The measurement of structural differences between economies: An axiomatic characterization

Transcription

1 Economic Theory 16, (2000) The measurement of structural differences between economies: An axiomatic characterization Guang-Zhen Sun 1,2 and Yew-Kwang Ng 1 1 Department of Economics, Monash University, Clayton, Victoria 3168, AUSTRALIA 2 Max Planck Institute for Research into Economic Systems, Kahlaische Strasse 10, Jena, GERMANY ( s: sun@mpiew-jena.mpg.de; kwang.ng@buseco.monash.edu.au) Received: September 3, 1999; revised version: November 9, 1999 Summary. In empirical studies concerning comparison of economic structures and/or structural changes of economies, it is quite useful to employ an aggregate index to describe the structural difference (similarity). This paper offers an axiomatic characterization of the measurement of structural difference between economies that leads to some difference (similarity) index which is practically useful in empirical studies. Keywords and Phrases: Axiomatic approach, Structural difference (similarity) index, Invariance of proportional sub-classification. JEL Classification Numbers: C43; C40. 1 Introduction In empirical studies concerning the international or inter-regional comparison of economic structures (either in terms of production, consumption, exports or whatever) or intertemporal comparison of a given economy in terms of structure, one often needs to employ an aggregate index to describe the structural difference (similarity). In this short note, we offer an axiomatic characterization of the measurement of structural difference between economies that gives us a unique difference (similarity) index. 1 We are grateful to an anonymous referee and Meng-hun Liu for valuable comments. Correspondence to: Ghuang-Zhen Sun 1 In an interesting study of the measurement of diversity and its value function, Weitzman (1992) starts with the given dissimilarity distance function between any pair of species. Solow et al s

2 314 G.-Z. Sun and Y.-K. Ng Take a hypothetical example with two economies, country 1 and country 2, for which the agricultural output as a percentage of GDP is denoted as x i1 (i =1, 2), industry output as x i2 (i =1, 2),...,service output as x in (i =1, 2) with j x ij =1. Let x i =(x i1,...,x in ), i =1, 2. Some authors, e.g., UNIDO (1979, p.72), used the so- called similarity coefficient (the cosine of the two vectors in terms of mathematics), i.e., j x 1j x 2j /( j x 1j 2 )1/2 ( j x 2j 2 )1/2 to measure the structural similarity of economies. Some variance- type measurements, such as j (x 1j x 2j ) 2 /2, are also used (see, e.g., Landesmann and Szekely 1995). A further examination of the above measurements, however, suggests that both of them have flaws in empirical studies. For convenience of discussion, consider the structural difference of Arthur s and Betty s daily consumption on food and drink. Suppose that Arthur spends 40% of his consumption expenditure on food (10% on bread and 30% on beef) and 60% on drink (30% on whiskey and 30% on milk) while Betty spends 60% of her consumption expenditure on food (15% on bread and 45% on beef) and 40% on drink (20% on whiskey and 20% on milk). We surely can measure their structural difference (similarity) of consumption [ based on either ] the 2-class (food and drink) consumption structure matrix or the 4-class (bread, beef, whiskey and milk) consumption structure matrix Intuitively, in both cases, the measurement of structural difference should yield the same result because the 4-class case is only a further proportional classification of the 2-class case. But both the similarity and variance-type measurements mentioned above give different results for these two cases. It is true that, if the two persons spend different proportions on the subdivided categories, one expects the use of the finer 4-class case to reveal more differences, as some of the finer differences were hidden within the broader 2- class case. However, if the proportions between the two persons are the same, no difference should arise. This note offers a set of axioms from which reasonable measurements of structural differences could be derived. 2 An axiomatic characterization of indices measuring structural differences between economies Consider economies with n sectors. The structures of economies, ε x and ε y, are described as x [x 1,...,x n ] and y [ y 1...,y n ] with j x j = j y j = 1, x j, y j 0 respectively. The information regarding the structural difference between ε x and ε y is contained in (x, y). To derive a measurement which universally applies to any number of economic sectors, n, we define a function, denoted as D, (1993, esp. pp ) work on measuring biological diversity similarly takes the pair-wise distance as given. The purpose of this paper is rather to derive the dissimilarity distance function in an axiomatic manner.

