## Abstract

We develop a simple model of a speculative housing market in which the demand for houses is influenced by expectations about future housing prices. Guided by empirical evidence, agents rely on extrapolative and regressive forecasting rules to form their expectations. The relative importance of these competing views evolves over time, subject to market circumstances. As it turns out, the dynamics of our model is driven by a two-dimensional nonlinear map which may display irregular boom and bust housing price cycles, as repeatedly observed in many actual markets. Complex interactions between real and speculative forces play a key role in such dynamic developments.

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## Notes

Among the several examples, Poterba (1984, 1991) and Mankiw and Weil (1989) focus on the impact of the ‘user cost’ and on demographic changes within an asset-based rational expectations model of the housing market; Ortalo-Magné and Rady (1999) stress the role of income and credit market shocks within a perfect foresight ‘life-cycle’ model; Glaeser and Gyourko (2007) analyze how shocks on demand and construction costs affect house price and quantity adjustments within a no-arbitrage dynamic rational expectations model with endogenous housing supply.

A number of recent theoretical papers on housing market take heterogeneity into account somehow. For instance, Sommervoll et al. (2010) build a model with buyers, sellers and mortgagees with adaptive expectations, whereas Burnside et al. (2011) develop a model in which agents hold heterogeneous expectations about long-run fundamentals and may change their view because of “social dynamics”. Note, however, that the approaches adopted in these models, as well as the underlying concepts of heterogeneity, are very different from ours.

Other papers that apply a similar ‘heterogeneous interacting agent’ approach to the dynamics of housing prices are Leung et al. (2009) and Kouwenberg and Zwinkels (2010). These preliminary studies, however, do not provide analytical results and are mainly concerned with numerical simulation and model calibration.

Put differently, in our model \(D_t^R \) represents the desired stock of housing (for given levels of income and population) by people who maximize their utility from ‘housing services’ and from consumption of alternative goods (‘non-housing’ consumption), subject to a standard budget constraint. Of course, the possible selling price of houses in the future may, in principle, also be important for these people. This additional aspect would be properly taken into account by modeling agents’ utility maximization in a two-period setting (see, e.g. Follain and Dunsky 1997), and price expectations would then play a prominent role by affecting expected utility from second-period wealth. This component is formally shifted to \(D_t^S \) in our simplified setup.

Of course, an interesting extension of our model would be to consider

*S*_{ t }=*dS*_{ t − 1}+*eE*[*P*_{ t }] and (for instance)*E*[*P*_{ t }] =*P*_{ t − 1}, i.e. new constructions are already planned and executed in period t-1 based on the expected (selling) price for period t, and construction firms hold naïve expectations. Note that such delivery lags represent, in general, further sources of instability (see, e.g. Wheaton 1999). In this particular case, one would end up with a three-dimensional dynamical system which has the same steady states as the present model but an even richer bifurcation structure.Our numerical examination, focussing on price and quantity deviations from equilibrium ‘fundamental’ values, is not affected by parameter

*b*, representing the exogenous real demand term. As a consequence, this parameter can always be chosen such that in the original model the total demand for houses is positive in any time step.In a related paper, Kouwenberg and Zwinkels (2010) use the ‘discrete choice’ approach of Brock and Hommes (1997, 1998) to model the weights of two speculative demand strategies. According to this approach, agents are boundedly rational in the sense that they tend to select those strategies which have produced a high fitness (measured in terms of realized profits or forecasting errors) in the recent past.

Note that the rate of depreciation is 2 percent per time period in all of our simulations. Furthermore, assuming that a time period is given with one year, a depreciation rate of 2 percent implies a (reasonable) half-life of a housing unit of roughly 35 years. We thank an anonymous referee for this suggestion.

As Fig. 2 suggests, each of the two nonfundamental steady states changes into a more and more complex attractor, via a sequence of period-doubling bifurcations.

Note that these parameters capture the supply-side of the economy.

Note that for \(\frac{e}{1+d}-2\ge \mbox{\thinspace }\frac{1}{d}-1\), or, equivalently,

*e*≥ (1 +*d*)^{2}/*d*, the steady state is unstable for any combination (*c*,*f*). We do not consider this case in Fig. 3.The historical housing price data provided by Shiller (http://www.econ.yale.edu/~shiller/) share, in a qualitative sense, some phenomena with our simulated housing price data. In particular, the period from 1890 to 1975 seems to be characterized by “bull and bear market dynamics” whereas the period from 1975 to 2010 displays more pronounced “boom and bust cycles”. However, the disaggregated data presented in Case (2010) is more ragged than Shiller’s nationwide data.

