m_f model fixed e rate

8/18/2019 M_F Model Fixed E Rate

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where H gives the amount of high-powered money created in the period under consideration.

Dividing the two sides of the above equation by P , we rewrite it as

P H

K Y Y P

eP NX

*

*

;, (2.32)

Note that F P H

. Hence, substitutingP H

for F into (2.13A), we rewrite it as

Y r LP H P H ,0 (2.33)

The specification of the model is now complete. It is given by the four key equations, (2.10),

(2.31), (2.32) and (2.33) in four endogenous variables:P H

r Y ,, and K . The equilibrium value of

r is given by (2.10). Substituting (2.10) into (2.31) and (2.33), we rewrite them as

10 ;, ***

C Y Y P

eP NX Gr I T Y C Y (2.34)

and

Y r LP H

P

H ,*0 (2.35)

We can solve (2.34) for the equilibrium value of Y . Substituting the equilibrium value of Y in

(2.35), we get the equilibrium value ofP H

. Again, substituting the equilibrium value of Y and

P H

in (2.32), we get the equilibrium value of K . The solutions of (2.34) and (2.35) are


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Determination of Y andP H

under Fixed Exchange Rate

AD

45 0 Line

AD

P H 0 Y 0 Y

P H

0

* ,Y r L

Y r L ,*

L

Figure 2.7


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Determination of K under Fixed Exchange Rate

NX + K ,P H

K Y Y

PeP

NX *0*

;,

0

P H

H

K 0 K

Figure 2.8


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illustrated in Figure 2.7 where in the upper panel AD schedule gives the aggregate demand for

domestic output as given by the RHS of (2.34) corresponding to every different Y . The

equilibrium Y , obviously, corresponds to the point of intersection of the AD schedule and the 45 0

line. In the lower panel, Y r L ,* schedule represents the RHS of (2.35) and it gives the value of

L corresponding to every different value of Y , when r is fixed at r *. The equilibrium L

corresponds to the equilibrium value of Y on the

Y r L ,* schedule. The excess of the

equilibrium value of L over the initial stock of real balance given byP

H 0 gives the equilibrium

value ofP H

- see (2.35). The equilibrium value of K is shown in Figure 2.8 where

K NX schedule gives the value of K NX corresponding to every different K , when Y is fixed

at its equilibrium value Y 0. H schedule is horizontal at the equilibrium value ofP H

. The

equilibrium K , as follows from (2.32) corresponds to the point of intersection of these two

schedules. The equilibrium K is denoted by K 0.

We are now in a position to examine the impact of fiscal policy in this model.

2.2.2 Fiscal Policy: The Effect of an Increase in G

We shall now examine how an increase in G by Gd affects the endogenous variables in this

model. We shall first derive the results using Figures 2.9 and 2.10, where the initial equilibrium

values of Y , L and K are denoted by Y 0

0* ,Y r L and K 0 respectively. Let us now examine how

the AD and the

Y r L ,* schedules are affected by the given increase in G in Figure 2.9.


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Effect of an Increase in G on Y andP H

under Fixed Exchange Rate

AD

45 0 Line

AD 1

AD

P

H 0 Y 0 Y

1

P H

0

P H

0* ,Y r L

1

* ,Y r L

Y r L ,*

L

Figure 2.9


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Effect of an Increase in G on K under Fixed Exchange Rate

NX + K ,P H

K Y Y

PeP

NX *0*

;,

1

P H

1 H

0

P H

H

K Y Y

PeP

NX *1*

;,

K 0 K 1 K

Figure 2.10


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Corresponding to every different Y aggregate demand for domestic goods rises by Gd . Hence

AD schedule shifts upward. The new AD schedule is labeled AD 1 in Figure 2.9. The value of L

corresponding to any given Y , with r equal to *r remains unaffected following the increase in G ,

as it not a determinant of demand for real balance. So

Y r L ,* schedule remains unaffected.

Thus, it is clear from Figure 2.9 that in the new equilibrium Y is larger and so are L and P H

.

The new equilibrium values of Y , L andP H

are denoted/given by 1* ,, Y r LY and

1

P H

respectively. Now, focus on figure 2.10 where the initial equilibrium K denoted K 0

corresponds to the point of intersection of the

K Y Y

P

eP NX *0

*

;, schedule and

H schedule. Corresponding to any given K , the value of NX + K becomes less following the

increase in Y from its initial equilibrium value Y 0 to its new equilibrium value Y 1. Thus the

K Y Y

PeP

NX *1*

;, schedule that gives the value of NX + K corresponding to every K ,

when Y is fixed at Y 1. Clearly, it will be below the

K Y Y P

eP

NX *

0

*

;, schedule. H

schedule now corresponds to01

P H

P H

. The new H schedule labeled 1 H will,

therefore, be above the initial H line. The new equilibrium K , denoted K 1, will be

unambiguously larger.


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To sum up our finding, following an increase in G , all the three key endogenous variables Y , K

and P H

go up.

Mathematical Derivation of the Results

Let us now derive the results mathematically. For this, we have to take total differential of

(2.32), (2.34) and (2.35) treating all exogenous variables other than G as fixed. This yields the

following equations

dY NX Gd dY C dY Y (2.36)

dY LP H

d y

(2.37)

P H

d dK dY NX Y (2.38)

The three equations (2.36), (2.37) and (2.38) contain three unknowns: dY ,

P H

d and dK

representing changes in Y ,P H

and K respectively from the initial equilibrium to the new

equilibrium. (Explain (2.36), (2.37) and (2.38). Why do they contain only three unknowns?)

