monetary model

8/16/2019 Monetary Model

1/40

Cagan and Lucas Models

Presented by Carolina Silva

01/27/2005


2/40

Introduction

I will present two odels t!at deterine noinal e"c!ange rates#

$%!e onetary odel# Cagan odel

$Lucas Model

&ven t!oug! t!e 'irst one is an ad-hoc odel( any o' its

predictions are iplied by odels wit! solid icro'oundations( and

it is t!e basis 'or wor) in ot!er topics* %!e Lucas odel is one o'

t!ose solid icro'oundations e"c!ange rate deterination

odels*


3/40

I* Cagan Model o' Money and Prices

In !is 1+5, paper( Cagan studied seven cases o' !yperin'lation*

-e de'ined periods o' !yperin'lations as t!ose w!ere t!e price

level o' goods in ters o' oney rises at a rate averaging at

least 50. per ont!

%!ese !uge in'lations are not t!ings o' t!e past( 'or e"aple(

between pril 1+ and uly 1+5( 3olivia4s price level rose by

2(000.

%!is iplies an annual in'lation rate o' alost 1(000


4/40

Cagan Model

Let M denote a country4s oney supply and P its price level(

Cagan4s odel 'or t!e deand o' real oney balances M/P is#

inflation.expectedrespect to

with balancesrealfordemandof citysemielasti theisandlog

t, periodof endat theheld balanceseynominalmonof logwhere

)( 1

η

η

P p

m

p p E pm t t t t d

t

=

=

−−=− +

Cagan 6usti'ies t!e e"clusion o' real variables suc! as output and

interest rate 'ro t!e oney deand 'unction( arguing t!at during

!yperin'lation t!e e"pected 'uture in'lation swaps all ot!er

in'luences on oney deand*


5/40

Solving t!e Model

!ic! are t!e iplication o' Cagan4s deand 'unction to t!erelations!ip between oney and t!e price level8

ssuing an e"ogenous oney supply ( in e9uilibriu#

(1) )(

: becomesdemandmoneythethus,

1 t t t t t

t

d

t

p p E pm

mm

−=−=

+η

So( we !ave an e9uation e"plaining price:level dynaics in ters

o' t!e oney supply*


6/40

Solving t!e Model

(3) 11

1

:get thatwe

bubbles)especulativno(ie,zero betotermsecondthessuming1

lim

11

1

s

t s

t s

t

T t

T

T

s

t s

t s

t

m p

pm p

∑

∑

∞

=

−

+

∞→

∞

=

−

++

=

+

+

++

=

η η

η

η

η

η

η

η

..., 3! ++ t t p p;irst( 'or t!e nonstoc!astic per'ect 'oresig!t( ie(

by successive substitution o' we get t!at#

Is t!is a reasonable solution o'


7/40

Siple Cases

t mmt ∀= 1* Constant oney supply#

m pm p

m p p p pm

t

t s

s

t s

t

t t t t t

=⇒

++

=

=⇒−=−

∑∞

=

−

+

η

η

η

η

η

11

also,and

)( 1


8/40

Siple Cases

t mmt µ +=

µ

2* Constant percentage growt! rate#

>uessing t!at t!e price level is also growing at rate ( and

substituting t!is guess in e9uations


9/40

%!e Stoc!astic Cagan Model

>iven t!e linearity o' t!e Cagan e9uation( e"tending its solution

to a stoc!astic environent is straig!t'orward* ?nder t!e no

bubble assuption( we !ave t!at#

(") )(11

∑∞

=

−

++

=t s

st

t s

t m E pη

η

η

η


10/40

%!e Cagan Model in Continuous %ie

Soeties is easier to wor) in continuous tie* In t!is case( t!eCagan nonstoc!astic deand


