Prof. Dr. Gerhard Illing Winter term 2009/2010

Advanced Macro and Money

Advanced Master Class Munich Graduate School of Economics at LMU

Lecture notes

Part 1.3

Models with Money

Basic references for this part:

Blanchard /Fischer, chapter 4/5;

Gali or Woodford chapter 2

Additional reading: Walsh, chapter 2/3

65

1.3.1 Challenges A) Policy (this will be addressed in part 2 of the course): How does monetary policy affect nominal prices / real economy? What is the adequate design of monetary policy? What is the task of central banks? How to cope with incentives problems? How should we define price stability? Is there a role for active stabilisation policy? How can policy be implemented in the presence of uncertainty? Can money or interest rate targeting implement optimal policy? Basis: Modern New Neoclassical Synthesis - Key features: 1) AS curve, generated by sticky nominal prices. 2) AD curve, derived from intertemporal optimisation (Euler equation) 3) LM curve, motivated by central bank policy (money supply or interest rate rule). Reference point: Frictionless virtual economy (real business cycle approach/ Modern stochastic growth theory): Classical dichotomy: in the absence of nominal and real frictions: flexible price economy with efficient market outcome B) Theory: Microfoundations to motivate a role for money How to introduce a role for money in the intertemporal framework? Key challenges:

a) In neoclassical economies with complete set of markets: no need for money (all contracts arranged already at the start of the economy).

b) Money is a dominated asset (bonds and other assets earn positive return, whereas nominal return of money is fixed to zero, With inflation: real return is negative)

c) Money is intrinsically useless (no direct utility out of holding /consuming money), Crucial: self fulfilling nature of money: I hold money only if I can trust that later other agents will accept it. Money as social convention!

Trading frictions are crucial: There has to be a need for liquidity. Many competing concepts; Sidrauski Model: As short cut, include money in utility function. Cash-in-advance model and the shopping time model can be seen as motivating special cases of the MIU approach. First, we start with an old fashioned approach – just assuming there is a demand for money! We ask: how will the price path be determined?

66

1.3.2 Money and Inflation – Cagan Model of Money and Prices

Fisher equation: ))(1()1()1()1( 1tt

t

ttt Er

PPEri π++=+=+ +

Logarithmic approximation: tttttt ppErEri −+=+= +1)(π Cagan’s isoelastic money demand function (allows linear analysis):

( ) bt

t

t iCPM −+= 1 ;

b

t

t

t

t

PP

PM

−+⎟⎟⎠

⎞⎜⎜⎝

⎛= 1 or:

)( 1 tttdt ppEbpm −−=− + with d

tdt Mm ln=

Determination of the Price Level for a given stochastic money supply path {Mt+s} s=0,…,T}: Equilibrium on the money market: ;t

dt mm = gives linear

difference equation: )( 1 tttt ppEbpm −−=− +

Forward looking solution: 1111

+++

+= ttt pE

bbm

bp

Price level as forward looking variable (non predetermined)

Substitute 2111 111

++++ ++

+= ttttttt pEE

bbmE

bpE .

Use law of iterated expectations 221 +++ = ttttt pEpEE to get

2

2

1 1)

1(

11

++ ⎟⎠⎞

⎜⎝⎛+

++

++

= tttt pEb

bmb

bmb

p

67

Iterated substitution gives:

Tt

T

stTs

s

t pEb

bmEb

bb

p ++−= ⎟

⎠⎞

⎜⎝⎛+

+⎟⎠⎞

⎜⎝⎛++

= ∑1

)(11

1 10

st

s

ssts

s

t pEb

bmEb

bb

p +∞→

+∞= ⎟

⎠⎞

⎜⎝⎛+

+⎟⎠⎞

⎜⎝⎛++

= ∑1

lim)(11

10

Assume that the sum converges ∞<⎟⎠⎞

⎜⎝⎛+ +

−=→∞

∑ )(1

lim 10 st

Ts

s

smE

bb

Rule out hyperinflationary bubbles:

Impose restriction 01

lim =⎟⎠⎞

⎜⎝⎛+ +

∞→st

s

spE

bb ,

Solution: )(11

10 sts

sFt mE

bb

bp +

∞=∑ ⎟

⎠⎞

⎜⎝⎛++

=

So the price level is determined by “future fundamentals” Specify expected future money supply process! Examples:1 A) Constant money supply: mmm tst ==+

tts

s

t mmb

bb

p =⎟⎠⎞

⎜⎝⎛++

= ∑∞=0 11

1

B) Constant growth rate of money: smm tst µ+=+

µµ bmb

bsb

mb

bb

p ts

s

ts

s

t +=∑ ⎟⎠⎞

⎜⎝⎛

+++∑ ⎟

⎠⎞

⎜⎝⎛++

= ∞=

∞= 00 11

111

1

1 Note: bb

bs

s

+=⎟⎠⎞

⎜⎝⎛+

∑∞= 1

10 ; ∑ +=⎟⎠⎞

⎜⎝⎛

+∞=0 )1(

1s

s

bbb

bs

68

C) Anticipated increase in money supply at some future date T:

⎩⎨⎧

≥><

=+ TsmmTsm

m st12

1

∑ −⎟⎠⎞

⎜⎝⎛

+++∑ ⎟

⎠⎞

⎜⎝⎛++

= ∞=

∞= Ts

s

s

s

t mmb

bb

mb

bb

p )(11

111

11210 =

∑ −⎟⎠⎞

⎜⎝⎛

+⎟⎠⎞

⎜⎝⎛++

+= ∞=0 121 )(

1111

s

sT

t mmb

bb

bb

mp =

)(1 121 mm

bbm

T

−⎟⎠⎞

⎜⎝⎛+

+

D) Stochastic money supply with AR(1) process:

11 ++ += ttt mm ερ ; 10 ≤≤ ρ ; 01 =+ttE ε So ts

stt mmE ρ=+

So =⎟⎠⎞

⎜⎝⎛

++=⎟

⎠⎞

⎜⎝⎛

++= ∑∑ ∞

=∞= 00 11

111

1s t

s

s ts

s

t mb

bb

mb

bb

p ρρ

ttb

bt mb

mb

p)1(1

11

11

1

1ρρ −+

=−+

=+

D1) mt follows a random walk ( 1=ρ ): tt mp =

D2) mt serially uncorrelated: ( 0=ρ ): ttt bm

bp ε

+=

+=

11

11

Open issues:

A) Rational of money supply process B) Rational of ruling out hyperinflationary bubbles C) Rational of money demand (Microfoundation for LM

curve)

69

A) Rational of Monetary Policy: Create seigniorage (Hyperinflation as consequence of the need to finance government spending) Seigniorage from money creation:

t

t

t

tt

t

ttM P

MM

MMP

MMSt

11 −− −=

−=

Steady state government revenue from money creation? With constant money growth rate 1)1( −+= tt MM µ ,

Seigniorage is: t

tM P

MSt µ

µ+

=1

Cagan: Isoelastic Money demand b

t

t

t

t

PP

PM

−+⎟⎟⎠

⎞⎜⎜⎝

⎛= 1

Constant process of money growth: tt MM )1(1 µ+=+ generates: tt PP )1(1 µ+=+ so:

( ) ( ) )1(111

bbM t

S +−− +=++

= µµµµ

µ .

Maximum seigniorage at: 0=∂∂ µtMS : for

b1

=µ

( ) 0)1()1(1 )2()1( =++−+=∂∂ +−+− bbM bS

tµµµµ

µµ )1()1( b+=+⇔

Continuous time analysis: tb

t

t eP

M π−= ; t

t

btM eS ππ −=

Optimal supply for monopolist without cost 1=bµ B) Bubbles Forward looking solution gives first order difference equation:

1111

+++

+= ttt pE

bbm

bp

70

One solution of this difference equation is:

)(11

10 sts

sFt mE

bb

bp +

∞=∑ ⎟

⎠⎞

⎜⎝⎛++

=

Without additional anchor, that equation difference has many other solutions:

)(11

111 ++ +

++

+=+= t

Fttt

Ftt ypE

bbm

bypp

(not specified without some endpoint constraint) Assume the following stochastic process {yt}:2

111

++ ++

= ttt vyb

by with 01 =+ttvE , so 11 ++= ttt yE

bby

tFtt ypp += is also a solution to the difference equation.

With tt

T

t

T

TTtt

T

Tyy

bbE

bbyE

bb

=⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛+

=⎟⎠⎞

⎜⎝⎛+ ∞→

+∞→

11

lim1

lim

yt is a bubble, with yt+s exploding at the rate 1+1/b. Solution to the difference equation (constant present value).

