Bubbles and capital flows

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BUBBLES

AND

CAPITAL

FLOWS

Jaume

Ventura

Working

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02-36

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2002

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MASSACHUSETTS INSTITUTE

_OFTECHNOLOGY_

I

JAN

2

1 2003

Bubbles

and

Capital

Flows

Jaume

Ventura

CREI,

Universitat

Pompeu

Fabra

and

MIT

October

2002

Abstract:This

paper

presents a stylized

model

ofinternationaltrade

and

asset price

bubbles. Itscentral insightisthat

bubbles

tend to

appear

and

expand

incountries

where

productivity is low relative tothe restofthe world.

These

bubbles

absorb

local

savings, eliminating inefficientinvestments

and

liberating resourcesthatare in part

used

toinvest in high productivity countries.

Through

thischannel,

bubbles

act

as

a substitutefor international capital flows, improving the internationalallocation of

investment

and

reducing rate-of-return differentials acrosscountries. This

view

of

assetprice

bubbles

has

importantimplicationsforthe

way

we

think

about

economic

growth

and

fluctuations. Italso provides a simple

account

of

some

realworld

phenomenae

that

have

been

difficultto

model

before,

such

as therecurrence

and

depth offinancial crises ortheirpuzzling

tendency

to

propagate

acrosscountries.

Commentsarewelcomeatiaume(5)mit.edu orjaume. ventura@crei.upf.es. I thankPolAntrasfor

providing excellentresearchassistance.I

am

alsograteful tothemembersoftheFacultyMacro Lunchat

MITfor theiruseful comments. Ofcourse, noneofthemis responsibleforanyerrororomissioninthe paper.

This

paper

presents astylized

model

ofinternational trade

and

asset price

bubbles.1 Its central insight isthat

bubbles

tend to

appear and

expand

in countries

where

productivityis lowrelativetothe restofthe world.

These

bubbles

absorb

local

savings, eliminating inefficientinvestments

and

liberating resourcesthat are in part

used

to invest in high productivity countries.

Through

this channel, bubbles act as a substituteforinternational capitalflows, improving the internationalallocation of investment

and

reducing rate-of-return differentials acrosscountries.This viewof

asset price bubbles

has

important implicationsforthe

way

we

think

about

economic

growth

and

fluctuations. Italso provides a simple

account

of

some

realworld

phenomenae

that

have

been

difficultto

model

before,

such

as therecurrence

and

depth of financial crisesortheir puzzling

tendency

to

propagate

across countries.

Tirole[1985]

has argued

that

markets

createasset price bubbles toeliminate

inefficient investments.2His

argument

goes

as follows:

Consider an

economy

where

thegrowth rate

exceeds

therate of returnto capital. Inthis

economy,

a bubble

can

create its

own

demand

without outgrowing savings

by

offering a rate of price

appreciation

above

therate of return but

below

thegrowth rate.

The

bubble

absorbs

partofthe

economy's

savings,

crowding

outinvestment

and

reducing thecapital

stock

and

output. Sincetheresources

devoted

to investment(roughlygrowth times

the capital stock)

exceed

the resources obtained

from

such

activity(roughlythe rate

of return timesthecapital stock), the bubble raises

consumption

and improves

welfare.

The

key

insight ofTirole'stheory is thatthe bubble takes

away

resources

from

inefficientinvestors

and

puts

them

inthe

hands

of

consumers.

An

implicit

assumption

in Tirole's

argument

isthat all investors face the

same

rateof return.

Assume

insteadthat, as a resultof frictions in financial markets, the

economy

contains efficient investorsthat enjoy rates ofreturn in

excess

of thegrowth

Byabubbleorassetpricebubble,Irefertothe differencebetween anassetpriceand the netpresent

valueofitsdividend flow orfundamentalvalue.

Tirole'spaperbuildsonthepath-breakingworkofSamuelson[1958],who wasthefirsttonotethat

rate,

and

also inefficientinvestors that

do

not. Inthis

economy,

abubble

can

create

its

own demand

within the

group

ofinefficient investorswithout outgrowingtheir savings

by

offering arate ofprice appreciation

above

theirrate of return but

below

the

growth

rate.

The

bubble

crowds

outinefficient investments

and

liberates resources that

can

be used

notonlytoraise

consumption,

but alsotoincrease efficient investments.

Through

this channel, the bubble

can

now

leadto

an

increase in the

capital stock

and

output. Sinceinefficientinvestorsdevote

more

resourcesto investment than theyobtain

from

it, the bubblestill raiseswelfare.

The

key

insight

now

is thatthebubble takes

away

resources

from

inefficient investors

and

puts

them

in the

hands

ofboth

consumers

and

efficientinvestors.

The

goalofthis

paper

isto

examine

the

consequences

ofthis

view

ofasset

price bubblesforthetheories of

economic

growth

and

fluctuations.

To do

this, I

keep

two

maintained

hypotheses

throughout thepaper.

The

first

one

isthatinternational

goods

markets

are sufficiently integratedthat long-run rates of

economic

growth are positively linked acrosscountries.This impliesthatcountries that

save

more

and have

bettertechnologies

and

policiesare richer but

do

not

grow

faster.3

The

second

hypothesis isthat international financial

markets

have

limited abilityto arbitrage

cross-countrydifferences in rates of return.This might

be

due

toa varietyoffrictions

such as

policy-induced barriers,transaction costs, information

asymmetries

and

sovereign risk.

4

Iftrade in

goods

ensures

that all countries

share

the

same

long-run rate of

economic

growthwhilefrictions in assettrade allow countries to

have

different

rates ofreturn, asset price

bubbles

naturally arise in thosecountries

where

the rateof return is

below

the

common

orworld growth rate.5

3

Leavingafewmiraclesand disasters aside,thereisampleevidenceinsupportof thisview.Despite

largecross-country differencesineconomicpolicies,savingratesandtechnology,theworldincome

distributionhasbeen relativelystableinthesecond halfofthe twentieth century. Howitt [2000]has

arguedthat thisstability mightbeduetotechnologyspillovers,whileAcemoglu andI[2002]haveargued

thatitmightbeduetoterms-of-tradeeffects.Through anyofthese channels,countries with bad

characteristicsare abletogrowfastand keep upwiththerest ofthe world.

4

Thesefrictionsdonotprecludeallinternational capitalflows,althoughthelatteraremuchsmallerthan

whattheoriesbased onfrictionlessmarketspredict. SeeKraayetal. [2000]fora reviewoftheevidence.

5

Inan influentialpaper,Abeletal.[1989] noticedthatthecapitalshareexceedsinvestmentinindustrial countries.Sincethisimpliesthattheaveragerateofreturnisabovethegrowth rate, manyhave used

To

studythe implications of thisobservation, I construct a stylizedworld

equilibrium

model

in

which

the cost of trading

goods

is negligiblewhile the cost of trading assets is prohibitive. This is a

crude

but effective

way

tocapturethe

two

maintained

hypotheses

discussed above.

The

model has

equilibriawith bubbles, in

additiontothe bubblelessequilibriumthat

we

always

takeforgranted.