11/40

Seignorage

t

t t

P

M M 1eignorage −−

=

Definition# represents t!e real revenues a governent ac9uiresby using newly issued oney to buy goods and nononey

assets#

Most !yperin'lations ste 'ro t!e governent4s need 'or

seignorage revenues* !at is t!e seignorage:revenue:a"ii@ing

rate o' in'lation8 Aewriting seignorage as#

t

t

t

t t

P

M

M

M M 1eignorage −−=

we can see t!at( i' !ig!er oney growt! raises e"pected in'lation(

t!e deand 'or real balances M/P will 'all( so t!at a rise in oney

growt! does not necessarily augent seignorage revenues*


12/40

Seignorage

11

1−−

==+t

t

t

t

P

P

M

M µ

η −

+

=

t

t

t

t

P

P

P

M 1

;inding t!e seignorage:revenue:a"ii@ing rate o' in'lation iseasy i' we loo) only at constant rates o' oney growt!#

Bow( e"ponentiating Cagan4s per'ect 'oresig!t deand( we get#

Substituting t!ese in t!e second seignorage e9uation#

1)1()1(1

eignorage −−− +=++

= η η µ µ µ µ

µ


13/40

Seignorage

η µ µ η µ µ

µ

η η 1 #)1)(1()1(

:isrespect towith*+thehus,

max!1 =⇒=++−+ −−−−

Cagan was surprised because( at least in a portion o' eac!

!yperin'lation !e studied( governents see to put t!e oney

to grow at rates !ig!er t!an t!e optial one*

$ daptative e"pectations ay iply s!ort run bene'its 'roteporarily increasing t!e oney growt! rate*

$Proble# even under rational e"pectations( i' t!e governent

can not coit to aintain t!e optial rate( its revenues could

be lower*


14/40

Siple Monetary Model o' &"c!ange

Aates

output.realof logtheisand priceof logtheisrate,interestnominalthewith)1log(iwhere

,(1) i 1t

y pii

y pm t t t

+=

+−=− + φ η

(3) )1(1 -/and

(!) logsinor///hen

10

11

00

+=+⇒

+==⇒

+

++t

t

t t t

t t t t t t

E ii

pe p P P

ξ

ξ

ξ

t ξ

variant o' Cagan4s odel# a S& wit! e"ogenous real output

and oney deand given by#

Let be t!e noinal e"c!ange rate


15/40


Aates

(") ii 10

1t1t t t t ee E −+= +++

n appro"iation in logs o' ?IP is#

Substituting t!e log PPP and


16/40


Aates

&ven t!oug! data do not support generally t!is odel in non

!yperin'lation environent( t!is siple odel yields one

iportant insig!t t!at is preserved in ore general 'raewor)s#

The nominal exchange rate must be viewed as an asset priceThe nominal exchange rate must be viewed as an asset price

In t!e sense t!at it depends on e"pectations o' 'uture variables(

6ust li)e ot!er assets*


17/40

Monetary Policy to ;i" t!e &"c!ange

Aate

e

(1) )( 1 t t t t t ee E em −−=− +η

Consider a special case o' t!e S& Cagan e"c!ange rate odel#

Suppose t!e governent 'i"es t!e noinal e"c!ange rateperanently at ( t!en substituting in


18/40

Soe observations

Can t!e e"c!ange rate be 'i"ed and t!e governent still !avesoe onetary independence8

$ d6usting governent spending can relieve onetary policy o'

soe o' t!e burden o' 'i"ing t!e e"c!ange rate* 3ut in practice(

'iscal policy is not a use'ul tool 'or e"c!ange rate anageent(

because it ta)es too long to be ipleented*

$;inancial policies can !elp also t!roug! sterilized interventions:to )eep t!e e"c!ange rate 'i"( t!e governent ay !ave to buy

'oreign currency denoinated bonds wit! doestic currency* %o

Dsterili@eE t!is( t!e governent reverses its e"pansive ipact by

selling !oe currency denoinated bonds 'or !oe cas!*


19/40

II* Lucas Model

ne o' t!e probles o' Cagan odel is t!at t!e oney deand'unction upon w!ic! it rest !as no icro'oundations* n t!e

ot!er !and( Lucas4s neoclassical odel o' e"c!ange rate

deterination gives a rigorous t!eoretical 'raewor) 'or pricing

'oreign e"c!ange and ot!er assets*e will see t!ree odels#

$%!e barter econoy

$%!e one oney onetary econoy

$%!e two oney onetary econoy

In all t!ese( ar)ets !ave no iper'ections and e"!ibit no

noinal rigidities* gents !ave rational e"pectations and

coplete in'oration*


20/40

* %!e 3arter &conoy

-ere we will study t!e real part o' t!e econoy#

$%wo countries( eac! in!abited by a representative agent*

$%!ere is one D'irE in eac! country( w!ic! are pure endowent

streas t!at generate a !oogeneous nonstorable country:speci'ic good( using no labor or capital input FG 'ruit trees*

