Monetary Cash Credit

8/9/2019 Monetary Cash Credit

1/17

11

(2

)

(2

)

(2

)

Monetary Economics: Cash/Credit Goods Model

Keke Sun

Jordi Sebasti

Pedro Garca Ares


2/17

2

Theoretical Part

Section 1


3/17

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


4/17

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


5/17

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


6/17

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


7/17

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


8/17

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


9/17

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


10/17

10

Matlab study

Section 2


11/17

11

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


12/17

12

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


13/17

13

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


14/17

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:


15/17

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!


16/17

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


17/17

17

Matlab codes

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