ACTA UNIVERSITATIS DANUBIUS Vol 8, no. 5/2012

176

Macroeconomics and Monetary Economics

The Equilibrium Analysis of a Closed Economy Model with

Government and Money Market Sector - II

Catalin Angelo Ioan1, Gina Ioan

2

Abstract: In this paper, we will continue the study of the dynamic equilibrium solutions in the purpose of investigating the dependence limits (potential output and interest rate limit). We find also an interesting linear relation between the potential output and interest rate limit.

Keywords: equilibrium; demand; income

JEL Classification: R12

1 Introduction

The purpose of this paper is to continue the analysis of a closed economy model

when the net exports are zero. After a correction of some errors founded in [2] we

have analyzed the dependence of the potential output and the interest rate limit on

the depending variables presented in the model.

2 The Model Equations ([5])

For the beginning let remind the model equations:

(1) D=C+I+G

(2) C=cYV+C0, C00, cY(0, 1)

(3) V=Y+TR-TI, TR0

(4) TI=riYY+T0, riY(0, 1), T0R

1 Associate Professor, PhD, Danubius University of Galati, Faculty of Economic Sciences, Romania, Address: 3 Galati Blvd, Galati, Romania, Tel.: +40372 361 102, fax: +40372 361 290, Corresponding author: catalin_angelo_ioan@univ-danubius.ro. 2 Assistant Professor, PhD in progress, Danubius University of Galati, Faculty of Economic Sciences, Romania, Address: 3 Galati Blvd, Galati, Romania, Tel.: +40372 361 102, fax: +40372 361 290, e-

mail: gina_ioan@univ-danubius.ro.

AUDŒ, Vol 8, no 5, pp. 176-185

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177

(5) I=inYY+irr+I0, inY(0, 1), ir0, I00

(6) G= G

(7) D=Y

(8) MD=mdYY+mrr+M0M0, mdY0, mr0, M00

(9) MD=M

(10) dt

dY=(D-Y), 0

(11) dt

dr=(MD-M), 0

where:

D – the aggregate demand;

C – the consumer demand (a concave function of V);

I – the investment demand;

G – the government spending;

V – the disposable income;

Y – the aggregate supply (national income);

TR – the government transfers;

TI – taxes;

cY – the marginal propensity to consume, c=dV

dC(0, 1),

2

2

dV

cd0;

riY – the tax rate, riY(0, 1);

inY –the rate of investments, inY(0, 1);

ir – a factor of influence on the investment rate, ir0;

r – the interest rate;

MD – the money demand in the economy;

mdY – the rate of money demand in the economy;

ACTA UNIVERSITATIS DANUBIUS Vol 8, no. 5/2012

178

mr – a factor of influencing the demand for currency from the interest rate,

mr0;

M – themoney supply.

3 The Static Equilibrium

Let note for the beginning, the autonomous component:

(12) E=cY(TR-T0)+C0+I0+ G 0

In order to have the equilibrium, that is D=Y, from (1)-(6) we obtain:

(13)

YYYrYr

0rr

YYYrYr

YYYY0

in)ri1(c1mmdi

MMiEmY

in)ri1(c1mmdi

Emdin)ri1(c1MMr

We will note below, for simplification:

(14) = MMiEm 0rr

4 A result on the stability of solutions of a system of differential

equations of first order, linear, with constant coefficients satisfying

some conditions

Lemma

Let the system of differential equations:

f

e

Y

X

dc

ba

dt

dYdt

dX

, a, b, c, d, e, fR, a, b, d0, c, e, f0, X(0)=X0,

Y(0)=Y0.

