Discussion Paper No. 989

FLIPPING
IN THE HOUSING MARKET

Charles Ka Yui Leung
Chung-Yi Tse

January 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Flipping in the Housing Market
Charles Ka Yui Leung and Chung-Yi Tse
December 2016
Abstract
We add arbitraging middlemen -- investors who attempt to profit from buying low andselling high -- to a canonical housing market search model. Flipping tends to take place insluggish and tight, but not in moderate, markets. To follow is the possibility of multiple
equilibria. In one equilibrium, most, if not all, transactions are intermediated, resulting in
rapid turnover, a high vacancy rate, and high housing prices. In another equilibrium, few
houses are bought and sold by middlemen. Turnover is slow, few houses are vacant, andprices are moderate. Moreover, flippers can enter and exit en masse in response to thesmallest interest rate shock. The housing market can then be intrinsically unstable evenwhen all flippers are akin to the arbitraging middlemen in classical finance theory. In
speeding up turnover, the flipping that takes place in a sluggish and illiquid market tends
to be socially beneficial. The flipping that takes place in a tight and liquid market can bewasteful as the efficiency gain from any faster turnover is unlikely to be large enough tooffset the loss from more houses being left vacant in the hands of flippers. Based on ourcalibrated model, which matches several stylized facts of the U.S. housing market, we showthat the housing price response to interest rate change is very non-linear, suggesting
cautions to policy attempt to “stabilize” the housing market through monetary policy.
Key words: Search and matching, housing market, liquidity, flippers and speculators,
financing and bargaining advantage.
JEL classifications: D83, R30, G12
Acknowledgement:
We thank Enrique Schroth as discussant of an earlier version of the paper in the 2011 NYU
Economics PhD Alumni Conference, Angela Mak for excellent research assistance and Paul Anglin,
Kosuke Aoki, Kuang-Liang Chang, Nan-Kuang Chen, Patrick Ho, Min Hwang, Hirokazu Ishise,
Munechika Katayama, Yuichiro Kawaguchi, Fred Kwan, Chun Wah Liu, Steve Malpezzi, Kazuo
Ogawa, Jonathan Parker, Erwan Quintin, Tim Riddiough, Andrew Sim, Edward Tang, Hajime
Tomura, Takayuki Tsuruga, Isabel Yan, seminar participants of City University of Hong KongHKIMR workshop, GRIPS, Kyoto University, National Taiwan University, Osaka University,
University of Macau, University of Tokyo, University of Wisconsin-Madison, Waseda University fordiscussions. Tse gratefully acknowledges financial support from HK GRF grant HKU 751909H. The
usual disclaimer applies.
Correspondence:
Leung, Department of Economics and Finance, City University of Hong Kong, Kowloon Tong; (email)kycleung@cityu.edu.hk
Tse, School of Economics and Finance, University of Hong Kong, Pokfulam Road, Hong Kong; (email) tsechung@econ.hku.hk

1Discussion Paper No. 989

FLIPPING
IN THE HOUSING MARKET

Charles Ka Yui Leung
Chung-Yi Tse

January 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Flipping in the Housing Market
Charles Ka Yui Leung and Chung-Yi Tse
December 2016
Abstract
We add arbitraging middlemen -- investors who attempt to profit from buying low andselling high -- to a canonical housing market search model. Flipping tends to take place insluggish and tight, but not in moderate, markets. To follow is the possibility of multiple
equilibria. In one equilibrium, most, if not all, transactions are intermediated, resulting in
rapid turnover, a high vacancy rate, and high housing prices. In another equilibrium, few
houses are bought and sold by middlemen. Turnover is slow, few houses are vacant, andprices are moderate. Moreover, flippers can enter and exit en masse in response to thesmallest interest rate shock. The housing market can then be intrinsically unstable evenwhen all flippers are akin to the arbitraging middlemen in classical finance theory. In
speeding up turnover, the flipping that takes place in a sluggish and illiquid market tends
to be socially beneficial. The flipping that takes place in a tight and liquid market can bewasteful as the efficiency gain from any faster turnover is unlikely to be large enough tooffset the loss from more houses being left vacant in the hands of flippers. Based on ourcalibrated model, which matches several stylized facts of the U.S. housing market, we showthat the housing price response to interest rate change is very non-linear, suggesting
cautions to policy attempt to “stabilize” the housing market through monetary policy.
Key words: Search and matching, housing market, liquidity, flippers and speculators,
financing and bargaining advantage.
JEL classifications: D83, R30, G12
Acknowledgement:
We thank Enrique Schroth as discussant of an earlier version of the paper in the 2011 NYU
Economics PhD Alumni Conference, Angela Mak for excellent research assistance and Paul Anglin,
Kosuke Aoki, Kuang-Liang Chang, Nan-Kuang Chen, Patrick Ho, Min Hwang, Hirokazu Ishise,
Munechika Katayama, Yuichiro Kawaguchi, Fred Kwan, Chun Wah Liu, Steve Malpezzi, Kazuo
Ogawa, Jonathan Parker, Erwan Quintin, Tim Riddiough, Andrew Sim, Edward Tang, Hajime
Tomura, Takayuki Tsuruga, Isabel Yan, seminar participants of City University of Hong KongHKIMR workshop, GRIPS, Kyoto University, National University of Taiwan, Osaka University,
University of Macau, University of Tokyo, University of Wisconsin-Madison, Waseda University fordiscussions. Tse gratefully acknowledges financial support from HK GRF grant HKU 751909H. The
usual disclaimer applies.
Correspondence:
Leung, Department of Economics and Finance, City University of Hong Kong, Kowloon Tong; (email)kycleung@cityu.edu.hk
Tse, School of Economics and Finance, University of Hong Kong, Pokfulam Road, Hong Kong; (e-mail)tsechung@econ.hku.hk

1

1

Introduction

In many housing markets, the purchases of owner-occupied houses by investors who
attempt to profit from buying low and selling high rather than for occupation arecommonplace. For a long time, anecdotal evidence abounds as to how the presence of
these investors, who are popularly known as flippers in the U.S., in the housing market
can be widespread. More recently, empirical studies have began to systematically
document the extent to which transactions in the housing market are motivated bybuying and selling for short-term gains and how these activities are correlated withthe housing price cycle. In particular, Haughwout et al. (2011) report that the shareof all new purchase mortgages in the U.S. taken out by investors — individuals who
hold two or more first-lien mortgages — was as high as 25% on average during theearly to mid 2000s. At the peak of the housing market boom in 2006, the figures
reached 35% for the whole of the U.S. and 45% for the“bubble states”. Depken etal. (2009) report that for the same period, on average, 13.7% of housing market
transactions were for houses sold again within the first two years of purchase in themetropolitan Las Vegas area. At the peak in 2005, it reached a high of 25%. Bayeret al. (2011) report that for five counties in the LA metropolitan area, over 15% of
all homes purchased near the peak of the housing market boom in 2003-2005 were resold within two years. Even in the cold period in the 1990s, the percentage remained
above 5%.
Arguably, the central questions on flipping in the housing market are how it may
contribute to housing price volatility and whether it serves any useful purpose. In thispaper, we study a housing market search model along the lines of Arnold (1989) and
Wheaton (1990) in which houses are demanded by flippers in addition to end-userhouseholds to address the two questions.
In our model, the end-user households are liquidity constrained to the extent thateach cannot hold more than one house at a time. In this case, a household which
desires to move because the old house is no longer a good match must first sell it
before the household can buy a new house. The primary advantage of flippers in ourmodel is that, collectively, they can hold as many houses as the market needs them todo so. A mismatched and liquidity-constrained household can then sell the old housequickly to flippers to be able to buy a new house sooner. Thus, first of all, the flippersin our model are akin to the arbitraging middlemen in illiquid markets in classicalfinance theory, who help speed up turnover and improve liquidity in the market. Suchliquidity services, not surprisingly, are in greatest demand in an otherwise sluggish
and illiquid market in which it can take a long time for mismatched households tosell houses themselves.
Often times, investors in the housing market are cash-rich investors as well as
experienced flippers. As cash-rich investors, they tend to have lower opportunity
costs than others in holding vacant houses.1 As experienced flippers, they should
1Helbert et al. (2013) report that many investors in the housing market are all-cash buyers.

2

be more adept at bargaining than many end-user households, who only buy and sellhouses infrequently.2 To capture such financing and bargaining advantages, in ourmodel, we allow for the possibility that the flippers finance real estate investment
at a lower cost and possess a greater bargaining strength than others. With theseadvantages, flippers sell houses at a relatively high price, in which case mismatchedhouseholds can be better oﬀ letting flippers sell on their behalf, irrespective of how
quickly they can sell houses themselves, if the flippers are not buying houses fromthem at too big a discount. In a tight and liquid market where houses are soldquickly, there can only be a small “bid-ask” spread in house flipping. Then, flippers
must be buying houses from mismatched households at a similarly high price if they
are selling those houses later on at a high price. The main novelty in our analysis
is that we find that mismatched households may find it attractive to sell to flippers
not just in sluggish and illiquid markets, but also in tight and liquid markets forthe especially high price they receive from selling to flippers in such markets where
the flipper-sellers’ advantages are passed onto household-sellers to the fullest extent
possible.
Because flippers can also thrive in a tight and liquid market while the market tends
to be tight and liquid when flipping is prevalent, there can be multiple equilibria in ourmodel. In one equilibrium, most, if not all, transactions are intermediated, resulting
in rapid turnover and high housing prices. In another equilibrium, few houses arebought and sold by flippers. Turnover is slow and prices are moderate.
With the multiplicity of equilibrium, wide swings in price and transaction canhappen without any underlying changes in housing supply, preference, and interest
rate. Moreover, in our model, flippers can enter and exit en masse in response to
the smallest interest rate shock. Then, on top of the usual eﬀect of interest rates
on asset prices, any such shocks can have a significant indirect impact on housing
prices through their influences on the activities of flippers. In all, we find that evenin the entire absence of any kind of extrinsic uncertainty and less-than-fully rational
agents, flipping can still contribute to housing price volatility. A natural question
to follow up is how important such a channel of volatility can be. Our quantitative
analysis indicates that housing prices can diﬀer by up to 23 percent across steady-state
equilibria and vary by 26 percent in response to a seemingly unimportant interest rate
shock when the model is calibrated to several observable characteristics of the U.S.
housing market.
While more houses are being flipped and remain vacant in the hands of flippers,
more households have to seek shelter in rental housing. In the model housing market,
flipping can be excessive if the eﬃciency gains from the faster turnover fall short of
the rental expenditures households incur during which houses are being flipped andleft vacant. In an equilibrium where mismatched households are selling to flippers
primarily for the high price flippers oﬀer, they do not tend to be better oﬀ than if
2Bayer et al. (2011) document the prevalence of experienced flippers in the housing market andshow that they often buy houses at lower prices and sell them at higher prices than others.

3there were no flipping, for later on, the households will be buying new houses at asimilarly high price. In this case, there is little to gain from flipping to oﬀset its cost
and the strategic complementary that gives rise to flipping in tight markets can be aform of market failure.
Our model has a number of readily testable implications. First, it trivially predicts
a positive cross-section relation between housing prices and Time-On-The-Market(TOM) — mismatched homeowners can either sell quickly to flippers at a discountor to wait for a better oﬀer from an end-user buyer to arrive — which agrees withthe evidence reported in Merlo and Ortalo-Magne (2004), Leung et al. (2002) andGenesove and Mayer (1997), among others.3
An important goal of the recent housing market search and matching literature
is to understand the positive time-series correlation between housing prices and sales
and the negative correlation between the two and the average TOM.4 In our model,
across steady-state equilibria, a positive relation between prices and sales and a negative relation between the two and the average TOM also hold — in the equilibrium
in which more houses are sold to flippers, prices and sales are both higher, whereas
houses on average stay on the market for a shorter period of time. More importantly,
our analysis adds a new twist to the time-series empirics of the housing market, whichis that vacancies should increase together with prices and sales if the increase in sales
is due to more houses sold to flippers, who will just leave them vacant until they aresold to the eventual end users.
Insofar as the flippers in our model act as middlemen between the original homeowners and the eventual end-user buyers, this paper contributes to the literature on
middlemen in search and matching pioneered by Rubinstein and Wolinsky (1987).
Previously, it was argued that middlemen could survive by developing reputations
as sellers of high quality goods (Li, 1998), by holding a large inventory of diﬀerentiated products to make shopping less costly for others (Johri and Leach, 2002;
Shevchenko, 2004; Smith, 2004), by raising the matching rate in case matching issubject to increasing returns (Masters, 2007), and by lowering distance-related trade
costs for others (Tse, 2011). This paper studies the role of middlemen in the provisionof market liquidity and the eﬀects of any financing and bargaining advantages that
middlemen may possess on the nature of equilibrium.
Among papers in the housing market search and matching literature, perhaps
Moen and Nenov (2014) is closest to ours in that they study how decisions made bymismatched households, like those in our model, can lead to multiplicity. In theirmodel, households are permitted to hold up to two houses, but holding two houses is3
Albrecht et al. (2007) emphasize another aspect of the results reported in Merlo and OrtaloMagne (2004), which is that downward price revisions are increasingly likely when a house spends
more and more time on the market.
4
Stein (1995), who explains how the down-payment requirement plays a crucial role in amplifying
shocks, is an early non-search-theoretic explanation for the positive relation between prices and sales.
Hort (2000) and Leung et al. (2003), among others, provide recent evidence. Kwok and Tse (2006)
show that the same relation holds in the cross section.

