ࡱ> X cRoot Entry;m~` F`\xsRsnWordDocumenty v`9ObjectPool;* Yxs Yxs;SummaryInformationy (N*#08  !"#$%&'()*+,-./01234567rM>I;=B?@:EADGCFH<JKNOQRUV%LZ[\stuvwxyz{|}~CompObjjRoot Entry;m~` F`\xsRsnWordDocumenty v`S ObjectPool;* Yxs Yxs;SummaryInformationy (N*#08  !"#$%&'()*+,-./01234567r>I;=B?@:EADGCFH<JKTWY%^LZ[\_`befghijklmnopqstuvwxyz{|}~ FMicrosoft Word Document MSWordDocWord.Document.69q՜.+,0HPt|  Dell Computer CorporationG A eProperties of Additive Purchasing Power Parities, illustrated with reference to World Bank 1985 dataOh+'08DP` |ortional to country volumes, give relatively lower volume estimates for certain countries (typically advanced economies), than methods which adopt a more uniform weighting system. This arises because international prices for the former group of aggregation methods are closer to the individual price structures for the advanced group of countries: interacting with the negative price quantity correlations in the data structure, this gives rise to the observed effect under the first principal component. This is a typical Gerschenkron type effect Section 6 examines the second principal component: while only accounting for 5.4% of the variability in the data, this component is nevertheless of considerable interest. It turns out that this component is essentially a contrast between GGK and Van Izjeren type indices: and it is related to the degree of negative price quantity correlation for particular commodities. Countries which have a high proportion of their expenditure on commodities for which there is a strong negative price quantity correlation have relatively smaller volumes under GGK as compared with Van Izjeren indices. 1. Additive Indices 1.1) Let  EMBED Equation.2  denote the price of item i in country j, (i = 1,I: j = 1,..J), and let  EMBED Equation.2  be the quantity: we assume that prices and quantities are strictly positive. Any aggregation method which computes international prices,  EMBED Equation.2 , and then defines the volume measure of country j as  EMBED Equation.2  is said to be an additive method. 1.2) In almost all the additive methods used in practice, the international price vector  EMBED Equation.2  is computed by weighting together the (deflated) price structures of individual countries. In this paper, we shall restrict attention to additive methods where international prices can be computed as arithmetic averages of the deflated price structures of individual countries. More specifically, all of the aggregation methods to be considered are members of the following general class of aggregation methods:- Let EMBED Equation.2  be a matrix of positive weights, satisfying the normalisation condition that  EMBED Equation.2 , (where the individual elements  EMBED Equation.2  may be functions of the prices and quantities P and  EMBED Equation.2 ). Then the weight matrix A defines an additive aggregation method in terms of the following equations in  EMBED Equation.2  ,  EMBED Equation.2  and  EMBED Equation.2 , namely,  EMBED Equation.2  For any positive vectors  EMBED Equation.2  ,  EMBED Equation.2 , and positive scalar  EMBED Equation.2 , providing a solution to the simultaneous equations (1) and (2), then  EMBED Equation.2  defines international prices and e defines the corresponding vector of expenditure deflators, (the reciprocal of PPPs). Equation (1) means that international prices are constructed as a weighted average of deflated country prices: equation (2) says that the expenditure deflators are proportional to country real volumes divided by country expenditures. 1.3) It is relevant to make three comments about this general definition of additive aggregation methods:- a) First of all, it is clear that the solutions to equations (1) and (2) are not unique, up to a multiplicative constant. Thus, if  EMBED Equation.2  ,  EMBED Equation.2  and  EMBED Equation.2 are a set of positive solutions, so is  EMBED Equation.2  ,  EMBED Equation.2  and  EMBED Equation.2 , for any positive constant k. This is a trivial complication. The way round is to specify one particular country, say country b, as base country, and to take the unique solution of equations (1) and (2) which also satisfies the condition  EMBED Equation.2 . b) The requirement for the term  EMBED Equation.2 in equation (2) has nothing to do with the trivial scaling indeterminacy at a), but relates to a much deeper difficulty. If  EMBED Equation.2  is removed from equations (1) and (2) by substituting (1) into (2), then it follows that the vector e must be the maximal left eigenvector of a particular positive matrix, and  EMBED Equation.2 is the corresponding maximal eigenvalue. In general, this eigenvalue will not equal 1, (except in an important special case- see below). So in general, unless the term  EMBED Equation.2  is included in equation (2), no positive solutions to equations (1) and (2) will exist. c) The normalisation constraint, that the rows of A sum to one, is critical. Relaxing this condition has a commensurate scaling effect on the value of  EMBED Equation.2 : so, in effect,  EMBED Equation.2 is indeterminate unless an appropriate normalisation convention is adopted for A. Failure to be clear about the normalisation of A can be the source of much confusion. 1.4) An important question about the system of equations (1) and (2) is- under what conditions is the value of  EMBED Equation.2  equal to 1. This, for example, was effectively the condition imposed by Geary (1958), in his original paper defining the GK index: he regarded it as an essential property. An almost complete answer to the question of when  EMBED Equation.2  was given by Cuthbert, (1999). The two results proved by Cuthbert were as follows:- a) if EMBED Equation.2  is of the form EMBED Equation.2 , for some positive vector  EMBED Equation.2 , then  EMBED Equation.2 . And the more difficult partial converse result b) if A = A(Q) is a function of Q only, and if  EMBED Equation.2  in equations (1) and (2), then  EMBED Equation.2 for some positive vector  EMBED Equation.2 . Cuthbert denoted the condition that  EMBED Equation.2  as being strong additivity . he also denoted an aggregation system defined by weights where (3)  EMBED Equation.2  as being a Generalised Geary Khamis, (GGK), aggregation system. Cuthberts results thus imply that all GGK aggregation systems are strongly additive: and that all strongly additive methods where the weights are functions only of Q are GGK. 1.5) In this paper, two specific GGK methods are considered. These are defined as follows:- a) Geary Khamis, (GK) index:  EMBED Equation.2  for all j in equation (3). (This index was originally defined by Geary, (1958). b) Ikl index:  EMBED Equation.2  for all j in equation (3). (This index was originally defined by Ikl, (1972), using a different formulation: see Cuthbert, (1999), for a proof that Ikls original formulation is equivalent to the definition given here.) In the GK index, therefore, countries with larger volumes have larger weights when the deflated prices of the individual countries are weighted together to form international prices: in contrast, under the Ikl index, countries with large volumes have no intrinsically greater weight in constructing international prices than countries with smaller volumes. As a shorthand, we shall say that indices where large countries have larger weights are non-democratic, while indices where country weights are independent of volume scale effects are democratic. This shorthand notation of course is not meant to imply any value judgement. 