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Regressive Welfare Effects of Housing
Bubbles
WP 18-10
Andrew Graczyk
Wake Forest University
Toan Phan
Federal Reserve Bank of Richmond
Regressive Welfare Effects of Housing Bubbles
Toan Phan∗
Andrew Graczyk
April 13, 2018
Working Paper No. 18-10
Abstract
We analyze the welfare effects of asset bubbles in a model with income inequality
and financial friction. We show that a bubble that emerges in the value of housing, a
durable asset that is fundamentally useful for everyone, has regressive welfare effects.
By raising the housing price, the bubble benefits high-income savers but negatively
affects low-income borrowers. The key intuition is that, by creating a bubble in the
market price, savers’ demand for the housing asset for investment purposes imposes
a negative externality on borrowers, who only demand the housing asset for utility
purposes. The model also implies a feedback loop: high income inequality depresses
the interest rates, facilitating the existence of housing bubbles, which in turn have
regressive welfare effects.
Keywords: rational bubble; inequality; housing; financial friction
JEL classifications: E10; E21; E44
1
Introduction
Many countries have experienced episodes of bubble-like booms in asset prices. Examples
include the real estate booms in Japan in the 1980s, Southeast Asia in the 1990s, the U.S.
in the 2000s, and more recently in China and Vietnam.1 In general, when there is a high
demand for savings but limited investment outlet, the rates of returns from investment are
depressed and real estate investment can serve as a prominent store of value. Thus, a low
∗
Graczyk: Wake Forest University, USA; a.c.graczyk@gmail.com; Phan: The Federal Reserve Bank of
Richmond, USA (corresponding author); toanvphan@gmail.com. We thank the editor, two anonymous
referees, Julien Bengui, Ippei Fujiwara, Lutz Hendricks, Guido Menzio, and Andrii Parkhomenko for helpful
suggestions. The views expressed herein are those of the authors and not those of the Federal Reserve Bank
of Richmond or the Federal Reserve System.
1
See, e.g., Hunter (2005), Mian and Sufi (2014), Fang et al. (2015).
1
interest rate environment, as seen in the recent decade, provides a fertile ground for the
emergence of asset bubbles, including those in real estate. Given the prevalence of bubble
episodes, a central question arises for academics and policymakers: What are the welfare
effects of asset bubbles?
In this paper, we highlight the nuanced welfare effects of asset bubbles, especially those in
real estate prices. We develop a simple overlapping generations (OLG) model of bubbles with
intra-generation heterogeneity and financial friction. As described in Section 2 of the paper,
households have identical preferences over a perishable consumption good and a durable
and perfectly divisible housing asset in fixed supply. Young agents receive endowments,
and a fraction of them are savers, who are born with high endowments, and the remaining
fraction are borrowers, who are born with low endowments. Young borrowers, given their
low endowment, want to borrow to purchase the desired amount of housing that maximizes
their utility. In contrast, young savers, given their high endowment, would instead like to
save for old age. Thus, for savers, housing not only yields utility dividend, but also serves as
a savings or investment vehicle. To highlight the difference between these two motives, we
assume that the utility over housing has a satiation level h̄. Any additional unit of housing
above the satiation level yields no additional utility and thus purely serves as an investment
vehicle.
In an economy without financial friction, households can achieve their first-best allocations by borrowing and lending in the credit market. However, in the presence of financial
friction, such as imperfect contract enforcement, young borrowers face a binding credit constraint, modeled as an exogenous limit on borrowers’ debt capacity. In equilibrium, the
constraint effectively limits how much savers can store their income by investing in the
credit market. As we show in Section 3, in an economy with high income inequality, there
is a shortage of storage for savers, which can lead to an equilibrium interest rate that is
below the economy’s growth rate. The low interest rate environment in turn facilitates the
emergence of asset bubbles.
In Section 4, the main part of our paper, we study housing bubbles. In a housing bubble
equilibrium, young savers acquire housing in excess of the satiation level, because they would
like to use the housing asset as an investment vehicle to save for old age. This “speculative”
demand for the housing asset by savers is similar to the demand of a bubble asset in a
standard rational bubbles model: an agent purchases an asset because he or she expects to
be able to sell it to someone else in the future.
We then show that the housing bubble has opposite effects on borrowers and savers. The
housing bubble increases the return from real estate investment for high-income savers, who
demand storage of value, and hence increases their welfare (relative to the bubbleless bench2
mark). In contrast, the housing bubble reduces the welfare of borrowers, because it raises
the price of housing and the speculative demand of savers for the housing asset crowds out
the allocation of housing to borrowers, who in equilibrium have a relatively higher marginal
utility from housing. By positively affecting high-income savers and negatively affecting
low-income borrowers, the housing bubble thus has regressive welfare effects. Overall, our
results imply a feedback loop on inequality: high income inequality depresses the interest
rate, thereby facilitating the existence of housing bubbles, which in turn have regressive
welfare effects. The key insight is that, by creating a bubble in the market price of housing,
savers’ demand for the housing asset for investment purposes imposes a negative pecuniary
externality on borrowers, who only demand the housing asset for utility purposes.
In comparison, Section 5.1 shows that the regressive welfare implications are lessened if
the model considers pure bubbles, which are widely used in the rational bubble literature for
their simplicity. A pure bubble is an asset that has no fundamental value, but which is traded
at a positive price (such as fiat money or unbacked public debt). The pure bubble provides
an additional and separate investment vehicle for savers: besides lending and investing in
the housing market, savers can invest in the pure bubble asset (by purchasing the asset
when young and reselling it when old). Unlike the housing bubble equilibrium, the pure
bubble equilibrium is characterized by an endogenous segmentation in the bubble market.
This is because only savers purchase the bubble asset for investment purposes, while creditconstrained borrowers have no demand for the asset. Furthermore, the option to invest in
the pure bubble asset is a substitute for the option to use the housing asset purely as an
investment vehicle. As a consequence, the crowding effect on housing that was prevalent in
the housing bubble case is absent in the presence of a pure bubble. Consequently, the negative
externality on borrowers’ welfare is absent in the pure bubble equilibrium. Furthermore,
because the presence of the pure bubble asset effectively enriches the available menu of
investment vehicles, we can show that the allocations in the pure bubble equilibrium Paretodominates the allocations in the housing bubble equilibrium.
Related literature. Our paper is related to the rational bubble literature, which has a
long heritage dating back to Samuelson (1958), Diamond (1965), and Tirole (1985). For a
survey of this literature, see Miao (2014) and Martin and Ventura (2017).2 Much of the
literature has focused on a positive analysis of bubbles. A common theme in this literature
is that rational bubbles emerge to reduce some inefficiency in the financial market, such as
an aggregate shortage of assets for storage or a credit market imperfection, as in Miao and
2
There is a complementary literature that focuses on the role of heterogeneous information in coordinating
agents’ actions to purchase and sell bubbles. See, inter alia, Abreu and Brunnermeier (2003), Doblas-Madrid
(2012), Doblas-Madrid and Lansing (2014), and Barlevy (2014).
3
Wang (2011), Martin and Ventura (2012), and Hirano and Yanagawa (2017), Ikeda and Phan
(2018). By departing from the pure bubble assumption and modeling a bubble as attached
to a fundamentally useful durable asset such as housing, our paper is related to Arce and
López-Salido (2011), Miao and Wang (2012), Wang and Wen (2012), Hillebrand and Kikuchi
(2015), Zhao (2015), Kikuchi and Thepmongkol (2015), and Basco (2016).
