A heterogeneous boundedly rational expectation model for housing market

Abstract

This research aims to test the housing price dynamics when considering heterogeneous boundedly rational expectations such
as naive expectation, adaptive expectation and biased belief. The housing market is investigated as an evolutionary system
with heterogeneous and competing expectations. The results show that the dynamics of the expected housing price varies substantially
when heterogeneous expectations are considered together with some other endogenous factors. Simulation results explain some
stylized phenomena such as equilibrium or oscillation, convergence or divergence, and over-shooting or under-shooting. Furthermore,
the results suggest that variation of the proportion of groups of agents is basically dependent on the selected strategies.
It also indicates that control policies should be chosen carefully in consistence with a unique real estate market during
a unique period since certain parameter portfolio may increase or suppress oscillation.

Full text

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Appl. Math. Mech. -Engl. Ed. 30(10), 1305–1316 (2009)
DOI: 10.1007/s10483-009-1010-1
c ?Shanghai University and Springer-Verlag 2009
Applied Mathematics
and Mechanics
(English Edition)
A heterogeneous boundedly rational expectation model for
housing market∗
Andrew Y. T. LEUNG (???)1, Jia-na XU (???)2,Wing Shum TSUI (???)1
(1. Department of Building and Construction, City University of Hong Kong,
Tat Chee Avenue, Kowloon, Hong Kong SAR, P. R. China;
2. Postdoctoral Centre, China Merchants Group Limited, 27/F, China Merchants Tower,
Shun Tak Centre, 168-200 Connaught Rd. C., Hong Kong SAR, P. R. China)
(Communicated by Li-qun CHEN)
Abstract
heterogeneous boundedly rational expectations such as naive expectation, adaptive ex-
pectation and biased belief. The housing market is investigated as an evolutionary system
with heterogeneous and competing expectations. The results show that the dynamics of
the expected housing price varies substantially when heterogeneous expectations are con-
sidered together with some other endogenous factors. Simulation results explain some
stylized phenomena such as equilibrium or oscillation, convergence or divergence, and
over-shooting or under-shooting. Furthermore, the results suggest that variation of the
proportion of groups of agents is basically dependent on the selected strategies. It also
indicates that control policies should be chosen carefully in consistence with a unique real
estate market during a unique period since certain parameter portfolio may increase or
suppress oscillation.
This research aims to test the housing price dynamics when considering
Key wordsevolutionary system, housing price dynamics, heterogeneous expectations
Chinese Library Classification
2000 Mathematics Subject Classification
O29, O175.14
39A11, 35G25
Introduction
It is believed that oscillation is particularly prone to appear in the housing market due
to the long lag time between capital demand and delivery. Such endogenous market cycles
should not occur[1-2]under the rational expectation hypothesis (REH), introduced by Muth,
which dominates the paradigm of expectation formation in the neoclassic economics. Under the
condition of REH, agents are all rational and make their estimations based on perfect knowledge
about all available information. It is clear that the economic model can be analyzed more easily
if all the agents perform uniformly. Thus, the irrational agents will be driven out of the market
by rational traders ultimately.
However, it can be traced back to Keynes, who questioned the REH and proposed the sig-
nificant role of the investors’ sentiment. In the housing market, researchers have long suspected
that housing price oscillation is partly attributed to the “irrationality” of estate agents, which
is inconsistent with the REH.
∗ Received Oct. 5, 2008 / Revised Aug. 25, 2009
Corresponding author Andrew Y. T. LEUNG, Professor, E-mail:Andrew.leung@cityu.edu.hk

