Erdem Basci

Ismail Saglam

English

Research Papers in Economics

 
 
On Roots of Housing Bubbles 
 
 
 
 
Erdem Başçı and Ismail Saglam 
 
 
 
 
 
 
 
 
Department of Economics 
TOBB University of Economics and Technology 
 
 
 
 
 
Working Paper No: 08-01 
TOBB University of Economics and Technology 
Department of Economics 
 
 
 
 
 
 
 
 
June 2008 
 
 
 
 TOBB University of Economics and Technology Department of Economics Working Papers are published without a formal 
refereeing process, with the sole purpose of generating discussion and feedback.  
 On roots of housing bubbles∗
Erdem Ba¸ sçıa , Ismail Saglamb,†
a Central Bank of the Republic of Turkey, Istiklal Cad. 10, Ulus, 06100 Ankara, Turkey
b Department of Economics, TOBB University of Economics and Technology, Sogutozu Cad. 43,
Sogutozu, 06560 Ankara, Turkey
Abstract
This paper is about instability of equilibrium real rents in a pure exchange economy. Our
main result hinges upon a low degree of substitutability between housing services and other
commodities together with a low share of owner-occupied houses. We argue that regions or
economies with a low share of owner-occupied houses are more prone to housing bubbles that
are driven by locally unstable rent dynamics.
Keywords: Housing bubbles; rents; stability of equilibrium; CES utility; Leontief utility
JEL classiﬁcation: C62; D11; D51; R21; R31
1 Introduction
Although housing constitutes the biggest asset class, there is a surprisingly low num-
ber of general equilibrium asset pricing models that pay speciﬁc attention to housing.1
Apart from being an important asset class, housing diﬀers from many other commodi-
ties as its price indices are observed to exhibit boom and bust episodes. Although the
term ‘bubble’ has frequently been used to describe such phenomena, there are no exist-
ing theories that approach the issue from the perspective of (in)stability of competitive
equilibrium. This paper is a theoretical attempt in that direction.
As for housing services, we focus only on pure dwelling services, i.e. what the rent is
paid for, apart from costs of utilities like heating, water, electricity etc. The ﬁrst natural
assumption then is that of a low degree of substitutability between housing services and
other commodities like food and energy. But then we know from the pioneering results
on the stability of general competitive equilibrium that the key assumption of ‘gross
substitutability’ (Arrow and Hurwicz, 1958; Hahn, 1958) between all commodities is
likely to be violated by the presence of housing services. This observation leaves the
∗The authors thank Yavuz Arslan, Abdullah Yava¸ s, Mehmet Yörüko˜ glu, and the seminar participants
at TOBB University of Economics and Technology and the Central Bank of the Republic of Turkey for
helpful comments. All the views expressed in this paper belong to the authors and do not represent
those of the Central Bank of the Republic of Turkey or its staﬀ.
†Corresponding author. Tel.: + 90 312 292 4212; fax: + 90 312 292 4104.
E-mail address: ismail.saglam@etu.edu.tr (I. Saglam).
1Piazzessi et al. (2007) is a notable exception.
1room open for the presence of unstable equilibria in models where housing is modelled
separately.
Evidence against a low degree of substitution is poor. Expenditure shares of broadly
deﬁned housing services are observed to move in the same direction as real relative rents
in Piazzessi et al. (2007). This indicates a low degree of substitution. Likewise, their
post-war (WWII) estimates also point to a low elasticity of substitution, although in their
paper they assume a high degree of substitution to ensure the uniqueness and stability
of their equilibrium. There are no studies to our knowledge that directly estimate the
elasticity of substitution of pure dwelling services. Ogaki and Reinhart (1998) report
however that when services (including housing services) are included, elasticity estimates
between durable and non-durable goods shrink substantially.
As far as housing prices are concerned, assumptions about the present level and future
growth prospects of rents are among the key determinants (see Smith and Smith, 2006).
That is why any satisfactory model of housing prices needs to focus on the determination
and dynamics of rental rates, which is what this paper is about.
The most frequently cited and studied regions for bubble price dynamics in the US
are regions like California, Boston, Chicago, Los Angeles County, and Manhattan (Smith
and Smith, 2006; Shiller, 1990). These share the common characteristics of being tourist
and student destinations as well as having high job turnover, together with legal or
physical restrictions on supply of houses. Similar housing price behavior is reported and
studied for countries with similar characteristics like Ireland and UK (Kenny, 1999; Black
et al., 2006). The natural consequence of these characteristics would be a high share of
houses oﬀered for rent in these regions.
Another basic reason for heterogeneity in home ownership is the life cycle. Campbell
and Cocco (2007) provide evidence that old and young people tend to be long and short
in housing, respectively. Such heterogeneity comes together with a strong income eﬀect
for home owners who have a signiﬁcant degree of rent income. Empirical work trying
to identify a long term housing demand function ﬁnds a positive price coeﬃcient in the
unrestricted regression where income and price are obviously multicollinear in case of high
endowment heterogeneity in housing (Kenny, 1999). Here, in this paper, we elaborate on
the theoretical possibility of a positively sloped aggregate demand function for housing
services when there are strong income eﬀects due to endowment heterogeneity.
The organization of the paper is as follows. Section 2 introduces the model and
Section 3 presents our results.
2 Model
We consider a simple exchange economy involving two consumers, indexed by i = 1,2.
There are two goods, indexed by g = b,h with b and h denoting bread and housing
(services), respectively. Let w
g
i denote the endowment of good g owned by consumer i.
We assume that good h is initially endowed by consumer 1, while good b by consumer 2,
solely; i.e., wb
1 = 0, wh
1 > 0, wb
2 > 0, and wh
2 = 0. Clearly, the aggregate supply of good
2b is wb
2 and that of good h is wh
1.
Let c
g
i denote the consumption of good g by consumer i, who has the utility function
of the CES form
Ui(c
b
i,c
h
i ) = [(c
b
i)
1−1/￿i + (αic
h
i )
1−1/￿i]
1/(1−1/￿i),
where ￿i ∈ [0,1) is the elasticity of substitution and αi > 0.
We normalize the price of good b to 1, and denote by r the relative real rental rate
of good h. The choice problem of consumer i is given by
max
cb
i,ch
i
Ui(c
b
i,c
h
i )
s.t. c
b
i + rc
h
i = w
b
i + rw
h
i .
We denote the solution by ￿cb
i(r),ch
i (r)￿ for a given rental rate r.
Deﬁnition 1. Consumers have endowment bias if cb
1(r)/ch
1(r) < wb
2/wh
1 < cb
2(r)/ch
2(r)
for any positive r.
Consumers with endowment bias plan to consume relatively more from the endowed
good at each positive rental rate.2
3 Results
3.1 The case of no substitution (￿i = 0)
When there is no substitution between bread and housing for the consumers, the
utility functions become a Leontief function
Ui(c
b
i,c
h
i ) = min{c
b
i,αic
h
i}
for each i.
It is clear that an interior solution must satisfy
c
b
i(r) = αic
h
i (r) (1)
for consumer i.
When r = 0, we have corner solutions, requiring ch
1(0) ≥ 0 and ch
2(0) ≥ wb
2/α2.
Using the budget constraints, we ﬁnd the quantity of good h demanded by consumers
1 and 2 as
c
h
1(r) ∈



