Exchange Stabilization Once Again

Exchange Stabilization Once AgainAuthor(s): Harry C. Eastman and Stefan StykoltSource: The Canadian Journal of Economics and Political Science / Revue canadienned'Economique et de Science politique, Vol. 24, No. 2 (May, 1958), pp. 266-272Published by: Wiley on behalf of Canadian Economics AssociationStable URL: http://www.jstor.org/stable/138774 .

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Canadian Journal of Economics and Political Science

EXCHANGE STABILIZATION ONCE AGAIN

HARRY C. EASTMAN and STEFAN STYKOLT

University of Toronto

THE point of substance in Mr. Wonnacott's note is illustrated in his Figure 2 which reveals that, when a fund counters all movements in the observed rate, some stabilization is achieved.1 Where the path of the free rate is a sine curve, the stabilized path of the observed rate will also be a sine curve if the fund counters every change in the observed rate, no matter how small, and if the

magnitude of the reaction of the fund is a function of the difference between the observed rate and the free rate as it would have existed at that moment had no intervention taken place. As this difference tends to zero, so does the size of the fund's reaction. The peak and trough of the observed rate will be on the path of the free rate, but to the right of the latter's peak and trough and therefore below and above them respectively, as is shown in Figure 1.2

The free rate of exchange 3 is

(1) x = A sin wt.

The observed rate has to be determined in the simplest case from

d - k(x - y). dt

To find y requires the solution of the non-homogeneous differential equation

dt + ky = Ak sin oct, dt

which is

y = Cie-t + (w + _ 2) sin (wt - a),

1Paul Wonnacott, "Exchange Stabilization in Canada, 1950-4: A Comment" in this issue of the JOURNAL, pp. 262-5. The point Mr. Wonnacott raises with respect to net selling and buying at the peaks and troughs was made by our draughtsman when he protested that the shifted curve C in Figure 1 of our original article was not a true sine curve. See our article, "Exchange Stabilization in Canada, 1950-4," this JOURNAL, XXII, no. 2, May, 1956, Figure 1, p. 222.

2In the light of this analysis the point raised in Mr. Wonnacott's footnote 7 loses all relevance because, as is shown above, the rate is stabilized even if no delay occurs between the moment the fund ceases buying and begins selling (or vice versa). Even if it were relevant, this apparently difficult point does not throw doubt on our proposition that "stabilization of a rate of exchange implies knowledge of a normal rate" since the fund is acting as if it knew the normal rate in the case he discusses, whether or not it does in fact know. If it does not, it is being le medecin malgre' lui.

3Professor J. A. Steketee of the Department of Applied Mathematics, University of Toronto, translated this and the following part of our argument into mathematical terms. We wish to thank him for his interest in applying his skills to an economic problem and for his patient co-operation.

Vol. XXIV, no. 2, May, 1958

266



y(observed rate)

// 7-1 7-

TIME

(t

FIGURE 1

0




where a is determined from

C k sin a = /( k2) COS : = Co /(2 a k) V/(c + k ) (

2 + 2)

For large t, the exponential part can be omitted and we have the observed rate

Ak (2) y = (W- 2 k) sin (t - a).

As k <

V/("2 + k2)

the amplitude of y is always less than the amplitude of x. The points of intersection of the curves (1) and (2) are easily found from

dy - dt

which gives

ot - a = 2 + nr. (n = 1, 2,3,... )

The fund makes a profit because it is a net seller when the rate is high and a net buyer when the rate is low. This is because the turnover of foreign exchange by the fund is equal below and above the normal level of the rate, but the rate at which each quantity is bought or sold varies in such a way as to leave the fund with a profit in terms of Canadian dollars at the end of the fluctuation.

The Area A CB is F I.

+7r/2[ k F- = A s [) sin t (- a) - d(wt)

JaLor+ K1 L (CNC + k )

a+7r/2[ k(2 ( 1

= A 1 sin c - 2 k2) sin ( -a) d

_ k cs, - 2 = A -coS + COS (+o - a)

= [cos a - cos (a i) 2' k2)].

The Area ADC is Fn.