Note, however, that in the bottom panels of Fig. 5 the relation between house prices in subsequent periods is reversed from positive to negative when considering price ranges that deviate from the benchmark fundamental considerably.

Note that in the steady state solution of the model, the amount of new constructions exactly offsets the depreciation of the existing stock of houses. More generally, the stock of houses may decline also in the presence of new constructions, if the latter are not sufficient to compensate for depreciation.

We leave an analytical study as well as a systematic numerical investigation of this model for future work. However, as noted by one of the anonymous referees, it may be interesting to study this model in more detail, in particular how the steady states and their stability domain depend on the model parameters.

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## Author information

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### Corresponding author

## Additional information

This paper was presented at the “Workshop on Evolution and Market Behavior in Economics and Finance”, Scuola Superiore Sant’Anna, Pisa, October 2009 and at the “Conference on Heterogeneous Agents in Financial Markets”, Erasmus University Rotterdam, Rotterdam, January 2009. We thank the participants, in particular Larry Blume, David Easley, Cars Hommes, Alan Kirman, Klaus Reiner Schenk-Hoppé, Valentyn Panchenko and Jan Tuinstra, for stimulating discussions. We are also very grateful to Giulio Bottazzi, Pietro Dindo and two anonymous referees for valuable comments and suggestions.

## Appendices

### Appendix 1

In this appendix, we derive the two-dimensional nonlinear dynamical system of the full model, its fixed points, the parameter region for which the model’s fundamental steady state is locally asymptotically stable, and necessary conditions for the emergence of a flip, a pitchfork, and a Neimark-Sacker bifurcation, respectively. A theoretical treatment of linear and nonlinear dynamical systems is provided by Gandolfo (2009) and Medio and Lines (2001), among others.

Note first that, by setting \(\pi_t =P_t -\bar{{P}}\) and \(\zeta_t =Z_t -\bar{{Z}}\), the two-dimensional linear dynamical system 5 for the model without speculation may be rewritten in terms of deviations from the fundamental steady state as

By now including the speculative demand term, we easily obtain the following two-dimensional nonlinear dynamical system in (*π*
_{
t
}, *ζ*
_{
t
})

Inserting \(\left( {\pi_{t+1} ,\zeta_{t+1} } \right)=\left( {\pi_t ,\zeta _t } \right)=\left( {\bar{{\pi }},\bar{{\zeta }}} \right)\) into Eq. 25, the three fixed points

and

can be calculated. Since the denominator of \(\bar{{\pi }}_{2,3} \) is always positive, the fixed points \((\bar{{\pi }}_{2,3} ,\bar{{\zeta }}_{2,3} )\) only exist if (1 − *d*) (*f* − *c*) − *e* > 0.

The Jacobian matrix of our model, evaluated at the steady state \(\left( {\bar{{\pi }}_1 ,\bar{{\zeta }}_1 } \right)=\left( {0,0} \right)\), reads

where *tr* = 1 − *c* − *e* + *d* + *f* and \(\det =d(1-c+f)\) stand for the trace and determinant of *J*, respectively. A set of necessary and sufficient conditions for both eigenvalues of *J* to be smaller than one in modulus (which implies a locally asymptotically stable steady state) is given by (i) 1 + *tr* + *det* > 0, (ii) 1 − *tr* + *det* > 0 and (iii) 1 − *det* > 0, respectively. After some simple transformations, this yields

and

Observe that for *f* = 0, Eqs. 29–31 are identical to Eqs. 9–11. In this case, Eqs. 30 and 31 would always be fulfilled. For *f* > 0, however, Eq. 29 is less restrictive than Eq. 9, while Eqs. 30 and 31 impose stronger restrictions. Note also that Eqs. 29–31 are independent of parameters *b* and *h*.

Violation of the first, second and third inequality (the remaining two inequalities hold) represents a necessary condition for the emergence of a flip, pitchfork and Neimark-Sacker bifurcation, respectively. In connection with supporting numerical evidence, this is usually regarded as strong evidence. Figure 2 furthermore reveals that the flip bifurcation is of the subcritical case whereas the pitchfork and Neimark-Sacker bifurcations are of the supercritical type.

### Appendix 2

In this appendix, we outline a more general model that includes as particular cases both the simplified formulation in ‘stock’ variables (adopted in this paper) and a formulation in pure ‘flow’ variables (new home demand and new constructions). Here we denote by *x*
_{
t
} the demand for houses and by *y*
_{
t
} the supply of houses in period *t*. The price adjusts to the excess demand in the usual manner, i.e.