To the best of our knowledge, among papers that analyze the welfare effects of bubbles,
ours is the first to document regressive welfare effects of a housing bubble. Saint-Paul (1992),
Grossman and Yanagawa (1993), and King and Ferguson (1993) show that if there is a positive externality in the accumulation of capital, the emergence of bubbles on an unproductive
asset would inefficiently divert resources from investment. Similarly, Hirano et al. (2015)
show that oversized bubbles inefficiently crowd out productive investment. On the other
hand, Miao et al. (2015) show that bubbles can crowd in too much investment. Caballero
and Krishnamurthy (2006) show that bubbles can marginally crowd out domestic savings
and cause a shortage of liquid international assets in a small open economy framework. Focusing instead on risk, Ikeda and Phan (2016) show that rational bubbles financed by credit
can be excessively risky. The regressive welfare effects that we highlight are complementary
to the welfare effects highlighted by these papers.
2
Model
Consider an endowment economy with overlapping generations of agents who live for two
periods. Time is discrete and infinite, with dates denoted by t = 0, 1, 2, . . . . The population
of young households in each period is constant with population Lt = 1 for all t. There is
a consumption good and a housing asset. The consumption good is perishable and cannot
be stored. The housing asset is durable and perfectly divisible. The supply of housing is
fixed to one. The consumption good is the numeraire and the market price of a unit of the
housing asset is denoted by pt .
Heterogeneity. Each generation consists of two types of households – high-income savers
and low-income borrowers (or debtors) – denoted by i ∈ {s, d} correspondingly. Each group
has an equal unit measure population. Each young household is endowed with ei units of
the consumption good, where es > ed . In addition, each household receives an endowment
of e > 0 when old.3 Without loss of generality, we normalize ed = 1. Thus, any increase in
es leads to an increase in (within-generation) income inequality.
3
We thus focus on the heterogeneity of endowments in young age. One can interpret the young-age
endowment as wage income and the old-age endowment as payment from, e.g., social security. Furthermore,
we have effectively set the economic growth rate to be zero. It is straightforward to extend our framework
with exogenous endowment growth.
4
Preferences. Households derive utility from the housing asset and from the consumption
good, consumed both when young and old. They have a separable utility function of the
following form
U (cit,y , cit+1,o , hit ) ≡ u(cit,y ) + βu(cit+1,o ) + v(min{hit , h̄}),
where cit,y and cit+1,o denote consumption in young and old age of a household of type i ∈ {s, d}
born in period t, β is the discount factor, hit denotes the housing, and h̄ ∈ ( 21 , 1) is a
constant that denotes the satiation level of housing consumption. The utility functions u
and v : (0, ∞) → R satisfy the usual Inada conditions (u0 , v 0 > 0, u00 , v 00 < 0, limc→0+ u0 (c) =
limh→0+ v 0 (h) = ∞).
The key assumption behind the expression v(min{hit , h̄}) is that households receives no
additional marginal utility beyond h̄. This assumption helps clearly illustrate the speculative
motive: if a household only purchases housing for utility, then it should never purchase more
than h̄; however, if a household additionally wants to use housing as an investment vehicle,
then it may purchase in excess of h̄.
Credit market and credit friction. Households can borrow and lend to each other via a
credit market. Let Rt denote the gross interest rate for debt between period t and t + 1.
As in Bewley (1977), Huggett (1993), and Aiyagari (1994), we model credit friction in the
simplest possible way: an agent can commit to repay at most d¯ units of the consumption
good, where d¯ ≥ 0 is an exogenous debt limit. This imperfection in the financial market
will lead to a constraint on households’ ability to borrow, as manifested in the optimization
problem below.
Remark 1. The presence of the credit friction is important for the existence of asset price
bubbles, which requires dynamic inefficiency. If the credit market were frictionless (e.g.,
d¯ := ∞), then the economy would be dynamically efficient and bubbles cannot arise (see
Section 4).
Optimization. A household purchases housing, consumes, and borrows or lends when
young, and then sells their housing asset and consumes when old. The optimization problem
of a young household of type i ∈ {s, d} born in period t consists of choosing housing asset
position hit , net financial asset position ait , and old-age consumption cit+1 to maximize lifetime
utility:
max
U (cit,y , cit+1,o , hit )
(1)
i i
i
i
ht ,ct,y ,ct+1,o ,at
5
subject to a budget constraint in young age:
1 i
a + cit,y = ei ,
Rt t
(2)
cit+1,o = pt+1 hit + ait + e,
(3)
pt hit +
a budget constraint in old age:
nonnegativity constraints on housing and consumption:4
hit , cit,y , cit+1,o ≥ 0,
and the credit constraint:
¯
ait ≥ −d.
(4)
Throughout most of the paper, we assume for simplicity that:
d¯ = 0.
This assumption allows us to focus on the housing bubble’s effect on the housing market
rather on the credit market. We relax this assumption in Section 5.2 and show that, as long
as d¯ is sufficiently small so that the credit constraint binds for borrowers, our main results
carry through.
Finally, to close the model, without loss of generality assume that the old savers own the
entire supply of housing in the initial period t = 0. We define an equilibrium as follows:
Definition 1. An competitive equilibrium consists of allocation {hit , cit,y , cit+1,o , ait }t≥0 and
positive prices {pt , Rt }t≥0 such that:
1. Given prices, the allocations solve the optimization problem (1) for all i ∈ {s, d} and
t ≥ 0.
2. The consumption good market clears:
cst,y + cdt,y + pt (hdt + hst ) = es + ed + e, ∀t ≥ 0;
3. The credit market clears:
ast + adt = 0, ∀t ≥ 0;
4
Because of the Inada conditions, these constraints do not bind in equilibrium.
6
4. And the housing market clears:
hst + hdt = 1, ∀t ≥ 0.
We will focus on steady-state (or stationary) equilibria, where quantities and prices are
time-invariant.
For the analyses in subsequent sections, it is convenient to label a constrained economy
as an economy where all households face an additional constraint hi ≤ h̄, which effectively
prevents them from purchasing units of the housing asset purely as an investment vehicle.5
As mentioned earlier, given the preferences over housing, if a household purchases more
than h̄ in housing, the excess amount h − h̄ – which yields no additional housing utility to
the household – must be purchased purely for investment. Thus, we label a housing bubble
equilibrium as an equilibrium where at least one type of households purchases more housing
than h̄, i.e., max{hs , hd } > h̄. As shown in Section 4, this equilibrium can be mapped to
a rational bubble equilibrium, where the bubble arises in the market value of housing. In
contrast, we define a bubbleless equilibrium as an equilibrium where no type of household
purchases housing in excess of h̄, i.e., max{hs , hd } ≤ h̄.
3
Bubbleless equilibrium
We start with the case of a bubbleless equilibrium, where households only purchase housing
for utility and hi ≤ h̄ for both types of households. For simplicity, we focus on the most
interesting parameter space in which there is sufficient inequality (es sufficiently large) such
that in equilibrium savers can satiate the utility for housing (hs = h̄), while borrowers are
credit constrained and cannot achieve the satiation level (hd = 1 − h̄ < h̄).
Given hs = h̄, the conjectured equilibrium can be uniquely characterized by two key
variables: the price of housing p and the interest rate R. Specifically, from budget constraints
(2) and (3), the consumption profile ciy , cio for i ∈ {s, d} are functions of R and p. In turn, R
and p are uniquely determined by a system of two first-order conditions, the first of which
is the unconstrained savers’ intertemporal Euler equation, and the second is the borrowers’
5
This could be because the government imposes a regulation against speculative investment in housing.
Note that recently several governments, including the Chinese government, have expressed intentions to curb
speculation in the property market (e.g., Bloomberg, 2016).
