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Modeling House Price Dynamics with Heterogeneous Speculators


This paper investigates the impact of speculative behavior on house price dynamics. Speculative demand for housing is modeled using a heterogeneous agent approach, whereas ‘real’ demand and housing supply are represented in a standard way. Together, real and speculative forces determine excess demand in each period and house price adjustments. Three alternative models are proposed, capturing in different ways the interplay between fundamental trading rules and extrapolative trading rules, resulting in a 2D, a 3D, and a 4D nonlinear discrete-time dynamical system, respectively. While the destabilizing effect of speculative behavior on the model’s steady state is proven in general, the three specific cases illustrate a variety of situations that can bring about endogenous dynamics, with lasting and significant price swings around the ‘fundamental’ price, as we have seen in many real markets.


  • Heterogeneous expectations
  • Housing markets
  • Boom-bust cycles
  • Bifurcation analysis

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Fig. 2
Fig. 3
Fig. 4


  1. 1.

    For historical accounts and empirical evidence, see Eichholtz (1997); Eitrheim and Erlandsen (2004); Kindleberger and Aliber (2005); Shiller (2005) and Case (2010), amongst others.

  2. 2.

    These demand specifications are heavily inspired by recent work in agent-based financial market modeling in which chartists interact with fundamentalists, as surveyed in Chiarella, Dieci, and He (2009); Hommes and Wagener (2009); Lux (2009) and Westerhoff (2009). Laura Gardini contributed to this research area quite substantially, see, e.g. Chiarella, Dieci, and Gardini (20022005); Bischi, Gallegati, Gardini, Leonbruni, and Palestrini (2006) and Tramontana, Gardini, Dieci, and Westerhoff (2009), to name only a few of her works. It is typically Laura who miraculously accomplishes an otherwise “undoable” mathematical analysis.

  3. 3.

    As discussed in Glaeser et al. (2008), this assumption can be justified in terms of the existence of a continuum of homeowners, receiving a Poisson-distributed shock in each period that forces them to sell their homes and leave the area. Of course, in a more realistic setup, probability \(\lambda \) of the shock might itself depend on the current price or on expected price movements.

  4. 4.

    Thanks to this assumption and the following (4), a bidirectional relationship between housing stock and housing supply flow is established.

  5. 5.

    This is in fact what happens with the linear case used in our examples (see Sect. 3).

  6. 6.

    In this case, we use the \({}^{{\prime}}\) symbol to denote, as usual, the first derivative.

  7. 7.

    This fact will prove useful in the four-dimensional model studied in Sect. 3.3.

  8. 8.

    More precisely, if \(\ {\Phi }^{{\prime}}\) is negative and increases in modulus, under our restrictions (16), \(\left \vert Det({J}_{0})\right \vert\) and \(\left \vert Tr({J}_{L})\right \vert\) increase with \(\left \vert {\Phi }^{{\prime}}\right \vert\) only from certain thresholds onwards. We will not consider this situation in the forthcoming examples, since it is generally associated with strong (and unrealistic) overreaction by fundamental traders.

  9. 9.

    This represents the most typical case in which destabilization occurs due to extrapolative demand from speculators who bet on the persistence of bull or bear markets, as shown in the forthcoming examples.

  10. 10.

    It will even be useful in the four-dimensional model presented in Sect. 3.3.

  11. 11.

    Note in particular that the supply function (28) can be obtained from a standard profit maximization setup with a quadratic cost function. Consistent with this setup, the optimal amount \(I(p)\) of new constructions is positive iff \(p > {\theta }_{0}/\theta := {p}_{\min }\). Taking into account this constraint properly would result in a piecewise-smooth dynamical system. Similar natural constraints involving upper and lower bounds on the variables may even result in piecewise-continuous systems. We remark here that Laura largely contributed in recent years to developing completely new analytical and numerical tools to deal with these kinds of maps (more details are provided in the concluding section). We hope to be able to ‘exploit’ Laura’s great experience in this field and to collaborate with her in the future on a possible extension of this work. As for now, we implicitly assume in our numerical experiments that fixed parameter \({\theta }_{0}\) is such that price \(p\) never falls below the above threshold.

  12. 12.

    Parameter calibration would, of course, be important in the case of isoelastic demand and supply and if the laws of motion were specified in relative price and stock adjustments.

  13. 13.

    In particular, the model can then be rewritten in deviations from the FSS, via the change of variables \(\eta := h - {h}^{{_\ast}},\pi := p - {p}^{{_\ast}}\). The model in deviations with linear demand and supply is independent of parameters \({h}^{{_\ast}}\) and \({p}^{{_\ast}}\) (or \({\beta }_{0}\) and \({\theta }_{0}\)), as can be checked.