{ r
α1+rwh
1} if r > 0
[0,∞) if r = 0
(2)
2Endowment bias is used as in the theory of ﬁnance, in the sense of attributing a higher value to an
asset due to one’s ownership of it (Kahneman et al., 1991). An alternative interpretation of endowment
bias in our model would be the presence of transaction costs in trade.
3and
c
h
2(r) ∈



{ 1
α2+rwb
2} if r > 0
[wb
2/α2,∞) if r = 0
(3)
respectively. The aggregate demand for good h at r ≥ 0 becomes
D
h(r) = c
h
1(r) + c
h
2(r) ∈



{ r
α1+rwh
1 + 1
α2+rwb
2} if r > 0
[wb
2/α2,∞) if r = 0.
(4)
To characterize the equilibrium of the economy, we restrict ourselves, by Walras Law,
to the clearing of the housing market. The aggregate supply of good h is wh
1 whereas the
aggregate demand for it is Dh(r) at the rental rate r. Equating the two, we solve for the
equilibrium rent.
We immediately observe that r = 0 supports equilibrium if and only if wb
2/α2 ≤ wh
1.
Apparently, this is the case if consumers have endowment bias, since with Leontief utility
functions condition in Deﬁnition 1 reduces to α1 < wb
2/wh
1 < α2.
Now, equating Dh(re) and wh
1 over re > 0, we get
r
e = α1
α2wh
1 − wb
2
wb
2 − α1wh
1
. (5)
Proposition 1. Equilibrium with a positive rental rate (i) exists if consumers have en-
dowment bias, and (ii) is always unique, whenever it exists.3
Proof. Part (i) follows from (5) and Deﬁnition 1, while (ii) is obvious from (5).
Corollary 1. Assume consumers have endowment bias. Then the positive equilibrium
rental rate is increasing in wh
1/wb
2.
Proof. Rewriting (5) as
r
e = −α2 +
α2 − α1
1 − (α1wh
1/wb
2)
, (6)
and using Deﬁnition 1, we conclude.
Proposition 2. The equilibrium with a positive rental rate is unstable if consumers have
endowment bias.
3Here, one can prove that equilibrium exists if and only if consumers have either endowment bias or
unendowed-good bias (which can be deﬁned by reversing the direction of the inequalities in Deﬁnition 1).
Above, we omit to state this full characterization, since when consumers have a bias towards unendowed
good, equilibrium is observed to be stable.
4Proof. Using (4) and (5), we evaluate
D
h￿(r
e) =
￿
α2 − α1
1 − (α1wh
1/wb
2)
￿2 α1wh
1
wb
2
￿
1 −
α1wh
1
wb
2
￿
, (7)
which is positive if consumers have endowment bias.
The equilibrium at zero rental rate is stable while the one at positive rental rate is
unstable. Rental rates are normally sticky due to the presence of contracts. Therefore,
if for any reason the rental rate of housing is above its positive equilibrium level, the
excess aggregate demand for housing would inﬂate the rental rate further up, leading to
an unbounded increase in rents. Conversely, when the rental rate of housing is below its
positive equilibrium level, the excess aggregate supply of houses would pull the rental
rate all the way down to the stable equilibrium at zero.
Given the described dynamics, we can explain by supply shocks the emergence as
well as collapse of a housing bubble. Corollary 1 implies for example that starting from
an unstable equilibrium, a positive shock to the supply of bread in comparison to the
supply of housing would reduce the equilibrium rental rate below its current market rate,
initiating the bubble. While the rental rate keeps on rising, the supply of housing (that
we do not model explicitly) would increase pushing up the equilibrium rental rate above
the market rate; hence the collapse of bubble.
3.2 The case of low substitution elasticity (￿i ∈ (0,1))
Here, we consider the cases where the elasticity of substitution between bread and
