Das House-Kapital: A Long Term Housing & Macro … House-Kapital: A Long Term Housing & Macro Model....
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Das House-Kapital: A Long Term Housing & Macro Model
Volker Grossmann (University of Fribourg, CESifo, IZA, CReAM)
Thomas Steger (Leipzig University, IWH, CESifo)
European Summer Symposium in International Macroeconomics (ESSIM) 2017
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1. Introduction – Motivation
Long-Term, Time-Series Data Housing wealth (Piketty & Zucman, QJE 2014) (data)
• Largest private wealth component • As share of private wealth: US 44%, UK 57% (in 2010) • Growing over time more than income (housing wealth-income ratio)
House and land prices (Knoll, Schularick & Steger, AER 2017) (data) • On average in major OECD countries, house prices tripled since 1950 • Even larger increases of land prices
Long-Term Research Questions Future evolution of housing wealth-income ratio, house prices and land prices?
How does secular increase in housing demand affect distribution of wealth?
Dynamic consequences of rent control for inequality and welfare?
Dynamic consequences of housing property taxation?
Macroeconomic effects of zoning regulations and building restrictions?
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1. Introduction – Model Features
Two-sector Ramsey Growth Model Non-residential sector Housing sector
Three Premises Premise 1: Fixed Land Endowment (Stock) Overall amount of land that can be used economically is fixed in the long run Mark Twain: “Buy land, they're not making it anymore.”
Premise 2: Land Rivalry Land used in housing production is permanently withdrawn from alternative
use (in non-residential sector) Premise 3: Land in Housing Production
a) Setting up new housing projects (real estate development) requires land b) Investment in residential structures does not require land (e.g. building higher
houses or fix broken windows)
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1. Introduction – Related Literature
Knoll, Schularick & Steger (2017): long term evolution of real house prices and land prices in 14 countries
Piketty & Zucman (2014, 2015): data on wealth to income (house-capital and non-residential wealth)
Ronglie (2015): housing sector drives rise in capital income share
Stiglitz (2015): land prices important for rising wealth-to-income ratios and rising inequality of wealth and income
Davis & Heathcote (2005), Hornstein (2009), Iacoviello & Neri (2010), Favilukis, Ludvigson & Van Nieuwerburgh (2017), Borri & Reichlin (2016): canonical housing-macro model Suitable for business cycle phenomena
Less suited to think long term Limited land scarcity (land is a fixed flow used for residential investment) No land rivalry Doesn’t fit SNA concepts: missing wealth component (non-residential land)
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2. House-Kapital Model – Firms
Housing Sector
Firm type 1: Real estate development
Firm type 2: Construction
Firm type 3: Housing services (decreasing returns)
( )2
2, Z N N
t tt tN
t Cost P I IIN ξ= +=
land costtransformation
co
contr l
st
o
( ) ( )1,X X X X Xt t t t t t tX I X I B M L
