth in the model with a single discount factor
is 0.53, much lower than in the data.

In the same vein, the model predicts a mortgage debt to GDP
ratio that is roughly in line with the data (0.31 vs. 0.34) and a fraction of liquidity
constrained agents that is consistent with the available empirical
estimates. Following Hall (2011), we take a model agent to be
liquidity-constrained if the holdings of net liquid assets are less
than two months (16.67% on an annual basis) of income.^{18}
Using this definition, 45% of households are
liquidity constrained.^{19} Jappelli (1990) estimates the share
of liquidity constrained individuals to be 20%.
Studies that have combined self-reported measures of credit
constraints from the Survey of Consumer Finances with indirect
inference from other datasets (such as the PSID), have typically
found that 20 percent is more likely to be a lower
bound. For instance, using evidence on the response of spending to
changes in credit card limits, Gross and Souleles (2002) argue that
the overall fraction of potentially constrained households is over
two thirds.

We now illustrate the propagation mechanism of aggregate shocks.
There are two aspects of heterogeneity that matter for aggregate
dynamics: one is exogenous, and reflects the assumption that
individuals have different abilities, planning horizons, and
utility weights. Because other papers have studied these features
in life-cycle models with aggregate shocks, we do not explore them
in detail here.^{20} Instead, we focus on the endogenous
component of heterogeneity, which reflects the fact that
individuals with different ages and income histories accumulate
different amounts of wealth over time; in turn, heterogeneity in
wealth implies different individual responses to the same
shock.

**Workings of the Model.** We focus on the response of aggregate hours to a technology
shock, since movements in hours are the key element of the
propagation mechanism in models that rely on technology shocks as
sources of aggregate fluctuations. In particular, we study how the
wealth distribution and its composition shape agents' responses to
shocks. To fix ideas, consider a stripped-down version of the
budget constraint of a working individual that keeps wealth
constant between two periods:
and
.^{21} Abstracting from
taxes and pensions, this implies the following budget
constraint:

(22) |

where
measures the resources besides wages that can be used to finance
consumption:^{22} the term
is net interest income;
the term
is the maintenance cost required
to keep housing unchanged. Different values of map into different positions of the agents along the
wealth distribution. For a wealthy homeowner (negative ), is positive and large, and wage
income is a small fraction of consumption . For a
renter, ; in addition, assuming that the renter
is not saving, , so that
too. For a homeowner with a mortgage (positive ), is negative. Normalize
and set aside idiosyncratic
shocks, so that at all times. Assuming that
stays constant, the log-linearized
budget constaint becomes, denoting with
where
is the steady-state value of a variable:

. | (23) |

This constraint can be interpreted as an equation dictating how much the household needs to work to finance a given consumption stream, given the wage. The larger the desired consumption the larger the required hours needed to finance the consumption stream, with an elasticity of hours to consumption given by consumption-wage income ratio . For a wealthy individual, is high and larger than one, since labor income is a small share of total earnings; for a renter without assets, ; for an indebted homeowner, , reflecting the need to use part of the earnings to finance maintenance costs and to service the mortgage. In other words, a wealthy person needs to increase hours by more than 1 percent to finance a 1 percent rise in consumption, since labor income is less than consumption; an indebted homeowner needs to increase hours by less than 1 percent to finance a 1 percent rise in consumption, because of the leverage effect; a renter without assets needs to increase hours 1 for 1 with consumption.

The other key equation determining hours is the standard labor supply schedule. Letting denote the steady-state Frisch labor supply elasticity, this curve reads as

. | (24) |

Combining equations 23 and 24 yields:

. | (25) |

Take the wage as the exogenous driving force of the model, since an exogenous rise in productivity exerts a direct effect on the wage. Whether the rise in the wage leads to an increase in hours depends on whether the consumption-wage income ratio, is smaller or larger than one. In other words, all else equal, borrowers () are more likely to reduce hours following a positive wage shock, whereas savers () are more likely to increase them.

For the economy as a whole, the response of total hours to a
wage change will be an average of the labor supply responses of all
households. If individual labor schedules were linear in net
wealth, the aggregate labor supply response would be linear in
average wealth, and wealth distribution would not affect labor
supply. There are, however, two main forces that undo the
linearity. First, retirees do not work, so any transfer of wealth
to and from them could affect how the workers respond to wage
shocks. Second, the interaction between borrowing constraints and
housing purchases creates an interesting nonlinearity. Above, we
have assumed that households do not change wealth in response to a
shock in the wage. However, if households switch from renting to
owning (or if they increase their house size) in good times, they
typically need to save for the downpayment. This increases the
incentive to work: intuitively, if the individual wants to keep
consumption constant when he buys the house, he needs to work more
hours. This effect creates comovement between hours and housing
purchases.^{23} In particular, it reinforces the
correlation between hours and housing demand in periods when a
large fraction of the population has, all else equal, low net
worth.

**Business Cycle Statistics.** In HP-filtered U.S. data, the variability of housing investment
is large, with a standard deviation that is between three and four
times that of GDP (in the period 1952-1982). Also, housing
investment is procyclical, with a correlation with GDP around
0.9. Together, these two facts imply that
the growth contribution of housing investment to the business cycle
is larger than its share of GDP. Household mortgage debt is
strongly procyclical from 1952 to 1982, but it becomes less procyclical after, with a
correlation with GDP that drops from 0.78 to
0.43. Table 3 compares the
benchmark model with the data. Overall, our baseline model does a
good job in reproducing the relative volatility of each component
of aggregate demand. In particular, it can account for about three
quarters of the variance of housing investment. On the contrary,
the model overpredicts the volatility of aggregate consumption. The
volatility of business investment is only slightly lower than in
the data. As in many RBC models without an extensive margin of work
and without direct shocks to the labor supply, our model
underpredicts the volatility of hours (0.33 percent
in the model, 1.6 percent in the data).

Turning to debt, the model does well in reproducing its cyclical
behavior.^{24} The key to this result is that the
bulk of the debt holders (mostly impatients) upgrades housing in
good times by taking out a (larger) mortgage. At the same time, the
model overpredicts the volatility of debt itself: the standard
deviation of the model variable is about four times larger than in
the data. We suspect that the reason for the higher volatility of
debt in the model has to do with the simplifying assumption that
only one financial asset is available, whereas in the data some
households (especially the wealthy) own simultaneously a mortgage
and other financial assets. If debt of low-wealth households is
more volatile than debt of high-wealth households, our model
variable can exhibit more volatility than its data counterpart.

