Bubble on real estate: the role of altruism and fiscal policy

A.1 Households’ behaviour

We maximize the Lagrangian function:

with respect to (c1,t,c2,t+1,c3,t+2,kt+2,dty,dt+1m,ht+1,xt+2,λ1,t,λ2,t+1,λ3,t+2,λ4,t+1).

First-order conditions are given by:

From Eqs. (62)–(63), we get : Rt+2=Rt+2d. Physical capital and deposit are perfect substitutes. From Eqs. (61) and (63), one has:

We restrict our attention to the case of a binding borrowing constraint to be able to discuss our results according to the importance of the different types of collateral. Using (64) and the Kuhn and Tucker conditions, this is ensured by λ4,t+1>0. Substituting Eqs. (56)–(58) and (65) into Eqs. (59)–(63), one has :

A.2 Proof of Proposition 1.

One can easily show that R_<1 if and only if δ>δ^. The borrowing constraint is binding if c3>β2R_2c1. This inequality is equivalent to:

Since R_<1 under δ>δ^, inequality (70) is satisfied if

We can write A₁ and sR_+A3 as continuous functions of ϕ and θ₂:

Note that

For θ₂ = 0, Inequality (70) is satisfied if LHS(ϕ,0)>RHS(ϕ,0). Note that LHS(ϕ,0) is an increasing function of ϕ and RHS(ϕ,0) a decreasing function of ϕ. We have LHS(+∞,0)>RHS(+∞,0) and LHS(1,0)<RHS(1,0) if δ is sufficiently high. Therefore, it exists a threshold ϕ_0>1 such that for ϕ>ϕ_0, LHS(ϕ,0)>RHS(ϕ,0). Note that LHS(ϕ,θ2) and RHS(ϕ,θ2) are both increasing functions of θ₂. Moreover, for δ high enough, LHS(+∞,1)<RHS(+∞,1). By continuity, there exist thresholds θ¯2∈(0,1) and ϕ_>1 such that for θ2<θ2¯ and ϕ>ϕ_, LHS(ϕ,θ2)>RHS(ϕ,θ2). Therefore, if δ is sufficiently high such that δ>δ^, ϕ>ϕ_ and θ2<θ¯2, Proposition 1 follows.

A.3 Proof of Proposition 2.

A bubbly steady state R¯ is a solution of G(R)=H(R), where G(R) and H(R) satisfy (28)–(29).

We start by determining the admissible range of values for R. We have already seen that θ1<R<1. We remind that we restrict our attention to the existence of a productive bubble (i.e. R¯<R_). As p=RG(R)a, B1+RB3 must be negative to get a positive bubble.

As B1=θ1+(1+θ1)β/(β2+δ)>0, B1+RB3<0 implies R>−B1/B3 with B3=(β2θ1−δ)/(β2+δ)<0. Note that under Assumption 2 (δ>δ¯), B₃ < 0 and −B1/B3>θ1. Then, the possible admissible range of values for productive bubbles is R∈(−B1/B3,R_), with R_<1.

To prove the existence of a stationary solution R¯∈(−B1/B3,R_), we use the continuity of G(R) and H(R). Note that G(R) has a vertical asymptote at R=−B1/B3 and is a decreasing and continuous function of R when R increases to R_. H(R) has two vertical asymptotes at R=θ1<−B1/B3 and R=(1+A2)/θ1>1. From Eqs. (28) and (29), we determine the boundary values of G(R) and H(R):

We have limR→−B1/B3H(R)<limR→−B1/B3G(R). The existence of a steady state R¯∈(−B1/B3,R_), solving G(R)=H(R), is ensured by 0<H(R_), namely:

Note that (1−θ1R_)/(R_−θ1)>1 if and only if R_<1. Using the proof of Proposition 1, Inequality (76) is in accordance with Inequality (71) and satisfied by continuity argument if ϕ close but larger than ϕ_ and θ2∈(0,θ¯2), because it ensures that A₁ is larger but close to sR_+A3.

We ensure now that the set (−B1/B3,R_) is non-empty. The set (−B1/B3,R_) is non-empty if and only if −B1/B3<R_ with:

Note that LHS(δ,θ1) and RHS(δ) are decreasing functions of δ and we have that:

We deduce that limδ→+∞LHS(δ,θ1)<limδ→+∞RHS(δ) if and only if:

If θ1<θ¯1, there exists a threshold δ~>0 such that for δ>δ~, LHS(δ,θ1)<RHS(δ). Therefore, if θ1<θ¯1 and δ>δ~, the set (−B1/B3,R_) is nonempty.

To conclude, if ϕ is close but larger than ϕ_, θ2<θ¯2, θ1<θ1¯ and δ sufficiently high such that δ>max{δ~,δ¯,δ^}, there exists a steady state R¯∈(−B1/B3,R_) which coexists with the bubbleless one R_.

To prove uniqueness, we observe that the equation H(R)=G(R) is equivalent to a polynomial of degree 3, i.e. has at most three solutions. Since limR→(1+A2)/θ1H(R)=−∞ and limR→(1+A2)/θ1G(R) is finite, we have limR→(1+A2)/θ1H(R)<limR→(1+A2)/θ1G(R). We have established that limR→R_H(R)>limR→R_G(R). This means that an odd number of solutions to the equation H(R)=G(R) belong to (R_,(1+A2)/θ1). Since we also have limR→−B1/B3G(R)>limR→−B1/B3H(R), there are an odd number of solutions to H(R)=G(R) that belong (−B1/B3,R_). Since there are at most three solutions, the solution R¯ in this last interval is unique.

A.4 Households’ behaviour in the model with taxation

We maximize the Lagrangian function:

with respect to (c1,t,c2,t+1,c3,t+2,kt+2,dty,dt+1m,ht+1,xt+2,λ1,t,λ2,t+1,λ3,t+2,λ4,t+1). First-order conditions are given by:

From Eq. (83), we get Rt+2=Rt+2d. From Eqs. (82) and (83), one has:

We restrict our attention on the case where λ4,t+1>0, i.e. λ1,t>Rt+1dRt+2dλ3,t+2. Substituting Eq. (79) into Eqs. (80), (81) and (83), we deduce:

A.5 Proof of Proposition 5.

The existence of the steady state is obtained because GT(R)>HT(R) when R tends to −B~1/B~3 and GT(R)<HT(R) when R tends to R__T. Uniqueness allows to deduce that at the steady state, we have GT′(R)<HT′(R).

Using Eq. (53), we have:

We start by computing ∂GT(R¯T)/∂(1−τh). Note that B₂ and B₄ do not depend on τ_h, while we have:

because R>θ1 and δ>β/θ1. We deduce that:

because B~1+R¯TB~3<0.

Using Eq. (55), H_T(R) can be rewritten as follows:

Obviously, ∂HT(R¯T)/∂(1−τh) has the sign of its numerator, i.e.

Using Eq. (92), we deduce that dR¯T/d(1−τh)<0. This means that R¯_T increases with τ_h, whereas ā_T decreases with τ_h.