Solving (2.36), we get

0 1 Y NX C Gd

dY (2.39)

Substituting (2.39) into (2.37), we get


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y LP H

d

.

0 1 Y NX C

Gd (2.40)

Again, substituting (2.39) and (2.40) into (2.38) and solving for dK , we get

Y y NX LdK . 0

1 Y NX C Gd (2.41)

Let us explain (2.39). Focus on the RHS. The numerator gives the excess demand for domestic

goods that emerges at the initial equilibrium Y following the increase in G by Gd . Producers

will begin to increase Y to meet this excess demand. However, an increase in Y will raise demand

of domestic goods also. A unit increase in Y will raise aggregate planned consumption demand

by C . But the whole of C does not represent additional demand for domestic goods alone. It is

allocated between both domestic and foreign goods. A unit increase in Y raises import demand

by Y NX . Of C , therefore, Y NX represents additional demand for foreign goods. Hence, as

Y rises, per unit increase in Y demand for domestic goods goes up by Y NX C and,

therefore, excess demand for domestic goods falls by 1 Y NX C . As excess demand

falls by 1 Y NX C , when Y rises by 1 unit, excess demand will fall by Gd when Y

increases by Y NX C

Gd 1

. This explains (2.39). Since Y increases from the initial

equilibrium to the new one by Y NX C

Gd 1

and r remains unchanged at *r , demand for

real balance increases from the initial equilibrium to the new one by Y y NX C

Gd L

1. .

Accordingly supply of real balance from the initial equilibrium to the new one has to increase by


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Y y NX C Gd

L1

, i.e.,

P H

d has to be equal to Y y NX C

Gd L

1. This explains

(2.40). As Y rises from the initial equilibrium to the new one by

1 Y NX C

Gd and other

determinants of NX remains unchanged, the fall in NX from the initial equilibrium to the new one

is given by Y Y NX C Gd

NX 1

. On the other hand,P H

increases from the initial

equilibrium to the new one by Y y NX C Gd

L 1 . Hence, as follows from (2.32), K has to

increase by Y Y NX C

Gd NX

1+

Y y NX C Gd

L1

. This explains (2,41).

Adjustment Process

We shall now describe one plausible mode of behavior on the part of the economic agents as

implied by this model that will bring about the changes in Y ,P H

and K as given in (2.39),

(2.40) and (2.41) respectively following the given increase in G by Gd . The increase in G

will create excess demand of Gd in the goods market. Producers will expand output by Gd to

meet this excess demand. The increase in Y by Gd will produce two effects. First, import

demand will go up by Gd NX Y . Second, demand for real balance will increase by Gd L y . If

the domestic agents whose demand for real balance has gone up try to sell domestic bonds to

increase their holding of real balance, price of domestic bonds will fall raising the rate of return

on domestic bonds. This will make domestic bonds more attractive than foreign bonds. Hence,

they will sell foreign bonds and use the foreign currency received to buy domestic currency.


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Thus, importers will want to purchase an additional eP

Gd NX Y amount of foreign currency

and those who have sold foreign bonds will want to sell an additionaleP

Gd L y amount of

foreign currency. Importers know that if they sell PGd NX Y . amount of domestic currency to

secure eP

Gd NX Y amount of foreign currency, their stock of real balance will go down by

Gd NX Y and they will have to sell foreign bonds to replenish their stock of real balance. So,

they will sell off foreign bonds right at the beginning to finance their import. Thus, in the foreign

currency market there will emerge an excess supply of foreign currency ofeP

Gd L y . The central

bank will buy up this excess supply with PGd L y amount of domestic currency. Therefore, the

supply of real balance will increase by Gd L y . Domestic economic agents have sold

eP

Gd L y + eP

Gd NX Y worth of foreign bonds either to augment their stocks of domestic

money or to prevent their stocks of domestic money from declining below their desired levels.

Therefore, they have soldeP

Gd L y + eP

Gd NX Y worth of foreign bonds to invest their sales

proceeds in domestic money. HenceeP

Gd L y + eP

Gd NX Y constitutes an increase in the

inflow of capital in terms of foreign currency. Thus, K goes up by Gd L y + Gd NX Y . This is

the end of the first round. As Y increases by Gd in the first round, people’s disposable income

rises by Gd . Hence their consumption demand for domestic goods will go up by

Gd NX C Y . There will thus emerge an excess demand in the domestic goods market and

producers will increase Y by Gd NX C Y . This increase in Y , as before, will raise the


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supply of real balance by Gd NX C L Y y and K by Gd NX C NX L Y Y y . This

process of expansion will continue until the increase in Y falls to zero. When that happens, the

economy achieves the new equilibrium. Thus, the total increase in Y ,P H

and K are given

respectively by

dY Gd + Gd NX C Y + Gd NX C Y 2 + Gd NX C Y 3 + ……………

= Y NX C

Gd 1

P H

d = Gd L y + Gd NX C L Y y + Gd NX C L Y y 2 +…………………….

= Y y NX C

Gd L

1

dK Gd NX L Y y + Y y NX L Gd NX C Y +

Y y NX L Gd NX C Y 2 + ………………………………..

= Y y NX L Y NX C Gd

1

This explains the adjustment process.

m_f model fixed e rate

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