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Chapter

7

Endogenous

Growth

II:

R&D

and

TechnologicalChange

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Economic

Growth:

Lecture

Notes

7.1

Expanding

Product

Variety:

The

Romer

Model

There

are

three

sectors:

one

for

the

final

good

sector,

one

for

intermediate

goods,

and

one

for

R&D.

Thefinalgoodsectorisperfectlycompetitiveandthusmakeszeroprofits. Itsoutputisusedeither

for

consumption

or

as

input

in

each

of

the

other

two

sector.

The intermediate good sector is monopolistic. There is product differentiation. Each intermediate

producer

is

a

quasi-monopolist

with

respect

to

his

own

product

and

thus

enjoys

positive

profits.

To

become

an

intermediate

producer,

however,

you

must

first

acquire

a

blueprint

from

the

R&D

sector.

A

blueprint

is

simply

the

technology

or

know-how

for

transforming

final

goods

to

differentiated

intermediate

inputs.

The

R&D

sector

is

competitive.

Researchers

produce

blueprints.

Blueprints

are

protected

by

perpetual patents. Innovators auction their blueprints to a large number of potential buyers, thus

absorbingalltheprofitsoftheintermediategoodsector. ButthereisfreeentryintheR&Dsector,

which

drive

net

profits

in

that

sector

to

zero

as

well.

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Economic

Growth:

Lecture

Notes

Inwhatfollows,wewillassume=,whichimplies

Nt

Yt =

A(Lt)1

(Xt,j)

dj.0

Note

that

=

means

the

marginal

product

of

each

intermediate

input

is

independent

of

the

quantityofotherintermediateinputs:

Yt

Lt 1

=

A

.

Xt,j

Xt,j

More

generally,

intermediate

inputs

could

be

either

complements

or

substitutes,

in

the

sense

that

the

marginal

product

of

input

j

could

depend

either

positively

or

negatively

on

Xt.

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Wewillinterpretintermediateinputsascapitalgoodsandthereforeletaggregatecapitalbegiven

by

the

aggregate

quantity

of

intermediate

inputs:

Nt

Kt = Xt,jdj.0

Finally,notethatifXt,j

=

X

for

all

j

and

t,

then

Yt =

ALt1NtX and

Kt =

NtX,

implying

Yt =A(NtLt)1(Kt)

or,

in

intensive

form,

yt

=

ANt

1

kt

.

Therefore,

to

the

extent

that

all

intermediate

inputs

are

used

in

thesamequantity,thetechnologyislinearinknowledgeN andcapitalK.Therefore,ifbothN and

K growataconstantrate,aswewillshowtobethecase inequilibrium,theeconomywillexhibit

long

run

growth,

as

in

an

AK

model.

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Economic

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Lecture

Notes

7.1.2 FinalGoodSector

Thefinalgoodsectorisperfectlycompetitive. Firmsarepricetakers.

Finalgoodfirmssolve Ntmax Yt wtLt (pt,jXt,j)dj

0

wherewt isthewagerateandpt,j isthepriceofintermediategoodj.

Profits

in

the

final

good

sector

are

zero,

due

to

CRS,

and

the

demands

for

each

input

are

given

by

theFOCsYt Yt

wt

=

Lt

=

(1

)Lt

and

Yt

Lt

1

pt,j = =

A

Xt,j

Xt,j

for

all

j[0, Nt].

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7.1.3 IntermediateGoodSector

Theintermediategoodsectorismonopolistic. Firmsunderstandthattheyfaceadownwardsloping

demandfortheiroutput.

Theproducerofintermediategoodj solves

max

t,j

=

pt,j

Xt,j

(Xt,j

)

subject

to

the

demand

curve

A

11

Xt,j =Lt ,pt,j

where

(X)

represents

the

cost

of

producing

X

in

terms

of

final-good

units.

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Economic

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Lecture

Notes

We

will

let

the

cost

function

be

linear:

(X) =

X.

The

implicit

assumption

behind

this

linear

specification

is

that

technology

of

producing

intermediate

goods

is

identical

to

the

technology

of

producing

final

goods.

Equivalently,

you

can

think

of

interme

diategoodproducersbuyingfinalgoodsandtransformingthemtointermediateinputs. Whatgives

themtheknow-howforthistransformationispreciselytheblueprinttheyhold.

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TheFOCsgive1

pt,j =p

>

1

for

the

optimal

price,

and

Xt,j =xL

for

the

optimal

supply,

where

1

2

xA 1 1.

Theresultingmaximalprofitsare

t,j =L

where

1 2

(p

1)x

=

1

x

=

1

A1

1

.

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Economic

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Lecture

Notes

Note that theprice is higher thanthemarginalcost (p= 1/>(X)=1), the gaprepresenting

the

mark-up

that

intermediate-good

firms

charge

to

their

customers

(the

final

good

firms).

Because

there

are

no

distortions

in

the

economy

other

than

monopolistic

competition

in

the

intermediate-

goodsector, thepricethatfinal-goodfirmsarewillingtopayrepresentsthesocialproductofthat

intermediate

input

and

the

cost

that

intermediate-good

firms

face

represents

the

social

cost

of

that

intermediate

input.

Therefore,

the

mark-up

1/

gives

the

gap

between

the

social

product

and

the

social

cost

of

intermediate

inputs.

Hint: Thesocialplannerwouldliketocorrectforthisdistortion. How?

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7.1.4 TheInnovationSector

Thepresentvalueofprofitsofintermediategoodj fromperiodtandonisgivenby

Vt,j =q

,j or Vt,j = t,j +Vt+1,j

qt

1 +Rt+1

=t

Weknowthatprofitsarestationaryandidenticalacrossallintermediategoods: t,j

=

L

for

all

t,

j.

As

long

as

the

economy

follows

a

balanced

growth

path,

we

expect

the

interest

rate

to

be

stationary

as

well:

Rt =Rforallt.Itfollowsthatthepresentvalueofprofitsisstationaryandidenticalacross

all

intermediate

goods:

L LVt,j =V =

R/(1+R)

R.

Equivalently,

RV

=

L,

which

has

a

simple

interpretation:

The

opportunity

cost

of

holding

an

asset

which

has

value

V

and

happens

to

be

a

blueprint,

instead

of

investing

in

bonds,

is

RV

;

the

dividend

that

this

asset

pays

in

each

period

is

L;

arbitrage

then

requires

the

dividend

to

equal

the

opportunity

cost

of

the

asset,

namely

RV

=

L.

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Economic

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Lecture

Notes

Newblueprintsareproducedusingthesametechnologyasfinalgoods: innovatorsbuyfinalgoods

and

transform

them

to

blueprints

at

a

rate

1/.

It

follows

that

producing

an

amount

N

of

new

blueprints

costs

N,

where

>

0

measures

the

cost

of

R&D

in

units

of

output.

Ontheotherhand,thevalueofthesenewblueprintsisV N,whereV =L/R.

Itfollowsthatnetprofitsforaresearchfirmarethusgivenby

prof itsR&D = (V

) N

Freeentryinthesectorofproducingblueprintsimposesprof itsR&D =0,orequivalently

V =.

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7.1.5 Households

Householdssolve

tmax

u(ct)t=0

s.t.

ct

+

at+1

wt

+

(1

+

Rt)at

As

usual,

the

Euler

condition

gives

u(ct)=

(1

+

Rt+1).u(ct+1)

And

assuming

CEIS,

this

reduces

to

ct+1 = [(1

+

Rt+1)] .ct

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Economic

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Lecture

Notes

7.1.6 ResourceConstraint

Final goodsare used either for consumption byhouseholds (Ct), or forproduction of intermediate

goods

in

the

intermediate

sector

(Kt =j

Xt,j),orforproductionofnewblueprintsintheinnovation

sector

(Nt). Theresourceconstraintoftheeconomyisthereforegivenby

Ct

+

Kt

+

Nt

=

Yt,

where

Ct =ctL,Nt =Nt+1Nt,andKt =Nt Xt,jdj.0 Asalways,thesumofthebudgetsacrossagentstogetherwiththemarketclearingconditionsreduce

to

the

resource

constraint.

Question:

what

are

the

market

clearing

conditions

here?

Related:

what

are

the

assets

traded

by

the

agents?

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7.1.7 GeneralEquilibrium

Combining the formula for thevalueof innovationwiththe free-entry condition, we infer L/R=

V

=

.

It

follows

that

the

equilibrium

interest

rate

is

LR

= =

1A 11

12

L/,

which

verifies

our

earlier

claim

that

the

interest

rate

is

stationary.

TheEulerconditioncombinedwiththeequilibriumcondition forthereal interestrate impliesthat

consumption

grows

at

a

constant

rate,

which

is

given

by

1 2Ct+1 = 1 +

=

[1

+

R] =

1 +

1A

1L/

Ct 1

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Economic

Growth:

Lecture

Notes

Next,

note

that

the

resource

constraint

reduces

to

Ct Nt+1

Yt

Nt +

Nt

1 +

X

=

Nt =

AL1

X

,

where

X

=

xL

=

Kt/Nt.