Demand and supply *x*
_{
t
} and *y*
_{
t
} (that are now regarded as ‘flow’ variables) include, in general, part of *unsatisfied demand*
\(\left( {x_{t-1}^B } \right)\) and *unsold houses*
\(\left( {y_{t-1}^U } \right)\) from the previous period, respectively. We neglect the speculative demand term for the moment. Demand in period *t* is specified as

Demand *x*
_{
t
} thus consists of new demand \(\hat{{b}}-cP_t \) and *backlogged *demand, here simply modeled as a fraction *α*, 0 ≤ *α* ≤ 1, of the demand that has remained unsatisfied in the previous period. Supply (i.e. houses for sale) in period *t* is defined as:

including new constructions, *eP*
_{
t
}, and a fraction *β*, 0 ≤ *β* ≤ 1,of unsold houses from the previous period (note that the term \(\beta dy_{t-1}^U \) takes depreciation into account). By definition, in each period *t* we have:

where the term min(*x*
_{
t
}, *y*
_{
t
}) represents the amount of houses sold (or, equivalently, of demand satisfied) in period *t*. Equations 32–34, together with identities 35 and 36 form a dynamical system expressed in flow variables, which takes backlogged demand and unsold houses into account. This model can be transformed into an equivalent model where ‘stock’ variables (the existing stock of houses and the desired *holding* of houses), rather than flow variables, are matched in each period. Note first that the quantity:

or recursively

represents the cumulated amount of houses sold in the current and previous rounds, by taking depreciation into account. By defining demand and supply in terms of stock (denoted by *D*
_{
t
} and *S*
_{
t
}, respectively) as follows:

dynamical system 32–36 can be rewritten as a three-dimensional system in the state variables *P*
_{
t
}, *D*
_{
t
}and *S*
_{
t
}:

where *Q*
_{
t
} = min(*D*
_{
t
}, *S*
_{
t
}), which turns out to be non-differentiable.^{Footnote 20} Easy computations demonstrate that dynamical system 41–43 admits a unique steady state, the coordinates of which are specified as follows:

In order to check that the stationary levels (Eq. 45) of supply and demand, as well as the steady state price (Eq. 44), correspond in fact to those obtained in Eqs. 6 and 7, it is enough to change the coordinates of the autonomous demand term, by defining the new parameter *b* (the one we adopt in the paper) as follows,

as can be shown by simple computations.

Next, the model with speculation can be obtained by adding a demand term \(D_t^S \), identical to Eq. 14, to the right-hand side of Eq. 42. As numerical simulations suggest, also this more general model produces a transition to complex boom and bust cycles, once extrapolative demand becomes strong enough.

Finally, the following significant particular cases give rise to two simplified models. First, the case *α* = *β* = 0 (unfilled demand and unsold houses are not translated to the next period) can be reduced to the following one-dimensional model in ‘pure’ flows:

where the speculative demand \(D_{t-1}^S \) is itself a cubic-type function of *P*
_{
t − 1} (via Eqs. 12–15). As can be shown, this model generates a pitchfork scenario for the steady states, followed by a sequence of bifurcations leading to chaotic dynamics, very similar to that illustrated in the paper.

Second, the case *α* = *β* = 1 (unsold houses and unsatisfied demand are entirely shifted to the next period) leads to a three-dimensional model formed by a price adjustment equation identical to Eq. 41 and by the two equations

In Eq. 48 the real demand *b*
_{
t − 1} − *cP*
_{
t
}, regarded as a function of *P*
_{
t
}, has an ‘autonomous’ component that depends on the state of the system at time *t* −1, namely, \(b_{t-1} :=\hat{{b}}+D_{t-1} -\left( {1-d} \right)\min \left( {D_{t-1} ,S_{t-1} } \right)\). In order to reduce the dimension of the system and to preserve differentiability, in the paper we replace the time varying term *b*
_{
t − 1} in Eq. 48 with the constant parameter *b* defined by Eq. 46. The latter is nothing else than the steady-state value of *b*
_{
t − 1}, i.e. \(b:=\hat{{b}}+\bar{{S}}-\left( {1-d} \right)\bar{{S}}=\hat{{b}}+d\bar{{S}}\). Such a simplification results in the two-dimensional model studied in the paper.

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### Cite this article

Dieci, R., Westerhoff, F. A simple model of a speculative housing market.
*J Evol Econ* **22**, 303–329 (2012). https://doi.org/10.1007/s00191-011-0259-8

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DOI: https://doi.org/10.1007/s00191-011-0259-8

### Keywords

- Housing markets
- Speculation
- Boom and bust cycles
- Nonlinear dynamics

### JEL Classification

- D84
- R21
- R31