Ftt pp = is unique solution only if the no-bubble condition

holds: 01

lim =⎟⎠⎞

⎜⎝⎛+ +

∞→st

s

spE

bb

Economic Motivation? Unlike No Ponzi Game Condition: Self fulfilling inflationary expectations (“speculative hyperinflation”) may make money useless as a medium of exchange without violating aggregate resource constraints! Money has to be essential to rule out such a price path! Microeconomic foundations for the money demand function! 2 More generally: a stochastic bursting bubble can also be a solution, if:

⎪⎩

⎪⎨

⎧

−

++

=

+

++

pv

pvypb

by

t

ttt

1probwith

probwith11

1

11 with 01 =+ttvE

71

1.3.3 Money in the Utility: Brock/Sidrauski Model Extension of standard growth models to include money: (see: Blanchard /Fischer, chapter 4/5; Woodford chapter 2; Walsh, chapter 2; original papers: Sidrauski 1967; Brock 1974) Money is one of many assets (some financial, some real). It is intrinsically useless (it provides no direct utility). Its nominal return im,t is dominated by return it on safe bonds. But money yields transaction benefits, captured by preferences:

Per period payoff function: ),(PMCu

U is assumed to be concave and increasing in both arguments;

C≥0; M/P≥0; 0),(;0),( / ≥>PMCU

PMCU PMc

With PMm /= . Shortcut to model liquidity services (indirect utility, derived from transaction costs (shopping time models); cash in advance constraint; OLG) Frequently it is assumed: (1) Additive separability: )()(),( mVCumCu += (2) There is some point of satiation m for real money

balances: Satiation level m with 0),( =mCUm (and 0),( <mCUm for mm >

Representative Household maximises: ∑ == Tt

t

tt

t

PMCUu 0 ),(β

subject to (1) Intertemporal Budget constraint (2) Endpoint constraints; No-Ponzi game constraint Formulate the relevant budget constraint with a complete set of nominal assets; perfect foresight; pure endowment economy Initial wealth Wt given; Endowment stream of nominal net income Pt (Yt -Tt) given;

72

Choose Ct; Mt/Pt and the allocation of nominal and real assets Bt, Kt, for a given sequence of intertemporal prices {Pt } and {Qt, t+s}.

Nominal discount factor )1(

11,11,

ννννν i

QQ Tt

TtTt +

∏=∏= −=−+=

(define: Qt, t =1)

The riskless short term nominal interest rate is: )()1(

11, +=

+ ttt

QEi

(1) Intertemporal Budget constraint There are different timing conventions in discrete time analysis. We use here the notation of Woodford (2003)3

3 In an alternative timing convention, financial markets open at the beginning of period t, and agents have to choose Mt, Bt and Kt. Then, agents receive income Yt and can consume from liquid assets. Then, money balances at the end of period t are Mt + Pt (Yt -Tt )-Pt Ct . So here, money Mt providing liquidity is not the money held at the end of the period. How does this alternative convention affect the results? (see Woodford, Appendix A 16).

73

Define the beginning of period wealth as 1111 )1()1( −−−− +++= t

mtttt MiBiW + (1+rt-1) Pt Kt-1

(it: nominal return for riskless one year bonds Bt held between t and t+1 (resp. rt return for real bonds Kt) Relevant state variable: total wealth Wt given at t. Eliminate Bt to formulate the per period budget constraint t=1,…,T as the evolution of nominal wealth: (1)

ttttttt

ttt

tt

t

mttt WTYPKP

irPW

iM

iii

CP t +−≤++

−++

++

−+ ++ )()

11(

11

1 11

How to derive this equation (dynamic optimisation with many assets)? Assume first that all assets are invested in risk-less bonds or capital, none in money. Risk-less bonds pay interest it, so Qt, t+1 =1/(1+it). Both bonds and capital bought last period determine present wealth by

Wt = (1+it-1) Bt-1+(1+rt-1) PtKt-1. Buying today bonds Bt and capital Kt yields wealth tomorrow Wt+1 = (1+it) Bt +(1+rt) Pt+1Kt. So budget constraint can be formulated as:

Pt Ct + Bt + Pt Kt= Pt(Yt-Tt) + Wt or In terms of wealth (eliminating Bt) as

PtCt + Wt+1 /(1+it) +Kt (Pt -Pt+1 (1+rt)/(1+it))= Pt(Yt-Tt) +Wt If part of assets are held as money (with interest im,t):

PtCt+Bt+Mt + Pt Kt = Pt(Yt-Tt)+Wt with next period wealth:

Wt+1 = (1+it) Bt + (1+im,t) Mt+(1+rt) PtKt! Eliminating again Bt gives 1) with wealth Wt as state variable.