3 The measurement of structural differences 315 on 2n 1 n< R +, where R + 2n {x, y) R 2n n i=1 x i 1, n i=1 y i 1, x i 0, y i 0}. That is, the definition domain of D is the union of the product of closed simplexes of different dimensions. We introduce axioms A1. Continuity. We mean by continuity that for any n N /{0}, D(x, y) is continuous with respect to the attributes x j (y j ), j =1...,n. A2. Symmetry: for any n N /{0}, D(ε 1,ε 2 )=D(ε 2,ε 1 ) and D(x 1,...,x n ; y 1,...,y n )=D(x i1,...,x in ; y i1,...,y in ) where (i 1,...,i n ) could be any (but the same for x and y) permutation of {1,...,n}. A3. Boundedness: 0 D(ε 1,ε 2 ) 1 and D(x, x) = 0 for any economic structure x. Note = in the first formula are required to hold for some extreme cases. 2 A4. Strong separability: for any n N /{0}, any subset of n {1,...,n}, denoted as s, s = {i 1,...,i s } where 1 i 1 <...<i s n, D(x 1,...,x n ; y 1,...,y n ) could be expressed as the sum of some function of D(x i1,...,x is ; y i1,...,y is ) and some function of D(x j1,...,x jn s ; y j1,...,y jn s ), where {j 1,...,j n s } = n/ s, 1 j 1 <... < j n s n. This property literally means that structural differences of economies could be obtained by aggregating the structural differences of subeconomies somehow, without the cross effect of the different groups of economic sectors, based on whatever partition of the indices of the economic sectors. A5. Invariance with respect to proportional sub-classifications: for any n N /{0}, any positive integer m, and ρ 1,...,ρ m 0 with m s=1 ρ s =1, D(x 1,...,x i,...,x n ; y 1,...,y i,...,y n ) = D(x <i,ρ 1 x i,...,ρ m x i, x >i ; y <i,ρ 1 y i,..., ρ m y i, y >i ) holds for any i n. This property is economically meaningful as argued in Introduction. We now claim Lemma 1. As an immediate consequence of A4, D(x 1,...,x n ; y 1,...,y n ) can be expressed as n D(x, y) = g i (D(x i, y i )) (1) for some functions g i, i n. i=1 Proof. We prove (1) for the case n = 3 only. The general case could be easily dealt with in a similar manner. From A4, D(x, y) can be written as D(x 1, x 2, x 3 ; y 1, y 2, y 3 )=g 3 (D(x 3, y 3 )) + g 3 (f 1 (D(x 1, y 1 )) + f 2 (D(x 2, y 2 ))) for some functions g 3,g 3, f 1 and f 2. But D(x, y) can also be written as D(x 1, x 2, x 3 ; y 1, y 2, y 3 ) = g 1 (D (x 1, y 1 )) + g 1 (h 2 (D(x 2, y 2 )) + h 3 (D(x 3, y 3 ))). Therefore, g 3 (f 1 (D(x 1, y 1 )) +f 2 (D(x 2, y 2 ))) = g 1 (D(x 1, y 1 )) + H (D(x 2, y 2 ), D(x 3, y 3 )) where the function 2 If x = y, one naturally expects D(x, y) = 0. Furthermore, the continuity of D(x, y) and the compactness of feasible sets for x and y imply, according to Weierstrass Theorem, that there exist (x, y ) and (x, y ) such that D(x, y ) D(x, y) D(x, y ) for any (x, y), i.e., D(x, y )/D(x, y ) D(x, y)/d(x, y ) 1. Thus, D(ε X,ε Y ) can be easily normalized to satisfy this property once the non-negativeness of the measurement is required.

