Monetary Economics
Basic Flexible Price Models
Nicola Viegi
April 16, 2014
Modelling Money
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Cagan Model - The Price of Money
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A Modern Classical Model (Without Money)
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Money in Utility Function Approach
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Cash in Advance Models (not really)
The Cagan Model
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Originally a model of Hyperin‡ation
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Main Point: Expectation about future fundamentals
determine prices now
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Used extensively in Exchange rate analysis, assets prices
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A lot of modern models look like this
The Model
Money Demand
Mtd
= L (Yt , it )
Pt
Fisher Equation
(1 + it ) = (1 + rt )
Pt +1
Pt
log approximation and uncertainty
log (1 + it ) = log (1 + rt ) + Et pt +1
pt
Assume Yt and rt …xed, the money demand equation (in log):
mtd
pt =
η (Et pt +1
pt )
mt
pt =
η (Et pt +1
pt )
pt =
η
1
mt +
Et pt +1
1+η
1+η
or
Forward Solution
pt
=
=
1
η
mt +
Et pt +1
1+η
1+η
1
η
1
η
mt +
Et
mt + 1 +
Et +1 pt +2
1+η
1+η
1+η
1+η
2
1
η
η
mt +
Et m t + 1 +
Et pt +2
1+η
1+η
1+η
= .........
s
T
1 T 1
η
η
m
+
Et pt +T
=
t
+
s
1 + η s∑
1+η
1+η
=0
=
(1)
Law of Iterated Expectations
Et (Et +1 pt +2 ) = Et pt +2
No Bubble
lim
T !∞
1
T !∞ 1 + η
lim
η
1+η
T
1
∑
s =0
T
Et pt +T = 0
η
1+η
s
mt + s < ∞
Solution
Generic Solution
pt =
1
1+η
∞
∑
s =0
η
1+η
s
mt + s
p
et
Prices at time t depend on all future expected money supply.
Expected ”fundamentals” (not actual)
Constant Money Supply
mt
m
Et m t + s = m
pt
=
=
(s
0)
∞
1
η
∑
1 + η s =0 1 + η
1
1
m
1 + η 1 1 +η η
= m
s
m
Future One Time Money Increase at T>0
Consider an surprise announcement at time t = 0 of a change of
policy at some time in the future T > 0
mt =
m0
m
t<T
>m t T
Thus
pt =
m t<T
m0 t T
How would the transition look like?
Solution
In the period between 0
pt
pt
pt
=
1
1+η
= m+
= m+
∞
∑
s =0
t
η
1+η
1
1+η
η
1+η
η
1+η
T
T
s
m+
T
1
1+η
t ∞
∑
s =0
t
m0
m
∞
∑
s =T
η
1+η
s
η
1+η
t
m0
s
m0
m
m
log price level
Solution
m’
p0
m
0
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T
time
Prices "Jump" at the announcement, before the policy is
implemented
Expectations drive economic dynamics
A Classical Monetary Model (Gali Ch. 1)
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Why Classical?
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Perfect Competition
Fully Flexible Prices
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Real Business Cycle Structure + Nominal Prices
Determination
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Three Agents: Households, Firms, Central Bank (setting
nominal interest rate)
Households
Representative household solves
∞
max E0
∑ βt U (Ct , Nt )
t =0
subject to
Pt Ct + Qt Bt
Bt
1
+ Wt Nt + Dt
for t = 0, 1, 2, .... plus solvency constrains
Optimality Conditions
Un,t +
Wt
Uc ,t = 0
Pt
Qt
Uc ,t = βEt
Pt
Uc ,t +1
1
Pt +1
Households
Speci…cation of Utility
U (Ct , Nt ) =
Ct1 σ
1 σ
1 ϕ
Nt
1+ϕ
which gives the log-linear optimality conditions (aggregate
variables)
wt pt = σct + ϕnt
1
(it Et fπ t +1 g ρ)
σ
where it = log Qt is the nominal interest rate, ρ = log β is the
discount raate, and π t +1 = pt +1 pt is the in‡ation rate
c t = Et f c t + 1 g
Firms
Pro…t Maximization
max Pt Yt
Wt Nt
subject to
Yt = At Nt1
α
Optimality Conditions
Wt
= (1
Pt
α) At Nt
log linear version (ignoring constants)
wt
pt = at
αnt
α
Equlibrium
Equlibrium Value of Real Varialbes
yt = c t
Labour Market
σct + ϕnt = at
αnt
Asset market Clearing
bt
= 0
yt
= Et fyt +1 g
1
(it
σ
Et f π t + 1 g
Aggregate Production relationship
yt = at + (1
α) nt
ρ)
Equlibrium
Equilibrium Values of Real Variables
nt =
rt
1+ϕ
1 σ
a t ; yt =
at
σ (1 α ) + ϕ + α
σ (1 α ) + ϕ + α
σ+ϕ
at
wt pt =
σ (1 α ) + ϕ + α
σ (1 + ϕ ) (1 ρa )
at
it Et fπ t +1 g = ρ
σ (1 α ) + ϕ + α
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Real varialbes determined independently of monetary policy
(neutrality)
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Optimal Policy Undetermined
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Speci…cation of Monetary Policy Needed to determine
nominal variables
Monetary Policy and Price Level Determination
A Simple Interest Rate Rule
it = ρ + φπ π t
combined with the de…nition of real rate rt
φπ π t
it
Et f π t + 1 g ,
Et f π t + 1 g + b
rt
if φπ > 1, unique stationary solution
∞
πt
∑ φπ (k +1) fbrt +k g
k =0
if φπ < 1, any process π t satisfying
π t +1
φπ π t
b
rt + ξ t +1
where Et fξ t +1 g = 0 for all t is consistent with a stationary
equlibrium (Price Level Indeterminacy)
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Illustration of the "Taylor Principle"
Monetary Policy and Price Level Determination
Exogenous Path of Money Supply (Cagan Again)
mt
pt = yt
ηit
Combining Money Demand and Fisher Equation
rt
it Et fπ t +1 g ,
pt =
η
1+η
Et fpt +1 g +
1
1+η
mt + ut
where ut = 1 +1 η (ηrt yt ) evolves independently of m.