One dimension where it is useful to compare the model with the
data pertains to home sales. In our model, we count a sale as every
instance in which a household pays the transaction cost to change
its housing: this involves own-to-own, rent-to-own and own-to-rent
transitions. By this metric, the average turnover rate in the model
(the ratio of sales to total houses) is 4 percent, a
number that matches the 3.9 percent in the
data.^{25} Moreover, the model correlation
between turnover rate and GDP is 0.39, and the
standard deviation is 0.29. The corresponding
numbers from the data are 0.69 and 0.54. The positive correlation between sales and economic
activity that the model captures reflects the presence of liquidity
constraints: when the economy is in recession and household balance
sheets have deteriorated, the potential movers in the model find
their liquidity so impaired, whether they are owners or renters,
that they are better off staying in their old house rather than
attempting to move and paying the transaction cost.

Having shown above that the model roughly captures postwar U.S. business cycles, we now consider the implications of two experiments. In the first, we lower the downpayment from 25 to 15 percent. In the second, we increase the idiosyncratic risk faced by households, changing the unconditional standard deviation of income from 0.30 to 0.45. Our experiment is intended to mirror two of the main changes that have occurred in the U.S. economy since the mid 1980s. The model results are in Table 4.

**A Decline in Downpayments.** Lower downpayments (column 2 in Table 4) lead to an increase in
the homeownership rate (from 64 to 76 percent) and to a higher level of debt (from 31 to 50 percent of GDP). Smaller downpayments
allow more housing ownership among the portion of the population
with very little net worth. While debt is higher, the increase in
homeownership works to keep total wealth inequality unchanged:
financial wealth inequality is higher, but housing wealth
inequality is lower. Turning to business cycles, the rise in
tends to reduce the volatility of
housing investment, from 6.42 to 5.94 percent, for two reasons. The first reason has to do
with adjustment costs: on average, because of adjustment costs,
homeowners modify their housing little over time relative to
renters. The second motive operates through the interaction of
labor supply and housing purchases. As we explained above, indebted
homeowners are more likely, compared to renters, to reduce hours in
response to positive technology shocks, so their presence dampens
aggregate shocks. Therefore, the higher homeownership rate induced
by looser borrowing constraints reduces aggregate
volatility.^{26}

**An Increase in Individual Earnings
Volatility.** Column 3 in Table 4 shows that, following a rise in
, the homeownership rate falls
from 64 to 59 percent: higher
risk makes individuals more reluctant to buy an asset that is
costly to change. All else equal, the lower homeownership rate
would tend to increase the volatility of housing investment, since
renters change housing consumption more often. However, this effect
is more than offset by the behavior of those who remain homeowners:
these people are now more reluctant to change their housing
consumption (relative to a world with less individual risk). This
occurs because modifying housing, in the presence of transaction
costs, depletes holdings of liquid assets and increases the utility
cost of a negative idiosyncratic shock, thus increasing the option
value of not adjusting the stock for given changes in net worth.
Quantitatively, the higher earnings volatility reduces the standard
deviation of housing investment from 6.42 to
5.52 percent. Moreover, higher income
volatility also reduces the sensitivity of debt to aggregate
shocks, since debt is used to finance housing purchases, and
housing purchases respond less to shocks.

**Combining Lower Downpayments and
Higher Volatility.** The last column of Table 4 shows the effects
of combining lower downpayments and higher volatility. The two
forces together predict an increase in homeownership rates from
64 to 67 percent. The data
counterpart is a two percentage points rise, from 64
to 66 percent. Moreover, the joint effect of
these two forces makes debt less procyclical, as in the data. The
correlation between debt and output falls from 0.71 to 0.39, a change that is remarkably
similar to the data (from 0.78 to 0.43, see Table 1).^{27} Together, lower downpayments and high
idiosyncratic volatility reduce the standard deviation of GDP from
2.09 to 2.03 percent, and
the standard deviation of housing investment from 6.42 to 5.04. percent. When these numbers are
compared to the data, the two changes combined can account for
13 percent of the variance reduction in GDP
and about 60 percent of the variance reduction in
housing investment.

Our interpretation of these results is as follows: in response
to lower downpayments and higher income volatility, leveraged
households become more *cautious* in response to aggregate
shocks, thus changing less borrowing and housing demand when
aggregate productivity changes.^{28} This is especially
true for housing, relative to other categories of expenditure,
since housing is a highly durable good and is subject to adjustment
costs. Because individuals are reluctant to adjust their housing
consumption during uncertain times, the sensitivity of hours to
aggregate shocks falls too. As a consequence, even if the
volatilities of consumption and business investment are not
changing, total output is less volatile.

In Figure 7, each panel shows average debt, hours and housing positions by age in the lowest and the highest aggregate state. The top panel plots the calibration with high downpayments and low idiosyncratic risk (the period 1952-1982): changes in the aggregate state generate large differences in debt, housing and hours. The bottom panel plots the case with low downpayments and high idiosyncratic risk (the period 1983-2010): changes in the aggregate state generate smaller differences in debt, housing, and hours, thus illustrating how these variables become less volatile and less procyclical.

Figure 8 plots the model dynamics when technology switches from its average value to a higher value (about 1 percent rise) in period 1. The responses are larger in the earlier period. On impact, housing falls before rising strongly in period 1. This result is well known in the household production literature (see, for instance, Greenwood and Hercowitz 1991 and Fisher 2007). In models with housing and business capital, business capital is useful for producing more types of goods than housing capital. Hence, after a positive productivity shock, the rise in the marginal product of capital implies that there is a strong incentive to move resources out of the housing to build up business capital, and only later is housing accumulated. The key aspect to note here is that higher idiosyncratic risk and lower downpayment requirements dampen the incentive to adjust housing capital, so that housing investment becomes less volatile.

Our result that higher individual uncertainty reduces the volatility of aggregate housing investment echoes the results of papers that study how durable purchases respond to changes in income uncertainty in models resulting from transaction costs. Eberly (1994), using data from the Survey of Consumer Finances, considers automobile purchases in presence of transaction costs: she finds that higher income variability broadens the range of inaction, and that the effect is larger for households that are liquidity constrained. Foote, Hurst and Leahy (2000) find a similar result using data on car holdings from the Consumer Expenditure Survey, and offer an explanation that involves the presence of liquidity constraints and precautionary saving: adjusting the capital stock for people with low levels of net worth depletes holdings of liquid assets and increases the utility cost of a negative idiosyncratic shock, thus increasing the option value of not adjusting the stock for given changes in net worth.

