The Classical Monetary Model - CREI...The Classical Monetary Model Jordi Galí CREI, UPF and Barcelona GSE May 2018 Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model

The Classical Monetary Model

Jordi Galí

CREI, UPF and Barcelona GSE

May 2018

Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 1 / 22

Assumptions

Perfect competition in goods and labor markets

Flexible prices and wages

Representative household

Money in the utility function

No capital accumulation

No �scal sector

Closed economy


Households

Preferences

E0∞

∑t=0

βtU�Ct ,

Mt

Pt,Nt

�Budget constraint

PtCt +QtBt +Mt � Bt�1 +Mt�1 +WtNt +Dt � Tt

with solvency constraint:

limT!∞

Et fΛt ,T (AT /PT )g � 0

where Qt � expf�itg and At � Bt +Mt .


Households

Optimality Conditions

�Un,tUc ,t

=Wt

Pt

Qt = βEt

�Uc ,t+1Uc ,t

PtPt+1

�Um,tUc ,t

= 1�Qt

Interpretation: 1�Qt = 1� expf�itg ' it

) opportunity cost of holding money


Households

Assumption:

U�Ct ,

Mt

Pt,Nt

�=

8<:C 1�σt �11�σ � N 1+ϕ

t1+ϕ + χ (Mt/Pt )1�σ�1

1�σ for σ 6= 1logCt � N 1+ϕ

t1+ϕ + χ log Mt

Ptfor σ = 1

Remark: separable real balances assumed

Implied optimality conditions

Wt

Pt= C σ

t Nϕt

Qt = βEt

(�Ct+1Ct

��σ � PtPt+1

�)Mt

Pt= χ

1σCt (1� expf�itg)�

1σ


Households

Implied optimality conditions

Wt

Pt= C σ

t Nϕt

Qt = βEt

(�Ct+1Ct

��σ � PtPt+1

�)Mt

Pt= χ

1σCt (1� expf�itg)�

1σ

Log-linear versions:wt � pt = σct + ϕnt

ct = Etfct+1g �1σ(it � Etfπt+1g � ρ)

mt � pt = ct � ηit

where πt � pt � pt�1 and β � expf�ρgJordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 6 / 22

Firms

TechnologyYt = AtN1�α

t (1)

where at � logAt follows an exogenous process

at = ρaat�1 + εat

Pro�t maximization:max PtYt �WtNt

subject to (1), taking the price and wage as given

Optimality condition:

Wt

Pt= (1� α)AtN�α

t

Log linear version

wt � pt = at � αnt + log(1� α)


Equilibrium

Goods market clearingyt = ct

Labor market clearing

σct + ϕnt = wt � pt = at � αnt + log(1� α)

Asset market clearingBt = 0

Mt = MSt

Aggregate output:yt = at + (1� α)nt


Equilibrium

Equilibrium values for real variables

nt = ψnaat + ψn

yt = ψyaat + ψy

rt = ρ� σψya(1� ρa)at

ωt � wt � pt = ψωaat + ψω

where ψna � 1�σσ(1�α)+ϕ+α

; ψn �log(1�α)

σ(1�α)+ϕ+α; ψya �

1+ϕσ(1�α)+ϕ+α

;

ψy � (1� α)ψn ; ψωa �σ+ϕ

σ(1�α)+ϕ+α; ψω �

(σ(1�α)+ϕ) log(1�α)σ(1�α)+ϕ+α

Neutrality : real variables independent of monetary policy

Monetary policy needed to determine nominal variables


Price Level Determination

Example I: A Simple Interest Rate Rule

it = ρ+ π + φπ(πt � π) + vt

where φπ � 0. Combined with de�nition of real rate:

φπbπt = Etfbπt+1g+brt � vtCase I: φπ > 1

bπt =∞

∑k=0

φ�(k+1)π Etfbrt+k � vt+kg= �

σ(1� ρa)ψyaφπ � ρa

at �1

φπ � ρvvt

=) nominal determinacyJordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 10 / 22


Case II: φπ < 1

bπt = φπbπt�1 �brt�1 + vt�1 + ξt

for any fξtg sequence with Etfξt+1g = 0 for all t

) nominal indeterminacy

) illustration of "Taylor principle" requirement



Responses to a monetary policy shock (φπ > 1 case):

