-17-

protection against country N, that is, it is optimal to set this
country's basket weight equal to zero.

: Following the same line of argument as above, it can be shown
that it is optimal to peg to a single currency L, if PPP holds between
the home country and all it's trading partners, except Ltl/ The
explanation for this result is that in this case the export market is
only disturbed from country L, and it is therefore optimal to give a
maximal weight (= 1) to this country 22/

The results in Cases 1-3 suggest that two conditions are of
importance when the basket weight for a country, say country N, is
chosen: (i) the share (or the elasticty of trade) between the home
country and country N (ty)s and (ii) the relationship between the

relative price, p’ - p’, and the exchange rate ef. The stronger this

relationship is, the smaller is the optimal weight.

VI. Numerical Illustrations

It is possible to describe one or two special cases, but to
demonstrate more fully the implications of these formulas, we proceed to
actual calculations. It should be noted that our method of calculating
the weights differ somewhat from that of LS or BK (for elasticity
weights). They use regression analysis whereas we manipulate the
matrices of equation (20) directly.

Tables 3 to 5 illustrate the computations of the optimal
currency basket weights for each of the countries. The first column,
labelled trading partners, represents those countries present in the

existing baskets calculated by the monetary authorities. These lists may


-11-

directly linked to employment, the following exposition will utilize the
production of export goods as the target of exchange rate policy.£/

The previous choice needs some additional comments. First, in
the Nordic countries a substantial part of the traded goods produced is
import-competing and these should therefore be included in the target.
However, inclusion of these types of goods do not alter the subsequent
conclusions we arrive at and for the sake of simplicity are therefore
excluded. Second, we may use a relative unit cost of labor measure as an
alternative target for the exchange rate policy. This requires that wage
rates are brought explicitly into the analysis. Again this only adds
complexity to the model without altering the main results.

In accordance with the previous discussion, it is assumed that
the policymaker is concerned with maintaining the production of exports
at some target x, and that the weighting scheme in the currency basket
is a policy instrument. The policy problem is to find the weights at a
particular point in time which minimizes the cost of deviations from the
target. To this end the policymaker is assumed to possess the following

quadratic objective function.
*
(33) Le E(x - x)?

To simplify matters it is further assumed that the target is expected to
be achieved, that is x = E (x). According to equation (4), rewritten
below as equation (14), basket weights are shown to effect the home-
numeraire exchange rate. It has already been shown in equation (12) this

will in turn affect the level of export production.


Table 1:

(6)

(7)

(8)

(9)

(10)

Simple N+1 Country Trade Model

Export-supply:

So h h
x= slp - Pp) s, > 0
Export-demand:
N X,
de J (pJ - pd
xe dy Ze (py - po) a, <0
j=l
Export-prices:
ph = el + pl j=l,...
Reduced forms:
N X
- Iya 4 pd h
x= sk J (yees + p -p)
jel
N X +
pn = (1-k)pl+k £ — (ef +p)

jel


Furthermore, we will assume that the basket of any one country is
independent of any other country .2/ The immediate solution that each of
these countries has used has been different and they have each made at
Teast one change since 1978. In this paper we consider the selection of
weights when the objective is to stabilize the production of home
exports.

The point of departure for our analysis is the model presented
in Branson and Katseli (BK) (1981,1982). BK developed a trade model
which they used to select weights for an effective exchange rate index
(also known as elasticity weights). Following Lipschitz and Sundararajan
(LS) (1980,1982) we consider the weights derived by BK as the first step
in obtaining optimal weights for the currency basket. Only under special
circumstances are the two sets of weights equivalent. As we shall
describe later in this paper, this difference is observed because of
different objective functions and assumptions used to derive the
weights.

The distinction between and the properties of exchange rate
indices and currency baskets are investigated in Section II. In
particular, a geometric exchange rate index is assumed and a small
country stabilization policy is considered. Section III casts this paper
in light of other studies. Section IV contains the basic trade model
which is used to derive the elasticity weights. Section V derives the
optimal weights for a currency basket assuming the objective is to
minimize fluctuations in the production of exports. Section VI presents
the empirical results. Optimal weights are presented and some special

cases are examined. Section VII contains concluding remarks.


References

Asheim, Geir B. (1984), "Properties of a System of Currency Baskets,"
Journal of International Economics, 16, May, pp. 311-318.

Bhandari ,J.S. (1985), "Experiments with Optimal Currency Composite,"
Southern Economic Journal, January, pp 711-730.

Black, Stanley W. (1976), Exchange Policies for Less Developed Countries
in a World of Floating Rates, Princeton Essays in International
Finance, No. 119.