ηηδ−
= − =
( )t th x γ=tt tN h S=
𝒁𝒁 0 𝑁𝑁
𝑥𝑥=X/N
Land
Numeraire Sector
Technology
Market Clearing
Numeraire Sector
Housing Sector
( ) ( ) ( )1Y Yt t t t
YtY B K L Z
α β α β− −=
( )
2
2 tt t NK
t
I
K Nt t t t
K K
Y C M I Iδ ξ+
+ += +
𝑍𝑍𝑌𝑌
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2. House-Kapital Model – Households
Household Optimization
Private Wealth
( )
( )
0 0{ , }
0
max log log
s.t. (1 ) (1 )
lim exp (1 )(0) given 0
t t t
tt t
C S
t r t t w t t t t t t
t
t r stW
C S e dt
W rW w L C p S T
W r ds
ρθ
τ τ
τ
∞=
∞ −
→∞
+
= − + − − −
−=
+
− ≥
∫
∫
: total financial asset holding non-residential land housing wealth non-residential wealth
Ht t
t
P N
N X Z Y H Z Yt t t t t t t t t t t t t
A
W q N q X K P Z P N K P Z≡ + + + = + +
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2. House-Kapital Model – Dynamic System
7 dynamic equations…
plus a set of static equations…
( )
( )
( )
11 1
1
(1 )
1
1
X X
YZ Y
K Y
X
w B q
w LR Br Z
R p x
px
ηη η
α βα
γ
γ
η η
αα ββ δ
γ
π γ
− −
+
−
= −
= − − +
=
= −
# housing housing projects services
per firm
Y
X Y
N X Z YW
Xx
Z N Z
L L L
K q N q X P Z
N
N
S xγ
=
+ =
= + + +
=
=
+
11
11
1
YK
X
X X
K YY
Y
C pS
wK Lr
LM q B
r w ZBL
η
α α βα
θ
αβ δ
η
δα β
−
− −−
=
=+
=
+=
( )1X X X
N Z
X B M L X
q PN
ηη δ
ξ
−= −
−=
[ ](1 )r
W rW wL C pS
C C rτ ρ
= + − −
= − −
( )
N N
Z Z Z
X X X X
q rq
P rP R
q r q R
π
δ
−
−
=
= + −
=
Analytical steady states Transitional dynamics
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3. Thinking Long Term – Steady States
𝑃𝑃𝑍𝑍∗ and 𝑃𝑃𝐻𝐻∗ depend positively on 𝐵𝐵𝑌𝑌 and 𝐷𝐷 ≔ 𝐿𝐿/𝑍𝑍 (not on 𝐵𝐵𝑋𝑋)
Ricardo's (1817) principle of scarcity.
( ) ( )1 1
* *1 1 1 1( ) ( )Y H YZP a r PD b r DB Bβ β
α α α α∗ ∗− − − −
− −= =
Does not depend on 𝐵𝐵𝑋𝑋, 𝐵𝐵𝑌𝑌 or 𝐷𝐷
Same for other ratios, such as
• non-residential wealth-to-income ratio 𝐾𝐾∗+𝑃𝑃𝑍𝑍∗𝑍𝑍𝑌𝑌∗
𝑁𝑁𝑁𝑁𝑃𝑃∗
• House price-to-rent ratio 𝑃𝑃𝐻𝐻∗
𝑝𝑝∗ℎ∗.
Land price & house price
Housing wealth-to-income ratio 𝑃𝑃𝐻𝐻∗𝐻𝐻∗
𝑁𝑁𝑁𝑁𝑃𝑃∗
Number of houses per unit of land, 𝑁𝑁∗/𝑍𝑍
Does not depend on 𝐵𝐵𝑋𝑋, 𝐵𝐵𝑌𝑌 or 𝐷𝐷 Increases in housing demand do not affect long land allocation
Residential structures per unit of land, 𝑋𝑋∗/𝑍𝑍, are increasing in 𝐵𝐵𝑋𝑋, 𝐵𝐵𝑌𝑌 and 𝐷𝐷
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3. Thinking Long Term – Calibration Initial conditions Initial housing stock (𝑁𝑁0, 𝑋𝑋0) set to match empirical 𝑃𝑃0
𝐻𝐻𝑁𝑁0𝑁𝑁𝑁𝑁𝑃𝑃0
in 1955
Initial capital (𝐾𝐾0) set to match empirical 𝐾𝐾0+𝑃𝑃0𝑍𝑍𝑍𝑍0𝑌𝑌
𝑁𝑁𝑁𝑁𝑃𝑃0 in 1955
Growth in TFP (𝐵𝐵𝑡𝑡𝑋𝑋, 𝐵𝐵𝑡𝑡𝑌𝑌) Matches GDP growth between 1955 and 2010 Given observed growth in population size (𝐿𝐿𝑡𝑡)
Directly observable Depreciation rates 𝛿𝛿𝐾𝐾, 𝛿𝛿𝑋𝑋 Capital income tax rate 𝜏𝜏𝑟𝑟
Other observables set to match Expenditure share for housing services Labor income share Value-added of residential construction Sectorial employment shares Sectorial investment rates
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1960 1980 2000 2020 2040 2060 2080 2100t0
100
200
300
400
PHNNDP
FRA
1960 1980 2000 2020 2040 2060 2080 2100t0
100
200
300
400
PHNNDP
GER
1960 1980 2000 2020 2040 2060 2080 2100t0
100
200
300
400
PHNNDP
UK
1960 1980 2000 2020 2040 2060 2080 2100t0
100
200
300
400
PHNNDP
USA
Data Source: Piketty & Zucman (2014).