These

are country bubbles, sincethey

can

be

sold only withinthe country.6

Bubbles

appear

in

countries with lowproductivity, eliminating

domestic

investment

and

raising

domestic

consumption. This shiftin

demand

lowersthe price ofinvestment

goods

relativeto

consumption

goods

all overthe world, raising investment incountries with high

productivity. Since the transferof resources

from

low-to high-productivitycountriesis

done

via prices

and

without

any

actualor recordedcapital flow, this could

be

aptly

described

as

a theoryofcapitalflows with zerocurrent accounts.

Since bubblesact

as

asubstitutefor international capitalflows,

many

oftheir

effects are akin tothosethat

one

would

expect

from

financial integration.

By

improving the

average

efficiency of investment, asset price

bubbles

tend toraisethe world growth rate.

By

shifting investments

towards

countries with high productivity, assetprice bubbles tendto

make

theworld

economy

more

sensitiveto

shocks

in

these countries

and

less sensitiveto

shocks

in othercountries.

By

providing

low-productivitycountries with a bettersavingsvehicle, asset price

bubbles

alsotendto

improve

theworld

income

distribution.

By

expanding

atthe

end

of

booms

and

contractingatthe

end

ofrecessions,

bubbles

tendto amplifythe effects of

productivity

shocks on

investmentwhile

dampening

theireffects

on

consumption. All

ofthese effects,

which

are typically associatedwithfinancial integration, arise here as

thisobservationtoquestion the empiricalvalidityof Tirole's model. Butthisconclusionrestsonthe

assumptionthat financialmarketsarefrictionless. Eveniftheaveragerate ofreturnexceedsthegrowth

rate,theeconomymightcontainpocketsofinvestors withlowratesofreturns thatarewilling tobuya

bubble.Henceit isnotpossibleingeneralto ruleoutthepresenceofbubbles bycomparingthe

aggregatecapitalshareand investment.

6

InVentura[2002], Iconsiderthecaseinwhichfinancialmarketsarewellintegrated inthesensethat

domesticandinternationaltransactionsare subjecttothesamesort offrictions. Inthiscase,bubblesare tradedacrosscountries and Irefer tothemasglobalbubbles.

the sole result ofasset price bubblessince capitalflows are not possible

and

tradeis

always

balanced

by

assumption.

Bubbles

also

have

some

effectsthat aredifferent

from

those that

one

would

expect

from

financial integration. Forinstance,

domestic

and

foreign

shocks

lead to

movements

in the size of the bubblethatgenerate potentially largewealth effects.

Through

thischannel, the

presence

ofbubbles magnifies theeffects ofproductivity

and

other

shocks

on aggregate

activity.

Another

effectof

bubbles

is to

open

the

door

to

shocks

toexpectations

as

a

new

source of

macroeconomic

fluctuations.

Bubbles

are inherently unstable, since theyarise in

environments

where

agents

need

to

coordinateto

one

among

a variety of possible equilibria. Since bubbles

magnify

the effects of productivity

shocks

and

open

the

door

toexpectational shocks, low productivitycountries that

have

large bubbles tendto

be

more

volatilethan high

productivitycountries that

have

small bubbles. But the relationship

between

volatility

and

productivity is nonlinear. Countrieswith intermediatelevelsof productivitytendto

have

smallerbut

more

volatile bubbles. In particular, I find thatinthese countries

external

shocks generate

movements

inthesize oftheir

bubbles

that

have

potentially

largewealth effects. This observation

can

be used

to provideatheoreticaljustification

tothe notion that

middle-income

countriesarevery vulnerable tosmall

economic

fluctuations in high-incomecountries. It

can

also

be used as key

building block in

an

attempttosketch a theoryofthe internationalcontagion of financial crises.

The

paper

isorganized as follows: Section

one

presents a stylized

model

of trade

and

growth

and

describes theworld equilibriumwithout bubbles. Section

two

shows

thatthere areadditionalworld equilibria with bubbles

and

formallydescribes them. Sectionsthreeto

seven

presentfive

examples

designed

toillustratethe

macroeconomic

effects of bubbles.

The

first

two

emphasize

therole ofbubbles as a substitutefor international capital flows.

The

remaining threedealwith the

concepts

offinancialfragility, vulnerabilitytoexternal

shocks

and

contagion. Section eight

1 .

A

Productivity-Based

View

of

the

Growth Process

In thissection I present a simple

model

oftrade

and

growth thatwill serve as a

toolor vehicle tostudythe conditions

under which

asset price bubbles

can

exist

and

their

macroeconomic

effects.

The

model

has

some

unrealistic features

such

as the prediction thatfactorprices are equalized acrosscountries orthe

assumption

that the

lawof

one

priceappliestoall goods.

These

aspectsofthe

model

are notimportant

however

fortheresultsofthis paper,

and have

been

chosen

onlytostreamline the discussion.

The

two

criticalfeaturesofthe

model have

already

been

discussed in the

introduction: whiletrade in

goods

ensures

that long-run rates of

economic

growth are

equalized acrosscountries, frictions in assettrade preclude theelimination of rate-of-return differentials across countries.

There

arevarious

models

oftrade

and

growth thatshare these

two

criticalfeatureswithout predicting that

goods

and

factorprices

areequalized acrosscountries. Forinstance,

Acemoglu

and

I [2002]wrote a

model

withthesecharacteristics recently.

Consider

a world

economy

with J countries, indexed byj=1,2 J.

There

are

two

factors of production: labor

and

capital,

and

they are

used

inthe production of

two

intermediates: K-

and

L-products.

These

intermediates are inturn

used

to

produce

two

final goods:

consumption

and

investment.

The

costs oftransporting factorsare prohibitive, whilethe costsof transporting

goods

are negligible.

In all countries thereare competitivefirms with

access

toa

common

technologyto

produce

intermediates.

To

produce

one

unitofthe K-product, these

firms require

one

unit ofcapital.

To

produce

one

unit ofthe L-product, they

need

one

unitof labor. Sincefinal

goods

production only requires intermediates, full

employment

offactors impliesthat both intermediates are

produced

in all countries. Perfectcompetition

ensures

that therental

and

the

wage

areequal tothe prices of

K-and

L-products in all countries, while internationaltrade

ensures

thatthe prices of

equalized acrosscountries

as

well.7 Definert

and

w,asthe

common

rental (orpriceof

K-products)

and

wage

(orprice ofL-products) atdatet.

In all countriesthereare also competitivefirms with

access

toa

common

technologyto

produce

finalgoods.

To

produce

one

unit ofthe

consumption

good,

firms

have

a

Cobb-Douglas

technologywith a L-product (orlabor) share equal to a,

with

0<a<1

.

To

produce

one

unit ofthe investment good, firms

have

a technologythat

requires

one

unitofthe K-product. Since all firmsin all countriesface the

same

price of intermediates,the costsof producingfinal

goods

are alsothe

same

in all countries. Perfect competitionthen

ensures

that

goods

prices equalthese production costs:

(1) 1

=

r

t

1-a

-w?

