Secular Stagnation, Land Prices, and Collateral Constraints Zhifeng - - PowerPoint PPT Presentation

Secular Stagnation, Land Prices, and Collateral Constraints

Zhifeng Cai

UMN

November 29, 2016

1 / 33

Motivation

◮ The 2007-2008 Financial Crisis differs considerably from

• ther postwar recessions

◮ Larger declines of macro variables ◮ Slower recovery 2 / 33

Motivation

Housing Price GDP

• .15
• .1
• .05

% deviation from trend

5 10 15 20

Quarters after recessions start

The Great Recession Previous Recessions

GDP

• .4
• .3
• .2
• .1

.1

% deviation from trend

5 10 15 20

Quarters after recessions start

Investment

• .15
• .1
• .05

% deviation from trend

5 10 15 20

Quarters after recessions start

Labor

• .4
• .3
• .2
• .1

.1

% deviation from trend

5 10 15 20

Quarters after recessions start

Housing Price

3 / 33

Overview

◮ Question:

• 1. Why slow recovery following the Great Recession?
• 2. What role did the real-estate(land) sector play?

◮ Proposes a standard neoclassical model with a land sector where land

serves dual roles:

• 1. As consumption for the households
• 2. As collateral for the firms to finance borrowing and working capital

◮ Results:

• 1. Theory: existence of multiple steady states.
• 2. Quantitative: substantial persistence upon large recessions

◮ Large recessions trigger transitions across steady states 4 / 33

Mechanism

𝐻ood Steady State

𝐼𝑗𝑕ℎ 𝐷𝑏𝑞𝑗𝑢𝑏𝑚 𝐵𝑑𝑑𝑣𝑛𝑣𝑚𝑏𝑢𝑗𝑝𝑜

𝐼𝑗𝑕ℎ 𝐼𝑝𝑣𝑡𝑓ℎ𝑝𝑚𝑒𝑡 𝑋𝑓𝑏𝑚𝑢ℎ

𝐼𝑗𝑕ℎ 𝑀𝑏𝑜𝑒 𝑄𝑠𝑗𝑑𝑓

𝑆𝑓𝑚𝑏𝑦𝑓𝑒 𝐺𝑗𝑠𝑛 𝑋𝑝𝑠𝑙𝑗𝑜𝑕 𝐷𝑏𝑞𝑗𝑢𝑏𝑚 𝐷𝑝𝑜𝑡𝑢𝑠𝑏𝑗𝑜𝑢

𝐼𝑗𝑕ℎ 𝐹𝑛𝑞𝑚𝑝𝑧𝑛𝑓𝑜𝑢

𝐶𝑏𝑒 Steady State

𝑀𝑝𝑥 𝐷𝑏𝑞𝑗𝑢𝑏𝑚 𝐵𝑑𝑑𝑣𝑛𝑣𝑚𝑏𝑢𝑗𝑝𝑜

𝑀𝑝𝑥 𝐼𝑝𝑣𝑡𝑓ℎ𝑝𝑚𝑒𝑡 𝑥𝑓𝑏𝑚𝑢ℎ

𝑀𝑝𝑥 𝑀𝑏𝑜𝑒 𝑄𝑠𝑗𝑑𝑓

𝑈𝑗𝑕ℎ𝑢𝑓𝑜𝑓𝑒 𝐺𝑗𝑠𝑛 𝑋𝑝𝑠𝑙𝑗𝑜𝑕 𝐷𝑏𝑞𝑗𝑢𝑏𝑚 𝐷𝑝𝑜𝑡𝑢𝑠𝑏𝑗𝑜𝑢

𝑀𝑝𝑥 𝐹𝑛𝑞𝑚𝑝𝑧𝑛𝑓𝑜𝑢

5 / 33

Related Literature

◮ The paper relates to macro models with collateral constraints

(Kiyotaki and Moore, 1997)

◮ Typical model features a unique steady state

◮ Recent extensions still feature a unique steady state:

• 1. Consumption role of land: Liu, Wang, and Zha (2013), Iacoviello

(2005)

• 2. Working capital: Jermann and Quadrini (2012), Mendoza (2010)

◮ The paper incorporates both the consumption role of land and

working capital → Multiple steady states

6 / 33

Road Map

◮ Start with a stylized model

◮ Isolate the key complementary forces ◮ Characterize conditions steady state multiplicity arises

◮ Extended model

◮ Sensitivity check ◮ The mechanism generates substantial persistence 7 / 33

Model

8 / 33

A Stylized Model

◮ Discrete time. Infinite horizon ◮ A continuum identical households: ◮ Consume consumption goods and land, supply labor, and

accumulate capital

◮ Owns a single private firm

◮ Constant-returns-to-scale production technology ◮ Working capital subject to collateral constraint