3. Thinking Long Term – Housing Wealth-to-NDP Ratio
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3. Thinking Long Term – House Price & Land Price
Land price (1955 – 2010): Model: factor 7.5; Data: factor 5.2 (France, UK, USA)
House price (1955 – 2010): Model: factor 3.4; Data: factor 3.2 (France, UK, USA)
Calibrated Model
1960 1980 2000 2020 2040 2060 2080 2100
2
4
6
8
10
12
Land price PZ
House price PH
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Caselli-Ventura (AER 2000) structure with 𝐽𝐽 groups of HH.
Heterogeneity in initial wealth holding (percentiles, US): 𝑊𝑊𝑗𝑗(0) with 𝑗𝑗 ∈ 1, … , 𝐽𝐽 .
Representative consumer, despite non-homothetic preferences.
If 𝜙𝜙 > 0, poorer HH has higher housing expenditure share
Two-stage numerical solution procedure.
( )
0
11
0{ , }
capital non-residential land houses
:
(
( ) 1max
1
s.t. (1 ) (1
0) given, NPGC holds
)
j j t
j j t
c s
j r j w j j j
Z Y Hj j j j
j
c s se dt
W rW wL c ps T
W K P Z P N
W
σθ θ
ρφ
σ
τ τ
∞=
−−
∞ −
=
− − −
= − + − − − +
= + +
∫
4. Dynamics of Wealth Inequality – Households Grossmann, Larin, Loefflad & Steger (2017)
1 (:
1 )
jj
j j
j
pse
c pss s
θφθ
= =+ − −
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4. Dynamics of Wealth Inequality – Results (1)
How does economic growth affect the distribution of wealth?
Experiment / calibration strategy Exogenous TFP growth and endogenous capital accumulation (population constant). State variables such that wealth-to-income ratios (2010) are matched. Portfolio structure of HH equals aggregate portfolio structure. Initial wealth distribution (percentiles) from World Wealth and Income Database.
Amplification of increase in wealth inequality if expenditure shares differ.
Rent increase affects the poor more, suppresses their ability to accumulate wealth.
(percent)
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4. Dynamics of Wealth Inequality – Welfare
How does economic growth affect the HH-specific welfare?
EV measure: 𝜓𝜓𝑗𝑗 is factor of the ideal consumption index each period in baseline scenario (0) such that HH j is indifferent to alternative scenario (1).
All prefer the growth scenario Rich prefer it more than the poor Difference more pronounced if
housing expenditure shares differ!
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4. Dynamics of Wealth Inequality – Rent Control
Effect of rent control in growing economy on HH-specific welfare: 𝑝𝑝 ≤ 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚
Poor prefer alternative scenario, rich dislike alternative scenario!
