Sectoral Bubbles and Endogenous Growth · 2012. 10. 28. · Abstract Introduction Empirical...

AbstractIntroduction

Empirical EvidenceThe Model

Equilibrium CharacterizationSymmetric Bubbly Equilibrium

Asymmetric Bubbly EquilibriumConclusion

Sectoral Bubbles and Endogenous Growth

Jianjun Miao & Pengfei Wang

ZHAO Yue, Kyoto University

Jan.2012

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Abstract

• Two-sector endogenous growth model with credit-driven stockprice bubbles.

• Bubbles’s effectI Credit easing effect(CEE): bubbles relax collateral constraints and

improve investment efficiency.

I Capital reallocation effect(CRE): bubbles in a sector attract morecapital to be allocated to that sector.

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Introduction

• Question: How do bubbles affect long-run economic growth?

• Object: Credit-driven bubbles• Accompanied by credit expansion.

I Optimistic beliefs → positive feedback loop → looser credit standardsand bubbles.

• Have real effects and affect market fundamentals.I Large elasticity of substitution → Negative capital allocation>

Positive capital allocation →Bubbles retard growth.

• Appear in a particular sector or industry.I Housing bubbles. internet bubble...

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Introduction

• Method:

• Base paper: Miao and Wang(2011)

• Two-sector endogenous growth model with credit-driven stock pricebubbles

I Capital in one sector has positive externality effect on human capital.

• Credit constraintI loan repayment ≤ stock market value with the collateral.

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Literatures

• Impact of bubbles on endogenous economic growth.• OLS model with endogenous growth due to externality in capital

accumulation.

• Saint-Paul (1992), Grossman and Yanagawa (1993),King andFerguson (1993)

• Key assumption: Bubbles are on intrinsically useless assets.• Conclusion: Bubbles crowd out investment and reduce the growth

rate of the economy.

• Infinite horizon endogenous growth AK model with financial frictions.

• Hirano and Yanagawa (2010)• Key assumption: Bubbles are also on intrinsically useless assets, but

can be used to relax collateral constraints;investment heterogeneity.• Conclusion: The degree of pledgeability is relatively low,bubbles

enhance growth.

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Literatures

• Bubbles in production economies with exogenous growth.• Introduce capital adjustment costs to the Diamond-Tirole model

and study the episodes of speculative growth

• Caballero, Farhi, and Hammour (2006)

• The effects of bubbles in the presence of financial frictions.• Caballero and Krishnamurthy (2006) and Farhi and Tirole (2010)• Conclusion: Bubbles can provide liquidity and crowd in investment.

• Kocherlakota(2009), Martin and Ventura (2011a), and Miao andWang (2011)

• Conclusion: Bubbles can relax collateral constraints and improveinvestment efficiency.

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Innovation

• Bubbles are attached to productive assets.

• Growth rate of a bubble here equals to the interest rate minus thedividend yield generated by the bubble,not the interest rate in theTirole (1985) model.

• Two production sectors.

• Sectoral bubbles have a capital reallocation effect betweenproduction sectors,which has not been studied in the literature.

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Credit easing effect

• Gan (2007b): Every 10 percent drop in collateral value, theinvestment rate of an average firm is reduced by 0.8 percentagepoints.

• Chaney, Sraer, and Thesmar (2009): The sensitivity ofinvestment to collateral value is stronger the more likely a firm is tobe credit constrained.

• Goyal and Yamada (2004): Investment responds significantly tostock price bubbles.And asset price shocks primarily affect firms thatrely more on bank financing and not necessarily those that useequity financing.

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Credit easing effect

Figure 1: Credit easing effect using China’s data

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Capital reallocation effect

Figure 2: Capital reallocation effect using China’s data

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Environment

• Households, final goods producers, capital goods producers, andfinancial intermediaries.

• Infinite; continues-time; no aggregate uncertainty.

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3.1 Household

∫ ∞

0

e−ρt log(Ct)dt

• Household earn labor income, trade firm stocks and make depositsto the financial intermediaries.

