# Abstract

This paper analyzes global dynamics in an overlapping generations general equilibrium model with housing-wealth effects. It demonstrates that monetary policy cannot burst rational bubbles in the housing market. Under monetary policy rules of the Taylor-type, there exist global self-fulfilling paths of house prices along a heteroclinic orbit connecting multiple equilibria. From bifurcation analysis, the orbit features a boom (bust) in house prices when monetary policy is more (less) active. The paper also proves that booms or busts cannot be ruled out by interest-rate feedback rules responding to both inflation and house prices.

# Acknowledgments

We are very grateful to an anonymous referee for many valuable comments and remarks. We also thank Paolo Canofari, Andrea Ferrero, Maurizio Fiaschetti, Ricardo Reis and participants to the 6th Annual Conference of the Portuguese Economic Journal, University of Porto, to the 54th Annual Conference of the Italian Economic Association, University of Bologna, and to the 2nd Macro Banking and Finance Workshop, University of Rome Tor Vergata, for very useful comments and suggestions. We gratefully acknowledge financial support from MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) and FCT (Fundação para a Ciência e a Tecnologia). The usual disclaimers apply.

## Appendix A. Solution of the consumer’s problem

In the intertemporal optimization problem, the representative consumer born at time *s* chooses the optimal time path of total consumption, *z*̅(*s*, *t*), to maximize the lifetime utility function (2), given (6) and the constraints (3) and (4). Using the definition of total consumption, *z*̅(*s*, *t*)≡*c*̅(*s*, *t*)+*R*(*t*)*m*̅(*s*, *t*), and the optimal intratemporal condition (6), we can write

where

subject to

and given *a*̅(*s*, 0). The optimality conditions are

Therefore, the individual budget constraint (51) can be expressed as

Integrating forward (55), using the transversality condition (54) and (52), total consumption turns out to be a linear function of total wealth:

where *k*̅(*s*, *t*) is human wealth, defined as the present discounted value of after-tax labor income,

where *L*(*R*(*t*))≡1+*R*(*t*)/Γ(*R*(*t*)). Time-differentiating (57) yields

Therefore, the dynamic equation for individual consumption is

## Appendix B. Aggregation

The per capita aggregate financial wealth is given by

Differentiating with respect to time yields

where *a*̅(*t*, *t*) is equal to zero by assumption. Using (3) yields

Using (11), the per capita aggregate consumption is given by

where

Note that *c*̅(*t*, *t*) denotes consumption of the newborn generation. Since *a*̅(*t*, *t*)=0, from (11) we have

Using (12), (63) and (65) into (64) yields the time path of per capita aggregate consumption:

## Appendix C. Proof of Proposition 1

Any equilibrium steady states, (*R*^{*}, *q*^{*}), are bounded points (*R*, *q*)∈(ℝ_{+}, ℝ_{++}) such that *R*^{*}–*P*(*R*^{*})>–*μ*. From equation (28), *q*=Ψ(*R*). From equation (29), *R*=0, because *L*′(0)=∞, or *q*=Φ(*R*). Therefore, there are two candidates for equilibrium steady states. First, we have *R*=0 and *P*(0)<0. In this case, the transversality condition is verified, because the condition –*P*(0)>–*μ* always holds. Second, from the condition *q*=Ψ(*R*)=Φ(*R*), there is a second candidate if the set {*R*≥0:Ψ(*R*)=Φ(*R*)>0} is non-empty. We have

where

and *r*=*R*–*P*(*R*)>–*μ* from the transversality condition. Because *β*(1–*α*)(*ρ*+*μ*)>0, then *r*_{+}>*ρ*>0>*r*_{–}. As *L*(*R*)>1, then the condition *r*>*ρ* must hold, which implies that the transversality condition is always met. Therefore, an equivalent condition for the existence of a second steady state is *R*–*P*(*R*) is monotonically increasing in *R*, which means that *R*–*P*(*R*)∈[–*P*(0), +∞). Therefore, the steady state *P*(0)+*r*_{+}≥0, where *P*(0)≥–*r*_{+}, there is only one steady state *P*(0)>0, and there is only steady state *P*(0)+*r*_{+}<0. We set *r*^{*}=*r*_{+}.