1.6) The next sub-class of additive indices we shall consider are the so-called Van Izjeren indices: (Van Izjeren (1987)). In equations (1) and (2), if, in equation (1), (4)  EMBED Equation.2  , then the aggregation method is said to be a Van Izjeren type index. Clearly, an  EMBED Equation.2  which satisfies equation (4) cannot satisfy equation (3): so the Van Izjeren indices are a distinct sub-class from the GGK indices, and have  EMBED Equation.2 . The three specific Van Izjeren indices considered for the purposes of the present study are:- a) the equal weighted Van Izjeren index, (referred to here as the VY index). This is defined by taking  EMBED Equation.2  for all j in equation (4). The VY is thus a democratic index. b) the Standardised Structure, (SS) index, introduced, (using a different formulation), by Sergueev (2000).It can be shown that the SS index is equivalent to a Van Izjeren index, with  EMBED Equation.2  , (where  EMBED Equation.2 stands for is proportional to). In words, this means that the SS has weights which are inversely proportional to total world volumes evaluated at the deflated prices of country j. Usually, these weights are likely to be fairly uniform, so one would expect the SS normally to be a fairly democratic type index. c) the Sakuma Rao Kurabayashi (SRK) index, introduced, (using a different formulation), by Sakuma Rao Kurabayashi, (2000). It can be shown that the SRK index is equivalent to a Van Izjeren index, with weights  EMBED Equation.2  . These weights will tend to apportion weights to individual countries almost in proportion to their individual volumes: so the SRK will be a non-democratic index. [Note that in their original paper, Sakuma Rao Kurabayashi introduced their index as an example of an index with eigenvalue  EMBED Equation.2 : that is, as an index which, in terms of the notation used here, is strongly additive. However, the fact that the eigenvalue in the SRK paper is apparently equal to 1 is an artefact of lack of precision about normalisation of the weight matrix: see the comment at para 1.3c above.] 1.7) The final additive index we consider is not a member of either the GGK or Van Izjeren sub-classes. This index is the Own Weights, (OW) index: see, for example, Dikhanov, (1997). The OW index is defined as follows: let  EMBED Equation.2 , that is, the expenditure share of item  EMBED Equation.2 in country j.The weights  EMBED Equation.2  in equation (1) are then defined as  EMBED Equation.2 . The OW index is likely to be fairly democratic. 2. The Structure of the Data Set 2.1) The data set used comprise data on expenditures, and prices relative to the USA, for 57 countries and 139 items, provided by the World Bank, and relating to 1985. For the purposes of the present study, three items (changes in stocks, net purchases abroad, and net exports of goods and services) were excluded from the calculation - since these are balancing items for which expenditures may be positive or negative. The indices were therefore computed on a data set comprising 136 items and 57 countries. The first step in the analysis was to convert the data to prices and quantities by dividing the expenditures by the relative prices: thus the prices used in the study are actually relative prices to the USA, and the quantity units for each item are the amounts of that item which can be bought for one US dollar in the USA. This particular choice of unit has no effect, of course, on the calculation of any of the indices. 2.2) The purpose of this section is to examine the structure of the basic set of price and quantity data so that, later on, the behaviour of the different indices can be related to features of the data set structure. Examining the structure of a set of price and quantity data is a non-trivial problem, however, since what is required is an approach which is independent of (a) changes in the units in which the quantities of individual items are measured: and (b) changes in the currency units in which prices are measured in different countries. This was a problem considered by Cuthbert (2000): the approach developed in that paper was to define indicators of price structure and quantity structure which were independent of both (a) and (b) above - and which essentially measured the extent to which a given  EMBED Equation.2  or  EMBED Equation.2  deviated from a simple multiplicative model defined in terms of item and country effects. 2.3) An improved version of Cuthberts (2000) approach is used in this paper. More specifically, let  EMBED Equation.2  and  EMBED Equation.2  be the international prices and price deflators arising from the Ikl direct volume aggregation method. Then the price structure indicator, IP, is defined as (5)  EMBED Equation.2  . The quantity structure indicator is defined symetrically, as follows, in terms of the Ikl direct price aggregation method. More specifically, consider the direct price analogue of the Ikl method, which defines an international basket,  EMBED Equation.2 , and volume deflator  EMBED Equation.2 , as positive solutions to the following sets of equations:  EMBED Equation.2  EMBED Equation.2  for all i EMBED Equation.2   EMBED Equation.2  for all j  EMBED Equation.2  for all j. Then the quantity indicator IQ is defined as (6)  EMBED Equation.2  . 2.4) There is thus a direct symmetry between the indicators of price and quantity structure defined in equations (5) and (6): this is a desirable property. Moreover, both sets of indicators are independent of the quantity units in which individual items are measured, of changes in currency units, and of choice of base country. The Ikl was chosen as the reference index for calculating these indicators because the Ikl direct price index is independent of changes in the currency units in individual countries - unlike many other direct price indices. However, as the work reported in Cuthbert (2000), makes clear, the types of indicator defined in equations (5) and (6) are robust in terms of choice of a different reference index for computation purposes. 2.5) As in Cuthbert, (2000), the internal structure of the price data set was examined by calculating the correlations between countries for the values of logIP. One way of summarising the structure of the resulting correlation matrix is to draw a dendrogram: see Figure 1. The dendrogram has been constructed on a nearest neighbour basis: the simplest way to interpret it is as follows. The scale of the vertical axis represents correlations: for any two countries, find the lowest point on the path on the dendrogram joining the two countries: then it is possible to find a chain of countries which joins the two countries in question, and where every correlation between neighbouring countries along the chain is larger than the given lowest point. Figure 1 indicates that there is indeed considerable structure in the matrix of price data, with similar and neighbouring countries tending to have highly positively correlated price structures. The figure indicates that there are three main groupings of countries, as classified on their price structures: these might loosely be categorised as advanced economies, Africa, and non-advanced Asia. 2.6) A similar process of constructing a nearest neighbour dendrogram was also carried out based on the logIQ correlations, and this is shown in Figure 2. The correlations here tend to be lower, but it is interesting that this quantity data gives rise to a broadly similar set of country groupings. 