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1306Andrew Y. T. LEUNG, Jia-na XU, and Wing Shum TSUI
Capozza and Seguin[3]studied the expectations of capital appreciation in the housing mar-
ket and proved the over-action of the participants in the housing market. Clayton[4]set up
a model that requires the expected return of a housing investment to be equal to the rate of
return available on alternative investments under the joint null hypothesis of rational expec-
tations, perfect markets and no risk premium. The results provided evidence of the irrational
expectations since such assumption cannot be satisfied. Riddel[5]and Chan et al.[6]both had
revealed speculative behavior in the housing market. In particular, Seslen[7]pointed out that
the households exhibit rational responses to returns on the upside but quite irrational responses
to downturns in the housing market.
As an alternative to the above empirical analysis, Wheaton proposed the impacts of bound-
edly rational expectations on real estate cycles based on computational analysis[8]. Computer
simulation has become a key technique for the study of boundedly rational choices in eco-
nomics[8]. Agent-based simulations investigate the market behaviors of interacting agents with
bounded or unbounded expectations and thus provide new insight to study the housing price
dynamics by losing the restrictions of analytic methods.
This paper is an extension of the work done by Wheaton on the following aspects: First,
this paper incorporates additional essential boundedly rational expectations than in Wheaton’s
research. Second, it tests the impacts of coexistence and interaction of heterogeneous agents
instead of those of the single type of agents on the housing price. Third, it sets up a proportional
evolutionary model of groups of heterogeneous agents based on the accumulated profits as the
fitness criterion. Finally, it illustrates the impacts of some important factors, such as the
demand elasticity, supply lag, and depreciation rate on the occurrence of oscillation under the
condition of coexistence and interaction of heterogeneous expectations in the housing market.
This paper is organized as follows. Section 1 discusses the evolutionary housing price dynam-
ics model with heterogeneous expectations. Section 2 shows the simulations results, including
simulations on the effects of the heterogeneous expectations distribution, the variation of the
proportion and the parametric study of demand and supply elasticity, supply lag and depreci-
ation rate of boundedly rational expectations. This paper ends with some conclusions.
1Housing market as an evolutionary system
The cobweb model started with the naive expectation assumption that the current expected
price is equal to the previous actual bargain price. Nerlove first introduced the adaptive price
expectation assumption into the cobweb model. He assumed that the current expected price
is equal to the weighted average of the previous actual price and the previous expected price.
An adaptive belief system was introduced by Brock and Hommes[9-10], who assumed that the
traders would choose their trading strategy according to an evolutionary “fitness measure”, and
the agents are rationally bounded. Housing price dynamics assuming heterogeneous backward-
looking expectations under an unanticipated economic shock is analyzed based on the cobweb
model in this paper.
1.1The cobweb model in the housing market
The cobweb model is one of the simplest economic models. It was given by Kaldor (1934)
with the following three assumptions: (i) The quantity demanded at time t is the function of
the actual price, that is, Dt= d(pt); (ii) the supply at time t is the function of expected price,
that is, St= s(ˆ pt); and (iii) the market clears in each period, Dt= St. The demand curve will
be constructed in a specific functional form to simplify the analysis. The quantity demanded
in the real estate market is assumed to be proportional to an exogenous economic variable and
respond to price at time t. Therefore, the demand in a real estate market is characterized by
Dt= α1
1
Utp−β1
t
,
(1)