## Appendix D. Proof of Proposition 2

For the steady-state equilibrium

and

because

yielding the eigenvalues *λ*_{1}=2*r*^{*}–*ρ*>0 and

Evaluating the trace and the determinant for equilibrium

and

As

we have

Recall that this steady state exists only if *P*(0)<0, and observe that *L*′(0)=∞ and *L*″(0)/(*L*′(0))^{2}=–∞. Then

which means that the eigenvalues of the Jacobian are infinite and the steady state is singular. If *P*(0)+*r*_{+}>0, the steady state is multiple (case (*b*) in Proposition 2) and is a kind of non-regular saddle point, and, if *P*(0)+*r*_{+}<0, the steady state is unique (case (*c*) in Proposition 2) and is a kind of non-regular source. In any case, the flow approaches or diverges from

In order to study local dynamics, we can use several methods, such as, first, finding a de-singularized projection and studying its local dynamics, and, second, studying global dynamics.

If we use the first method, the natural way to remove the singularity introduced by *L*′(*R*) at *R*=0, would be to recast the system in variables (*L*, *q*),

where *L*≥1, *R*=*R*(*L*) is increasing and *R*(1)=0, *R*′(1)=0. However, in this case there is an unique steady state *P*(0)+*r*_{+}>0 (i.e. case (*b*) in Proposition 2) behaves as a generalized saddle point.

## Appendix E. Proof of Proposition 3

As the system (28)–(29) does not have an explicit solution, we must employ qualitative methods in order to study global dynamics. One possible method is to find a first integral of system (28)–(29), that is, a Lyapunov function *V*(·) such that *V*(*R*, *q*)=constant. We could not find this function. Another method is to determine a trapping area for the heteroclinic orbit. The rationale is the following: as the steady state _{+}; therefore there is an intersection of the unstable manifold of the first point and of the stable manifold of the second which is non-empty. In order to prove that it exists, and to characterize it, we consider a trapping area for the heteroclinic orbit. In order to prove this, we start by determining the slopes of the heteroclinic orbit in the neighborhoods of the two equilibria, we build a trapping area enclosing the heteroclinic, and demonstrate that all the trajectories starting inside the trapping area escape from it, with the exception of those starting at any point along the heteroclinic orbit.

### Appendix E.1. Slopes of the eigenspaces associated to the two equilibria

The unstable eigenspace

The unstable eigenspace *V*_{1}, 1)^{⊤} and (*V*_{2}, 1)^{⊤} which are associated to eigenvalues *λ*_{1} and *λ*_{2}, respectively, where

In general, *V*_{2}>0 and the sign of *V*_{1} is ambiguous. Observe that the the slope of *λ*_{1}>*λ*_{2}, and therefore *V*_{1}<0 and *V*_{2}>0 and the slope associated to the dominant eigenvalue is negative and the slope associated to the non-dominant eigenvalue is positive; (2) if *λ*_{1}<*λ*_{2}, which is equivalent to *V*_{1}>*V*_{2}>0 and the slope associated to the both eigenvalues are both positive, but the one associated with

The stable manifold associated to steady state

However, we saw that the projection of the steady state *R*′(1)=0, then a naïve calculation for the slope of the stable manifold in the neighborhood of the singular equilibria could be *dq*/*dR*=(*r*_{+}–*ρ*)/(*αβ*(*ρ*+*μ*)*R*′(1))=∞.

Instead, observe that along the singular surface *R*=0 we have *R*>0 will be deflected away from the equilibrium point on hitting the surface *R*=0, with the exception of the one which converges to the equilibrium point

In order to determine the slope of the trajectory which converges to the singular steady state, we determine the slope of the vector field hitting the surface *R*=0 using equation (67). We have

because *L*′(0)=∞. Therefore, the stable manifold associated to the singular steady state

The heteroclinic orbit,

### Appendix E.2. Trapping area

Next we consider the case in which *λ*_{1}<*λ*_{2}, which is depicted in Figure 1. Recall that Ω is tangent to a line *dq*/*dR*=0 in the neighborhood of point *λ*_{1}) and of the isocline

### Figure 7:

As the heteroclinic Ω is tangent to that eigenspace in the heighborhood of that equilibrium point, it will lie between the isocline

This allows to consider the trapping area whose sides are given by the segment of the isocline *q*-axis and the previous eigenvector-line.

