8/11/2019 cirmes

1/6

On Buying Cheap and Selling Dear: Another NoteAuthor(s): T. L. PowrieSource: The Canadian Journal of Economics and Political Science / Revue canadienned'Economique et de Science politique, Vol. 31, No. 4 (Nov., 1965), pp. 566-570Published by: Blackwell Publishingon behalf of Canadian Economics AssociationStable URL: http://www.jstor.org/stable/139832.

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8/11/2019 cirmes

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NOTES

ON

BUYING

CHEAP AND SELLING DEAR: ANOTHER

NOTE*

T. L. POWRIE

University of Alberta

How

do transactions which

resist

all

movements

in a

flexible exchange

rate

affect

the

stability

of the rate? Professor Eastman

has

improved

the

answer

to

this question

in

a recent note.' This note

is

an extension

of

the

same topic,

in order to show that the effect of official intervention in the market depends

on

the behaviour

of

private

short-term

capital

movements.2

It will be shown

that,

to achieve

the most efficient stabilization,

the type of official inter-

vention

chosen

must

depend

on the

behaviour of

private

funds.

Let

excess demand

in a

foreign

exchange market

be described

by

(1)

g-hR + k sin

wt-(m,

+

n,)(R-N)-(m,

+

n,)DtR.

R

is the

exchange rate,

and

g,

h,

k,

w,

m., ne,

ms,

and

n.

are

constants, each

not less

than zero.

The term

(g

-

hR)

describes excess demand

in the

absence

of fluctuations in the market. Sinusoidal fluctuations in demand are intro-

duced

by (k

sin

wt),

where

t is

time,

and k and w are

respectively

the

ampli-

tude

and

the

frequency

of the fluctuations.

The

term

(m,

+

n,)

(R

-

N)

introduces

transactions

which resist deviations of the

exchange

rate from its

normal

or

average

value

N.

The

constants

m,

and

n,

are

the strengths

with

which private

short-term

capital

movements

and official

intervention

re-

spectively

resist

(R

-

N).

Finally, (mi, + n8)DtR

describes transactions which

resist

all

changes

in the

exchange

rate,

with m8 and

n,

being

the

strengths

respectively

of

private

short-term

capital

movements

and of official

inter-

vention in this direction. By defining each of m ,

m8,

n., and n, to be non-

negative,

we

are

in

effect

excluding any

discussion

of

short-term

capital

movements

which

aggravate

exchange

rate fluctuations.

*I am

much

indebted to my colleague

Dr. W. Haque,

for

patient

guidance

to the

mathematics

required

for

this

note.

'H. C.

Eastman,

On

Buying

Cheap

and

Selling

Dear: Professor

Powrie's

Paradox,

this

JOURNAL,

XXX,

no.

3

(Aug.

1964),

431-5.

2The second

last

paragraph

of

Eastman's

note

is

based

on the

incorrect

premise that the

behaviour

of private

short-term capital

has no relevance

for

the effect of official

intervention.

There

also seems to

be a small ambiguity

in Eastman's

note,

in that his distinction

between

the

observed,

stabilized

rate

of

exchange (call

it

R)

and the rate which

would have existed

in

the

absence

of

stabilizing

influences

(call

it

R*)

is not

consistently expressed.

In

paragraph

two,

rates

of

exchange

must

mean

R* to be

correct;

in

the

next several

paragraphs,

an

unqualified

reference

to the rate

clearly

means

R;

in the third

last

paragraph

there is

the

implication

that the rate meant

R*

on

page

221 of Eastman

and

Stykolt,

Exchange Stabi-

lization

in Canada, 1950-54

(this JOURNAL,

May

1956),

for

the

paragraph

does not make

sense

otherwise.

Such

a minor

lapse

from

clarity

could

pass

unnoticed, except

that it

suggests

a

theory

to

explain

an

otherwise

puzzling

point.

Eastman's

note

validates

and

extends my

discussion

of the

topic,

but attributes

to

my

treatment

such features

as unreal distinction

and error.

The

puzzle is,

why

was he

saying

I was

wrong

while

he

was

proving

I

was

right?

The explanatory

hypothesis

is

that where

I wrote the rate,

he sometimes

read

the rate

that would

have

existed

in

the absence

of stabilization.

8/11/2019 cirmes

3/6

Notes

567

To

find

the

equilibrium

exchange

rate, set

the

excess

demand

function

equal

to zero

and

note

that

N

=

g/lh.

Then

(2)

R

=

N

+

fkl(h

+

m.

+

ne)

I

sin wt-{

(m.

+

nz)/(h

+

m,

+

n,)IDR.

The

solution

of

this

differential

equation

is

(3)

R-N +

A sin(wt-

a)

where

(4)

A

-

\k/V{

(h

+

mc

+

n

)2

+

(ms

+

n,)2

W2}

and

a

is

determined

by

(5)

tan

a

=

(min + n8)w/(h

+ mc + n0).

(The

solution

also contains

a

transient

term which

approaches

zero as t in-

creases

and

which

is

ignored.)

Let

S

be

the net

total sales

of

foreign

currency

arising from all

private

short-term capital

movements

and official transactions.

(6)

S =

mi(R

-

N) +

miDjR

+

n,(R

-

N) +

n3DgR

(7)

=

G

sin(wt

-

a

+ ,B')

+ H

sin(wt

-

a

+

3 )

..