$&volution o' output#

agents. bynownare processesstochastic

itsandrandomare and where, and 010

1 t t t t t t t t g g y g y x g x −− ==

$&ac! 'ir issues a per'ectly divisible s!are o' coon stoc)

w!ic! is traded in a copetitive ar)et*


21/40

%!e 3arter &conoy

t x

)()(0

11 t yt yt t xt e yqwe xwW

t t +++=

−−

$;irs pay out all o' t!eir output as dividends to s!are!olders(w!ic! are t!e sole source o' support 'or individuals*

$e will let be t!e nueraire good*

$?nder t!is 'raewor)( t!e wealt! a doestic agent brings toperiod t is#

$ nd t!e agent !as to allocate t!is wealt! between consuptionand new s!are purc!ases#

t t t t yt x yt xt t cqcweweW +++= 0


22/40

%!e 3arter &conoy

(1) )()( 0011 t yt yt t x yt xt yt x

e yqwe xwwewecqct t t t t t

+++=+++−−

(1) .

),( #

st

ccu E Max j

y x

j

t jt jt

∑

∞

=++β

&9uating t!e last two e9uations we get t!e budget constraint 'ordoestics#

In t!is way( doestic agents !ave to c!oose se9uencesto solve#{ }∞

=++++ #,,,

j y x y x jt jt jt jt

wwcc


23/40

%!e 3arter &conoy

(") )%)(,(&),( :

(3) )%)(,(&),( :

(!) ),(),( :

0

11111

0

1111

!1

11

11

+++

++

+=+=

=

++

++

t t t y xt y xt y

t t y xt y xt x

y x y xt y

e yqccu E ccuew

e xccu E ccuew

ccuccuqc

t t yt t

t t yt t

yt t t t

β

β

%!us( t!e doestic &uler e9uations are#

I' we put an H over t!e variables in t!e

doestic agent proble and in t!e doestic &uler e9uations( we

get t!e 'oreign agent proble and 'oreign &uler e9uations*

{ }∞=++++ #

,,, j

y x y x jt jt jt jt wwcc


24/40

%!e 3arter &conoy

e need to add 'our ore constraints to clear t!e ar)ets#

(4)

(5) () 1

(2) 1

0

0

0

0

t y y

t x x

y y

x x

ycc

xccww

ww

t t

t t

t t

t t

=+

=+=+

=+


25/40

%!e 3arter &conoy

)4( ),5( .

),(!

1),(

!

1 00

st

ccuccu Maxt t t t y x y x

+

>iven t!at we !ave coplete and copetitive ar)ets( we canapply t!e wel'are t!eore and solve t!e social planner proble#

and t!e solution will be an copetitive e9uilibriu#

!

!),(

!

1),(

!

1

),(!1),(

!1

: 00

00

!!

00

11t

y yt

x x

y x y x

y x y x ycc

xcc

ccuccu

ccuccu

FOC t t t t

t t t t

t t t t

==∧==⇒

=

=


26/40

%!e 3arter &conoy

!

100 ====t t t t y y x x

wwww

Bow we !ave to loo) 'or t!e prices and s!ares t!at support t!ise9uilibriu*

$S!ares# a stoc) port'olio t!at ac!ieves coplete insurance o'

idiosyncratic ris) is(

$Prices# to get an e"plicit solution we need to give a 'unction 'or

to t!e utility( let

γ

γ θ θ

−==

−−

1),( and

11 t

y x y xt

C ccuccC

t t t t


27/40

%!e 3arter &conoy

?nder all w!at we !ave seen and assued( t!e &uler e9uationsiply#

+

=

+

=

−=

+

+

−

+

+

+

−

+

1

0

1

1

1

0

1

1

1

1

1

1

1

t t

t

t

t t

t t

t

t

t

t

t t

t

t

t

t t

yq

e

C

C E

yq

e

x

e

C

C E

x

e

y

xq

γ

γ

β

β

θ

θ


28/40


29/40

%!e ne:Money Monetary &conoy

* centrali@ed securities ar)et opens( w!ere agents allocatet!eir wealt! toward stoc) purc!ases and t!e cas! t!ey will need