(2) qt

=r

t

where

_qtisthe price ofinvestment

and consumption

isthe numeraire.

The

demographic

structure ofcountries isthatofa two-period overlapping generations

model

withconstant population.All generations

have

sizeone. In

each

date, a

new

generation of

consumers

is born that lives

two

periods:

young

and

old.

The

consumers'

goal in life isto

maximize

the

expected

valueof old

age

consumption.

When

young,

consumers

work

and save

their

wage.

The

only decision

intheir life is

what

to

do

with thesesavings.

When

old,

consumers

use

the returnto theirsavings to

purchase

consumption

goods. Thisformulation is nothing buta stark versionofthe popularlife-cycle

model

of savings.

7

ThisresultisduetoSamuelson[1948]and appliestoawide rangeofmodels(SeeKrugmanand

Helpman[1985]).Whatisperhapssurprisinghere isthatfactorpricesareequalizedforany

cross-countrydistribution ofcapital-laborratios.Thisisa special butveryconvenientpropertyof this specific production structurethatI borrow from Ventura[1997].

In

each

generation,

some

ofthe

young

create

and

operatefirms that

purchase

investment

goods

in theiryouth

and

convert

them

into capitalthat

can be used

in their

old age. Forsimplicity, I

assume

that thisentrepreneurial activityrequires

no

effort

and

thatthere is

enough

talent in thecountryso astodrive the

wage

of

entrepreneurs tozero.

The

entrepreneursofcountryj atdatet are

capable

of

producing 7ij

+1 units of capitalwith

each

investment good. I refer to n)

+1

as

the

productivityofcountryj

and

assume

that it

can

varystochasticallyover time within a

support that is strictlypositive

and

bounded

above. Let ij

and

Kj

be

the investment

and

capitalstock ofthe country.

Assuming

that capitalfullydepreciatesin

one

generation,

we

have

that:

(3) Ki+1 =7i!+1-l{

Given

these assumptions,

each

unit of

income

investedin country_jyields a

r 71^

rate ofreturn equalto

—

—

—

.

To

financetheir

purchases

ofinvestmentgoods,

qt

firms issueshares

and

sell

them

in the

domestic

stock market. Afteroutput

has

been

produced

and

distributed in the

form

of dividend,thefirm

has

no

assets

and

thevalue

ofits shares dropstozero.

A

key assumption

is thatthe costs oftrading in the

domestic

stock

market

are negligible,while thecostsoftrading in foreign stock

markets

are prohibitive. This

asymmetry

incosts could

be

due

toa policy-induced barrier

such as

prohibitive capitalcontrols, oritcould

be

due

tothe inabilityofcountriesto

commit

notto

expropriate foreign investments.

Whatever

thereason, the

young

are forced toinvest

allof theirsavings in the

domestic

stock market,

and

this implies the following:

(4)

q

t

-l{=w

t

Equation (4) statesthat investment is equal to laborincome, sincethe

young

save

the entire

wage.

Equation (5)

shows

that

consumption

isequal to capital income, sincetheold

have no bequest

motive.

Define world

averages

by omitting thecountry subscript. For instance,the

-i

world

average

capital stock is K,

=

-

VK{

. Applying

Shepard's

lemma

to Equations

J

i

(1)

and

(2),

we

findtheworld

average

demands

for

K

and

L-products

as

1 1

—

(X ex

—

•

q, •I,

+

C.

and

C., respectively.Sincethe

average

supplies ofthese

r, r

r w,

products are

K

t

and

1, international

commodity

markets

clearif

and

only if:

(6) r

t

K

t

=q

t

l

t+

(1-a)C

t

(7)

w

t

=a-C,

The

competitive equilibrium ofthisworld

economy

is aset of prices

and

quantities

such

that

consumers

optimize

and markets

clear.

Together

with

an

initial

distribution ofcapitalstocks, Equations (1)-(7) provide a

complete

description ofthis

equilibrium.

By

Walras' law,

one

among

Equations(4)-(7) is

redundant

and can be

dropped. I

show

next

some

properties of thisworld equilibrium.

Let R{

be

the rate ofreturnto theportfolio ofcountryj.

Since

this portfolio

containsonly

domestic

investment,

we

can use

Equations (1), (2)

and

(7), to findthat:

(8)

R!=g,

1-«-7r|

Q

where

_gt is thegrowth rateofworldconsumption, i.e. g

t

=

—

_—

.

Consider

the

case

in

the rate ofreturn

remains

constant

because

the price of investment

goods

also declines atthe

same

rate. Thisfollows

from

Equation (2)

and

isa direct

consequence

ofthe

assumption

that only capital is

used

inthe productionof investment goods.

The

ideathatthereexists a coreofcapital

goods

that

can be produced

using onlythis

same

coreofcapital

goods

isthe basic tenet underlying the lineargrowth

model

of

Rebelo

[1991].

Here

we

have adopted

the

extreme

AK

version of this

model

in

which

there isonly

one

capital

good

inthis core.

Itfollows

from

Equations (3), (5)

and

(7) thattheworld growth rate of

consumption

isgiven by:

(9) gt

=oc-R,

The

world growth rate

depends

positively

on

a

and

R

t.

The

higheris a,the

higheristhe labor

share

in

income and

the higherisworld savings.

The

higheris

R

t,

the higher isthe

average

return obtained

from

these savings. Allcountriessharea

common

growth rate inthe long run.

The

reason is simple: growth in high-return countries

improves

the

terms

oftrade oflow-return

ones

and

this

keeps

theworld

income

distribution stable. Since

wages

are equalized internationally, allcountries

save

the

same, and

differencesin

consumption

reflect differences in ratesof return.

C

j

R

j

To

see

this, notethat Equations (3)-(5) implythat

—

-

=

—

-.

8

C

t

R

t

Solving Equations (8)

and

(9),

we

obtainthis stylized description ofthe growth process as afunction ofthe

exogenously

specified productivityprocesses:

(10)

g,=(a-7i

t)

1-a

Itwould bestraightforwardtointroducelabor-augmentingproductivitiesdifferencesacrosscountriesas

World

growth

depends

on average

productivity, while

each

country's position

in the worlddistribution of

consumption

orwealth

depends

on

itsrelative productivity. This world thereforeencapsulates a traditional or productivity-basedviewof

economic

growth

and

fluctuations. This

view

is reasonable

and has

the potential toexplain

some

ofthe

most

interesting growth experiences,

such as

the existenceofgrowth

miracles orworldwide

changes

in rates of

economic

growth. Butthe

productivity-based

viewis notwell suited to explain otherrealworld

phenomenae

such

asthe

recurrence

and

depth offinancialcrises ortheirpuzzling

tendency

to propagate acrosscountries.

A

key

question istherefore:

How

can

we

model

these aspectsof

realitywithout losing thebasic insightsofthe productivity-based

view?

A

central claim ofthis

paper

isthat theimplicit(and unjustified)

assumption

that asset prices reflect their

fundamental

value isthe obstacle that stands

on

the

way

ofdoing this.