◮ Land supply is fixed

9 / 33

Households problem

max

c,l,n,nd,k ∞

• t=1

βtU(ct, nt, lt−1) Subject to: ct + ptlt + kt ≤ wtnt + πt + ptlt−1 + (1 − δ)kt−1 πt = A(nd

t )1−αkα t−1 − wtnd t

wtnd

t

≤ ξptlt + κkt

Timeline

0 ≤ nt ≤ n0, ct, lt, kt ≥ 0, l0, k0 given

10 / 33

Preference

U(c, l, h) =

• c − χ n1+1/ν

1 + 1/ν 1−1/σ 1 − 1/σ + ω l1−1/σ 1 − 1/σ

◮ No wealth effect on labor supply (GHH preference) ◮ σ is both:

◮ Intratemporal elasticity of substitution (Matters) ◮ (Inverse of) intertemporal elasticity of substitution (Not matter) 11 / 33

Competitive Equilibrium

Definition

A competitive equilibrium is {ct, kt+1, lt+1, nt, nd

t }∞ t=1 and

{pt, wt, qt}∞

t=1 such that:

• 1. Given prices, allocations solve the households problem.
• 2. Land and labor market clears every period: l = l0, n = nd

◮ A steady state is a competitive equilibrium where capital stock

kt is time invariant.

12 / 33

Theorem

Suppose that

• 1. Consumption and land are complementary (Low σ)
• 2. Labor supply is elastic (High ν)

Then there exists an open set U ∈ R2 such that for any combinations

• f loan to value ratios (κ, ξ) ∈ U, there exists more than one

locally-stable steady states.

13 / 33

Warm-Up: the Frictionless Case

◮ Suppose there is no credit constraint ◮ The steady state (c, k, n, w, p) is fully characterized by: c + δk = Akαn1−α n

1 ν = w

(1 − α)A w 1

α

k = n β

• Aα(k/n)α−1 + (1 − δ)
• = 1

(Resources Constraint) (Labor supply) (Labor demand) (Capital FOC) ωl−σ

• c − χ n1+ 1

ν

1 + 1

ν

1/σ + βp − p = 0 (Land FOC) ◮ The four equations in the box solve real allocations independent of

land price p.

14 / 33

Frictional Case

◮ The steady state (c, k, n, w, p) is fully characterized by: c + δk − Akαn1−α = 0 n

1 ν = w

min

• ξpl0 + κk

w , (1 − α)A w 1

α

k

• = n

β

• Aαkα−1 (n)1−α + (1 − δ) +

(1 − α)A(k/n)α w − 1

• κ
• = 1

F(p, c, k, w, n) := ωl−σ

• c − χ n1+ 1

ν

1 + 1

ν

1/σ − (1 − β)p+ (1 − α)A(k/n)α w − 1

• ξp = 0

(Land FOC) ◮ Real allocations cannot be solved independent of land price p

15 / 33

Strategy

◮ Resources Constraint, labor demand, labor supply, capital FOC

jointly define a mapping from land price p to (c, k, w, n)

◮ This mapping is constant in the frictionless case.

◮ Write the land FOC as F(p, c, k, w, n) = 0 ◮ Plug the mapping into the land FOC ⇒ 1 equation 1 unknown:

f(p) := F(p, c(p), k(p), w(p), n(p)) = 0

16 / 33

Frictional Case

Willingness to buy function

◮ ’Willingness to buy’ function:

f(p) = ωl−σ

• c(p) − χl(p)1+ 1

ν

1 + 1

ν

1/σ + (1 − α)Ak(p)αl(p)−α w(p) − 1

• ξp
• Benefit

− (1 − β)p

• Net cost

◮ Land price p is part of steady state iff f(p) = 0. ◮ Crucial: f is nonmonotonic

17 / 33

f(p) is Nonmonotonic

Proof

p

2 4 6 8 10 12

f(p)

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02 18 / 33

Why Nonmonotonic?

f(p) :=

Land First Order Condition

• F(p, c(p), k(p), w(p), n(p)
• Other conditions

) = 0 ∂f(p) ∂p = ∂F ∂p

• Direct Price Effect−

+ ∂F ∂c ∂c ∂p

Indirect Collateral Effect+

+ other terms (1) Two opposing forces:

◮ Direct Price Effect: Land gets expensive, less willing to buy. ◮ Indirect Collateral Effect:

Land Price ⇑= ⇒ Consumption ⇑ = ⇒ More willing to buy land

19 / 33

Collateral Effect in Detail

• 1. Land Price ⇑
• 2. Firm working capital constraint relaxed
• 3. Employment ⇑, Capital ⇑
• 4. Households wealth ⇑, consumption ⇑
• 5. Willingness to buy land ⇑ (Land-consumption complementarity)

20 / 33

Discussion of Assumptions

𝑀𝑏𝑜𝑒 𝑄𝑠𝑗𝑑𝑓 ⇑ 𝐷𝑝𝑜𝑡𝑣𝑛𝑞𝑢𝑗𝑝𝑜 ⇑ 𝑋𝑗𝑚𝑚𝑗𝑜𝑕𝑜𝑓𝑡𝑡 𝑢𝑝 𝑐𝑣𝑧 𝑚𝑏𝑜𝑒 ⇑ Labor Supply Elastic (Need High 𝜉) C and Land Complementary (Need Low 𝜏)