ItfollowsthatCt/Nt isconstantalongthebalancedgrowthpath,andthereforeCt, Nt, Kt,andYt all

grow

at

the

same

rate,,where,again,

1 2

11 +=

1 +1

A 1L/

The equilibrium growth rate of the economy decreases with , the cost of producing new knowl

edge.ThegrowthrateisalsoincreasinginL,oranyotherfactorthatincreasesthescale(size)ofthe

economy,

and

thereby

raises

the

profits

of

intermediate

inputs

and

the

demand

for

innovation.

This

is

the

(in)famous

scale

effect

that

is

present

in

many

models

of

endogenous

technological

change.

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Economic

Growth:

Lecture

Notes

TheFOCwithrespecttoXt gives

Xt =xL,

where

1 1

x =A 1 1

represents

the

optimal

level

of

production

of

intermediate

inputs.

TheEulercondition,ontheotherhand,givestheoptimalgrowthrateas

1 1

1 +

=

[1

+

R] =

1 +

1

A 1 1L/

,

where

R =

1

A 1

1

1

1

L/

representsthatsocialreturnonsavings.

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Note

that

1

x =

x 1 > x

That

is,

the

optimal

level

of

production

of

intermediate

goods

is

higher

in

the

Pareto

optimum

than

in

the

market

equilibrium.

This

reflects

simply

the

fact

that,

due

to

the

monopolistic

distortion,

productionof intermediategoods is inefficiently low inthe market equilibrium. Naturally, thegap

x/xisanincreasingfunctionofthemark-up1/.

Similarly,

1

R =

R 1 >

R.

That

is,

the

market

return

on

savings

(R)

falls

short

of

the

social

return

on

savings

(R),

the

gap

again

arising

because

of

the

monopolistic

distortion

in

the

intermediate

good

sector.

It

follows

that

1 +

>

1 +

,

so

that

the

equilibrium

growth

rate

is

too

low

as

compared

to

the

Pareto

optimal

growth

rate.

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Economic

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Lecture

Notes

Policyexercise: Considerfourdifferenttypesofgovernmentintervention:

asubsidyontheproductionoffinalgoods

asubsidyonthedemandforintermediateinputs

asubsidyontheproductionofintermediateinputs

asubsidyonR&D.

Which

of

these

policies

could

achieve

an

increase

in

the

market

return

and

the

equilibrium

growth

rate?

Which

of

these

policies

could

achieve

an

increase

in

the

output

of

the

intermediate

good

sector?

Which

one,

or

which

combination

of

these

policies,

can

implement

the

first

best

allocation

as

a

market

equilibrium?

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7.1.9 BuildingontheShouldersofGiants

In the original Romer (1990) model, the innovation sector uses a different technology than the

one

assumed

here.

In

particular,

the

technology

for

producing

a

new

blueprint

is

linear

in

the

effective

labor

employed

by

the

research

firm,

where

effective

means

amount

of

labor

(number

of

researchers)

times

the

existing

stock

of

knowledge.

Hence,

for

research

firm

j,

Nj,t =

Lj,tNt

Theaggregaterateofinnovationisthusgivenby

=

LR&DNt t Nt

where

LR&Dt isthetotalamountoflaboremployedintheR&Dsector. Marketclearinginthelabor

marketisnowLfinalt +LR&Dt =L. Theprivatecostofinnovationisnowproportionaltowt,whilethe

valueofinnovationremainsasbefore. Therestofthemodelisalsoasbefore.

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Economic

Growth:

Lecture

Notes

7.2.1 R&DTechnology

Let

V

j denote

the

value

of

an

innovation

for

individual

j

realized

in

period

t

and

implemented

in

t+1

period

t

+

1.

Let

z

jt denotetheamountofskilledlaborthatapotentialinnovatorjemployesinR&D

and

q(z

jt ) the probability that such R&D activity will be successful. q :R [0,1] represents the

technology

of

producing

innovations

and

satisfies

q(0)

=

0, q >

0

> q, q(0)

=, q() = 0.

Thepotentialresearchermaximizes

q(z

jt ) V

j

t+1wt jz .t

It

follows

that

the

optimal

level

of

R&D

is

given

byq(zjt )Vt

j+1 =wt or

z

jt =

g

Vtj+1/wt

where the function g(v) (q)1(1/v) satisfies g(0) = 0, g > 0, g() = . Note that z will be

stationary

only

if

both

V

and

w

grow

at

the

same

rate.

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7.2.2 TheValueofInnovation

Whatdeterminesthevalueofaninnovation? Forastart,letusassumeaverysimplestructure. Let

A

jt represent

the

TFP

of

producer

j

in

period

t.

The

profits

from

his

production

are

given

by

jt

=

A

jt

where

represents

normalized

profits.

We

can

endogenize

,

but

we

wont

do

it

here

for

simplicity.

When a producer is born, he automatically learns what is the contemporaneous aggregate level of

technology.

That

is,

A

jt

=

At

for

any

producer

born

in

period

t.

In

the

first

period

of

life,

and

only

in

that

period,

a

producer

has

the

option

to

engage

in

R&D.

If

his

R&D

activity

fails

to

produce

an

innovation,

them

his

TFP

remains

the

same

for

the

rest

of

his

life.

If

instead

his

R&D

activity

is

successful,

then

his

TFP

increases

permanently

by

a

factor

1

+

,

for

some

>

0.

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Lecture

Notes

Thatis,foranyproducerj borninperiodt,andforallperiods t+1inwhichheisalive,

At ifhisR&DfailsAj =

(1+)At ifhisR&Dsucceeds

It

follows

that

a

successful

innovation

generates

a

stream

of

dividends

equal

to

At per periodfor

all

> t

that

the

producer

is

alive.

Therefore,

1

n

Vt+1 =

(At) =vAt (7.1)1 +R

=t+1

where

where

R

is

the

interest

rate

per

period

and

1

n

v

. 1 +

R

R

+

n

=1

Note

that

the

above

would

be

an

exact

equality

if

time

was

continuous.

Note

also

that

v

is

decreasing

in

both

R

and

n.

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Remark: Weseethattheprobabilityofdeathreducesthevalueof innovation,simplybecause it

reduces

the

expected

life

of

the

innovation.

Here

we

have

taken

n

as

exogenous

for

the

economy.

But

later

we

will

endogenize

n.

We

will

recognize

that

the

probability

of

death

simply

the

probability

thattheproducerwillbedisplacedbyanothercompetitorwhomanagestoinnovateandproducea

better

substitute

product.

For

the

time

being,

however,

we

treat

n

as

exogenous.

7.2.3 TheCostofInnovation

Suppose

that

skilled

labor

has

an

alternative

employment,

which

a

simple

linear

technology

of

pro

ducing

final

goods

at

the

current

level

of

aggregate

TFP.

That

is,

if

lt

labor

is

used

in

production

of

final

goods,

output

is

given

by

Atlt.Sincethecostoflaboriswt,inequilibriumitmustbethat

wt

=

At.

(7.2)

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Economic

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Lecture

Notes

7.2.4 Equilibrium

Combining(7.1)and(7.2),weinferthat

Vt+1

v=

wt

It

follows

that

the

level

of

R&D

activity

is

the

same

across

all

new-born

producers:

zj =

zt =g(v).t

TheoutcomeoftheR&Dactivityisstochasticfortheindividual. BytheLLN,however,theaggregate

outcomeisdeterministic. Theaggregaterateofinnovationissimply

t

=

q(zt) =(v)where

(x)q(g(x)).

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Notes


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7.2.5 BusinessStealing

Consideraparticularmarketj,inwhichaproducerjhasmonopolypower. Supposenowthatthere

is

an

outside

competitor

who

has

the

option

to

engage

in

R&D

in

an

attempt

to

create

a

better

product

that

is

a

close

substitute

for

the

product

of

producer

j.

Suppose

further

that,

if

successful,

the

innovation

will

be

so

radical

that,

not

only

it

will

increase

productivity

and

reduce

production

costs,

but

it

will

also

permit

the

outsider

to

totally

displace

the

incumbent

from

the

market.

Remark:

Here

we

start

seeing

how

both

production

and

innovation

may

depend

on

the

IO

structure.

In

more

general

versions

of

the

model,

the

size

of

the

innovation

and

the

type

of

competition

(e.g.,

Bertrand

versus

Cournot)

determine

what

is

the

fraction

of

monopoly

profits

that

the

entrant

can

graspandhencetheprivateincentivesforinnovation.

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g

What is the value of the innovation for this outsider? Being an outsider, he has no share in the

market

of

product

j.

If

his

R&D

is

successful,

he

expects

to

displace

the

incumbent

and

grasp

the

whole

market

of

product

j.

That

is,

an

innovation

delivers

a

dividend

equal

to

total

market

profits,

(1+)At, in each period of life. Assuming that the outsider also has a probability of death (ordisplacement)

equal

to

n,

the

value

of

innovation

for

the

outsider

is

given

by

Vout =1

n

[(1+)At

vAt]=(1 +)

t+1

1 +R

=t+1

Nowsupposethattheincumbentalsohastheoptiontoinnovateislaterperiodsoflife. Ifhedoesso,

he

will

learn

the

contemporaneous

aggregate

level

of

productivity

and

improve

upon

it

by

a

factor

1 +.Thevalueofinnovationinlaterperiodsoflifeisthusthesameasinthefirstperiodoflife:

V

in =

1

n

[At] =vA t.t+1 1 +

R

=t+1

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Notes


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Obviously,

V

out > V

int+1.Thisisbecausetheincumbentvaluesonlythepotentialincreaseinproduc t+1

tivity

and

profits,

while

the

outsider

values

in

addition

the

profits

of

the

incumbent.

