Download Money in the utility model

Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Money in the utility model
Michaª Brzoza-Brzezina
Warsaw School of Economics
1 / 50
Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Plan of the Presentation
1
Motivation
2
Model
3
Steady state
4
Short run dynamics
5
Simulations
6
Extensions
7
Conclusions
2 / 50
Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Plan of the Presentation
1
Motivation
2
Model
3
Steady state
4
Short run dynamics
5
Simulations
6
Extensions
7
Conclusions
3 / 50
Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Introduction
The neoclassical growth model by Ramsey (1928) and Solow
(1956) provides the basic framework for modern
macroeconomics.
But these are models of nonmonetary economies.
In order to explain monetary policy we have to introduce a
monetary policy instrument.
Sidrauski (1967) introduced money into the neoclassical
growth model.
At this time money was considered the monetary policy
instrument.
The model derives from Ramsey (1928): utility maximization
by the representative agent
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Motivation
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Steady state
Short run dynamics
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Extensions
Conclusions
Introduction (cont'd)
How can we introduce money?
- nd explicit role for money (e.g. money is necessary to make
transactions - Cash in Advancec (CIA) approach, Clower
(1967))
- assume money yields utility (Money in the Utility Function
(MIU) - Sidrauski (1967))
- treat money as asset allowing to shift resources
intertemporally (Samuelson (1958))
MIU is simple though not very appealing. Can be rationalized
by transaction dedmand for money.
The model allows us studying:
- impact of money (i.e. monetary policy) on the real economy,
- impact of money on prices,
- optimal rate of ination.
Derivation follows (approximately) Walsh (2003)
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Plan of the Presentation
1
Motivation
2
Model
3
Steady state
4
Short run dynamics
5
Simulations
6
Extensions
7
Conclusions
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Household's problem - objective
Suppose the utility function of the representative household is:
Ut = u (ct , mt )
(1)
where ct = NCtt and mt = PMt Nt t
and utility is increasing and strictly concave in both arguments.
The representative household seeks to maximize lifetime utility:
∞
U = E0 ∑ βt u (ct , mt )
t =0
(2)
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Constraint
subject to budget constraint:
it −1 Bt −1 Mt −1
+
Pt
Pt
Mt Bt
Ct + Kt + +
Pt Pt
Yt + τNt + (1 − δ)Kt −1 +
and the production function:
=
(3)
Yt = F (Kt −1 , Nt ) or in per capita form (assuming constant
population N and production function with constant returns to
scale):
yt = f (kt −1 )
This means that households earn income which can be spent on
consumption, invested as capital, saved as bonds or kept as money.
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Motivation
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Simulations
Extensions
Conclusions
Constraint per capita
We can substitute production function and rewrite the budget
constraint in per capita form:
=
f (kt −1 ) + τt + (1 − δ)kt −1 +
ct + kt + mt + bt
it −1 bt −1 + mt −1
πt
(4)
where πt ≡ Pt /Pt −1 .
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Motivation
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Extensions
Conclusions
Lagrangian
How can this problem be solved?
Bellman equation
Lagrangian.
Walsh proposes the former. We choose the latter:
∞
max E0 ∑ βt u (ct , mt ) + λt [f (kt −1 )
c ,m ,k , b
t =0
+τt + (1 − δ)kt −1 +
−ct − kt − mt − bt ]
it −1 bt −1 + mt −1
πt
(5)
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
First order conditions
First order conditions for this problem are:
ct :
kt :
mt :
bt :
βt u 0 (ct ) = λt
(6)
−λt + Et λt +1 [f 0 (kt ) + (1 − δ)] = 0
(7)
βt u 0 (mt ) − λt + Et λt +1
−λt + Et λt +1
it
π t +1
1
π t +1
=0
=0
(8)
(9)
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Motivation
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Simulations
Extensions
Conclusions
Equilibrium conditions - Euler equation
FOCs can be transformed into formulae describing allocation
choices in equilibrium.
Substitute (6) into (9):
u 0 (ct ) = βEt u 0 (ct +1 )
it
π t +1
(10)
This is the Euler equation - key intertemporal condition in general
equilibrium models. It determines allocation over time.
Utility lost from consumption today equals utility from consuming
tomorrow adjusted for the (real) gain from keeping bonds.
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Conclusions
Equilibrium conditions - capital
Substitute (9) into (7):
Et u 0 (ct +1 )[f 0 (kt ) + (1 − δ)] = Et u 0 (ct +1 )
it
π t +1
(11)
Dene real interest rate (Fisher equation):
rt ≡ Et
it
π t +1
(12)
The marginal product of capital (net of depreciation) equals the
real interest rate.
