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A Theory of Demand for Products Distinguished by Place of Production (Une thorie de lademande de produits diffrencis d'aprs leur origine) (Una teora de la demanda de productosdistinguindolos segn el lugar de produccin)Author(s): Paul S. ArmingtonReviewed work(s):Source: Staff Papers - International Monetary Fund, Vol. 16, No. 1 (Mar., 1969), pp. 159-178Published by: Palgrave Macmillan Journals on behalf of the International Monetary FundStable URL: http://www.jstor.org/stable/3866403 .Accessed: 12/02/2012 08:07

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A Theory of Demand for ProductsDistinguished by Place of Production

Paul S. Armington *

I. Introduction and Summary

INTERNATIONAL TRADE flows are commonly dentified and clas-sified on the basis of three characteristics: he kind of merchandise

involved, the country (or region) of the seller, and the country (orregion) of the buyer. In theories of demand for tradable goods, it isfrequently assumed hat merchandise f a given kind supplied by sellersin one country s a perfect substitute or merchandise f the same kindsupplied by any other country. This assumption mplies-leaving asideany factors that lead buyers o spend more for a given item than neces-sary-that elasticities of substitution etween these supplies are infiniteand that the corresponding rice ratios are constants. While the impor-

tance of lags in buyers' responses, and other such "imperfections" nbuyers' behavior, need not be overlooked, an appeal to them as the solebasis for changes in relative prices of directly competing merchandisewould appear to be neither realistic nor attractive theoretically. Apreferable approach would be to recognize explicitly that any worldmodel of feasible dimensions would identify few, if any, kinds ofmerchandise or which the perfect-substitutability ssumption s tenable.

Accordingly, this paper presents a general theory of demand forproducts hat are distinguished ot only by their kind-e.g., machinery,

chemicals-but also by their place of production. Thus French machin-ery, Japanese machinery, French chemicals, and Japanese chemicalsmight be 4 different products distinguished n the model. Such productsare distinguished rom one another n the sense that they are assumedto be imperfect substitutes n demand. Not only is each good, such aschemicals, different rom any other good but also each good is assumedto be differentiated from the buyers' viewpoint) according to thesuppliers' area of residence. If the model distinguished 10 goods and

* Mr.Armington,

economist in the Current Studies Division of the ResearchDepartment, is a graduate of Swarthmore College and the University of Californiaat Berkeley. Before joining the Fund in 1965, he was a Research Fellow inEconomics at the Brookings Institution.

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20 supplying areas, the number of products distinguished n the modelwould be 200.1

The geographic areas that serve as a basis for distinguishing roductsby origin are also used as a basis for identifying different sources ofdemand. For example, if France is identified as 1 of the 20 supplyingareas, the model would contain a function expressing French demandfor each of the 200 products. There would be 10 French demands ordomestic products and 190 French import demands. Conversely, oreach French product, here would be 1 domestic demand and 19 exportdemands.

The problem onfronted n this paper s that of systematically implify-ing the product demand functions o the point where they are relevantto the practical purposes of estimation and forecasting.2 Starting withthe general Hicksian model, the exposition runs through a sequence ofprogressively more restrictive assumptions eading to a specification ofthe product demand unctions which, though highly simplified, preservesthe relationships etween demand, ncome, and prices that are apt to bequantitatively ignificant.

The fundamental modification of the basic Hicksian model is theassumption of independence-an assumption whose implications have

already been explored in other branches of demand theory and incapital theory.3 In its present application, he assumption of indepen-dence states, roughly, that buyers' preferences or different products ofany given kind (e.g., French chemicals, Japanese chemicals) are inde-pendent of their purchases of products of any other kind.4 By thisassumption, or example, an increase n purchases of French machinerydoes not change the buyers' relative evaluation of French chemicalsand Japanese chemicals. Given the assumption of independence, hequantity of each good demanded by each country (e.g., French demand

1 In the second paragraph of Section I and throughout the rest of the paper, adistinction is made between "goods" and "products." "Goods" are distinguishedonly by kind (that is, by the kinds of wants or needs they serve), whereas"products" are distinguished both by kind and by place of production. The geo-graphic and commodity dimensions of the model are spelled out more formally inSection II.