74

(2) Endpoint constraints 2a) W0 given. 2b) 01 ≥+TW Solvency constraint: at the terminal date,

assets left cannot be negative. Or “no Ponzi Game” constraint 011,0 ≥++ TT WQ

For ∞→T 0)(lim , ≥∞→

TTttTWQE implies:

∑∞

+=+++ −−≥

1,111 )((

tTTTTTttt TypQEW Current debt cannot

exceed present value of future net income. Solve the optimisation problem using the standard Lagrangian approach:

∑ == Tt

t

tt

t

PMCUu 0 ),(β +

])11(

1[ 111,

0tt

t

ttt

t

mttttttttt

T

tt KP

irPM

iii

WQCPWYP t

+++= +

+−−

+

−−−−+∑λ

+ 00 Wλ - 11,1 +++ TTTT WQλ This gives as FOC:

(a) ttt

ttc

t PP

MCu λβ =),(

(b) 11, ++ = tttt Q λλ → Arbitrage equation (c) tttt PPri /)1(1 1++=+ Fisher equation

(otherwise: either Bt =0 or Kt=0) [ 0;0 ≥≥ tt KB !]

(d) t

mtt

t

t

t

ttPM i

iipp

Mcu t

+

−=

11);(/ λβ → Money demand (LM curve)

(e) Transversality condition: 011,011,1 == +++++ TTTTTT WQWQλ or 0)(lim 11,0 =++

∞→TTtT

WQE

75

(a) gives: t

ttt

tt

tt

PPQ

CuCu 1

1,1

1

)(')(' +

++

+=

ββ or, using (c):

(1) tt

t

t

t

ttc

t

ttc

rPP

iP

MCu

PMCu

++

=++

= ++

++

11

11

),(

),(11

11 ρρ

(2) t

mt

t

ttc

t

ttPM

iii

PMCu

PMCu

t

+

−=

1);(

);(/ Money demand curve

Interpretation of FOC conditions: (1) MRS between consumption today and tomorrow equal to real discount factor

t

t

ttc

t

ttc

t

tr

PMCu

PMCu

CdCd

+=

+=− +

++

+ 11

),(

),(

11 1

11

1 ρ

(2) MRS between consumption today and consumption of real money balances equal to opportunity cost of holding money balances (present value of interest lost next period due to holding money):

t

mt

t

ttc

t

ttPM

tt

tiii

PMCu

PMCu

PMdCd

t

+

−==−

1);(

);(

/

/

(Microbased) version of the LM curve From FOC for optimal real balances, we can derive a money demand function (which, in general, also depends on the consumption level [because of ))((' tcU ]

76

1.3.3 Appendix: Fisher equation under uncertainty When inflation is stochastic, we need to modify the Fisher equation.

Euler equation for nominal bonds: 11,,111 );;(

);;(+++++

=t

t

ttttttc

tttc

PP

Qmcumcu β

ξξ

or ttt

t

tttc

ttttc QPP

mcumcu

,11

,111

);;();;(

++

+++ =ξξ

β

Define );;(

);;( ,111,1

tttc

ttttctt mcu

mcuq

ξξ

β ++++ = ,

Then for holding nominal bonds for all states of the world tt ,1+ξ :

FOC: ttt

ttt Q

PPq ,1

1,1 +

++ = , so expected return of nominal bonds:

ttt

t

ttt i

QEPPqE

+== +

++ 1

1)()( ,11

,1

Euler equation for real bonds (Real price of real bonds):

t

ttttt

tttc

ttttc

PPQq

mcumcu 1

,1,1,111

);;();;( +

+++++ ==

ξξ

β

Does the Fisher equation hold under uncertainty? We argued:

ttt

ttt

t

ttt

ttt rP

PEqEPPqE

iQE

π+⋅

+=⋅==

+=

++

+++ 1

11

1)()()(1

1)(1

,1

?

