EXPANDED MIYAZAWA FRAMEWORK: LABOR

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1 The Regional Economics Applications aboratory (REA) is a cooperative venture between the University of Illinois and the Federal Reserve Bank of Chicago focusing on the development and use of analytical models for urban and regional economic development. The purpose of the Discussion Papers is to circulate intermediate and final results of this research among readers within and outside REA. The opinions and conclusions expressed in the papers are those of the authors and do not necessarily represent those of the Federal Reserve Bank of Chicago, Federal Reserve Board of Governors or the University of Illinois. All requests and comments should be directed to Geoffrey J. D. Hewings, Director, Regional Economics Applications aboratory, 607 South atthews, Urbana, I, , phone (217) , FAX (217) Web page: EXPANDED IYAZAWA FRAEWOR: ABOR AND CAPITA INCOE, SAVINGS, CONSUPTION, AND INVESTENT INS by ichael Sonis and Geoffrey J.D. Hewings REA 00-T-14 December, 2000

2 Expanded iyazawa Framework: abor and Capital Income, Savings, Consumption, and Investment inks. ichael Sonis Regional Economics applications aboratory, University of Illinois, Urbana, I 61801, USA and Bar Ilan University, Ramat Gan, Israel. sonism@mail.biu.ac.il Geoffrey J.D. Hewings Regional Economics applications aboratory, University of Illinois, Urbana, I 61801, USA and Bar Ilan University, Ramat Gan, Israel. hewings@uiuc.edu 1. Introduction In two seminal publications of iyazawa (1960, 1968), summarized in iyazawa (1976), the author developed a generalization of the eynesian income multiplier in the form of a matrix of inter-relational income multipliers. The matrix iyazawa model has a form: x A C x f Y V 0 Y g (1) Here A is a matrix of direct input coefficients; x is the gross output, f is the final demand, the vector Y represents total income, 1 the matrix V represents the value-added ratios of households; the vector g represents the exogenous income; the matrix C represents the consumption expenditures. The following analysis holds, (see Hewings et al. (1999), pp ) iyazawa considered the following 2x2 block-matrix A C V 0 (2) 1 In Pyatt (1998), discussion is raised of the distinction between factor income (iyazawa) and institutional income (social accounting models) in more extensive input-output/social accounting systems. It is not clear in the iyazawa system what types of other income are included in g in equation (1), referred to as exogenous income. In all probability, it will be less inclusive that the institutional income found in the social accounting system.

3 Expanded iyazawa Framework 2 The eontief inverse for the iyazawa matrix (2) has a form: ( ) ( ) B I I BC I B B + BCVB BC 0 I 0 VB I VB I 0 0 I C C V I 0 I 0 I VB I + V C (3) where B ( I A) is the eontief interindustrial inverse matrix, VBC is the matrix of interincome groups coefficients, and: ( ) ( ) 1 + (4) I I VBC I V C is the iyazawa interrelational income multiplier or generalized eynesian multiplier, and: ( ) 1 I A CV B + BCVB (5) is an enlarged eontief inverse. Also the iyazawa fundamental equations of income formation from capital hold: V VB C BC (6) In this paper we extend the iyazawa scheme of matrix inter-relational income multiplier by incorporating in it the labor and capital income, savings, consumption and investments. Applications to the construction of Sraffa-eontief-iyazawa model and to the Pyatt-Round Social Accounting odel are considered in detail. 2. Extension considering Wages, Savings, Consumption and Investment The concept of extension presented in this paper is stimulated by the matrix presentation of Sonis and Hewings (1988) of what may be referred to as an onion-skin approach to demographiceconomic impact analysis (see Stone, 1981; Batey and adden, 1983; and Batey, 1985). Such an extension can be achieved by consideration of the block-matrices W V S and C ( C! I! )

