Interest Rates and Housing Market Dynamics in a
Housing Search Model (Preliminary and
Incomplete, Do Not Distribute)
Elliot Anenberg
Edward Kung
January 13, 2017
Abstract
We introduce mortgages into a dynamic equilibrium, directed search model
of the housing market. Mortgage rates play their natural role in our model
by affecting the share of per-period income that a homeowner devotes to their
mortgage payment rather than consumption. We estimate the model using detailed micro data on home listings, and by exploiting the insights of Menzio et
al. (2010) to maintain tractability despite the significant amount of heterogeneity that even basic mortgage contracts introduce into a search model. We use
the estimated model to show that housing market conditions are substantially
more responsive to changes in mortgage rates than a reduced-form analysis of
house prices suggests. Our results have implications for monetary policy and
the literature on the interactions between the housing market and credit supply.
1
1
Introduction
Fluctuations in housing values have significant consequences for the macroeconomy
because they influence consumption decisions, residential investment, and financial
stability. It is therefore important to understand the main economic forces driving the
dynamics of housing markets. A growing literature has focused on the role of interest
rates. Interest rates are a natural factor to examine because most home purchases
are financed with large, fixed-rate loans. In addition, this is a particularly important
relationship to study because it helps us to understand the transmission of monetary
policy to the housing market.
Much of the empirical literature that attempts to identify the effect of interest
rates on housing valuations uses regressions of house prices on interest rates. This is
a natural starting point because if house prices could be characterized by a simple
frictionless asset pricing model, then the valuation for housing would would be fully
reflected by the price. These studies tend to find that interest rates have only a
modest, negative effect on house prices. Estimates of the semi-elasticity range from
1-9, implying that a 1 percentage point increase in the mortgage rate decreases house
price growth by 1-9 percentage points.1 Taken at face value, these results suggest
that interest rates are not a major factor driving changes in housing valuations.2 One
important policy implication of these findings is that in practice, controlling housing
market fluctuations may be largely beyond the reach of conventional monetary policy
tools.3
These empirical findings seem puzzling in two respects. First, the mortgage balance associated with the typical house purchase is in the hundreds of thousands of
dollars. Therefore, even small changes in mortgage rates can have large effects on
the share of income that a borrower must devote to debt service, which should have
1
See Kuttner [2012] and Fuster and Zafar [2015] for a review of these existing studies. While
the endogeneity of interest rate changes may bias these estimates toward zero (i.e. if the Federal
Reserve engages in countercyclical monetary policy), even studies that exploit plausibly exogenous
variation in interest rates still find small effects. See Adelino et al. [2012].
2
Favara and Imbs [2015], Maggio and Kermani [2015]emphasize that broader measures of credit
supply that including borrowing limits (e.g. LTV limits) are more important drivers of housing
valuations.
3
For example, based in part on the empirical evidence, Fed Chair Yellen argues in a
2014 speech that macroproduential policies are generally better suited than traditional monetary policy to prevent excessive house price movements and financial instability.
See
https://www.federalreserve.gov/newsevents/speech/yellen20140702a.htm#f5.
2
meaningful effects on willingness to pay. Indeed, Himmelberg et al. [2005] calibrates
a simple, frictionless user cost model of house prices, which uses the assumption that
the net present costs of renting and owning should be equal in equilibrium, and generates an implied semi-elasticity of house prices with respect to interest rates of about
20–a much larger elasticity than what is typically estimated in the data.4
Second, in contrast to house prices, other housing market variables, such as existing home sales and new construction activity, are actually relatively interest rate
elastic. These elasticities have received less attention in the literature. If interest
rates have only a modest effect on willingness to pay for housing, as the empirical
relationship between house prices and rate changes seems to suggest, then why are
existing home sales and new construction activity–which should also be sensitive to
willingness to pay–relatively rate elastic?
In this paper, we show that interest rates do have sizable effects on housing market
valuations, and that the sales price response vastly understates the effect of rate
changes on latent buyer valuations. In contrast to the reduced form approach used
by much of the existing empirical literature, our results come from a structural model
of the housing market. Our model features search frictions and simple mortgage
contracts, and we estimate our model using detailed micro data from the San Diego
housing market. Our estimated model predicts that a 100 basis point increase in
rates will decrease buyer willingness to pay by 13 percent for the typical home. This
is three times larger than the effect of the rate increase on average sale prices implied
by the model, which is 4 percent. The model generates these predictions while also
matching many features of the data, including the high interest rate elasticity of new
construction permits.
The model is a simple search model with heterogeneous homeowners, who are
differentiated by their interest rate and mortgage amount and who face exogenous
moving shocks. Interest rates matter in the model because they influence the share of
per-period income that a homeowner must devote to their mortgage payment rather
than consumption. Homeowners who receive moving shocks become sellers and direct
their search for a buyer into one of many submarkets, which are effectively list price
levels for a given quality of housing. The basic tradeoff faced by the seller is that
4
More recently, Glaeser et al. [2012] emphasizes that when the user cost model is extended
to include refinancing and volatile interest rates, then the implied semi-elasticity can be reduced
substantially.
3
searching at a higher price level will typically result in a higher price but slower match,
whereas the opposite would be true for buyers. A distribution of submarkets can
emerge in equilibrium because sellers are heterogeneous in their outstanding mortgage
amount and mortgage rate, which causes some sellers to prefer low prices and faster
sales while others prefer the opposite.
We also incorporate new construction into our model. Builders buy up depreciated homes and optimally choose when to start construction to replace the depreciated structure. The optimal construction timing depends on current and expected
future interest rates, as well as idiosyncratic factors. Once construction is completed,
builders face a similar home selling problem as the one faced by sellers of existing
homes. They face search frictions, and they compete with other sellers–both new construction sellers and existing home sellers–for buyers. Builders choose different list
prices in equilibrium because they have concave utility over profit and have heterogeneous construction costs. In this way, construction costs for builders play a similar
role as outstanding mortgage balances for existing homeowners in the home selling
problem.
Introducing even simple mortgage contracts and construction costs into a search
model is complicated because it introduces heterogeneity that is difficult to accommodate computationally. Generally, one would need to keep track of the entire distribution of heterogeneous agents when computing the equilibrium of the model, which
makes estimating or even solving the model once difficult. To deal with this computational challenge, we note that because buyers can freely visit any submarket, expected
buyer utility across submarkets needs to be equalized. Menzio and Shi [2010, 2011]
showed that a similar condition in directed search models of the labor market allow us
to compute the equilibrium of our model without keeping track of the distributions
of buyer and seller heterogeneity. Intuitively, the equilibrium conditions force the
equilibrium match rate to depend only on the expected utility in a submarket, and
not on the distribution of agents across submarkets. As a result, we are able to solve
our model in and out of steady state, and we are able to fit our model using moments
across different years and market conditions.
The key effects in the model that produce our main results are as follows. Consider
a negative shock to interest rates. Because there are search frictions, higher buyer
demand from lower rates is cleared through a combination of sellers choosing to list
at higher prices, and sellers experiencing higher sale hazards. Absent search frictions,
4
there is no channel through the sale hazard, so prices would have to increase by
more in order to clear the shock. How much more depends on the parameters of the
model, which we estimate from rich micro data on list prices choices and sale hazards
conditional on list prices. Our parameter estimates imply that seller utility is highly
concave in net wealth (i.e. sale price less outstanding mortgage amount), which causes
sellers to choose prices that leave them with a fairly consistent level of net wealth.