2 See Section VI, pages 170-71.3 Seminal contributions were made by Robert M. Solow, "The Production Func-tion and the Theory of Capital," The Review of Economic Studies, Vol. XXIII(1955-56), pp. 101-108, and by R. H. Strotz, "The Empiiical Implications of aUtility Tree," Econometrica, Vol. 25 (April 1957). A fuller development is givenby I. F. Pearce, Chapters 4 and 5, in his A Contiibution to Demand Analysis

(Oxford University Press, 1964), pp. 133-230. H.A.J. Green provides a goodreview of the relevant literature in his book, Aggregation in Economic Analysis(Princeton University Press, 1964).4 The assumption of independence s stated in more precise terms in Section III,page 164.

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A THEORY OF DEMAND FOR PRODUCTS

for chemicals-in-general) an in principle be measured unambiguously.5In other words, there exist demands or groups of competing products.Following conventional erminology, each such demand can be calleda market. There would be, for example, the French market or chemi-cals; and chemicals supplied by different countries or areas (includingFrance, of course) could be said to compete n that market. Moreover,demand for any particular product (e.g., French demand for Japanesechemicals) can be rigorously xpressed as a function of the size of thecorresponding market (e.g., French demand for chemicals-in-general)and of relative prices of the competing products.

It is next assumed hat each country's market share is unaffected bychanges in the size of the market as long as relative prices in thatmarket remain unchanged. On this additional assumption, he size ofthe market is a function of money income and of the prices of thevarious goods (e.g., the price of chemicals-in-general, he price ofmachinery-in-general) 6 Combining his function with he product demandfunction described above, the demand or any product becomes a func-tion of money income, the price of each good, and the price of thatproduct relative o prices of other products n the same market. (Pricesof products competing n other markets are influential only insofar as

they determine he prices of goods.)If there are a large number of products competing in the market(in other words, if the number of supplying areas identified in themodel is large), then further ways of simplifying he product demandfunctions are needed if they are to be of much relevance to practicalresearch. The approach suggested in this study (Section IV) is toassume that (a) elasticities of substitution etween products competingin any market are constant-that is, they do not depend on marketshares, and (b) the elasticity of substitution between any two products

competing n a market s the same as that between any other pair ofproducts competing in the same market. These assumptions yield aspecific form for the relation between demand for a product, the sizeof the corresponding market, and relative prices; and the only priceparameter n this function is the (single) elasticity of substitution nthat market.

Differentiation f the demand unctions yields an analysis of changesin demand for any given product (Section V). The percentage changein demand for any product depends additively on the growth of the

market in which it competes and on the percentage change in theproduct's share in that market. The change in the product's market5 See page 164.6 See pages 165-66.

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share will depend n a specific way on the change n the product's pricerelative o the average change in prices of products n the market. Thegrowth of the market will depend mainly on the change in income andon the income elasticity of demand for the respective good (i.e., theclass of products of which the given product is a member). In orderthat market growth depend exclusively on this income effect (and noton the prices of goods), certain additional assumptions are needed(see p. 170). The study concludes with a brief discussion f the relevanceof demand heory to some of the current research n the area of tradeanalysis and forecasting.

II. Geographic and Commodity Dimensions of the Model

Any large model of the world economy would make use of somevector of countries or other geographic areas, C=(C1,C2, . . . ,Cy),as well as some vector of goods, X= (X1,X2, . . . ,Xn). The presentdemand model stipulates, n addition, hat each good is differentiated nuse (e.g., in demand) according o where it is produced. Any "good,"Xi, refers o a group of "products," ach supplied by a different ountryor area; that is, X= (XI,X,i2 . . ,Xim), where Xij is assumed o be

an imperfect ubstitute or Xk(ji k) from the viewpoint of buyers nany country or area, Ci. For later reference, t will be useful to set outthese specifications n the following ormat:

X= (X11,X2, . . . ,XlmnX2,X22,

... .,AX2m . ? ,Xnl,Xn2, * * .,Xnm) (1)

- (X1,X2, . . . ,Xn), where

Xi=- (Xil, Xi2, . . . ,Xm), for i= 1,2, . . . ,n.

Thetop

line of this formulation, whichmay

be called theproduct

vector,can also be presented s a product matrix:

supplying ountries _

goods 1 2 ........ m

1 Xll X12 ........XmI 2 X21 X22 ........ X2

. . . . . . . . . . .? * * . . . . .. . ..

n IX'L., Xn. ........ X.

162

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The country vector C= (C1,C2, . . ,Cm) also lists the varioussources of demand. The demand of buyers n any country, CG, or any

product, Xij is here called a product demand, and the demand sideof the model is described by all such functions. Since there are mdemands or each product, and since there are mn products, he demandside of the model is comprised f m2n product demands, of which mn aredomestic demands and mn(m-1) are export (or import) demands.(Import demands are not residual demands, depending on domesticsupply functions, as is the case in models which presume hat goods arehomogeneous over different sources of supply. In the present model,the analysis of ex ante demand-domestic, import, and export-requires

no particular ssumptions bout supply unctions.)