2.7) Finally, for each country, the correlations between the logIP and logIQ values were calculated. This calculation shows clear evidence of negative correlation between the price and quantity structures within each country, with the correlations lying between -0.14 and -0.45 for all countries. This confirms the accepted wisdom that goods which are relatively high volume in a given country tend to be relatively low priced, and vice versa. This is an expected feature of the data set which, as we will see, is of great significance as regards the comparative behaviour of the different aggregation methods. 3. A Metric for Measuring Distances between Price Vectors. 3.1) This section represents a short, but important, digression from the main flow of the paper. One of the things we will want to do later is to have a suitable measure of the distance between different price vectors. Such a measure, however, in addition to having the usual properties of a distance measure, should also have a number of special properties appropriate for dealing with price data. In particular, we would want the distance between two vectors of prices to be independent of (a) changes in the units in which the quantities of individual items are measured: and (b) changes in the currency units in which prices are measured in different countries. 3.2) More formally, if x and y are positive vectors of prices, and D is a distance measure between x and y, then we want to find a measure D satisfying the following properties:- a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d) for vector  EMBED Equation.2 , define  EMBED Equation.2  : then  EMBED Equation.2  (This property means that the distance between price vectors is invariant to the choice of quantity unit for measuring the quantities of individual items). e) For scalars  EMBED Equation.2 ,  EMBED Equation.2  (This property means that the distance between price vectors is invariant to changes in the unit of currency in either country.) f)  EMBED Equation.2 . (This property is the standard triangle equality which all good distance measures should satisfy.) It is possible to define a distance measure satisfying all these properties. The author has devised the Mean Absolute Deviation (MAD) measure, defined as follows:-  EMBED Equation.2  , where x and y are n vectors, and mean is the arithmetic mean of the terms  EMBED Equation.2 . It may be possible to define other distance measures, as well as MAD, satisfying all the above properties. In the rest of this paper, when it is required to calculate a measure of distance between price vectors, it is this MAD distance measure which is used. 4. The Results of the Different Aggregation Methods 4.1) The aggregation methods defined in section 1 were applied to the data set of 136 items and 57 countries, to give volume estimates for each method and country: (the USA was taken as base country). International price factors and expenditure deflators for each method were, of course, also obtained. 4.2) It is useful to obtain a first impression of the relationships between the different indices by deriving a simple measure of distance between any pair of indices. If  EMBED Equation.2 and  EMBED Equation.2  represent the volume estimates for country j produced by indices 1 and 2, then a simple measure of distance between indices 1 and 2 is defined as the larger of  EMBED Equation.2 or  EMBED Equation.2 , where in both cases the maxima are taken over all countries j. (Note that this simple distance measure is not an ideal measure - for example, it is not invariant to choice of the base country when the indices are being calculated. But for present purposes it is adequate for giving an initial impression of how the different aggregation methods compare to one another.) 4.3) Table 1 shows the volume distances between the different indices using the above measure. For convenience, the indices have been ordered with the less democratic indices, the GK and SRK, coming first, and the more democratic indices later. Also included in the table in the final column is the Gini Elteto Koves Schultz (GEKS) index: this is not an additive index, so it is not considered in detail in this paper. But since it is a standard index used in international comparisons, it is useful to have this indication of how it compares with the other indices. 4.4) The first point to note about Table 1 is that the distances between the indices are typically large. For example, the distance of 0.65 between the Ikl and the GK means that for one country, there is a 65% difference between these two indices: (in fact, this country is SLE, the larger of the two indices in this case being the GK). This is an important point: it means that, for as heterogeneous a set of data as considered here, choice of aggregation method matters a great deal, at least for some countries. 4.5) The second point to notice about Table 1 is that the democratic / non-democratic dimension has a big effect on the distances between the different indices. The two non-democratic indices, the GK and the SRK, differ by at most 10.8% for some country. The group of democratic indices, (Ikl, VY, SS, OW), differ by at most 26.4% for some country. The big differences, however, are between the two groups. 4.6) Finally, bringing in the GEKS index, the Table shows that there are substantial differences between the GEKS and all the additive indices considered. The GEKS is closest to the Ikl, (but there is still a difference of 22.5% for one country, BGD, the larger of the two indices in this case being the Ikl). The distance from the GEKS to the other indices is much larger for the non-democratic rather than the democratic indices: (in fact, the largest distance in the Table is between the SRK and the GEKS: this occurs for NPL, where the SRK estimate of volume is double the GEKS estimate.) This last feature is predictable, in that the GEKS, while non-additive, is nevertheless based on a democratic weighting system. 4.7) After this initial exploratory work, we now want to consider in a rather more formal fashion the differences between the volume estimates produced by the different aggregation methods. We can imagine the volume estimates produced by the different methods arranged in an array of dimension 6 by 57, denoted by V, with the rows corresponding to the different aggregation methods, and the columns to the different countries. Since the USA has been (arbitrarily) taken as base country, each row has been scaled so that the column corresponding to the USA is constant. What we want to do is to examine the structure of this method by country array V, but using a technique which is invariant under multiplicative scaling of the rows of the array, (and so is independent of choice of base country): and also which is independent of scale effects in the columns, (that is, we do not want our analysis of the structure of V to be affected by the fact that some countries are bigger than others). 