Then X~

)t(Xlimt

, Y~

)t(Ylimt

, Y~

,X~

R if and only if:

1. (a-d)2+4bc=0 with the solution:

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179

2

t2

da

00

t2

da

20

2

t2

da

20

t2

da

00

)da(

afce4te

b2

da

da

)da(ebf2bYX

2

dae

da

ceaf4YY

)da(

debf4e

)da(

bfde4Xte

da

)da(ebf2bYX

2

daX

2. (a-d)2+4bc0 and 12 are roots of the equation:

2-(a+d)+(ad-bc)=0:

e, fR with the solution:

bcad

afceek

b

aek

b

aY

bcad

bfdeekekX

t2

2t1

1

t2

t1

21

21

where:

12

0101

2

12

0202

1

bcad

afcebbY

bcad

bfdeaXa

k

bcad

afcebbY

bcad

bfdeaXa

k

3. (a-d)2+4bc0 and 1=+i, 2=-i, 0 are the roots of the equation:

2-(a+d)+(ad-bc)=0: e, fR with the solution:

bcad

afectsine

)bcad(2

)bfde(c2)ceaf)(ad(Y

2

adcX

1tcose

bcad

ceafYY

bcad

debftsine

)bcad(2

)ceaf(b2)debf)(ad(X

2

adbY

1tcose

bcad

bfdeXX

t00

t0

t00

t0

5 The Dynamic Equilibrium

Let the system of first order differential equations:

(15)

)MMD(dt

dr

)YD(dt

dY

, , 0

which becomes after (1)-(9):

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180

(16)

)MM(

E

r

Y

mmd

i

dt

drdt

dY

0rY

rY

where we note Y=1-cY(1-riY)-inY0

Using the above lemma, it follows that: Y~

)t(Ylimt

, r~)t(rlimt

, r~,Y~

R+ if

and only if:

1. (Y+mr)2+4irmdY=0 then:

2rY

0YY

t2

m

r

rY

rY

rY0r0r0

rY

t2

m

2

rY

Y0Y0

2rY

r0rt

2

m

2rY

0rr0

t2

m

rY

rY0r0r0

rY

)m(

MMEmd4

tei2

m

m

)m(EMMi2riY

2

m

em

EmdMM4rr

)m(

EmMMi4e

)m(

MMiEm4Y

tem

)m(EMMi2riY

2

mY

rY

rY

rY

rY

and:

YrrY

0YY

YrrY

0rr

mdim

MMEmdr~

mdim

MMiEmY~

2. (Y+mr)2+4irmdY0 and 12are roots of the equation:

2+(Y -

mr)-(Ymr+irmdY)=0 then:

YrrY

0YYt2

r

Y2t1

r

Y1

YrrY

0rrt2

t1

mdim

MMEmdek

iek

ir

mdim

MMiEmekekY

21

21

where:

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181

12

YrrY

0rrY10Y1

YrrY

0YYr0r

2

12

YrrY

0YYr0r

YrrY

0rrY20Y2

1

mdim

MMiEmY

mdim

MMEmdiri

k

mdim

MMEmdiri

mdim

MMiEmY

k

and:

YrrY

0YY

YrrY

0rr

mdim

MMEmdr~

mdim

MMiEmY~

3. (Y+mr)2+4irmdY0 and 1=+i, 2=-i, 0 are roots of the

equation: 2+(Y-mr)-(Ymr+irmdY)=0 then:

YrrY

0YY

t

YrrY

0rrYY0YYr0

Yr0

t

YrrY

Y0Y0

YrrY

r0r

t

YrrY

Y0Yrr0rYr0

Yr0

t

YrrY

0rr0

mdim

)MM(Emd

tsine)mdim(2

))MM(iEm(md2)Emd)MM()(m(r

2

mcY

1

tcosemdim

Emd)MM(rr

mdim

Em)MM(i

tsine)mdim(2

)Emd)MM((i2)Em)MM(i)(m(Y

2

mbr

1

tcosemdim

)MM(iEmYY

and:

YrrY

0YY

YrrY

0rr

mdim

MMEmdr~

mdim

MMiEmY~

We will call Y~

- the potential output and r~ - the interest rate limit.

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182

6 The Analysis of Variation Limits

Therefore again:

(17)

YrrY

0YY

YrrY

0rr

mdim

MMEmdr~

mdim

MMiEmY~

From the above relations, we obtain:

(18) Er~iY~

rY

or, in original terms:

(19) r~iY~

inri1c1 rYYY =cY(TR-T0)+C0+I0+ G

We can easily write the relation (18) as:

(20) YY

r Er~

iY~

Because Y

ri

0 it follows that the dependence of the output potential of the interest

rate limit is inverse.