Otherwise ◮ If labor inelastically supplied (ν → 0) ⇒Equilibrium labor not affected by working capital constraint ⇒Output and consumption not affected by land price ◮ If consumption and land perfect substitutes ( σ → ∞) ⇒ Linearity structure implies level of consumption irrelevant

21 / 33

Discussion of Assumptions

𝑀𝑏𝑜𝑒 𝑄𝑠𝑗𝑑𝑓 ⇑ 𝐷𝑝𝑜𝑡𝑣𝑛𝑞𝑢𝑗𝑝𝑜 ⇑ 𝑋𝑗𝑚𝑚𝑗𝑜𝑕𝑜𝑓𝑡𝑡 𝑢𝑝 𝑐𝑣𝑧 𝑚𝑏𝑜𝑒 ⇑ Labor Supply Elastic (Need High 𝜉) C and Land Complementary (Need Low 𝜏)

◮ When labor supply is perfectly elastic (ν → ∞), multiple

steady states exist if and only if σ < 1.

22 / 33

Quantitative Analysis

23 / 33

The Extended Model: Summary

Details

◮ Two types of agents: households and firms. Households(HHs)

are owners of the firms.

◮ There is a rental market for land ◮ HHs choose to own land (residential land) or rent it ◮ Firms accumulate land (commercial land) and can allocate it to

rental or production use

◮ Land and capital can be used as collateral

24 / 33

The households problem

max

c,lh,n ∞

• t=1

βtU(ct, nt, lt−1) ct + ptlht + rtlr

ht−1

≤ ptlht−1 + dt + wtnt lt−1 = l1t−1 + l2t−1 h10 given, lt ≤ ¯ l, l1t−1 + l2t−1 ≥ 0

◮ lht: residential land; lr ht: land rent by households; ◮ pt: land purchase price; rt: land rental rate; ◮ U (c, h, l) =  

• ω(c−χ n1+ 1

ν 1+ 1 ν

)1−1/σ+(1−ω)l1−1/σ 1/(1−1/σ) 

1−η

1−η

25 / 33

The Firm’s Problem

Back

max

i,d,hf,l ∞

• t=1

Mtdt dt + kt + ptlft ≤ F(kt−1, nt, lp

ft−1) + ptlft−1 − wtnt + rtlr ft + (1 − δ)kt−1

wtlt ≤ ξ(ptlft + kt) k0, lf0 given

◮ lft: commercial land; ◮ lr

ft: land rent to households;

◮ lp

ft: land used for production;

◮ Cobb-Douglas Production Function: F(k, n, l) = k1−γ−αnγlα

26 / 33

Competitive Equilibrium

Definition

A competitive equilibrium is {ct, nt, lht, lr

ht, kt, lft, lr ft, dt, nd t }∞ t=1 and

{pt, wt, rt}∞

t=1 such that:

• 1. Given prices, allocations solve the households problem.
• 2. Land, labor, and land rental market clears every period:

lh + lf = h0, n = nd, lr

h = lr f

27 / 33

An Equivalence Result

Suppose the land’s share in production α = 0. Then the equilibrium allocations (consumption, labor, capital, and investment) and prices in the extended model are the same as those in the stylized model.

28 / 33

Calibration

Parameters Value Source Discount factor β 0.99 Quarterly model Intertemporal elasticity η 2 Standard

• Pref. weight on land

ω 0.9 Housing Price/GDP= 1.7 Disutility of working χ 8.2 Steady state labor equal to .33 Depreciation δ 2.5% Standard Labor share γ 0.64 Standard Land share α 0.03

Liu, Wang, and Zha (2013), Iacoveillo (2005)

Productivity A 1 Normalization Aggregate land stock h0 1 Normalization Elasticity of sub. c&l σ .33 Middle of micro estimates Frisch elasticity of labor ν 5 Macro Studies Loan to value ratio ξ .04 Collateral cons. binds in large rec. only

Table 1: Calibration

Elasticity of sub. c&h Loan to value ratio 29 / 33

S-shaped Law of Motion for Capital

Transitional Dynamics

kt

11 11.2 11.4 11.6 11.8 12 12.2 12.4 12.6

kt+1-kt

#10-3

• 0.5

0.5 1 1.5 2 2.5

Capital

30 / 33

Accounting for the Slow Recovery

◮ Feed in unexpected shock to loan-to-value ratio ξ that lasts

seven quarters from 2007q4 to 2009q2...