Branson, William H. and Louka T. Katseli Papaefstratiou (1980), "Income
Instability, Terms of Trade, and the Choice of an Exchange Rate
Regime," Journal of Development Economics 7, March, pp. 49-70.

(1981), "Exchange Rate Policy for Developing
Countries," in S. Grassman and E. Lundberg, The World Economic Order:

Past and Prospects, London, Macmillan, 1981.

(1982), "Currency Baskets and the Real Effective
Exchange Rates," in M. Gersovitz, et. al., The Theory and Experience
in Honour of Sir W. Arthur Lewis, London, George Allen and Unwin.

Flanders, M. June and Elhanan Helpman (1979), "An Optimal Exchange Rate
Peg in a World of General Floating," Review of Economics Studies,
July, pp. 533-42.

Ilkka, P. (1978), "The Amendment of the Currency Act," Bank of Finland
Monthly Bulletin, No. 2.

Kredit-och Valutaoversikt (1981), "Svenska Kronan Devalverad," Sveriges
Riksbank, No. 3.

Lipschitz, Leslie and V. Sundararajan (1980), "The Optimal Basket in a
World of Generalized Floating," IMF Staff Paper, pp. 80-100.

(1982), "The Optimal Currency Basket in a World of
Generalized Floating with Price Uncertainty" in Michael B. Connolly
ed., The International Monetary System: Choices for the Future, New
York, Praeger, 1982.

Lundstrom, H. (1978), "Ett Ar i Valutakorgen-Nagra Erfarenheter och
Synspunkter," Ekonomisk revy No. 8.

Skanland, H. (1983), "Norsk Valutakurspolitikk," Penger og Kreditt No. 2,
(Norges Bank).

Solheim, J.A. and T.0. Sporastoyl (1983), "Valutakursindekser-
Konstruksjon og Anvendelse," Penger og Kreditt No. 3, (Norges Bank).

Turnovsky, S.J. (1982), "Determination of the Optimal Currency Basket: A
Macroeconomic Analysis," Journal of International Economics, May,
pp. 333-54.



~18-

exclude important countries, but we decided to limit our investigation to
those countries now used in each currency basket. The column labelled
actual basket weights presents the t;'s used in each country. Note that

the U.S. dollar is used as the numeraire currency and its weight is

N
considered as the residual (that is, 1 - & Wy ) and is therefore reported
2

below the total of the other weights. The columns labelled 1-3 refer to
calculations using equation (20) and actual basket weights. Following
these columns is another column of ty weights which represent actual
export shares used to evaluate equation (20). The columns labelled 4-6
correspond to those in columns 1-3 but export share weights (1;'s) are
used.

The evaluation of the optimal weights is performed using data
over the time period January 1973 to March 1982. Columns labelled 1 and
4 report calculations using matrix manipulation which correspond directly
to equation (20). In order to ascertain the impact of third-party
effects we recalculate the optimal weights by imposing two restrictions:
(i) the Q matrix is forced to be diagonal, that is the covariance between-
exchange rate is zero -- Covles, 4) = 0 for j #q (see columns labelled 2
and 5), (ii) and in addition the column vector T = 0 - the covariance
between numeraire exchange rate and relative price between numeraire and
home country is zero (see columns labelled 3 and 6). This latter
assumption is used in LS. In fact, LS argue that there is no reason to

-1
expect r2 to be stable and therefore set it to zero.


Table 4: Optimal Basket Calculations for Finland

Trading Basket Share
Partners Weights (1) (2) (3) Weights (4) (5) (6)
% Tj

Belgium -027 -.06 -024 ~=—.026 -023 ~.027 -024 =.022
Switzerland 025 -.06 -019 = .018 -02 -.048 +032 = 015
Denmark +05 +305 -045 = .051 -064 +20 -066 -064
United Kingdom .147 35 -153 «116 -186 38 -174 4147
France -061 -.09 +055 = .062 061 -.07 +063 =.062
Germany 189 58 18 153 149 54 «157.121
Holland -052 -.35 043.046 -056 -.33 -063 = .050
Italy -039 189 037 +04 +033 17 -033 033
Japan -05 -08 048 = .049 +02 -01 021 -019
Norway +056 -.163 051 = 054 .07 -.16 :072 = .068
Sweden -20 152 +194 .202 191 18 +193 .193
Total 896 * 849.817 873 * 898 = .794
United States .104 * «151.183 127 * +102 .206

Note to Table: Trading partners listed above are those currently used in
Finland's basket. The column labelled basket weights represent weights used
in the calculation of Finnish basket while the column labelled export share
are derived from direction of trade statistics. Columns labelled (1) and (4)
represent weights derived using equation (20), the other columns are special
cases of equation (20).