4.8) The following procedure has been adopted for examining the structure V in a way which is independent of row and column scaling effects: namely:- a) Take logs: i.e., define  EMBED Equation.2  b) Consider the residuals,  EMBED Equation.2 , on fitting a main effects model to these quantities: that is  EMBED Equation.2  (where, in an obvious notation,  EMBED Equation.2  represents arithmetic mean over the dotted subscript). c) Then analyse the principal components of the covariance matrix of the matrix R of residuals: (effectively, this means extracting the leading eigenvalues and associated eigenvectors of  EMBED Equation.2 , since the columns of R sum to zero and are already centred on their means). 4.9) Table 2 shows the first three principal components of the matrix  EMBED Equation.2 and their associated eigenvalues. Effectively, almost all (over 98%) of the variability in the R matrix is accounted for by the first two principal components - with the first principal component alone accounting for 93% of the variability. We concentrate, therefore, on explaining these two principal components. There is a clear pattern to the coefficients of the first principal component: most of the countries which we have described as falling in the advanced group have positive coefficients on the first principal component, and most countries in the Africa and non-advanced Asia groups have negative coefficients. The first principal component, therefore, effectively represents a difference between the advanced and other groups. There is no obvious pattern to the coefficients of the second principal component. 4.10) The next step was to calculate the scores of the different aggregation methods on these two principal components and to plot the scores: this plot is shown in Figure 3. The picture which emerges from Figure 3 is interesting. Looking at the first principal component, (the horizontal axis), it can be seen that the two non-democratic indices, the GK and the SRK, have large negative scores, while the other more democratic indices have positive scores on this component. Referring back to the coefficients of the first principal component in Table 2, this implies that countries like LUX, USA, CAN, JPN, DEU, NDR, IRL, FRA, FIN, which have large positive coefficients on the first principal component, do relatively better under th    eProperties of Additive Purchasing Power Parities, illustrated with reference to World Bank 1985 dataAPreferred Customer@NormalPreferred Customer29Microsoft Word for Windows 95@/hY@s@Λq@snLmJ iJ4nJ Cov(IP iܥhc Te:›    lllll|X(lo1(!!!=^XQ__1124115197V\F@o|s@o|s_1124115257F@o|s|s_1124115337Y[F|s|s_1124115728F|s 8|s_1124084142')FOxsOxs_1124084264FOxsOxs_1124084338:FOxsxs_1124084353Fxsxs_1124083767#%Fxs'xs_1124084033F'xs'xs_1124084077$(F'xs'xs_1124084126F'xsOxs_1124170938_aiF_|s|s_1124171116cF|s|sOle PIC beL_1124116094Z^F 8|s 8|s_1124116231{F 8|s 8|s_1124170708]`uF 8|s_|s_1124170798oF_|s_|s_1124107482ATF;ys7Cys_1124107481F7Cys_Lys_1124107480SQF_Lys_Lys_1124107479F_LysTys_1124107486NJF*ysn2ys_1124107485Fn2ysn2ys_1124107484OMFn2ys;ys_1124107483F;ys;ys_1124105755GRFys!ys_1124107489F!ys!ys_1124107488KIF!ys*ys_1124107487F*ys*ys_1124113285LXFTys`|s_1124113356F`|s μ|s_1124113428UWF μ|s μ|s_1124113510F μ|s@o|s_1124104385;=AF`Kxs sxs_1124104414;F sxs sxs_1124104513<@5F sxs sxs_1124104554/F sxs@ys_1124105040BEF< ys ys_1124105321 F ys ys_1124105384DHF ysys_1124105594Fysys_1124104689?C)F@ys@ys_1124104930>F#F@ys@ys_1124104981F@ys< ys_1124104898F< ys< ys_112409638135qF`xsxs_1124096396kFxsxs_112409731448eFxsxs_1124097641_Fxs@xs_112409842479YF@xs@xs_1124098622SF@xs@xs_11240986462PMF@xs`Kxs_1124098686GF`Kxs`Kxs_1124095393/1Fxsxs_1124095634Fxsxs_1124095766.6}Fxs`xs_1124095879wF`xs`xs_1124084454+-Fxsxs_1124084558Fxsxs_1124094441,0Fxsxs_1124094573Fxsxs_1124082996!Fxsxs_1124083104Fxsxs_1124083138&Fxsxs_1124083382Fxsxs_1124082554Fནxs_xs_1124082711F_xs_xs_1124082791 F_xs_xs_1124082891F_xsxs_1124082334F xs xs_1124082387F xsནxs_1124082386" Fནxsནxs_1124082385Fནxsནxs"RsVp VvKProperties of Additive Purchasing Power Parities. Dr J.R. Cuthbert Analytical Consulting Ltd. Edinburgh* 0. Introduction 0.1) The purpose of this paper is to consider the comparative properties of a number of different additive methods of constructing purchasing power parities (PPPs) when applied to a real world data set - and to relate the properties of the PPPs to the underlying structure of the data set. The research reported here is an extension of earlier work by Cuthbert, (1999, 2000, 2001), and has been carried out following a suggestion made at the meeting at the World Bank in 2001, on Recent Advances in PPPs, that further consideration of additive methods would be useful. The data set used is 1985 data provided by the World Bank: while now fairly aged, this data set was chosen because it was regarded as being both extensive and of high quality. I am grateful to the World Bank for making this data set available. Responsibility for the opinions expressed here is entirely the authors. 0.2) The structure of the paper is as follows:- Section 1 defines what is meant by an additive PPP volume index, and then introduces the particular additive indices to be studied in this paper. Two specific sub-classes are defined first: first of all, the sub-clerass of strongly additive indices, (essentially equivalent to the class of Generalised Geary Khamis (GGK) indices defined by Cuthbert (1999)): and secondly the sub-class of Van Izjeren indices. The specific indices studied in the paper are as follows: the Geary Khamis (GK) and Ikl indices, both belonging to the GGK sub-class: the equal weighted Van Izjeren (VY) index, the standardised structure (SS) index, and the Sakuma/Rao/Kurabayashi (SRK) index, which are all members of the Van Izjeren sub-class: and finally the own-weights (OW) index. Section 2 considers the data set used in the study, that is, World Bank data relating to 1985. The structure of the data set is examined by using appropriate indicators of price and quantity structure. It is shown that the 57 countries in the data set fall into natural groupings based on the similarities of their price structures. It is also shown that the data set exhibits strong negative price quantity correlation. Both of these features will be shown to be very relevantܥhc De:     NNNNN^X(N1l(=4hXpaQ_@ sVp VvKProperties of Additive Purchasing Power Parities. Dr J.R. Cuthbert Analytical Consulting Ltd. Edinburgh* 0. Introduction 0.1) The purpose of this paper is to consider the comparative properties of a number of different additive methods of constructin to understand the comparative behaviour of the different indices being studied. Section 3 is a brief digression, introducing an appropriate distance metric for use on price vectors, which will be required in the subsequent analysis. Section 4 considers the results of applying the six additive index methods to the data set. It also develops an appropriate methodology for examining the structure of the resulting method-by-country matrix of volume estimates, in terms of a form of principal component analysis. It is shown that 98% of the variability in the