4

a particularly costly state, and so is holding no house. The strategic complementaryin their model is that when other mismatched households are choosing to buy a
new home first before selling, there will be a tight market, in which an individual
mismatched household is also better oﬀ to buy first before selling. To do otherwise
can leave the household in the costly state of holding no house for an extended periodof time. Conversely, when other households choose selling first, there will be a slow
market, in which it is optimal for an individual household to also choose selling first to
avoid being stuck in the costly state of holding two houses for a long time. Like theirmodel, decisions made by mismatched households in ours impart on market tightness,through which the strategic complementary works. Unlike their model in which prices
do not appear to play any role in the strategic complementary, how prices vary withmarket tightness is crucial to the strategic complementary in ours. It is this last
mechanism through which our model gives rise to the implication that housing prices
are positively correlated with sales and vacancies but negatively correlated with TOM
in the time series.
A simple model of housing market flippers as middlemen is also in Bayer et al.
(2011). The model though is partial equilibrium in nature and cannot be used toanswer many of the questions we ask in this paper. Intermediaries who buy up mismatched houses from households and then sell them on their behalf are also present
in the model of the interaction of the frictional housing and labor markets of Headand Lloyd-Ellis (2012). But there the assumption is merely a simplifying assumption
and the presence of these agents in the given setting appears inconsequential. Analyzes of how middlemen may serve to improve liquidity in a frictional market also
include Gavazza (2012) and Lagos et al. (2011). These studies do not allow end-userhouseholds a choice of whether to deal with the middlemen and for the multiplicity
of equilibrium like we do though. Multiple equilibria in a search and matching modelwith middlemen can also exist in Watanabe (2010). The multiplicity in that model
is due to the assumption that the intermediation technology is subject to increasing
returns to scale, whereas the multiplicity in ours arises from a particular strategic
complementary. Moreover, only one of the two steady-state equilibria in that modelis stable, whereas there can be more than one stable steady-state equilibria in ours.
The next section presents the model, the detailed analysis of which follows in Section 3. In section 4, we study the planner’s problem of optimal flipping in the model
housing market. Section 5 explores various empirical implications of the model. In
section 6, we test the time-series implications of our model as pertain to especially the
behavior of the vacancy rate. In Section 7, we calibrate the model to several observable characteristics of the U.S. housing market to assess the amount of volatility thatthe model can generate. Section 8, which draws on results we present in a technicalnote (Leung and Tse, 2016a) to accompany the paper, discusses several extensionsof the model. Section 9 concludes. All proofs are relegated to the Appendix. For
brevity, we restrict attention to analyzing steady-state equilibria in this paper. A
second companion technical note (Leung and Tse, 2016b) covers the analysis of the5dynamics for the special case in which all agents possess the same bargaining power.5

2

Model

2.1

BasicsThere is a continuum of measure one risk-neutral households, each of whom discounts
the future at the rate . There are two types of housing: owner-occupied, the supplyof which is perfectly inelastic at   1 and rental, which is supplied perfectly elastically for a rental payment of  per time unit. A household staying in a matchedowner-occupied house enjoys a flow utility of   0 whereas a household either ina mismatched house or in rental housing none. A household-house match breaks up
exogenously at a Poisson arrival rate , after which the household may continue tostay in the house but it no longer enjoys the flow utility . In the meantime, thehousehold may choose to sell the old house and search out a new match. An important assumption is that households are liquidity constrained to the extent that eachcan hold at most one house at a time. Then, a mismatched homeowner must firstsell the old house and move to rental housing before she can buy a new house. Our
qualitative results should hold as long as there is a limit, not necessarily one, on thenumber of houses a household can own at a time.6 The one-house-limit assumption
simplifies the analysis considerably.
The end-user market Households buy and sell houses in a frictional market inwhich the flow of matches falls out from a concave and CRS matching function ( ), with  and  denoting, respectively, the measures of buyers and sellers in the market. Let  =  denote market tightness. Then, the rate at which aseller finds a buyer is ( ) () ≡=  ( 1) whereas the buyer’s matching rate is  () =  ()  Given that  is increasing andconcave in  and ,


 0
 0


We impose the usual regularity conditions on  forlim  () = lim  () = ∞lim  () = lim  () = 0

→0

→∞

→∞

5

→0

Not for publication, available for download in http://www.sef.hku.hk/~tsechung/index.htm
In Section 8 and in Leung and Tse (2016a), we explain how this is the case when households
can hold up to two houses at a time.
6

6

The Walrasian investment market Instead of waiting out an end-user buyer tocome along, a mismatched household may sell the old house right away in a Walrasianmarket to specialist investors — agents who do not live in the houses they have bought
but rather attempt to profit from buying low and selling high. Because homogeneous
flippers do not gain by buying and selling houses among themselves, the risk-neutral
flippers may only sell in the end-user market and will succeed in doing so at the samerate  () that any household-seller does in the market. We assume that flippers
discount the future at the same rate  that end-user households do but allow for
the possibility that they finance real estate investment at a diﬀerent rate  . In thecompetitive investment market, prices adjust to eliminate any excess returns on realestate investment.
Of course, it is hard to envisage that there are two distinct markets — one frictional and one Walrasian — for a household-seller to choose between in reality. The
assumption is but a convenient modelling fiction for where the seller chooses between
selling right away at a discount to investors and waiting for a better oﬀer from an
end-user buyer to arrive.
The assumption of a Walrasian investment market is, by all means, a simplifying
assumption. A more general assumption is to model the market as a frictional market
too, but possibly one in which the frictions are less severe than in the end-user market.
If flippers are entirely motivated by arbitrage considerations and do not care if thehouses to be purchased are good matches for their own occupation, search should be amuch less serious problem. Moreover, if flippers enter the housing market and searchat a zero entry cost, in equilibrium, the market will be populated by an infinite massof flipper-buyers, in which case, any household-sellers who bother to search for aninvestor to trade with will be able to find one instantaneously for any usual matching
function. In this way, a Walrasian investment market can be thought of as the limitof a frictional market in which the costs of entry and search for flippers tend to zero.
In Section 8 below, we discuss how our qualitative results should hold in the generalcase in which the market is populated by only a finite measure of flipper-buyers whomeet household-sellers in a frictional market.

2.2

Stocks and flows

Accounting identities At any one time, a household can either be staying in amatched house, in a mismatched house, or in rental housing. Let  ,  , and 
denote the measures of households in the respective states, which must sum to theunit measure of households in the market; i.e., +  +  = 1(1)Each owner-occupied house must be held either by an end-user household or by aflipper. Hence,(2) +  +  = 7where  denotes both the measures of active flippers and houses held by these agents.
If each household can hold at most one house at any moment, the only buyers in
the end-user market are households in rental housing; i.e., =  (3)

On the other hand, sellers in the market include both mismatched homeowners andflippers, so that(4) =  +  

Housing market flows In each unit of time, the inflows into matched owneroccupied housing are comprised of the successful buyers among all households inrental housing ( ()  ), whereas the outflows are comprised of those who become
mismatched in the same time period ( ). In the steady state,· = 0 ⇒  ()  =  (5)Households’ whose matches just break up may choose to sell their old houses rightaway to flippers in the investment market or to wait out a buyer to arrive in the enduser market. Let  denote the (endogenously determined) fraction of mismatchedhouseholds who sell in the investment market and 1− the fraction who sell in the enduser market. In each time unit then, the measure of mismatched homeowners sellingin the end-user market increases by (1 − )  , whereas the exits are comprised of
the successful sellers ( ()  ) in the meantime. In the steady state,· = 0 ⇒ (1 − )  =  ()  (6)

and  =  , imply that
It can be shown that (5) and (6), together with  = + +  ()  =  ()   =  ()  
which
say
³·
´ that, respectively, the flows into and out of rental housing are equal
 = 0 and the measure of houses bought by flippers are matched by the mea´³·sure sold  = 0 . Figure 1 depicts the flows of households into and out of thethree states.
(figure 1 about here)

8

2.3

Market tightness

Solving (1)-(6), we can show the following.
Lemma 1a. At  = 0,  = 0.
b. As  increases from 0 toward 1,




 0
 0
 0
 0




c. At  = 1,  = 0.
Parts (a) and (c) of the Lemma say that, respectively, at one extreme, when no
mismatched households are selling to flippers, there cannot be any vacant housesin the hands of these agents, and at the other extreme, when all households sell
to flippers immediately after becoming mismatched, there cannot be any household
remaining in a mismatched house in the steady state. In general, by part (b), as 
increases from 0 toward 1, there are more houses in the hands of flippers and fewer
in the hands of mismatched homeowners.
What is less obvious in the Lemma is that as  increases, there would also be
more households in rental housing, as well as being matched in the steady state.
Intuitively, when flippers hold a greater fraction of the owner-occupied housing stock,fewer households can stay in owner-occupied houses and therefore more must be
accommodated in rental housing. In the meantime, there are fewer mismatchedhouseholds spending any time at all selling their old houses in the end-user market
before initiating search for a new match. Then, there should only be more matchedhouseholds in the steady state.
When more mismatched households sell to flippers right away, there are not only
more buyers in the end-user market but also fewer sellers as well.7 This is because as
houses are sold faster where there are more buyers, there can only be a smaller stockof houses left for sale in the steady state. The two tendencies should then conspireto give rise to a tighter market with a larger
=



=


 + 

Lemma 2 The stock-flow equations (1)-(6) can be combined to yield a single equation,
 +  () (1 − ) − (1 −  + )  = 07

By (2),  =  +  =  −  , which is decreasing in  given that    0.

9(7)

in  and , from which an implicit function  =  () for  ∈ [0 1], can be defined,
and that    0. Both the lower and upper bounds, given by, respectively,  (0)and  (1), are strictly positive and finite for any  ∈ (0 1). For each ,  () is
decreasing in , and that lim→1  (0) = 0.
The last part of the Lemma says that a larger housing stock tends to give rise toa slower end-user market. Other things equal, there will be more houses for sale witha larger housing stock and as houses are sold more slowly as a result, there can only
be fewer mismatched households entering the end-user market as buyers. Altogether
then, market tightness  falls. Indeed, if there is one house for each household ( = 1)
and if no mismatched households are selling to flippers at all ( = 0), no householdsellers can rid themselves of the old mismatched houses to start searching for a newhouse in which case, with  =  = 0, market tightness falls to zero.

2.4

Asset values and housing prices

Lemmas 1 and 2 above describe how the measures of households in the three states,
the stock of houses in the hands of flippers, and the market tightness depends on
. The value of  — for the fraction of mismatched households selling to flippers —depends on the households’ comparison of the payoﬀs of selling in the investment
versus the end-user markets.
Asset values for households A household staying in a mismatched house andtrying to sell it to an end user has asset value  satisfying, =  () ( +  −  ) (8)where  denotes the price the household expects to receive for selling the house to
an end user and  the value of being in rental housing. Under (8), the mismatchedhomeowner is entirely preoccupied with disposing the old house while she makes noattempt to search for a new match. This is due to the assumption that a household
cannot hold more than one house at a time. Once the household manages to sell theold house and only then, it moves to rental housing and enters the end-user market as awould-be buyer, who may eventually be buying either from a mismatched homeownerat price  or from a flipper at a price we denote as   .
Lemma 3 In the steady state, the fraction of flipper-sellers among all sellers in theend-user market is equal to the fraction of mismatched households selling to flippers
in the first place; i.e.,=  + 
In this case, = − +  () ( − (  + (1 − )  ) −  ) 10(9)where  is the exogenously given flow rental payment and  the value of staying ina matched house, given by =  +  (max { +     } −  ) (10)The owner-occupier enjoys the flow utility  while being matched. The match will
be broken, however, with probability  after which the household may sell the houseright away in the investment market at a price we denote as   and switch torental housing immediately thereafter. Alternatively, the household can continue
to stay in the house while trying to sell it to an end user. Note that under (10)and (8), the mismatched household has one chance only to sell the old house in
the investment market, at the moment the match is broken. Those who forfeit thisone-time opportunity must wait out a buyer in the end-user market to arrive. The
restriction is without loss of generality though in a steady-state equilibrium, in which
the asset values and housing prices stay unchanging over time.8
Asset values for flippers If a flipper expects to receive   for selling a house in
the end-user market and has previously paid   for it in the investment market, theflipper has asset value  satisfying, = −   +  () (  −   −  ) where  is the flipper’s cost of funds. With free entry in house flipping,  = 0 andtherefore, ()  =
  
(11)
 () + 
Bargaining When a household-buyer in rental housing is matched with a flipperseller, the division of surplus in the bargaining satisfies  ( −   −  ) = (1 −   ) (  −   ) (12)where   denotes the flipper-seller’s share of the match surplus. When the same buyer
is matched with a household-seller in a mismatched house, the division of surplus in
the bargaining satisfies  ( −  −  ) = (1 −   ) ( +  −  ) (13)where   denotes the household-seller’s share of the match surplus. If flippers areagents specializing in buying and selling, it is most reasonable to assume that   ≥ .
8Equation (9) assumes that the rental household is better oﬀ buying a house either from amismatched homeower or a flipper rather than continuing to stay in rental housing. Equations(10) and (8) assume that the mismatched household is better oﬀ selling the old house either in theinvestment or end-user market rather than just staying in the mismatched house. Lemma 14 in the
Appendix verifies that all this holds in equilibrium.

