Key Sector Analysis: A Case of the Transited Polish Economy Henryk Gurgul Paweł Majdosz The transition process from a centrally planned economy to a market economy started in Poland at the beginning of the 1990s. In this paper we try to answer the question in which direction has the structure of Polish economy changed, if indeed it has. By means of the key sector analysis applied to the Polish input-output tables that come from the period 1990–2000, we find that the structure of the Polish economy still remains characteristic of a centrally planned economy rather than a market economy. Although, in the last year of the period under study, the first improvement symptoms could be observed (the increased significance of services in the Polish economy) but there is still a lot of work to be done. An inefficient operation in the case of some sectors reaches a considerable level. This is reflected by the structure of the most important input-output coefficients, of which, the most important inputs are located on the diagonal of the sensitive matrix. Key Words: input-output tables, transition, key sector analysis jel Classification: C67, p21 Introduction In the early 1990s the centrally planned Polish economy started transforming into a market economy. Without any economic theory on how to carry out such a process, this task appeared to be very difficult. In the first stage of the transition, the economy in Poland suffered from the two opposing trends in the macroeconomic statistics. On the one hand, industrial output, wages and salaries dropped considerably and on the other, inflation and unemployment rose. Although, Balcerowicz’s plan based on the three nominal anchors allowed inflation to be kept under control, the other macroeconomic statistics still remained below an advisable level. The first improvement symptoms could be observed in Henryk Gurgul is a Professor at the Department of Applied Mathematics, University of Science and Technology, Poland. Paweł Majdosz is an Assistant at the Department of Quantitative Methods, School of Economics and Computer Science, Poland. Managing Global Transitions 3 (1): 95–111 96 Henryk Gurgul and Paweł Majdosz trade, and later also in services. Manufacturing, especially in the case of heavy industry, appeared to be extremely resistant to change. The transition process did not avoid the basic principles used by the Central Statistical Office in Poland when preparing the input-output tables. For example, the Polish input-output table from 1990 was compiled according to the material product system (mps), while the table for 1995 and later was made according to the system of national accounts (sna). Therefore, the first task was to transform the input-output matrix from 1990 into the sna system to assure the comparability of the matrices coming from different periods and the results of the computations based on these matrices. Under this study, we decided to analyse the changes of the economic structure by means of methods based on the entropy theory (key sector analysis). In 1967 Theil published his work on the entropy decomposition analysis, which provided a useful way of examining errors or changes in input structures. Theil (1967) suggested that a change could be decomposed into a set of additive components and he formulated a maximum entropy principle. On the basis of this principle, so-called key sector analysis can be performed. Key sector analysis provides empirical evidence regarding the economic structure of sectors within an economy (Chen-ery and Watanabe 1958; Hewings and Romanos 1981; Hewings 1982; De-fourny and Thorbecke 1984; Białas and Gurgul 1998). The main aim of the key sector investigations is to find the sectors whose structure has the greatest impact on the rest of the economy. A key sector analysis of backward and forward linkages is related to the so-called multiplier product matrix (mpm). This matrix is based on the maximum entropy criterion. By means of this matrix, the probable future course of economic development can be identified. The interaction of the different firm strategies towards innovation explains the dynamics of the entries in the input-output matrix. Rasmussen (1956) introduced to economics the notions of backward and forward linkages. These two indices allow us to find the key sectors of an economy. The Entropy Decompositions A static 10 model (Cmiel and Gurgul 1996a; 1996b; 