Miyazawa meet Christaller: Spatial multipliers within a Triple Decomposition of Input-output Central Place Systems.

Transcription

1 Miyazawa meet hristaller: Spatial multipliers within a Triple Decomposition of nput-output entral Place Systems. Michael Sonis Department of Geography, Bar lan University, 52900, Ramat-Gan, srael and Regional Economics Applications Laoratory, University of llinois, 607 South Mathews, #220, Urana, llinois, sonism@mail.iu.ac.il Geoffrey J. D. Hewings Regional Economics Applications Laoratory, University of llinois, 607 South Mathews, #220, Urana, llinois, hewings@uiuc.edu ABSTRAT: A new triple UDL-factorization (U represents an upper-triangular lock matrix, L represents a lower-triangular lock matrix, and D represents the diagonal lock matrix) of the Leontief inverse is ased on the Schur lock-inversion of matrices. This factorization is applied to the decomposition of the Leontief inverse for input-output systems within a central place hierarchy of the hristaller-lösch, Bekmann-McPherson type. Such a factorization reflects the process of gradual complication of the central place hierarchy and the parallel augmentation of ackward and forward linkages within it. n this scheme the classical Miyazawa interrelation income multipliers play the role of spatial ackward and forward linkages multipliers within developing hierarchical central place systems of towns, cities and central capital. 1. ntroduction: The prolem posed. The idea to investigate the input-output relationship within the central place system is not a new one. The necessity to comine together the hierarchical structure of central place system with the input-output structure of the transaction flows within one unifying framework was stressed in the programmatic paper of sard (1960, p.141). The next attempt to mention some ideas of central place theory useful for the description of the regional economic interaction was made y halmers et al., 1978; however, no attention was directed to the intricate hierarchical structure of central place systems. A more systematic treatment of this prolem was undertaken y Roison and Miller (1991). They used the rudimentary structure of (capital and neighoring cities) an intercommunity central place system, without paying attention to the fine structure of the central place hierarchy. The complexity of mathematical presentation was limited to the level of a simple two-community two-order su region level with one dominant central place. n this paper, the theory of central place hierarchies and multi-regional input-output analysis will e integrated. As a result, it will e possile to show in what way the decomposition of the Leontief inverse for input-output central place systems reflects the process of complication of the evolving hierarchy of central places. The main feature of the discussion is the interpretation of

2 the classical Miyazawa interrelational multipliers as spatial multipliers in the developing structure of ackward and forward linkages within a three-level hierarchical central place systems of towns, cities and central capital. n the next section, the description of the central place system will e provided. Section 3 will apply a triangular decomposition of the Leontief inverse for an input-output ased central place system. Sections 4 and 5 provide the decomposition for a top-down and a ottom-up system. Section 6 introduced the Miyazawa income-consumption distriution framework that is then integrated into the central place system in section 7. The fine structure of this distriution is revealed in section 8 and the paper concludes with some reflections on challenges for empirical implementation. 2. Structure of the lassical entral Place System. The spatial description of the original hristaller central place model is ased on three generic geometric properties of central places associated this central place system (see Sonis, 1986): 1. The first property is that all hinterland areas of the central places at the same hierarchical level form a hexagonal covering of the plane with the centers on the homogeneous triangular lattice; 2. The second property is that the size of the hinterland areas increases from the smallest (on the lower tier of the central place hierarchy) to the largest (on the highest tier of hierarchy) y a constant nesting factor k. This nesting factor expresses one of the hristaller s three principles, namely, marketing (k=3), transportation (k=4) and administrative (k=7) principles; 3. The third property is that the center of a hinterland area of a given size is also the center of an hinterland of each smaller size (hristaller, 1933, Sonis, 1985) The Beckmann-McPherson (1970) central place model differs from the hristaller framework y applying variale nesting factors and y using the Löshian principle of all possile coverings of the plane y hexagons of variale integer sizes. Their centers are the vertices of the initial hristaller triangular lattice (Lösch, 1940).