This

business

stealing

effect

implies

that,

ceteris

paribus,

that

innovation

will

originate

mostly

in

outsiders.

Remark: In the standard Aghion-Howitt model, as opposed to the variant considered here, only

outsiders engage in innovation. Think why this isthe case in that model, and whythis mightnot

be the case here. Then, find conditions on the technology q and the parameters of the economy

that

would

ensure

in

our

model

a

corner

solution

for

the

insiders

and

an

interior

solution

for

the

outsiders.

(Hint:

you

may

need

to

relax

the

Inada

condition

for

q.)

We

will

henceforth

assume

that

only

outsiders

engage

in

innovation.

Remark: Things could be different if the incumbent has a strong cost advantage in R&D, which

could

be

the

case

if

the

incumbent

has

some

private

information

about

the

either

the

technology

of

the

product

or

the

demand

of

the

market.

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AssumingthatonlyoutsidersengageinR&D,andusingVout v,weinferthattheoptimalt+1/wt =(1+)level

of

R&D

for

an

outsider

is

z

out

=

zt =

g

((1

+

)v).tand

therefore

the

aggregate

rate

of

innovation

is

t =q(zt) =((1+)v)We

conclude

that

the

growth

rate

of

the

economy

is

yt+1

At+1

yt

=

At

= 1 +

((1

+

)v)

.

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Economic

Growth:

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Notes


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Wecannowreinterprettheprobabilityofdeathassimplytheprobabilityofbeingdisplacedbya

successful

outside

innovator.

Under

this

interpretation,

we

have

n

=

((1

+

)v)andvsolves

v=

R+((1+)

v)

Notethatanincreaseinwillnowincreasevbylessthanone-to-one,becausethedisplacementratewill

also

increase.

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7.2.6 EfficiencyandPolicyImplications

Discussthespillovereffectsofinnovation... Bothnegativeandpositive...

Discussoptimalpatentprotection... Trade-offbetweenincentivesandexternalities...

7.3 RamseyMeetsSchumpeter: TheAghion-HowittModel

notes

to

be

completed

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Notes


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7.4

Romer

Meets

Acemoglu:

Biased

Technological

Change

7.4.1 Definition

Consideratwo-factoreconomy,with

Yt

=

F

(Lt, Ht, At)

where

L

and

H

denote,

respectively,

unskilled

labor

and

skilled

labor

(or

any

two

other

factors)

and

A

denotes

technology.

WesaythattechnologyisHbiasedifandonlyif

F

(L,

H,

A)

/H

>

0

A

F

(L,

H,

A)

/L

NotethatthisisdifferentfromsayingthattechnologyisHaugmenting.

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7.4.2 A simplemodelofbiased technologicalchange

WeonsideravariantoftheRomermodelwherewesplitthefinalgoodsectorintwosub-sectors,one

that

is

intensive

in

L

and

another

that

is

intensive

in

H.

Aggregateoutputisgivenby

1 1 1

Yt

=

(YLt) +(1) (YHt )

where

NLt

1YLt

=

L (xLt) dj0 NHt

YHt = H

(xHt )1

dj

0

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Giventhetechnologies,theskillpremiumisgivenby

wH

NH

1

H

1

wL=

const

NL L

where

(1)(1).

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Economic

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Notes


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Therelativevalueofinnovationsisgivenby

VH

pH

1

H

=

VL pL L

NH

1 H

1

=

const

.

NL

L

and

the

equilibrium

innovation

rates

satisfy

NH H

1=

const

.

NL

L

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Hence, oncewetake intoaccounttheendogeneityoftechnologies,theequilibriumskillpremium is

given

by

H

2

=

const

.

L

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Notes


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Finally,itiseasytoshowthatthegrowthrateisgivenby

1

1

1 (1) (HH)

1 +

(LL)1

1

.

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Chapter3

The

Neoclassical

Growth

Model

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In theSolowmodel, agents inthe economy(orthe dictator) followa simplistic linearrule forcon

sumption and investment. In the Ramseymodel, agents(or the dictator)choose consumptionand

investmentoptimallysoastomaximizetheirindividualutility(orsocialwelfare).

3.1 TheSocialPlanner

Inthissection,westarttheanalysisoftheneoclassicalgrowthmodelbyconsideringtheoptimalplan

of

a

benevolent

social

planner,

who

chooses

the

static

and

intertemporal

allocation

of

resources

in

the

economy

so

as

to

maximize

social

welfare.

We

will

later

show

that

the

allocations

that

prevail

in

adecentralizedcompetitivemarketenvironmentcoincidewiththeallocationsdictatedbythesocial

planner.

Togetherwithconsumptionandsaving,wealsoendogenizelaborsupply.

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3.1.1 Preferences

Preferences

are

defined

over

streams

of

consumption

and

leisure,

x

=

{xt}t=0, where xt = (ct, zt),

and

are

represented

by

a

utility

function

U

:

X R,

where

X

is

the

domain

of

xt,

such

that

U(x) =U(x0, x1,...)

Wesaythatpreferencesarerecursive ifthere isafunctionW :XRR(oftencalledtheutility

aggregator)suchthat,forall{xt}t=0,

U

(x0, x1,...) =W[x0,U(x1, x2,...)]

Wecanthenrepresentpreferencesasfollows: Aconsumption-leisurestream{xt} inducesautilityt=0

stream{Ut} accordingtotherecursiont=0

Ut

=W(xt,Ut+1).

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Wesaythatpreferencesareadditivelyseparable iftherearefunctionst

:X Rsuchthat

U(x) =

t(xt).t=0

Wetheninterprett(xt)astheutilityenjoyedinperiod0fromconsumptioninperiodt+1.

Throughoutouranalysis,wewillassumethatpreferencesarebothrecursiveandadditivelyseparable.

In other words, we impose that the utility aggregator W is linear in ut+1 : There is a function

U :R RandascalarRsuchthatW(x,u) =U(x) +u.Hence,

Ut

=U(xt) +Ut+1.

or,equivalently,

=U(xt+)Ut

=0

iscalledthediscountfactor,with(0,1).

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U is sometimes called the per-period utility or felicity function. We let z > 0 denote the maxi

mal amount of time per period. We accordingly let X = R+ [0, z]. We finally impose that U is

neoclassical,bywhichwemeanthatitsatisfiesthefollowingproperties:

1. U iscontinuousand(althoughnotalwaysnecessary)twicedifferentiable.

2. U isstrictlyincreasingandstrictlyconcave:

Uc(c,z) > 0> Ucc(c,z)

Uz(c,z) > 0> Uzz(c,z)

U2 < UccUzzcz

3. U satisfiestheInadaconditions

limUc

= and limUc

= 0.c 0

c

zlim

0Uz

= andzlim

z

Uz

= 0.

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3.1.2 TechnologyandtheResourceConstraint

Weabstractfrompopulationgrowthandexogenoustechnologicalchange.

The

time

constraint

is

given

by

zt

+lt

z.

We usually normalize z = 1 and thus interpret zt

and lt

as the fraction of time that is devoted to

leisureandproduction,respectively.

Theresourceconstraintisgivenby

ct

+it

yt

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LetF(K,L)beaneoclassicaltechnologyandletf() =F(,1)betheintensiveformofF.Output

intheeconomyisgivenby

yt =F(kt, lt) =ltf(t),

wherekt

t =lt

isthecapital-laborratio.

Capitalaccumulatesaccordingto

kt+1

=(1)kt

+it.

(Alternatively,interpretlaseffectivelaborand astheeffectivedepreciationrate.)

Finally,weimposethefollowingnaturalnon-negativitlyconstraints:

ct

0, zt

0, lt

0, kt

0.

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Combiningtheabove,wecanrewritetheresourceconstraint as

ct+kt+1 F(kt, lt)+(1)kt,

andthetimeconstraintas

zt

= 1lt,

with

ct

0, lt

[0,

1], kt

0.

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3.1.3 TheRamseyProblem

The social planner chooses a plan {ct, lt, kt+1} so as to maximize utility subject to the resourcet=0

constraint

of

the

economy,

taking

initial

k0 as

given:

maxU0

=

tU(ct,1lt)t=0

ct+kt+1 (1)kt+F(kt, lt), t0,

ct

0, l

t

[0,1], k

t+1

0., t0,

k0

>0 given.

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3.1.4 OptimalControl

Let t

denote the Lagrange multiplier for the resource constraint. The Lagrangian of the social

planners

problem

is

L0 =

tU(ct,1lt) +

t[(1)kt+F(kt, lt)kt+1ct]t=0

t=0

Lett

tt

anddefinetheHamiltonian as

Ht

=H(kt, kt+1, ct, lt, t)U(ct,1lt) +t

[(1)kt

+F(kt, lt)kt+1

ct]

WecanrewritetheLagrangianas

tL0

= {U(ct,1lt) +t

[(1)kt

+F(kt, lt)kt+1

ct]}=tHt

t=0

t=0

or,inrecursiveform,Lt

=Ht

+Lt+1.

