Asset Bubbles and Bailout
Tomohiro Hirano (University of Tokyo)
Noriyuki Yanagawa (University of Tokyo)
May 28-29, 2012
CIGS Conference on Macroeconomic Theory andPolicy 20121

Motivations• Bubblebursts cause a free-fall in real economic activity.

• To

mitigate it, government takes various
types of bailouts:
Capital injection→recapitalizations of the net
worth

Although bailouts may mitigate adverse eﬀects
of the bubble bursts, what happens if bailout policies are anticipated?
2

Boom and Bust of Asset Bubbles
Yt

Bubbly Dynamics

Free FallBail-Out
Bubbleless Economy

Bubble Burst3t

Questions• How does the anticipated bailout aﬀect theemergence of bubbles?

• To

what extent is ex-post bailout eﬃcientfrom ex-ante perspective?

• How does the anticipated bailout aﬀect boom-bust cycles?

Finally, we discuss the eﬀects of bailout on tax
payers (households)?

4

Bailout we consider:
Government levies a lump sum tax on households when bubbles burst.
Direct transfer to entrepreneurs with bubble assets→Households

bursts.

pay direct costs for the bubble

5Bailout(Conti):
A fraction λ ∈ [0, 1] of entrepreneurs with bubble assets is rescued.
· λ = 1 means all entrepreneurs are rescued.
· λ = 0 means no entrepreneurs are rescued.
An increase in λ means an expansion of bailout,i.e., more entrepreneurs are rescued.
From ex-ante perspective, each entrepreneur anticipates government bailout with probability λ .
6Related Literature on bailout (monetary policy)
・Three period model:
Farhi and Tirole(2009, 2011), Diamond and Rajan(2011)

・Dynamic macro model:
Gertler, Kiyotaki, and Queralto(2011)

We analyze the eﬀect of bailout on stochasticbubbles in a dynamic macro model.

7

Related Literature on rational bubbles
Recent developments:Woodford(1990), Caballero/Krishnamurthy(2006), Kiyotaki/Moore(2008), Farhi and Tirole (2009), Kocherlakota(2009), Hirano/Yanagawa(2010), Martin/Ventura(2010),
Aoki/Nikolov(2011)

We consider bailout in a rational bubbles model.

8Model: A extension of Kiyotaki (1998)
・Inﬁnitely-lived agents model・Two types of goods: consumption and capital goods
・Agents: entrepreneurs and households (workers)·entrepreneurs with H-projects (H-entrepreneurs)·entrepreneurs with L-projects (L-entrepreneurs)
・Investment technologies followikt+1= αti zti

(1)iis the capitalzti (≥ 0) is the investment at date t, and kt+1
at date t + 1. αti is the marginal productivity of investment
at date t. αH > αL .

9

In this economy, there are competitive ﬁrms
which produce consumption goods using capitaland labor.
• Production function
yt = ktσ nt1−σ ,(2)kt and nt are capital goods and labor input at
date t. yt is output at date t.

For simplicity, we assume that capital fully depreciates in one period.

10Factors of production are paid their marginal
product:
qt = σKtσ−1 and wt = (1 − σ)Ktσ .

(3)

Kt is aggregate capital goods. wt is wage rate.

Each household (worker) supplies one unit oflabor inelastically in each period, and earns wage
income.
In equilibrium,cut = wt − Tt
Tt > 0 if bubbles collapse.
11Two Technical assumptions• At

each date, each entrepreneur meets
H-projects with probability p, and Lprojects with probability 1 − p. The entrepreneur knows his own type of date t.
→Type of entrepreneurs changes over time.

• Borrowing constraint→The entrepreneur cannot borrow as muchas they want.

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t-1 periodt period

HHsell bubbles

LLpurchase bubbles

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Entrepreneur’s Maximization Problem
M ax∞{cit , zti , bit , xit }t=0E0 [P∞

tiβlogct]t=0

cit + zti + Pt xit = qt kti − rt−1 bit−1 + bit + Pt xit−1 ,(4)rt bit ≤ θqt+1 αti zti ,(5)xit ≥ 0,(6)xit is the level of bubble assets purchased by a type ientrepreneur at date t, and (5) is the borrowing constraint,and (6) is the short sale constraint. Pt is the per unit priceof bubble assets in terms of consumption goods on survive
at date t.
14• Stochastic Bubbles (Weil 1989)

Here we assume the following Markov chain:
Pr(Pt > 0 | Pt−1 > 0) = π,Once bubbles burst, they never arise again.
• Bailout Scheme
Each entrepreneur anticipates
bailout with probability λ.
15

government

For H-types:• If q t+1 αH > rt , BC and SSC bind.
zti =1iiiβ(qk−rb+P
xt tt−1 t−1t t−1 ),Hθqt+1 α1−rt(7)eit ≡ yti − rt−1 bit−1 + Pt xit−1 .