11

2.5

Households’ optimizationWrite ( ) ≡  +   −  

(14)

as the diﬀerence in payoﬀ for a mismatched household between selling in the investment market ( +   ) and in the end-user market ( ). If  ( )  0 ( 0), forall  ∈ [0 1], the household strictly prefers to sell in the investment (end-user) market
at the given  no matter what others choose to do. For certain , there may exist
some  () ∈ [0 1] such that  (  ()) = 0, in which case equilibrium requires
a fraction  () of mismatched households selling in the investment market and therest selling in the end-user market. In sum, we can define a relation,⎧ ( 1) ≥ 0⎨ 1
 ()  (  ()) = 0 
 () =⎩0 ( 0) ≤ 0

between market tightness  and the fraction  of mismatched households selling in
the investment market from the households’ optimization.

2.6

Equilibrium

We now have two steady-state relations between  and : the  () function in
Lemma 2 from the stock-flow equations and the  () relation from mismatchedhouseholds’ optimization. A steady-state equilibrium is any { } pair that simultaneously satisfies the two relations.9

3

Analysis

3.1

Flippers’ advantages and the nature of equilibrium

To proceed with the analysis of equilibrium, we begin with the characterization ofthe  ( ) function that underlies the  () relation. By (11)-(14),µ¶   +    ()−1 ( ) = (1 −   )(15)
  −     ()For   6= 1, (15) has the same sign as  −  ()  
 ()   +   (16)Recall that a mismatched household receives   right away if it sells in the investment market, whereas it will receive  at some uncertain future date if it oﬀers the9

Proposition 6 in the Appendix establishes the existence of equilibrium.

12

house for sale in the end-user market. By (16), the comparison in (14) is likened to acomparison between the instantaneous reward   and an appropriately discounted
future reward  of selling the house in the two markets.
In a tighter end-user market with a larger , houses, on average, are sold fasterin the market. Then,  in (16) should be discounted less heavily in the comparisonof the payoﬀs between selling in the two markets. Indeed, for given   and  , theexpression in (16) is decreasing in , whereby households find selling in the investment
market less attractive when they can sell their mismatched houses faster in the tighter
end-user market.
Both   and  , however, can also depend on . How  ( ) behaves as afunction of  can only be resolved by also checking how the two prices vary with .
Solving   and  , together with those of the various asset values, from (8)-(13)and substituting in the solutions to (15),  ( ) is seen to have the same sign as¶µb−   −    () + (1 −   −  (  −   ))  () − ( + )  ( ) ≡  

(17)
10
where  = .
3.1.1

Inventory advantageb ( ) is indeed everywhere decreasing in  for
It is straightforward to verify that 
 ≥

 
≡ 
 1 + (18)

Lemma 4 For  ≥ ,  () is everywhere non-increasing, given by,⎧ ≤ ¡1⎨ 1¢
 () = () ∈ 1  0 ⎩0 ≥ 0¡¢¡¢b 1  1 = 0 and b 0  0 = 0, and that
where 1  0 are defined by, respectively,  ()   011
(figure 2 about here)
10
The solutions are presented in Lemma 13 in the Appendix. There are two sets of prices andasset values, one derived under the assumption that  ≤ 0 and the other  ≥ 0. In either case, b in (17).
is seen to have the same sign as 11b is decreasing in  Likewise, the superscript
The superscript “d” denotes the  is at where b“u” is used later on to denote the  is at where  is increasing in .

13

Panel A of Figure 2 illustrates an example of the  () function in the Lemma,
under which mismatched households only prefer selling in the investment market(  0) for small  to avoid the possibly lengthy wait in selling in a slow end-user
market. The condition of the Lemma, by (18), is met for  =  and   =   . Thismeans that in the absence of any financing or bargaining advantage over end-userhouseholds, flippers may survive only when the end-user market is relatively slow
and illiquid (small ) and on the basis of helping mismatched households overcome
the liquidity problems that they face in the market.
To pin down the equilibrium  and , we invert  () in Lemma 2 to define ≡ −1
 , whereby  : [  (0)    (1)] → [0 1]. In the three panels of Figure 2,
we superimpose a diﬀerent  () onto the same  (). Given a strictly increasing () and a non-increasing  (), equilibrium is guaranteed unique. In Panel A, at =  (0),  () = 0; then { } = {0  (0)} is the unique steady-state equilibrium. In this equilibrium, which we refer to as the no-intermediation equilibrium, allsales and purchases are between two end users. In Panel C, at  =  (1),  () = 1;
then { } = {1  (1)} is the unique steady-state equilibrium. In this equilibrium,
which we refer to as the fully-intermediated equilibrium, all transactions are interb ( ()  ) = 0,
mediated by flippers. In between, there can be equilibria at where 
as illustrated in Panel B. In such partially-intermediated equilibria, with mismatchedhouseholds indiﬀerent between selling in the investment and the end-user markets, afraction, but only a fraction, of all transactions are intermediated.
Lemma 5 For each  ≥ , there exists some  ( )  1, such that   0 in
equilibrium if and only if  ∈ ( ( )  1], and that  ( )   0.
The Lemma says that we can divide - space, for each  ≥ , into two
subsets bordered by an upward-sloping  ( ), whereby   0 ( = 0) in the uniqueequilibrium for {  } to the right (left) side of the border, as illustrated in Figure 3.
Intuitively, when flippers are obliged to finance real estate investment at a larger  ,they can only aﬀord to pay a lower price. Households then still find it attractive to sell
to flippers when selling in the end-user market is becoming more diﬃcult. By Lemma
2, as the housing stock  increases, there will be a more sluggish end-user market
in which houses are sold more slowly. In this way, as  increases, the indiﬀerenceb = 0 holds only for larger and larger . Perhaps of particular interest iscondition b  0 can hold even for an arbitrarily large  if therethat the Lemma implies that 
is also a large enough . That is, there can be a liquidity provision role for flippers
even when they are handicapped by a very large financing cost if, in the meantime,
there is a large enough housing stock to make selling in the end-user market very
diﬃcult for mismatched households.
(figure 3 about here)143.1.2Financing/Bargaining advantageb ( ) in (17) is at first decreasing in  — there is less of a liquidityFor all   0, 
provision role for flippers as the end-user market gets tighter and more liquid. For  , however, which holds for  suﬃciently below  and/or for   suﬃcientlyb ( ) is eventually increasing in . Altogether, b ( ) is U-shaped
above   ,12 b ( )  changes sign just once, which is guaranteed to be the caseif moreover  under a fairly weak condition on  () — a condition we assume holds in the following.13 That is, if the flipper-seller possesses some suﬃciently large financing and/or
bargaining advantages over the household-seller, there is a force, other than for liquidity reason, that makes selling in the investment market attractive for mismatchedhouseholds, and such a force strengthens as the market gets tighter.
Where   , the flipper-seller has a lower opportunity cost of holding thevacant house and therefore a more favorable outside option in bargaining than thehousehold-seller. Where      , the flipper-seller can extract a greater share of
the match surplus than the household-seller can. With one or both advantages, theflipper-seller would be able to bargain for a price   above the price  that themismatched homeowner can bargain for herself in the sale to the same end-user buyer,
other things being equal. In a tighter and more liquid market, by (11), the “bid-ask”
spread in house flipping narrows as houses are sold more quickly. Then, as the market
tightens, if the flipper is able to sell a house at a relatively high   in the end-user
market, the flipper must be paying a more than proportionately higher   for thehouse earlier in the investment market. In all, for   , mismatched households
may also find selling to flippers attractive for the especially high price flippers oﬀer
when the market is tight and liquid as the flipper-seller’s advantages are passed onto
the household-seller to the fullest extent possible in such a market.
Lemma 6 There exist some 0 and 00 , with 0  0  00 ≤ , such thata. For  ∈ (0 0 ],b. For  ∈ (0  00 ), () = 1 for  ≥ 0⎧
⎨ 1
 () =
 ()
⎩ 
1

 ≤ ¡1¢ ∈ 1  1  ≥ 1

12
Given that the condition   , by (18), is a condition on by how much    is below   ,it appears that it may be more illuminating to state results in terms of the comparison of the two
ratios instead of stating results in terms of the level of  . The conditions to follow in Lemma 6,
however, cannot be stated just in terms of one and/or both ratios. For coherence, we think it isleast confusing to state conditions
throughout
in terms of the level of  .
³´The condition is 2   () −  
isoelastic.
132+  2  () ≤ 0, which is guaranteed to hold if  () is

15

¡
¢
b 1  1 = 0 and b (1  1) = 0 over where
where 1  1 are defined by b ( ) is decreasing and increasing in , respectively. Here,  () is firstdecreasing, reaching a minimum above zero, and then increasing toward 1.

c. For  ∈ [00  ) 

⎧
1
⎪
⎪
⎪
⎪
⎨  () () =
0
⎪
⎪
⎪
 ()
⎪
⎩ 
1

 ≤ ¡1¢ ∈ £ 1 ,0 ¤ ∈ 0  0 
 ∈ (0 ,1 ) ≥ 1¡¢b 1  1 = 0 andwhere 1  0  0  1 are defined by, respectively, ¢¡b ( ) is decreasing in , and b (0  0) = 0 andb 0  0 = 0 over where ¡¢b (1  1) = 0 over where b ( ) is increasing in . For  ∈ 1 ,0 ,  ()  
0, whereas for  ∈ (0 ,1 ),  ()   0.
b ( ) function
Part (a) of the Lemma covers the situation in which the U-shaped stays above zero for all  and , meaning that the mismatched household’s payoﬀ ofselling in the investment market exceeds that of in the end-user market at any market
tightness. Then, with  () = 1 for all , as illustrated in Panel A of Figure 4, theunique equilibrium must be at  = 1 for any values of , as indicated in Figure
4. In this case, flippers’ financing/bargaining advantage is so overwhelming that theunique equilibrium must be a fully-intermediated equilibrium.
(figure 4 about here)
Lemma 6(b) covers the situation for  above a first threshold 0 but below a second threshold 00 , whereby  (), though can fall below one, never falls all the way
down to zero for any , as illustrated in Panel B of Figure 4, so that any equilibrium
must be at where   0, as indicated in Figure 3. In this case, flippers’ financing/bargaining advantage, though remains operative, only suﬃces to allow them tooﬀer a price attractive enough to lure all mismatched households to sell in the investment market when the end-user market is suﬃciently tight.
As  reaches and rises above the 00£ threshold,
Lemma 6(c) says that  () now¤ 
does fall to zero over a given interval 0  0 of market tightness, as illustrated inPanel C of Figure 4
00
Lemma 7 For each£  ∈ [  ),¤a no-intermediation equilibrium with  = 0 exists if
and only if  ∈  ( )   ( ) , for some  ( ) and  ( ), where 0   ( ) ≤ ( )  1, and that  ( )   0 and  ( )   0. Otherwise, any
equilibrium must either be a partially- or fully-intermediated equilibrium with   0

16

Lemma 7 says that we can divide the - space for  ∈ [00  ) in Figure 3 into
three subsets with the middle, but only the middle, subset made up of all {  }for which there exists a no-intermediation equilibrium. Conversely, for small , withwhich the end-user market tends to be suﬃciently tight and liquid to cause flippers
to pass onto their advantages to household-sellers to a large enough extent, and forlarge , with which the end-user market tends to be suﬃciently slow and illiquid topresent enough diﬃculties for households to sell houses themselves, any equilibrium
must involve at least a fraction of households selling in the investment market. Thatthe border in Figure 3 dividing the second and the third subsets is upward-sloping
has the same interpretation as to why the border in the Figure for  ≥  is upwardsloping: an increase in  hinders flippers’ ability to oﬀer a price attractive enoughto allow them to carry out their liquidity-provision role so that flipping can continue
to take place only when the end-user market is more illiquid due to a larger housing
stock, whereas the border dividing the first and second subsets slopes down as theweakened financing advantage of flippers caused by a larger  must be compensatedby more of the advantage being passed onto households as resulting from the market
becoming tighter due to a smaller housing stock for households to still find selling toflippers advantageous.
3.1.3

Summing up

In the model housing market, flippers possess two advantages over end-user households: (1) inventory advantage and (2) financing/bargaining advantage. For thesmallest  and largest   , the financing/bargaining advantage suﬃces to enable
flippers to oﬀer a good enough price to lure all mismatched households to sell in theinvestment market at any level of market tightness in the end-user market resulting
from any level of housing supply. As the advantage weakens, flippers may only survive in sluggish markets on the basis of helping mismatched households overcome theliquidity problems and in tight markets on the basis of oﬀering a very attractive price.
Finally, when the financing/bargaining advantage weakens further and disappears altogether, flippers may only survive on the basis of providing liquidity services. In thiscase, equilibrium is guaranteed unique and may involve flipping only for a relatively
sluggish market resulting from a large housing supply.

