Article

The Divergence Indicator: A Technical Note

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1981
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In july 1978, the heads of government of the members of the European Economic Community agreed to the establishment of a “scheme for the creation of a closer monetary cooperation (European monetary system) leading to a zone of monetary stability in Europe.” 1 On March 13, 1979, the European Monetary System (EMS) came into existence and replaced the previous snake arrangement.2 More than a regional zone of fixed parities, the EMS is intended to further economic alignment among participating countries. An element in achieving this alignment is the operation of the divergence indicator, “designed to detect Community currencies that happen to deviate upward or downward from the Community average as represented by the ECU.” 3 When a currency crosses its threshold of divergence, there is the presumption that the authorities will implement corrective measures. Appropriate responses include diversified intervention in the foreign exchange market and domestic economic policy measures.

In principle, the divergence indicator allows the burden of balance of payments adjustment to be distributed more symmetrically under the EMS than it was under the snake arrangement. Under the latter, two countries were obliged to intervene mutually to prevent the rate of exchange between their currencies from deviating from parity by more than 2.25 per cent, but unless the central rate was to be changed, the prospective depletion of the reserves of the deficit country would require it to accommodate its policies to the rate of inflation in the surplus country. While the requirement to stay within the parity band continues under the EMS, the associated bias toward the policy stance of low-inflation countries is tempered by the operation of the divergence indicator, which is supposed to favor the policy stance most widely adopted by member countries. Accordingly, under EMS rules a country—either surplus or deficit—with a “divergent” currency must accommodate its policies to those of its partners.

Although the basic principle underlying the divergence indicator is clear enough, it is less clear how the actual specification of the indicator leads to the identification of “divergent” currencies under various circumstances. Here, there are two questions of general interest that this paper addresses. First, does the design of the divergence indicator lead to unbiased treatment of all member countries? Second, how does the design further the EMS objective of greater “economic alignment” among participating currencies? The issue of bias is handled by comparing the divergence indicator’s treatment of large versus small countries. Specifically, while it has been argued that the method of computation ensures that the triggering of the indicator is “independent of the relative weight of each currency in the ECU,” 4 it is not clear in what sense this is true. The paper investigates whether and precisely how the conditions that trigger the divergence indicator can vary with the size of the country.

The efficacy of the divergence indicator in promoting greater economic alignment is assessed in terms of the divergence indicator’s treatment of coalitions, that is, two or more member currencies whose exchange rates tend to move together vis-à-vis other member currencies. Presumably, some favoring of coalitions—in the sense of allowing them relatively more divergence—would be consistent with the aim of the EMS to foster greater monetary cooperation. Based on this view, individual currencies moving in tandem might be “rewarded” by being allowed more divergence than the sum of their weights would otherwise warrant. The associated diminution in the requirement that coalition members take adjustment measures would be mirrored in an increase in this requirement among noncoalition members, and would, in effect, give coalition members a greater voice in setting the pace of economic activity within the EMS.

The sequence in which these questions are analyzed is as follows. Section I reviews the basic elements of the divergence indicator, that is, how the country weights are determined, how divergence is measured, and how a country’s threshold of divergence is defined. Section II considers the proposition that the triggering of a country’s divergence indicator is independent of relative country weights. To make intercountry comparisons of the conditions under which a country’s divergence indicator is triggered, a method for quantifying these conditions is developed. The analysis in Section II indicates that while some conditions would trigger the divergence indicator of either a large country or a small country, other conditions would trigger the divergence indicator for a small country but not for a large country. Section III considers the implicit treatment of coalitions of member currencies and shows that the formulation of the divergence indicator favors coalitions. Section IV summarizes the analysis and briefly sketches its implications.

I. Mechanics of the Divergence Indicator

At the center of the EMS is the European Currency Unit (ECU). The ECU is a composite unit consisting of specified amounts of the currencies of all member countries. The following equation illustrates the valuation of the ECU in terms of Netherlands guilders. (See Table 1 for notation.)

Table 1.Composition of the European Currency Unit
CurrencyUnits of CurrencyWeights1
Deutsche mark (DM)a1 = 0.828w1 = 34.50
French franc (F)a2 = 1.15w2 = 18.63
Pound sterling (£)a3 = 0.0885w3 = 14.582
Netherlands guilder (f.)a4 = 0.286w4 = 10.74
Italian lira (Lit)a5 = 109.0w5 = 8.38
Belgian franc (BF)a6 = 3.66w6 = 8.98
Danish krone (DKr)a7 = 0.217w7 = 2.74
Irish pound (£Ir)a8 = 0.00759w8 = 1.11
Luxembourg franc (Lux F)a9 = 0.14w9 = 0.34

As percentage shares, on the basis of central rates.