This causes some sellers to favor a higher sale hazard over a higher list price, even in
response to an interest rate shock, thus mitigating the price effect of the shock. The
effect of the shock on prices is mitigated further by sticky prices, which we incorporate
into our model to reflect the empirical reality that sellers infrequently change their
list price. When list prices are sticky, even more of the shock will be cleared through
higher sale hazards.5 In contrast, new construction activity remains fully sensitive to
the interest rate shock. This is because the builders’ expected profit function when
determining whether or not to build is sensitive to both higher prices and higher sale
hazards, since they also experience frictions search frictions when selling their newly
built homes. Higher sale hazards could actually be more attractive for builders than
for regular sellers, because they typically have higher effective holding costs as the
home remains vacant during the marketing period. For these reasons, construction
permits are more sensitive to the elasticity of buyer valuations with respect to interest
rates than are realized prices.
In addition to the empirical literature on the effects of credit supply on the housing
market mentioned above, our paper contributes to the growing literature that uses
search models to study the housing market.6 One contribution we make to this
literature is that we introduce mortgage contracts into a search and matching model.
This allows us to study how interest rates affect housing markets through the debt
service burden channel.7 Another contribution of our paper to this literature is that
we take a microeconomic, structural approach to estimating our model; that is, we
estimate the parameters our model using detailed micro data on the list price choices
5
This effect is similar to an effect present in labor search models where more of an aggregate
demand shock will be cleared through unemployment when wages are sticky.
6
See Burnside et al. [2011], Head et al. [2014], Piazzesi and Schneider [2009], Ngai and Tenreyro
[2014], Krainer [2001], Carrillo [2012], Albrecht et al. [2007], Novy-Marx [2009], Diaz and Jerez
[2013], Caplin and Leahy [2011], Genesove and Han [2012], Wheaton [1990], Guren [2013]
7
Ngai and Sheedy [2016] studies the effects of interest rates on the housing market in a search
model, but in their model the interest rate effect operates through the discount factor—there are no
mortgages in their model.
5
and selling outcomes of individual sellers. Much of the existing literature calibrates
their models using external estimates or estimates parameters by targeting broad,
aggregate moments. An advantage of our estimation approach is that our precise
quantative predictions from our model are more closely supported by the data. Other
papers that estimate related models of the home selling problem using micro data
include Merlo et al. [2015], Carrillo [2012], Anenberg [2016], Horowitz [1992]. Relative
to these papers, we are unique in that buyer valuations and prices are endogenously
determined, and that we incorporate new construction into our model.8
Methodologically, we are similar to Hedlund [2016] in that he also applies the
insights of Menzio and Shi [2010, 2011] to estimate a directed search model of the
housing market. As a result, like us, he is able to solve his model both in and out of
steady state. Hedlund [2016] also incorporates mortages and new construction into
his search model, though his main focus is on foreclosures and the cyclical dynamics
of macroeconomic and mortgage market variables.
Finally, we contribute to the literature on non-market valuation that builds off
the seminal work of Rosen [1974], Bayer et al. [2007], Berry et al. [1995]. Bayer et al.
[2016], Bishop and Murphy [2011], Caetano [2012] focus on estimating willingness to
pay for amenities (e.g. school quality or crime) when households behave dynamically
and are forward looking with respect to the value of the amenity. Ouazad and Rancière [2013], Kung and Mastromonaco [2015]focus on estimating willingess to pay in
the presence of credit constraints. We provide a framework for estimating willingness
to pay for amenities in the presence of search frictions. While our focus in this paper
is on interest rates, our model could be used to obtain more accurate estimates of
willingess to pay in the presence of search frictions for a more general set of amenities
in future research. 9
8
Paciorek [2013], Murphy [2008] also model new construction, but they do not consider search
frictions in the home selling problem as we do in this paper. Even among the macro housing search
papers, most of the literature generally treats the housing stock as fixed, and/or considers only
steady states. Head et al. [2014] is a recent and notable exception in the macro literature.
9
For example, if neighborhoods with better schools have higher sale hazards and higher prices,
then our framework would suggest that estimates of willigness to pay for school quality using standard methods would be biased downward.
6
2
Motivating Empirical Facts
In this section, we show that home sales and home construction are generally much
more rate elastic than home prices. We provide evidence that sales and construction
are rate elastic mainly because homebuyer demand is rate elastic. Taken together, the
facts in this section present a puzzle. If rates influence sales and construction mainly
through the buyer demand channel, then why aren’t house prices–which also should
be sensitive to buyer demand–also rate elastic? In the remainder of the paper, we
turn to a model with search frictions to help us better understand these relationships.
2.1
General Patterns
In this section, we examine the correlation between 30 year fixed rate mortgage rates
and several housing market variables of interest. Both mortgage rates and the housing market variables are measured at the national level. We estimate simple linear
regressions of the following form:
log(yt ) − log(yt−4 ) = α0 + α1 (rt−1 − rt−5 ) + α2 (Xt−1 − Xt−5 ) + t
(1)
where y is the housing market variable of interest, r is the mortgage rate (measured
in percentage points), X are other covariates and t indexes the quarter-year. We
lag the right hand side variables by one quarter to reflect the fact that housing
market variables are measured with some delay (?). We have over 30 years of data,
though the precise number of observations varies with the outcome variable. The
mortgage rate is the average quarterly rate in percentage points as reported in the
Freddie Mac primary mortgage market survey. For our main results, we include the
national unemployment rate in X. α1 represents the semielasticity of y with respect to
mortgage rates–that is, the percentage point change in y in response to a 1 percentage
point increase in mortgage rates.
Because changes in interest rates in equation 1 are likely endogenous, we do not
emphasize the magnitude of the individual estimates of α1 . Rather, we emphasize
the relative estimates of α1 across outcome variables. We think that these relative
estimates are informative because it is reasonable to assume that the magnitude of
the bias due to the endogeneity of interest rate changes is similar across specifications
with different outcome variables.
7
Column 1 of Table 4 displays results for quality-adjusted house prices, as measured
by the Corelogic repeat sales index. The estimated semielasticity is close to zero.
Column 2 shows the semielasticity when total sales volume of new and existing
homes is included as the outcome variable in equation 1. The estimated semielasticity
is -6.8, suggesting that sales volume is more rate elastic than house prices. When the
number of homes available for sale is divided by sales volume—often referred to as
“months supply” in the industry—the estimated semielasticity, reported in column 3,
is a bit higher in absolute value. This result suggests that the rate elasticity of sales
volume is affected by changes in buyer demand—i.e. through homes selling faster or
slower—and not just by changes in the number of sellers putting homes on the market.
This point is reinforced by column 4, which shows that changes in the number of new
listings coming onto the market is actually slightly positively associated with changes
in mortgage rates.10
Column 5 shows the semielasticity when building permits is included as the outcome variable in equation 1. The estimated semielasticity is -11.5, suggesting that like
sales volume, permits are more rate elastic than house prices. Permits could be sensitive to rates for a couple of reasons. First, rates could affect the demand for housing,
which should affect the revenue side of a builder’s profit function. Second, rates could
affect the builder’s financing of construction costs, which should affect the cost side
of a builder’s profit function.11 To better understand the mechanism through which
interest rates influence permit activity, we include interest rates of shorter maturities
on the right side of equation 1. The motivation for these additional specifications is
that if the demand channel is more important, then permits should be most sensitive to longer maturity rates, as most borrowers finance home purchase with 30 year
fixed rate mortgages. If the cost channel is more important, than permits should be
more sensitive to shorter maturity rates, assuming that most builders finance their
construction costs with shorter maturity loans and rates. Column 6 show the results.