4 Expanded iyazawa Framework 3 where the matrix W is the matrix of wages, the matrix S is the matrix of the savings coefficients, the matrix C! is the matrix of households consumption and the matrix I! is the matrix of investment coefficients. In this case the iyazawa block-matrix (2) obtains the form of 3x3 block-matrix: A C! I! W 0 0 S 0 0 (7) The formulae (3)-(6), thus, imply that the iyazawa matrix of inter-income groups coefficients multiplier obtains the form: W WBC! WBI! B( C! I! ) S SBC! SBI! (8) and the iyazawa inter-relational income multiplier or generalized eynesian multiplier is (see Sonis and Hewings, 2000a): W ( I ) I B( C! I! ) S W I + W C! W I! I + ( C! I! ) S S C! S I! (9) and the enlarged eontief inverse is: W I A ( C! I! ) ( I ACW! IS! ) S W B+ B( C! I! ) B S (10) Also the iyazawa fundamental equations of income formation hold: W W B S S ( C! I! ) B( C! I! ) (11)

5 Expanded iyazawa Framework 4 3. Extensions inking Capital and abor Introducing the division of total income, Y, into wages of labor and profits of capital with the help of a series of block-column and block-row vectors. The first block-column vector, y Y y, represents total income, the second, V V V, with the components V, V, represents the income from capital profits and labor wages, the block-column vector g g g represents the exogenous income from capital and labor, and the block-row vector C ( C C ) represents the consumption expenditures from wages and profits. We will obtain from the iyazawa scheme (1) - (2) the following model (imura et. al.,2000 ): x A C C x f y V 0 0 y + g y V 0 0 y g (12) The model (12) includes two building blocks representing separately the income generation from capital and labor: x A C x f + y V 0 y g x A C x f + y V 0 y g (13) (14) These building blocks have the original iyazawa form of income generation. Therefore, the following analysis holds (see Hewings et al. 1999, pp ). Consider, at first, the iyazawa model (13) for income generation connected with capital. iyazawa evaluated the following block matrix A C V 0 (15) The eontief inverse for the iyazawa matrix (15) has a form:

6 Expanded iyazawa Framework 5 ( ) ( ) 1 B I I BC I 0 B 0 B+ BCVB BC 0 I 0 VB I VB I 0 0 I C C V I 0 I 0 I V B I + V C (16) where B ( I A) is the eontief inter-industrial inverse matrix and, ( ) 1 I V BC I + V C (17) is the iyazawa capital inter-relational income multiplier or generalized eynesian multiplier, and ( ) 1 I A C V B+ BC V B (18) is an enlarged eontief inverse. Also the iyazawa fundamental equations of income formation from capital hold: V VB C BC (19) Analogously, it is possible to obtain similar results for the iyazawa model (14) for income generation connected with labor, replacing the index by the index in formulae (16)-(19). Returning to the iyazawa model (12), we will present the results of the analysis using the substitutions into (15)-(19) of the block vectors V V V and C ( C C ) instead of V, C (see Hewings et al, 1999, p. 24): for the iyazawa matrix A C C V 0 0 V 0 0 (20) the eontief inverse may be shown to have the form:

7 Expanded iyazawa Framework 6 ( ) ( ) 1 B I I I C C V I 0 0 I 0 0 I 0 V 0 I0 0 I0 0 I C C V I + V C V C V V C I V C + (21) where 1 V V ( I A CV) I A ( C C ) ( I A C V C V ) (22) is the enlarged eontief inverse the model (12), and 1 V VBC VBC V VBC VBC ( I VBC) I B( C C ) I I + V C V C I + V C V C I + V C (23) is the iyazawa labor-capital inter-relational income multiplier or generalized eynesian multiplier for labor-capital. The fine structure of the iyazawa labor-capital inter-relational income multiplier,, can be presented with the help of the interlaced inter-relational income multipliers ( ) I V C I V C ; + ( ) I V C I V C + (24) This fine structure of the iyazawa labor-capital inter-relational income multiplier,, has a following form: V BC VBC (25) where the iyazawa fundamental equations of income formation have a form: C C ; V C ; C C ; V C (26)