4 316 G.-Z. Sun and Y.-K. Ng H (D(x 2, y 2 ), D(x 3, y 3 )) g 1 (h 2 (D(x 2, y 2 )) + h 3 (D(x 3, y 3 ))) g 3 (D(x 3, y 3 )). Thus the value of H (D(x 2, y 2 ), D(x 3, y 3 )) is independent of D(x 3, y 3 ). We are done. Lemma 2. A2 and A4 imply for some function g. D (x 1,...,x n ; y 1,...,y n ) = n g (D (x i, y i )) (2) Proof. From Lemma 1, D(x, y) = n i=1 g i (D(x i, y i )). For any w 1,w 2 [0, 1], consider (x, y) (w 1, 0, 1 w 1, 0,...,0; w 2, 0, 1 w 2, 0,...,0). Thus, by A4 and A2, D(x, y) = g 1 (D (w 1,w 2 )) + g 2 (D (0, 0)) + g 3 (D (1 w 1, 1 w 2 )) + g i (D(0, 0)) i>3 = g 1 (D(0, 0)) + g 2 (D (w 1,w 2 )) + g 3 (D (1 w 1, 1 w 2 )) + g i (D(0, 0)) i>3 i=1 Hence, g 2 (D(w 1,w 2 )) = g 1 (D(w 1,w 2 )) + g 2 (D(0, 0)) g 1 (D(0, 0)). Similarly, it could be derived that, g i (D(w 1,w 2 )) = g 1 (D(w 1,w 2 )) + g i (D(0, 0)) g 1 (D(0, 0)) for any i 3. Let [ n ] g(d(x i, y i )) = g 1 (D(x i, y i )) + 1 g i (D(0, 0)) (n 1)g 1 (D(0, 0)) n i=2 It is easy to verify D(x, y) = n i=1 g i (D(x i, y i )) = n i=1 g(d(x i, y i )). Note that in formulae (1) and (2), functions g i (i n) and g are dependent of n, denoted as g ni and g n respectively in what follows. As is shown below, a much stronger result than Lemma 2 could be obtained from A2, A4 and A5. Lemma 3. A2 A5 imply D(x, y) = n D (x i, y i ) (3) i=1 Proof. By (2), which is derived from A2 and A4, D(0,...,0; 0,...,0) = ng n (D(0, 0)). But D(0,...,0; 0,...,0) = 0 by A3. Hence, g n (D(0, 0)) = 0 for any n 1. For any w 1,w 2 0, by Lemma 2 and A5, D(nw 1, nw 2 ) = ng n+1 (D(w 1,w 2 ))+g n+1 (D(0, 0)) = ng n+1 (D(w 1,w 2 )). On the other hand, D(nw 1, nw 2 ) = ng n (D(w 1,w 2 )). Therefore, ng n+1 (D(w 1,w 2 )) = ng n (D(w 1,w 2 )), i.e., g n+1 (D(w 1,w 2 )) = g n (D(w 1,w 2 )). But again by A5, D(w 1,w 2 )=D(0,w 1 ;0,w 2 )= g 2 (D(0, 0))+g 2 (D(w 1,w 2 )) = g 2 (D(w 1,w 2 )). Thus, for any n > 2,g n (D(w 1,w 2 )) = g 2 (D(w 1,w 2 )) = D(w 1,w 2 ). (3) obviously holds true for the case n =1.Weare done.

5 The measurement of structural differences 317 Lemma 4. A1 A5 imply that D(rw 1, rw 2 )=rd(w 1,w 2 ), w 1,w 2 0, r > 0, provided that (w 1,w 2 ), (rw 1, rw 2 ) R 2 +. (4) Proof. Step 1 (4) holds for any positive integer r. From Axiom 5 and Lemma 3, D(rw 1, rw 2 )=D(w 1,...,w }{{} 1 ; w 2,...,w 2 )=rd(w }{{} 1,w 2 ) r r Step 2 (4) holds for any positive rational number r. Suppose r = p/q, with p and q being positive integers. From Step 1, D(w 1,w 2 )=qd( w1 ). Hence q, w1 q D( pw1 q, pw2 w1 q )=pd( q, w2 q )=p 1 q D(w 1,w 2 ). Step 3 (4) holds for any positive real number r. Consider a sequence of positive rational numbers {r i } that converges to r. From Step 2, D(r i w 1, r i w 2 )= r i D(w 1,w 2 ), i. Let i, from the continuity of D (A1), D(rw 1, rw 2 ) = rd(w 1,w 2 ). As an immediate consequence of Lemmas 1 4, we have, Theorem 1. A1 A5 imply that the structural difference between economies ε X = (x 1,...,x n ) and ε Y =(y 1,...,y n ) with x i = y i =1, x i, y i 0, i n, D (ε 1,ε 2 ) D (x 1,...,x n ; y 1,...,y n ) = D (x i, y i ) i = x i D ( ) 1, y i /x i + y i D ( ) 1, x i /y i i {τ x τ >y τ } i {τ y τ >x τ } where D is a homogenous function with degree one. (5) 3 Further analyses By taking an axiomatic approach, we derive a class of indices measuring the structural differences of economies, as shown in Theorem 1. Theoretically, there nevertheless exist many candidates for D(w 1,w 2 ) in (5). For instance, 1/3 D(w 1,w 2 )=(w 1 w 2 ) 1/3 w1 w 1/3 2 and D(w 1,w 2 )=(w 1/2m 1 w 1/2m 2 ) 2m for any m 1 satisfy A1 A5. For the practical purpose in empirical studies, we need to narrow down the possible candidates for D(w 1,w 2 ) and find some specific measurement. To this end, we need to introduce some additional condition, besides that stated in A1 A5, on the particular function g(s) D(1, s), s [0, 1]. But, intuitively speaking, what is the appropriate interpretation of D(1, s) in terms of economic structure? A natural interpretation is that D(1, s) can be justly used to measure the structural difference between a pair of two 2-sector economies, ε 1 :(1, 0) and ε 2 :(s, 1 s). The structure of a 2-sector economy is determined by the share of the first sector and hence D(1, s) indeed captures the structural difference[ between ε 1 ] and ε 2. Let us focus on the economic structure matrix 1 s Q(s). Clearly, if Det(Q(s)) = 0, i.e., s = 1, the economic 0 1 s