1997) is given by the equation y=(I-A)x, (1) Managing Global Transitions Key Sector Analysis: A Case of the Transited Polish Economy 97 where A denotes the 10 matrix, x the output vector, and y the final de- mand vector. Therefore, x =(I - A)-1y = By. (2) Define n Bi• = a bij, (3) j=1 n B•j = a bij, and (4) i=1 n V = a bij. (5) The input-output multiplier matrix (mpm) (Sonis and Hewings 1989) is given by the formula: M = V Bi.B.j] = 1 B1• B2• . Bn• [B.1,B.2,...,Bn] = [mij]. (6) Thus, the structure of the mpm is essentially connected with the properties of the sectoral backward and forward linkages defined below. Assuming that B.j0 stands for the largest column multiplier and Bi 0. for the largest row multiplier, then the element located at (i0, j0) is given by the formula: 1 mi0 j0 = Bi0•B• j0 , V (7) which is called the largest cross. If this cross is excluded from M, then the second largest cross can be found. After the exclusion of row i0 and j0, the second largest cross mi 0 j 0 can be found and so on. The number of crosses is equal to the rank of the matrix mpm. Following Rasmussen (1956), there are two types of indices of the Leontief inverse, which are called backward linkages n1 ani=1 Bij nB•j BLj =1n 2 aB n i,j=1 ij and forward linkages n1 anj=1 Bij FLi =1n n2 ai,j=1 Bij = V nBi• = . V (8) (9) , Volume3 · Number 1 · Spring2005 98 Henryk Gurgul and Paweł Majdosz A backward linkage greater than 1 (BLj > 1) means that a unit change in final demand in sectorj will create an above average increase in activity in the economy, and analogously if forward linkage is greater than i (FLi > 1), it is taken for granted that a unit change in all sectors of the final demand will create an above average increase in sector i. Definition i. Sector k is called key sector if both indices are greater than1. Definition 2. Sector k is forward linkage oriented if FL is above 1 and BL is below 1. Definition 3. Sector k is backward linkage oriented if FL is below 1 and BL is above 1. Definition 4. Sector k is called weak oriented if both indices are less than1. Assuming B = (I - A)-1 = [bij] to be Leontief inverse, also known as the matrix of total inputs. For each location (i0,j0) we define a matrix F(i0, j0) (Sonis and Hewings 1989; 1992) of the form: F(i0,j0)= B 1i0 B 2i0 [Bj01, Bj02,..., Bj 0n] = [bii0bj 0j]. (10) This matrix is referred to as the first order field of influence of change. The economic interpretation of this matrix is related to the Sherman-Morrison (1950) formula. By using this formula, changes of entries in matrix B can be investigated. If the change e is located in position (i0, j0) in matrix A, following the above mentioned Sherman-Morrison formula, we have e( b e(i0 ,j0) bb e ii0 j0 j = bi j + j, (11) 0 j0 j , 1 - bj0i0 e or in matrix form e ?(i0,j0) B( i 0,j0) - B = F(i0,j0). (12) The sum of all components of the matrix F(i0, j0) is given below: S(F(i0, j0)) = Xbii 0 bj 0j = B• i0Bj 0• and («) M = 1 [S(F(j,i))]. (14) Managing Global Transitions Key Sector Analysis: A Case of the Transited Polish Economy 99 These formulas allow us to investigate the importance of direct inputs, which means the impact of the coefficients of A on B = (I - A)-1. Below we demonstrate that mpm has the property of maximum entropy (Shannon and Weaver 1964; Theil 1967; Kullback 1970). Let Q = [Qij] be a positive matrix and YJQij = Bi•, ^Qij = B•j and ^Qij = V. (15) j i ij Consider the probability spaces (X, F, Pi) i = 1,2, that is a basic set of elements x?Xanda collection F of all possible events (sets) made up of elements of the sample space X for which a probability measure Pi, i = 1,2 has been defined. Assuming that the probability measures P1 and P2 are absolutely continuous with respect to one another, then there exists a probability measure \i (for example /1 = (P1 + P2)/2) and functions fi(x), i = 1,2 called the generalized probability densities (Radon-Nikodym derivatives), unique up to sets of probability zero in /1, measurable in /1,0 < fi (x) < 00 almost everywhere in /1 such that, for all A?F, Pi(A) = fAfi(x)dn(x), i = 1,2. Applying the Taylor expansion logx = x - 1 - 1 (x- 1) 2 ,c ? (min{1,x},max{1,x}), one can see that 2f 1( x )log!dMx) = [ f2(x)dp(x)- /(f 1(x)dM( x ) =1-1- 1 c 2/f 1(x) (f 2( x ) - 1 ^( x ) ? 