The rate elasticity of permits appears to be entirely driven by the 30 year mortgage
rate, suggesting that permits are rate elastic because buyer demand is rate elastic. In
our model below, we will assume the rates influence building activity solely through
10
New listings is for the San Diego CBSA and comes from the CoreLogic listings data that we use
to estimate our model and describe below.
11
In the small literature that structurally models the building decision and includes a role for
interest rates, interest rates typically affect the building decision through the demand channel. See
Murphy [2007], Paciorek [2013].
8
the demand channel.
One way to rationalize the findings in Table 4 is to suggest that house prices are
not sensitive to rates because the housing supply curve is elastic. However, in Table 2
we show that prices are relatively rate insensitive even in metropolitan areas where ?
measures that the local housing supply is relatively inelastic. In particular, we split
metropolitan areas into three groups of equal size based on the ? measure of housing
supply elasticity. Table 1 shows results where we estimate the regressions shown in
Table 1 separately by supply elasticity group. CBSA fixed effects are included. The
results are similar in both high and low housing supply elasticity metros suggesting
that elastic housing supply alone cannot rationalize the motivating facts that we
present in this section.
2.2
Micro Evidence for Rate Elasticity of Home Buyer Demand
In this subsection, we use a novel microdataset to provide further evidence that 1)
mortgage rates affect homebuyer demand and 2) a shock to homebuyer demand from
a shock to the mortgage rate is partly cleared through the probability of sale. In a
model without search frictions, all homes sell instantenously and there is no role for
2).
Our dataset–provided by a private vendor called Optimal Blue–records applications for mortgage rate locks at a daily frequency. Buyers who apply for rate locks
usually do so between the sale agreement date (i.e. when the buyer and sell tentatively agree upon a price and other terms) and the sale closing date (i.e. when the
buyer pays the seller and takes ownership of the home). A mortgage rate lock is a
guarentee by a lender to a borrower that the borrower can obtain mortgage financing
at the locked in mortgage rate, regardless of what happens to mortgage rates subsequently.12 Our dataset covers the time period from 2013-2016; about 25 percent of
originated purchase mortgages over this time period appear in our dataset.
We estimate the following regression on our locks dataset:
log(N umLockst ) − log(N umLockst−2 ) = α0 + α1 (rt+L − rt+L−2 ) + t
12
(2)
A rate lock is usually valid for a specified number of days. Typically, the buyer choose a rate
lock contract that expires after the closing date occurs.
9
where N umLockst is the total number of rate lock applications on day t and r is
the market mortgage rate on day t. 13 Table 3 presents the results. Two-day changes
in mortgage rates are strongly negatively associated with the two-day change in the
number of rate lock applications. A 10 basis point increase in the mortgage rate is
associated with a 13 percent drop in applications. Interestingly, the correlation only
holds when the changes are contemporaneous: columns 2-4 show that for L 6= 0, the
correlation is zero. This result strongly suggests that the response in applications for
L = 0 is due to movements in the mortgage rate, rather than some unobserved factor.
Why do the number of applications drop (rise) when mortgage rates rise (drop) ?
There are a couple possibilities. One, purchase agreements may be less frequent when
rates rise and some buyers may apply for rate locks on (or just after) the agreement
date. Two, buyers may back out of purchase agreements when rates rise, possibly
because the buyer can’t obtain financing at the higher rate or no longer wants to pay
the negotiated price because the debt service burden is too high under the higher
rate.
In either case, the results provide strong evidence that buyer demand responds
to changes in mortgage rates. It is highly unlikely that supply–i.e. sellers adjusting
their decision to list their home for sale or new construction–would respond at such a
high, 2-day frequency. Furthermore, the demand response featured here is through a
quantity channel. The number of home sales is effectively responding strongly to the
change in mortgage rates. 14 A framework for measuring the effect of mortgage rates
on the housing market that is motivated by a frictionless model and only analyzes
house prices would miss this response.
3
Model
We consider a directed search model of a local housing market where there are h = 1, 2
types of housing units (new and old). New homes are produced by builders using
undeveloped land, and old homes are held by existing homeowners. New homes
become old homes after one ownership spell, and old homes will sometimes depreciate
13
Daily market mortgage rates are obtained from Loansifter.
Unfortunately, we cannot properly gauge the price response because although our data contains
the purchase price assocatied with the mortgage rate lock, we do not have any house characteristics
that we would need to adjust prices for house quality. When we simply put average purchase price
as the dependent variable in 2, α1 is negative and significant.
14
10
into undeveloped land.
Each period, some builders and some owners will become sellers. Sellers list their
houses at p = p1 , . . . , pL possible price levels. Buyers will choose which type of house
(new or old) to search for, and at what list price level. We define a house type and
list price pair, (h, p) as a submarket.
Within submarkets, buyers meet sellers via a frictional matching process. Let
θ = b/s be the ratio of buyers to sellers in the submarket, often referred to as market
tightness. Then, the probability that a buyer meets a seller is qb (θ) and the probability
that a seller meets a buyer is qs (θ). We assume that qs and qb are continuous, that
qb (0) = 1 and qs (0) = 0, and that qb is strictly decreasing while qs is strictly increasing.
In equilibrium, the basic tradeoff faced by the buyer is that searching at higher list
prices will typically result in a faster match, whereas the opposite is true for sellers.
3.1
Buyers
Buyers are ex-ante homogeneous. In each period, they may freely enter or exit the
local housing market. Let V b (x) be the value function of entering the housing market
when the aggregate state of the economy is x. The aggregate state can take on x =
x1 , . . . , xN possible values, and evolves according to a first order Markov transition
matrix Π. The aggregate state variable encapsulates variables in the economy which
affect the housing market, such as mortgage interest rates.
Let k be the present value of the buyer’s utility if he does not enter the housing
market. k can be thought of as the outside option of living somewhere else, or of
renting forever. k can be considered an aggregate state variable contained in x. In
equilibrium,
V b (x) = k
(3)
as buyers will freely enter the market until the point in which the marginal buyer is
indifferent between entry or exit.
Buyers enter the housing market as renters. The per-period cost to renting is rent.
We assume that rental units come from a separate housing stock than owner-occupied
units, and are owned by absentee landlords. Buyers then decide which submarket (i.e.
house type and list price) to search in. In submarket (p, h), when the aggregate state
is x, the buyer meets a seller with probability qb (θ(p, h, x)).
11
After the buyer meets a seller, he discovers an idiosyncratic preference shock for that particular house, which is drawn from Gh (). The idiosyncratic preference
shock is additive in utility and is consumed at the time of purchase.15 After drawing
the preference shock, the buyer decides whether or not to purchase the house at the
listed price p. If purchased, he finances the home with a 100% LTV, interest-only,
fixed-rate loan that is paid off at the time of resale. The interest rate r is one of the
aggregate state variables, and can take on one of r = r1 , . . . , rN possible values. The
buyer then becomes an owner.
An owner can be described by the price he purchased the house at, p, and the
interest rate on his loan r. Let V o (p, r, x) be the value function of an owner when the
aggregate state is x. A buyer searching at price p when the interest rate is r and the
aggregate state is x will therefore purchase if and only if:
V o (p, r, x) + ≥ k
(4)
as k is the value of returning to the market as a buyer or of exiting the market.
We can now write the value function of being a buyer as:
V b (x) = u(y − rent) − cb + max βEx0 ,|x,h k + . . .
p,h
n
o
0
. . . + qb (θ(p, h, x)) max 0, V (p, r, x ) + − k
o
(5)
To understand equation (5), u(y − rent) is the flow utility from consumption, where y
is per-period income and rent is the rental rate. cb is the cost of searching the market
as a buyer, and may be thought of as the time investment of searching through listings
and visiting homes. Buyers choose which house type h and list price p to search at.