8 Expanded iyazawa Framework 7 Hence, the eontief inverse (20) can be presented in the form: C C B ( ) V V BC V V BC (27) 4. The Sraffa-eontief-iyazawa Income Distribution odel The Sraffa-eontief income distribution model can be presented in a form of Sonis, Hewings (2000b): px (1 + r) pax + w* p( I A) x (28) where x is a gross output, p is the vector of prices, A the matrix of direct input coefficients, f is the final demand, r is the uniform rate of capital profits and w* is the wage rate of labor. Sraffa found the linear trade of relation between the rate of profits r and the wage rate w*, (see Sraffa, 1960, p.22, and also Pasinetti, 1977, p. 115): r µ µ ( 1 w* ) 1 (29) where µ is the Perronean maximal simple eigenvalue ( 0< µ < 1) corresponding to the left hand p and right hand x eigenvectors of positive matrix A: pa µ p; Ax µ x (30) Therefore, the capital income V rpa rµ p (31) and the labor income ( ) ( µ ) V w* p I A w* 1 p (32) generate the block-matrix of income V rµ p ( 1 ) 1 w V * µ p V w*1 ( µ ) p w* (33)

9 Expanded iyazawa Framework 8 Next let us introduce the block-matrix of consumption C ( C C ) whose components are the capital and labor consumption. With such specifications of block-matrices V and C, the iyazawa model (12) with matrix (20) represents the Sraffa-eontief-iyazawa income generation model and formulae (21) -(28) represent its analysis. 5. Interconnection between the iyazawa 3x3 Block-atrix odel of Income Propagation and the Pyatt-Round Social Accounting odel It is possible to prove the following: Statement 1: The 3x3 block Pyatt-Round Social Accounting model SA, (see details and references in Pyatt, Round (1985)) y1 A11 0 A13 y1 x1 y A 0 0 y + x y 3 0 A32 A 33 y 3 x 3 (34) is equivalent to 3x3 block iyazawa model: y1 A11 0 A13 y1 x1 y1 A11 0 A13 y1 x1 y2 A y2 + x2 y 2 A y2 + x2 y 3 B3 A32 A y 3 x β y3 BA 3 32A y 3 x β (35) where ( ) 1 B I A ; x B A x + B x (36) 3 33 β The structure of the Pyatt-Round accounting multiplier 1 for the system (34) and the structure of the iyazawa multiplier 2 for the system (35) can be ascertained with the help of an analytical technique developed for nxn block eontief multipliers in the subsection 3 of Sonis and Hewings (2000a).

10 Expanded iyazawa Framework 9 I A11 0 A13 1 A12 I 0 0 A32 I A 33 B B A B A B A B B22 A21B1 B22 B22 A21B1 A13B3 BA A B BA B B (37) I A A B B A 2 A21 I 0 B22 A21B1 I B22 A21B1 A13B3 BA 3 32A21 0 I BA 3 32A21B11 0 B 33 (38) where B B B B B ( I A11) ; ( I A11 A13B3 A32 A21) ; ( I A21B1 A13B3 A32 ) ; ( I B3 A32 A21B1 A13 ) ; ( I A33 A32 A21B1 A13 ) B33B3 (39) Further, it is possible to compare the SA with original 2x2 iyazawa system. First of all, we consider a statement, which is implicitly stated in paper of Pyatt (1998). Statement 2: The 3x3 block SA model (34) is equivalent to the pair of 2x2 block iyazawa models ( α ),( β ) : y1 A11 A13B3 A32 y1 xα + y A 0 y x ( α ) where xα x1 + A13B3x and 3 y1 A11 A13 y1 x1 + y B A A 0 y x β ( β ) ( β ) where xβ B3A32x2 + B3x3 The iyazawa multipliers for the systems ( α ),( β ) are:

11 Expanded iyazawa Framework 10 α I A A B A B B A B A A I B A B B (40) β B11 B11A I A A BA A I BA A B B (41) The analogous Statement 3 can be formulated: Statement 3: The 3x3 block SA model (34) is equivalent to the pair of 2x2 block models ( γ ),( δ ),: y1 A11 A13 y1 x1 + y A A A y x γ ( γ ) where xγ A32x2 + x3 and y2 0 A21B1A13 y2 xδ + y A A y x ( δ ) where xδ A21B1 x1 + x2 The Schur-iyazawa multipliers for the systems ( γ ),( δ ) are, cf. subsection 2 from Sonis and Hewings (2000a): γ I A A B B A B A A I A B A A B B (42) δ I A B A B B A B A B A I A B A B B (43) Statements 1, 2, and 3 enable interpretations that are compatible with conventional economic reasoning. oreover, the important fact is that the exclusion of the first column and row from the Pyatt-Round accounting multiplier, 1, (see (40)) yeilds the Schur-iyazawa multiplier δ ; the exclusion of the second column and row gives the Schur-iyazawa multiplier γ ; and the exclusion of the third column and row gives the iyazawa multiplier α.