• Thanks to log-utility,
cit = (1 − β)(qt kti − rt−1 bit−1 + Pt xit−1 )16(8)Eﬀects of bubbles on investment
Two competing eﬀects• Crowd-in Eﬀect
Bubbles→increase net worth→ﬁnance moreinvestment⇒Bubbles generate “balance sheet eﬀects”.
• Crowd-out Eﬀect(Traditional Tirole Eﬀect)
Which one of these dominates depends on αH .
17For L-types: BC and SSC do not bind.
They decide optimal portfolio allocations.
Risk averse L-types allocate their savings between L-projects(safe), lending to H-types(safe),and bubble assets(risky).
zti + Pt xit − bit = βeit

(9)They allocate their savings, so that marginal expected utility is equalized between three assets.
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F.O.Cs with respect to zti , bit , and xit ,(zti ) :1q t+1 αL
q t+1 αLq t+1 αLi=πβ+(1−π)λβ+(1−π)(1−λ)β+µt,i,πi,(1−π)λi,(1−π)(1−λ)
citct+1ct+1ct+1(10)(bit ) :rtrtrt1=πβ+(1−π)λβ+(1−π)(1−λ)β, (11)i,πi,(1−π)λi,(1−π)(1−λ)
citct+1ct+1ct+1(xit ) :mit+11Pt
Pt+1= πβ i,π + (1 − π)λβ i,(1−π)λ,i
cit
xct+1ct+1t(12)ii
L i· ci,πt+1 = (1 − β)(qt+1 α zt − rt bt + Pt+1 xt )i,(1−π)λ· ct+1= (1 − β)(qt+1 αL zti − rt bit + mit+1 )i,(1−π)(1−λ)· ct+1= (1 − β)(qt+1 αL zti − rt bit )We analyze a bailout that fully recovers the net worth, mit+1 = Pt+1 xit .

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From this portfolio decisions,−rδ PPt+1tt

Pt xit = Pt+1
Pt− rtβeit

Pt+1(1 − δ) Pt
iiβeit .
zt − bt = Pt+1
Pt − rt(13)(14)δ ≡ π + (1 − π)λ.
A rise in λ induces L-types to take on morerisks, i.e., they are willing to buy more bubbleassets.
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In equilibrium,• If α1H < αH < α2H , collateral value is low.
ztL > 0.
• If αH ≥ α2H , collateral value is high.
ztL = 0.
L-types invest in L-projects for risk-hedge whencollateral value is low.

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Maximization Problem of Household
Maxu u u ∞{ct ,bt ,xt }t=0E0"∞X#β t log cut

(15)

t=0

cut + Pt xut = wt − rt−1 but−1 − Tt + but + Pt xut−1 ,(16)rt but ≤ 0,(17)

xut ≥ 0.

(18)

Each worker is endowed with one unit labor endowment at each period, which is supplied inelastically in the labor market and earns wage rate.

22The Deﬁnition of Competitive Equilibrium
Competitive equilibrium is a set of prices© H L u H L u H∞{rt , qt , wt }t=0 and quantities Ct , Ct , Ct , Bt , Bt , Bt , Ztª∞
L, Zt , Gt , Xt , Kt+1 , Yt t=0 such that(i) Each entrepreneur (as well as each household) maximizes his/her expected discounted utility under the constraint, choosing {cit , bit , zti , xit }∞t=0 .
(ii) Each ﬁrm maximizes its proﬁt.
(iii) All markets clear.

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・Credit Market Clearing Condition:
BtH + BtL + Btu = 0PPiHi
Lb
≡
B,b
≡
Btt ,i∈Ht ti∈Lt t(19)・Goods Market Clearing Condition:
CtH + CtL + Ctu + ZtH + ZtL = Yt ,
(20)
PP
PPi
Hi
Li
Hiz
≡
Z,z
≡
Z,c≡
C,
cttt
i∈Ht ti∈Lt ti∈Ht ti∈Lt t ≡CtL .