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Motivation
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Extensions
Conclusions
Equilibrium conditions - Euler for money
Substitute (6) into (8):
u 0 (mt ) + βEt u 0 (ct +1 )
1
π t +1
= u 0 (ct )
(13)
Lost utility from consumption today equals utility from money
today and additional utility from consumption tomorrow.
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Motivation
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Short run dynamics
Simulations
Extensions
Conclusions
Equilibrium conditions - Opportunity cost of money
Merge (6), (9) and (8):
βt u 0 (mt ) − βt u 0 (ct ) +
βt u 0 (ct )
it
Divide by βt and rearrange:
u 0 (mt )
u 0 (ct )
= 1−
1
it
=0
(14)
This equates the marginal rate of substitution between money and
consumption to their relative price
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Motivation
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Conclusions
Further equations
In order to solve the model we have to assume a production and
utility function.
Production function:
yt = e z ktα−1
(15)
z t = ρ z z t − 1 + ε z ,t
(16)
t
where
is a stochastic TFP shock.
Utility function:
Ut = ln ct + ln mt
(17)
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Motivation
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Conclusions
Monetary policy
Monetary policy is very simple. We assume that money follows a
stochastic process:
Mt = e θ Mt −1
t
which yields:
mt =
where:
mt −1
πt
eθ
t
θt = ρθ θt −1 + ε θ,t
(18)
(19)
Is a stochastic money supply shok.
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Extensions
Conclusions
Budget constraint
Assume that bonds are in zero net supply:
bt = 0
and that the government budget is balanced every period:
N τt Pt = Mt − Mt −1
or:
τt = mt −
Substitute these into (4) to get:
mt −1
πt
yt + (1 − δ)kt −1 = ct + kt
(20)
(21)
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Motivation
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Conclusions
The model
We now have 9 equations:
(10), (11), (12), (14), (15), (16), (18), (19), (21)
to solve for 9 endogeneous variables:
yt , kt , mt , ct , it , rt , πt , zt , θt .
This will be done in two steps:
1 Find the model's steady state,
2 Derive log-linear approximation arround it.
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Plan of the Presentation
1
Motivation
2
Model
3
Steady state
4
Short run dynamics
5
Simulations
6
Extensions
7
Conclusions
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Motivation
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Steady state
Short run dynamics
Simulations
Extensions
Conclusions
The steady state
The steady state is where the model economy converges to
(stabilizes) in the absence of shocks.
In our model there is no population or productivity growth, so in
the steady state output, consumption etc. will be constant.
How do we solve for the steady state?
1 assume shocks are zero,
2 drop time indices (xt −1 = xt = x ss ).
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Motivation
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Steady state
Short run dynamics
Simulations
Extensions
Conclusions
The steady state - capital
The steady state is where the model economy converges to
(stabilizes) in the absence of shocks.
In our model there is no population or productivity growth, so in
the steady state output, consumption etc. will be constant.
From (11), (12) and (15) we have:
α (k ss )α−1 =
1
β
−1+δ
or
α−1 1
1 1
ss
k =
−1+δ
α β
(22)
This determines capital (and output) in the SS.
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Motivation
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Short run dynamics
Simulations
Extensions
Conclusions
The steady state - consumption
Take (21) and (15). This yields:
c ss = (k ss )α − δk ss
(23)
Since k ss is given by (22), this determines steady state consumption.
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Extensions
Conclusions
The steady state - money
Take (14) and (10). The former yields:
mss = c ss
The latter
i ss
i ss − 1
π ss = βi ss
so that:
mss = c ss
π ss
π ss − β
(24)
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Motivation
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Extensions
Conclusions
Neutrality of money
Denition of (long run) neutrality: in the long run real
variables do not depend on money nor ination.
This is true for all variables but real money holdings.
But the latter are usualy excluded from the denition.
So, in the MIU money is neutral in the long run.
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Plan of the Presentation
1
Motivation
2
Model
3
Steady state
4
Short run dynamics
5
Simulations
6
Extensions
7
Conclusions
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Short run dynamics
Steady state tells us about the long run
But most interesting in most applications are short run
dynamics
We have a set of dierence equations that describe it
But they are non-linear: dicult to solve, to estimate etc.
The traditional way of analyzing the dynamics of a model is to
assume that the economy is always in the viscinity of the
steady state
Then we take a log-linear approximation around the steady
state which is relatively simple to handle
The procedure of log-linearization is explained in Walsh (p.
85-87)
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Log-linearization
Log-linear approximation. Dene x̂t ≡ ln xxsst as log-deviation
from steady state.