method-by-country matrix can be described in terms of two principal components. The next two sections examine the nature of these principal components and relate them to the underlying structure of the data. Section 5 considers the first principal component, which accounts for 93% of the variability in the data. Aggregation methods (like the GK and SRK) , which compute international prices by weighting together country price vectors by weights which are propg purchasing power parities (PPPs) when applied to a real world data set - and to relate the properties of the PPPs to the underlying structure of the data set. The research reported here is an extension of earlier work by Cuthbert, (1999, 2000, 2001), and has been carried out following a suggestion made at the meeting at the World Bank in 2001, on Recent Advances in PPPs, that further consideration of additive methods would be useful. The data set used is 1985 data provided by the World Bank: while now  "$%&'(*/123457:;>@ABCDEFGHIJLQSTUVX]_`abdiklmnoqtuxz{|}~fairly aged, this data set was chosen because it was regarded as being both extensive and of high quality. I am grateful to the World Bank for making this data set available. Responsibility for the opinions expressed here is entirely the authors. 0.2) The structure of the paper is as follows:- Section 1 defines what is meant by an additive PPP volume index, and then introduces the particular additive indices to be studied in this paper. Two specific sub-classes are defined first: first of all, the sub-cl_1124040460 1Fxs xs_1124041260+F xs xs_1124082336%F xs xs_1124082335F xs xs_1124040464IFS{xsS{xs_1124040463CFS{xsxs_1124040462=Fxsxs_11240404617Fxsxs_1124040798 aF ,rxsS{xs_1124040468[FS{xsS{xs_1124040466 UFS{xsS{xs_1124040465OFS{xsS{xs_1124039957yFjxsjxs_1124040096sFjxsjxs_1124040469 mF ,rxs ,rxs_1124040673gF ,rxs ,rxsass of strongly additive indices, (essentially equivalent to the class of Generalised Geary Khamis (GGK) indices defined by Cuthbert (1999)): and secondly the sub-class of Van Izjeren indices. The specific indices studied in the paper are as follows: the Geary Khamis (GK) and Ikl indices, both belonging to the GGK sub-class: the equal weighted Van Izjeren (VY) index, the standardised structure (SS) index, and the Sakuma/Rao/Kurabayashi (SRK) index, which are all members of the Van Izjeren sub-class: and finally the own-weights (OW) index. Section 2 considers the data set used in the study, that is, World Bank data relating to 1985. The structure of the data set is examined by using appropriate indicators of price and quantity structure. It is shown that the 57 countries in the data set fall into natural groupings based on the similarities of their price structures. It is also shown that the data set exhibits strong negative price quantity correlation. Both of these features will be shown to be very relevant to understand the comparative behaviour of the different indices being studied. Section 3 is a brief digression, introducing an appropriate distance metric for use on price vectors, which will be required in the subsequent analysis. Section 4 considers the results of applying the six additive index methods to the data set. It also develops an appropriate methodology for examining the structure of the resulting method-by-country matrix of volume estimates, in terms of a form of principal component analysisComparative 1OWeliminated symmetricallyIklIkl. 143 .A ,hence 45 12 .A16 14 .A September 2003 2. It is shown that 98% of the variability in the method-by-country matrix can be described in terms of two principal components. The next two sections examine the nature of these principal components and relate them to the underlying structure of the data. Section 5 considers the first principal component, which accounts for 93% of the variability in the data. 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Since the former group of countries contains the larger, advanced economies, the first principal component appears to be describing a classic Gerschenkron effect. 4.11) Looking at the vertical axis in Figure 3, relating to scores on the second principal component, what we see is that the two GGK indices, (the GK and the Ikl), have positive scores: the three Van Izjeren indices have negative scores: and the OW (which does not fall into either the GGK or Van Izjeren sub-classes), falls in the middle. This suggests that underlying the second principal component is the fact that some countries do relatively better under GGK as compared with Van Izjeren indices: and vice versa. 5. Understanding the Effect Underlying Principal Component 1 5.1) To illustrate how the effect underlying principal component 1 arises, we concentrate on two particular indices, one non-democratic (the GK), and the other democratic (the Ikl), and consider what factors explain the relative variation between these two indices. As we have already noted from section 2, there is strong negative price quantity correlation for each individual country in the data set. So countries for which country prices are relatively close to GK international prices are likely to have relatively small GK volumes: and countries which have country prices relatively close to Ikl prices are likely to have relatively small Ikl volumes. If we consider the ratio of the Ikl to GK volumes for each country, we could hypothesise that this ratio will be related to a measure of the relative closeness of that countrys prices to GK as compared with Ikl prices. 5.2) To measure the distance between country prices and international prices, we use the MAD distance measure described in section 3. Figure 4 shows, for each country, the distance from that countrys prices to GK international prices, and Ikl international prices. The ordering of the countries on the horizontal axis of the chart corresponds to the ordering of the countries in the logIP correlation dendrogram in Figure 1. Thus countries up to YUG correspond to what I have called the advanced group: the next set up to TZA, corresponds to the Africa group: and the last set to non-advanced Asia. 5.3) Figure 4 shows some interesting features: a) Most advanced countries are closer to the GK than to the Ikl: African and non-advanced Asian countries are the other way round: and there is a group of intermediate countries, (in the tail of the advanced block, from HKG to YUG) where distances from the Ikl and GK are about the same. This overall pattern is much as we might expect - we would expect GK prices to be heavily dominated by the advanced block, which contains not just the largest economies, but a large number of economies with similar price structures. Nevertheless, it is fascinating to see this confirmed so strikingly in the chart: and also that the turnover point, where the GK becomes further from country prices than the Ikl, occurs exactly at the boundary of the advanced grouping that emerged from the logIP dendrogram. b) As expected, the Ikl is much closer to equidistance from all countries than the GK, reflecting its more democratic nature. But note that the Ikl is still closer to most advanced countries than to most African countries - and that most non-advanced Asian countries are even further distant from Ikl prices. Again, this is a feature which could have been predicted - but one that it is nevertheless interesting to see working through in practice. Although the Ikl is democratic in that it is not affected by individual country scale effects, nevertheless it will reflect the relative numbers of countries in the different country groups - and the relative homogeneity of the price structures within these groups. Hence Ikl international prices will reflect both the size and homogeneity of the advanced grouping of countries. This is an important point. 