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G.M.Angeletos

( )


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Givenkt, ct

andlt

enteronlytheperiodtutilityandresourceconstraint;(ct, lt)thusappearsonlyin

Ht.Similarly,kt,enteronlytheperiodtandt+1utilityandresourceconstraints;theythusappear

onlyinHt andHt+1.

Lemma

9 If{ct, lt, kt+1}

istheoptimumand{t} theassociatedmultipliers,thent=0 t=0

Ht

(ct, lt)=argmaxH(kt, kt+1,c,l,t)c,l

taking

(kt, kt+1)asgiven,and

Ht + Ht+1

kt+1 =argmaxH(kt, k, ct, lt, t) +H(k

, kt+2, ct+1, lt+1, t+1)k

taking

(kt, kt+2)

as

given.

Wehenceforthassumeaninteriorsolution. Aslongaskt

>0,interiorsolution is indeedensuredby

theInadaconditionsonF andU.

85


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G.M.Angeletos

C bi i th b t


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Combiningtheabove,weget

Uz(ct, zt)=FL(kt, lt)

Uc(ct, zt)

and Uc(ct, zt)

Uc(ct+1, zt+1)= 1+FK(kt+1, lt+1).

Bothconditionsimposeequalitybetweenmarginalratesofsubstitutionandmarginalrateoftransfor

mation. Thefirstconditionmeansthatthemarginalrateofsubstitutionbetweenconsumptionand

leisure

equals

the

marginal

product

of

labor.

The

second

condition

means

that

the

marginal

rate

of

intertemporalsubstitutioninconsumptionequalsthemarginalcapitalofcapitalnetofdepreciation

(plusone). ThislastconditioniscalledtheEulercondition.

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Theenvelopecondition fortheParetoproblemis

(maxU0)=

L0

=0 =Uc(c0, z0).k0

k0

Moregenerally,

t

=Uc(ct, lt)

representsthemarginalutilityofcapitalinperiodtandwillequaltheslopeofthevaluefunctionat

k

=

kt in

the

dynamic-programming

representation

of

the

problem.

88

G.M.Angeletos

Suppose for a moment that the horizon was finite T < Then the Lagrangian would be L


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Suppose for a moment that the horizon was finite, T


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Proposition

10 Theplan{ct, lt, kt} isasolution tothesocialplanner sproblem ifandonly ift=0

Uz(ct, zt)=FL(kt, lt), (3.1)

Uc(ct, zt)

Uc(ct, zt)

Uc(ct+1, zt+1)= 1+FK

(kt+1, lt+1), (3.2)

kt+1

=F(kt, lt)+(1)kt

ct, (3.3)

forallt0,and

k0 >

0

given,

and

lim

tUc(ct, zt)kt+1 = 0.

(3.4)t

Remark: Weprovednecessityof(3.1)and(3.2)essentiallybyaperturbationargument,and(3.3)is

justtheconstraint. Wedidnotprovenecessityof(3.4),neithersufficiencyofthissetofconditions.

SeeAcemoglu(2007)orStokey-Lucasforthecompleteproof.

Note

that

the

(3.1)

can

be

solved

for

lt

=

l(ct, kt),

which

we

can

then

substitute

into

(3.2)

and

(3.3).

Wearethen leftwitha system of two difference equations in two variables, namely ct

and kt. The

intitialconditionandthetransversalityconditionthengivetheboundaryconditionsforthissystem.

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G.M.Angeletos

3 1 5 Dynamic Programing


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3.1.5 DynamicPrograming

Consideragainthesocialplannersproblem. Foranyk >0,define

V(k)max

tU(ct,1lt)t=0

subjectto

ct

+kt+1

(1)kt

+F(kt, lt), t0,

ct, lt,

(1

lt), kt+1

0,

t

0,

k0

=k given.

V iscalledthevaluefunction.

91


The Bellman equation for this problem is


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TheBellmanequationforthisproblemis

V(k)=maxU(c,1l) +V(k)

s.t.

c

+

k

(1

)k

+

F

(k,

l)

k 0, c[0, F(k,1)], l[0,1].

Let

[c(k), l(k), G(k)]

=

arg max{...}.

Thesearethepolicyrules. ThekeypolicyruleisG,whichgivesthedynamicsofcapital. Theother

rulesarestatic.

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G.M.Angeletos

Define k by the unique solution to


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Definekbytheuniquesolutionto

k=(1)k+F(k,1)

andnotethatkrepresentsanupperboundonthelevelofcapitalthatcanbesustainedinanysteady

state. Withoutseriouslossofgenerality,wewillhenceforthrestrictkt

[0, k].

Let B be the set of continuous and bounded functions v : [0, k] R and consider the mapping

T

:

B

B

defined

as

follows:

Tv(k)=maxU(c,1l) +v(k)

s.t. c+k (1)k+F(k,l)

k [0, k], c[0, F(k,1)], l[0,1].

The conditions we have imposed on U and F imply that T is a contraction mapping. It follows

thatT hasauniquefixedpointV =TV andthisfixedpointgivesthesolution.

93


TheLagrangianfortheDPproblemis


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g g p

L=U(c,1l) +V(k) +[(1)k+F(k,l)k c]

TheFOCswithrespecttoc,landk give

c

L= 0

Uc(c,z) =

L= 0 Uz(c,z) =FL(k,l)

l

L= 0 =Vk(k

)k

TheEnvelopeconditionis

Vk(k) =L

=[1+FK

(k,l)]k

94

G.M.Angeletos

Combining,weconclude


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g,Uz(ct, lt)

=Fl(kt, lt)Uc(ct, lt)

and

Uc(ct, lt)

Uc(ct+1, lt+1)=[1+FK(kt+1, lt+1)],

whicharethesameconditionswehadderivedwithoptimalcontrol. Finally,notethatwecanstate

theEulerconditionalternativelyas

Vk(kt)

Vk(kt+1)=

[1

+

FK(kt+1, lt+1)].

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3.2 DecentralizedCompetitiveEquilibrium


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p q

3.2.1 Households

Householdsareindexedbyj[0,1].Forsimplicity,weassumenopopulationgrowth.

Thepreferencesofhouseholdj aregivenby

tU(cUj0

jt , z

jt )=

t=0

Inrecursiveform,Ujt =U(cjt , z

jt ) +U+1.

jt

Thetimeconstraintforhouseholdj canbewrittenas

zjt = 1lj

t.

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G.M.

Angeletos

Thebudgetconstraintofhouseholdj isgivenby


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denotestherentalrateofcapital,wt

denotesthewagerate,Rt

denotestheinterestrateon

jl+wt tjt

jt

jt

jt

jt

jt +

jt,+i =rtk +Rtb+xc

wherert

y

risk-freebonds. Householdj accumulatescapitalaccordingto

k +1

=(1)kjtjt

jt+i

andbondsaccordingto

b +1jt =b

jt +x

jt

Inequilibrium,firmprofitsarezero,becauseofCRS.It followsthatt =0. Combiningtheabove

wecanrewritethehouseholdbudgetas

jt

jt+k +1+b +1 (1+rt)k

jt

jt +(1+Rt)b

jt+wtl

jt .c

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Economic

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Notes

Thenaturalnon-negativityconstraint


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ktj

+1

0

is

imposed

on

capital

holdings,

but

no

short-sale

constraint

is

imposed

on

bond

holdings.

That

is, household can either lend or borrow in risk-free bonds. We only impose the following natural

borrowingconstraint:

q

(1+Rt+1)bjt+1

(1+rt+1)ktj

+1

+

w

.qt+1=t+1

where1

qt

(1+R0)(1+R1)...(1+Rt)

= (1 +Rt)qt+1.

This constraint simply requires that the net debt position of the household does not exceed the

presentvalueofthelaborincomehecanattainbyworkingalltime.

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G.M.

Angeletos

Simple arbitrage between bonds and capital implies that, in any equilibrium, the interest rate on


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risklessbondsmustequaltherentalrateofcapitalnetofdepreciation:

Rt =

rt

.

If Rt

< rt

, all individuals would like to short-sell bonds, and there would be excess supply of

bonds. IfRt

> rt

,nobodyintheeconomywouldinvestincapital.

Households

are

then

indifferent

between

bonds

and

capital.