-20-

been arbitrarily calculated, in the sense that we do not replicate the
existing baskets, by using 1980 as the base year. It is clear from the
charts that during most of the time period the optimal baskets are above
the actual baskets which suggest weaker home currencies. The patterns of
the country indices themselves are quite different reflecting differences

in home currency historical experiences.

VII. Conclusion

The purpose of this paper is to derive optimal weights for a
currency basket taking into consideration the objective of the policy
makers in the Nordic countries. Our analysis is based on the work of
Branson-Katseli and Lipschitz-Sundararajan. The existing literature in
this area often interchange the terms effective exchange rate index and
currency basket. We carefully distinguish between the two terms. Both
are defined as weighted average of bilateral exchange rates. However,
the weights used in their calculations are different. Generally, the
weights in an effective index are some form of trade weights, while the
weights in a basket are more broadly defined.

In this paper we derive both export share weights by using a
simple multi-country model, and basket weight by assuming that the
objective of the policymakers is to minimize fluctuations in the
production of exports. We show that only under special circumstances are
the two weights the same. The basket weights are a function of these
export share weights but also other factors such as the covariances of
relative prices and exchange rates. Using the formulas we have derived,

various calculations are made for Norway, Finland and Sweden.


-R2-

Willtamson, John H. (1982), "A Survey of the Literature on the Optimal
Peg," Journal of Development Economics, Vol. 11, pp. 39-62.



-15-
j 1 oi, _ _ _
(21) Cov(p* - p, e) = 0 j=h,2,...,N and q=2,...,N

and it is assumed that r=0 then from (20) the optimal weights are

equivalent to the trade weights:2/
OL
Wq * 1 for all q.

It also follows from (18) that if Cov(p4 - py ey) #0 for at
least one j and q, that is if we have a relationship between exchange
rates and relative domestic prices, then the trade weights are not the

optimal basket weights. The importance of these results are that they

set a benchmark: Only when all Cov(py

weights the optimal basket weights 2/

- p!, e) = 0 are the trade

Let us consider an extreme case of a relationship between

exchange rates and the relative domestic price:

Case 2
If we have "purchasing-power-parity" (PPP) between the home

country and all other countries in the basket, that is

(22) p+ ef = pf gel,...,Ny-

there does not exist a unique optimal weighting scheme.
This result is easily explained by substituting (22) into (9).
We obtain x(w) = 0, that is the production of the export equals zero no

matter what are the values of the chosen weights.


-7-

demand and supply aspects of the market of traded goods. BK show that
the weights are functions of trade shares and elasticities, which are
much like the weights described by Black. These weights are also very
similar, if not identical, to those used in an effective exchange rate
index, for example, the Federal Reserve's trade weighted dollar exchange
rate.

Lipschitz and Sundararajan (1980,1984) expand the calculations
for basket weights by incorporating additional factors. LS's
contribution allows for correlation between relative prices, domestic
prices relative to foreign prices, and exchange rates while also
including trade elasticities. In their latter paper LS synthesize the
BK-FH approach with their own. Our work is an extension of LS.

This overview of the optimal currency basket literature would
not be complete if we did not mention another strand which has been more
recently considered. Turnovsky (1982) analyzes the choice of optimal
currency basket using a general equilibrium macro model of a small open
economy with perfect capital mobility. The optimality criterion he
considers is the stabilization of domestic real income. Furthermore, he
assigns a central role to international capital flows and exchange rates.
Bhandari (1985) has reported on experiments using the Turnovsky model as
his basic framework. The aim of the Bhandari paper is to analyze the
sensitivity of the weights to the various assumptions rather than to
estimate the weights relevant for a particular country. Note that the
approach we have adopted, a partial equilibruim approach, should yield
weights that can be derived as special cases of Turnovsky. It is also
much easier to estimate optimal weights from our formation. Perhaps in a

later paper we will consider this more general alternative.


-F2-

6. In the subsequent table references are given to articles written by
"authorities" on the exchange rate policy specific for the countries in
question.