5.5) We can now examine the question raised at the end of paragraph 5.1: namely, how do the relative values of the Ikl and GK for a country relate to the relative closeness of that countrys prices to GK and Ikl international prices respectively. This question is examined in Figure 5. As a measure of relative closeness of country prices to GK as compared with Ikl prices, I have taken D(Ikl prices, country prices) - D(GK prices, country prices). This measure is plotted on the horizontal axis in Figure 5, against the Ikl/GK ratio on the vertical axis. The figure shows a striking picture. Note first of all how the country groupings of advanced, intermediate, non-advanced Asia, and Africa are spread out almost perfectly along the x-axis, (with some anomalies- like PAK lying with the African countries.) But above all, note the very strong linear relationship: in other words, almost all, (83%), of the variability in the Ikle/GK ratio is explained by a linear relationship with the measure of the relative closeness of country prices to the GK as compared with the Ikle. The fact that there is such a relationship is not unexpected: but the fact that it is of such a simple form, and so strong, is interesting. 5.6) What this section has illustrated, therefore, is how it is the distance from international prices to country prices, ( interacting with the observed negative price quantity correlation in the data set), which drives the type of effect underlying the first principal component: and, moreover, how the distance from country prices to international prices reflects the interaction of the particular aggregation method being used with the underlying structure of price and quantity data. An important ancillary point to remember is the point made at the end of paragraph 5.3(b): even a democratic index like the Ikl may have international prices which are consistently closer to one group of countries than another, depending on the group structure of the data. Hence, given negative price quantity correlation, there will be some consistent group effects even with democratic indices like the Ikl. 6. Understanding the Effect Underlying Principal Component 2 6.1) As already noted in section 4, the second principal component can be interpreted as implying that some countries, (those with large positive coefficients for this principal component) do relatively better under indices in the GGK class compared with the Van Izjeren class: and vice versa for indices with large negative coefficients. 6.2) We can formulate an intuitive hypothesis to suggest what sort of features in the data might account for this effect. It will be recalled from equation 3 that GGK international prices are formed by summing the following terms over countries, namely:  EMBED Equation.2  Now suppose that, for a given item, prices and quantities were, (in some sense), negatively correlated over countries. Then, as the terms in the above expression are summed over countries, countries which have DocumentSummaryInformation8L_1124039638F Yxs@caxs_1124039790F@caxsjxs_1124039895Fjxsjxslarge weights, (large  EMBED Equation.2 ), will tend to be associated with relatively low prices: and conversely, low prices will tend to be associated with high weights. So, (although this argument is loose and intuitive), we might expect GGK international prices to be relatively low for those items for which there is strong negative price quantity correlation, compared to a weighting scheme which did not depend directly on  EMBED Equation.2 . 6.3) This intuitive argument suggests the following hypothesis: namely that, if a particular country has a high proportion of its expenditure on items which have high negative price quantity correlation, then that country will tend to have relatively low volumes under GGK compared to, say, Van Izjeren type aggregation methods. 6.4) To test this hypothesis, the indicators IP and IQ were used to calculate, for each item i, the covariance  EMBED Equation.2 . It is interesting that, for almost all items, these covariances were negative: (in fact, the covariances were negative for 132 out of the 137 items in the data set). For any chosen cut-off value, those items can be selected where the price quantity covariance is less than the cut-off, and for each country, the proportion of expenditure in that country on selected items was computed. Figure 6 shows the result, (for cut-off value -0.9) of plotting the country coefficients on the second principal component, (vertical axis), against the proportion of expenditure on selected items: (SLE was omitted from the plot as an outlier). This particular cut-off value was chosen because it is around this value that the strongest effect was observed: so there is an element of data dredging going on here. Nevertheless, the strength of the linear relationship in Figure 5 is striking, with a correlation coefficient of -0.66 between the coefficient in principal component 2, and the proportion of expenditure on selected items. 6.5) There is therefore strong evidence that the above hypothesis is consistent with the data. What seems to be underlying the second principal component is that there are differences between items in the degree of negative price quantity correlation: and that countries which have a high proportion of expenditure on highly negatively correlated items will tend to have relatively smaller volumes under GGK indices compared with indices in the Van Izjeren class. 7. Conclusions 7.1) In conclusion, (and without restating the results in detail), this paper has shown a) That there is a good deal of structure in the data set considered, with negative correlation between prices and quantities, (both for countries and for items): and with countries grouped into natural country blocks on the basis of their price, (and also quantity), characteristics. b) It has been shown that there are two main effects which describe the relative values of the group of additive indices considered here: first of all, a Gerschenkron type effect, accounting for 93% of the residual variability in the method by country volume matrix: and secondly, a consistent difference between GGK and Van Izjeren class indices, accounting for 5.4% of the residual variability. c) It has been shown how the effects observed at (b) relate to the features of the data set described at (a). References. Cuthbert, J.R. (1999).: Categorisation of additive purchasing power parities: Review of Income and Wealth, series 45, no.2. Cuthbert, J.R. (2000).: Theoretical and practical issues in purchasing power parities, illustrated with reference to the 1993 Organisation for Economic Co-operation and Development data: Journal of Royal Statistical Society, Series A, 163. Cuthbert, J.R. (2001).: Using price and quantity indicators to explore data structure: World Bank/OECD Seminar on Recent Advances in Purchasing Power Parities: Washington. Dikhanov, Y. (1997) Sensitivity of PPP-Based Income Estimates to Choice of Aggregation Procedures. The World Bank, Washington. Geary, J. (1958) A note on the comparison of exchange rates and purchasing power between countries. Journ. Roy. Stat. Soc., 121, 97-99. Ikl, D. M. (1972) A new approach to index number theory. Quarterly Journal of Economics, 86, 188-211. Sakuma, I., Rao, D.S., Kurabayashi,Y., (2000) Additivity, Matrix Consistency, and a new Method for International Comparisons of Real Income and Purchasing Power Parities. IARIW 26th General Conference, Cracow. Sergueev, S. (2000) Development of Multilateral Methods for International Price and Volume Comparisons: Method of Standardised Structure. Vienna. Van Izjeren, J. (1987) Bias in international index numbers: a mathematical elucidation. Eindhoven: CIP-Gegevens Koninklijke Bibliotheek. * Address for correspondence: 42 Cluny Dr., Edinburgh, United Kingdom EH10 6DX. 