The United Kingdom does not participate in the exchange rate and intervention mechanism of the EMS, but sterling is included in the calculation of the ECU. Its weight is based on the notional central rate agreed to on October 4, 1981.

As percentage shares, on the basis of central rates.

The United Kingdom does not participate in the exchange rate and intervention mechanism of the EMS, but sterling is included in the calculation of the ECU. Its weight is based on the notional central rate agreed to on October 4, 1981.

All currencies are expressed in terms of their central values in guilders. Thus, in equation (1) and in all subsequent equations, ECU stands for the central guilder value of the ECU, DM for the guilder value of the deutsche mark, F for the guilder value of the French franc, etc. The central value of the guilder in terms of guilders is, of course, one.

The a’s in equation (1) represent the number of units of currency included in the valuation of the ECU. By dividing both sides of equation (1) by ECU, it is possible to derive the EMS weights, that is, the shares of the respective currencies in the ECU. Thus,

where the country weights (i.e., the w’s) are defined as follows:

While the a’s (i.e., the number of units of each currency) are subject to review every five years, they are not adjusted in response to changes in the configuration of central rates of member currencies. Consequently, the w’s rise and fall pari passu with the currencies’ central values. Table 1 gives the a’s and w’s for the participating currencies. The same schedule of a’s has been in effect since the inception of the EMS in March 1979. The schedule of w’s included in the table was agreed to on October 4, 1981.

Under the snake arrangement, there was no explicit treatment of divergence. Rather, the intervention rules allowed a currency to deviate from parity vis-à-vis any and all other participating currencies by just under 2.25 per cent; when the parity band was crossed, currency intervention was required. Under the EMS, the obligation to intervene in support of individual intrasnake parities is explicitly buttressed by a limitation on the sum of a currency’s deviations from parity vis-à-vis the other member currencies.

Formally, a currency’s actual divergence can be expressed as the weighted sum of the percentage deviations from parity of each participating currency with the currency in question. Letting Ai stand for the ith country’s actual divergence, one has

where the dj’s are the respective percentage deviations from par. The rules of the EMS require corrective action in the event that a currency’s actual divergence, as calculated by equation (4), is greater than or equal to its threshold of divergence (Ti), as calculated by equation (5).5

Note that the formulation of the threshold of divergence in equation (5) implies a measurable restriction on currency behavior beyond that embodied in the snake arrangement. Specifically, under snake rules, if a currency were to have departed from parity vis-à-vis all other member currencies by just under 2.25 per cent, its actual divergence as calculated on the basis of equation (4) would have been

and as long as the parity band vis-à-vis another snake currency was not penetrated, no corrective action was required. Visual comparison of the right-hand sides of equations (5) and (6) indicates that the maximum amount of divergence tolerated by EMS rules is 25 per cent less than the maximum that was implicitly tolerated by snake rules. Moreover, since the EMS retains the parity-grid requirements of the snake arrangement, it is unambiguously more restrictive than its predecessor.

II. Importance of a Currency’s Weight in Signaling Divergence

As quoted in the introductory part of this paper, 6 it has been argued that the method of calculating the threshold of divergence ensures that the appearance of the divergence signal is independent of the relative weight of each currency in the ECU. On the face of it, this assertion appears fairly straightforward in that the formula for calculating a currency’s threshold of divergence—equation (5)—excludes its own weight, as, under certain circumstances, does the formula for calculating a currency’s actual divergence—equation (4).7 Thus, for example, if any currency moves against all other participating currencies by 75 per cent of 2.25 per cent, its divergence indicator will ring regardless of the size of its weight. Nevertheless, the right-hand side of equation (6) (i.e., the formula for calculating the maximum potential divergence that was consistent with snake rules) shares the property that the currency’s own weight is excluded, suggesting that perhaps all such formulas are characterized by this property. This line of reasoning raises two questions about the meaning of the statement that the triggering of the indicator is “independent of the relative weight of each country in the ECU.” First, could an indicator be designed that would be more biased in favor of large countries? Second, does the design of the divergence indicator remove the entire influence of a currency’s weight from the conditions under which its indicator is triggered?