The probability that he meets a seller is qb (θ(p, h, x)), and if is high enough, he
will purchase the house, getting a surplus of V o (p, r, x0 ) + − k starting next period.
Otherwise, he returns to the market as a buyer, or exits the market, both of which
give present value k.16
15
This assumption simplifies our notation, but is equivalent to a model in which the preference
shock is additive and consumed over the period of living in the house.
16
We have implicitly assumed something about the timing of the purchase, which is that the price
and mortgage are locked in at time t, when the search decision is being made, even though the buyer
does not start becoming an owner until time t + 1. This is realistic for our empirical implementation,
where the time period is a month, as prices and mortgage contracts are usually locked in a month
or two in advance of the closing date.
12
3.2
Owners
Owners stay in their homes until they receive an exogenous moving shock, which
happens with probability λ each period. If an owner does not receive a moving shock,
she simply consumes her income minus interest payments, y − rl, where l is the loan
amount, and moves on to the next period as an owner. If an owner receives a moving
shock, there is an additional probability α that the house depreciates to undeveloped
land. In this case, the owner immediately receives a liquidation value pc for the
depreciated home, pays off the loan amount l, and transfers ownership of the unit to
a builder. The terminal utility for the owner in this situation is U (pc − l), where U
is the utility function used to evaluate net wealth at the time of a move.
If the house does not depreciate, the owner becomes a seller. Let V o (l, r, x) and
V s (l, r, x) be the value functions of owners and sellers, respectively. We can write:
V (l, r, x) = u(y − rl) + βEx0 |x (1 − λ)V o (l, r, x0 ) + . . .
o
0
s
. . . + λ(1 − α)V (l, r, x ) + λαU (pc − l)
(6)
Sellers continue to consume their income minus mortgage payments each period. In
addition, they pay a per-period search cost cs , which can be interpreted as including
both the time investment of selling a home (listing, showing, etc) and the penalty for
not selling the home fast enough (as in the case of moving to a new job.)
When an owner first becomes a seller, she chooses a list price to market her home
at. In subsequent periods, the seller can only change her list price with probability
ρ. Let V s (l, r, x) be the value function of a seller free to change her list price and
let W s (l, r, x, p) be the value function for a seller currently listing at price p. We can
write:
V s (l, r, x) = max
W s (l, r, x, p)
(7)
p
and
s
W (l, r, x, p) = u(y − rl) − cs + βEx0 |x κ(p, h, x)U (p − l) + . . .
. . . + (1 − κ(p, h, x))ρV s (l, r, x0 ) + . . .
s
0
. . . + (1 − κ(p, h, x))(1 − ρ)W (l, r, x , p)
(8)
Here, U (p − l) is the utility over net wealth at the time of a move and κ(p, h, x) is the
13
probability that the seller meets a willing buyer in submarket (p, h). The probability
that the seller meets a willing buyer is given by:
o
0
κ(p, h, x) = qs (θ(p, h, x))Ex0 ,|x,h V (p, r, x ) + − k ≥ 0
(9)
i.e. it is the probability that the seller meets a buyer with preference shock high
enough to warrant purchase of the home at price p. Since owners have old houses,
h = 2 in the seller’s problem.
3.3
Builders
Builders can be in three stages of development: 1) sitting on a plot of depreciated
land, deciding whether or not to begin development, 2) under construction, and 3)
ready to market a completed home. In the first stage, builders start with a plot of
land which they acquire for price pc from owners whose homes depreciate. They are
not required to begin development immediately. Rather, this is a decision they make
each period based on the aggregate state and on idiosyncratic shocks to startup cost.
In the second stage, when the builder has begun development, there is a probability
φ of completing development each period. When development is complete, builders
choose a list price to market the home at, much like sellers of existing homes. Builders
have heterogeneous construction costs C, which for simplicity we assume is paid at
the time of sale. We also assume that the price of land, pc , is paid by the builder at
the time of sale.
Let V 1 (C, x) be the value function of a builder sitting on an undeveloped plot of
land, let V 2 (C, x) be the value function of a builder in development, and let V 3 (C, x)
be the value function of a builder who is listing her property for sale. For builders
with undeveloped land, there is an additively separable, idiosyncratic cost η to begin
development each period. Denote the cdf of η as Fη . The builder’s value functions
are therefore given by:
n
V 1 (C, x) = βEη,x0 |x max 0, V 2 (C, x0 ) − η
o
V 2 (C, x) = βEx0 |x (1 − φ)V 2 (C, x0 ) + φV 3 (C, x0 )
V 3 (C, x) = max
W 3 (C, x, p)
p
14
(10)
(11)
(12)
3
W (C, x, p) = −cc + βEx0 |x κ(p, h, x)U (p − C) + . . .
. . . + (1 − κ(p, h, x))ρV 3 (C, x0 ) + . . .
3
0
. . . + (1 − κ(p, h, x))(1 − ρ)W (C, x , p)
(13)
Builders selling completed homes behave similarly to sellers of existing homes. In
(13), cc is the listing and marketing cost for developers. The probability of meeting a
willing buyer, κ(p, h, x), is the same as in (9). Since developers sell new homes, h = 1
in equation (13).
3.4
Equilibrium
An equilibrium in the housing market consists of value functions V o , W s , V 1 , V 2 , W 3 ,
and a market-tightness function θ that satisfies equations (3) thru (13). From the
value functions and market-tightness, we can derive all the decision rules for agents
in the economy, at any aggregate state. In the Appendix, we prove the existence of
an equilibrium using Brouwer’s fixed point theorem.
A special feature of our model is that the equilibrium value functions and markettightness do not depend on the distribution of agents already present in the economy,
nor on the transition dynamics of these distributions. This is not a general feature
of equilibrium search models, but rather arises out of the indifference condition of
the buyers. To develop some intuition for why this is, let us suppose that a positive
number of buyers search in submarket (p, h) when the state is x. This implies that
(p, h) maximizes (5) in state x. Since V b (x) = k, we can rewrite (5) to give:


θ(p, h, x) = qb−1 


(1 − β)k − u(y − rent) + cb
n

o 

(14)
βEx0 ,|x,h max 0, V o (p, r, x0 ) − k + This shows that, for any submarket in which buyers are willing to search, the equilibrium market-tightness is a function only of p, h, x, and not of the distribution of
agents, as long as the owner value function only depends on p and x.17 Intuitively,
the market-tightness will vary over submarkets in such a way as to make buyers indifferent between searching at a higher list price with lower match probability, or a
lower list price with higher match probability. Multiple submarkets can be active at
17
In the Appendix, we prove that an equilibrium where V o depends only on p and x exists.
15
any one time, though the buyers will be indifferent between them. Directed search is
important for this feature to hold: if search were undirected, buyers would have to
integrate over the distribution of seller types in order to make an entry decision.
The structure of our model is closely related to the directed search models of
the labor market in Menzio and Shi [2010, 2011]. Menzio and Shi were the first
to study the implications of indifference conditions in models of directed search. In
their model, firms face a free entry condition on job postings, which regulates the
market-tightness to depend only on the aggregate state, and not on the distribution
of worker-firm matches within the economy. This is the same role that the free entry
condition of buyers plays in our model.
