McCandless]Prof. McCandless
UCEMA
Prof. McCandless
UCEMA
Macroeconomics II
Money in the Utility function
[
October 24, 2006
1
Money in the Utility function
Money in the Utility function
Alternative way to add money to model
Utility approximates bene…ts from using money in transactions
Use sub-utility of the form
u(cit ;
mit i
mi
; lt ) = u(cit ; t ; 1
Pt
Pt
hit );
Households maximize
i
; hit
Households choose sequences of cit ; mit ; kt+1
E0
1
X
t
u(cit ;
t=0
mit
;1
Pt
1
t=0
to maximize
hit );
subject to the sequence of period t budget constraints,
i
cit + kt+1
+
(gt
1)Mt
1
mit
= wt hit + rt kti + (1
Pt
)kti +
mit 1
+ (gt
Pt
is a lump-sum transfer of money
we use
u(cit ;
mit
;1
Pt
hit ) = ln cit + D ln
Household’s FOCs
1
mit
Pt
+ Bhit
1)
Mt 1
Pt
First order conditions for the household are
1
cit
1
cit
1
cit
Pt
i
ct+1 Pt+1
=
Et
=
Et
=
B
wt
1
cit+1
+
DPt
mit
[rt+1 + (1
)]
another equation from teh budget constraint
i
cit + kt+1
+
mit
= wt hit + rt kti + (1
Pt
)kti +
mit 1
+ (gt
Pt
1)
Prices and money
Aggregate version of the …rst FOC is
1
1
1
= Et
+D
Pt Ct
Ct+1 Pt+1
Mt
But
Et
1
1
1
= Et
+D
Pt+1 Ct+1
Ct+2 Pt+2
Mt+1
substituting this into the …rst FOC gives
1
Pt Ct
=
=
1
1
1
+D
+D
Ct+2 Pt+2
Mt+1
Mt
1
1
1
2
Et
+ Et D
+D
Ct+2 Pt+2
Mt+1
Mt
Et
Et
Repeated substitutions give
1
X
1
= DCt
Pt
j=0
j
Et
1
Mt+j
Prices and money
The money growth rule is
Mt+1 = gt+1 Mt
prices in terms of a sequence of growth rates of money are
1
1
DCt X
=
Pt
Mt j=0
2
j
Et
j
Y
k=1
1
gt+k
Mt 1
Pt
Prices are forward looking and depend on the entire expected future
growth rates of money
We assume money growth follows the process
ln gt = (1
) ln g + ln gt
1
+ "gt
The production sector
Production is standard and competitive
The production function is
Yt =
Ht1
t Kt
;
with costs of
costs = wt Ht + rt Kt :
Competitive factor market conditions are
rt =
1
t Kt
Ht1
;
and
wt = (1
)
The stochastic shocks for technology,
ln
t
=
ln
t Kt
t
Ht :
follow the process
t 1
+ "t ;
Equilibrium conditions
Given that there is a unit mass of identical agents, aggregate variables are
= cit
= mit
= hit
Ct
Mt
Ht
and
Kt = kti
Stationary states
Stationary state version of the FOCs are
1
C
=
r
=
C
=
Pt
Pt
+D
M
CPt+1
t
1
(1
w
B
3
)
The stationary state budget constraint is simply
Y =C+ K
The factor market conditions in a stationary state are
r
w
=
=
1
K
(1
1
H
)K H
The production function is
Y =K H
1
Stationary states
The rental on capital is
r=
1
(1
)
From the factor market conditions, get
1
w = (1
)
r
with this w, get
w
B
C=
The …rst …rst order condition can be written as
1=
g
+D
C
M=P
to give
M=P = D
gC
g
Stationary states
Use the factor market conditions to get H as a function of K, put this
into the production function to get
r(1
w
Y =K
)
1
Use the stationary state household budget constraint to get
K=h
r(1
w
4
C
i1
)
Use the above equation to get output
Using the factor market conditions, get
H=
r(1
w
)
K
Stationary states
For the standard economy with = :99, = :025, = :36; B = 2:5805;
and set D = :01 to get same money holdings as in the cia model when
g = 1.
Variable
Stationary
state
value
:035101
2:3706
:9187
12:6707
:3335
1:2354
r
w
C
K
H
Y
:009187g
M=P
g :99
Log-linear version of the model
0
et +
= C
"
g
+
DC
M=P
#
Pet
g
et+1
Et C
g
Et Pet+1
[r + (1
)] Et w
et+1 ;
= w
et + rEt ret+1
et ;
0 = w
et C
e
e t+1 wH w
e t rKe
0 = C Ct + K K
et wH H
rt
e t (1
et;
0 = Yet et
K
)H
e t (1
et;
0 = ret et (
1) K
)H
0
0 = w
et
f
0 = Mt
et
get
et + H
et;
K
f
Mt 1 :
Log-linear version of the model
h
i
e t+1 ; M
ft ; Pet ,
the three "state" variables, xt = K
h
i
et ; Yet ; H
et
the …ve "jump" variables, yt = ret ; w
et ; C
h
i
the stochastic variables zt = et ; get
5
DC f
Mt ;
M=P
[r + (1
et;
)] K K
Solve the model as
0 = Axt + Bxt 1 + Cyt + Dzt ;
0 = Et [F xt+1 + Gxt + Hxt 1 + Jyt+1 + Kyt + Lzt+1 + M zt ] ;
zt+1 = N zt + "t+1 ;
to get
xt+1 = P xt + Qzt ;
and
yt = Rxt + Szt :
Linear policy function
The solution to the model are linear policy functions
xt+1 = P xt + Qzt
and
yt = Rxt + Szt
where
Linear policy function
and
and
3
0:9418 0 0
0
1 0 5;
P =4
0:5316 1 0
3
2
0:1552
0
5;
0
1
Q=4
0:4703 1:6648
2
2
6
6
R=6
6
4
0:9450
0:5316
0:5316
0:0550
0:4766
2
6
6
S=6
6
4
1:9417
0:4703
0:4703
1:9417
1:4715
3
0 0
0 0 7
7
0 0 7
7;
0 0 5
0 0
0
0
0
0
0
3
7
7
7:
7
5
Notice the coe¢ cient on prices of a money growth shock
Money and prices
6
0.02
0.018
0.016
money and prices
0.014
money
0.012
0.01
prices
0.008
0.006
0.004
0.002
0
0
5
10
15
periods
20
25
30
Figure 1: Response of money and prices to money growth shock
Recall that prices are forward looking in this model
The response of money and prices to a money growth shock are
Real variables and a technology shock
These are the same as the responses in Cooley-Hansen
Seigniorage
Government budget constaint is
gt = gbt g =
Mt
Mt
Pt
1
let g be the average government de…cit …nanced with money issue and the
random variable gbt follows the process
where "gt
N (0;
g ),
with
ln gbt =
g
ln gbt
1
+ "gt
as the standard error.