6 318 G.-Z. Sun and Y.-K. Ng structures are the same for ε 1 and ε 2. Note that Det(Q(s)) = 0 also means that the minimal absolute value of eigenvalues of Q(s) is zero. On the other hand, both the determinant and the minimal absolute value of eigenvalues of Q(s) are (1 s). This suggests we formally introduce, Assumption 1 D(1, s) =α(1 s), s [0, 1], where α>0 is a constant. Now we have, Theorem 2. The measurements of structural differences satisfying Axioms 1 to 5 and Assumption 1 must assume the form D(x, y) = j x j y j /2. (6) Proof. From Theorem 1, we only need to show that the value of α occurring in Assumption 1 must be 1/2. Consider an extreme case in which x j y j =0, j. Thus, D(x, y) =α( j x j + j y j )=2α. Axiom 3 requires α 1/2. On the other hand, for any x and y, D(x, y) = j α x j y j α j (x j + y j )=2α. It in turn requires that α 1/2 to ensure that = in Axiom 3 hold true for some cases. Thus, α =1/2. In fact, the index derived in Theorem 2 has been widely used in practice for long. For examples, UN (1981, ch.4) employed it to measure the structural change of output and employment in Western European manufacturing industry; it was also used in UNIDO (1983, ch.3, esp. p65) to compare patterns of change in developed market economies, developing countries and centrally planned economies for the periods and Finger and Kreinin (1979), in studying the export similarity, proposed an export similarity index, S (x, y) j Min{x j, y j }, to compare only patterns of exports across product categories (p.906). Although Finger and Kreinin have noticed that the (export) similarity index should, and indeed is, not influenced by the relative sizes or scales of total exports (p.906), they do not seem to be aware of the fact that the similarity index, compared to other indices, has another important advantage that it satisfies Axiom 5. It is easy to show, as UN (1981, p.189) correctly points out, that S (x, y) =1 D(x, y), where D(x, y) is the index we derive in Theorem 2. As far as we know, no attempt has been made in deriving the structural similarity (or difference) index such as Finger and Kreinin s S (or D in Theorem 2) by taking an axiomatic approach. A more general property, hence a weaker condition than Assumption 1, regarding D(1, s) could be given in terms of elasticity of the structural difference with respect to the perturbation [ ] of the relevant attribute in the economic structure matrix Q(s). Let t =1 s. Define η 1 s 0 1 s f,t D(1,1 t) t t D(1,1 t). Assumption 1 is equivalent to η f,t = 1. More generally, we have Assumption 1 η f,t = β>0.