Letting

atj

=

btj

+

ktj

denote

total

assets,

thebudgetconstraintreducesto

cj +ajt+1

(1+Rt)a

j +wtlt

j,t t

and

the

natural

borrowing

constraint

becomes

a

j

t+1

at+1,

where

1

a

qwqt

t+1

=t+1

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Growth:

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Notes

Weassumethat{Rt, wt} satisfiest=0


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1

q

w

< M


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lifetimeutilitysubjecttoitsbudgetconstraints

tU(cj jt jt )

,

1

l

max

U =

0

t=0

jt

jt

jt

jt

jt+1

(1+Rt)a

[0,1], +1

ajt

+wtls.t. +ac

jt 0, lc a t+1

Let

jt

jt=

t be

the

Lagrange

multiplier

for

the

budget

constraint,

we

can

write

the

Lagrangian

as

t

U(c

(1+Rt)ajt +wtl

jt

jta

jt+1

cj jt

jt

jt

tHjt,1l

jt,1l

) +L = =0

t=0 t=0

where

jt (1+Rt)ajtHjt = jt +wtljt jta +1cjtU(c ) +

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Economic

Growth:

Lecture

Notes

Usingjt

=Uc(cjt , z

jt ),wecanrestatetheEulerconditionas


72/192

Uc(cjt , z

jt )[1+Rt]Uc(c +1, z+1),

jt

jt

with equality whenever a +1

jt > at+1. That is, as long as the borrowing constraint does not bind,

households equate their marginal rate of intertemporal substitution with the (common) return on

capital. Ontheotherhand,iftheborrowingconstraintisbinding,themarginalutilityofconsumption

todaymayexceedthemarginalbenefitofsavings: thehouseholdwouldliketoborrow,butitcant.

Forarbitraryborrowinglimitat,thereisnothingtoensurethattheEulerconditionmustbesatisfied

withequality. But if at isthenaturalborrowing limit, andtheutilitysatisfiestheInadacondition

Uc

0, then a simple argument ensures that the borrowing constraint can never bind. as c

jt

jt=0forall t,implyingUc(c +1, z+1) =andthereforeSupposethatat+1 =at+1.Thenc

j

j=z

necessarily

Uc(cj

t , z

j

t )

< [1

+

Rt]Uc(cj

t

j

t ),

unless

also

c

j

t =

0

which

in

turn

would

be

optimal

only

, z

ifat

=at.ButthiscontradictstheEulercondition, provingthata0 > a0 suffices forat > a forallt

dates,andhencefortheEulerconditiontobesatisfiedwithequality.

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G.M.

Angeletos

Moreover,iftheborrowingconstraintneverbinds,iteratingjt

=[1+Rt]jt+1

impliestjt

=qtj

0.


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Wecanthereforerewritetheterminalconditionas

lim

ttj

atj

+1 =

lim

ttj

at+1 =

0j

lim

qtat+1t t t

Butnotethat

qtat+1 =

q

w

=t

and qw


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j jmax

tU(ct

, zt

)t=0

j

js.t.

qtct

+

qtwtzt

xt=0

t=0

where

xq0(1+R0)a0

+

qtwt

0 is Lagrange multiplier associated to the intertermporal budget. You can check that

theseconditionscoincidewiththeonederivedbefore.

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Economic

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Notes

3.2.2 Firms


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ThereisanarbitrarynumberMt

offirmsinperiodt,indexedbym[0, Mt].Firmsemploylaborand

rentcapitalincompetitivelaborandcapitalmarkets,haveaccesstothesameneoclassicaltechnology,

andproduceahomogeneousgoodthattheysellcompetitivelytothehouseholdsintheeconomy.

LetKtm andLmt

denotetheamountofcapitalandlaborthatfirmmemploysinperiodt.Then,the

profitsofthatfirminperiodtaregivenby

mt =

F

(Ktm, Lmt )

rtKtm

wtLtm.

Thefirmsseekstomaximizeprofits. TheFOCsforan interiorsolutionrequire

FK

(Ktm, Lt

m) = rt.

FL(Ktm, Lt

m) =

wt.

108

G.M.

Angeletos

Asweshowedbefore intheSolowmodel,underCRS,an interiorsolutiontothefirmsproblemto

exist if and only if r and w imply the same Km/Lm This is the case if and only if there is some


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exist if and only if rt and wt imply the same Ktm/Lmt . This is the case if and only if there is some

Xt (0,)suchthat

rt

= f(Xt)

wt

= f(Xt)f(Xt)Xt

wheref(k)F(k,1).Providedso,firmprofitsarezero,m =0,andtheFOCsreducetot

Km =XtLm.t t

That is,theFOCspindownthecapital laborratio foreachfirm(Ktm/Lmt ),butnotthesizeofthe

firm(Lmt

). Moreover,becauseallfirmshaveaccesstothesametechnology,theyuseexactlythesame

capital-labor

ratio.

(See

our

earlier

analysis

in

the

Solow

model

for

more

details.)

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Lecture

Notes

3.2.3 MarketClearing


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Thereisnosupplyofbondsoutsidetheeconomy. Thebondmarket thusclearsifandonlyif

Lt

0 =

btj

dj.

0

Thecapitalmarket clearsifandonlyif

Mt 1

Km

jdm= k djt t0 0

Km jEquivalently,Mt dm=kt,wherekt

=Kt

1

kt

dj istheper-capitacapital.0 t 0

Thelabormarket,ontheotherhand,clears ifandonlyif

Mt Lt Lmt

dm

=

ltjdj

0 0

Equivalently,Mt Lmdm=lt

wherelt

=Lt

Lt ljdj istheper-headlaborsupply.0

t 0

t

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Economic

Growth:

Lecture

Notes

Proposition

13 Thesetofcompetitiveequilibriumallocationsfor themarketeconomycoincidewith the

set of Pareto allocations for the social planner.


80/192

setofParetoallocationsfor thesocialplanner.

Proof.

j

b+ 0I

will

sketch

the

proof

assuming

that

(a)

in

the

market

economy,

k

j

0 is

equal

across

allj; and(b) the social planner is utilitarian. For the more general case, we need to allow for an

unequal initial distribution of wealth across agents. The set of competitive equilibrium allocations

coincideswiththesetofParetooptimalallocations,eachdifferentcompetitiveequilibriumallocation

corresponding to a different point in the Pareto frontier (equivalently, a different vector of Pareto

weights

in

the

objective

of

the

social

planner).

For

a

more

careful

analysis,

see

Stokey-Lucas

or

Acemoglu.

112

G.M.

Angeletos

a. We first consider how the solution to the social planners problem can be implemented as a

competitive equilibrium.


81/192

competitiveequilibrium.

Thesocialplannersoptimalplanisgivenby{ct, lt, kt} suchthatt=0

Uz(ct,1lt)=FL(kt, lt), t0,

Uc(ct,1lt)

Uc(ct,1lt)

Uc(ct+1,1lt+1)=[1+FK

(kt+1, lt+1)], t0,

ct

+kt+1

=(1)kt

+F(kt, lt), t0,

k0

>

0

given,

and

lim

tUc(ct,

1

lt)kt+1

= 0.t

Lett kt/lt andchoosethepricepath{Rt, rt, wt} givenbyt=0

Rt

= rt

,

rt

=

FK

(kt, lt) =

f(t),

wt

= FL(kt, lt) =f(t)f(t)t,

113

Economic

Growth:

Lecture

Notes

Trivially,thesepricesensurethattheFOCsaresatisfiedforeveryhouseholdandeveryfirmifweset

cjt =ct,ltj

=lt andKtm/Lmt =kt foralljandm.Next,weneedtoverifythattheproposedallocation


82/192

t t, t t t / t t j , y p p

satisfiesthebudgetconstraintofthehouseholds. Fromtheresourceconstraint,

ct

+kt+1

=F(kt, lt)+(1)kt.

FromCRSandtheFOCsforthefirms,F(kt, lt) =rtkt

+wtlt.Combining,weget

ct+

kt+1 =

(1

+

rt)kt+

wtlt.

Thebudgetconstraintofhouseholdj isgivenby

ctj +kt

j+1

+btj

+1

=(1+rt)ktj +(1+Rt)b

jt

+wtltj,

Forthistobesatisfiedattheproposedpriceswithctj

=ct

and ltj

=lt, it isnecessaryandsufficient

thatktj

+bjt

=kt

forallj,t.Finally, itistrivialtocheckthebond,capital,andlabormarketsclear.

114

G.M.

Angeletos

b. Wenextconsidertheconverse,howacompetitiveequilibriumcoincideswiththeParetosolution.

Becauseagentshavethesamepreferences,facethesameprices,andareendowedwithidenticallevel


83/192

g p , p ,

of initial wealth, and because the solution to the individuals problem is essentially unique (where

essentially

means

unique

with

respect

to

ctj

, ltj

,

and

atj

=

ktj

+

btj

but

indeterminate

with

respect

to

theportfoliochoicebetweenktj

andbtj

),wehavethatctj

=ct, ltj

=lt andatj

=at forallj,t.Bythe

FOCstotheindividualsproblem,itfollowsthat{ct, lt, at} satisfiest=0

Uz(ct,1lt)

Uc(ct,1lt)=wt, t0,

Uc(ct,

1

lt)=[1+rt],

Uc(ct+1,1lt+1)t0,

ct

+at+1

=(1+rt)at

+wtlt, t0,

a0 >0 given, and limtUc(ct,1lt)at+1 = 0.

t

From

the

market

clearing

conditions

for

the

capital

and

bond

markets,

the

aggregate

supply

of

bonds

iszeroandthusat

=kt.

115

Economic

Growth:

Lecture

Notes

Next,bytheFOCsforthefirms,


84/192

rt = FK(kt, lt)

wt =

FL(kt, lt)

andbyCRS

rtkt

+wtlt

=F(kt, lt)

CombiningtheabovewiththeFOCsandthebudgetconstraintsgives

ct

+kt+1

=F(kt, lt)+(1)kt, t0,

which is simply the resource constraint of the economy. Finally, and limttUc(ct,1lt)at+1 =

0 with at+1 = kt+1 implies the social planners transversality condition, while a0 = k0 gives the

initialcondition. Thisconcludestheproofthatthecompetitiveequilibriumcoincideswiththesocial

plannersoptimalplan.