The finding that housing and debt are less sensitive to
aggregate shocks when downpayments are low and idiosyncratic risk
is high can account for part of the Great Moderation, but is at
odds with the events of the 2007-2009 financial crisis, when both
housing and debt fell substantially. Explaining the crisis is
beyond the scope of this paper, but in this section we show that
our model expanded to take into account the "credit crunch" can
generate, at least qualitatively, the observed response of housing
and debt in the Great Recession. We extend the stochastic structure
of the model so that, when the worst technology shocks hit, credit
standards get tighter too, in the form of lower loan-to-value
ratios and higher costs of financial intermediation (higher
borrowing interest rates). In other words, consistent with the
post-2007 evidence,^{29} recessions are now a combination of
negative financial and negative technology shocks occurring
simultaneously. We implement this scenario by assuming that the
maximum loan-to-value ratio changes over
time as a function of total factor productivity, : formally,
Moreover,
we also introduce an additional cost of financial intermediation in
the form of an interest rate premium
to be paid by debtors. The
budget constraint for a home buyer become respectively:

(26) |

with

(27) |

where
is the indicator
function equal to 1 if the household is a net
debtor, 0 otherwise. The state vector remains
unchanged with respect to the benchmark model, and so does the
equilibrium definition. In the calibration, we let drop by 6 percentage points in
correspondence of the two lowest values of ,
and leave it constant for all other values of .^{30} We set the values of the interest
rate premium at 0.75% for the two lowest aggregate
productivity realizations, in both periods ( is
equal to zero for all other values of ).

We find that this simple modification of the model can
qualitatively account for the behavior of housing and debt in the
most recent events. Figure 9 shows the impulse
responses to positive and negative productivity shocks, comparing
the early period with the late period (defined as in the baseline
exercise). In the late period, debt, housing and GDP respond less
to positive shocks, so that one finds evidence of the Great
Moderation so long as the economy is lucky enough not to be hit by
(too negative) negative shocks. When the worst recessionary shocks
hit, however, the decline in debt and in housing purchases are
considerably larger in the late period than in the early period. In
other words, when leverage is high, the housing sector can better
absorb "small" business-cycle shocks, but becomes more
vulnerable to large negative shocks that result in a credit crunch:
these shocks cause highly-leveraged households to sharply reduce
their debt and housing purchases.^{31}

We discuss in this section four alternative versions of the model where we modify the calibration used in our benchmark.

**Discount Factor.** To analyze the model with homogeneous discounting, we modify the
calibration for the discount factor (
) and for the relative utility
from renting (
) in order to achieve the same
homeownership rate and interest rate as in our baseline. As shown
in Table 5, the volatilities of housing investment
and output are now slightly higher than in the baseline
calibration, but the correlations of housing investment and of
hours with output fall: this result occurs because fewer people are
close to the borrowing limit (only 15 percent of households are
liquidity-constrained) and in need of increasing hours to finance
the downpayment in good times. In addition, with a single discount
factor, very few people hold debt in equilibrium, and the
distribution of wealth is more egalitarian than in the data: the
Gini coefficient for wealth is 0.53, lower than in
the data and in the benchmark model. The model predicts, unlike the
data, a negative correlation between turnover and GDP: with a
single discount rate, more housing capital reallocation occurs in
bad times.

**Persistence of the Income Process** One key parameter is the persistence of income shocks. Our
benchmark sets
. The robustness analysis in
Table 5 shows that, holding total income risk constant, some of the
model properties are a non-monotonic function of . When the shocks are not very persistent (
), the equilibrium level of
debt is relatively low, fewer people are at the liquidity
constraint, and debt and housing investment are less volatile and
slightly less cyclical. Conversely, when income shocks are highly
persistent (
), more people are liquidity
constrained, but more people are lucky for a spell long enough to
afford the downpayment for a house and to keep housing and debt
relatively unchanged in response to shocks.^{32} In other
experiments not reported in the Table, we have found that only for
intermediate values of the persistence coefficient (between
0.85 and 0.92), can the model
account for both the high volatility of housing investment and the
high correlation of debt with economic activity. Moreover, for
values of above 0.95
housing turnover is negatively correlated with GDP, and housing is
negatively correlated with business investment.

**Housing Transaction Costs.** We consider two polar cases, zero and high transaction costs.
With no transaction costs, the standard deviation of housing
investment, which is 6.42 percent in the
baseline, rises to 10.42 percent (see Table
5).^{33} Because houses are less risky,
homeownership rises, from 64 to 68 percent. Aggregate volatility falls: housing and
nonhousing capital become closer substitutes as means of saving,
and the higher volatility of housing investment is offset by the
reduced covariance between housing and nonhousing investment. The
correlation between housing and non-housing investment, which is
0.18 in the baseline (0.36 in the data), becomes -0.40 in absence of
transaction costs. It is interesting to relate this result to the
household production literature, which models adjustment costs
either as convex or using a time-to-build specification.^{34}
Fisher (2007) argues that the household production model predicts
that housing and business investment are negatively correlated,
unless one assumes that household capital is complementary to
business capital and labor in market production. Here, we note that
our baseline model with nonconvex housing adjustment costs
reproduces (unlike the model with no transaction costs) the
positive correlation between housing and business investment that
one finds in the data: sooner or later these costs must be paid in
order to consume more housing, and it is better to pay them in good
times, when the marginal utility of consumption is low. Moreover,
impatient renters cannot wait to become homeowners, thus
effectively buying houses and borrowing (i.e. selling claims on
capital) after a positive productivity shock.

Table 5 also reports the results for the high adjustment cost case (). The high model predicts low housing turnover (2.1 percent) relative to the data (4 percent), and an acyclical behavior of housing sales (sales are procyclical both in the data and in the benchmark model). Such model severely underpredicts the volatility of housing investment. We conjecture that moving shocks (when combined with income shocks) could restore the level of housing turnover that is observed in the data even in the presence of high transaction costs. It is not clear, however, whether moving shocks could make turnover procyclical, unless they are more likely to happen in good times.

**Housing Depreciation.** The last column of Table 5 reports the results when the housing
depreciation rate is lowered from 5 to
3 percent. The performance of some of the
model's second moments worsens considerably. Housing investment
becomes too volatile, the cyclicality of housing investment is much
lower than in the data, and the model fails to match the comovement
of housing with business investment.

In this paper, we develop an equilibrium business cycle model where houses can be used as collateral, purchased or rented, and adjusted at a large cost. The resulting dynamics of housing investment and household debt are realistic not only at the macroeconomic level, but also at the level of individual household behavior: even if agents only infrequently adjust their housing choice, housing investment is the most volatile component of aggregate demand in our model, a result that is mirrored in the data. Our model accounts for the procyclicality and volatility of housing investment, as well as for the procyclicality of household debt. The model can also explain why housing investment has become relatively less volatile, and household debt less procyclical, as a consequence of increased household-level risk and lower downpayment requirements, two structural changes that have occurred in the U.S. economy around the mid-1980s. We further extend the model to account for a "Great Recession" episode characterized by negative technology shocks coupled with tighter credit conditions. This simple modification generates an interesting nonlinearity which is consistent with recent events: when leverage is high, housing, debt and output respond less to positive shocks (as in the Great Moderation) but are relatively more vulnerable to negative shocks, making a recession worse (as in the Great Recession).