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A heterogeneous boundedly rational expectation model for housing market1307
where Utis the after-tax cost of housing ownership[11-12], β1is the demand elasticity, and α1is
the constant.
Ut= (1 − Tt) × it,
where T is the normal income tax rate and i is the nominal interest rate. The demand shock
is assumed constant, it= i, Tt= T. The impacts of interest rate change on the housing price
are widely documented[13].
The housing market is often modelled within a stock-flow framework, i.e.,
(2)
St
St−1
= (1 − δ) +Ct−n
St−1,
(3)
where δ is the depreciation rate and Ct−nis the new space delivery at time t. The consideration
of the supply lag is due to the longer production cycle in the real estate market, i.e., the decision
in acquiring new space is made at n periods before delivery time and the quantity depends on
the estimate of price at delivery time t, ˆ pt,
Ct−n
St−1
= α2ˆ pβ2
t,
(4)
where, ˆ ptis the expected price at time t when the agent made the decision at time t − n, β2is
the supply elasticity, and α2is the constant. Therefore, Eq. (3) can be rewritten as
St
St−1
= (1 − δ) + α2ˆ pβ2
t.
(5)
It is clear that the supply at current time depends on the supply at the previous time
period as well as the expected price. The expectations on price include the boundedly rational
ones, such as the naive expectation, adaptive expectation, biased (optimistic or pessimistic)
expectation, and trendy following expectation.
1.2Heterogeneous expectations
It is incredibly difficult to model all diverse beliefs of agents and their interactions working
in the market. Even though by using several simple types of representative heterogeneous
boundedly rational agents, some important features of the financial market can be matched,
including fat tails, excess relative returns, the persistence of volatility and trading volume,
which cannot be explained by the traditional analytical methods under the REH[14]. There
are an increasing number of researchers attempting to build the evolutionary model of the
heterogeneous agents in the financial market[15]. Although it is believed to be particularly
prone to oscillation due to the long supply lag and high product value, few papers on the study
of computational agents can be found in describing the housing market behavior.
According to Dt= St, a set of difference equations with time delay can be used to describe
the expected price dynamics in the real estate market. The specific form of the difference
equation depends on the assumption made for the heterogeneous expectations.
It is assumed that the forecasting price at time t obeys the simple linear trading rules with
a time lag,
ˆ pt= gpt−n+ h.
Although the linear backward-looking forecasting rule looks simple, it can be easily applied
in the real estate markets[16]. This paper considers three main forecasting ways listed below,
which are based on the previous actual price at time t − n.
Naive expectation
The traditional cobweb model is based on the naive expectation as-
sumption, stating that the expected price at delivery time t is equal to the actual price at
decision time t − n given that ˆ pt= pt−n. The price dynamics is governed by
(
ˆ pt+n−1)−β1= (1 − δ) + α2ˆ pβ2
(6)
ˆ pt+n
t.
(7)

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1308Andrew Y. T. LEUNG, Jia-na XU, and Wing Shum TSUI
Biased belief
economic theory. They have unique views on the expected price in a biased form:
Estate agents have different preferences to the risks and profits in the
ˆ pt= pt−n+ h,
(8)
where h is the bias coefficient. If the agents have a continuous optimistic expectation on the
price, then h > 0; if they are continuously pessimistic, then h < 0. In particular, when h = 0,
the model will be reduced to the traditional naive expectation model. The dynamical model
based on the biased expectation model can be described as
(
ˆ pt+n− h
ˆ pt+n−1− h)−β1= (1 − δ) + α2ˆ pβ2
Trend follower phenomenon is one of the most famous
topics of bounded rationality that has been investigated for a long time. It can be written as
t.
(9)
Trend following expectation
ˆ pt= pt−n+ g × (pt−n− ˆ p∗).
(10)
It implies that the belief of agents on the new price always follows the previous price trend and
the change rate is equal to g for g > 0. With the trend following model, agents believe that
the increasing or decreasing trend at decision time t − n will be kept at delivery time t. The
dynamical model can be characterized as
(
ˆ pt+n− g × ˆ p∗
ˆ pt+n−1− g × ˆ p∗)−β1= (1 − δ) + α2× ˆ pβ2
As an alternative to the above assumption that the expected price
is based only on previous actual price, another important backward-looking forecasting rule is
based on the fitness of the previous actual price and the previous expected price. Adaptive
expectations are introduced into the cobweb model:
t.
(11)
Adaptive expectation
ˆ pt= ωpt−n+ (1 − ω)ˆ pt−n,
0 ≤ ω ≤ 1.
(12)
It shows that the expected price at delivery time t is the weighted average of expected and
actual prices at decision time t−n by the expectation weight factor ω. When ω = 1, the model
reduces to the traditional model with naive expectation. When ω = 0, ˆ pt= ˆ pt−n, the expected
price at delivery time t is equal to the expected price at decision time t−n. Equation (12) can
also be written as
ˆ pt= ˆ pt−n+ ω(pt−n− ˆ pt−n),
It shows that the expected price at delivery time t is a correction to the expected price at
decision time t − n according to the previous expectation bias. The constant ω controls the
degree of correction. The dynamical adaptive expectation model can be characterized as
0 ≤ ω ≤ 1.
(13)
(
ˆ pt+n− (1 − ω)ˆ pt
ˆ pt+n−1− (1 − ω)ˆ pt−1)−β1= (1 − δ) + α2ˆ pβ2
Deductive model for the structure variation
Hommes et al.[17]and Brock et al.[18]utilized discrete choice mechanism to describe the
probability when an agent chooses a strategy . The higher the fitness of trading strategy, the
more the traders will select it. In addition, the accumulated “realized profits” is tested as the
fitness criteria. This paper assumes that the choice of the agent is affected by the proportion
of specific group number to the population number. In other words, the agent will choose
the expectation method which is adopted by more agents with higher probability. The Polya
deductive model is used to describe the choice of the agents[19].
t.
(14)
1.3