divergence indicator could be more biased in favor of large countries

It would be possible to construct indicators that tolerated relatively more divergence by larger countries. For example, if the divergence threshold were defined by

for all participating countries, a country would be more constrained than under snake rules if its weight were less than 0.25 (compare the right-hand sides of equations (6) and (7)) but less constrained than if its weight were greater than 0.25. Compared with EMS rules, equation (7) would permit greater divergence by all member countries, but with a larger country being allowed a disproportionate increase. This becomes apparent if equations (5) and (7) are compared. For all currencies with nonzero weights (i.e., all participating currencies), Ti*>Ti Moreover, the larger the currency’s weight, the larger is the amount by which Ti*>Ti The bias toward large size associated with formulations that do not allow for the exclusion of a country’s own weight arises from the principle that a currency cannot depart from parity vis-à-vis itself. Accordingly, leaving a currency’s own weight in the formula for determining the threshold of divergence increases the tolerated amount of divergence vis-à-vis other countries; furthermore, for currencies with larger weights, the never-to-be-used allowance for divergence vis-à-vis themselves translates into a relatively greater scope for divergence vis-à-vis other currencies.8

does divergence indicator remove entire influence of a currency’s weight?

Having shown that other formulations of the threshold of divergence would have been possible that would have afforded greater divergence to currencies with larger weights, the question now is whether the entire effect of size is removed by the particular formulation that the EMS employs. Whether or not a situation, in which two currencies depart from parity with each other by just under 2.25 per cent, will trigger either or both countries’ divergence indicators depends on the magnitude and direction of the accompanying movement in third-country exchange rates.9 For example, if third currencies depreciate fully with the “weak” currency, the divergence indicator of the strong currency will ring. If third currencies appreciate fully with the “strong” currency, the divergence indicator of the weak currency will ring. And if third currencies depreciate somewhat vis-à-vis the strong currency and appreciate somewhat vis-à-vis the weak currency, in principle, the divergence indicator of either, both, or neither of the two original currencies could ring. Whether or not a currency’s divergence indicator does ring under these circumstances depends on the actual magnitude of the associated movement in third-currency values. Hence, to show how a currency’s own weight affects the conditions under which its divergence indicator is triggered, this section investigates how the magnitude of the movement in third-currency values required to trigger a given currency’s divergence indicator varies with the given currency’s weight in the EMS.

To this end, let currencies A and B differ from parity with each other by just under 2.25 per cent.10 For currency A, the actual divergence (Aa) is calculated as follows from equation (4):

where wb is the weight of the currency with which it diverges the most, wc is the combined weight of all third currencies, and x is the weighted average percentage departure of these third currencies from their central values vis-à-vis currency A. A’s threshold of divergence (Ta) can be calculated from equation (5) as follows:

and its indicator will ring if AaTa. Such is the case if

or in simplified terms if

To see how A’s weight influences the magnitude of x, differentiate equation (11) with respect to wa

The sign of dx/dwa is determined by the sign of the expression in parentheses, which, in turn, depends on whether the posited increase in wa is at the expense of wb, in which case dx/dwa > 0, or at the expense of wc, in which case dx/dwa < 0. However, for the purpose at hand, it is appropriate to assume that ∂wc/dwa = 0, 11 since the concern here is only with the implications of variations in the magnitudes of A’s and B’s weights.

The resulting positive sign for dx/dwa implies that the larger A’s weight—and the smaller B’s weight—the greater is the required departure from parity between A and C in order for A’s threshold of divergence to be crossed. This suggests that, under the circumstances in which A and B diverge by just under 2.25 per cent, third currencies would have to move relatively more against the large currency for its divergence indicator to ring than they would have to move against the smaller currency to cause its divergence indicator to ring. For example, it is possible that a movement of 1.5 per cent in third-currency values away from the smaller currency would trigger its divergence indicator, while the same percentage movement of the third currencies away from the larger currency would be insufficient to trigger its indicator. While in both situations the respective currencies depart from parity by 2.25 per cent vis-à-vis each other and by 1.5 per cent vis-à-vis third currencies, for the smaller country the relative weight attached to its 2.25 per cent difference from parity with the larger country is larger than the respective relative weight attached to the larger country’s 2.25 per cent difference from parity with it. Moreover, this conclusion follows directly from the fact that the country weights are not all equal.