Formally, the trapping area [*A*, *B*, *C*] is defined by the vertices

Next we have to demonstrate that the (global) direction of the vector field, given by equation (67), which is generated by equations (28)–(29), allows us to prove that all the trajectories hitting the three boundaries of the trapping exit the trapping area.

At side (*A*, *B*) we have

given the fact that –*r*_{–}<*R*–*P*(*R*)<*r*_{+} if

then the vector field points globally out of the trapping area within (*A*, *B*) with a horizontal slope.

At side (*A*, *C*), the vector field corresponds to a horizontal line between point *A* and point *C*, which is in the intersection of a horizontal line passing through the *q*-axis and the direction defined by the eigenvector associated to the dominant eigenvalue at point *B*. These two lines meet at point *C*. Along (*A*, *C*), we have

then the vector field points globally out of the trapping area, at all points located at (*A*, *C*).

At side (*B*, *C*), which is a segment of the eigenspace *V*_{1}, then the trajectories exit the trapping area if the slope of the vector field is less steep than *V*_{1}, that is, if and only if

The numerator is equivalent to

It is negative because *R*–*P*(*R*)>*r*_{+}, *q*>Ψ(*R*), and because we assume *λ*_{1}<*λ*_{2}. Then, all the trajectories reaching segment (*B*, *C*) will exit the trapping area.

Therefore, there is an unique trajectory starting from point *B*, *A*,

## Appendix F. Proof of Proposition 4

We use the same methods as for the proof of Propositions 1, 2 and 3.

First, the steady-state conditions are *q*=Ψ(*R*, *q*) and *R*=0 or *q*=Φ(*R*, *q*), and a steady state is an equilibrium if *R*^{*}–*P*(*R*^{*}, *q*^{*})+*μ*>0. A steady state exists and is an equilibrium if there is a *q*^{*}>0 such that *q*^{*}=Ψ(0, *q*^{*}) and –*P*(0, *q*^{*})>–*μ*. Using the Taylor rule (21), the equilibrium condition is equivalent to (*q*–*q*_{+})(*q*–*q*_{–})=0, where

As *q*_{–}<0<*q*_{+}, then

which holds for any *P*(0, 0). As

then the transversality condition holds without further conditions.

An interior steady state is determined from the non-negative values of (*R*, *q*) such that *q*=Ψ(*R*, *q*)=Φ(*R*, *q*)>0. If we define *r*(*R*, *q*)≡*R*–*P*(*R*, *q*), this condition is equivalent to (*r*–*r*_{+})(*r*–*r*_{–})=0. As a necessary condition for a positive *q* is *r*(*R*, *q*)>*ρ*, then *r*(*R*, *q*)=*r*_{+}>–*μ*, which means that the transversality condition is automatically verified. This is equivalent to *γR*+ϵ*q*=(1+*γ*)(*P*(0, 0)+*r*_{+}). Substituting in *q*=Ψ(*R*, *q*), we get the equation

After some algebra, we can prove that this equation has a non-negative solution for *R* if and only if *r*_{+}+*P*(0, 0)≥*BP*(0, 0)>0.

The local dynamics are determined in the same way as in the proofs of Propositions 2 and 3. But, in this case, the eigenvalues for the steady state

for the values of the parameters such that there are two steady states.

## Appendix G. Proof of Proposition 5

It is easy to see that the steady state conditions are exactly the same as in the case of the conventional Taylor rule. The only thing that may change is related to the local and global dynamics of the model.

Applying the same methods as for the proof of Proposition 2, we find the eigenvalues for the steady state

where sign(*λ*_{2})=sign(1+*γ*–*δ*).

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**Published Online:**2016-3-1

**Published in Print:**2016-6-1

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