0. The inequality Lf1( x )log$dWx)?0 X f1(x) is known as the basic information inequality. Volumej · Number 1 · Spring2005 loo Henryk Gurgul and Paweł Majdosz Applying this inequality to the two dimensional distributions with the density function fXY(x, y) and the product of one dimensional distribution fX(x)fY(y) we have f fXY(x, y) log fX ( x ) fY (y) dß(x, y) < 0, and as a consequence I fXY (x, y) log fXY (x,y)dp(x,y) x < - f fX(x)logfX(x)d^x)- [ fY(y)logfY(y)d^y). Jx x The above inequality can be written in the form H(X,Y) < H(X) + H(Y), where H(X) = - f fX(x)logfX(x)d^x) x is called the entropy of random variable X (or its distribution). Similarly H(X,Y) = - f fXY(x,y)logfXY(x,y) d (x,y) x is called the entropy of a two dimensional random variable (X,Y) (or its distribution). For the discrete two dimensional distribution we have (in a special case) the inequality pi• p• j pij i, j a pp pi,j log i•• j ? 0. Hence - I pi,j log pi,j < - I p» log p1. - X p.j log p.j. (16) i,j i j Applying this result to the probabilistic distribution Q(pij = Qij/V) and the product M of its marginals (pit = Bit/V and p.j = B.j/V) and taking into account h(Q) = - X p( i , j)ln p(i,j) = - X QV ln QV (l7) i,j i,j we have (ln V + ln V) Managing Global Transitions Key Sector Analysis: A Case of the Transited Polish Economy 101 lni • 2• j = H(M). (18) Bi•B• j Bi•B• j ln V2 V2 i, j The multiplier product matrix M depends on the column and row multipliers. Therefore, mpm does not take into account the interactions of each sector with other sectors. From (18) it follows that mpm has the property of maximal entropy in the class of all matrices with fixed marginals. For the case where simultaneous changes occur in two places (i0, j0) and (i 1, j1) in a direct inputs matrix, a formula similar to (11) can be derived (see Hewings and Romanos 1981). Notice that the sum of all the elements of M is equal to the sum of all the elements of B. Matrix M represents the maximum entropy tendency. Thus, matrix M may be considered to represent the most homogenous distribution of the components of the column and row multipliers of the Leontief inverse B (which represents total inputs). From the economic point of view, the mpm matrix stands for the equalisation tendency of total inputs in an economy (i. e. in all industries the same - in monetary approach - output needs approximately the same input). Define D = diag(B - M) (diagonal), R = B - M - D, S = 2-1 (R + RT) (symmetric with null diagonal), Sa = 2-1 (R - RT) (asymmetric with null diagonal). Therefore, R = 2-1(R+RT) + 2-1(R-RT) = S + Sa. Thus B = M + D + S + Sa, where M represents the maximum entropy tendency and the diagonal matrix D stands for the so-called additional sectoral scale effects. The symmetric matrix S and asymmetric matrix Sa represent the symmetric and asymmetric tendency. Application to the Polish Input-Output Tables from the Period 1990–2000 The statistical data used here come from the Polish Statistical Yearbooks (published by The Central Statistical Office) covering the period 1990– 2000. Since input-output tables are published on a basis of a five-year-period in Poland, we focus our attention on the three years in the period Volume 3 · Number 1 · Spring 2005 102 Henryk Gurgul and Paweł Majdosz table i Key Sector analysis of the Polish economy over the period 1990–2000 (aggregation 6×6) Sector 1990 1995 2000 FL BL FL BL FL BL 1. Manufacturing 2.295 1.070 2.646 I.258 1.556 I.O38 2. Construction 0.536 0.988 0.526 I.087 0.731 1.081 3. Agriculture and forestry 0.852 1.122 0.854 I.O45 O.854 I.I67 4. Transportation and 0.736 1.057 0.691 1.022 O.832 O.980 communication 5. Trade 0.687 0.894 0.536 O.780 O.979 O.853 6. Service 0.894 0.869 0.747 0.808 I.O47 0.881 of interest, namely: 1990,1995, and 2000. We began by using an aggregation 6×6. Using a notion of backward and forward linkages, the taxonomy of the Polish economy is carried out. This taxonomy characterises changes in the economic structure over the period under consideration (Sonis and Hewings 1981; Durand and Markle 1984). The results of key sectors analysis are summarised in table 1 and figures 1, 2, and 3. Over the whole analysed period, the key sector was manufacturing. This indicates that a unit change in final demand in this sector will create an above average increase in activity in the economy, and unit change in all sectors of the final demand will create an above average increase of output in this sector. It is worth noting that both the backward and forward linkages of this sector tend to increase