.(by

(3))

where

(8)

G

=

kV(M

2

+

mi22w2)/\/f

(h +

Mc

+ n,)2

+

(Mn,

+

n,)2

w2}

and

(9)

H

=

kV\(n.2

+

n,2

w2)/v{

(h

+ mi,

+

n)2

+

(mi,

+

n,)2

w2A

and

,B'

and

p3

are determined

by

tan

,3'

=

m,w/m,,

tan

p

=

n,w/n,.

An alternative equation for S, which consolidates official and private tran-

sactions

into one

net

expression,

is

(10)

S

=

Fsin(wt-a

+,)

where

(11)

F =

kV/{(m,

+

nC)2

+

(m.,

+

n,)2w2/2/{(h +

min

+

n,)2

+

(ms

+

ns)2

UP)

and

3

is

determined

by

(12)

tan 3

=

(mi,

+

n,)w/(m,,

+ n,).

Now

we need

a

measure of

the

efficiency,

as

stabilizers of the

exchange

rate,

of

these

S transactions.

Eastman

has

provided

the

conceptual

basis

for

the

measure:

the smallness

of the

capital

flow

that

achieves

a

given

reduction

in the

amplitude

of

fluctuations in

the rate. 3

To

adapt

this concept

to the

present

problem,

first

set

up

a

special

model x

as

a standard

of efficiency. In

36'OnBuying

Cheap

and Selling

Dear,

434.

8/11/2019 cirmes

4/6

568

T.

L. POWBIE

model

x,

all short-term capital

movements

resist

deviations

of

the exchange

rate

from its

normal value.

Excess

demand

in model

x is

(13)

g - hR

+

k sin wt

-

e(R

-

N),

where

the subscript

x

identifies

the model

and

e is the strength

of

short-term

capital

movements.

The amplitude

of Rx

is

(14)

Ax

=

k/(h

+

e)

and

the

amplitude

of

short-term

capital

flows

in the

model

is

(15)

F_

=

ke/(h + e).

Now

set

A

(from

equation

4) equal

to

Ax

(equation

14)

and

solve

for e to

find

(16)

e

= -/{ (h + m, +

nC)2

+

(ms

+

n)2 w2}

-

h,

which

is the value

of

e

required

to make

A.

= A. Putting

this

value

of

e

into

equation

15,

we

get

(17)

Fx

=

[kV{

(h

+

mc

+

nc)2

+

(ms

+

nf)2

W2}

-

kh]/{ (h +

m,

+

nf)2

+

(mi

+

ns)2

w2},

which

is the value

of

F,

required

to make Ax

= A.

Let

the

measure

of the

efficiency

of

stabilizing

short-term

capital

movements

be

(18) E = Fx/(G + H) = [N/ (h

+

mc

+

nf)2

+ (m, +

nS)2

w2}

-

h]/[V/(mc2

+ Min2

w2) +

V(n 2

+ n82

W2)].

Terms

G

and

H

come

from

equations

8 and

9. Their

sum

is

used

instead

of

F

from equation

11 because

F

conceals

a

form

of

inefficiency

in

the

general

model.

In the

general

model,

G is the

amplitude

of net

private

short-term

transactions

and

H is the

amplitude

of net official intervention.

Since

these

two

sets

of transactions

may

have different

phasing,

they

may

in

part

merely

offset

each other

without

affecting

the

exchange

rate.

This

partial

cancellation

of

effect

is

inefficient,

but

the wasted transactions

are not reflected

in the size

of

F,

the net

amplitude

of all

short-term capital

flows.

To include

these

wasted

transactions

in the measure

of

efficiency,

G

+

H,

the

gross

amplitude

of

short-

term

capital

flows,

is used

in the measure.

(The

term

gross

amplitude

is

used

for

want

of a

better,

but

note that

its

components,

G

and

H,

are

both

net

amplitudes.)

The

larger

is

E,

the more

efficient

is

the

model.

In

words,

efficiency

is

greater

if

exchange

rate fluctuations

are

limited to

any

given

amplitude

by

smaller

gross

short-term

capital

flows. The

index

of

efficiency

E

equals

one

when

all

short-term

capital

flows resist

deviations

of the

exchange

rate from

its

normal

value.

Let

m

=

m,

+

Ms,

m being

the total strength

of all

private

short-term

capital

flows.

Similarly,

let n

=

n,

+

n8,

n

being

the total

strength

of

official

intervention.

Now

we can consider

a few

particular

cases

of

the above

general

model.

First consider

a

situation

in

which

all

private

short-term

capital

movements

8/11/2019 cirmes

5/6

Notes

569

resist (R

-

N),

that

is,

where

m,

= m and

m,

=

0.

In

model

1,

let

official

transactions

also

be

entirely

devoted

to

resisting (R

-

N),

that

is,

let

n0

=

n

and

n,

=

0.

In model

2,

let

official

transactions

resist

only

DgR,

that

is,

let

n.

=

n and n0

=

0.

The

values

of A

and

E

can

be obtained for each

model

simply by putting

the

appropriate

values

of

m,

mi8,

nc,

and n8 into

equations

4 and

18

(subscript

numbers

identify

the

model):

Al

=

k/(h

+

m

+

n)

E1

=

[V/

(h

+

m

+

n)21 -hI/(m

+ n)

A2

=

k/Vt

(h

+ m)2

+

n2W2}

E2

=

[V/t(h

+

M)2

+

n2W2I

-

h]/(m + nw)

AI

is

greater

or less

than

A2

as

w2

is

greater

or less than

1

+ 2(h +

m)/n.

However,

E1

is

always greater

than

E2

since

E1

=

1 and

E2