116

G.M.

Angeletos

Theequivalencetotheplannersproblemthengivesthefollowing.


85/192

Corollary

14

(i)Anequilibriumexistsforanyinitialdistributionofwealth. Theallocationofproduction

across

firms

is

indeterminate,

and

the

portfolio

choice

of

each

household

is

also

indeterminate,

but

the

equilibrium is unique as regards prices, consumption, labor, and capital. (ii) If initial wealth k0

j +b0

j is

equalacrossallagentj,thencjt =ct, ltj

=lt andktj

+bjt =ktforallj.Theequilibriumisthengivenbyan

allocation{ct, lt, kt} suchthat,forallt0,t=0

Uz(ct,1lt)

=

FL(kt, lt),Uc(ct,

1

lt)

Uc(ct,1lt)=[1+FK

(kt+1, lt+1)],Uc(ct+1,1lt+1)

kt+1

=F(kt, lt)+(1)kt

ct,

with

k0 >

0

given

and

limtt

Uc(ct,

1

lt)kt+1 = 0.

Finally,

equilibrium

prices

are

given

by

Rt

=R(kt)f(kt), rt

=r(kt)f(kt), wt

=w(kt)f(kt)f(kt)kt.

117

Economic

Growth:

Lecture

Notes

3.3 SteadyState

( ) ( )


86/192

Proposition15 There exists a unique (positive) steady state (c, l, k) > 0. The steady-state values of

the

capital-labor

ratio,

the

productivity

of

labor,

the

output-capital

ratio,

the

consumption-capital

ratio,

the

wage rate, the rental rate of capital, and the interest rate are all independent of the utilityfunction U

and are pinned down uniquely by the technology F, the depreciation rate , and the discount rate . In

particular,thecapital-laborratio k/l equatesthenet-of-depreciationMPKwiththediscountrate,

f

(

)

=

,

and isadecreasingfunctionof+,where1/1.Similarly,

R =, r =+, w =FL(,1)=

Uz(c,1l),

Uc(c,1l)

y

y

c

y

=

f(),

=

(),

= k ,l k k

wheref()F(,1)and()f()/.

118

G.M.

Angeletos

Proof. (c, l, k)mustsolve

U (c 1 l)


87/192

Uz(c ,1l )=FL(k

, l),Uc(c,1l)

1 =

[1

+

FK(k, l)],

c =F(k, l)k .

Letk/ldenotethecapital-laborratioatthesteadstate. ByCRS,

F

(k,

l) =

f()l FK

(k,

l) =

f()

FL(k,l) =f()f()

wheref()F(,1). TheEulerconditionthenreducesto1=[1+f()]orequivalently

f()=

where1/1. That is,thecapital-laborratio ispinneddownuniquelybytheequationofthe

119

Economic

Growth:

Lecture

Notes

MPK, net of depreciation, with the discount rate. It follows that the gross rental rate of capital

andthenet interestratearer =+ andR =,whilethewagerate isw =FL:(,1). Labor


88/192

productivity(outputperworkhour)andtheoutput-capitalratioaregivenby

y=f() and

y=(),

l k

where()f()/. Finally,bytheresourceconstraint,theconsumption-capitalratioisgivenby

c y

=

(

)

=

k

.

k

Thecomparativestaticsaretrivial. Forexample,anincreasein leadstoanincreasein,Y /L,and

s=K/Y. WecouldthusreinterprettheexogenousdifferencesinsavingratesassumedintheSolow

modelasendogenousdifferencesinsavingratesoriginatinginexogenousdifferencesinpreferences.

Homework: consider the comparative statics with respect to exogenous productivity or a tax on

capital

income.

120

G.M.

Angeletos

3.4 TransitionalDynamics

Consider the condition that determined labor supply:


89/192

Considertheconditionthatdeterminedlaborsupply:

Uz(ct,

1

lt) =FL(kt, lt).Uc(ct,1lt)

We can solve this for lt

as a function of contemporaneous consumption and capital: lt

= l(ct, kt).

SubstitutingthenintotheEulerconditionandtheresourceconstraint,weconclude:

Uc(ct,

1

l(ct, kt))

Uc(ct,1l(ct, kt))=

[1

+

FK(kt+1, l(ct+1, kt+1))]

kt+1 = F(kt, l(ct, kt))+(1)ktct

Thisisasystemoftwofirst-orderdifferenceequationinct

andkt.Togetherwiththeinitialcondition

(k0

given)andthetransversalitycondition,thissystempinsdownthepathof{ct, kt}t=0.

121

Economic

Growth:

Lecture

Notes

3.5 Exogenous laborandCEI

Suppose that leisure is not valued or that the labor supply is exogenously fixed Either way let lt = 1


90/192

Supposethatleisureisnotvalued,orthatthelaborsupplyisexogenouslyfixed. Eitherway,letlt = 1

forallt.Supposefurtherthatpreferencesexhibitconstantelasticityofintertemporalsubstitution:

cU(c) =

11/ 1

,1

1/

where >0istheelasticityofintertemporalsubstitution.

The

Euler

condition

then

reduces

to

ct+1

= [(1+Rt+1)],

ct

orequivalently ln(ct+1/ct)(Rt+1

).Thus,controlsthesensitivityofconsumptiongrowthto

the

rate

of

return

to

savings

122

G.M.

Angeletos

Proposition

16 Theequilibriumpath{ct, kt} isgivenbytheuniquesolutiontot=0

ct+1{[1 f (k ) ]}


91/192

t+1

={[1+f(kt+)]},

ct

kt+1 =

f(kt)

+

(1

)kt

ct,

forallt,with initialconditionk0 >0givenand terminalcondition

limkt

=k,t

wherek isthesteadystatevalueofcapital, that is,f(k) =+.

Remark. That the transversality condition reduces to the requirement that capital converges to

the

steady

state

will

be

argued

later,

with

the

help

of

the

phase

diagram.

It

also

follows

from

the

following

result,

which

uses

information

on

the

policy

function.

123

Economic

Growth:

Lecture

Notes

Proposition

17 For any initial k0

< k (k0

> k), the capital stock kt

is increasing (respectively, de

creasing) over time and converges to asymptotically to k. Similarly, the rate of per-capita consumption


92/192

growth ct+1/ct is positive and decreasing (respectively, negative and increasing) over time and converges

monotonically

to

0.

Proof. The dynamics are described by kt+1 = G(kt), where G is the policy rule characterizing the

planners problem. The policy rule is increasing and satisfies k = G(k) if and only if k = 0 or

k=k,k < G(k)< k forallk(0, k),andk > G(k)> k forallk > k. (SeeStokey-Lucas for

the

proof

of

these

properties.)

The

same

argument

as

in

the

Solow

model

then

implies

that

{kt}

t=0

is monotonic and converges to k. The monotonicity and convergence of {ct+1/ct} then followst=0

immediately fromthemonotonicity and convergenceof {kt}

t=0 togetherwith the fact that f(k) is

decreasing.

Wewillshowthisresultalsographicallyinthephasediagram,below.

124

G.M.

Angeletos

3.6 ContinuousTimeandPhaseDiagram

TakinglogsoftheEulerconditionandapproximatingln=ln(1+ ) andln[1 +f(kt)]


93/192

g g pp g ( ) [ f ( t)]

f(kt),wecanwritetheEulerconditionas

lnct+1

lnct

[f(kt+1)].

Thisapproximationisexactwhentimeiscontinuous.

Proposition

18

Consider

the

continuous-time

version

of

the

model.

The

equilibrium

path

{ct, kt}t[0,)

istheuniquesolutionto

ct=[f(kt)] =[Rt

],ct

kt =f(kt)kt ct,

forallt,withk0

>0givenand limtkt =k,wherek isthesteady-statecapital.

125

Economic

Growth:

Lecture

Notes

Wecannowusethephasediagramtodescribethedynamicsoftheeconomy. SeeFigure3.1.

Thek =0locusisgivenby(c,k)suchthat

(Figure not shown due to unavailable original.)


94/192

k =

f(k)

k

c

= 0

c

=

f(k)

k

Ontheotherhand,the c=0locusisgivenby(c,k)suchthat

c =c[f(k)] = 0 k=k or c= 0

Thesteadystateissimplytheintersectionofthetwoloci:

c =k = 0 {(c,k) = (c, k) or (c,k)=(0,0)}

where

k

(f)1(

+

)

and

c

f(k)

k.

Wehenceforthignorethe(c,k)=(0,0)steadystateandthec=0partofthe c=0locus.

126


95/192


96/192

G.M.

Angeletos

Let the function c(k)represent the saddle path. Interms ofdynamicprogramming, c(k) issimply

the optimal policy rule for consumption given capital k. Equivalently, the optimal policy rule for

capital accumulation is given by


97/192

capitalaccumulationisgivenby

k =

f(k)

k

c(k),

withthediscrete-timeanaloguebeing

kt+1

=G(kt)f(kt)+(1)kt

c(kt).