III. Market Demands and Product Demands: AnApplication of the Assumption of Independence

Ex ante demand functions state relationships hat must exist amongcertain variables if buyers are to be satisfied. Buyers' satisfactionentails getting the most for their money, given the available selectionof products and their prices. Demand functions may thus be viewed asstatements of conditions under which an index of buyers' satisfaction sas high as limited ncomes and given prices permit. Given such an index,U, these conditions or demand unctions an be derived by maximizing Usubject o a budget constraint.7

The general approach o the derivation of the product demand func-tions identified n the previous section is to express U as a function ofall mn products; that is, using (1), U= U(X).8 Then, given a cor-responding rice vector,

P= P11,P12, . . . ,Pim,P2l,P22, . . . ,P2m, . . .,Pnln2 . . . ,Pnm, (2)

and national money expenditure, D, U(X) is maximized ubject to thebudget constraint D=PX'. Once U is specified, the first-order ondi-

7 Each of the four simplifying assumptions introduced in Section I can be inter-preted as a certain restriction on, or specification of, the index U. Thus, by usingthe maximization procedure to derive the product demand functions, a formal link

is establishedbetween

generaldemand functions and the

highly simplified relation-ships mentioned in Section I.8 The rest of this paper deals with the demand of any single country and does not

consider the aggregation of demands across markets. Hence, no notation is attachedto U, or to other variables mentioned later, to identify the country whose demandis referred to.

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tions, together with the budget constraint, imply this country's mndemand unctions, each one having the general orm

Xij-= X(D,Pll,P12 . . . ,PlM,P21,P22, . .

P2n, .? *PnlXPn2, . . .? ,nm, (3)for all i and all j.

Of course, the close association between products of the same kindis not reflected at all in the general orm of (3). The problem at handis to specify U in such a way that the information mplicit n the productclassification cheme s fully utilized, o the end that the product demandfunctions

maybe

appropriately implified.The first and most funda-

mental step is to specify U in such a way that the demand or any good,Xi, can be measured nambiguously.

Professor Solow asked the question (albeit in a rather different con-text),9 Under what condition can U be "collapsed" n the followingway?

U= U(Xl,Xl2, . . . ,Xm,X21,X22

. . . ,X2m . . . ,Xnl,Xn2, . . . ,Xnm) (4)

=U'(X

2,X2,. . . ,X), where

Xi-= i(XilX2, . . . ,Xim), for i= 1,2, . . . ,n10

If U can be so collapsed, all combinations of Xil,Xf2, . . . ,Xim whichyield any given value of Xi are equally good, and that given value is aspecific quantity of Xi-in-general. n other words, f (4) is true, demandsfor goods-here called market demands-can be measured unambigu-ously. The necessary and sufficient condition for collapsing U is thatmarginal ates of substitution etween any two products of the same kindmust be independent f the quantities f the products of all other kinds."In other words, buyers' relative evaluation (at the margin) of differentproducts ompeting n a given market must not be affected by their pur-chases n other markets. This is the assumption f independence.12

9 See Solow, op. cit.10 Compare with equation (1).11The proof is in Solow, op. cit., and in an early work by Leontief referred to in

the Solow article.12 In theory, the assumption of independence might be viewed as tautological;

for independence could well be taken as a defining characteristic of productsdistinguished by their kind (that is, by the kind of want or need they serve).

Following this approach, an alternative, more basic assumption would be neces-sary, namely, that products can be rigorously classified by kind in the first place.In practice, however, goods must be identified within the framework of some avail-able classification scheme (such as the Standard International Trade Classification).Given this constraint, independence is not necessarily tautological. How far the

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Given the assumption of independence, which leads to (4), demandfor any particular product Xij can be written as a function of Xi andof relative product prices in the ith market.13 f the demand or Xi canin turn be related to income and appropriate price variables, then amanageable pecification f (3) is within reach.

The price variables on which Xi will depend are, naturally, he pricesof goods, or Pi (i= 1,2, . . . ,n), and Pi is a function of prices ofproducts n the ith market, ust as Xi is a function of quantities of theseproducts. But Pi cannot be just any function of product prices; theprices of goods must be such that the demand for the ith good, whichthey explain, s consistent with the optimum election of products n the

ith market. More exactly, the demand for Xi as determined by incomeand prices of goods must be the same as the value of p1 mplied by alldemands or products n the ith market as determined y direct referenceto income and product prices.