◮ Such that land price drops by 25%

2004 2006 2008 2010 2012 2014 2016

• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

0.05 Loan-to-value Ratio 9t

31 / 33

Accounting for the slow recovery

2004 2006 2008 2010 2012 2014 2016

• 0.3
• 0.2
• 0.1

0.1 Output

Data Model

2004 2006 2008 2010 2012 2014 2016

• 0.8
• 0.6
• 0.4
• 0.2

0.2 Investment 2004 2006 2008 2010 2012 2014 2016

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 Land Price 2004 2006 2008 2010 2012 2014 2016

• 0.4
• 0.3
• 0.2
• 0.1

0.1 Labor 32 / 33

Conclusion

◮ I propose a theory where slow recoveries follow deep recessions. ◮ Crucial ingredient: the dual role of land as households

consumption and firm collateral

◮ The model generates persistent recessions comparable to

post-Great Receession data

33 / 33

Timeline

Back

Firm Produces, Borrows to Finance Wage Market Opens Firm Revenue Realizes Firm Repays Loan If Default Lender seizes fraction

• f capital and land

t-1

t

t+1 Cash Flow Mismatch

1 / 66

Existence and Uniqueness of Recursive Competitive Equilibrium

◮ Define an operator T mapping [K′(K), P(K)]n to

[K′(K), P(K)]n+1

◮ Existence (Schauder’s Theorem)

◮ Problem is continuous and general fixed point theorem

apply.

◮ Uniqueness

◮ Peudo-Concavity ◮ x0 monotonicity 2 / 66

Elasticity of Substitution between Housing and Consumption

Back

◮ Little consensus in the literature

◮ Most micro estimates between 0.13 and 0.6 ◮ Flavin and Nakagawa(2008), Hanushek and Quigley(1980),

Siegel (2008), Stokey(2009), Li, Liu, Yang, and Yao(2016)

◮ Piazzesi, Schneider, and Tuzel(2007): > 1 ◮ Bajari, Chan, Krueger, and Miller(2012): > 6

◮ Set η = 0.33 as benchmark

3 / 66

Loan to value ratio

Back

◮ Constraint occasionally binding ⇒ cannot estimate using

steady state targets

◮ Here: Constraint binds with 7.5% drop in output ◮ ξ = 0.04

4 / 66

Accounting for the labor wedge

Back 2004 2006 2008 2010 2012 2014 2016

• 0.02

0.02 0.04 0.06 0.08 0.1 0.12 Data 2004 2006 2008 2010 2012 2014 2016

• 0.2

0.2 0.4 0.6 0.8 1 1.2 Model

Labor Wedge Firm Component

5 / 66

Accounting for the slow recovery

2004 2006 2008 2010 2012 2014 2016 0.6 0.7 0.8 0.9 1 Credit shock (9) 2004 2006 2008 2010 2012 2014 2016 0.97 0.98 0.99 1 1.01 1.02 1.03 Productivity shock (A) 6 / 66

Accounting for the slow recovery

2004 2006 2008 2010 2012 2014 2016

• 0.3
• 0.2
• 0.1

0.1 Output

Data Model

2004 2006 2008 2010 2012 2014 2016

• 0.8
• 0.6
• 0.4
• 0.2

0.2 Investment 2004 2006 2008 2010 2012 2014 2016

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 Land Price 2004 2006 2008 2010 2012 2014 2016

• 0.4
• 0.3
• 0.2
• 0.1

0.1 Labor 7 / 66

Accounting for the slow recovery

Back 2010 2011 2012 2013 2014 2015 2016

• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

0.05 Output

Data Benchmark model Credit Shock only

2010 2011 2012 2013 2014 2015 2016

• 0.8
• 0.6
• 0.4
• 0.2

0.2 Investment 2010 2011 2012 2013 2014 2015 2016

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 Land Price 2010 2011 2012 2013 2014 2015 2016

• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

0.05 Labor 8 / 66

Policy Function

Back K 8 8.5 9 9.5 K0-K

• 0.04
• 0.02

0.02 0.04 0.06 0.08 w-=wss K 8 8.5 9 9.5 K0-K

• 0.04
• 0.02

0.02 0.04 0.06 0.08 w-=0.98wss K 8 8.5 9 9.5 K0-K

• 0.04
• 0.02

0.02 0.04 0.06 0.08 w-=0.97wss K 8 8.5 9 9.5 K0-K

• 0.02

0.02 0.04 0.06 0.08 w-=0.95wss

Neoclassical Constant-p Benchmark

9 / 66

Computation

◮ Aggregate State X = (K, W−) where W− is previous

period wage.