Once

we

push

itaside, it ispossibleto catch

an

eye-opening

glimpse

of

what

the

productivity-based

view

has

been

leaving behind.

2.

Country

Bubbles

If R{

>

_g, for allj

and

t, theworld equilibrium described in the previoussection

is unique. But if R{

<

_g, for

some

j

and

t,there might

be

otherequilibria in

which

useless assetsorfirmsarevalued

and

traded in the stockmarket. Followingstandard convention, I refertothese useless assetsorfirms

as

pure bubbles.

9

To

be

clear, these bubbles are rational sincetheirexistence

does

not rely

on

the

presence

of

individuals thatare incapable ofprocessing publicly available information orthat

behave

in a

manner

that is inconsistent with the usual

axioms

ofchoicetheory. In the

bubblyequilibria studied here, the old sell bubblestotheyoung.

The

laterunderstand

Thedistinction betweenproductiveassetsand pure bubblesisusefulasatheoreticaldevice. However intherealworldassetpricebubblessometimeseemtoappearinproductive assets.Olivier[2000]

showshowto modifythetheorytoallowforbubblesthatare "attached"toproductive assets.

thatthese bubbleswill neverdeliveradividend, buttheystill

buy

them

because

they

rationallyexpect a return inthe

form

of price appreciation.

Since the costs of internationaltrade in assetsare still prohibitive, bubbles are onlytraded withinthecountry.

That

is,they are country bubbles. Let BJ

be

the sizeof

the bubble in countryj atdatet, i.e. thevalueof allthe uselessfirms traded inthe stock

market

ofcountryj. This bubble

grows

according to this process:

(11) B{₊₁

=

_mIi-B{

where

u.j₊₁ is thegrowth rate ofthe bubbleor itsrateof price appreciation.

Note

that

u.{₊₁ might not

be

known

as ofdatet, since in general the valueofthe bubbleatdate

t+1

depend

on

informationthat is not available atdate t. I definea country bubble

as

an

initial value, i.e.

B

J

>

and

a price appreciation process, i.e. u.j₊₁ for all t>0.

The

presence

of a bubble enlarges the portfoliochoiceofthe

young

sincethey

now

can

choose

whether

to hold thecapital stock, thecountry bubble, or both.

We

must

thereforereplace Equation (4) by this pair:

(12) qt

-lj+B!=w

t if

E

_t

RJ<E

t

K^l

j B|

J[0,w

t] if

E

t

{uU=E

t

j^l^

I 9t

w

t if

E

t

RJ>E

t

j^!±i

Equation (12)

shows

thatthe

young

now

distribute theirsavings

between

the

stockofcapital

and

the bubble. Equation (13) describesthe choice ofcountry

portfolio

as

a function of assetreturns.

The

return to holding a bubble is its

growth

r .jr^

rate, i.e. u.{

+1; whilethe returnto holding capital is the dividend, i.e.

-^

—

^-. Since

q.

the

young

are risk-neutral, they

choose

theasset thatoffersthe highest

expected

return. If both assetsofferthe

same

expected

return, the

young

are indifferent

between

holding capital, the bubbleor both.

The

presence

ofthebubble also impliesthatthe

consumption

ofthe old is

now

given

by

theircapital

income

plusthe

proceeds

ofselling the bubble. Therefore,

we

must

replace Equation (5) by:

(14)

C{=r

t

-K{+B{

The

restofthe

model

remains

the

same.

For a given initial distributionof

capital stocks, a competitive equilibrium oftheworld

economy

consists of a set of

country

bubbles

such

that Equations (1)-(3), (6)-(7)

and

(11)-(14) holdfor all possible

realizations oftheworld

economy.

Once

again, by Walras' law

we

can drop

one

among

Equations(6), (7), (12)

and

(14).

The

productivity-basedvieworbubbleless

equilibrium obtains by setting

Bj,=0

forallj;

and

E

_t^i{₊₁

}<E,j—

—

—

[ forall_j

and

t. Thisequilibrium

always

exists,

as

shown

in theprevious section. Butit might not

be

unique.

Since the country portfolio

now

might containthe bubble, itsrateof return

need no

longer

be

equaltothe rate ofreturn to capital. Define bj

as

the

share

ofthe

B

j

bubblein thecountryportfolioorsavings, i.e. b{

=

—

r.

The

rate of return to

qt -lj+Bi

countryj's portfolio is

now

given by:

(15)

R^g^tt'-a-bj^

+

^b^

A

bitof algebra

shows

that Equation (9) still describestheworld growth rate.

However,

it is not possibleto provide a

complete

description ofthegrowth process as afunction of

exogenous

impulses using Equations (9)

and

(15) alone.

To

do

so

we

need

tospecify

exogenously

notonly the pathof productivity, but also

how

countries

coordinate toa particularequilibrium. Forinstance,the derivation ofEquation (10)

implicitly

assumes

that: (i) thebubblelessequilibrium is unique or, alternatively, (ii) all

countriescoordinate to thebubblelessequilibrium with probabilityone. Either ofthese

two assumptions

orviews is sufficient tojustifythe productivity-based

view

ofthe

growth process. But they are not theonly possibilities.

As

a prelude to

making assumptions on

how

countriescoordinate toa given

equilibrium, itis useful toobtain a simplecharacterization ofallthe sets ofcountry bubblesthat are feasible in world equilibrium.

The

following proposition

does

this:

Proposition#1

:

Consider

J

sequences

{bj |~™

such

that bj

e

[0,1] for allt

and

j.

The

set ofcountry bubbles defined bythese

sequences

isfeasible if

and

onlyif, for all

j

andt, (C1)

E

t-jgt+1

--^-l>EJg£f

-jijL with

b|=1

ifthe inequality isstrict;

and

b{ f

v-

i ,. Li *V-a

(C2)g

_t+i

a

v

2>i

+i-(1-bj)

The

proof simplyconsists

on

substituting Equation (11) into (13)

and

using

equilibrium pricestoobtain (C1),

and

to substitute Equation (15) into Equation (9)to

obtain (C2).

The

proposition is useful

because

it allows usto

check whether

a

proposed

set ofcountry

bubbles

isfeasible by solving a small

system

ofequations.

To

develop

intuitions

about

the

macroeconomic

effectsof bubbles, I develop

nextfive

examples.

Together, they reveala

new

and

broader

view

ofthegrowth process in

which

both productivity

and

asset price bubblesplay central roles,

and

interact in interesting

and

somewhat

surprisingways.

A

recurring

theme

isthat bubbles substituteforcapitalflows and, as a result, their

main

effects areakin or

similar tothoseof financial integration.

Another

theme

isthat

movements

in the size

of

bubbles

createlarge wealth shocks.

These

movements

might

be

driven by

shocks

to productivity. Butthe

presence

of

bubbles

also

opens

the

door

for

shocks

to expectationsto

become

part ofthe

macroeconomic

landscape.

3.

Bubbles as

a

Substitute

for

Long-Term

Capital

Flows

Assume

theworld contains infinitely

many

countries of

two

types. Half of

them

have

low productivity, i.e. n\

= n

l;whilethe rest

have

high productivity, i.e.

n|

=

JtH

>

7t

L

.

