Heterogeneous expectations, housing bubbles and tax policy Carolin - - PowerPoint PPT Presentation

Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Heterogeneous expectations, housing bubbles and tax policy

Carolin Martin, Noemi Schmitt and Frank Westerhoff

Department of Economics, University of Bamberg

11th Conference on Nonlinear Economic Dynamics September 4-6, Kyiv, Ukraine

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Starting point and goal

History is replete with dramatic housing market bubbles that had serious effects for the real economy. We therefore seek to develop a plausible housing market model that helps us to understand such dynamics. We also explore to which extent policy makers can stabilize housing markets by adjusting housing market related taxes.

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Related literature

Poterba’s (1984, 1991) housing market model, including a rental market, a housing capital market and perfect foresight, reveals that housing market related taxes affect the user cost

• f housing. His main attention rests on the model’s steady

state and not on its dynamics. Dieci and Westerhoff (2016) add heterogeneous expectations (switching depends on market circumstances) to Poterba’s model and show that nonlinear interactions between real and speculative forces may lead to complex housing market dynamics.

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Related literature

In our paper, we combine the housing market model by Poterba (1984, 1991) with the heuristic switching approach by Brock and Hommes (1997, 1998). Our model produces endogenous boom-bust dynamics, which can be tamed via housing market related taxes. Currently, we see a boom in this line of research: Bolt et al. (2019), Burnside et al. (2016), Campisi et al. (2018), Dieci and Westerhoff (2012), Diks and Wang (2016), Eichholtz et

• al. (2015), Glaeser and Nathanson (2017), Kouwenberg and

Zwinkels (2014), Schmitt and Westerhoff (2019), ...

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

A behavioral housing market model

According to Poterba’s (1984, 1991) rental market setup, the market clearing condition for housing services is defined as (1) Dt = St The demand for and the supply of housing services is described as (2) Dt = a − bRt and (3) St = cHt By inserting (2) and (3) in (1), the rent level Rt is given by (4) Rt = α − βHt, where α = a

b > 0 and β = c b > 0

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

A behavioral housing market model

Poterba’s (1984, 1991) housing capital market setup entails a market clearing condition for housing stock (5) Zt = Ht The development of the housing stock is given by (6) Ht = It + (1 − δ)Ht−1 Since housing investments in period t are described as (7) It = γPt−1, we obtain the evolution of the housing stock as (8) Ht = γPt−1 + (1 − δ)Ht−1

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

A behavioral housing market model

For a hypothetical house price level Pt at time t, investor i’s end-of-period wealth is formalized as (9) W i

t+1 = (1 + r)W i t + Z i t(Pt+1 + Rt − (1 + r + δ)Pt) − ci

Investor i’s mean-variance optimization problem is modeled by (10) maxZ i

t

• E i

t[W i t+1] − λi 2 V i t [W i t+1]

• Investor i’s solution to the above maximization problem

yields (11) Z i

t = E i

t [Pt+1]+Rt−(1+r+δ)Pt

λiV i

t [Pt+1] 7 / 28

Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

A behavioral housing market model

Introducing some ”standard” assumptions allows us to express investors’ total housing demand as (12) Zt = Et[Pt+1]+Rt−(1+r+δ)Pt

λσ2

From the market clearing condition for housing stock (5), it then follows that (13) Pt = Et[Pt+1]+Rt−Htλσ2

1+r+δ

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

A behavioral housing market model

Extrapolative and regressive expectations are given by (14) E E

t [Pt+1] = Pt−1 + χ(Pt−1 − Pt−2)

(15) E R

t [Pt+1] = Pt−1 + φ(P − Pt−1)

The market shares of extrapolators and fundamentalists are formalized as (16) NE

t = exp[νAE

t ]

exp[νAE

t ]+exp[νAR t ]

(17) NR

t = exp[νAR

t ]

exp[νAE

t ]+exp[νAR t ] 9 / 28

Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

A behavioral housing market model

The fitness of extrapolative and regressive expectations is modeled by (18) AE

t = (Pt−1 + Rt−2 − (1 + r + δ)Pt−2)Z E t−2

(19) AR

t = (Pt−1 + Rt−2 − (1 + r + δ)Pt−2)Z R t−2 − c

Investors’ average house price expectations are defined by (20) Et[Pt+1] = NE

t E E t [Pt+1] + NR t E R t [Pt+1]

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Stability analysis

Proposition 1 The model’s unique steady state, implying, amongst others, that P =

αδ (r+δ)δ+(β+λσ2)γ and H = αγ (r+δ)δ+(β+λσ2)γ , is locally

asymptotically stable if and only if (i) N

Eχδ + γ(β+λσ2)N

E χ

1+r+δ−N

E χ

< N

Rφ + 2δ+r 1−δ

and (ii) N

Rφ + γ(β+λσ2) 2−δ

< 2 + r + δ + 2N

Eχ,

where N

E = 1 1+exp[−νc] and N R = 1 1+exp[νc], respectively.

