The Classical Monetary Model - CREI...The Classical Monetary Model Jordi Galí CREI, UPF and...
Transcript of The Classical Monetary Model - CREI...The Classical Monetary Model Jordi Galí CREI, UPF and...
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 2 / 22
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 .
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 3 / 22
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 4 / 22
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 5 / 22
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)
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 7 / 22
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 8 / 22
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 9 / 22
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
Price Level Determination
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 11 / 22
Price Level Determination
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 12 / 22
Price Level Determination
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 13 / 22
Price Level Determination
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.
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 14 / 22
Price Level Determination
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 15 / 22
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 16 / 22
The Case of Non-Separable Real Balances
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)]
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 17 / 22
The Case of Non-Separable Real Balances
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)
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 18 / 22
The Case of Non-Separable Real Balances
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 19 / 22
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 20 / 22
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
Jordi Galí (CREI, UPF and Barcelona GSE) Classical Monetary Model May 2018 22 / 22