1,1,1

Obviously, this is only an approximation disregarding covariance!

Assume that ttq ,1+ , t

t

PP 1+ is jointly lognormal distributed

For x being normal, we have: ( )][][exp][exp 21 xVarxExE +=

So for x and y being lognormal, we get:

( )][ln][lnexp][ 21 xVarxExE +=

( )]ln,[lnexp][][][ yxCovyExEyxE =

77

tttt

t PPP

P π=−= ++ lnlnln 11

)lnlnexp()()(ln)(ln1

,11

,11

,1+

++

++

+ =t

ttt

t

ttt

t

ttt P

PqCovPPEqE

PPqE

)ln()(ln)(ln ,11

,1 tttt

ttt qCov

PPEqE π++

+ −+= =

)ln(][ln][lnexpln)(ln ,11

21

1,1 ttt

t

t

t

ttt qCov

PPVar

PPEqE π+

+++ −⎟⎟

⎠

⎞⎜⎜⎝

⎛++=

)ln(][)()1(ln ,12

1tttttt qCovVarEr πππ +−+−+−=

)1ln()(ln1

,1 tt

ttt i

PPqE +−=+

+ - So:

)(ln][)( ,121

ttttttt qCovVarEri πππ ++−+= So we need to take into account an inflation risk premium:

)(ln][ ,121

tttt qCovVar ππ ++−

But we usually disregard inflation risk premium, approximating the Fisher equation:

)( ttt Eri π+=

78

1.3.4 Model frictions motivating indirect utility of money: → Cash in advance and shopping time models 1.3.4.1 Cash in advance constraint Extreme transaction technology: Some goods - cash goods xt - can be bought only by paying in cash: ttt Mxp ≤ .

→ Consumption of cash goods is taxed by t

mt

iii

t

+

−

1

Constraint is a special cases of the MIU approach. Representative household has no direct utility from money holding. She maximises:

Max ),()1(

10

ttt

t xcU∑∞

= + ρ s. t.

(1) Flow budget constraint:

ttttttt

ttt

tt

t

mttttt WTYPKP

irPW

iM

iii

XPCP t +−≤++

−++

++

−++ ++ )()

11(

11

1 11

(2) Cash in advance constraint for cash-goods

ttt MXp ≤ for all t Lagrangian multipliers for both constraints: t

t λβ and tt νβ

79

][

11]

11[

1])([

),(

0

1110

0

ttttt

t

tt

ttt

ttt

t

mtttttttttt

t

t

ttt

t

XPM

Wi

KPirPM

iiiXPCPWTYP

lcUMax

−+

⎥⎦

⎤⎢⎣

⎡+

−++

−−+−

−−−+−

+

∑

∑

∑

∞

=

+++

∞

=

∞

=

νβ

µβ

β

FOC: (1) wrt ct ttttc PxcU µ=),( (2) wrt xt )(),( tttttx PxcU νµ +=

(3) wrt Mt tt

mtt i

iit νµ =

+

−

1

(4) wrt Wt ( ) 11 −=+ ttt i µµβ

(5) wrt Kt 0]11[ 1 =++

− +tt

ttt P

irPµ or

t

t

t

tP

Pri 1

11 +=++

(5) t

t

t

tP

Pri 1

11 +=++ Fisher equation

(1), (2) t

mt

t

tt

ttC

ttx

iii

xcUxcU

t

+

−+=

+=

11

),(),(

µνµ

(1), (4) tt

t

tttc

ttcrP

PiXCu

XCu++

=++

= +++

11

11

),(),( 111 ρρ

(2), (4) )(

)(),(

),( 11111

ttt

ttt

ttX

ttXP

PXCuXCu

νµνµ

++

= +++++

For

tmt ii > , constraint ttt Mxp ≤ is binding (giving a positive demand for money). Substitute tttt pMmx /== in the objective function: optimization problem formally equivalent to max (indirect) utility of real money and credit goods.

80

1.3.4.2 Shopping time model

),( tt mcT transaction technology: real resources used up when tc is consumed with real money balances tm available.

Money reduces transaction costs (shopping time required for purchasing goods). ),( tt mcT with 0>

tcT ; 0<tmT

Transform so as to maximize indirect utility function )),((),( tttt mxfUmxV = .