3 n this paper a stylized example of the Beckmann-McPherson three tier entral place system will e used, including an uran hierarchy of single largest entral place K and the hierarchical levels including towns T = { T, T, T, T, T, T } and intermediate cities {,,,,, } = (see figure 1). (The reader should not associate the properties of real towns and cities to the central places in this stylized example.). Finally, the system is assumed to e for a closed economy with just one complete hierarchy; hence, there is no external trade. <<insert figure 1 here>> 3. The Application of the Triangular Decomposition of Leontief nverse for the Analysis of the nput-output entral Place System To egin, the central place spatial organization of settlements is set aside and only the hierarchy and the flow of intermediate goods etween these three hierarchical levels will e considered. n such a case, the matrix of direct inputs can e presented in the form: A A A A A A A TT T TK = T K A A A KT K KK (1) For the analysis of such an input-output system, the following structure of the Leontief inverse B = ( A) 1 will e used together with the a triangular decomposition (cf Sonis and Hewings, 2000): ( ) ( ) B TT ATB T ATK BKK T B = ( A) =B ( T) AT B ( T) AK BK 2 3 BKK ( T) A B KT BK AK K (2) where the Leontief inverses of the economies on the three separate hierarchical levels of the Beckmann-McPherson entral place system are: ( ); ( ); ( ) B = A B = A B = A (3) T TT K KK

4 Further, the input-output partial central place system, including only two hierarchical levels of central city K and intermediate cities and not including the level of towns T, corresponds to the lock matrix of direct inputs ( ) AT A A K = AK A KK (4) ts Leontief inverse ( ) ( ) 2 2 ( ) ( ) ( ) ( ) 1 B T B T AK B K 2 2 BKK T AK B BKK T BT = AT = (5) includes the components ( ) [ ] ; ( ) [ ] K K K KK KK K K B T = A A B A B T = A A B A (6) The triangular decomposition (see Sonis and Hewings, 2000, p.571) ( ) BT ( T) ( T) B A B 2 K K = BKAK = B K 2 B U K = L B K K (7) 2 allows interpretation of the component B ( ) T as an extended Leontief inverse of the hierarchical level of intermediate cities under the influence of the central city K. The ackward and forward linkages together generate the feedack loop of the form: K K < insert figure 1 here> The augmented inputs representing the coinfluence of different hierarchical levels (see Sonis and Hewings, 1998) have a form: A = A + A B A ; A = A + A B A 3 3 T T TK K K TK TK T K A = A + A B A ; A = A + A B A 3 3 T T K K KT K K T T TK A = A + A B A ; A = A + A B A 3 3 T KT K T K K KT T T (8)

5 Moreover, the component ( ) ( ) BTT = ATT ATB T AT ATK BKK T A KT (9) represents the extended Leontief inverse of the hierarchical level of towns, T, under the influence of the central city K and the intermediate cities. n the triangular decomposition (2) the lower triangular matrix L= B ( T) AT LT = 2 3 BKK ( T) A L KT BK AK KT LK (10) represents the ackward linkages of all three hierarchical levels such that the second lock column of L: 0 0 = BK A K L K represents the new ackward linkages of the susystem of two hierarchical levels of central city K and intermediate cities appearing as a result of adding to the economics of central city K the economics of intermediate cities ; the first column of the matrix L : 2 3 B ( T) A T L = T 2 3 BKK ( T) A KT L KT represents the extension of the ackward linkages within the system appearing as a result of further extension of to the economics of central city K and intermediate cities with the help of adding the economy of hierarchical level of the towns T. Analogously, the upper triangular matrix ( ) ( ) ATB T ATK BKK T UT UTK U = AK B K U = K (11)

6 represents the growing system of forward linkages within the augmenting system of all three hierarchical levels. The ackward and forward augmentation of linkages can e presented with the help of following spatial scheme: <nsert here scheme> n this scheme three feedack loops of spatial economic dependencies: T K T K K K The diagonal lock matrix 3 B TT 2 D= B ( T) B K (12) represents the Leontief inverse of the hierarchical level of the central city K and the extended Leontief inverses corresponding to the augmentation of the level K to the two hierarchical levels K, and to the three hierarchical levels K,, T. The augmentation of the ackward and forward linkages in three-tier hierarchical system is presented in figure 2. This figure presents the stages of development of the entral Place system starting from one central place (capital) and gradually expending to two hierarchical levels (capital and neighoring cities) and further, to three hierarchical levels (capital, neighoring cities and surrounding towns) < insert figure 2 here> n the two following sections, the decomposition will e presented for a top-down and ottom up central place model respectively. 4. Decomposition of Leontief nverse for nput-output top-down entral Place Model