◮ Need to solve:

• 1. Law of motion for capital: K′ = Φ (K, W−)
• 2. Labor function: L = L (K, W−)
• 3. Land price function: P = P (K, W )
• 4. Law of motion for wage: W = W (K, W−)

10 / 66

Policy Functions

K

8 8.5 9 9.5

K0-K

• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01

K

8 8.5 9 9.5

L

0.22 0.24 0.26 0.28 0.3 0.32 0.34

wt-1=wss wt-1=1wss wt-1=12wss

K

8 8.5 9 9.5

P

3 4 5 6 7 8

K

8 8.5 9 9.5

W

1.98 1.99 2 2.01 2.02 2.03

11 / 66

Typical Dynamics

𝐿𝑢+1 𝐿𝑢 𝜂2 𝑥𝑡𝑡 𝜂3 𝑥𝑡𝑡 45𝑝 𝜂𝑥𝑡𝑡 𝜂4 𝑥𝑡𝑡

12 / 66

Typical Dynamics

Back

𝐿𝑢+1 𝐿𝑢 𝐵 𝐶 𝐷1 𝐷2 𝐷3 𝐷4 𝜂2 𝑥𝑡𝑡 𝜂𝑥𝑡𝑡 𝜂3 𝑥𝑡𝑡 𝐷 𝐷 𝐶 𝐶1 𝐶2 𝑇𝑛𝑏𝑚𝑚 𝑡ℎ𝑝𝑑𝑙 𝐶𝑗𝑕 𝑡ℎ𝑝𝑑𝑙 𝜂4 𝑥𝑡𝑡

13 / 66

The Extended Model

◮ Representative households:

◮ Consume consumption goods and land, supply labor,

receive dividends from the representative firm.

◮ Representative firm

◮ Hire labor and produce; accumulate capital and land; pay

• ut dividend to households

◮ Subject to working capital constraint

◮ Land supply is fixed

14 / 66

Real GDP

Back

8.5 9 9.5 10 1975 1980 1985 1990 1995 2000 2005 2010 2015

Real GDP and its linear trend 1970-2007

15 / 66

House Price Index

Back

1 1.5 2 2.5 1975 1985 1995 2005 2015 Real House Price Index Constant 2% Growth Trend

S&P/Case-Shiller U.S. National Home Price Index, deflated by GDP deflator 16 / 66

Land Price Index

Back

1 2 3 4 5 6 1975 1985 1995 2005 2015 Real Land Price Index Constant Growth Trend

Lincoln Institute of Land Policy, Davis and Heathcote(2007) 17 / 66

Cross sectional evidence

Question: Is there a systematic relation between the extent of housing price drop and pace of recovery at the MSA level?

• .2
• .15
• .1
• .05

.05 Log Deviation From Trend 1990 1995 2000 2005 2010 2015

Employment

MSA with smallest housing price deline MSA with medium housing price deline MSA with biggest housing price deline

18 / 66

Labor Wedge

◮ Construct firm component of the labor wedge following

Karabarbounis(2013)

◮ Large and persistent spike after Lehman’s bankruptcy 0.38 0.4 0.42 0.44 0.46 0.48 0.5

Data

19 / 66

Financial Market Indicators

1 2 3 4 Percentage 2005q1 2008q1 2011q1 2014q1 2017q1

baa-aaa bond yield spread

• 50

50 100 Net percentage 2005q1 2008q2 2011q3 2014q4 2018q1

Lending Standards

20 40 60 80 2005q1 2008q2 2011q3 2014q4 2018q1

VIX index

• 2

2 4 6 2005q1 2008q2 2011q3 2014q4 2018q1

• St. Louis Fed Financial Stress Index

20 / 66

Equilibrium Labor

Kt

1 2 3 4 5 6 7 8 9

Lt

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Frictionless Credit constraint only Sticky wage only dual friction

21 / 66

Drop of labor in percentage terms

Kt

1 2 3 4 5 6 7 8 9

(Lt

constrained-Lt unconstrained)/Lt unconstrained

• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1

Frictionless Sticky wage

22 / 66

Drop of labor in percentage terms

Kt

1 2 3 4 5 6 7 8 9

(Lt

constrained-Lt unconstrained)/Lt unconstrained

• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1

Frictionless Sticky wage

23 / 66

Recursive households problem with sticky wage

V (k, h, b; K) = max

c,h′,k′,b′,l,ld u (ct, ht) + βV (k′, h′, b′; K′)

Subject to: c + p(K)h′ + b + k′ ≤ w(K)l + π + p(K)h + q(K)b′ + (1 − δ)k 0 ≤ l ≤ min{l0, l(K)}, c, h′, k′ ≥ 0 K′ = Φ(K) Where π = max

ld

A(ld)γkα − w(K)ld w(K)ld ≤ ξp(K)h′ ld ≤ l(K)

24 / 66

Recursive Competitive Equilibrium

Definition

A sticky-wage recursive competitive equilibrium is functions (h′, k′, b′, l, ld)(k, h, b; K), p(K), w(K), q(K), l(K), Φ(K) such that:

• 1. Given the price functions, the decision rules solve the households

problem

• 2. Housing and bond market clears every period: h = h0, b = 0
• 3. w(K) = w0 for some w0.

l(K) = min{l(K, h0, 0; K), ld(K, h0, 0; K)}

• 4. Consistency: Φ(K) = k′(K, h0, 0; K);

25 / 66

Functional equations of RCE

The system of equations characterizing (C (K) , P (K) , Φ (K) , l (K)) are:

θh−η + βp (Φ (K)) (C (Φ (K)))−σ = (C (K))−σ   1 −

• γAKα (l (K))γ−1 − wss
• ξ

wss 1 wss

γA

• 1

γ−1 K α 1−γ > ξp(K)h0 wss

• 

  p (K) (C (K))−σ = β (C (Φ (K)))−σ αA (Φ (K))α−1 (l (Φ (K)))γ + 1 − δ

• l (K)

= min wss γA

• 1

γ−1

K

α 1−γ , ξp (K) h0

wss , l0

• C (K)

= AKα [l (K)]γ + (1 − δ) K − Φ (K) + y0

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Numerical Example

Parameters σ 2 η 2 δ 10% γ 0.65 α 0.3 β 0.96 A 1 θ .1

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Dynamics

Kt

1 2 3 4 5 6 7 8 9

Kt+1-Kt

#10-3

• 2
• 1

1 2 3 4 5 6 7

Sticky wage model + credit constraint

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Dynamics

Kt

1 2 3 4 5 6 7 8 9

Kt+1-Kt

0.05 0.1 0.15 0.2 0.25 0.3

Frictionless model + credit constraint

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Recursive households problem

V (k, h, b; K) = max

c,h′,k′,b′,l,ld

c1−σ 1 − σ + θ h1−η 1 − η + βV (k′, h′, b′; K′) Subject to: c + p(K)h′ + b + k′ ≤ w(K)l + π + p(K)h + q(K)b′ + (1 − δ)k 0 ≤ l ≤ l0, c, h′, k′ ≥ 0 K′ = Φ(K) Where π = max

ld

A(ld)γkα − w(K)ld w(K)ld ≤ ξp(K)h′ ⇒ Decision rules: (h′, k′, b′, l, ld)(k, h, b; K)

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Recursive Competitive Equilibrium

Definition

A flexible-wage recursive competitive equilibrium is a collection of functions (h′, k′, b′, l, ld)(k, h, b; K), p(K), w(K), q(K), Φ(K) such that:

• 1. Given price functions, decision rules solve the hh’s problem
• 2. Housing, labor, bond market clears: h = h0, ld = l, b = 0
• 3. Consistency: k′(K, h0, 0; K) = Φ(K)

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Local Stability

Theorem

Suppose ξ ∈ (ξss, ξ0] for some ξ0, α + γ < 1 and σ sufficiently

• big. There exists multiple locally stable steady states.

Proof: Still working in progress

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Step 1: the continuity of f

• 1. Identify the ’kink’ ¯

p: ξ¯ ph0 = wssl0

• 2. f(p) is continous for p less than ¯

p and for p greater than ¯ p: f (p) =      θh−η

0 cσ ss − (1 − β) p, for p ¯

p a0p

γ 1−α σ + a1p γ 1−α − a2p, for p < ¯

p

• 3. f(p) is continous at ¯

p.

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Step 2: existence of multiple s.s. at ξss

◮ when ξ = ξss = wssl0 pssh0 , ¯

p = pss, thus f(¯ p) = f(pss) = 0.

◮ Assumptions required: σ sufficiently big, α + γ < 1. ◮ Steps:

• 1. f(0; ξss) = 0 and f ′+(0; ξss) > 0

⇒ ∃p1 > 0s.t.f(p1, ξss) > 0

• 2. f(pss; ξss) = 0 and f ′−(pss; ξss) > 0

⇒ ∃p2 > p1s.t.f(p2, ξss) < 0

• 3. By continuity of f, we are done.

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Step 3: extend the result to ξ > ξss

◮ Pick ε > 0 sufficiently small such that for any

ξ ∈ (ξss, ξss + ε]:

◮ f(¯

p, ξ) > 0 where ¯ p > p2 > p1

◮ f(p1, ξ) > 0 ◮ f(p2, ξ) < 0

◮ Slight complication: need to show f is continuous w.r.t ξ

for ξ sufficiently close to ξss, at p = p1 and p = p2.

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Introduction

◮ ”Secular stagnation” episodes:

◮ Follow large crises: Great Recession, Japanese 1990 crisis,

and Great Depression..

◮ Feature slow recovery of output, employment, and

investment.

◮ Existing theories: focus on insufficient demand arising from

zero lower bound.

◮ Eggertsson and Mehrotra(2015), Eggertsson, Mehrotra, and

Summers(2016), Michau(2015), Garrec and Touze (2016)

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Endogenous Land Allocation

◮ Land price is volatile whereas residential and commercial land

growth is largely uniform (Davis and Jonathan 2007, Davis 2009)

◮ Introduce adjustment cost for land

φ(hf

t , hf t−1) = τ(hf t − hf t−1

hf

t−1

)2hf

t−1

◮ Set τ to ∞.

37 / 66

Conclusion

◮ This paper proposes a theory of how land price dynamics

contributes to secular stagnation

◮ Slower recovery speed upon large negative shocks. ◮ Crucial elements: wage rigidities and working capital

constraint on firm.

◮ Future work: more serious quantitative experiment.