Despite its complexity, the model precludes an examination of
certain aspects of housing behavior that may be relevant for
understanding business cycle fluctuations. One limitation is that
we have not endogenized house prices.^{35} There are two main
reasons for our choice. First, allowing for variable house prices
would require specifying a two-sector model with housing and
nonhousing goods that are produced using different technologies, or
a model with different price stickiness in housing and nonhousing
goods; and would probably require a rich array of shocks in
addition to productivity shocks, since we know from existing
studies that technology shocks alone cannot quantitatively explain
observed movements in house prices: all of this would considerably
increase computational costs. Second, although movements in house
prices are economically important, cyclical fluctuations in the
price of housing are smaller than the corresponding fluctuations in
its quantity, which are the focus of our paper: for example, over
the period 1970-2008, the standard deviation of year-on-year growth
in real housing investment is 14 percent, while the corresponding
number for real house prices is 3.7 percent.^{36}

A second aspect of our model is that it does not explicitly
consider mortgage default. Under the assumption that all debt is
collateralized, and given that no shock is large enough to cause
agents to owe on their house more than they are worth, agents would
not find it optimal to default on their debts, even if they had
this option. However, default is an important device against risk
in an economy where housing values decline in recessions. In
Appendix C,^{37} we sketch an extension of our model
that dispenses from aggregate productivity shocks and features
large housing depreciation shocks as the main source of business
cycles. The model allows debtors to default on their mortgage, at
the cost of losing their house and being excluded from the mortgage
market. We assume that lenders cannot observe individual borrowers'
characteristics, but can charge a higher interest rate on all loans
in states of the world where default rates are higher to satisfy a
zero profit condition. In this setup, indebted households will
weigh the utility premium benefit of being homeowners against the
cost of servicing their debt in states where they have negative
equity. When a depreciation shock destroys part of the housing
capital, borrowing rates rise, and highly leveraged individuals
find themselves underwater, and decide to default on their debt,
becoming renters. The model can be used to study how shocks to
housing values interact with the mortgage default rate, interest
rates, debt and the housing stock. For plausibly calibrated values,
a shock that destroys 20 percent of the existing
housing stock leads to a rise in defaults (from 0 to 10 percent), a rise in borrowing premia (from 0 to
1.5 percent), and a sharp decline in debt,
output and housing investment.

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We solve for the model equilibrium using a computational method similar to the one used in Krusell and Smith (1998). The value and policy functions are computed on grids of points for the state variables, and then approximated with linear interpolation at points not on the grids (with the exception of the policy functions for housing, that are defined only on points on the grid). The algorithm consists of the following steps:

- Specify grids for the state space of individual and aggregate
state variables.
The number of grid points was chosen as follows: 7 points for the aggregate shock, 3 values for the idiosyncratic shock, 25 points for the housing stock, and 500 points for the financial asset.

^{38}For aggregate capital, we choose a grid of 15 equally spaced points in the initial range where denotes the average value of this variable in the simulations. The range is then updated at each iteration consistently with the simulated , assigning as its boundaries the minimum and the maximum simulated values. - Guess initial coefficients
for the linear functions that approximate the laws of motion of
capital and labor:

(28) . (29) Because factor prices (wages and interest rates) only depend on aggregate capital and labor in equilibrium, this approach is equivalent to assuming that individuals forecast these factor prices using a function of for each value of the aggregate state .

- Starting from age backward, compute optimal
policies as a function of the individual and aggregate states,
solving first the homeowner's and renter's problems
separately.
^{39}Notice that the intra-temporal optimal value for labor hours as a function of consumption and productivity shock for ages is the following:^{40}

(30) which allows one to derive consumption before age directly from the budget constraint. For the homeowner:

(31) so that the per-period utility function for can be transformed as follows:

(32)

For the tenant, taking into consideration the intra-temporal condition for optimal house services to rent:

(33)

so that the per-period utility function for can be transformed as follows:

(34)

As a consequence, the homeowner's dynamic optimization problem entails solving for policy functions for and only, while the renter's one consists in solving for only. The problems of the retired people ( ) are similar to the above, where we set

- Draw a series of aggregate and idiosyncratic shocks according
to the related stochastic processes. Draw a series of "death"
shocks according to the survival probabilities. Use the
(approximated) policy functions and the predicted aggregate
variables to simulate the optimal decisions of a large number of
agents for many periods. In the simulations, we perform linear
interpolation between grid points for
but we restrict the choices of
to lie on the grid. We simulate
90,000 individuals for 5,000 periods, discarding the first 200
periods.
^{41}Compute the aggregate variables and at each .

- Run a regression of the simulated aggregate capital and the simulated aggregate labor on lagged aggregate capital, retrieving the new coefficients for the laws of motion for and . We repeat steps 3 and 4 until convergence over the coefficients of the regressions. We measure goodness of fit using the of the regressions: they are always equal to 0.997 or higher at convergence for and around 0.95 for ; the corresponding wage rate and interest rate functions are also very accurate: the of the regression of the wage rate on aggregate is 0.999, the of the regression of the interest rate on aggregate is 0.992.

The (parsimonious) process for individual income productivity that we specify in the model is:

. | (35) |

We want to pick values for and that are in line with evidence.

- Floden and Lindé (2001) estimate an AR(1) process for wages of the form in (35) and estimate (using PSID data covering the 1988-1992 period), after controlling for observable characteristics and measurement error, values of (and thus implying ).
- Heathcote, Storesletten and Violante (2010) estimate an ARMA(1,1) process for wages using PSID data. Their estimate of the autoregressive component is 0.97.
- Scholz, Seshadri, and Khitatrakun (2006) specify and estimate a model of household log labor earnings (not wages) that controls for fixed effects, a polynomial in age, and autocorrelation in earnings. Their sample is the social security earnings records. Their estimates for married, no college, two-earners are (and ).

Several studies document the increase in the cross-sectional dispersion of earnings in the United States between the 1970s and the 1990s. This increase is often decomposed into a rise in permanent inequality (attributable to education, experience, sex, etc.) and a rise of the persistent or transitory shocks volatility. Despite some disagreement on the relative importance of these two components, the literature finds that both play a role in explaining the increase in income dispersion.