It follows that:
xt = x ss e xˆ
t
' x ss e 0 + x ss e 0 (xˆt − 0) = x ss (1 + xˆt )
This can be used to derive the approximation. Some useful tricks:
xt yt
xt
yt
xta
ln xt
=
=
=
=
x ss y ss (1 + x̂t + ŷt )
x ss
(1 + x̂t − ŷt )
y ss
(x ss )a (1 + ax̂t )
ln x + x̂t
(25)
(26)
(27)
(28)
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Motivation
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Conclusions
LL Euler
ct−1 = βEt
rt
ct +1
r ss
(c ss )−1 (1 − ctˆ ) = β ss Et (1 + r̂t − ĉt +1 )
c
Steady state;
Divide:
r ss
(c ss )−1 = β ss
c
(29)
cˆt = Et (ĉt +1 − r̂t )
(30)
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Motivation
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Extensions
Conclusions
LL budget constraint
yt + (1 − δ)kt −1 = ct + kt
y ss (1 + ŷt ) + (1 − δ)k ss (1 + k̂t −1 ) = c ss (1 + ĉt ) + k ss (1 + k̂t )
Steady state:
y ss + (1 − δ)k ss = c ss + k ss
Substract:
y ss ŷt + (1 − δ)k ss k̂t −1 = c ss ĉt + k ss k̂t
Rearrange:
k̂t = (1 − δ)k̂t −1 +
y ss
c ss
ŷ
−
ĉ
t
k ss
k ss t
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Motivation
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Conclusions
LL production function
yt = e z ktα−1
t
ln yt
ss
ln y + ŷt
y ss
= zt + α ln kt −1
= zt + α ln k ss + αk̂t −1
= (k ss )α
ŷt = αk̂t −1 + zt
(31)
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Motivation
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Extensions
Conclusions
LL Fisher
rt ≡ Et
it
π t +1
r̂t ≡ ı̂t − Et π̂t +1
(32)
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Motivation
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Extensions
Conclusions
LL money rule (money supply)
mt =
mt −1
πt
eθ
t
ln mss + m̂t = ln mss + m̂t −1 − ln π ss − π̂t + θt
m̂t
=
m̂t −1 − π̂t + θt
(33)
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Motivation
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Simulations
Extensions
Conclusions
LL opportunity cost (money demand)
ct
mt
Steady state:
= 1−
1
it
c ss
1
(1 + ĉt − m̂t ) = 1 − ss (1 − ît )
ss
m
i
c ss
mss
=
i ss − 1
i ss
ĉt − m̂t =
ît
i ss − 1
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Motivation
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Extensions
Conclusions
LL equilibrium condition for capital
1
z
t +1 α − 1
Et
[αe kt + (1 − δ)] = Et
r
ct +1
ct +1 t
Denote xt ≡ αe zt ktα−−11 + (1 − δ) = α kty−t 1 + 1 − δ
1
Et
1
ct +1
xt +1
= Et
1
ct +1
rt
r ss
x ss
Et (1 − ĉt +1 + x̂t +1 ) = ss Et [1 − ĉt +1 + r̂t ]
ss
c
c
ss
ss
Divide by steady state xc ss = cr ss
Et x̂t +1 = r̂t
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Motivation
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Extensions
Conclusions
LL equilibrium condition for capital cont'd
Now go back to xt ≡ α kty−t 1 + 1 − δ
x ss Et (1 + x̂t +1 ) = α
substitute for Et x̂t +1 and x ss
r ss Et (1 + r̂t ) = α
substract steady state
y ss
E (1 + ŷt +1 − kˆt ) + 1 − δ
k ss t
y ss
E (1 + ŷt +1 − kˆt ) + 1 − δ
k ss t
r ss = α
r ss r̂t = α
y ss
k ss
+1−δ
y ss
E (ŷ − k̂ )
k ss t t +1 t
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Motivation
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Extensions
Conclusions
LL conclusions
Again, we have 9 equations in 9 variables,
We have some additional parameters (r ss , π ss ,i ss ) that have
to be calculated,
From (29): r ss = β−1
From (18): π ss = 1
From (12): i ss = π ss r ss = β−1
Now we can solve the model e.g. in DYNARE and check its
properties.
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Plan of the Presentation
1
Motivation
2
Model
3
Steady state
4
Short run dynamics
5
Simulations
6
Extensions
7
Conclusions
38 / 50
Motivation
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Short run dynamics
Simulations
Extensions
Conclusions
Simulations in DYNARE
Simulations will be done in DYNARE
What is DYNARE?