Finland

Puro (1978)

Norway Sweden

Lundstrom (1978)
Kredit-och valuta-
oversikt (1981)
Sweden's ability to
compete interna-
tionally

Skanland (1983)
Solheim and
Sporastoyl (1983)
Competitiveness of
Norwegian industry

General goal {Competitiveness of

Finnish industry

Specific
measures

Stabilizing cost
and price levels

Relative unit cost
cost of labor,
relative export
prices, relative

Unit labor cost,
relative consumer
prices, balance of
trade

consumer prices
The target we consider in this paper, to minimize fluctuations in the
production of exports, accords with the general goal suggested by the
authorities listed in the table above.

7. For the sake of comparison, the matrix notation in (19) is adopted
from LS(1982).

8. LS(1982), p. 130, show that if the relative domestic prices, pJ - p!,
are fixed, the optimal basket weights are equal to the trade weights.
Clearly, this proposition is a special case of our result in case 1.

9. If (21) holds, we see from (15) that the variance of the production
of the export can be separated into two terms: (i) the variance of the
relative domestic prices and (ii) a variance term related to exchange
rates. The first of these is not affected by the basket weights. Since
the second term becomes zero if the trade weights are chosen as basket
weights, the variance of the production of the export will also be
minimized by the choice of these weights.

10. The proof of the proposition in case 3 is as follows:
By substituting (23) into (9), we obtain:

1
h
T5°j

N-l og N-
x(w) = (syek)| pi t3(py - pt) + r |.


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~4-

N Ww.
(1) 1" (tw) = om (EM (tye (0)) J
jal J j

In this basket, there are N currencies with weights, Wy j=1,...,N and
the sum of the weights equals one, © wal. When (tw) is greater than
1(0,w) this implies a depreciation of the home currency. In later
sections of this paper logarithms are used so (1) is rewritten in log
form where lower case letters denote logs of the variables (with the

exception of the weights and parameters)

(2) in (t,w) = ' W. (en (t) - ef (0))

jar JS j
Further note that in subsequent discussions we shall drop the time
subscript t. It is assumed, henceforth, that everything is at time t
unless otherwise indicated.

Two attributes of an effective exchange rate are considered
which are particularly relevant for a country with a small open economy.
The first attribute applies when the home country can only alter one
bilateral exchange rate. For the sake of convenience, we use the home-
numeraire country exchange rate denoted as en If the monetary authority
in the home country attempts to fix the exchange rate index, that is
i=0,2/ then from (2) it can be shown that the sum of the weighted

bilateral exchange rates must be zero.2/

N
(3) = ow, eM = 0.


II. Definition and General Properties of Exchange Rate Indices

Often in the literature on optimal currency baskets the two
terms effective exchange rate index and currency basket are used
interchangeably. In this paper we will distinguish between the two
terms. An effective exchange rate index is simply the weighted average
of various bilateral exchange rates. Typically the weights associated
with an effective exchange rate index represent “elasticity weights"
derived to maintain a trade balance goal of some sort. Similarly,

a currency basket is the weighted average of bilateral exchange rates.
However, the weights for a currency basket are derived for any type of
macroeconomic objective. Therefore, these currency basket weights tend
to be more general. Note that the differences are because different
objectives are used. Basket weights often utilize effective exchange
rate weights, but will not be, in general, equal to them because
additional factors are used in their derivation. Only in special cases
will the two weights be identical. It is necessary to keep this
distinction in mind when interpreting our further results.

The rest of the discussion in this section concerns the general
properties of exchange rate indices, whether they are “effective” or
"basket". In this paper the nominal exchange rate index is denoted as
I"(t,w) where I, r, t, ware the exchange rate index, the reference
currency, time, and weights respectively. The home country's (denoted by
h) bilateral exchange rate with respect to currency j at time t is
written as ey(t). The effective exchange rate index for the home

currency is defined as a geometric average.


~16-

Another interesting case is the following:

Case 3

If PPP holds between the home country and another country, say
country N,
(23) pt - py = en

and if there is no correlation between relative domestic prices, p! - pt,

and exchange rates, ef,

for the additional N-1 countries,
(24) cov(p§ - pi, ef) = 0 J.a-l,..,NeL,

then the optimal weights simplify tol0/

(25) wy 0 we = t/(L-ty) > ty g=1,...,Nel.

a

In other words, (23)-(24) are sufficient conditions for the optimal
basket weight of one country, N, to be zero.

An intuitive explanation for the result in case 3 is as
follows: According to our discussion related to equation (11), country N
affects the production of the export through the relative domestic price

Ny 4) - pt. If this relative price rises so will the production of

the export. If (23) holds, the relative price, (p + el) - pt, is always

(p

zero, and no distortion will be transmitted from country N to the export

market of the home country. Consequently, the home country needs no


-6-

III. Previous Studies

In a recent article Williamson (1982) surveyed many of the
issues on currency baskets. Most of the existing literature on currency
baskets have been applied to developing countries, in particular Latin
America. This does not imply that it is not relevant for developed
countries, because it is, especially for those countries with small open
economies. This section highlights some of the important aspects of the
literature and casts our research into the existing framework.