7
interior (0 < hd < h̄) first-order condition with respect to housing:6
u0 (csy ) = βR u0 (cso ),
p u0 (cdy ) = v 0 (hd ) + βp u0 (cdo ),
(5)
(6)
where the consumption allocations . The first condition equates the marginal cost (in terms of
utility for a saver) of saving a unit of consumption to the marginal gain. The second equates
the marginal cost (in terms of utility for a borrower) of investing in one unit of housing to
the marginal gain, which consists of the marginal utility dividend v 0 and the marginal return
from subsequently reselling the housing unit. We denote Rn and pn as the solution to this
system of equations (the subscript n refers to “no bubble”). Similarly, {ciy,n , cio,n }i∈{s,d} and
Uni denote the associated consumption profiles and lifetime utility. It is straightforward to
show that all else equal, Rn is decreasing and ps is increasing the endowment of savers es .
Figure 1 illustrates the determination of the interest rate Rn and housing price pn . The
blue curve labeled “foc savers” plots the lending first-order condition (5) of savers, while
the red curve labeled “foc borrowers” plots condition (6) of borrowers.7 Their intersection
determines Rn and pn . The dashed line plots what happens when there is an exogenous
increase in the endowment es of savers while all other parameters stay the same (and thus
an increase in inequality). The curve associated with the Euler equation of savers shifts to
the left, leading to a decrease in the interest rate Rn (as there is an increase in the demand
for storage from savers). Thus, higher inequality is associated with a lower interest rate.
For convenience, we implicitly define ē as the savers’ endowment threshold such that
Rn |es =ē = 1.
(7)
It is then immediate that Rn ≥ 1 if and only if es ≤ ē. In other words, the interest rate Rn
is at least as large as the growth rate of the economy if and only if there is not too much
inequality.
For the conjectured prices and allocations characterized above to indeed constitute a
bubbleless equilibrium, savers must find it optimal to not invest in housing in excess of h̄.
When is this the case? By purchasing each marginal unit of housing in excess of h̄ at price
pt and reselling it at price pt+1 in the subsequent period, the individual gets a return rate of
In the conjectured allocation, savers are consuming housing at the kink hs = h̄, where the housing utility
term v(min{hs , h̄}) is not differentiable. Hence, we do not have a first-order condition for housing for savers
that corresponds to (6).
7
The foc borrowers curve is flat because of the assumption d¯ = 0, which makes equation (6) independent
of R. When d¯ > 0, the foc borrowers curve will be downward sloping, and all of our analysis will continue
to hold. See Section A.8 and Figure 6 for more details.
6
8
Figure 1: Determination of bubbleless steady state’s interest rate and housing price
pt+1 /pt , which is equal to 1 in steady state. The alternative method of saving via the credit
market yields the return rate of R. Hence, for each individual saver to not have an incentive
to use housing as an investment vehicle, it must be that Rn ≥ 1, i.e., the interest rate Rn
is at least as large as the growth rate of the economy. The following lemma formalizes this
insight:
Lemma 1 (Bubbleless steady state). The bubbleless steady state exists if and only if Rn ≥ 1,
or equivalently, es ≤ ē, where ē is implicitly defined by (7). This steady state is characterized
by interest rate Rn and housing price pn , which are implicitly defined by (5) and (6) with
housing allocation hs = h̄. All else equal, Rn is strictly decreasing in es .
Proof. Appendix A.1.
Remark 2. Note that the bubbleless equilibrium as described above is the only equilibrium
in the constrained economy (recall Definition 1). This is because even when es > ē and
Rn < 1, the no-speculation constraint hi ≤ h̄ binds for savers, which prevents them from
making a profit from investing in the housing asset in excess of h̄.
What happens when es > ē and Rn < 1? As we show in the next section, the low interest
rate environment provides a fertile ground for a bubble to arise in the housing market.
9
4
Housing bubble equilibrium
When Rn < 1, as we showed in the previous section, the bubbleless equilibrium does not
exist, as savers find it attractive to invest in housing even if they do not derive additional
utility. This section characterizes the housing bubble equilibrium that will instead arise.
In this equilibrium, savers will purchase housing in excess of h̄. In the region hs > h̄, the
housing utility function is differentiable. Hence, unlike in the bubbleless case, the first-order
condition with respect to housing of savers must hold with equality, leading to the following
equation:
pu0 (csy ) = v 0 (hs ) + βpu0 (cso ).
Combining the savers’ housing first-order condition above with their lending first-order condition (5) yields a familiar asset pricing equation:
p=
v 0 (hs )
u0 (csy )
| {z }
+
marginal utility dividend
p
R
|{z}
.
(8)
resale value
Equation (8) states that the price of each unit of housing is equal to the marginal housing
0 s)
, plus the resale value discounted
dividend, captured by the marginal rate of substitution vu0(h
(csy )
by the interest rate p/R. Alternatively, using time subscript on pt , equation (8) can be
0 s)
written as pt = vu0(h
+ pt+1
, whose recursion leads to a standard forward-looking asset
(csy )
R
P
v 0 (hs )/u0 (csy )
p
pricing equation pt = j≥0
+ limj→∞ t+j+1
, where the first term on the rightRj
Rj
hand side captures the discounted value of the marginal utility dividend stream, and the
second term captures the “bubble component.”
Since any amount in excess of h̄ yields no marginal housing utility, the term v 0 (hs )/u0 (csy )
in equation (8) is equal to zero, leading to:
p=
p
,
R
(9)
i.e., the price of housing must grow at the interest rate. Because p > 0, it then necessarily
follows from (9) that R = 1 in the housing bubble equilibrium.
There is another intuitive way to understand the identity R = 1 as a no-arbitrage condition for savers. For these agents, the investment in each additional unit of housing in
excess of h̄ yields a marginal return rate of 1 in steady state, as savers purchase each unit of
housing and subsequently resell it for a return rate of p/p = 1. In equilibrium, savers must
be indifferent between investing in housing and lending. For this to be the case, the interest
rate on lending R must be equal to 1.
10
In summary, the conjectured housing bubble equilibrium can be characterized as follows.
Savers purchase hs > h̄ (instead of hs = h̄ as in the bubbleless case). The interest rate is
determined by savers’ no-arbitrage condition Rb = 1. Other variables are uniquely characterized by the housing allocation hs and the housing price p via budget constraints (2) and
(3). In turn, hs and p are determined by two equations: savers’ intertemporal Euler (5),
which given R = 1 is simplified to
u0 (csy ) = βu0 (cs0 ),
(10)
and borrowers’ housing first-order condition (6). We denote by hsb and pb the solution to this
system of equations. Similarly, {ciy,b , cio,b }i∈{s,d} and Ubi denote the associated consumption
profiles and lifetime utility.
Figure 2 illustrates the determination of hsb and pb . Here, the curve labeled “foc borrowers”
plots the housing first-order condition (6) of borrowers on the hs × p plane, while the curve
labeled “foc savers” plots the lending first-order condition (10) of savers. Their intersection
determines hsb and pb .
Furthermore, as in Figure 1, the dashed line in Figure 2 plots what happens when there
is an exogenous increase in es while all other parameters stay the same. The curve associated
with the Euler equation of savers shift to the right on the hs × p plane, leading to an increase
in the housing price pb and in hsb . This is because savers have more resources to invest into
the housing market.
As formalized in the following lemma, the conjectured allocations and prices characterized
above indeed constitute a housing bubble steady state if and only if es > ē:
Lemma 2 (Housing bubble steady state). The housing bubble steady state exists if and only
if Rn < 1, or equivalently es > ē. This steady state is characterized by interest rate Rb = 1,
housing allocation hs > h̄, and housing price pb , which are implicitly defined by (6) and (10).
All else equal, hsb and pb are strictly increasing in es .
Proof. Appendix A.2.