Although it is true that the smaller country’s divergence indicator can ring under conditions in which the larger country’s divergence indicator cannot ring, it is not possible for the divergence indicator of the smaller currency to ring when third currencies are closer to it than to the larger country’s currency. To pursue the example cited above, if third currencies are 1.5 per cent away from parity with the larger currency, they will be 0.75 per cent from the smaller currency, 12 and this will be insufficient to trigger its divergence indicator. This property is due to the configuration of weights within the EMS, which makes it impossible for a currency’s divergence indicator to ring if third currencies are closer to it than to the other currency;13 since third currencies must be at least 1.125 per cent away from a currency for its divergence indicator to ring, there can be no reversals in which the divergence indicator rings for the currency to which third currencies are closer.

This line of reasoning also suggests that the divergence indicator cannot ring simultaneously for two opposing currencies. Because the divergence indicator cannot ring for a currency if third currencies are within 1.125 per cent of parity and because third currencies must be within 1.125 per cent of one of the currencies that depart from each other by 2.25 per cent,14 the divergence indicator cannot ring for both at the same time.15

III. Coalitions of Currencies

In this section, the divergence indicator is examined from the perspective of coalitions of currencies. As in Section II, the basic scenario posits two currencies A and B whose exchange rate vis-à-vis each other differs from parity by just under 2.25 per cent. In contrast to Section II, however, the other currencies participating in the EMS are assumed to move in tandem with either currency A or currency B. It is shown that the design of the divergence indicator favors coalitions, in the sense that exchange rate behavior that would trigger the divergence indicator for a single country would not necessarily do so for a coalition whose weights sum to the weight of the single country.

To this end, let wa and wb denote the weights of A and B, respectively, and wa and wb denote the respective weights of A’s allies and B’s allies. A’s threshold of divergence is given by

If A moves by y per cent against B and its allies, but stays constant against its own allies, the actual divergence registered by A will be

Its threshold of divergence will be penetrated if

Substituting in wb+wb+wa for 1 − waequation (15) gives

Note that if wa=0 (i.e., A has no allies), y is smaller than if wa>0. This implies that A can diverge by more vis-à-vis the B currencies without its divergence indicator being triggered if it is in a coalition of currencies than if it stands alone. This is so because ((0.75) (2.25) = 1.69) (1 − wa) is the minimum percentage of overall divergence for which A’s indicator is triggered. If it has allies, then it can diverge more vis-à-vis the others (i.e., its rivals) without crossing its overall threshold. It is likewise true that a coalition with the same total weights as A would be allowed more divergence—in the sense that y is larger—than would be allowed A by itself.

Equation (16) is interesting in several other respects as well. First, the divergence indicator, if it is to ring for any currency in a given situation, will ring for the one for which y is the smallest. In turn, this will be the currency for which wa/(wb+wb) is the smallest (i.e., the currency for which the ratio of its allies’ weights to the sum of its rivals’ weights is the smallest). Thus, it rings for the largest member of one of the coalitions. However, once this currency, A for example, responds to such adjustment measures as are required and its value moves closer to parity with B’s coalition, the divergence indicator of one or more of its former allies may ring.

Second, if 2.25 is substituted for y in equation (16), it can be solved for the relationship between the weights of A’s and B’s allies that must prevail if A’s threshold of divergence is ever to be triggered by movements in the values of the currencies in the A and B coalitions. If y > 2.25, A’s divergence can never ring, since the parity-grid constraint will require bilateral currency intervention before the threshold of divergence is ever crossed. Making the afore-mentioned substitution, one can simplify equation (16) to

Thus, A’s threshold can be penetrated only if the combined weight of B plus its allies is at least three times the weight of A’s allies. If the combined weights of B plus its allies is less than three times the weight of A’s allies, A’s threshold of divergence can never be crossed by interbloc currency movements, since the parity-grid intervention point would be crossed before A’s divergence indicator were ever triggered.

Third, because of the omission of A’s weight from the right-hand side of equation (17), there is some built-in preference for coalitions of countries with equal weights. For example, if the sum of the weights both in A’s coalition and in B’s coalition were to equal 0.5, A’s indicator could be triggered if its weight were larger than 0.33 but could never be triggered if its own weight were less than 0.33. At the same time, it should be understood that the more equal in size the members are, the more likely it is that the divergence indicator, if it rings at all, will ring simultaneously for more than one country in the coalition. Thus, for example, if A’s coalition were to include itself and another country of equal weight, keeping constant the sum of the weights in A’s coalition, it would be less likely for A’s divergence indicator to ring than if A’s share in the coalition were larger; on the other hand, if A’s divergence indicator were to ring under these circumstances, it would also ring for A’s ally.