'or consuption*

* ecentrali@ed goods trading now ta)es place in t!e Ds!opping

allE*

5*%!e cas! value o' goods sales is distributed to stoc)!olders as

dividends( w!o carry t!ese noinal payents into t!e ne"t

period*

Observation:Observation: t!e state o' t!e world is revealed be'ore trading( t!us

agents )now e"actly !ow uc! cas! t!ey need to 'inance t!e

current period consuption plan* So( it is no necessary to carry

cas! 'ro one period to t!e ne"t( and t!ey won4t do it i' t!enoinal interest rate is positive*


30/40


transfer money

t

t

value saredividend ex

t yt x

dividends

t

t t yt xt

t P

M ewew

P

yqw xw P W

t t

t t

01111

!

)(

11

11 ∆

++++

=−

−−−−

−−

−−

0

t yt x

t

t t ewew

P

mW

t t ++=

>iven t!ese assuptions( doestic agent4s period t wealt! is#

nd in t!e security ar)et( t!e agent allocate !is wealt! between#

ssuing a positive noinal interest rate( t!e cas! in advanceconstraint binds#

)(t t yt xt t

cqc P m +=


31/40


?sing t!e last t!ree e9uations( we get t!at t!e doestic agentproble is#

c

!)( .

),(6ax

0

y

0

1111

#

t

1111

t yt xt x

t yt x

t

t t t yt x

t

t

j

y x

j

t

ewewqc

ewew P

M yqw xw

P

P st

ccu E

t

t t t t

jt jt

+++=

++∆

++

−−−−

++

−−−−

∞

=∑β


32/40


%!e doestic agent proble iplies t!e 'ollowing &uler

e9uations#

(") )%)(,(&),( :

(3) )%)(,(&),( :

(1) ),(),( :

0

111

1

11

0

11

1

11

!1

11

11

+++

+

+++

+=

+=

=

++

++

t t t

t

t y xt y xt y

t t

t

t y xt y xt x

y x y xt y

e yq P

P ccu E ccuew

e x P P ccu E ccuew

ccuccuqc

t t yt t

t t yt t

yt t t t

β

β

%!e 'oreign agent !as t!e sae proble and &uler e9uations

but wit! an H over t!e variables t!at !e c!ooses


33/40


and

1 1

0

00

00

t t t

t y yt x x

y y x x

mm M

ycc xcc

wwww

t t t t

t t t t

+=

=+∧=+

=+∧=+

!

!00 t y yt x x

ycc

xcc t t t t ==∧==

%o clear t!e ar)ets we need to add t!e constraints#

%!e e9uilibriu o' t!e barter econoy is still t!e per'ect ris):

pooling e9uilibriu#

!

100==== t t t t y y x x wwwwand

%!e only t!ing t!at !as c!anged is t!e e9uity pricing 'orulae(

w!ic! now include t!e Din'lation preiuE*


34/40


?sing t!e sae constant relative ris) aversion utility 'unction weused in t!e barter econoy( we !ave t!at#

+

=

+

=

=

−=

+

+

+

−

+

+

+

+

−

+

+

++

1

0

1

1

1

1

0

1

1

1

1

1

1

11

1

t t

t

t

t

t

t t

t t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t

t t

yq

e

M

M

C

C E

yq

e

x

e

M

M

C

C

E x

e

x

x

M

M

P

P

y

xq

γ

γ

β

β

θ

θ


35/40

%!e ne:Money Monetary &conoy(

pricing ot!er assets

payoff of utilitymarginal

11

bondthe buyingof costutility

1 )$),(($),( 11 +++= t y xt t t y x P ccu E P bccu t t t t β

1)1( −+= t t ib

t e9uilibriu( t!e price b o' a noinal bond t!at pays 1 dollar at

t!e end o' t!e period ust satis'y#

I' is t!e noinal interest rate( t!en

%!us( using t!e usual utility 'unction( noinal interest rate will be

positive in all states i' t!e endowent growt! rate and onetary

growt! rates are positive*

t i


36/40

C* %!e %wo:Money Monetary &conoy

Let t!e !oe currency be t!e DdollarE( and t!e 'oreign( t!e DeuroE*Bow( t!e !oe good " can only be purc!ased wit! dollars( and y