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FMicrosoft Equation 2.0 DS Equation Equation.29q4 u .1  &` & MathTypePTimes New Roman4- 2 `@D(k~ 2 `Cx,` 2 `k 2 `4y)~ 2 `= 2 ` D(x,~` 2 ` y) ~` 2 ` k` 2 `  k`Times New Roman%- 2 1 2 }2 2 1 2 Q2Symbol- 2 ` "Times New Roman4- 2 `z,` 2 `.` & "Systemn-L4D @<mJ iJ4nJ k 1 , k 2 , >0 FMicrosoft Equation 2.0 DS Equation Equation.29q4  .1  & & MathTypePTimes New Romanr- 2 `@k 2 `,` 2 `S k` 2 `c,` 2 ` ` 2 `> 2 `0Times New Roman- 2 1 2 2 & "Systemn-L4@XmJ iJ4nJ D(ax,ay) =D(x,y)  " a,x,y. FMicrosoft Equation 2.0 DS Equation Equation.29q4 G .1  &` & MathTypePTimes New Roman- 2 `@D(~ 2 `x,` 2 `y) ~` 2 `= 2 `5D(x,~` 2 ` y) ~`` 2 ` ` 2 ` x,` 2 `_y.`Symbol- 2 `a 2 ` a 2 `|a 2 ` "Times New Roman- 2 `|,` & "Systemn-L4 @PmJ iJ4nJ ax = {a i x i }  " i FMicrosoft Equation 2.0 DS Equation Equation.29q 4 ; .1  ` &  & MathTypePSymbol- 2 `1a 2 `aTimes New RomanA- 2 `x ` 2 `= 2 ` {` 2 `zx 2 `} `` 2 `O  i`kTimes New Roman- 2 iP 2 UiPSymbol- 2 `? " & "Systemn-@L 4@mJ iJ4nJ a>0 FMicrosoft Equation 2.0 DS Equation Equation.29q"  .1  & & MathType0Symbol- 2 @1a 2 @>Times New Roman%- 2 @0 & "Systemn-L"XpmJ iJ4nJ D(x,y)=0  x=ky for some scalar k. FMicrosoft Equation 2.0 DS Equation Equation.29ql4  .1  & & MathTypePTimes New Roman- 2 `@D(x,~` 2 `%y)~ 2 `= 2 `0 ` 2 `  x` 2 ` =2 `G ky for som`~~`+2 ` e scalar k`k~` 2 `U.`Symbol- 2 ` & "Systemn-Ll4@LiJmJtiJ D(x,y)= D(y,x)   "x,y. FMicrosoft Equation 2.0 DS Equation Equation.29q4  .1  @& & MathTypePTimes New Romane- 2 `@D(x,~` 2 `%y)~2 `  D(y,`~`2 `A x) ~``` 2 ` x,` 2 `y.`Symbol- 2 `= 2 ` " & "Systemn-L4 @< iJmJiJ D(x,y) 0   "x,y. FMicrosoft Equation 2.0 DS Equation Equation.29q 4  .1   &  & MathTypePTimes New Romanl- 2 `@D(x,~` 2 `%y)~2 ` 0 ```` 2 `O x,` 2 ` y.`Symbol- 2 ` 2 `?" & "Systemn-L 4l@ dmJ iJ4nJ IQ ij  = f j q ij c i FMicrosoft Equation 2.0 DS Equation Equation.29q^  7 .1  &@? & MathType "-<H<6Times New Roman)- 2 4IQw 2  ` 2 \= 2  ` 2 ahf~ 2 ayqTimes New Roman)- 2 ijPP 2 jP 2 `ijPP 2 (iPSymbol- 2  c & "Systemn-HL^ O pmJ iJ4nJ b j  = [p ij c i  ] -1i  FMicrosoft Equation 2.0 DS Equation Equation.29q E K .1   &  & MathTypeSymbol- 2 )b 2 .cTimes New Roman=- 2 @9jP 2 @|ijPP 2 @ iP 2 4| -1^ 2 WiPTimes New Roman - 2  ` 2 k= 2  [` 2 p 2   ]`Symbol- 2 * & "Systemn-GL EXl ֘mJ iJ4nJ f j  = p ij c ii  p ij q iji  FMicrosoft Equation 2.0 DS Equation Equation.29q g  .1   & F & MathType "-s Times New RomanI- 2 `@f~ 2 `] ` 2 `)= 2 `m ` 2 Hp 2 1p 2 qTimes New Roman- 2 jP 2 Q#ijPP 2 QiP 2 iP 2  ijPP 2 ijPP 2 ^iPSymbol- 2 cSymbol- 2 ;D 2 - & "Systemn-L g mJ iJ4nJ  FMicrosoft Equation 2.0 DS Equation Equation.29q=4 T .1   & & MathTypeP & L=4@ ָiJ|mJxiJ c i  = f j q ij b j p ij b k p ikk  j  FMicrosoft Equation 2.0 DS Equation Equation.29q #%&'()+.13456789:;=BDEFGHJOQRSTUW\^_`abdiklmnoqtuxz{|}~  .1  @& & MathType "-<# <Symbol- 2 <c 2 av b 2 ( bTimes New Roman)- 2 +iP 2 jP 2 %ijPP 2  jP 2  ijPP 2 o k 2 o ikP 2  k 2 ojPTimes New Roman5- 2  ` 2 o= 2  ` 2 -f~ 2 >q 2 a p 2  pSymbol- 2 Y;  2 ) & "Systemn-L  mJ iJ4nJ  FMicrosoft Equation 2.0 DS Equation Equation.29q=4 T .1   & & MathTypeP & L=4@ mJ iJ4nJ f j FMicrosoft Equation 2.0 DS Equation Equation.29q{  .1  @&@ & MathTypepTimes New Roman)- 2 `@f~Times New Roman)- 2 jP & "Systemn-L{h mJ iJ4nJ c i FMicrosoft Equation 2.0 DS Equation Equation.29q4  .1  & & MathTypePSymbol- 2 `<cTimes New Roman- 2 +iP & "Systemn-L4@ dmJ iJ4nJ IP ij  = e j p ij p i FMicrosoft Equation 2.0 DS Equation Equation.29q  7 .1  @&? & MathType "-<<Times New Roman- 2 4IPw 2 $ ` 2 = 2 4 ` 2 ae 2 a=pTimes New RomanQ- 2 nijPP 2 jP 2 ijPP 2 (iPSymbol- 2 p & "Systemn-L ( mJ iJ4nJ e j FMicrosoft Equation 2.0 DS Equation Equation.29q{  .1  @&` & MathTypepTimes New Roman- 2 `4eTimes New Roman- 2 jP & "Systemn-L{h mJ iJ4nJ p i FMicrosoft Equation 2.0 DS Equation Equation.29q4  .1  & & MathTypePSymbol- 2 `=pTimes New Roman)- 2 /iP & "Systemn-L4@ mJ iJ4nJ q ij FMicrosoft Equation 2.0 DS Equation Equation.29q4{  .1  @& & MathTypepTimes New Roman- 2 `4qTimes New RomanI- 2 ijPP & "Systemn-L4{@h mJ iJ4nJ p ij FMicrosoft Equation 2.0 DS Equation Equation.29q4{  .1  @& & MathTypepTimes New Roman- 2 `@pTimes New Roman)- 2 ijPP & "Systemn-L4{@h(hiJmJtiJ a ij  = q ij q ikk    FMicrosoft Equation 2.0 DS Equation Equation.29q  X .1   &  & MathType "-<<Symbol- 2 1a 2 aq 2 qTimes New Roman- 2 FijPP 2 ijPP 2 oikP 2 fkTimes New Roman- 2  ` 2 = 2   ` 2   `Symbol- 2 Y & "Systemn-L (mJ iJ4nJ a ij FMicrosoft Equation 2.0 DS Equation Equation.29qW{  .1  @ & & MathTypepSymbol- 2 `1aTimes New Roman- 2 FijPP & "Systemn-LW{Th( mJ iJ4nJ i FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  &` & MathType Times New Romanl- 2 `4ik & "Systemn-L(ΐ,iJmJiJ q ij  = p ij q ij p ij q iji   J FMicrosoft Equation 2.0 DS Equation Equation.29q  ~ .1   &  & MathType "-<< Symbol- 2 0qTimes New Roman)- 2 ijPP 2 xijPP 2  ijPP 2 ovijPP 2 o ijPP 2 QiPTimes New Romani- 2  ` 2 = 2  ` 2 ap 2 a"q 2 p 2  q 2 %  `Symbol- 2 Y & "Systemn-L (mJ iJ4nJ l=1 FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  &@ & MathType0Symbol- 2 `7l 2 `}=Times New Roman!- 2 `1 & "Systemn-L0( iJmJpiJ a j    p ij q iji  [p kj (q k1 +...+q kJ ) k  ] FMicrosoft Equation 2.0 DS Equation Equation.29qg  .1  &F & MathType "-lSymbol- 2 `1aTimes New Roman}- 2 jjP 2 Q ijPP 2 Q ijPP 2  iP 2 kjP 2  k1 2 kJs 2 ^7kTimes New Roman!- 2 ` `` 2 `k ` 2 , p 2  q 2 [ 2 p 2  (q~ 2 E +.` 2  .` 2  .+q`Symbol- 2 `Symbol- 2 ;(  2 Times New Roman}- 2 b)~ 2 ] & "Systemn-Lg (mJ iJ4nJ "" FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&@ & MathTypeTimes New Romane- 2 @(" 2 @"Symbol- 2 @ & "Systemn-L֨iJ|mJhiJ a j    1[e j p kj (q k1 +...+q kJ ) k  ] FMicrosoft Equation 2.0 DS Equation Equation.29q V  .1  &v & MathType "-Symbol- 2 1aTimes New Roman!- 2 jjP 2 /|jP 2 / kjP 2 /< k1    !&()*+,.123689:;<=>?@ABCDEFHKNPQRSTUVWY\_abcdefgilmprstuvwxyz{|~ 2 /\kJs 2 ~7kTimes New Roman!- 2  `` 2 k ` 2 [ 2 e 2 p 2  (q~ 2  +.` 2 .` 2 5.+q`Symbol- 2 Symbol- 2 Times New Roman!- 2 j 1 2 )~ 2 -] & "Systemn-L V (4mJ iJ4nJ a j  = )1J FMicrosoft Equation 2.0 DS Equation Equation.29q*+  .1  &@ & MathTypep "-\*Symbol- 2 1aTimes New Roman5- 2 `jjPTimes New Romanx- 2  ` 2 = 2  ` 2 |1 2 wJ & "Systemn-L*+(mJ iJ4nJ l1 FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  &@ & MathType0Symbol- 2 `7l 2 `|Times New Roman!