A consequence of dependence only on p, h, x is that the model becomes much more
tractable to solve, while still allowing for rich price and volume dynamics. Moreover,
the assumption of directed search is quite reasonable for housing markets: home
buyers certainly do not choose which listings to visit at random. The assumption
of free entry and exit of buyers is also reasonable for local housing markets, and we
note that k may be contained in the aggregate state variable, thus allowing for a
time-varying indifference condition that could represent changes to the attractiveness
of the local market (i.e. through higher wages or amenities).
4
4.1
Estimation
Parametric Assumptions
For estimation, we make the following parametric assumptions. We assume that the
matching function is Cobb-Douglass with exponent γ so that the probability that a
buyer meets a seller is:
qb (θ) = min(1, Aθ−γ )
(15)
and the probability that a seller meets a buyer is:
qs (θ) = min(1, Aθ1−γ )
(16)
where A is a scaling parameter. We also assume that per period utility, u, is CRRA
with risk aversion parameter, σ:
16
c1−σ
(17)
1−σ
We assume that sellers and builders evaluate terminal wealth as if they consume
a constant amount of their wealth each period for T periods and that unconsumed
wealth grows at some constant per-period rate. Under this interpretation, we can
write the terminal wealth utility function as
u(c) =
U (w) = Bu(bw)
(18)
where B and b are parameters to be estimated. This is a fairly flexible way to model
terminal utility over wealth, as it nests the situation in which final net wealth is
amortized over either a finite or infinite number of future periods. To accommodate
negative values of wealth, which can occur when the seller sells for a price below the
mortgage balance, we allow the utility to become linear for bw < 1.
We assume that the match quality draws, h , for h = 1, 2 the idiosyncratic development cost, η, and construction costs, C, are all iid and normally distributed with
means µh , µη , µC and standard deviations σh , ση , σC . We also impose that in equilibrium, the price of land pc is such that the expected value of the builders is equal
to a constant kc reflecting the builders’ opportunity cost:
h
EC V 1 (C, x) = kc
i
If pc is observed, then kc can be computed directly from the model estimates.
Finally, we assume that the buyer’s outside option of living somewhere else, k, is
equal to the value of renting forever:
k=
u(y − rent)
1−β
(19)
This amounts to a normalization because the effect of k on buyer decisions is not
separately identifiable from the effect of their search cost, cb .
We assume that a model period is one month and that the mortgage rate is the
only aggregate state with any uncertainty.18 Agents in the economy expect that
monthly mortgage rates will evolve according to a random walk, where the variance
18
Therefore, any changes to income or rent will be interpreted as unexpected shocks. We will
relax this assumption in later versions of the paper.
17
of this process is calibrated to match the variance of monthly mortgage rate changes
in the data.
To normalize the model and because some parameters are not well identified given
the moments we use as described below, we set some parameters ex-ante. In particular, we set A = 0.5, µ2 = 0 , µη = 0. We set the discount factor β equal to 0.951/12 . We
set λ, the probability of a moving shock, equal to 0.008 to reflect that moving occurs
once every 10 years on average. We set α, the probability of depreciation conditional
on moving shock, equal to 0.05. We set φ = 61 so that the average construction time
from start to completion is 6 months, to match the average time-to-build reported
by the Census. We set rent and income equal to the annual averages for San Diego
as computed from Zillow and BEA data. We calibrate ρ = 0.37 using an auxiliary
dataset on San Diego home listings from 2008-2013 from Altos Research. These data
report the list price for each week that a home is on the market. In these data, for
all homes that are on the market for at least one month, we find that 37 percent of
the time, the average list price in a particular month is different from the average list
price in the previous month.
4.2
Data and Moments Used for Estimation
We estimate the remaining parameters of the model by simulated method of moments.
Our main dataset that we use to compute the empirical moments is a dataset provided
by Corelogic on home listings in the San Diego MSA from 2001-2003. For each listing,
we observe the date the home was initially listed for sale, the date it was delisted,
and the outcome of the listing episode—i.e. did it result in a sale or a withdrawal.
If the outcome is a sale, we observe the sales price. We also observe initial list prices
and list prices at the time the home is delisted from the MLS. We also obtain a
dataset recording all sales transactions in San Diego dating back to 1988, provided
by Dataquick. We use the sales dataset to merge on the initial purchase price for
each home listing (assuming the home was purchased subsequent to 1988). We also
use the sales dataset to identify listings that are new construction. The sales dataset
has a flag for new construction sales, and so if a listing can be linked to a recent new
construction sale, we classify the listing as new construction.
We assume that all new construction home listings in our data are of similar house
quality. For existing home listings, in order to focus on a set of homes that are of
18
similar quality, we use only listings that are between the 30th and 60th percentile of a
predicted price distribution. The predicted price is computed by running a regression
of list price on house characteristics, neighborhood fixed effects, and time fixed effects,
and then predicting each home’s price from the regression estimates, excluding time
fixed effects.
We include four sets of moments. The first set of moments are the empirical
counterparts to κ(p, h, x), which is the sale hazard for each list price, house type, and
year. To compute these, we group list prices into bins of size $10k and compute the
average probability that a listing sells in each month by list price bin, house type,
and year. Our second set of moments describe the list price policy functions.
For existing homes, we are also able to compute the empirical counterpart to
the list price policy function, ps (l, r, x), which is defined as the list price maximizing
equation (7). To do this, we compute the average list price for each outstanding
loan amount bin (again of width $10k), outstanding mortgage rate, and year, using
our matched list price data. In our model, outstanding loan amount is equal to the
initial purchase price, and we obtain the initial purchase price for each listing as
described above. For simplicity, we assume that all listings within a year have the
same outstanding mortgage rate, which we set equal to the average mortgage rate in
the month of initial home purchase across listings.
We are not able to compute the empirical counterpart to the list price policy
function for new constructions, pc (C, x), because we do not observe the construction
costs of builders. Instead, we use the data to compute the mean and variance of new
construction list prices, and use those as estimation moments. The model counterparts to these empirical momemnts are the mean and variance of pc (C, x) integrated
across the distribution of C.
Finally, we include the empirical counterpart to the overall probability of building,
i.e. EC [V 2 (C, x) − η ≥ V 1 (C, x)]. We again must integrate out construction costs
because we do not observe them in our data. Collecting the empirical probability of
building required some creativity with the data. From the California Statewide Infill
Study conducted by the Institute of Urban and Regional development, we obtain an
estimate of the stock of potential infill parcels in San Diego in 2004. Infill parcels are
those in developed, residential areas of San Diego that are economically underutilized
(e.g. have a low improvement-value-to-land-value ratio) or in some cases are vacant.
These parcels would not include undeveloped land in the periphery of San Diego.
19
One can roughly equate this estimate to the stock of depreciated homes in our model
that have not yet been developed by builders. To get an estimate of the stock in
other years, we assume that the the stock of undeveloped homes by builders, L,
t
evolves according to Lt = Lt−1 − P ermits
+ λαH where λα as defined above is the
4
depreciation rate, H is an estimate of the size of the housing stock in SanDiego
from the 2000 census, and P ermits is total building permits in San Diego from the
census. Since permits reflect all types of new construction, we divide permits in
each year by four to reflect an estimate from the Infill Study of the share of total
construction in San Diego that is infill construction. Finally, to get the empirical
counterpart to EC [V 2 (C, x) − η ≥ V 1 (C, x)] in each year, we compute P ermitst /4Lt
for t = 2001 − 2003. The computation of these empirical moments is admittedly quite
rough, but in practice, we find that only our parameter estimates of the idiosyncratic
startup costs, η, are particularly sensitive to these moments. As such, we do not put
much emphasis on the interpretation of these particular parameter estimates.