  14. 14.

    Equivalently, these inequalities can be directly derived from the 2-D Jacobian matrix of system (32).

  15. 15.

    In particular, this condition is always satisfied (under parameter restriction (16)) if \(\omega = 1\), i.e. if no exogenous upper bound is imposed on the market impact of extrapolators, because in this case \({\Phi }^{{\prime}} = \gamma \) does not depend on parameter \(\psi \).

  16. 16.

    On the contrary, it turns out from the comparison of (34) and (35) that the (absolute) slope \(\beta \) of the ‘real’ demand curve has no specific influence on the type of bifurcation occurring when \(\gamma \) increases.

  17. 17.

    Recall that parameters \({\beta }_{0}\) and \({\theta }_{0}\) (or, alternatively, \({h}^{{_\ast}}\) and \({p}^{{_\ast}}\)) can be arbitrarily chosen without affecting the numerical results presented below.

  18. 18.

    See Sect. 3.3 for a brief discussion of the relationship between demand parameters and price expectations of the two types of agents.

  19. 19.

    More generally, the trend signal may be modeled as the deviation of the latest observation from a time average computed over the last N periods, or even as the deviation between short-term and long-term moving averages. However, these more realistic specifications would increase the dimension of the dynamical system considerably. See, e.g. Chiarella, He, and Hommes (2006).

  20. 20.

    In fact, the chartist demand component in function (36) can again be written as \(\overline{w}{\mu }_{t}({p}_{t} - {p}_{t-1})\), where the trend extrapolation coefficient \({\mu }_{t}\) is now state-dependent and attains its maximum, \(\mu \), when the trend signal \(\left \vert {p}_{t} - {p}_{t-1}\right \vert \rightarrow 0\), whereas \({\mu }_{t}\) decreases as \(\left \vert {p}_{t} - {p}_{t-1}\right \vert\) becomes larger. Unlike a linear function with constant slope \(\mu \), this demand function thus partly ‘levels off’ if larger price movements are observed.

  21. 21.

    Intuitively, at a non-fundamental steady state, fundamentalist demand would be different from zero, whereas trend-based chartist demand vanishes at any steady state solution. This situation of permanent excess demand would set in motion price corrections towards the FSS.

  22. 22.

    If the parameter \(\lambda \) is small, the bifurcation value \({\mu }_{NS}\) is indeed very close to the upper bound of the interval, \(1/(\alpha \overline{w})\).

  23. 23.

    For applications to evolutionary finance see, e.g. Brock and Hommes (1998); Hommes (2001); Chiarella and He (2002); Westerhoff (2004) and De Grauwe and Grimaldi (2006). Applications to (macro)economic dynamics include Brock and Hommes (1997); Lines and Westerhoff (2012) and De Grauwe (2010).

  24. 24.

    A very similar interpretation of the speculative demand function in terms of expected unit profits applies also to the models studied in the previous sections.

  25. 25.

    Note that the forecast errors in (40) and (41) can also be interpreted as the difference between the expected and the actual price change in period \(t - 1\). For instance, in the case of chartists: \({p}_{t}^{e,C} - {p}_{t} = ({p}_{t}^{e,C} - {p}_{t-1}) - ({p}_{t} - {p}_{t-1}) =\widehat{ \mu }({p}_{t-1} - {p}_{t-2}) - ({p}_{t} - {p}_{t-1})\).

  26. 26.

    See, e.g. Lines and Westerhoff (2012) for a discussion of this point within a macro-model with heterogeneous inflationary expectations.

  27. 27.

    Note that the set of conditions (26) turns out to be extremely useful in all cases studied in the present paper.

  28. 28.

    Again we assume that the second and fourth inequalities in (37) are satisfied for any \(\mu \in V\), which is the case in the following numerical example.

  29. 29.

    For instance, under the same parameter setting of Fig. 4, coexisting attractors can be numerically observed by means of bifurcation diagrams against parameter \(\theta \), for \(\mu = 8\) and \(\theta \) ranging between \(0.5\) and \(0.8\).


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This work was carried out with the financial support of MIUR (Italian Ministry of Education, University and Research) within the PRIN Project “Local interactions and global dynamics in economics and finance: models and tools”. We are grateful to Carl Chiarella for his comments and suggestions on an earlier draft of the paper.

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Dieci, R., Westerhoff, F. (2013). Modeling House Price Dynamics with Heterogeneous Speculators. In: Bischi, G., Chiarella, C., Sushko, I. (eds) Global Analysis of Dynamic Models in Economics and Finance. Springer, Berlin, Heidelberg.

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