Remark 3. An interesting feature of our model that differs from the standard rational
bubbles models (e.g., Samuelson, 1958; Diamond, 1965; Tirole, 1985) is that the housing
asset is not a pure bubbly asset – defined as an asset that does not have any fundamental
value but is traded at a positive price (see Section 5.1). In our framework, the housing asset
is fundamentally useful as it yields utility to households. However, the pricing equation
(8) of the housing asset is similar to that of a pure bubbly asset in the sense that, from
the perspective of savers, each marginal unit of additional investment in the housing asset
beyond the satiation point h̄ does not yield any additional marginal utility.
11
Figure 2: Determination of housing bubble steady state’s interest rate and housing price
Remark 4. As is standard in the rational bubbles literature, the housing bubble steady
state is saddle-path stable, as shown in Appendix A.7.
Remark 5. The low interest rate condition (Rn < 1) for the existence of bubbles is related
to the dynamic inefficiency condition in Diamond (1965) and Tirole (1985). There has been
an active debate over the empirical validity of this condition. For instance, Abel et al. (1989)
argue that the U.S. (between 1929 and 1985) and six other advanced economies (between 1960
and 1985) satisfy a sufficient condition for dynamic efficiency that aggregate investment falls
short of capital income. However, using more updated data on mixed income and land rents,
Geerolf (2017) finds evidence to the contrary, providing strong support for the hypothesis
that these major economies are dynamically inefficient.8 Recent theoretical models have also
pointed out that dynamic inefficiency is not always a necessary condition for the existence
of rational bubbles if there are financial frictions (e.g., Miao and Wang, 2011; Farhi and
Tirole, 2012; Martin and Ventura, 2012) or international capital inflows (e.g., Ikeda and
Phan, 2018).
12
Figure 3: Equilibrium housing allocation, interest rate, and housing price as functions of
savers’ endowment
13
4.1
Comparative Statics
We now conduct a comparative statics exercise and compare the bubble and the bubbleless
equilibria. Throughout, we vary the endowment of savers es while keeping all other parameters fixed. Recall that an increase in es leads to an increase in the income of the top earners
(savers) relative to the bottom earners (borrowers).
Figure 3 plots the equilibrium interest rate R, housing price p, and housing allocation
s
h as functions of es .9 It also plots hn = h̄, Rn , pn (the dotted lines) and hsb , Rb , pb (the
dashed lines). From Lemma 1 and Lemma 2, we know that when es ≤ ē, the equilibrium is
bubbleless and savers only purchase housing up to the satiation level hs = h̄ (the top panel).
The interest rate and housing price are determined by Rn and pn (the middle and bottom
panels). As shown and as consistent with Lemma 1, Rn is decreasing while pn is increasing
in es .
When es > ē, we know from the lemmas that the housing bubble equilibrium arises and
replaces the bubbleless equilibrium, explaining the kinks in the functions for the equilibrium
quantity and prices. Because of the investment motive, savers purchase housing beyond the
satiation level h̄ (the top panel). The interest rate is equal to the marginal return rate of
buying and selling a unit of housing: Rb = 1 (the middle panel). Consistent with Lemma
2, the housing price pb and housing allocation hsb are increasing in es (the bottom and top
panels).
Most importantly, note that in the region es > ē, not only do savers purchase more
housing (hs > h̄), but also the equilibrium interest rate Rb is higher than Rn (the middle
panel) and the equilibrium housing price pb is higher than pn (the bottom panel). In other
words, the presence of the bubble in the housing market when es > ē raises the equilibrium
interest rate and the housing price. The following lemma summarizes this observation:
Lemma 3 (Housing bubble prices). If es > ē (so that the housing bubble equilibrium exists),
then pb > pn and Rb > Rn .
Proof. See Appendix A.3.
Intuitively, by providing an additional investment vehicle for savers, the housing bubble
raises savers’ demand for the housing asset, leading to an increase in its price. Furthermore,
8
See also Giglio et al. (2016), who find no evidence of bubbles that violate the transversality condition
in housing markets in the U.K. and Singapore. Also see Engsted et al. (2016) who provide econometric
evidence for explosive housing bubbles in OECD countries.
9
To be consistent with our analysis, the minimum value of es in the figure is set to be sufficiently large
(relative to ed , which was normalized to one) so that borrowers are credit constrained and they cannot
achieve the satiation housing level h̄ in equilibrium.
14
the investment by savers in the bubbly housing market crowds out their lending in the credit
market, raising the interest rate.
4.2
Welfare Analysis
We can now establish the main result of the paper that the presence of the housing bubble
has regressive welfare effects and exacerbates welfare inequality. Recall that Uni , i ∈ {s, d},
denotes the lifetime utility of households from bubbleless consumption profile {ciy,n , cio,n } and
housing allocation hs = 1 − hd = h̄ derived in Section 3.10 Similarly, Ubi denotes the lifetime
utility from consumption profile {ciy,b , cio,b } and housing allocation hs = 1 − hd = hb derived
in Section 4. We will compare Uni and Ubi .
The presence of the housing bubble has heterogeneous effects on savers and borrowers.
For savers, who want to save for old age, the housing bubble improves their welfare by
improving the return from investing in the housing asset. In contrast, the housing bubble
has a negative effect on the welfare of borrowers. This is because it increases the price of
housing, hence reducing the amount of housing that borrowers purchase and consequently
their housing utility. The following proposition summarizes these regressive welfare effects
of the bubble and the main result of our paper:
Proposition 4 (Housing bubble benefits savers but not borrowers). If es > ē (so that the
housing bubble equilibrium exists), then Ubs > Uns but Ubd < Und .
Proof. Appendix A.4
Figure 4 illustrates the welfare effects of the housing bubble. It plots Ubi and Uni as
functions of es . When es > ē, the housing bubble arises and it raises the welfare of savers
(reflected by the fact that the solid line representing savers’ lifetime utility Ubs lies above the
dotted line representing Uns ) but reduces the welfare of borrowers (reflected by the fact that
the solid line representing borrowers’ lifetime utility Ubd lies below the dotted line representing
Und ).
Furthermore, the figure also illustrates that the housing bubble exacerbates the welfare
inequality in the economy. In the low-inequality region es ≤ ē, a higher value of es is
(obviously) associated with a higher welfare level for savers in equilibrium, as reflected by
an upward sloping solid curve representing Uns . However, in this region, in the absence of
speculative demand for the housing asset, an increase in es does not lead to a higher demand
Also recall that when es > ē, even though the bubbleless equilibrium no longer exists, Uni can still
be thought of as the lifetime utility in the constrained economy where households cannot use housing for
speculation.
10
15
Figure 4: Comparative statics on welfare
for housing, nor a higher housing price. Thus, borrowers are not affected, as reflected by a
flat solid line representing Und .
In the high-inequality region es > ē, the equilibrium features a housing bubble and the
equilibrium lifetime utility switch from Uni to Ubi (represented by the kinks at ē in the solid
curves). In this region, an increase in es raises savers’ speculative demand for and the price
of the housing asset, crowding out the allocation of housing to borrowers. Hence, borrowers’
lifetime utility decreases, as reflected by the downward sloping portion of the solid curve
representing Ubd . Therefore, an interesting implication arises on the feedback loop between
inequality and bubble: high income inequality facilitates the existence of a housing bubble,
which in turn has regressive welfare effects and exacerbates welfare inequality.
