On Buying Cheap and Selling Dear: Another Note

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: Wiley on behalf of Canadian Economics AssociationStable URL: http://www.jstor.org/stable/139832 .

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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 intervention 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 amplitude 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 respectively 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 intervention 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."



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 n2 + 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 transactions 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'On Buying Cheap and Selling Dear," 434.



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



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 < 1. Thus, for any given n, R2 may have a smaller amplitude than R1 if w is large enough, but stabilization in model 2 can never be as efficient as in model 1. If private short-term capital movements resist (R - N), achievement of maximum efficiency requires that official intervention also resist (R - N).

Now turn to another pair of models, 3 and 4. In both of these, all private short-term capital flows resist DIR. That is mc = 0, min = m. In model 3, nc = n, n, = 0; in model 4, n, = 0, n, = n.

A3 = k/Vf (h + n)2 + M2W2}

E3 = [V/{(h + n)2 + M2w21 - h]/(mw + n) A4= ki Vt h2 + (m + n)2w2}

E4 = [V/{h2 + (m + n)2w2} - h]/(m + n)w A3 t A4 as W2 t (2h + n)/(2m + n).

Both E3 and E4 are less than one, but the condition determining which is larger apparently does not reduce to a simple statement in terms of h, m, n, and w. Numerical examples show that either efficiency may be the greater one. For instance, if w = 1, E3 t E4 as h i m. The conclusion is, if private short-term funds resist DIR, an efficiency of unity is not possible, but attain- ment of the highest possible efficiency may require that official transactions resist either (R - N) or DIR.

One reason for this conclusion lies in the problem of minimizing the waste of official transactions on the cancellation of other stabilizing influences. If official trading resists (R - N), part of it will be wasted in offsetting some of the private short-term capital movements resisting DIR. On the other hand, if official trading resists DIR, part of it will be wasted in offsetting some of the stabilizing effect of the private transactions, described by -hR, which are related to the absolute level of the exchange rate. In general, the waste will be minimized if the official trading reinforces whichever is the stronger sort of stabilizing behaviour in the private sector of the market. There is another possible reason for the above conclusion. Eastman has pointed out that the most efficient stabilization occurs when all short-term flows resist (R - N), because only then are movements of these funds exactly in phase with fluctuations of the rate which would have existed in the absence of stabilization. In



570 T. L. POWIRE

the present case where private funds resist DIR, their movement is out of phase with the unstabilized rate. The phase lag, (B - a), of net total short- term capital flows depends on whether official intervention is related to (R - N) or to DIR. (See equations 5 and 12.) Depending on the sizes of the parameters, either kind of official intervention could be more favourable or less unfavourable with respect to the phasing of short-term capital flows.

In correspondence, Professor Eastman has expressed reservations, which I share, about the relevance of model 4 as a simplified description of the Canadian foreign exchange market under a flexible rate. Private short-term capital movements were statistically related to changes in the exchange rate, so one might suppose that model 4's mDzR is a useful simulation of their behaviour. However, the real world private capital movements may not have been caused by changes in the exchange rate. To the extent that they were not so caused, they must be reflected, in the present model, in the parameter k, not in m. A model in which m = 0 might be a better approximation of reality than is model 4. Another important empirical question is the size of h. As h and m both approach zero, E3 and E4 both approach one, in which case both forms of official intervention would be about equally efficient.

FIXED FACTOR PROPORTIONS IN CLASSICAL ECONOMICS: A NOTE

BRIN BIXLEY University of Toronto

It is common to find in modern critical commentaries of classical economics the assertion that members of the Classical School assumed "fixed factor proportions." Professor Blaug has claimed that "Ricardo, Mill and Marx treated all commodities as produced under conditions of constant costs and fixed technical coefficients."' In a recent article on Mill, Professor Hollander has written: "I have interpreted Mill's comments on equipment and materials as referring not merely to the notion that production requires the presence of inputs in addition to labour, but more specifically to the assumption that factors must be used in fixed proportions."2

The great contribution of marginal productivity analysis was to make factor proportions continuously variable in given states of technological knowledge; indeed, without some such assumption the concept of marginal productivity is meaningless.3 Tlis is contrasted with supposed classical assumptions:

This formulation of the principle of variation of proportions as a general rule govern- ing all resources is one of Menger's greatest achievements. ... Classical theory

1M. Blaug, Economic Theory in Retrospect (Homewood, 1962), 277. 2S. Hollander, "Technology and Aggregate Demand in J. S. Mill's Economic System," this JOURNAL, XXX, no. 2 (May 1964), 177. 3If we are thinking of the marginal productivity of a factor in a particular line of employ- ment. If alternative occupations are open to the factor then it is possible to derive a marginal productivity curve (demand curve) for that factor, even though the factor is used in fixed (but different) proportions in the alternative occupations.



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On Buying Cheap and Selling Dear: Another Note