## Abstract

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 2*D*, a 3*D*, and a 4*D* 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.

### Keywords

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

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## Notes

- 1.
- 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 (2002, 2005); 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.
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.
Thanks to this assumption and the following (4), a bidirectional relationship between housing stock and housing supply flow is established.

- 5.
This is in fact what happens with the linear case used in our examples (see Sect. 3).

- 6.
In this case, we use the \({}^{{\prime}}\) symbol to denote, as usual, the first derivative.

- 7.
This fact will prove useful in the four-dimensional model studied in Sect. 3.3.

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

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

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

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

- 27.
Note that the set of conditions (26) turns out to be extremely useful in all cases studied in the present paper.

- 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.
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 \(0.8\).

## References

Agliari, A., Dieci, R., & Gardini, L. (2007). Homoclinic tangles in a Kaldor-like business cycle model.

*Journal of Economic Behavior and Organization,62*, 324–347.Bischi, G., Gallegati, M., Gardini, L., Leonbruni, R., & Palestrini, A. (2006). Herding behaviours and non-fundamental high frequency asset price fluctuations in financial markets.

*Macroeconomic Dynamics,10*, 502–528.Bischi, G., Gardini, L., & Merlone, U. (2009). Impulsivity in binary choices and the emergence of periodicity.

*Discrete Dynamics in Nature and Society*Article ID 407913.Brock, W., & Hommes, C. (1997). A rational route to randomness.

*Econometrica,65*, 1059–1095.Brock, W., & Hommes, C. (1998). Heterogeneous beliefs and routes to chaos in a simple asset pricing model.

*Journal of Economic Dynamics Control,22*, 1235–1274.Case, K. (2010). Housing, land and the economic crisis.

*Land Lines,22*, 8–13.Chiarella, C., & He, X.-Z. (2002). Heterogeneous beliefs, risk and learning in a simple asset pricing model.

*Computational Economics,14*, 95–132.Chiarella, C., Dieci, R., & Gardini, L. (2002). Speculative behaviour and complex asset price dynamics: A global analysis.

*Journal of Economic Behavior and Organization,49*, 173–197.Chiarella, C., Dieci, R., & Gardini, L. (2005). The dynamic interaction of speculation and diversification.

*Applied Mathematical Finance,12*, 17–52.Chiarella, C., Dieci, R., & He, X.-Z. (2009). Heterogeneity, market mechanisms, and asset price dynamics. In T. Hens & K. Schenk-Hoppé (Eds.),

*Handbook of financial markets: Dynamics and evolution*(pp. 277–344). Amsterdam: North-Holland.Chiarella, C., He, X.-Z., & Hommes, C. (2006). A dynamic analysis of moving average rules.

*Journal of Economic Dynamics and Control,30*, 1729–1753.Day, R., & Huang, W. (1990). Bulls, bears and market sheep.

*Journal of Economic Behavior and Organization,14*, 299–329.De Grauwe, P. (2010). The scientific foundation of dynamic stochastic general equilibrium (DSGE) models.

*Public Choice,144*, 413–443.De Grauwe, P., & Grimaldi, M. (2006). Exchange rate puzzles: A tale of switching attractors.

*European Economic Review,50*, 1–33.De Grauwe, P., Dewachter, H., & Embrechts, M. (1993).

*Exchange rate theory – chaotic models of foreign exchange markets*. Oxford: Blackwell.Dieci, R., & Westerhoff, F. (2010). Heterogeneous speculators, endogenous fluctuations and interacting markets: A model of stock prices and exchange rates.

*Journal of Economic Dynamics and Control,34*, 743–764.Dieci, R., & Westerhoff, F. (2012). A simple model of a speculative housing market.

*Journal of Evolutionary Economics, 22*, 303–329.Eichholtz, P. (1997). A long run house price index: The Herengracht index, 1628–1973.

*Real Estate Economics,25*, 175–192.Eitrheim, O., & Erlandsen, S. (2004). House price indices for norway 1819–2003. In O. Eitrheim, J. T. Klovland & J. F. Qvigstad (Eds.),

*Historical monetary statistics for Norway 1819–2003*(pp. 349–375). Norges Bank Occasional Paper No. 35. Oslo: Norges Bank.Farebrother, R.W. (1973). Simplified Samuelson conditions for cubic and quartic equations.

*The Manchester School,41*, 396–400.Glaeser, E., Gyourko, J., & Saiz, A. (2008). Housing supply and housing bubbles.

*Journal of Urban Economics,64*, 198–217.Hommes, C. (2001). Financial markets as nonlinear adaptive evolutionary systems.

*Quantitative Finance,1*, 149–167.Hommes, C., & Wagener, F. (2009). Complex evolutionary systems in behavioral finance. In T. Hens & K. Schenk-Hoppé (Eds.),

*Handbook of financial markets: Dynamics and evolution*(pp. 217–276). Amsterdam: North-Holland.Kindleberger, C., & Aliber, R. (2005).

*Manias, panics, and crashes: A history of financial crises*(5th ed.). Hoboken: Wiley.Kouwenberg, R., & Zwinkels, R. (2011). Chasing trends in the us housing market, Technical report. Working Paper, Erasmus University Rotterdam, available at: http://ssrn.com/abstract=1539475.

Leung, A., Xu, J., & Tsui, W. (2009). A heterogeneous boundedly rational expectation model for housing market.

*Applied Mathematics and Mechanics,30*, 1305–1316.Lines, M., & Westerhoff, F. (2012). Effects of inflation expectations on macroeconomic dynamics: Extrapolative versus regressive expectations.

*Studies in Nonlinear Dynamics and Econometries*, in press.Lux, T. (2009). Stochastic behavioural asset-pricing models and the stylized facts. In T. Hens & K. Schenk-Hoppé (Eds.),

*Handbook of financial markets: Dynamics and evolution*(pp. 161–216). Amsterdam: North-Holland.Medio, A., & Lines, M. (2001).

*Nonlinear dynamics: A primer*. Cambridge: Cambridge University Press.Parke, W. R., & Waters, G. A. (2007). An evolutionary game theory explanation of ARCH effects.

*Journal of Economic Dynamics and Control,31*, 2234–2262.Piazzesi, M., & Schneider, M. (2009). Momentum traders in the housing market: Survey evidence and a search model.

*American Economic Review,99*, 406–411.Shiller, R. (2005).

*Irrational exuberance*(2 ed.). Princeton: Princeton University Press.Shiller, R. (2008).

*The subprime solution*. Princeton: Princeton University Press.Stewart, G. W. (2001).

*Matrix Algorithms – volume II: Eigensystems*, Philadelphia: SIAM.Sushko, I., Gardini, L., & Puu, T. (2010). Regular and chaotic growth in a Hicksian floor/ceiling model.

*Journal of Economic Behavior and Organization,75*, 77–94.Tramontana, F., Gardini, L., Dieci, R., & Westerhoff, F. (2009). The emergence of “bull and bear” dynamics in a nonlinear model of interacting markets.

*Discrete Dynamics in Nature and Society*, Article ID 310471.Tramontana, F., Westerhoff, F., & Gardini, L. (2010). On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders.

*Journal of Economic Behavior and Organization,74*, 187–205.Westerhoff, F. (2004). Multiasset market dynamics.

*Macroeconomic Dynamics,8*, 596–616.Westerhoff, F. (2009). Exchange rate dynamics: A nonlinear survey. In J.B. Rosser (Ed.),

*Handbook of research on complexity*(pp. 287–325). Cheltenham: Edward Elgar.

## Acknowledgements

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. https://doi.org/10.1007/978-3-642-29503-4_2

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