3.2

Multiplicity

In Panels B and C of Figure 4, where  () becomes an increasing function over arange of , there can well be multiple equilibria given that  () is upward sloping
throughout. Figures 5 and 6 illustrate two examples of multiplicity. In both examples,
there are as many as three equilibria. Full intermediation, with  = 1, is equilibriumin the examples since at  =  (1), flippers will be paying a price attractive enoughto lure all mismatched households to sell in the investment market, while if all mismatched houses are sold in the investment market right away, the rapid turnover will17

indeed give rise to a tight market with  =  (1). In a slower market, flippers will
no longer be paying a   attractive enough to lure all mismatched households tosell in the investment market. But precisely because fewer or none at all mismatchedhouses are sold in the investment market, a relatively sluggish end-user market will
emerge from the slower turnover. As a result, a smaller  and a smaller  is alsoequilibrium in Figures 5 and 6.

(figures 5, 6 about here)

Consider a small perturbation from the middle equilibrium in Figures 5 and 6 thatb ( )  0 since
knocks the { } pair oﬀ to the right of the  () function. Then, b   0 for  ≥ 0 , after which all mismatched households will find it better to sellin the investment market. The increase in turnover will raise  further. Eventually themarket should settle at the  = 1 equilibrium. Conversely, a perturbation that knocks
the { } pair oﬀ to the left of  () function from the middle equilibrium in Figures
5 and 6 should send the market to a smaller  equilibrium. In general, an equilibrium
at where  () is increasing should be unstable. By analogous arguments, the otherequilibria in the two examples should be locally stable.14 Hence, there are not justmultiple steady-state equilibria but also multiple locally stable steady-state equilibria.
A necessary condition for multiplicity, by Lemmas 4 and 6, is that  ∈ (0  )
over which  () becomes increasing over a range of . The two examples above
suggest that for  ∈ [00  ), suﬃce for the existence of multiple equilibria is that (0)  0  1 ≤  (1) (19)where the first inequality ensures the existence of at least one   1 equilibrium andthe last inequality the existence of the  = 1 equilibrium. Lemma 15 in the Appendix
presents a set of conditions that guarantee that (19) holds.
With multiple steady-state equilibria, how much flipping takes place in the market can be fickle, especially when the equilibrium the market happens to be in isunstable. In general, where there are multiple equilibria, any seemingly unimportantshock can dislocate the market from one equilibrium and move it to another, causing
catastrophic changes in flippers’ market share, turnover, and sales. To follow suchdiscrete changes in flipping can be significant fluctuations in housing price, a subject
we shall address in Section 6. And then in Section 7, we will calibrate the model toseveral observable characteristics of the U.S. housing market to assess the quantitative importance of such a channel of volatility. But next, we should first study the
problem of optimal flipping in the model housing market.
14

A more rigorous local stability analysis is in Leung and Tse (2016b).

18

4

Eﬃciency

The planner, who is subject to the same trading and financing frictions that agents inthe model housing market face, chooses the fraction of mismatched households using
the services of flippers to maximize the utility flows over time that households derivefrom matched owner-occupied housing net of the rental expenditures incurred; i.e.,
½Z ∞
¾
−
max
 (  −  )  
(20)


0

subject to (1)-(4), the equations of motions for  and  given by the diﬀerences between the LHS and RHS of (5) and (6), respectively, and some given initial conditions
for the two state variables.15
Lemma 8 In the steady state, the planner chooses⎧⎨ 1  ( (1)) ≥ 0
  ( ( )) = 0 =⎩0  ( (0)) ≤ 0

where

µ
¶


 () =−  +  +  () − 




(21)

(22)

The function  (), which can be thought of as the planner’s incentives to havemismatched households selling to flippers,16 starts oﬀ equal to positive infinity at
 = 0, is everywhere decreasing in , and ends up equal to some negative valueas  → ∞. Then, with  ()   0, one and only one of the three cases in(21) applies for a given parameter configuration. In particular, the third line implies
that the necessary and suﬃcient condition for any flipping to be optimal is that
 ( (0))  0, which holds for small  (0) with  () a decreasing function. Other
things equal, a small  (0) will arise out of a large housing supply, with which there
will be a sluggish and illiquid market. Conversely, in a tight and liquid market witha large  (0) due to a small housing supply, there will be rapid turnover no matter
what, in which case there is little to gain from any increase in turnover to oﬀset theadditional resources incurred in the provision of rental housing while flipping takes
b inplace. In comparing households’ private incentives to sell to flippers as given by 
15

The equations of motion for  and  do not constitute independent restrictions given (1)-(4)and the equations of motions for  and  as the two equations can be shown to be implied by
the former set of equations. Likewise, the initial values for  and  are not free variables but are
restricted by the initial conditions for  and  and (1) and (2).
16
Fornally,  () denotes the sign of the steady-state shadow value of  in the planner’s optimization. When it is positive, the objective function in (20) increases in value as the planner sends
one more mismatched household to rental housing by making the household sell to a flipper.

19

(17) with the planner’s incentives as given by  in (22) in Proposition 1 below, wefind that such excessive flipping is indeed possible in equilibrium.
Proposition 1
a. Equilibrium is eﬃcient if
 
≡ 
 () 
¶
µ
−
 
 = 1 −≡  
 ()    − 
 = 1 −at the optimum .
b ( )   () for  at which
b. 
and
 1
≤2
  ()

∙µ

b

0


¸12
¶

− 1 (1 + )  



(23)

In case  = , Proposition 1(a) says that   =   and the Hosios (1990) condition
holds exactly. The congestion externalities and the eﬀects of imperfect appropriability
in the present model just cancel out and eﬃciency is obtained when the buyer’s shareof the match surplus, whoever the buyer is matched with, is just equal to the elasticityof the matching function with respect to the measure of buyers. For   ,     though. This means that in particular, if    but   =   =   , the price thatflippers oﬀer will be too attractive to give rise to excessive incentives for mismatchedhouseholds to sell in the investment market.
The conditions in part (b) are general suﬃcient conditions for excessive incentivesbfor flipping. If equilibrium is at where  
 0, households sell to flippers primarily
for the high price in the investment market. The RHS of (23) is a positive real numberfor   , which will exceed the LHS of the condition for large  given the concavityof  (). The Proposition can then be interpreted to say that in a tight and liquidmarket, possibly arising from a small housing supply, there is little eﬃciency gainfrom flipping even though mismatched households may find the high price in theinvestment market attractive, which in itself does not contribute to eﬃciency sincehouseholds would be paying a similarly high price to buy new houses later on when
they are selling at a high price in the investment market earlier.
Given that eﬃciency does not always increase with  and  and that the market incentives to sell to flippers can be suboptimal or excessive, there is no reason to expectthat when there exist multiple equilibria, the more active equilibria are necessarily
20

more eﬃcient. Most of all, the equilibria cannot be pareto ranked under all circumb ( ) = 0, asset values
stances. Specifically, in a steady-state equilibrium where 
for matched and mismatched homeowners and renters are given by, respectively,17 =( +    ())  ( +  +    ()) =   ()  ( +  +    ()) =(   −   )  ()   ( +  +   )

It is straightforward to verify that both  and  are increasing in . Any homeowners — matched or mismatched — benefit from the higher housing prices in a tighter
market. But the asset value for households in rental housing  is decreasing in 
if       , which is a necessary condition for the multiplicity of equilibrium(equation (18) and Figure 3). In this case, would-be buyers are made worse oﬀ by thehigher housing prices in the tighter market. In the comparison between two steadyb ( ) = 0, homeowners are better oﬀ whereas rentersstate equilibria both at where 
are worse oﬀ in the larger  equilibrium than in the smaller  equilibrium. Any twosuch equilibria cannot then be pareto ranked. The same conclusion can be shownb ( )  0 equilibrium and a b ( ) = 0
to carry over to comparisons between a b ( ) = 0 equilibrium and a b ( )  0 equilibrium.
equilibrium and between a 

5Empirical implications

In this section, we shall explore several empirical implications of the model. To thisend, we begin with characterizing how the model housing market’s vacancy rate,
trading volume, and the turnover of houses and households vary with .

5.1

Vacancy, trading volume, TOM, and TBM

In the model housing market, the entire stock of vacant house is comprised of houses
held by flippers. With a given housing stock, the vacancy rate is simply equal to . A direct corollary of Lemma 1(b) is that:
Lemma 9 In the steady state, the vacancy rate for owner-occupied houses is increasing in .
17

The asset values and housing prices referred to hereinafter are special cases of those in Lemma
13 in the Appendix. In particular, the equations for  and  are from (42) and (43), respectively,
whereas the equation for  is from (41), evaluated at  ( ) = 0

21

Housing market transactions per time unit in the model are comprised of (i) houses households sell to flippers, (ii)  ()  houses flippers sell to households, and(iii)  ()  houses sold by one household to another, adding up to an aggregatetransaction volume,  =  +  ()  +  ()  
(24)
Lemma 10 In the steady state, TV is increasing in .
Given that houses sold in the investment market are on the market for a vanishingly small time interval and houses sold in the end-user market for a length of timeequal to 1 () on average, we may define the model’s average TOM as () ( +  )
1

×0+
×



 ()

(25)

Lemma 11 In the steady state, on average, TOM is decreasing in .
TOM is a measure of the turnover of houses for sale. A more household-centricmeasure of turnover is the length of time a household (rather than a house) has tostay unmatched. We define what we call Time-Between-Matches (TBM) as the sumof two spells: (1) the time it takes for a household to sell the old house, and (2) thetime it takes to find a new match afterwards. While the first spell (TOM) on average
is shorter with an increase in , the second is longer as the increase in  to accompany
the increase in  causes the buyer’s matching rate to fall. The old house is sold more
quickly. But it also takes longer on average to find a new match in a market withmore buyers and fewer sellers. To examine which eﬀect dominates, write the model’s
average TBM as
µ
¶
1
1
1

+ (1 − )
+

(26)
 ()
 ()  ()where 1 is the average TBM for households who sell in the investment market18
and 1 + 1 for households who sell in the end-user market.19
Lemma 12 In the steady state, on average, TBM is decreasing in .
Lemma 12 may be taken as the dual of Lemma 1(b) (   0). Whenmatched households are more numerous in the steady state, on average, they mustbe spending less time between matches.
18

The household sells the old house instantaneously. Given a buyer’s matching rate , the average
TBM is then 1
19
Let 1 denote the time it takes the household to sell the old house in the end-user market and2 − 1 the time it takes the household to find a new match³ after the old house ´is sold. Then theR∞R∞household’s TBM is just 2 . On average,  [2 ] = 0 −1 1 2 −(2 −1 ) 2 1 = 1 + 1

22

5.2

Housing prices

No-intermediation equilibrium In the no-intermediation equilibrium, all housing market transactions are between pairs of end-user households at price20
 =  ( () + ) − (1 −   )  ()+ ( +  +    ()) (27)

evaluated at  =  (0).
Fully-intermediated equilibrium In the fully-intermediated equilibrium, all houses
are first sold from mismatched households to flippers at price  =   () ( + )( +    ())  + ( + (1 −   )  ()) (28)

  ( () +  ) ( + )( +    ())  + ( + (1 −   )  ()) (29)

in the investment market and then at price  =

from flippers to end-user households in the end-user market, both evaluated at  = (1). With houses sold by households to flippers on the market for a vanishingly
small time interval and houses sold by flippers to households for, on average, 1 () 
0 units of time, prices and TOM in the model housing market, as in the real-world
housing market, are positively correlated in the cross section, given that by (28) and(29),      . Besides, with      , the model trivially predicts that houses
bought by flippers are at lower prices than are houses bought by non-flippers. Both
Depken et al. (2009) and Bayer et al. (2011) find evidences of such flipper-buy
discounts in their respective hedonic price regressions.
Partially-intermediated equilibrium In a steady-state equilibrium in which mismatched households sell in both the investment and end-user markets, in addition to
the two prices   ()  =(30) ( +  +    ())   () +   (31)  =
 ( +  +    ())for transactions between a flipper and an end-user household, there will also be transactions between two end-user households, carried out at price =   () +    ( +  +    ())

20

(32)

Equation (27) is from (38) evaluated at  = 0; (28) and (29) are from (46) and (45), respectively,
evaluated at  = 1; (30), (31), and (32) are from (40), (39), and (38), respectively, all evaluated at
 ( ) = 0.