Fn = A sin 2k sin ( - a) d Ja-r/2 [ V/(0 + k )

A [- cos 0 + / k2- cos ( C -oSa) 2 A[cos(a \I/() +

- kc ) a-ra/2+ 2 .

A [COS - 2--COS0 v( 2 + k2) i -

268



Notes and Memoranda

But k

COS a = /(2 + k2)

and

cos ( +a)s = - - c -os (a -

from which it follows that FI = FI.

If the turnover is

dy dt

or

k(x - y)

and the price is constant p + y, then the integral

Jrdt(p +y)d(t)

would be about the total amount of profit or loss. This integral has to be computed.

t'~)r = "+'4,/2dydy p= if d- (p + y)d(wt). Pi = y)d( Pt).

da dt -,/2dt

Now dy Akco

dt -/(, + k2) cos (ot - a). Let

Ak

P-V = + k2) Then

A 2k2 , a+Cr/2 P I 2 +k24- j [p* + sin (4 - a)] cos (4> - a)d4. co + k a!

2 2 Akw pii 2 22k^ [p?* + sin ((- a)] cos (~ - a)do. Ct + J a-T/2

This is

r/2 7r/2

+ [p* + sin 0]cos d0 d = p* sin 0 + ? sin2 0 = P* -+ [p*0 + sin 0] cs 0 dn p* ir, + si 0

J [p* + sin 0] cos d = p* sin 0+ sin2 0 = p* - ? . _-r/2 --r12

So we have 2 2

PI = A 2(P + o);

P A2 (PkE 2 Pii- 2+k2(P - )'

269




A2k20 PI - PI = 2

This shows that the profit is proportional to A2 and also depends on

k2w 2 + k2

which is practically equal to w for big values of k (k>w). In the real world, the problem facing a fund cannot be resolved so simply.

Although the fund does not know from moment to moment what the free rate is, it does know the direction in which the free rate is moving. The fund obtains an impression of the behaviour of the free rate by comparing, over a

period of time, the rate of change in the observed rate and the amount of the fund's own interference. The more it takes to limit the rate of change in the observed rate, the greater is the presumed movement in the free rate.

The fund cannot allow itself to be guided by the difference between the observed rate and what it knows of the free rate since at any moment a change in the free rate may reflect, not a random movement, but a change in the trend of the normal rate. In the latter case, intervention would result in chronic purchases or sales of foreign exchange and interference with the trend. Hence, in the real world, the fund ought never to intervene strongly. The best it can do, unless, of course, it can forecast the normal rate, is to keep the observed rate quite close to the free rate. It will reduce fluctuations only slightly and the size of the profits will indicate the amount of stabilization performed.

Thus our original point that an effective policy of stabilization requires a

knowledge of the normal rate stands. Our claim that no stabilization takes

place if all movements are countered by the fund needs to be qualified: some stabilization is achieved by such operations, as is described above. However, because of the danger of going against the trend, as actually happened in Canada in 1951-2, the fund is unable to intervene strongly and therefore

profits are highly relevant in evaluating the effectiveness of its operations. This

explains the seeming paradox of the fund's behaving as if it were one of a

large number of competing traders even though its resources are sufficient to

give it some monopoly power. The other questions raised by Mr. Wonnacott refer either to assumptions or

matters of exposition and do not advance the argument. His discussion of the transition from one normal level to another raises the problem of defining the normal rate and suggests that definition is a matter of personal inclination, whereas it consists of selecting a period coextensive with the time horizon of

operators in the foreign exchange market.4 If the rates K and N in Mr. Wonnacott's Figure 1 are normal rates according to our definition, they are not

subject to stabilization in the sense of reducing fluctuations. If they are one- half of a long-term fluctuation, then stabilizing action would consist of net

selling when the rate was above normal and net buying when it was below.

4Eastman and Stykolt, "Exchange Stabilization in Canada, 1950-4," 221.

