A simple model of a speculative housing market

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.

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

$$ \left\{ {\begin{array}{l} \pi_{t+1} =\left( {1-c-e} \right)\pi_t -d\zeta_t \\ \zeta_{t+1} =e\pi_t +d\zeta_t \\ \end{array}} \right.. $$

(24)

By now including the speculative demand term, we easily obtain the following two-dimensional nonlinear dynamical system in (π _t , ζ _t )

$$ \left\{ {\begin{array}{l} \pi_{t+1} =\left( {1-c-e} \right)\pi_t +\frac{f\pi_t -gh\pi_t^3 }{1+h\pi_t^2 }-d\zeta_t \\ \zeta_{t+1} =e\pi_t +d\zeta_t \\ \end{array}} \right.. $$

(25)

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

$$ (\bar{{\pi }}_1 ,\bar{{\zeta }}_1 )=(0,0) $$

(26)

and

$$ (\bar{{\pi }}_{2,3} ,\bar{{\zeta }}_{2,3} )=\left( {\pm \sqrt {\frac{(1-d)(f-c)-e}{h(e+(1-d)(c+g))}} \;,\quad \frac{e}{1-d}\bar{{\pi }}_{2,3} } \right) $$

(27)

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

$$ J=\left( {{\begin{array}{*{20}c} {1-c-e+f} \hfill &\quad {-d} \hfill \\ e \hfill & \quad {d} \hfill \\ \end{array} }} \right), $$

(28)

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

$$ f>c+\frac{e}{1+d}-2, $$

(29)

$$ f<c+\frac{e}{1-d}, $$

(30)

and

$$ f<c+\frac{1}{d}-1. $$

(31)

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.

$$ P_t =P_{t-1} +a\left( {x_{t-1} -y_{t-1} } \right),\quad \quad a=1. $$

(32)

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

$$ x_t =\hat{{b}}-cP_t +\alpha x_{t-1}^B . $$

(33)

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:

$$ y_t =eP_t +\beta dy_{t-1}^U , $$

(34)

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:

$$ x_t^B =\max (x_t -y_t ,0),\mbox{i.e.}x_t =x_t^B +\min (x_t ,y_t ), $$

(35)

$$ y_t^U =\max (y_t -x_t ,0),\mbox{ i.e.}y_t =y_t^U +\min (x_t ,y_t ), $$

(36)

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:

$$ Q_t :=\sum\limits_{k=0}^t {\min (x_k ,y_k )d^{t-k}} , $$

(37)

or recursively

$$ Q_t =\min \left( {x_t ,y_t } \right)+dQ_{t-1} =x_t -x_t^B +dQ_{t-1} =y_t -y_t^U +dQ_{t-1} , $$

(38)

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:

$$ D_t :=x_t +dQ_{t-1} \mbox{\thinspace }=x_t^B +Q_t , $$

(39)

$$ S_t :=y_t +dQ_{t-1} \mbox{\thinspace }=y_t^U +Q_t , $$

(40)

dynamical system 32–36 can be rewritten as a three-dimensional system in the state variables P _t , D _t and S _t :

$$ P_t =P_{t-1} +a\left( {D_{t-1} -S_{t-1} } \right),\quad \quad a=1, $$

(41)

$$ D_t =\hat{{b}}-cP_t +\alpha D_{t-1} -(\alpha -d)Q_{t-1} , $$

(42)

$$ S_t =eP_t +\beta dS_{t-1} \mbox{\thinspace }+(1-\beta )dQ_{t-1} , $$

(43)

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

$$ \bar{{P}}=\frac{\hat{{b}}}{c+e}, $$

(44)

$$ \bar{{D}}=\bar{{S}}\left( {=\bar{{Q}}} \right)=\frac{e\bar{{P}}}{1-d}=\frac{\hat{{b}}e}{\left( {c+e} \right)\left( {1-d} \right)}. $$

(45)

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,

$$ b:=\hat{{b}}+d\bar{{S}}=\hat{{b}}+\frac{\hat{{b}}de}{\left( {c+e} \right)\left( {1-d} \right)}=\hat{{b}}\frac{c\left( {1-d} \right)+e}{\left( {c+e} \right)\left( {1-d} \right)}, $$

(46)

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:

$$ P_t =P_{t-1} +a\left( {D_{t-1} -S_{t-1} } \right)=P_{t-1} +a\left( {\hat{{b}}-\left( {c+e} \right)P_{t-1} +D_{t-1}^S } \right),\quad \quad a=1, $$

(47)

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

$$ D_t =(b_{t-1} -cP_t )+D_t^S , $$

(48)

$$ S_t =dS_{t-1} \mbox{\thinspace }+eP_t . $$

(49)

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.