housing services is so low that the two commodities remain to be imperfect complements,
i.e. ￿1,￿2 ∈ (0,1). Unfortunately, for these cases, the closed-form equilibrium solutions
are not available. So, we simulate below an artiﬁcial economy.
Let α1 = 0.25, α2 = 4, wh
1/wb
2 = 1, and ￿1 = ￿2. For diﬀerent values of ￿i, the
calculated equilibrium rental rates are reported in Table 1. (The last row corresponds
to the already analyzed Leontief case).
Table 1
Equilibrium Rental Rates under CES Utilities
￿i re
1 re
2 re
3
1/2 n.a. 1 (stable) n.a.
1/3 5.5728 × 10−2 (stable) 1 (unstable) 1.7944 × 101(stable)
1/6 2.4966 × 10−4 (stable) 1 (unstable) 4.0055 × 103 (stable)
1/11 2.3843 × 10−7 (stable) 1 (unstable) 4.1941 × 106 (stable)
1/51 2.8585 × 10−13 (stable) 1 (unstable) 5.0706 × 1030 (stable)
1/101 1.3468 × 10−13 (stable) 1 (unstable) 6.4278 × 1060 (stable)
0 0 (stable) 1 (unstable) ∞ (stable)
5In the above table, when the elasticity of substitution is suﬃciently high (￿i = 1/2),
there exists a unique and stable equilibrium. For suﬃciently low elasticities of substi-
tution, we obtain three equilibria, as illustrated in Figure 1, where Sh(r) denotes the
aggregate supply of housing which is always equal to wh
1.
r
w1
h
Sh(r)
Dh(r)
1
2
3
re
3
re
2
re
1
Figure 1: Housing Equilibria
Table 1 shows that as the utility functions converge to a Leontief function, the lowest
of the equilibrium rental rates converges to zero while the highest one goes to inﬁnity.
Moreover, the lowest and highest of the equilibrium rental rates are found to be stable
while the one in between is unstable. Here, we can characterize the set of values of the
parameters ￿i, for which the middle of these three equilibrium rental rates is unstable.
Under CES utilities with positive elasticities of substitution, the aggregate demand for
housing is given by
D
h(r) =
r1−￿
α1 + r1−￿w
h
1 +
r−￿
α2 + r1−￿w
b
2. (8)
Assume that the equality
w
h
1/w
b
2 =
1 + (α1)−1
1 + α2
(9)
is satisﬁed. Then Dh(1) = wh
1, implying r = 1 is an equilibrium rental rate. Now
evaluating the derivative of (8) at the rental rate r = 1, we conclude that Dh￿(1) > 1 if
and only if
1 − ￿1
1 + α1
+
1 − ￿2
1 + (α2)−1 > 1. (10)
6One can check that for the economy simulated in Table 1, r = 1 is an unstable equilibrium
if and only if ￿1 +￿2 < 0.75. In the extreme case where individuals have full endowment
bias, i.e. α1 = 1/α2 = 0, condition (10) reduces to ￿1 + ￿2 < 1.
Finally, we note that in the case of multiple equilibria, the amplitude of a possible
bubble is determined by the range between the two stable equilibria which in turn are
mainly determined by the elasticity of substitution and ownership parameters.
References
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from micro data. Journal of Monetary Economics 54, 591-621.
Hahn, F.H., 1958. Gross substitutes and the dynamic stability of general equilibrium,
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Kahneman, D., Knetsch, J.L., Thaler, R.H., 1991. The endowment eﬀect, loss aversion,
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7

Bubble, bubble, where’s the housing bubble?

Gross substitutes and the dynamic stability of general equilibrium,

House prices, fundamentals and bubbles.

Housing, consumption and asset pricing.

How do house prices aﬀect consumption? Evidence from micro data.

Intertemporal substitution and durable goods: Longrun data.

Modelling the demand and supply sides of the housing market: evidence from Ireland.

On Roots of Housing Bubbles

Abstract

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