The

goal ofthis

example

isto determine

how

the

presence

ofbubbles

affects boththeworld growth rate

and

thedistribution ofwealth acrosscountries.

Itis usefulto start

by

describing the bubblelessequilibrium. Itfollows

from

Equations(8)

and

(9) that inthis equilibrium the following applies:

(16)

71+71

a

V 2 1-a -a (17)

R

J

=g

1 -«Tt'

The

world growth rate

depends

only

on average

productivity, while the

cross-country distribution of rates of return (and

consumption

or wealth)

depends

on

how

dispersed this productivityis.

As

discussed already,this isthe

essence

ofthe productivity-based

view

of

economic

growth.

Butthis might not

be

the only equilibrium ofthis world

economy.

Consider

the

possibilitythat countryj deviates

from

the others

and

coordinatesto

an

equilibrium

with astationary bubble, i.e. b{

=

b

forallt. Isthis possible? Since the world

growth

rate isconstant

and

there is

no

uncertainty, condition (C1) for this bubble

can be

written asfollows:

1

(18) g1

"_a

>n

j

To

createits

own

demand,

the bubble

must

grow

ata rate thatis

no

lowerthan the

-a rate ofreturnto capital, i.e. _g1_a Tt

j

.

To

ensure

that it

does

not

outgrow

savings, the

bubble

must

grow

ata ratethatis

no

higherthan thegrowth rate, i.e. g. Therefore, Equation (19)

says

thatthere is

room

fora bubble in countryj if

and

onlyifthe growth

rate

exceeds

therate of return to capital.

Except

forthe knife-edge

case

in

which

inequality (18) holdsstrictly,

we

have

that:

(19) b

=

1

That

is, thebubble

crowds

out all theinvestmentofthe country. Ifcountryj isa

high-productivity countrythis bubble is not feasible

because

condition (18) never holds (To

check

this, substitute Equation (16) into condition (18) with 7ii=7t

H

). Ifcountryj isa

low-productivitycountry the bubble might

be

feasible

because

condition (18) holdsfor

some

parameter

values.10

Ifithas lowproductivity,countryjmightalsohaveanyofacontinuumofnon-stationarybubbles

definedbyaninitialvaluebo'e(0,1]and sequenceofb\suchthat(C1)holds withstrictequality.These

bubbles vanishgraduallyover timeandwillbeignoredintherest of thispaper.

Consider

the

case

in

which

afraction $ofthe lowproductivity countries

coordinatetothebubblyequilibrium, whilethe restcoordinatetothe bubbleless one. I

referto<j)asthe size of the world bubble. Itfollows

from

Equations(9)

and

(15) that:

( (1-_-<t>) _L . _H_> 71

+

Tt 1-a (20) g

=

a-V

2-

a<|> , fg if b{

=

1 (21)

R'~|

.-a

g

1^a •7Tj if bj

=

Equations(20)

and

(21)describe theworld growth rate

and

the return tothe

country portfolios in the bubblyequilibrium. Substituting Equation (20) into condition

(18),

we

findthat this equilibrium exists if

and

onlyifcross-countrydispersion in

71 Ot

productivityis sufficientlyhigh, i.e. if

—

n-< •

Assume

this

from

now

on.

n

2

-a

Using Equation (20), it is straightforward to

check

that theworld growth rate increaseswiththe size ofthe bubble.

The

key

intuition isthatthe bubble eliminates

inefficient investments in low-productivitycountries

and

raisesefficient investments in

high-productivityones.

An

increaseof

one

percent intheworld bubble increases

consumption

(or

reduces

investment spending)

by

one

percent in countries

where

average

productivity is

n

l

. This lowers the price of investment

goods

and

raises the

wage

all overthe world.

A

lowerprice ofinvestment

goods

means

that

each

unit of

savings

buys

more

investment goods, while a higher

wage means

that savings increases all overthe world.

A

one

percent increase in theworld bubble leadsto

an

increase ofinvestment

spending

of

a

percent inthe rest oftheworld

where

average

productivityis -

—

—

(See

Equations(7)

and

(12)). Since Equations(18)

and

2-a-<(>

(A i\\\ TT^~_l_TT^

(20)

ensure

thatthe bubble isfeasible if

and

onlyif

n

L

< a--

—

—

,

we

have

2-a<|)

16

that

an

increasein the size of thebubble increasestheworld capitalstockfor a given levelofproduction

and

this raisesthe world growth rate.

The

maximum

world growth rate applies inthe limit

where

the bubble is as largeas it

can

be, i.e. <j>-»1.

Using Equation (21),

we

alsofindthatthe

presence

ofbubbles raisesthe rate

ofreturn oftheportfolio ofthe countries that hold thebubble relative tothe restofthe world. This

means

thatthesecountries

move

up

in theworlddistribution of

consumption and

wealth. Since countriesthat

have

a bubble also

have

low

productivity,the

presence

ofbubbles tendsto

reduce

rate-of-return differentials

and

C

R

j

improve

theworld distribution of

consumption and

wealth.

(Remember

that

—

-

=

—

_-).

C

t R,

Infact, the

most

equalitariandistribution applies in thelimit

when

the bubble is

as

large

as

it

can

be, i.e. _<J>->1.

This

example

illustratestherole of bubbles

as

a substitutefor long-term capitalflows. Sincethelater are not possible, stock

markets generate

bubbles

as

an

alternativeorsecond-best

mechanism

to

improve

theworldallocation of investment.

Bubbles

arise in low-productivity countries, eliminating inefficientinvestments

and

liberating resourcesthat are

used

in part to increase investmentin high-productivity countries.Thistransfer is

made

via

changes

in prices

and

without actual capitalflows,

since

we

have

assumed

throughoutthattrade is balanced

and

current

accounts

are

zero.

By

increasingtheefficiency ofworld investment, bubbles raise theworldgrowth rate.

By

reducing rate-of-return differentials across countries, bubbles

improve

the

world

income

distribution.

4.

Bubbles

as

a

Substitute

for

Short-Term

Capital

Flows

Consider

nexta worldwith infinitely

many

countries ofasingle type. Productivityfluctuates

between

a high

and

a low state, i.e. n\

e

{n:

L ,7t H jwith 7i H

> n

L . I

assume

that productivity is

known

before

making

investments, i.e.

E

t{nj+1}=n{+1.

The

probability oftransitioning

from

one

statetothe otheris equaltoA.

and

is

independent

acrossj

and

t.

The

world startswith halfofthecountries in

each

state. Sincethis is

the invariant distribution,the proportion of countries in

each

stateis constantover

time.

The

goal ofthis

example

istodetermine

how

the

presence

of

bubbles

determines

the

way

countries react to productivityshocks.

The

bubblelessequilibrium ofthis world

economy

isthe

same

as

inthe previous

example,

i.e.

as

described in Equations (16)-(17).

The

only difference isthat

now

all countriesare ex-ante identical

and each

of

them

spends

half ofthetime being

low-productivity

and

halfofthetime being high-productivity.

