bermperoglou garcia parado izvolenkiy
TRANSCRIPT
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Cash Credit Goods in CIA model
Cash / Credit goods model
Macroeconomics IIAssignment 2
Dimitrios BermperoglouAndres Fernando Garcia Parrado
Anton Izvolenskiy
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Cash Credit Goods in CIA model
Literature Review
Money and Interest in a Cash-in-Advance Economy
Lucas & Stokey, Econometrica 1987
Aggregate general equilibrium model A cash-in-advance constraint on a subset of goods (cash goods) Existence, characterization and calculation of equilibria Classical approach: if changes in stock of money do not affect the distribution of future money
growth, then only prices adjust and no effects on real side of economy.
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Cash Credit Goods in CIA model
The model
Consumers
[ ]
t
ttt
t
t
ttttttt
t
tt
t
tkmdc
mc
m
kAkmdcts
dcEtttt
++
+++=+++
+
=
1
1..
lnlnmax
1
1
1
0,,,
Production process
1= ttt kAy
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Cash Credit Goods in CIA model
Productivity shock
( )2
,1
,0~
1
AA
tAtt AA
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Cash Credit Goods in CIA model
SSSSSS
SS
SSSS mmSSin
=
++=
11..
Governments transfers
ssss
ss
SS
t
t
t
t
tt
t
mSSin
mm
P
MM
+=
+=
=
1..
1
11
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Cash Credit Goods in CIA model
Solving the model
Consumers Problem
[ ]
t
ttt
t
t
ttttttt
t
tt
t
tkmdc
mc
mkAkmdcts
dcEtttt
++
+++=+++
+
=
1
1..
lnlnmax
1
1
1
0,,,
[ ]
=
+++
+++++=
0
11
111
lnlnt
t
t
ttttttt
t
ttttttt
t
tt c
mkmdc
mkAdcEL
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Cash Credit Goods in CIA model
[ ]
[ ]
[ ]
[ ] [ ] )4(
)3(1
)2(
)1(1
...
1
11
1
11
++
+
++
=
+
+=
=
+=
tttt
tt
t
tt
ttt
t
t
t
tt
t
t
kAEk
Em
dd
cc
COF
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Cash Credit Goods in CIA model
Optimal cash - credit good choice
From (1)-(3) and Fisher equation...
+=+
+ )1(1
1
t
t
t i
c
dE
Imposing steady state.
( )
)9()(
)8()1(
)31(
)7()()(
)6()4( 11
SSSS
SS
SS
SS
SSSSSSSS
SS
mcCIA
c
d
kkdcBC
k
=
+=
=++
=
Equations (6), (7), (8), (9) jointly determine the steady state values cSS
, dSS
, mSS
, kSS
.
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Cash Credit Goods in CIA model
+=
+=
)1()1()8(
SS
SS
SS
c
d
Or equally,
+=+=
+= )1()1(
)1()8(
SSSSSSSS
SS
SS
iRc
d
Implications
SS
SSSSSS
c
digrowthmoneyhigher
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Cash Credit Goods in CIA model
A little economic intuition
Nominal interest rate is the cost of holding money for purchasing cash goods
Increase in the interest rate means increase in the cost of holding money Substitution away from the cash goods towards the credit goods
ShortShortShortShort----run and longrun and longrun and longrun and long----runrunrunrun superneutralitysuperneutralitysuperneutralitysuperneutrality failsfailsfailsfails !!!!
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Cash Credit Goods in CIA model
Superneutrality review...
Cash credit good model
idcu
dcu
d
c+=1),(
),(
Money growth shocks affect the nominal interest rate Direct effect: consumption of cash goods is taxed, while credit goods not Long and short-run superneutrality fails
Standard stochastic CIA model
nl
c
f
i
lcu
lcu +
=
1
),(
),(
Money growth shocks affect the nominal interest rate Direct effect: consumption is taxed by a nominal interest rate, while leisure not Long and short-run superneutrality does not hold
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Cash Credit Goods in CIA model
Optimal rate of inflation
Solve for the steady state
( )
SSSS
SS
SS
dc
d
k
=
++
+=
=
)1(
)1(
1
1
Where ( ) 01)( 1 >=
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Cash Credit Goods in CIA model
Maximizing steady state utility
( SSSSSSSS dcdcu lnln,max
+=
F.O.C.
( )( )
( )
=
+=++ +
+
=
=++
=
1
1)1()1(
.1
01
1
01
*
2
SS
SSSS
SS
SSSS
d
cd
d
cd
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Cash Credit Goods in CIA model
In MIU model the optimal rate of inflation was:
rFisher
ilmcu
MIU
SS
m
=
==
)(
00),,(
Notice that
CIAMIUr
=>> 0(*),0
The cost of inflation in MIU model (with superneutrality!) implies a loss in utility due to money balances
reductions.
In cash/credit CIA model, the cost of inflation is mitigated somehow since more inflation implies less utility
through a cut in cash goods and more utility through an increase in credit goods.