If 𝜙𝜙 > 0, representative agent may prefer alternative scenario: No equity-efficiency trade-off
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5. Summary & Conclusions
New Housing & Macro Model: Think Long Term Real estate development (extensive margin) withdraws land from alternative uses Investment in structures (intensive margin) does not require land Consistent with evidence that 80 percent of house price increase since WWII is
associated with rising land prices (Knoll et al., 2017) Rising wealth-to-income ratio over time (housing vs. non-residential wealth)
• In long run (for baseline calibration): 𝟒𝟒𝟒𝟒𝟒𝟒 + 𝟑𝟑𝟑𝟑𝟒𝟒 = 𝟕𝟕𝟑𝟑𝟒𝟒 (percent) • Sensitive w.r.t. long run interest rate • But independent of TFP and population size
Wealth inequality and welfare Secular increase in housing demand associated with rising wealth inequality Amplified by endogenous heterogeneity in housing expenditure shares
Poor prefer rent ceiling, while rich dislike rent ceiling
Rent control possibly efficiency-enhancing
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Supplement – Wealth-to-NDP ratios: Long Run Implications
𝟒𝟒𝟒𝟒𝟒𝟒 + 𝟑𝟑𝟑𝟑𝟒𝟒 = 𝟕𝟕𝟑𝟑𝟒𝟒
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Housing wealth-to-NDP ratio Non-residential wealth-to-NDP ratio
Source: Piketty & Zucman (2014). Note: Time series smoothed over decades.
Non-residential wealth: private wealth ./. housing wealth ./. net foreign assets.
Housing wealth: gross housing assets (not reduced by mortgages and other financial liabilities).
Supplement – Private Wealth
100%120%140%160%180%200%220%240%260%280%300%
1950 1960 1970 1980 1990 2000 2010
USA
50%
70%
90%
110%
130%
150%
170%
190%
210%
230%
1950 1960 1970 1980 1990 2000 2010
GER
50%
100%
150%
200%
250%
300%
1950 1960 1970 1980 1990 2000 2010
FRA
50%
100%
150%
200%
250%
300%
1950 1960 1970 1980 1990 2000 2010
UK
back
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Supplement – House Prices & Land Prices
Source: Knoll et al. (2017) back
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Source: Knoll et al. (2017)
Supplement – House Prices (country by country)
back
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Numeraire good
𝑌𝑌𝑡𝑡 = 𝐵𝐵𝑡𝑡𝑌𝑌 𝐾𝐾𝑡𝑡𝑌𝑌 𝛼𝛼 𝐿𝐿𝑡𝑡𝑌𝑌 1−𝛼𝛼
non-residential land missing
Construction
𝑋𝑋𝑡𝑡 = 𝐵𝐵𝑡𝑡𝑋𝑋 𝐾𝐾𝑡𝑡𝑋𝑋 𝛾𝛾 𝐿𝐿𝑡𝑡𝑋𝑋 1−𝛾𝛾
intermediate input
Housing services
𝐵𝐵𝑡𝑡𝐻𝐻𝑋𝑋𝑡𝑡𝛽𝛽�̅�𝑍1−𝛽𝛽
gross investment
= 𝐻𝐻𝑡𝑡�̇net
investment
+ 𝛿𝛿𝐻𝐻𝐻𝐻𝑡𝑡replacementinvestment
𝒁𝒁� is (time-invariant) flow variable
Housing market clearing
𝑆𝑆𝑡𝑡 = 𝑝𝑝𝑡𝑡 𝐻𝐻𝑡𝑡
housing consumption
Capital market clearing
𝐾𝐾𝑡𝑡𝑋𝑋+𝐾𝐾𝑡𝑡𝑌𝑌 = 𝐾𝐾𝑡𝑡
Labor market clearing
𝐿𝐿𝑡𝑡𝑋𝑋+𝐿𝐿𝑡𝑡𝑌𝑌 = 𝐿𝐿𝑡𝑡
Supplement: Canonical Model
Davis and Heathcote (2005), Hornstein (2009), Iacoviello and Neri (2010), Favilukis et al. (2015), and Borri and Reichlin (2016)
Merits & Features Suitable for business cycle phenomena Limited land scarcity / No land rivalry
Long-run inconsistency: Replacement investment require land: ∫ �̅�𝑍𝑑𝑑𝑡𝑡∞0 = ∞!