7 The measurement of structural differences 319 Theorem 2. The measurements of structural differences satisfying Axioms 1 to 5 and Assumption 1 must assume the form ( D(x, y) = x i 1 y ) β ( i /2+ y j 1 x ) β j /2 (7) x i y j i {τ x τ >y τ } j {τ x τ <y τ } D(1,1 t) t t D(1,1 t) Proof. From Assumption 1, i.e., = β, with β>0wecan easily solve the differential equation, D(1, s) D(1, 1 t) =ct β = c(1 s) β, where c > 0. Thus, from Theorem 1, D(x, y) = i {τ x τ >y τ } cx i (1 yi x i ) β + j {τ x τ <y τ } cy j (1 xj y j ) β. Analogous to the proof of Theorem 2, we consider an extreme case in which x j y j =0, j. Thus, D(x, y) =c( j x j + j y j )=2c. Axiom 3 requires c 1/2. On the other hand, for any x and y, D(x, y) i {τ x τ >y τ } cx i + j {τ x τ <y τ } cy j 2c. It in turn requires c 1/2 to ensure that = in Axiom 3 hold true for some cases. Thus, c =1/2. Remark 1. We obtain Theorem 2 by letting β = 1. In other words, Theorem 2 is a special case of Theorem 2. It should be pointed out that β 1 seems to be an appealing requirement. Consider the following example, again with two economic sectors. Suppose the total output of economies is the same with each other in both cases below. Case 1. Economic Structure Matrix [ ] Case 2. Economic Structure Matrix [ ] Intuitively, the structural difference in case 1 is larger, at least not less, than that in case 2 because in case 1 the output of the first sector in economy two is three times of that in economy one but the output of the second sector in economy one is only 2% ( 2/97) more than in economy two, while in case 2, their output in both sectors is nearly of the same level. But D(case 1) > D(case 2) requires ( ) β> 1. 3 In fact, that β is required to be no less than unity could be justified ( ) in a general manner once one notices that the concept of Schur- convexity is applicable to measurements of structural differences. 4 At the first glance, it may 3 It is easy to show from 3(1 1 3 )β /2 + 99( )β /2 [51( )β /2] 2 that β 1. 4 A function f : R n m L R + is said to be Schur-convex if X R n m L, f (BX ) f (X ) holds for any n n bi-stochastic matrix B. The notion of Schur-convexity has found extensive applications in studies of economic and social measurements. In the literature of multidimensional inequality measures, Schur-convexity is a natural extension of the Pigou- Dalton principle from the unidimensional to multidimensional contexts (see, e.g., Kolm, 1977; Tsui, 1995). In a recent study, Li, Sun and Yang (1998) find that this notion is quite powerful in studies of measurements of the social division of labor as well.

8 320 G.-Z. Sun and Y.-K. Ng appear that the example above suggests that β should be strictly larger than one, as the difference in structure between the two economies seems to be larger in Case 1 than in Case 2. However, this appearance is more apparent than real. It is true that the difference for sector 1 (row 1) is larger for Case 1 than Case 2 at least relatively speaking. However, this bigger relative difference also applies to a relatively unimportant sector in Case 1 (accounting for no more than one or a few percentages of the economy for both economy 1 and economy 2). On the other hand, the same absolute difference and hence smaller relative difference in Case 2 is for an important sector, accounting for about half of the economy. If we not only recognize the difference in relative terms but also weight in accordance to the relative importance of the respective sector in reckoning the structural difference, the two may just offset each other, making β equal one and our formula in Theorem 1 acceptable. It is worthy to note that Theorems 2 and 2 certainly do not exclude other possible assumptions, rather than assumptions 1 or 1 presented here, which, in combination with some or all axioms above, might be able to characterize the measurements of structural differences. For instance, as far as the property of D(w 1,w 2 ) is concerned, we might replace Assumption 1 (1 ) with Assumption 2 D(w 1,w 2 )(w 1 w 2 ) can be seen as a function of (w 1 w 2 ), i.e., D(w 1,w 2 )=g( ). That is, in reckoning the sector-specific difference between economies, only the difference between the percentage shares of this sector matters. Then we can establish Theorem 3. The measurements of structural differences satisfying Axioms 1 to 5 and Assumption 2 must assume the form D(x, y) = j x j y j /2 Proof. From Lemma 4 and Assumption 2, for any r 0,g(r ) =D(rw 1, rw 2 )= rd(w 1,w 2 )=rg( ). Thus for any 0,g( ) = g(1). That is, D(w 1,w 2 )= (w 1 w 2 )g(1)=(w 1 w 2 )γ. Using a similar argument to that for Theorem 1 one can easily prove γ =1/2. Theorem 2 and Theorem 3 suggest that Theorem 2 can be generalized in more than one ways. References Finger, J. M., Kreinin, M. E.: A measure of export similarity and its possible uses. The Economic Journal 89, (1979) Landesmann, M. A., Szekely, I. P.(eds.): Industrial restructuring and trade reorientation in Eastern Europe. New York: Cambridge University Press 1995 Li, C., Sun, G.-Z., Yang, X.: A note on the measurement of the extent of division of labor for simple production economies. Working Paper, Department of Economics, Monash University (1998)

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