IV. Summary and Conclusions

The principal contribution of this paper is its construction of a measure of the conditions under which a currency’s divergence indicator is triggered. This measure can be used to make intercountry comparisons of the degree to which the divergence indicator impinges on a country’s ability to pursue its desired economic course. The idea behind the measure is to posit a specified percentage deviation from parity between two member currencies and to calculate, for each of these currencies, how large a departure from parity vis-à-vis third currencies would be required for each currency’s divergence indicator to be triggered.

In comparing the divergence indicator’s effect on a large versus a small country, it was found that relatively more disparate third-currency movement is tolerated for large countries than for small. Specifically, there exist configurations of deviations from central rates that would trigger the divergence indicator of the small country if the deviations were vis-à-vis its currency, and that would not trigger the divergence indicator of the large country if percentage deviations of the same magnitude prevailed vis-à-vis its currency. However, it was also shown that, with the present distribution of currency weights, the divergence indicator cannot be triggered for a country—even the smallest—if third currencies are closer to its currency than to its opposite’s currency—even if its opposite is the largest country in the EMS. Moreover, this conclusion, in conjunction with the requirement derived from the cross-currency arbitrage condition that the sum of third-currency deviations from parity vis-à-vis two currencies that differ from parity by 2.25 per cent must also be 2.25 per cent, implies that the divergence indicator can ring for at most one of two currencies on opposite sides of the parity grid.

Although it was found to be true that the divergence indicator of a small country can be triggered under conditions that would not trigger a larger country’s divergence indicator, it was also established that there are situations—if all member currencies deviate from parity with the currency by the same critical percentage—under which the divergence indicator would be triggered for a country regardless of its size. Furthermore, it was concluded that the design of the divergence indicator is less biased in favor of large countries than it might have been, with the existing bias being a natural consequence of the differences in the weights of EMS participants.

The analysis also established the existence of a certain preference for coalitions of currencies. One interesting finding in this respect was that for two or more currencies moving in tandem vis-à-vis another currency or group of currencies, greater third-currency deviation from parity is tolerated than if the coalition were a single country whose weight equaled the sum of the weights of coalition members. Another finding is that if the divergence indicator were to ring for any coalition member, it would ring for the largest. However, the inference that this entails a bias in favor of small countries that counteracts the afore-mentioned bias against them is valid only in a formalistic sense, since the small country’s ability to pursue its “divergent” economic course depends on its being shielded by its larger ally. If the ally’s divergence indicator is triggered (necessitating a change in its policies), the small country’s policies will similarly have to change—either to prevent its divergence indicator from being triggered or in response to the subsequent triggering of its divergence indicator.

That the divergence indicator favors the policy stance adopted by larger countries and by coalitions of countries seems appropriate in view of the composition and the objectives of the EMS. Larger countries account for a larger share of the economic activity in the European Economic Community, and it seems quite reasonable that they have a larger vote in determining the pace of economic activity within the EMS. Moreover, in view of the role that the EMS is to play in furthering the course of European economic integration and the political difficulties inherent in achieving monetary coordination, it seems wise to reward coalitions of countries achieving such coordination by a relatively greater vote as well.

Ms. Salop, economist in the Eastern European Division of the European Department, is a graduate of the University of Pennsylvania and Columbia University. Before coming to the Fund, she was on the staff of the Board of Governors of the Federal Reserve System and the Federal Trade Commission.

The author would like to thank Patrick Honohan, of the Central Bank of Ireland, for his useful comments on an earlier version of this paper.

In the language of the “Conclusions of the Presidency of the European Council on 6 and 7 July 1978,” issued after the meeting in Bremen and reprinted in Bulletin of the European Communities (No. 6, 1978), pp. 17–18.

See “The European Monetary System: Structure and Operation,” Deutsche Bundesbank, Monthly Report, Vol. 31 (March 1979), pp. 11–18.

Henri Baquiast, “The European Monetary System and International Monetary Relations,” in The European Monetary System: Its Promise and Prospects, ed. by Philip H. Trezise, The Brookings Institution (Washington, 1979), pp. 49–59, especially p. 52.

Ibid., p. 52.

The calculations of equations (4) and (5) include all countries in the EMS, even though sterling does not participate in the exchange arrangement and the lira is allowed to differ from parity with other participating currencies by 6.0 per cent rather than 2.25 per cent. The influence exerted by a currency’s movement vis-à-vis these two currencies on its divergence indicator is taken into account in interpreting the currency’s measured divergence. Essentially, deviations from parity with the lira and from the notional parity with sterling are limited to 2.25 per cent for the purposes of equation (4). See “A Guide to the Arithmetic of the EMS Exchange-Rate Mechanism,” Central Bank of Ireland, Quarterly Bulletin (Autumn 1979), pp. 76–100, especially pp. 92–93.