5
5.1
Discussions
Comparison with a pure bubble
To appreciate the welfare results established in the previous section, we compare them against
the welfare effects of a pure bubble, which is an asset that pays no dividend but has a positive
16
market price and which has been analyzed extensively in the literature (e.g., Samuelson, 1958;
Diamond, 1965; Sargent and Wallace, 1982; Tirole, 1985). The pure bubble asset can be
useful as a savings instrument, however, unlike housing, the pure bubble asset does not give
households any direct utility. As a consequence, there will be an endogenous segmentation
of the pure bubble market, as only savers purchase the asset as an investment vehicle.
Consequently, the bubble will have much less effect on borrowers, as we show below.
5.1.1
Housing asset does not yield any utility
First, let us consider the benchmark where households do not derive utility from housing,
that is, v(h) ≡ 0. Then the housing asset provides no fundamental value. The model is then
similar to the models of pure rational bubbles (e.g., fiat money or unbacked government
debt) of Samuelson (1958), Diamond (1965), and Sargent and Wallace (1982).
As is standard and well known in this environment, there is always a bubbleless equilibrium where the housing asset is not traded. The lifetime utility in this equilibrium is given
by Uns = u(es ) + βu(e) and Und = u(ed ) + βu(e). As usual, the bubbleless interest rate is
u0 (cs )
u0 (es )
determined by the Euler equation of savers: Rn = βu0 (cys ) = βu
0 (e) .
o
Furthermore, when Rn < 1, the economy is dynamically inefficient and there exists
another bubble equilibrium, where the housing asset is traded at a positive price. The key
difference compared to the equilibrium in Section 4 is that, because here households do not
derive utility from housing, borrowers will not have any incentive to purchase the housing
asset. Thus, the equilibrium features an endogenous segmentation in the housing market:
hd = 0 and hs = 1. As investing in the bubble allows savers to transfer more resources from
young age to old age, it improves their welfare: savers’ lifetime utility with the bubble is
Ubs = u(es − pb ) + βu(e + pb ) > Uns , where pb is the bubble equilibrium price of the housing
asset. As borrowers do not participate in the housing market, the rise in housing price
due to the bubble does not affect them: borrowers’ lifetime utility with the bubble is also
Ubd = u(ed ) + βu(e) = Und .
In summary, when households do not derive utility from housing (v ≡ 0), the negative
effect on borrowers of the increase in the price of housing due to the bubble is absent. The
endogenous segmentation of the housing market effectively shields borrowers from the effect
of the housing price bubble.
5.1.2
Pure bubble as another asset
In this subsection, we will show that the insight from the previous subsection also carries
through to an alternative environment where households still derive utility from housing
17
(i.e., v satisfies the same properties as in Section 2) but an additional pure bubble asset is
introduced.
For clarity, in this section we focus on the constrained economy where households face a
constraint hi ≤ h̄ for i ∈ {s, d}, which prevents them from using units of the housing asset
purely as an investment vehicle. This assumption effectively prevents the possibility of a
housing bubble coexisting with a pure bubble in equilibrium and thus allows us to clearly
distinguish between the housing bubble equilibrium and the pure bubble equilibrium to be
derived below.
Formally, assume there is an asset in fixed unit supply that pays no dividend but is traded
at price bt per unit. Given prices, each household of type i chooses its holding xit ≥ 0 of the
bubble asset. Their optimization problem is:
max
hit ,cit,y ,cit+1,o ,xit ,ait
U (hit , cit,y , cit+1,o ),
(11)
subject to budget constraints:
pt hit + bt xit + ait
1
+ ciy,t = ei ,
Rt
cio,t+1 = pt+1 hit + bt+1 xit + ait + e,
(12)
(13)
the nonnegativity constraints:
xit , hit , cit,y , cit+1,o ≥ 0,
the credit constraint (4), and the constraint on housing investment:
hit ≤ h̄.
To close the model, assume that old savers own the entire supply of housing and the bubble
in the initial period t = 0.
The definition of a pure bubble equilibrium is similar to Definition 1, except that we have
an additional condition that the pure bubble price is positive:
bt > 0, ∀t ≥ 0,
and an additional market clearing condition of the bubble asset:
xst + xdt = 1, ∀t ≥ 0.
As before, we focus on steady-state equilibria where the quantities and prices are time18
invariant.
Existence and Characteristics. Similar to the housing bubble case, savers in equilibrium
must be indifferent between investing in the pure bubble asset and lending. The former
yields a return of b/b = 1 in steady state, and the latter yields R. Hence, the interest rate
in the pure bubble steady state must also be R = 1 as in the housing bubble case. It is also
straightforward to show, as in Lemma 2, that the pure bubble exists if and only if Rn < 1,
or equivalently, es > ē (see Appendix A.5). Furthermore, given es > ē, it is straightforward
to show that the constraint hs ≤ h̄ binds for savers, leading to allocation hs = 1 − hd = h̄.
Given that borrowers are credit constrained, and that the return from the pure bubble
asset is equal to the interest rate, it is not optimal for borrowers to invest in the pure bubble
asset. Thus, the equilibrium will feature an endogenous segmentation of the market into
participants (savers) and nonparticipants (borrowers), i.e., xs = 1 and xb = 0.
Given hs = h̄ and xs = 1, other variables in the conjectured equilibrium are uniquely
characterized by two key variables: the price of housing p and the price of the pure bubble
asset b. From budget constraints (12) and (13), the consumption profile ciy , cio are functions
of p and b. In turn, p and b are determined by the first-order conditions (6) and (10). The
following lemma summarizes our discussion above. It also shows that the price of housing in
the pure bubble equilibrium is the same as the price of housing in the bubbleless equilibrium:
Lemma 5 (Pure bubble equilibrium). The pure bubble steady state exists if and only if
es > ē. The steady state is characterized by interest rate R = 1, housing allocation hs = h̄,
pure bubble asset allocation xs = 1, along with housing price p = pn and pure bubble price b
that is implicitly determined by the Euler equation of savers:
u0 (es − pn h̄ − b) = βu0 (e + pn h̄ + b)
(14)
Proof. Appendix A.5.
Welfare Analysis. Assume es > ē so that the pure bubble equilibrium exists. Are the welfare
implications of a pure bubble different from those of a housing bubble? Let Uxi denote the
lifetime utility in the pure bubble equilibrium.
Like the housing bubble, the pure bubble allows savers to store their income into old
age more efficiently and hence improves their welfare relative to the bubbleless case, i.e.,
Uxs > Uns .
However, as the pure bubble market absorbs savers’ demand for storage, savers no longer
need to use the housing asset for an investment purpose. Thus, the pure bubble does not
affect the housing price (as stated in Lemma 5). As a consequence, it does not affect
19
the welfare of borrowers, i.e., Uxd = Und . In summary, the negative externality of savers’
investment in the housing market is absent in the pure bubble equilibrium.
Furthermore, we can compare welfare across the pure bubble equilibrium and the housing
bubble equilibrium. On the one hand, because the negative externality on borrowers is absent
in the pure bubble case, it is straightforward to see that borrowers are better off in this case:
Uxd > Ubd . On the other hand, because the pure bubble is similar to the housing bubble in
providing storage for savers, it can be shown that savers get the same welfare in the two
equilibria: Uxs = Ubd . The intuition above is summarized by the following result:
Corollary 6. The pure bubble steady state Pareto-dominates the housing bubble steady state:
Uxd > Ubd and Uxs = Ubs .
Proof. Appendix A.6
5.2
Role of the credit constraint
So far, we have assumed that the credit limit is d¯ = 0. However, our main result (in
particular Proposition 4 of the regressive welfare effects) carries through if d¯ > 0, as long
as we continue to assume es is sufficiently large so that the credit constraint (4) binds for
borrowers in equilibrium. In fact, the negative effect of the housing bubble on borrowers will
be strengthened when d¯ > 0. This is because the bubble exerts an additional negative effect
on borrowers through the credit market. To see this, recall that the budget constraint of
credit-constrained young borrowers is given by
cdy + phd = ed +
d¯
.