• Labor supply: Inelastic; normalized to 1.·CC

= rt − ρ (1)

I rt: interest rate,and equal to the rate of return on each stock.

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3.2 Final Goods Producers

• Each final goods producer hires labor and rents two types of capitalgoods to produce output

Yt = Aωα−1

[ω

1σ k

σ−1σ

it + (1− ω)1σ k

σ−1σ

2t

] ασσ−1

(K1tNt)1−α (2)

• View sector 1 as the sector producing human capital, sector 2 asthe manufacturing sector.

• Knowledge has a positive spillover effect through human capital.

I kit: the stock of type i =1;2 capital goods rented by a final goods producer

I Kit: the aggregate stock of type i capital

I A: TFP

I σ: the elasticity of substitution between the two types of capital

I ω: share parameter

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3.2 Final Goods ProducersFinal Goods Producers’problem:

maxkit,k2t,Nt

Aωα−1

[ω

1σ k

σ−1σ

it + (1− ω)1σ k

σ−1σ

2t

] ασσ−1

(K1tNt)1−α−ωtNt−R1tk1t−R2tk2t

(3)F.O.C

(1− α)Yt

Nt= wt (4)

Aαω1σωα−1(K1tNt)

1−α

[ω

1σ k

σ−1σ

it + (1− ω)1σ k

σ−1σ

2t

] ασσ−1−1

k− 1

σit = R1t (5)

Aα(1− ω)1σωα−1(K1tNt)

1−α

[ω

1σ k

σ−1σ

it + (1− ω)1σ k

σ−1σ

2t

] ασσ−1−1

k− 1

σ2t = R2t

(6)

I wt: wage rateI Rit: rental rate of type i capital

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3.3 Capital Goods Producers

• Ex ante identical but heterogeneous ex post because they faceidiosyncratic investment opportunities.

• This assumption captures firm-level investment lumpiness andgenerates ex post firm heterogeneity.

• Law of motion for capital of firm j in sector i between time t andt+ dt

Kjit+dt =

{(1− δdt)Kj

it + Ijit with probability πdt

(1− δdt)Kjit with probability 1− πdt

(7)

• Capital Goods Producers’problem

Vi0

(Kj

i0

)= max

Ijit

∫ T

0

e−∫ t0 rsds

(RitK

jit − πIjit

)dt+ e−

∫T0 rsdsViT

(Kj

iT

)(8)

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3.3 Capital Goods Producers• Financial frictions

Ijit ≤ RitKjit + Lj

it (9)

• Firms do NOT raise new equity,Loans do NOT pay interest paymentsI Lj

it: bank loans

I RitKjit: internal funds

Ljit ≤ Vit

(ξKj

it

)(10)

• Interpretation• Collateral constraint: following Kiyotaki and Moore (1997),the bank

never allows the loan repayment Ljit to exceed the stock market

value Vit(ξKjit) of the pledged assets.

• Enforcement constraint: when the investment opportunity arrives atdate t; the continuation value to the firm of not defaulting is notsmaller than the continuation value of defaulting.

Vt

(Kj

t + Ijt

)− Lj

t ≥ Vt

(Kj

t + Ijt

)− Vt

(ξKj

t

)16 / 35





3.3 Capital Goods Producers

Note

• Kiyotaki and Moore (1997)’s collateral constraint: liquidation valuerules out bubbles

Ljit ≤ ξQitK

jit (11)

I Qit: shadow price of the capital produced in sector i.

• Here allow the collateralized assets to be valued in the stock marketas the going-concern value when the new owner can use theseassets to run the reorganized firm after default.e.g.