Table 4 presents our preliminary parameter estimates. Our estimate of the risk
aversion parameter, σ, at around 2 is sensible and in line with typical estimates of the
coefficient of risk aversion in CRRA utility functions. Figure 1 shows the estimated
terminal wealth utility function, U (w). The estimated slope is relatively steep for
levels of wealth below $20k, and then flattens out significantly.
4.3
Identification
In this section, we provide a discussion of the main features of the data identify each
of our parameters. The mean and variance of the match quality draws are mainly
identified by the mean and range of list prices for which the empirical sale hazards
are positive. For example, when there is more variance in match quality draws, sellers
will realize positive sale hazards at a larger range of list prices. The seller’s search
cost for existing homes, cs , is partly identified through the minimum list price level
that is associated with an active submarket. For example, if search costs are large,
then submarkets with low list prices would not be active, because sellers would never
choose to list at such prices. The mean and variance of construction costs, µC and
σC , are identified by the mean and variance of list price choices for new construction
sellers. The shape of the empirical sale hazard for new construction homes relative to
existing homes helps to identify seller search cost for builders, cc . The variance of the
20
building startup cost shock, ση is identified by the probability of building moments.
For example, a larger variance of startup costs would imply a smaller probability of
building. The parameters describing the terminal wealth utility function, b,B, σ, are
identified by empirical list price policy function for existing homes. For example,
without risk aversion (σ → 0), the list price choice would be flat with respect to the
purchase price.
It is worth noting that although in practice we use moments from three separate
years to estimate our model, the variation just described that mainly identifies our
parameters could be obtained from a single cross section of data. Therefore, we do not
require an exogenous change in mortgage rates to estimate our model. However, once
we have parameter estimates, we can exploit the structure of our model to simulate
the response of the model economy to an exogenous change in mortgage rates. Thus,
there are two main advantages of our structural approach over the reduced form
approach used in much of the existing empirical literature. One, we do not require an
exogenous change in mortgage rates, which is challenging to find, to obtain results.
Two, the structure of our model allows us to characterize housing valuations, which we
argue cannot be easily observed or proxied for by observable housing market variables
because of search frictions.
5
Model Fit and Discussion
Figure 2 shows the model fit for the sale hazards. We do a good job of fitting the
level of sale hazards for both types of homes and in each year. We also generate sale
hazards that are downward sloping and concave in the list price, which is consistent
with the data as is shown most carefully in Guren [2014].
Figure 2 shows the model fit for the list price policy function for existing homes.
Our model matches the optimal list prices in the data quite well. These results
suggest that the optimal list price is generally increasing in the initial purchase price,
but the rate at which it increases with purchase price is less than 1 for 1. The
increasing relationship between list price and purchase price has been documented
in previous literature (see Genesove and Mayer [2001], Anenberg [2011]), and reflects
risk aversion or loss aversion by sellers; i.e. if sellers were risk-neutral, then the initial
purchase price should not affect list price, conditional on house quality. List price
does not increase with purchase price at a 1-for-1 rate because the probability of sale
21
is decreasing in list price. Thus, the tradeoff for the seller is between a higher price
and a lower sale probability vs. a lower price and a higher sale probability. The
optimal tradeoff depends on the local curvature of the seller’s utility function.
Finally, Table Table 5 shows that we also match the other moments that we use
for estimation quite well.
6
Model Simulations
In this section, we describe model predictions of a housing market response to an
unanticipated 100 basis point permanent increase in mortgage rates. The results are
largely symmetric for an interest rate decrease. In these simulations, we use the 2001
market as the initial conditions and shock the interest rate after 24 periods.
Figure 4 plots how average sale prices respond to the interest rate increase for
both new and existing homes. The effect of the shock is to reduce transaction prices
by about 4.3 percent over a period of about 3 months. The response is not instantaneous because list prices are sticky, and it takes some time for sellers to adjust.19
Interestingly, for existing homes, there is a persistent price growth trend of about 2
percent per year, both before and after the interest rate shock, which is unrelated
to any changes in fundamentals. This result supports existing research showing the
persistence in house price growth that results from frictions in the housing market,
including search frictions that we focus on (see Guren [2014] for a review of these
studies). We will return to a discussion on house price momentum in a few paragraphs.
Figure 5 shows that the price response vastly understates the true response in
actual housing valuation. We define housing valuation as the price which would give
the buyer an expected utility of k if the probability of matching with a seller was
100%. We still assume that the buyer receives an idiosyncratic preference shock and may reject the purchase if is too low. Figure 5 shows that housing valuation
declines by about 13 percent in response to the 100 basis point rate shock. This is
three times larger than the response in expected sale price.
Finally, we plot the responses of transaction volume and permits to the 100 basis
point rate shock. Figure 6 plots transaction volume over #sellers for existing homes.
19
In currently unreported results, we show that the results are basically similar even without sticky
list prices. The price response is only slightly larger and only takes one period to play out.
22
We decide to plot the ratio rather than the raw transaction volume because the
number of sellers also evolves over time, and is fairly sensitive to initial conditions.
By plotting the ratio, we get a better sense of the liquidity of housing transactions.
Figure 6 shows that the transaction rate declines by 9 percent in response to the 100
basis point rate shock. Figure 7 plots permits over #depreciated plots, again focusing
on the ratio because the denominator is sensitive to initial conditions. Figure 7 shows
that permits decline by about 8 percent in response to the rate hike.
Our simulation results are consistent with the motivating empirical facts that
permits and sales volume are more sensitive to interest rate changes than house
prices. However, our simulation results show that underlying housing valuations
are still very sensitive to the interest rate. To explain the mechanism, we turn to
Figures 8 and 9, which plot the list price policy functions for existing homes and
new constructions respectively. Figure 8 shows that for most of the purchase price
distribution, the optimal choice of list price is not particularly sensitive to the interest
rate environment. The main effect of increasing interest rates on the list price policy
function is that owners who purchased at very high prices can no longer list at a very
high price and still expect to find a buyer—they therefore have to reduce their optimal
list price significantly. However, there are not many such owners. Most owners with
lower purchase prices do not change their list prices much in response to the interest
rate hike. Our model estimates, particularly of U (w), therefore suggest that when
rates change, sellers are willing to tradeoff time-to-sell for a level of capital gain that
leaves them in a fairly consistent spot on the terminal wealth utility function. Figure
10 reinforces this point, as it shows that most sellers optimally choose a lower sale
hazard in response to the rate hike, rather than decreasing list prices. Figure 11 plots
the actual tradeoff faced by sellers in the high interest rate environment, and their
optimal choices in both the high and low rate environments, for a seller with purchase
price of $300k. Figure 11 shows that under the high interest rate environment, the
seller is not willing to wait an additional half-month (on average), in order to receive
the same capital gain that she would have under the low-rate environment (about 5
percentage points more).
In addition to illustrating the intuition for why sale prices are less responsive to
interest rate changes than valuations, Figure 8 also demonstrate why our model can
generate house price momentum for existing homes. The model is in steady-state
when all sellers have purchase prices equal to the point in which the list price policy
23
function crosses the 45-degree line. When most of the sellers have purchase prices that
are below the steady-state price level, then they will list at a higher price, resulting in
the next cohort of sellers having a higher purchase and list prices, until the point in
which the purchase price distribution converges to the steady state. There can thus be
momentum in existing home prices without any change in fundamentals. Momentum
is not observed in the price of new constructions, because construction costs are not
tied to purchase prices.20
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27
Figure 1: Estimated Terminal Wealth Utility Function
Terminal wealth is measured in 1000s.