R
Since the bubble raises the interest rate R (relative to the bubbleless rate), it reduces the
¯
total resources available for young borrowers ed + Rd and thus reduces their ability to consume
and purchase housing. In short, the housing bubble not only raises the cost of housing, but
also raises the cost of borrowing for credit-constrained borrowers. On the flip side, by raising
the interest rate on lending, the bubble provides an additional benefit to savers who want
save for old age. Thus, the presence of a positive binding credit constraint d¯ amplifies the
regressive welfare effect of the housing bubble. Appendix A.8 provides more details of the
analysis.
5.3
Policy discussions and rental market
Given the externality of savers’ demand in housing for investment purposes, policy interventions may be warranted. In fact, recent policy debates in China and other emerging
20
economies (e.g., Bloomberg, 2016; Nikkei Asian Review, 2017; The Economist, 2018), can
be discussed within the context of our model.11 The well-documented shortage of assets
for storage of wealth, such as safe government bonds, in these economies could be a reason
why bubbles tend to arise in the real estate markets. A restriction on speculative investment in housing could prevent housing bubbles but would entail trade-offs, because it would
help low-income households but hurt high-income households, as highlighted in our welfare
analysis.
Furthermore, we have so far abstracted away from the presence of a rental market. In
Appendix A.9, we extend the model to relax this assumption. We show that if the rental
market is frictionless, then borrowers can rent from savers, reducing the demand for borrowing and thus raising the interest rates. As a consequence, the housing bubble may not arise.
However, the housing bubble will arise again just as in the benchmark model if there are
sufficient frictions in the rental market. In practice, many problems, including the presence
of moral hazard and adverse selection, are prevalent in the rental markets. It has been documented that such markets are quite underdeveloped in emerging economies, including even
in major cities in China (The Economist, 2018). An implication of the model is that the
development of a functioning rental market, as discussed in recent proposals by the Chinese
government to develop a market for good-quality rental housing, could help mitigate the
negative externality of the housing bubble on low-income households.
6
Concluding remarks
We have shown that a housing bubble, or, more generally, a bubble attached to a fundamentally useful asset, has heterogeneous welfare effects on households, depending on their
demand for savings and borrowing. By providing an additional investment vehicle, it raises
the returns from investment for savers and thus improves their welfare. However, by raising
the interest rate on debt and raising the housing price, the housing bubble negatively affects
the welfare of borrowers, who need debt to finance their purchase of housing. Our model also
implies a feedback loop on inequality: high income inequality leads to an environment with
low interest rates, which facilitate housing bubbles, which in turn have regressive welfare
effects. The key insight is that savers’ demand for the housing asset for investment purposes
imposes a negative externality on borrowers, who only demand the housing asset for utility
purposes, by creating a bubble in the market price.
11
As an anecdote, the Chinese leader Xi Jinping said in his address at the 19th Party Congress in Beijing
that “Houses are built to be inhabited, not for speculation” (Bloomberg, 2017).
21
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24
A
A.1
Appendix
Proof of Lemma 1
Recall that Rn and pn solve the two first-order conditions (5) and (6) with hs = h̄, which can be
rewritten as:
u0 (es − pn h̄) = βRn u0 (e + pn h̄)
| {z }
| {z }
csy,n
(15)
cso,n
pn u0 (ed − pn (1 − h̄)) = v 0 (1 − h̄) + βpn u0 (e + pn (1 − h̄)).
|
|
{z
}
{z
}
cdy,n
(16)
cdo,n
Equation (16) uniquely determines pn . To see this, define the difference between the two sides
of (16) as:
∆(p) ≡ pn u0 ed − p(1 − h̄) − v 0 (1 − h̄) − βpu0 e + p(1 − h̄) .
Because u is strictly concave, this function is strictly increasing in p. Also, ∆(0) < 0. Furthermore,
since limc→0+ u0 (c) = ∞, it follows that limp→ed /(1−h̄)− ∆(p) = ∞. Hence, by continuity, there
exists a unique pn ∈ (0, ed /(1 − h̄)) such that ∆(pn ) = 0. Then, given pn , equation (15) uniquely
u0 (es −pn h̄)
determines Rn = βu0 e+p h̄ .
( n )
We now show that all else equal, Rn is increasing in es . Since an increase in es does not affect
equation (16), it follows that pn is not affected by a change in es . As a consequence, an increase in
u0 (es −pn hs )
es lowers Rn = βu
0 (e+p hs ) .
n
Finally, we show that the bubbleless steady state exists if and only if es ≤ ē. First, suppose
on the contrary that a bubbleless steady state exists but es > ē. Because the interest rate Rn is
decreasing in es , by the definition of ē, it follows that Rn < 1. However, this would imply that a
saver would prefer to unilaterally deviate from purchasing hs = h̄ to purchasing hs > h̄, because
investing in additional housing yields a marginal return rate of 1 that dominates the return Rn from
the credit market. Thus, the existence of a bubbleless steady state requires es ≤ ē.
Second, suppose es ≤ ē. Then Rn ≥ 1. Thus, no saver would have an incentive to unilaterally
deviate from purchasing hs = h̄ to purchasing hs > h̄, because investing in additional housing yields
a marginal return rate of 1 that is weakly dominated by the return Rn from the credit market. It
then straightforwardly follows that the allocations and prices associated with hs = h̄, Rn , and pn
satisfy the market clearing and individual optimization problems and thus constitute a bubbleless
steady state.
A.2
Proof of Lemma 2
Recall that hsb and pb solve (10) and (6), which can be rewritten respectively as:
u0 (es − phs ) = βu0 (e + phs )
| {z }
| {z }
csy,b
(17)
cso,b
pu0 (ed − p(1 − hs )) = v 0 (1 − hs ) + βpu0 (e + p(1 − hs )).
|
{z
}
|
{z
}
cdy,b
(18)
cdo,b
Using similar arguments to those used in Section A.1, it is straightforward to show that hsb and pb
are unique.
25
Now, we show hsb and pb are increasing in es . Consider two arbitrary endowment levels es1 > es2 ,
with the corresponding solutions (hsb,1 , pb,1 ) and (hsb,2 , pb,2 ). We need to show that hsb,1 > hsb,2 and
pb,1 > pb,2 .
Because of the concavity of u, equation (17) determines phs as an increasing function of es ,
implying pb,1 hsb,1 > pb,2 hsb,2 . Equation (18) can be further rewritten as:
h
i
p u0 ed − p + phs ) − βu0 (e + p − phs )) = v 0 (1 − hs ).
Thus,
h
i
pb,1 u0 ed − pb,1 + pb,1 hsb,1 ) − βu0 e + pb,1 − pb,1 hsb,1 )
v 0 (1 − hsb,1 )
h
i = 0
.
v (1 − hsb,2 )
pb,2 u0 ed − pb,2 + pb,2 hsb,2 ) − βu0 e + pb,2 − pb,2 hsb,2 )
(19)
Suppose on the contrary that pb,1 ≤ pb,2 . Then, because pb,1 hsb,1 > pb,2 hsb,2 , it must be that hsb,1 >
hsb,2 . Then the left-hand side of (19) is smaller than 1, while the right-hand side is larger than 1,
leading to a contradiction. Therefore, pb,1 > pb,2 . Similarly, hsb,1 > hsb,2 . In other words, both hsb
and pb as implicitly determined by (17) and (18).
Now we show that the housing bubble steady state exists if and only if es > ē. By definition, the
conjectured allocations and prices constitute a housing bubble steady state if and only if hsb > h̄.