In his seminal article, Black (1976) suggests that exchange
rate index weights should be selected so as to reflect bilateral trade
using direction of trade measures or to reflect elasticities derived from
a multilateral trade model. According to our use of terminology, 31ack
describes appropriate weights for an exchange rate index, not a currency
basket which policymakers use for their exchange rate goals. Black in
this work only describes these weights and does not analytically derive
them.

Various elasticity weights in several articles have been
suggested as "optimal" in the sense that if policymakers adopt them when
creating their exchange rate index they may be able to achieve their
exchange rate target. Most notably, Flanders and Helpman (1981)
(henceforth FH) derive such weights within a Keynesian model framework
using a small open economy. One of the results FH derive is as follows:
The weights used should reflect demand elasticity for traded goods
because this allows changes in prices to have a small impact on the
balance of trade which is considered a target in their analysis.

Another example in the literature of elasticity weights is

BK (1981). This approach differs from FH because they consider both the


Table 2: Definitions of Variables

Lower case letters indicate that the variable is in logarithm

e - Exchange rate (home currency in term of a unit of
currency j)

of - Domestic price of exports

pn - Domestic price index

pd - Foreign country j index of prices

x 08 x4.x5) - Exports (supply, demand, exports for country j)

Sy - Export elasticity of supply with respect to relative
price term

4, - Export elasticity of demand with respect to relative
price term

d

k - Ratio — )
x x


International Finance Discussion Papers
Number 266

October 1985

OPTIMAL CURRENCY BASKET IN A WORLD OF GENERALIZED FLOATING:
AN APPLICATION TO THE NORDIC COUNTRIES

by

Hali J. Edison and Erling Vardal

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
acknowledgment by a writer that he has had access to unpublished material)
should be cleared with the author or authors.


-9-

xX .
=) (ef + pi)

2
73
xs
a
wae

(
jel

This result suggests that changes in the home country price, p”, does not
have any effect on the price of exports.

From (9) it also follows that a 1% devaluation of currency j
will, ceteris paribus, reduce the production of the export by
sy K(X; /X)%. This result follows since a devaluation of currency j means
a lower export-demand for country j, and consequently the equilibrium
price of export decreases. This lower price leads to lower exports
supplied. The larger country j's share of the export demand, the larger
is the effect.

According to equation (9), the size of the production of the

export varies with

X,
(11) r (gbyiet + pd - ph),

which is a weighted average of foreign prices measured in the home

currency, ef + py, relative to the home domestic price, p’. If the

measure in (11) increases, this then leads to an increase in the
Production of export. Equation (11) may be used as a measure of the
overall competitiveness of the home country's export industry. Moreover,

the relative domestic price between the home country and country j, ef +

pl - p, may be taken as a measure of the competitiveness between these

two countries.


~8-

IV. The Trade Model

A conventional trade model along the lines of Branson-Katseli
(1981) serves as our basic framework for deriving and evaluating the
optimal weights in a currency basket. Because our objective is to
minimize fluctuations in the production of exports we report only the
export side of the BK model. Note that the home country's export: good
is demanded by N foreign countries.

Table 1 reports the basic N+1 country trade model, and table 2
provides a summary of the definition of the variables used in this model.
Equation (6) gives the home country's supply of the export good, xs, as a
function of the relative home currency prices. Here on is the home
country's price index of exportables and po is the index of home domestic
prices. Equation (7) presents the total foreign demand of home country
good x4, os is the price of the export in country j currency and pl is
an index of domestic prices in each j country. For simplicity it is
assumed that the price elasticity of export demand, dy. does not differ
between countries.2/ Equation (8) embodies the assumption of the "law of
one price", that is that the home price of exports is equal to the
foreign price of exports converted into home currency at the current
exchange rate. Equations (9)-(10) are two reduced form equations, for
the domestic price of exports (pn) and for the quantity of exports (x),
and are derived using the market equilibrium condition.