28
Figure 2: Model Fit: Sale Hazards
The x-axis is the list price in 1000s. The y-axis is the sale hazard. The x’s denote
the average sale hazard at each list price level that we compute from the data. The
solid line denotes the model predictions at our estimated parameters.
29
Figure 3: Model Fit: List Price Policy Function
The x-axis is the initial purchase price in 1000s. The y-axis is the list price choice
in 1000s. The x’s denote the average list price at each initial purchase price level
that we compute from the data. The solid line denotes the model predictions at our
estimated parameters.
30
Figure 4: Impulse Response of Average Sale Price
5.9
avg log sales price
existing homes
new constructions
5.85
5.8
5.75
0
5
10
15
20
25
t
31
30
35
40
45
50
Figure 5: Impulse Response of Frictionless Housing Valuation
6.45
existing homes
new constructions
log frictionless WTP
6.4
6.35
6.3
6.25
6.2
0
5
10
15
20
25
t
32
30
35
40
45
50
Figure 6: Impulse Response of Transaction Rate (Existing Homes)
-0.78
-0.8
log avg. sale hazard
-0.82
-0.84
-0.86
-0.88
-0.9
-0.92
-0.94
-0.96
-0.98
0
5
10
15
20
25
t
33
30
35
40
45
50
Figure 7: Impulse Response of Permit Rate
-4.35
-4.36
log (permits/lots)
-4.37
-4.38
-4.39
-4.4
-4.41
-4.42
-4.43
-4.44
0
5
10
15
20
25
t
34
30
35
40
45
50
Figure 8: List Price Policy Functions for Existing Homes by Interest Rate
6.6
6.4
# 10-3
1.6
List Price Policy Functions for Existing Homes
low rate
high rate
sellers at t=24
1.4
6.2
1.2
1
5.8
0.8
5.6
5.4
0.6
5.2
0.4
5
0.2
4.8
4.6
4.6
4.8
5
5.2
5.4
5.6
5.8
log loan amt
35
6
6.2
6.4
0
6.6
mass of sellers
log price
6
Figure 9: List Price Policy Functions for New Construction by Interest Rate
6.4
List Price Policy Functions for New Constructions
low rate
high rate
builders selling at t=24
4.5
4
6.2
6
log price
# 10-5
5
3.5
5.8
3
5.6
2.5
5.4
2
5.2
1.5
5
1
4.8
4.6
4.6
0.5
4.8
5
5.2
5.4
5.6
5.8
log constr cost
36
6
6.2
6.4
0
6.6
mass of builders selling
6.6
Figure 10: Optimal Choice of Sale Hazrd (Existing Homes) by Interest Rate
Optimal Choice of Sale Hazard for Existing Homes
low rate
high rate
sellers at t=24
0.45
1.4
1.2
0.4
sale hazard
# 10-3
1.6
1
0.35
0.8
0.3
0.6
0.25
0.4
0.2
0.15
4.6
0.2
4.8
5
5.2
5.4
5.6
5.8
log loan amt
37
6
6.2
6.4
0
6.6
mass of sellers
0.5
Figure 11: Tradeoff Between Price and Expected Time-on-Market (Existing Homes)
6.5
6
Tradeoff
Choice (high rate)
Choice (low rate)
Expected time on market
5.5
5
4.5
4
3.5
3
2.5
2
0.05
0.1
0.15
0.2
0.25
Capital gain
38
0.3
0.35
0.4
0.45
Table 1: Effects of Interest Rates on Housing Variables
4-qtr chg in 30yr FRM
House Prices
0.0115**
(0.0050)
Sales Volume
-0.0680***
(0.0113)
4-qtr % change in:
Months Supply
Permits
-0.0977***
-0.1146***
(0.0339)
(0.0131)
4-qtr chg in 10yr Treasury
Permits
-0.0705***
(0.0208)
Permits
-0.1890***
(0.0498)
0.0615
(0.0413)
0.0194
(0.0421)
216
175
-0.1004***
(0.0212)
4-qtr chg in 2yr Treasury
Observations
158
*** p<0.01, ** p<0.05, * p<0.1
Permits
106
106
176
175
Changes are 4-quarter changes. Interest rates are in percentage points. Housing
market variables are in logs. Prices come from Corelogic. Sales volume and months
supply comes from the National Association of Realtors. Permits come from the
Census Bureau. All variables reflection national averages or totals. Newey west
standard errors with lag length equal to five quarters in parenthesis.
39
Table 2: Interaction with Housing Supply Elasticity
Metro Housing Supply Elasticity
Low
Medium
High
Dependent variable: 4-qtr % chg in house prices
4-qtr chg in 30yr FRM 0.0083*** 0.0067***
(0.0005)
(0.0004)
Observations
13572
13728
0.0072***
(0.0003)
13884
Dependent variable: 4-qtr % chg in sales volume
4-qtr chg in 30yr FRM -0.0832*** -0.0506***
(0.0081)
(0.0105)
Observations
5287
5317
-0.0047
(0.0129)
5251
Depndent variable: 4-qtr % chg in permits
4-qtr chg in 30yr FRM -0.0905*** -0.0870***
(0.0050)
(0.0057)
-0.1118***
(0.0056)
Number CBSAs
87
88
89
*** p<0.01, ** p<0.05, * p<0.1
Changes are 4-quarter changes. Interest rates are in percentage points. Housing
market variables are in logs. Prices come from Corelogic. Sales volume and months
supply comes from the National Association of Realtors. Permits come from the
Census Bureau. All variables reflection national averages or totals. Newey west
standard errors with lag length equal to five quarters in parenthesis. CBSA fixed
effects are included in each regression.
40
Table 3: Effect of Mortgage Rates in Mortgage Rate Locks Data
L = 0
FRMt+L‐FRMt+L‐2
Ln(NumLockst)‐Ln(NumLockst‐2)
L = 1
L = 2
L = 3
L = ‐5
‐1.3403*** ‐0.4130 0.1006 0.1524 ‐0.0299
(0.2840) (0.2885) (0.2891) (0.2866) (0.2887)
Observations
R‐squared
*** p<0.01, ** p<0.05, * p<0.1
742
0.029
741
0.003
740
0.000
739
0.000
737
0.000
Table 4: Parameter Estimates
Parameter
Description
Value Estimated or Fixed?
γ
exponent on matching function
0.5979
Estimated
A
scale parameter on matching function
0.5
Fixed
µ1
mean of match quality draws, new homes
0.6011
Estimated
µ2
mean of match quality draws, old homes
0
Fixed
σ1
s.d. of match quality draws, new homes
33.55
Estimated
σ2
s.d. of match quality draws, old homes
28.50
Estimated
λ
rate of moving shocks
0.008
Fixed
cb
buyer search cost
1.5795
Estimated
cc
seller search cost (builders)
1.9661
Estimated
cs
seller search cost (owners)
0.8256
Estimated
ρ
probability of being able to change list price
0.37
Fixed
σ
coefficient of relative risk aversion
2.0703
Estimated
b
scale parameter on terminal wealth
0.0286
Estimated
B
scale parameter on terminal utility
2.0825
Estimated
β
subjective discount factor
0.9957
Fixed
ση
s.d. of startup cost shock
3.8055
Estimated
µC
mean of construction costs
$82.72k
Estimated
σC
s.d. of construction costs
$59.02k
Estimated
41
Table 5: Model Fit
Moment
Mean new construction list price, 2001
Mean new construction list price, 2002
Mean new construction list price, 2003
S.D. new construction list price, 2001
S.D. new construction list price, 2002
S.D. new construction list price, 2003
Probability of starting construction, 2001
Probability of starting construction, 2002
Probability of starting construction, 2003
42
Data
$300k
$340k
$410k
$45k
$44k
$55k
0.0103
0.0094
0.0103
Simulated
$341k
$349k
$360k
$47k
$48k
$48k
0.0091
0.0103
0.0115
A
Existence of equilibrium
Here, we prove the existence of an equilibrium in which the value functions and
policy functions depend only on p, r, and x, and do not depend on the distribution
of agents in the economy. We consider a computational loop represented by operator
T , which acts on V o and W s , which are vectors in RL×N ×N and RL×N ×N ×L . The
3
4
operator T takes an initial value for V = (V o , W s ) ∈ RL ×N , and computes new
values using equations (3)-(9). If T has a fixed point, then this proves the existence
of an equilibrium. Note that the owner/seller’s problem is separable from the builder’s
problem (i.e. equations (3)-(9) do not depend on the value functions of the builders),
so we can first prove the existence of an equilibrium in the owner’s problem, then
prove the existence of an equilibrium in the builder’s problem. The existence proof
for an equilibrium in the builder’s problem is similar so we omit it.