This condition s fulfilled f

pp P4 p .?, .P+i= ilaXil aXi2

im aXi,,, (5)for i=1,2, . . . ,n14

Note that (5) implies hat

P aanX P~2 ax1--=1a-X-, and Pia =a' etc.,

DXi2 aXi3which are the first-order onditions or optimum mix of products n theith market.

It is not clear from inspection of (5) that Pi depends only on product

prices. In fact, to ensure hat Pi is independent f Xi, it must be assumedthat pi is linear and homogeneous. Then the partial derivatives n (5)depend only on ratios of quantities of products demanded n the ith

assumption restricts the realism of the model depends on the extent to which thedifferent items in the goods vector correspond to more or less discrete wants orneeds. Within the limitation imposed by the available classification scheme, theanalyst may attempt to select a vector of goods that renders the independenceassumption as realistic as possible. It should be noted, however, that supply factorsmay suggest to him a rather different choice of goods. The identification of goods,for purposes of a complete model, cannot be guided in practice by any singlecriterion.

13The function may be derived from the conditions for minimizing the moneycost of purchasing any given volume of Xi, given some specification of '0. The

derivation is analogous to that of the demand for a factor of production as afunction of output and relative factor prices.

14 See Solow, op. cit.

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market, and these ratios in turn depend only on ratios of the productprices; hence, Pi is only a function of P

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By substituting 6) into (7), a particular reformulation f (3) isobtained. In this reformulation he role of prices is handled by only

m+n variables, compared with mn variables in (3): product pricesoutside the ith market affect Xij only insofar as they determine pricelevels in other markets (that is, PI,P2, . . . ,P-1,Pi+1, . . . ,P,). This,specifically, s the sense in which the two restrictions on U serve toutilize the information mplicit n (1). And these restrictions-both verymild-open the door to more powerful simplifying ssumptions, uch asthose discussed below.

IV. Product Demand Functions Assuming a Single,Constant Elasticity of Substitution in Each Market

If many countries or areas were identified n the model, equations 7),in the above form, would probably be too complicated o be of practicaluse. A way to simplify them is to introduce the assumptions that(a) elasticities of substitution n each market are constant and (b) theelasticity of substitution between any two products competing in amarket s the same as that between any other pair of products competing

in the same market. n terms of the index U (equation 4), these assump-tions are equivalent o the specification hat the p5's re constant-elasticity-of-substitution CES) functions, having he general orm

Xi= --i, (Xi,Xi,2, . .. A im)1 16

= [blXj-Pf + bi2Xi2-P' +. + bimXim Pi] . (8)

Given (8), it can be shown that equations (7) have the form

Xij= bij X ) , (9)where ao is the elasticity of substitution n the ith market and bqi is aconstant.17

In equation (9) and in all those that follow, Xij can be interpretedeither as the demand for the ith good supplied by country j or as thedemand for the ith good supplied by the jtl group of countries. Forexample, he jth group could be all foreign countries; n this instance, (9)would express the demand or total imports of the it' good (Xij) as afunction of demand or the ith good wherever produced Xi) and of theratio of the average mport price (Pij) to the average price level in themarket (Pi). This flexible interpretation f the variables n (9) is the

16 This function is linear and homogeneous, as required.17 The derivation of (9) from (8) is in Part I of the Appendix.

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consequence f two properties f Ht s specified n (8): (a) the marginalrate of substitution between any pair of products competing in theith market s independent of demand for any other product(s) in thatmarket, so that ~ in (8) can be "collapsed" n the manner of equation(4); (b) 0i in (8) is linear and homogeneous, o that the index functionsappearing n any "collapsed" orm of (8) must be linear and homogene-ous also. Hence there exists an unambiguous emand or any subset ofproducts n the ith market, and this demand can be related o the over-allmarket demand just as Xi, under strictly analogous conditions, can berelated o total income.