De…ne 't as the gross growth rate of money that is required to …nance
the budget de…cit in period t.
Then,
Mt = ' t M t
and
gt = gbt g =
('t
1) Mt
Pt
7
1
1
=
('t
1) Mt
't
Pt
0.02
Y
0.015
K
w and C
0.01
0.005
0
-0.005
H
r
p
-0.01
0
20
40
60
80
100
periods
Figure 2: Responses to a :01 impulse in technology
Seigniorage
Households do not receive direct transfers from government
Household budget constraint is
i
cit + kt+1
+
mit
= wt hit + rt kti + (1
Pt
)kti +
mit 1
Pt
Full model is
1
Ct
1
wt
wt
Mt
Ct + Kt+1 +
Pt
Yt
rt
wt
Stationary states
gbt g
=
=
=
Pt
DPt
+
;
Ct+1 Pt+1
Mt
1
Et
[rt+1 + (1
)] ;
wt+1
BCt ;
Et
= wt Ht + rt Kt + (1
=
=
=
=
t Kt
Ht1
;
Ht1 ;
(1
) t K t Ht ;
Mt Mt 1
:
Pt
t Kt
1
Notice how output changes to cover seigniorage
8
)Kt +
Mt 1
;
Pt
0.01
real seigiorage
0.008
0.006
0.004
0.002
0
1
1.1
1.2
1.3
g ross stationary state inflation rate
1.4
1.5
Figure 3: Real seigniorage
Annual in‡ation
0%
Corresponding '
1
rental
0:0351
wages
2:3706
consumption
0:9187
real balances =M=P
0:9187
output
1:2354
capital
12:6707
hours worked
0:3335
seigniorage = g
0
Stationary states (Bailey curve)
10%
1:024
0:0351
2:3706
0:9187
0:2767
1:2441
12:7601
0:3359
0:0065
100%
1:19
0:0351
2:3706
0:9187
0:0547
1:2472
12:7910
0:3367
0:0087
400%
1:41
0:0351
2:3706
0:9187
0:0308
1:2475
12:7943
0:3368
0:0090
Relation between real seigniorage and in‡ation rate
Log-linearization
Only two equations need changes
0
et + K K
e t+1 + M=P M
ft
= CC
wH w
et
et
wH H
rKe
rt
the seigniorage equation
0 = ge
gt + M=P
1
1
'
Pet
M=P
[r + (1
1
1
'
Pet
et
)] K K
ft +
M=P M
M=P f
Mt
'
M=P f
Mt
'
1
1
where get
ln gbt ln 1 = ln gbt is the log of the shock to the government
de…cit that is …nanced by seigniorage.
9
Next slide gives matrices for ' = 1 and ' = 1:19
[5cm]
5cm
P
2
0:9418 0 0
4
0
1 0
0:5316 1 0
2
3
0:1552 0
4
0
0 5
0:4703 0
2
0:9450 0 0
6 0:5316 0 0
6
6 0:5316 0 0
6
4 0:0550 0 0
0:4766 0 0
3
2
1:9417 0
6 0:4703 0 7
7
6
6 0:4703 0 7
7
6
4 1:9417 0 5
1:4715 0
=
Q =
R
=
S
=
5cm
P
=
Q =
R
=
S
=
2
0:9418
4 0:3243
2:0213
2
0:1547
4 0:9330
5:8433
2
0:9477
6 0:5331
6
6 0:5331
6
4 0:0523
0:4807
2
1:9359
6 0:4735
6
6 0:4735
6
4 1:9359
1:4624
3
5
3
7
7
7
7
5
3
0 0
1 0 5
1 0
3
0:0005
0:2178 5
0:3644
3
0 0
0 0 7
7
0 0 7
7
0 0 5
0 0
3
0:0011
0:0006 7
7
0:0006 7
7
0:0011 5
0:0017
Response of real variables to a seigniorage shock
Response of nominal variables to a technology shock
10
x 10
2
-5
H
1.5
r
1
0.5
0
Y
-0.5
w and C
-1
0
10
20
30
40
50
Figure 4: Response to seigniorage shock, ' = 1:19
0
-0.05
-0.1
money
-0.15
prices
-0.2
-0.25
-0.3
-0.35
-0.4
0
20
40
60
80
100
periods
Figure 5: Responses of money and prices to a technology shock when seigniorage
is collected in the stationary state, ' = 1:19
11