Because YrrY mdim 0 in order to have Y~0 it must that 0 that is:

(21) MMiEm 0rr 0

The first partial derivatives of Y~

are:

(22)

2YrrY

0YYr

r mdim

MMEmdi

m

Y~

0

(23) YrrY

r

mdim

m

E

Y~

0

(24)

2YrrY

Y0Yr

r mdim

MMEmdm

i

Y~

0

(25)

2YrrY

r

2

YrrY

0rrr

Y mdim

m

mdim

MMiEmmY~

0

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183

But Y=1-cY(1-riY)-inY imply: 1cY

Y

, Y

Y

Y cri

, 1

inY

Y

, from where:

(26)

2YrrY

r

2

YrrY

0rrr

Y

Y

YY mdim

m

mdim

MMiEmm

c

Y~

c

Y~

0

(27)

2YrrY

rY

2

YrrY

0rrrY

Y

Y

YY mdim

mc

mdim

MMiEmmc

ri

Y~

ri

Y~

0

(28)

2YrrY

r

2

YrrY

0rrr

Y

Y

YY mdim

m

mdim

MMiEmm

in

Y~

in

Y~

0

(29) YrrY

r

0 mdim

i

MM

Y~

0

(30)

2YrrY

r

2

YrrY

0rrr

Y mdim

i

mdim

MMiEmi

md

Y~

0

Also, for

YrrY

0YY

mdim

MMEmdr~

0 we have:

(31)

2YrrY

0YYY

r mdim

MMEmd

m

r~

0

(32) YrrY

Y

mdim

md

E

r~

0

(33)

2YrrY

0YYY

r mdim

MMEmdmd

i

r~

0

(34)

2YrrY

Y

2

YrrY

0rrY

Y mdim

md

mdim

MMiEmmdr~

0 from where:

(35)

2YrrY

Y

2

YrrY

0rrY

Y

Y

YY mdim

md

mdim

MMiEmmd

c

r~

c

r~

0

(36)

2YrrY

YY

2

YrrY

0rrYY

Y

Y

YY mdim

cmd

mdim

MMiEmcmd

ri

r~

ri

r~

0

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184

(37)

2YrrY

Y

2

YrrY

0rrY

Y

Y

YY mdim

md

mdim

MMiEmmd

in

r~

in

r~

0

(38) YrrY

Y

0 mdimMM

r~

0

(39)

2YrrY

Y

2

YrrY

0rrY

Y mdimmdim

MMiEm

md

r~

0

7 Conclusions

As a general conclusion, we obtain the following:

the increase in absolute value of mr – the factor of influencing the demand

for currency from the interest rate imply the increasing of the potential output

and the decreasing of the interest rate limit;

the potential output and the interest rate limit are increasing at an increase of

the autonomous component;

the increase in absolute value of ir – the factor of influence on the investment

rate imply the decreasing of the potential output and the decreasing of the

interest rate limit;

the increase of cY – the marginal propensity to consumeimply the increasing

of the potential output and the increasing of the interest rate limit;

the increase of riY – the tax rate imply the decreasing of the potential output

and the decreasing of the interest rate limit;

the increase of inY – the rate of investments imply the increasing of the

potential output and the increasing of the interest rate limit;

the increase of M – the money supply imply the decreasing of the potential

output and the decreasing of the interest rate limit;

the increase of mdY – the rate of money demand in the economy imply the

decreasing of the potential output and the increasing of the interest rate limit.

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8 References

Dornbusch, R.; Fischer, S. & Startz, R. (1997). Macroeconomics. Richard D. Irwin Publishers.

Ioan, G. & Ioan, C.A. (2011). The Equilibrium Analysis of a Closed Economy Model with Government and Money Market Sector. Acta Universitas Danubius. Oeconomica, Vol. 7, No. 5, pp. 127-143.

Mankiw, N.G. (2010). Macroeconomics. Worth Publishers.

Romer, D. (1996). Advanced Macroeconomics. McGraw-Hill.

Stancu, S. & Mihail, N. (2009). Macroeconomie. Modele statice și dinamice de comportament. Teorie

și aplicații/Macroeconomics. Static and dynamic models of behavior. Theory and Applications. Bucharest: Economică.