Equation (9) can be written n a variety of useful ways; for example,

PiJXij bfij(.Xj) (pi 1-, (10)

which relates money demand for Xij to the size of the correspondingmarket measured n value terms, and

Xi = bjt Pij )' or (11)

pi,_\

Pi/ ,-jbX

Ib (j i) 1-04 (12)

which expresses he market share as the dependent variable. As is clearfrom (12), value shares are constant if , = 1. (In this special case,i, is a Cobb-Douglas unction with parameters ik.) If oi > 1, a relative

fall (or increase) in Pij yields an increase (or decrease) in the marketshare of Xj. The reverse s true f oi < 1. Ordinarily, t would be expectedthat ei exceeds unity: an "improvement n competitiveness" hould yieldan increased hare, and vice versa. But the logic of the model (as distinct

from some particular nterpretation f the variables) places no suchrestriction n this elasticity.

V. Analysis of Changes 18

Total differentiation f the market demand function (6) and theproduct demand unction (9) yields the following relation between thepercentage change in demand for Xi>, n value terms, and percentagechanges n income and price variables:

18 The argument of this section is more fully developed in Part II of the Ap-pendix.

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d (PqXij dD dPi I dPk

DPV= e

D- (-1 )

p~ +

P ,/k

Pk

- (f-1) (dPj dPi) 19

where ei is the income elasticity of demand for Xi, i is the direct priceelasticity of demand for Xi, and rji/k is the cross elasticity of demand forXi with respect to Pk (k=1,2, . . . ,i-l,i+l, . . . ,n). The firstthree terms together measure the growth of the market (in value) forXij, while the fourth term measures the percentage change in Xj'sshare of the market.

The change in demand for Xij can also be analyzed in more traditionalterms by replacing the price level in the ith market with product pricesand market shares. As is demonstrated in the Appendix,20 one of theproperties of Pi is that the percentage change in this variable is equal toan average of changes in component product prices, weighted by marketshares; specifically,

dPi __m Si dPik S ik = PiXikP-

=/k'

where S= PiXi

dP.Substituting this summation for -i in the second and fourth terms of

Pi(13), one obtains (after a little shuffling of terms) the following result:

d(PijXij) dD -PX = D---- ( 1-Sij)(ri-l)+Sjj(r--1) dP~jri,j U_ -ij

+2 Sik(i-I)-Sik(i-l) dpi 2 _/-. (14)k#j _ ik k#i Pk

Here, the growth of demand for Xij is divided into the following com-ponents: an income effect (first term), an "own price" effect (secondterm), the effect of prices of closely related products (third term), andthe effect of all other prices (fourth term). The bracketed coefficient of

dP- is the direct price elasticity of demand for Xij, in value terms, whilePo

dPithe bracketed coefficient of pik represents the cross elasticity of demand^ikfor Xqi with respect to the price of any other product competing in the

19 E indicates summation over all k, for k = 1,2, . . - 1,i + 1, . . ,n.kai20See page 174, ootnote 29.

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same market. Conversely, of course, the cross elasticity of demand forXik with respect o Pij would be given by [S1j(ai-1 ) -Sj(n,- 1)].

The analysis of changes into market-expansion actors and share-adjustment actors (equation 13) is of greater relevance to currentresearch than the more conventional breakdown shown in (14)-seeSection VI. On the other hand, (14) is useful because it focuses atten-tion on how changes in individual product prices affect trade, and inparticular n the role played by market shares.21

In the event that equations (13) or (14) are yet too complicatedto suit practical purposes, the feasibility of introducing wo furthersimplifying ssumptions may be considered. The first of these is that theelasticity of demand or Xi (q) equals unity, and the second is that thethird term of (13), or the fourth term of (14), is small enough to beignored. The first of these assumptions, which implies that expenditureon the ith class of products is independent of price changes in theith market, ocuses attention on the effects of changes n relative pricesin the market and abstracts rom any (presumably small) effects ofchanges n the general evel of prices n the market. The second assump-tion would not be unreasonable f changes n price evels in other marketsare very small, or if such changes are apt to have offsetting effects ondemand for Xi.22 On these additional assumptions, (13) and (14)reduce o the relatively imple ormula

d(PXij dD 1) dP15)DZ- -((^ dPidP^ (15)

VI. Conclusion

In much of the current research n the area of trade analysis andforecasting, he change n any particular rade flow is viewed as the sum

21 See pages 174-75 of the Appendix. This role of market shares is examinedin detail by the author in "The Geographic Pattern of Trade and the Effects ofPrice Changes," which will be published in a subsequent ssue of Staff Papers. Thepartial elasticities in (14) formally resemble the formulas derived by Hicks andAllen. (See J. R. Hicks and R. G. D. Allen, "A Reconsideration of the Theory ofValue: Part II.-A Mathematical Theory of Individual Demand Functions,"Economica, New Series, Vol. I (1934), pp. 201-202 and 208-11.) Similarformulas have been derived and applied in the context of international trade byProfessor P. J. Verdoorn. For his early work in this area, see Annex A of hiscontribution to the annual papers of the Dutch Economic Society, 1952, and alsoAppendix A of "The Intra-bloc Trade of Benelux," in Economic Consequences ofthe Size of Nations, ed. by E. A. G. Robinson (London, 1960), pp. 319-21.