12 Expanded iyazawa Framework 11 In addition, the extensions of the of the SA model through the introduction of different assets (including human capital) that provide factor services and including different types of institutions, lead to an nxn SA model, whose accounting multiplier can be constructed by the analytical technique used for nxn Block Input-Output analysis (see Sonis and Hewings, 2000a). 6. Expansion by ayers of Wages, Profits, Savings, Consumption, Investment by abor and Capital This extension can be achieved by the consideration of following block-matrices: W S V ; C C I C I P S ( ) (44) where W, S, C, I are the wages, savings, consumption and investments from labor, P, S, C, I are the profits, savings, consumption and investments from capital. In this case, the iyazawa matrix has the form: A C I C I W S P S (45) The iyazawa matrix of inter-income groups coefficients multiplier obtains the form: W WBC WBI WBC WBI S SBC SBI SBC SBI B( C I C I) P PBC PBI PBC PBI S S BC S BI S BC S BI (46) and the iyazawa inter-relational income multiplier or generalized eynesian multiplier is:

13 Expanded iyazawa Framework 12 W S ( I ) I + ( C I C I) P S I + W C W I W C W I S C I + S I S C S I P C P I I + P C P I S C S I S C I S I + (47) with the enlarged eontief inverse : ( I A C W I S C P I S ) W S I A ( C I C I) P S (48) Also the iyazawa fundamental equations of income formation hold: W W S S P P S S ( C I C I ) B( C I C I ) (49) The components of formulae (45) (49) represent the different layers of the income formation and expenditure and savings and investment in the expended iyazawa model. 7. Conclusion This paper explores the possibility to enlarge the original iyazawa 2x2-block income generation scheme to the case of many layers of labor and capital income, savings, consumption and investments. This onion-skin approach facilitates the inclusion of the iyazawa income generation and propagation mechanism into the Sraffa-eontief model and thus provides an explanation of the conceptual and structural interconnection of iyazawa ideas with those independently elaborated within the Pyatt-Round Social Accounting scheme. Of course, as Pyatt

14 Expanded iyazawa Framework 13 (1998) has noted, there are some important differences in the nature of the income propagation process in the iyazawa and more traditional SA systems, with the latter containing a more inclusive form of income (especially from non-wage and salary sources). However, these differences can still be handled within the general framework provided in this paper. REFERENCES Batey, P.W.J. (1985) Input-output models for regional demographic-economic analysis: some structural comparisons, Environment and Planning A 13, Batey, P.W.J. and. adden (1983) The modeling of demographic-economic change within the context of regional decline: analytical procedures and empirical results Socio-Economic Planning Sciences 17, Hewings, G.J.D.,. Sonis,. adden, Y. imura (eds) (1999) Understanding and Interpreting Economic Structure. Advances in Spatial Sciences, Springer-Verlag, New York, Heidelberg. imura, Y., G.J.D. Hewings,. Sonis (2000), Some addenda to the Sraffa-eontief odel, (unpublished manuscript). iyazawa,. (1960) Foreign trade multiplier, input-output analysis and the consumption function, Quarterly Journal of Economics 74, iyazawa,. (1968) Input-output analysis and interrelational multiplier as a matrix, Hitotsubashi Journal of Economics 7, iyazawa,. (1976), Input-Output Analysis and the Structure of Income Distribution. New York, Springer-Verlag. Pasinetti,.. (1977), ectures on Theory of Production. New York, Columbia University Press. Pyatt, G. and J.I. Round (eds), (1985), Social Accounting atrices: a basis for planning. Washington, DC: The World Bank.

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