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 1 / 28
Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions 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. 2 / 28
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 of 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. 3 / 28
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), ... 4 / 28
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) D t = S t The demand for and the supply of housing services is described as (2) D t = a − bR t and (3) S t = cH t By inserting (2) and (3) in (1), the rent level R t is given by (4) R t = α − β H t , where α = a b > 0 and β = c b > 0 5 / 28
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) Z t = H t The development of the housing stock is given by (6) H t = I t + (1 − δ ) H t − 1 Since housing investments in period t are described as (7) I t = γ P t − 1 , we obtain the evolution of the housing stock as (8) H t = γ P t − 1 + (1 − δ ) H t − 1 6 / 28
Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions A behavioral housing market model For a hypothetical house price level P t 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 ( P t +1 + R t − (1 + r + δ ) P t ) − c i Investor i’s mean-variance optimization problem is modeled by � � t +1 ] − λ i E i t [ W i 2 V i t [ W i (10) max Z i t +1 ] t Investor i’s solution to the above maximization problem yields t = E i t [ P t +1 ]+ R t − (1+ r + δ ) P t (11) Z i λ i V i t [ P t +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) Z t = E t [ P t +1 ]+ R t − (1+ r + δ ) P t λσ 2 From the market clearing condition for housing stock (5), it then follows that (13) P t = E t [ P t +1 ]+ R t − H t λσ 2 1+ r + δ 8 / 28
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 [ P t +1 ] = P t − 1 + χ ( P t − 1 − P t − 2 ) (15) E R t [ P t +1 ] = P t − 1 + φ ( P − P t − 1 ) The market shares of extrapolators and fundamentalists are formalized as exp [ ν A E t ] (16) N E t = exp [ ν A E t ]+ exp [ ν A R t ] exp [ ν A R t ] (17) N R t = exp [ ν A E t ]+ exp [ ν A R 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) A E t = ( P t − 1 + R t − 2 − (1 + r + δ ) P t − 2 ) Z E t − 2 (19) A R t = ( P t − 1 + R t − 2 − (1 + r + δ ) P t − 2 ) Z R t − 2 − c Investors’ average house price expectations are defined by (20) E t [ P t +1 ] = N E t E E t [ P t +1 ] + N R t E R t [ P t +1 ] 10 / 28
Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions Stability analysis Proposition 1 The model’s unique steady state, implying, amongst others, that αδ αγ P = ( r + δ ) δ +( β + λσ 2 ) γ and H = ( r + δ ) δ +( β + λσ 2 ) γ , is locally asymptotically stable if and only if E χ E χδ + γ ( β + λσ 2 ) N R φ + 2 δ + r (i) N < N E χ 1 − δ 1+ r + δ − N and R φ + γ ( β + λσ 2 ) E χ , (ii) N < 2 + r + δ + 2 N 2 − δ E = R = 1 1 where N 1+ exp [ − ν c ] and N 1+ exp [ ν c ] , respectively. Moreover, violation of the first (second) inequality is associated with a Neimark-Sacker (Flip) bifurcation. 11 / 28
Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions Interpretation of FSS The model’s unique fundamental steady state implies that αδ αγ P = ( r + δ ) δ +( β + λσ 2 ) γ , H = ( r + δ ) δ +( β + λσ 2 ) γ , R = α − β H , E = R = 1 1 1+ exp [ − ν c ] , N N 1+ exp [ ν c ] Effects of real parameters: r ↑ : P ↓ , H ↓ , R ↑ γ ↑ : P ↓ , H ↑ , R ↓ Effects of behavioral parameters: c ↑ : N E ↑ , N R ↓ χ ↑ : no effects 12 / 28
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 N E ↑ ) Effects of real parameters: r ↑ : beneficial for stability γ ↑ : harmful for stability 13 / 28
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 N E = 0 . 731 Steady state is unstable due to a Neimark-Sacker bifurcation, e.g. χ = 1 . 1 exceeds χ NS crit = 1 . 08 One time step corresponds (roughly) to one year Next: time series examples and bifurcation diagrams 14 / 28
Introduction Model Setup Analytical insights Numerical insights Tax policy Conclusions Dynamics for base parameter setting 20.1 1.05 1.00 20.0 H P 0.95 19.9 1 5 10 15 20 25 30 1 5 10 15 20 25 30 time time 0.95 0.31 N^E 0.30 0.50 R 0.29 0.05 1 5 10 15 20 25 30 1 5 10 15 20 25 30 time time 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 1.35 21.1 1.00 20.0 H P 0.65 18.9 1 10 20 30 40 50 60 1 10 20 30 40 50 60 time time 0.95 0.40 N^E 0.30 0.50 R 0.20 0.05 1 10 20 30 40 50 60 1 10 20 30 40 50 60 time time 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. 19 / 28
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