Note that when es = ē, the systems (15-16) and (17-18) yield the same solution: Rn = Rb = 1,
hsn = hsb = h̄, and pn = pb . Furthermore, we already proved that hsb is increasing in es . Hence,
hsb > h̄ if and only if es > ē. This completes the proof.
A.3
Proof of Lemma 3
The fact that if es > ē, then Rb > Rn immediately follows from Lemma 1 and Section A.1. From
Lemma 2 and the proof in Section A.2, we know that pb is increasing in es , and that pb = pn when
es = ē. Hence, it immediately follows that if es > ē, then pb > pn .
A.4
Proof of Proposition 4
We start by showing that Ubs > Uns . Recall that
Ubs = u(es − pb hsb ) + βu(e + pb hsb ) + v(h̄),
| {z }
| {z }
csy,b
cso,b
where the three terms on the right-hand side correspond to the utility over consumption in young
age, the discounted utility over consumption in old age, and the utility over (satiated) housing
consumption, respectively. Similarly, recall that
Uns = u(es − pn h̄) + βu(e + pn h̄) +v(h̄).
| {z }
| {z }
csy,n
cso,n
Thus, after canceling out the v(h̄) term (as savers satiate their housing utility in both steady-state
allocations), it suffices to show that
u(es − pb hsb ) + βu(e + pb hsb ) > u(es − pn h̄) + βu(e + pn h̄),
or equivalently
F (pb hsb ) > F (pn h̄),
26
(20)
where F (x) ≡ u(es − x) + βu(e + x). Note that the solution to maxx F (x) is the solution to the
first-order condition u0 (es − x) = βu0 (e + x). From (17), it follows that x = pb hsb solves maxx F (x).
From (15) and the fact that Rn < 1, it follows that x = pn h̄ does not solve maxx F (x). Therefore,
(20) automatically follows. This completes the proof of Ubs > Uns .
Now we show Und > Ubd . Recall that
Ubd = u(ed − pb hdb ) + βu(e + pb hdb ) + v(hdb ),
| {z }
| {z }
cdy,b
cdo,b
and
Und = u(ed − pn hdn ) + βu(e + pn hdn ) + v(hdn ),
| {z }
| {z }
cdy,n
cdo,n
where hdb = 1 − hsb and hdn = 1 − h̄ < h̄. Also recall that hdb < hdn . Hence, under the bubbleless
steady-state prices, the housing allocation hdb (along with the net asset position adb = 0) is feasible for
borrowers. Thus, by the definition of hdn and adn = 0 as the solution to the optimization problem of
borrowers given the bubbleless steady-state prices, it follows that borrowers must be at least better
off with the bubbleless steady-state allocations than with the bubble steady-state allocations:
Und ≥ u(ed − pn hdb ) + βu(e + pn hdb ) + v(hdb ).
(21)
Furthermore, combining the fact that pn hdb < pb hdb (because the housing price is higher in the
housing bubble steady state) and the fact that u0 (ed − pb hdb ) > βu0 (e + pb hdb ) (borrowers in the
housing bubble steady state are constrained in their ability to tilt consumption forward) yields
u(ed − pn hdb ) + βu(e + pn hdb ) > u(ed − pb hdb ) + βu(e + pb hdb ).
(22)
Inequalities (21) and (22) then imply Und > Ubd , as desired.
A.5
Proof of Lemma 5
It is straightforward that given the conjectured prices, the conjectured allocations solve the optimization problems of individual borrowers and savers. Thus, by definition, to verify whether the
allocations and prices specified in the lemma constitute a pure bubble steady state, it is equivalent
to verify that b > 0 (the pure bubble has a positive price). Recall that b solves equation (14).
Because u is strictly concave, it follows from this equation that b > 0 if and only if
u0 (es − pn h̄) < βu0 (e + pn h̄).
Because of equation (15), this inequality is equivalent to
Rn < 1,
as desired.
A.6
Proof of Corollary 6
Recall from Proposition 4 that Und > Ubd . Furthermore, from Lemmas 1 and 5, we know that
the bubbleless and pure bubble steady states yield the same allocation to borrowers, and hence
Uxd = Und . Thus, it immediately follows that Uxd > Ubd .
27
It remains to show that Uxs = Ubs . Since the interest rate is R = 1 in both the housing bubble
and the pure bubble steady states, the Euler equation for savers in both steady states is
u0 (csy ) = βu0 (cso ).
Furthermore, in both cases, we have
csy + cso = es + e.
In other words, both (csy,b , cso,b ) and (csy,x , cso,x ) are solutions to the system of two equations above.
As u is strictly concave, the system has only one solution. Thus the two consumption allocations are
the same. Furthermore, savers achieve the same housing utility v(h̄) in both steady states. Hence,
Uxs = Ubs , as desired.
A.7
Stability of the housing bubble steady state
The housing bubble steady state values (pb , hb , Rb ) constitute the fixed point of the following dynamic system, which can be summarized by two difference equations in the price of bubble pt and
the allocation of housing to savers hst :
v 0 (1 − hst )
u0 (e + pt+1 (1 − hst ))
+
β
pt+1
s
u0 (ed − pt (1 − ht ))
u0 (ed − pt (1 − hst ))
u0 (e + pt+1 hst )
pt+1 ,
pt = β 0 s
u (e − pt hst )
pt =
(focD)
(focS)
along with the following Euler equation that determines the interest rate Rt :
Rt =
u0 (es − pt hst )
.
βu0 (e + pt+1 hst )
Equations (focD) and (focS) are simply the first-order conditions of borrowers and savers with
respect to housing, where we have substituted the interest rate Rt by the previous Euler equation. Given the state variable pt , these two equations implicitly define hst and pt+1 , denoted
by H(pt ) and P(pt ), respectively. From hst and pt+1 , we also obtain Rt as a function R(pt )
of pt . Let zt := (pt , hst , Rt ). Then these functions define a dynamical system zt+1 = Φ(zt ) ≡
(P(pt ), H(P(pt )), R(P(pt ))). It can be verified from the assumptions on the utility functions that
P and R are increasing functions of pt . Figure 5 illustrates these two equations. Fixing pt at an
arbitrary value, the blue and red solid lines plot the curves corresponding to equations (focD) and
(focS), respectively, which determine pt+1 and hst for a given pt . The dashed lines plot how the
curves shift when the state variable pt increases.
We now show that the housing bubble steady state is saddle-path stable. Suppose p0 > pb .
Then, because of the monotonicity of R, we have R0 = R(p0 ) > R(pb ) = 1. It then follows that
p1 = p0 R0 > p0 . Hence p1 > p0 > pb . By recursion, we get pt > · · · > p1 > p0 > pb , for all t > 0.
Hence, {pt } does not converge to pb . Similarly, suppose p0 < pb . Then pt < · · · < p1 < p0 < pb ,
for all t > 0 and {pt } again does not converge to pb . Only when p0 = pb does the system converge
to the housing bubble steady state. Hence, the housing bubble steady state is saddle-path stable.
This saddle-path stability is standard in the rational bubbles literature (e.g., Tirole, 1985).
28
Figure 5: Illustrations of (focD) and (focS).
A.8
Derivation for Section 5.2
With a strictly positive debt limit d¯ that is binding for borrowers, the bubbleless steady-state
equations (15) and (16) that determine pn and Rn are instead given by:
d¯
0
s
u e −
− pn h̄
= βRn u0 e + d¯ + pn h̄
Rn
¯
d
0
d
− pn (1 − h̄)
= v 0 (1 − h̄) + βpu0 e − d¯ + pn (1 − h̄) .
pn u e +
Rn
Figure 1 is replaced by a similar Figure 6, where the horizontal line representing the first-order
condition for borrowers in the former figure is replaced by the downward sloping line in the latter.