270



Notes and Memoranda

Mr. Wonnacott's remaining analysis follows and extends the criticisms of Professor Kindleberger5 which are treated below. Professor Kindleberger agrees that a speculator in a market in which pure competition exists and in which expectations are independent (and, presumably, have an elasticity of zero or at least of less than unity) will stabilize price most when he makes most profits. Though Professor Kindleberger cannot imagine a stabilization fund in that position, it may nevertheless be useful to treat the situation as if it were such in order to answer certain questions. Indeed, the assumption made in our original article that the Canadian Exchange Fund Account

operated as if it were a competitor was made in order to make the model conform to the information available about the real world. A good deal of the point of the article was to explain that this self-denial on the part of the fund was the result of a mistaken strategy.6 Mr. Wonnacott does not grasp this reason for making the assumption that the fund does not influence the rate appreciably. In addition, he falls victim to the fallacy of composition. Although an individual firm in perfect competition is held to have no influence on price, its adjustment to price, in aggregation with other similar adjustments, does of course, influence price. It is in this sense that a successful speculator in a competitive market can be said to have a stabilizing influence.

Professor Kindleberger continues by contrasting the case of pure competition and independent expectations with one in which expectations are "linked," and in which the speculator who makes the most money is the one who destabilizes most. This raises the question of the usefulness in this context of the concept of expectations that have an elasticity exceeding unity or that are "linked."

If expectations have an elasticity of more than one, the rate will move

exponentially as a consequence of any change unless some special assumption is made to prevent the disorganization of the market. Such models lack

generality. Professor Kindleberger's first model, in which such explosive movements are set off by the transactions of one speculator is even more

special. What "linked" expectations mean other than this is not clear. The

phrase "... if he is therefore conscious of his impact on prices" suggests that the speculator can somehow influence psychologically the expectations of others so as to cause price movements in the direction he desires.

In his discussion of expectations, Mr. Wonnacott follows Professor Kindle-

berger in attempting to make the arbitrary and unrealistic assumption that the operations of the fund have a different and more stabilizing effect on the

psychology of the market than have the operations of other traders; but he then concludes in favour of the general assumption which, as he rightly recognizes, we did make: that the observed rate returns to the path of the free rate when official interference ceases. In his third paragraph, Professor Kindleberger writes: "With a trend, as Eastman and Stykolt have pointed out, the profits criterion is meaningless because of the revaluation of initial assets

5C. P. Kindleberger, "Exchange Stabilization Further Considered: A Comment," this JOURNAL, X(III, no. 3, Aug., 1957, 408.

6See Eastman and Stykolt, "Exchange Stabilization in Canada, 1950-4," 223-5.

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Canadian Journal of Economics and Political Science Canadian Journal of Economics and Political Science

and the arbitrariness of the accounting period." This overstates our position, but in any case, the feasibility of a statistical calculation of profits from ex

post data which include the revaluation of assets should not be confused with a priori reasoning about the behaviour of speculators necessary to stabilize a rate and with the a priori meaning of profits. While we discussed the difficul- ties of the former, we did not cast doubts on the latter.

Professor Kindleberger makes three statements about profits from stabilization: that profits are misleading or irrelevant, that they are zero when stabilization is at a maximum, and that a stabilization fund that seeks to maximize

profits must permit or encourage instability. Profits are not irrelevant and they may be misleading only to the extent explained in our note "Exchange Stabilization Further Considered."7 Maximum profits do not mean maximum stabilization when the fund exercises monopoly power and zero profits mean either no stabilization or complete stabilization. The last case should not be difficult to identify, but neither should it be given undue importance since the reason for adopting the Canadian Exchange Fund Account's policy of

countering movements in the rate is precisely that it proved impossible to

gauge the normal level of the rate. Stabilization with losses is only possible if the fund pushes the rate to the opposite side of the normal rate from which it would otherwise be, but not as far. This is only accomplished if the fund reverses the movement in the free rate, not if it only counters all changes.

In his last paragraph, Professor Kindleberger attributes to us a criterion of stabilization that we must disclaim. We held that stabilization consists in

buying when prices are low and selling when they are high. The criterion he

adopts is "whether a ... fund operates contrary to the tendency of the market which reverses itself. .. ." This is presumably the case in which the intervention of a fund causes the expectations of private speculators to reverse themselves, the fund being cast in the role of a deus ex machina.