23

For      ,        . Just as in the fully-intermediated equilibrium,
a positive relation between prices and TOM holds in the cross section and houses
bought by flippers are at lower prices. Moreover, here houses sold by flippers are soldat a premium over houses sold by one end-user household to another. Such flipper-sell
premiums are also found to exist in Depken et al. (2009) and Bayer et al. (2011).
Across equilibria Across steady-state equilibria,  is largest in the equilibrium
where flippers are most numerous. Then, prices should be highest in the given equilibrium where the competition among buyers is most intense.
Proposition 2 Across steady-state equilibria in case there exist multiple equilibria,
housing prices in both the end-user and investment markets are highest in the equilibrium with the tightest market and lowest in the equilibrium with the most sluggish
market.

5.3

Correlations among prices, TV, vacancy, TOM, and TBM

Now, a direct corollary of Proposition 2 and Lemmas 9-12 is that:
Proposition 3 Across steady-state equilibria in case there exist multiple equilibria,
prices, TV, and vacancies increase or decrease together from one to another equilibrium, whereas the average TOM and TBM move with the former set of variables in
the opposite direction.
Interest rate shocks In a typical asset pricing model, the price of an asset falls
when the interest rate goes up. The same tends to hold in the present model. Specifically, in the no-intermediation equilibrium, an increase in , by (27), leads to a lower
 for suﬃciently large  (0) and/or . Similarly, in the fully-intermediated equilibrium, by (28) and (29), respectively, both   and   are decreasing in  . But
in either equilibrium, with  remaining fixed at 0 or 1, market tightness, vacancies,
turnover, and sales are all invariant to the respective interest rate shocks.
In a partially-intermediated equilibrium, prices in the end-user market,   and , as well as in the investment market   , are decreasing in  , just as they arein the fully-intermediated equilibrium. Housing prices in a partially-intermediatedequilibrium, however, can also vary to follow any movements in  triggered by thegiven interest rate shock — when the market becomes tighter in particular, prices arealso higher. Hence, if a given positive (negative) shock to  should cause  andtherefore  to decrease (increase), there will be lower (higher) housing prices to follow
because of a direct negative (positive) eﬀect and of an indirect eﬀect due to the exit
(entry) of flippers. When the two eﬀects work in the same direction, the interest rate
shock can cause significantly more housing price volatility than in a model that only
allows for the usual eﬀect of interest rates on asset prices.
24

A positive shock to  need not cause  and  to fall though. In case there exist
multiple equilibria, the shock can possibly dislocate the market from a given equilibrium and send it to another equilibrium. In case the direct eﬀect of an interest rate
shock and the indirect eﬀect via the movements in  aﬀect housing prices diﬀerently,
in what direction housing prices will move cannot be unambiguously read oﬀ fromb ( ) = 0 for  and substitute the result into(30)-(32). To proceed, we solve (30)-(32), respectively,   () + ((  −   )  () − (1 −   ))  ()+ 
 ( +  +    ())(33)  =   () +    + ((  −   )  () − (1 −   ))  ()+  ( +  +    ())


(34)

 =


   () +    + ((  −   )  () − (1 −   ))  ()+ 
 ( +  +    ())(35)  =The three expressions are independent of  — whatever eﬀects a given change in will have on housing prices are subsumed through the eﬀects of the change in b ( ) = 0. To evaluate thethat follows the change in  obtained from holding eﬀects of  on housing prices is to simply check how these three expressions behave
as functions of .
b ( ) = 0, a shock to
Proposition 4 Across steady-state equilibria and holding  , whether positive or negative, will cause housing prices to increase (decrease), as
long as to follow the interest rate shock are increases (decreases) in  and .
By Proposition 4, the indirect eﬀect of an interest rate shock on housing prices
through the entry and exit of flippers and then in market tightness always dominates
the direct eﬀect shall the two be of opposite directions. A surprising implication isthat housing prices can actually go up in response to an increase in flippers’ cost offinancing, if to follow the higher interest rate is also a heightened presence of flippers
in the market. In any case, a direct corollary of Lemmas 9-12 and Proposition 4 isthat:b ( ) = 0, a shock to 
Proposition 5 Across steady-state equilibria and holding will cause housing prices, TV, and vacancies to move in the same direction, whereas
the average TOM and TBM will move in the opposite direction.
In the above, we have restricted attention to analyzing how changes in  alone
may aﬀect housing prices. It turns out that many of the implications continue tohold for equiproportionate increases or decreases in  and  . Proposition 7 in the
Appendix contains the details.

25

6Time-series Relations among Housing price, TV,
and Vacancy

By Propositions 3, 5, and 7 in the Appendix, any movement from one to anothersteady-state equilibrium would involve housing prices, TV, and vacancies all goingup or down together. The positive time-series relation between housing prices andthe volume of transaction is well known and numerous models have been constructedto account for it. In Kranier (2001), for instance, a positive but temporary preference
shock can give rise to higher prices and a greater volume of transaction, whereas Diazand Jerez’s (2013) analysis implies that an adverse shock to construction will shorten
TOM, and may possibly lead to higher prices and a greater volume of transaction. The
paper by Ngai and Tenreyro (2014) studies the comovement in prices and sales overthe seasonal cycle and they argue that increasing returns in the matching technology
play a key role in generating such cycles.
Unique to our analysis is that vacancies should also move in the same direction
with prices and TV. In contrast, in both the Kranier and the Diaz and Jerez’s models,the increase in sales should be accompanied by a decline in vacancy — given that whena house is sold, it is sold to an end-user, who will immediately occupy it, vacancies
must decline, or at least remain unchanged. In Ngai and Tenreyro’s model, households
are assumed to move out of their old houses and into rental housing immediately after
they become mismatched. Then, any and all houses on the market are vacant houses
and given the assumed increasing-returns-to-scale matching technology, vacancies riseand fall with prices and the volume of transaction in the seasonal cycle. A mismatchedhousehold in their model, however, could well have stayed in the old house and avoided
rental housing and the payment thereof until it has successfully sold the old house.
In this alternative setup, the stock of vacant houses only includes houses held bypeople who have bought new houses before they manage to sell their old ones. Then,
it is no longer clear that vacancies must rise and fall with prices and the volume oftransaction in the seasonal cycle of Ngai and Tenreyro.
Figure 7 depicts the familiar positive housing price-transaction volume correlation
for the U.S. for the 1981Q1 to 2011Q3 time period.21 The usual housing market searchmodel predicts that vacancies should decline in the housing market boom in the late
1990s to the mid 2000s and rise thereafter when the market collapses around 2007.
Figures 8 and 9 show that any decline in vacancy is not apparent in the boom.22 In
21
Housing Price is defined as the nominal house price, which is the transaction-based house priceindex from OFHEO (http://www.fhfa.gov), divided by the CPI, from the Federal Reserve Bank at
St. Louis. We set Housing Price at 1981Q1 equal to 100. Transaction is measured by the quarterly
sales in single-family homes, apartment condos, and co-ops, normed by the stock of such units. The
sales data are from the Real Estate Outlook by the National Assoication of Realtors, complied by
Moody’s Analytics. The housing stock is defined as the sum of owner-occupied units and vacantand for-sale-only units. The data are from the Bureau of Census’s CPS/HVS Series H-111 available
at http://www.census.gov/housing/hvs/data/histtabs.html.
22
Vacancy rate is obtained by dividing the number of vacant and for-sale-only housing units by

26

fact, if there is any comovement between vacancies on the one hand and prices andsales on the other hand in the run-up to the peak of the housing market boom in 2006,
vacancies appear to have risen along with prices and the volume of transaction. In
a literal interpretation of our model, vacancies should fall very significantly to follow
the market collapse since 2007. Apparently, the decline in vacancy in Figures 8 and 9
in the post-2007 period is modest, compared to the increase in the boom years. Twoforces absent in our analysis — the massive amount of bank foreclosures and unsold
new constructions in the market bust — may have accounted for the slow decline invacancy since 2007.

(figures 7, 8, 9 about here)

In a more systematic analysis, we first verify that in the 1981Q1 to 2011Q3 sample
period, all three variables are (1) at conventional significance levels. Next, we test
for cointegration. Assuming the absence of any time trends and intercepts in the
cointegrating equations, both the Trace test and the Max-eigenvalue test indicate
two such equations, whose normalized forms readPrice — 6045.51 × Vacancy = 0,
Transaction — 0.74 × Vacancy = 0,
which together imply that the three variables do tend to move in the same direction
from one to another long-run equilibrium over time.23

7

Quantitative predictions on volatility

Given the possible multiplicity of equilibrium and that an interest rate shock may haveimportant eﬀects on the extent of intermediation, the model can be consistent witha volatile housing market. The question remains as to how important quantitatively
such channels of volatility can be. In this section, we calibrate the model to several
the housing stock as defined in the prevous note.
23
With other time trend and intercept assumptions, either one or both of the tests suggest that
there exist only one or as many as three cointegrating equations. In a single cointegrating equation
with non-zero coeﬃcients for all three variables, at least two of the three coeﬃcients must be of
the same sign. Then, the two variables concerned must move in opposite directions across long-run
equilibria. With as many cointegrating equations as the number of variables, there exist definite
long-run values for the three variables, which rules out the possibility of the system moving from
one to another long-run equilibrium altogether. Restricting a priori to two cointegrating equationsin the estimation, however, we always obtain two equations whose coeﬃcients have the same signs
as those in the system above whatever the trend and intercept assumptions are. Then, any long-run
movements of the three variables must be in the same direction.

27

observable characteristics of the U.S. housing market and study by how much housing
prices can fluctuate across steady-state equilibria and in response to interest rate
shocks.
To begin, we take a time unit in the model to be a quarter of a year and assume a
Cobb-Douglas matching function with which  () =  for some   0 and  ∈ (0 1).
We set a priori the mismatch rate  = 0014 to calibrate a two-year mobility rate of
11.4% for owner-occupiers reported in Ferreira et al. (2010) and  = 084, which isthe elasticity of the seller’s matching hazard with respect to the buyer-seller ratio
reported in Genesove and Han (2012). Next, the parameters  and  and the shareof mismatched households selling to flippers  are chosen to calibrate:
1. a quarterly transaction rate of owner-occupied houses of 1.78%
2. a vacancy rate of owner-occupied houses of 1.84%
3. the share of houses bought by flippers among all transactions of owner-occupied
houses equal to 19%
The first two targets are, respectively, the average quarterly transaction rate and theaverage vacancy rate for the period 2000Q1-2006Q4, calculated from our dataset forthe plots in Figures 7~9 and the estimations in Section 6. Estimates of the shareof houses bought by flippers come from two sources: the 25% investors’ share of all
new purchase mortgages in the whole of the U.S. in Haughwout et al. (2011) inthe 2000Q1-2006Q4 time period and the 13.7% housing market transactions sharefor houses sold again within the first two years of purchase in the metropolitan Las
Vegas area in Depken et al. (2009) in the same time period. Because an investor in
Haughwout et al. (2011) may intend to hold the house as a long-term investment,
the 25% share is probably an overestimate of the true flippers’ share. Because not allhouses bought for short-term flips can actually be sold within two years, the 13.7%
share in Depken et al. (2009) is probably an underestimate of the true flippers’sshare. Our 19% target is obtained by taking a simple average of the two estimates.
Given the targets, denoted as  ,  = 1 2 3, respectively, we choose , , and 
to
( 3 µ
¶2 )
X  − 
b
min


=1
where the b ’s are the model’s calibrated values of the corresponding targets24 , yielding  = 0085,  = 0865, and  = 025 at which the calibrated values of the threetargets are reported in the second column of Table 1.
24

The 
bs are equal to     , and    for the model’s transaction rate, vacancy
rate, and the flippers’ transaction share, respectively. The minimization is carried out via a gridsearch with a grid size of 0.005 for each of , , and , subject to  ≤ 12 and  ≥ 06. The first
constraint is for expediency in the grid search and is not binding. Given that  in the model is thestock of owner-occupied houses relative to the population of households demanding such housing,
anything near the bound of the second constraint is probably unreasonable.