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Cash Credit Goods in CIA model
Short run dynamics Impulse responses
Log-linearized model
( )
( )
1,1
1,1
1
1
11
11
)(
)(
0)(
0)(
0)(
0)2(
0)4(
0)3(),1(
++
++
++
++
++=
+=
=+
=++
=
=+
=+
=++
tuttt
tAtt
tttt
ttSS
SS
tSS
SS
tSS
SS
ttt
tt
ttttt
ttt
t
Auumoneyshock
AAprodshock
mumhmoneygrowt
yky
kd
y
dm
y
cBC
kAyPF
d
kyE
mE
Note: we used the log-linearized CIA constraint ( mc = ) to eliminate the variable c from the system
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Cash Credit Goods in CIA model
Solving the rational expectation model using Uhligs code
Variables
Endogenous state x = [k, m]
Endogenous control y = [d, y, , ]
Exogenous z = [A, u]
Initial Form
[ ] [ ]
11
1111
1
00
++
+++
+=
=++++++=+++
ttt
ttttttt
tttt
Nzz
MzLzKyyEJHxGxxEFDzCyBxAx
Final Form
equationnobservatioSzRxy
equationtransitionQzPxx
ttt
ttt
+=
+=
1
1
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Cash Credit Goods in CIA model
In Uhligs code
QZ decomposition Impulse responses for 5 years Money growth shock u Responses of : cash goods consumption (c=m)
credit goods consumption (d)
output (y)
Calibrated parameters from Table 2.2 (Walsh) Sensitivity analysis w.r.t. utility weight of credit goods (0.5, 2) Sensitivity analysis w.r.t. autoregressive coefficient of money growth (0.2, 0.9)
tuttt Auu ,11 ++=
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Cash Credit Goods in CIA model
Summarizing basic resultsIn the cash/credit CIA model superneutrality of money fails both in the short and the long run,
similarly to the stochastic CIA model (with leisure). The purchase of cash goods is taxed by a
positive nominal interest rate. Monetary policy shocks trigger higher expected inflation, thus a
substitution effect between cash and credit goods.
In contrast to the standard CIA model, this model yields a well-defined SS utility-maximizing
inflation rate which is higher than the optimal inflation of the MIU model.
Impulse responses capture our models results. An expansionary monetary policy (positive money
growth shock) results to a substitution from cash towards credit goods and a rise in output.
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Cash Credit Goods in CIA model
Summarizing basic results
In the cash/credit CIA model superneutrality of money fails both in the short and the long run,
similarly to the stochastic CIA model (with leisure). The purchase of cash goods is taxed by a
positive nominal interest rate. Monetary policy shocks trigger higher expected inflation, thus a
substitution effect between cash and credit goods.
In contrast to the standard CIA model, this model yields a well-defined SS utility-maximizing
inflation rate which is higher than the optimal inflation of the MIU model.
Impulse responses capture our models results. An expansionary monetary policy (positive money
growth shock) results to a substitution from cash towards credit goods and a rise in output.
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Cash Credit Goods in CIA model
Summarizing basic results
In the cash/credit CIA model superneutrality of money fails both in the short and the long run,
similarly to the stochastic CIA model (with leisure). The purchase of cash goods is taxed by a
positive nominal interest rate. Monetary policy shocks trigger higher expected inflation, thus a
substitution effect between cash and credit goods.
In contrast to the standard CIA model, this model yields a well-defined SS utility-maximizing
inflation rate which is higher than the optimal inflation of the MIU model.
Impulse responses capture our models results. An expansionary monetary policy (positive money
growth shock) results to a substitution from cash towards credit goods and a rise in output.
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Cash Credit Goods in CIA model
Summarizing basic results
In the cash/credit CIA model superneutrality of money fails both in the short and the long run,
similarly to the stochastic CIA model (with leisure). The purchase of cash goods is taxed by a
positive nominal interest rate. Monetary policy shocks trigger higher expected inflation, thus a
substitution effect between cash and credit goods.
In contrast to the standard CIA model, this model yields a well-defined SS utility-maximizing
inflation rate which is higher than the optimal inflation of the MIU model.
Impulse responses capture our models results. An expansionary monetary policy (positive money
growth shock) results to a substitution from cash towards credit goods and a rise in output.
Greater money growth persistence () leads to larger movements in expected inflation in response
to a monetary shock. This, in turn, produces larger adjustments of consumption and output.
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Cash Credit Goods in CIA model
Summarizing basic results
In the cash/credit CIA model superneutrality of money fails both in the short and the long run,
similarly to the stochastic CIA model (with leisure). The purchase of cash goods is taxed by a
positive nominal interest rate. Monetary policy shocks trigger higher expected inflation, thus a
substitution effect between cash and credit goods.
In contrast to the standard CIA model, this model yields a well-defined SS utility-maximizing
inflation rate which is higher than the optimal inflation of the MIU model.
Impulse responses capture our models results. An expansionary monetary policy (positive money
growth shock) results to a substitution from cash towards credit goods and a rise in output.
Greater money growth persistence () leads to larger movements in expected inflation in response
to a monetary shock. This, in turn, produces larger adjustments of consumption and output.
The more the utility enjoyed from consuming credit goods (higher ), the more inelastic are the
responses of the credit good consumption and more elastic are the responses of the cash good.
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Cash Credit Goods in CIA model