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Supplement – Comparison of Models Land Price in Response to Housing Stock Destruction
0 20 40 60 80 100 120 140t
1.05
1.10
1.15
1.20
1.25
PZa Canonical Model: H0 0.8H
0 20 40 60 80 100 120 140t
0.950.960.970.980.991.00
PZb House Capital Model: N0 0.8N , X0 0.8X
0 20 40 60 80 100 120 140t
0.990
0.992
0.994
0.996
0.998
1.000PZ
c House Capital Model: N0 0.8N , X0 X
0 20 40 60 80 100 120 140t
0.96
0.97
0.98
0.99
1.00PZ
d House Capital Model: N0 N , X0 0.8X
Two models differ on isolated impact of housing stock destruction on 𝑃𝑃𝑍𝑍 dynamics.
The reason is that the land price determination is different!
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A general equilibrium consists of sequences of quantities, prices, and operating profits of housing services producers
for initial conditions (𝐾𝐾0, 𝑁𝑁0, 𝑋𝑋0) > 0 and 𝐵𝐵𝑡𝑡𝑌𝑌 ,𝐵𝐵𝑡𝑡𝑋𝑋 ,𝐵𝐵𝑡𝑡ℎ , 𝐿𝐿𝑡𝑡 𝑡𝑡=0∞ such that
1. representative household maximizes lifetime utility;
2. representative firm in 𝑋𝑋 sector and 𝑌𝑌 sector, representative real estate developer, and housing services producer maximize PDV of infinite profit stream, taking prices as given;
3. land market, labor market, bond market and market for structures clear: 𝑍𝑍𝑡𝑡𝑌𝑌 + 𝑁𝑁𝑡𝑡 = 𝑍𝑍, 𝐿𝐿𝑡𝑡𝑋𝑋 + 𝐿𝐿𝑡𝑡𝑌𝑌 = 𝐿𝐿𝑡𝑡, 𝐾𝐾𝑡𝑡𝑌𝑌 = 𝐾𝐾𝑡𝑡, 𝑋𝑋𝑡𝑡 = 𝑁𝑁𝑡𝑡𝑥𝑥𝑡𝑡;
4. land price is the PDV of rental rates per unit of land in 𝑌𝑌 sector;
5. financial asset market clears: 𝐾𝐾𝑡𝑡 + 𝑞𝑞𝑡𝑡𝑁𝑁𝑁𝑁𝑡𝑡 + 𝑞𝑞𝑡𝑡𝑋𝑋𝑋𝑋𝑡𝑡 + 𝑃𝑃𝑡𝑡𝑍𝑍𝑍𝑍𝑡𝑡𝑌𝑌 = 𝑊𝑊𝑡𝑡;
6. market for housing services clears: 𝑆𝑆𝑡𝑡 = 𝑁𝑁𝑡𝑡ℎ𝑡𝑡;
7. market for Y good clears: 𝑌𝑌𝑡𝑡 = 𝐶𝐶𝑡𝑡 + 𝐼𝐼𝑡𝑡𝐾𝐾 + 𝐼𝐼𝑡𝑡𝑁𝑁 + 𝑀𝑀𝑡𝑡 (redundant due to Walras’ law)
Supplement: General Equilibrium
This image cannot currently be displayed.
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Symbol Meaning Y final output of numeraire good
KY=K physical capital
LX, LY labor in 𝑋𝑋 and 𝑌𝑌 production
ZY land in 𝑌𝑌 sector
X, x=X/N residential buildings (aggregate and per house)
M materials (in terms of numeraire)
δK, δX depreciation rates (physical capital and buildings)
N number of housing projects
qN, , qX shadow price of N and X
RZ, PZ rental rate and price of land
h housing services (per housing project)
S=Nh total housing services‘ demand and supply
τr , τw , T policy parameters
Supplement – Notation