28

Table 1: Calibration Targets and Calibrated Model Values
Targeted value Calibrated value
Transaction rate (quarterly)
001780016
Vacancy rate
001840018
Flippers’ share in transactions 019
019Thus far in the calibration, we have eﬀectively identified  = 025 as equilibrium. For equilibrium to be indeed at  = 25, we need to pick the values for{          } to force  = 025 as well. Since only the ratio  = , but not
the levels of the two parameters, matters for the value of  and the comparison ofprices, we first normalize  = 1. We then obtain an estimate of  (or equivalently) equal to 1.43 from the results in Anenberg and Bayer (2013). The details are in
Appendix 10.2. Next, we set  = 002 for an annual rate of 8% to match the usual
30-year fixed-rate mortgage rate. Lastly, for the lack of any obvious empirical counterpart, we set the household-seller’s bargaining strength   = 05. Then, for eachof   = 05 06 065 07, and 08, we look for the value of  at which  = 025.
The results are shown in Table 2.2526
For the last two pairs of   and  in Table 2, the  = 025 equilibrium is theunique equilibrium. For the first three pairs, there are two other equilibria each besidethe  = 025 equilibrium. Table 3 reports the prices in these equilibria. For instance,for   = 05 and  = 31% per annum, the three equilibria are at  = 0, 025, and1, respectively.27 The price , on the next row, is the average of     , and   ,weighted by the shares of transactions taking place at the respective prices, with in the smallest- equilibrium set equal to 1. Evidently, the volatility arising fromthe multiplicity is non-trivial, with average prices diﬀering by up to 23% across theequilibria.
Table 2: Calibrated   and  for  = 025
25
A  below  by a few percentage points can make sense if flippers, but not end-user households,
tend to be all-cash investors. Herbert et al. (2013) report that the majority of investors acquiring
foreclosures are indeed all-cash buyers. Even though no comparable evidence is available for otherproperties, it would not be surprising that cash is often used too. Moreover, investors may also makeuse of mortgages with zero initial or negative amortization, short interest rate reset periods, or low
introductory teaser interest rates. Such mortgages obviously are ideal for flippers who plan to sellquickly for short-term gains. Amromin et al. (2012) find that borrowers who take out such “complex”
mortages are usually high income individuals with good credit scores. Foote et al. (2012) find that
periods of interest rate resets do not tend to trigger significant increases in defaults, consistent withthe finding of Amromin et al. that the borrowers of such mortgages are sophisticated investors.
26
Notice that the model does not require    for flippers to survive or for the multiplicity ofequilibria. For smaller values for , we can force  to be equal to 0.25 for much larger  .
27
At  = 1, in the steady state, one half of all sales are purchases made by flippers. This is justabout equal to the peak investor share in the “bubble states” reported in Haughwout et al. (2011).

29



 (annual basis) (annual basis)



05
06
065
07
08
00077 00091 00099 00106 0012
31%
37%
4%
423% 48%

Table 3: Multiple Equilibria
05
06
31%
37%
0 025 1
0 025 1
1 11
12 1 111 123065
4%025 071 1
1
108 111

Table 4: Housing Prices and Interest Rates,   = 07
 (annual basis) 35% 421% 422% 423% 427% 6%
=0
089
089
 = 0251
 = 063107
=1113 112
To study the response of housing prices to interest rate shocks, we report in Table
4 average housing prices  for various small deviations of  from a benchmark of = 423% and   = 07 at which equilibrium is unique at the calibrated valueof  = 025. Fixing   = 07, for all values of  under consideration, equilibrium
remains unique. The entries in the table are normed by the average equilibrium priceat the benchmark  . Here, housing prices hardly move to follow a given interest rate
shock if the shock has not caused any changes in equilibrium . But when the given
interest rate shock does cause  to change significantly, it also leads to significant
changes in housing prices. Specifically, a decline in  from 6% per annum to 4.27%
per annum causes no noticeable change in  when the given movement in  has noeﬀect on . But a further decline in  from 4.27% per annum to 4.21% per annum
now causes  to increase by 26% as  rises from 0 to 1 in the meantime. Thereafter, remains essentially unchanged from any additional decline in  as  has alreadyreached the upper bound of 1. All this suggests that the response of housing prices tointerest rate shocks can appear erratic and unpredictable. Before a given threshold is reached, the response is at most moderate. When  crosses the threshold totrigger the entry of flippers, the housing market can become significantly tighter andhousing prices significantly higher as a result.

30

8Extensions

The model we studied in this paper is, by all means, a very special model. In Leungand Tse (2016a), we study how the major results of the paper may be aﬀected undercompetitive search, a two-house-limit liquidity constraint for households, allowing forinvestors choosing between short-term flips and long-term investments and letting
mismatched households sell to housing market intermediaries right before buying a
new house. Below are the summaries of the findings.

8.1

Competitive search

In our model, the multiplicity of equilibrium arises out of flipper-sellers passing onto
household-sellers their advantages in bargaining to the fullest extent possible in a tightand liquid market. A natural question to ask is whether the multiplicity is special toprice determination by bargaining as we have assumed or whether similar conclusions
hold under the alternative assumption of price determination via competitive search.
In the competitive search version of our model, based on Mortensen and Wright(2002), the end-user market is segmented into submarkets, each of which is controlled
by a competitive market maker, who charges entry fees for other agents for buying
and selling in his submarket in return for regulating the transaction price and market
tightness at some pre-specified levels. The household-buyers, household-sellers andflipper-sellers choose which submarket — among the options available — to enter into
to maximize the respective expected returns of buying and selling. In this setting,
we find that in equilibrium,⎧⎨ 1  ( (1)) ≥ 0
  ( ( )) = 0 =⎩0  ( (0)) ≤ 0
where

¶
µ
¶µ



− (1 + ) −  ( + ) +  () −  () =
can be thought of as the incentives of mismatched households to sell in the investb ( ) in (17) under
ment market under competitive search — the counterpart to 
bargaining and  () in (22) for eﬃciency.
First notice that if  =  ,  () =  () and hence, as is well known, competitive
search is eﬃcient. For  6= , however,  () 6=  (). Most importantly for our
purpose, for

 

1+
b ( ) for   , is U-shaped, first decreasing but eventually increasing.
 (), like This, of course, opens up the possibility of multiplicity. Indeed, this necessary condition for multiplicity is the same necessary condition for multiplicity in Lemma 6 and31

Figure 3 for   =   . Thus, as long as flippers possess a suﬃciently large financing
advantage, they can attract mismatched households to sell in the investment market
when the end-user market is particularly tight, in addition to when the end-user market is particularly sluggish, whether prices in the end-user market are determined bybargaining or via competitive search.

8.2

Two-house-limit liquidity constraint

In some version of the housing market search model, most notably in Wheaton (1990),
there is not any rental housing but rather a mismatched household must stay in its oldhouse while searching for a new match, and then the old house will be put up for sale
only after the household has found a new house to move into. In this environment, ahousehold will be holding as many as two mismatched houses if the household is hit
by a moving shock again before it is able to sell the previously mismatched house. If
there is a two-house-limit liquidity constraint, the household will then be prevented
from entering the end-user market as a buyer until it is able to sell one of the twomismatched houses that it is now holding. A liquidity provision role for flippers arises.
Indeed, a household may wish to sell to flippers right after it is moving to a new housefrom the old mismatched house. By doing so, the household will never be holding
two houses at any moment in time. In all, we find that when more households sell toflippers, either right after finding new houses to move into or right after being hit by
two successive moving shocks, there will be a tighter end-user market.
In this setup, flippers, like those in the present model, may be able to lure households to sell in the investment market not just when the end-user market is sluggish
but also when it is tight if they possess a large enough financing/bargaining advantage. In this model though, there is not any relationship between the activities offlippers and the vacancy rate since the latter is simply equal to the diﬀerence between
the housing stock and the population of households, in the entire absence of rental
housing. Furthermore, unlike the present model, for eﬃciency, all households should
use the services of flippers since nobody would be incurring any rental expendituresduring which flipping takes place. And in case there exist multiple equilibria, a more
active equilibrium should pareto dominate a less active one, with any household owning at least one house at any moment.

8.3

Short-term flips versus long-term investment

In reality, there can be two strategies for housing market investments — short-term
flip versus long-term investment in which an investor holds the house for an extended
period of time, earning the rental revenue in the interim and in anticipation for acertain capital gain in the medium to long term. In a summary of case studies of four
metropolitan areas in the U.S., Herbert et al. (2013) report that both investment
strategies were commonly adopted by housing market investors in the wake of the32

collapse of the housing market in the U.S. in 2007. In the early phase when prices
appeared to have reached the lowest level, most investments were found to be shortterm flips, whereas in the latter phase when the market appeared to have stabilized,
most investments were found to be medium- to long-term investments.
In the present model, the supply of rental housing is assumed perfectly elastic atsome exogenously given rental. We could have chosen to assume the arguably more
realistic setting in which the stock is exogenously given while the market rental isdetermined in equilibrium. All the same, underlying either setting is the presumptionthat the rental and owner-occupied housing stocks are two separate stocks. By allmeans, a richer analysis would allow for the same housing stock to serve as bothrental and owner-occupied housing. In this revised model, just as in the present
model, mismatched households choose between selling in the investment market oroﬀering their houses for sale in the end-user market. Unlike the present model, thespecialist investors in the revised model may choose to oﬀer their properties for sale
and/or for rent. It turns out that in this setup, the only kind of steady state is one inwhich at least a fraction of mismatched households choose to sell to investors while
investors choose to oﬀer their properties both for sale and for rent. Any such steadystate, however, can be equilibrium only under selected values of the housing stock
and investors’ cost of financing, just as a fully- or partially-intermediated equilibrium
exists only for some subset of the parameter space in the present model. Whenthe conditions are not met, no steady-state equilibrium exists in the revised model.
This result is perhaps highly suggestive for in reality, housing market investors do
predominantly choose whether to flip or to invest long term during diﬀerent phasesof the housing price cycle as reported in Herbert et al. (2013). No matter, to analyze
a model that allows for investors choosing between the two investment strategies, it
becomes imperative to study the full dynamics. Undoubtedly, this can be a very
fruitful exercise towards a fuller understanding of the dynamics of housing market
investment but is best to be left for future research.28

8.4

Selling to housing market intermediaries right before
buying a new house

We have in this paper assumed that a mismatched household must either sell to aflipper or to an end-user household before it can start looking for a new house. Strictly
speaking, given the assumption of instantaneous sale in the investment market, thehousehold can choose to stay in the old house while searching for a new one and thensell the old house to a flipper only right before the household buys the new house.29
28

Indeed, Head et al. (2014) have explored a number of interesting implications of a model thatallows households and real estate developers a choice between the two strategies. They do not allow
for specialist investors in their model like we do though.
29
The households who move within a given city in Head and Lloyd-Ellis (2012) can do just that.
There, households do not face any liquidity constraints and the assumption of having housing market

33This means that, a one-house-limit liquidity constraint notwithstanding, mismatchedhouseholds should be able to enter the end-user market as buyers before selling their
old houses.
Given that all mismatched houses are for sale in the end-user market, whoever
their owners are, had we allowed for any and all mismatched households to enter theend-user market as buyers right away, the tightness in the end-user market would notdepend in any way on how many mismatched households choose to sell to flippers
in the first instance, if any mismatched households choose to do that at all. In thisalternative setting, market tightness depends solely on the housing stock, among
other factors, and is completely isomorphic to price determination in equilibrium.
Moreover, flippers’ liquidity provision role is wholly fulfilled so long as they are in themarket ready to buy up any houses would-be buyers need to sell just before buying.
Then, unless flippers possess some large enough financing/bargaining advantage, must be just equal to zero in equilibrium. In sum, equilibrium is guaranteed uniqueand any sale to flippers that takes place at the moment households first becoming
mismatched must arise out of flippers’ financing/bargaining advantage.
True, with a frictionless investment market in place, our assumption that mismatched households must rid themselves of their old houses first before entering theend-user market as buyers is ad hoc. We could have included some additional technical details to better justify the assumption.30 But we think it is more constructive
to simply note that the assumption is a well-motivated assumption. In reality, theinvestment market for the housing asset is by no means completely free of frictions.
Sales in the market are certainly not instantaneous. If a household is not able tosell the old house quickly enough to pay oﬀ the mortgage for the house, it can faceconsiderable diﬃculties in getting a mortgage for the new house. A more realistic
and arguably more rigorous analysis is to model the investment market as a frictional
market too to formally motivate the assumption that households must first sell before
they enter the market as buyers.
Of course, the questions of whether and how our qualitative results would survive
in the more general setting of a frictional investment market remain. Suppose inparticular, there is but a finite measure of flipper-buyers in a frictional investment
market whom mismatched households meet randomly. Where there is not a conscious choice of selling in which market for households, the asset value of a matchedhousehold simplifies to, =  +  ( −  ) Once becoming mismatched, a household is a seller in both the end-user and theintermediaries is merely a simplifying assumption to facilitate analysis.
30
For example, consider a discrete time version of the model. Say a period is divided into two
subperiods where the investment market is open in the first subperiod only and the search market
is open next in the second subperiod. In Leung and Tse (2016a), we propose three other alternative
justifications.