38 / 66

Fritionless Case: GHH

◮ Consider an environment with no working capital constraint. ◮ there exists a unique steady state {lss, kss, wss, css, pss} where:

lss =  (1 − α) A 1

β − 1 + δ

Aα

• α

α−1 



ν

kss = ((1 − α) A)ν 1

β − 1 + δ

Aα 1+αν

α−1

wss = γA 1

β − 1 + δ

Aα

• α

α−1

39 / 66

The model with perfectly sticky wage

◮ Discrete time. Infinite horizon ◮ A continnum identical households/enterprenuer: ◮ Consume consumption goods and housing, supply labor

inelastically, and accumulate capital

◮ Owns a single private firm

◮ CD production technology ◮ Working capital subject to collateral constraint

◮ Land supply is fixed

40 / 66

Households problem

max

c,h,l,ld,k ∞

• t=1
• ωc1−1/η

t

+ (1 − ω) h1−1/η

t

1/(1−1/η)1−σ 1 − σ − χ l

1+ 1

ν

t

1 + 1

ν

subject to ct + ptht + bt + kt ≤ wtlt + πt + ptht−1 + qtbt+1 + (1 − δ)kt−1 πt = max

ld

t

A

• ld

t

γ kα

t−1 − wtld t

qtbt+1 + θwtld

t ≤ ξptht + κkt

≤ lt ≤ l0, ct, ht, kt ≥ 0, h0, k0 given

41 / 66

Competitive Equilibrium

Definition

A competitive equilibrium given sticky wage w0 is {ct, kt+1, ht+1, lt, ld

t , bt}∞ t=1 and {pt, wt, qt}∞ t=1 such that:

• 1. Given prices, allocations solve the households problem.
• 2. Housing and bond market clears every period: h = h0, b = 0
• 3. w(K) = w0 for some w0. Equilibrium labor is determined by the

minimum of labor demand and labor supply.

◮ A steady state a competitive equilibrium where capital stock kt

is time invariant.

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Prestep: Fritionless Case

◮ Consider an environment with no working capital constraint, and

wage is flexible.

◮ there exists a unique steady state {lss, kss, wss, css, pss} given by:

ωh−1/η c1/η + βp = p β

• αAlγkα−1 + 1 − δ
• =

1 ωc−1/η ωc1−1/η + (1 − ω) h1−1/η (1−σ)/(1−1/η)−1 = wχl

1 ν

c = Alγkα − δk w = γAlγ−1kα

◮ focus on equilibrium with wt = wss (Shimer, 2012)

43 / 66

Theorem

Back

Suppose η < 1. Then there exists an ¯ η(η) < 1 such that for any α + γ > ¯ η and κ sufficiently small, there exists an interval U ξ such that, if ξ ∈ U ξ, then there exists more than 1 locally stable sticky-wage steady states given wss.

44 / 66

Steady state characterization

◮ The steady state (c, k, p) is fully characterized by:

1 − ω ω h

−1 η

+ βpc

−1 η +

γAkαl(k, p)γ−1 wss − 1

• ξpc

−1 η

= pc

−1 η

β

• Aαkα−1 (l(k, p))γ + (1 − δ) +

γAkαl(k, p)γ−1 wss − 1

• ξ
• =

1 c + δk − Akαl(k, p)γ = where l(k, p) = min

• ξ(ph0 + k)

wss , γA wss

• 1

γ−1

k

α 1−γ , l0

• ◮ The last two equations define a mapping from p to k, c.

45 / 66

Willingness to pay function

◮ Define a ’willingness to pay’ function in consumption units:

f(p) = 1 − ω ω h

−1 η

0 c(p)

1 η

+ γAk(p)αl(k(p), p)γ−1 wss − 1

• ξp

− (1 − β)p

◮ (c, k, p) is a steady state if and only if the associated p

satisfies f(p) = 0

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Graphic Illustration of f(p)

p

0.5 1 1.5 2 2.5 3 3.5 4

f(p)

• 0.01
• 0.005

0.005 0.01 0.015

pss

47 / 66

Outline of the proof:

◮ f(p) is continuous w.r.t. p for any ξ ◮ Define ξss =

wssl0 pssh0+kss . Show that given ξss, there exists multiple

nontrivial steady states:

• 1. f(0; ξss) > 0
• 2. f(pss; ξss) = 0 and f ′−(pss);

◮ Show that the case extends to ξ ∈ (ξss, ξ0] for some ξ0.

48 / 66

Graphic Illustration of f(p)

p

0.5 1 1.5 2 2.5 3 3.5 4

f(p)

• 0.015
• 0.01
• 0.005

0.005 0.01 0.015

pss 9>9ss 9=9ss

49 / 66

Dynamics with perfectly sticky wage

Kt

1 2 3 4 5 6 7 8 9

Kt+1-Kt

0.05 0.1 0.15 0.2 0.25 0.3

Frictionless model + credit constraint

50 / 66

Dynamics with perfectly sticky wage

Kt

1 2 3 4 5 6 7 8 9

Kt+1-Kt

#10-3

• 2
• 1

1 2 3 4 5 6 7

Sticky wage model + credit constraint

51 / 66

Wage adjustment rule II

◮ wage adjustment rule:

wt ≥ ζwt−1

◮ set ζ = 0.995 ◮ Nondecreasing nominal wage in an economy with 2%

inflation.