As

X—

>0 differences in

productivity

become

permanent and

we

approach

theworld

we

analyzed in the previous subsection.

Consider

the possibility thatcountryjdeviates

from

theothers

and

coordinates

to

an

equilibrium with a stationary bubble, i.e. bj

=

bH if n\

= n

H

and

bj

= b

L if n\

=

n

l .

Inthis case, condition (C1) impliesthefollowing:

(22)

b

L 1

g

1 -a >7tL

and

1-X+X-—TT

bH 1 g1 ~_a >tih 1 Since ti h

>

_g1 a

the bubble

must

expand

atthe

end

ofhigh productivityperiodsto

create its

own

demand,

i.e. bL

>b

H. This in turn impliesthatthe bubble

must

contractat

the

end

oflow productivity periods.

Except

fortheknife-edge

case

in

which

both

inequalities hold strictly, itfollows

from

₍₂₂₎ that:

(23) b L

=

1 b H

=

-X +

—

-

—

1 ,1-a

Now

eitherallthe countries

can

have

thebubble, or

none

of

them

can have

it.

Note

also that

shocks

toproductivitygenerate

movements

in the bubble. Transitions to the high state leadto contractionsofthe bubble, whiletransitions tothe lowstate lead toexpansions. Equation (23)

shows

thatthe size ofthese

movements

is

positively relatedtotheworld

growth

rate.

The

largeristhe latterthe

more

attractive

is thebubble inside high productivity periods,

and

thesmallerarethe

expansions

at

the

end

ofthese periodsthatare requiredtosupport the

demand

forthe bubble.

Equation (23) also

shows

thatthere isa positiverelationship

between

the sizeof these

movements

in the bubble

and

the persistenceofshocks.

The

smalleristhe

transition probabilitythelower isthe likelihood thatthebubble

expands

and,

as

a

result,the largerare the

expansions

thatare required atthe

end

of high productivity

periods.

Consider

the

case

in

which

afraction _tyofthe countriescoordinatetothe stationarybubble, whilethe restcoordinate tothe bubbleless equilibrium.11 Itfollows

from

Equations (9)

and

(15)that, inthis case:

(24) g

(1-(}))-7tL +(1-(|)-bH)-7lH

2-a-<j>-(1

+ b

H)

Notethatnow§isthefractionofallcountries with a bubble, whileinthepreviousexample$wasthe fractionoflow-productivity countries with a bubble.Sincethepreviousexampleobtainsasthelimit in

which\^>0 andbH

—

>0,alltheformulasintheprevious section arevalidunderbothdefinitions.

g1-«-7tH

-(1-b

H)

+

g-b

H if n|

=

n{ +1

=

n

H -a (25)

R

i

=

j9' r_° TtH (1

-

bH )

+

9 if 7i|

=

it H

and

ti| +1

= n

L g if Tt{

=

7t{ +1

=

n

L g bH if n{

= n

l

and

nj₊₁

=

n

H

where

I

have used

the finding that b

L

=1. Equations (23)-(25) provide a full description ofthe bubblyequilibrium, ifit exists.

The

latterrequires a sufficiently high dispersion

of productivities. Infact, it is

easy

to

show

thatthis restriction is

more

stringent

now

than in the previous

example.

Assume

that it issatisfied

from

now

on.

The

equilibriumvaluesforbH

and

g areobtained by crossing Equations (24)

and

(25). Figure 1

shows

how

these values

depend

on

<|)

and

A..

An

increase in the

world bubble raises theworld growth rate and, forthe

reasons

discussed, it increases

the sizeofthe bubble in high productivity countries

(compare

points

A

and

B).

An

increase in thetransition probability raisesthe size ofthe bubble in high productivity

countries and,

as

a result, it

reduces

efficientinvestment

and

lowerstheworldgrowth rate

(compare

points

A

and

C).

Inthis

example

the

presence

of a bubble not only raises the long-run rateof growth, but also affects the nature of

economic

fluctuations.

A

firsteffect ofthe

bubble is hat increasesthevolatility ofinvestment. Incountries that

do

not

have

a

bubble, investmentis not sensitiveto

domestic

productivity

and

is

determined

solely bythe savings oftheyoung. Incountries that

do

have

a bubble, investmentfluctuates with

domestic

productivity.

When

productivity is high, the

bubble

contracts

and

investmentexpands.

When

productivity is low, the bubble

expands

and

investment

contracts. This is

how

bubbles

breakthe link

between

changes

in

domestic

investment

and

thesavings oftheyoung, justas international capitalflows

would

do

if

they

were

possible.

A

second

effect ofthe bubble istocreate potentially largewealth effectsthat impact

on

consumption. In countrieswithout a bubble,

consumption

fluctuates

one-to-one

with

domestic

productivity. This is notthe

case

in countries with a bubble, since

their

consumption

depends

on

both the returnto capital

and

the rate ofgrowth ofthe bubble. Conditional

on

being in normal times(i.e. periods in

which

a country

does

not

transition

from

one

state totheother), countries with a bubble

have

a

smoother

consumption process

thanthose without a bubble.

The

reason

isthat therate of

return to their portfolio is higherinthe lowstate but lowerinthe highstate. Ifthis

were

all,

we

would conclude

thatthe bubble

reduces

the volatility of

consumption

just

as

international capitalflows

would do

inthis context ifthey

were

possible. But bubbles also generate a source ofvolatilityin

consumption

that

we

would

not

observe

ina financially integrated world. During thetransitions

from

one

statetotheother, the

movements

in thesize ofthebubble generate wealth effects that might potentially

have

a large impact

on consumption

and

welfare. In particular,

we

have

that atthe

end

ofhigh-productivityperiods thereis a

consumption

boom,

while atthe

end

of

low-productivity periodsthere is a

consumption

bust.

Thisworld

economy

providesanother

example

of

how

stock

markets

create bubbles as a substitutefor international capitalflows.

Bubbles

expand

when

the

economy

transitions tothelowproductivity state, absorbing savings

and

eliminating

inefficient investments.

Bubbles

contract

when

the

economy

transitions tothe high

productivity state, releasing savings

and making

room

for efficientinvestments.

These

movements

in the bubble shiftinvestments

from

lowto high productivitycountries,

improving theworld allocation ofinvestment

and

reducing rate-of-return differentials across countries.

As

a resultofthe

movements

in the bubble, investment

becomes

more

volatilethan

what

standard

models would

predict, while

consumption

becomes

smoother

in

normal

times but subjectto potentially largewealth

shocks

associated

with

expansions

and

contractions in the bubble.

Some

ofthe implicationsofthe theoryare

becoming

clear

now.

The

same

factors that

would generate

a capitaloutflow(i.e. the realizationthat a high

productivityperiod is ending)generate growth inthebubble

as

a preludeto a collapse of investment. Conversely, the

same

factors that

would

generate acapital inflow(i.e.

therealization thata low productivity period isending) generate a collapse in the

bubble

as

a preludeto

an

investment

boom.