∂πt∂εvt

= � 1φπ � ρv

< 0

∂it∂εvt

= � ρvφπ � ρv

< 0

∂mt∂εvt

=ηρv � 1φπ � ρv

7 0

∂yt∂εvt

= 0

Discussion: liquidity e¤ect and price response



Example II: An Exogenous Path for the Money Supply fmtg

Combining money demand and the de�nition of the real rate:

pt =�

η

1+ η

�Etfpt+1g+

�1

1+ η

�mt + ut

where ut � (1+ η)�1(ηrt � yt ). Solving forward:

pt =1

1+ η

∞

∑k=0

�η

1+ η

�kEt fmt+kg+ ut

where ut � ∑∞k=0

�η1+η

�kEtfut+kg

) price level determinacy



In terms of money growth rates:

pt = mt +∞

∑k=1

�η

1+ η

�kEt f∆mt+kg+ ut

Nominal interest rate:

it = η�1 (yt � (mt � pt ))

= η�1∞

∑k=1

�η

1+ η

�kEt f∆mt+kg+ ut

where ut � η�1(ut + yt ) is independent of monetary policy.



Assumption∆mt = ρm∆mt�1 + εmt

rt = yt = 0

Price response:

pt = mt +ηρm

1+ η(1� ρm)∆mt

) large price response

Nominal interest rate response:

it =ρm

1+ η(1� ρm)∆mt

) no liquidity e¤ect


The Case of Non-Separable Real Balances

Labor supply a¤ected by monetary policy ) non-neutrality

Example:

U (Xt ,Nt ) =X 1�σt � 11� σ

� N1+ϕt

1+ ϕ

where

Xt �"(1� ϑ)C 1�ν

t + ϑ

�Mt

Pt

�1�ν# 11�v

for ν 6= 1

� C 1�ϑt

�Mt

Pt

�ϑ

for ν = 1



Optimality conditions:

Wt

Pt= Nϕ

t Xσ�νt C ν

t (1� ϑ)�1

Qt = βEt

(�Ct+1Ct

��ν �Xt+1Xt

�ν�σ � PtPt+1

�)

Mt

Pt= Ct (1� expf�itg)�

1ν

�ϑ

1� ϑ

� 1ν

Log-linearized money demand equation:

mt � pt = ct � ηit

where η � 1/[ν(expfig � 1)]



Log-linearized labor supply equation:

wt � pt = σct + ϕnt + (ν� σ)(ct � xt )= σct + ϕnt + χ(ν� σ) (ct � (mt � pt ))= σct + ϕnt + ηχ(ν� σ)it

where χ � km (1�β)1+km (1�β)

2 [0, 1) with km � M/PC =

�ϑ

(1�β)(1�ϑ)

� 1ν

Equivalently,wt � pt = σct + ϕnt +vit

where v � kmβ(1� σν )

1+km (1�β)



Labor market clearing:

σct + ϕnt +vit = at � αnt + log(1� α)

which combined with aggregate production function:

yt = ψyaat + ψyi it

where ψyi � �v(1�α)

σ(1�α)+ϕ+αand ψya �

1+ϕσ(1�α)+ϕ+α

) long run non-superneutrality


Assessment of size of short-run non-neutralities

Calibration: β = 0.99 ; σ = 1 ; ϕ = 5 ; α = 1/4 ; ν = 1/ηi "large"

) v ' kmβ

1+ km(1� β)> 0 ; ψyi ' �

km8< 0

Monetary base inverse velocity: km ' 0.3 ) ψyi ' �0.04M2 inverse velocity: km ' 3 ) ψyi ' �0.4

) small output e¤ects of monetary policy


Optimal Monetary Policy

Social Planner�s problem

maxU�Ct ,

Mt

Pt,Nt

�subject to

Ct = AtN1�αt

Optimality conditions:

�Un,tUc ,t

= (1� α)AtN�αt

Um,t = 0

Optimal policy (Friedman rule): it = 0 for all t

Intuition

Implied average in�ation: π = �ρ < 0Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 21 / 22

Implementationit = φ(rt�1 + πt )

for some φ > 1. Combined with the de�nition of the real rate:

Etfit+1g = φit

whose only stationary solution is it = 0 for all t.

Implied equilibrium in�ation:

πt = �rt�1


Documents

The Classical Monetary Model - CREI...The Classical Monetary Model Jordi Galí CREI, UPF and Barcelona GSE May 2018 Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model