The proof will proceed in two steps. First, we show the existence of a closed,
3
4
bounded, and convex set V ⊂ RL ×N such that if V ∈ V then T V ∈ V. We then
show that T is continuous at all V ∈ V. T and V therefore satisfy the conditions of
Brouwer’s Fixed Point Theorem, and there exists a fixed point of T in V.
It is useful first to define a group of operators representing each individual step
within the computational loop. The first step is to compute the expected surplus
from buyer search, conditional on finding a seller. This is represented by the operator
3
4
E : RL ×N → RL×N given by:
EV (p, x) = Ex0 ,|x,h=2 max{0, V o (p, r, x0 ) − k + }
(20)
The second step is to calculate equilibrium market tightness using the buyer’s indif3
4
ference condition (14). This is represented by the operator Θ : RL ×N → RL×N ,
defined as:
!
−1 (1 − β)k − u(y − rent) + cb
(21)
ΘV (p, x) = qb
βEV (p, x)
Although qb−1 is not formally defined for arguments greater than 1, we allow Θ to
return 0 when the term inside qb−1 is greater than 1. This will ensure continuity of Θ
and ensure that sellers will not search in those submarkets.
The third step is to calculate the probability that a seller meets a buyer using
43
3 ×N 4
equation (9). This is represented by the operator K : RL
→ RL×N :
KV (p, x) = qs (ΘV (p, x))Ex0 ,|x,h=2 V o (p, r, x0 ) − k + ≥ 0
(22)
The fourth step is to compute the new value of T s , using equation (8). This is
3
4
2
2
represented by the operator T s : RL ×N → RL ×N , defined as:
s
T V (l, r, x, p) = u(y − rl) − cs + βEx0 |x KV (p, x)U (p − l) + . . .
. . . + (1 − KV (p, x))ρ max
W s (l, r, x0 , p0 ) + . . .
0
p
s
0
. . . + (1 − KV (p, x))(1 − ρ)W (l, r, x , p)
(23)
The final step is to compute the new value of V o , using equation (6). This is repre3
4
2
sented by the operator T o : RL ×N → RL×N , defined as:
T V (l, r, x) = u(y − rl) + βEx0 |x (1 − λ)V o (l, r, x0 ) + . . .
o
s
0
. . . + λ(1 − α) max W (l, r, x , p) + λαU (pc − l)
p
3 ×N 4
And the operator T : RL
3 ×N 4
→ RL
(24)
is simply defined by:
T V = (T o V, T s V )
A.1
(25)
Proof of boundedness
The first step is to show the existence of a closed, bounded, and convex set V such
that if V ∈ V then T V ∈ V. Let us define:
(
W
s
W̄ s
)
u(y − r̄p̄) − cs
= u(y − r̄p̄) − cs + β min
, U (p − p̄)
1−β
(
)
u(y − rp) − cs
= u(y − rp) − cs + β max
, U (p̄ − p)
1−β
44
(26)
(27)
and
u(y − r̄p̄) + λ(1 − α)W s + λαU (pc − p̄)
1 − β(1 − λ)
u(y − rp) + λ(1 − α)W̄ s + λαU (pc − p)
=
1 − β(1 − λ)
Vo =
(28)
V̄ o
(29)
where r̄ and p̄ are the highest possible interest rates and price levels, respectively, and
2
2
2
r and p are the lowest. Let V = [V o , V̄ o ]L×N × [W s , W̄ s ]L ×N . V is closed, bounded,
and convex. Since KV (p, x) ∈ [0, 1] for any p, x and any V , it is easy to show using
equation (23) that if V ∈ V then T s V (l, r, x, p) ∈ [W s , W̄ s ] for any l, r, x, p. Similary,
we can use equation (24) to show that if V ∈ V, then T o V (l, r, x) ∈ [V o , V̄ o ] for any
l, r, x. Therefore, if V ∈ V then T V ∈ V.
A.2
Proof of continuity
The second condition of Brouwer’s Fixed Theorem is that T is continuous at each
V ∈ V. To show this, we simply need to show that the operators given in equations
(20)-(24) are continuous. As a reminder, the definition of continuity for a mapping
T : X → Y from normed vector space X to normed vector space Y is:
Definition 1. T : X → Y is continuous at x ∈ X if for all ρ > 0, there exists δ > 0
such that for any y ∈ X, kx − yk < δ implies kT x − T yk < ρ. If T is continuous at
all x ∈ X then we simply say T is continuous in X.
Let k·k be the sup-norm. We will simply demonstrate that the operator E is
continuous. The continuity of the rest of the operators follow
from
the continuity of
E. Given ρ > 0, we let δ = ρ/2. Let V, Ṽ ∈ V such that V − Ṽ < δ. We write:
EV (p, x) =
X
x0 |x
πx0 |x
(Z
)
∞
o
k−V o (p,r,x0 )
45
0
[ − k + V (p, r, x )] g()d
(30)
and so:
EV − E Ṽ
X
=
πx0 |x



x0 |x
Z k−min{V o ,Ṽ o } h
k−max{V o ,Ṽ o }
Z ∞
+
i
− k + max{V o , Ṽ o } g()d
k−min{V o ,Ṽ o }
h
i
V o − Ṽ o g()d
(31)
Now note that:
Z k−min{V o ,Ṽ o } h
k−max{V o ,Ṽ o }
≤
Z k−min{V o ,Ṽ o } h
≤ δ
and:
k−max{V o ,Ṽ o }
Z k−min{V o ,Ṽ o }
k−max{V o ,Ṽ o }
i
− k + max{V o , Ṽ o } g()d
i
max{V o , Ṽ o } − min{V o , Ṽ o } g()d
g()d ≤ δ
(32)
Z k−min{V o ,Ṽ o } h
i
− k + max{V o , Ṽ o } g()d ≥ 0 > −δ
k−max{V o ,Ṽ o }
(33)
Further note that:
Z ∞
h
k−min{V o ,Ṽ o }
o
o
i
V − Ṽ g()d ≤ δ
Z ∞
k−min{V o ,Ṽ o }
g()d ≤ δ
(34)
g()d ≥ −δ
(35)
and:
Z ∞
k−min{V o ,Ṽ o }
h
i
V o − Ṽ o g()d ≥ −δ
Z ∞
k−min{V o ,Ṽ o }
Together, this implies that if V − Ṽ < δ, then EV − E Ṽ < 2δ = ρ, as desired.
46