only in the first half of the analysed period, while in 2000 their level is even below the one observed in 1990. This means that the influence of the changes in the final demand of this sector on the whole economy was initially becoming stronger with time, but then the trend was reversed. It should also be emphasised that there is no forward linkage oriented sector at this level of aggregation except for services in 2000. The backward oriented sectors were agriculture and forestry as well as transportation and communication, but in the case of the second one, only over the first half of the analysed period. Also, construction could be considered as a backward linkage oriented sector since 1995, when its BL reached the greatest value. Apart from the key sectors, it is also interesting which of the selected sectors can be categorised as a weakly oriented sector. It results from our computations that trade alone is a weakly oriented sector over the whole period under consideration. Managing Global Transitions Key Sector Analysis: A Case of the Transited Polish Economy 103 figure i Visualisation of the results of key sector analysis for 1990 Backward Linkage Oriented Sectors 3 4 Key Sectors 1 O 2 5 6 O Weakly Oriented Sectors Forward Linkage Oriented Sectors 0,7 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 Forward Linkages (FL) figure 2 Visualisation of the results of key sector analysis for 1995 1,3 1,2 1,1 1,0 0,9 0,8 Backward Linkage Oriented Sectors 2 4 3 Key Sectors o 1 6 5 Weakly Oriented Sectors Forward Linkage Oriented Sectors 0,7 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 Forward Linkages (FL) figure 3 Visualisation of the results of key sector analysis for 2000 1,3 1,2 1,1 1,0 0,9 0,8 0,7 Backward Linkage Oriented Sectors 3 O Key Sectors 2 O 1 4 5 O 6 O Weakly Oriented Sectors Forward Linkage Oriented Sectors 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 Forward Linkages (FL) Volumej · Number 1 · Spring2005 104 Henryk Gurgul and Paweł Majdosz The investigations of the most important coefficients were performed by means of the Sherman-Morrison formula defined by (n), which gives the change in the entries of the Leontief inverse, caused by a change in one component of the direct coefficients matrix A. For any fixed position (i0,j0) we perturbed the corresponding element of A by replacing the element ai0,j 0 by ai0,j 0 (1 + e). Let us denote the direct coefficients matrix perturbed in this way by Ae( i 0,j 0). The number can be interpreted as the relative perturbation of ai0, j 0. Then the inverse error matrix given by BL(i 0, j 0) = [bj00 )] = (I - a ( i 0, j 0) ) -1 - (I -A) -1 , the relative inverse error matrix ?(i0,j0) RB = ?(i0,j0) b ij bij , and its norm ?(i0,j0) b ||RBL(i 0, j 0)|| = V \Jd___. i, j =1 b i ,j were computed. In the same way we computed the above elements for all possible positions (i 0, j0), so we finally obtained the sensitivity matrix SM = h|RBL(i0, j0)||" . Then we assigned ranks in descending order for the elements of this matrix. Rank i in position (i0, j0) indicates that ai0,j0 is the most sensitive (according to the inverse) element of matrix A, i. e. the relative change of this element has the greatest influence on B = (I - A)-1. The analysis was done, in the first step, for the years 1990, 1995 and 2000 in an aggregation 6×6. The results are presented in table 2. One will find that the most important input-output coefficients are manufacturing products used as input by manufacturing itself, agriculture and forestry products used by agriculture and forestry itself or by manufacturing. Among the most important coefficients in the analysed periods are also the coefficients of manufacturing products used by agriculture and forestry or by transportation and communication or by services, and service products used by manufacturing, and products of trading used by manufacturing. The input coefficients whose changes are less important for the economy under study are also interesting. The least important coefficients in the considered periods were those of construction products used by construction itself or trade, agriculture and Managing Global Transitions Key Sector Analysis: A Case of the Transited Polish Economy 105 table 2 Inverse important coefficients over the period 1990–2000 (aggregation 6×6) 1990 1995 2000 1. 2. 3. 4. 5. 6. 1. 2. 3. 4. 5. 6. 1. 2. 3. 4. 5. 6. 1. 