Software (free :-)) for solving, simulating and estimating
general equilibrium models
www.dynare.org
But requires a computing platform: MATLAB (expensive) or
Octave (free)
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Conclusions
Parameters
We have to assume parameter values. These are taken from the
literature
β = .99 (Note that in steady state this is the reciprocal of the
(annualised) real interest rate of 4%)
α = .33 (Capital share)
δ = .025 (Depreciation approx. 10% on annual basis)
ρz = ρθ = .9 (Some inertia)
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Conclusions
DYNARE code for MIU
14.12.11 14:57
F:\Wyklady\Monetary_Theory\MIU\Dynare\MIU.mod
1 of 1
// MIU model, basic setting
var y, c, k, m, i, r, pi, z, theta;
varexo e_z, e_t;
parameters alpha, beta, delta, rho_z, rho_t, y_ss, k_ss, c_ss, r_ss, i_ss, pi_ss;
alpha = 0.33;
delta = 0.025;
beta=0.99;
rho_z
= 0.9;
rho_t = 0.9;
k_ss=(1/alpha*(1/beta-1+delta))^(1/(alpha-1));
y_ss=k_ss^alpha;
c_ss=y_ss-delta*k_ss;
r_ss=1/beta;
i_ss=r_ss;
pi_ss=1;
model;
c=c(+1)-r;
k=(1-delta)*k(-1)+(y_ss/k_ss)*y-(c_ss/k_ss)*c;
y=alpha*k(-1)+z;
i=pi(+1)+r;
m=m(-1)+theta-pi;
m=c-(1/(i_ss-1))*i;
r_ss*r=alpha*(y_ss/k_ss)*(y(+1)-k);
z=rho_z*z(-1)+e_z;
theta=rho_t*theta(-1)+e_t;
end;
initval;
y = 0;
c = 0;
k = 0;
m = 0;
i = 0;
r = 0;
pi = 0;
z = 0;
theta = 0;
end;
shocks;
var e_z; stderr 0.01;
var e_t; stderr 0.01;
//var e_z, e_t = phi*0.009*0.009;
end;
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Conclusions
IRFs to a productivity shock
−3
y
0.02
4
0.01
2
0
10
−3
4
x 10
20
m
30
40
5
c
k
0.01
0.005
10
−4
2
0
0
x 10
x 10
20
r
30
40
20
z
30
40
10
20
30
40
−5
10
−3
5
0
10
0
x 10
20
pi
30
40
20
30
40
0
10
20
30
40
−5
10
0.02
0.01
0
42 / 50
Motivation
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Conclusions
IRFs to a money supply shock
−3
m
0
1
−0.02
0.8
−0.04
0.6
−0.06
0.4
−0.08
0.2
−0.1
10
20
30
40
0
i
x 10
10
pi
20
30
40
30
40
theta
0.1
0.015
0.08
0.01
0.06
0.04
0.005
0.02
0
10
20
30
40
0
10
20
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Motivation
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Conclusions
IRFs - conclusions
Productivity shocks have an impact on the real economy and
drive the business cycle in the MIU model
Monetary shocks have an impact on nominal variables:
ination and nominal interest rate
But monetary shocks aect only real money balances, but no
other real variables
In the MIU model money is neutral also in the short run.
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Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Plan of the Presentation
1
Motivation
2
Model
3
Steady state
4
Short run dynamics
5
Simulations
6
Extensions
7
Conclusions
45 / 50
Motivation
Model
Steady state
Short run dynamics
Simulations
Extensions
Conclusions
Optimal rate of ination
One of the hot topics in monetary economics is the optimal
rate of ination.
This is not only a theoretical but also a very practical question:
Central Banks have to choose ination targets
What does the MIU model tell us about the optimal rate of
ination?
Utility depends on consumption and real money balances.
The only impact ination has on utility is via real money
balances.
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Motivation
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Extensions
Conclusions
Optimal rate of ination cont'd
For easiness of exposition let's restrict attention to steady
state. Look at (24)
mss = c ss
π ss
β
= c ss (1 + ss
)
ss
π −β
π −β
We now that c ss is independent of ination. So ination
reduces real money balances and, hence, utility.
Optimal is minimal rate of ination. Given (12) and the
requirement i ≥ 0 we have:
π OPT = −r
So the optimal rate of ination (in the MIU model) is deation
at rate −r . This was postulated by M.Friedman.
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Extensions
Conclusions
Nonneutrality in the MIU model
Our model is neutral,
But it is possible to generate nonneutrality in the MIU model,
One possibility: add labour-leisure choice and nonseparable
preferences,
Walsh (2003) shows the solution and simulations,
Fig 2.3 and 2.4 from textbook,
But this model cannot, under standard calibration, generate
reaction functions and moments similar to those observed in
the data.
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Motivation
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Conclusions
Plan of the Presentation
1
Motivation
2
Model
3
Steady state
4
Short run dynamics
5
Simulations
6
Extensions
7
Conclusions
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Motivation
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Extensions
Conclusions
MIU - conclusions
We solved and simulated a simple monetary dynamic
stochastic general equilibrium model (DSGE),
This was able to reect some desired business cycle features in
reaction to productivity shocks,
But monetary shocks have only nominal eects (except for
impact on real money balances),
This is inconsistent with stylised facts presented in Lecture 1,
We have to look for other models ,
E.g. the sticky price new Keynesian model - next lecture.
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