Vit

(Kj

it

)= Qit

(ξKj

it

)+Bit

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3.4 Competitive Equilibrium

Let I1t =∫Ijitdj and K1t =

∫Kj

itdj denote aggregate investment andaggregate capital in sector i.A competitive equilibrium consists of trajectories(Ct) , (Kit) , (Iit) , (Yt) , (rt) , (wt),and (Rit),i = 1, 2,such that:

(i) Households optimize so that equation (1) holds.(ii) Each firm j solves problem (8) subject to (7), (9) and (10).(ii) Rental rates satisfy:

R1t = Aαω1σωα−1(K1tNt)

1−α

[ω

1σ k

σ−1σ

it + (1− ω)1σ k

σ−1σ

2t

] ασσ−1−1

k− 1

σit

(12)

R2t = Aα(1− ω)1σωα−1(K1tNt)

1−α

[ω

1σ k

σ−1σ

it + (1− ω)1σ k

σ−1σ

2t

] ασσ−1−1

k− 1

σ2t

(13)

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3.4 Competitive Equilibrium

(iii) The wage rate satisfies (4) for Nt = 1.(iv) Markets clear in that:

Ct + π (I1t + I2t) = Yt = Aωα−1

[ω

1σ k

σ−1σ

it + (1− ω)1σ k

σ−1σ

2t

] ασσ−1

(K1t)1−α

(14)To write equations (12), (13), and (14), we have imposed themarket-clearing conditions kit = Kitand Nt = 1 in equations (5), (6), and(2).

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4.1 A Single Firm’s Decision ProblemProposition 1: Suppose Qit > 1,then:(i) The market value of the firm is given by (15);

Vit

(Kj

it

)= QitK

jit +Bit (15)

Bit > 0,thenLj

it ≤ Vit

(ξKj

it

)= ξQitK

jit +Bit (16)

(ii) Optimal investment is given by:

Ijit = (Rit + ξQit)Kjit +Bit (17)

(iii) (Bit;Qit) satisfy the following differential equations:

rtQit = Rit + (Rit + ξQit)π (Qit − 1)− δQit +•Qit

(18)

rtBit = π (Qit − 1)Bit +•Bit

(19)

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4.1 A Single Firm’s Decision Problem

(vi) And the transversality condition:

limT→∞

exp

(−∫ T

0

rsds

)QiTK

jiT = 0, lim

T→∞exp

(−∫ T

0

rsds

)QiTB

jiT = 0

(20)

I Qit: determined variable;the shadow value of capital.

I Bit: determined variable:bubble component.

•Bit

Bit+ π (Qit − 1) = rt for Bt > 0 (21)

• Bubbles are on productive assets and their growth rate is less thanthe interest rate, therefore they cannot be ruled out by thetransversality condition.

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4.2 Equilibrium System

Proposition 2: Suppose Qit ¿ 1, then the equilibrium dynamicsfor(Bit, Qit,Kit, Iit, Ct, Yt)satisfy the following system of di.erentialequations:

•Kit

= −δKit + πIit, Ki0 given (22)

Iit = (Rit + ξQit)Kit +Bit (23)

together with (14), (18)-(19), and the transversality condition:

limT→∞

exp

(−∫ T

0

rsds

)QiTKiT = 0, lim

T→∞exp

(−∫ T

0

rsds

)QiTBiT = 0

(24)where R1t and R2t satisfy (12) and (13), respectively, and rt satisfies (1).

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4.3 Bubbleless Equilibrium Bit = 0• Without financial frictions: Qit = 1

R1t = R2t = R∗ ≡ αA (25)

ω

1− ω=

K1t

K2t(26)

K1t = ωKtK2t = (1− ω)Kt if define Kt = K1t +K2t (27)

Yt = Aωα−1K1−α1t

[ω

1σ k

σ−1σ

it + (1− ω)1σ k

σ−1σ

2t

] ασσ−1

= AKt (28)

g0 = αA− ρ− δ < A− ρ− δ (first-best).• With financial frictions: Qit > 1, collateral constraints are binding.

g∗ =

·Kit

Kit= −δ + π (Rit + ξQit) (29)

(r + δ)Q∗ = R∗ + π (R∗ + ξQ∗) (Q∗ − 1) (30)

Q∗ =(1− π)αA

ρ+ πξ(31)