- Moffitt and Gottschalk (2008) study changes in the variance of permanent and transitory component of income volatility using data from the PSID from 1970 to 2004. They find that the non-permanent component (transitory) variance of earnings (for male workers) increased substantially in the 1980s and then remained at this new higher level through 2004. They report (see Figure 7 in their paper) that the variance of the transitory component rose from around 0.10 to 0.22 between the 1970s and the 1980s-1990s. This corresponds to a rise in the standard deviation from 0.32 to 0.47. Their estimate of the autocorrelation of the transitory shocks is 0.85.
- Using PSID data, Heathcote, Storesletten, and Violante (2010) decompose the evolution of the cross-sectional variance of individual earnings over the period 1967-2000 into the variances of fixed effects, persistent shocks, and transitory shocks. They find that the variance of persistent shocks roughly doubles during the 1975-1985 decade.
- Haider (2001) finds that increases in earnings instability over the 1970s and increases in lifetime earnings inequality in the 1980s account in equal parts for the increase of inequality in the data. To measure the magnitude of earnings instability in year , he uses the cross-sectional variance of the idiosyncratic deviations in year . His estimate of is 0.64. He finds that the unconditional standard deviation of the instability component rises from around 0.23-0.24 to about 0.35-0.37 during the 1980s.
- Krueger and Perri (2006) model log income as an ARMA process of
the kind

(36)

From this brief review, we conclude that a plausible value for the persistence of the productivity shock is around 0.9. We set the standard deviation of income to be equal to 0.3 in the early part of the sample, which is the lower bound of the estimates reported above. We set the standard deviation to 0.45 in the second part of the sample: a change of 0.15 is in the range of estimates reported by Moffitt and Gottschalk (2008).

Table 1: U.S. Economy. Cyclical Statistics and Housing Market Facts

Early Period 1952.I -1982.IV | Late Period 1983.I -2010.IV | Whole Sample 1952.I -2010.IV | |
---|---|---|---|

Standard Dev: GDP | 2.09 | 1.62 | 1.88 |

Standard Dev: C | 0.93 | 0.83 | 0.88 |

Standard Dev: IH | 7.12 | 4.45 | 6.00 |

Standard Dev: IK | 4.90 | 5.36 | 5.11 |

Standard Dev: Debt | 2.23 | 2.20 | 2.21 |

Standard Dev: Hours | 1.60 | 1.37 | 1.49 |

Standard Dev: Housing Turnover | 0.54 (68.I-82.IV) | 0.29 | 0.40 |

Correlations: IH,GDP | 0.89 | 0.75 | 0.84 |

Correlations: Debt,GDP | 0.78 | 0.43 | 0.63 |

Correlations: Hours,GDP | 0.82 | 0.86 | 0.83 |

Correlations: Turnover,GDP | 0.69 | 0.10 | 0.46 |

Correlations: IH,IK | 0.36 | 0.40 | 0.36 |

Correlations: Debt,C | 0.72 | 0.37 | 0.56 |

Averages: Homeownership | 64% | 66% | 65% |

Averages: Debt to GDP | 34% | 59% | 46% |

Averages: Housing Turnover | 3.9% | 4.3% | 3.2% |

Averages: Gini wealth | 0.79 | 0.83 | 0.81 |

Averages: Gini labor income | 0.40 | 0.46 | 0.83 |

Averages: Gini consumption | 0.23 | 0.26 | 0.25 |

**Notes**: *C*, *IH* and *IK* are consumption, residential fixed
investment and business fixed investment respectively, divided by
the GDP deflator (sources: BEA). *GDP* is the sum of
the three series. Durables expenditures are included in *IH*. *Debt* is the stock of Home mortgages held
by households and nonprofit organizations (source: Flow of Funds
Accounts), divided by the GDP deflator. *Hours* are
total hours worked for the entire economy from Francis and Ramey
(2009). Cyclical statistics (standard deviations and correlations)
for all series refer to the series logged and detrended with
HP-filter (smoothing parameter 1,600). Data on inequality are from
Wolff, 2010 (wealth); http://www.census.gov/hhes/www/income/data/ (income); and
from Krueger and Perri, 2006 (consumption). Housing Turnover is the
ratio of total home sales divided by the existing housing stock
(see text for the source).

Table 2: Parameter Values for the Benchmark Model Economy

Parameter | Value | Target/Source | |
---|---|---|---|

Preferences: Discount factor, patients | 0.999 | ||

Preferences: Discount factor, impatients | 0.941 | Hendricks (2007) | |

Preferences: Fraction of impatient agents | - | 2/3 | Gini coefficient of Wealth: 0.73 |

Preferences: Weight on leisure in utility | 1.65 | - | |

Preferences: Productive time | 2.65 | Time worked: 40% | |

Preferences: Weight on housing in utility | 0.15 | ||

Preferences: Utility, renting vs. owning | 0.838 | Home ownership rate = 64% | |

Preferences: Utility weights (family size) | see text | Cagetti (2003) | |

Life, Retirement: Survival probabilities | see text | Decennial Life Tables | |

Life, Retirement: Retirement period | 46 | Retirement age 65 years | |

Life, Retirement: Pension | 0.4xinc. | 40% average income | |

Technology: Capital share | 0.26 | ||

Technology: Capital depreciation rate | 0.09 | ||

Technology: Housing depreciation rate | 0.05 | ||

Technology: Autocorrelation, technology shock | 0.925 | King and Rebelo (1999) | |

Technology: Standard dev., technology shock | 0.0148 | ||

Technology: Housing transaction cost | 0.05 | National Association Realtors (2005) | |

Technology: Minimum House Size | 1.5xinc. | See text | |

Borrowing: Max debt, fraction lifetime wage | 0.25 | See text | |

Borrowing: Maximum debt, fraction of house | 0.75 | See text | |

Individual Income Process: Autocorrelation, earnings shock | 0.90 | Floden and Linde (2001) | |

Individual Income Process: Standard deviation, earnings shock | 0.30 | See appendix B | |

Individual Income Process: Age-dependent earnings ability | see text | Hansen (1993) |

Table 3: U.S. Economy and Baseline Model. Comparison for the Early Period

1952.I -1982.IV (Early Period) | Model | |
---|---|---|

Standard Dev: GDP | 2.09 | 2.09 |

Standard Dev: C | 0.93 | 1.63 |

Standard Dev: IH | 7.12 | 6.42 |

Standard Dev: IK | 4.90 | 4.16 |

Standard Dev: Debt | 2.23 | 8.34 |

Standard Dev: Hours | 1.60 | 0.33 |

Standard Dev: Housing Turnover | 0.54 (68.1-82.IV) | 0.29 |

Correlations: IH,GDP | 0.89 | 0.66 |

Correlations: Debt,GDP | 0.78 | 0.71 |

Correlations: Hours,GDP | 0.82 | 0.65 |

Correlations: Turnover,GDP | 0.69 | 0.39 |

Correlations: IH,IK | 0.36 | 0.18 |

Correlations: Debt,C | 0.72 | 0.85 |

Averages: Homeownership | 64% | 64% |

Averages: Debt to GDP | 34% | 31% |

Averages: Housing Turnover | 3.9% | 4.0% |

Averages: Gini wealth | 0.79 | 0.73 |

Averages: Gini labor income | 0.40 | 0.41 |

Averages: Gini consumption | 0.23 | 0.26 |

Averages: Liquidity constrained | NA | 0.45 |

**Notes**: The model moments are based on statistics from a
simulation of 5,000 periods. Liquidity constrained agents in the
model are those who own liquid assets less than 16.67 percent (two
months in a year) of annual income.