1 7 8 5 11 4 1 7 4 6 11 10 1 5 7 8 17 4 2. 15 34 24 26 23 18 13 16 30 27 33 15 20 14 33 28 26 10 3. 1 33 3 36 30 35 1 35 3 36 26 34 1 36 3 35 30 34 4. 10 21 31 14 12 25 8 24 29 9 21 19 15 29 32 16 13 22 5. 6 29 20 19 32 27 12 25 31 20 32 18 6 23 19 21 25 24 6. 9 28 22 16 17 13 5 23 28 17 22 14 18 27 31 12 11 9 Note that inverse important coefficients are numbered according to importance. The ten top inverse important coefficients are in bold. forestry products used by construction or by transportation and communication or by services. The authors also provided the results of the decomposition of the Leontief inverse into the M (called mpm), diagonal, symmetric and asymmetric matrices in the years under consideration (see tables 3 and 4). The additional effect greater than one can be observed in the case of such sectors as: agriculture and forestry (1990, 1995, and 2000), manufacturing (1995) and also transportation and communication (1995). The smallest effect can be identified in the case of the trade sector. From matrix S it follows that the largest (by absolute value) bilateral balances occur in the pair of sectors: agriculture and forestry and transportation and communication (1990), manufacturing and services (1995) as well as agriculture and forestry and services (2000). From matrix Sa it follows that the largest (by absolute value) bilateral imbalances were in the pair of sectors construction and manufacturing (1990–2000). The authors also performed the computations for two different aggregations: 10×10 and 24×24. Unfortunately, since 2000 the Central Statistical Office has published the Polish input-output tables on the basis of different schema that makes it impossible to prepare an aggregation 10 × 10 and 24 × 24. Therefore, we have used an input-output matrix updating technique based on the sum of squared differences. More details can be found in Jackson and Murray (2004). The selected results are summarized below (see tables 5, 6, and 7). One will notice that the superscripts of top inverse important coefficients (obtained from 100 and 576 coefficients) are strongly related to the Volume3 · Number 1 · Spring2005 io6 Henryk Gurgul and Paweł Majdosz Managing Global Transitions Key Sector Analysis: A Case of the Transited Polish Economy 107 table 5 Key sector analysis of the Polish economy over the period 1990–2000 (aggregation 10 × 10) Sector 1990 1995 2000 FL BL FL BL FL BL 1. Group of industries of fuels and energy 1.372 1.061 1.420 1.027 1.073 0.893 2. Group of industries of raw material 1.711 1.048 1.922 1.170 1.050 0.991 3. Electromachine 0.963 0.994 1.004 I.I69 0.753 0.958 4. Food 0.656 1.073 0.711 1.239 0.820 1.292 5. Group of light industries 0.947 1.029 1.140 1.093 0.887 0.979 6. Construction 0.561 0.960 0.545 O.984 0.797 I.O54 7. Agriculture and forestry 0.968 1.091 0.911 0.955 1.026 I.I67 8. Transportation and communication 0.860 1.027 0.812 0.920 O.989 0.956 9. Trade 0.839 0.871 0.585 0.711 1.299 O.845 10. Services 1.122 0.846 0.949 0.731 1.306 O.865 key sectors numbers. Therefore our conjecture is that the parameters of location of inverse important coefficients determine approximately the same key sectors as indicators BLj and FLi. We also find that the most important input coefficients come from the main diagonal of the input-output matrix (large intraindustry flows). This means that this structure of the most important inputs was still typical for centrally planned economies (domination of raw materials and fuels). Large coefficients airi can be a signal of the inefficiency of economy (intraindustry flows dominate interindustries flows - i. e. in some branches their production is mainly devoted to their input, therefore, some branches producing final goods can experience a shortage of intermediate goods needed for their own inputs). In highly developed market economies these coefficients are lower. In the considered periods we also find some important input-output coefficients connected with interindustries flows, for example, agriculture and forestry products used as input by the food industry, food products used as input by trade, metallurgical products used as input by the electromachine industry. In these three cases, the importance of these coefficients is somewhat natural. Other conclusions (analogous to the case of aggregation 6×6) are left to the reader. Volume 3 · Number 1 · Spring 2005 io8 Henryk Gurgul and Paweł Majdosz table 6 