7 f the forward linkages in the system K,, T are negligile, i.e. the matrix of direct inputs will have the form ATT 0 0 A= AT A 0 AKT AK A KK (13) then the triangular representation (2) will include only ackward linkages: B T B= BAT B BK( AKT+ AKBAT) B B KAK K (14) This decomposition corresponds to the circulation of flows of intermediate flows within the central places and to the case of top-down transaction flows where each central place includes the same structure of industries and is sending transaction flows of direct inputs to each dependent central place within its own hinterland area and exchanging production with the central places of the same hierarchical tier. 5. Decomposition of the Leontief nverse for nput-output ottom-up entral Place Model The ottom-up system of transaction flows corresponds to the case of negligile ackward linkages with the following lock-matrix of direct inputs: ATT AT ATK A= 0 A A K 0 0 A KK (15) The corresponding triangular decomposition of the Leontief inverse will include only forward linkages: B T ( AT + ATK BK AK ) B ATK B K B = B AKBK B K (16)

8 The lock-structure form of the triangular decomposition (2) allows the incorporation of the fine structure of the spatial economic dependencies etween central places within different hierarchical levels. Each such specifications of this fine structure can e incorporated into the triangular decomposition (2). Next, consideration will e given to the income generated from production and the impact of its expenditures on goods and services produced in the system. While exchange in production need not e hierarchical, it will e assumed that consumption of goods is made according to the usual central place principles of seeking the good at the nearest place. 6. Miyazawa ncome-onsumption Distriution in the nput-output Model. Miyazawa (1968, and 1976 for the most complete exposition) introduced the matrix model: X A X f = Y + V 0 Y g (17) A with a lock matrix M = V 0 for the analysis of the interrelationships among various income groups in the process of income formation. n the Miyazawa interpretation, the matrix A represents the inter industry direct inputs; vector x is gross output, vector f is final demand excluding consumption expenditures), vector Y represents the total income, the matrix V represents the value-added ratios of household sectors; and the matrix represents the coefficients of consumption expenditures. Miyazawa considered the usual Leontief interindustry inverse B=( - A) 1 ; the matrix multiplier VB showing the induced income earned from production activities among industries; the matrix multiplier B showing the induced production due to endogenous consumption (per unit of income) in each household sector; the matrix multiplier L = VB showing interrelationships among incomes through the process of propagation from consumption expenditures. -1 g 1 is interpreted as the interrelational income multiplier, and The matrix K = - L =( - VB) the matrix multiplier KVB is interpreted as the matrix multiplier of income formation. Further, the following Froenius-Schur form:

9 B A V e =( - - ) 1 (18) may e referred to as the enlarged Leontief inverse; its properties are: KVB = VB ; BK = B (19) e e For the Miyazawa income generation scheme ased on lock matrix M = A V 0 L NM O QP the Leontief lock-inverse has the following decomposition that separates the ackward and forward linkages effects: g L N M -1 Be Be B( M) = - M = = VB K e O QP L NM Be KVB O BK K QP (20) The structure of the interconnection etween the interrelational income multiplier K and the enlarged Leontief inverse B e can e seen and interpreted from the following formula: K = + VB e (21) where the enlarged matrix multiplier L e = VB shows the interrelationships among incomes through the process of enlarged propagation from consumption expenditures. Next, the fundamental UDL-factorization g L N M -1 Be BK B( M) = - M = = KVB K O QP L NM 0 e O B B QP L N M 0O K Q P L N M 0O 0 VB Q P separates multiplicatively the induced income earned from production activities among industries VB and the induced production due to endogenous consumption (per unit of income) in each household sector B from the industrial production activities. n this factorization, the interrelational income multiplier, K, appears explicitly and the enlarged Leontief inverse, B e, appears implicitly. The structure of the interconnection etween the enlarged Leontief inverse, B e, and the interrelational income multiplier, K, can e seen and interpreted from the following formula: B B BKVB e = + (23) (22)