Table 4: Model Predictions, Changing Downpayment Requirements and Income Volatility

(1) Baseline Early Period | (2) | (3) | (4) Late Period | |
---|---|---|---|---|

Standard Deviation: GDP | 2.09 | 2.08 | 2.05 | 2.03 |

Standard Deviation: C | 1.63 | 1.63 | 1.66 | 1.68 |

Standard Deviation: IH | 6.42 | 5.94 | 5.52 | 5.04 |

Standard Deviation: IK | 4.16 | 4.05 | 4.21 | 4.16 |

Standard Deviation: Debt | 8.34 | 3.04 | 2.61 | 1.44 |

Standard Deviation: Hours | 0.33 | 0.32 | 0.31 | 0.31 |

Standard Deviation: Housing Turnover | 0.29 | 0.44 | 0.21 | 0.21 |

Correlations: IH, GDP | 0.66 | 0.69 | 0.55 | 0.54 |

Correlations: Debt, GDP | 0.71 | 0.63 | 0.50 | 0.39 |

Correlations: Hours, GDP | 0.65 | 0.64 | 0.47 | 0.42 |

Correlations: Turnover, GDP | 0.39 | 0.77 | 0.42 | 0.28 |

Correlations: IH, IK | 0.18 | 0.24 | 0.08 | 0.09 |

Correlations: Debt, C | 0.85 | 0.77 | 0.68 | 0.58 |

Averages: Homeownership | 64% | 76% | 59% | 67% |

Averages: Debt to GDP | 31% | 50% | 23% | 35% |

Averages: Housing Turnover | 4.0% | 3.0% | 5.1% | 5.6% |

Averages: Gini wealth | 0.73 | 0.73 | 0.73 | 0.73 |

Averages: Gini labor income | 0.41 | 0.41 | 0.48 | 0.48 |

Averages: Gini consumption | 0.26 | 0.26 | 0.31 | 0.31 |

Averages: Liquidity constrained | 0.45 | 0.45 | 0.39 | 0.38 |

**Notes:** Baseline calibration and sensitivity analysis.
(1) is the baseline
calibration that is targeted to the U.S. data for the period
1952-1982. In (2), we increase the loan-to-value ratio from 0.75 to
0.85. In (3), we increase earnings volatility from 0.3 to 0.45. In
(4), we increase both loan-to-value ratio and earnings volatility
so to calibrate the U.S. economy for the period 1983-2010.

Table 5: Robustness Analysis

Data |
Model | One- | Persistence =.7 =.95 |
Transaction cost =0% =8% |
Low =3% |
|||
---|---|---|---|---|---|---|---|---|

Standard Dev: GDP |
2.09 |
2.09 | 2.16 | 2.08 | 2.02 | 2.05 | 2.01 | 2.05 |

Standard Dev: C |
0.93 |
1.63 | 1.69 | 1.69 | 1.69 | 1.69 | 1.72 | 1.68 |

Standard Dev: IH |
7.12 |
6.42 | 6.72 | 4.99 | 4.73 | 10.42 | 3.45 | 11.33 |

Standard Dev: IK |
4.90 |
4.16 | 4.83 | 4.24 | 4.12 | 4.99 | 3.95 | 5.17 |

Standard Dev: Debt |
2.23 |
8.34 | 14.78 | 2.68 | 2.11 | 1.68 | 2.11 | 0.68 |

Standard Dev: Hours |
1.60 |
0.33 | 0.39 | 0.32 | 0.27 | 0.36 | 0.27 | 0.30 |

Standard Dev: Housing Turnover |
0.54 |
0.29 | 0.40 | 0.16 | 0.22 | 2.14 | 0.13 | 0.16 |

Correlations: IH,GDP |
0.89 |
0.66 | 0.58 | 0.61 | 0.49 | 0.34 | 0.54 | 0.30 |

Correlations: Debt,GDP |
0.78 |
0.71 | 0.72 | 0.60 | 0.58 | 0.69 | 0.39 | 0.11 |

Correlations: Hours,GDP |
0.82 |
0.65 | 0.60 | 0.50 | 0.43 | 0.45 | 0.34 | 0.45 |

Correlations: Turnover,GDP |
0.69 |
0.39 | -0.32 | 0.18 | -0.15 | 0.67 | -0.08 | 0.10 |

Correlations: IH,IK |
0.36 |
0.18 | 0.08 | 0.19 | 0.03 | -0.40 | 0.19 | -0.44 |

Correlations: Debt,C |
0.72 |
0.85 | 0.83 | 0.78 | 0.72 | 0.82 | 0.54 | 0.24 |

Averages: Homeownership |
64% |
64% | 64% | 66% | 71% | 68% | 74% | 70% |

Averages: Debt to GDP |
34% |
31% | 9% | 17% | 42% | 40% | 37% | 46% |

Averages: Housing Turnover |
3.9% |
4.0% | 3.3% | 4.7% | 2.9% | 42.0% | 2.1% | 3.8% |

Averages: Gini wealth |
0.79 |
0.73 | 0.53 | 0.68 | 0.73 | 0.73 | 0.72 | 0.72 |

Averages: Gini labor income |
0.40 |
0.41 | 0.42 | 0.45 | 0.39 | 0.41 | 0.41 | 0.42 |

Averages: Gini consumption |
0.23 |
0.26 | 0.24 | 0.23 | 0.26 | 0.26 | 0.26 | 0.26 |

Averages: Liquidity constrained |
NA |
0.45 | 0.15 | 0.30 | 0.49 | 0.47 | 0.45 | 0.45 |

**Notes:** In the one- model, we
recalibrate and the average so that the homeownership rate is 64% and the interest
rate is 3%, as in the baseline model. No parameter changes are made
in the other models, except those noted in row 2 of the Table.