Key sector analysis of the Polish economy over the period 1990–2000 (aggregation 24×24) Sector 1990 1995 2000 FL BL FL BL FL BL 1. Products of the coal and fuel industry 1.826 1.116 2.067 1.014 1.261 0.797 2. Energy, gas, hot water 1.273 1.031 1.223 I.O49 1.212 1.042 3. Metal ores, products from metallurgic irons and non-ferrous industry, recycling of metals 1.921 1.107 2.004 I.409 1.288 I.254 4. Products of metal industry 0.863 0.992 0.880 1.113 O.876 1.061 5. Machinery and device 1.017 0.972 1.031 1.123 O.76I 1.053 6. Products of precise industry 0.554 0.921 0.797 1.199 O.716 1.023 7. Products of the transport industry and the transportation trade 0.782 1.044 0.952 0.975 0.975 1.008 8. Products of electrotechnical industry 0.833 1.063 0.624 I.I48 0.599 0.880 9. Chemicals and chemical products and products manufactured with other non-metal materials 1.765 1.072 2.703 1.104 1.227 1.002 10. Products of the wood industry but not including furniture 0.944 1.056 0.700 I.089 0.813 1.100 11. Products of the paper and printing industry, data carriers, remaining products and material services 1.109 1.017 1.279 1.168 0.921 0.986 Continued on the next page Conclusions The main aim of this paper is to carry out a taxonomy of the Polish economy in transition. A further interest is to identify the most important input-output coefficients and also answer the question as to whether or not the structure of the Polish economy is still characteristic for a centrally planned economies. Applying the methods based on the entropy theory, we identified the sectors which can be considered as a key, and examined the additional Managing Global Transitions Key Sector Analysis: A Case of the Transited Polish Economy 109 table 6 (continued) Sector 1990 1995 2000 FL BL FL BL FL BL 12. Textiles 0.821 1.066 0.862 1.173 0.656 O.889 13. Clothes and products manufactured from fur, skin or products manufactured with skin 0.592 1.001 0.561 1.000 0.587 O.892 14. Food 0.879 1.094 0.819 1.248 0.962 1.343 15. Production and services of construction 0.717 0.980 0.733 1.000 1.125 1.102 16. Agriculture, hunting, forestry and fishing 1.251 1.112 1.074 0.956 1.165 1.211 17. Transportation 1.264 1.085 0.975 0.958 1.130 0.976 18. Communication 0.567 0.826 0.581 0.721 0.905 1.048 19. Trade 1.368 0.890 0.544 0.692 1.859 0.870 20. Municipal services, water and its distribution 0.604 1.145 0.467 0.846 0.760 0.937 21. Housing services 0.502 0.924 0.857 0.810 0.747 0.997 22. Education, medical services, social services 0.604 0.823 0.450 0.688 0.647 0.742 23. Services for people (hotels, restaurants, tourism, financial agency, leasing of machines and services) 0.803 0.916 1.447 0.841 2.244 1.021 24. Government administration, organizations 1.139 0.745 0.369 0.675 0.564 O.766 scale effects as well as the symmetric and asymmetric tendency in the economy. The striking empirical finding based on the most important input-output coefficient analysis is that the greatest importance is associated with the input coefficients which come from the diagonals of input-output matrices. It can be concluded that the structure of most important inputs was still typical for centrally planned economies even in 2000. In addition, large input coefficients aiti may be a signal of an inefficiency of the Polish economy over the period under consideration. It is worth noting that there exist also positive tendencies in the Polish economy. For example, the increased importance of services. In our opinion, this trend will continue also in the future. Volume3 · Number 1 · Spring2005 no Henryk Gurgul and Paweł Majdosz table 7 Ten top inverse important coefficients Aggregation Year Coefficients 10×10 1990 a2,2; a1,1; a7,7; a5,5; a7,4; a4,9; a2,3; a1,2; a3,3; a9,2 1995 a2,2; a1,l; a5,5; a3,3; a4,4; a7,7; a2,3; a7A; a8,8; a10,2 2000 a1,1; a3,1; a3,3; a1,6; a1,2; a5,1; a1,3; ai,4; a6,6; a2,6 24×24 1990 a3,3; aM; a 9,9; a16,16; a11,11; a16,14; a1,2; a12,12; ai4,i9; ai6,io 1995 a3,3; a9,9; a11,11; a1,1; a1,2; a12,12; a14,14; a16,16; a16,14; a5,5 2000 a3,3; a23,23; a16,16; a16,14; a14,14; a1,2; a3,4; a7,17; a13,18; a9,2 Acknowledgments The authors thank the anonymous referee for helpful comments. The autors also thank the managing director Artur Satora and Mrs Mał-gorzata Wegłowska from Central Statistical Office in Warsaw for supplying data. References Białas, S., and H. Gurgul. 1998. 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