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(2
)
(2
)
(2
)
Monetary Economics: Cash/Credit Goods Model
Keke Sun
Jordi Sebasti
Pedro Garca Ares
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2
Theoretical Part
Section 1
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The Model
3
(3
)
(3
)
subject to a budget constraint
the CIA constraint
and non-negativity conditions0uU
g
!
0
1,,,
)ln(lnmaxi
itit
tttt
dc
kmdc
F
11
1 )1(1
k
4
e tt
tttttttt kAkdc HX
t
t
tttt adc |
4e
1
1XU
s.t
a
Representative households problem
B.C
CIA
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Value Function problem
4
(3
)
(3
)
subject to a budget constraint
the CIA constraint
and non-negativity conditions
0uU
t
t
tttt adc |
4e
1
1XU
s.t
_ a
4
4
!
tttt
t
tttt
t
tt
ttt
tttt
tt
mdckm
k
m
VEdckmdc
kaV
11
1
1
1
1,,,
1,
11
,1
lnln)( maxHX
X
Fa
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The Lagrangian
5
(1
)
(2
)
(3
)
(3
)
_ a
4
4
4
UNUXQ
HX
X
F
ttt
t
ttt
tttt
t
tttt
t
tt
ttt
tttt
dc
dckkA
dc
kdc
1
11
,1
lnln
1
11
1
1
1
1,,,
maxa
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The FOCs
6
with respect to dt
with respect to mt
0),(1 1 ! tttktt
kaVEc
QF
0),(1 1 ! tttktt
kaVEd
UQF
0),(1
1),( 1
1
1 !4
-
ttkt
ttat kaVkaVEF
tttkttta kaVEkaV QF ! ),(),( 11
! HF 1),(),( 111 tttktttk a kkaVEkaV
ct
dt
mt
at
kt-1
By the Envelope Theorem:
a-1
1 + Real Interest Rate
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At the steady state
7
ka !41
1*
RVV kk **F!
QF ! ka VV *
From
From
From
From and RVV ka **
1
1* F!
4 4! 1*** RVV ka F
From 4! 1**** RVV kk FQF
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Solution (i)
8
44
! 111
1
i
VkF
Q
4
!
1
1 iR
ii
Vk!! 11
F
Q
Since
CIA constraint binds ( >0), the nominal interest rate >0
CIA constraint doesnt bind ( =0), the nominal interest rate =0
Q
Q
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Solve for optimal
Solution (ii)
9
U
U 0* e ttt d NQ
Slackness Condition:
( = 0 if > 0)U
0!UN t
As long as the nominal interest is positive, > 0 is positive
and > 0 and this implies that > 0 from whichthe condition implies that
tQ
tt d*Q tN0!UN t 0!U
So, as long as the nominal rate of interest is positive,
the household will never use cash to purchase d
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Matlab study
Section 2
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Uhligs Toolkit
11
21211111
2111
0
0
=!=
!=
!=
ttt
tttttttt
tttt
N
MKXXJEHXGXXFE
DCXBXX
I
? A? A? Att
ttttt
tt
uZ
ydcY
kX
,
,,,
,
,
p
p
p
TP
Initial Form:
Final Form:
ttt
ttt
SZRXY
QZPXX
!
!
1
1
State Variables
Control Variables
Shocks
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Procedure
Calculate
FOC s1.
? A
? A
? A
? A ? A)1(1
1
1
111
1
11
HEPFP
TQPFP
PUQ
QP
E !
!
!
!
tt
tt
tt
t
tt
ttt
tt
t
t
tt
t
t
kAEk
Em
dd
cc
Calculate
Steady States2.
Introduce the steady states on the Uhlig Code
Capital, output, cash good, credit good, r, m, inflation, lambda
Settting
parameters
values3.
Introduce the parameters values onto the model
alpha = 0.35
delta = 0.019
inflation = 0.0125
g_bar = 1.0125
rho_z = 0.95
xi = - 0.15
beta = 0.989
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Procedure (ii)
Log linearizedmodel4.
Calculate
matrices5.
1,1
1,1
1
1
11
1
0
0
0
0
0)1(1
01
1
)1(
0)(1
1
!
!
!
!
!
!
!
!
!
tuttt
tAtt
tttt
t
SSSSSS
ttSS
SS
tSS
SS
tSS
SS
ttt
tSS
SS
tSS
tSS
SS
t
SS
SSSS
SS
tt
t
ttSSSSSS
tt
Auu
AA
mum
mmddcc
yky
kd
y
dc
y
c
kAy
cc
dd
kk
yyykk
yE
cc
E
IJK
IV
T
U
E
UUPP
HEHEPFP
TPT
FP
Introduce the coefficients of the matrices into the Uligh Code
A, B ,C ,D, F, G, H, J, K , M, N
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Output (i)
14
ttt
ttt
SRXY
XX
!
!
1
1
0.19088 0
0.31941 0!P !Q 0.35365 -0.10095
0.477 -0.0 116
!S!R
Recursive equilibrium law of motion fory(t) on z(t)
0.3 411 0
0.31 68 0
0.0373 1 0
-0. 8615 0
-0.31941 1
0.5 506 -0.070741
0.40871 0.047473
0.591 9 -0.047473
-0.13883 -0.3 167
-0.477 1.0 1
Recursive equilibrium law of motion fory(t) on x(t-1)
Recursive equilibrium law of motion forx(t) on z(t)
Recursive equilibrium law of motion forx(t) on x(t-1)
Final Form:
7.0!U
7.0!V
For:
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Output (ii)
15
-1 0 1 2 3 4 5-0.08
-0.06
-0.04
-0.02
0
0.02
0.04Impuls
s po ns
s
o a sh ock in money growt h
Years aftersh ock
ercentdeviation
from
stea
dy
state
output
cas h good
c redit good
-1 0 1 2 3 4 5-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0. 1
0.15Impulse res po nses to a sh ock in money growt h
Years aftersh ock
ercentdeviation
from
stead
y
state
output
c ash good
c redit good
1.0!U 2.0! 1.0!U 7.0!
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Output (iii)
16
-1 0 1 2 3 4 5-0. 1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06Impu
se
es po nses to a sh ock in money g
o
th
ea
s afte
sh ock
Pe
centdeviation
fom
s
ead
y
s
ate
output
c ash good
c
ed i
good
7.0!U 7.0!7.0!U 2.0!
-1 0 1 2 3 4 5-0 .03
-0 .02 5
-0 .02
-0 .01 5
-0 .01
-0 .00 5
0
0 .00 5
0 . 01I
s
r
s
s
s t
s h
c k
!
r
w"
h
Y
rs
ft
r s h
c k
P
#
rc
#
$
t
%
#
&
(
t
)
$
fr)
m
st
#
(
%
0
st
(
t#
t
t
c
s h!
1
c r
1
"
!
1
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Matlab codes
Appendix
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