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A heterogeneous boundedly rational expectation model for housing market1309
The proportion vector is Xt= (X1
number with expectation i to the population number at time t. The probability that an agent
chooses the expectation i at time t is qi
written as
?
0,
t,X2
t,··· ,XI
t), where Xi
tis the proportion of the group
t(Xt); αi
t(Xt) is a stochastic variable, which can be
αi
t(Xt) =
1,qi
1 − qi
t(Xt),
t(Xt).
(15)
Bt= (B1
at time t. Thus, the total number of the agents at time t = 1 is
t,B2
t,··· ,BI
t) is the amount vector denoting the amount of agents with expectation i
B = B1
1+ B2
1+ ··· + BI
1.
(16)
The increased amount of agents with expectation i at time t + 1 follows Bi
the increased amount of agents with expectation i at time t is
t+1= Bi
t+ αi
tand
Bi
t= (B + t − 1)Xi
t.
(17)
Now the proportion function of agent number with expectation i can be rewritten as
Xi
t+1= Xi
t+
1
B + t(αi
t(Xt) − Xi
t),
(18)
where Xi
Since E[μi
1= Bi
t|(Xt)] = 0,
1/B. Considering E[αi
t(Xt)|(Xt)] = qi
t(Xt), the stochastic error is μi
t= αi
t− qi
t.
Xi
t+1= Xi
t+
1
B + t(qi
t(Xt) − Xi
t) +
1
B + tμi
t.
(19)
So the expectation of the stochastic system is
E[(Xi
t+1)|(Xt)] = Xi
t+
1
B + t(qi
t(Xt) − Xi
t),
(20)
which equals the determined system
Xi
t+1= Xi
t+
1
B + t(qi
t(Xt) − Xi
t).
(21)
It is obvious that the choice probability qi
strategy. Three cases will be considered in this paper, in the situations when the agent chooses
the strategy randomly, linearly with the proportion, and based on the accumulated profits.
Case 1
Agent chooses the expectation randomly. It presents that almost no difference in
utility for the agents if the agent chooses the expectation randomly. However, obviously, it is
an uncommon phenomenon for the agents to trade in the housing market because of the very
high price and quite different quality of the products.
Case 2
Agent chooses the expectation linearly varying with the proportion. In the housing
market, the agents have preference to the different products and the choice of the agents follows
a certain probability distribution when the agent number goes to infinity. In this part, it assumes
that the choice probability is linearly varying with the proportion, i.e., qi
is the preference matrix.
Case 3
Agent chooses the expectation using both the proportion and the accumulated
profits. The accumulated profits of group i at time t can be described as
tis a key factor, which is dependent on the diverse
t(Xt) = AXt, where A
Wi
t= Wi
t−1+ (pt− Ci
t) × Si
t,
(22)