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Dynamics with wage adjustment rule II

K

8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8

K0-K

• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01

w=wss w=1wss w=12wss

53 / 66

Impulse Response

t

5 10 15 20 25 30 35 40 45

K

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

Big shock Medium shock Small shock

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Wage adjustment rule

◮ wage adjustment rule:

log(w) − log(wss) = εw(log(y) − log(yss))

◮ set εw = 0.45

◮ Huo and Victor (2016),Gornemann, Kuester, and Nakajima

(2012)

◮ Steady state multiplicity arises when η ≤ 0.3

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Equilibrium Functions

k 11 11.5 12 12.5 k0-k #10-3

• 0.5

0.5 1 1.5 2 2.5 Capital k 11 11.5 12 12.5 l 0.24 0.26 0.28 0.3 0.32 0.34 Labor k 11 11.5 12 12.5 p 3 4 5 6 7 8 9 Land price k 11 11.5 12 12.5 w 2.2 2.25 2.3 2.35 2.4 Wage 56 / 66

Dynamics with different η

K

11 11.2 11.4 11.6 11.8 12 12.2 12.4 12.6

K0-K

#10-3

• 3
• 2
• 1

1 2 3 4 5 6 7

Li etc.(2016): 2=0.487 Benchmark: 2=0.3 FN(2008): 2=0.13

57 / 66

Accounting for the slow recovery

2004 2006 2008 2010 2012 2014 2016

• 0.7
• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1

Model

2004 2006 2008 2010 2012 2014 2016

• 0.7
• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1

Data Investment Land price Labor Output

58 / 66

Accounting for the labor wedge

◮ Labor wedge spikes during the recession and remains high

after the recession: labor wedge = log(MPN) − log(MRS)

2005 2006 2007 2008 2009 2010 2011 2012 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

Data

2005 2006 2007 2008 2009 2010 2011 2012

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Model 59 / 66

Outline of the proof:

◮ f(p) is continuous w.r.t. p for any ξ ◮ Solve the unique unconstrained steady state

(css, kss, pss, wss, lss).

◮ If κ is sufficiently small, define ξss = wsslss−κkss

pssh0

. Show that given ξss, there exists multiple nontrivial steady states:

• 1. f(0; ξss) > 0
• 2. f(pss; ξss) = 0 and f ′−(pss);

◮ Show that the case extends to ξ ∈ (ξss, ξ0] for some ξ0.

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Graphic Illustration of f(p)

Back

p

2 4 6 8 10 12

f(p)

• 0.025
• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

pss 9>9ss 9=9ss

61 / 66

Frictional Case

◮ The steady state (c, k, p, w, n) is fully characterized by:

c + δk = Akαn1−α n

1 ν

= w min ξph0 + κk w , (1 − α)A w 1

α

k = n β

• Aαkα−1 (n)1−α + (1 − δ) +

(1 − α)Akα (l)−α w − 1

• κ
• =

1 ωh−σ

• c − χ n1+ 1

ν

1 + 1

ν

1/σ + βp+ (1 − α)Akα (l)−α w − 1

• ξp

= p

62 / 66

Frictional Case

◮ The steady state (c, k, p, w, l) is fully characterized by:

c + δk = Akαn1−α n

1 ν

= w min

• ξpl0 + κk

w , (1 − α)A w 1

α

k

• =

n β

• Aαkα−1 (n)1−α + (1 − δ) +

(1 − α)Akα (n)−α w − 1

• κ
• =

1 ωl−σ

• c − χ n1+ 1

ν

1 + 1

ν

1/σ + βp+ (1 − α)Akα (n)−α w − 1

• ξp

= p

63 / 66

Transitional Dynamics

Back

Mild Recession

5 10 15 20 25

K

0.995 0.996 0.997 0.998 0.999 1 neoclassical benchmark

Large Recession

5 10 15 20 25

K

0.92 0.94 0.96 0.98 1 neoclassical benchmark 64 / 66

View of This Paper

◮ Economy exhibits multiple ”regimes” (locally stable steady states)

◮ Good regime: high capital accumulation⇐

⇒high land price

◮ Bad regime: low capital accumulation ⇐

⇒ low land price

◮ Thus asymmetric response to small and large shocks:

◮ Mild recession ⇒ Quick recovery ◮ Severe recession ⇒ Regime switch ⇒ Permanent impact 65 / 66

Model Overview

Standard Neoclassical model with a land sector

◮ Land serves dual roles:

◮ Consumption for the households ◮ Collateral for the firm to finance its working capital

◮ Result:

The equilibrium law of motion for capital is S-shaped with multiple locally stable steady states.

66 / 66