It is straightforward to show that the comparative statics result that Rn is decreasing in es continues
to hold.
Similarly, the housing bubble steady-state equations (17) and (18) that determine hsb and pb are
instead given by:
d¯
0
s
s
u e −
− pb h
= βu0 e + d¯ + pb hsb
Rb
¯
d
0
d
s
pb u e +
− p(1 − hb )
= v 0 (1 − hsb ) + βpu0 e − d¯ + pb (1 − hsb ) .
Rb
It is straightforward to extend the proof of Lemma 2 to show that hsb is increasing in es , and thus
hsb > h̄ if and only if es > ē, and we skip the details for brevity. It follows that a housing bubble
steady state exists if and only if es > ē, or equivalently Rn < 1, as in Lemma 2. Figure 2 also does
not qualitatively change with d¯ > 0.
We now show Ubd < Und (i.e., the housing bubble reduces lifetime utility of borrowers). Since
hdb = 1 − hsb < 1 − h̄ = hdn , it follows that under the bubbleless steady-state prices, the housing
¯ is feasible for borrowers. Thus,
allocation hdb (along with the net asset position adb = adn = −d)
d
d
by the definition of hn and an as the solution to the optimization problem of borrowers given
the bubbleless steady-state prices, it follows that borrowers must be at least better off with the
bubbleless steady-state allocations:
¯ n − pn hd ) + βu(e − d¯ + pn hd ) + v(hd ).
Und ≥ u(ed + dR
b
b
b
(23)
Furthermore, combining the fact that not only pn hdb < pb hdb (because the housing price is higher
¯
¯
in the housing bubble steady state), but also Rdn < Rdb (because the interest rate is higher in the
29
Figure 6: Determination of bubbleless steady state’s interest rate and housing price with
d¯ > 0.
housing bubble steady state, negatively affecting the ability of borrowers to raise debt) and the fact
¯ b − pb hd ) > βu0 (e − d¯ + pb hd ) (borrowers in the housing bubble steady state are
that u0 (ed + dR
b
b
constrained in their ability to tilt consumption forward) yields
u(ed +
d¯
d¯
− pn hdb ) + βu(e − d¯ + pn hdb ) > u(ed +
− pb hdb ) + βu(e − d¯ + pb hdb ).
Rn
Rb
(24)
Inequalities (23) and (24) then imply Und > Ubd , as desired.
Similarly, we can show Ubs > Uns (that the housing bubble benefits savers).
A.9
Extension with rental market
Frictionless rental market
In this extension, we introduce a perfectly competitive and frictionless rental market to the model
in the main text. Let hi ≥ 0 continue to denote a household of type i’s purchase of the housing
asset, and let ĥi denote the household’s net rental housing, where a positive position means the
household is a renter and a negative position means the household is a landlord. The purchasing
and renting prices of housing are denoted by p and p̂, respectively.
Taking (steady-state) prices as given, the optimization problem of a representative household
¯ to nonnegative
of type i is to maximize U (ciy , cio , hi + ĥi ) subject to the credit constraint ai ≥ −d,
constraints ciy , cio , hi , hi + ĥi ≥ 0, and to the following budget constraints:
phi + p̂ĥi +
ai
+ ciy = eiy
R
cio = phi + as + e.
30
In an equilibrium with the competitive rental market, the rental market must clear: ĥs + ĥd = 0. We
again focus on the parameter region where es is sufficiently high such that in equilibrium, borrowers
will be credit constrained.
The first-order condition with respect to rental housing of a representative borrower yields
p̂ = v 0 (hd + ĥd )/u0 (cdy ).
(25)
Intuitively, the price of housing must be equal to the marginal rate of substitution between housing
services and consumption for borrowers.
As in the main model, the first-order conditions with respect to borrowing/lending of the unconstrained savers and the constrained borrowers yield:
1
βu0 (cs )
βu0 (cd )
= 0 so > 0 do .
R
u (cy )
u (cy )
(26)
Furthermore, by letting hsc ≡ hs + ĥs denote the consumption of housing services of a representative saver (the amount of housing that enters the utility function), we can rewrite the optimization
problem of savers as maximizing U (csy , cso , hsc ) subject to the following budget constraints
phs +
as
+ csy = esy + p̂(hs − hsc )
R
cso = phs + as + e,
and the nonnegativity constraints csy , cso , hs , hsc ≥ 0. The first-order condition with respect to hs of
this problem then yields:
(p − p̂)u0 (csy ) = βpu0 (cso ),
or equivalently:
p
.
(27)
R
Intuitively, the price of housing is equal to the dividend, measured by the rental price p̂, plus the
resale value Rp . In summary, the equilibrium prices p, p̂, and R must satisfy (25), (26), and (27).
Equations (25), (26), and (27) imply two results. First, the fact that the credit constraint binds
for borrowers implies that borrowers do not buy housing and are net renters. To see this, let λd be
the Lagrange multiplier associated with the nonnegativity constraint hd ≥ 0. Then the first-order
condition with respect to hd yields:
p = p̂ +
p =
v 0 (hd + ĥd )
βu0 (cdo )
+
p
+ λd
u0 (cdy )
u0 (cdy )
= p̂ + p
βu0 (cdo )
+ λd .
u0 (cdy )
Combined with the inequality in (26) due to the binding credit constraint, we get:
p < p̂ +
p
+ λd .
R
Combined with asset pricing equation (27), we get
0 < λd ,
31
i.e., the constraint hd ≥ 0 binds. Thus, in equilibrium hd = 0 and hs = 1. Because v satisfies the
Inada conditions, it follow that dˆ > 0, i.e., borrowers will rent in equilibrium.
Second, since p̂ and p must be positive in any equilibrium with the rental market, equation (27)
implies that R > 1. Therefore, the presence of the rental market effectively rules out the possibility
of bubbles, including housing bubbles. In fact, equation (27) implies that in equilibrium the price
of housing is simply equal to the present value of rental dividends:
p=
p̂
1−
1
R
,
i.e., the housing price is equal to the fundamental value of housing, and the bubble component is
necessarily zero.
Frictional rental market
The previous subsection assumes a frictionless rental market. However, in practice, rental markets
tend to have many frictions, especially in developing economies. In this subsection, we show a
simple way to further extend the model to capture, in a reduced-form, rental market frictions.
Specifically, assume that there is a transaction (or maintenance) cost Ψ associated with renting,
and the budget constraint of young households would then become:
phi + 1 + Ψ(ĥi ) p̂ĥi + ciy = eiy ,
where for simplicity we assume Ψ(h) ≡ ψ2 h2 , with ψ ≥ 0 being an exogenous constant. Also assume
for simplicity that the maintenance cost is a dead-weight loss to the economy (i.e., the cost is not
paid to anyone).
When ψ = 0, the model collapses to the frictionless rental market in the previous subsection.
Obviously, when ψ → ∞, the cost of renting is too high and the model collapses to the main model
with no rental market in the main text. However, we can find a tighter upper bound for ψ. If there
were an equilibrium with rental market clearing, then from households’ first-order conditions, the
asset pricing equation of housing would become:
p = (1 − ψ ĥd )p̂ +
p
.
R
(28)
For the equilibrium price p to be positive, it is necessary that the dividend term (1 − ψ ĥd )p̂ in
equation (28) above is positive. It is then straightforward to show that for ψ sufficiently large (a
sufficient condition is ψ > 1/(1 − h̄)), an equilibrium with a rental market does not exist.
In summary, if the rental market is sufficiently frictional, then the equilibrium with rental
described in the previous subsection breaks down, and we are back to the model without rental in
the main text.
32
@FedFRASER