24• Labor Market Clearing Condition:
Nt = 1.

(21)• Capital Market Clearing Condition:KtH + KtL = Kt .

(22)

• Bubble Market Clearing Condition:
Xt = X.

25(23)

Evolution of Capital• If rt = qt+1 αL ,

Ã!

βpAt
βpAt
L
Kt+1 = α+ α βAt −− Pt X .
θαHθαH
1 − αL
1 − αL| {z }
{z}
|H

ZtH

ZtL

• If qt+1 αL < rt < qt+1 αH ,Kt+1 = αH [βAt − Pt X].
|{z}
ZtH

At ≡ σKtσ + Pt X is aggregate wealth of en-trepreneurs.

26

WhenαH is relatively low
Kt+1booms

bubbly dynamics
bustdynamics inbubbleless economy

Kt
Dynamic Effect of Stochastic Bubbles on the saddle path27

Sustainability Condition:
In order that stochastic bubbles can exist,
At+1
Pt+1
≥.
AtPt
At+1At
Pt+1
Pt

(24)

: wealth’s growth rate
: bubbles’ growth rate

Otherwise, the size of bubbles explodes and theeconomy eventually cannot sustain bubbles.
Pt+1
A change in λ aﬀects both AAt+1and.
Ptt28

Main Result 1: The existence condition ofstochastic bubbles when government bailout isanticipated with probability λ isθ < δβ(1 − p).

and1 − δ(1 − p)βα >α≡ α1H ,(1 − δβ)θ + pβδHLδ ≡ π + (1 − π)λ.
→ Bubbles cannot arise in low αH economies.

29Givenθand π,
Bubble Regions

0α1H(λ)αH

α1H is a decreasing function ofλ.
Bubbles cannot occur in lowαH economies,
because wealth’s growth rate is too low.

30

Result 1 suggests
The more government bailout is expected, bubble regions become wider and wider.
Bubbles are more likely to occur, the more government bailout is expected.
Note:
We can also characterize bubble regions with πor θ.

31

Why?
At+1Pt+1≥.
AtPt(25)• Required Rate of Return:Pt+1rt (1 − p − φt )=.
Pt
δ(1 − p) − φt(26)where δ ≡ π + (1 − π)λ and φt ≡ Pt X/βAt
When λ rises, Pt+1 /Pt declines.
economies can sustain bubbles.

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Even low αHAllen and Gale (2007):· · · In the 1930’s the market was the problem and government intervention through regulation or direct ownership of banks was the solution. Today many argue
that inconsistent government macroeconomic policies
or moral hazard in the ﬁnancial system caused by government guarantees is at the root of recent crises. · · ·Quote from their book titled “UnderstandingFinancial Crisis”

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Question 2:
To what extent is ex-post bailout eﬃcient fromex-ante perspective?
We consider ex-ante eﬃciency in terms of output.
Main result 2 is the following Figure.

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Yt (1≦t≦s) At date s, bubbles collapse.

J H1 < J H < J H2λ↑→ZtL↓, ZtH↑λ↑→ZtH↓

ZL=00λ＝λ*λ=1λValue of λwhere ex-ante efficiency is maximized.

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Yt (1≦t≦s)

At date s, bubbles collapse.

J H ³ J H2λ↑→ZtH↓

λ*=0λ=1λValue of λwhere ex-ante efficiency is maximized.

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Question 3:Suppose that government chooses λ∗ , and entrepreneurs believe it.
How does the anticipated bailout aﬀect macrodynamics?
Main Result 3: Impulse Response in the Figure.

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Pt

ZHt

birth

burst
t=0

t=st

Yt

t=0

t=s

t=0

t=st

Ct

t=0

λ=λ*

t=st

λ=038

t

bubbleless steady-statedtwt

birthburstt=0

t=st

t=0

t=sts Ltd t ¯ P t X/KAt : size of bubbless Lt ¯ Z Lt /KAt : share of L-projects
t=0

λ=λ*

t=st

λ=039

bubbleless steady-stateFinally, we discuss how does bailout aﬀect tax
payers (households)?
• Negative eﬀect:

They pay direct costs of the bubble bursts.

• Positive eﬀect:

1. Wage rate after the bursts increases thanks
to the bailout.
2. Wage rate before the bursts increases when
the bailout is anticipated.

40