10 7. Miyazawa ncome-onsumption Distriution in nput-output entral Place Model The Miyazawa income-consumption distriution in the input-output hierarchical central place model can e presented with the help of the following matrix: A A A TT T TK T A A A T K A M = = A KT A K A KK K V V V V T K (24) where the matrix A is the lock matrix of direct inputs on different hierarchical levels of towns, cities and capital city, V ( V V V ) = represents the income corresponding to the different T K sectors in towns, cities and capital city and T = represents the consumption in the central K place hierarchy. Analogously to formula (2), the following formula provides the following triple decomposition: ( ) B M = DT UT UTK T LT D U K = = L L D KT K K K VT V VK (25) L D U U = VL where induced income and induced production are

11 LKT LK T T U = UT UTT = + UKT UK K UKTT + UK + K VL = ( V V V ) L = ( V + V L + V L V + V L V ) T K T T T K KT K KT K (26) Therefore, the Miyazawa interrelation income multiplier (see 23) will have the form: DT T K = + ( VT V V K) LT D UT = L L D U U KT K K KT K K DT T = ( VT + VLT + VKLKT V + VKLKT V K) D UTT + D U + U + K KT T K K (27) Analogously to the spatial feedack loop interpretation of the formula (2), the Miyazawa interrelation income multiplier can e interpreted as spatial multiplier in the central place inputoutput system. 8. The Fine Structure of the Miyazawa ncome-onsumption Distriution in nput-output entral Place Model Next, a simple case of the structure will e presented that includes, the circulation of flows of intermediate goods within the central places and the top-down structure of the transaction flows etween the dependent central places only. Such a structure is presented in figure 2. <insert figure 2 here> This scheme generalizes slightly the usual assumption of classical central place theory demanding that oth income earned from production and consumption expenditures from this income will e concentrated in same location. The corresponding lock-matrix of direct inputs reflecting the central place hierarchy of such system has a form:

12 ATT AT ATK A= AT A A K = AKT AK A KK att 11 att 2 2 att 3 3 att 4 4 att 5 5 att 6 6 = at a 1 1 T a at a 2 1 T a at a 3 2 T a at a 4 3 T a at a 5 4 T a at a 6 5 T a a a a a a a a a a a a a A KT1 KT2 KT3 KT4 KT5 KT6 K1 K2 K3 K4 K5 K6 KK (28) The LD-sustructure of the Leontief inverse for this input-output central place system has a form drawing on (14) that is as follows: B T LD = BAT B BK( AKT+ AKBAT) B B KAK K where B T TT 1 1 TT 2 2 TT 3 3 =, (29) TT 4 4 TT 5 5 TT 6 6

13 B = (30) where ( ) ( ) 1 1 = a ; = a ; i= 1, 2,...,6 (31) TT i i TT i i i i i i and B A T a 1 1 T 1 1 a 1 1 T 1 1 a 1 1 T 1 1 a 1 1 T 1 1 a 1 1 T 1 1 a 1 1 T 1 1 =, (32) a 1 1 T 1 1 a 1 1 T 1 1 a 1 1 T 1 1 a 1 1 T 1 1 a 1 1 T 1 1 a 1 1 T 1 1 ( ) B A = B ( A + A B A ) = 3 K KT K KT K K KT1 KT2 KT3 KT4 KT5 KT6 ( ) ( ) where = B a + a a + a a, i= 1, 2,...,5, KTi K KTi Ki ii iti Ki i i i Ti and = B a + a a + a a KT6 K KT6 K1 11 T 1 6 K6 66 6T6 (33) ( ) BA = Ba Ba Ba Ba Ba Ba (34) K K K K1 K K2 K K3 K K4 K K5 K K6 So the places of non-zero locks in the matrix L of the ackward linkages can e seen on the following scheme:

14 (35) corresponding to the scheme of ackward linkages that was descried in figure 2. The structure of the income generation in this input-output central place system is given y following diagonal matrix reflecting the income generation within towns, cities and the central city: V vtt 11 vtt 2 2 vtt 3 3 vtt 4 4 vtt 5 5 vtt 6 6 v = 1 1 v 2 2 v 3 3 v 4 4 v 5 5 v 6 6 v KK (36) The consumption in this central place system is given y the following matrix whose lock structure reflects the ottom-up spatial organization of consumption in which the income generated in the towns, cities and capital is spent also in the neighoring central places:

15 ctt c 11 TT c 12 TT c 16 T c 1 1 T c 1 6 TK 1 ctt c 2 1 TT c 2 2 TT c 2 3 T c 2 1 T c 2 2 TK 2 ctt c 3 2 TT c 3 3 TT c 3 4 T c 3 2 T c 3 3 TK 3 ctt c 4 3 TT c 4 4 TT c 4 5 T c 4 3 T c 4 4 TK 4 ctt c 5 4 TT c 5 5 TT c 5 6 T c 5 4 T c 5 5 TK 5 c TT c 6 1 TT c 6 5 TT c 6 6 T c 6 6 TT c 1 1 TK 6 = c T c 11 T c 12 c 1 1 K 1 ct c 2 2 T c 2 3 c 2 2 K 2 ct c 3 3 T c 3 4 c 3 3 K 3 ct c 4 4 T c 4 5 c 4 4 K 4 ct c 5 5 T c 5 6 c 5 5 K 5 ct c 6 1 T c 6 6 c 6 6 K 6 ckt c 1 KT c 2 KT c 3 KT c 4 KT c 5 KT c 6 K c 1 K c 2 K c 3 K c 4 K c 5 K c 6 KK (37) The formulae (24-37) include all locks of the Miyazawa interelational income multiplier (27) and can e used for its analysis. 9. onclusions n this paper the attempt is descried to unify two central theories in the regional science: the classical input-output theory of Leontief and the classical hristaller-lösch central place theory. The considerations are presenting the elaoration of complete theoretical framework for such unification. t is expected that the proposed theoretical methodology will e useful for the analysis of organization of the production economics in geographical space. References: Beckmann, M.J. and J.. McPherson ity size distriution in the central place hierarchy: an alternative approach, Journal of Regional Science, 10, halmers J A, E J Anderson, T Beckhelm and W Hannigan Spatial nteraction in sparsely populated regions: An hierarchical Economic Base Approach, nternational Regional Science Review, vol.3, no. 1, pp hristaller, W Die Zentralen Orte in Suddeutscland, Fischer, Jena (English translation, 1966). entral places in Southern Germany, Prentice Hall, Englewood liffs, NY. sard W, The Scope and Nature of Regional Science: hannels of Synthesis.. Reprinted in. Smith (ed.) Practical Methods of Regional Science and Empirical Applications, Selected Papers of Walter sard, vol. 2, pp

16 Lösch A Die Raeumliche Ordnung der Wirtschaft, Fischer, Jena. (English translation, 1954 The Economics of Location, Yale University Press, New Haven, onnecticut. Roison, M.H. and J.R. Miller entral place theory and intercommunity input-output Analysis. Papers in Regional Science, 70, Sonis, M Hierarchical structure of central place System: The arycentric calculus and decomposition principle. Sistemi Urani, 1, 1985, Sonis, M A ontriution to the central place Theory: Super-imposed Hierarchies, Structural Staility, Structural hanges and atastrophies in central place Hierarchical Dynamics. n R. Funck and A. Kuklinsky (eds), Space-Structure- Economy: A Triute to Alfred Lösch, Karlsrhue Papers in Economic Policy Research, pp Sonis, M. and G.J.D. Hewings Economic omplexity as Network omplication: Multiregional input-output Structural Path Analysis, Annals of Regional Science, 32, Sonis, M. and G.J.D. Hewings LDU-factorization of the Leontief inverse, Miyazawa income multipliers for multiregional systems and extended multiregional Demo-Economic analysis, Annals of Regional Science, 34,