Figure 1: Mortgage Debt, Housing Investment and GDP

**Note:** Variables are inflation-adjusted, HP-filtered (
) and expressed in percent
deviation from their trend.

Figure 2: Efficiency and Preference Profiles

Figure 3: Homeowner's Housing Investment Decision as a Function of Initial House Size and Liquid Assets

**Note:** The figure illustrates, for each combination of
initial house and liquid assets, the homeowner's housing decision
for next period. It is plotted for a patient agent who is 65 years
old, when aggregate productivity and the average capital labor
ratio are equal to their average value.

Figure 4: A Typical Life-Cycle Profile

**Note:** This figure plots life-cycle choices of a randomly
chosen impatient agent from birth (age 21) to death (age 90). In
panel 1, the thin line denotes the maximum debt limit given the
housing choice. In panel 3, the "x" symbol denotes the amount
rented when the individual is renting, whereas the solid line
denotes the amount owned when the individual owns a house.

Figure 5: Comparison Between Model (Baseline Calibration) and Data

**Note:** The data come from the summary statistics of the
1983 Survey of Consumer Finances, as reported in Kennickell and
Shack-Marquez (1992). For each age, the model variable is the
product of the fraction of households in that age holding housing
or debt, times the median holding of housing or debt. The data
variable is constructed in the same way.

Figure 6: Lorenz Curves for Total Wealth and Housing Wealth

**Note:** The Lorenz curves for total wealth and housing
wealth in the data are from Díaz and Luengo-Prado (2010)
using data from 1998 Survey of Consumer Finances.

Figure 7: Comparison between Early and Late Period: Debt, Hours and Housing by Age

**Note:** The top panel plots model variables in the baseline
calibration (low individual risk and high downpayment
requirements), where housing, debt and hours worked are relatively
more volatile (the difference between a boom and a recession is
larger). The bottom panel plots the calibration with high
individual risk and low downpayment requirements.

The thin/thick line shows the reading of each variable by age
when the economy is in the lowest/highest aggregate state
(recession/boom). Housing and Debt are expressed as a ratio of
average GDP. Hours are normalized in each age by their age
average.

Figure 8: Impulse Responses to a Positive Technology Shock: Early and Late Period Calibration

**Note:** Model dynamics following an exogenous switch in
aggregate productivity (in period zero) from the
median state to next higher value (a 1 percent increase) lasting
four periods. Each variable is displayed in percent deviation from
the unshocked path.

Figure 9: Impulse Responses to Positive and Negative Technology Shocks: Comparison between the Early and Late Period Calibration, Model with Cyclical Loan-to-Value Ratios and Interest Rate Premia

**Note:** Model dynamics following an exogenous switch in
productivity in period zero. The thick lines plot a
1 percent increase in productivity that does not change financial
conditions in the early (solid lines) and late (dashed lines)
period calibration. The thin lines plot a 1 percent decrease in
productivity together with a worsening in financial conditions.
Each variable is displayed in percentage deviation from the
unshocked path.

** We thank Massimo Giovannini and Joachim Goeschel for their invaluable research assistance. We thank Chris Carroll, Kalin Nikolov, Dirk Krueger, Makoto Nakajima, as well as various seminar and conference participants for helpful comments on various drafts of this paper. Pavan aknowledges financial support from the Spanish Ministry of Education (Programa de Movilidad de Jovenes Doctores Extranjeros). Supplementary material is available at the website https://www2.bc.edu/~ iacoviel/. Return to text

+ Matteo Iacoviello, Division of International Finance, Federal Reserve Board, 20th and C St. NW, Washington, DC 20551. Email: matteo.iacoviello@frb.gov. Return to text

++ Marina Pavan, Universitat Jaume I & LEE, Castellón, Spain. E-mail: pavan@eco.uji.es. Return to text

1. Campbell and Hercowitz (2005) and Gerardi, Rosen and Willen (2010) discuss the role of financial reforms, and Dynan, Elmendorf and Sichel (2007) discuss the evolution of household income volatility. Return to text

2. If one excludes the 2008-2010 period from the time-series, the decline in the volatility of housing investment and the decline in the correlation between debt and GDP are slightly larger. Return to text

3. Under convex costs, housing adjustment takes the form of a series of small adjustments over a number of periods. Under our specification, the homeowner's housing stock follows an rule, remaining unchanged over a long period and ultimately changing by a potentially large amount. See Carroll and Dunn (1997) for an early partial equilibrium model with behavior for housing. Return to text

4. Krusell and Smith (1998) explore a heterogeneous-agents setting with discount rate heterogeneity which replicates key features of the data on the distribution of wealth. Return to text

5. We crudely assume that the pension is the same for everyone. Allowing pensions to mimic something that looks like the actual Social Security system in the U.S. would make our model computationally intractable, since it would enlarge the state variables in the household problem to encompass their entire income history. Return to text

6. We refer to as financial liabilities, and to as financial assets. Because bonds are claims on aggregate capital, their return varies with the aggregate state. Return to text

7. To compute , we fix interest and wages at current values. To compute we assume for . Return to text

8. In the United States, lending institutions typically send a "Verification of Employment" (VOE) form to the borrower's employer to determine start date of employment, current and previous salary, and the probability of continued employment among other things. Return to text

9. One can interpret the marginal cost of one house to be 1 for the financial sector, since loanable funds can be converted into housing costlessly; and the marginal benefit to be the sum of the current rental income, , plus expected return next period, , where is the opportunity cost of funds for the financial sector. Equating costs and benefits yields equation (5) . Return to text

10. We have examined the robustness of our results by letting agents use both the aggregate capital stock and the housing stock in forecasting future prices, with nearly identical results, but at a higher computational cost. It is possible that higher moments of the wealth distribution could be both relevant in predicting future prices and yield different aggregate dynamics, so that our decision rules would describe a bounded rationality equilibrium, rather than a good approximation to the rational expectations equilibrium. Yet the evidence that adding to the set of the state variables does not change aggregate dynamics leads us to be skeptical of this interpretation. See Young (2010) for an insightful discussion of these issues. Return to text

11. Queisser and Whitehouse (2005) report that average pensions for males in the United States are 40 percent of the economy-wide average earnings. Return to text

12. The NIPA Fixed Asset Tables indicate depreciation rates for housing ranging from 1.2 to 4.5 percent, depending on the type of structure and its use (see Fraumeni, 1997). We choose a slightly higher value because we want to account for unmeasured labor time that is used to repair, renovate, or maintain or improve the quality of housing at a given location (Peek and Wilcox, 1991); because higher values are typically considered in the existing literature, especially when housing is broadly interpreted to include consumer durables (Chambers, Garriga and Schlagenhauf, 2009, Gervais 2002, and Díaz and José Luengo-Prado, 2010); and because a higher depreciation rate (5 percent instead of 2 percent, say) reduces the extent to which aggregate housing tends to decrease on impact following a positive aggregate technology shock in a model with two capital goods. Return to text

