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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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2/6
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
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