- 2 `1 & "Systemn-L0(hJmJ`iJ a ij FMicrosoft Equation 2.0 DS Equation Equation.29qW{  .      !"#$&6'()*+,-./0123457G89:;<=>?@ABCDEFHWIJKLMNOPQRSTUVXiYZ[\]^_`abcdefghjyklmnopqrstuvwxz{|}~1  @ & & MathTypepSymbol- 2 `1aTimes New Roman- 2 GijPP & "Systemn-LW{Th(mJ iJ4nJ a ij  = a j  for all i, where a j 0 and a j  = 1 j - FMicrosoft Equation 2.0 DS Equation Equation.29q   .1   &` & MathType Symbol- 2 1a 2 la 2 a 2 aTimes New Roman- 2 @FijPP 2 @jP 2 @jP 2 @jP 2 jPTimes New Roman!- 2  ` 2 = 2   `2   for all i`~~`kk`k 2  ,`2 `  where `~`2  and `` 2 [ ` 2 '= 2 k 1`Symbol- 2 Symbol- 2 *Times New Roman!- 2 0 & "Systemn-L (DhJmJ`iJ b j  = v j  -1 FMicrosoft Equation 2.0 DS Equation Equation.29q*  .1  &@O & MathTypepSymbol- 2 )bTimes New Roman%- 2 9jP 2 jP 2 )k-1^Times New Roman!- 2  ` 2 k= 2  v` & "Systemn-@ @"DL*((mJ iJ4nJ b j  = 1 FMicrosoft Equation 2.0 DS Equation Equation.29q_{  .1  @& & MathTypepSymbol- 2 `)bTimes New Romanma- 2 9jPTimes New Roman- 2 ` ` 2 `k= 2 ` 1` & "Systemn-L_{ h(ΈhJ|mJXiJ  a ij =b j q ij b j q ijj    FMicrosoft Equation 2.0 DS Equation Equation.29qL  s .1  @@ &  & MathType "-<<G Times New Roman- 2 @ ` 2 = 2 aq 2 q 2   `Times New Roman- 2 ijPP 2 2jP 2 sijPP 2 o0jP 2 oqijPP 2 jPSymbol- 2 a 2 a"b 2  bSymbol- 2 Y3 & "Systemn-LL h(mJ iJ4nJ l1 FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  &@ & MathType0Symbol- 2 `7l 2 `|Times New Roman=- 2 `1 & "Systemn-L0( mJ iJ4nJ b FMicrosoft Equation 2.0 DS Equation Equation.29q`4  .1  @& & MathTypePSymbol- 2 `)b & "Systemn-HL`4@(Έ(iJmJiJ  a ij =b j q ij b j q ijj    FMicrosoft Equation 2.0 DS Equation Equation.29qL  s .1  @@ &  & MathType "-<<G Times New Roman - 2 @ ` 2 = 2 aq 2 q 2   `Times New Roman- 2 ijPP 2 2jP 2 sijPP 2 o0jP 2 oqijPP 2 jPSymbol- 2 a 2 a"b 2  bSymbol- 2 Y3 & "Systemn-LL h(mJ iJ4nJ l1 FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  &@ & MathType0Symbol- 2 `7l 2 `|Times New Roman9- 2 `1 & "Systemn-L0(mJ iJ4nJ l1 FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  &@ & MathType0Symbol- 2 `7l 2 `|Times New Roman9- 2 `1 & "Systemn-L0( mJ iJ4nJ b FMicrosoft Equation 2.0 DS Equation Equation.29q`4  .1  @& & MathTypePSymbol- 2 `)b & "Systemn-HL`4@(ΈhJ|mJXiJ  a ij =b j q ij b j q ijj    FMicrosoft Equation 2.0 DS Equation Equation.29qL  s .1  @@ &  & MathType "-<<G Times New Roman- 2 @ ` 2 = 2 aq 2 q 2   `Times New Roman- 2 ijPP 2 2jP 2 sijPP 2 o0jP 2 oqijPP 2 jPSymbol- 2 a 2 a"b 2  bSymbol- 2 Y3 & "Systemn-LL h( 0iJmJiJ  a ij FMicrosoft Equation 2.0 DS Equation Equation.29q{  .1  @&@ & MathTypepTimes New Roman- 2 `@ `Times New Roman- 2 ijPPSymbol- 2 `a & "Systemn-L{h(mJ iJ4nJ l=1 FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  &@ & MathType0Symbol- 2 `7l 2 `}=Times New Roman-- 2 `1 &     !"#$&+-./0279:;<=?DFGHIKPRSTUWZ[^`abcdefghikprstuw|~"Systemn-L0( mJ iJ4nJ l FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&  & MathType0Symbol- 2 `7l & "Systemn-L( mJ iJ4nJ l FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&  & MathType0Symbol- 2 `7l & "Systemn-L( mJ iJ4nJ l FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&  & MathType0Symbol- 2 `7l & "Systemn-L( mJ iJ4nJ l FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&  & MathType0Symbol- 2 `7l & "Systemn-GL(mJ iJ4nJ l -1 FMicrosoft Equation 2.0 DS Equation Equation.29qW{  .1  @ & & MathType0Symbol- 2 7lTimes New Roman- 2 4-1^ & "Systemn-LW{Th( hJmJ`iJ p FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  ``&  & MathType0Symbol- 2 =p & "Systemn-L( mJ iJ4nJ l FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&  & MathType0Symbol- 2 `7l & "Systemn-L(΀mJ iJ4nJ p i q ib  = p ib q ibi  i  FMicrosoft Equation 2.0 DS Equation Equation.29qE \ .1   &  & MathTypeSymbol- 2 9pTimes New Roman2- 2 @+iP 2 @~ibP 2 @ ibP 2 @ ibP 2 iP 2 iPTimes New Roman- 2 q 2  ` 2 Q= 2  ` 2  p 2  qSymbol- 2 *  2 *8 & "Systemn-LElS mJ iJ4nJ l FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&  & MathType0Symbol- 2 `7l & "Systemn-HL(hJ|mJXiJ ke FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  &` & MathType Times New Roman- 2 @@ke & "Systemn-GL,(hJ|mJXiJ kp FMicrosoft Equation 2.0 DS Equation Equation.29qW  .1   & & MathType0Times New Roman!- 2 `@kSymbol- 2 `p & "Systemn-DLWTcS mJ iJ4nJ l FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&  & MathType0Symbol- 2 `7l & "Systemn-HL iJ|mJxiJ e FMicrosoft Equation 2.0 DS Equation Equation.29q=`  .1  @ & ` & MathType Times New Roman- 2 4e & "Systemn-4IlHL=` iJ|mJxiJ p FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  ``&  & MathType0Symbol- 2 =p & "Systemn-L iJ|mJxiJ p FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  ``&  & MathType0Symbol- 2 =p & "Systemn-LS mJ iJ4nJ l FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&  & MathType0Symbol- 2 `7l & "Systemn-HL iJ|mJxiJ e FMicrosoft Equation 2.0 DS Equation Equation.29q=`  .1  @ & ` & MathType Times New Roman- 2 4e & "Systemn-4IlHL=` iJ|mJxiJ p FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  ``&  & MathType0Symbol- 2 =p & "Systemn-L(LhJmJ iJ (1)                                p i =e j p ij a ijj            for all i(2)                                e j =lp i q iji  p ij q iji            for all j        FMicrosoft Equation 2.0 DS Equation Equation.29qO&:   .1   "&" & MathType "-\Times New Roman-2 4 (1) ~~```````2  ```    !"#%*,-./0279:;<>CEFGHIKNQSTUVWXYZ[]`cefghijklmnoqvxyz{}```````2 P ``````````2   ````` 2 = 2 e 2 Gp2  ``````````2 T for all i~~`kk`k2 4 (2) ~~```````2  ``````````2 P ``````````2   e````` 2 = 2 4q 2 p 2 q2  ``````````2 d for all j ~~`kk`k`2 )  ``````Times New Roman- 2 OiP 2 OjP 2 O"ijPP 2 OijPP 2 jP 2  jP 2 iP 2 qijPP 2 iP 2 R ijPP 2 R ijPP 2 iPSymbol- 2  p 2 a 2 l 2 4,pSymbol- 2 9 2 ~+ 2 <  & "Systemn-LO&: S mJ iJ4nJ l FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  `&  & MathType0Symbol- 2 `7l & "Systemn-HLSmJ iJ4nJ e j FMicrosoft Equation 2.0 DS Equation Equation.29q{  .1  @&` & MathTypepTimes New Romanx- 2 `4eTimes New Roman- 2 jP & "Systemn-L{hSmJ iJ4nJ p i FMicrosoft Equation 2.0 DS Equation Equation.29q4  .1  & & MathTypePSymbol- 2 `=pTimes New Roman- 2 /iP & "Systemn-HL4@S mJ iJ4nJ Q FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  &` & MathType@Times New Roman- 2 @4Q+ & "Systemn-LmJ iJ4nJ a ij FMicrosoft Equation 2.0 DS Equation Equation.29qW{  .1  @ & & MathTypepSymbol- 2 `1aTimes New Roman-- 2 FijPP & "Systemn-LW{ThXmJ iJ4nJ a ij  = 1 for all i j  FMicrosoft Equation 2.0 DS Equation Equation.29q  ) .1   @ &  & MathType Symbol- 2 -aTimes New Roman- 2 @BijPP 2  jPTimes New Roman- 2  ` 2 =2  1 for all``~~`kk 2 "  i`kSymbol- 2 *8 & "Systemn-IL DiJ|mJxiJ A = A[P,Q]=[a ij ] FMicrosoft Equation 2.0 DS Equation Equation.29qT{ _ .1  @ &  & MathTypepTimes New Roman)- 2 `@A 2 `A 2 `\P 2 `Q+Times New Roman9- 2 `T ` 2 ` = 2 `d ` 2 `[ 2 `K,` 2 `] 2 `= 2 ` [ 2 `O ]Times New Roman8- 2  ijPPSymbol- 2 `r a & "Systemn-LT{h mJ iJ4nJ p FMicrosoft Equation 2.0 DS Equation Equation.29q  .1  ``&  & MathType0Symbol- 2 =p & "Systemn-LTmJ iJ4nJ v j  = p i q iji  FMicrosoft Equation 2.0 DS Equation Equation.29q E % .1   & & MathTypeTimes New Romand- 2 @v 2 s= 2 qTimes New Roman- 2 @?j PH 2 @iP 2 @ijPP 2 iiPSymbol- 2 pSymbol- 2 * & "Systemn-L ElmJ iJ4nJ p i FMicrosoft Equation 2.0 DS Equation Equation.29q4  .1  & & MathTypePSymbol- 2 `=pTimes New Roman)- 2 /iP & 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