13. The National Association of Realtors estimates that average commission rates (excluding houses sold without brokers, which account for about 10 to 25 percent of existing home sales, according to media reports, reports of the National Association of Realtors, and academic studies) range from 4.3 to 5.4 percent, based on 2004 data documenting a $65 billion brokerage industry and an existing home sales volume of $1.35 trillion. Return to text

14. According to the 2009 American Housing Survey, only 20 percent of total owner-occupied units have a ratio to current income less than 1.5. Return to text

15. The figure is plotted for a patient agent who is entering retirement (65 years old), when aggregate productivity and the capital-labor ratio are equal to their average value. Return to text

16. In the model, renters change their housing position every period, since they face no cost in doing so. This assumption is in line with the data, that show that on average renters move about every two years. Return to text

17. We are aware, of course, of the difficulty in comparing the model with the data along this dimension: in the data, 15 percent of the moves are associated with a move to a different state, and 35 percent of the moves are associated with a move to a different county. Most of these moves are probably "moving shocks" rather than movements along the housing ladder. Return to text

18. Liquid assets are defined as According to this definition, an owner is not liquidity constrained so long as it saves sufficiently more (borrows less) than the minimum downpayment in the house ( ); a renter ( ) is not constrained if financial assets are sufficiently large ( ). Return to text

19. The baseline model predicts that 70 percent of renters and 31 percent of homeowners are liquidity constrained; and that 67 percent of impatient agents and 2 percent of patient agents are liquidity constrained. Return to text

20. See for instance the work of Ríos-Rull (1996) and Gomme et al. (2004). Return to text

21. Obviously, the optimal decisions involve the joint choice of (1) consumption, (2) housing, (3) debt and (4) hours worked. By assuming that housing and debt remain constant across two subperiods, we can study the joint determination of consumption and hours by focusing on the budget constraint and the Euler equation for labor supply only. This is a reasonable assumption for small shocks (such as aggregate shocks). Return to text

22. Renters have constant shares of housing and nonhousing consumption, so that where is the ratio of housing expenditure to nondurable consumption. With minor modifications, the arguments in the text carry over to this case, since cannot be negative for renters Return to text

23. The limiting case of zero forced savings would be the case in which no downpayment is needed to buy a house. In that case the individual can keep consumption constant at the time of the purchase without increasing hours worked if transaction costs are zero. If the individual has to pay the transaction cost, this provides an incentive to work more at the time of the purchase. Campbell and Hercowitz (2005) propose a similar argument to discuss the relationship between hours and durable purchases. Return to text

24. We define household debt as (that is, the average of the household liabilities). Return to text

25. The turnover rate in the data is constructed as the sum of sales of existing single-family homes (source: National Association of Realtors) plus new single-family homes sold (from Census Bureau), divided by the total housing stock (from Census Bureau). The series starts in 1968. Return to text

26. A similar intuition has been proposed in Campbell and Hercowitz (2005), who show that financial innovation alone can explain more than half of the reduction in aggregate volatility in a model with borrowers and lenders and downpayment constraints. Aside from modeling differences (our model considers the owning/renting margin and addresses issues related to life cycle, lumpiness and risk that are absent in their setup), the intuition they offer for their result carries over to our model, but we find that the effect of lower downpayment requirements is quantitatively smaller. We conjecture that the differences depend on one modeling assumption: in our setup, indebted homeowners mitigate aggregate volatility, but this effect is partly offset by the wealthier homeowners (the creditors) who tend to increase aggregate volatility by working relatively more in response to positive aggregate shocks; instead, Campbell and Hercowitz assume that labor supply of wealthy homeowners is constant, thus killing this offsetting mechanism. Return to text

27. Likewise, the correlation between debt and consumption falls in the model from 0.85 to 0.58, a decline similar to the data (from 0.72 to 0.37). Return to text

28. Higher uncertainty in itself reduces the willingness to borrow, whereas lower downpayments lead to an increase in debt. In our baseline calibration, the second effect dominates - as shown in table 4, the ratio of debt to GDP rises from 0.31 to 0.35 when both changes are present. As a consequence, in the late period individuals are more cautious, even if they hold more debt. For this reason, the fraction of liquidity constrained households in the model falls from 45 to 38 percent. Return to text

29. Jermann and Quadrini (forthcoming) document that credit shocks have played an important role in capturing U.S. output during the last decades. Return to text

30. Total factor productivity is discretized using a 7-state Markov chain (see Appendix). For the lowest two aggregate productivity levels: in the period 1952-1982, , and in the period 1983-2010, . Return to text

31. Incidentally, we note that the volatility of GDP is still smaller in the late than in the early period calibration. Return to text

32. To keep our experiments simple and easier to interpret, we do not attempt here at recalibrating some of the other parameters in order to match the same targets as in the benchmark model. Return to text

33. Thomas (2002) argues that lumpiness of fixed investment at the level of a single production unit bears no implications for the behavior of aggregate quantities in an otherwise standard RBC model. Her argument rests on the representative household's desire to smooth consumption over time, a desire that undoes any lumpiness at the level of the individual firm. Our sensitivity analysis shows that there are differences between the models with and without adjustment cost. Adjustment costs imply smaller housing adjustment at the aggregate level, but larger housing adjustments (when they occur) at the individual level. Return to text

34. See for instance Gomme, Kydland and Rupert (2001). Return to text

35. The recent papers by Kiyotaki, Michaelides and Nikolov (2011), Favilukis, Ludvigson and Van Nieuwerburgh (2009), and Ríos-Rull and Sánchez-Marcos (2008) are steps in this direction. Return to text

36. For house prices, we use the Conventional Mortgage Home Price Index (adjusted for inflation). Return to text

37. Appendix C is available at https://www2.bc.edu/~iacoviel/. Return to text

38. The upper bound for the housing grid and the lower bound for debt are chosen wide enough so that they never bind in the simulations. Return to text

39. In computation, we exploit the strict concavity of the value function in the choice for assets as well as the monotonicity of the policy function in assets (for the homeowner problem, the monotonocity is for any given choice of the housing stock). Return to text

40. We prevent individuals from choosing negative hours. Return to text

41. We enforce the law of large numbers by making sure that the simulated fractions of ages and of labor productivity shocks correspond to the theoretical ones, by randomly adjusting the values of the shocks. Return to text

This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to text