34

investment markets, in which case  satisfies, =  () × max { +  −   0} +  ( ) × max { +   −   0} 

(36)where  denotes the rate at which the household-seller meets a flipper-buyer in africtional investment market and  the ratio of flipper-buyers to household-sellers
in the market. On the other side of the investment market, a flipper-buyer meetsa household-seller at the rate  ( ) =  ( )  , whereby the flipper’s asset value  satisfies,  = − +  ( ) × max {  −   −    0} with  ≥ 0 denoting the flow search cost of the flipper and   , given by   =  () (  −   ) (37)

the asset value of a flipper-seller in the end-user market, or what is the same thing
the value of a vacant house to a flipper.
If flippers enter the market at an entry cost of , in equilibrium   ≤ . Like
prices in the end-user market, the price in the investment market   is determinedby Nash bargaining. Our model is a special case of this more general model for which → ∞ as  and  tend to zero.
In this model, there should always be a positive surplus in a household-seller andhousehold-buyer match to result in a positive  +  −  in (36) since flippers
must be buying at a lower price   than end-user households do at  to makeany profit at all from house flipping. Then, the household-seller should always findit optimal to sell to an end user should he be lucky to meet one before he ever meets
any flipper-buyer. Whether or not there is a non-negative surplus in a match between
a household-seller and a flipper-buyer can depend on , the tightness in the end-user
market, in much the same way whether or not household-sellers prefer to sell in the
Walrasian investment market in our model depends on . Specifically, other things
equal, there can be a non-negative surplus in a household-seller and flipper-buyer
match to result in a non-negative  +   −  in (36) only when the outside optionof the seller  is relatively unfavorable due to a sluggish end-user market. Absent
any financing/bargaining advantage on the part of flippers then, flipping should only
take place in slow markets. In case flipper-sellers do possess some suﬃciently large
financing/bargaining advantage, they tend to sell houses at a relatively high   andby (37), the high   should be followed by a disportionately high   in a tightend-user market in which vacant houses are quickly sold. In turn, if flippers placea relatively high valuation on vacant houses, there tend to be a large surplus in ahousehold-seller and flipper-buyer match to result in a non-negative  +   − in (36). But precisely because all or at least a fraction of end users sell to flippers,
the end-user market tightens. In all, results similar to Lemmas 4-7 of the paper,
which form the core of our qualitative results, should hold in the generalization to africtional investment market.
35

9Concluding remarks

Housing market flippers can be arbitraging middlemen like those we model in thispaper, intermediaries who survive on the basis of superior information, or momentum
traders blindly chasing the market trend. Our analysis suggests a number of empiricalimplications to distinguish between these theories of flipping. First and foremost, if
flippers are predominantly momentum traders instead of specialist middlemen, there
should not be any significant flipper-buy discounts and flipper-sell premiums. But
such discounts and premiums do exist and they are sizeable, as reported in Depken
et al. (2009) and Bayer et al. (2011). Second, in a housing market boom fed by
the entry of momentum traders, there can and usually will be sales and purchases
among flippers. While in a more elaborate theory of intermediation as in Wright andWong (2014), this can also happen. But this does not seem like a robust implicationof a theory of middlemen in the housing market. Third, a theory of housing pricespeculation should imply that prices should stay at the peak for at most a shortwhile and then fall right afterwards. In our model, the market can conceivably movefrom a low-price to a high-price steady-state equilibrium and then just stays at the
new equilibrium for any length of time. Relatedly, speculators should only buy inan up market, whereas in our model, there can be multiple locally stable steadystate equilibria involving flipping. Indeed, in our model, the gross returns to flipping,    , by (11), is actually higher in a lower-price less active equilibrium than in ahigher-price more active equilibrium. Lastly, if housing market middlemen are mostly“market edge” investors who survive on the basis of informational advantage rather
than investors who have more flexible and less costly financing like those in our model,
they should use ordinary mortgages as much as end-user households do. Herbert etal. (2013) report that there was little evidence of bank lending to investors acquiring
foreclosed properties in the wake of the 2007 housing market meltdown in the U.S.
Instead, the great majority of acquisitions were bought with cash. In addition, inthe housing market boom in the U.S. before 2007, there is ample evidence, as weremarked in note 27, that investors took advantage of “complex mortgages” morethan ordinary households did. Other than observations on the choices of financing,
the two theories may also be distinguished by what market conditions under which
flipping takes place. In our model, flipping tends to occur in particularly tight aswell as particularly sluggish markets — a prediction that is hard to envisage to comefrom a theory of housing market intermediaries who survive on the basis of superior
information.
In the U.S., house flipping is thought to often involve renovating before sellingrather than simply buying and then putting up the house for sale right after. In thisline of thinking, the returns to flipping are more about the returns to the renovations investment than the returns to holding the houses on behalf of the liquidityconstrained owners. The question then is what prevents the original owners themselves from earning the returns on the investment. A not implausible explanation is36

that many original owners lack the access to capital to undertake the investment, justas the original owners in our model lack the access to capital to hold more than one
house at a time. Thus, at a deeper level, our model is not just a model of buy-and-sell
flips but should also encompass, with suitable modifications, buy-renovate-sell flips.

37

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40

A

Appendix

A.1

Lemma and Proposition

Lemma 13 If max { +     } =  , = {(  ( + ) ( +   ) − (1 −   ) ((1 − )    + (1 −   )  ) ) (38)
+ ( +  +   ) ( +   ) }  ( +    −  (1 − ) (1 −   ))  + ( +  +   ) ( +   −  (1 − ) (1 −   ))  + ( +  +   )   =     =   ( +  )

(39)
(40)

 = {((1 − ) (1 −   )    + (1 −   )    + (1 −   −  (  −   ))(41)
× )  − ( +  +   ) ( +   ) }  ( ) ( +   ) 
 ( +  +   )   = ( +  +   )

 =

(42)
(43)

where

 = ( +  +   ) (( +   )  + (1 −   )  ) 
If max { +     } =  +   , = {(  ( + ) ( +   ) −  (1 −   ) ( + (1 − )    −    )) (44)
+ ((1 −   )  + ( +   ) ( +   )) }    ( +  ) (( +    − (1 −   ) (1 − ) )  + ( +   ) )  ((   +  −  (1 − ) (1 −   ))  + (   + ) )  =   =(45)(46)

 = {(((1 − )    +  ) (1 −   )  + (1 −   )    − (  −   )  )(47)
× − ( +   ) (  +  +    ) }  ( ) ( +   ) ((( +   )  + (1 −   )  )  −  ) ((( +   )  + (1 −   )  )  −  ) = 


 =

(48)
(49)

where
 = ( +   ) (  +  +    + (1 −   )  ) − (1 −   ) (1 − )  41

Lemma 14
a. Write =  −  −    =  −  
respectively, as the match surpluses in matches between a buyer in rental housing
and a flipper-seller and a mismatched household-seller. If  ( ) ≥ 0,  ≥0. If  ( ) ≤ 0,  ≥ 0.

b. max { +     } ≥ 0
Lemma 15 Assume that  () =  for some   0 and  ∈ (0 1). The firstinequality of (50) below is the condition for   00 , whereas the second inequality
and (51) together guarantee that 1  1 and   .
µ³


1− 

´

( + ) 

 
1
¶ 1−

 

(50)

(1 − ) + (1 + )  



 


( + )  − (1 −   ) + (1 + )  ≥ (1 −   ) (51)
The two conditions above and that  be suﬃciently close to 1 guarantee that (19)holds.
( + )Proposition 6

Equilibrium exists for all {            } tuple.

Proposition 7
a. In the fully-intermediated equilibrium, equiproportionate increases in  and lower housing prices. The same eﬀect is felt in the no-intermediation equilibrium for suﬃciently large  (0) and/or .
b. In a partially-intermediated equilibrium,i. equiproportionate increases in  and  , holding  fixed, lower housing
prices;b ( ) = 0, equiproportionate
ii. across steady-state equilibria and holding changes in  and  , whether positive or otherwise, cause   to increase(decrease) as long as to follow the interest rate shocks are increases (decreases) in  and  for   =   = 12 and  ∈ [0 1 + ]; the sameeﬀect is felt on   and  for  in neighborhoods of  = 0, 1, and1 + .

42

A.2

Calibrating  = In the model housing market, the flow payoﬀs for matched owner-occupiers, mismatched owner-occupiers, and buyers in rental housing are equal to , 0, and −,
respectively. In this case then, the diﬀerence between the flow payoﬀs of matched
and mismatched owner-occupiers is equal to , and that between mismatched owneroccupiers and buyers is . Anenberg and Bayer (2013) report estimates on
1. mean flow payoﬀ for matched owner-occupiers 0.0273
2. flow payoﬀ for owner-occupiers mismatched with their old houses 0.024, comprising 30% of all mismatched households
3. flow payoﬀ for owner-occupiers mismatched with the metro area 0.0014, comprising 70% of all mismatched households
while normalizing the flow payoﬀ of buyers to 0. We may thus equate  = 00273 −0024 = 000033 and  = 0024 so that  = 8 for households who are mismatched with
their old houses and  = 00273 − 00014 = 00259 and  = 00014 so that  = 0054
for households who are mismatched with the metro area. Taking a weighted averageof the two estimates gives a value for  equal to 249. Alternatively, one may take theweighted average first before taking the ratio:  = 00273−003 ×0024 −07 ×00014
and  = 003 × 0024 + 07 × 00014, so that  = 043. Since it is not clear in thecontext of our model which method is conceptually better than the other, we resort
to taking a simple average of 2.49 and 0.43 to obtain a value of 1.43 for .

A.3

Proofs

Proof of Lemmas 1 and 2 Solve (1)-(6) for
 =



+

(52)

 =



+

(53)

(1 − ) 

+

(54)

 (1 − ) − (1 − )  + 

+

(55)

 =
 =

Equation (7) in Lemma 2 is obtained by substituting (53)-(55) into= + 

The LHS of (7) is equal to (1 − (1 − ) )   0 at  = 0 but is negative for arbitrarily
large  given the concavity of . A solution is guaranteed to exist. Diﬀerentiatingwith respect to  yields
(1 − ) − 43which is positive for small  but negative otherwise. The solution to (7) must thenbe unique, and that the LHS is decreasing in  at where it vanishes. Given that the
LHS of the equation is increasing in ,    0. The lower and upper bounds (0) and  (1) are given by the respective solutions to +  (1 − ) − (1 + )  = 0(56)

 +  (1 − ) −  = 0(57)both of which are seen to be positive and finite. That    0 holds as the LHSof (7) is decreasing in . That lim→1  (0) = 0 falls out from (57).
Solve (7) for − (1 − ) ( + )

(58)
=

and substitute into (53)-(55), respectively, = − (1 − ) ( + )

+

(59)

 (1 − ) +  (1 − )

(60)
+

 =

(61)
+
The comparative statics Lemma 1 can be obtained by diﬀerentiating (52) and (59)(61), respectively, with respect to , and then noting that    0. The boundary
values for  and  are obtained for setting  = 0 and  = 1 in (53) and (54),respectively.
 =Proof of Lemma 3 Substituting from (53) and (54) into +
yields the result
of the Lemma.