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4.3 Bubbleless Equilibrium Bit = 0Proposition 3: Suppose

αA− ρ− δ > 0 (32)

(i) If

ξ >αA (1− π)

π− ρ

π(33)

then consumption, capital, and output on the balanced growth path growat the rate

g0 = αA− ρ− δ (34)

(ii) If

ξ <αA (1− π)

π− ρ

π(35)

αAπ (ρ+ ξ)

ρ+ πξ> δ (36)

then the balanced growth path grow at the rate

g∗ =αAπ (ρ+ ξ)

ρ+ πξ− δ < g0 (37)

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4.3 Bubbleless Equilibrium Bit = 0To ensure a balanced growth path when collateral constraintsbind,ξ should satisfy:

ρ (δ − αAπ)

π (αA− δ)< ξ <

αA (1− π)

π− ρ

π

Intuition behind the determinant of growth:

g = −δ +π (I1t + I2t)

K1t +K2t= −δ + s

Yt

Kt(38)

where the aggregate investment rate or the aggregate saving rate

s =π (I1t + I2t)

Yt=

απ (ρ+ ξ)

ρ+ πξ

• Both s and the Y/Kare constant along a balanced growth path.• Larger π =⇒ more investment opportunities =⇒ larger g∗

• larger ξ =⇒ looser collateral constraints =⇒ raising the investmentrate.

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5.Symmetric Bubbly Equilibrium,Bit > 0On the balanced growth path, the growth rate gb satisfies:

r = gb + ρ (39)

Becauser = gb + π (Qit − 1) (40)

Therefore,the capital price on the balanced growth path Qb

Q1t = Q2t = Qb =r − gb

π+ 1 =

ρ

π+ 1 (41)

Therefore,the interest rate Rb satisfies:

(r + δ)Qb = Rit + π(Rit + ξQb)(Qb − 1) (42)

BecauseR1t = R2t = Rb = αA (43)

• Two sectors face the same degree of financial frictions =⇒ thepresence of bubbles in both sectors does not distort capitalallocation across the two sectors. 26 / 35





5.Symmetric Bubbly Equilibrium,Bit > 0Proposition 4: Suppose condition (36) and the following condition hold:

ξ <αA (1− π)

ρ+ π− ρ

π(44)

Then, on the balanced growth path,(i) Both the bubbleless equilibrium and the symmetric bubbly equilibriumexist;(ii) The economic growth rate in the symmetric bubbly equilibrium isgiven by:

gb =(1 + ρ)αA

ρ+ π+ ρξ − ρ− δ (45)

(iii) g∗ < gb < g0 Because

gb =

•Kit

Kit= −δ + π

(Rb + ξQb +

Bit

Kit

)(46)

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6.1 Bubbles in the Sector with ExternalityIf B1t > 0, and B2t = 0 ,then on the balanced growth path

r = g1b + ρ (47)

r = g1b + π (Q1 − 1) (48)

ThereforeQ1 =

ρ

π+ 1 > 1 (49)

Because(r + δ)Q1 = R1t + π (R1t + ξQ1) (Q1 − 1) (50)

Therefore

R1 =1

π

ρ+ π

1 + ρ[ρ (1− ξ) + δ + g1b] (51)

From the F.O.C of final goods producers,(R1t

R2t

)σ

=ω

1− ω

K2t

K1t(52)

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6.1 Bubbles in the Sector with ExternalityThen

R1 = Aαω1σ+α−1

[ω

1σ +

1− ω

ωσ−1σ

(R1

R2t

)σ−1]ασ−σ+1

σ−1

(53)

R1t = R1 and R2t = R2 are constant. Because

g1b =

•K1t

K1t= −δ + π

(R1 + ξQ1 +

B1t

K1t

)(54)

g1b =

•K2t

K2t= −δ + π (R2 + ξQ2t) (55)

Thus, Q2t = Q2 is also equal to a constant. Sinceρ+ g1b︸︷︷︸r

+δ

Q2 = R2 + π (R2 + ξQ2) (Q2 − 1) (56)

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6.1 Bubbles in the Sector with ExternalityTherefore,

Q2 =1− π

πξ + ρR2 (57)

Substituting this equation into (55) yields

R2 =1

π

πξ + ρ

ξ + ρ(δ + g1b) (58)

(51),(58) and (53) yields a nonlinear equation for g1b.