Consider continuous time problem:

dtetntcuV t

tmtc

ρ−∞∫ −= 0)();(

)](1);([)0(max

Modified budget constraint including transaction costs:

),( ttttttttt mcTPnWcPMi +−+ or:

dtetmtcTtntw

PM

PB

dtetmtrtdtetc

ttr

ttrttr

)(0

0

0

0

0

)(0

)(0´

))](),(()()([

)()]()([)(

−∞

−∞−∞

−∫++

=∫ ++∫ π

Redefine gross consumption as: ))(),(()()( tmtcTtctx += . )(tx increasing in )(tc , invert function to ))(),(()( tmtxftc = .

Transform into indirect utility function: dtetntmtxfuV t

tmtc

ρ−∞∫ −= 0)();(

)](1);(),(([)0(max s.t.

dtetntw

pM

pB

dtetmtrtdtetx

ttr

ttrttr

)(0´

0

0

0

0

)(0

)(0´

])()([

)()]()([)(

−∞

−∞−∞

∫++

=∫ ++∫ π

Formally equivalent to MIU approach. For exogenous labor supply tt yn =

)),(()(),( ttttt mxfUcUmxV == , with xt specified by: ),( tttt mcTcx += .

81

1.3.6 Determination of the Price Level Analyse the following questions:

a) What determines initial price level: money supply? b) Introduce government budget constraint c) Analyse conditions for superneutrality of money (growth

model) d) Conditions for inflationary bubbles (Cagan model) e) Stochastic economy: Analyse dynamic reaction to shocks f) Introduce price rigidities (requires price setting)

Real effects of money are limited as long as prices are flexible. Here: look at determination of initial price level and bubbles Bubbles and self fulfilling inflationary expectations Impose additive separable utility: )()(),( mVCumCu += and normalize 1)( =Cuc Consider isoelastic utility function:

[ ] σαα

σ

1111

)/(1

1),(−−

−== PMC

PMCUU

FOC give as LM curve: tt

t

t

t Ci

iP

M +−=

11αα

More generally, for the CES utility function:

11

1111

1 )1(),(−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−+=

−−

bb

b

bbb

PMC

PMCU αα

FOC give as LM curve: t

b

t

t

t

t Ci

iP

M−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=

11αα

82

Determinacy of the price level Determine the price level for a given money supply process. Dynamic relation, because real money balances depend on the expected price path:

a) Expectation about future monetary policy has impact on current money holding

b) possible: self fulfilling bubbles for price path How to provide a nominal anchor? Note: Fixing nominal rate of interest leaves price level indeterminate (money supply endogenous) FOC for real money balances can be reformulated as:

( )1

111),(1),(),(+

++=−t

ttct

ttmttc PmCu

PmCumCu β

Analyse a steady state for real economy: YCC tt ==+1 (with ρ=r ). 1.3.6.1 Policy of constant money growth Consider tt MM )1(1 µ+=+ . Rewrite FOC as:

( ) 111 ),(1

),(),( ++++=− tttctttmttc mmCummCumCu

µβ

Equilibrium characterised by the difference equation:

(A) )(1

)( 1++= tt mGmF

µβ

with ( )mmYumYumF mc ),(),()( −= and mmYumG c ),()( =

and transversality condition 0)1(

1lim 1 =+

+∞→ T

TTT P

Wr

(B) 0)(lim =∞→

TT

TmGβ

with t

tt

t

tttc

tt

t

PM

rPMmcUmG

)1(1),()(+

== ββ

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For additive separable utility )()(),( mVCumCu += , we have: ( )mmVYumF mc )()()( −= and mYumG c )()( =

A steady state m* with *mmt = and *)(mVm constant, so

*)(1

*)( mGmFµ

β+

= or µ

β+

−=1

1)(*)(

YumV

c

m

exists for ρ

ρβµ+

−=−≥1

1

For ρ

ρµ+

−=1

, mm =* with 0)( =mVm (if satiation level exists)

→ Deflationary price path: ttt PPP )1/(1)1(1 ρµ +=+=+ [zero nominal interest rate]

Note: steady state m* is unstable, because 0)(('' <tmV , so 0)( >∆ tm for m>m*

With forward looking behaviour (rational expectations), the price level 0p and price path tneptp )(

0)( −= µ are pinned down uniquely if the unstable paths can be ruled out as violating some constraints!