Proof of Lemmas 4-7 and the construction of Figure 3 Given that lim→0  =0 and lim→∞  = 0,b = (1 −   −  (  −   ))  − ( + )  = ∞
lim 

→0

Diﬀerentiating,µ¶b = lim −   −    − ( + ) 
→∞


¶
b µ




= 
+ (1 −   −  (  −   ))−   −  





(62)

(63)

For  ≥ , the expressions in both (62) and (63) are negative for sure. In this case,b starts out equal to positive infinity and falls continuously below
as a function of , ¡¢¡¢b 1  1 = 0 and b 0  0 = 0,zero. Then, there exist some 1 and 0 that satisfy 
b is decreasing in  and in ,
respectively. Given that 
44

1. 1  0 b ( ) ≥ b ( 1)  0 for   1 
2. for any  ∈ [0 1], b ( ) ≤ b ( 0)  0 for   0 
3. for any  ∈ [0 1], ¢¡b ( ) = 0 holds at some  =  (), where  ()   0.
4. for  ∈ 1  0 , This completes the proof of Lemma 4.
For  ≥ , by the three Panels of Figure 2, there will be flipping in the uniqueequilibrium if and only if (0)  0 (64)
By (56),  (0) is decreasing in , with limiting values
lim  (0) = ∞lim  (0) = 0

→0

→1

By (17), 0 satisfies
¶
µ
¡ ¢
¡ ¢

−   (1 + )  0 + (1 −   )  0 − ( + )  = 0whereby it is decreasing in  . The limiting values←−
lim 0 =  0→−lim 0 =  0 

 →∞

 →

are the respective solutions to
³←
³←
−´
−´
−  (1 + )   0 + (1 −   )   0 − ( + )  = 0
(1 −   ) ³−→ ´
 0 − ( + )  = 0

both of which are positive and finite. Then, condition (64) cannot be met for 
close to 0 but must be satisfied for  close to 1. With  (0) decreasing in  and 0
decreasing in  , the combinations of  and  under which (64) holds as an equality
define an upward-sloping relation  ( ) between the two parameters. This proves
Lemma 5 and explains the part of Figure 3 for  ≥ .
b in (62) is equal to positive infinity, whereas if the condition
For   , lim→∞ bin note 14 is met,  in (63) is at first negative, reaches zero, and becomes positiveb
thereafter. In this case, , as a function of , is U-shaped and starts oﬀ and ends up
equal to positive infinity.
Writeb min (;  ) ≡ min b ( ) b min   0 and  b min   0. Notice also
where by the Envelope Theorem,  b min (;  ) = ∞ and lim → b min (;  ) = −∞. Then, there exist
that lim →0 45

b min (1; 0 ) = 0 and b min (0; 00 ) = 0, respectively, wheresome 0 and 00 that satisfy 
000
0      .
b min (1;  ) ≥ 0. Given that b ( ) ≥ b min (;  ), b ( )  0 for
For  ≤ 0 , 
all   1. This proves Lemma 6(a) and explains the part of Figure 3 for  ≤ 0 .
b min (1;  )  0. Hence, there exists two , namely 1 and 1
For  ∈ (0  00 ), b ( 1) = 0, over where  bb
at which  0 and  
 0, respectively. Within¡  ¢b (  ()) = 0 For
the interval 1  1 , the exists some  () ∈ (0 1) such that £  ¤b ( 1)  0. Moreover, since b min (0;  )  0 for   00 ,  ()  0
∈ 1  1 , 
for all . This proves Lemma 6(b) and explains the part of Figure 3 for  ∈ (0  00 ).
b ( 0) = 0. Call this ∗ . For  6= ∗ ,At  = 00 , there is one  at which b ( 0)  0 Then, an  = 0 equilibrium exists if and only if  (0) = ∗ . Given thatas  varies from 0 to 1,  (0) spans the entire positive real line, there exists one andonly one  at which  (0) = ∗ . This explains how the U-shaped border in Figure
3 is tangent to the  = 00 line.
b min (  )  0. Then, there are two , namely 0 and 0 at
For  ∈ [00  ), b ( 0) = 0, over where   0 and   0, respectively. Within thewhich ¡¢bb ( 0)  0. As  increases from 00 to , given that  interval 0  0 ,  0,¡  ¢
bb
and that    0 at  = 0 and    0 at  = 0 , the interval 0  0
expands, whereby 0   0 and 0   0. This proves Lemma 6(c).
As to the construction of the U-shaped border in Figure 3, first notice that in thiscase, an  = 0 equilibrium exists if and only if£¤ (0) ∈ 0  0 
Given that 0   0 and  (0)   0, the condition  (0) = 0 defines anb in theupward-sloping border  ( ) in the - space. Given the continuity of ,limit as  → , 0 tends to the same 0 at which  =  just hold. Then, there is thesame  that keeps  (0) = 0 at  = . On the other hand, with 0   0, thecondition  (0) = 0 defines a downward-sloping border  ( ) in the - space.

Proof of Lemma 8 Only two of the four equations of motion constitute independent restrictions. By utilizing (1)-(4), we can reduce the system to that of two statevariables, the equations of motion of which are given by, respectively,· =  ( −  ) −  ·

 =  −  ( − (1 − )) and an equation for  given by= − 
Write the Hamiltonian of maximizing (20) subject to (65)-(67) as

(65)
(66)
(67)

 = − (  −  ) + Γ ( ( −  ) −  ) + Γ ( −  ( − (1 − ))) (68)
46

where Γ and Γ are the respective co-states for  and  . Then, we have= Γ  

µ
µ
¶
¶
·

  − (1 − )−Γ = −  − Γ − −  + − Γ  −   − µ¶·  − (1 − )−+ Γ  +
Γ =   − Γ  − 

(69)
(70)
(71)

The last two equations imply that the co-states must be growing at the rate −in the steady state. We can then writee − Γ = Γe − Γ = Γ


(72)
(73)

e that are stationary over time. Substitute (72), (73) and thee and Γfor some Γsteady-state values of  and  from (52) and (55), respectively, into (70) and (71),
µ
¶
¶
µ


e
e
e
+ Γ   − Γ =  − Γ  +  − 


µ
¶


e = − + Γe  +  
e
Γ
−Γ





the solutions of which are

¢
¡
¢
¡




−


−



+

+


e =  ¡
¢

Γ



 +  −   +   +  ( + )
¡
¢


+

+

−


e =  ¡
¢

Γ



 +  −   +   +  ( + )−e above. This explains
By (69), the optimal value for  depends on the sign of Γ
 () in (22).
Proof of Lemma 10 By (56), (0)

1−
=
≤


 +  ( (0))
 +  ()(74)

since  ≥  (0). Substituting (2) into (24) and then from (58) and (52),  =  + ( −  )  =47

(1 + ) −  (1 − ) 
+

(75)

Diﬀerentiating and simplifying,µ¶  (1 + )  1 − 

 
= 
+
−

 ( + )2
 
+
µ
¶
  (1 + ) 

≥ 
+
 0
−
 ( + )2
+
+where the first inequality is by (74) and the second inequality by the concavity of .
But then    0; hence     0.
Proof of Lemma 11 Substituting from (2), (75), and (52) and simplifying, (25)becomesµ¶−11−
(1 + )  − ( + )Diﬀerentiating with respect to  yields an expression having the same sign asµ¶µ¶ 1 −  

 
− +
−
( + 2) − +−( + 2) =   +  
µ
µ
¶¶


−
+−
 0
+


where the first inequality is by (74) and the second by the concavity of . The Lemma
follows given that    0.
Proof of Lemma 12 Substituting from (5), (6), and then (1), (26) becomes
1 1−
1 − 

+
=



a decreasing function of  . But where    0, there must be a smaller average
TBM.
Proof of Lemma 13 Setting max { +     } =  in (10) and solving (8)(13) for the three prices and three asset values yield the solutions in the first part of
the Lemma. Setting max { +     } =  +   before solving (8)-(13) yield
the solutions in the second part.
Proof of Lemma 14 In case   0, by the second part of Lemma 13,  has thesame sign as(1 + ) (   + )  0
If  = 0, by the first part of Lemma 13,  has the same sign as
 + (1 + )    −  (1 − ) (1 −   ) + ( + ) 
48

(76)

But if  = 0, by (7),µ¶−   −    + (1 −   −  (  −   ))  = ( + ) Substituting into (76) for ( + ) ,( +   )  + (1 −   )   0

If  ≤ 0, by the first part of Lemma 13,
 =


 0
 +  +  This completes the proof of Part (a) of the Lemma.
For Part (b), if  ≥ 0, the condition to check is +   ≥ 0
By the second part of Lemma 13, the condition becomes  + (1 −   )  − ( + )  ≥ 0 b ≥ 0. If  ≤ 0, we want to show thatwhich holds whenever  ≥ 0

which holds by the first part of Lemma 13.

Proof of Lemma 15 By (17), 1 satisfies,µ¶− (1 + )    (1 ) + (1 −   )  (1 ) − ( + )  = 0at where the above is increasing in . Let  () =  . Then, 1  1 if− (1 + )   + 1 −   − ( + )  0
The second inequality of (50) follows under (51), which serves to ensure that   .
By the proof of Lemmas 4-7, 00 satisfies,on³ ´min   00 − (1 + )    () + (1 −   )  () − ( + )  = 0Evaluating the above and solving for 00 yield the far left term in (50).
With  (1) implicitly defined by (57), it can be shown that lim→1  (1) = 1,
whereas by Lemma 2, lim→1  (0) = 0. Under (51) and (50), there exists a strictly
positive 0 . Then, the first inequality of (19) would hold for  suﬃciently close to
1. Under the same conditions, the last inequality of (19) holds too with 1  1 but
lim→1  (1) = 1.
49

b ( ) and  (), given
Proof of Proposition 1 Part (a) is obtained by setting by (17) and (22) respectively, equal to each other and solving for   and   . Forb ( )   0,part (b), if  ¶  2µ
(1 −   −  (  −   ))   −   −   0
 −  

Then,
¶
µ



−   −    + (1 −   −  (  −   ))


µ
¶


− + −



! µ
¶Ã
¶
µ

 
1
 1
+
1
−

−
−   −  




 
 
1 −  ∙µ¶ ¸12
 1
 2 −   −   
−

 

b ( ) −  () =The second inequality comes from minimizing   −   −  + with respect to . Rewriting the last line gives the condition in the Proposition.
Proof of Proposition 2 We begin with showing end-user market price   inthe full-intermediation equilibrium, given by (29), is higher than   in a partiallyintermediated equilibrium, given by (31). Now, at where  = 1 ,  ( 1) = 0, thetwo   are by construction equal. Second, with   in the first equation increasingin  by the concavity of  and the   in the second equation increasing in  for  , which is a necessary condition for multiplicity,   in the full-intermediation
equilibrium must exceed   in the partially-intermediated equilibrium, since in thiscase  (1) ≥ 1 whereas   in the partially-intermediated equilibrium has a  ≤ 1 .
Lastly, with     in the partially-intermediated equilibrium, the single end-user
market price in the full-intermediation equilibrium exceeds the two end-user market
prices in the partially-intermediated equilibrium. Next, in a comparison between  in two partially-intermediated equilibria, given that   in (31) is increasing in ,
there must be higher   in the larger  equilibrium. The same ranking applies tothe two  , given that  in (32) is similarly increasing in  in case   . The
final comparison is between  in a partially-intermediated equilibrium and  in the
no-intermediation equilibrium, given by (27). At where  = 0 ,  ( 0) = 0, the two are by construction equal. With the first  known to be increasing in , the proof
is completed by noting that  , given by (27), is likewise is increasing in  given the
concavity of . This completes the proof that end-user market housing prices across
steady-state equilibria can be ranked by the value of . Given that   = +  ,

investment market housing prices are ranked in the same order as in end-user market
housing prices.
50

Proof of Proposition 4 By diﬀerentiating (33)-(35) with respect to  and notingthat  ≤ 1 and    0.
Proof of Proposition 6 Define Φ () ≡  ( ()), a continuous function mapping [0 1] into itself. A steady-state equilibrium is any fixed point of Φ. By Brouwer’s
Fixed Point Theorem, a continuous function mapping the unit interval into itself mustpossess a fixed point.
Proof of Proposition 7 Write  =  , which remains constant amid any
equiproportionate changes in  and  . Substituting  = −1 into (28) and (29)and diﬀerentiating proves the first part of (a). In a no-intermediation equilibrium, 
is given by (27), which is independent of  but decreasing in  for suﬃciently small (0) and/or . This proves the second part of (a). For (b), substituting  = −1
into (30)-(32), respectively, yields, −1 ( +  + 



(77)

   +   −1  =
−1 ( +  +   )(78)

  = )   +   −1−1 ( +  +   )
b = 0 from (17) for
all of which are decreasing in . Solving  =(   −   −   )  + (1 −   −  (  −   )) − and substituting into (77)-(79), respectively, gives=  =

  2



(79)

(80)

(81)

((   −   )  + (1 −   −  (  −   )) )    − (  +  − )   


(82)
((   −   )  + (1 −   −  (  −   )) )    − (  (  + ) −   ) 

 =

(83)
where
 = (((  −   )  − (1 −   −  (  −   )) )  + ( +   ) )(84)
× ((  −   )  − (1 −   −  (  −   )) )   =

Diﬀerentiating (81) with respect to , evaluating the resulting expression at   =  = 12 yields an expression whose sign is given by that ofµ¶2 
 ( − 1 + 1 − ) − 2 − ( − 1 + 1) ( − 1 + 1 − ) −  51The expression is strictly positive at  = 0 and  = 1 +  if the RHS of (80) at  =   = 12 is positive. And then diﬀerentiating twice with respect to  yields
−22


 0


Thus,   in (81) must be increasing in  for  ∈ [0 1 + ]. For   and  ,diﬀerentiating (82) and (83) with respect to  and evaluating at   =   = 12 and = 0, 1, and 1 + , respectively, all yield a strictly positive expression as long as the
RHS of (80) is positive at   =   = 12. Then,   and  in (82) and (83) must
be increasing in  for  in neighborhoods of 0, 1, and 1 + 

52

Figures