As pointed out in Branson and Katseli, k in (9)-(10) is an
index of market power. In the small country case, k tends to unity as 4,

tends to minus infinity. Observe that when this is the case equation

(10) simplifies to


Footnotes:

* The first author is an Economist in the International Finance

Division of the Board of Governors of the Federal Reserve System and the
second author is an Associate Professor at the Institute of Economics,
University of Bergen, Bergen, Norway. Work was done on this project
while the second author was a visiting scholar in the Division of
International Finance of the Board of Governors. An earlier version of
this paper was presented at the 5th International Symposium on
Forecasting, Montreal, Canada, June 9-12, 1985. The views expressed
herein are solely those of the authors and do not necessarily represent
the views of the Federal Reserve System or any members of its staff.

1. A paper by Asheim (1984) examines the properties of a system of
currency baskets.

2. According to (1), at time 0 the index equals 1. We assume that the

monetary authority keeps the index constant. Therefore, I"(t,w) = 1 and
iM(tyw) = 0.

3. fithout loss of generality, we assume that the exchange rates at time
0, E3(0), are scaled such that they equal 1. Therefore, e7(0) = 0, and
(3) Ys immediately obtained from (2).

4. The result of equation (4) can be more clearly seen by noting

hos wl + ef
e} = w+ ej

i=(1- Wy > Wo ++ eWy) ef + wy (ef + el)
+ wale} + ey

And since I = 0 then

Ws

j e

he 1
ef = - j

uz

5. Without changing the results we derive in any significant way, this
assumption simplifies the formulaes in the text.


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Further, if it is assumed that triangular arbitrage between

exchange rates hold (that is, that et = ef

j

shown that the authorities have the ability to fix the exchange rate

+ e}) then from (3) it can be

index by using the home-numeraire rate to offset the movement of the sum

of the weighted numeraire-bilateral exchange rates’/

N

hee -
(4) = je Wy ej

1

Equation (4) can be interpreted as a policy rule. The home-
numeraire exchange rate is manipulated to maintain a constant exchange
rate index. This property may be true of either an exchange rate index
or a currency basket. The only difference between the two is Wye

The second attribute relates to the home country's exchange

rate for any country j relative to the numeraire. This relationship can

be obtained by manipulating equation (4)

n
he gt _ 1
(5) eg =e ar e;
If currency q appreciates by 1% against the numeraire (se, = 0.01, ae} =0

j #q), then the home currency will depreciate against currency q by (1 -
Wq)% and will appreciate by We against all the other currencies (that is

assuming i=0).


~19-

It is quite difficult to generalize, but some observations can
be gleaned from tables 3-5 and other experiments we have tried:
a) The historical export share indices for each country tend
to be higher (weaker home currency) than the actual basket.
b) In most cases the calculated optimal weights for the
United States tend to be higher and those weights for United
Kingdom and Germany lower.
c) There is a complicated interaction of the effects of
covariances of exchange rates which yields a rather different
pattern to the weights making them quite difficult to
interpret.
d) The interpretation of negative weight for currency basket
is important. In the tables a zero is assigned to the weight,
if the calculated weight optimal is negative. However, for
Finland and Sweden we were unable to interpret our results for
the weights when no restrictions were imposed on equation (20).
Both Turnovsky and BK argue that negative weights have an
economic interpretation. Negative weights imply that the index
will move in the opposite direction as the currency movement
and at least one other country currency will have a larger
weight (perhaps greater than unity) to compensate for this

feature. We have not interpreted our results in this light.

Charts 1-3 show the time paths of the actual currency basket
indices and two 'optimal' indices using the optimal weights from columns

(2) and (3) of the corresponding country tables 3-5. The indices have


Table 5:

Optimal Basket Calculations for Sweden

Trading packet Share”
Partners Weights (1) (2) (3) Weights (4) (5) (6)
tj Tj

Austria -015 31 012 = .014 014 321 -012 = .013
Belgium -038 -.22 -035  =—.037 052 -201 05 -050
Canada 011 -.014 = -.09 -009 .014 -022 -106 = .012
Switzerland 022 038 017.016 021 -044 -021 ~=—-.016
Denmark -082 37 -077.—.083 .107 -40 104.108
United Kingdom .132 12 +137 £104 .137 119 .137 108
France 055 05 -049 .056 -063 -059 059 = .064
Finland 072 -05 -065 .073 .074 .047 069 = =.075
Germany 162 -.17 154 131 144 218 141.116
Holland 051 -074 -043 = 045 -06 -093 -056 053
Italy -036 -.02 -034 ~=.037 041 -022 -040 042
Japan 027 -.03 025 027 018 -041 017 017
Norway 094 192 -089 = .092 128 241 125.125
Spain .012 10 01 012 015 -093 -014 = .015
Total -809 * 747 £736 -888 * -845 =. 84
United States .191 * 253.264 112 * -155 «186

Note to Table:
Swedish basket.