Finally,notethat,nomattertheformofU(c),youcanwritethedynamicsintermsofkand:

t

= f(kt)t

kt

= f(kt)kt

c(t),

where c() solves Uc(c) = , that is, c() Uc1(). Note that Ucc


98/192

Suppose

that

each

household

receives

an

endowment

e >

0

from

God,

so

that

its

budget

becomes

cj +kj =wt

+rtkj

+(1)kj +et t+1 t t

Addingupthebudgetacrosshouseholdsgivesthenewresourceconstraintoftheeconomy

kt+1kt =f(kt)kt ct +e

Ontheotherhand,optimalconsumptiongrowthisgivenagainby

ct+1

=

{[1

+

f

(kt+1)

]}ct

130

G.M.

Angeletos

Turningtocontinuoustime,weconcludethatthephasediagrambecomes

ct=[f(kt)],

ct

kt =f(kt)ktct+e.


99/192

Inthesteadystate,k isindependentofeandc movesonetoonewithe.

Considerapermanent increase inebye. This leadstoaparallelshift inthe k =0 locus,butno

change in the c=0 locus. If theeconomywas initially at the steady state, thenk staysconstant

andcsimplyjumpsbyexactlye.Ontheotherhand, iftheeconomywasbelowthesteadystate,c

will

initially

increase

but

by

less

that

e,

so

that

both

the

level

and

the

rate

of

consumption

growth

will

increase

along

the

transition.

See

Figure

3.2.

131

}

ehigh

elowe

e{

c

k

Figure 3.2

Figure by MIT OCW.

Economic

Growth:

Lecture

Notes

3.7.2 TaxationandRedistribution

Suppose that the government taxes labor and capital income at a flat tax rate (0,1). The


100/192

governmentthenredistributestheproceedsfromthistaxuniformlyacrosshouseholds. LetTt

bethe

transfermadeinperiodt.

Thehouseholdbudgetis

cj

+

ktj

+1 =

(1

)(wt+

rtkj

)

+

(1

)kj

+

Tt,t t t

implyingUc(ct

j)

Uc(ctj

+1)=[1+(1)rt+1

].

Thatis,thetaxratedecreasestheprivatereturntoinvestment. Combiningwithrt =f(kt)weinfer

ct+1

={[1+(1)f(kt+1)]} .ct

132

G.M.

Angeletos

Addingupthebudgetsacrosshouseholdgives

ct+kt+1 =(1)f(kt+1)+(1)kt +Tt


101/192

Thegovernmentbudgetontheotherhandis

Tt

= (wt

+rtkt

j) = f(kt)

j

Combiningwegettheresourceconstraintoftheeconomy:

kt+1

kt

=f(kt)kt

ct

Observethat,ofcourse,thetaxschemedoesnotappear intheresourceconstraintoftheeconomy,

for itisonlyredistributiveanddoesnotabsorbresources.

133

Economic

Growth:

Lecture

Notes

Weconcludethatthephasediagrambecomesct

=[(1)f(kt)],ct

kt =f(kt)ktct.


102/192

Inthesteadystate,k andc aredecreasingfunctionsof.

A.

Unanticipated

Permanent

Tax

Cut

Consideranunanticipatedpermanenttaxcutthatisenactedimmediately. Thek =0locusdoesnot

change,butthe c=0locusshiftsright. Thesaddlepaththusshiftsright. SeeFigure3.3.

A

permanent

tax

cut

leads

to

an

immediate

negative

jump

in

consumption

and

an

immediate

positive

jumpin investment. Capitalslowly increasesandconvergestoahigherk.Consumption initiallyis

lower,

but

increases

over

time,

so

soon

it

recovers

and

eventually

converges

to

a

higher

c.

134

c

k

Thigh Tlow

Figure 3.3

Figure by MIT OCW.

G.M.

Angeletos

B.

Anticipated

Permanent

Tax

Cut

Consider

a

permanent

tax

cut

that

is

(credibly)

announced

at

date

0

to

be

enacted

at

some

date


103/192

t >0.Thedifferencefromthepreviousexerciseisthatc=0locusnowdoesnotchangeimmediately.Itremainsthesamefort t.Therefore,thedynamicsofcandk willbedictatedbytheoldphasediagram(theonecorrespondingtohigh)fort t,

At t=

t and on, the economy must follow the saddle path corresponding to the new low , which

will

eventually

take

the

economy

to

the

new

steady

state.

For

t 0


135/192

F

(K,

L)

F

(K,

L),

>

0.

2. Positiveanddiminishingmarginalproducts:

FK(K,L)>0, FL(K,L)>0,

FKK(K,

L)

0>f(k) limk0

f(k)= limk

f(k)=0

FK

(K,L)=f(k) FL(K,L),=f(k)f(k)k

Example: Cobb-Douglastechnology

In

this

case,

K =

,

L =

1

,

and

F(K,L)=KL1

f(k)=k

19

Economic

Growth:

Lecture

Notes

2.1.3 The Resource Constraint, and the Law of Motions for Capital and

Labor(Population)

Thesumofaggregateconsumptionandaggregateinvestmentcannotexceedaggregateoutput. That

is the social planner faces the following resource constraint:


138/192

is,thesocialplannerfacesthefollowingresourceconstraint:

Ct

+It

Yt

(2.3)

Equivalently,inper-capitaterms:

ct +

it

yt (2.4)

Weassumethatpopulationgrowthisn0perperiod:

Lt

=(1 +n)Lt1 =(1 +n)tL0

(2.5)

WenormalizeL0

=1.

20

G.M.

Angeletos

Suppose that existing capital depreciates over time at a fixed rate [0,1]. The capital stock in

the beginning of next period is given by the non-depreciated part of current-period capital, plus

contemporaneousinvestment. Thatis, the lawofmotionforcapital is

Kt+1 = (1 )Kt + It. (2.6)


139/192

Kt+1 (1 )Kt +It. (2.6)

Equivalently,inper-capitaterms:

(1+n)kt+1 =(1)kt +it

Wecanapproximatelywritetheaboveas

kt+1

(1n)kt

+it

(2.7)

The sum +n can thus be interpreted as the effective depreciation rate of per-capita capital.

(Remark: Thisapproximationbecomesexactinthecontinuous-timeversionofthemodel.)

21

Economic

Growth:

Lecture

Notes

2.1.4 TheDynamicsofCapitalandConsumption

Inmost of thegrowth models thatwewill examine in thisclass, the keyof theanalysiswill beto

derive a dynamic system that characterizes the evolution of aggregate consumption and capital in

the

economy;

that

is,

a

system

of

difference

equations

in

Ct and

Kt (or

ct and

kt).

This

system

is


140/192

verysimpleinthecaseoftheSolowmodel.

Combiningthelawofmotionforcapital(2.6),theresourceconstraint(2.3),andthetechnology(2.1),

wederivethedifferenceequationforthecapitalstock:

Kt+1 Kt F(Kt, Lt)Kt Ct (2.8)

That is, the change in the capital stock is given by aggregate output, minus capital depreciation,

minusaggregateconsumption.

kt+1

kt

f(kt)(+n)kt

ct.

22

G.M.

Angeletos

2.1.5 FeasibleandOptimalAllocations

Definition1 Afeasibleallocationisanysequence{ct, kt}

t=0 R

2

thatsatisfiestheresourceconstraint+

kt+1

f(kt)

+

(1

n)kt

ct.

(2.9)


141/192

The setof feasibleallocations representsthe choiceset for thesocial planner. Theplanner then

usessomechoiceruletoselectoneofthemanyfeasibleallocations.

Later,wewillhavetosocialplannerchooseanallocationsoastomaximizewelfare(Paretoefficiency).

Here,weinsteadassumethatthedictatorfollowsasimplerule-of-thump.

Definition2 ASolow-optimalcentralizedallocationisanyfeasibleallocationthatsatisfiestheresource

constraintwithequalityand

ct =(1s)f(kt), (2.10)

forsomes(0,1).

23

Economic

Growth:

Lecture

Notes

2.1.6 ThePolicyRule

Combining(2.9)and(2.10)givesasingledifferenceequationthatcompletelycharacterizesthedy

namicsoftheSolowmodel.

Proposition

3

Given

any

initial

point

k0

>

0,

the

dynamics

of

the

dictatorial

economy

are

given

by

the


142/192

path{kt} such thatt=0

kt+1 =G(kt), (2.11)

forallt0,where

G(k)

sf(k)

+

(1

n)k.

Equivalently, thegrowthrate isgivenby

t kt+1 kt

=(kt), (2.12)kt

where

(k)s(k)(+n), (k)f(k)/k.

24

G.M.

Angeletos

Gcorrespondstowhatwewillcallthepolicyrule intheRamseymodel.Thedynamicevolutionof

theeconomy isconciselyrepresentedbythepath{kt} thatsatisfies(??),orequivalently(2.11),t=0

forallt0,withk0 historicallygiven.

The

graph

of

G

is

illustrated

in

Figure

2.1.


143/192

Remark. Think of G more generally as a function that tells you what is the state of the economy

tomorrowasafunctionofthestatetoday. HereandinthesimpleRamseymodel,thestateissimply

kt.Whenwe introduceproductivityshocks,thestate is(kt, At).Whenwe introducemultipletypes

of capital, the state is the vector of capital stocks. And with incomplete markets, the state is the

wholedistributionofwealthinthecross-sectionofagents.