Moreover, violation of the first (second) inequality is associated with a Neimark-Sacker (Flip) bifurcation.

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Interpretation of FSS

The model’s unique fundamental steady state implies that P =

αδ (r+δ)δ+(β+λσ2)γ , H = αγ (r+δ)δ+(β+λσ2)γ , R = α − βH,

N

E = 1 1+exp[−νc], N R = 1 1+exp[νc]

Effects of real parameters:

r ↑: P ↓, H ↓, R ↑ γ ↑: P ↓, H ↑, R ↓

Effects of behavioral parameters:

c ↑: NE ↑, NR ↓ χ ↑: no effects

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Interpretation of NS condition

Effects of behavioral parameters:

χ = 0, 0 < φ < 1: FSS is always stable χ ↑: stability may be lost φ ↑: stability may be regained c or ν ↑: stability may be lost (via NE ↑)

Effects of real parameters:

r ↑: beneficial for stability γ ↑: harmful for stability

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Base parameter setting

Our base parameter setting implies that P = 1, H = 20, R = 0.3 and NE = 0.731 Steady state is unstable due to a Neimark-Sacker bifurcation, e.g. χ = 1.1 exceeds χNS

crit = 1.08

One time step corresponds (roughly) to one year Next: time series examples and bifurcation diagrams

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Dynamics for base parameter setting

1 5 10 15 20 25 30 1.00 1.05 0.95 time P 1 5 10 15 20 25 30 20.0 20.1 19.9 time H 1 5 10 15 20 25 30 0.50 0.95 0.05 time N^E 1 5 10 15 20 25 30 0.30 0.31 0.29 time R Figure 1: Model dynamics for base parameter setting. 15 / 28

Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Dynamics for alternative parameter setting

1 10 20 30 40 50 60 1.00 1.35 0.65 time P 1 10 20 30 40 50 60 21.1 20.0 18.9 time H 1 10 20 30 40 50 60 0.50 0.95 0.05 time N^E 1 10 20 30 40 50 60 0.40 0.30 0.20 time R Figure 2: Model dynamics for χ = 1.35, φ = 0.75, ν = 1.3. 16 / 28

Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Effects of behavioral parameters

Figure 3: Effects of behavioral parameters. 17 / 28

Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Effects of real parameters

Figure 4: Effects of real parameters. 18 / 28

Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Tax policies

We consider the following tax policies: (1) Tax on purchase of houses (2) Tax on rental income (3) Tax on owning housing stock (4) Revenue tax for housing constructors (5) Tax on wealth of investors (6) Tax deductibility of information cost Our focus is on the model’s fundamental steady state and on the Neimark-Sacker stability condition, summarized by Propositions 2-7 and illustrated via bifurcation diagrams.

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Policy 1: Tax on purchase of houses

A tax on house purchases affects investor i’s wealth equation: W i

t+1 = (1 + r)W i t + Z i t(Pt+1 + Rt − (1 + r + δ + τ)Pt) − ci

Proposition 2

The model’s new unique steady state is given by P =

αδ (r+δ+τ)δ+(β+λσ2)γ

and H = γ

δ P and implies that N E = 1 1+exp[−νc] and N R = 1 1+exp[νc].

If N

Eχδ + γ(β+λσ2)N

E χ

1+r+δ+τ−N

E χ < N

Rφ + 2δ+r+τ 1−δ

is violated, the steady state undergoes a Neimark-Sacker bifurcation and becomes unstable.