Trading partners listed above are those currently used in
The columns labelled basket weights represent weights used

in the calculation of Swedish basket while the column labelled export share
Columns labelled (1) and (4)
represent weights derived using equation (20), the other columns are special

are derived from direction of trade statistics.

cases of equation (20).


I. Introduction

Since the collapse of the fixed exchange rate regime in the
early 1970s, countries have been faced with the question whether to peg
their currencies and, if so, how this should be accomplished. Several
countries have chosen to tie their currencies to unilaterally designed
baskets of other currencies. European examples include Austria, Finland,
Greece, Norway, Portugal, Spain, and Sweden. This paper considers the
objectives of the policymakers and attempts to derive the optimal weights
used in a currency basket. Given the formulas derived in the paper,
optimal currency baskets are considered for the Nordic countries:

Norway, Sweden, and Finland.

By the end of 1978 each of these countries had chosen to define
its own currency basket and to stabilize the value of its currency in
reference to this basket. Prior to 1978, Norway and Sweden had
participated in the European Community's currency arrangement often
referred:to as the "snake". Under this arrangement both countries
experienced appreciation of its kroner value measured by a general
exchange rate index which was not consistent with their other
macroeconomic goals. Therefore, to gain more control of the exchange
rates, their authorities opted for currency baskets. The use of currency
baskets introduced new and different problems.

There are a number of decisions which the policymaker must make
when using a currency basket as a method of stabilization. Some of these
are: (i) the choice of weights to be assigned to the different
currencies, (ii) the choice of a geometric or an arithmetic index, and
(iii) the choice of time period to be used for the base. In this paper

we limit the analysis to the first problem: the choice of weights.


-12-
N
ho. 1
(14) e, = Ew e;,

A direct expression for this dependency is obtained by substituting (14)
into (12):

N j_on ,% 1
(15) x(w) = srk "5 (py - p) v2 (t5 - wy) e;]

Equation (15) gives the expression x(w) which is used in (13).
For our empirical work we separate out the numeraire-home relative price
from the numeraire-country j prices by adding and subtracting pt from the

right hand side of (15). This yields
N ; N
a ae j_ al . 1
(16) x(w) = syek[(p> - p) + : % (p? - p-) +2 (x, w;) ej]

Note that x(w) is a function of both 1;'S and wy'S, the export
shares and the optimal basket weights, respectively. It will be shown
that in only special cases will the basket weights equal the export
shares. The problem of choosing weights can now be seen as minimizing

the objective function with respect to the weights
(17) Min L = EGx(w) = ECx(w)))?

wort. (w)


-14-

From (19), the solution with respect to the optimal weights,
w, can be derived as:

(20) wettotne realty,

By utilizing the BK-model, LS(1982) derive optimal basket
weights when the assumed target is to minimize the variance of the home
country's terms of trade. Surprisingly, the optimal weights LS derive
are similar to those in (20). In fact, the optimal weights are identical
except that the trade weight, t, in (20) are export shares while in LS, rt
reflects total shares.2/

As LS note, terms of trade is not a meaningful exchange rate
policy target when the home country is small (k=1). The basket weights
derived by LS therefore only apply for economies that have some power in
the markets for traded goods (k<1). On the other hand, the basket
weights we derive in (20) is based on minimizing the variance of the
production of exports. This target is relevant (or particularly
relevant) for economies with no power in the market for export goods.

Our result is striking because we derive the same weights as LS except
using a different policy objective function.

Three special cases for the weights derived in equation (20)

are considered below:

Case 1

If relative domestic prices, p? - pl, are uncorrelated with

exchange rates, that is,


Abstract

The purpose of this paper is to derive optimal weights for a
currency basket taking into consideration the objective of the
policymaker. We carefully distinguish between the two terms: effective
exchange rate index and currency basket, which are often used
interchangeably in the literature. In general, our analysis is an
extension of the work of Branson-Katseli and Lipschitz-Sundararajan and
then applied to the Nordic countries. We use the policy objective of
minimizing fluctuation in export production and illustrate our results
using Norway, Sweden and Finland. The weights we derive create optimal
currency baskets which are different from the ones used in the

countries.