25


144/192

G.M.

Angeletos

2.1.7 SteadyState

Asteadystateoftheeconomyisdefinedasanylevelk suchthat,iftheeconomystartswithk0

=k,

thenkt =k forallt1.That is,asteadystate isanyfixedpointk ofG in(2.11). Equivalently,

a

steady

state

is

any

fixed

point

(c

, k

)

of

the

system

(2.9)-(2.10).


145/192

Atrivialsteadystateisc=k=0:Thereisnocapital,nooutput,andnoconsumption. Thiswould

notbeasteadystate if f(0)>0.Weare interested forsteady statesatwhichcapital, outputand

consumptionareallpositiveandfinite. Wecaneasilyshow:

Proposition

4

Suppose

+

n

(0,

1)

and

s

(0,

1).

A

steady

state

(c, k)

(0,

)2for

the

dictatorial

economy exists and is unique. k and y increasewith s and decreasewith and n,whereas c isnon-

monotonicwithsanddecreaseswithandn.Finally,y/k = (+n)/s.

Proof. k isasteadystateifandonlyifitsolves

0 =sf(k)(+n)k,

27

Economic

Growth:

Lecture

Notes

Equivalently+n

(k)= (2.13)s

where(k) f(k).Thefunctiongivestheoutput-to-capitalratio intheeconomy. Thepropertiesk

of

f

imply

that

is

continuous

and

strictly

decreasing,

with


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(k) =f(k)kf(k) FL


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exogenousshockperturbstheeconomyandmovesawayfromthesteadystate.

The following uses the properties of G to establish that, in the Solow model, convergence to the

steadyisalwaysensuredandismonotonic:

Proposition

5

Given

any

initial

k0

(0,

),

the

dictatorial

economy

converges

asymptotically

to

the

steady state. The transition ismonotonic. The growth rate is positive and decreases over time towards

zero ifk0 < k; itisnegativeand increasesover time towardszero ifk0 > k

.

29

Economic

Growth:

Lecture

Notes

Proof. From the propertiesof f,G(k) =sf(k)+(1n)>0 and G(k) =sf(k) k forallk < k andG(k)< k forallk > k. It followsthatkt < kt+1 < k whenever

kt (0, k)

and

therefore

the

sequence

{kt} is

strictly

increasing

ifk0 < k

.Bymonotonicity,ktt=0

converges asymptotically to some k k. By continuity of G, k must satisfy k = G(k), that is k


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mustbeafixedpointofG.ButwealreadyprovedthatGhasauniquefixedpoint,whichprovesthat

k =k.Asymmetricargumentprovesthat,whenk0 > k,{kt}

is

stricttly

decreasing

and

againt=0

convergesasymptoticallytok.Next,considerthegrowthrateofthecapitalstock. Thisisgivenby

kt+1

kt =

s(kt)

(

+

n)

(kt).t

kt

Notethat(k)=0iffk=k, (k)>0iffk < k ,and(k) k .Moreover,bydiminishing

returns, (k) =s(k) (kt+1) > (k) = 0 whenever kt (k

,). This proves that t is positive and decreases

towards

zero

if

k0 < k and

it

is

negative

and

increases

towards

zero

if

k0 > k .

30

G.M.

Angeletos

Figure2.1depictsG(k),therelationbetweenkt andkt+1.TheintersectionofthegraphofGwiththe

45o linegivesthesteady-statecapitalstockk.Thearrowsrepresentthepath{kt} foraparticulart=

initialk0.

Figure

2.2

depicts

(k),

the

relation

between

kt and

t.

The

intersection

of

the

graph

of

with

the

45o line gives the steady-state capital stock k The negative slope reflects what we call conditional


149/192

45 linegivesthesteady-statecapitalstockk .Thenegativeslopereflectswhatwecall conditional

convergence.

Discuss localversusglobalstability: Because(k)


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32

0 ktk*

(k)

- (+n)

kt+1- ktk

t

= (kt)Figure 2.2. The growth rate in the Solow model.

Figure by MIT OCW.

G.M.

Angeletos

2.2 DecentralizedMarketAllocations

Intheprevioussection,wecharacterizedthecentralizedallocationdictatedbyasocialplanner. We

nowcharacterizethecompetitivemarketallocation

2 2 1 Households


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2.2.1 Households

Householdsaredynasties,livinganinfiniteamountoftime. Weindexhouseholdsbyj[0,1],having

normalizedL0 = 1.

The

number

of

heads

in

every

household

grow

at

constant

rate

n

0.

Therefore,

the

size

of

the

population inperiod t isLt =(1 +n)t andthenumberofpersons ineachhousehold inperiod t is

alsoLt.

Wewritecjt , kjt , b

jt , i

jt fortheper-headvariablesforhouseholdj.

33

Economic

Growth:

Lecture

Notes

Each person in a household is endowed with one unit of labor in every period, which he supplies

inelastically in a competitive labor market for the contemporaneous wage wt. Householdj is also

endowedwithinitialcapitalk0j. Capitalinhouseholdj accumulatesaccordingto

(1

+

n)kj =

(1

)kj +

it,t+1

t


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whichweapproximateby

ktj

+1 =(1n)kj

+it. (2.14)t

Householdsrentthecapitaltheyowntofirmsinacompetitivemarketfora(gross)rentalratert.

The household may also hold stocks of some firms in the economy. Let tj be the dividends (firm

profits) that householdj receive in period t. It is without any loss of generality to assume that

thereisnotradeofstocks(becausethevalueofstockswillbezeroinequilibrium). Wethusassume

that

household

j

holds

a

fixed

fraction

j

of

the

aggregate

index

of

stocks

in

the

economy,

so

that

tj

=jt,wheret

areaggregateprofits. Ofcourse,

jdj= 1.

34

G.M.

Angeletos

Thehouseholdusesitsincometofinanceeitherconsumptionorinvestmentinnewcapital:

cjt +ijt =y

jt .

Total

per-head

income

for

household

j

in

period

t

is

simply


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yjt =wt +rtkjt +

jt . (2.15)

Combining,wecanwritethebudgetconstraintofhouseholdj inperiodtas

cjt +ijt =wt +rtk

jt +

jt (2.16)

Finally,theconsumptionandinvestmentbehaviorofhouseholdisasimplisticlinearrule. Theysave

fractionsandconsumetherest:

jt =(1s)y

jt and i

jt

i=syt. (2.17)c

35

Economic

Growth:

Lecture

Notes

2.2.2 Firms

ThereisanarbitrarynumberMt offirmsinperiodt,indexedbym[0, Mt].Firmsemploylaborand

rentcapitalincompetitivelaborandcapitalmarkets,haveaccesstothesameneoclassicaltechnology,

and

produce

a

homogeneous

good

that

they

sell

competitively

to

the

households

in

the

economy.


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LetKt

m andLmt

denotetheamountofcapitalandlaborthatfirmmemploysinperiodt.Then,the

profitsofthatfirminperiodtaregivenby

mt =

F

(Ktm, Lmt )

rtKtm

wtLtm.

Thefirmsseektomaximizeprofits. TheFOCsforaninteriorsolutionrequire

FK(Ktm, Lt

m) = rt. (2.18)

FL(Ktm, Lt

m) = wt. (2.19)

36

G.M.

Angeletos

Rememberthatthemarginalproductsarehomogenousofdegreezero;thatis,theydependonlyon

thecapital-laborratio. Inparticular,FK

isadecreasingfunctionofKtm/Lmt andFL isanincreasing

functionofKtm/Lt

m.Eachoftheaboveconditionsthuspinsdownauniquecapital-laborratioKtm/Lt

m.

For

an

interior

solution

to

the

firms

problem

to

exist,

it

must

be

thatrt andwt areconsistent,that

is,

they

imply

the

same

Km/Lm.

This

is

the

case

if

and

only

if

there

is

some

Xt

(0,

)

such

thatt t


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rt

= f(Xt) (2.20)

wt = f(Xt)f(Xt)Xt (2.21)

where

f(k)

F

(k,

1);

this

follows

from

the

properties

FK(K,

L) =

f(K/L)

and

FL(K,

L) =

f(K/L)f(K/L) (K/L),whichweestablishedearlier. That is,(wt, rt)mustsatisfywt =W(rt)

whereW(r)f(f1(r))rf1(r).

If(2.20)and(2.21)aresatisfied,theFOCsreducetoKtm/Lmt =Xt,or

Ktm =XtL

mt . (2.22)

37

Economic

Growth:

Lecture

Notes

That is,theFOCspindownthecapital-laborratioforeachfirm(Ktm/Lmt

),butnotthesizeofthe

firm(Lmt ). Moreover,allfirmsusethesamecapital-laborratio.

Besides,(2.20)and(2.21)imply

rtXt

+

wt

=

f(Xt).

(2.23)


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Itfollowsthat

rtKtm +wtLt

m = (rtXt +wt)Ltm =f(Xt)Lt

m =F(Ktm, Lt

m),

andtherefore

m =

Lm[f(Xt)

rtXt

wt]

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