Providedthatthecountry always

coordinatestothe stationary bubble, the

movements

ofthelateraretherefore closely

tiedto

movements

in productivity. But

why

would

the

economy

always coordinate to

the stationary

bubble?

The

answer, ofcourse, is that it

does

not

have

to.

5.

Financial

Fragility

Consider

the

same

worldas intheprevious

example,

but

now assume

thatall

countries coordinateto

an

equilibrium using a country-specific sunspotvariablethat

says

BUBBLE

with probability<|>

and

NO

BUBBLE

with probability

1-<J>.

These

sunspots are

independent

acrosscountries

and

overtime.

When

a country switches fromthe bubblytothe bubblelessequilibrium, theold find thatthevalueofthebubble they are holding collapsestozero

and

theysuffera loss.

When

a countryswitches

from

the bubblelesstothebubbly equilibrium, theoldfind

themselves

abletosell a

uselessobjecttothe

young

and

experience a windfall.

The

world startswith afraction

$ ofall the countrieshaving a bubble. Sincethis isthe invariant distribution, the proportion of countries in

each

state is constantovertime.

The

goal ofthis

example

is

to

show

how

the

presence

ofbubbles

makes

countriesfragileto

shocks

to

expectations. In fact, itis tempting tothinkaboutthis

example

as a

model

offinancial

fragility.

This

change

in

assumptions

implies a

few minor

adjustments in the

argument

ofthe previoussubsection. Ifthe

j

th

sunspot

variable

says

NO

BUBBLE,

we

have

that

b{

=0.

Ifthe

j

th

sunspot

variable

says

BUBBLE,

we

have

then a stationary bubble very similar tothe

one

in the previous section.Sincethere isa probability thatthe

bubble bursts, condition (C1)

now

impliesthat:

(26)

1-X

+

X-

b^

bL

g

1-ct -())>7i L

and

1-A.

+

X-bH g 1-" -<t)>7i r

Except

forthe knife-edge

case

in

which

both inequalities hold strictly, itfollows

from

Equation (26) that:

(27) bL

=

1

X +

-n

4>-g1-cc

Conditional

on

remaining in the bubblyequilibrium, the behavior ofthe bubble is

basicallythe

same

as

intheprevious

example.

Of

course,

now

the bubble also bursts

and reappears

with

some

probability.Also,

we

can

show

thatthe dispersion of

productivity requiredtosustainthe bubblyequilibrium is higher here, sincethe

probabilityof bursting

makes

thebubble less attractive.

The

presence

ofthebubble

opens

the

door

for

shocks

toexpectationsto affect real variables

such as

investment

and

consumption. Conditional

on

not

changing

the equilibrium, thebehaviorof

consumption

and

investmentin countries with

and

without a bubble is the

same

as

in thepreviousexample. But

now

all

countries

go

through periods in

which

they

have

a bubble

and

periods in

which

they

do

not.

When

thebubble bursts,

consumption

declines as thewealth oftheold collapses

and

investment increases. Inthe aftermath ofthe bursting ofthe bubble,

consumption

increases in high productivity countries but declines in low productivity ones.

When

the bubble appears,

consumption

increases

and

investment declines. In

the aftermathofthe

appearance

ofthe bubble,

consumption

declines in high productivitycountries but increases in lowproductivityones.

An

interestingfinding relatestothe interaction

between shocks

to productivity

and shocks

to expectations. Since countries tend to

have

a largebubble

component

in theirwealth

when

productivity is low,

shocks

toexpectations tend to

have

more

extreme

effects

on

macroeconomic

aggregates

and

welfare during recessions.

Conversely, during

booms

countries are lessfragileor

more

resilientto

shocks

to expectations

because

thebubble

component

ofwealth is smaller. Therefore,

we

find

that

shocks

toproductivity

and

expectations multiplytheeffects of

each

other

and

tendto increasetheamplitude of

economic

fluctuationsiftheyare positively correlated.

Although the

model

cannot predicttheonset ofexpectations-driven crises,12it

stilloffers fresh insights intoold problems.

Consider

for instancethe popularviewthat

countries should

impose

a Tobin tax or

some

sort ofcapital controlsto

reduce

the probability of costlyfinancial crises

and

help stabilize small

open

economies.

The

example

presented heredepicts a

case

in

which

financialfragilityarises precisely

as

a resultof

impediments

to assettrade. In the

absence

ofthese impediments, allworld

investment

would be

located in the high productivity countries. In

such

aworld, there

would

be

no

room

for

shocks

to expectations

and

theirwealth effects.

The

theory

therefore

suggests

theopposite

view

thatin orderto eliminate

bubbles and

the

financialfragility that

accompanies them one

should not

add

frictions infinancial markets, but

remove them

instead.

12

Thiswouldcertainlybetoomuchtoask fromthemodel.Afterall,the modelcannotpredict theonset

ofproductivity-driven criseseither.Todeterminethe equilibriumpathoftheworld economy,we must

specify a pathforbothproductivityandexpectations. Bothvariables aretakenas given hereandthisis

why Ilabelthemshocks.

6.

Vulnerability to

External

Shocks

Consider

a world

economy

with

two

countries. North's productivity fluctuates

between

a high

and

a lowstate, i.e. tc^

g

{ti

l ,7i; h }with

n

H

>

n

L; while South'sis

constant but low:

n

s

<n

l.

As

in previous

examples,

this productivity is

known

before

investments are

made,

i.e.

E

t{Tt^+1}=7i^+1. Inthis world

economy,

all productivity

fluctuations

have

theirorigin in North.

The

goal ofthis

example

isto

show

thatthe

presence

ofbubbles

makes

low

and middle-income

countries particularlyvulnerable toproductivityfluctuationsin rich countries.

Ifthe North-South productivity

gap

is small, the uniqueequilibrium ofthis world

economy

is thebubbleless one. In thisequilibrium, theworld allocates halfof its

investment in North

and

the restin South.

As

a result,

we

have

that:

(28) 9t

=

71,

+

71

a-—

V 2 J 1-a (29)

R?"

Equations (28)

and

(29)describethe evolution ofworld growth

and

theworld distribution of

consumption

orwealth in the bubbleless equilibrium.

The

top panelof

Figure2

shows

this evolution graphically.

The

world growth ratefluctuates as a result ofproductivity

shocks

in North. Since

each

countryinvestsat

home,

North tendsto suffer

more

from

its

own

shocks

than South.

IftheNorth-South productivity

gap

is large,there is a bubblyequilibriumin

which South has

a stationary bubble defined by bf

=

1 forallt.

Assume

South

coordinates tothis equilibrium with probabilityone.

Then,

allthe investmentis

done

in

North and, as a result,

we

have

that:

(30) g,

(31)

*L

=

1^

Rf

a

The

middle panel of Figure2

shows

the evolution oftheworld growth rate

and

relative

consumptions

in the bubblyequilibrium.

The

presence

ofthebubble

makes

the worldgrowth rate

more

sensitiveto productivity

shocks

in North, since

now

these

shocks

notonly raisethe rate ofreturn to North'sinvestment but alsothegrowth rate ofSouth'sbubble.