Table 3: Optimal Basket Calculations for Norway

Export
Trading Basket Share
Partners Weights (1) (2) (3) Weights (4) (5) (6)

Tj Fj

Austria -015 -085 -013° 014 -009 041 +010 =.009
Belgium 024 240 -022 = =.023 023 31 024 +022
Canada -036 -.041 -.064 .030 -009 -.003 -.087 .007
Switzerland -012 046 -013 -009 .018 -051 029.014
Denmark -068 -.037 -066 .069 104 -.069 -106 = .105
United Kingdom .147 -160 +147 «116 176 +177 +165 .139
France -092 -030 -089 = .094 047 -013 +049 = -.048
Finland -03 -022 -027-— 031 -035 028 -037 036
Germany 177 -120 +175 6143 164 139 172.132
Holland 046 -037 +045 -041 063 -050 -069 = =.056
Italy -033 -030 -032 = .034 -032 -019 -032 = .032
Japan -06 ~.068 059 = .059 -021 -.103 021 -020
Sweden 15 -190 147 +152 +216 +266 +218 ~=.218
Total -89 1.00 -835 815 -917 684 +932 .838
United States .11 0 +165 = .185 --83 316 -068 162

Notes to Table: Trading partners listed above are those currently used in
Norway's basket. The column labelled basket weights represents weights used
in the calculation of the Norwegian basket while the column labelled export
shares are derived from direction of trade statistics. Columns (1) and (4)
represent weights derived using equation (20). Columns (2), (3), (5), (6)
are special cases of equation {0}.


Optimal Currency Basket in a World of Generalized Floating:

An Application to the Nordic Countries

*
Hali J. Edison
and
wk
Erling Vardal
September 1985

Economist, Division of International Finance, Board of Governors of
the Federal Reserve System, Washington, D.C. 20551.

Visiting Scholar, Division of International Finance, Board of
Governors of the Federal Reserve System, Washington, D.C. 20551 and
Associate Professor, Institute of Economics, University of Bergen, 19
Jonas Reinsgate, N5000 Bergen, Norway.


-F3-

Because of (24)

Var(x(w)) = (5y2k)2[var("E ey(p3 - phy) + var("E y ef) |.

Since ef = ef + ef, by using (14)

N-1 N
he 1
Pty ef = Bley - wg(l = tye} - (1 - cydmy sh

N-1
Consequently, Var( 5 Fj ef) = 0 when the weighting scheme in (25) is
chosen, and Var(x(wi) will be minimal.

11. An additional requirement is that cov(p! - pt, ef) =0.

12. LS(1980), p. 88, have a PPP-proposition that is different from cases9
2-3. They show that if PPP holds between the trading partners of the
home country, j.e. p> - pJ = ed, j=2,....N, and if in addition

cov(p" - pl, e;) = 0, j=2,....5N, then it is optimal for the home country
to peg to the numeraire.


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-10-

Using the triangular arbitrage condition, ef = en + ej (9) can

be rewritten as

"

(12) x= sak tle} + pd - p”) +e
where Ty X5/X, that is the trade (export) share of country j.

If the home country is small, it can not influence e and pi.
Under this circumstance the home country can affect the production of the
export either by manipulating the home-numeraire exchange rate or the
domestic price level. This holds even if the home country is so smail
that it exerts no market power, that is when k = 1.

The variable T; in (12) is a trade weight, and can be used to
define an exchange rate index and under certain circumstances optimal
currency basket weights. In the next section we shall define a policy
objective function which allows us to derive weights that incorporate

other factors.

V. The Optimal Weighting Scheme

In this section we focus on the type of exchange rate policy
targets used in the Nordic countries. Typically, in these countries,
exchange rate policy is aimed at promoting and maintaining the
competitiveness of the industries which produce traded goods. The
background is that the level of employment is viewed as the most
important overall target in the economy and a substantial fraction of the

labor force is employed in the traded-goods sector. Since production is


-13-

where x(w) is given in (16), and the weights satisfying this problen
w? = (Wy sees sli) are defined as optimal.
The first-order conditions after minimizing (17) are as

follows:

. Covip! - ot, e;)

N 11 N
(18) z (ws - 1; )Covles,e,) - z T;

j-2
Covi p! - a) = 0 for q = 2,...,N.

To simplify the expression in (18) we switch our notation to

matrices and vectors. The N-1 equations in (18) are rewritten as:

(19) Q(w- t) -—(t) -T

1
f)

where Q = [coves .e4) an (N-1) by (N-1) variance covariance

matrix of exchange rates.

T= |cov(p)-p' ed) | = a (N-1) by (N-1) covariance matrix
between the numeraire-country q
exchange rates and the relative price
of the (N-1) foreign countries and the
numeraire country.

Te. |cov(p'-p",e0) = a (N-1) column vector of the covariance

between the numeraire-country q
exchange rate and the relative price
between the numeraire and home

country.