Bermperoglou Garcia Parado Izvolenkiy

8/9/2019 Bermperoglou Garcia Parado Izvolenkiy

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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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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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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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Productivity shock

( )2

,1

,0~

1

AA

tAtt AA


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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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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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[ ]

[ ]

[ ]

[ ] [ ] )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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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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+=

+=

)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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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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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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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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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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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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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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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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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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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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Summarizing basic results

In the cash/credit CIA model superneutrality of money fails both in the short and the long run,









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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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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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