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Can we rule out divergent paths? A) Explosive real money paths (deflationary price paths).

...* 10 <<< mmm Real balances are exploding: 0; →∞→ TT Pm cannot be an equilibrium, if m is bounded from above! Transversality constraint rules out deflationary price paths.

0lim)(lim ==∞→∞→

TT

TTT

TmmG ββ

But implicit assumption here: there are no government bonds (See Woodford, chapter 2.4) B) Hyperinflationary paths Self fulfilling hyperinflation: If money is not crucial, the economy converges to a barter equilibrium: 0)(lim =

∞→tm

t

For 0→Tm , 0)()( →= mYumG c and ( ) mmVmmVYumF mmc )(0)()()( −→−=

Key: What happens to ))(()(lim

0)(tmVtm mtm →

?

For ))((')(lim

0)(tmVtm

tm →=0, any path leading to 0lim =

∞→TT

m can

be an equilibrium Bubbles ruled out if 0))((')(lim

0)(>

→tmVtm

tm,

Because 0)(0))(( →<∆ tmfortm violates law of motion: real money growth cannot be strictly negative at m(t)=0. Money must be essential: Marginal utility of money increases faster than the rate at which money goes to zero.

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Specific payoff functions:

A) CES payoff function σ

σ/11

/111)( −

−= mmV :

law of motion: σρµ /11)( −−+= mmm&

10lim)('lim /11

00<>= −

→→σσ formmV

mm

B) Indirect utility for the Cagan demand function:

ibcpMm −=−= lnlnln

bmcmV ln)(' −

= =i >0 for mc ln>

bmcmmV 1ln)( +−

= ; 01)('' <−=mb

mV

mmmVmm

ln)('lim0

−=→

=0

)()()(ln tpbctptM &−=−

For constant rate of money growth:

]ln)([ln1)( 0 tMctpb

tp µ−+=& ,

thus for steady state: )()( 0 tMCeMCtP t == µ (saddle point stability) Self fulfilling explosive price paths might not be ruled out for that case (just the border case). Now: look at alternative at different specifications of an interest rate policy.

86

1.3.6.2 Policy of a pure interest rate peg (Sargent/Wallace) Fix nominal interest rate: { } tuii tt ∀+= with 0)( =tuE Money supply is endogenous a) All real variables are well specified: Real balances are

uniquely determined by: i

imVm +=

1)(

b) Rate of inflation is uniquely determined (Fisher equation): riEEptpE tt −==−+ )()()( 1 π

No speculative bubble for the price path. But: Initial price level P0 is not specified - there is nominal indeterminacy:

If 0m (with 0P ) is a solution to i

imVm +=

1)( 0 , then also any

0mλ (with 0Pλ ) Difference equation: [ ] constantr-i)( 1 ==−+ tt ptpE not sufficient to pin down the initial price level: If tPln is a solution (with tMln ), then also α+tPln with α+tMln Key Lesson: Pure interest rate peg leaves price level indeterminate In contrast, feedback interest rate rules are able to control price level: 1.3.6.2 Interest Rate Feedback Rules Example: Inflation Targeting/ or Price Level Targeting Announced (Credible) Central Bank Rule:

ttt ufii +−+= *)( ππ Under what conditions do we get a determinate price level? Expected interest rate: tttttt uEEfiiE 111 *)( −−− +−+= ππ Implied dynamics of inflation, using Fisher equation

)()( 111 +−− += tttt EriE π :

87

)(*)( 1111 tttttt uEEfriE −−+− +−+−= πππ First order difference equation for inflation. Forward Looking Solution:

( )))(1* 1 riEf tt −−+= +πππ

Analyse perfect foresight solution, using 0)(*; 1 ==− − tt uEri π : ( ))**1 ππππ −=−+ tt f

*)(1* ππππ −+= +StSt f

Uniquely determined dynamics of inflation only if f>1! In that case, always reversal to π* after deviations from the inflation target: *ππ →t ! For f<1, infinite number of solutions depending on expected inflation )( TtE +π

For f>1, the price level is also uniquely determined (using

1−−= ttt PPπ , we get the second order difference equation, *)(* 11 ππ −−+=− ++ tttt PPfPP