22 One might be tempted to assume that the cross elasticities, 7/k, are zero.However, if /k= 0 and 7n= 1, then et must equal unity, since, by definition,-7 = t/k + ei. See Paul Anthony Samuelson, Foundations of Economic Anal-

k?iysis (Harvard University Press, 1961), p. 105. Ordinarily, t would be intolerablyrestrictive o assume that the income elasticity is perforce equal to 1.

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of two components: the change that would occur if the given sellercountry were to maintain ts share in the market (that is, its share oftotal sales to the given buyer country n the given commodity lass), andthe deviation of actual sales from constant-shares ales. Within thisframework, forecasting of trade is essentially a two-step process inwhich (a) forecasts of growth n the various markets, ogether with abase-period matrix, yield a constant-shares matrix for the projectionperiod, and (b) this constant-shares matrix s modified o take accountof factors expected to yield gains or losses in shares. In retrospectiveanalysis, he role of these factors s evaluated by comparing actual salesin particular markets-or, more frequently, groups of markets-with

constant-shares ales. This general approach o trade analysis and fore-casting might conveniently be called the modified-shares pproach.23Is the breakdown f changes n trade flows into the two components

merely a matter of accounting hat seems useful for certain purposesbut which has no causal significance, no roots in buyers' behavior?Or can the traditional heory of buyers' behavior provide a satisfactoryrationalization f the modified-shares pproach? Can a few assumptionstie theory and practice ogether?

This paper shows that the modified-shares pproach does not require

any radical departures rom the traditional heory of buyers' behavior.Starting with the fundamental assumption that products of differentcountries competing n the same market are imperfect substitutes, hestudy shows how a powerful and reasonably realistic specialization ofthe function describing buyers' behavior-the specialization ndicatedby equations 4) and (8)24-leads to quite simple demand relationshipsembodying he constant-share nd share-adjustment omponents. Themodified-shares pproach may find foundations n the demand side thatare different rom those proposed in this paper, including the marketimperfections eferred o at the outset. But the assumption hat productsare distinguished by place of production s a very convenient pointof departure oward a rigorous theory of market growth and shareadjustment.

23For an introduction to the research in this area, together with an appraisalof alternative frameworks for the analysis of international trade, see Grant B.Taplin, "Models of World Trade," Staff Papers, Vol. XIV (1967), pp. 433-55.Forecasting methods involving the modified-shares approach are currently beingdeveloped at the International Monetary Fund and elsewhere. Regarding retro-spective analysis, studies of export performance measured by changes in averageexport shares appear from time to time in recent Fund Annual Reports.

24See pages 164 and 167.

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172 INTERNATIONAL MONETARY FUND STAFF PAPERS

MATHEMATICAL APPENDIX

I. Derivation of Product Demands Assuming a Single, ConstantElasticity of Substitution n Each Market

Section IV introduces the simplifying assumption that any given quantity-indexfunction, 0

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notation, letI a , the elasticity of substitution in the ith market. Equation1 + pi

(17) then becomesXlk = X( k = 1,2, . . . ,m, (18)

where 0 < at < 00. (The limiting cases have the following interpretations: if-= 0, the products are perfect complements; if ao = 00, the products are perfectsubstitutes.

Now, substituting (18) into (8), and writing pi in terms of oa

{X{= bik[ (Xi) ) ] } (19)

--= \

PX1, -iPi^1b'i-

b -Xt I bk as-I O'

-I

Then, solving (19) for X,

X4=-- btj'X, X b,ti' P\ ,) 1-'T. (20)

Equation (20) may be viewed as a particular specification of (7).Simplification of (20) can be effected by relating the complex, relative-price

term to the price level in the market, P,. Pi has a particular form, correspondingto the specification of qp4. sing (5) and (8),

Pti= Pt+ X =- Pijbij-lXij i .

Then, substituting rom (19)

-n in - 1

Ps=Pijbis-lX^ ,b^ f(p)3~1

1- (21)

= Pt 2, bik (P)

Therefore,

k Pt / \ Pi-k '( p') b i ( P(J) i es (22)Substituting (22) into (20), equation (9) is obtained. To repeat,

Xwis bfr ay Pru e e (9)

where Xq is any country's demand for any product, expressed in volume.

