the classical monetary model - crei...the classical monetary model jordi galí crei, upf and...
Post on 25-Apr-2020
9 Views
Preview:
TRANSCRIPT
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
top related