wit! euros* 3esides( "4s dividends are paid in dollars and y4s in

euros* gents can get t!e 'oreign currency during security ar)et

trading*Currencies evolve according to#

1 7

0

1

:

:

t t t

t t t

! ! euro

M M dollar

λ

λ

=

= −

Bow we will !ave a new product# clais to 'uture dollar and euro

trans'ers* It will be assued t!at initially t!e !oe agent is

endowed wit! t!e w!ole strea o' dollars and t!e 'oreign( wit! t!e!ole strea o' euros* %!en t!e can trade*


37/40

%!e %wo:Money Monetary &conoy

uritiesof valuemar"et

t ! t M t yt x

transfersmoney

t

t t !

t

t M

dividends

t y

t

t t t x

t

t t

r r ewew

P

! #

P

M yw

P

P # xw

P

P W

t t t t

t t

t t

sec

00

1

0

11

1

1111

11

11

−−−−

−−

−−

++++

∆+

∆++= −

−−

−

ψ ψ

ψ ψ

%!en( we !ave t!at t!e !oe agent current:period wealt! is#

nd t!is wealt! will be allocated according to#

t

t t

t

t t ! t M t yt xt

P

# n

P

mr r ewewW

t t t t +++++= 00 ψ ψ


38/40


:e's8ulerfollowingimply the and sconstraint

advanceincashtheande'uationslast two by theimplied9+the before,s

0

t t yt t xt t c P nc P m ==

)%)(,(&),( :

)%)(,(&),( :

)%)(,(&),( :

)%)(,(&),( :

),(),( :

0

1

1

1111

0

1

1

116

0

11

1

0

111

0

11

1

11

!1

0

11

11t

11

11

+

+

++

+

+

++

+

+

++

+

+∆

=

+

∆

=

+=

+=

=

++

++

++

++

t

t

t t y xt y xt !

t

t

t y xt y xt

t t

t

t t y xt y xt y

t t

t

t y xt y xt x

y x y x

t

t t y

r P

# ! ccu E ccur

r P

M ccu E ccur

e y P

P # ccu E ccuew

e x P P ccu E ccuew

ccuccu P

P # c

t t yt t

t t t t

t t yt t

t t yt t

yt t t t

β ψ

β ψ

β

β

nd again t!e 'oreign agent !ave a syetric set o' &uler e9s*


39/40


1 1

00

00

00

t t t t t t

t y yt x x

y y x x

nn ! mm M

ycc xcc

wwww

t t t t

t t t t

+=∧+=

=+∧=+

=+∧=+

!

!

00 t y y

t x x

ycc

xcc

t t t t ==∧==

%oget!er wit! t!e &uler e9s* e !ave t!e clear ar)et conditions#

it! t!ese e9s* e !ave t!e 'ollowing e9uilibriu#

!

10000 ========t t t t t t t t ! ! M M y y x x

wwww ψ ψ ψ ψ

and


40/40


;ro t!e 'irst &uler e9uation( we get t!at t!e noinal e"c!ange

rate is#

t

t

t

t

y x

y x

t

x

y

!

M

ccu

ccu#

t t

t t

),(

),(

1

!=

ConclusionConclusion# as in t!e onetary approac!( t!e deterinants o' t!e# as in t!e onetary approac!( t!e deterinants o' t!e

noinal e"c!ange rate are relative oney supply and relativenoinal e"c!ange rate are relative oney supply and relative

>Ps* %wo a6or di''erences are t!at in t!e Lucas odel#>Ps* %wo a6or di''erences are t!at in t!e Lucas odel#

$S depends on pre'erencesS depends on pre'erences

$S does not depend e"plicitly on e"pectationsS does not depend e"plicitly on e"pectations

monetary model

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