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II. Analysis of Changes

Total differentiation of (9) and (6) yields the relationship between changes inXij and changes in the explanatory variables. Starting with (9),

dX.i = ?X7 dX. + a--p dP,j + -aAj dPi

_= a dXi - aiXiiPif'dPi + aiXjPi-'dP4.

Dividing through by Xij,dX,j DXxjXi dX, dP4j dP(at x4 x -LT(23)Xij XiX,ij Xi ip pI

dX, ( dPis dP,\=X

-p -pI,

noting that the partial elasticity of Xij with respect to Xi equals unity. The firstterm represents the growth of the market for Xij; the second term represents thepercentage change in Xij's share of the market. The growth of the market can, ofcourse, be analyzed by differentiating 6).

dXi dD dPi , dPk 28X,i= D-- p + /k p , (24)

where et is the income elasticity of demand for Xi, 7 is the direct price elasticityof demand for Xi, and 77/k is the cross elasticity of demand for Xi with respect toPk

(k=

1,2,. . .,i - 1,i

+ 1,. . .

,n).And

substituting (24)into

(23),dX4, dD dP, dPk IdP,4 P,j \ 25X. L D Pi- 7 +47 L/K- - p)L' i. (25)

dPi Pik PikXiIt can be shown that = Sik pik, where Sk -p- = the market share

of XLk in value terms.29 Thus the effects on Xif of changes in prices of products

28 l indicates summation over all k, for k= 1,2 ..i - + 1, .... ,n.kli

29 Since -p in (8) is linear and homogeneous,

X Xz. Xik = ? p X,*, using (5).

Therefore,

p, = XkPik.

Then

dP, _ P4kXk dP4k m PLXL8 Xd , nP, ' =LX

PLX,PLXL~

-X

X,X

Xi

the second term, a weighted average of percentage changes in market shares, iszero.

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competing in the ith market depend not only on aoand n, but also on market shares.This role of market shares can be seen more clearly if the second and fourth termsof (25) are expanded in the following way:

(dP,j dP\ dP, (dPi - dPk\) dPik- a'-77, -\- Pi

Js- SP

S.kpik

[ dPj dP dP\ d dPk\ 30= - as^"D Sik D " Sj + Sik Id,,sS )"--d,- (dp,j

Si-'p\k

dPitv dPik dPij - dPik

= - ( d - Stf ap + SJ + Skatf - SQ7itkdPik

--=-1-S,)~- + o . v'- s -S {p Pik

Substituting nto (25),

Xij Et-

(dD 1Sd,--S1] p, + Sik

,,Sik, pdk

+ I '7/7p. Pk (26)

The bracketed coefficient of dPi in (26) is the direct, partial elasticity of

demandorandheracketedPik i the cross elasticity ofemand for X,j, and the bracketed coefficient of - is the cross elasticity ofPikdemand for Xij with respect to the price of any other product in the ith class. Thedirect elasticity of demand for X4s s inversely related to its market share (assumingthat o, > 7n). Intuitively, the more important is Xij in the market, the smaller willbe the percentage gain or loss from the substitution associated with a given changein its price-and the larger the percentage change in demand for all other productsin the market. By the same token, the cross elasticity of demand for Xij, withrespect to the price of any other product in the same market, is directly related tothe market share of that product (again, if n > ,/).

The two bracketed coefficients in (26) each have two terms, reflecting the factthat a change in the price of a product affects demand for that product, or anyother product in the same market, in two different ways. First, the price changealters relative prices of products in that market, bringing about a substitution effectmeasured by the first term. Second, the price change alters the price level in themarket, bringing about a change in the size of the market itself, and this market-expansion effect is measured by the second term. In (25), market-expansion actorsand share-adjustment factors are clearly separated, while in (26) they arescrambled in such a way as to focus attention on the effects of particular pricechanges.

The percentage change in demand for Xij in value terms may be obtained byadding P to both sides of (25) or (26). After considerable manipulation it is

found that this change, (pi,jj) is the same as shown in (25) or (26) except

that aoand 7 are replaced by (as - 1) and (. - 1), respectively.30

I indicates summation over all k, for k = 1,2, . . .j-l,j + 1, . ,m.k#j

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176 INTERNATIONAL MONETARY FUND STAFF PAPERS

In some practical applications, it may be feasible to introduce the furthersimplifying assumptions that 1 = 1 and that the fourth term of (26), or the thirdterm of (25), is quantitatively nsignificant. In this event, the percentage change indemand for Xo in value terms reduces to a relatively simple formula. Starting from(26)

d(Pv,Xu1) dD dPj dP k

= .sdDD _ S,)---(dPi -Sk (27)

The irsterm represents the growth of the market for Xi in value terms and theThe first term represents the growth of the market for Xio, in value terms, and thesecond term represents the percentage change in the product's value share of themarket. Of course, this result can also be derived directly from (25).