7This JOURNAL, XXIII, no. 3, Aug., 1957, 404-8.

ENERGY SOURCES IN CANADA: A FURTHER COMMENT

IRENE M. SPRY

London, England

FOR a comprehensive picture of the importance of water-power as compared with other sources of energy in Canada, of the extent of electrification of industries and homes, and of the uses to which electricity is put, the following government publications give essential materials: Dominion Bureau of Statistics, Industry and Merchandising Division, Reference Paper No.

69, Energy Sources in Canada: Commodity Accounts for 1948 and 1952 (1956). D.B.S., Industry and Merchandising Division, Reference Paper No. 74, Energy Sources in

Canada: Commodity Statements for 1926, 1929, 1933 and 1939 (1957). D.B.S., "Canada's Changing Energy Pattern: Part 1, The Supply and Distribution of

Individual Commodities in 1952; Part 2, The Pattern in 1952; Part 3, Changes from 1926 to 1956," Canadian Statistical Review, XXXII, no. 7, July, 1957, i-v; XXXII, no. 8, Aug., 1957, i-vi; XXXII, no. 10, Oct., 1957, i-v. (These three articles are now available together in a special reprint.)


and the arbitrariness of the accounting period." This overstates our position, but in any case, the feasibility of a statistical calculation of profits from ex

post data which include the revaluation of assets should not be confused with a priori reasoning about the behaviour of speculators necessary to stabilize a rate and with the a priori meaning of profits. While we discussed the difficul- ties of the former, we did not cast doubts on the latter.

Professor Kindleberger makes three statements about profits from stabilization: that profits are misleading or irrelevant, that they are zero when stabilization is at a maximum, and that a stabilization fund that seeks to maximize

profits must permit or encourage instability. Profits are not irrelevant and they may be misleading only to the extent explained in our note "Exchange Stabilization Further Considered."7 Maximum profits do not mean maximum stabilization when the fund exercises monopoly power and zero profits mean either no stabilization or complete stabilization. The last case should not be difficult to identify, but neither should it be given undue importance since the reason for adopting the Canadian Exchange Fund Account's policy of

countering movements in the rate is precisely that it proved impossible to

gauge the normal level of the rate. Stabilization with losses is only possible if the fund pushes the rate to the opposite side of the normal rate from which it would otherwise be, but not as far. This is only accomplished if the fund reverses the movement in the free rate, not if it only counters all changes.

In his last paragraph, Professor Kindleberger attributes to us a criterion of stabilization that we must disclaim. We held that stabilization consists in

buying when prices are low and selling when they are high. The criterion he

adopts is "whether a ... fund operates contrary to the tendency of the market which reverses itself. .. ." This is presumably the case in which the intervention of a fund causes the expectations of private speculators to reverse themselves, the fund being cast in the role of a deus ex machina.

7This JOURNAL, XXIII, no. 3, Aug., 1957, 404-8.

ENERGY SOURCES IN CANADA: A FURTHER COMMENT

IRENE M. SPRY

London, England

FOR a comprehensive picture of the importance of water-power as compared with other sources of energy in Canada, of the extent of electrification of industries and homes, and of the uses to which electricity is put, the following government publications give essential materials: Dominion Bureau of Statistics, Industry and Merchandising Division, Reference Paper No.

69, Energy Sources in Canada: Commodity Accounts for 1948 and 1952 (1956). D.B.S., Industry and Merchandising Division, Reference Paper No. 74, Energy Sources in

Canada: Commodity Statements for 1926, 1929, 1933 and 1939 (1957). D.B.S., "Canada's Changing Energy Pattern: Part 1, The Supply and Distribution of

Individual Commodities in 1952; Part 2, The Pattern in 1952; Part 3, Changes from 1926 to 1956," Canadian Statistical Review, XXXII, no. 7, July, 1957, i-v; XXXII, no. 8, Aug., 1957, i-vi; XXXII, no. 10, Oct., 1957, i-v. (These three articles are now available together in a special reprint.)


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Exchange Stabilization Once Again