Baquiast (cited in footnote 3).

If all currencies deviate from parity with currency i by an equal percentage, equation (4) can be written as

An alternative example of an indicator that would be biased in favor of larger countries is given by Ti** in equation (7′):

Note that, like Ti* in equation (7), Ti** is identical for all participating currencies, and that a country’s threshold of divergence would be greater than under EMS rules only if its weight were larger than the average weight of all participating countries. This is apparent if equations (5) and (7′) are compared. Thus,

which simplifies to

Note also that the formulation in equation (7′), while biased toward currencies with larger weights, permits the same overall amount of potential divergence as the EMS, in that for both equations (5) and (7′) the sum of the respective thresholds over all participating currencies is given by

The cross-currency arbitrage conditions require that the values of third currencies change when the exchange rate between any two currencies changes. For example, in terms of French francs, deutsche mark, and Netherlands guilders, it must always be the case that F/DM = F/f. ⋅ f./DM. Thus, if the franc appreciates vis-à-vis the deutsche mark, the guilder can either appreciate vis-à-vis the deutsche mark, depreciate vis-à-vis the franc, or move in some intermediate fashion.

While the exercises are carried out in the text under this assumption, it is only to simplify the algebraic presentation. The same qualitative results obtain if A and B differ from parity by any amount that exceeds 1.69 per cent (i.e., the product of (2.25) and (0.75)).

Under the alternative assumption that ∂wb/∂dwa = 0 and ∂wc/∂wa < 0, the sign of dx/dwa becomes negative. Accordingly, as A’s weight rises at the expense of C’s weight, less departure from parity between A and C is required to penetrate A’s divergence threshold when A and B diverge from parity vis-à-vis each other by just under 2.25 per cent. In this case, since B’s weight stays fixed and A and B depart by almost the full 2.25 per cent, less and less additional divergence is needed from C in order for A’s divergence indicator to be triggered.

Taking logs and differentiating the cross-currency arbitrage conditions cited earlier, one has

d log F/DM = d log F/f. + d log f./DM

In words, the percentage change in the franc/deutsche mark rate equals the sum of the percentage change in the franc/guilder rate plus the percentage change in the guilder/deutsche mark rate. Thus, if the franc depreciates by 2.25 per cent vis-à-vis the deutsche mark, the sum of the percentage depreciations of the guilder vis-à-vis the deutsche mark must equal 2.25 per cent.

The proof of this proposition is as follows. Assume that the exchange rate between the currencies of the two countries with the largest weights in the EMS (i.e., France and the Federal Republic of Germany) differs from parity by just under 2.25 per cent. One can then calculate for each currency the minimum percentage deviation from parity vis-à-vis third currencies needed to trigger its divergence indicator, by following the steps laid out in equations (8)(11) in the text. Letting the French franc be A, the deutsche mark be B, and all other currrencies be C, and substituting the weights from Table 1 into equation (11), one has

To show that xi > 1.125 for all i, differentiate equations (11′) and (11′′) with respect to wa. For this calculation, wb is held constant but wc can vary. (Essentially, neither wa nor wb can rise, since they are the largest in the EMS. Hence, one can consider only decreases in wa or wb that are offset by increases in wc. Moreover, since the algebraic equivalents of xa and xb are symmetrical with respect to wa and wb, both

Hence, one needs to see only how the x’s change with respect to either wa or wb.) Differentiating and simplifying, one has

and

This indicates that the xa and xb calculated in equations (11′) and (11′′) are minima for the EMS, since if the calculation is performed for smaller countries (involving reductions in wa and wb as in equations (12′) and (12′′), the respective x’s rise.

See footnote 12.

This point was made in “The European Monetary System: Structure and Operation” (cited in footnote 2), p. 15. Moreover, the same point can be made directly from footnote 13. Specifically, the sum of xa + xb > 2.25 when the French franc is substituted for currency A and the deutsche mark for currency B. Moreover, it was found that for these countries xa and xb are at a minimum. Accordingly, for all A and B, xa + xb > 2.25. This means that the divergence indicator cannot ring simultaneously for two currencies on opposite sides of the parity grid, since the actual sum of third-currency percentage deviations from parity vis-à-vis the two currencies A and B must be exactly 2.25. To meet this condition, for at least one currency A or B, the third-currency deviation from parity with it must be below the level that would trigger its divergence indicator.

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