The

presence

ofthe bubblealsotendstosynchronize

economic

fluctuations in North

and

South. Inthe bubbleless equilibrium, thereturn to

each

countryportfolio

depends

more

on

the country's

own

productivitythan in the

productivity oftheother country. In thebubblyequilibrium, the returnstoboth

countries' portfolios

depend

only

on

North's productivity.

As

a result,the bubble generatesa perfect correlation in countryportfolios

and

wealth. Despite North having

more

volatile productivitythan South, both countries experiencethe

same

volatility in

their

consumption

and

wealth.

IftheNorth-South productivity

gap

is small,

South

is betteroffinvesting. Ifthe

gap

is large,

South

is betteroff not investing

and

instead holding a bubble. Ifthe

gap

is intermediate

however,

South

might

be

betteroffinvesting

when

North's productivity

is low, butmight

be

betteroffholding a bubble

when

North's productivityis high. Is

such

behavior possible?

The answer

is 'almost'. IftheNorth-South productivity

gap

is

intermediate, thereis a bubbly equilibrium in

which

South's bubble

expands

with

an

increase in North's productivity

and

contracts with a decrease.

To

see

this,

assume

that

South

has

a stationary bubble, i.e. _bf

=

b

H if 7$

= n

H

and

bf

=

bL if n"

= n

L .

26

Assume

alsothatthe growth rate

has

fourstates, i.e. _g,

=

_gHH if n"

=

ji^₊₁

=

n

H , gt

=

g HL if

<

= n

H

and

<

₊₁

= n

l, gt

=

g LL if

n

1 ?

=

<

+1

= n

l

and

g,

=

g HL if it*

=

rc L

and

jt J J +1

=

ji h

(thisturns outto

be

the rightguess, ofcourse). With these definitions,

condition (C1) impliesthat:

(32)

(1-X)g

LL

+X-^-g

LH

_>

0-X)-g»»

+ X

.* -gHL

>

(1-A)-(gLL

)^+A-(g

LH

)^

(1-A)-(gHH

)^

+

X-(gHL

)^

7i

and

Exceptforthe knife-edge

case

in

which

both inequalities arestrict, itfollows

that: (33) bL

g

LM -x (1-X)-(gLL

)^+X-(g

LH

)^

and

bH

=

1

n

&

-g

LL -(1-A)

This bubble issimilar tothose

found

in previous

examples.

When

North productivityis

high, the bubble

expands

and

reallocates investments

from

South

toNorth.

When

North productivity is low, the bubble contracts

and

reallocates investments

from

North toSouth. Using condition (C2),

we

alsofindthe following processforthe

growth

rate as a function ofthe bubble:

( _h a1-a (34) g HH

_.

a-71

2-a

-g

HL

=

x1-a

a-2-ab

L

,g

LL

₌

( „H

a

L\

„LA

7t"

+(1-b

L)-7t'

2-ab

L 1-a

and

gL"

=

( „H

a

L\ Jl.\ 7l"+

(1-b

L)-7t'

2-a

1-o

27

where

I

have used

the finding that b

H

=1. Equations (33)

and

(34) implicitlydefine the

growth process

and

the behaviorofthe bubble. Ingeneral, these equations

cannot

be

solved analytically, although itis quite straightforwardtosolve the

system

numerically.

The

bottom

panel ofFigure

2

shows

theevolution ofthe world growth rate

and

theworlddistribution of

consumption

inthis equilibrium. Conditional

on

being in

normal

times (i.e. periods in

which

a country

does

nottransition

from

one

statetothe

other),

South

experiences a

smoother consumption process

than North. During a

boom,

South

holdsthe bubbleto take

advantage

ofthe high productivityofNorth.

During a recession,

South

shifts its portfolio

towards

investmentto

moderate

thefall in its return. But,

as

in previous

examples, bubbles generate

potentially largewealth

effectsduring thetransitions. In particular, the onsetof a recession leads to a

collapsein the bubblewhile the beginning of a

boom

leadsto

an expansion

inthe bubble.

On

impact, recessions

and

booms

are felt

much

more

strongly in

South

even

ifthey

have

theirorigin in North. Afterthedustclears,

changes

in North productivity

affect relatively

more

North than South. This world

economy

therefore illustrates

how

bubbles create a

channel

forsmall productivity

shocks

in North tocreate

sharp

changes

in asset prices in South.

These

movements

in asset prices in turn

generate

largeeffects

on

consumption, wealth

and

welfare.

This

example

also provides a

new

perspective

on

theconnection

between

economic

reforms

and

vulnerability to external crises.

The

key

observation is that successful

economic

reforms in

South

lead to a progressivereduction in the

North-South

productivitygap.

Assume

initiallythatthis

gap

is large

and

South,

perhaps

with

some

encouragement

from

North, decidesto

implement

reforms. In theirfirststage,

these reforms might

be

successful at raising South's productivity, butthey

have no

impact

on

its

consumption

orwelfare since

South

still preferstohold the bubble

ratherthanto invest.

Assume

that, despite this apparent lack ofsuccess,

South

decidesto

pursue

reforms

even

further untilthe productivity

gap

becomes

intermediate. Inthis

second

stage,

South

notices

some

convergence towards

North's

average

levels of

consumption

and

wealth. Butatthe

same

time,

South

finds itself

more

vulnerableto North shocks. Small

drops

in North productivitycreate

sharp

financial crises. Small increases in North productivity lead toasset price

booms.

The

reforms

have

made

consumption

higher

on

average, but

much

more

volatile.

Assume

that

South

reactstothisby furtherimplementing reforms untiltheproductivity

gap

is

small. Atthis point.

South

finally enjoys high

and

stable consumption.

The

bubble is

gone,

and so

arethewealth effectsthat are atthe rootofthe sharpcrises

and

booms.

The

bubble also

determines

thedistributiveeffects of reforms.

Whenever

we

reach stage

two

ofthe reforms, a conflictarises

between

the

young

and

theold. Successful

economic

reforms

make

investment

more

attractive, reducing the

demand

forthebubble

and

leading toitscontraction or bursting. This leadsto afall in

consumption

and

the welfare oftheold. Afterthis initialdecline in wealth

and

consumption, all futuregenerations in

South

(and North) are betteroff

as

aresult of

the

economic

reforms.

To

the extenthath transferring resources across different

groups

is costly,the

presence

ofa bubble

makes

more

difficult toreach a

consensus

on

the

need

to

implement

reforms.

The

key reason

isthatthese

economic

reforms

harm

the

owners

ofthe bubble ifthey are successful.

7.

Contagion

Consider

a world

economy

with three countries,

West,

East

and

South;with

productivities equal to7i

w

>jiE>7ts==0, respectively.

With

this assumption,

we

already

know

that

West

cannot

have

a bubble

and

that

South

can.

Whether

East

can

have

a

bubble or not

depends

on parameter

values. I

assume

that

South

uses

a

sunspot

variable tocoordinatetoa given equilibrium. This sunspot variable

has two

states:

BUBBLE

and

NO

BUBBLE;

and

the probability to transition

from

one

statetothe