Hence: τ ↑: P ↓, H ↓, R ↑, stability domain increases

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Effects of taxes

Figure 5: Effects of taxes on purchases of houses, rental income, owning housing stock and revenue of housing constructors, respectively. 21 / 28

Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Policy 2: Tax on rental income

A tax on rental income affects investor i’s wealth equation: W i

t+1 = (1+r)W i t +Z i t(Pt+1 +(1−τ)Rt −(1+r +δ)Pt)−ci

Proposition 3

The model’s new unique steady state is given by P =

(1−τ)αδ (r+δ)δ+((1−τ)β+λσ2)γ and H = γ δ P and implies that N E = 1 1+exp[−νc]

and N

R = 1 1+exp[νc]. If N Eχδ + γ((1−τ)β+λσ2)N

E χ

1+r+δ−N

E χ

< N

Rφ + 2δ+r 1−δ is

violated, the steady state undergoes a Neimark-Sacker bifurcation and becomes unstable.

Hence: τ ↑: P ↓, H ↓, R ↑, stability domain increases

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Policy 3: Tax on owning housing stock

A tax on housing stock affects investor i’s wealth equation: W i

t+1 = (1 + r)W i t + Z i t(Pt+1 + Rt − τ − (1 + r + δ)Pt) − ci

Proposition 4

The model’s new unique steady state is given by P =

(α−τ)δ (r+δ)δ+(β+λσ2)γ

and H = γ

δ P and implies that N E = 1 1+exp[−νc] and N R = 1 1+exp[νc].

If N

Eχδ + γ(β+λσ2)N

E χ

1+r+δ−N

E χ

< N

Rφ + 2δ+r 1−δ is violated, the steady state

undergoes a Neimark-Sacker bifurcation and becomes unstable.

Hence: τ ↑: P ↓, H ↓, R ↑, no effect on stability domain

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Policy 4: Revenue tax for housing constructors

A revenue tax for housing constructors affects their profit maximization problem: maxIt{(1 − τ)Et−1[Pt]It − Ct} Proposition 5

The model’s new unique steady state is given by P =

αδ (r+δ)δ+(β+λσ2)(1−τ)γ and H = (1−τ)γ δ

P and implies that N

E = 1 1+exp[−νc] and N R = 1 1+exp[νc].

If N

Eχδ + (1−τ)γ(β+λσ2)N

E χ

1+r+δ−N

E χ

< N

Rφ + 2δ+r 1−δ is violated, the steady state

undergoes a Neimark-Sacker bifurcation and becomes unstable.

Hence: τ ↑: P ↑, H ↓, R ↑, stability domain increases

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Policy 5: Tax on wealth of investors

A wealth tax affects investor i’s wealth equation: W i

t+1 = (1−τ)((1+r)W i t +Z i t(Pt+1+Rt −(1+r +δ)Pt))−ci

Proposition 6

The model’s new unique steady state is given by P =

αδ (r+δ)δ+(β+(1−τ)λσ2)γ and H = γ δ P and implies that N E = 1 1+exp[−νc]

and N

R = 1 1+exp[νc]. If N Eχδ + γ(β+(1−τ)λσ2)N

E χ

1+r+δ−N

E χ

< N

Rφ + 2δ+r 1−δ is

violated, the steady state undergoes a Neimark-Sacker bifurcation and becomes unstable.

Hence: τ ↑: P ↑, H ↑, R ↓, stability domain increases

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions Figure 6: Effects of wealth taxes and partial deductibility of information costs. 26 / 28

Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Policy 6: Deductibility of information cost

Deductibility of information cost, wealth of fundamentalists:

W R

t+1 = (1−τ)((1+r)Wt +Zt(Pt+1+Rt −(1+r +δ)Pt))−c(1−τx),

where x denotes fraction of deductibility Proposition 7

The model’s new unique steady state is given by P =

αδ (r+δ)δ+(β+(1−τ)λσ2)γ and H = γ δ P and implies that

N

E = 1 1+exp[−(1−τx)νc] and N R = 1 1+exp[(1−τx)νc].

If N

Eχδ + γ(β+(1−τ)λσ2)N

E χ

1+r+δ−N

E χ

< N

Rφ + 2δ+r 1−δ is violated, the steady state

undergoes a Neimark-Sacker bifurcation and becomes unstable.

Hence: x ↑: P, H, R remain constant, stability domain increases via N

E ↓

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Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions

Conclusions

We develop a novel housing market model that rests on Poterba’s (1984, 1991) user cost framework and Brock and Hommes’ (1997, 1998) heuristic switching approach. Our model generates endogenous boom-bust dynamics, provided that investors extrapolate past price changes sufficiently strongly. However, the housing market’s real side also matters. Our model allows to investigate the effects of housing market related taxes on the model’s steady state, stability and

• ut-of-equilibrium behavior.

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