Assuming η > 1 and solving forward
pt =
where ut = ∑k∞=0
0
1
1+η
η
1 +η
∞
∑
k =0
k
η
1+η
Et fut +k g
k
0
Et fmt +k g + ut
Monetary Policy and Price Level Determination
Exogenous Path of Money Supply (Cagan Again)
Rewrite in term of expected future money growth rates
pt = mt +
1
1+η
∞
∑
k =1
η
1+η
k
0
Et f∆mt +k g + ut
which implies the following nominal interest rate:
it
= η
1
it
= η
1
where υt
η
1
h
[yt
1
1+η
( mt
∞
∑
k =1
pt )]
η
1+η
k
Et f∆mt +k g + υt
i
0
yt + ut is independent of policy
Money in The Utility Function
Money in the Utility function is just a modelling trick to determine
money demand inside the private sector optimization
The household’s optimization problem (with …xed labour supply)
∞
max
∑ βi u
fC t ,M t ,K t +1 gt∞=0 t =0
Ct ,
Mt
Pt
subject to the real budget constraint
Kt + 1 +
Mt
Mt 1
= (1 + rt ) Kt +
+ wt
Pt
Pt
Ct
Tt
Production is given by a neoclassical production function with
constant return to scale
Yt = F (Kt , Lt )
Equilibrium Conditions
uc
uc
Mt
Pt
Mt
Ct ,
Pt
Ct ,
= β (1 + rt +1 ) uc
Ct +1 ,
Mt
Pt
+ βuc
= uM /P
Ct ,
Mt +1
Pt +1
(2)
Ct +1 ,
Mt +1
Pt +1
Pt
(3)
Pt +1
Substituting for βuc (Ct +1 , Mt +1 /Pt +1 ) from (2) in (3) and
rearrange, we get:
uc
Ct ,
Mt
Pt
1
1
Pt
(1 + rt +1 ) Pt +1
= uM /P
Ct ,
Mt
Pt
(4)
Equilibrium Conditions
Remember the Fisher Equation that relates real and nominal
interest rates
(1 + it ) = Et (1 + rt +1 )
Pt +1
Pt
under perfect foresight, (4) can than be written as:
it
= uM /P
1 + it
LM Equation
Ct ,
Mt
Pt
/uc
Ct ,
Mt
Pt
(5)
General Equilibrium in MIU
Need to Specify:
Government Bahaviour
Ts =
Ms
Ms
Ps
1
Factor prices
rt
wt
=
∂F (K , 1)
∂K
= F (K , 1)
Kt
∂F (K , 1)
∂K
where the second relation uses the property of constant return to
scale (F = L∂F /∂L + K ∂F /∂K )
Steady State
To be consistent with the data a model including money should
have the following
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Neutrality of Money: the real equilibrium is independent of
the money stock
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Superneutrality of money: the real equilibrium is independent
of the money growth rate
Nominal Equilibrium
=
Pt /Pt 1 =
Ct /Ct 1 =
Mt /Pt /Mt 1 /Pt 1 =
π =
Mt /Mt
1
1+σ
1+π
1
1
σ
Real Equlibrium
from
uc
Ct ,
Mt
Pt
= β (1 + rt +1 ) uc
Ct +1 ,
Mt +1
Pt +1
In steady state
uc
M
P
C,
= β (1 + rt +1 ) uc
C,
M
P
(1 + r ) = 1/β
Combining with rt =
stock must solve
∂F (K ,1 )
,,
∂K
gives that the steady state capital
FK (K , 1) = r = 1/β
1
which depends only on the technology and the real discount rate,
not on the money stock or growth. Because capital stock is
constant, so consumption in steady state is uniquely determined by
the real side of the economy
Monetary Equilibrium
In steady state, in this model, money is neutral and superneutral
With a value for the steady state consumption, we can solve for a
value of the steady state real money balances by combining the
Fisher equation
(1 + i ) = Et (1 + r ) Pt +1 /Pt
( 1 + i ) = Et ( 1 + r ) ( 1 + π )
and
i
= uM /P
1+i
C,
M
P
/uc
C,
M
P
would give and equation in steady state real money and of known
parameters
Optimal Rate of In‡ation
Friedman (1969)
Social Marginal cost of producing money = 0
Private marginal cost of money = i
= SMC
i = 0
PMC
( 1 + i ) = Et ( 1 + r )
= r +π = 0
π =
r
i
De‡ation
Pt +1
Pt
Cost Of In‡ation
Money Demand Function
i
= uM /P
1+i
C,
M
P
/uc
C,
M
P
Conclusions
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Basic Flexible Price Models Highlight:
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The Role of Expectations
The Importance of Policy Design for Price Determination
Cost of In‡ation (even when we have neutrality of money)
Problems:
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Money has no e¤ect on real variables (only welfare)
Friedman (1968) not realistic (De‡ation is costly, but why?)