Proposition 5: Suppose that there exists a unique solution (R1, R2, g1b)

to the system of equations (51), (53) and (58).Suppose that:

g1b >(ξ + ρ) (π + ρ)

1− π− δ > 0 (59)

Then the steady-state asymmetric bubbly equilibrium with B1t > 0 andB2t = 0 exists and the economic growth rate is g1b.NoteB1t/K1t > 0,Q2 > 1 and Q1 > 1 =⇒ R1 < R2

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6.2 Bubbles in the Sector without ExternalityIf B1t = 0,and B2t < 0 ,then on the balanced growth path,

Proposition 6 Suppose that there exists a unique solution (R1, R2, g2b)

to the following system of equations:

R1 =1

π

πξ + ρ

ρ+ ξ(δ + g2b) (60)

R2 =1

π

π + ρ

ρ+ 1[ρ (1− ξ) + δ + g2b] (61)

together with (53). Suppose that

g2b >(ξ + ρ) (π + ρ)

1− π− δ > 0 (62)

Then the steady-state asymmetric bubbly equilibrium with B1t = 0, andB2t < 0 exists and the economic growth rate is g2b.

NoteB2t/K2t > 0,Q2 > 1 and Q1 > 1 =⇒ R2 < R1 31 / 35





6.3 Do Bubbles Enhance or Retard Growth?

Proposition 7 Suppose that the conditions in Propositions 3(ii) and 4-6hold.(i) If σ > 1/(1− α), then

g2b < g∗ < gb < g1b

(ii) If 0 < σ < 1/(1− α), then

g∗ < g1b < gbg∗ < g2b < gb

(iii) If σ = 1/(1− α),then

g2b = g∗ < gb = g1b

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Define β = K1t/Kt,then

Yt

Kt= Aωα−1

[ω

1σ β

σ−1σ + (1− ω)

1σ (1− β)

σ−1σ

] ασσ−1

β1−α (63)

g = −δ +π (I1t + I2t)

K1t +K2t= −δ + s

Yt

Kt(38)

• Credit easing effect (CEE) relaxes the collateral constraints andraises the aggregate saving rate s

• Capital reallocation effect (CRE) influences capital allocationbetween the two sectors represented by β

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• Bubbleless and the symmetric bubbly equilibria, β = ω but s goesup, therefore no CRE but only CEE,=⇒ gb > g∗.

• Asymmetric bubbles, β > ω.

• B1t > 0,then the externality effect raises the Y/K. CRE let g1b go up.Butsince CEE decrease, then g1b goes down.Only when σ > 1

1−α, CRE

dominates, the gb < g1b.

• B2t > 0,

• CRE is negative compare to symmetric bubbly equilibria due to theless productive sector 2. And CEE decrease,therefore g2b < gb.

• CRE is negative,but CEE is positive compare to bubbleless equilibriadue to the bubble in sector 2.Therefore only σ > 1

1−α,CRE’s effect

dominates, g2b > g∗.

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This paper showed...• Bubbles have a credit easing effect in that they relax collateralconstraints and improve investment efficiency.

• Sectoral bubbles also have a capital reallocation effect in thatbubbles in one sector attracts capital to be reallocated to thatsector.

• If the elasticity of substitution between the two types of capitalgoods is relatively large, then the capital reallocation effect willdominate the credit easing effect.

• In this case, the existence of bubbles in the sector that does notgenerate externality will reduce long-run growth.Bubbles may occurin the other sector that generates positive externality or in bothsectors. In these cases, the existence of bubbles enhances economicgrowth.

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Sectoral Bubbles and Endogenous Growth · 2012. 10. 28. · Abstract Introduction Empirical...

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