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Une th6orie de la demande de produits differencies d'apresleur

origineResume

Cette etude offre un appui heorique a certaines pratiques de rechercheselon lesquelles a variation d'un flux commercial particulier ntre paysest consideree comme la resultante de deux elements : la modificationqui se produirait i le pays fournisseur onne devait conserver a part dumarche, et l'ecart entre les ventes effectives et celles qui s'effectueraientsi sa part du marche demeurait constante. Ces pratiques comprennent

la methode de prevision des echanges dans laquelle : 1) les previsionsde l'expansion des divers marches, ombinees avec une matrice relative ala periode courante, fournissent une matrice pour la periode futureetablie sur la base de parts du marche constantes; ) cette matrice etabliesur la base de parts du marche constantes st modifiee pour tenir comptede facteurs censes engendrer des gains ou des pertes de parts. Dans lapresente etude, on fait valoir que l'analyse des modifications des fluxcommerciaux n deux elements representant, 'un des parts du marcheconstantes et l'autre des parts du marche modifiees, n'interesse pas

seulement a comptabilite, mais qu'elle peut certainement tre rattacheepurement t simplement la theorie traditionnelle u comportement esconsommateurs. On part de l'hypothese que les produits ont differenciesnon seulement d'apres eur espece mais egalement d'apres eur origine.Autrement dit, on suppose que des produits originaires de differentspays et offerts concurrement ur e meme marche ne sont pas susceptiblesde se remplacer parfaitement. I est demontre ensuite qu'une specialisa-tion poussee et suffisamment ealiste de la fonction de bien-etre de Hickspermet d'obtenir des equations assez simples de la demande englobantles deux elements mentionnes

plushaut. Cette

specialisatione fonde

sur l'hypothese "d' independance" telle qu'elle a 6te formulee parR. M. Solow, R. H. Strotz et d'autres) ainsi que sur celle selon laquelleles elasticites du remplacement ntre produits offerts concurremment urun marche quelconque donne sont constantes et egales.

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Una teoria de la demanda de productos distinguiendolossegun el lugar de produccion

Resumen

Este estudio ofrece un apoyo te6rico a ciertas practicas de investiga-cion segun las cuales a la variacion en una corriente determinada deintercambio ntre paises se la considera como la suma de dos compo-nentes: la variacion que ocurriria i un pais vendedor dado mantuvierasu participacion n el mercado, y la desviaci6n de las ventas efectivascon respecto a las ventas que tendrian ugar de permanecer onstantes

las participaciones. Dichas practicas ncluyen a previsi6n de los inter-cambios, en la cual 1) los pronosticos del crecimiento en diversosmercados, unto con la matriz de un periodo de base, resultan en unamatriz de participaciones onstantes para el perfodo futuro, y 2) semodifica sta matriz de participaciones onstantes para tener en cuenta alos factores que se espere que produzcan perdidas o ganancias en lasparticipaciones. En este trabajo se mantiene que el analisis de lasvariaciones en las corrientes de intercambio comercial, separando elcomponente de participaciones onstantes el de ajuste de las participa-

ciones, es mas que una simple cuestion de contabilidad, yen

realidadse le puede ligar sencilla y rigurosamente la teoria tradicional delcomportamiento el consumidor. El punto de partida es el supuesto deque se establecen distinciones no solamente segun la clase de losproductos sino tambien segun el lugar de produccion de los mismos.Es decir, se supone que los productos de distintos paises que compitenen el mismo mercado son sustitutos mperfectos. Se demuestra uegoque una especializacion ficaz y bastante realista de la funci6n hicksianadel bienestar lleva a relaciones de demanda muy sencillas en las quese incluyen os componentes de la participacion onstante y del ajustede las participaciones. Esta especializacion xige el supuesto de "inde-pendencia" (que formularan R. M. Solow, R. H. Strotz, et al.) y elsupuesto de que las elasticidades de sustitucion ntre los productos quecompiten en un mercado determinado on constantes e iguales.

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